A fuzzy basis function vector-based multivariable adaptive controller ...
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H
and the corresponding controller parameters are K1
=
224:78 0
0
01 71 :
;
K2
=
2624:8 0
0
01 73 :
:
(42)
The simulation shows that the pendulum can be well balanced for initial conditions in the range of x1 (0) = (089 ; 89 ) and x2 (0) = 0. The angle response of the pendulum control system (37) with control law (42) for initial condition x1 = 85 ; x2 = 0 is shown in Fig. 4.
controller design of fuzzy [19] Z. X. Han and G. Feng, “State feedback dynamic systems using LMI techniques,” in Fuzzy—IEEE'98, 1998, pp. 538–544. [20] I. R. Petersen, “A stabilization algorithm for a class of uncertain linear systems,” Syst. Contr. Lett., vol. 8, pp. 351–357, 1987. [21] P. Apkarian, G. Becker, P. Gahinet, and H. Kajiwara, “LMI techniques in control engineering from theory to practice,” presented at the Workshop Notes, IEEE CDC, Kobe, Japan, 1996. [22] P. Gahinet, A. Nemirovski, A. Laub, and M. Chilali, The LMI Control Toolbox. Natick, MA: The Mathworks, Inc., 1995.
VII. CONCLUSION In this paper, a number of dynamic output feedback controller design methods are developed for fuzzy dynamic systems based on quadratic stability theory and LMI techniques. It has been shown that the global stability of the nonlinear fuzzy control system can be guaranteed if a set of linear matrix inequalities are satisfied. Constructive algorithms for obtaining controllers are also proposed.
A Fuzzy Basis Function Vector-Based Multivariable Adaptive Controller for Nonlinear Systems Zhang Huaguang, Lilong Cai, and Zeungnam Bien
REFERENCES [1] L. A. Zadeh, “Outline of a new approach to the analysis of complex systems and decision processes,” IEEE Trans. Syst., Man, Cybern., vol. SMC-3, pp. 28–44, 1973. [2] E. H. Mamdani and S. Assilian, “Applications of fuzzy algorithms for control of simple dynamic plant,” Proc. Inst. Elect. Eng. C., vol. 121, pp. 1585–1588, 1974. [3] M. Sugeno, Industrial Applications of Fuzzy Control. New York: Elsevier, 1985. [4] M. Sugeno and G. T. Nishida, “Fuzzy control of model car,” Fuzzy Sets and Systems, vol. 16, pp. 103–113, 1985. [5] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its application to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. SMC-15, pp. 116–132, 1985. [6] H. Nakanishi, I. B. Turksen, and M. Sugeno, “A review and comparison of six reasoning methods,” Fuzzy Sets Syst., vol. 57, pp. 257–294, 1993. [7] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Sets and Systems, vol. 45, pp. 135–156, 1992. [8] K. Tanaka and M. Sano, “A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer,” IEEE Trans. Fuzzy Syst., vol. 2, pp. 119–134, 1994. [9] K. Tanaka, T. Ikeda, and H. O. Wang, “Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: Quadratic stabilizability, control theory, and LMI,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 1–13, 1996. [10] S. G. Cao and N. W. Rees, “Identification of dynamic fuzzy models,” Fuzzy Sets and Systems, vol. 74, pp. 307–320, 1995. [11] S. G. Cao and G. Feng, “Modeling of complex control systems,” in Proc. IFAC Symp. Nonlinear Control Systems Design, Davis, CA, 1995, pp. 935–938. [12] S. G. Cao, N. W. Rees, and G. Feng, “Analysis and design for a class of continuous-time fuzzy control systems,” Int. J. Contr., vol. 64, pp. 1069–1087, 1996. [13] S. G. Cao, N. W. Rees, and G. Feng, “Analysis and design for a class of complex control systems—Part I: Fuzzy modeling and identification,” Automatica, vol. 33, no. 6, pp. 1017–1028, 1997. [14] S. G. Cao, N. W. Rees, and G. Feng, “Analysis and design for a class of complex control systems—Part II: Fuzzy controller design,” Automatica, vol. 33, no. 6, pp. 1029–1039, 1997. [15] G. Feng, S. G. Cao, N. W. Rees, and C. K. Chak, “Design of fuzzy control systems with guaranteed stability,” Fuzzy Sets Syst., vol. 85, no. 1, pp. 1–10, 1997. [16] S. G. Cao, N. W. Rees, and G. Feng, “Further results about quadratic stability of continuous time fuzzy control systems,” Int. J. Syst. Sci., vol. 28, no. 4, pp. 397–404, 1997. [17] G. Feng, S. G. Cao, and N. W. Rees, “An approach to control of a class of nonlinear systems,” Automatica, vol. 32, no. 10, pp. 1469–1474, 1996. [18] S. G. Cao, N. W. Rees, and G. Feng, “Quadratic stability analysis and design of continuous time fuzzy control systems,” Int. J. Syst. Sci., vol. 27, pp. 193–203, 1996.
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Abstract—In this paper, a new fuzzy basis function vector (FBFV) approach for the adaptive control of multivariable nonlinear systems is presented. With this method, the nonlinear plant is first linearized. The linearized bias and uncertainties as well as disturbances are assumed to be included in the model structure and their upper bound will be adaptively learned by the FBFV method. The output of the FBFV is used as the parameters of the robust controller in the sense that both the robustness and the asymptotic error convergence can be obtained for the multivariable nonlinear system. The effectiveness of the proposed analysis and design method is illustrated with a simulated example. Index Terms—Fuzzy basis function vector, multivariable control, nonlinear system, stability.
I. INTRODUCTION Fuzzy logic controllers are now being widely used in many engineering applications. As such, there is increasing need for a theoretical analysis on the stability, consistency, and robustness to clarify the control performance of fuzzy control system. Because there are only few studies on theoretical analysis of fuzzy control, a user might hesitate to use a fuzzy controller in a critical environments [1]. In recent years, there have been some attempts to design fuzzy controllers and explain their performance based on variety of nonlinear control theories. A sufficient and necessary condition of closed-loop stability in T-S fuzzy model was presented in [13]. The control law must be expressed by a T-S fuzzy model, which is hardly used to other kinds of nonlinear systems. To solve this problem, Kiriakidis [14] studied the quadratic stability analysis method in which the T-S fuzzy model was analyzed as a linear Manuscript received May 12 1998; revised October 31, 1999. This work was supported by the Liaoning Provincial Natural Science Foundation, the National Natural Science Foundation of China, and the Hong Kong Research Grant Council Foundation. This paper was recommended by Associate Editor L. O. Hall. Z. Huaguang is with the Department of Automatic Control, Northeastern University, Shenyang, 110006, China. L. Cai is with the Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. Z. Bien is with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea. Publisher Item Identifier S 1083-4419(00)01399-6.
1083–4419/00$10.00 © 2000 IEEE
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B, VOL. 30 , NO. 1, FEBRUARY 2000
system, subject to a class of nonlinear perturbations. The linear states feedback control law was also used. The solution of the relative Lyapunov equation must be a positive definite matrix. It is sometimes difficult to meet this condition. A robust controller for T-S fuzzy system was presented in [15] and an excellent analysis on stability and robustness was conducted. The main result of this work is that the closed-loop is globally stable and robust with respect to unstructured uncertainty, which may include modeling error and disturbances. The main limitations are that the system and model must not have strong nonminimum phase effects and the unstructured uncertainty must be relatively small compared to the inputs and outputs. Also, there are some closely related neural control approaches [4]. Lately, fuzzy basis function networks for model-reference adaptive control have been used to resolve the above problems [2], [3], [11]. In [2], a fuzzy basis function network is used to approximate an unknown system parameter vector. The weights of the fuzzy basis function network are adaptively adjusted. The proof of stability is not provided. In [3] and [11], on the other hand, a stable adaptive control approach is given in which fuzzy forms and neural networks are used. Its control is composed of a bounding-control term, a sliding-mode-control term, and a certainty-control term. This hybrid control approach is so complicated that it is hardly practicable. Moreover, the emphasis for both is placed on the control of single-input single-output (SISO) plants. In this paper, a fuzzy basis function vector (FBFV)-based adaptive control scheme is proposed for control of multi-input multi-output (MIMO) square nonlinear systems. In the proposed scheme, the nonlinear system is first linearized and then treated as a partially known system. The partially known dynamics is used to design a nominal feedback controller to stabilize the nominal system, and a FBFV-based robust controller is designed to compensate for the effects of the system uncertainties. A FBFV in this work is introduced to learn the upper bound of the system uncertainty, and its output is used as the parameters of the robust controller. By proper design of a Lyapunov function, which consists of the vector of output tracking error and the parameter matching error vector of the robust controller, we prove the stability of the closed-loop nonlinear control system and obtain both robustness with respect to the unknown dynamics and asymptotic error convergence for the system in the meantime. This paper is organized as follows. In Section II, formulation of the problem concerning the affine nonlinear system is offered. The FBFV is proposed in Section III. In Section IV, a robust adaptive controller is designed and the stability is proved. A simulation example is provided in Section V. Section VI concludes the paper.
II. PROBLEM FORMULATION OF AN AFFINE NONLINEAR SYSTEM A. Notations The following notations will be used extensively throughout the paper. jAj denotes the modulus matrix of matrix A, i.e., a matrix with every element of jAj has the absolute value of original matrix A. kAk is a l2 norm of matrix A. jAj < jB j means jaij j < jbij j; 8aij 2 A and 8bij 2 B , where A = [aij ]; B = [bij ]. For A = [aij ]; jAjm is the matrix, all of whose elements are equal to amax , where amax = maxi;j jaij j. sign(A) is called a sign matrix of the vector A, which is diagonal and meets the condition of sign(A)A = jAj. tr(A) is the trace of the matrix A. Notice that the above definitions about jAj; jAj < jB j; jAjm are also suitable for the case when the matrix is replaced by a vector. We often handle the vector case in this paper as well.
