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European Journal of Control (2009)5:534–544 # 2009 EUCA DOI:10.3166/EJC.15.534–544
A New Decentralized Fuzzy Model Reference Adaptive Controller for a Class of Large-scale Nonaffine Nonlinear Systems Reza Ghasemi, Mohammad Bagher Menhaj and Ahmad Afshar Department of Electrical Engineering, AmirKabir University of Technology, Tehran, Iran
This paper proposes a new method to design a decentralized fuzzy adaptive controller (FAC) for a class of large-scale nonaffine nonlinear systems in which functions of the systems and interactions are unknown. Comparing to previous paper which mainly concentrates on affine large-scale system (LSS), the proposed method is on nonaffine nonlinear LSS. The stability of the closedloop system is guaranteed based on Lyapunov theory. The proposed controller is robust against uncertainty and external disturbance. To show the effectiveness of the proposed method, an illustrative example is given. The simulation results are very promising. Keywords: Lyapunov Stability, Robust Adaptive Control, Non-affine Nonlinear Systems, Fuzzy Systems, Large Scale System
1. Introduction In the recent years, control design for large-scale systems (LSSs) and effort to extend it have attracted much attention. Research in a control of LSSs is motivated by many emerging applications that employ novel actuation devices for active control of industrial automation, cooperating robotic systems, power systems, and aerospace processes. Centralized control for the LSSs is usually impractical due to the requirement of a large Correspondence to: R. Ghasemi, E-mail: [email protected] and [email protected] E-mail: [email protected] E-mail: [email protected]
amount of information exchanges between subsystems and the lack of computing capacity [14]. Nowadays, fuzzy adaptive controller (FAC) has attracted many researchers to develope appropriate controllers for nonlinear systems especially for LSSs because of the following reasons. 1. Due to its tunable structure, the performance of the FAC is superior to that of the fuzzy controller. 2. Instead of using adaptive controller, FAC can use the knowledge of the experts in the controller. In the recent years, FAC has been fully studied as follows: 1. The Takagi–Sugeno (TS) fuzzy systems have been used to model nonlinear systems and then TS-based controllers have been designed with guaranteed stability [5, 6]. In ref. [9], modeling of affine nonlinear system is discussed and subsequently a stable TSbased controller is developed. Designing of the sliding mode FAC for a class of multivariable TS fuzzy systems are presented in ref. [2]. In refs. [7, 20], the nonaffine nonlinear function are first approximated by the TS fuzzy systems, and then stable TS fuzzy controller and observer are designed for the obtained model. In these papers, modeling and controller have been designed simply, but the systems must be linearizable around some operating points. 2. The linguistic fuzzy systems have been used to design controllers for nonlinear systems. Received 16 July 2008; Accepted 21 December 2009 Recommended by E. Mosca, EF. Camacho
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Refs. [12, 24, 26, 29] have considered linguistic fuzzy systems to design stable adaptive controller for affine systems based on feedback linearization and furthermore in [26] the zero dynamic has been considered to be stable. Stable FAC based on sliding mode is designed for affine systems in ref. [16]. Designing of the FAC for affine chaotic systems are presented in refs. [1, 23]. How to design stable FAC and linear observer for class of affine nonlinear systems is presented in refs. [8, 21, 25, 33]. Fuzzy adaptive sliding mode controller is presented for class of affine nonlinear time delay systems in refs. [3, 13, 32]. The output feedback FAC for class of affine nonlinear MIMO systems is suggested in ref. [30]. The main incompetency of these papers are those restricted conditions on their functions. References [17, 18] are involved in stable FAC for class of nonaffine nonlinear systems. The inefficiency of these papers is bad performance of the controller when the controller has not been adjusted. Stable adaptive controller for class of linear LSS is proposed in refs. [10, 19, 22, 31]. Ref. [4] deals with designing FAC based on sliding mode for class of large-scale affine nonlinear systems. Ref. [34] presents decentralized sliding mode fuzzy adaptive tracking for a class of affine nonlinear systems in LSSs. Ref. [28] designed FAC for a class of affine nonlinear time-delayed systems. These papers have many restricted conditions. Compared to previous paper, which mainly concentrates on affine systems LSS, the proposed method is a nonaffine nonlinear LSS. In this paper, we propose a new method to design a decentralized robust adaptive controller based on fuzzy systems for a class of large-scale nonaffine nonlinear systems with guaranteed stability. The capability of stabilizing closedloop system, robustness against external disturbance and uncertainty and also convergence of the tracking error and boundedness of the estimation errors are advantages of the proposed controller. The rest of the paper is organized as follows. Section 2 gives problem statement. General concept of the fuzzy systems is formulated in Section 3. The designing of FAC is proposed in Section 4. Section 5 shows simulation results of the proposed controller and Section 6 concludes the paper.
where xi ¼ ½xi;l ; . . . ; xi;ni T 2 Rni is the state vector of the system which is assumed available for measurement, ui 2 R is the control input, yi 2 R is the system output, fi ðxi ; ui Þ is an unknown smooth nonlinear function, mi ðx1 ; x2 ; . . . ; xN Þ is an unknown nonlinear interconnection term, and di ðtÞ is bounded disturbance. The control objective is to design an adaptive fuzzy controller for system (1) such that the system output yi ðtÞ follows a desired trajectory yd ðtÞ while all signals in the closed-loop system remain bounded. In this paper, we will make the following assumptions concerning the system (1) and the desired trajectory yd ðtÞ. Assumption 1: Without loss of generality, it is assumed that the nonzero function fu ðxi ; ui Þ ¼ @fðxi ; ui Þ=@ui satisfies the following condition: fu ðxi ; ui Þ fmin > 0
for all
ðxi ; ui Þ 2 Rni R ð2Þ
dfu ðxi ; ui Þ fdm dt fdm 2 R is known and constant. Assumption 2: The desired trajectory yd ðtÞ and its time ðjÞ derivatives yd ðtÞ; j ¼ 1; 2; . . . ; ni , are all smooth and bounded. Assumption 3: The interconnection term satisfies the following: jmi ðx1 ; x2 ; . . . ; xN Þj i ðkxi kÞ
ð3Þ
where i ðkxi kÞ is an unknown nonlinear function. Assumption 4: The disturbance in the above equation is bounded by: di ðtÞ dmax
ð4Þ
Define the tracking error vector as: ei ¼ ½ei;1 ; ei;2 ; . . . ; ei;ni T 2 Rni
ð5Þ
where
2. Problem Statement Consider the following large-scale nonaffine nonlinear system. 8 l ¼ 1; 2; :::; ni 1 < x_ i;l ¼ xi;lþ1 x_ 1;ni ¼ fi ðxi ; ui Þ þ mi ðx1 ; x2 ; . . . ; xN Þ þ di ðtÞ : yi ¼ xi;1
ei;1 ¼ yd yi
ð6Þ
Taking the nth i derivative of both sides of the equation (6) we have
i ¼ 1; 2; . . . ; N
ð1Þ
536 ðn Þ
R. Ghasemi et al. ðn Þ
ðni Þ
ei;1i ¼ yd i yi ðn Þ
¼ yd i fi ðxi ; ui Þ mi ðx1 ; x2 ; . . . ; xN Þ di ðtÞ ð7Þ Use equation (5) to rewrite the above equation as: ðn Þ e_ i ¼ Ai0 ei þ bi fyd i
fi ðxi ; ui Þ mi ðx1 ; x2 ; . . . ; xN Þ di ðtÞg where Ai0 and bi 2 0 1 60 0 6 6. . . . Ai0 ¼6 6. . 6 40 0
are defined below. 3 0 0 1 07 7 .. . . .. 7 ni ni . . .7 72R 7 0 15
ð8Þ
Invoking the implicit function theorem, it is obvious that the nonlinear algebraic equation fi ðxi ; ui Þ vi ¼ 0 is locally soluble for the input ui for an arbitrary ðxi ; vi Þ. Thus, there exists some ideal controller ui ðxi ; vi Þ satisfying the following equality for a given ðxi ; vi Þ 2 Rni R: fi ðxi ; ui Þ vi ¼ 0
ð14Þ
As a result of the mean value theorem, there exists a constant in the range of 0 < < 1, such that the nonlinear function fi ðxi ; ui Þ can be expressed around ui as: and
0 0 0 0 2 3 0 607 6 7 ni 7 bi ¼6 6 .. 7 2 R 4.5
ð9Þ
fi ðxi ; ui Þ ¼ fi ðxi ; ui Þ þ ðui ui Þfiu ¼ fi ðxi ; ui Þ þ eui fiu
ð15Þ
where fiu ¼ @fðxi ; ui Þ=@ui jui ¼ui and ui ¼ ui þ ð1 Þui . Substituting equation (15) into the error equation (12) and using (14), we get
1 Consider the vector ki ¼ ½ki;1 ; ki;2 ; . . . ; ki;ni T be coefficients of LðsÞ ¼ sni þ ki;ni sni 1 þ . . . þ ki;1 and chosen so that the roots of this polynomial are located in the open left-half plane. This makes the matrix Ai ¼ Ai0 bi kTi be Hurwitz. Thus, for any given positive definite symmetric matrix Qi , there exists a unique positive definite symmetric solution Pi for the following Lyapunov equation: ATi Pi þ Pi Ai ¼ Qi
ð10Þ
e_ i ¼ Ai ei bi feui fiu þ mi ðx1 ; x2 ; . . . ; xN Þ þ di ðtÞ þ tanhðbTi Pi ei ="Þ þ v0i g
ð16Þ
However, the implicit function theory only guarantees the existence of the ideal controller ui ðxi ; vi Þ for system (14), and does not recommend a technique for constructing solution even if the dynamics of the system are well known. In the following, a fuzzy system and classic controller will be used to obtain the unknown ideal controller.
