JAMRIS 2010 Vol 4 No 4

JOURNAL of AUTOMATION, MOBILE ROBOTICS & INTELLIGENT SYSTEMS

Editor-in-Chief Janusz Kacprzyk

Executive Editor: Anna Ładan aladan@piap.pl

(Systems Research Institute, Polish Academy of Sciences; PIAP, Poland)

Associate Editors: Mariusz Andrzejczak (PIAP, Poland) Katarzyna Rzeplińska-Rykała (PIAP, Poland)

Co-Editors: Dimitar Filev (Research & Advanced Engineering, Ford Motor Company, USA)

Kaoru Hirota

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(Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, Japan)

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Editorial Office: Industrial Research Institute for Automation and Measurements PIAP Al. Jerozolimskie 202, 02-486 Warsaw, POLAND Tel. +48-22-8740109, office@jamris.org

Copyright and reprint permissions Executive Editor

Editorial Board: Chairman: Janusz Kacprzyk (Polish Academy of Sciences; PIAP, Poland) Plamen Angelov (Lancaster University, UK) Zenn Bien (Korea Advanced Institute of Science and Technology, Korea) Adam Borkowski (Polish Academy of Sciences, Poland) Wolfgang Borutzky (Fachhochschule Bonn-Rhein-Sieg, Germany) Oscar Castillo (Tijuana Institute of Technology, Mexico) Chin Chen Chang (Feng Chia University, Taiwan) Jorge Manuel Miranda Dias (University of Coimbra, Portugal) Bogdan Gabryś (Bournemouth University, UK) Jan Jabłkowski (PIAP, Poland) Stanisław Kaczanowski (PIAP, Poland) Tadeusz Kaczorek (Warsaw University of Technology, Poland) Marian P. Kaźmierkowski (Warsaw University of Technology, Poland) Józef Korbicz (University of Zielona Góra, Poland) Krzysztof Kozłowski (Poznań University of Technology, Poland) Eckart Kramer (Fachhochschule Eberswalde, Germany) Andrew Kusiak (University of Iowa, USA) Mark Last (Ben–Gurion University of the Negev, Israel) Anthony Maciejewski (Colorado State University, USA) Krzysztof Malinowski (Warsaw University of Technology, Poland)

Andrzej Masłowski (PIAP, Poland) Tadeusz Missala (PIAP, Poland) Fazel Naghdy (University of Wollongong, Australia) Zbigniew Nahorski (Polish Academy of Science, Poland) Antoni Niederliński (Silesian University of Technology, Poland) Witold Pedrycz (University of Alberta, Canada) Duc Truong Pham (Cardiff University, UK) Lech Polkowski (Polish-Japanese Institute of Information Technology, Poland) Alain Pruski (University of Metz, France) Leszek Rutkowski (Częstochowa University of Technology, Poland) Klaus Schilling (Julius-Maximilians-University Würzburg, Germany) Ryszard Tadeusiewicz (AGH University of Science and Technology in Kraków, Poland)

Stanisław Tarasiewicz (University of Laval, Canada) Piotr Tatjewski (Warsaw University of Technology, Poland) Władysław Torbicz (Polish Academy of Sciences, Poland) Leszek Trybus (Rzeszów University of Technology, Poland) René Wamkeue (University of Québec, Canada) Janusz Zalewski (Florida Gulf Coast University, USA) Marek Zaremba (University of Québec, Canada) Teresa Zielińska (Warsaw University of Technology, Poland)

Publisher: Industrial Research Institute for Automation and Measurements PIAP

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JOURNAL of AUTOMATION, MOBILE ROBOTICS & INTELLIGENT SYSTEMS VOLUME 4, N° 4, 2010

CONTENTS REGULAR PAPER

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Improving the intensification and diversification balance of the tabu solution for the Robust Capacitated International Sourcing problem (RoCIS) H.J. Fraire Huacuja, J.L. González-Velarde, G. Castilla Valdez

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Positive realizations of hybrid linear systems described by the general model using the state variable diagram method T. Kaczorek

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Optimization of a reactive fuzzy logic controller for a mobile robot using evolutionary algorithms A. Meléndez, O. Castillo, A. Alanis

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Motion prediction of moving objects in a robot navigational environment using fuzzy-based decision tree approach V.S. Rajpurohit, M.M.M. Pai 19

Control imrovement of shunt active power filter using an optimized-PI controller based on ant colony algorithm and swarm optimization B. Berbaoui, B. Ferdi, C. Benachaiba, R. Dehini

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EVENTS

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Throwable tactical robot – description of construction and performed tests R. Czupryniak, M. Trojnacki

SPECIAL ISSUE SECTION 35

Hybrid Intelligent Systems for Control and Automation - Part I Guest Editors: Oscar Castillo and Patricia Melin 37

Recurrent neural identification and control of a continuous bioprocess via first and second order learning I. Baruch, C.-R. Mariaca-Gaspar 53

Novel genetic optimization of membership functions of fuzzy logic for speed control of a direct current motor for hardware applications in FPGAs Y. Maldonado, O. Castillo, P. Melin

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POSITIVE REALIZATIONS OF HYBRID LINEAR SYSTEMS DESCRIBED BY THE GENERAL MODEL USING THE STATE VARIABLE DIAGRAM METHOD Received 14th June 2010; accepted 12th July 2010

Tadeusz Kaczorek

Abstract: The realization problem for linear hybrid systems described by the general model is formulated and solved. Sufficient conditions for the existence of positive realizations are established. A procedure based on the state variable diagram method for computation of a positive realization of a given transfer matrix is proposed. Effectiveness of the procedure is demonstrated on two examples.

Concluding remarks are given in section 4.

In the paper the following notation will be used. The set of n´m real matrices will be denoted by Ân´m and Ân=Ân´1. The set of n´m real matrices with nonnegative entries will be denoted by and Â+n´m and Â+n=Â+n´1. The n´n identity matrix will be denoted by In and the transpose will be denoted by T.

2. Preliminaries and the problem formulation Keywords: positive realization, hybrid, general model, state variable diagram.

Consider a hybrid linear system described by the equations [4] (1a)

1. Introduction In positive systems inputs, state variables and outputs take only non-negative values. Examples of positive systems are industrial processes involving chemical reactors, heat exchangers and distillation columns, storage systems, compartmental systems, water and atmospheric pollution models. A variety of models having positive linear behaviour can be found in engineering, management science, economics, social sciences, biology and medicine, etc.

Positive linear systems are defined on cones and not on linear spaces. Therefore, the theory of positive systems is more complicated and less advanced. An overview of state of the art in positive systems theory is given in the monographs [3], [4]. The realization problem for positive discrete-time and continuous-time without and with delays was considered in [1], [4-9], [15] and for positive fractional linear systems in [13]. The reachability, controllability and minimum energy control of positive linear systems with delays have been considered in [2].

A new class of positive 2D hybrid linear systems described by two vector equations has been introduced in [10] and of fractional positive hybrid systems in [11]. The realization problem for positive linear hybrid systems has been investigated in [12], [16], [17]. Structural decomposition of transfer matrix of positive normal hybrid systems has been proposed in [14].

In this paper a method for computation of positive realizations of linear hybrid system described by the general model will be proposed.

The paper is organized as follows. In section 2 fundamentals of positive hybrid linear systems are recalled and the realization problem is formulated. The main result is presented in section 3. In subsection 3.1 the proposed state variable diagram method is presented for singleinput single-output (SISO) linear hybrid systems. An extension of the method for multi-input multi-output (MIMO) systems is presented in subsection 3.2.

(1b)

where are the state, input and output vectors and .

(1c) Boundary conditions for (1a) have the form (2) Definition 1. The hybrid system (1) is called internally positive if and for arbitrary boundary conditions

(3) and any inputs .

(4)

The transfer matrix T(s,z) of the hybrid system (1) is given by

(5) where Âp´m(s,z) is the set of p´m real matrices in s and z with real coefficient.

Theorem 1. [4] The hybrid system (1) is internally positive if and only if Articles

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(6)

(7) (10) where Mn is the set of n´n Metzler matrices (with nonnegative off-diagonal entries).

From (5) we have (8) since

which by definition is the ratio of Y(s,z) and U(s,z) for zero boundary conditions, where , and and are the zet and Laplace operators.

Using (8) and (9) we can find

.

(11)

Knowing the matrix D we can find the strictly positive transfer matrix

and the strictly proper transfer function

(9) Definition 2. The matrices (1c) satisfying the conditions (6), (7) and (5) are called the positive realization of the transfer matrix .

The problem under considerations can be stated as follows.

Given a rational matrix T(s,z)ÎÂp´m(s,z). Find its positive realization, i.e. a realization (1c) satisfying the conditions (6) and (7).

In this paper sufficient conditions for the existence of a positive realization will be established and a procedure for computation of a positive realization for a given transfer matrix T(s,z) will be proposed.

(12)

where .

Multiplying the numerator and denominator of (12) by we obtain

3. Problem solution 3.1. SISO systems First we shall solve the problem for SISO hybrid systems using the state variable diagram method [16].

Let a given transfer function of the SISO hybrid system have the form

Fig. 1. State variable diagram for transfer function (13).

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(13) Defining

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(14)

From (14) and (13) we have (15) and (16) Using (15) and (16) we may draw the state variable diagram shown in Fig. 1.

The number of integration elements 1/s is equal to q1 and the number of delay elements 1/z is equal to 2q2. The outputs of the integration elements are chosen as the state variables and the outputs of the delay elements as the state variables . Using the state variable diagram we may write the equations

(17a)

(17b)

(17c) where

(17d) Substituting in the equations (17a) i by i+1 and differentiating with respect to t the equations (17b) we obtain the equations (1) with

(18a) where

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(18b) Theorem 2. There exists a positive realization (18) of the transfer function (10) if the following conditions are satisfied 1) 2) Proof. If the condition 2) is met then and the coefficients of the strictly proper transfer function (12) are nonnegative. From (18) it follows that if the conditions 1) and 2) are satisfied then and by Theorem 1 the realization (18) is positive. ÂŁ From (18) we have the following corollary.

Corollary 1. If the conditions 1) and 2) of Theorem 2 are satisfied then there exists a positive realization of the transfer function (10) with A0 = 0 and B0 = 0 and .

Example 1. Find a positive realization of the transfer function (19) Using (8) and (9) we obtain (20) and the strictly proper transfer function (21) In this case (15) and (16) have the form (22) and (23) 6

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Using (22) and (23) we may draw the state variable diagram shown in Fig. 2.

Fig. 2. State variable diagram for transfer function (21).

The outputs of the integration elements are chosen as the state variables x1(s,z), x2(s,z) and the outputs of the delays elements as the state variables x3(s,z),..., x6(s,z). From the state variable diagram we have the equations

(24) and (25) The equations (24) and (25) can be written in the form (1), where

(26) The desired positive realization of (19) is given by (20) and (26).

3.2. MIMO systems First we shall consider linear hybrid m-inputs and one-output systems with the transfer matrix (27) where (28)

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Fig. 3. State variable diagram for transfer function (36).

It is assumed that the minimal common denominator d(s,z) satisfies the assumption (29)

Theorem 4. There exists a positive realization (32) of the transfer matrix (27) if all coefficients of the numerator nk(s,z), k = 1,…,m are nonnegative and all coefficient of the denominators dk(s,z), k = 1,…,m are nonpositive except the leading coefficient equal to 1.

Using (8) and (9) we can find the matrix D and the strictly proper transfer matrix Tsp(s,z). Applying the approach presented above for SISO systems to MIMO system with (27) we may find a realization of each transfer function (28). A realization of the transfer function (27) can be found by the use of the following theorem.

Proof. If the assumptions are satisfied then by Theorem 2 the realization (30) of the transfer function (27) is a positive one. From (32) it follows that in this case all matrices (32) have nonnegative entries and by Theorem 1 the realization of the transfer matrix is positive. £

Theorem 3. Let

Example 2. Given the transfer matrix (30)

be a realization of the transfer function (28). Then a realization of the strictly proper transfer matrix

T(s,z) = [T1(s,z) T2(s,z)]

(33)

where T1(s,z) is given by (19) and (34) Using (8) and (9) from (33), (19) and (34) we have

(31) (35) is given by and

(32) Proof. Using (8), (31) and (32) we obtain

(36) The state variable diagram corresponding to the transfer function Tsp1(s,z) is shown in Fig. 2. and the positive realization is given by (26) i.e. £ 8

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Fig. 4. Connection of state variable diagrams.

By Theorem 3 the desired realization of the transfer matrix (33) is given by

(37)

(40)

The state variable diagram corresponding to Tsp2(s,z) is shown in Fig. 3.

Using this state variable diagram we can write the equations

where the submatrices A11, A12, B11, A12, C1 are given by (37) and submatrices A21, A22, B21, A22, C2 are given by (39). The realization is positive since all entries of the matrices (40) are nonnegative.

Remark 1. If the assumption (29) is not satisfied and

and (41) (38) From those equations we have the realization of Tsp2(s,z) in the form

then to decrease the dimension of a realization of (27) it is recommended to find d(s,z) and write the transfer matrix (27) in the form (42) where degs d(s,z) (degz d(s,z)) denotes the degree of the minimal common denominator with respect to s(z).

Note that the m-inputs and p-outputs systems can be considered as the sequence of p m-inputs and one-output systems. In this way the presented approach can be extended for m-inputs and p-outputs linear systems.

4. Concluding remarks

(39) The state variable diagram corresponding to the transfer matrix (36) can be obtained as the connection of the state variable diagrams shown in Fig. 2 and Fig. 3 (see Fig. 4).

The problem of computation of positive realizations of hybrid linear systems described by the equations (1) by the use of the state variable diagram method has been addressed. It has been shown that there exists a positive realization of a given transfer matrix if all coefficients of the numerator of each transfer function are nonnegative and all coefficients of the denominator are nonpositive except the leading one equal to 1. The presented method enable us to find a positive realization with zero A0, B0 matrices. If the condition (41) is satisfied then it is recommended to find first minimal common denominator for each row of the transfer matrix. Those considerations can be extended to linear hybrid systems with delays and to linear fractional hybrid systems.

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ACKNOWLEDGMENTS This work was supported by Ministry of Science and Higher Education in Poland under work No NN514 1939 33.

AUTHOR Tadeusz Kaczorek - Professor at the Faculty of Electrical Engineering, Bialystok University of Technology, Bialystok, Poland. E-mail: kaczorek@ee.pw.edu.pl.

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Benvenuti L., Farina L., “A tutorial on the positive realization problem”,, IEEE Trans. Autom. Control, vol. 49, no. 5, 2004, pp. 651-664.

Busłowicz M., Kaczorek T., “Reachability and minimum energy control of positive linear discrete-time systems with one delay“. In: 12th Mediterranean Conference on Control and Automation, 6th-9th June, 2004, Kusadasi, Izmir, Turkey.

Farina L., Rinaldi S., Positive Linear Systems; Theory and Applications, J. Wiley, New York, 2000.

Kaczorek T., Positive 1D and 2D systems, Springer Verlag, London 2001.

Kaczorek T., “Realization problem for positive discretetime systems with delay”, System Science, vol. 30, no. 4, 2004, pp. 117-130.

Kaczorek T., “Positive minimal realizations for singular discrete-time systems with delays in state and control”, Bull. Pol. Acad. Sci. Techn., vol. 53, no. 3, 2005, pp.

293-298.

Kaczorek T., “A realization problem for positive continuous-time linear systems with reduced numbers of delays”, Int. J. Appl. Math. Comp. Sci., vol. 16, no. 3, 2006, pp. 325-331.

Kaczorek T., “Realization problem for positive multivariable discrete-time linear systems with delays in the state vector and inputs”, Int. J. Appl. Math. Comp. Sci., vol. 16, no. 2, 2006, pp. 101-106.

Kaczorek T., “Positive minimal realizations for singular discrete-time systems with one delay in state and one delay in control”, Bull. Pol. Acad. Sci. Techn., vol. 52, no. 3, 2005, pp. 293-298.

Kaczorek T., “Positive 2D hybrid linear systems”, Bull.

Pol. Acad. Sci. Techn., vol. 55, no. 4, 2007, pp. 351-355.

Kaczorek T., “Positive fractional 2D hybrid linear systems”, Bull. Pol. Acad. Sci. Techn., vol. 56, no. 3, 2008, pp. 273-277.

Kaczorek T., “Realization problem for positive 2D hybrid systems”, COMPEL, vol. 27, no. 3, 2008, pp. 613-623.

Kaczorek T., “Realization problem for positive fractional discrete-time linear systems”, Int. J. Factory Autom.

Robotics and Soft Comput., vol. 3, 2008, pp. 76-86.

Kaczorek T., “Structural decomposition of transfer matrix of positive normal hybrid systems”, Archive of Control Sciences, vol. 18, no. 4, 2008, pp. 399-413.

Kaczorek T., Busłowicz M., “Minimal realization problem for positive multivariable linear systems with delays”, Int. J. Appl. Compt. Sci., vol. 14, no. 2, pp. 181-187.

Kaczorek T., Sajewski Ł., “Computation of positive realizations of MIMO hybrid linear systems with delays using the state variable diagram method”, System Science,

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Sajewski Ł., “Positive linear hybrid systems Realization in the form of two-dimensional general model”, Automation 2010, Pomiary Automatyka Robotyka 2/2010.

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MOTION PREDICTION OF MOVING OBJECTS IN A ROBOT NAVIGATIONAL ENVIRONMENT USING FUZZY-BASED DECISION TREE APPROACH Received 10th January 2010; accepted 5th July 2010

Vijay S. Rajpurohit, M.M. Manohara Pai

Abstract: In a dynamic robot navigation system the robot has to avoid both static and dynamic objects on its way to destination. Predicting the next instance position of a moving object in a navigational environment is a critical issue as it involves uncertainty. This paper proposes a fuzzy rulebased motion prediction algorithm for predicting the next instance position of moving human motion patterns. Fuzzy rule base has been optimized by directional space approach and decision tree approach. The prediction algorithm is tested for real-life bench- marked human motion data sets and compared with existing motion prediction techniques.

Results of the study indicate that the performance of the predictor is comparable to the existing prediction methods.

Keywords: short term motion prediction, fuzzy rule base, rule base optimization, fuzzy predictor algorithm, directional space approach, decision tree approach.

1. Introduction For an autonomous mobile robot, performing a navigation-based task in an unknown environment to detect and avoid encountered obstacles is an important issue.

It is also a key function for the robot body safety, as well as for the task continuity. Generally, the architecture for the vision-based robotic systems with the ability of obstacle detection and avoidance are relatively complicated. This may be attributed to the extraction of information from a stream of the site images consisting of the static and dynamic obstacles. In a dynamic robot navigation system, the robot has to acquire the information on moving objects and predict their future positions in order to make path planning efficient. Short term object motion prediction in a dynamic robot navigation environment refers to the prediction of next instance position of a moving object based on the previous history of its motion. The living beings and vehicles characterize the dynamic environment and exhibit motion in various directions with different velocities.

Real-life data often suffer from inaccurate readings due to environmental constraints, sensors, size of the objects and possible change in motion pattern of the moving objects. This needs the system to be robust to handle these uncertainties and predict next instance object position as accurate as possible within a short duration. As a result, object motion prediction still continues to be an active field of research. Research literature has addressed solutions to the short term object motion predictions with different methods such as: curve

fitting or regression methods [7], [18], neural network based approaches [1], [2], [4], Hidden Markov stochastic models [19], Bayesian Occupancy Filters [5], Extended Kalman Filter [9], [12], Stochastic prediction model [17], regression methods [18], [7] proposed in the literature, sample the positions of moving object at definite time intervals, and fit the information to the regression equation. With the current sampling positions, the regression model predicts the position of the object for the next sampling duration. The main drawback of this method is the estimation of model coefficients in real-life environment, which makes the system complex. Amalia Foka et al. [1],[2] have proposed a Polynomial Neural Network (PNN) architecture for object motion prediction. The PNN uses a second order polynomial equation as a transfer function at each node. Training is done using evolutionary method. The algorithm needs huge amount of data sets for training and the performance of the algorithm is poor in case of unseen datasets. Relative Error Back Propagation neural network [4] for object motion prediction considers rectilinear motions of moving objects. The algorithm needs huge dataset for training and quality of results depend on the training data set used. Statistical methods for estimating obstacle locations using statistical features have been proposed such as Hidden Markov Model [19] to predict object motion. The method is computationally intensive. The method proposed by R. Madhavan et al. [12] uses Extended Kalman Filter. Each prediction step is dependent on the previous sequence of observations made and the quality of prediction reduces with increase in time and space horizon. C. Laugier and S. Petti [5] have proposed Baysean programming framework to predict the future position of moving object. The navigational environment is represented as a four dimensional occupancy grid. The method is not suitable for large scale environment because of intrinsic complexity and numerical computations. R. Irajit et al. [15] in their work have proposed a methodology based on Artificial Potential Fields (APF) method which provides simple and effective motion planners for practical path planning in fully dynamic environments. They have exploited the fuzzy modeling to define Fuzzy Artificial Potential Fields (FAPF) which provides a real-time and flexible path planning. It is shown that FAPF paves a way to merge both global and local path planning strategies. Simulations show that the planner is both very fast and capable of handling the local minima which can trap mobile robots before reaching the goal. Based on the literature survey on motion prediction models it is observed that i)The existing models lack flexibility in handling the uncertainties of the real-life situations; ii) Probabilistic Articles

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models sometimes fail to model the real-life uncertainties; iii) The existing prediction techniques show poor response time due to their complex algorithmic structure; iv) Most of the approaches validate the results with simulated data or simple navigational environments.

The present work overcomes these difficulties with a novel solution for short term motion prediction using fuzzy rule-based prediction technique. History of moving object motion positions is captured in the form of fuzzy rule base, and the next instance object position is predicted using fuzzy inference process. Because of the multivalued nature of fuzzy logic, this approach enjoys high robustness in dealing with noisy and uncertain data.

However, direct implementation of the rule base is not suitable for real-life navigation systems due to the formation of huge number of rules. The total number of fuzzy rules to be used are directly proportional to the number of fuzzy sets defined for the application and the number of fuzzy members present in each fuzzy set.

Inconsistent and redundant rules identified in the rule base are optimized by defining directional space within navigational space and decision tree approach.

The authors in their previous work [16] have implemented the extraction of objects of interest within the robotic navigational environment from the stereo vision system. Hence the focus of the present work is only limited to the prediction of the moving object's motion within the navigational environment.

The paper is organized as follows. In section 2, fuzzy rule-based object motion prediction process is explained.

Sections 3 and 4 discuss the optimization of the fuzzy rule-base using directional space approach and decision tree approach.

In section 5 the fuzzy rule-base implementation details are presented. Experimental results are presented in section 6. Finally, concluding remarks are given in section 7.

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knowledge of the obstacles and a through observation of the history of motion of the moving objects. Fuzzy logic is an important branch of intelligent robotics. It does not need to establish accurate mathematical models and it is easy to construct its control structure with good robustness.

In the proposed work, the navigational environment is modeled as a fuzzy world model. The robot is capable of visualizing the navigation environment in front (about 180 degrees in semi circular range). Fuzzy regions in front of the robot are defined according to the visualization capability of the sensors. Each object detected has a distance variable from the Robot. This range data has a different membership in each of the 7 range subsets defined as Very very far (VVFAR), Very far (VFAR), Far (FAR), Moderate (MOD), Near (NEAR), Very near (VNEAR), Very very near (VVNEAR). The direction of universe is divided into 7 subsets. The linguistic variables that describe the angle heading are Very very left (VVLEFT), Very left (VLEFT), Left (LEFT), Front (FRONT), Right (RIGHT), Very right (VRIGHT), Very very right (VVRIGHT). The fuzzy representation of the environment is shown in Figure 1 with numerical notation for each region. The fuzzy representation divides the whole navigation environment into different regions like VVFAR-VLEFT (61), FAR-RIGHT(44) and NEAR-FRONT(23) etc.

2. Fuzzy Rule-based Object motion prediction The difficulty of dynamic obstacle motion prediction lies on the uncertainty of obstacle motions. In the proposed work we have considered intentional motion model for the moving objects within the navigational environment. Motion state of an obstacle at time t is generally represented by (p(t),v(t),a(t)) which represent the position, velocity and acceleration of the object at time t. In this model an obstacle moves in a scheduled route, such as a predetermined destination, or a programmed route.

The obstacle may also try to avoid collision with others.

In this case we have, a(t) = a(t - dt) + be(t)

(1)

Where e(t) represents the variations in the accelerations resulting from any internal or external forces of the obstacle a and b are any two constants that specify the tendency of acceleration change. The function e(t) depends on the particular environmental conditions. It differs from the random motion model in the way that, e(t) cannot be described by any probability distribution. The acquisition of e(t) relies very much on the background 12

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Fig. 1. Division of Navigation Space into Fuzzy subsets of Range and Direction.

As the regions defined are fuzzy in nature, there can be overlaps from one region to another region. For simplicity these overlaps are not shown in the figure. The range and angle information need to be represented by a suitable membership function. Many authors have addressed critical issues relating to the selection and per-

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formance of fuzzy membership functions for various realtime robot control applications [6], [8], [14]. In most of the cases triangular membership function has proved superior over other membership functions like trapezoidal, Gaussian, bell shaped, polynomial-PI and sigmoidal.