211
B. Problem Formulation Consider the following kind of MIMO square nonlinear system (i.e., a system with as many inputs as outputs) given by:
X_ = f (X ) + y1 (t) = h1 (X )
m
j =1
gj (X )uj (t)
111
ym (t) = hm (X )
(1)
where
X 2 Rn plant state vector; U = [u1 ; u2 . . . ; um ] 2 Rm control-input vector; Y = [y1 ; y2 . . . ; ym ] 2 Rm output vector. f (X ) = [f1 (X ); f2 (X ); . . . ; fn (X )]T : Rn ! Rn , is a smooth function vector, fi (X ); i = 1; 2; . . . ; m is a smooth function of X ; gi (X ) = [gi1 (X ); . . . ; gin (X )]T : Rn ! Rn , is a smooth function vector too; hi (1) : Rn ! R; i = 1; 2; . . . ; m is smooth function.
The nonlinear system of (1) is input linear and usually called an affine system. For convenience, the above equation will be rewritten in a condensed form
X_ = f (X ) + G(X )U Y = h(X )
(2) (3)
where
U = [u1 ; u2 ; . . . um ]T Y = [y1 ; y2 ; . . . ym ]T G(X ) = [g1 (X ); g2 (X ); . . . gm (X )] h(X ) = [h1 (X ); h2 (X ); . . . hm (X )]T : To proceed conveniently, some necessary notations and terminology are introduced and recalled in the following. The Lie derivative of a scalar function along a vector is defined as
Lf (X ) =
@ @ @ ;...; f (X ) = @X @x1 @xn n
= i=1
@ f (X ) @xi i
f1 (X )
111
fn (X ) (4)
where X = [x1 ; x2 ; . . . ; xn ]T . The derivative of taken first along f and then along a vector g is
Lg Lf (X ) =
@ (Lf ) g(X ): @X
(5)
If is being differentiated j times along f , the notation Lfj is used with L0f (X ) = (X ). Definition 1 [7]: A MIMO nonlinear system described by the following equations:
X_ = f (X ) +
m
j =1
gj (X )uj (t)
yi = hi (X ) i = 1; . . . ; m has a general (vector) relative degree [r1 ; . . . ; rm ] at X0 if Lg Lkf hi (X ) = 0 for all 1 i; j m; k < ri Lg Lrf 01 hi (X ) 6= 0 for all X in a neighborhood of X0 .
(6)
0 1 and
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The coupling matrix
J (X; t) =
Assumption A3: The plant has a general (vector) relative degree
Lg Lrf Lg Lfr Lg Lfr
01 h1 (X ) 1 1 1 Lg 01 h2 (X ) 1 1 1 Lg 111 111
01 hm (X ) 1 1 1 Lg
Lfr Lfr
01 h1 (X ) 01 h2 (X )
Lfr
01 hm (X )
[r1 ; . . . ; rm ], and its zero dynamics are exponentially stable.
(7)
is nonsingular at X = X0 . Remark 1: The system (6) is said to have a strong (vector) relative degree [r1 ; . . . ; rm ] if Lg Lkf hi (X ) 0 for all 1 i; j m; k < ri 0 1 and Lg Lfr 01 hi (X ) 6= 0 for 8X 2 Rn , and (7) holds. Differentiating the output yi with respect to time t in (6), we have
y_ i
= Lf hi (X ) +
m j =1
Lg hi (X )uj (t)
The Assumption A3 is needed to guarantee state boundedness under state feedback linearization. In order to understand the reason for A3, we should first introduce the concept of zero dynamics of square MIMO systems. There exists a diffeomorphic coordinate transformation T (X ) = (; ), which transforms system (1) or equivalently (6) to the following form: _i = i
1
_2i
111
_ri
(8)
yi(r
) = Lr hi (X ) + f
j =1
Lg Lfr
01 hi (X )uj (t)
where (9)
for Lg Llf hi (X ) = 0; l < ri 0 1 and Lg Lfr 01 hi (X ) 6= 0. In this way, the MIMO system of (1) can be linearized and considered as a collection of MISO systems of (9). Therefore, we may rewrite the plant's input-output equation as Lfr h1 y1(r ) .. .. . . = (r ) ym Lfr hm Y (t) B (X; t) Lg Lfr 01 h1 (X ) 1 1 1 Lg Lfr 01 h1 (X ) .. .. .. + Lg Lfr 01 hm (X ) 1 1 1 Lg Lfr 01 hm (X ) J (X; t) u1 (t) . 2 .. : (10) um (t) U (t) An ideal static-state feedback linearizing control law can be obtained by
U3
= J 01 (0B + V ):
= Lfr hi (X ) +
yi = 1i _ i = qi (; ) i = 1; 2; . . . ; m
where Lf hi (X ) and Lg hi (X ) are the Lie derivatives of hi (X ) with respect to f (X ) and gj (X ), respectively. Since Lg hi (X ) = 0, which implies y_ i = Lf hi (X ), we obtain m
2
= 3i
(11)
Note that, for convenience, the references to X and t are dropped in (11). The term V in (11) is designated below. In order for U 3 to be defined, some assumptions about the plant have to be met. In particular, we need the following assumptions. Assumption A1: The matrix J as defined above is nonsingular, i.e., J 01 exists and has bounded norm for all X 2 Sx ; t 0, where Sx 2 Rn is a compact set that allows state trajectories. This is equivalent to assuming that
p (J ) min > 0
(12)
kJ k2 = 1 (J ) max < 1
(13)
where p (J ) and 1 (J ) are, respectively, the smallest and largest singular values of J . Assumption A2: The reference signal V is bounded.
m j =1
Lg Lfr
01 hi (X )uj (t)
(14) T
= 1i ; . . . ; ri T _ = _r1 ; . . . ; _rm = y1(r ) ; . . . ; ym(r ) = [1 ; 2 ; . . . ; m ]T q (; ) = [q1 (; ); . . . ; qm (; )]T : i
T
The combined form of (14) is rewritten as
_ = B (X; t) + J (X; t)U (t) _ = q (; ):
(15)
If the control law is chosen as in (11), it linearizes the normal part of (15) as
_ = V _ = q (; ):
(16)
It is apparent that is an observed state, and is an unobserved state. Definition 2: The dynamics
_
= q(0; )
(17)
is referred to as the zero dynamics of the system (1). Lemma 1: Assume that q (; ) in (15) or (16) satisfies the Lipschitz condition and the zero dynamics of nonlinear system (1) _ = q (0; ) is exponentially stable and is bounded. Then the state and X 2 Rn are bounded. Proof: Because _ = q (0; ) is exponentially stable when the states move outside a ball j j > B , then there exist some positive constants
1 ; 2 ; 3 ; 4 ; B and a function ( ) to meet the following conditions:
1 j j2
() 2 jj2 d
q (0; ) 0 3 j j2 if j j > B d d
d
where
4 jj
(18) (19) (20)
jj =1 T jj2 =1 T d
d
=1
d
d d
; ;... : d1 d2 dm
(21)
Since we have bounded reference signals V in (16), j j k1 where k1 is a positive constant.
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Using (19), we have
_ = dd _ = dd q(; ) 0 3 jj2 + dd [q(; ) 0 q(0; )] if jj > B: (22) If q (; ) is Lipschitz in , then jq (; ) 0 q (0; )j k2 j j for some
positive k2 . By using this condition, if j j > B , we now have
_ 0 3 jj2 + dd j[q(; ) 0 q(0; )]j 0 3 jj2 + 4 k2 jjjj (23) 0 3 jj2 + 4 k1 k2 jj: _ Therefore, 0 if j j max(B; 4 k1 k2 = 3 ). This ensures
boundedness of and . The system states are therefore bounded. III. FUZZY SYSTEM FORMATION
Fig. 1.
From (10) we can obtain:
( ) = F (X ) + U
E X Y
(24)
where
( ) =1 J 01 (X ) 2 Rm2m (25) 1 0 1 m F (X ) = J (X )B (X ) 2 R : (26) 0 1 E (X ) and F (X ) are assumed to be bounded by the following unknown positive function P1 (X ) and function vector Q1 (X ): 0 kE (X )01k < P1 (X ) (27) 0 jF (X )jm < Q1 (X ): (28) In practical situations where E (X ) and F (X ) are not exactly known, E (X ) and F (X ) may be expressed as (29) E (X ) = E0 (X ) + 1E (X ) F (X ) = F0 (X ) + 1F (X ) (30) where E0 (X ) (nonsingular) and F0 (X ) are known parts, and 1E (X ) and 1F (X ) are unknown parts that contain linearized bias, uncertainty, and disturbance. Remark 2: According to the bounded properties of E (X ) and F (X ) in (25) and (26), the uncertain dynamics 1E (X ) and 1F (X )
Structure of the control system.
= [v1 ; v2 ; . . . ; vm ]T
where V Namely, let
vi
E X
are also bounded by
( )
k1E (X )01k < P2 (X ) j1F (X )j < Q2 (X )
( )
where P2 X and Q2 X are a positive function and function vector of X , respectively. Based on the analysis, (24) can be written as:
( ) = F0 (X ) + U + (t)
(33)
( ) = 1F (X ) 0 1E (X )Y
(34)
E0 X Y where
t
is defined as the system uncertainty vector. The following system without uncertainty is called the nominal system:
( ) = F0 (X ) + U:
E0 X Y
(35)
For the nominal system in (35), let the nominal controller be (see Fig. 1)
U0
= E0 (X )V 0 F0 (X )
= yd(r ) 0 air 01"i(r 01) 0 1 1 1 0 ai1 "_i 0 ai0 "i
(37)
=4 yi 0 yd :
(38)
where
"i
Here, "i is the output tracking error and yd are the desired reference trajectories of yi , while the parameters air 01 ; ai1 ; ai0 in (37) are chosen in such a way that the characteristic polynomial of (37) sr air 01 sr 01 air 02 sr 02 1 1 1 ai1 s ai0 (39) i s
...
1 ( )=
+
+
(36)
+ +
+
is Hurwitz. In this way, we find that the error dynamics of the nominal system has the following form: "i(r ) air 01 "i(r 01) air 02 "i(r 02) 1 1 1 ai1 "i ai0 "i : (40)
+
+
+
_+
=0
Equation (40) implies that the output tracking error, "i , will converge asymptotically to zero. Consider now the system in (33) with uncertain dynamics. In this case, the control input of the closed-loop system can be set in the following form:
U
(31) (32)
is chosen to provide stable tracking.
