Design of Fuzzy gain scheduled PI Controller for a nonlinear SISO Process

GRD Journals | Global Research and Development Journal for Engineering | International Conference on Innovations in Engineering and Technology (ICIET) - 2016 | July 2016

e-ISSN: 2455-5703

Design of Fuzzy Gain Scheduled PI Controller for A Nonlinear SISO Process 1S.

Nagammai 2G. Swathi Lakshmi 3A. MahaLakshmi 2,3 U.G student 1,2,3 Department of Electronics and Instrumentation Engineering 1,2,3 K.L.N. College of Engineering, Pottapalayam, Sivagangai 630 612, India Abstract The typical nonlinear process such as conical tank and spherical tank process has the difficulty in controller design because of a change in system dynamics and valve nonlinearity. All industrial processes are almost nonlinear with wide operating range; the use of conventional PID controller becomes inefficient. For this reason the controller design for nonlinear system has become a very determinative and significant field of research. In a gain scheduled PI controller the controller gains are allowed to vary within a predetermined range and therefore eliminates the problem faced by the conventional PI controller. Gain Scheduling is the process through which the multiple local linear controllers are combined in order to control the process over the entire operating range. A main problem on the design of gain scheduling controller (GSC) is the design and implementation of the switching logic to have a smooth transition in plant response as the operating point changes. This difficulty is solved with the application of fuzzy logic controller (FLC). Fuzzy gain-scheduled (FGS) PI controller is a nonlinear controller which utilizes fuzzy rules and reasoning for determining the PI controller parameters. In this paper the illustration of FGS-PI controller is proposed to single input single output (SISO) nonlinear system namely conical tank process. The performances of the controllers are compared based on servo tracking using simulation results.

Keyword- Conical Tank Process, GSC, FGS, SISO __________________________________________________________________________________________________

I. INTRODUCTION It is evident that, nonlinear tanks are used as processing units in process industries such as Petro Chemical industries, metallurgical industries, food processing industries, paper industries and mixing industries, wherein the control of liquid level in tanks is a basic problem. The level control of the nonlinear tank process with varying cross sectional area is very cumbersome. In many industrial processes such as distillation columns, evaporators, Fluidized catalytic cracking unit and boiler, control liquid in the vessel at a desired value is of great significance. The proportional integral and derivative controllers are used extensively in industries because of its easy implementation. Chidambaram et al proposed a method of designing a PID controller for time delayed stable system based on co efficient matching technique. Dave et al discussed about the difficulty in designing conventional controllers for nonlinear systems and higher order systems.

Many researchers have proved that, conventional feedback controller is well suited for linear system. Since 1980s fuzzy system has been applied in engineering applications and Mamdani (M) implemented FLC for industrial applications the very first. In 1985 fuzzy modelling nonlinear system has been discussed by Takagi and Sugeno (TS). Genetic algorithm based fuzzy controller for a conical tank was proposed by Madhubala et al. Nithya et al studied the performance of M-FL and TS-FL for controlling liquid levels in conical tank process in real time.

The tuning of conventional PID controller for nonlinear process is still a emerging research area. Hence a controller which can accommodate system nonlinearity, parameter variations, uncertainties need to be designed. The GSC is a kind of robust and nonlinear controller which guarantees stability and robustness to changes in parameters and external disturbances acting on the system. It is also clear that; implementation of GSC is relatively easier compared to other types of nonlinear controllers.

This paper is organized as follows. The section 2 gives description of nonlinear conical tank process. In Section 3 the concepts and design of Gain Scheduling controller is discussed. Section 4 explains an implementation of Fuzzy gain-scheduled PI controller for the nonlinear process. The simulated results are given in section 5. The conclusion is given in section 6.

II. PROCESS DESCRIPTION The system shown in Fig 1 consists conical tank connected with manual valve in the outlet pipe. The control objective is to maintain water level of the tank namely h1 (t) at the desired value by manipulating the inlet water flow namely q1 (t) to the tank.

Using Bernoulli’s law the mass balance equation for the tank is written and given in equation [1].