Let vi be defined as ðn Þ
vi ¼ yd i þ kTi ei þ tanhðbTi Pi ei ="Þ þ v0i
ð11Þ
where tanhð:Þ is the hyperbolic tangent function, is a large positive constant, and " is a small positive constant. By adding and subtracting the term kTi ei þ tanhðbTi Pi ei ="Þ þ v0i from the right-hand side of equation (8), we obtain e_ i ¼ Ai ei bi ffi ðxi ; ui Þ vi þ mi ðx1 ; x2 ; . . . ; xN Þ þ di ðtÞ þ tanhðbTi Pi ei ="Þ þ v0i g ð12Þ Using assumption (1), equation (11) and the signal vi which is not explicitly dependent on the control input ui , the following inequality is satisfied: @ðfi ðxi ; ui Þ vi Þ @fi ðxi ; ui Þ ¼ >0 @ui @ui
ð13Þ
3. Fuzzy Systems Fig. 1 shows the basic configuration of the fuzzy systems considered in this paper. Here, we consider a multi-input, single-output fuzzy systems: x 2 U Rn ! y 2 V R. Consider that a multi-output system can be separated into a group of single-output systems. The fuzzifier performs a mapping from a crisp input vector x ¼ ½x1 ; x2 ; :::; xn T to a fuzzy set, where the label of the fuzzy set are such as ‘‘small’’, ‘‘medium’’, ‘‘large’’, etc. The fuzzy rule base is consisted of a collection of fuzzy IF–THEN rules. Assume that there are M rules, and the lth rule is Rl ðuÞ: ifðx1 is Al1 . . . xn is Al1 Þthen ðy is Bl Þl ¼ 1; 2; . . . ; M
ð17Þ
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Fig. 1. Configuration of fuzzy system.
where x ¼ ½x1 ; x2 ; . . . ; xn T and y are the crisp input and output of the fuzzy system, respectively. Alj and Bl are fuzzy membership function in Uj and V, respectively. The fuzzy inference performs a mapping from fuzzy sets in U to fuzzy sets in V, based on the fuzzy IF– THEN rules in the fuzzy rule base. The defuzzifier maps fuzzy sets in V to a crisp value in V. The configuration of Fig. 1 represents a general framework of fuzzy systems, because many different choices are allowed for each block in Fig. 1, and various combinations of these choices will construct different fuzzy systems [26]. Here, we use the sum–product inference and the center–average defuzzifier. Therefore, the fuzzy system output can be expressed as M P
yðxÞ ¼
y
n Q l
i¼1 l¼1 M Q n P l¼1 i¼1
Ali ðxi Þ
ð18Þ
Ali ðxi Þ
where Ali ðxi Þ is the membership degree of the input xi to fuzzy set Alj and yl is the point at which the membership function of fuzzy set Bl achieves its maximum value. The fuzzy systems in the form of (18) are proven in [27] to be a universal approximator if their parameters are properly chosen. Theorem 1 [26]: Suppose fðxÞ is a continuous function on a compact set U. Then, for any " > 0, there exists a fuzzy system like (18) satisfying: supjfðxÞ yðxÞj "
ð19Þ
x2U
where ¼ y1 y2 . . . y M is all consequent parameters,
n Q
wi ðxÞ ¼
i¼1 M Q n P l¼1 i¼1
Ali ðxi Þ
ð20Þ a vector grouping and wðxÞ ¼ ½w1 ðxÞ
ð21Þ
Ali ðxi Þ
The fuzzy system (18) is assumed to be well defined so M Q n P Ali ðxi Þ 6¼ 0 for all x 2 U. that l¼1 i¼1
4. FAC Design In Section 2, it has been shown that there exists an ideal control for achieving control objectives. In this section, we show how to develop a fuzzy system to adaptively approximate the unknown ideal controller. The ideal controller can be represented as: ui ¼ fi ðzÞ þ upid þ "iu
ð22Þ
where fi ðzÞ ¼ i1 wi1 ðzÞ, and i1 and wi1 ðzÞ are consequent parameters and a set of fuzzy basis functions, respectively. "iu is an approximation error that satisfies j"iu j "max and "max > 0. The upid is the primary controller that developed properly to initially control the underlying system and parameters i1 are determined through the following optimization. i1 ¼ arg min supTi1 wi1 ðzÞ fi ðzÞ i1
The output given by (18) can be rewritten in the following compact form: yðxÞ ¼ wðxÞT
w2 ðxÞ . . . wM ðxÞT is a set of fuzzy basis functions defined as:
ð23Þ
Denote the estimate of i1 as i1 and uirob as a robust controller to compensate approximation error, uncertainties, disturbance and interconnection term to rewrite the controller given in (22) as: ui ¼ Ti1 wi1 ðzÞ þ upid þ uirob
ð24Þ
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In which uirob is defined below. T b P i ei i ðT wi2 ðxÞ þ fmin uicom þ fmin uir þ v^0i Þ uirob ¼ fmin bTi Pi ei i2 ð25Þ In the above, Ti1 wi1 ðzÞ approximates the ideal controller, Ti2 wi 2 ðxÞ tries to compensate the interconnection term, uicom compensates for approximation errors and uncertainties, uir is designed to compensate for bounded external disturbances, and v^0i is estimation of v0i . Define error vector ~i1 ¼ i1 i1 and use (24) and (25) to rewrite the error equation (16) as: e_ i ¼ Ai ei bi fð~Ti1 wi1 ðzÞ þ uirob "iu Þfiu þ mi ðx1 ; x2 ; . . . ; xN Þ þ di ðtÞ þ
tanhðbTi Pi ei ="Þ
þ
v0i
ð26Þ
g
Consider the following update laws.