= U0 + U1
(41)
where U0 is given by (36) for stable control of the nominal system (35), while U1 is a robust control term used to control and deal with the effects of the uncertainty, as shown in Fig. 1. From (33), we have
( ) = F0 (X ) + U0 + U1 + (t):
E0 X Y
From (37), we form V Yd(r) 0 Ar01 "(r01) 0 1 1 1 A1 " 0 A0 "
=
where
_
(42)
(43)
= max fri g i (i = 0; 1; . . . ; r 0 1) Ai = diag a1i ; a2i ; . . . ; am i T i "(i) = "1i ; "2i ; . . . ; "m (i = 1; 2; . . . ; r 0 1) T Yd(r) = yd(r ) ; yd(r ) ; . . . ; yd(r ) : r
Substitute (36) into (42) to obtain
( )( 0 V ) = U1 + (t)
E0 X Y
(44)
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while Y
0 V = Y 0 Y + A 0 " 0 + 1 1 1 + A ": (r )
1
r
d
(r
1)
(45)
0
Then, the error dynamics of the closed-loop system with uncertainty becomes "(r) Ar01 "(r01)
+
+1 1 1+A " = E (X )0 (t)+E (X )0 U : 1
0
0
1
0
1
(46)
Fig. 1 shows the closed-loop structure of the control systems. The function of the Normal Controller U0 and Robust Controller U1 can be seen from (36) and (44). The step called Dynamics-Settings performs the function of (43). The mechanism FBFV forms a fuzzy basis function vector. The Adaptive mechanism is used to adaptively tune the _ weight matrix T so that the closed-loop stability can be guaranteed. The concrete method will be given in the next subsection. For the bounded property of system uncertainty vector t in (34), we have the following Lemma: Lemma 2: If the control input U is designed in such a way that the modulus vector of the control signal is upper bounded by a positive function vector Umax X , that is,
()
( )
jU (t)j
m
( )
< Umax X
(47)
() ( )
Fig. 2. Fuzzy membership functions of the premise part.
as defined in (34). Akl and Cij are linguistic terms characterized by fuzzy membership functions A xk and C i t , respectively.
( )
then the modulus vector of the system uncertainty vector t in (34) is also upper bounded by some positive function vector X , that is
j(t)j = j1F (X ) 0 1E (X )Y j < (X ):
(48)
Proof: Equation (24) can be written in the following form: Y
= E (X )0 F (X ) + E (X )0 U (t): 1
1
(49)
Using (49) in (34), we have
( ) = 1F (X ) 0 1E (X )(E (X )0 F (x) + E (X )0 U (t)) = 1F (X ) 0 1E (X )E (X )0 F (x) 0 1E (X )E (X )0 U (t): 1
1
t
1
1
1
1
2
1
( ) = exp 0 (x 0 )
1
Ai;j
Ai;j
( ) = [ ( ) ( ) . . . ( )]
( )
:
111
:
( ( ))
( ( ))
... ...
M RiM if x1 is AM 1 ; x2 is A2 M then i is CiM i t
( = 1 2 ... ) ( = 1 2 ... )
( ( ))
...x
n
i
(53)
Ai;j
0
=
Ai;j +1
j=c
Ai;j +1
j
0
1
=c j =c 2
3
05 1 = 1; 2; . . . ; M ;
j
i
i
=(
( ) 1 2 ...
( ) = (X; ) =
...
()
where xi i ; ; ; n are the plant state variable, and i t i ; ; ; m are the i-th element of the plant uncertainty vector
)
M
=
( ) = (X ) (i = 1; 2; . . . ; m) (54) where is the value of variable (t) of the center point at which ( (t)) achieves its maximum value, i.e., ^ = [ ; ; . . . ] 2 R is the weight vector of the fuzzy basis expansion vector (X ); and the j -th fuzzy basis expansion in (X ) = [ (X ); (X ); . . . ; (X )] is defined as (x ) (X ) = (55) (x ) _
i X
_
i
j =1
ij
i
C
i1
_T
j X ij
i
i
i
i2
i
M
iM
1
2
j
(52)
= 1; 2; . . . ; n
where c1 ; c2 ; c3 are some predetermined constants. _ Given an input vector X x10 ; x20 ; ; xn0 , the output i X can be inferred by taking the weighted average of the ji ; j ; ; ;M
T
M
n i=1
is AM n
= 1; 2; . . . ; m
( )
Ai;j Ai;j Ai;j +1 : < c3
max
()
2
Ai;j
2
( )
j
Remark 3: It follows from Lemma 2 that the bounded property of the system uncertainty, t , depends on the system structure and the form of the controller. It will be shown in the next section that the control signal satisfies the condition in (47), and therefore, the bounded condition of the system uncertainty in (48) always holds. In the following, the fuzzy basis function network is adopted to learn X ; m X T , the upper bound of the 1 X ; 2 X ; system uncertainty vector. Consider a fuzzy basis expansion system of i X , which consists of M fuzzy rules Ri1 if x1 is A11 ; x2 is A12 ; xn is A1n 1 then i is Ci1 i t xn is A2n Ri2 if x1 is A21 ; x2 is A22 ; 2 then i is Ci2 i t
:
i
A xi
m
2
i=1
1 2 ... )
m
1
Mi :
n
( = 1 2 ... ) = ( = 1 2 ...
( ) + P (X )P (X )Q (X )+ P (X )P (X )U (X ) (51) = (X ): < Q2 X
2
where A xi is the membership function of xi in fuzzy set Aji ; Ai;j is the center of A xi ; Ai;j is the width of A xi . The following conditions have to be met:
1
1
1
Equation (52) is a T-S fuzzy model of zero order, in fact, and can be solved as follows. We divide the value domain of xi into Mi i ; ; ; n equal individuals, as shown in Fig. 2. In Fig. 2, Aji j ; ; ; Mi ; i ; ; ; n are the premise fuzzy subset in (52). Mi can be chosen according to the range scale and uncertainty degree of variable xi . In this paper, the following Gaussian membership function is used:
( )
j(t)j j1F (X )j + j1E (X )E (X )0 F (x)j + j1E (X )E (X )0 U (t)j j1F (X )j + k1E (X )kk1E (X )0 kjF (x)j + k1E (X )kkE (X )0 kjU(t)j
= M M ...M =
M
(50) Therefore, we have
( ( ))
n
M j =1
i
A
n i=1
A
i
where M is the number of fuzzy rules. Therefore, no training of the premise fuzzy subsets in (52) takes place. Only the consequences should be adaptively identified.
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For further analysis, the following assumptions are made: Assumption A4: Given an arbitrary small positive constant vector "~3 and a continuous function vector (X ) defined in (51) in a compact set 6, there exists an optimal weight matrix 3 such that the output vector of the optimal fuzzy network satisfies
3T 3 max X 26 j"~(X )j = max X 26 j (X ) 0 (X )j < "~
(56)
"~(X ) = 3T (X ) 0 (X )
(57)
with
3 ]T where 3T = [13 ; 23 ; . . . ; m
with ` >
215
0 and arbitrary positive initial values matrix of
_ M 2m 2 R
then the output tracking errors vector " converges asymptotically to the zero vector. Proof: Considering following Lyapunov function
@ = 12 S T S + 21 `01 tr[~T ~]
where
_
~ = 3 0 ;
2 Rm2M .
Lemma 3: If the modulus vector of the system uncertainty and its upper bound satisfy the following relationship in the compact set 6
(X ) 0 j(t)j = "0 > "~3
(59)
Proof: From (58), we can get
(X ) > "~3 + j(t)j
"~(X ) = 3T (X ) 0 (X ) < 3T (X ) 0 (~ "3 + j(t)j) 3 T 3 (60) = (X ) 0 j(t)j 0 "~ : 3T (X ) 0 j(t)j > "~ + "~3 > O
= 0`01 tr[3T ^_ ] + `01 tr[^T ^_ ] = 0`01 tr 3T ` (X )jE T C T j 3
= 3E + 9E0 (X )01 (t) + 9E0 (X )01 U1
3=
I O
O I
111 111
_
+ tr
111 0Ar02 0A0 0A1 0A2 1 1 1 0Ar02 9 = [O O O 1 1 1 I ]T (63) mr 2 1 mr 2 mr mr 2 m m 2 m m 2 m E 2R ; 32R ; 92R ; O2R ; I 2R . For the design of the adaptive robust controller using the FBFV and analysis of error convergence of the closed-loop system, we have the following theorem. Theorem: Consider the error dynamics in (62). If (a) the robust controller U1 is designed as follows:
_T
U1 = E0 (X)(C 9)01 [0C 3E 0 sign(S T )kC 9E0 (X )01 k3
__
= ` (X )jE T C T j
C 9E001(X )
(65)
_T
(X )jE T C T j C 9E001(X )
:
(71)
@_ = E T C T C 9E0 (X )01(t) 0 tr 3T (X )jE T C T j 3 C 9E001(X ) = E T C T C 9E0 (X )01(t) 0 jE T C T j C 9E001(x) 3T (X ) jE T C T j C 9E001(x) (j(t)j 0 3T (X )) < 0:
(72)
Remark 4: It is not necessary for the weight matrix of the fuzzy network to converge to its optimal values, and the values of the weight matrix are adaptively adjusted until the variable S converges to the zero vector. The weight matrix will then become a constant matrix to guarantee that the output tracking error vector converges asymptotically to the zero vector after S = CE = 0. V. SIMULATION STUDY A simulation model is given by
X_
8(X)
(64) where the vector S = CE = [s1 ; . . . ; sm ]T ; S 2 Rm ; C 2 Rm2mr ; (b) the matrix C should be chosen such that the polynomial Si is Hurwitz about "i (i = 1; 2; . . . ; m); _ (c) the matrix in (64) is updated by the following adaptive mechanism:
C 9E001(X )
Therefore, we have
(62)
O O
_
+ `01 tr ` (X )jE T C T j C 9E001(X ) = 0tr 3T (X )jE T C T j C 9E001(X )
Let E = ["; "; _ . . . ; "r01 ]T , then (46) can be written as the following state space form:
O O
(69)
_
IV. ROBUST CONTROLLER DESIGN
where
@_ = S T S_ + `01 tr[~T ~_ ]:
`01 tr[~T ~_ ] = 0`01 tr[~T ^] = 0`01 trf[3 0 ^]T ^g
(61)
where O means zero vector. The objective of this paper is to design a robust adaptive controller, U1 , using the fuzzy basis function network in (55) so that the closed-loop system has strong robustness and the output tracking error is guaranteed to converge asymptotically to zero.