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Design of Fuzzy Gain Scheduled PI Controller for A Nonlinear SISO Process (GRDJE / CONFERENCE / ICIET - 2016 / 077)

Fig. 1: Schematic diagram of a single capacity Conical Tank Process 2

2

dh  π  dh π  R  dh π  R  dh3 q1 - q2 = A =  r 2     h2    dt  3  dt 3  H  dt 3  H  dt

(1)

Where

q2 = C h

q1 ---In flow rate to tank 1 ( cm3 / sec ) q2 --- Outflow rate of tank 1 ( cm3 / sec ) C is the discharge Co efficient of drain valve r is radius at any height ‘h’ (in cm) A. Experimental Determination of ‘C’ In order to obtain the state space model of the bench mark problem that agrees with the experimental set up, the valve discharge coefficient has been determined experimentally. The valve discharge coefficient C is determined by making the following arrangements in the experimental setup shown in Fig.1. Initially tank is filled to the maximum level and then the inlet valve to tank is made shut for determining C. i.e., q1  0 After these arrangements the mass balance equation given in equation (1) become, •

•

-q2 = Ah  -C h = Ah Rearranging the above equation,

-

C dh dt = A h

Integrating over the limits hini to hfin for the period 0 to T, we get T

h fin

C dh -  dt =  A0 h hini Solving the above equation we get an expression for valve coefficient ‘C’

C=

2A  hini - h fin  T 

(2)

2

1 1 R A= π r 2 = π   h2ini 3 3 H Where, The initial and final level of tank is varied in steps by opening the drain valve of tank partially and the time taken to reach the final level is noted down. The readings in Table 1 are substituted in equation (2) to determine the valve discharge coefficient C.

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Design of Fuzzy Gain Scheduled PI Controller for A Nonlinear SISO Process (GRDJE / CONFERENCE / ICIET - 2016 / 077)

Time taken to decrease Valve discharge coefficient (in sec) 35.5 30 35 5.1 30 20 26 10.2 20 15.5 15 4.2 15.5 10 8 6.84 10 5 7.5 3.63 Caverage 6 Table 1: Data’s For Determination of C Variable Description Value C Discharge coefficient of drain valve 6 R Top radius of the tank 22.5 cm H Total height of the tank 60 cm

hini (in cm)

hfin (in cm)

hs

Steady state water level of tank

25 cm

Table 2: Process Variables and Steady State Values

III. DESIGN OF GAIN SCHEDULED PI CONTROLLER The concept of Gain Scheduling was initially applied to the development of flight control system. The nonlinear process possesses the characteristics that, the process dynamics changes with the operating conditions of the process. It is feasible to reduce the effects of parameter variations simply by changing the controller parameters as functions of the auxiliary variables.

Gain scheduling controller (GSC) is a type of adaptive controller in which the controller gains are adjusted based on parameter variations. The advantage of GSC is that, the controller parameters can be changed very quickly in response to process changes.

In order to design GSC the variables that can be used as scheduling variables is to be determined. The auxiliary variables that are measured must reflect the operating conditions of the plant. Ideally simple expressions that relate the controller parameters and the scheduling variables are derived. In level control application the tank level or head is used as the scheduling variable.

Considering a conical tank system with cross section area ‘A’ which varies with height ‘h’ the mass balance equation is,

q1 - C h = A(h)

dh dt

dh 1  = f(h,q)= q1 - C h  dt A(h)  The model is linearized at an operating point h s and the transformed equation is,

Q1(S)-

C H(S)= SA  h  H(S) 2 hs

1/A  h  H(S) β   C Q1 (S) S + S +α 2A  h  hs

β = 1/A  h 

and

α=

C 2A  h  hs

The closed loop transfer function of the system in presence of PI controller is,

K  β  Kp + i   C(s) S +α  S  = K  β  R(s) 1+ Kp + i   S +α  S 

β  K p S + Ki  C(s)  2 R(s) S + S  α + K p β  + Ki β

(3)