For our application as the prediction needs to be more accurate and the strength of the rule/ rules fired can make remarkable difference in the prediction, selection of triangular membership function for representing angle and range values is inevitable. The selection of 07 fuzzy subsets for range and angle is moderate as selecting 05 categories will have less number of fuzzy rules but, quality of prediction may reduce if navigation space is large, selecting 09 or more number of categories will increase the number of fuzzy rules as well as the complexity of the system which could reduce the response time of the predictor. Both range and angle subsets are normalized between 0-1.

In the rule base formation phase, rules are defined and added to the rule base using real-life data consisting of human motion patterns with velocity in the range 2-10 kmph. At time t1, the position (angle and range) of the moving object from the robot is read. Using fuzzification the observed data is converted to fuzzy value. At time t2 (t2 > t1 and t2 - t1 > d, where d is threshold time difference greater than or equal to 1 sec) the sensor reads the position of the same object. The reason for considering d Âł 1 sec is that, the time needed to process the captured image to identify the objects of interest by the vision system needs at least 01 sec or more as per the current literature. The maximum value of the d considered was 04 seconds. This is because, as the time gap between the measurements increases the quality of the prediction reduces as well as the prediction looses its significance.

The read value of the object position is converted to fuzzy value. The same process is followed at time t3 (t3 > t2 and t3 - t2 = t2 - t1) to get the fuzzy value of the location of the same object under observation. A fuzzy rule with the positions of the moving object at time t1 and t2 as the antecedent and the position of the object at time t3 as the consequent is formed and added to the rule-base.

Each rule in the rule-base is represented as

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Fig. 2. Short term motion prediction.

3. Optimization of the Rulebase

by Partitioning the Navigational Space Many of the rules defined in the system look inconsistent such as 1. IF object at t1 is VFAR,VVLEFT and object at t2 is FAR,VVLEFT THEN object predicted at t3 is MOD,VVLEFT. 2. IF object at t1 is VFAR,VVLEFT and object at t2 is FAR,VVLEFT THEN object predicted at t3 is FAR,VVLEFT.

IF (R1, q1) and (R2, q 2) THEN (R3, q 3) where R1 and q1 represent the range and the angle respectively of the object at time t1, R2 and q 2 represent the range and the angle respectively of the object at time t2, and R3 and q 3 represent the range and the angle respectively of the object at time t3. Similar rules are added to the rule-base for different objects observed at various positions in the navigation environment.

In the implementation phase of the predictor, the robot observes the moving object at time t1 and t2 and sends the data to the fuzzy predictor algorithm. With the application of fuzzy inference process, prediction of the next instance position of the moving object is carried out. The complete process of short term motion prediction is represented in Figure 2.

Fig. 3. Division of navigational space into Directional Space.

Where the antecedents are same but the consequents are different. The reason for the inconsistency is due to the direction of traversal of the object. Future motion of the object is dependent on the history of the direction of the traversal of the object. To overcome this type of inconsistency, while defining the rule base, partitioning of the navigational space is done. Considering the navigational space that is tessellated in eight geographical Articles

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directions, the sensor readings of the object positions taken at previous two time intervals forms a trajectory in one of these directions. {SW(d1),S(d2),SE(d3),E(d4),NE(d5),N(d6), NW(d7),W(d8)} A separate directional space is created for each direction (Figure 3) and rules are clustered based on the direction of traversal object. Depending on the direction of traversal of the object, only those rules which belong to that directional space will be selected for processing.

4. Rulebase Optimization using Decision tree Approach The proposed fuzzy predictor algorithm has to process all the rules in a sequential form. The time complexity of the algorithm is linear and is of the order O(n). This is reduced by reordering the rules in the form of a decision tree. Each group of rules in the directional space is reorganized and IF-ELSE statements are written in the form of a decision tree. The decision tree is a classifier in the form of a tree structure, where each node is either a leaf node - indicates the value of the target attribute (class) of examples or a decision node specifies some test to be carried out on a single attribute-value, with one branch and sub-tree for each possible outcome of the test.

Considering the basic organization of the fuzzy rulebase (which is a sequential set of rules) for two rules Rule1: IF ((R1==2 , q1 == 2)) and IF ((R2==1, q2 == 1)) THEN R3,q3 = 21; Rule2: IF ((R1==2, q1 == 2)) and IF ((R2==1, q2 == 2)) THEN R3,q3 = 22; We can have rules i) starting with R1=2 and q1, R2, and q2 with any values from 0-6 ii) starting with R1=2 and q1 = 2 and R2 and q2 with any values from 0-6 iii) starting with R1=2, q1 = 2, R2=1 and q2 with any values from 0-6 iv) starting with R1=2, q1 = 2, R2=1, and q2 with any values from 0-6 These set of rules when organized in sequential order form a huge number of rules and consequently increasing the size of the rule base for processing. Using decision tree approach the two rules defined previously can be reorganized as follows.

1) if(R1==2) 2) { 3) if(q1 == 2) 4) { 5) if(R2==1) 6) { 7) if( q2 == 1) 8) {R3, q3 = 21;} 9) if(q 2 == 2) 10) {R3, q3 = 22; } 11) } 12) } 13) } 14

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In the above expression if R1š2 , no expression within the if block of R1 is executed. Similarly all the rules in the fuzzy rulebase can be reorganized in the form of a decision tree. For the developed rule-base, Figure 4 gives the partial representation of the decision tree for IF-ELSE statements. The input read by the fuzzy predictor algorithm classifies the input set to one of the directional spaces (1 to 8) defined in Section 3. Each internal node v is labeled with an integer 1 to 8 indicating the direction of traversal of the moving object and one of the directions will be selected based on the history of object motion. Each level in the decision tree corresponds to a fuzzy set indicating either the range or direction subsets (R1,q1,R2,q2). Each item in the fuzzy antecedent is processed as and when it receives inputs at each level in the decision tree and each input is a partial information of the position of the object in the navigational environment. Each interior node in the decision tree corresponds to a variable; an arc to a child represents a possible value of that variable. A leaf represents a possible value of target variable given the values of the variables represented by the path from the root. Based on the input one of the outgoing edges will be selected. The outgoing thick edge represents the selected fuzzy subset and remaining dotted edges represent the other unselected nodes.

Fig. 4. Optimization of Fuzzy rule-base using Decision tree.

To execute the algorithm, the process starts at the root node v, follows the edge labeled f(nv), and continues recursively. Thus, the execution of the algorithm gives a path from the root to some leaf. Each leaf has an integer label; when the execution reaches a leaf, its label is returned as the algorithm's output.

Let T(A, f) be the length of the root-to-leaf path in decision tree A traversed when the input is f(rule). The complexity T(A) of any decision tree algorithm A is its depth and the complexity of the problem is the depth of the shallowest decision tree. For the prediction algorithm, the time complexity of the decision tree representation of the rule-base system is given by T(n) = O(n)

(2)

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where n is the depth of the tree. μC(y) = max{μC'1 (y), μC'2 (y),….. μC'L(y)} Table 1 represents the selection of the nodes of the decision tree at various levels.

Table 1. Decision Tree Analysis for Short Term Motion Prediction.

Level 0 1 2

Nodes DS={1,2,3,4,5,6,7,8} IF (R1 and q1) and (R2 and q2) R1={0,1,2,3,4,5,6}

3

q1={0,1,2,3,4,5,6}

4

R2={0,1,2,3,4,5,6}

5

q2={0,1,2,3,4,5,6}

6

R3={0,1,2,3,4,5,6} and q3={0,1,2,3,4,5,6}

Type Directional Space Rule Antecedent

Defuzzification of the final output is done to get the crisp value. Three most commonly used defuzzification techniques are considered: i) Fuzzy OR method/Min-Max, ii) Center Of Area (COA) and iii) Mean Of Maximum (MOM) methods. These methods operate on range and angle output subsets separately to generate the final crisp value, indicating the range and angle of the final output.

6. Experimental Results Fuzzy Object distance at time t1 Fuzzy Object angle at time t1 Fuzzy Object distance at time t2 Fuzzy Object angle at time t2 Rule Consequent: Predicted Fuzzy Region at time t3

5. Fuzzy Rule-base Implementation for Motion Prediction The rule-base implementation comprises of the observation of the moving objects in the navigational environment at equal time intervals and prediction of their future position using the Fuzzy rule-base. This step involves the fuzzy inference process. The Fuzzy inference process comprises five parts: fuzzification of the input data, application of the fuzzy operator (AND or OR) in the antecedent, implication from the antecedent to the consequent, aggregation of the consequents across the rules and defuzzification. The fuzzy inference process adopted the Mamdani model. The Mamdani model uses rules whose consequent part is a fuzzy set.

Ri: if x1 is Ai1 and x2 is Ai2 and x3 is Ais then Y is Ci i = 1,2,3 ...M

(3)

where M is the number of fuzzy rules, xj Î Uj (j = 1, 2, 3 ... s) are the input variables, y Î V is the output variable and Aij and Ci are fuzzy sets characterized by membership functions μAij(xij) and μc'i, respectively.

Given the inputs of the form x1 is A'1, x2 is A'2, ... Xr is A'r where A'1, A'2, ... A'r are Fuzzy subsets of U1, U2, ... Ur. The contribution of rule Ri to Mamdani model's output is a Fuzzy set whose fuzzy membership function is computed by μ

(5)

μc'i(y) = μA'1 (x1) A'2 (x2) ... μA'j(xj)}

(4)

where ^ denotes the 'min' operator. The final output of the model is the aggregation of outputs from all the rules using the max operator.

Table 2. Evolution of Short term predictor.

Development Stage

Basic Unoptimizeed Fuzzy predictor

Number of Rules to be processed in the Worst case 1200

Predictor with Directional Space Approach

140 (Approx)

Predictor with Decision Tree Approach

43(Approx)

Average Time Complexity O(n) where n is the number of Fuzzy Rules O(n') where n' is the number of Fuzzy Rules and n'< n O(log n')

Table 2 represents the evolution of the Fuzzy predictor algorithm. The table is parameterized by the stage of the algorithm development, the number of rules to be processed and the time complexity. The unoptimized fuzzy predictor consists of all the rules identified during the formation of the rule-base. As the rule-base is large and consists of inconsistent rules, its response time and relative error is high. All the rules are processed in a linear order, the time complexity of the predictor is O(n) where n is the number of rules. The directional space approach clusters the rules in different directions which reduces inconsistency, as well as response time. The decision tree approach reorganizes the rule-base and reduces the response time and time complexity of the predictor to O(logn') where n' is the number of rules processed by the predictor algorithm. The fuzzy predictor algorithm is developed in C++ language.

The algorithm is tested on 1.66 GHz machine in VC++ environment. The tests are carried out for real-life benchmarked datasets[3], [11], [13]. These data sets are gathered through i) INRIA Labs with data captured at INRIA Labs at Grenoble, France (A wide angle camera lens in the entrance lobby of the INRIA Labs at Grenoble, France. The resolution is half-resolution PAL standard); ii) Motion Capture Web group of Univ. of S. California (Consisting of Human Motion Patterns); iii) CMU Graphics Lab dataset.

(Vicon motion capture system consisting of MX-40 cameras with images of 4 megapixel resolution).

The data sets consist of different human motion patterns. These include people walking alone, running, meeting with others, window shopping, entering and exiting shops (average speed in the range 2-10 kmph). The position of the moving objects within the navigational Articles

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environment at any instant of time is given separately as a database so that any prediction algorithm can be tested and analyzed for any number of objects. These motions exhibit intentional motion and predicting the next instance position of objects in such scenario is an important task as it can find applications in keeping track of human motion patterns in hospitals, shopping complex and in exhibition halls etc.

Figure 5 represent the movement of the objects from left to right direction and the corresponding short term motion prediction path. Pi and Ai represent the predicted and the actual path traversed by the moving object.

Pi(G) and Ai(G) represent the predicted goal and the actual goal of the object. A1 is the actual path observed and A1(G) is the actual goal reached by the object A1.

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position as a fuzzy region than as a (x,y) coordinate. This helps in the robot to classify the predicted region as a danger zone or the region of interest.

Fig. 6. Average response time and relative error of the Short term predictor at prediction step: 02 seconds.

Fig. 5. Prediction graphs showing few of the path prediction solutions for Short term motion prediction.

We define the Relative Error (RE) for M sample test data (sum of the number of predicted positions for a specific object in motion) as RE =

(da-dp) /M da

(6)

Where da is the actual position, dp is the predicted position of the moving object in the navigational environment.

The average relative error is calculated for various test cases using Min Max, MOM and COA defuzzification techniques. For each test case the average response time is also calculated to find its suitability to real-life environment. For measuring the performance of the system the standard parameters like prediction steps and relative error are used. The prediction algorithm is tested with prediction steps 02 seconds (Fig. 6), 03 seconds (Fig. 7), 04 seconds (Fig. 8).

Table 3 represents the results of the Short term predictor at various stages of development. Each stage in the evolution of the fuzzy predictor is parameterized by the relative error and average response time. These prediction steps indicate the in between time gap for each successive measurement (of the object position) by the vision system.

Variations in the velocity and directions of motion of the moving objects in these test cases are the sources of uncertainty in predicting the next instance position of the moving object. Tests are carried out to measure the relative error between the actual and predicted positions when minute variations in velocity and directions of the moving objects are observed (Fig. 9).

The proposed predictor generates the next instance 16

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Fig. 7. Average response time and relative error of the Short term predictor at prediction step: 03 seconds.

Defuzzification of the output generates the predicted coordinate position of the moving object. The response time of the algorithm with Min Max defuzzification varied in the band from 1.45 milliseconds to 2.9 milliseconds and the relative error in the band from 0.04 to 0.4.

The response time of the algorithm with COA defuzzification varied in the band from 3 milliseconds to 7 milliseconds and the relative error in the band from 0.01 to 0.1. The response time of the algorithm with MOM defuzzification varied in the band from 1.95 milliseconds to 3.37 milliseconds and the relative error in the band from 0.04 to 0.1. From the graphs it is observed that the predictor with MOM defuzzification performs better in terms of response time with less relative error.

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Table 3. Results of Short term predictor at various stages of development.

Development Stage

Relative Error

Basic Unoptimized Fuzzy predictor Predictor with Directional Space Approach Predictor with Decision Tree Approach

1-20% 1-15%

1-10%

Average Response time in millisec 500 15-20

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Table 4. Comparison of Short term predictors.

Short Term Predictor Neural Network predictor[8] Bayesian Occupancy Filters[10] Extended Kalman Filter[12] Fuzzy Predictor with MOM

Relative Error 6-17%

Response time in seconds 560x 10-3 sec

1-10%

100x 10-3 sec

1-20%

0.1 sec

1-10%

02x 10 sec to 05x 10-3 sec

A few of the well known motion prediction techniques are re-implemented and are compared with the developed fuzzy predictor in respect of response time and relative error (Table 4). From the table it can be observed that the performance of the predictor is comparable with regard to relative error but better than the other prediction methods as far as response time is concerned.

7. Conclusion

Fig. 8. Average response time and relative error of the Short term predictor at prediction step: 04 seconds.

In a dynamic navigation system the robot has to avoid stationary and moving objects to reach the final destination. Short term motion prediction for moving objects in such an environment is a challenging problem.

This paper proposes a simplified approach for predicting the future position of a moving object (human motion patterns) using fuzzy inference rules derived from experts knowledge and real-life data. The rule-base has been optimized by directional space approach and decision tree approach. Fuzzy based prediction is more flexible, can have more real life parameters, comparable to the existing approaches and suited for real-life situations.

The results of the study indicate that, the fuzzy predictor algorithm gives comparable accuracy with quick response time when compared to existing techniques.

ACKNOWLEDGMENTS The authors are thankful to the benchmark dataset provided by EC Funded CAVIAR project, CMU Graphics lab and Motion capture web group. We are indebted to AICTE, Government of India, for funding our project.

AUTHORS Vijay S. Rajpurohit* - Department of Computer Science and Engineering, Gogte Institute of Technology, Belgaum, 590008, India. E-mail: vijaysr2k@yahoo.com.

M.M. Manohara Pai - Department of Information and Communication Technology, Manipal Institute of Technology, Manipal, 576104,India.

E-mail: mmm.pai@manipal.edu. * Corresponding author

References [1]

Fig. 9. Relative Error when measured with variations in velocity and direction of motion of moving objects.

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Foka Amalia, Trahanias Panos E., “Predictive Autonomous Robot navigation”. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, EFPL, Lausanne, Switzerland, Oct. 2002, pp. 490-494.

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Chang Charles C., Song Kai-Taj, “Dynamic motion planning based on Real-Time object Prediction”. In: Proceedings of the 1996 IEEE Intl. Conference on Robotics and Automation, Minnesota, USA. Laugier C., Petti S., “Navigation in Open and Dynamic Environments”. In: IEEE International Conference on Robotics and Automation (ICRA'05), 2005, Barcelona, Spain.

Huang Hong, Chang Liuchen, “Simulation study on the membership functions of the basic logic controller for the electric vehicle propulsion systems”. In: Canadian Conference on Electrical and Computer Engineering, Calgary, Canada, May 1996, pp. 1004-1007.

Yu Huiming, Su Tong, “A Destination Driven Navigator with Dynamic Object Motion Prediction”. In: International Conference on Robotics 2L Automation, Seoul, Korea.

21st-26th May, 2001, pp. 2692-2697.

Zhao Jin, Bose Bimal K., “Evaluation of membership functions for fuzzy logic controlled induction motor drive”. IECON Proceedings (Industrial Electronics Conference), volume 1, 2002, pp. 229-234.

Kalman R., “A New Approach to Linear Filtering and Prediction Problem”, Transactions of the ASME-Journal of Basic Engineering, vol. 82, 1960, pp. 35-45.

Webb P., Fayad C., Brettenbatch C., “The integration of an optimized Fuzzy logic navigation algorithm into a semi-autonomous Robot Control System”. In: International Workshop on Recent Advances in Mobile Robots, Leicester, UK, 29th June 2000.

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Madhavan R., Schlenoff C., Moving Object Prediction for Off-road Autonomous Navigation. SPIE Aerosense conference, Orlando, 21st-25th April, 2003.

Fisher Robert, Santos-Victor Jose, James Crowley (2001), CAVIAR Video Sequence Ground Truth, Retrieved Jan 2008,fromhttp://homepages.inf.ed.ac.uk/rbf/ CAVIAR/, funded by EC Funded CAVIAR project/IST 2001 37540.

Berkan Riza C., Trubatch Sheldon L., Fuzzy Systems Design Principles: Building Fuzzy If-Then Rule Bases.

John Wiley & Sons Inc., 1997.

Irajit R., Manzuri-Shalmanit M. T., “A New Fuzzy-Based Spatial Model for Robot Navigation among Dynamic Obstacles”. In: IEEE International Conference on Control and Automation, Guangzhou, China, 30th May - 1st une, 2007, pp. 1323-1328.

Rajpurohit Vijay S., Pai Manohara M. M., “Feature Extraction Learning For Stereovision Based Robot Navigation System”. In: 14th IEEE International Conference on Advanced Computing Communications - ADCOM 2006, pp. 362-365, ISBN-1-4244-0715-X, NITK, Surathkal,

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CONTROL IMROVEMENT OF SHUNT ACTIVE POWER FILTER USING AN OPTIMIZED-PI CONTROLLER BASED ON ANT COLONY ALGORITHM AND SWARM OPTIMIZATION Received 6th February 2010; accepted 26th July 2010

Brahim Berbaoui, Brahim Ferdi, Chellali Benachaiba, Rachid Dehini

Abstract: In the last years, there has been a increase currents harmonics on electrical network injected by nonlinear loads, such as rectifier equipment used in telecommunication system, power suppliers, domestic appliances, ect.

This paper makes a comparison of the effectiveness of the two methods on particular optimization problem, namely.

The tuning of the parameters for PI DC link voltage to a shunt active power filter. The simulation results demonstrates that the optimized PI controller by ant colony (ACO) presents a advantage of little response time and best control performances compared to the optimized PI with Particle swarm (PSO). This comparison is shown on reducing harmonic current supply (THD).

Keywords: ant colony optimization, particle swarm optimization, shunt active power filter, harmonic compensation, PI controller.

1. Introduction The advancements of electronic devices technology and its use in the industry has produced harmonic in line distribution network [1]. In order to eliminate theses troubles, researchers has proposed new technique to eliminate theses harmonics. One of the theories is the instantaneous power theory (p-q theory). This theory was introduced by Akagi, Kanazawa and Nabae in 1983 [2] in Japanese.

By tradition, passives filters have been used to eliminate the current harmonic distortion and compensate the reactive power, but can resonate with supply impedance.

The correct implantation of PI controller DC link voltage of (SAPF) depends two parameters proportinnal gain (Kp) and integral gain (Ki) which are tuned by trial and error. In the other hand has a problem in the time needed to accomplish this task.

To trounce this problem many methods have been developed, such as Ziegler-Nichols [3].

An improvement in tuning can be achieved using optimization techniques, and in particular those based on artificial intelligence.

In this paper, we formulate the problem of design DC link voltage PI controller as an optimization problem. The problem formulation adopts three performances indexes, the maximum overshoot, the rise time and the integral absolute error of step response as the objective function to determine the PI control parameters for getting a well performance under a given system, the primary design goal is to obtain good load disturbance response by mini-

mizing the integral absolute control error. At the same time, the transient response is assured by minimizing the others three performance indexes.

Two approach methods has been used to show its impact on SAPF the ant colony algorithm and particle swarm algorithm.

2. Ant colony optimization The main idea of ACO is to model the problem as the search for a minimum cost path in a graph that base the evolutionary meta-heuristic algorithm. The behavior of artificial ants is inspired from real ants. They lay pheromone trails and choose their path using transition probability. Ants prefer to move to nodes which are connected by short edges with a high among of pheromone. The algorithm has solved traveling salesman problem (TSP), quadratic assignment problem (QAP) and job-shop scheduling problem (JSSP) and so on [4]-[5].

The problem must be mapped into a weighted graph, so the ants can cover the problem to find a solution. The ants are driven by a probability rule to choose their solution to the problem (called a tour). The probability rule (called Pseudo-Random-Proportional Action Choice Rule) between two nodes i and j.

(1) The heuristic factor hij or visibility is related to the specific problem as the inverse of the cost function. This factor does not change during algorithm execution; instead the metaheuristic factor tij (related to pheromone which has an initial value t0) is updated after each iteration. The parameters a and b enable the user to direct the algorithm search in favor of the heuristic or the pheromone factor. These two factors are dedicated to every edge between two nodes and weight the solution graph.

The pheromones are updated after a tour is built, in two ways: firstly, the pheromones are subject to an evaporation factor (r), which allows the ants to forget their past and avoid being trapped in a local minimum (equation 2). Secondly, they are updated in relation to the quality of their tour (equations 3 and 4), where the quality is linked to the cost function.

(2) (3) belong otherwise

(4) Articles

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Where m is the number of ants, L represents the edges of the solution graph, and Ck is the cost function of tour Tk, built by the kth ant.

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4. Organization of objective function In this work, the optimized parameters objects are proportional gain Kp and integral gain Ki, the transfer function of PI controller is defined by:

3. Particle swarm optimization Particle swarm optimization (PSO) is a population based stochastic optimization technique inspired by social behavior of bird flocking or fish schooling [6]. PSO learns from the scenario and uses it to solve the optimization problems. In PSO, each single solution is a "bird" in the search space.

We call it "particle". All particles have fitness values which are evaluated by the fitness function to be optimized, and have velocities which direct the flying of the particles. The particles fly through the problem space by following the current optimum particles.

PSO is initialized with a group of random particles (solutions) and then searches for optima by updating generations. In each iteration, every particle is updated by following two "best" values. The first one is the best solution (fitness) it has achieved so far. (The fitness value is also stored.) This value is called Pbest. Another "best" value that is tracked by the particle swarm optimizer is the best value, obtained so far by any particle in the population. This best value is a global best and called gbest.

For example, the i.th particle is represented as xi = (xi1, xi2, ..., xid) in the d-dimensional space. The best previous position of the i.th particle is recorded and represented as: Pbesti = (Pbesti1, Pbesti2, ..., Pbestid)

(5)

The index of best particle among all of the particles in the group is gbestd. The velocity for particle i is represented as vi = (vi1, vi2, ..., vid).

The modified velocity and position of each particle can be calculated using the current velocity and the distance from Pbestid to gbestd as shown in the following formulas [8].

(8) The gains Kp and Ki of PI controller are generated by the ACO and PSO algorithm for a given plant. As shown in Fig. 1. The output u(t) of PI controller is given by (equation 9): ACO _ PSO r (t )

e(t )

Gs (s )

u (t )

G p (s )

y (t )

Fig.1. PI control system.