E_
(68)
The first term in (69) is shown in (70) at the bottom of the next page. The second term in (69) is
and (60) can be deduced from (57)
Therefore,
__
Then
3T (X ) 0 j(t)j > 0:
(67)
~_ = 0 :
(58)
then the following relation holds:
(66)
= f (X ) + G(X )U
(73)
= h(X )
(74)
Y where
f (X ) = h(X ) =
x4 0 x1 0 x2 x3 1 + x3 ; G(X ) = x22 x3 x3 0 x3 0 x2 u1 2x4 ; U = u2 :
0 2 2 0
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Calculating the relative degree, we obtain
= 1; r2 = 2: Taking a transform that 1 = x2 0 x3 ; 2 = x4 ; 3 = x3 ; = x1 ,
PARAMETERS
r1
we get
_ x22 0 x2 x3 01 0 + 2x3 4 = 2x22 () B (X; t) J (X; t) _ = 2 0 :
y1 y2 Y t
u1 u2 U t
TABLE I AND INITIAL VALUES SETTINGS
(75)
()
(76)
Thus, the zero dynamics of the given system is
_ =0
:
(77)
Obviously, the zero dynamics of the given system is exponentially stable at the equilibrium point. From (25), and (26), we obtain
( ) = J 01 (X ) = 0:05x13 0:025 0x22 + x2 x3 F (X ) = E (X )B (X ) = 2 0:5x2x3 0 0:5x2 x23 + 0:5x22
E X
If we take E0
3 = 001 0x:25
then we have
;
F0
= 0:5x22x300x20:5x2 x23 2
1E = 0:50x3 00x3 1F = 0x:25xx322 : = ["1 ; "2 ; "_2], the error
By taking E closed-loop system is
state equation of the
_ = 3E + 9E0 (X )01(t) + 9E0 (X )01U1
E
where
3=
0 0 0 0 1 0 0a20 a21
0a10
:
;
Membership functions of variable x
;x ;x ;x
.
(78)
1 0 9= 0 0 0 1
and the parameters aij can be chosen by (39). Table I gives all relative parameters in the simulation study, which can be chosen subjectively according to the above method. Fig. 3 shows the membership functions of variable x1 ; x2 ; x3 ; x4 , which consist of small, medium, and large. Therefore, the number of . the fuzzy rules should be 4 Figs. 4 and 5 give the response curves of the output y1 t ; y2 t respectively. And it can be observed that the nonlinear control system
3 = 81
Fig. 3.
() ()
Fig. 4. Response curve of the output y (t).
will reach the stable state after 15 s. The greater ` is, the faster the response is. But too great an ` may cause a small vibrating motion in the meantime.
_ = E T C T C E_ = E T C T C [3E + 9E0 (X )01(t) + 9E0 (X )01U1 ] = E T C T C 3E + E T C T C 9E0 (X )01(t) + E T C T C 9E0 (X )01U1 = E T C T C 3E + E T C T C 9E0 (X )01(t) + E T C T C 9E0 (X )01E0 (X )(C 9)01[0C 3E _T 0 sign(S T )kC 9E0 (X )01k 8(X )] _T = E T C T C 9E0 (X )01(t) 0 E T C T sign(S T )kC 9E0 (X )01k3 8(X ) _T = E T C T C 9E0 (X )01(t) 0 jE T C T jkC 9E0 (X )01k 8(X ):
ST S
(70)
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217
On the Optimal Design of Fuzzy Neural Networks with Robust Learning for Function Approximation Hung-Hsu Tsai and Pao-Ta Yu
Fig. 5.
Abstract—A novel robust learning algorithm for optimizing fuzzy neural networks is proposed to address two important issues: how to reduce the outlier effects and how to optimize fuzzy neural networks, in the function approximation. This algorithm is able to reduce the outlier effects by cooperating with a conventional robust approach, and then to optimize fuzzy neural networks by determining the optimal learning rates which can minimize the next-step mean error at each iteration of our algorithm.
Response curve of the output y (t).
Index Terms—Function approximation, fuzzy neural networks, optimization, robust learning algorithm.
VI. CONCLUSION An adaptive controller using a fuzzy basis function vector (FBFV) is proposed in this paper. Our theoretical analysis demonstrates that the fuzzy basis function expansion vector can be used to learn the MIMO system uncertainty bounds in the Lyapunov sense, and a FBFV-based adaptive hybrid controller can be designed to eliminate the effects of dynamical uncertainties and guarantee that the output tracking errors converge asymptotically to zero. In addition, it has a better robustness with respect to unstructured uncertainty. Computer simulation shows that this method is strong, robust, fast convergence, and easy to design and use. ACKNOWLEDGMENT The authors would like to thank the reviewers for their constructive comments. REFERENCES [1] R. Yager and D. P. Filev, Essentials of Fuzzy Modeling and Control. New York: Wiley, 1994. [2] M. Zhihong, X. H. Yu, and Q. P. Ha, “Adaptive control using fuzzy basis function expansion for SISO linearizable nonlinear systems,” in Proc. 2nd ASCC, Seoul, Korea, 1997, pp. 695–698. [3] J. T. Spooner and K. M. Passino, “Stable adaptive control using fuzzy systems and neural networks,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 339–359, 1996. [4] G. A. Rovithakis and M. A. Christodoulou, “Adaptive control of unknown plants using dynamical neural networks,” IEEE Trans. Syst., Man, Cybern., vol. 24, pp. 400–412, 1994. [5] C. Wen and Z. Huaguang, “Input/output linearization for nonlinear systems with uncertainties and disturbances using TDC,” Cybern. Systems, vol. 28, pp. 625–634, 1997. [6] D. Changguo, Linear Algebra in Control Systems, China: Southeastern Univ. Press, 1993. (in Chinese). [7] A. Isidori, Nonlinear Control Systems: An Introduction, Berlin, Germany: Springer-Verlag, 1989. [8] S. S. Sastry and A. Isidori, “Adaptive control of linearizable systems,” IEEE Trans. Automat. Contr., vol. 34, pp. 1123–1131, 1989. [9] Y. S. Lu and J. S. Chen, “A self-organizing fuzzy sliding-mode controller design for a class of nonlinear serve systems,” IEEE Trans. Ind. Electron., vol. 41, pp. 492–502, 1994. [10] C. Y. Su and Y. Stepanenko, “Adaptive controller of a class of nonlinear systems with fuzzy logic,” IEEE Trans. Fuzzy Syst., vol. 2, pp. 285–294, 1994. [11] D. Driakov, Advances in Fuzzy Control. Berlin, Germany: SpringerVerlag, 1998, pp. 155–188. Physica. [12] Z. Bien and W. Yu, “Extracting core information from inconsistent fuzzy control rules,” Fuzzy Sets Syst., vol. 71, pp. 95–111, 1995. [13] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Sets Syst., vol. 45, pp. 135–156, 1992. [14] K. Kiriakidis et al., “Quadratic stability analysis of the Takagi-Sugeno fuzzy model,” Fuzzy Sets Syst., vol. 98, pp. 1–14, 1998. [15] T. A. Johansen, “Fuzzy model based control: Stability, robustness, and performance issues,” IEEE Trans. Fuzzy Syst., vol. 2, pp. 221–234, 1994.
I. INTRODUCTION Function approximation has been successfully applied in diverse applications such as signal processing, image restoration, pattern recognition, control system, system identification, etc. [1]–[4]. The main task of the function approximation is to approximate a desired function or an input-output relation from a set of input-output patterns. In several applications, the most desired functions are highly nonlinear. The highly nonlinear desired functions are difficult to be specified or modeled by the precisely mathematical formulas. Hence, how to find approximated functions for the highly nonlinear desired functions is an crucial issue in many applications. When imprecise or erroneous training patterns are collected and employed during the approximation process, the learning mapping may oscillate sharply such that the gross errors, caused by these erroneous training patterns, will be enlarged. Chen and Jain have proposed a robust back-propagation learning algorithm to resist the noise effects (or alternatively called outlier effects in this correspondence) such as to reduce the gross errors [5]. However, Chen and Jain’s algorithm still suffers from the problem of slow convergence. In [6], Wang et al. have exploited the fuzzy-neural networks (FNN’s) and B-spline membership functions (constructed by control points) to realize the robust learning with the outlier rejection for the function approximation. However, the training complexity of Wang et al.’s algorithm increases greatly as the number of control points is getting large due to the dimension of the matrix proportional to the number of control points. Another limitation of Wang et al.’s algorithm is to collect a set of empirical data in advance for the construction of B-spline membership functions. This leads to the difficulty, in real applications, collecting various sets of empirical data in advance for the various problems. Furthermore, Wang et al.’s algorithm suffers from the problem that the analysis of the training (learning) behavior of the FNN’s is extremely difficult due to the use of a complicated membership function—B-spline functions. Therefore, in this correspondence we analyze the training behavior of the FNN’s with common membership functions (such as triangular shapes) to circumvent the difficulty that Wang et al.’s algorithm suffers from. Note that most FNN’s we concentrate to investigate are trained by the conventional learning algorithms based on the gradient decent method [4], Manuscript received January 24, 1999; revised October 31, 1999. This work was supported in part by the National Science Council, R.O.C., under Grant NSC88-2213-E-194-011. This paper was recommended by Associate Editor S. Lakshmivarahan. H.-H. Tsai is with the Department of Information Management, Nan Hua University, Chiayi, Taiwan, R.O.C. P.-T. Yu is with the Department of Computer Science and Information Engineering, Chung Cheng University, Chiayi, Taiwan, R.O.C. Publisher Item Identifier S 1083-4419(00)01401-1.