i
1 T f_u ei Qi ei þ 2i eTi Pi ei fui fui 1 2 min ðQi Þfmin kei k2 þmin ðPi Þfd m kei k2 fui ð32Þ Use (29) and (32) to have the following which completes the proof. . f_u 1 T ei Qi ei þ 2i eTi Pi ei þ fui bTi Pi ei 0 fui fui
ð27Þ
ð33Þ Q.E.D
Lemma 2: Based on lemma 1and equation (10), the following inequality holds. svdmax ðAi Þ
_i1 ¼ 1 bTi Pi ei wi1 ðzi Þ _i2 ¼ 2 bTi Pi ei wi2 ðxi Þ u_ ir ¼ uir bTi Pi ei u_ icom ¼ uicom bTi Pi ei v^0 ¼ v^0 bT Pi ei i
After some algebraic manipulations, the following inequality is obtained.
fdm min ðPi Þ 2fmin max ðPi Þ
ð34Þ
Proof: Using equation (10) and after some algebraic manipulations, the following inequality is obtained. ð35Þ kQi k ATi Pi þ kPi Ai k ¼ 2kPi Ai k
i
where 1 ¼ T1 > 0; 2 ¼ T2 > 0; uir > 0; uicom > 0; v^0i > 0 are constant parameters. In following equation, maxð:Þ and svdmax ð:Þ are maximum eigenvalue and maximum singular value decomposition, respectively. Lemma 1: The. following inequality max ðQi Þ fdm fmin max ðPi Þ.
holds
1 T f_u ei Qi ei þ 2i eTi Pi ei þ bTi Pi ei 0 fui fui fui
if
ð28Þ
Proof: From assumption (1) and > 0, we can have the following inequality has been satisfied. . fui bTi Pi ei 0 ð29Þ From assumption (1) and the above lemma, it is obvious that ðmin ðQi Þfmin þ min ðPi Þfdm Þ 0
ð30Þ
This in turn leads to the following inequality. 1 min ðQi Þfmin þ min ðPi Þfdm Þkei k2 0 f 2ui
ð31Þ
Using the above equation, we get kQi k 2kPi kkAi k ¼ 2max ðPi Þsvdmax ðAi Þ
ð36Þ
Use (30) and (36) to have the following which completes the proof. svdmax ðAi Þ
fdm min ðPi Þ 2fmin max ðPi Þ
ð37Þ Q.E.D
Theorem 2: Consider the error dynamical system given in (26) for the LSS (1) satisfying assumption (1), interconnection term satisfying assumption (3), the external disturbances satisfying assumption (4), and a desired trajectory satisfying assumption (2), then the controller structure given in (24), (25) with adaptation laws (27) makes the tracking error converge asymptotically to a neighborhood of origin and all signals in the closed-loop system be bounded. Proof: To prove convergence of the tracking error and boundedness of parameters error, we must apply the tracking error and the parameters error in Lyapunov function. Thus, the parameters such as ei , ~i1 , ~i2 , u~ir , u~icom , and v~0i are applied in Lyapunov function. Consider the following Lyapunov function.
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Decentralized Fuzzy MRAC
V¼
N X 1
1 T ~T 1 ~ ~ ei Pi ei þ ~Ti1 1 1 i1 þ i 2 2 i 2 2 f u i i¼1 ! 2 2 u~ir u~icom v~i 02 þ þ þ uir uicom v^0i ð38Þ
where ~i1 ¼ i1 i1 , ~i2 ¼ i2 i2 , u~ir ¼ uir dmax = fmin , u~icom ¼ uicom "max max =fmin , and v~0i ¼ v^0i v0i . The time derivative of the Lyapunov function becomes. V_ ¼
f_u 1 T 1 e_i Pi ei þ eTi Pi e_i þ 2i eTi Pi ei fui 2 fui fui
N X 1 i¼1
~T 1 _ _ þ ~Ti1 1 1 i1 þ i2 2 i2 þ
!
u~ir u_ ir u~icom u_ icom v~0i v^_ 0i þ þ uir uicom v^0i
Use (26), to rewrite above equation as: f_u 1 T T ei ðAi Pi þ Pi Ai Þei þ 2i eTi Pi ei 2 fui fui i¼1 ! þ bTi Pi ei tanhðbTi Pi ei ="Þ þ fui 1 T þ b Pi ei ðv0i þ ð~Ti1 wi1 ðzÞ þ uirob "iu Þfui fui i þ mi ðx1 ; x2 ; . . . ; xN Þ þ di ðtÞÞ þ
N X 1
~T 1 _ _ þ ~Ti1 1 1 i1 þ i2 2 i2 þ þ
i ðkxi kÞ ¼ T 2 wi2 ðxi Þ þ i
u~ir u_ ir uir
u~icom u_ icom v~0i v^_ 0i þ uicom v^0i ð40Þ
Use bTi Pi ei tanhðbTi Pi ei ="Þ ¼ bTi Pi ei and (10), to rewrite (40) as follows: ! N _u X T 1 1 f i T T V_ ¼ ei Qi ei 2 ei Pi ei bi Pi ei þ fui 2 fui fui i¼1 1 T þ b Pi ei v0i þ ð~Ti1 wi1 ðzÞ þ uirob "iu Þfui fui i þ mi ðx1 ; x2 ; . . . ; xN Þ þ di ðtÞ þ u~ir u_ ir u~icom u_ icom v~0i v^_ 0i ~T 1 _ _ þ þ þ ~Ti1 1 1 i1 þ i2 2 i2 þ uir uicom v^0i ð41Þ
ð42Þ
where i is approximation error and satisfying ji j i max . Using assumption (1) yields 1=fui 1=fmin and by assumptions (3), (4) and equations (42), to rewrite (41) as follow. V_
ð39Þ
V_ ¼
Function i ðkxi kÞ is smooth and it can be approximated with the fuzzy system as the following equation.
! 1 T T f_ui T ei Qi ei 2 ei Pi ei b Pi ei þ fui 2 fui fui i i¼1 T T b P i ei b Pi ei i i 0 T T ~ þ vi bi Pi ei i1 wi1 ðzÞ Ti2 wi2 ðxÞ fmin fmin T b P i ei T T i bi Pi ei uicom bi Pi ei uir v^0i fmin T T b Pi ei b Pi ei T i i T þ bi Pi ei "max þ i2 wi2 ðxi Þ þ max fmin fmin T b P i ei i ~T 1 _ _ þ dmax þ ~Ti1 1 1 i1 þ i2 2 i2 fmin u~ir u_ ir u~icom u_ icom v~0i v^_ 0i þ þ þ uir uicom v^0i
N X 1
ð43Þ The equation (43) can be rewritten as below. ! N _u X 1 1 f eTi Qi ei 2i eTi Pi ei bTi Pi ei þ V_ fui 2 fui fui i¼1 T b P i ei v^0i v0i bTi Pi ei ~Ti1 wi1 ðzÞ i fmin |fflfflfflfflfflffl{zfflfflfflfflfflffl}
T b P i ei i
fmin
v~0i
Ti2 T i 2 wi2 ðxÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} ~Ti2
. bTi Pi ei uicom "max max fmin |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} u~icom
. bTi Pi ei uir dmax fmin |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} u~ir
u~ir u_ ir u~icom u_ icom v~0i v^_ 0i ~T 1 _ _ þ ~Ti1 1 þ þ 1 i1 þ i 2 2 i 2 þ uir uicom v^0i ð44Þ To derive updates law, the above equation has been rewritten as below.