(9) For a given plant, the problem of designing a PI controller is to adjust the parameters Kp and Ki for getting a desired performance of the considered system. Both the amplitude and time duration of the transient response must be kept within tolerable or prescribed limits, for this condition, three key indexes performance of the transient response are utilized to characterize the performance of PI control system. These key indexes maximum overshoot, rise time and integral absolute control error are adopted to create objective function which is defined as: (10) The maximum overshoot is defined as:

t+1) t) t + c1rand()( Pbesti,m - xi,m v(i,m = wv(i,m ) + c2rand() (t) ( gbestm - xi,m) t+1) (t) (t+1) + vi,m x(i,m = xi,m

(11) (6)-(7)

i = 1,2,...,n; m = 1,2,...,d; Where: - Number of particles in the group, n - Dimension, d - Pointer of iterations (generations), t t) - Velocity of particle i at iteration t, v(i,m - Inertia weight factor, w c1, c2 - Acceleration constant, rand() - Random number between 0 and 1, - Current position of particle i at iterations, v(i,dt) Pbesti - Best previous position of the i.th particle, Gbest - Best particle among all the particles in the population.

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ymax characterize the maximum value of y and yss denote the steady-state value of y.

For yrt represent the function of the rise time is defined as the time required for the step response.

In the other hand, the integral of the absolute magnitude of control error is written as: (12)

5. Configuration of shunt active power filter The most important objective of the APF is to compensate the harmonic currents due to the non linear load.

Exactly to sense the load currents and extracts the harmonic component of the load current to produce a reference current Ir as shown in Fig. 2, The reference current consists of the harmonic components of the load current which the active filter must supply [7].

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This reference current is fed through a controller and then the switching signal is generated to switch the power switching devices of the active filter such that the active filter will indeed produce the harmonics required by the load. Finally, the AC supply will only need to provide the fundamental component for the load, resulting in a low harmonic sinusoidal supply.

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(15) The harmonic component of the total power can be extracted as: (16) Where, pL : The DC component pL : Harmonic component

Current Detection

Similarly,

Control Circuit

(17) Finally, we can calculate reference current as:

(18) Here,

(19)

Fig. 2. Equivalent schematic of shunt APF.

7. ACO and PSO applied to optimize PI parameters of dc-link capacitor

6. Instantaneous active and reactive p-q power theory

In this paper, we present the SAPF as controlled plant, the SAPF diagram is shown in Fig. 3.

The identification theory that we have used on shunt APF is known as instantaneous power theory, or PQ theory.

It is based on instantaneous values in three-phase power systems with or without neutral wire, and is valid for steady-state or transitory operations, as well as for generic voltage and current waveforms.

Load

Inputs: Vector of tension: va (t), vb (t) and vc (t) Vector of current: ia (t), ib (t) and ic(t)

Converter

The PQ theory consists of an algebraic transformation (Clarke transformation) of the three phase voltages and current in the abc coordinates to the ab coordinates [8].

ref current generator

ACO-PSO (13) Fig. 3. Control diagram of APF system.

(14) The instantaneous power is calculated as:

The estimation of the reference currents from the measured DC bus voltage is the basic idea behind the PI controller based operation of the SAF. The capacitor voltage is compared with its reference value v*da in order to maintain the energy stored in the capacitor constant.

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The DC link voltage discretely at the positive zero-crossing point of respective phase source voltage, computes the variation of power according to difference of DC link voltage between two sampling points. The regulation of the error between the capacitor voltage and its reference is assured by The PI controller which its output is multiplied by the mains voltage waveform Vs1, Vs2, Vs3 in order to obtain the supply reference currents.

The equivalent schematic diagram of system which is used to maintain the DC link voltage constantly is shown in Fig.4.

In this work, the objective of an optimal design of PI controller DC-Link for given plant is to find a best parameters Kp and Ki of PI control system such that the performance indexes on the transient response is minimum.

For ACO approach, each parameter of Kp and Ki is hinted by 100 nodes respectively and there is resolution 0.0001 among each node, one node represents a solution value of parameters Kp and Ki. Thus, the more accuracy trails are updated after having constructed a complete path and the solution found.

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Table 1. Initial values parameters of ACO. Ant Number Maximum Cycle Time Initial Value of Nodes Trail Intensity Coefficient r Relative Important Parameter of Trail Intensity a Relative Important Parameter of Visibility b

20 100 0.1 0.6 3 2

And for the particle swarm the parameters values are presented Table 2.

Table 2. Parameters of PSO algorithm.

Population Size Number of Iterations wmax wmin c1 = c2 Min-offset

60 150 0.7 0.1 1.5 200

9. Simulation result Load

The proposed PI controller of DC link capacitor designed by ACO and PSO on filtering system that was set in Matlab Simulink environment to predict performance of the proposed method. The SAPF model parameters are shown in the following Table 3.

Table 3. SAPF parameters.

Converter

ref current generator

Fig. 4. Equivalent schematic diagram system.

For PSO approach, the evolution procedure of PSO Algorithms is presented as follow Fig. 5. Producing initial populations is the first step of PSO. The population is composed of the chromosomes that are real codes. The corresponding evaluation of a population is the “fitness function” which is the performance index of a population.

The fitness value is bigger, and the performance is better. After the fitness function has been calculated, the fitness value and the number of the generation determine whether or not the evolution procedure is stopped (Maximum iteration number reached?). After this, calculate the Pbest of each particle and Gbestid of population (the best movement of all particles). The update the velocity, position, gbest and Pbest of particles give a new best position.

8. Design of optimizing algorithm In this work, we have used the following parameters values for the ant colony optimization which is step in the Table 1.

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Supply phase voltage U Supply frequency fs Filter inductor Lf Dc link capacitor Cf Smoothing inductor Sample time Ts

220 V 50 Hz 1mH 4.4 mF 0.1 mH 4 μs

A number of simulation results were developed with different cases. The SAPF is connected in parallel with nonlinear load, the first case is the PI-classical using on the system to allow us to see the regulation of DC link voltage and its effect for damping harmonics current and reducing total harmonic distortion (THD). For the second case the proposed optimized PI-controller with ACO and PSO has been introduced in order to improve a SAPF performance and meet the requirements of harmonic elimination and reactive compensation.

10. A Case of classical PI-controller In the conventional PI controller the parameters Kp and Ki has been determined by classical method which is Ziegler-Nichols method for tuning PI controller. This procedure is now accepted as standard in control system and is based on plant step responses.

The method used in this work known as the continuous cycling which integration and derivative terms of the controller are disabled and the proportional gain is increased until a continuous oscillation. Considering Ku and its related oscillating period Tu, the PI parameters can be calculated from the following equation:

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(20) The PI control scheme involves regulation of the dc link voltage to set the amplitude of reference current for harmonic and reactive power compensation [9], [10]. In this study, we have simulated only the network supply connected on the nonlinear load, the total harmonic distortion found is 20.90 % which indicate the harmonic presence in the current source caused by nonlinear load.

In the first case, the SAPF has been introduced in order to compensate these harmonics and has reduced the THD from 20.90% to 0.99%. The results founded are shown in the following figures.

Fig. 7. Injected current waveform.

Fig. 6. Load current waveform Table 4. Harmonic supply current phase-a-component with traditional PI controller method.

Fig. 8. DC link voltage waveform.

Harmonic supply current components Isa(n)/Isa(1) [%] N 5 0.18 7 0.34 11 0.24 13 0.21 17 0.21 19 0.17 23 0.18 25 0.13 29 0.16 31 0.11 35 0.12 37 0.11 0.12 41 43 0.10 47 0.11 0.10 49 0.99 THD

Fig. 9. Harmonic spectrum of supply current.

11. B Case of optimized PI-controller

Table. 5. THD results.

Without filtering THDi (%)

20.90

FSAPF filtering with classical PI DC link voltage 0.99

Robustness

3.90

In the second case, the shunt active power filter was examined using optimized PI - controller DC link voltage, the optimal parameters has been determined by using ant colony optimization (ACO) and particle swarm optimization (PSO). The main objective is to minimize the fitness Articles

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function that is defined by the equation (10).

In this paper, we have based on the minimizing integral absolute error, so it has been multiplied by coefficient a. The objective function is returned by the following equation: (21) In this case, we have fixed a value: a = 2.5 and that to give an importance for the integral error in formulation function. Simulation studies are carried out to predict performance of the proposed method. The Fig. 10 shows the DC link voltage response curves of system used primal PI parameters and optimized PI parameters, and the value of system indexes are compared in Tab. 5. The source voltage, current, load current, harmonic order and Dc link voltage waveforms are shown in the following figures after adopted the optimized system.

In Fig. 10, the stability convergence and robustness.

Hence, the high performance can be achieved.

Table. 5 comparisons of SAPF indexes between used and unused ant colony algorithm and particle swarm algorithm.

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Table 6. Harmonic supply current phase-a-component with optimized PI controller methods.

N 5 7 11 13 17 19 23 25 29 31 35 37 41 43 47 49 THD

Harmonic supply current components Isa(n)/Isa(1) [%] Isa(n)/Isa(1) [%] using PSO using ACO 0.14 0.13 0.33 0.33 0.21 0.20 0.21 0.20 0.19 0.19 0.15 0.15 0.18 0.17 0.11 0.11 0.15 0.14 0.09 0.09 0.11 0.11 0.09 0.09 0.10 0.10 0.09 0.09 0.10 0.09 0.10 0.09 0.93 0.91

Parameter PI non PI PI and indexes optimized with ACO with PSO Proportional gain 120 190 180 Integral gain 1.05 0.0004 0.00028 Overshoot (%) 85.66 88.52 87.99 Rise time (sec) 0.0009 0.000869 0.00087 Integral 1.0182e+001 6.8543e+000 7.0013e+000 absolute error

Fig. 11. Source voltage waveform.

Fig. 10. DC link voltage response curve of SAPF used ant colony optimization and particle swarm optimization.

The results we obtained demonstrate that a low THD value can be reached by using the optimized system studied in this paper.

The current source represented by Fig. 12 takes the sinusoidal form, as well as the spectral analysis Table 6 shows the absence of the more share of the harmonics rows which implies the good performances of the optimized PI-controller compared with classical PI. 24

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Fig. 12. Supply current waveform of single phase.

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Table 7. THD results.

Without SAPF using SAPF using Robustness filtering optimized PI optimized PI with PSO with ACO THDi 20.90 0.93 0.91 4.25/ACO (%) 4.09/PSO

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AUTHORS Brahim Berbaoui*, Brahim Ferdi, Chellali Benachaiba, Rachid Dehini - Department of Electrical Engineering, Bechar University, B.P 417 BECHAR (08000), Algeria.

Email: b_berbaoui@yahoo.fr. * Corresponding author

References

12. C Comparative Study In this paper we have presented a comparative between two optimization approach ant algorithm and swarm algorithm for design PI controller DC link voltage of SAPF. The PI-ACO control method has improved the active power filter performance compared with PI-PSO and traditional PI controller, and it can be seen in the Supply current filtering result Fig. 12.

The deformations have been clearly reduced and also the harmonic distortion has been decreased compared with the SAPF filtering using traditional PI controller method, so this comparison can been shown in the following figure.

[1] [2]

[3]

[4]

[5]

13. Conclusion According to the results of the computer simulations, the optimized PI using ACO is better than the traditional PI and also PI with PSO. The PI with ACO algorithm is the best controller which presents satisfactory performances, less overshoot and minimal rise time compared with classical PI and optimized PI with PSO. Furthermore, results has demonstrated that the control strategy with ACO for DC link voltage is efficient for compensating the current harmonics, and the proposed system has reduced the THD with 8 % less than primal system and 2% less than system using particle swarm optimization ( SAPF without ACO ) as shown in Fig. 13.

So we can say that the PI-ACO is the best controller which presented satisfactory performance and good robustness.

[6]

[7]

[8]

[9]

ACKNOWLEDGMENTS This work was supported by the AGH University of Science and Technology under Grant No. 11.11.120.612.

[10]

Fang Zheng Peng, ”Application Issues of Active Power Filters”. IEEE Industry Application Magazine, 1998.

Akagi H., Kanazawa Y., Nabae A., "Generalized Theory of Instantaneous Reactive Power and Its Applications".

Transactions of the IEE-Japan, Part B, vol. 103, no.7, 1983, pp. 483-490. (in Japanese) Ziegler J.G., Nichols N.B., „Optimum settlings for automatic controllers”. ASME Transactions, Vol. 64, 1942, pp. 759-768.

Dorigo M., Gambardella L.M., “Ant Colony System: A Cooperation Learning Approach to the Traveling Salesman Problem”. IEEE Trans. Evolutionary Computation, vol. 1, no. 1, 1997, pp. 53-66.

Maniezzo V., Colorni A., “The Ant System Applied to the Quadratic Assignment Problem”. IEEE Transactions on Knowledge and Data Engineering, vol. 11, issue 5, 1999, pp. 769-778.

Kennedy J., Eberhart R.C., “Particle Swarm Optimization”. In: Proceedings of the 1995 IEEE international conference on neural networks, vol. 4. Piscataway, NJ: Inst. of Electrical and Electronics Engineers; 1995, pp. 1942-8.

Wada K., Fujita H., Akagi H., “Considerations of a Shunt Active Filter Based on on Voltage Detection for Installation on a Long Distribution Feeder'. In: Proc. Conf.

IEEE-IAS Ann. Meeting, 2001, pp. 157-163.

Akagi H., Kanazawa Y., Nabae A., ”Generalized Theory of the lnrtrntansour Reactive Power in Three-Phssc Circuirs”. In: Proc. IPEC Tokyo '93 Int. CO Power Electronics, Tokyo, 1983, pp. 1375-1186.

Gu J.J., Xu D.G., “Active power filters technology and its development”, Electric Machines And Control, vol. 7, no. 2, 2003, pp. 126-132.

Jou H.L., Wu J.C., Chu H.Y., “New single-phase active power filter”. Proc. Inst. Elect. Eng., Electr. Power Appl., vol. 141, no. 3, May 1994, pp. 129-134.

Fig. 13. Comparative harmonic spectrum of supply current for both with classical PI, PI-AO and PI-PSO. Articles

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THROWABLE TACTICAL ROBOT – DESCRIPTION OF CONSTRUCTION AND PERFORMED TESTS Received 27th July 2010; accepted 30th August 2010

Rafał Czupryniak, Maciej Trojnacki

Abstract: This paper concerns a throwable tactical robot (TTR) for special purposes. The necessity of use of that kind of robots and the existing design solutions are discussed. There are also described construction, parameters, and principles of operation of the robot and the control panel, as well as the conducted robot tests.

Keywords: throwable robot, special robot, teleoperation.

1. Introduction According to increasing rate of terror acts the character and methods of setting up explosive charges has changed. Therefore, there appeared a need to change conception, construction and the scale of pyrotechnic robots and even their tasks. In the last years a bigger emphasis was placed on development of small robots which purpose is not neutralization but acquiring information.

Thanks to acquiring significant information, the services are able to react adequatly to the incident. The role of small robots is and will be to perform preliminary reconnaissance of incident place, acquiring information for intervention squads in open or secret manner. The other vital attribute of small robots is their independent work as information sources, tracing and simple neutralization machines. The main advantage should be the possibility

to reach every target virtually unnoticeably either in a combat mission or rescue action. Their small sizes should assure secrecy and possibility of free penetration of very small spaces. [1], [2], [5]

2. Throwable robots The review of existing design solutions shows the most desired direction of development of this kind of robots for active teleobservation and tracing. The main characteristics which small robots should have is the possibility to place them by hand throwing at the operator's interest area. That is in the simplest and fastest way, decreasing the danger to indispensable minimum, allowing to penetrate spaces inaccessible for the operator, i.e. beyond an obstacle. [1], [2] Nowadays we can find information about different commercial and non-commercial solutions of throwable robots. One of non-commercial robots of this kind is the robot presented in the Figure 1. It consists of two modules connected with each other by a common driven axis.

It has also 4 balls (Omni-Ball) positioned at the end of modules that have one active and two passive axes of rotation. Such a solution of the robot construction enables both its moving, in a collapsed form, in narrow areas using balls drive, and moving, for instance, in debris area using then additionally the drive providing the mutual rotation of two modules. [3]

Fig. 1. Throwable Tethrahedral Mobile Robot an its use conception [3].

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There are several throwable robots sold on the market, among others SpyBowl 360, Eye Ball, Recon Scout and EyeDrive. Almost each of those devices has a different construction idea of making teleobservation and tracing possible. [1], [5] SpyBowl (Fig. 2a) is a device thrown or rolled towards the target. The device is made as aluminium body covered with rubber coating in form of a ball with 115 mm in diameter. Such a construction allows for the transport of large, repeated loads. It is equipped with four cameras allowing for the acquisition of static images (Fig. 2b) and with microphones transmitting the sound. The device can rotate about its vertical axis with speed of 0.22 rad/s, which allows us to watch all environment in dynamic way.

Additionally the image can be seen from each camera independently. The range of the radio transmission varies between 20-30 meters inside building and 100-300 m outside. Entire device weighs 1 kg and can be thrown at the distance of 30 meters or thrown up to the height of 6 meters. Operating time on a battery is 45 min. The main application place of the SpyBowl device are closed rooms and buildings in the action zone of military and police special forces. [1], [5]

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place, straight before entering upon an intervention. The device has one camera providing a good quality picture to 23 meters. In order to collect complete information about the environment the device rotates about its own axis with the speed of 4 turns/min. Thanks to an extra software it is possible to acquire a panoramic view. Besides, the device has near infrared illuminators of the range of 8 meters and thanks to them the camera is able to see in the darkness. The microphone has the range of 5 meters. Operating time on battery is 2 hours, in standby mode - 24 hours. Radio and video transmission takes place at a distance up to 125 meters depending on the environment. [1], [4], [7]

(a)

Fig. 3. Throwable robot Eye Boll [7].

(b)

Fig. 4. Robot EyeDrive [7].

Fig. 2. Throwable robot SpyBowl (a) and the view of one the robot`s cameras (b) [5].

A similar device, regarding design, is Eye Ball R1 (Fig. 3). It is designed for throwing at the distance of 50 meters, rolling or dropping. It provides an audio and video transmissions in real time. The device is used in tactic operations, where special forces take the advantage of newest information about situation in given

The third interesting device for teleobservation and tracing is the Recon Scout robot. It is a mobile two-wheeled robot with titanic body and wheels from the urethane plastic. Such a construction allows throwing the robot at the distance up to 31.5 meters and dropping from the height of 9.1 meter. Moving forward is enabled by the socalled tail, which is the robot's support. Robot's parameters are following: width 187 mm, wheels diameter 76 mm, speed 1,1 km/h, range inside the building to 30 meters, outside 76 meters, working time 1 hour. The robot is equipped with black&white camera with sensitivity of 0,0003 lux. Due to small size it succeeded to obtain a total weight of the device - 0.544 kg. [1],[6] The last presented robot is EyeDrive (Fig. 4). This is a four-wheeled robot produced in Israel (with the possibility to use a caterpillar track) operated by a single man.

The robot can be thrown up to 3 meters high. The system of cameras allows for obtaining a panoramic view with image definition of 2500x570 pixels. The microphone transmits sound from the distance of 10 m. The robot's Articles

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range inside the building is 70 meters and outside to 300 meters. Operating time on battery is 3 hours and in stand-by mode - 24 hours. The robot`s weight is 2.3 kg, there is a possibility to carry additional loads (sensors, explosives, etc.) weighing up to 3 kg. [1], [7] Summing up the review of throwable mobile robots we can presume that the future of robotics and consequently of mobile robots looks promising. According to the development of technology and electronics we can expect a wider use of remotely controlled and autonomous devices. They assure safe accomplishment of the task without endangering people`s life. The only eventual loss can be damaged technical unit. [1]

Discarding height (limiting) Range inside building Range in open space Standard equipment

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9m 30 m 100 m Camera, microphone

3. Throwable tactical robot 3.1. Device conception Throwable tactical robot (TTR) has been designed for active teleobservation in military, police and rescue applications. It is a solution for threats which brings reconnaissance done by special forces before starting the action.

TTR is a device, which can be placed at the target area from a considerable distance and then survey it while being teleoperated. The camera and microphone placed inside the robot and its mobile abilities result in it being a perfect reconnaissance device limiting significantly the risk of health or life loss of a group members performing actions in the dangerous area. In assumption the TTR can be equipped with additional external device, the so-called rucksack, which enables to carry specialised charges: flashbangs, deafening and explosives. Additional equipment enables TTR use for explosive charges neutralization by pyrotechnic troops or making disorganisation and panic in the aggressors group.

The construction of the robot has been developed as a part of the developement project for which Table 1 contains the assumption data. The design work over the robot has been partially supported by analisys done with the usage of MD Adams and Ansys software.

Two different models of the device have been evolved.

According to the first solution (Fig. 5), the robot was supposed to have a trunk, being at the same time the running gear.

The first idea has been abandoned, because of the forseen technological and technical difficulties with mounting and exploatation.

The second conception of the robot's construction developed in the project is desribed in point 3.2. It is characterized by greater compactness and funcionality thanks to the ability to join additional load to the trunk.

It was not possible in the previous version of the device.

Fig.5. The first conception of TRM's construction (section).

3.2. Robot construction Robot's body (Fig. 6a) is a specially formed cylinder which is an assembling base for all construction elements both outside and inside. Inside the body there are made ribs for double purpose. They are the elements which strengthen the shell of the body against deformation and fix the components of the robot. Thanks to the internal ribs it is possible to easily mount electronic boards on one side (Fig. 6b) and the battery on the other (Fig.6c). In the central part of the body the camera and microphone are fixed. A special micro junction is used for connecting the robot to the additional operational load, the so-called rucksack, its detection and releasing. (a)

(b)

(c)

Table. 1. The desired parameters of the throwable tactical robot.

Parameter Robot's weight in standard version Weight of additional load Weight of control panel Maximum speed Throwing range 28

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Desired value 1 - 1.5 kg 0.1 kg 6-8 kg 3 km/h 20 m

Fig. 6. Construction design of TTR: additional device „rucksack” (a), position of electronic plates (b) and battery (c).

The engines are seated in properly formed sleeves attached to the body. Wheel rims are fixed to external extreme parts of the body by means of the bearings. Here,

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the kind of the bearing is very important because of its wear as a result of an impact, friction and price. Drive transmission is a result of meshing of the rack embedded in engine axis and inside gear embedded in the rim. As an overload coupling for the protection of gears and the engine there are applied micro rubber blocks placed in cut-outs of the ring with internal teeth. They enable gradual angular shift against the rim.

The made prototypes of the device and control panel are shown in Fig. 7 and the most important parameters of the device are presented in Table 2. (a)

(b)

Fig. 7. Prototype TTR (a) and control panel (b).

Table. 2. The most important parameters of the throwable tactical robot.

Parameter Robot's weight in standard version Weight of additional load Weight of control panel Robot size (width/ height/length) Control panel size Maximum speed Maximum ramp angle Throwing range Discarding height (limiting) Range inside building

Value 1.3 kg 0.16 kg 7 kg 205/100/210 mm (with tail) 360x340x194 mm 3.3 km/h 25 deg 15 - 20 m 7 m (9 m) 30 - 110 m

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Range in open space Standard equipment

120 - 150 m camera, microphone Radio transmission telemetry, vision Operating time »1h Control panel operating time »4h Maximal number of controlled devices 3 The range of robot mentioned in Table 2 directly depends on environmental condition of the radio waves propagation.

Shock absorption on contact with the ground after throwing is assured by appropriately cut T-shape tread on the wheel. The additional sideways shock absorption is assured by rubber straps.

High maximal robot's speed is 3.3 km/h and enables to perform inside reconnaissance efficiently and time sparing. Higher speed value would be a significant obstruction for the operator so the reached result seems to be optimal.

The device is switched on in non-standard way, that is, by turning the wheel and switched off remotely from the console or after longer inactivity time. Such a switching on should make it service easier by the operator wearing gloves. In order to save energy the device automatically limits power consumption by unused components.

The robot is provided with an internal connection which helps to communicate with other robots and initiate explosives.

The electronic part of the device is divided into following functional blocks: supply module, micro controller with peripheries managing the model's functions, engine drive programmer, vision transmitter, telemetry receiver, interface scheduling CAN BUS. Cameras with good optical characteristic, working also in infrared band are applied in the device. The advantage of such a solution is operating the robot in insufficient lightning and in the darkness what results with better picture quality and makes the work more comfortable.

3.3. Control panel Control unit has form of an unfolded box of 330x234 x170 mm and weight 7 kg. A special construction assures high stiffness, shock and bending resistance. The box has a special seal which effectively protects it from sand and dust and assures complete water resistance and waterproofness up to 10 meters. Material, which the box is made from, is very durable in temperatures from -33°C to +90°C and resistant to oils, lubricants and other aggressive substances.

There are positioned antenna, monitor and two loudspeakers in the box lid. Application of directional antenna enables to acquire high power gain and directivity of radio beam. Additionally small size of the antenna enabled to build it in completely in the box lid what protects it from mechanical damage and increases functionality of the control panel. The operator observes the picture from the device camera on the monitor and through loud speakers, that are built in on monitor sides, can hear sound from the microphone installed on the device. The use of the monitor with a TFT matrix allows for obtaining a picture of better quality and brightness than the usual LCD matrix. The Articles

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TFT matrix assures lower power consumption as well, which is essential in case of battery use. The control desk is positioned at the bottom of the box. It is equipped with one joystick and push-buttons responsible for operating different device functions. Behind the control desk (synoptics) are placed all the electronics controlling the control panel and the exchangeable battery package.