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H
and the corresponding controller parameters are K1
=
224:78 0
0
01 71 :
;
K2
=
2624:8 0
0
01 73 :
:
(42)
The simulation shows that the pendulum can be well balanced for initial conditions in the range of x1 (0) = (089 ; 89 ) and x2 (0) = 0. The angle response of the pendulum control system (37) with control law (42) for initial condition x1 = 85 ; x2 = 0 is shown in Fig. 4.
controller design of fuzzy [19] Z. X. Han and G. Feng, “State feedback dynamic systems using LMI techniques,” in Fuzzy—IEEE'98, 1998, pp. 538–544. [20] I. R. Petersen, “A stabilization algorithm for a class of uncertain linear systems,” Syst. Contr. Lett., vol. 8, pp. 351–357, 1987. [21] P. Apkarian, G. Becker, P. Gahinet, and H. Kajiwara, “LMI techniques in control engineering from theory to practice,” presented at the Workshop Notes, IEEE CDC, Kobe, Japan, 1996. [22] P. Gahinet, A. Nemirovski, A. Laub, and M. Chilali, The LMI Control Toolbox. Natick, MA: The Mathworks, Inc., 1995.
VII. CONCLUSION In this paper, a number of dynamic output feedback controller design methods are developed for fuzzy dynamic systems based on quadratic stability theory and LMI techniques. It has been shown that the global stability of the nonlinear fuzzy control system can be guaranteed if a set of linear matrix inequalities are satisfied. Constructive algorithms for obtaining controllers are also proposed.
A Fuzzy Basis Function Vector-Based Multivariable Adaptive Controller for Nonlinear Systems Zhang Huaguang, Lilong Cai, and Zeungnam Bien
REFERENCES [1] L. A. Zadeh, “Outline of a new approach to the analysis of complex systems and decision processes,” IEEE Trans. Syst., Man, Cybern., vol. SMC-3, pp. 28–44, 1973. [2] E. H. Mamdani and S. Assilian, “Applications of fuzzy algorithms for control of simple dynamic plant,” Proc. Inst. Elect. Eng. C., vol. 121, pp. 1585–1588, 1974. [3] M. Sugeno, Industrial Applications of Fuzzy Control. New York: Elsevier, 1985. [4] M. Sugeno and G. T. Nishida, “Fuzzy control of model car,” Fuzzy Sets and Systems, vol. 16, pp. 103–113, 1985. [5] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its application to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. SMC-15, pp. 116–132, 1985. [6] H. Nakanishi, I. B. Turksen, and M. Sugeno, “A review and comparison of six reasoning methods,” Fuzzy Sets Syst., vol. 57, pp. 257–294, 1993. [7] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Sets and Systems, vol. 45, pp. 135–156, 1992. [8] K. Tanaka and M. Sano, “A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer,” IEEE Trans. Fuzzy Syst., vol. 2, pp. 119–134, 1994. [9] K. Tanaka, T. Ikeda, and H. O. Wang, “Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: Quadratic stabilizability, control theory, and LMI,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 1–13, 1996. [10] S. G. Cao and N. W. Rees, “Identification of dynamic fuzzy models,” Fuzzy Sets and Systems, vol. 74, pp. 307–320, 1995. [11] S. G. Cao and G. Feng, “Modeling of complex control systems,” in Proc. IFAC Symp. Nonlinear Control Systems Design, Davis, CA, 1995, pp. 935–938. [12] S. G. Cao, N. W. Rees, and G. Feng, “Analysis and design for a class of continuous-time fuzzy control systems,” Int. J. Contr., vol. 64, pp. 1069–1087, 1996. [13] S. G. Cao, N. W. Rees, and G. Feng, “Analysis and design for a class of complex control systems—Part I: Fuzzy modeling and identification,” Automatica, vol. 33, no. 6, pp. 1017–1028, 1997. [14] S. G. Cao, N. W. Rees, and G. Feng, “Analysis and design for a class of complex control systems—Part II: Fuzzy controller design,” Automatica, vol. 33, no. 6, pp. 1029–1039, 1997. [15] G. Feng, S. G. Cao, N. W. Rees, and C. K. Chak, “Design of fuzzy control systems with guaranteed stability,” Fuzzy Sets Syst., vol. 85, no. 1, pp. 1–10, 1997. [16] S. G. Cao, N. W. Rees, and G. Feng, “Further results about quadratic stability of continuous time fuzzy control systems,” Int. J. Syst. Sci., vol. 28, no. 4, pp. 397–404, 1997. [17] G. Feng, S. G. Cao, and N. W. Rees, “An approach to control of a class of nonlinear systems,” Automatica, vol. 32, no. 10, pp. 1469–1474, 1996. [18] S. G. Cao, N. W. Rees, and G. Feng, “Quadratic stability analysis and design of continuous time fuzzy control systems,” Int. J. Syst. Sci., vol. 27, pp. 193–203, 1996.
H
H
Abstract—In this paper, a new fuzzy basis function vector (FBFV) approach for the adaptive control of multivariable nonlinear systems is presented. With this method, the nonlinear plant is first linearized. The linearized bias and uncertainties as well as disturbances are assumed to be included in the model structure and their upper bound will be adaptively learned by the FBFV method. The output of the FBFV is used as the parameters of the robust controller in the sense that both the robustness and the asymptotic error convergence can be obtained for the multivariable nonlinear system. The effectiveness of the proposed analysis and design method is illustrated with a simulated example. Index Terms—Fuzzy basis function vector, multivariable control, nonlinear system, stability.
I. INTRODUCTION Fuzzy logic controllers are now being widely used in many engineering applications. As such, there is increasing need for a theoretical analysis on the stability, consistency, and robustness to clarify the control performance of fuzzy control system. Because there are only few studies on theoretical analysis of fuzzy control, a user might hesitate to use a fuzzy controller in a critical environments [1]. In recent years, there have been some attempts to design fuzzy controllers and explain their performance based on variety of nonlinear control theories. A sufficient and necessary condition of closed-loop stability in T-S fuzzy model was presented in [13]. The control law must be expressed by a T-S fuzzy model, which is hardly used to other kinds of nonlinear systems. To solve this problem, Kiriakidis [14] studied the quadratic stability analysis method in which the T-S fuzzy model was analyzed as a linear Manuscript received May 12 1998; revised October 31, 1999. This work was supported by the Liaoning Provincial Natural Science Foundation, the National Natural Science Foundation of China, and the Hong Kong Research Grant Council Foundation. This paper was recommended by Associate Editor L. O. Hall. Z. Huaguang is with the Department of Automatic Control, Northeastern University, Shenyang, 110006, China. L. Cai is with the Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. Z. Bien is with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea. Publisher Item Identifier S 1083-4419(00)01399-6.
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system, subject to a class of nonlinear perturbations. The linear states feedback control law was also used. The solution of the relative Lyapunov equation must be a positive definite matrix. It is sometimes difficult to meet this condition. A robust controller for T-S fuzzy system was presented in [15] and an excellent analysis on stability and robustness was conducted. The main result of this work is that the closed-loop is globally stable and robust with respect to unstructured uncertainty, which may include modeling error and disturbances. The main limitations are that the system and model must not have strong nonminimum phase effects and the unstructured uncertainty must be relatively small compared to the inputs and outputs. Also, there are some closely related neural control approaches [4]. Lately, fuzzy basis function networks for model-reference adaptive control have been used to resolve the above problems [2], [3], [11]. In [2], a fuzzy basis function network is used to approximate an unknown system parameter vector. The weights of the fuzzy basis function network are adaptively adjusted. The proof of stability is not provided. In [3] and [11], on the other hand, a stable adaptive control approach is given in which fuzzy forms and neural networks are used. Its control is composed of a bounding-control term, a sliding-mode-control term, and a certainty-control term. This hybrid control approach is so complicated that it is hardly practicable. Moreover, the emphasis for both is placed on the control of single-input single-output (SISO) plants. In this paper, a fuzzy basis function vector (FBFV)-based adaptive control scheme is proposed for control of multi-input multi-output (MIMO) square nonlinear systems. In the proposed scheme, the nonlinear system is first linearized and then treated as a partially known system. The partially known dynamics is used to design a nominal feedback controller to stabilize the nominal system, and a FBFV-based robust controller is designed to compensate for the effects of the system uncertainties. A FBFV in this work is introduced to learn the upper bound of the system uncertainty, and its output is used as the parameters of the robust controller. By proper design of a Lyapunov function, which consists of the vector of output tracking error and the parameter matching error vector of the robust controller, we prove the stability of the closed-loop nonlinear control system and obtain both robustness with respect to the unknown dynamics and asymptotic error convergence for the system in the meantime. This paper is organized as follows. In Section II, formulation of the problem concerning the affine nonlinear system is offered. The FBFV is proposed in Section III. In Section IV, a robust adaptive controller is designed and the stability is proved. A simulation example is provided in Section V. Section VI concludes the paper.
II. PROBLEM FORMULATION OF AN AFFINE NONLINEAR SYSTEM A. Notations The following notations will be used extensively throughout the paper. jAj denotes the modulus matrix of matrix A, i.e., a matrix with every element of jAj has the absolute value of original matrix A. kAk is a l2 norm of matrix A. jAj < jB j means jaij j < jbij j; 8aij 2 A and 8bij 2 B , where A = [aij ]; B = [bij ]. For A = [aij ]; jAjm is the matrix, all of whose elements are equal to amax , where amax = maxi;j jaij j. sign(A) is called a sign matrix of the vector A, which is diagonal and meets the condition of sign(A)A = jAj. tr(A) is the trace of the matrix A. Notice that the above definitions about jAj; jAj < jB j; jAjm are also suitable for the case when the matrix is replaced by a vector. We often handle the vector case in this paper as well.