540
R. Ghasemi et al.
! N _u X T 1 1 f i T T V_ ei Qi ei 2 ei Pi ei bi Pi ei þ fui 2 fui fui i¼1 ! T b Pi ei v^_ 0 i _ i ~Ti1 bTi Pi ei wi1 ðzÞ 1 v~0i 2 i2 v^0i fmin |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼0 ¼0 T bi P i e i _i1 ~Ti2 wi2 ðxÞ 1 1 fmin |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼0 T u_ icom T u_ ir u~icom bi Pi ei ~ uir bi Pi ei uicom uir |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼0
¼0
ð45Þ Using (27), the above inequality rewrites as: V_
N X i¼1
1 1 T f_u ei Qi ei þ 2i eTi Pi ei þ bTi Pi ei fui 2 fui fui
!
ð46Þ Use the lemma 1, V_ 0 are satisfied. Using Barbalat’s lemma, the tracking error asymptotically to the neighborhood of the origin is guaranteed. Further-
which is unavoidable, the proposed adaptive laws (27) is modified it by introducing a modification term as follows: _i1 ¼ 1 bTi Pi ei wi1 ðzi Þ 1 i1 _i2 ¼ 2 bTi Pi ei wi2 ðxi Þ 2 i2 u_ ir ¼ uir bTi Pi ei uir uir u_ icom ¼ uicom bTi Pi ei uicom uicom v^_ 0 ¼ v^0 bT Pi ei v^0 v^0 i
i
i
i
ð47Þ
i
5. Simulation Results In this section, we apply the proposed decentralized fuzzy model reference adaptive controller to a two-inverted pendulum problem [14] in which the pendulums are connected by a spring as shown in Fig. 2. Each pendulum may be positioned by a torque input ui applied by a servomotor and its base. It is assumed that the angular position of pendulum and its angular rate are available and can be used as the controller inputs. The pendulum dynamics are described by the following nonlinear equations.
8 x_ 11 ¼ x12 > > < m1 gr kr2 kr
kr2 sinðx11Þ þ x_ 12 ¼ ðl bÞ þ satðu1 Þ þ sinðx21 Þ > j1 2j1 j1 4j1 j1 > : y1 ¼ x11
ð48Þ
8 _ x22 > < x21 ¼ m2 gr kr2 kr
kr2 _ 22 ¼ sinðx x ðl bÞ þ satðu sinðx12 Þ 21 Þ þ 2Þ þ > 4j2 j2 j2 2j2 j2 : y2 ¼ x21 more, boundedness of the coefficient parameters is Q.E.D guaranteed. It completes the proof. Remark 1: The term tanhðbTi Pi ei ="Þ in the signal vi compensates for the error modeling and disturbance and it is a smooth approximation of the discontinuous term signðbTi Pi ei Þ usually used in robust controllers. The " is chosen so that the sign(.) function can be approximated by tanh(.). The sign(.) function is not used in the paper due to avoiding chattering in the response. Remark 2: It is very important to select properly the controller parameters to gain a satisfactory performance. Here at this stage, number of the rules and the input membership functions are obtained by trial and error. Remark 3: To guarantee the boundedness of the parameters in the presence of the approximation error,
where y1 ; y2 are the angular displacements of the pendulums from vertical position. m1 ¼ 2 kg; m2 ¼ 2:5 kg are the pendulum end masses, j1 ¼ 0:5 kg; j2 ¼ 0:62 kg are the moment of inertia, k ¼ 100 N=m is spring constant, r ¼ 0:5 m is the height of the pendulum, g ¼ 9:81 m=s2 shows the gravitational acceleration, l ¼ 0:5 m is the natural length of spring,
1 ; 2 ¼ 25 are the control input gains and b ¼ 0:4 m presents distance between the pendulum hinges. We consider the desired value of the outputs be zero (yid ¼ 0 for i ¼ 1, 2). As discussed in Section 4 the following primary PID controller are obtained after some trials and errors. Z t ei d þ e_i Þ ð49Þ upid ¼ 40ðei þ 1=4 0
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Decentralized Fuzzy MRAC
Figs. 3 and 4 present the outputs of the system where only the controller defined in equation (49) is applied to the system. Obviously the primary controller by itself is not admissible. Now we applied the proposed controller defined in (24), (25). Initially the PID controller keeps the states of system xi1 ; xi2 in the range of ½1; 1; ½5; 5. Let xi ¼ ½xi1 ; xi2 T ; zi ¼ ½xi1 ; xi2 ; vi T
Fig. 2. Two inverted pendulum connected by a spring.
Fig. 3. Performance of the PID controller in first subsystem.
Fig. 4. Performance of the PID controller in second subsystem.
and vi are defined over ½45; 45. For each fuzzy system input, we define 6 membership functions over the defined sets. Consider that all of the membership functions are defined ! by the Gaussian function j ð Þ ¼ exp
ð cÞ2 , where c is center of the 22
membership function and is its variance. We assume
542
that the initial value of i1 ð0Þ; i2 ð0Þ; uir ð0Þ; uicom ð0Þ, and v^0i ð0Þ be zero. Furthermore, it has been assumed that fmin ¼ 1, 1 ¼ 10, 2 ¼ 10, ucom ¼ 5, ur ¼ 5, v^0i ¼ 5. In equation (47) and remark (1), we assume that ¼ 0:01, " ¼ 0:01. The parameters fdm ; fmin and
Fig. 5. Performance of the proposed controller in first subsystem.
Fig. 6. Performance of the proposed controller in second subsystem.
Fig. 7. Control input u1.
R. Ghasemi et al.
the vector ki ¼ ½ki;1 ; ki;2 ; . . . ; ki;ni T has been chosen so that the lemma 2 holds. As shown in Figs. 3–6), it is obvious that the performance of the proposed controller is promising. Figs. 7 and 8 shown the total input of each subsystem.
Decentralized Fuzzy MRAC
543
Fig. 8. Control input u2.
6. Conclusion In this paper, we propose a new decentralized fuzzy model reference adaptive output tracking controller for a class of large-scale nonaffine nonlinear systems. Fuzzy systems used to approximate the part of controller let us to use the knowledge of the experts in the controller design procedure. Using the Lyapunov’s stability analysis, it has been shown that the derived adaptation laws guaranty the stability of closed-loop system, and asymptotic convergence of the tracking error to a neighborhood of zero. Robustness against external disturbances and approximation errors, relaxing the conditions and using knowledge of experts are the merits of the proposed controller.