The control panel is divided into modules of synoptics, mainboard and transmission. It should largely facilitate servicing and operating the control panel. The control panel is supplied by exchangeable eight-cell lithium-polymer battery package. It is characterized by low self-discharging so it can be stored for a longer time without recharging. Additionally there is no memory-effect in its case. The exchangeable battery pack is integrated with the side grip of the operating panel what enables fast battery change. In the front part of the control panel are audio-video junctions enabling recording and reproduction of registered actions.

Three devices can be controlled from one control panel independently. It increases the functionality of the set and reduces the risk of the device uselessness in case of radio contact loss between the device and control panel or device damage. Possibility of switching among three devices allows also for having three independent observation points.

Special electronic systems built on base of single-system processors are designed for the control panel needs.

Those systems are optimized due to EMC interferences, thermal overload or errors of transmission and adequate standard signal.

The control panel is designed as a general-purpose one. On the PCB panel of the control panel are placed 2 joysticks and 28 push-buttons, where 16 are illuminated. Additionally there are 3 diode lines. All elements can be used in an arbitrary way according to the needs.

3.4. Tests of the robot and the control panel For the robot prototype was done number of tests in various conditions.

The first test consisted in performing a series of 20 throws of the device at a distance of 15-20 meters in a straight line on concrete base. Pending those throws there was no construction damage, it means, the device was still fully operative and ready to follow commands from the control panel.

The second test consisted in throwing the device into rooms through the open window (Fig. 8a). It enabled to work out the throwing technique, its evaluation and precision. In that test there were no damages, neither mechanical nor control system.

The third test consisted in series of dropping from the height of the 2nd floor, that is, from about 9 meters (Fig. 8b). The fall from such a height enables to acquire speed of 48 km/h. It follows from the tests that this case is the most difficult and demanding. In the worst case the full load is received and transferred by one wheel and bearing to the body construction. In result of the tests there were changed the rubber strap on the rim dispersing and absorbing the fall as well as the ball bearings to crosswise roller bearings.

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(a)

(b)

Fig. 8. Throwing tests into a room (a) and dropping from the second floor (b).

The aim of next tests was to determine limiting range values of telemetry and vision transmitter. One of the most essential questions was to study possibilities to move inside buildings. It is well-known that constructions of most buildings are made of reinforced concrete.

Such a building construction causes both strong suppression and dispersion of electromagnetic waves. The most frequent result is the loss of radio communication which makes further control of the device impossible. In most cases the vision data are lost firstly, which prevents the operator from visual control over drive direction. The next is loss of telemetry range. After losing the image there is possible to transmit the sound. In some cases only listening is sufficient, however it limits evidently possibilities to acquire valuable information about dangerous situation.

The telemetry and vision range was satisfactory up to 100-110 meters. Both transmissions had continuous and non-disrupted character enabling efficient task performance. The result is impressive regarding small size of the device, considerable proximity of ground introducing noise and just coincidental shading of a direct view between the panel and the device. Tests in open area demonstrated that the transmission range is not shorter than 110-150 meters. It offers great opportunities for the device application not only to inspect car`s chassis but all kind of reconnaissance around buildings or other device of special purpose. The reconnaissance zone is large enough to recommend usage of this device by military pyrotechnic troops, police and rescue units. Application of additional load allows for taking a counter-load and its detonation in justified cases.

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During further tests there was done a drive down the stairs to the basement in the building with limited illumination and space reconnaissance with use of the camera working in infrared mode. All stairsteps were surmounted autonomously without operator's intervention. Both orientation in the corridor and reconnaissance were performed at moderate light which allowed for testing the quality of the picture sent by the device and determining the device usability in such conditions.

At moderate light it was possible to observe the basement and identify detailed furnishings. The robot's camera was not equipped with illuminators so for observation of places like a basement an infrared radiation must be provided. Nowadays works on attaching an illuminator are conducted what can enable to do a reconnaissance in total darkness. There will be tested diode illuminators of visible radiation and infrared radiation.

The next tests consisted in driving down the stairs (Fig. 9a). They were the simplest due to easy access to the staircase, its large space and good illumination. During the decent the picture observation is not possible due to fast frame changing and variation of direction of camera view. The descent itself has rather random character. It does not cause any danger neither for the device nor the environment. The worst case that can happen is rotation around the longitudinal axis and driving the stairs sideways. In this case the device gets pretty large rotation speed and large driving down speed and the operator is not able to control the drive till the moment when the device stops.

In the next tests the throw range of the device was examined (Fig. 9b). Because the construction weight is 1 kg the throw range depends on a large measure on the thrower. The obtained results of 15-20 meters depends additionally on the way of falling, that is, if the device starts to overturn at touchdown.

From obtained results we can conclude that the thrower should have no problems by delivering the device both to a long distance and i.e. roof of a one-floor building.

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There were performed tests in the arrival hall at OkÄ&#x2122;cie Airport as well (Fig. 10a) in co-operation with the Border Guard. The hall is characteristic due to its construction, which is reinforced strongly and has a lot of steel construction elements. Such conditions cause a special problem for radio modems due to strong damping of radio signal and a large number of signal rebounds from those constructions. Test drives were performed in time of normal flight service in order to show real operational use closer and to obtain additional factors that can influence the communication quality with the device.

Performed tests allowed to cover the distance of 80 meters on the way between screening plan of terminals T1 and T2. Additionally tests drives were done between separate transporters due to maximize quantity and level of interferences both from the transporter construction and its work. The reconnaissance proceeded without significant problems although the device was screened directly through protective elements of the luggage transporter. (a)

(b)

(a)

Fig. 10. Device tests in the arrival hall of OkÄ&#x2122;cie Airport (a) and on the board of Boeing 737 (b). (b)

Fig. 9. Tests of driving downstairs (a) and distance throwing (b).

Special tests were performed on the board of Boeing 737 of Polish Airlines LOT (Fig. 10b). Their task was to present possibilities to do reconnaissance in a very difficult place like passenger cabin. The problem results from small device size and large number of elements generating disturbances. Support elements of the seats are a very dense obstacle for propagation of waves and generate a large number of reflections diminishing primary radio signal. In such special circumstances the radio communication and possibility to access each corner of the fuselage of the aeroplane construction was tested. The tests Articles

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of the device were performed in this Boeing model due to its significant popularity and in connection with great probability of necessity to reconnaissance just this construction.

During tests the reliability of load release mechanism was tested (Fig. 11), both on the mechanism and the software. The use of explosive charges should, depending on the character of the special forces actions, introduce chaos and panic amongst aggressors and lead to neutralization of dangerous and dubious packages on spot.

It enables the large spectrum of using the device in potential events and scenarios.

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ACKNOWLEDGMENTS The presented work has been financed from the research project sponsored by the Ministry of Science and Higher Education called “Ultradurable robotized devices to active teleobservation, usable in military, police and rescue purposes”(R00-00058/3).

AUTHORS Rafał Czupryniak, Maciej Trojnacki* - Industrial Research Institute for Automation & Measurements, Al. Jerozolimskie 202, PL-02-486 Warsaw, Poland.

E-mail: mtrojnacki@piap.pl. * Corresponding author

References [1]

[2] [3]

Fig. 11. Reliability tests of the mechanism releasing explosive charge.

Additional load can carry different materials, including explosives, deafening and blinding charges. The character of applied material depends in every case on the kind of activity and usage context. You should take into consideration that in case of using explosive material the device will be destroyed without chance to repair it. The cost that it will entail is not of great importance due to the fact of using it in special, rescue or other actions.

Films presenting selected versions of the robot are at the link [8].

4. Summary In this paper the construction and working mode of the throwable tactical robot and its control panel is presented.

Test results testify that the robot is resistant to the downfall caused by throwing down, descent or dropping from the second floor. The shape and mass of the robot enables to throw it at the distance of 15-20 meters or to the roof of one-floor building. Application of the camera working in the infrared mode enables a reconnaissance by meanlight. The robot copes very well in surveying not easily accessible spaces and places where the radio communication is difficult. The communication range enables teleoperating the device both in open area and in rooms adequately to 110 and 150 meters. The transmission ranges obtained in these cases are highly satisfying considering the size of the device.

Such attributes of the robot predispose it to active teleobservation in military, police and rescue use. Additional equipment allows to use TTR to neutralization of explosive charges by special forces. The robot enables to limit the risk of health or life loss or group members performing actions in dangerous areas.

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[4]

[5] [6] [7] [8]

R. Czupryniak, P. Szynkarczyk, Trojnacki M., “Tendencies in the development of mobile ground robots (2) New trends in mobile robotics”, PAR, no. 7-8, 2008, pp. 10-13. (in Polish).

P. Szynkarczyk, R. Czupryniak, “Mobile robots and security”, PAR, no. 2/2008, appending on CD, pp. 441-450.

K. Tadakuma, R. Tadakuma, K. Nagatani, K. Yoshida, M. Aigo, M. Shimojo, K. Iagnemma, “Throwable Tetrahedral Robot with Transformation Capability”. In: 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems, 11th-15th October, 2009 St. Louis, USA, pp. 2801-2808.

E. Ackerman, Throwable EyeBall R1 Surveillance Robot, 20 of August 2009 available at: http://www.botjunkie.com.

Defense Update Online Defense Magazine: http://www.defense-update.com.

Official web page of ReconRobotics Inc.: http://www.recon-scout.com.

Official web page of ODF Optronics Ltd.: http://www.odfopt.com.

Video clips presenting some PIAP's mobile robots in action : http://www.youtube.com/user/osmpiap.

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SPECIAL ISSUE

Editorial to the Special Issue Section on Hybrid Intelligent Systems for Control and Automation Part I Guest Editors: Oscar Castillo and Patricia Melin

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Journal of Automation, Mobile Robotics & Intelligent Systems

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Editorial Oscar Castillo*, Patricia Melin

Editorial to the Special Issue Section on “Hybrid Intelligent Systems for Control and Automation - Part I” The special issue on hybrid intelligent systems for control and automation comprises four contributions, which are selected and extended versions of papers previously presented at the International Seminar on Computational Intelligence held at Tijuana, Mexico on January of 2010. The papers describe different contributions to the area of hybrid intelligent systems with application on control and automation. In the papers, an optimal combination of intelligent techniques is applied to solve in an efficient and accurate manner a problem in a particular area of application.

In the first paper, by Ieroham Baruch and Carlos-Roman Mariaca-Gaspar, Recurrent Neural Identification and Control of a Continuous Bioprocess via First and Second Order Learning is presented. This paper applied a new Kalman Filter Recurrent Neural Network (KFRNN) topology and a recursive Levenberg-Marquardt (L-M) learning algorithm capable to estimate parameters and states of highly nonlinear unknown plant in noisy environment. The proposed KFRNN identifier, learned by the Backpropagation and L-M learning algorithm, was incorporated in a direct and indirect adaptive neural control schemes. The proposed control schemes were applied for real-time recurrent neural identification and control of a continuous stirred tank bioreactor model, where fast convergence, noise filtering and low mean squared error of reference tracking were achieved.

In the second paper, by Yazmin Maldonado et al., a Novel Method for Genetic Optimization of Membership functions of Fuzzy Logic for Speed Control of a Direct Current Motor for Hardware Applications in FPGAs is presented. This paper proposes a novel method for genetic optimization of triangular and trapezoidal membership functions of fuzzy systems, for hardware applications such as the FPGA (Field Programmable Gate Array). This method consists in taking only certain points of the membership functions, with the purpose of giving more efficiency to the algorithm. The genetic algorithm was tested in a fuzzy controller to regulate engine speed of a direct current (DC) motor, using the Xilinx System Generator (XSG) toolbox of Matlab, which simulate VHDL (Very High Description Language) code.

In the third paper, by Héctor Joaquín Fraire Huacuja et al., a method for Improving the Intensification and Diversification Balance of the Tabu Solution for the Robust Capacitated International Sourcing Problem is presented. This paper addresses the robust capacitated international sourcing problem (RoCIS), which consists of selecting a subset of suppliers with finite capacity, from an available set of potential suppliers internationally located.

In the fourth paper, by Abraham Meléndez et al., the Optimization of a Reactive Controller for a Mobile Robot using Evolutionary Algorithms and Fuzzy Logic is presented. This paper describes an evolutionary algorithm used for the optimization of a reactive controller applied to a particular mobile robot. The algorithm optimizes the Fuzzy Inference System and the position and number of the sensors on the robot, while trying to use the minimum amount of power possible.

In conclusion, this special issue represents a contribution to the state of the art in the area of hybrid intelligent systems with application on control and automation.

Guest Editors: Oscar Castillo and Patricia Melin Tijuana Institute of Technology, Tijuana, Mexico ocastillo@tectijuana.mx, pmelin@tectijuana.mx

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RECURRENT NEURAL IDENTIFICATION AND CONTROL OF A CONTINUOUS BIOPROCESS VIA FIRST AND SECOND ORDER LEARNING Ieroham Baruch, Carlos-Roman Mariaca-Gaspar

Abstract: This paper applies a new Kalman Filter Recurrent Neural Network (KFRNN) topology and a recursive Levenberg-Marquardt (L-M) learning algorithm capable to estimate parameters and states of highly nonlinear unknown plant in noisy environment. The proposed KFRNN identifier, learned by the Backpropagation and L-M learning algorithm, was incorporated in a direct and indirect adaptive neural control schemes. The proposed control schemes were applied for real-time recurrent neural identification and control of a continuous stirred tank bioreactor model, where fast convergence, noise filtering and low mean squared error of reference tracking were achieved.

Keywords: backpropagation learning, continuous stirred tank bioreactor, direct adaptive neural control, indirect adaptive sliding mode control, Kalman filter recurrent neural network identifier, Levenberg-Marquardt learning.

1. Introduction The universal approximation abilities of the artificial neural networks to approximate complex non-linear relationships without prior knowledge of the model structure, makes them a very attractive alternative to the classical modeling and control techniques [1], [2], [3]. This property has been proved by the universal approximation theorem [3]. Among several possible network architectures the ones most widely used are the Feedforward (FFNN) and the Recurrent Neural Networks (RNN). In a feedforward neural network the signals are transmitted only in one direction, starting from the input layer, subsequently through the hidden layers to the output layer, which requires applying a tap delayed global feedbacks and a tap delayed inputs to achieve a nonlinear autoregressive moving average neural dynamic plant model.

A recurrent neural network has local feedback connections to some of the previous layers. Such a structure is suitable alternative to the first one when the task is to model dynamic systems, and the universal approximation theorem has been proved for the recurrent neural networks too. The preferences given to recurrent neural network identification with respect to the classical methods of process identification are clearly demonstrated in the solution of the “bias-variance dilemma” [3]. Furthermore, the derivation of an analytical plant model, the parameterization of that model and the Least Square solution for the unknown parameters have the following disadvantages: (a) the analytical model did not include all factors having influence to the process behavior; (b) the analytical model is derived taking into account some

simplifying suppositions which not ever match; (c) the analytical model did not described all plant nonlinearities, time lags and time delays belonging to the process in hand; (d) the analytical model did not include all process and measurement noises which are sensor and actuator dependent. In (Sage, [4]) the method of invariant imbedding has been described. This method seemed to be a universal tool for simultaneous state and parameter estimation of nonlinear plants but it suffer for the same drawbacks because a complete nonlinear plant model description is needed. Furthermore, the managing of noisy input/output plant data is required to augment the filtering capabilities of the identification RNNs, [5].

Driven by these limitations, a new Kalman Filter Recurrent Neural Network (KFRNN) topology and the recursive Backpropagation (BP) learning algorithm in vector-matrix form has been derived [6] and its convergence has been studied [6], [7]. But the recursive BP algorithm, applied for KFRNN learning, is a gradient descent first order learning algorithm which does not allow to augment the precision and accelerate the learning [5], [7]. Therefore, the aim of this paper was to use a second order learning algorithm for the KFRNN, as the Levenberg-Marquardt (L-M) algorithm is, [8]. The KFRNN with L-M learning was applied for Continuous Stirred Tank Reactor (CSTR) model identification [9], [10]. The application of KFRNNs together with the recursive L-M could prevent all the problems caused by the use of the FFNN, thus improving the learning and the precision of the plant state and parameter estimation in presence of noise. Here, the parameters and states, obtained from the KFRNN identifier will be used in order to design a Direct and Indirect Adaptive Neural Control (DANC and IANC) of CSTR bioprocess plant model.

2. Kalman Filter RNN This section is dedicated to the KFRNN topology, the recursive Backpropagation and the recursive LevenbergMarquardt algorithms for the KFRNN learning. The KFRNN is applied as a state and parameter estimator of nonlinear plants.

2.1. Topology of the KFRNN Let us consider the linearized plant model (1), (2), represented in a state-space form: Xd.(k+1) = Ad(k) Xd(k) + Bd(k) U(k) + Q1(k)

(1)

Yd(k) = Cd(k) Xd(k) + Q2(k)

(2)

Where: E [.] means mathematical expectation; the proArticles

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Fig. 1. Block-diagram of the KFRNN topology. cess and measurement noises Q1 (.), Q2 (.) are white, with Q1(k), Q2(s) and the initial state Xd(k0) independent and zero mean for all k, s, with known variances E [Xd(k) XdT(k)] = P0, E[Q1(k) Q1T(k)] = Q(k) d (k-t), E[Q2(k) Q2T(k)] = R(k) d (k-t), where d (k-t) = 1 if k = t, and 0 otherwise. The optimal Kalman filter theory is completely described in [4], and we would not repeat it here. For us the Kalman Filter (KF) is a full rank optimal state estimator capable to estimate the system states, to filter the process and measurement noises, taking in hand all plant information available like: input/ output plant data, all parameters of the plant model (1), (2), and the given up noise and initial state statistics (mean and variance). The basic Kalman filter equations for the estimated state and output variables are given by:

Xe.(k+1) = Ad(k) Xe(k) - Ke(k) Ye(k) + B2(k) Ue(k) (6) B2 = [Bd ; Ke]; UeT = [U ; Yd]

(7)

Z(k) = Cd(k) Xe(k)

(8)

Ye(k+1) = A2 Ye(k) + Z(k)

(9)

The obtained new KF RNN topology is given in Fig. 1.

The first layer of the KFRNN represented the plant model (equations (10)-(13)) and the second layer represented the output noise filtering model (equations (14)-(18)). The KF RNN topology is described by the following equations: X(k+1) = A1X(k) + BU(k) - DY(k)

(10)

B = [B1 ; B0]; UT = [U1 ; U2]

(11)

A1= block-diag (A1,i), | A1,i | < 1

(12)

Z1(k) = G[X(k)]

(13)

C = [C1 ; C0]; ZT = [Z1 ; Z2]

(14)

V1(k) = CZ(k)

(15)

V(k+1) = V1(k) + A2V(k)

(16)

A1 = block-diag (A2,i), |A2,i | < 1

(17)

Y(k) = F[V(k)]

(18)

Xd.(k+1) = Ae(k) Xe(k) + Ke(k) Yd(k) + Bd(k) U(k) (3) Ad (k) = Ad (k) - Kd (k) Cd (k)

(4)

Ye (k) = Cd (k) Xd (k)

(5)

Where: Xe (k) is the estimated state vector with dimension Ne; Ae (k) is a (Ne x Ne) closed-loop KF state matrix; Ye (k) is the estimated plant output vector variable with dimension L; Ke(k) is the optimal Kalman filter gain matrix with dimension (Ne x L). This gain matrix is computed applying the optimal Kalman filtering methodology given in [4]. So, the KF performed noise filtration by means of an optimal closed-loop feedback which has the drawback that the feedback amplified the noise components of the error, especially when the feedback gain is high. The second drawback is that the KF design needs complete plant parameter and noise information, which means that if the plant data are incomplete the process noise level is augmented. To overcome this we need to take special measures like to augment the filtering capabilities of the KF. The third drawback is that the KF could not estimate parameters and states in the same time processing noisy measurements with unknown noise statistics, and it will be our task. To resolve this task we need to derive the topology and the BP learning algorithm of a new recurrent KF-like neural network subject of learning and capable to estimate parameters and states in the same time. First of all we could rewrite the equation (3) defining a new extended input vector, containing all available input/output information issued by the plant, and second - we could modify the output equation (5), so to convert it to an output noise filter. After that we obtain: 38

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Where: X, Y, U are vectors of state, output, and augmented input with dimensions N, L, (M+1), respectively, Z is an (L+1) - dimensional input of the feedforward out-put layer, where Z1 and U1 are the (Nx1) output and (Mx1) input of the hidden layer; the constant scalar threshold entries are Z2 = -1, U2 = -1, respectively; V is a (Lx1) pre-synaptic activity of the output layer; the super-index T means vector transpose; A1, A2 are (NxN) and (LxL) block-diagonal weight matrices; B and C are [Nd (M+1)] and [Lx(N+1)]- augmented weight matrices; B0 and C0 are (Nx1) and (Lx1) threshold weights of the hidden and output layers; F[.], G[.] are vector-valued tanh(.) or sigmoid(.) -activation functions with corresponding dimensions. Here the input vector U and the input matrix B of the KF RNN are augmented so to fulfill the specifications (7) and the matrix D corresponded to the feedback gain matrix of the KF.

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Fig. 2. Block-diagram of the adjoint KFRNN topology.

E(k) = Yd(k) - Y(k); E1(k) = F'[Y(k)] E(k)

(20)

matrices A1, A2 are denoted by Vec (A1(k)), Vec (A2(k)), respectively, where (24), (30) represented their learning as an element-by-element vector products; E, E1, E2, E3, are error vectors (see Fig. 2), predicted by the adjoint KF RNN model. So, the KF RNN is capable to issue parameter and state estimations for control purposes, thanks to the optimization capabilities of the BP learning algorithm, applying the “correction for error” delta rule of learning (see Haykin, [3]). The stability of the KF RNN model is assured by the activation functions [-1, 1] bounds and by the local stability weight bound conditions given by (12), (17). The stability of the KF RNN movement around the optimal weight point has been proved by one theorem and the rate of convergence lemma, (see the Ph.D. thesis of Mariaca [7]). It is stated below.

Theorem of stability of the BP KF RNN used as a plant identifier [7]: Let the KF RNN topology is given by equations (10)-(18) (see Fig.1) and the nonlinear plant model, is as follows:

F'[Y(k)] = [1-Y2(k)]

(21)

Xd.(k+1) = G[ Xd(k), U(k)]

(31)

C(k) = E1(k) ZT(k)

(22)

Yd(k) = F[ Xd(k)]

(32)

DA2(k) = E1(k) VT(k)

(23)

Vec(DA2(k)) = E1(k) × X(k)

(24)

E3(k) = G'[Z(k)] E2(k); E2(k) = CT(k) E1(k)

(25)

G'[Z(k)] = [1-Z2(k)]

(26)

B(k) = E3(k) UT(k)

(27)

D(k) = E3(k) YT(k)

(28)

Where: {Yd(.), Xd(.), U(.)}are output, state and input variables with dimensions L, Nd , M, respectively; G(.), F(.) are vector valued nonlinear functions with respective dimensions.

Under the assumption of KF RNN identifiability made, the application of the BP learning algorithm for C, A1, A2, B, D, in general vector-matrix form, described by equation (19)-(30), and the learning rates h(k), a(k) (here they are considered as time-dependent and normalized with respect to the error) are derived using the following Lyapunov function:

A1(k) = E3(k) × XT(k)

(29)

L(k) = L1(k) + L2(k)

Vec(A1(k)) = E3(k) × X(k)

(30)

Where: L1(k) and L2(k) are given by:

The dimension of the state vector of the KF RNN is chosen using the simple rule which is: N=L+M. From Fig.1 we could see that here we have a two layer Jordan canonical topology with a global feedback which filtered the process noise better then a two layer feedforward topology containing input and output tap delays representing a successive noise sensitive NARMA model, [6].

2.2. BP Learning of the KFRNN So the KF RNN topology corresponded functionally to the KF definition (6)-(9) and ought to be learnt applying the BP learning algorithm derived using the adjoint KF RNN (see Fig. 2) based on KF RNN topology applying the diagrammatic method, [11].

The BP learning algorithm, expressed in vector-matrix form is as follows: W(k+1) = W(k) + W(k) + W(k-1); |Wij | < W0 (19)

Where: F'[.], G'[.] are derivatives of the tanh(.) activation functions; W is a general weight, denoting each weight matrix (C, A1, A2, B, D) in the KF RNN model, to be updated; W (C, A1, A2, B, D), is the weight correction of W; Yd is an L-dimensional output of the approximated plant taken as a reference for KF RNN learning; h, a are learning rate parameters; DC is an weight correction of C; DB is an weight correction of B; DD is an weight correction of D, DA1 is the weight correction of A1 , DA2 is the weight correction of A2; the diagonals of the

(33)

~ ~ ~ ~ L2(k) = tr (WA1(k)WAT1(k)) + tr (WA2(k)WAT2(k)) + ~ ~ ~ ~ tr (WB(k)WBT(k)) + tr (WC(k)WCT(k)) + ~ ~ tr (WD(k)WDT(k)) Where: ~ ~ WA1(k) = A1(k) - A*1, WA2(k) = A2(k) - A*2, ~ ~ WB(k) = B(k) - B*, WC(k) = C(k) - C*, ~ WD(k) = D(k) - D* Ù

Ù

Ù

Ù

Ù

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are vectors of the weight estimation error and (A*1, A2*, B*, C * , D* ) and (A1(k), A2(k), B(k), C(k), D(k)) denoted the ideal neural weight and the estimated neural weight at the k-th step, respectively, for each case.

