Adaptive control design for high order mimo nonlinear time delay systems based on neural network by menez

Scientific Journal of Control Engineering April 2014, Volume 4, Issue 2, PP.43-50

Adaptive Control Design for High-order MIMO Nonlinear Time-delay Systems Based on Neural Network Jimin Yu#, Zhixu Peng, Linqin Cai, Baohua Wu College of Automation, Chongqing University of Post and Telecommunications, Chongqing 400065, P.R. China #

Email: zdyjm@163.com

Abstract An adaptive controller for a class of high-order MIMO nonlinear time-delay systems in block-triangular form is proposed in the paper. The radial basis function neural network is chosen to approximate the unknown nonlinear functions in the system dynamics at first. Lyapunovâ&#x20AC;&#x201C;Krasovskii functionals are used to compensate the influence of delay terms. Then an adaptive neural network output tracking controller is designed by using the back-stepping recursive method. Based on Lyapunov stability theory, the proposed controller can guarantee all closed-loop signals are globally, uniformly and ultimately bounded, while the output tracking is able to converge to a neighborhood of the origin. Finally, a simulation example is given to illustrate the correctness of the theoretical results. Keywords: Adaptive Control; High-order Nonlinear Delay Systems; MIMO; Output Tracking; Back-stepping Design; RBF Neural Network

1 INTRODUCTION In practice, many control systems are multi-input multi-output (MIMO) multivariable systems. As we all known that it is difficult to analyze and control these nonlinear uncertain systems in view of the complex coupling between input and output variables and the time-delay. In [1], the integral Lyapunov functions were used to obtain adaptive fuzzy controller for the first uncertain MIMO nonlinear systems with the block-triangular functions. In [2], a fuzzy adaptive control scheme based on observer was proposed for a class of MIMO nonlinear systems with immeasurable states, and the output feedback control laws and parameter adaptive laws were derived. In [3], hyperbolic tangent functions were used to solve the singularity problem of Lyapunov synthesis in MIMO nonlinear time delay systems. However, most of the exiting work mainly focuses on the first-order MIMO nonlinear systems. Little attention has been paid to the high-order nonlinear system with time-delay. It is difficult to get the analytical solutions of highorder nonlinear system because of the large inertia. The traditional approaches to study the high-order systems mainly include the fast order reduction of model [5], the approximation of first or second order inertia and pure delay link, sliding mode control [6], fuzzy adaptive control as well as robust adaptive control [7]. In [8], for a class of highorder nonlinear dynamic systems, a composite control strategy was proposed based on adaptive terminal sliding mode control and disturbance observer theory, which achieves sliding mode controllerâ&#x20AC;&#x2122;s switching gain. In [9], for the tracking control for a class of high order nonlinear systems with input delay, the fuzzy logic approach was used to estimate the unknown continuous functions, and a state conversion method was presented to eliminate the delayed input items. Finally, the proposed closed 1oop system was concluded to be semi-globally uniformly ultimately bounded and stable. In this paper, based on the back-stepping and neural network approximation, we further take into account the tracking control problem for a class of MIMO high-order nonlinear time delay systems with unknown nonlinear functions in block-triangular form. To design an adaptive controller for theses MIMO high-order nonlinear systems, we apply the RBF (Radial Basis Function) neural network to approximate the unknown nonlinear functions, which - 43 http://www.sj-ce.org

reduces the difficulty of the controller design. Moreover, we adjust the norm of the neural network weight vector instead of the weight vector. As a result, the adjusting parameters are greatly reduced, and the simulating verification is simplified.