References 1. Chen L, Chen G, Lee YW. Fuzzy modeling and adaptive control of uncertain chaotic systems. Inf Sci 1999; 121: 27–37 2. Cheng CC, Chien SH. Adaptive sliding mode controller design based on T–S fuzzy system models. Automatica 2006; 42: 1005–1010 3. Chiang CC. Adaptive fuzzy sliding mode control for time-delay uncertain large-scale systems. Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, Seville, Spain, December 12–15, 2005, pp. 4077–4082 4. Chiang CC, Lu WH. Decentralized adaptive fuzzy controller design for uncertain large-scale systems with unknown dead-zone. IEEE Conf, 2007, pp. 1–6 5. Feng G. An approach to adaptive control of fuzzy dynamic systems. IEEE Trans Fuzzy Syst 2002; 10(2): 268–275 6. Feng G, Cao SG, Rees NW. Stable adaptive control of fuzzy dynamic systems. Fuzzy Sets Syst 2002; 131: 217–224 7. Golea N, Golea A, Benmahammed K. Stable indirect fuzzy adaptive control. Fuzzy Sets Syst 2003; 137: 353–366 8. Ho HF, Wong YK, Rad AB, Lo WL. State observer based indirect adaptive fuzzy tracking control. Simul Model Pract Theory 2005; 13: 646–663
9. Hsu YC, Chen G, Tong S, Li HX. Integrated fuzzy modeling and adaptive control for nonlinear systems. Inf Sci 2003; 153: 217–236 10. Ioannou P, Ponte Jr MD. Adaptive control techniques for a class of large-scale systems. IEEE Conference, 1988 11. Ioannou P, Sun J. Robust Adaptive Control. PrenticeHall, Inc., Englewood Cliffs, NJ, 1996, Electronic copy at 12. Jagannathan S. Adaptive fuzzy logic control of feedback linearization discrete time dynamical systems under persistence of excitation. IEEE Automatica 1998; 34(11): 1295–1310 13. Jiang X, Xu W, Han QL. Observer-based fuzzy control design with adaptation to delay parameter for timedelay systems. Fuzzy Sets Syst 2005; 152: 637–649 14. Karimi B, Menhaj MB, Saboori I. Decentralized adaptive control of large scale nonaffine nonlinear systems using radial basis function neural network. IEICE Trans Fundam 2007; E90-A(10): 2239–2247 15. Khalil H. Nonlinear Systems, 2nd edn. Prentice Hall, 1996 16. Labiod S, Boucherit MS, Guerra TM. Adaptive fuzzy control of a class of MIMO nonlinear systems. Fuzzy Sets Syst 2005; 151: 59–77 17. Labiod S, Guerra TM. Adaptive fuzzy control of a class of SISO non-affine nonlinear systems. Fuzzy Sets Syst 2007; 158: 1126–1137 18. Liu YJ, Wang W. Adaptive fuzzy control for a class of uncertain non-affine nonlinear systems. Inf Sci 2007; 177(8): 3901–3917 19. Pagilla PR, Dwivedula RV, Siraskar NB. A decentralized model reference adaptive controller for large-scale systems. IEEE/ASME Trans Mechatronics 2007; 12(2): 154–163 20. Park CW, Park M. Adaptive parameter estimator based on T–S fuzzy models and its applications to indirect adaptive fuzzy control design. Inf Sci 2004; 159: 125–139 21. Shaocheng T, Bin C, Yongfu W. Fuzzy adaptive output feedback control for MIMO nonlinear systems. Fuzzy Sets Syst 2005; 156: 285–299 22. Shi L, Singh SK. Decentralized adaptive controller design for large scale systems with higher order interconnections. IEEE Trans Autom Control 1992; 37(8): 1106–1113
544 23. Tang Y, Zhang N, Li Y. Stable fuzzy adaptive control for a class of nonlinear systems. Fuzzy Sets Syst 1999; 104: 279–288 24. Tong SC, Li Q, Chai T. Fuzzy adaptive control for a class of nonlinear systems. Fuzzy Sets Syst 1999; 101: 31–39 25. Tong S, Li HX, Wang W. Observer-based adaptive fuzzy control for SISO nonlinear systems. Fuzzy Sets Syst 2004; 148: 355–376 26. Wang LX. A Course in Fuzzy Systems and Control. Prentice Hall PTR, 1997 27. Wang LX, Mendel JM. Fuzzy basis function, universal approximation, and orthogonal least square learning. IEEE Trans Neural Network 1993; 3–5: 807–814 28. Wu H. Decentralized adaptive robust control for a class of large-scale systems including delayed state perturbations in the interconnections. IEEE Trans Autom Control 2002; 47(10): 1745–1751 29. Ying-guo P, Hua-guang Z. Design of fuzzy direct adaptive controller and stability analysis for a class of nonlinear system. In: Proceedings of the American Control conference, Philadelphia, Pennsylvania, June 1998, pp. 2274–2275
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30. Yiqian T, Jianhui W, Shusheng G, Fengying Q. Fuzzy adaptive output feedback control for nonlinear MIMO systems based on observer. In: Proceedings of the 5th World Congress on Intelligent Control and Automation Hangzhou, PR China, June 15–19, 2004, pp. 506–510 31. Yousef H, Simaan MA. Model reference adaptive control for large scale systems with application to power systems. IEE Proceedings-D 1991; 138(4): 321–327 32. Yu WS. Model reference fuzzy adaptive control for uncertain dynamical systems with time delays. IEEE Int Conf Syst Man Cybern 2004: 5246–5251 33. Zhang L. Stable fuzzy adaptive control based on optimal fuzzy reasoning. In: IEEE Proceedings of the Sixth International Conference on Intelligent Systems Design and Applications (ISDA’06), 2006 34. Zhang Y, Chen B, Liu S, Zhang S. Fuzzy is direct adaptive sliding mode tracking control for a class of nonlinear similar composite large-scale systems. In: Proceedings of the American Control Conference Anchorage, AK, May 8–10, 2002, pp. 2961–2967 35. Zimmermann HJ. Fuzzy Set Theory and Its Application. Kluwer Academic Publishers, 1996
A New Decentralized Fuzzy Model Reference Adaptive Controller for a Class of Large-scale Nonaffine Nonlinear Systems Reza Ghasemi, Mohammad Bagher Menhaj and Ahmad Afshar Department of Electrical Engineering, AmirKabir University of Technology, Tehran, Iran
This paper proposes a new method to design a decentralized fuzzy adaptive controller (FAC) for a class of large-scale nonaffine nonlinear systems in which functions of the systems and interactions are unknown. Comparing to previous paper which mainly concentrates on affine large-scale system (LSS), the proposed method is on nonaffine nonlinear LSS. The stability of the closedloop system is guaranteed based on Lyapunov theory. The proposed controller is robust against uncertainty and external disturbance. To show the effectiveness of the proposed method, an illustrative example is given. The simulation results are very promising. Keywords: Lyapunov Stability, Robust Adaptive Control, Non-affine Nonlinear Systems, Fuzzy Systems, Large Scale System
1. Introduction In the recent years, control design for large-scale systems (LSSs) and effort to extend it have attracted much attention. Research in a control of LSSs is motivated by many emerging applications that employ novel actuation devices for active control of industrial automation, cooperating robotic systems, power systems, and aerospace processes. Centralized control for the LSSs is usually impractical due to the requirement of a large Correspondence to: R. Ghasemi, E-mail: [email protected] and [email protected] E-mail: [email protected] E-mail: [email protected]
amount of information exchanges between subsystems and the lack of computing capacity [14]. Nowadays, fuzzy adaptive controller (FAC) has attracted many researchers to develope appropriate controllers for nonlinear systems especially for LSSs because of the following reasons. 1. Due to its tunable structure, the performance of the FAC is superior to that of the fuzzy controller. 2. Instead of using adaptive controller, FAC can use the knowledge of the experts in the controller. In the recent years, FAC has been fully studied as follows: 1. The Takagi–Sugeno (TS) fuzzy systems have been used to model nonlinear systems and then TS-based controllers have been designed with guaranteed stability [5, 6]. In ref. [9], modeling of affine nonlinear system is discussed and subsequently a stable TSbased controller is developed. Designing of the sliding mode FAC for a class of multivariable TS fuzzy systems are presented in ref. [2]. In refs. [7, 20], the nonaffine nonlinear function are first approximated by the TS fuzzy systems, and then stable TS fuzzy controller and observer are designed for the obtained model. In these papers, modeling and controller have been designed simply, but the systems must be linearizable around some operating points. 2. The linguistic fuzzy systems have been used to design controllers for nonlinear systems. Received 16 July 2008; Accepted 21 December 2009 Recommended by E. Mosca, EF. Camacho
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Refs. [12, 24, 26, 29] have considered linguistic fuzzy systems to design stable adaptive controller for affine systems based on feedback linearization and furthermore in [26] the zero dynamic has been considered to be stable. Stable FAC based on sliding mode is designed for affine systems in ref. [16]. Designing of the FAC for affine chaotic systems are presented in refs. [1, 23]. How to design stable FAC and linear observer for class of affine nonlinear systems is presented in refs. [8, 21, 25, 33]. Fuzzy adaptive sliding mode controller is presented for class of affine nonlinear time delay systems in refs. [3, 13, 32]. The output feedback FAC for class of affine nonlinear MIMO systems is suggested in ref. [30]. The main incompetency of these papers are those restricted conditions on their functions. References [17, 18] are involved in stable FAC for class of nonaffine nonlinear systems. The inefficiency of these papers is bad performance of the controller when the controller has not been adjusted. Stable adaptive controller for class of linear LSS is proposed in refs. [10, 19, 22, 31]. Ref. [4] deals with designing FAC based on sliding mode for class of large-scale affine nonlinear systems. Ref. [34] presents decentralized sliding mode fuzzy adaptive tracking for a class of affine nonlinear systems in LSSs. Ref. [28] designed FAC for a class of affine nonlinear time-delayed systems. These papers have many restricted conditions. Compared to previous paper, which mainly concentrates on affine systems LSS, the proposed method is a nonaffine nonlinear LSS. In this paper, we propose a new method to design a decentralized robust adaptive controller based on fuzzy systems for a class of large-scale nonaffine nonlinear systems with guaranteed stability. The capability of stabilizing closedloop system, robustness against external disturbance and uncertainty and also convergence of the tracking error and boundedness of the estimation errors are advantages of the proposed controller. The rest of the paper is organized as follows. Section 2 gives problem statement. General concept of the fuzzy systems is formulated in Section 3. The designing of FAC is proposed in Section 4. Section 5 shows simulation results of the proposed controller and Section 6 concludes the paper.