Then the identification error is bounded, i.e.:

DY[A2ij(k)] = D1, i(k)Vj(k)

(42)

D1, i(k) = Fi'[Y(k)]

(43)

DY[A1ij(k)] = D2, i(k)Xj(k)

(44)

L(k+1) = L1(k+1) + L2(k+1) < 0

(34)

DY[Bij(k)] = D2, i(k)Uj(k)

(45)

DL(k+1) = L(k+1) - L(k)

(35)

DY[Dij(k)] = D2, i(k)Yj(k)

(46)

D2, i(k) = Gi'[Zi(k)]Ci D1, i(k)

(47)

Ù

Ù

Ù

Ù

Ù

Where the condition for L1(k+1) < 0 is fulfilled when the maximum learning rate is chosen in the limits, given below: 1 ö

æ 1æ 1+ ç ÷ ç è 2 ø < h max < è

1 ö

÷

For L2(k+1) < 0 fulfillment we have the condition: DL2(k+1) < -h maxïe(k+1)ï2 - a maxïe(k)ï2 +d(k+1)

Therefore, the Jacobean matrix could be formed as: DY[W(k)] = [DY(Cij(k)), DY(A2ij(k)), DY(Bij(k)), DY(A1ij(k)), DY(Dij(k))] (48)

2ø ymax

ymax

(36)

Note that h max changes adaptively during the learning process of the network, where: 5

h max = max({hi}

The P(k) matrix was computed recursively by the equation: P(k) = a-1(k){P(k-1)-P(k-1).

W[W(k)]S-1[W(k)]WT[W(k)]P(k-1)}

(49)

Where the S(.), and W(.) matrices were given as follows: S[W(k)] = a(k)L(k) + WT[W(k)]P(k-1)W[W(k)] (50)

i=1

Here all: the unmodeled dynamics, the approximation errors and the perturbations, are represented by the dterm, and the complete proof of that theorem and the convergence lemma for (36) are given in the Appendix A and can be seen also with more details in [7].

2.3 . Recursive Levenberg-Marquardt Learning of the KFRNN The general recursive L-M algorithm of learning, [5], [7], [8] is given by the following equations:

WT[W(k)] = L(k)-1 =

1 0

0 ¼

ÑY T[W(k)] ¼ 0 ; 1

0 ; 10-4 £ r £ 10-6 ; r

0,97 £ a(k) £ 1; 103 £ P(0) £ 106

(51)

W(k+1) = W(k) + P(k)ÑY[W(k)]E[W(k)]

(37)

The matrix W(.) had dimension (Nwx2), whereas the second row had only one unity element (the others were zero). The position of that element was computed by:

Y[W(k)] = g[W(k), U(k)]

(38)

i = k mod (nw) + 1; k > nw

E2[W(k)] = {Yp(k)- g[W(k), U(k)]}2

(39)

¶ g[W, U(k)] ¶W W = W(k)

After this, the given up topology and learning were applied for the CSTR system identification.

(40)

DY[W(k)] =

40

N° 4

(52)

3. Recurrent Trainable NN

Where: W is a general weight matrix (A1, A2, B, C, and D) under modification; P is the covariance matrix of the estimated weights updated; DY [.] is an Nw-dimensional gradient vector; Y is the KFRNN output vector which depends of the updated weights and the input; E is an error vector; Yp is the plant output vector, which is in fact the target vector. Using the same KFRNN adjoint block diagram (see Fig.2), it was possible to obtain the values of the gradients DY [.] for each updated weight, propagating the value D(k) = I through it. Following the block diagram of Fig. 2, equation (37) was applied for each element of the weight matrices (A1, A2, B, C, D) in order to be updated. The corresponding gradient components (40) are obtained as follows:

X(k+1) = AX(k) + BU(k)

(53)

DY[Cij(k)] = D1, i(k)Zj(k)

B = [B1 ; B0]; UT = [U1T ; U2T]

(54)

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(41)

This section is dedicated to the topology, the BP and the L-M algorithms of RTNN learning. The RTNN could be obtained from the KFRNN removing the output local and global feedbacks. The RTNN was used as a feedback/feedforward controller.

3.1. Topology of the RTNN The RTNN model and its learning algorithm of dynamic BP-type, together with the explanatory figures and stability proofs, are described in [6], [7], so only a short description will be given here. The RTNN topology, derived in vector-matrix form, was given by the following equations:

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A = block-diag (Ai), |Ai| < 1

(55)

Z1(k) = G[X(k)]

(56)

C = [C1 ; C0]; ZT = [Z1T ; Z2T]

(57)

V(k) = CZ(k)

(58)

Y(k) = F[V(k)]

(59)

Where: X, Y, U are vectors of state, output, and augmented input with dimensions N, L, (M+1), respectively, Z is an (L+1) dimensional input of the feedforward output layer, where Z1 and U1 are the (Nx1) output and (Mx1) input of the hidden layer; the constant scalar threshold entries are Z2 = -1, U2 = -1, respectively; V is a (Lx1) pre-synaptic activity of the output layer; the super-index T means vector transpose; A is (NxN) block-diagonal weight matrix; B and C are [Nx(M+1)] and [Lx(N+1)]augmented weight matrices; B0 and C0 are (Nx1) and (Lx1) threshold weights of the hidden and output layers; F[.], G[.] are vector-valued tanh(.) or sigmoid(.) -activation functions with corresponding dimensions. Equation (55) represents the local stability condition imposed on all blocks of A. The dimension of the state vector X of the RTNN is chosen using the simple rule of thumb which is: N=L+M.

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Mariaca [7] for more details).

Theorem of stability of the BP RTNN used as a direct system controller [7]: Let the RTNN with Jordan Canonical Structure is given by equations (53)-(59) and the nonlinear plant model is given by (31), (32). Under the assumption of RTNN identifiability made, the application of the BP learning algorithm for A(.), B(.), C(.), in general matricial form, described by equation (19), (63)(65) without momentum term, and the learning rate h(k) (here it is considered as time-dependent and normalized with respect to the error) are derived using the following Lyapunov function: L(k) = L1(k) + L2(k)

(66)

Where: L1(k) and L2(k) are given by: 1

L1(k) = —2 e2(k) ~ ~ ~ ~ L2(k) = tr (WA(k)WAT(k)) + tr (WB(k)WBT(k)) + ~ ~ tr (WC(k)WCT(k)) Where: ~ ~ ~ WA(k) = A(k) - A* , WB(k) = B(k) - B*, WC(k) = C(k) - C* Ù

Ù

Ù

E(k) = Yd(k) - Y(k); E1(k) = F'[Y(k)] E(k)

(60)

are vectors of the estimation error and (A* , B*, C*) and (A(k), B(k), C(k)) denoted the ideal neural weight and the estimate of the neural weight at the k-th step, respectively, for each case.

Let us define: Ymax = maxk ïçY(k)÷ï, and Jmax = maxk ïçJ(k)÷ï, where Y(k) = ¶o(k)/¶W(k), and J(k) = ¶y(k)/¶u(k), where W is a vector composed by all weights of the RTNN, used as a system controller, and ïç×÷ï is an Euclidean norm in Ân.

Then the identification error is bounded, i.e.:

DC(k) = E1(k) ZT(k)

(61)

L(k+1) = L1(k+1) + L2(k+1) < 0

(67)

E3(k) = G'[Z(k)] E2(k); E2(k) = CT(k) E1(k)

(62)

DL(k+1) = L(k+1) - L(k)

(68)

DB(k) = E3(k) UT(k)

(63)

Where the condition for L1(k+1) < 0 fulfillment is that the maximum rate of learning is inside the limits:

DA(k) = E3(k) XT(k)

(64)

Vec(DA(k)) = E3(k) × X(k)

(65)

3.2. BP Learning of the RTNN The same general BP learning rule (19) was used here.

Following the same procedure as for the KFRNN, it was possible to derive the following updates for the RTNN weight matrices:

Ù

Ù

Ù

0 < hmax <

Where DA, DB, DC are weight corrections of the of the learned matrices A, B, and C, respectively; E, E1, E2, and E3 are error vectors; X is a state vector; F'(.) and G'(.) are diagonal Jacobean matrices, whose elements are derivatives of the tanh activation functions (see equations (21) and (26)). Equation (64) represents the learning of the full feedback weight matrix of the hidden layer. Equation (65) gives the learning solution when this matrix is diagonal vA, which is the present case. The initial values of the weight matrices during the learning are chosen as arbitrary numbers inside a small range. The stability of the RTNN model used as a direct controller is assured by the activation functions [-1, 1] bounds and by the local stability weight bound condition given by (55). The stability of the RTNN movement around the optimal weight point has been proved by one theorem (see the Ph.D. thesis of

2 J2maxY2max

and for L2(k+1) < 0, we have: DL2(k+1) < -hmax½e(k+1)½2 + b(k+1)

(69)

Note that hmax changes adaptively during the learning process of the network, where: 3

h max = max({hi} i=1

Here all: the unmodelled dynamics, the approximation errors and the perturbations, are represented by the b-term, and the complete proof of that theorem and the rate of convergence lemma, are given in [7].

3.3.

Recursive Levenberg-Marquardt Learning of the RTNN The general recursive L-M algorithm of learning [5], Articles

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[7], [8] is given by equations (37)-(40), where W is the general weight matrix (A, B, C) under modification; Y is the RTNN output vector; E is an error vector; Yp is the plant output vector. Using the RTNN adjoint block diagram [5], it was possible to obtain the values of DY [.] for each updated weight propagating D=I. Applying equation (40) for each element of the weight matrices (A, B, C), the corresponding gradient components are obtained as:

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Wp(z) = Cp(zI-Ap)-1Bp

(76)

Pi(z) = (zI-Ai)-1Bi

(77)

Q1(z) = Cc(zI-Ac)-1B1c

(78)

Q2(z) = Cc(zI-Ac)-1B2c

(79)

DY[Cij(k)] = D1, i(k)Zj(k)

(70)

The control systems z-transfer functions (76)-(79) are connected by the following equation, which is derived from Fig. 3, and is given in z-operational form:

D1, i(k) = Fi'[Yi(k)]

(71)

Yp(z) = Wp(z)[I+Q1(z)Pi(z)]-1Q2(z)R(z)+q(z)

(80)

DY[Aij(k)] = D2, i(k)Xj(k)

(72)

q(z) = Wp(z)q1(z)+q2(z)

(81)

DY[Bij(k)] = D2, i(k)Uj(k)

(73)

D2, i(k) = Gi'[Zi(k)]Ci D1, i(k)

(74)

Where: q(z) represented a generalized noise term. The RTNN and the KFRNN topologies were controllable and observable, and the BP algorithm of learning was convergent, [5], [7], so the identification and control errors tended to zero:

Therefore the Jacobean matrix could be formed as: DY[W(k)] = [DY(Cij(k)), DY(Aij(k)), DY(Bij(k))] (75)

Ei(k) = Yp(k)-Y(k) ® 0; k ®¥

(82)

The P(k) matrix was computed recursively by equations (49)-(52). Next, the given up RTNN topology and learning were applied for CSTR system control.

Ec(k) = R(k)-Yp(k) ® 0; k ®¥

(83)

4. Adaptive Control Systems Design This section is dedicated to the design of direct and indirect (sliding mode) adaptive control system using the KF RNN as a nonlinear plant identifier. The RTNN was used as a feedback/feedforward controller in the case of direct adaptive neural control.

4.1. Direct Adaptive Neural Control Scheme This section described the direct adaptive control using KFRNN as plant identifier and RTNN as a plant controller (feedback / feedforward). The block-diagram of the control system is given in Fig. 3. The following study described the linearized model of that closed-loop control system.

This means that each transfer function given by equations (76)-(79) was stable with minimum phase. The closed-loop system was stable and the feedback dynamical part of the RTNN controller compensated the plant dynamics. The feedforward dynamical part of the RTNN controller was an inverse dynamics of the closed-loop system one, which assured a precise reference tracking in spite of the presence of process and measurement noises.

4.2.

Indirect Adaptive Control Scheme (Sliding Mode Control) The indirect adaptive control using the RTNN as plant identifier has been described in, [5]. Later the proposed indirect control has been derived as a Sliding Mode Control (SMC) and applied for control of unknown hydrocarbon biodegradation processes, [6], using the KF RNN identifier with BP learning. Here we applied the KF RNN identifier with L-M learning. The block diagram of the indirect adaptive control scheme is shown in Fig. 4. It contains identification and state estimation KF RNN and a sliding mode controller.

Fig. 3. Block-diagram of the closed-loop neural control system.

Let us present the following z-transfer function representations of the plant, the state estimation part of the KFRNN, and the feedback and feedforward parts of the RTNN controller: 42

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Fig. 4. Block diagram of the closed-loop system containing KF RNN identifier and a SMC. The stable nonlinear plant is identified by a KF RNN model with topology, given by equations (10)-(18) lear-

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ned by the stable BP-learning algorithm, given by equations (19)-(30), or using the second order LM-learning algorithm, given by equations (37)-(52). The simplification and linearization of the neural identifier equations (10)-(18), omitting the DY (.) term, leads to the next local linear plant model, extracted from the complete KF RNN model: X(k+1) = A1X(k) + BU(k)

(84)

Z(k) = HX(k); H = CG'(Z)

(85)

Where G'(.) is the derivative of the activation function and L = M, is supposed.

In [12], the sliding surface is defined with respect to the state variables, and the SMC objective is to move the states form an arbitrary space position to the sliding surface in finite time.

In [13], the sliding surface is also defined with respect to the states but the states of the SISO systems are obtained from the plant outputs by differentiation. In [14], the sliding surface definition and the control objectives are the same. The equivalent control systems design is done with respect to the plant output, but the reachability of the stable output control depended on the plant structure.

In [6], the sliding surface is derived directly with respect to the plant outputs which facilitated the equivalent SMC systems design. Let us define the following sliding surface equation as an output tracking error function: P

S(k+1) = E(k+1) + ågi E(k-i+1);ïgï< 1

(86)

i=1

Where: S(.) is the Sliding Surface Error Function (SSEF) defined with respect to the plant output; E(.) is the systems output tracking error; gi are parameters of the desired stable SSEF; p is the order of the SSEF. The tracking error in two consecutive moments of time is defined as:

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The substitution of (89) in (90) gives: P

R(k+1) - FAX(k) - FBU(k) + ågi E(k-i+1) = 0 (91) i=1

As the local approximation plant model (84), (85), is controllable, observable and stable (see [6], [7]), the matrix A1 is diagonal, and L = M, then the matrix product (HB), representing the plant model static gain, is nonsingular, and the plant states X(k) are smooth nonincreasing functions. Now, from (91) it is easy to obtain the equivalent control capable to lead the system to the sliding surface which yields: P

Ueq(k) = (FB)-1 -FAX(k)+R(k+1)+ ågi E(k-i+1) i=1

(92) Following [12], the SMC avoiding chattering is taken using a saturation function instead of sign one. So the SMC takes the form: Ueq(k) = U*(k) =

if ïçUeq(k)÷ï < U0

-U0Ueq(k)/ïçUeq(k)÷ï if ïçUeq(k)÷ï ³ U0 (93) The SMC substituted the multi-input multi-output coupled high order dynamics of the linearized plant with desired decoupled low order one.

5. Description of the CSTR Bioprocess Plant The CSTR model given in [9], [10] was chosen as an example of RNN applications in system identification and control of biotechnological plants. Numerical values for the parameters and nominal operating conditions of this model are given in Table 1.

Table 1. Parameters and operating conditions of the CSTR.

S(k+1) = R(k) - Z(k); E(k+1) = R(k+1) - Z(k+1)

(87)

Where R(k), Z(k) are L-dimensional reference and output vectors of the local linear plant model. The objective of the sliding mode control systems design is to find a control action which maintains the systems error on the sliding surface which assure that the output tracking error reaches zero in P steps, where P < N. So, the control objective is fulfilled if: S(k+1) = 0

Parameters

Parameters

Q = (L/min) CAf = 1.0 (mol/L) Tf = TfC = 350 (K) V = (L) E/R = 9.95x103 (K) -DH = 2x105 (cal/mol)

r×rc = 1000 (g/L) CpCpc = 1 (cal/gK) Qe0 = 103.41 (L/min) hA = 7x105 (cal/min K) T0 = 440.2 (K) k0 = 7.2x1010 (l/min)

The CSTR is described by the following continuous time nonlinear system of ordinary differential equations:

(88)

Now, let us to iterate (85) and to substitute (84) in it so to obtain the input/output local plant model, which yields: Z(k+1) = FX(k+1) = F[AX(k) + BU(k)]

(89)

dCA(t) Q E = (CAf - CA(t)) - k0CA(t)exp dt V RT(t)

dT(t) Q E (-DH)CA(t) = (Tf - T(t)) + exp dt V RT(t) rCp -hA rcCpcCA(t) Qc(t) 1 - exp Qc(t)rcCpc rCpV

From (86)-(87), and (89) it is easy to obtain:

(94)

(Tef - T(t))

P

R(k+1) - Z(k+1) + ågi E(k-i+1) = 0

(90)

(95)

i=1

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In this model it is enough to know that within the CSTR, two chemicals are mixed and that they react in order o produce a product compound A at a concentration CA(t), and that the temperature of the mixture is T(t). The reaction is exothermic and it produces heat which slows down the reaction. By introducing a coolant flow-rate Qc(t), the temperature can be varied and hence the product concentration can be controlled. Here CAf is the inlet feed concentration; Q is the process flow-rate; Tf and Tef are the inlet feed and coolant temperatures, respectively; all of which are assumed constant at nominal values. Likewise, k0, E/R, V, DH, r, Cpc, Cp, and rc are thermodynamic and chemical constants related to this particular problem. The quantities Qc0, T0, and CA0, shown in Table 1, are steady values for a steady operating point in the CSTR. The objective was to control the product compound A by manipulating Qc(t). The operating values were taken from [9] and [10], where the performance of a NN control system is reported.

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6.1.

Simulation Results of Bioprocess Plant Neural Identification Results of detailed comparative graphical simulation of CSTR KFRNN plant identification by means of the BP and the L-M learning are given in Fig. 5 and Fig. 6. A 10% white noise with different variance (SEED parameter) for each run was added to the plant inputs and outputs and the behavior of the plant identification was studied accumulating some statistics of the final MSE% (xav) for KFRNN BP and L-M learning. The results for 20 runs are given in Tables 3 and 4.

The mean average cost for all runs (e) of KFRNN plant identification, the standard deviation (s) with respect to the mean value, and the deviation (D) are presented in Table 2 for the BP and L-M algorithms. They were computed by the formulas: 1

n

e = n 책xavk, s = k=1

1 n

책D

2 i

, D = xav- e

(96)

6. Simulation Results Some simulation results of the CSTR biotechnological plant neural identification and control are summarized in this part.

The numerical results given in Tables 2, 3, and 4 are illustrated by the bar-graphics in Figures 7a)and b).

Fig. 5. Graphical results of identification using BP KFRNN learning. a) Comparison of the plant output (continuous line) and KFRNN output (pointed line); b) state variables; c) comparison of the plant output (continuous line) and KFRNN output (pointed line) in the first instants; d) MSE% of identification.

Fig. 6. Graphical results of identification using L-M KFRNN learning. a) Comparison of the plant output (continuous line) and KFRNN output (pointed line); b) state variables; c) comparison of the plant output and KFRNN output in the first instants; d) MSE% of identification.

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Table 2. Standard deviations and mean average values of identification validation using the BP and L-M algorithms of KF RNN learning.

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The comparative results showed inferior MSE%, e, and s for the L-M algorithm with respect to the BP one.

6.2.

BP algorithm

L-M algorithm

e = 0.9457 s = 0.0416

e = 0.8264 s = 0.0188

Table 3. MSE% of 20 runs of the identification program using the KFRNN BP algorithm.

No MSE% No MSE% No MSE% No MSE%

1 0.9559 6 0.9444 11 0.8523 16 0.9688

2 0.9654 7 0.9591 12 0.8105 17 0.8630

3 0.8821 8 0.9700 13 0.9863 18 0.8624

4 0.9614 9 0.9685 14 0.9038 19 0.8521

5 0.8798 10 1.0034 15 1.0122 20 0.8898

Simulation Results of Bioprocess Plant Adaptive Neural Control The graphical simulation results of DANC using the L-M algorithm of learning are presented in Fig.8 where the final MSE% was 0.854% for the L-M algorithm of learning.

Similar results are given on Fig.9 for the indirect SMC control. The final value of the MSE% obtained for the indirect SMC using the L-M algorithm of learning for the KFRNN identifier is of 0.434%.

The graphical results and the obtained final MSE% showed that the indirect SMC control is about twice times more precise that the DANC due to the utilization of the estimated states and parameters in that case, and also due to the SMC algorithm of control which substitute the plant dynamics by a decoupled lower order one.

7. Conclusions Table 4. MSE% of 20 runs of the identification program using the KFRNN L-M algorithm.

No MSE% No MSE% No MSE% No MSE%

1 0.8123 6 0.8072 11 0.8659 16 0.8628

2 0.8001 7 0.8072 12 0.8105 17 0.8226

3 0.8553 8 0.8285 13 0.8269 18 0.8514

4 0.8360 9 0.8236 14 0.8218 19 0.8288

5 0.8149 10 0.8037 15 0.8118 20 0.8280

This paper proposed a new KFRNN model for system identification and state estimation of nonlinear plants.

The KFRNN is learnt by the first order BP and by the second order L-M recursive learning algorithms. The validating results of system identification reported here gave priority of the L-M algorithm of learning over the BP one which is paid by augmented complexity. The estimated states and parameters of the plant, obtained by this Kalman filter recurrent neural network model are used for direct and indirect adaptive trajectory tracking control system design. The applicability of the proposed neural

Fig. 7. Comparison between the final MSE% for 20 runs of the identification program: a) using BP algorithm of learning, b) using L-M algorithm of learning.

Fig. 8. Detailed graphical simulation results of CSTR plant DANC using L-M learning. a) comparison between the plant output and the reference signal; b) comparison between the plant output and the reference signal in the first instants; c) control signal; d) MSE% of control.

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Fig. 9. Detailed graphical simulation results of CSTR plant Sliding Mode Indirect Control using L-M KFRTNN learning. a) comparison between the plant output and the reference signal; b) comparison between the plant output and the reference signal in the first instants; c) control signal; d) MSE% of control. control system, learnt by the BP and L-M algorithms, was confirmed by simulation results with a CSTR plant. The results showed good convergence of the two algorithms applied. The graphical and numerical validation identification results showed that the L-M algorithm of learning is more precise but more complex then the BP one. The control results of DANC and IANC (SMC) showed a great precision of reference tracking (the final MSE% is 0.854% for the DANC and 0.434 for the indirect SMC). The better results obtained with the indirect SMC are due to the utilization of the estimated states and parameters in that case, and also due to the SMC algorithm of control which substitute the plant dynamics by a decoupled lower order one.

[4]

ACKNOWLEDGMENTS

[7]

[5]

[6]

The graduated Ph.D. student Carlos-Roman Mariaca-Gaspar is thankful to CONACYT for the scholarship received during his studies at the Department of Automatic Control, CINVESTAVIPN, MEXICO. [8]

AUTHORS Ieroham Baruch*, Carlos-Roman Mariaca-Gaspar Department of Automatic Control, CINVESTAV-IPN, Av. IPN No 2508, 07360 Mexico City, Mexico. Phone: (+52-55)5747-3800/ext. 42-29. E-mails: {baruch;cmariaca}@ctrl.cinvestav.mx. * Corresponding author

[9]

[10]

References [1]

[2]

[3]

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Narendra K.S., Parthasarathy K., “Identification and Control of Dynamical Systems Using Neural Networks”, IEEE Transactions on Neural Networks, vol. 1, no. 1, 1990, pp. 4-27.

Hunt K.J., Sbarbaro D., Zbikowski R., Gawthrop P.J., “Neural Network for Control Systems (A survey)”, Automatica 28 (1992), pp. 1083-1112.

Haykin S., Neural Networks, a Comprehensive Foundation, Second Edition, Section 2.13, 84-89; Section 4.13, 208-213. Prentice-Hall, Upper Saddle River, New Jersey 7458, 1999.

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Sage A.P., Optimum Systems Control, Prentice-Hall Inc., Library of Congress Catalog Number 68-20862, Englewood Cliffs, New Jersey, 1968.

Baruch I. S., Mariaca-Gaspar, C. R., “A Levenberg-Marquardt Learning Applied for Recurrent Neural Identification and Control of a Wastewater Treatment Bioprocess”, International Journal of Intelligent Systems, Wiley Periodicals, Inc., vol. 24, 2009, pp. 1094-1114.