2 PROBLEM FORMULATION AND PRELIMINARIES Consider the n-input n-output continuous-time MIMO nonlinear system in block-triangular form with unknown time delays described by (1). xT y 

p p



1 q

(1)

p  1 , q  1 are the delay state variables of the j subsystem. ( p  1)(q  1)  1 with i  [ x j ,1 , x j ,2 , , x j ,i j ]T  R j is the vector of delay-free states for the first i j differential equations of the j

For   0 , x j ,i j

subsystem; u j  [u1 , u2 ,

is the inputs for the first j subsystems; y  [ y1 , y2 ,

, u j ]T

, yn ]T  Rn is the outputs;

 j ,i and  j , m are time delays ;  j ,i  0 ,  j ,m  0 ; f j ,i () , g j ,i () and y j are unknown smooth functions. j

The control objective: (1).Design an adaptive NN feedback controller to ensure closed-loop systems globally bounded; (2).The output y j follows the specified desired trajectory ydj . Assumption 1: For j  1, 2, , n , 0  c j ,i j ()  g j ,i j ()  c j ,i j () , x j ,m j 1  u j . Assumption 2: Let p j ,1 , p j ,2 ,

i j  1, 2,

, mj  1

.The

functions

c j ,i j ()

and

satisfy

, p j ,m j be odd positive integers.

Assumption 3: The desired trajectories ydj , j  1, 2, , n , and their time derivatives up to the nth order, are continuous and bounded.Young inequality: for any two vectors x and y ,the following inequality holds: p p 1 q xT y  x  p y where   0 , p  1 and q  1 are constants, such that ( p  1)(q  1)  1 .Gromwell p q

 j ,12

inequality: V j ,1  Vz j ,1  VU j ,1 

2w j ,1

, if the continuous function u (t ) satisfies. u (t )  M  k t u (s)ds in [t0 , t1 ] , 0

then wj ,1  0 ,  j ,1  0, l j ,1  0 . Lemma 1: For any real numbers a , b and p  1 , the following inequality holds:  j ,i j  W j ,i j

3 ADAPTIVE NN CONTROLLER DESIGN In order to reduce the adjusting number of neural network adaptive parameters in the simulation, we define a new 2 unknown constant  j ,i j  W j ,i j .In this paper, we estimate the  j ,i j instead of the ideal weights of network W j ,i j with Lyapunov method. Thus, we can only adjust one parameter in each system. Define ˆ is the estimate of  j ,i , so the estimate error i ,i j  i ,i j

j ,i j

 ˆi ,i j :

Step 1: ( j,1) Let z j ,1  x j ,1  ydj for j  1, 2,

, n , consider the Lyapunov function as follows: Vz  j ,1

z j ,12 2

Then, the time derivative of Vz j ,1 is given by (2) Vz j ,1  z j ,1 ( f j ,1  g j ,1 x j ,2

With Young inequality, We have z j ,1h j ,1 ( x j ,1 (t   j ,1 ))  yields VZ j ,1  z j ,1 ( f j ,1  g j ,1 x j ,2

p j ,1



p j ,1

 h j ,1  ydj )

(2)

1 2 1 2 z j ,1  h j ,1 ( x j ,1 (t   j ,1 )) Substituting this inequality into (2) 2 2

1 1 z j ,1  ydj )  h j ,12 ( x j ,1 (t   j ,1 )) . To deal with the delay term, consider the 2 2 - 44 http://www.sj-ce.org

Lyapunov-Krasovskii functional as follows: VU j ,1 

1 t U j ,12 ( x j ,1 (s))ds Differentiating VU j ,1 with respect to time, 2 t  j ,1

we obtain (3).

1 1 VU j ,1  U j ,1 ( x j ,1 (t ))  U j ,1 ( x j ,1 (t   j ,1 )). 2 2 2 U j ,1 ( x j ,1 (t   j ,1 ))  h j ,1 ( x j ,1 (t   j ,1 )), U j ,1 ( x j ,1 (t ))  h j ,12 ( x j ,1 (t )). Then

choose

(3) Suppose

h j ,12 ( x j ,1 (t ))  z j ,12  j ,12 ( x j ,1 (t )), where  j ,12 ( x j ,1 (t )) is a known function. Putting (2) and (3) together, we have (4).