where xi ¼ ½xi;l ; . . . ; xi;ni T 2 Rni is the state vector of the system which is assumed available for measurement, ui 2 R is the control input, yi 2 R is the system output, fi ðxi ; ui Þ is an unknown smooth nonlinear function, mi ðx1 ; x2 ; . . . ; xN Þ is an unknown nonlinear interconnection term, and di ðtÞ is bounded disturbance. The control objective is to design an adaptive fuzzy controller for system (1) such that the system output yi ðtÞ follows a desired trajectory yd ðtÞ while all signals in the closed-loop system remain bounded. In this paper, we will make the following assumptions concerning the system (1) and the desired trajectory yd ðtÞ. Assumption 1: Without loss of generality, it is assumed that the nonzero function fu ðxi ; ui Þ ¼ @fðxi ; ui Þ=@ui satisfies the following condition: fu ðxi ; ui Þ fmin > 0
for all
ðxi ; ui Þ 2 Rni R ð2Þ
dfu ðxi ; ui Þ fdm dt fdm 2 R is known and constant. Assumption 2: The desired trajectory yd ðtÞ and its time ðjÞ derivatives yd ðtÞ; j ¼ 1; 2; . . . ; ni , are all smooth and bounded. Assumption 3: The interconnection term satisfies the following: jmi ðx1 ; x2 ; . . . ; xN Þj i ðkxi kÞ
ð3Þ
where i ðkxi kÞ is an unknown nonlinear function. Assumption 4: The disturbance in the above equation is bounded by: di ðtÞ dmax
ð4Þ
Define the tracking error vector as: ei ¼ ½ei;1 ; ei;2 ; . . . ; ei;ni T 2 Rni
ð5Þ
where
2. Problem Statement Consider the following large-scale nonaffine nonlinear system. 8 l ¼ 1; 2; :::; ni 1 < x_ i;l ¼ xi;lþ1 x_ 1;ni ¼ fi ðxi ; ui Þ þ mi ðx1 ; x2 ; . . . ; xN Þ þ di ðtÞ : yi ¼ xi;1
ei;1 ¼ yd yi
ð6Þ
Taking the nth i derivative of both sides of the equation (6) we have
i ¼ 1; 2; . . . ; N
ð1Þ
536 ðn Þ
R. Ghasemi et al. ðn Þ
ðni Þ
ei;1i ¼ yd i yi ðn Þ
¼ yd i fi ðxi ; ui Þ mi ðx1 ; x2 ; . . . ; xN Þ di ðtÞ ð7Þ Use equation (5) to rewrite the above equation as: ðn Þ e_ i ¼ Ai0 ei þ bi fyd i
fi ðxi ; ui Þ mi ðx1 ; x2 ; . . . ; xN Þ di ðtÞg where Ai0 and bi 2 0 1 60 0 6 6. . . . Ai0 ¼6 6. . 6 40 0
are defined below. 3 0 0 1 07 7 .. . . .. 7 ni ni . . .7 72R 7 0 15
ð8Þ
Invoking the implicit function theorem, it is obvious that the nonlinear algebraic equation fi ðxi ; ui Þ vi ¼ 0 is locally soluble for the input ui for an arbitrary ðxi ; vi Þ. Thus, there exists some ideal controller ui ðxi ; vi Þ satisfying the following equality for a given ðxi ; vi Þ 2 Rni R: fi ðxi ; ui Þ vi ¼ 0
ð14Þ
As a result of the mean value theorem, there exists a constant in the range of 0 < < 1, such that the nonlinear function fi ðxi ; ui Þ can be expressed around ui as: and
0 0 0 0 2 3 0 607 6 7 ni 7 bi ¼6 6 .. 7 2 R 4.5
ð9Þ
fi ðxi ; ui Þ ¼ fi ðxi ; ui Þ þ ðui ui Þfiu ¼ fi ðxi ; ui Þ þ eui fiu
ð15Þ
where fiu ¼ @fðxi ; ui Þ=@ui jui ¼ui and ui ¼ ui þ ð1 Þui . Substituting equation (15) into the error equation (12) and using (14), we get
1 Consider the vector ki ¼ ½ki;1 ; ki;2 ; . . . ; ki;ni T be coefficients of LðsÞ ¼ sni þ ki;ni sni 1 þ . . . þ ki;1 and chosen so that the roots of this polynomial are located in the open left-half plane. This makes the matrix Ai ¼ Ai0 bi kTi be Hurwitz. Thus, for any given positive definite symmetric matrix Qi , there exists a unique positive definite symmetric solution Pi for the following Lyapunov equation: ATi Pi þ Pi Ai ¼ Qi
ð10Þ
e_ i ¼ Ai ei bi feui fiu þ mi ðx1 ; x2 ; . . . ; xN Þ þ di ðtÞ þ tanhðbTi Pi ei ="Þ þ v0i g
ð16Þ
However, the implicit function theory only guarantees the existence of the ideal controller ui ðxi ; vi Þ for system (14), and does not recommend a technique for constructing solution even if the dynamics of the system are well known. In the following, a fuzzy system and classic controller will be used to obtain the unknown ideal controller.
Let vi be defined as ðn Þ
vi ¼ yd i þ kTi ei þ tanhðbTi Pi ei ="Þ þ v0i
ð11Þ
where tanhð:Þ is the hyperbolic tangent function, is a large positive constant, and " is a small positive constant. By adding and subtracting the term kTi ei þ tanhðbTi Pi ei ="Þ þ v0i from the right-hand side of equation (8), we obtain e_ i ¼ Ai ei bi ffi ðxi ; ui Þ vi þ mi ðx1 ; x2 ; . . . ; xN Þ þ di ðtÞ þ tanhðbTi Pi ei ="Þ þ v0i g ð12Þ Using assumption (1), equation (11) and the signal vi which is not explicitly dependent on the control input ui , the following inequality is satisfied: @ðfi ðxi ; ui Þ vi Þ @fi ðxi ; ui Þ ¼ >0 @ui @ui
ð13Þ
3. Fuzzy Systems Fig. 1 shows the basic configuration of the fuzzy systems considered in this paper. Here, we consider a multi-input, single-output fuzzy systems: x 2 U Rn ! y 2 V R. Consider that a multi-output system can be separated into a group of single-output systems. The fuzzifier performs a mapping from a crisp input vector x ¼ ½x1 ; x2 ; :::; xn T to a fuzzy set, where the label of the fuzzy set are such as ‘‘small’’, ‘‘medium’’, ‘‘large’’, etc. The fuzzy rule base is consisted of a collection of fuzzy IF–THEN rules. Assume that there are M rules, and the lth rule is Rl ðuÞ: ifðx1 is Al1 . . . xn is Al1 Þthen ðy is Bl Þl ¼ 1; 2; . . . ; M
ð17Þ
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Fig. 1. Configuration of fuzzy system.