Baruch I.S., Mariaca-Gaspar C.R., Barrera-Cortes J., “Recurrent Neural Network Identification and Adaptive Neural Control of Hydrocarbon Biodegradation Processes. In: Hu Xiaolin, Balasubramaniam P. (eds.), Recurrent Neural Networks, I-Tech Education and Publishing KG, Vienna, Austria, ISBN 978-953-7619-08-4, 2008, Chapter 4, pp. 61-88.

Mariaca Gaspar C.R., Topologies, Learning and Stability of Hybrid Neural Networks, Applied for Nonlinear Biotechnological Processes, Ph. D. Thesis (in Spanish), Baruch, I.S., Martinez-Garcia, J.C. (thesis directors), Department of Automatic Control, CINVESTAV-IPN, Mexico City, 3rd July 2009.

Ngia L.S., Sjöberg J., “Efficient Training of Neural Nets for Nonlinear Adaptive Filtering Using a Recursive Levenberg Marquardt Algorithm”, IEEE Trans. on Signal Processing, vol. 48, 2000, pp. 1915-1927.

Zhang T., Guay M., “Adaptive Nonlinear Control of Continuously Stirred Tank Reactor Systems”. In: Proceedings of the American Control Conference, Arlington, 25th27th June, 2001, pp. 1274-1279.

Lightbody G., Irwin G.W., “Nonlinear Control Structures Based on Embedded Neural System Models”, IEEE Trans. on Neural Networks, no. 8, 1997, pp. 553-557.

Wan E., Beaufays F., “Diagrammatic Method for Deriving and Relating Temporal Neural Network Algorithms”, Neural Computations, vol. 8, 1996, pp. 182-201.

Young K. D., Utkin V.I., Ozguner U., “A Control Engineer's Guide to Sliding Mode Control”, IEEE Transactions on Control Systems Technology 7 (3), 1999, pp. 328-342.

Levent A., “Higher Order Sliding Modes, Differentiation and Output Feedback Control”, International Journal of Control, Special Issue Dedicated to Vadim Utkin on the Occasion of his 65th Birthday (Guest Editor: Fridman L.M.), ISSN 0020-7179, vol. 76, no. 9/10, 15th June-10th

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[14]

July 2003, pp. 924-941.

Eduards C., Spurgeon S.K., Hebden, R.G., “On the Design of Sliding Mode Output Feedback Controllers”, International Journal of Control, Special Issue Dedicated to Vadim Utkin on the Occasion of his 65th Birthday (Guest Editor: Fridman L.M.), ISSN 0020-7179, vol.76, no.

9/10, 15th June-10th July 2003, pp. 893-905

Appendix A: Stability proof of the theorem for KF RNN topology and BP learning.

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Where: the term u(k) = u(k) + OF; Q1,2,3,4 the are the higher order terms in the Taylor series approximation; Dh(x(k), u(k)) = u(k) + OF is the unmodeled dynamics; OF is an offset.

If Assumptions 1 and 2 fulfill, the learning algorithm for the RTNN is given by (8) and the learning parameters hk, ak are normalized and depended on the output error structure.

Then, the approximation error is bounded.

Consider a Lyapunov candidate function as: L(k) = L1(k) + L2(k)

(A.3)

In which L1(k) and L2(k) are given by: Let the Extended Recurrent Trainable Neural Network with Jordan Canonical Structure given by (1), (2), (3), (4), (5), (6), (7) and the nonlinear plant model as follows: x(k+1) = f[x(k), v(k)] y(k) = h[x(k)]

1

L1(k) = 2 e2(k)

(A.4)

(A.1) (A.2) (A.5)

and the plant and activation functions fulfill the following assumptions: Assumption 1: The plant dynamics is locally Lipchitz, so the nonlinear functions f(×), h(×) are as: f: = {f ½f = s+Df, ïçDf÷ï £ f0+f1ïçx(k)÷ï} h: = {h ½h = s+Dh, ïçDh÷ï £ h0+h1ïçx(k)÷ï} and Df, Dh are modeling errors, which reflex the effect of unmodelled dynamics.

Assumption 2: The activation functions have the following Taylor approximation:

Where: are vectors of the estimation error and (A*1 , A2* , B*, C*, D*) and (A1(k), A1(k), B(k), C(k), D(k)) denoted the ideal neural weight and the estimate of neural weight at the k-th step, respectively, for each case.

Let us consider the equation (A.4). The change of the Lyapunov function in two consecutive samples due to the training process is obtained by: Ù

Ù

Ù

Ù

Ù

DL1(k) = L1(k+1)-L1(k) = [e(k+1)-e(k)][e(k)+ +

1 2

e(k+1)-

1 2

e(k)]

(A.6)

Then, defining De(k) as the difference between two consecutive error samples, then the equation (A.6) becomes: with the approximation error bound given by:

1

DL1(k) = De(k)[e(k)+ 2 De(k)]

(A.7)

Where: can be defined as: and the output signal error is defined by: De(k) = e(k) = y(k)-y(k)

¶e(k) DW ¶W

(A.8)

Putting all weights into one vector as W = [[A1] T [A2] T [B] T [C] T [D] T]T Now, let us define the state estimation error, add and subtract the RTNN to the last equation and apply the Assumption 2, then: D(k) = x(k)-x(k)

(A.9)

Where: A1 = [[A1n] T [A1n] T ¼ [A1n] ]T, A2 = [[A21] T [A22] T ¼ [A2l] ]T, B = [[B1] T [B2] T ¼ [Bm] ]T, C = [[C1] T [C2] T ¼ [Cn] ]T, D = [D1] T which represents the weight vectors constructed by their columns. Also let:

Let us now define the output identification error and put it in terms of the state estimation error as:

(A.10)

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Where: (hA1, hA2, hB, hC, hD) and (aA1, aA2, aB, aC, aD) represented the learning rate matrices, the momentum rate matrices corresponding to the matrix weights (A1, A2, B, C, D), respectively, and.................................................................................. . Moreover, hi(i = 1,...,5) and ai(i = 1,...,5) are two positive constants, and IZ is an identity matrix, where the general symbol Z is substituted by A1, A2, B, C, D, respectively. We could define DW as: DW = hDW(k)+aDW(k-1)

(A.11)

(A.12)

(A.13) Let:

(A.14) Then: De(k+1) = -ge(k+1)-le(k)

(A.15)

and 1

1

1

DL1(k+1) = De(k+1)[e(k+1)+ 2 De(k+1)] = - 2 e2(k+1)[2g-g2] + e(k+1)e(k)[g-1]l + 2 l2e2(k)

(A.16)

Proposing: l = g-1, then: DL1(k+1) = -

1 2

e2(k+1)[-2g2+4g-1] -

1 2

l2[De(k)]2

(A.17)

According to the Lyapunov stability theory, if convergence must be guaranteed, then DL(k+1) < 0, thus -2g2+4g-1> 0, and: 1

1

1- Ö2 < g < 1+ Ö2 (A.18) That is:

3

(A.19)

Let: hmax = max {hi}. Thus, as long as: i=1

(A.20)

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Note that

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is the Euclidean norm, therefore:

(A.21) Now let: Y(k) = 1

1- Ö2 Ymax

¶e(k) ¶y(k) =, and Ymax = maxkïçY(k)÷ï, then: ¶W ¶W 1

< hmax <

1+ Ö2 Ymax

(A.22)

Now, working with equation (A5), we have:

If we consider the change of the Lyapunov function in two consecutive samples due to the training process, we obtain:

(A.23) Now substituting the following quantities:

We could obtain:

(A.24) And if the updated learning law is given by (19), then:

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Now we could use the followings trace properties: tr(AB) = tr(BA); tr(A)+ tr(B)+ tr(C) = tr(A+B+C); (AB)T = BTAT; tr(AAT) = tr(ATA) = ïçA÷ï22 ; tr(AT) = tr(A); tr(aA) = atr(A) Note that ïç÷ï2 is the Euclidean norm, a is a constant and (A1, A2, B, C, D) are weight matrices. Then:

So, due to the learning matrix law given by the equations (19)-(29), and collecting the errors as a common factor, using trace properties, we can rewrite DL2(k) as:

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Due to the error definition, collecting and put the following equation as a function of e(k), we get:

First and second traces gave us four terms as: , and Using the following inequality [6], [7]: nite matrix , we obtained:

, which is valid for any

, and for any positive defi-

Analyzing term by term and applying the Rayleigh inequality:

we could obtain:

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Now, making inner terms equal to one as in the unit circle condition for discrete time, as:

At last we get the final condition: (A.25) Where: the unmodeled dynamics and/or perturbations term d(k) is given by: (A.26) Applying the Lemma of the KF RNN rate of convergence, [6], [7], for the result (A.26) we could conclude that: the d - term must be bounded by the weight matrices and the learning parameter, in order to obtain the final result: As a consequence: From equations (A.22) and (A.25) we easily could get the equation (20). Therefore the boundedness of the

is guaranteed.

Lemma of KF RNN rate of convergence.

Applying the limit's definition, the identification error bound condition is obtained as:

Proof.

Starting from the final result of the Theorem of BP KF RNN stability: After an analysis of the iterations from k=0, we get:

Dividing by k and applying the limit's definition, the identification error bound condition is obtained in the final form:

From here we could see that the term d must be bounded by weight matrices and the learning parameter, in order to obtain:

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NOVEL GENETIC OPTIMIZATION OF MEMBERSHIP FUNCTIONS OF FUZZY LOGIC FOR SPEED CONTROL OF A DIRECT CURRENT MOTOR FOR HARDWARE APPLICATIONS IN FPGAS Yazmin Maldonado, Oscar Castillo, Patricia Melin

Abstract: This paper proposes a novel method for genetic optimization of triangular and trapezoidal membership functions of fuzzy systems, for hardware applications such as the FPGA (Field Programmable Gate Array). This method consists in taking only certain points of the membership functions, with the purpose of giving more efficiency to the algorithm. The genetic algorithm was tested in a fuzzy controller to regulate engine speed of a direct current (DC) motor, using the Xilinx System Generator (XSG) toolbox of Matlab, which simulate VHDL (Very High Description Language) code.

Keywords: genetic algorithms, fuzzy, controller, Matlab, Simulink, Xilinx System Generator, VHDL, FPGA.

functions of a fuzzy logic system for hardware applications such as FPGA (Field Programmable Gate Array). This method involves taking only a small number for points of the membership functions in order to give greater efficiency to the algorithm. The GA has been tested in a fuzzy logic controller to regulate the speed engine direct current (DC) using the Matlab [12] platform and XSG [13] with good results.

This paper is organized as follows: in section 2 we present an introductory explanation of Genetic Algo rithms, Fuzzy Inference Systems and FPGAs, section 3 describes the novel method for genetic optimization of membership functions for FLC in FPGAs, the test and result the novel genetic optimization of membership functions for FLC for speed regulate the motor DC are shown in section 4. Finally, section 5 presents the conclusions.

1. Introduction

2. Preliminaries

Fuzzy logic controllers are used successfully in many application areas, and these include control, classification, etc. [1],[2],[3],[4],[5],[6],[7]. These systems based on rules incorporate linguistic variables, linguistic terms and fuzzy rules. The acquisition of these rules is not an easy task for the expert and is of vital importance in the operation of the controller.

The process of adjusting these linguistic terms and rules is usually done by trial and error, which implies a difficult task, and for this reason there have been methods proposed to optimize those elements that over time have taken importance, such as genetic algorithms [8],[9],[10].

A Genetic Algorithm (GA) [9],[10] is a stochastic optimization algorithm inspired by the natural theory of evolution. From a principle proposed by Holland [9], GAs have been used successfully to manage a wide variety problems such as control, search, etc. [11].

This paper proposes a novel method for genetic optimization of the triangular and trapezoidal membership

The Genetic Algorithm is an optimization and search technique based on the principles of genetics and natural selection. A GA allows a population composed of many individuals to evolve under specified selection rules to a final state that maximizes the “fitness” (i.e. minimizes the cost function) [14].

A GA is inspired by the mechanism of natural selection where stronger individuals are likely the winners in a competitive environment. Here the GA uses a direct analogy of such natural evolution. Through the genetic evolution method, an optimal solution can be found and represented by the final winner of the genetic game [15].

Throughout a genetic evolution, the fitter chromosome has a tendency to yield good quality offspring, which means a better solution to any problem. In a practical GA application, a population pool of chromosomes has to be installed and these can be initially randomly set. The size of this population varies from one problem to another. In each cycle of genetic operations, termed as an evolving process, a subsequent generation is created from the

Fig. 1. GA cycle.

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chromosomes in the current population. This can only succeed if a group of these chromosomes, generally called “parents is selected via a specific selection routine.

The genes of the parents are mixed for the production of offspring in the next generation. It is expected that from this process of evolution, the better chromosome will create a larger number of offspring, and thus has a higher chance of surviving in the subsequent generation, emulating the survival of the fittest mechanism in nature.

Figure 1 shows the GA cycle.

Of course, the GA is not the best way to solve every problem, GAs have proven to be a good strategy because of its optimal results in several areas of application [3], [16].

The GA has applications in a wide variety of fields to develop solutions to complex problems, including optimization of fuzzy systems, offering them learning and adaptation, they are commonly called genetic fuzzy systems or fuzzy system hybrids.

Fuzzy systems have been used more and more, because they tolerate imprecise information and can be used to model nonlinear functions of arbitrary complexity.

A fuzzy system (FIS) consists of three stages: Fuzzification, Inference and Defuzzification [17]. We describe below these stages.

Fuzzification: Is the interpretation of input values (numeric) by the fuzzy system, and the obtained output are fuzzy values.

Definition 1. Let x Î X be a linguistic variable and Ti(x) a fuzzy set associated with a linguistic value Ti. The translation of a numeric value x corresponds to a linguistic value associated with a degree of membership, x®uTi(x), and this is known as Fuzzification. The membership degree uTi(x) represents a value of membership to a fuzzy set [18].

Inference: Is basically like the brain of the system, here the rules of the form if-then that describe this behavior are used [2]. For example: If x1 is A1 and ¼ and xn is An Then y is B

(1)

where x1¼ xn are the inputs, A1¼ An, B are linguistic terms and y is the output.

Defuzzification Consists in obtaining a numeric value for the output. This stage basically selects a point that is the most representative of the action to perform [2]. There are several methods to calculate the Defuzzification, such as the Center of Height (COH), Center of Gravity (COG), etc. The COG is shown in Equation 2.

(2)

where h1 is the maximum height of the consequent from rule i to rule N [2].

In Figure 2, the fuzzy system information processing is illustrated.

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Fig. 2. Fuzzy System.

A fuzzy system can be implemented on a general purpose computer, or by a specific use of microelectronics realization. The first offers a great versatility in terms of ease of development in various high level programs, the second device is performed in high scale integration, such as the ASICs [19].

The ASICs offer great advantages for high performance and price reduction is concerned, however, also have the disadvantage of requiring a high level of production of the same design to actually be affordable and time to get in the market is large relative to that of using a FPGA. The applications of the FPGAs go beyond the simple implementation of digital logic. The FPGAs can be used to implement specific architectures to accelerate a particular algorithm. Systems based on FPGAs provide better performance than their corresponding implementations in software platforms for general use. A specific architecture for an algorithm can have a yield of 10 to 1000 times higher than an implementation on a DSP (Digital Signal Processor). Applications that require a great number of simple operations are suitable for implementation on FPGA, a processing element can be designed to perform this operation and several instances of it can be played to perform parallel processing [20].

An FPGA is a semiconductor device that contains in its interior components such as gates, multiplexers, etc.

These are interconnected with each other, according to a given design. These devices use the VHDL programming language, which is an acronym that represents the combination of VHSIC (Very High Speed Integrated Circuit) and HDL (Hardware Description Language) [13].

Implementing an embedded fuzzy system on an FPGA is not as easy as it seems, since there are few appropriated design tools to achieve this task. Most of the time, the designer needs to construct every part of the inference system from scratch. Fortunately, there is an increasing interest in the development of designing platforms the easily achieve this task, such is the case of the Xfuzzy 2.1 and Xfuzzy 3.0; however, at the present time they cannot provide VHDL code for trapezoidal membership functions for arithmetic calculation. Other implementations of a FIS on an FPGA are reported in [21].

Any hardware implementation of an electronic system requires a complex methodology to test and validate every stage in the design process to guarantee its correct functionality; this is particularly true when the designer decides to use a HDL to make a design.

The FPGAs are good platforms for fast prototyping of digital hardware. FPGAs are very effective in implementing FLSs since they allow fast modeling and hardware verification. FPGAs can be programmed in the system,

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without shutting down the system. This functionality allows modification and tuning of rules and-or fuzzifiers to achieve better control performance. In order to accelerate the design of FLS hardware, it is helpful to have a design environment, which allows algorithmic specification of the FLS and eases the automatic synthesis and verification of FLS hardware. The topic of FLS implementation onto FPGAs has been investigated by several researchers. A brief overview of the work done by some researchers is presented next.

The design of an FPGA implementation is done by specifying the logic function to develop, either by a CAD (computer aided design) or through a hardware description language. Having defined the function to perform, the design is transferred to the FPGA. This process program the configurable logic blocks (CLBs) to perform a specific function (there are thousands of configurable logic blocks in the FPGA). The configuration of these blocks and their interconnections flexibility are the reasons why it can get very complex designs. The interconnections enable connecting the CLBs. Finally, it has configuration memory cells (CMC, Configuration Memory Cell) distributed throughout the chip, which store all information necessary for programming programmable elements mentioned. These cells usually consist of configuration RAM and are initialized in the process of loading of the configuration [22]. The programmable elements of an FPGA are: 1. Configurable Logic Blocks (CLBs) 2. In/Out Blocks (IOBs) 3. Programmable Interconnection - By fuse technology and be of OTP. - By antifusing or by type SRAM cells.

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The basic elements of an FPGA are: 1. Configurable logic blocks CLBs, and their structure and content is called architecture. There are many types of architectures, which vary largely in complexity (from a simple door to more complex modules or SPLD-like structures). They often include bi-stable or MOS (Metal Oxide Semiconductor) to facilitate the implementation of sequential circuits. Other important modules are the building blocks of I / O (IOB).

2. Interconnection resources, whose structure and content defines the routing architecture.

3. RAM, which is loaded during reset to configure and connect blocks.

Figure 4 shows these elements.

CLB

IOB

Interconection

Fig. 4. FPGA Basic Elements.

Figure 5 shows the CHIP Spartan Basic Elements.

Depending on the manufacturer we can find different solutions. FPGAs currently available on the market, depending on the structure adopted by the logical blocks that are defined, can be classified as belonging to four major families shown, in the Figure 3.

3. Novel Genetic Optimization of Membership Functions for Fuzzy Logic Controller in FPGAs The fuzzy logic controller is coded in VHDL, the FLC for the fuzzification stage, is able to instantly calculate the degree of membership, using a method to calculate the slopes [23][24][25], the inference is working with the max-min [26][27][28] and the defuziffication with the method of heights.

Figure 6 shows the block diagram in XSG of the FLC for the regulation of speed of a DC motor, the system inputs are reset, error e(t), error of change e'(t), and the parameters of each membership function for inputs and output are in total 11. The system only has an output Y(t).

The FLC has two inputs and one output, each input and output contains three membership functions, two trapezoidal one triangular. Figure 7 shows the triangular and trapezoidal membership functions (MF) that are used.

Fig. 3. Block logic a) Symmetrical Array (XILINX), b) Sea of Gates (ORCA), c) Row Based (ACTEL) and d) Hierarchical PLD (ALTERA and XILINX).

For the optimization of the FLC using GAs, you must define the chromosome that represents the information of the individual, which in this case is related to the universe of discourse and the linguistic terms. Figure 8 shows the chromosome of the GA.

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Fig. 5. CHIP Spartan of Xilinx Basic Elements.

Fig. 6. Block diagram in XSG of FLC. 56

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a)

b)

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The GA is of multiobjective type [15], which means that to determine the best individual three evaluations are performed: a) Minimum overshoot (3)

Fig. 7. Parameters of the Membership Functions. a) MF trapezoidal, b) MF triangular.

b) Minimum undershoot (4) c) Minimum output steady state error (sse) (5)

Fig. 8. GA chromosome.

In Table 1, shows the boundary parameters of the chromosome.

The FLC linguistic terms were optimized with the GA, but the fuzzy rules are not changed. The process of the GA is shown in Figure 11.

Parameters

Table 1. Boundary parameters of the chromosome.

Input 1 0 < a2 < 128 b1 = 128 128 < a1 < 255

Input Input 2 0 < a2 < 128 b1 = 128 128 < a1 < 255

Output 0 < a2 < 128 b1 = 128 128 < a1 < 255

Figure 9 shows the input of the FLC with fixed and variable parameters. Each input and output has a size or 8 bits.

Fig. 9. Points of membership functions input and output.

The blue points are fixed, the red dots are for parameter a2, the green dots are fixed (b1) and the yellow dots are for parameter a1.

Figure 10 shows the range of parameters membership functions.

Fig. 11. Optimization GA.

4. Test and Results the Novel Genetic Optimization of Membership Functions for FLC for Speed Regulate the Motor DC Fig. 10. Range of parameters membership functions.

To evaluate the ability of the GA, the FLC was simulated for speed control using a mathematical model of the plant in Matlab-Simulink [12], as shown in Figure 12.

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The FLC has the inputs, error (e(t)) and change of error (e'(t)), and the output is the control signal (y(t)). The inputs are calculated as follows: e(t)= r(t) - y(t)

(6)

e'(t)= e(t) - e(t-1)

(7)

where t is the sampling time.

The reference signal r(t), is given by: r(t) =

15 0

t>0 tÂŁ0

(8)

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Each input and output of the FIS has three linguistic terms. For the linguistic variable error and change of error, the terms are {NB, Z, PB} in this case NB is Negative Big, Z is Zero and PB is Positive Big. For the linguistic variable control signal the linguistic terms are {BD, H, BI}, in this case BD is Big Decrement, H is Hold and BI is Big Increment.

A series of experiments was performed that are listed in Table 2.

In experiment No. 17 the best FLC was found because this has the lower error value. Below are the FIS characteristics for experiments 14 and 17.

Figure 13 shows how the GA modified the parameters of the membership functions for the input e(t).

Fig. 12. Model.

Table 2. GA Parameters for different experiments.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

200 200 200 100 200 150 150 100 200 200 200 100 100 100 100 50 50 300 300 300 200

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Crossover (XOVSP) 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.7 0.7 07 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.8 0.8

Selection (SUS) 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 08 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8

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OvershootUndershoot 0.4000 0.9104 0.4000 0.0148 0.4124 0.9104 0.1920 0 0.9104 0.9104 0.6600 0.9104 0.0148 0.4124 0.9104 0.0148 0 0.9104 0.0256 0.0148 0.9104

SSE

Error

0.0425 0.1048 0.0425 0.0118 0.0198 0.1048 0.0110 0.0851 0.1048 0.1048 0.0637 0.1048 0.0118 0.0198 0.1048 0.0118 0.0814 0.1048 0.0205 0.0118 0.1048

0.0413 7.7489e-3 0.0413 0.0118 0.0021 0.0077 0.0027 0.0517 0.0077 0.0077 0.0345 7.75e-3 0.0118 5.7457e-3 7.75e-3 0.0118 2.0788e-3 7.75e-3 2.538e-3 0.0118 0.0077

Time (sec) 1161.761412 992.011800 769.952523 859.303842 1030.261653 896.357154 689.695587 754.005340 924.325968 839.401213 810.448215 711.466736 804.630357 671.710473 709.538318 822.235179 723.025902 960.829622 941.800589 1391.119402 936.360153

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Fig. 13. FIS for the experiment 14, input e(t).

Figure 14 shows the input e'(t) modified by the GA.

Fig. 14. FIS for the experiment 14, input e'(t).

Figure 15 shows the output y(t) of the FIS.

Fig. 15. Output FIS for the experiment 14.

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Figure 16 shows the control surface modified by the GA.

Fig. 16. Control Surface for experiment 14.

Figure 17 shows the output signal of the PD Incremental FLC for experiment 14.

Fig. 17. Velocity of the motor.

Figure 18 shows convergence the GA.

Fig. 18. GA convergence for experiment 14.

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Figure 19 shows the best experiment for the input e(t).

Fig. 19. Best FIS for the input e(t)).

Figure 20 shows the input e'(t) modified by the GA.

Fig. 20. Best FIS for the input e'(t).

Figure 21 shows the output y(t) of the FIS modified by the GA.

Fig. 21. Best FIS for the output y(t).

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Figure 22 shows the control surface modified by the GA.

Fig. 22. Control Surface.

Figure 23 shows the output signal close loop of the FLC for experiment 17.

Fig. 23. Velocity Figure 24 shows convergence the GA for the best experiment.

Fig. 24. Shows the GA convergence error for experiment 17.

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5. Conclusions We proposed a novel method for genetic optimization of a fuzzy logic controller in FPGA for the regulation of speed of a DC motor, were the method optimizes three triangular and trapezoidal membership functions for the two inputs and one output of the FLC. The genetic algorithm optimizes only three of the eleven parameters of the membership functions, the algorithm proved to be very efficient with good results. The objective function of the GA considers three characteristics: overshoot, undershoot and steady state error, so this makes it a multiobjective GA. Each FIS was simulated in an Incremental PD Fuzzy Con-troller for speed control of the DC motor. The best FLC was obtained in 50 generations with 70% crossover and 80% selection, with a result of zero of overshoot and under-shoot steady state error of 2.0788e-3, in a time of 723.025902 seconds with a speed of 15 rpm. MatlabSimulink and Xilinx System Generator was used to perform the simulations.