Vz j ,1  VU j ,1  z j ,1[ Fj ,1 (Z j.1 )  g j ,1 x j ,2

p j ,1

(4)

1 1 z j ,1  ydj  + z j ,1  j ,12 ( x j ,1 (t )), Z j ,1  [ x j ,1 , ydj , dydj ]T . Since the neural networks (NN) have 2 2 * good ability of approximation, we use NN to approximate Fj ,1 such that for given  j ,1  0 ,

where Fj ,1 (Z j ,1 )  f j ,1 

Fj ,1  W j ,1T S (Z j ,1 )   j ,1 (Z j ,1 ),

(5)

where | j ,1|   is an unknown constant. By Young’s inequality and the definition of  j ,i j , we have (6) (7). * j ,1

z j ,1W

T j ,1

S (Z j ,1 ) 

 j ,1 2v j ,12

T j ,1

z j ,1 j ,1 (Z j ,1 ) 

(Z j ,1 )S (Z j ,1 ) z

2 j ,1

v j ,12



(6)

1 2 1 *2 z j ,1   j ,1. 2 2

(7)

By utilizing (5)-(7), (4) can be rewritten in the following form. Vz j ,1  VU j ,1 

 j ,1  j ,12 1 2 1 *2 p T 2 S ( Z ) S ( Z ) z   z j ,1   j ,1  z j ,1 g j ,1 x j ,2 . j ,1 j ,1 j ,1 j ,1 2 j ,12 2 2 2 j ,1

We further consider the Lyapunov function as follows: V j ,1  Vz j ,1  VU j ,1  w j ,1

ˆ j ,1 

2 j ,12

 j ,12

(8)

. Design NN adaptive law as follows:

2w j ,1

S T (Z j ,1 )S (Z j ,1 ) z j ,12  l j ,1ˆ j ,1 ,

(9)

where wj ,1  0 ,  j ,1  0, l j ,1  0 are constants to be selected. From (8) and (9), we have V j ,1 

ˆ j ,1 T  j ,12 1 2 1 *2 1 p 2 S ( Z ) S ( Z ) z   z j ,1   j ,1  z j ,1 g j ,1 x j ,2   j ,1l j ,1ˆ j ,1 j ,1 j ,1 j ,1 2 j ,12 2 2 2 w j ,1 j ,1

As we know l j ,1 w j ,1

 j ,1ˆ j ,1  

l j ,1 2w j ,1

 j ,12 

l j ,1 2w j ,1

 j ,12 .

(10)

Design virtual smooth controller: n

 j ,2  ( z j ,1 By assumption 1 and  z j ,1

p j  p j ,1 1

 j ,2

p j ,1

From (10)-(12), we have V j ,1  nz Step 2 : ( j , i j ( j  1, 2,

c j ,1 ( x j ,1 )

)

1 p j ,1

(11)

p j ,1

(12)

 0 , we arrive at

c j ,1 z j ,1 2 j ,1

ˆ j ,1 T 1  S ( Z j ,1 ) S ( Z j ,1 ) 2 2 j ,12



l j ,1 2w j ,1

, n , i j  2,



p j  p j ,1 1

2 j ,1

 j ,2

p j ,1

  g j ,1 z j ,1

 c j ,1 z j ,1 x j ,2

p j ,1

p j  p j ,1 1

  j ,2

p j ,1



x j ,2

 j ,12

l j ,1 1   j ,1*2   j ,12 . 2 2 2w j ,1

, m j  1 )).Let z j ,i j  x j ,i j   j ,i j , construct the Lyapunov function as - 45 http://www.sj-ce.org