where x ¼ ½x1 ; x2 ; . . . ; xn T and y are the crisp input and output of the fuzzy system, respectively. Alj and Bl are fuzzy membership function in Uj and V, respectively. The fuzzy inference performs a mapping from fuzzy sets in U to fuzzy sets in V, based on the fuzzy IF– THEN rules in the fuzzy rule base. The defuzzifier maps fuzzy sets in V to a crisp value in V. The configuration of Fig. 1 represents a general framework of fuzzy systems, because many different choices are allowed for each block in Fig. 1, and various combinations of these choices will construct different fuzzy systems [26]. Here, we use the sum–product inference and the center–average defuzzifier. Therefore, the fuzzy system output can be expressed as M P
yðxÞ ¼
y
n Q l
i¼1 l¼1 M Q n P l¼1 i¼1
Ali ðxi Þ
ð18Þ
Ali ðxi Þ
where Ali ðxi Þ is the membership degree of the input xi to fuzzy set Alj and yl is the point at which the membership function of fuzzy set Bl achieves its maximum value. The fuzzy systems in the form of (18) are proven in [27] to be a universal approximator if their parameters are properly chosen. Theorem 1 [26]: Suppose fðxÞ is a continuous function on a compact set U. Then, for any " > 0, there exists a fuzzy system like (18) satisfying: supjfðxÞ yðxÞj "
ð19Þ
x2U
where ¼ y1 y2 . . . y M is all consequent parameters,
n Q
wi ðxÞ ¼
i¼1 M Q n P l¼1 i¼1
Ali ðxi Þ
ð20Þ a vector grouping and wðxÞ ¼ ½w1 ðxÞ
ð21Þ
Ali ðxi Þ
The fuzzy system (18) is assumed to be well defined so M Q n P Ali ðxi Þ 6¼ 0 for all x 2 U. that l¼1 i¼1
4. FAC Design In Section 2, it has been shown that there exists an ideal control for achieving control objectives. In this section, we show how to develop a fuzzy system to adaptively approximate the unknown ideal controller. The ideal controller can be represented as: ui ¼ fi ðzÞ þ upid þ "iu
ð22Þ
where fi ðzÞ ¼ i1 wi1 ðzÞ, and i1 and wi1 ðzÞ are consequent parameters and a set of fuzzy basis functions, respectively. "iu is an approximation error that satisfies j"iu j "max and "max > 0. The upid is the primary controller that developed properly to initially control the underlying system and parameters i1 are determined through the following optimization. i1 ¼ arg min supTi1 wi1 ðzÞ fi ðzÞ i1
The output given by (18) can be rewritten in the following compact form: yðxÞ ¼ wðxÞT
w2 ðxÞ . . . wM ðxÞT is a set of fuzzy basis functions defined as:
ð23Þ
Denote the estimate of i1 as i1 and uirob as a robust controller to compensate approximation error, uncertainties, disturbance and interconnection term to rewrite the controller given in (22) as: ui ¼ Ti1 wi1 ðzÞ þ upid þ uirob
ð24Þ
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In which uirob is defined below. T b P i ei i ðT wi2 ðxÞ þ fmin uicom þ fmin uir þ v^0i Þ uirob ¼ fmin bTi Pi ei i2 ð25Þ In the above, Ti1 wi1 ðzÞ approximates the ideal controller, Ti2 wi 2 ðxÞ tries to compensate the interconnection term, uicom compensates for approximation errors and uncertainties, uir is designed to compensate for bounded external disturbances, and v^0i is estimation of v0i . Define error vector ~i1 ¼ i1 i1 and use (24) and (25) to rewrite the error equation (16) as: e_ i ¼ Ai ei bi fð~Ti1 wi1 ðzÞ þ uirob "iu Þfiu þ mi ðx1 ; x2 ; . . . ; xN Þ þ di ðtÞ þ
tanhðbTi Pi ei ="Þ
þ
v0i
ð26Þ
g
Consider the following update laws.
i
1 T f_u ei Qi ei þ 2i eTi Pi ei fui fui 1 2 min ðQi Þfmin kei k2 þmin ðPi Þfd m kei k2 fui ð32Þ Use (29) and (32) to have the following which completes the proof. . f_u 1 T ei Qi ei þ 2i eTi Pi ei þ fui bTi Pi ei 0 fui fui
ð27Þ
ð33Þ Q.E.D
Lemma 2: Based on lemma 1and equation (10), the following inequality holds. svdmax ðAi Þ
_i1 ¼ 1 bTi Pi ei wi1 ðzi Þ _i2 ¼ 2 bTi Pi ei wi2 ðxi Þ u_ ir ¼ uir bTi Pi ei u_ icom ¼ uicom bTi Pi ei v^0 ¼ v^0 bT Pi ei i
After some algebraic manipulations, the following inequality is obtained.
fdm min ðPi Þ 2fmin max ðPi Þ
ð34Þ
Proof: Using equation (10) and after some algebraic manipulations, the following inequality is obtained. ð35Þ kQi k ATi Pi þ kPi Ai k ¼ 2kPi Ai k
i
where 1 ¼ T1 > 0; 2 ¼ T2 > 0; uir > 0; uicom > 0; v^0i > 0 are constant parameters. In following equation, maxð:Þ and svdmax ð:Þ are maximum eigenvalue and maximum singular value decomposition, respectively. Lemma 1: The. following inequality max ðQi Þ fdm fmin max ðPi Þ.
holds
1 T f_u ei Qi ei þ 2i eTi Pi ei þ bTi Pi ei 0 fui fui fui
if
ð28Þ
Proof: From assumption (1) and > 0, we can have the following inequality has been satisfied. . fui bTi Pi ei 0 ð29Þ From assumption (1) and the above lemma, it is obvious that ðmin ðQi Þfmin þ min ðPi Þfdm Þ 0
ð30Þ
This in turn leads to the following inequality. 1 min ðQi Þfmin þ min ðPi Þfdm Þkei k2 0 f 2ui
ð31Þ
Using the above equation, we get kQi k 2kPi kkAi k ¼ 2max ðPi Þsvdmax ðAi Þ
ð36Þ
Use (30) and (36) to have the following which completes the proof. svdmax ðAi Þ
fdm min ðPi Þ 2fmin max ðPi Þ
ð37Þ Q.E.D
Theorem 2: Consider the error dynamical system given in (26) for the LSS (1) satisfying assumption (1), interconnection term satisfying assumption (3), the external disturbances satisfying assumption (4), and a desired trajectory satisfying assumption (2), then the controller structure given in (24), (25) with adaptation laws (27) makes the tracking error converge asymptotically to a neighborhood of origin and all signals in the closed-loop system be bounded. Proof: To prove convergence of the tracking error and boundedness of parameters error, we must apply the tracking error and the parameters error in Lyapunov function. Thus, the parameters such as ei , ~i1 , ~i2 , u~ir , u~icom , and v~0i are applied in Lyapunov function. Consider the following Lyapunov function.