The general second order transfer function of the system is,

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Design of Fuzzy Gain Scheduled PI Controller for A Nonlinear SISO Process (GRDJE / CONFERENCE / ICIET - 2016 / 077)

 C (s) n  R( s) S 2  2 S   2 n n 2

(4)

Comparing equation (3) & (4) we get,

Kp =

2ζω - α n β

ω 2 Ki  n β It is desired to have overshoot of less than 5% and settling time of 200 seconds, which corresponds to

n 

4  0.02 and ts

  0.7 & n  0.03 2

1 R A(hs )= π   hs 2 = 92cm2 3 H We have, β = 1/A  hs   0.011 α=

C 2A  hs  hs

 0.0065

As the numerical value of α << 2ζω , the scheduler is simplified to n

K p (h)=

2ζω n  2ζω A  h   0.04 A  h  n β

ω 2 Ki (h)= n = ω 2 A  h   0.0009 A  h  n β

(5)

(6)

The PID controller gains at a steady state operating level of 25 cm is obtained as,

K p = 2ζω A  hs   4 n

Ki = ω 2 A  hs   0.09 n Equation (5) & (6) is used to implement a gain scheduled PI controller which is shown in Fig.2

Fig. 2: Simulink diagram of gain scheduled PI controller

IV. DESIGN OF FUZZY GAIN SCHEDULED PI CONTROLLER Analysing nonlinear systems with conventional mathematics is very difficult and physical implementation requires intense computational effort. Complex logical decision making is required to design conventional PI controller. In order to overcome these difficulties intelligent technique namely FLC is used. Fuzzy logic is flexible with relatively low computing power; nonlinear control can be performed effectively. Fuzzy logic system is based on logical reasoning it has the ability to fuzzily any

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Design of Fuzzy Gain Scheduled PI Controller for A Nonlinear SISO Process (GRDJE / CONFERENCE / ICIET - 2016 / 077)

crisp data which helps in easy implementation of control logic. In many industrial processes, FLC is used which takes the control action like human for controlling process. FLC is implemented to non-linear systems which capitulate better results.

For designing FLC the membership Function and rules are selected based on the process knowledge. For control of level in conical tank process, FLC is designed with level as the input function and PI controller gains as output function. In order to fuzzy based gain scheduling, linearized models for four different in flow rates are obtained by MATLAB linear Analysis using the Simulink model shown in Fig.3. The PI controller parameters are manually tuned using auto tuner for each flow condition.

The process transfer function and the controller parameters are given in Table.3.

In flow rate (Lph) 36 50 72 108

Process Transfer function

0.88 S  1.6 0.23 S  0.296 0.055 S  0.05 0.01 S  0.0065

Steady state level

Kp

Ki

2.77

0.857

4.65

5.44

0.61

0.61

11.1

0.43

0.073

25

0.31

0.0067

Table 3: Process Transfer Function and Controller Parameters

Fig. 3: Simulink model for linearization of plant In TS-FLC design the antecedent component is the level of the tank which is a linguistic variable whereas the consequent components are PI controller gains which are crisp variables. The linguistic variables assigned for level are namely, Very small (Vs), Small(S), Large (L) & very Large (VL). The level input range is considered from 0 to 40 and the output range is considered from 0 to 1.The type of the membership function used is triangular and the plot of membership function is shown in Fig.4.

Fig. 4: Choice of membership function for level

The TS FL outputs are proportional gain Kp & Integral gain Ki consists of four singleton membership functions namely Very small (Vs), Small(S), Large (L) & very Large (VL) as shown in Fig.5.

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Design of Fuzzy Gain Scheduled PI Controller for A Nonlinear SISO Process (GRDJE / CONFERENCE / ICIET - 2016 / 077)

Fig. 5: Singleton membership functions for Kp & Ki

The rules are framed based on the knowledge of the process and the four rules that are used is given below: If level is VS then Kp is VL and Ki is VL If level is S then Kp is L and Ki is L If level is L then Kp is S and Ki is S If level is VL then Kp is VS and Ki is VS The Simulink diagram of Fuzzy gain scheduled PI controller is shown in Fig.6. As the controller gains are normalized in the execution of FLC, appropriate demoralization is done for simulation.