Vz j ,i  j

z j ,i j 2 2

. A direct calculation gives Vz j ,i j  z j ,i j ( f j ,i j  g j ,i j x j ,i j 1

p j ,i j

 h j ,i j   j ,i j ). Similar to step 1, we have

1 1 z j ,i 2 + h j ,i 2 ( x j ,i j (t   j ,i j )). Note that  j ,i j can be expressed as 2 j 2 j ij ij ij  j ,i j  j ,i j  j ,i j ( k 1) i j  j ,k p  ( f j ,k  g j ,k x j ,k 1 j ,k )   h j ,k ( x j ,k )   y  ˆ j ,k . ( k ) dj ˆ  x  x  y   k 1 k 1 k 0 k 1 j ,k j ,k dj j ,k

z j ,i j h j ,i j ( x j ,i j (t   j ,i j )) 

 j ,i

We have: Vz j ,i  z j ,i j [ f j ,i j  g j ,i j x j ,i j 1

p j ,i j



ij ij  j ,i j  j ,i j ( k 1) 1 p z j ,i j   ( f j , k  g j , k x j ,k 1 j ,k )   y ( k） dj 2 k 1 x j , k k  0 ydj

i i  j ,i j 2 1  j , k ˆ 1 j 1 j   j , k   z j ,i j ( ) ]  h j ,i j 2 ( x j ,i )   h j , k 2 ( x j ,k ). j ˆ 2 k 1 x j , k 2 2 k 1 k 1  j , k ij

(13)

To deal with the delay term, consider the Lyapunov-Krasovskii functional as follows: VU j ,i  

i 1

t  j ,i j

1 1 j t  j ,i j ( x(s))ds     j ,k ( x(s))ds. Differentiating VU j ,i j yields 2 2 k 1 t  j ,k i 1

VU j ,i  j

i 1

1 1 1 j 1 j  j ,i j ( x(t ))   j ,i j ( x(t   j ,i j ))    j ,k ( x(t ))    j ,k ( x(t   j ,k )). 2 2 2 k 1 2 k 1 ij

(14)

If we choose  j ,i j ( x(t   j ,i j ))    j , k ( x(t   j , k ))  h j ,i j 2 ( x(t   j ,i j ))   h j , k 2 ( x(t   j ,k )), k 1

k 1

then  j ,i j ( x(t ))    j , k ( x(t ))  h j ,i j 2 ( x(t ))   h j , k 2 ( x(t )). Suppose: ij

k 1

h j ,i j 2 ( x(t ))   h j ,k 2 ( x(t ))  z j ,i j 2 (  j ,i j 2 ( x(t ))    j , k 2 ( x(t ))),

where

 j ,i ( x(t )) ,  j ,k ( x(t )) are known functions. From (13)-(15), we can get j

Vz j ,i  VU j ,i  z j ,i j ( Fj ,i j (Z j ,i j )  g j ,i j x j ,i j 1 j

where: Fj ,i j  f j ,i j 

p j ,i j

ij i j 1  j ,i j ( k 1) i j  j ,i j  j ,i j p j  p j ,i j 1 1 p j ,k z j ,i j  y  ( f  g x )  ˆ j ,k   j ,k j , k j , k 1 ( k ) dj ˆ 2 k  0 ydj k 1 x j .k k 1  j , k

ij ij p j  p j ,i j 1  j ,i j 2 1 1   z j ,i j ( )  z j ,i j (  j ,i j 2 ( x(t ))    j , k 2 ( x(t ))). x j , k 2 k 1 2 k 1

Similar to step 1, we can get Fj ,i  W j ,i T S (Z j ,i )   j ,i (Z j ,i ), j j j j j Vz j ,i  VU j ,i  j

T where Z j ,i j  [ x j ,i j , ydj ,

 j ,i

2 j ,i j

p j ,i 1 1 1 S T (Z j ,i j )S (Z j ,i j ) z j ,i j 2   j ,i j 2   j ,i j *2  z j ,i j 2  z j ,i j g j ,i j x j ,i j 1 j , 2 2 2

, ydj j ,  j ,1 , i

, ˆj ,i j 1 ]. Consider the Lyapunov function as follows:

V j ,i j  V j ,i j 1  Vz j ,i  VU j ,i  j

Since

V j ,i j 1  (n  i j  2)( z j ,12  i j 1

2 w j ,i j i j 1

 z j ,i j 12 )  

 j ,k 2

 ( k 1

 j ,i 2 l j ,k

k 1 2 w j , k

 j ,k 2

l j ,k 2 p p 1   j ,k *2   j ,k )  c j ,i j 1 z j ,i j 1 x j ,i j j ,i j 1   j ,i j j ,i j1 2 2 2w j ,k - 46 http://www.sj-ce.org

(15)

Young inequality and Lemma 1 imply the existence of a smooth function  j ,i j ()  0 , such that. c j ,i j 1 z j ,i j 1 x j ,i j

p j ,i j 1

  j ,i j

n  ij 1

 j ,i 1  ( z j ,i j

p j ,i j 1

i j 1

  z j ,l 2  z j ,i j 2  j ,i j () Then a virtual smooth controller of the form l 1

ˆ j ,i

2 j ,i j 2

S T ( Z j ,i j ) S ( Z j ,i j ) 

1   j ,i j 2

)

c j ,i j

V j ,i j  (n  i j  1) z j ,k   2

k 1

l j ,k 2w j , k



is such that:

 j ,k 2

l j ,k p p 1   j ,k *2   j ,k 2 )  c j ,i j z j ,i j x j ,i j 1 j ,i j   j ,i j 1 j ,i j 2 2 2w j , k

 (

2 j ,k

1 p j ,i j

k 1

, n) : let z j ,m j  x j ,m j   j ,m j , construct the Lyapunov function V z j ,m 

Step j, m j ( j  1,

calculation gives Vz j ,m j  z j ,m j ( f j , m j  g j , m j u j

follows: ˆ j ,m j 

w j ,m j 2 j , m j 2

t  j ,m j

A straightforward

 h j , m j   j , m j ) .To deal with the delay term, consider the Lyapunov-

p j ,m j

Krasovskii functional as follows: VU j ,m  

z j ,m j 2

1  ( x(s))ds  2

m j 1



t  j ,k

k 1

1  ( x(s))ds Design NN adaptive law as 2

S T ( Z j , m j ) S ( Z j , m j ) z j , m j 2  l j, m j ˆ j, mj Similar to step 2, choose Lyapunov function as follows

V j ,m j  V j 1,m( j1)  V j ,m j 1  Vz j ,m  VU j ,m  j

 2j ,m

(16)

2w j , m j

Then, the time derivative of V j , m j is given by mj

V j , m j  (n  m j  1) z j ,l 2   l 1



li , k 2wi , k

i 1 k 1

i ,k 2 )  c j ,m z j ,m x j ,m 1 j

p j ,m j

li , m j 2wi ,i j

j m  2 1  i2, k   ( i , k   i , k *2 j

i 1 k 1

  j ,m j 1

p j ,m j

After step n, mn : the last step for the nth subsystem, it can be shown that the derivative of n, mn along the closedloop trajectories satisfies the following inequality: mj

Vn, mn   ( z j ,k 2  j 1 k 1

l j ,k

2w j , k

cn, mn zn, mn xn, mn 1

n m  2 1 l  2j ,k )   ( j ,k   j ,k *2  j ,k  j ,k 2 )

pn ,mn

j 1 k 1

  n, mn 1

2w j , k

(17)

pn ,mn

From assumption 1, we know xn,mn 1  un . Combined with the formula (12), we can obtain adaptive designed controller and adaptation law are as follows 1 un  ( zn,mn

ˆn,m  n

ˆn,m 1 S T ( Z n, m ) S ( Z n,m )    n, m 2 2n, m 2 n

cn, mn wn, mn 2n, mn 2

)