211
B. Problem Formulation Consider the following kind of MIMO square nonlinear system (i.e., a system with as many inputs as outputs) given by:
X_ = f (X ) + y1 (t) = h1 (X )
m
j =1
gj (X )uj (t)
111
ym (t) = hm (X )
(1)
where
X 2 Rn plant state vector; U = [u1 ; u2 . . . ; um ] 2 Rm control-input vector; Y = [y1 ; y2 . . . ; ym ] 2 Rm output vector. f (X ) = [f1 (X ); f2 (X ); . . . ; fn (X )]T : Rn ! Rn , is a smooth function vector, fi (X ); i = 1; 2; . . . ; m is a smooth function of X ; gi (X ) = [gi1 (X ); . . . ; gin (X )]T : Rn ! Rn , is a smooth function vector too; hi (1) : Rn ! R; i = 1; 2; . . . ; m is smooth function.
The nonlinear system of (1) is input linear and usually called an affine system. For convenience, the above equation will be rewritten in a condensed form
X_ = f (X ) + G(X )U Y = h(X )
(2) (3)
where
U = [u1 ; u2 ; . . . um ]T Y = [y1 ; y2 ; . . . ym ]T G(X ) = [g1 (X ); g2 (X ); . . . gm (X )] h(X ) = [h1 (X ); h2 (X ); . . . hm (X )]T : To proceed conveniently, some necessary notations and terminology are introduced and recalled in the following. The Lie derivative of a scalar function along a vector is defined as
Lf (X ) =
@ @ @ ;...; f (X ) = @X @x1 @xn n
= i=1
@ f (X ) @xi i
f1 (X )
111
fn (X ) (4)
where X = [x1 ; x2 ; . . . ; xn ]T . The derivative of taken first along f and then along a vector g is
Lg Lf (X ) =
@ (Lf ) g(X ): @X
(5)
If is being differentiated j times along f , the notation Lfj is used with L0f (X ) = (X ). Definition 1 [7]: A MIMO nonlinear system described by the following equations:
X_ = f (X ) +
m
j =1
gj (X )uj (t)
yi = hi (X ) i = 1; . . . ; m has a general (vector) relative degree [r1 ; . . . ; rm ] at X0 if Lg Lkf hi (X ) = 0 for all 1 i; j m; k < ri Lg Lrf 01 hi (X ) 6= 0 for all X in a neighborhood of X0 .
(6)
0 1 and
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The coupling matrix
J (X; t) =
Assumption A3: The plant has a general (vector) relative degree
Lg Lrf Lg Lfr Lg Lfr
01 h1 (X ) 1 1 1 Lg 01 h2 (X ) 1 1 1 Lg 111 111
01 hm (X ) 1 1 1 Lg
Lfr Lfr
01 h1 (X ) 01 h2 (X )
Lfr
01 hm (X )
[r1 ; . . . ; rm ], and its zero dynamics are exponentially stable.
(7)
is nonsingular at X = X0 . Remark 1: The system (6) is said to have a strong (vector) relative degree [r1 ; . . . ; rm ] if Lg Lkf hi (X ) 0 for all 1 i; j m; k < ri 0 1 and Lg Lfr 01 hi (X ) 6= 0 for 8X 2 Rn , and (7) holds. Differentiating the output yi with respect to time t in (6), we have
y_ i
= Lf hi (X ) +
m j =1
Lg hi (X )uj (t)
The Assumption A3 is needed to guarantee state boundedness under state feedback linearization. In order to understand the reason for A3, we should first introduce the concept of zero dynamics of square MIMO systems. There exists a diffeomorphic coordinate transformation T (X ) = (; ), which transforms system (1) or equivalently (6) to the following form: _i = i
1
_2i
111
_ri
(8)
yi(r
) = Lr hi (X ) + f
j =1
Lg Lfr
01 hi (X )uj (t)
where (9)
for Lg Llf hi (X ) = 0; l < ri 0 1 and Lg Lfr 01 hi (X ) 6= 0. In this way, the MIMO system of (1) can be linearized and considered as a collection of MISO systems of (9). Therefore, we may rewrite the plant's input-output equation as Lfr h1 y1(r ) .. .. . . = (r ) ym Lfr hm Y (t) B (X; t) Lg Lfr 01 h1 (X ) 1 1 1 Lg Lfr 01 h1 (X ) .. .. .. + Lg Lfr 01 hm (X ) 1 1 1 Lg Lfr 01 hm (X ) J (X; t) u1 (t) . 2 .. : (10) um (t) U (t) An ideal static-state feedback linearizing control law can be obtained by
U3
= J 01 (0B + V ):
= Lfr hi (X ) +
yi = 1i _ i = qi (; ) i = 1; 2; . . . ; m
where Lf hi (X ) and Lg hi (X ) are the Lie derivatives of hi (X ) with respect to f (X ) and gj (X ), respectively. Since Lg hi (X ) = 0, which implies y_ i = Lf hi (X ), we obtain m
2
= 3i
(11)
Note that, for convenience, the references to X and t are dropped in (11). The term V in (11) is designated below. In order for U 3 to be defined, some assumptions about the plant have to be met. In particular, we need the following assumptions. Assumption A1: The matrix J as defined above is nonsingular, i.e., J 01 exists and has bounded norm for all X 2 Sx ; t 0, where Sx 2 Rn is a compact set that allows state trajectories. This is equivalent to assuming that
p (J ) min > 0
(12)
kJ k2 = 1 (J ) max < 1
(13)
where p (J ) and 1 (J ) are, respectively, the smallest and largest singular values of J . Assumption A2: The reference signal V is bounded.
m j =1
Lg Lfr
01 hi (X )uj (t)
(14) T
= 1i ; . . . ; ri T _ = _r1 ; . . . ; _rm = y1(r ) ; . . . ; ym(r ) = [1 ; 2 ; . . . ; m ]T q (; ) = [q1 (; ); . . . ; qm (; )]T : i
T
The combined form of (14) is rewritten as
_ = B (X; t) + J (X; t)U (t) _ = q (; ):
(15)
If the control law is chosen as in (11), it linearizes the normal part of (15) as
_ = V _ = q (; ):
(16)
It is apparent that is an observed state, and is an unobserved state. Definition 2: The dynamics
_
= q(0; )
(17)
is referred to as the zero dynamics of the system (1). Lemma 1: Assume that q (; ) in (15) or (16) satisfies the Lipschitz condition and the zero dynamics of nonlinear system (1) _ = q (0; ) is exponentially stable and is bounded. Then the state and X 2 Rn are bounded. Proof: Because _ = q (0; ) is exponentially stable when the states move outside a ball j j > B , then there exist some positive constants
1 ; 2 ; 3 ; 4 ; B and a function ( ) to meet the following conditions:
1 j j2
() 2 jj2 d
q (0; ) 0 3 j j2 if j j > B d d
d
where
4 jj
(18) (19) (20)
jj =1 T jj2 =1 T d
d
=1
d
d d
; ;... : d1 d2 dm
(21)
Since we have bounded reference signals V in (16), j j k1 where k1 is a positive constant.
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Using (19), we have
_ = dd _ = dd q(; ) 0 3 jj2 + dd [q(; ) 0 q(0; )] if jj > B: (22) If q (; ) is Lipschitz in , then jq (; ) 0 q (0; )j k2 j j for some
positive k2 . By using this condition, if j j > B , we now have
_ 0 3 jj2 + dd j[q(; ) 0 q(0; )]j 0 3 jj2 + 4 k2 jjjj (23) 0 3 jj2 + 4 k1 k2 jj: _ Therefore, 0 if j j max(B; 4 k1 k2 = 3 ). This ensures
boundedness of and . The system states are therefore bounded. III. FUZZY SYSTEM FORMATION
Fig. 1.
From (10) we can obtain:
( ) = F (X ) + U
E X Y
(24)
where
( ) =1 J 01 (X ) 2 Rm2m (25) 1 0 1 m F (X ) = J (X )B (X ) 2 R : (26) 0 1 E (X ) and F (X ) are assumed to be bounded by the following unknown positive function P1 (X ) and function vector Q1 (X ): 0 kE (X )01k < P1 (X ) (27) 0 jF (X )jm < Q1 (X ): (28) In practical situations where E (X ) and F (X ) are not exactly known, E (X ) and F (X ) may be expressed as (29) E (X ) = E0 (X ) + 1E (X ) F (X ) = F0 (X ) + 1F (X ) (30) where E0 (X ) (nonsingular) and F0 (X ) are known parts, and 1E (X ) and 1F (X ) are unknown parts that contain linearized bias, uncertainty, and disturbance. Remark 2: According to the bounded properties of E (X ) and F (X ) in (25) and (26), the uncertain dynamics 1E (X ) and 1F (X )
Structure of the control system.
= [v1 ; v2 ; . . . ; vm ]T
where V Namely, let
vi
E X
are also bounded by
( )
k1E (X )01k < P2 (X ) j1F (X )j < Q2 (X )
( )
where P2 X and Q2 X are a positive function and function vector of X , respectively. Based on the analysis, (24) can be written as:
( ) = F0 (X ) + U + (t)
(33)
( ) = 1F (X ) 0 1E (X )Y
(34)
E0 X Y where
t
is defined as the system uncertainty vector. The following system without uncertainty is called the nominal system:
( ) = F0 (X ) + U:
E0 X Y
(35)
For the nominal system in (35), let the nominal controller be (see Fig. 1)
U0
= E0 (X )V 0 F0 (X )
= yd(r ) 0 air 01"i(r 01) 0 1 1 1 0 ai1 "_i 0 ai0 "i
(37)
=4 yi 0 yd :
(38)
where
"i
Here, "i is the output tracking error and yd are the desired reference trajectories of yi , while the parameters air 01 ; ai1 ; ai0 in (37) are chosen in such a way that the characteristic polynomial of (37) sr air 01 sr 01 air 02 sr 02 1 1 1 ai1 s ai0 (39) i s
...
1 ( )=
+
+
(36)
+ +
+
is Hurwitz. In this way, we find that the error dynamics of the nominal system has the following form: "i(r ) air 01 "i(r 01) air 02 "i(r 02) 1 1 1 ai1 "i ai0 "i : (40)
+
+
+
_+
=0
Equation (40) implies that the output tracking error, "i , will converge asymptotically to zero. Consider now the system in (33) with uncertain dynamics. In this case, the control input of the closed-loop system can be set in the following form:
U
(31) (32)
is chosen to provide stable tracking.