[15] [16] [17] [18] [19]

[20]

[21]

[22]

AUTHORS Yazmin Maldonado, Oscar Castillo*, Patricia Melin Tijuana Institute of Technology, Tijuana, Mexico. E-mail: ocastillo@tectijuana.mx. * Corresponding author

[23]

[24]

References Cirstea M.N., Khor J.G., McCormick M., Neural and fuzzy logic control of drives and power system, Newnes, 2002.

[2] Driankov D., Hellendorn H., Reinfrank M., An Introduction to Fuzzy Control, Springer, 1996.

[3] Goldberg D.E., Genetic Algorithm in Search, Optimization & Machine Learning, 1989.

[4] Jantzen J., Tunning of Fuzzy PID Controllers, 1998, pp. 1-22.

Klir G., Yuan B., Fuzzy Sets and Fuzzy Logic Theory and [5] Applications, Prentice Hall, 1995.

McNeilland D., Freiberger P., Fuzzy Logic: The Revolutio[6] nary Computer Technology that is Changing our World, Simon & Schuster, 1993.

Tsoukalas L., Ohrig R.E., Fuzzy and Neural Approaches in [7] Engineering, WileyI-Interscience, 1997.

Tommiska M., Vouri J.,”Implementation of genetic [8] algorithms with programmable logic devices”. In: Proc 2nd Nordic Workshop Genetic Algorithm, 1996, pp. 71-78.

Holland J. H., Adaptation in Natural and Artificial Sys[9] tems, Cambrige, MA: MIT Press, 1992.

[10] Golberg D.E., Genetic Algorithms in Sear Optimization and Machine Learning, Boston, MA: Addison-Wesley, 1989.

[110] Grefenstette J.J., Gopal R., Rosmaita B.J., Van Gucht D., “Genetic Algorithms for the traveling salesman problem”. In: Proc. 1st Int. Conf. Genetic Algorithms, 1985, pp. 160-168.

[12] Web page of Matlab-Simulink, available in www.mathworks.com, 2010.

[13] Web page of Xilinx system Generator and FPGAs, available in www.xilinx.com, 2010.

[14] Haupt Randy L., Haupt Sue E., Practial Genetic Algo[1]

[25]

[26]

[27]

[28]

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rithms, Wiley, 2004.

Man K.F., Tang K.S., Kwong S., Genetic algorithms, Springer, 2000.

Koza J.R., Genetic Programming: on the Programming of Computers by means of natural selection, 1992.

Zadeh L.A., “Fuzzy Sets”, Information and Control 8, 1965, pp. 338-353..

Zdenko K., Stjepan B., Fuzzy Controller Design Theory and Applications, Taylor and Francis, 2006.

Montiel O., Maldonado Y., Sepulveda R., Castillo O., “Simple Tuned Fuzzy Controller Embedded into an FPGA”. In: IEEE NAFIPS Conference Proceedings, 2008, pp. 1-6.

Montiel O., Maldonado Y., Sepulveda R., Castillo O., “Development of an Embedded Simple Tuned Fuzzy Controller”. In: IEEE World Congress on computational Intelligence WCCI-2008, 2008, pp. 555-561.

Velo F.J.M., Baturone L., Solano S.S., Barriga A., “Rapid Design of Fuzzy Systems with XFUZZY”, IEEE International Conference on Fuzzy Systems FUZZ-IEEE-2003, 2003, pp. 342-347.

Buj Gelonch R.A., Sancho Francisco C., Procedimiento de Diseńo de Circuitos Digitales Mediante FPGAs, Universidad de Lleida, Escuela politécnica superior Escuela técnica en informática de sistemas, 2007. (in Spanish) MaldonadoY., Montiel O., Sepulveda R., Diseńo y validación de la etapa de fuzzificacion para sistemas difusos en FPGAs, ERA-08, 2008, pp. 1-7. (in Spanish) Maldonado Y., Montiel O., Sepulveda R., Castillo O., “Design and simulation of the fuzzification Stage through the Xilinx System Generator”, Soft Computinf for Hybrid Intelligent systems, Springer, 2008, pp.

297-305.

Montiel O., Sepulveda R., Maldonado Y., Castillo O., Design and simulation of The Type-2 Fuzzification Stage: Using Active Membership Functions, Evolutionary Design of Intelligent systems in Modeling, Simulation and Control, Springer, 2009, pp. 273-293.

Olivas J.A., Sepulveda R., Montiel O., Validacion y Prueba de una Maquina de Inferencia Difusa Mediante Xilinx System Generator, ERA-08, 2008, pp. 1-6. (in Spanish) Olivas J.A., Sepulveda R., Montiel O., Castillo O., Methodology to Test and Validate a VHDL Inference Engine through the Xilinx System Generator, Soft Computinf for Hybrid Intelligent Systems, Springer, 2008, pp. 325-331.

Sepulveda R., Oscar Montiel O., Olivas J., Castillo O., Methodology to Test and Validate a VHDL Inference Engine of a Type-2 FIS, through the Xilinx System Generator, Evolutionary Design of Intelligent systems in Modeling, Simulation and Control, Springer, 2009, pp. 295- 308.

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IMPROVING THE INTENSIFICATION AND DIVERSIFICATION BALANCE OF THE TABU SOLUTION FOR THE ROBUST CAPACITATED INTERNATIONAL SOURCING PROBLEM (ROCIS) Héctor Joaquín Fraire Huacuja, José Luis González-Velarde, Guadalupe Castilla Valdez

Abstract: This paper addresses the robust capacitated international sourcing problem (RoCIS), which consists of selecting a subset of suppliers with finite capacity, from an available set of potential suppliers internationally located.

This problem was introduced by González-Velarde and Laguna in [1], where they propose a deterministic solution method based on tabu search memory strategies. The process consists of three steps: build an initial solution, create a neighborhood of promising solutions and perform a local search in the neighborhood. In this work we propose improving the construction of the initial solution, the construction of the neighborhood, the local search, and the intensification and diversification balance. Experimental evidence shows that the improved tabu solution with diversification outperforms the best solutions reported for six of the instances considered, increases by 18% the number of best solutions found and reduces by 44% the deviation of the best solution found, respect to the best algorithm reported.

Keywords: RoCis, heuristic approach, optimization, initial solution, tabu search, memory strategies.

1. Introduction The international sourcing problem consists of selecting a subset of suppliers, with a finite production capacity, from an available set of potential suppliers located internationally. In this paper we analyze the variant proposed in [1], which considers only a product in a single period and uncertainty on the demand and the exchange rate are modeled via a set of scenarios. In the formulation of this problem it is assumed that the costs depend on the economic conditions in the countries where the suppliers and the plants are located and that the production capacity of suppliers is finite. The robust formulation considers that a solution is feasible if and only if it is feasible in all the scenarios. The objective function minimizes the expected value of the costs and penalizes the solutions whose optimal cost in some scenario surpasses the expected value of the optimal costs in all the scenarios. Through this mechanism the associated risk is incorporated.

The rest of our paper is organized as follows: related work, problem formulation, solution proposal and experimental results.

2. Related work Now we summarize the most relevant works from the literature about the plant location problem, because it is 64

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closely related to the international sourcing problem.

Jucker and Carlson solve a single product, single period problem, with price and demand uncertainty [2]. Hodder and Jucker present a deterministic single period, single product model [3]. Hodder and Jucker optimally solve a single period, single product model, setting the plants quantity [4]. Haug approaches the deterministic problem with a single product and multiple periods with discount factors [5]. Louveaux and Peters solve a scenario-based problem in which the capacity is a first stage decision [6]. Gutierrez and Kouvelis explore the generation of scenarios to model price uncertainty and solve a simple plant location problem [7]. Kouvelis and You propose an un-capacitated version robustness approach based on a minimax regret criterion [8] .

Now we describe the most relevant work about the international capacitated sourcing problem. The robust formulation of the international capacitated sourcing problem was proposed by Gonzalez-Velarde and Laguna [1]. In this work they propose a solution method based on the Benders paradigm, incorporating Tabu Search (TS) mechanisms. The process consists of building an initial solution, creating a neighborhood of promising solutions and performing a local search on the neighborhood. As the choice of the initial solution determines the efficiency of the process, this solution is constructed by applying a heuristic that gives preference to suppliers with lower fixed costs and greater production capacity.

González-Velarde and Martí propose a non-deterministic solution method based on GRASP, without incorporating the adaptive element, so the algorithm is classified as adaptive memory programming (AMP) type and path relinking is used to post processing the built solutions [9]. In the heuristic used to build a set of initial solutions, the shipping cost of each supplier to all plants is considered. The authors suggest that this way of incorporating the shipping cost seems too pessimistic.

In this work we propose to modify the TS based solution by improving the construction of the initial solution, the construction of the neighborhood, the local search, and the intensification and diversification balance.

3. Problem formulation AThe robust capacitated international sourcing problem (RoCIS) consists of selecting a set of suppliers to satisfy the demand for products at several plants located in different countries. The model deals with a single item in a single period. The uncertainty in the demand and the exchange rates are modeled via a set of scenarios. The model uses the following definitions:

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Parameters N: international plants set {1, 2,..., n}. international suppliers set {1, 2,..., m}.

M: scenarios set.

S: fi: fixed cost associated with supplier i. total unit cost for delivering items from supplier i cij: to plant j. bi: capacity of supplier i. demand at plant j under scenario s. djs: exchange rate at supplier's i location under eis: scenario s. ps: occurrence probability of scenario s.

Variables xijs: product shipment from supplier i to plant j under scenario s.

1 if supplier i is contracted and 0 otherwise. yi: Given a supplier selection y=[yi] i=1,2,...m, then the problem becomes separable, and the following transportation problem must be solved for each scenario s:

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4.1. Improving the initial solution construction The reported solutions to RoCIS problem considers two strategies to select the suppliers that must be incorporated into an initial solution [1, 9]. The first one gives priority to the smaller fixed cost suppliers and greater production capacity the second one incorporates the expected value of the products shipment cost from the selected supplier to all plants. The main limitation of the first strategy is that it does not consider the shipment cost, and even though this factor is considered the second one, the mechanism used is too pessimistic. In this work we propose to modify the incorporation mechanism of the shipment cost, to include only the plants towards which the products shipment from the site of the supplier is less expensive.

To describe this proposal, let Ci={cij | j=1,2,...n}the shipment costs set from supplier i to all the plants and BCi a threshold cost defined on Ci. Then, the set of plants toward which the products shipment from the site of the supplier i is less expensive, can be defined as:

Minimize (1)

Now if cmin and cmax are the minimum and maximum of the shipment costs in Ci, then BCi can be modeled as:

subject to: (2) To include in Gi only the plants in Pi+ , must be defined

(3) as: (4) Then, the problem consists of minimizing:

(5) where

and

The a a value locates the initial solution on different regions of the search space, and it can be used as a long term diversification mechanism in the local search by dynamically changing its values.

.

4. Solution proposal The solution method reported in [1] is a heuristic search based on Benders decomposition paradigm. An initial solution is constructed giving priority to the suppliers of smaller fixed cost and larger production capacity. For each supplier selection, the problem is decomposed into transportation subproblems, one for each scenario. The optimal dual solution for each sub problem is used to find a promissory neighborhood and a local search in the neighborhood is carried out. The method uses several short term tabu memories, to monitor the suppliers used in the visited solutions [3]. As the search goes, the best found solution is updated and continues until finishing the exploration of the neighborhood. When the search stops, a new search begins in the best found solution neighborhood, the procedure continues during a certain number of iterations (50). Figure 1 shows the detailed algorithm for this solution method. As we can see the tabu solution, except for the diversification induced by the short term memory, it does not include a long term diversification process.

4.2. Improving the neighborhood construction To improve the quality of generated neighbors the mechanism used to determine the relative cost of the three types of movements that are applied to generate the neighborhood (insert, delete and suppliers exchange) is modified. With the current ri definition, the movements are selected based on their impact on the growth rate of the objective function, which could return in some cases an inappropriate choice. Currently ri is defined as:

where fi is the fixed cost of supplier i,

is

the expected dual price of supplier i, ps is the occurrence probability of scenario s and pis is the supplier i dual price in scenario s.

In opposite would be more appropriate to select the movements based on the net increase of the objective function generated when the movements are applied. Then Articles

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ri is redefined as:

where bi is the production capacity of supplier i.

4.3. Improving the neighborhood local search To improve the local search process is proposed to apply path re-linking on the two best global solutions found when each iteration ends, after the second iteration. The path re-linking strategy used is basically the described in [9]. The algorithm used to perform the process is shown in Figure 1.

Data structures used: y' prior best global solution found. y'' actual best global solution found.

For each pair of solutions y' and y'': Step 1: Determine the yÇ and yÈ solutions considering: a. yÇi = 1 if y'i = 1 and y''i = 1, otherwise yÇi = 0, b. yÈi = 1 if y'i = 1 or y''i = 1, otherwise yÈi = 0.

Step 2: Determine the S' set of selected suppliers in y' but not selected in y'' Step 3: Determine the S'' set of not selected suppliers in y' and selected in y'' Step 4: Add to the yÇ solution the suppliers in S' in appropriated order to reach y' Step 5: Alternate between delete of y' a supplier of S' and add a supplier of S'' until reach the y'' solution Step 6: Append to y'', one by one the suppliers of set S' in the appropriate order to reach yÈ.

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done. Table 1 shows the experimental results obtained with the improved tabu solution (ITS a), solving the instances with different a values. The first column contains an identifier of the instance solved. The second one contains the best solution reported with AMP [9]. The following 8 columns contain the best solution found and the number of iterations required to find it with each algorithm used (a=0.2, 0.4, 0.6, and 0.8). The last column contains the best global solution found by the algorithms evaluated. For each algorithm, the best solution found is emphasized when this is the best overall solution. As we can see on average approximately 12 iterations are needed for all a? values. As we can see the instances 25 and 29 for a=0.6 are which consume the largest number of iterations without reach the best known, and for a=0.4 the best known are achieved in few iterations. Then if we discard these instances the average iterations to achieve the best solution is reduced as shown in Table 10. As we can see the iterations required for a=0.2 and 0.4 remains around 12, but for a=0.6 it is reduced to 10. Therefore the sequence of a values considered to use in the diversification process is the following: S {a0=0.6, a?=0.2, a2=0.4} This sequence takes advantage of the speed to reach the greatest number of best solutions with a= 0.6 and helps to refine the results with the following values a=0.2, 0.4.

Data structure used: Supplier selection: Hashing solution representation:

Fig. 1. Path re-linking algorithm.

Incorporating a long term diversification process The evaluation of the ROCIS problem objective function requires applying |S| times the linear optimizer. This constitutes the main source of computational cost of the TS method reported in [1]. To reduce this cost, the candidate solutions evaluated and their objective values are recorded. Each time that a candidate solution must be evaluated, the record is reviewed and if the candidate already is recorded, the objective value is retrieved; otherwise, the candidate solution evaluation is performed and recorded. This record can be used to determine the behavior of local search carried out in the neighborhood built in the iteration. If the number of candidate solutions that were recorded is counted in each iteration, then we can determine what neighborhood percentage has already been explored. High values of this percentage indicate that the local search is stangned and that a diversification process is required. When iteration ends, we can determine if the neighborhood percentage that has been explored, exceeds a specified level (50% in our case). If this limit is reached, a new initial solution is built by assigning to the a parameter, a value that has not been used; otherwise, the iteration continues building a new neighborhood.

To determine the sequence of a values used in the diversification process a preliminary experimentation was

List of evaluated solutions: coded_sol[H[y]] Tabu suppliers lists: insertion, delete and swap

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Main algorithm Step 1: Building initial solution y 1.1

Calculate

1.2

Build a list of suppliers in ascending order by Gi =

1.3

1.4 1.5

fi bi

Build the solution y, selecting suppliers in the sorted list until the sum of the capacities of the selected suppliers is greater than D. Determine F[y], solving the distribution subproblems generated for all scenarios.

Record the solution y and its objective value F[y] in the list of evaluated solutions coded_sol[H[y]]

Step 2: Repeat until reach 50 iterations 2.1 Generate a promising solution neighborhood of y 2.1.1 Determine the expected value of shadow prices (pis) linked to the constraint corresponding to each supplier, in the solution of the |S| sub problems (for all the scenarios).

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2.1.2 Calculate the relative cost of the suppliers (ri.)

2.1.3 For each of the possible insertions, deletions and swaps of suppliers that can be made from the solution, validate the appropriate feasible configuration with respect to the maximum demand D. 2.1.4 Build lists of candidates movements of insertion, deletion and swap, with the movements identified in the previous step that are not stored in the tabu list for each type of movement.

The list of candidates for insertion contains the suppliers with the 3 lowest values of ri.

The list of candidates for deletion contains the suppliers with the 3 highest values of ri.

The list of candidates for swap contains the suppliers with the

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As part of the diversification mechanism the following variables and constants are defined: RC: The counter of candidate solutions generated in the current neighborhood already registered in the hashing list.

The number of candidate solutions in the current neighborhood: NCS = 3 + 3+ The percentage of the current neighborhood had already been reviewed in previous iterations (stagnation level observed): SL = RC / NCS Function InitialSolution(a) Step 1: Calculate Step 2: Build a suppliers list in ascending sorted by

lowest values of rj -

ri corre-sponding to the swap between supplier i by supplier j in configuration y.

2.2 Do local search 2.2.1 For each configuration y' generated from the movements of the candidate lists of insertion, deletion and swapping:

where

and

where 2.2.1.1

The suppliers involved in the movement used to generate the configuration y' are: appended to the insertion tabu list (if the movement was for deletion), or removed (if the movement was to insertion). This two tabu lists are used too when a swap movement is applied, cnsidering that a swap movement requires a deletion and an insertion. The number of iterations during which a supplier involved in a movement is considered tabu are: for insertions and eliminations and for swaps.

2.2.1.2

If the solution y' is already saved in the list of evaluated solutions coded_sol[H[y]], its objective value F[y'] is retrieved, otherwise F[y'] is calculated and appended to the list.

Update the best solution found ybest.

2.2.1.3 2.2.2 y = ybest

Step 3: Build the solution y, selecting suppliers in the sorted list until the sum of the capacities of the selected suppliers is greater than D. Step 4: Determinate F[y], solving the transportation sub problems generated in all scenarios.

Step 5: Record the y solution and its objective value F[y] in the list of evaluated solution coded_sol[H[y]] Step 6: Return the actual solution y.

Main algorithm Step 1: change = 0 Step 2: y = SoluciónInicial(a=0.6) Step 3: Repeat until 50 iterations 3.1 Generate a promising solution neighborhood of y 2.1.1 Determine the expected value of shadow prices (pis) linked to the constraint corresponding to each supplier, in the solution of the 27 sub problems (for all the scenarios).

Fig. 2. Tabu solution algorithm TS [1].

3.1.2 Calculate the suppliers relative cost (ri.)

Data structures used: Supplier selection: Hashing solution representation: List of evaluated solutions: coded_sol[H[y]] Tabu suppliers lists: insertion, delete and swap

3.1.3 For each of the possible insertions, deletions and swaps of suppliers that can be made from the solution, validate the feasibility of configuration with Articles

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respect to the maximum demand D. 3.1.4 Build lists of candidate movement for insertion, deletion and swap, with aspiration criteria.

The list of candidates for insertion contains the suppliers with the 3 lowest values of ri.

The list of candidates for deletion contains the suppliers with the 3 highest values of ri.

The list of candidates for swap contains the suppliers with the

nations and

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for swaps.

If the solution y' is already saved in the list of evaluated solutions, its objective value F[y'] is retrieved and the repeated candidate solutions counter RC is incremented, otherwise F[y'] is calculated and appended to the list.

3.2.1.3 Update the best solution found ybest.

3.2.2 Path relinking process 3.2.2.1 From the second iteration, update the two best solutions found.

3.2.2.2 Apply path relinking to the two best solutions found and update ybest 3.2.3 Diversification process 3.2.3.1 Calculate the stagnation level SL=RS/NCS 3.2.3.2 If ( SL > 0.5 ) then change=change+1 If (change=1) then y=InitialSolution(a=0.2) If (change=2) then y=InitialSolution(a=0.4) Else y= ybest 3.2.1.2

lowest values of rj

- ri corresponding to the swap between supplier i by supplier j in configuration y 3.2 Local search process 3.2.1 For each configuration y' generated from the movements of the candidate lists of insertion, deletion and swapping: 3.2.1.1 The suppliers involved in the movement used to generate the configuration y' are: appended to the insertion tabu list (if the movement was for deletion), or removed (if the movement was to insertion). This two tabu lists are used too when a swap movement is applied, cnsidering that a swap movement requires a deletion and an insertion. The number of iterations during which a supplier involved in a movement is considered tabu are: for insertions and elimi-

Fig. 3. Improved tabu solution with diversification ITSD algorithm.

Table 1. Performance of the improved tabu solution ITS (with a= 0.2, 0.4, 0.6, 0.8 and 50 iterations).

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 68

AMP 33178.63458 44181.48210 39558.82437 47120.47639 41515.93392 41285.57388 42015.04585 55627.07483 46055.98672 57188.41647 60692.58875 55603.79858 67389.80329 65420.80667 78184.02415 38094.86669 34109.31059 34127.48022 40558.79816 32210.96759 41551.65039 38833.67675 44391.63693 41831.94585 53709.18863 61377.26091 69464.05787 75482.59766 61818.89140 68193.73131

Articles

ITS 0.2 33178.63458 44181.48210 39558.82437 47120.47639 41515.93392 41285.57388 42015.04585 55627.07483 46055.98672 57188.41647 60692.58875 55603.79858 67389.80329 65595.87523 78184.02415 37809.00955 34109.31059 33814.09910 40558.79816 31496.84804 41741.15551 38833.67675 44391.63693 41831.94585 53709.18863 61377.26091 69541.17681 75482.59766 62170.32689 68073.37865

12 12 12 10 12 20 7 10 6 32 6 11 14 9 6 22 10 10 9 11 11 8 7 18 7 11 27 5 28 11

ITS 0.4 33178.63458 44181.48210 39558.82437 47120.47639 41515.93392 41285.57388 42015.04585 55627.07483 46055.98672 57188.41647 60692.58875 55617.16356 68158.76152 65427.00810 78184.02415 37809.00955 34109.31059 33814.09910 40558.79816 31496.84804 41527.77087 38833.67675 44391.63693 41831.94585 53709.18863 61377.26091 69496.30247 75482.59766 61818.89140 68193.72023

10 10 9 7 8 17 12 9 5 28 11 9 13 24 7 12 12 9 11 12 7 25 12 6 8 10 9 20 5 24

ITS 0.6 33178.63458 44181.48210 39558.82437 47120.47639 41515.93392 41285.57388 42015.04585 55627.07483 46055.98672 57188.41647 60692.58875 55603.79858 68158.76152 65420.80667 78184.02415 37809.00955 34109.31059 33814.09910 40558.79816 31496.84804 41741.15551 38833.67675 44391.63693 41831.94585 54180.96605 61377.26091 69464.04654 75952.11365 61963.37426 68193.72023

9 9 7 9 10 5 7 11 10 2 9 10 9 10 21 11 9 9 11 9 7 11 13 6 43 8 26 4 44 18

ITS 0.8 33178.63458 44181.48210 39558.82437 47120.47639 41515.93392 41285.57388 42015.04585 55627.07483 46055.98672 57188.41647 60692.58875 55617.16356 68158.76152 65420.80667 78184.02415 37820.65015 34109.31059 33814.09910 40570.84487 31496.84804 41741.15551 38833.67675 44391.63693 41831.94585 53709.18863 61377.26091 69464.04654 75482.59766 61851.60940 68292.71756

8 10 6 16 13 13 11 7 9 3 20 23 7 10 36 11 9 12 30 12 10 10 11 28 10 19 45 20 30 17

Best 33178.63458 44181.48210 39558.82437 47120.47639 41515.93392 41285.57388 42015.04585 55627.07483 46055.98672 57188.41647 60692.58875 55603.79858 67389.80329 65420.80667 78184.02415 37809.00955 34109.31059 33814.09910 40558.79816 31496.84804 41527.77087 38833.67675 44391.63693 41831.94585 53709.18863 61377.26091 69464.04654 75482.59766 61818.89140 68073.37865

Journal of Automation, Mobile Robotics & Intelligent Systems

As it is showed in Figure 3, the InitialSolution(a) function is used to diversify the search and is initially invoked with a0. When the observed stagnation level reaches 50%, after a complete exploration of the current neighborhood, the InitialSolution(a) function is invoked with a?. The search continues and when the stagnation level reaches 50% again, the InitialSolution(a) function is called with a2. Then the search continues until finish, without changing the a value.

Figure 3 shows the detailed proposed tabu solution algorithm, which incorporates the improvements in the construction of the initial solutions, in the construction of neighborhoods, in the local search, and the long term diversification mechanism.