1 pn ,mn

S T (Z n,mn )S (Z n,mn ) zn,mn 2  ln,mn ˆn,mn ,

(18) (19)

such that n

Vn,mn   ( j 1 k 1

where C   ( j 1 k 1

 j ,k 2 2

  j ,k 

l j ,k 2w j , k

 2j ,k  z j ,k 2 ) C ,

(20)

 j ,k ) is a constant. Theorem 1 Closed-loop systems (1) that satisfy the 2

- 47 http://www.sj-ce.org

assumptions 1-4, there exists a state feedback controller u such as (18) and NN adaptation law ˆ such as (19), such that all the closed-loop trajectories remain bounded and the output tracking error converges to a neighbourhood of the origin. Proof. From (19), we have Vn.mn  cVn,mn   , where c  min{2, l j ,k } , n

   (

 j ,k 2

j 1 k 1

  j ,k 

l j ,k 2w j , k

 j ,k )   2

t  j ,i j

i 1

1 1 j t  j ,i j ( x( s))ds     j ,k ( x(s))ds. 2 2 k 1 t  j ,k

By Gromwell inequality, we can obtain: Vn, mn 

 c

 exp ct [Vn, mn ( z (0)) 

 c

]  Vn, mn ( z (0)) 

 c

(21)

Since Vn,mn ( z (0)) is bounded, then z j ,i j , x j ,i j and ˆ j ,i j are bounded. From (16) and (21), we have zn, mn 2

 Vn, mn  Vn,mn ( z (0)) 



, then z j ,1  x j ,1  ydj  2(Vn, mn ( z (0)) 

2 c to a neighborhood of the origin. The proof is finished.

 c

). So, the output tracking error converges

4 SIMULATIONS In this section, a simulation study is presented to verify the effectiveness of the adaptive NN controller for the general case. In this case, the bounds on the functions of delayed states are not known. Consider the following twoinput, two-output system in block-triangular form  x11  x11  (1  sin 2 ( x11 )) x123  x11 (t   11 )  2 2 5  x12  x11 x12  x12  x22  (1  sin ( x11 )  cos ( x12 ))u1  x12 (t   12 )  3  x21   x21  x22  x21 (t   21 )  5  x22  ( x12  x21 ) x22  x11u1  (2  sin( x21 x22  x11 )u2  x22 (t   22 ) y  x 11  1  y2  x21

The desired trajectories yd1  cos(t ) , yd 2  sin(t ) . The simulation results shown in from Fig.1 to Fig.6 are based on the following parameters. w11  w12  w21  w22  8 ; l11  l12  l21  l22  0.0001 ; 11  12  8 ; 21  22  5 ; c11 

1 1 1 ; c12  ; c21  ; 42 5 8

1 ; [ ,  ,  ,  ]T  [0,0,0,0]T , 12  0.5 ,  21  1.5 ,  22  0.5 . 7 11 12 21 22 The initial conditions: [11 , 12 , 21 , 22 ]T  [0,0,0,0]T , [ x11 (0), x12 (0), x21 (0), x22 (0)]T  [0,0,0,0]T . The NN simulation results clearly show that the proposed controller can guarantee the bounded of all the signals in the closed-loop system From Fig.1 and Fig.2, we can get that the output signal y1 and y1 (t ) can effectively follow the reference yd 1 (t ) and y2 (t ) respectively. At the same time, Fig.3 and Fig.4 clearly show the sate signal yd 2 (t ) and x21 are globally bounded. From Fig.5 and Fig.6, we can get the tracking error converges to a relatively small neighborhood. c22 

FIG.1. OUTPUT ("- -") AND REFERENCE SIGNAL ("–") - 48 http://www.sj-ce.org

FIG.2 OUTPUT("- -") AND REFERENCE SIGNAL("-")

FIG.4.STATE SIGNAL x22

FIG.3 STATE SIGNAL x12

FIG.5. TRACKING ERROR y1 (t ) ď&#x20AC; yd 1 (t )