= U0 + U1
(41)
where U0 is given by (36) for stable control of the nominal system (35), while U1 is a robust control term used to control and deal with the effects of the uncertainty, as shown in Fig. 1. From (33), we have
( ) = F0 (X ) + U0 + U1 + (t):
E0 X Y
From (37), we form V Yd(r) 0 Ar01 "(r01) 0 1 1 1 A1 " 0 A0 "
=
where
_
(42)
(43)
= max fri g i (i = 0; 1; . . . ; r 0 1) Ai = diag a1i ; a2i ; . . . ; am i T i "(i) = "1i ; "2i ; . . . ; "m (i = 1; 2; . . . ; r 0 1) T Yd(r) = yd(r ) ; yd(r ) ; . . . ; yd(r ) : r
Substitute (36) into (42) to obtain
( )( 0 V ) = U1 + (t)
E0 X Y
(44)
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while Y
0 V = Y 0 Y + A 0 " 0 + 1 1 1 + A ": (r )
1
r
d
(r
1)
(45)
0
Then, the error dynamics of the closed-loop system with uncertainty becomes "(r) Ar01 "(r01)
+
+1 1 1+A " = E (X )0 (t)+E (X )0 U : 1
0
0
1
0
1
(46)
Fig. 1 shows the closed-loop structure of the control systems. The function of the Normal Controller U0 and Robust Controller U1 can be seen from (36) and (44). The step called Dynamics-Settings performs the function of (43). The mechanism FBFV forms a fuzzy basis function vector. The Adaptive mechanism is used to adaptively tune the _ weight matrix T so that the closed-loop stability can be guaranteed. The concrete method will be given in the next subsection. For the bounded property of system uncertainty vector t in (34), we have the following Lemma: Lemma 2: If the control input U is designed in such a way that the modulus vector of the control signal is upper bounded by a positive function vector Umax X , that is,
()
( )
jU (t)j
m
( )
< Umax X
(47)
() ( )
Fig. 2. Fuzzy membership functions of the premise part.
as defined in (34). Akl and Cij are linguistic terms characterized by fuzzy membership functions A xk and C i t , respectively.
( )
then the modulus vector of the system uncertainty vector t in (34) is also upper bounded by some positive function vector X , that is
j(t)j = j1F (X ) 0 1E (X )Y j < (X ):
(48)
Proof: Equation (24) can be written in the following form: Y
= E (X )0 F (X ) + E (X )0 U (t): 1
1
(49)
Using (49) in (34), we have
( ) = 1F (X ) 0 1E (X )(E (X )0 F (x) + E (X )0 U (t)) = 1F (X ) 0 1E (X )E (X )0 F (x) 0 1E (X )E (X )0 U (t): 1
1
t
1
1
1
1
2
1
( ) = exp 0 (x 0 )
1
Ai;j
Ai;j
( ) = [ ( ) ( ) . . . ( )]
( )
:
111
:
( ( ))
( ( ))
... ...
M RiM if x1 is AM 1 ; x2 is A2 M then i is CiM i t
( = 1 2 ... ) ( = 1 2 ... )
( ( ))
...x
n
i
(53)
Ai;j
0
=
Ai;j +1
j=c
Ai;j +1
j
0
1
=c j =c 2
3
05 1 = 1; 2; . . . ; M ;
j
i
i
=(
( ) 1 2 ...
( ) = (X; ) =
...
()
where xi i ; ; ; n are the plant state variable, and i t i ; ; ; m are the i-th element of the plant uncertainty vector
)
M
=
( ) = (X ) (i = 1; 2; . . . ; m) (54) where is the value of variable (t) of the center point at which ( (t)) achieves its maximum value, i.e., ^ = [ ; ; . . . ] 2 R is the weight vector of the fuzzy basis expansion vector (X ); and the j -th fuzzy basis expansion in (X ) = [ (X ); (X ); . . . ; (X )] is defined as (x ) (X ) = (55) (x ) _
i X
_
i
j =1
ij
i
C
i1
_T
j X ij
i
i
i
i2
i
M
iM
1
2
j
(52)
= 1; 2; . . . ; n
where c1 ; c2 ; c3 are some predetermined constants. _ Given an input vector X x10 ; x20 ; ; xn0 , the output i X can be inferred by taking the weighted average of the ji ; j ; ; ;M
T
M
n i=1
is AM n
= 1; 2; . . . ; m
( )
Ai;j Ai;j Ai;j +1 : < c3
max
()
2
Ai;j
2
( )
j
Remark 3: It follows from Lemma 2 that the bounded property of the system uncertainty, t , depends on the system structure and the form of the controller. It will be shown in the next section that the control signal satisfies the condition in (47), and therefore, the bounded condition of the system uncertainty in (48) always holds. In the following, the fuzzy basis function network is adopted to learn X ; m X T , the upper bound of the 1 X ; 2 X ; system uncertainty vector. Consider a fuzzy basis expansion system of i X , which consists of M fuzzy rules Ri1 if x1 is A11 ; x2 is A12 ; xn is A1n 1 then i is Ci1 i t xn is A2n Ri2 if x1 is A21 ; x2 is A22 ; 2 then i is Ci2 i t
:
i
A xi
m
2
i=1
1 2 ... )
m
1
Mi :
n
( = 1 2 ... ) = ( = 1 2 ...
( ) + P (X )P (X )Q (X )+ P (X )P (X )U (X ) (51) = (X ): < Q2 X
2
where A xi is the membership function of xi in fuzzy set Aji ; Ai;j is the center of A xi ; Ai;j is the width of A xi . The following conditions have to be met:
1
1
1
Equation (52) is a T-S fuzzy model of zero order, in fact, and can be solved as follows. We divide the value domain of xi into Mi i ; ; ; n equal individuals, as shown in Fig. 2. In Fig. 2, Aji j ; ; ; Mi ; i ; ; ; n are the premise fuzzy subset in (52). Mi can be chosen according to the range scale and uncertainty degree of variable xi . In this paper, the following Gaussian membership function is used:
( )
j(t)j j1F (X )j + j1E (X )E (X )0 F (x)j + j1E (X )E (X )0 U (t)j j1F (X )j + k1E (X )kk1E (X )0 kjF (x)j + k1E (X )kkE (X )0 kjU(t)j
= M M ...M =
M
(50) Therefore, we have
( ( ))
n
M j =1
i
A
n i=1
A
i
where M is the number of fuzzy rules. Therefore, no training of the premise fuzzy subsets in (52) takes place. Only the consequences should be adaptively identified.
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B, VOL. 30 , NO. 1, FEBRUARY 2000
For further analysis, the following assumptions are made: Assumption A4: Given an arbitrary small positive constant vector "~3 and a continuous function vector (X ) defined in (51) in a compact set 6, there exists an optimal weight matrix 3 such that the output vector of the optimal fuzzy network satisfies
3T 3 max X 26 j"~(X )j = max X 26 j (X ) 0 (X )j < "~
(56)
"~(X ) = 3T (X ) 0 (X )
(57)
with
3 ]T where 3T = [13 ; 23 ; . . . ; m
with ` >
215
0 and arbitrary positive initial values matrix of
_ M 2m 2 R
then the output tracking errors vector " converges asymptotically to the zero vector. Proof: Considering following Lyapunov function
@ = 12 S T S + 21 `01 tr[~T ~]
where
_
~ = 3 0 ;
2 Rm2M .
Lemma 3: If the modulus vector of the system uncertainty and its upper bound satisfy the following relationship in the compact set 6
(X ) 0 j(t)j = "0 > "~3
(59)
Proof: From (58), we can get
(X ) > "~3 + j(t)j
"~(X ) = 3T (X ) 0 (X ) < 3T (X ) 0 (~ "3 + j(t)j) 3 T 3 (60) = (X ) 0 j(t)j 0 "~ : 3T (X ) 0 j(t)j > "~ + "~3 > O
= 0`01 tr[3T ^_ ] + `01 tr[^T ^_ ] = 0`01 tr 3T ` (X )jE T C T j 3
= 3E + 9E0 (X )01 (t) + 9E0 (X )01 U1
3=
I O
O I
111 111
_
+ tr
111 0Ar02 0A0 0A1 0A2 1 1 1 0Ar02 9 = [O O O 1 1 1 I ]T (63) mr 2 1 mr 2 mr mr 2 m m 2 m m 2 m E 2R ; 32R ; 92R ; O2R ; I 2R . For the design of the adaptive robust controller using the FBFV and analysis of error convergence of the closed-loop system, we have the following theorem. Theorem: Consider the error dynamics in (62). If (a) the robust controller U1 is designed as follows:
_T
U1 = E0 (X)(C 9)01 [0C 3E 0 sign(S T )kC 9E0 (X )01 k3
__
= ` (X )jE T C T j
C 9E001(X )
(65)
_T
(X )jE T C T j C 9E001(X )
:
(71)
@_ = E T C T C 9E0 (X )01(t) 0 tr 3T (X )jE T C T j 3 C 9E001(X ) = E T C T C 9E0 (X )01(t) 0 jE T C T j C 9E001(x) 3T (X ) jE T C T j C 9E001(x) (j(t)j 0 3T (X )) < 0:
(72)
Remark 4: It is not necessary for the weight matrix of the fuzzy network to converge to its optimal values, and the values of the weight matrix are adaptively adjusted until the variable S converges to the zero vector. The weight matrix will then become a constant matrix to guarantee that the output tracking error vector converges asymptotically to the zero vector after S = CE = 0. V. SIMULATION STUDY A simulation model is given by
X_
8(X)
(64) where the vector S = CE = [s1 ; . . . ; sm ]T ; S 2 Rm ; C 2 Rm2mr ; (b) the matrix C should be chosen such that the polynomial Si is Hurwitz about "i (i = 1; 2; . . . ; m); _ (c) the matrix in (64) is updated by the following adaptive mechanism:
C 9E001(X )
Therefore, we have
(62)
O O
_
+ `01 tr ` (X )jE T C T j C 9E001(X ) = 0tr 3T (X )jE T C T j C 9E001(X )
Let E = ["; "; _ . . . ; "r01 ]T , then (46) can be written as the following state space form:
O O
(69)
_
IV. ROBUST CONTROLLER DESIGN
where
@_ = S T S_ + `01 tr[~T ~_ ]:
`01 tr[~T ~_ ] = 0`01 tr[~T ^] = 0`01 trf[3 0 ^]T ^g
(61)
where O means zero vector. The objective of this paper is to design a robust adaptive controller, U1 , using the fuzzy basis function network in (55) so that the closed-loop system has strong robustness and the output tracking error is guaranteed to converge asymptotically to zero.