To analyse the robustness of the proposed controllers a disturbance in the valve opening at various instants after steady state level has been reached.

Fig. 6: Simulink diagram of Fuzzy gain scheduled PI controller

V. SIMULATION RESULTS The performance of the proposed control algorithms are analysed by simulating the conventional PI controller, GS-PI controller & FGS-PI controller structure shown in Fig.2 and Fig.6 using MATLAB. The performance of above controllers is shown in Fig.7 for step change in level variation. The servo regulatory response of proposed controllers is shown in Fig.8. It shows that the disturbance rejection capability of FGS-PI controller is better than other controllers. The variation of in flow rate of proposed controllers is presented in Fig.9, which shows a smooth transition in flow rate whenever the operating point changes. The variation of controller parameters Kp & Ki in case of FGS-PI controller for an operating level of 20 cm is shown in Fig.10 & Fig.11.

It clearly shows the convergence of parameters at a steady value for level variations.

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Design of Fuzzy Gain Scheduled PI Controller for A Nonlinear SISO Process (GRDJE / CONFERENCE / ICIET - 2016 / 077)

Fig. 7: Servo response of proposed controllers

Fig. 8: Servo-Regulatory response of proposed controllers

Fig. 9: Variation of in flow rate of proposed controllers

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Design of Fuzzy Gain Scheduled PI Controller for A Nonlinear SISO Process (GRDJE / CONFERENCE / ICIET - 2016 / 077)

Fig.10: Variation of proportional gain in FGS-PI controller

Fig. 11: Variation of integral gain in FGS-PI controller

VI. CONCLUSIONS The conventional PI controller, GS-PI controller & FGS-PI controller algorithms are demonstrated for a nonlinear conical tank process. The time response shows that the settling time, overshoot and rise time constraints are superior for the FGS-PI controller compared to other controllers. The performance summary given in Table 4 indicates that, the FGS-PI controller offers less ISE and IAE value. Further FGS-PI controller exhibits no overshoot compared to other type of controllers.

Parameter

Conventional PI Controller

Gain scheduled PI Controller

Peak time Rise time Settling time % overshoot

43 7.27 151 6

74 17.22 126 38

Fuzzy Gain scheduled PI Controller 200 24 115 0

ISE

133810

117700

88467

IAE

8939 8399 Table 4: Time response specifications

725.5

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Design of Fuzzy Gain Scheduled PI Controller for A Nonlinear SISO Process (GRDJE / CONFERENCE / ICIET - 2016 / 077)

REFERENCES [1] Karl Johan Astrom, Bjorn Wittenmark Adaptive Control (2nd Edition) [2] Zhen-Yu Zhao, Masayoshi Tomizuka, and Satoru Isaka “ Fuzzy gain Scheduling of PID controllers”, IEEE transactions on Systems, Man and Cybernetics,1392-1398,1993.

[3] Madhubala, T.K., M. Boopathy, J. Sarat Chandra Babu and T.K. Radhakrishnan “ Development and tuning of fuzzy controller for a conical level system” Proceedings of International Conference Intelligent Sensing and Information Processing, Tiruchirappalli, India,pp. 450-455,2004.

[4] R.Anandanatarajan and M. Chidambaram. ”Experimental evaluation of a controller using variable transformation on a hemi-spherical tank level process”, Proceedings of National Conference NCPICD, pp. 195-200, 2005.

[5] Leehter Yao and Chin-Chin Lin, “Design of Gain Scheduled Fuzzy PID Controller”, World Academy of Science, Engineering and Technology, Vol. 1 ,pp.714-718,2007.

[6] S.Nithya, N.Sivakumaran , T.K.Radhakrishnan and N.Anantharaman, “ Soft Computing based controllers implementation for nonlinear process in real time”,Proceedings of the World Congress on Engineering and Computer Science , October 2022, SanFrancisco, USA, 2010.

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