5 CONCLUSIONS In this paper, we design the NN controller and adaptive laws for a class of MIMO high-order nonlinear time delay systems, by using back-stepping and the neural network approximation theory. Based on Lyapunov function, the designed adaptive controller can ensure that all the signals of the closed-loop system remain bounded. Moreover, the tracking errors are able to converge to a neighborhood of the origin. Simulation results further show the proposed adaptive controller scheme is feasible. The MIMO high-order nonlinear systems with variable time delay are more common in the control fields and they are much more complex. Yet we only discuss the situation with constant time delay in the paper. For the future work, we plan to study these systems with variable time delay.

ACKNOWLEDGMENT This work was supported in part by the Natural Science Foundation of Chongqing (CSTC) under Grant No. 2012jjA1642 and the Project of Science and Technology of Chongqing Municipal Education Commission under Grant No. KJ130522

REFERENCES [1] B. Chen, X.P. Liu, S.C. Tong. Adaptive Fuzzy Output Tracking Control of MIMO Nonlinear Uncertain Systems. IEEE transactions on Fuzzy systems Vol.15, NO.2, April 2007 [2] Tong Shaocheng, Qu Lianjiang. Fuzzy adaptive output feedback control for a class of nonlinear MIMO system. Vol.20 No.7, July 2005. [3] Ge, S. S., & Tee, K. P., Approximation-based control of nonlinear MIMO time-delay systems. Automatica, 43(2007): 31-43 [4] Qigong Chen, Lisheng Wei, and Ming Jiang. Stability Analysis of a Class of Nonlinear MIMO Networked Control Systems. Second International Symposium on Knowledge Acquisition and Modeling, 2009 [5] Wang Xiaojuan, Wu Xuguang, Yao Weigang. Technology on fast order reduction of model. Computer aided engineering, Vol.19 No.3, 2010.9 [6] Liu Jinkun, Sun Fuchun. Research and development on theory and algorithms of sliding mode control. Control theory & Application, 2007, 24(3): 407-418 [7] Li Qioali, Yu Jun, Mao Beixing, Bu Chunxia. The Robust Adaptive Control of Nonlinear Time-Delay Systems. Hen Nan Science, Vol.29, No.7, Jul. 2011 - 49 http://www.sj-ce.org

[8] Li Wei, Duan Jianmin. Composite Terminal Sliding M ode Control for a Class of High-order Nonlinear Dynamic Systems. Computer Simulation, 1006-9348(2011)09-0207-04 [9] Qing Zhu, Ai-Guo Song, Tian-Ping Zhang, Yue-Quan Yang. Fuzzy Adaptive Control of Delayed High Order Nonlinear Systems. International Journal of Automation and Computing, 9(2), April 2012, 191-19

AUTHORS 1

Professor, He received the Ph.D. in applied

D, Professor, Tutor of master students. He

mathematics from Zhengzhou Unversity in

received the Ph.D. degree in Pattern

2003. He is currently a professor with the

Recognition and Intelligence System from

key lab of industrial wireless networks and

the School of Information, University of

Networked Control of the Ministry of

Science and Technology of China, Hefei,

Jimin YU was born in 1964, Male, Ph.D,

Linqin CAI was born in 1973, Male, Ph.

Education and the College of Automation,

China, in 2007. He is currently a Professor

Chongqing University of Post and Telecommunications. His

at the School of Automation, Chongqing University of Posts and

research interests include fraction order dynamic systems,

Telecommunications, Chongqing, China. His current research

numerical methods, and stability theory of functional equations.

interests include Complex System Modeling and Simulation, Multi-agent Systems and Intelligence Computation.

Master. His research interest is the model

Master. Her research interest is the neural

predictive control.

network, nonlinear control.

Email: peng.zhixu@163.com

Email: wbhflefriend@126.com

Zhixu Peng was born in 1987, Male

- 50 http://www.sj-ce.org

Baohua WU was born in 1986. Female