E_
(68)
The first term in (69) is shown in (70) at the bottom of the next page. The second term in (69) is
and (60) can be deduced from (57)
Therefore,
__
Then
3T (X ) 0 j(t)j > 0:
(67)
~_ = 0 :
(58)
then the following relation holds:
(66)
= f (X ) + G(X )U
(73)
= h(X )
(74)
Y where
f (X ) = h(X ) =
x4 0 x1 0 x2 x3 1 + x3 ; G(X ) = x22 x3 x3 0 x3 0 x2 u1 2x4 ; U = u2 :
0 2 2 0
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Calculating the relative degree, we obtain
= 1; r2 = 2: Taking a transform that 1 = x2 0 x3 ; 2 = x4 ; 3 = x3 ; = x1 ,
PARAMETERS
r1
we get
_ x22 0 x2 x3 01 0 + 2x3 4 = 2x22 () B (X; t) J (X; t) _ = 2 0 :
y1 y2 Y t
u1 u2 U t
TABLE I AND INITIAL VALUES SETTINGS
(75)
()
(76)
Thus, the zero dynamics of the given system is
_ =0
:
(77)
Obviously, the zero dynamics of the given system is exponentially stable at the equilibrium point. From (25), and (26), we obtain
( ) = J 01 (X ) = 0:05x13 0:025 0x22 + x2 x3 F (X ) = E (X )B (X ) = 2 0:5x2x3 0 0:5x2 x23 + 0:5x22
E X
If we take E0
3 = 001 0x:25
then we have
;
F0
= 0:5x22x300x20:5x2 x23 2
1E = 0:50x3 00x3 1F = 0x:25xx322 : = ["1 ; "2 ; "_2], the error
By taking E closed-loop system is
state equation of the
_ = 3E + 9E0 (X )01(t) + 9E0 (X )01U1
E
where
3=
0 0 0 0 1 0 0a20 a21
0a10
:
;
Membership functions of variable x
;x ;x ;x
.
(78)
1 0 9= 0 0 0 1
and the parameters aij can be chosen by (39). Table I gives all relative parameters in the simulation study, which can be chosen subjectively according to the above method. Fig. 3 shows the membership functions of variable x1 ; x2 ; x3 ; x4 , which consist of small, medium, and large. Therefore, the number of . the fuzzy rules should be 4 Figs. 4 and 5 give the response curves of the output y1 t ; y2 t respectively. And it can be observed that the nonlinear control system
3 = 81
Fig. 3.
() ()
Fig. 4. Response curve of the output y (t).
will reach the stable state after 15 s. The greater ` is, the faster the response is. But too great an ` may cause a small vibrating motion in the meantime.
_ = E T C T C E_ = E T C T C [3E + 9E0 (X )01(t) + 9E0 (X )01U1 ] = E T C T C 3E + E T C T C 9E0 (X )01(t) + E T C T C 9E0 (X )01U1 = E T C T C 3E + E T C T C 9E0 (X )01(t) + E T C T C 9E0 (X )01E0 (X )(C 9)01[0C 3E _T 0 sign(S T )kC 9E0 (X )01k 8(X )] _T = E T C T C 9E0 (X )01(t) 0 E T C T sign(S T )kC 9E0 (X )01k3 8(X ) _T = E T C T C 9E0 (X )01(t) 0 jE T C T jkC 9E0 (X )01k 8(X ):
ST S
(70)
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217
On the Optimal Design of Fuzzy Neural Networks with Robust Learning for Function Approximation Hung-Hsu Tsai and Pao-Ta Yu
Fig. 5.
Abstract—A novel robust learning algorithm for optimizing fuzzy neural networks is proposed to address two important issues: how to reduce the outlier effects and how to optimize fuzzy neural networks, in the function approximation. This algorithm is able to reduce the outlier effects by cooperating with a conventional robust approach, and then to optimize fuzzy neural networks by determining the optimal learning rates which can minimize the next-step mean error at each iteration of our algorithm.
Response curve of the output y (t).
Index Terms—Function approximation, fuzzy neural networks, optimization, robust learning algorithm.
VI. CONCLUSION An adaptive controller using a fuzzy basis function vector (FBFV) is proposed in this paper. Our theoretical analysis demonstrates that the fuzzy basis function expansion vector can be used to learn the MIMO system uncertainty bounds in the Lyapunov sense, and a FBFV-based adaptive hybrid controller can be designed to eliminate the effects of dynamical uncertainties and guarantee that the output tracking errors converge asymptotically to zero. In addition, it has a better robustness with respect to unstructured uncertainty. Computer simulation shows that this method is strong, robust, fast convergence, and easy to design and use. ACKNOWLEDGMENT The authors would like to thank the reviewers for their constructive comments. REFERENCES [1] R. Yager and D. P. Filev, Essentials of Fuzzy Modeling and Control. New York: Wiley, 1994. [2] M. Zhihong, X. H. Yu, and Q. P. Ha, “Adaptive control using fuzzy basis function expansion for SISO linearizable nonlinear systems,” in Proc. 2nd ASCC, Seoul, Korea, 1997, pp. 695–698. [3] J. T. Spooner and K. M. Passino, “Stable adaptive control using fuzzy systems and neural networks,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 339–359, 1996. [4] G. A. Rovithakis and M. A. Christodoulou, “Adaptive control of unknown plants using dynamical neural networks,” IEEE Trans. Syst., Man, Cybern., vol. 24, pp. 400–412, 1994. [5] C. Wen and Z. Huaguang, “Input/output linearization for nonlinear systems with uncertainties and disturbances using TDC,” Cybern. Systems, vol. 28, pp. 625–634, 1997. [6] D. Changguo, Linear Algebra in Control Systems, China: Southeastern Univ. Press, 1993. (in Chinese). [7] A. Isidori, Nonlinear Control Systems: An Introduction, Berlin, Germany: Springer-Verlag, 1989. [8] S. S. Sastry and A. Isidori, “Adaptive control of linearizable systems,” IEEE Trans. Automat. Contr., vol. 34, pp. 1123–1131, 1989. [9] Y. S. Lu and J. S. Chen, “A self-organizing fuzzy sliding-mode controller design for a class of nonlinear serve systems,” IEEE Trans. Ind. Electron., vol. 41, pp. 492–502, 1994. [10] C. Y. Su and Y. Stepanenko, “Adaptive controller of a class of nonlinear systems with fuzzy logic,” IEEE Trans. Fuzzy Syst., vol. 2, pp. 285–294, 1994. [11] D. Driakov, Advances in Fuzzy Control. Berlin, Germany: SpringerVerlag, 1998, pp. 155–188. Physica. [12] Z. Bien and W. Yu, “Extracting core information from inconsistent fuzzy control rules,” Fuzzy Sets Syst., vol. 71, pp. 95–111, 1995. [13] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Sets Syst., vol. 45, pp. 135–156, 1992. [14] K. Kiriakidis et al., “Quadratic stability analysis of the Takagi-Sugeno fuzzy model,” Fuzzy Sets Syst., vol. 98, pp. 1–14, 1998. [15] T. A. Johansen, “Fuzzy model based control: Stability, robustness, and performance issues,” IEEE Trans. Fuzzy Syst., vol. 2, pp. 221–234, 1994.
I. INTRODUCTION Function approximation has been successfully applied in diverse applications such as signal processing, image restoration, pattern recognition, control system, system identification, etc. [1]–[4]. The main task of the function approximation is to approximate a desired function or an input-output relation from a set of input-output patterns. In several applications, the most desired functions are highly nonlinear. The highly nonlinear desired functions are difficult to be specified or modeled by the precisely mathematical formulas. Hence, how to find approximated functions for the highly nonlinear desired functions is an crucial issue in many applications. When imprecise or erroneous training patterns are collected and employed during the approximation process, the learning mapping may oscillate sharply such that the gross errors, caused by these erroneous training patterns, will be enlarged. Chen and Jain have proposed a robust back-propagation learning algorithm to resist the noise effects (or alternatively called outlier effects in this correspondence) such as to reduce the gross errors [5]. However, Chen and Jain’s algorithm still suffers from the problem of slow convergence. In [6], Wang et al. have exploited the fuzzy-neural networks (FNN’s) and B-spline membership functions (constructed by control points) to realize the robust learning with the outlier rejection for the function approximation. However, the training complexity of Wang et al.’s algorithm increases greatly as the number of control points is getting large due to the dimension of the matrix proportional to the number of control points. Another limitation of Wang et al.’s algorithm is to collect a set of empirical data in advance for the construction of B-spline membership functions. This leads to the difficulty, in real applications, collecting various sets of empirical data in advance for the various problems. Furthermore, Wang et al.’s algorithm suffers from the problem that the analysis of the training (learning) behavior of the FNN’s is extremely difficult due to the use of a complicated membership function—B-spline functions. Therefore, in this correspondence we analyze the training behavior of the FNN’s with common membership functions (such as triangular shapes) to circumvent the difficulty that Wang et al.’s algorithm suffers from. Note that most FNN’s we concentrate to investigate are trained by the conventional learning algorithms based on the gradient decent method [4], Manuscript received January 24, 1999; revised October 31, 1999. This work was supported in part by the National Science Council, R.O.C., under Grant NSC88-2213-E-194-011. This paper was recommended by Associate Editor S. Lakshmivarahan. H.-H. Tsai is with the Department of Information Management, Nan Hua University, Chiayi, Taiwan, R.O.C. P.-T. Yu is with the Department of Computer Science and Information Engineering, Chung Cheng University, Chiayi, Taiwan, R.O.C. Publisher Item Identifier S 1083-4419(00)01401-1.
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