5. Experimental results The experiments were done in a computer Dell Optiplex 160L with a Pentium IV processor to 2.4 GHZ and 1 GB ram. The source code was compiled using Visual C 6.0 and the operating system Windows XP. For the solution of the transportation sub problems LINDO API 2.0 was used. To evaluate the performance of the algorithms, the larger instances reported in [9] were used. The instances were generated with 20 plants, 40 suppliers and 27 scenarios, and constitute a representative sample of instances with the same size and different hardness degrees. As the optimal solutions for these instances are not known, the results obtained in this work are compared with the best solutions reported in [9]. To evaluate the impact of the proposed improvements on the TS performance, four types of experiments were carried out.

In the first one the proposed improvements for the initial construction, the construction of the neighborhood and local search were evaluated. Table 1 shows the results obtained with the improved tabu solution (ITS a), for different values. The table shows that the improved TS found better solutions than those reported for instances 16, 18, 20, 21, 27 and 30. As we can observe the best global solutions found by AMP are also found by ITS using one or more of the a values. However there is not a single value which allows finding the best global solution for all instances. Experimental evidence confirms that the value operates as a diversification mechanism on the search process.

Table 2 shows a summary of the experimental results, including: the average cost of the found solutions, the number of overall best solutions found, the error rate over the average cost of best solution and the average time used to solve each instance (in CPU seconds).

In other hand, in Table 1 we can observe that 4 ITS algorithms obtains the best known solutions for 19 instances, and similarly 3 ITS algorithms for 3 instances, 2 ITS algorithms for 4 instances and 1 ITS algorithm for 4 instances. Then if four groups of instances are considered: I1, I2, I3 and I4, where the group In contains all the instances for which n ITS algorithms obtains the best known solutions. We could consider that for n>m, the instances in In are easiest than the instances in Im. Then the question is ¿ a structural and landscape analysis of the instances can help us to explain the relative hardness observed?

In the second experiment the structural analysis of the instances was done. For all instances the sparsity, the

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variation coefficient and the skewness of the instance parameters (shipment cost matrix, fixed cost vector, capacity supplier vector, plants demand matrix and currency rate matrix) were calculated. The sparsity measures the percentage of parameter structure elements that are equal to zero; the main interest in this measure is that according to Mitchell and Borchers, it has a strong influence on algorithm behavior [11]. The variation coefficient (VC) is defined as s/X where s is the standard deviation and X the mean of the structure elements. VC gives an estimate of the variability of the structure elements, independent of their size. The skewness is the third moment of the mean normalized by the standard deviation; it gives an indication of the degree of asymmetry of the structure elements. In the experiment all the instances were considered grouped in I1, I2, I3 and I4. Then the sparsity, the variation coefficient and skewness were calculated for the five components of each instance: shipment cost matrix, fixed cost vector, capacity supplier vector, plants demand matrix and currency rate matrix. For all the instances and components the observed sparsity percentage was zero.

Table 3 contains the obtained results for the variation coefficient and Table 4 the results for the skewness. As we can observe the four instances groups shows a similar structure, because the differences between the values of the variation coefficient and of the skewness are minimal.

In the third experiment a ruggedness analysis of the landscape was done. The central idea of the landscape analysis in combinatorial optimization is to represent the space searched by an algorithm as a landscape formed by all feasible solutions and the objective value assigned to each solution [12]. The information generated with the landscape analysis is used to gain knowledge about: the search space characteristics and their relation with the behavior of local search or metaheuristic algorithms [13],[14], problem or problem instance hardness [15], [16], or useful parameterizations of local search algorithms [17]. A search landscape is considered rugged if there is a low correlation between neighboring points.

To measure this correlation a random walk of length m, is performed in the search landscape to interpret the resulting series of m points {f(xt), t=1,…, m} as a time series. The autocorrelation r(s) of the points in the series that are separated by s steps is defined as:

where s2(f) and f are the variance and the mean of the points in the series. Now the search landscape correlation length is defined as:

where |r(1)| ¹ 0. Then the lower is the value of l, the more rugged is the landscape [18].

Previously to the determination of the search space correlation length (l) values for the instances in the considered groups, we must determine the length of the random walk to be applied. For this purpose were calculated five times the average of the l values of all the instances with a random walk length given. The obtained results, Articles

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Table 2. Comparative summary of the performance of the improved tabu solution ITS with 50 iterations.

AMP Val. 50359.15105 # Bests 24 Dev. 0.10% CPU secs 218.23

TS 51730.25 1 2.93% 381.25

ITS 0.2 50337.79759 26 0.05% 374.22

ITS 0.4 50341.93472 25 0.06% 392.92

Table 3. Structural information of the instances groups I1, I2, I3 and I4 (variation coefficient).

Shipment cost Min Median Max Mean Supplier capacity Min Median Max Mean Fixed cost Min Median Max Mean Demand Min Median Max Mean Exchange rate Min Median Max Mean

I1 0.45 0.48 0.48 0.47 I1 0.10 0.12 0.12 0.12 I1 0.10 0.12 0.12 0.12 I1 0.15 0.16 0.16 0.16 I1 0.10 0.10 0.10 0.10

I2 0.46 0.48 0.49 0.48 I2 0.11 0.11 0.12 0.11 I2 0.11 0.11 0.12 0.11 I2 0.15 0.16 0.16 0.16 I2 0.10 0.10 0.10 0.10

I3 0.46 0.47 0.49 0.47 I3 0.11 0.12 0.12 0.12 I3 0.11 0.12 0.12 0.12 I3 0.16 0.16 0.16 0.16 I3 0.10 0.10 0.10 0.10

I4 0.45 0.47 0.50 0.47 I4 0.10 0.12 0.13 0.11 I4 0.10 0.12 0.13 0.11 I4 0.15 0.16 0.17 0.16 I4 0.10 0.10 0.10 0.10

Table 5. Average l value obtained in five random walks with two lengths: 1000 and 50000 steps.

Walk

l average (Length=1000) 0.25 0.26 0.28 0.28 0.25

1 2 3 4 5

l average (Length=50000) 0.19 0.18 0.20 0.19 0.20

Table 6. Minimum, median, maximum and mean of the search landscape correlation length (l value) obtained with a random walk of 50000 steps, for the groups I1, I2, I3 and I4.

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Search landscape correlation length (l value) I1 I2 I3 I4 0.17 0.16 0.13 0.14 0.20 0.18 0.16 0.19 0.22 0.21 0.17 0.23 0.19 0.18 0.15 0.19

ITS 0.6 50383.51267 24 0.14% 371.75

ITS 0.8 50352.94571 23 0.08% 368.97

Best 50310.56363 30 0%

Table 4. Structural information of the instances groups I1, I2, I3 and I4 (skewness).

Shipment cost Min Median Max Mean Supplier capacity Min Median Max Mean Fixed cost Min Median Max Mean Demand Min Median Max Mean Exchange rate Min Median Max Mean

I1 0.08 0.17 0.23 0.16 I1 -0.10 -0.03 0.50 0.09 I1 -0.10 -0.03 0.50 0.09 I1 -0.05 -0.04 0.03 -0.02 I1 -0.03 0.00 0.09 0.01

I2 0.13 0.17 0.25 0.18 I2 0.00 0.13 0.49 0.19 I2 0.00 0.13 0.49 0.19 I2 -0.10 -0.01 0.12 0.00 I2 -0.08 -0.01 0.05 -0.01

I3 0.17 0.25 0.38 0.27 I3 -0.30 0.00 0.22 -0.03 I3 -0.30 0.00 0.22 -0.03 I3 -0.06 0.02 0.05 0.00 I3 0.05 0.06 0.07 0.06

I4 -0.03 0.19 0.35 0.17 I4 -0.56 -0.05 0.64 -0.01 I4 -0.56 0.08 0.64 0.09 I4 -0.12 -0.03 0.10 -0.01 I4 -0.06 0.02 0.06 0.01

Graph 1. Average l value obtained in five random walks with two lengths: 1000 and 50000 steps.

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Table 7. Performance of the improved tabu solution with diversification ITSD and 50 iterations.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

AMP 33178.63458 44181.48210 39558.82437 47120.47639 41515.93392 41285.57388 42015.04585 55627.07483 46055.98672 57188.41647 60692.58875 55603.79858 67389.80329 65420.80667 78184.02415 38094.86669 34109.31059 34127.48022 40558.79816 32210.96759 41551.65039 38833.67675 44391.63693 41831.94585 53709.18863 61377.26091 69464.05787 75482.59766 61818.89140 68193.73131

ITS 0.2 33178.63458 44181.48210 39558.82437 47120.47639 41515.93392 41285.57388 42015.04585 55627.07483 46055.98672 57188.41647 60692.58875 55603.79858 67389.80329 65595.87523 78184.02415 37809.00955 34109.31059 33814.09910 40558.79816 31496.84804 41741.15551 38833.67675 44391.63693 41831.94585 53709.18863 61377.26091 69541.17681 75482.59766 62170.32689 68073.37865

12 12 12 10 12 20 7 10 6 32 6 11 14 9 6 22 10 10 9 11 11 8 7 18 7 11 27 5 28 11

ITS 0.4 33178.63458 44181.48210 39558.82437 47120.47639 41515.93392 41285.57388 42015.04585 55627.07483 46055.98672 57188.41647 60692.58875 55617.16356 68158.76152 65427.00810 78184.02415 37809.00955 34109.31059 33814.09910 40558.79816 31496.84804 41527.77087 38833.67675 44391.63693 41831.94585 53709.18863 61377.26091 69496.30247 75482.59766 61818.89140 68193.72023

10 10 9 7 8 17 12 9 5 28 11 9 13 24 7 12 12 9 11 12 7 25 12 6 8 10 9 20 5 24

ITS 0.6 33178.63458 44181.48210 39558.82437 47120.47639 41515.93392 41285.57388 42015.04585 55627.07483 46055.98672 57188.41647 60692.58875 55603.79858 68158.76152 65420.80667 78184.02415 37809.00955 34109.31059 33814.09910 40558.79816 31496.84804 41741.15551 38833.67675 44391.63693 41831.94585 54180.96605 61377.26091 69464.04654 75952.11365 61963.37426 68193.72023

9 9 7 9 10 5 7 11 10 2 9 10 9 10 21 11 9 9 11 9 7 11 13 6 43 8 26 4 44 18

ITSD 33178.63458 44181.48210 39558.82437 47120.47639 41515.93392 41285.57388 42015.04585 55627.07483 46055.98672 57188.41647 60692.58875 55603.79858 67433.60472 65420.80667 78184.02415 37809.00955 34109.31059 33814.09910 40558.79816 31496.84804 41527.77087 38833.67675 44391.63693 41831.94585 53709.18863 61377.26091 69464.04654 75482.59766 61818.89140 68073.37865

9 9 7 9 10 5 7 11 10 2 9 10 19 10 18 11 9 9 11 9 30 11 13 6 38 8 49 15 40 24

Best 33178.63458 44181.48210 39558.82437 47120.47639 41515.93392 41285.57388 42015.04585 55627.07483 46055.98672 57188.41647 60692.58875 55603.79858 67389.80329 65420.80667 78184.02415 37809.00955 34109.31059 33814.09910 40558.79816 31496.84804 41527.77087 38833.67675 44391.63693 41831.94585 53709.18863 61377.26091 69464.04654 75482.59766 61818.89140 68073.37865

Table 8. Comparative summary of the performance of the improved tabu solution with diversification ITSD and 50 iterations.

AMP Val. 50359.15105 # Bests 24 Dev. 0.10% CPU secs 218.23

TS 51730.25 1 2.93% 381.25

ITS 0.2 50337.79759 26 0.05% 374.22

with random walk lengths of 1000 and 50000 steps, are showed in Table 5 and Graph 1. As we can observe, with 1000 steps the average of the l values varies from 0.25 to 0.28 and for 50000 steps varies from 0.18 to 0.20. Given the high resource consumption required to solve the ROCIS instances, we consider that with 50000 steps the average of the l values shows an appropriated precision level and stability. Now we calculate the l values for the instances in each group using a random walk with 50000 steps. Table 6 shows the minimum, median, maximum and the mean of the average l values for each group (I1, I2, I3 and I4). As we can observe that do not exist a significant difference respect to the landscape ruggedness generated for the random walk with the instances of the different groups. All the average l values are very similar and closer to zero. The random walk algorithm seems perceive that all the instances have a high hardness level regardless of the group that they belong. Then it seems more appro-

ITS 0.4 50341.93472 25 0.06% 392.92

ITS 0.6 50383.51267 24 0.14% 371.75

ITS 0.8 50312.02403 29 0.003% 386.94

Best 50310.56363 30 0%

priate to incorporate in the tabu solution a long term diversification process to avoid to get stuck in the local optimums.

In the last experiment the performance of the improved tabu solution with diversification (ITSD) was evaluated. As we can see in the section 4.4. (Incorporating a diversification process), the sequence of a values used in the diversification process is the following: S {a0=0.6, a?=0.2, a2=0.4} The ITSD algorithm starts with a=0.6 and the first time that stagnation is detected, is changed to 0.2 and in the second one a switches to 0.4. The search continues with the last one a value, until reaching the stopping condition Table 7 shows the comparative performance of the ITSD algorithm with respect to ITS algorithm (for a=0.2, Articles

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0.4 and 0.6) and AMP algorithm. As we can see, the solution with diversification is able to find 29 of the 30 global best solutions, outperforming all the algorithms evaluated. Table 8 shows a summary of the experiment results. Tables 2 and 8 shows that with 50 iterations ITS (with a= 0.2 and 0.4) and ITSD are better in quality than AMP, but the last one is better than ITS and ITSD in efficiency.

To reduce the resources consumption of ITS and ITSD, the number of iterations was reduced to 30. Table 9 shows the obtained results and we can observe that now ITSD outperfoms in quality and efficiency to ITS (for a=0.2) and to AMP.

106096 respectively, for the research reported in this paper.

Table 9. Comparative summary of the performance of the improved tabu solution with diversification ITSD and 30 iterations.

References

ITS 0.2 ITSD AMP Val. 50,359.15105 50,344.08684 50,310.56363 # Bests 22 25 26 0.066% 0.056% Dev. 0.10% CPU secs 218.23 235.76 218.20

AUTHORS Héctor Joaquín Fraire Huacuja, Guadalupe Castilla Valdez - Instituto Tecnológico de Ciudad Madero. 1° de Mayo y Sor Juana I. de la Cruz S/N, Ciudad Madero Tamaulipas, México CP. 89440.

José Luis González-Velarde - Centro de Calidad y Manufactura, Tecnológico de Monterrey, Monterrey Nuevo León, México. * Corresponding author

[1]

[2]

[3]

Table 10. Average iterations needed for the improved tabu to reach the best solution without instances 25 and 29.

ITS 0.2 Average Iterations 12.10714 until the best reached

ITS 0.4 12.42857

ITS 0.6 10.00000

[4]

[5]

6. Conclusions and future work This paper approaches the robust capacitated international sourcing problem (RoCIS) which consists of selecting a subset of suppliers with finite capacity, from an available set of potential suppliers internationally located. The tabu solution proposed in [1] consists of three phases: build an initial solution, create a neighborhood of promising solutions and perform an extensive search in the neighborhood. In this work the construction of the initial solution, the construction of the neighborhood, and the local search were improved. Also the intensification and diversification balance of the tabu solution was improved, incorporating a long term diversification process. Experimental evidence shows that the improved tabu solution with diversification outperforms the best solutions reported for six of the instances considered, increases 18% the number of best solutions found and reduces 44% the deviation from the best solution found, respect to the best algorithm solution reported.

Future work includes improving the efficiency of the proposed solution, incorporating different diversification mechanisms and stopping conditions based on the stagnation detection.

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ACKNOWLEDGMENTS Authors thank the support received from Tecnológico de Monterrey, Consejo Nacional de Ciencia y Tecnología (CONACYT) and Consejo Tamulipeco de Ciencia y Tecnología (COTACYT) through projects CAT128, CONACYT-67032 and TAMPS-2007-C1572

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OPTIMIZATION OF A REACTIVE FUZZY LOGIC CONTROLLER FOR A MOBILE ROBOT USING EVOLUTIONARY ALGORITHMS Abraham MelĂŠndez, Oscar Castillo, Arnulfo Alanis

Abstract:

2. Mobile Robot

This paper describes an evolutionary algorithm application for the optimization of a reactive fuzzy controller applied to mobile robot navigation. The evolutionary algorithm optimizes the fuzzy inference system and the position and number of the sensors on the robot, while trying to use the less power possible.

The robot is based on the description for mobile robots presented in [21] and assumes a wheeled mobile robot consisting of one or two conventional, steered, unactuated and not-sensed wheels and two conventional, actuated, and sensed wheels (conventional wheel chair model). This type of chassis provides two DOF (degrees of freedom) locomotion by two actuated conventional nonsteered wheels and one or two un-actuated steered wheels. The Robot has two degrees of freedom (DOFs): y-translation and either x-translation or z-rotation [21].

Fig. 1 shows the robots configuration, it will have 2 independent motors located on each side of the robot and one castor wheel for support located at the form of the robot.

Keywords: fuzzy control, genetic optimization, genetic fuzzy systems, robotic systems.

1. Introduction Robots are being more commonly used in many areas of research and a reason for this is that they are becoming more accessible economically for researchers. In this paper we consider the optimization of a fuzzy controller; that gives the ability of reaction to the robot. This may be too general, so let's limit what in this paper will be described as ability of reaction - this is applied in the navigation concept, so what this means is that when the robot is moving, and at some point of its journey it encounters an unexpected obstacle, it will react to this stimulation avoiding and continuing on its path. The trajectory and path following are considered independent parts and are not consider on this paper [19].

There are many traditional techniques available to use in control, such as PD, PID and many more, but we took a different approach in the Control of the robot, using an area of soft computing which is fuzzy logic that was introduced by Zadeh [1]. Later this idea was applied in the area of control by Mamdami [2], where the concept of FLC (Fuzzy Logic Controller) originated. It is also important to mention that this is not the only area were the fuzzy concepts are applied but it is where the most work has been done, and were many people have contributed important ideas and methods like Takagi and Sugeno [2].

There are many recent papers on controlling mobile robots with intelligent techniques, in particular with fuzzy logic and genetic algorithms [3], [4], [5], [6], [7].

However, in this paper the proposed approach is to use an evolutionary algorithm to optimize the fuzzy logic reactive controller of a mobile robot. There are also several works on using fuzzy logic for tracking control and navigation of mobile robots [8], [9], [10], [11], [12], [13], [14], [15], [16], [19] ,[20].

This paper is organized as follows: in section 2 we describe the mobile robot used in these experiments, section 3 describes the development of the evolutionary method. Section 4 shows the simulation results. Finally, section 5 presents the Conclusions.

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Fig. 1. Kinematic coordinate system assignments [21].

The kinematic equations of the mobile robot are as follows: Equation 1 is the sensed forward velocity solution [21] VBx VBy wBz

=

R 2la

-lb -la -1

lb -la 1

wW1 wW2 (1)

Equation 2 is the Actuated Inverse Velocity Solution [21] wW1 wW2

=

R R(2lb2+1)

-lalb lalb

-lb2-1 -la -lb2-1 la

VBx VBy wBz

Where under the Metric system we have the following: VBx, VBy - Translational velocities [m/s], wBz - Robot z-rotational velocity [rad/s], wW1, wW2 - Wheel rotational velocities [rad/s], R - Actuated wheel radius [m], la, lb - Distances of wheels from robot's axes [m].

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3. Evolutionary Method Description In this section we will describe the Genetic Algorithm applied to the problem of finding the best fuzzy logic reactive controller for a mobile robot. The genetic algorithm optimizes the structure of the fuzzy system for control, which means tuning the membership functions and optimizing the number of fuzzy rules.

The purpose of using a evolutionary method, is to obtained the best reactive control possible, for the robot, but also taking into consideration other desirable characteristics on the robot that we want to improve making this a multi objective [17] problem, for this we will take advantage of the HGA (Hierarchy Genetic Algorithm) intrinsic characteristic to solve multi objective problems, now let us state the main goal of our HGA. The main goal is to optimize the Reactive Control taken in to consideration the following: Fine tune the Fuzzy Memberships Optimize the FIS if then fuzzy rules The mobile robot Power Usage In Fig. 2, we show the global cycle process of the GA, under the Evaluation of the each individual, is where we are going, to measure the goodness of each of the FIS (Fuzzy Inference System) represent by each Individual chromosome, in our test area, that will take place in a unknown environment (Maze [18]) to the robot where the robot's objective will be find the exit, avoiding hitting the walls and any other obstacle present.

Cover Distance (3)

Running Time (3)

Fitness (mamdani) 27 rules Fitness (5)

Power Level (3) Fig. 3. Fitness FIS. The chromosome architecture, is shown in Fig. 4, where we have encoded the membership functions type and parameters, we have set a maximum number of 5 membership functions for each of the outputs and input and output variables.

All the results obtain will get persisted on a Data Base, were we will store each step on the genetic cycle, keeping track of the genealogy of each chromosome, and with this we can examine each of the top individuals and can back track the behavior of the genetic algorithm.

Fig. 4. Chromosome Architecture.

4. Simulation Results

Fig. 2. Genetic Algorithm process.

Our criteria to measure the Fuzzy Inference System (FIS) global performance will take into consideration the following: Cover Distance, Time used to cover distance, Battery life.

All of these variables are the inputs of the Evaluation FIS that we will use to obtain the fitness value of each chromosome. In Figure 3 the structure of the fuzzy inference system is illustrated.

The FIS that we optimized is a Mamdani type fuzzy system, consisting of 3 inputs that are the distances obtain by the robots sensors describe on section 2, and 2 outputs that control de velocity of the servo motors on the robot, all this information is encoded on each chromosome.

We have worked on the reactive control for a mobile robot before, where we use a particular maze problem to test the effectiveness of each of the reactive controls, and we did not use any optimization strategy to fine tune the controllers as it was a manually process, and because of that experience we decided to apply GA to this problem. In Fig. 5 and Table I one can see the results we obtained in our prior experiments for fuzzy systems of 27 rules and 10 rules, respectively.

Fig. 5. Sample trajectories of the proposed approach.

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RAS LAS 27 Rules FIS 38.26 38.46 40.42 40.64 40.06 40.12

9 10 12 Average Standard Deviation 1 2 3 4 5 6 7 8 9 Average Standard Deviation

Time

LFE

60.34 59.50 60.70

Yes Yes Yes

39.58 39.74 1.14 1.16 10 Rules FIS 31.51 47.43 35.55 54.33 36.66 53.68 36.66 53.68 36.91 54.18 35.44 52.51 36.14 51.66 33.48 49.58 37.61 51.51

60.18 0.62

52.06 2.32

52.27 3.10

35.55 1.92

55.75 51.80 55.40 54.10 49.95 46.65 49.85 51.76 55.15

Yes Yes Yes Yes Yes Yes Yes Yes Yes

Preliminary results show promising data and as expect the HGA improves the overall performance of the controller for the mobile robot, and improves the results obtained previously. The best reactive controller obtained with the HGA with the same maze problem outperforms the best reactive controller obtained manually, which supports the idea that an evolutionary algorithm optimizes the structure and parameters of the fuzzy logic controller.

ACKNOWLEDGMENTS We would like to express our gratitude to the CONACYT, and Tijuana Institute of Technology for the facilities and resources granted for the development of this research.

AUTHORS Abraham Meléndez, Oscar Castillo*, Arnulfo Alanis Graduate Division, Tijuana Institute of Technology, Tijuana, BC 22379, Mexico. Email: ocastillo@hafsamx.org * Corresponding author

References

* LAS=Left Motor Average Speed * RAS= Right Motor Average Speed * FE= Found Exit

[1]

We show in Table 2 the results of two experiments with the evolutionary algorithm.

On the experiment #1 we can see that the top individual has a fitness value of 0.3568 and 40 active rules, comparing this with the experiment # 2 where the top individual that has a fitness of 0.3566 and only 12 active rules, we can conclude that the solution of 12 rules is preferred.

[2]

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Table 2. Reactive Control Optimized Results with the Evolutionary Algorithm.

Experiment # 1 Fitness 0.3568 0.3561 0.3561 0.3560 0.3560 0.3537 0.3494 0.3415 0.3415 0.3386 0.3386 0.3384 0.3384 0.3384 0.3384 0.3384 0.3384 0.3384 0.3379 0.3372 76

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5. Conclusions

Table 1. Reactive Control Non Optimized Results.

Experiment

N° 4

Experiment # 2 Num_Rules 40 18 12 48 27 3 48 12 12 6 9 2 2 2 12 12 12 12 18 36

Fitness 0.3566 0.3340 0.3340 0.3340 0.3340 0.3340 0.3339 0.3339 0.3339 0.3339 0.3338 0.3338 0.3327 0.3302 0.3302 0.3302 0.3302 0.3302 0.3302 0.3302

Generation 185 0 6 9 17 101 191 194 232 132 1 6 2 8 14 53 85 140 129 159

Num_Rules 12 6 24 8 18 12 4 18 16 6 32 18 2 12 8 16 8 9 16 24

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