539
Decentralized Fuzzy MRAC
V¼
N X 1
1 T ~T 1 ~ ~ ei Pi ei þ ~Ti1 1 1 i1 þ i 2 2 i 2 2 f u i i¼1 ! 2 2 u~ir u~icom v~i 02 þ þ þ uir uicom v^0i ð38Þ
where ~i1 ¼ i1 i1 , ~i2 ¼ i2 i2 , u~ir ¼ uir dmax = fmin , u~icom ¼ uicom "max max =fmin , and v~0i ¼ v^0i v0i . The time derivative of the Lyapunov function becomes. V_ ¼
f_u 1 T 1 e_i Pi ei þ eTi Pi e_i þ 2i eTi Pi ei fui 2 fui fui
N X 1 i¼1
~T 1 _ _ þ ~Ti1 1 1 i1 þ i2 2 i2 þ
!
u~ir u_ ir u~icom u_ icom v~0i v^_ 0i þ þ uir uicom v^0i
Use (26), to rewrite above equation as: f_u 1 T T ei ðAi Pi þ Pi Ai Þei þ 2i eTi Pi ei 2 fui fui i¼1 ! þ bTi Pi ei tanhðbTi Pi ei ="Þ þ fui 1 T þ b Pi ei ðv0i þ ð~Ti1 wi1 ðzÞ þ uirob "iu Þfui fui i þ mi ðx1 ; x2 ; . . . ; xN Þ þ di ðtÞÞ þ
N X 1
~T 1 _ _ þ ~Ti1 1 1 i1 þ i2 2 i2 þ þ
i ðkxi kÞ ¼ T 2 wi2 ðxi Þ þ i
u~ir u_ ir uir
u~icom u_ icom v~0i v^_ 0i þ uicom v^0i ð40Þ
Use bTi Pi ei tanhðbTi Pi ei ="Þ ¼ bTi Pi ei and (10), to rewrite (40) as follows: ! N _u X T 1 1 f i T T V_ ¼ ei Qi ei 2 ei Pi ei bi Pi ei þ fui 2 fui fui i¼1 1 T þ b Pi ei v0i þ ð~Ti1 wi1 ðzÞ þ uirob "iu Þfui fui i þ mi ðx1 ; x2 ; . . . ; xN Þ þ di ðtÞ þ u~ir u_ ir u~icom u_ icom v~0i v^_ 0i ~T 1 _ _ þ þ þ ~Ti1 1 1 i1 þ i2 2 i2 þ uir uicom v^0i ð41Þ
ð42Þ
where i is approximation error and satisfying ji j i max . Using assumption (1) yields 1=fui 1=fmin and by assumptions (3), (4) and equations (42), to rewrite (41) as follow. V_
ð39Þ
V_ ¼
Function i ðkxi kÞ is smooth and it can be approximated with the fuzzy system as the following equation.
! 1 T T f_ui T ei Qi ei 2 ei Pi ei b Pi ei þ fui 2 fui fui i i¼1 T T b P i ei b Pi ei i i 0 T T ~ þ vi bi Pi ei i1 wi1 ðzÞ Ti2 wi2 ðxÞ fmin fmin T b P i ei T T i bi Pi ei uicom bi Pi ei uir v^0i fmin T T b Pi ei b Pi ei T i i T þ bi Pi ei "max þ i2 wi2 ðxi Þ þ max fmin fmin T b P i ei i ~T 1 _ _ þ dmax þ ~Ti1 1 1 i1 þ i2 2 i2 fmin u~ir u_ ir u~icom u_ icom v~0i v^_ 0i þ þ þ uir uicom v^0i
N X 1
ð43Þ The equation (43) can be rewritten as below. ! N _u X 1 1 f eTi Qi ei 2i eTi Pi ei bTi Pi ei þ V_ fui 2 fui fui i¼1 T b P i ei v^0i v0i bTi Pi ei ~Ti1 wi1 ðzÞ i fmin |fflfflfflfflfflffl{zfflfflfflfflfflffl}
T b P i ei i
fmin
v~0i
Ti2 T i 2 wi2 ðxÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} ~Ti2
. bTi Pi ei uicom "max max fmin |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} u~icom
. bTi Pi ei uir dmax fmin |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} u~ir
u~ir u_ ir u~icom u_ icom v~0i v^_ 0i ~T 1 _ _ þ ~Ti1 1 þ þ 1 i1 þ i 2 2 i 2 þ uir uicom v^0i ð44Þ To derive updates law, the above equation has been rewritten as below.
540
R. Ghasemi et al.
! N _u X T 1 1 f i T T V_ ei Qi ei 2 ei Pi ei bi Pi ei þ fui 2 fui fui i¼1 ! T b Pi ei v^_ 0 i _ i ~Ti1 bTi Pi ei wi1 ðzÞ 1 v~0i 2 i2 v^0i fmin |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼0 ¼0 T bi P i e i _i1 ~Ti2 wi2 ðxÞ 1 1 fmin |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼0 T u_ icom T u_ ir u~icom bi Pi ei ~ uir bi Pi ei uicom uir |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼0
¼0
ð45Þ Using (27), the above inequality rewrites as: V_
N X i¼1
1 1 T f_u ei Qi ei þ 2i eTi Pi ei þ bTi Pi ei fui 2 fui fui
!
ð46Þ Use the lemma 1, V_ 0 are satisfied. Using Barbalat’s lemma, the tracking error asymptotically to the neighborhood of the origin is guaranteed. Further-
which is unavoidable, the proposed adaptive laws (27) is modified it by introducing a modification term as follows: _i1 ¼ 1 bTi Pi ei wi1 ðzi Þ 1 i1 _i2 ¼ 2 bTi Pi ei wi2 ðxi Þ 2 i2 u_ ir ¼ uir bTi Pi ei uir uir u_ icom ¼ uicom bTi Pi ei uicom uicom v^_ 0 ¼ v^0 bT Pi ei v^0 v^0 i
i
i
i
ð47Þ
i
5. Simulation Results In this section, we apply the proposed decentralized fuzzy model reference adaptive controller to a two-inverted pendulum problem [14] in which the pendulums are connected by a spring as shown in Fig. 2. Each pendulum may be positioned by a torque input ui applied by a servomotor and its base. It is assumed that the angular position of pendulum and its angular rate are available and can be used as the controller inputs. The pendulum dynamics are described by the following nonlinear equations.
8 x_ 11 ¼ x12 > > < m1 gr kr2 kr
kr2 sinðx11Þ þ x_ 12 ¼ ðl bÞ þ satðu1 Þ þ sinðx21 Þ > j1 2j1 j1 4j1 j1 > : y1 ¼ x11
ð48Þ
8 _ x22 > < x21 ¼ m2 gr kr2 kr
kr2 _ 22 ¼ sinðx x ðl bÞ þ satðu sinðx12 Þ 21 Þ þ 2Þ þ > 4j2 j2 j2 2j2 j2 : y2 ¼ x21 more, boundedness of the coefficient parameters is Q.E.D guaranteed. It completes the proof. Remark 1: The term tanhðbTi Pi ei ="Þ in the signal vi compensates for the error modeling and disturbance and it is a smooth approximation of the discontinuous term signðbTi Pi ei Þ usually used in robust controllers. The " is chosen so that the sign(.) function can be approximated by tanh(.). The sign(.) function is not used in the paper due to avoiding chattering in the response. Remark 2: It is very important to select properly the controller parameters to gain a satisfactory performance. Here at this stage, number of the rules and the input membership functions are obtained by trial and error. Remark 3: To guarantee the boundedness of the parameters in the presence of the approximation error,
where y1 ; y2 are the angular displacements of the pendulums from vertical position. m1 ¼ 2 kg; m2 ¼ 2:5 kg are the pendulum end masses, j1 ¼ 0:5 kg; j2 ¼ 0:62 kg are the moment of inertia, k ¼ 100 N=m is spring constant, r ¼ 0:5 m is the height of the pendulum, g ¼ 9:81 m=s2 shows the gravitational acceleration, l ¼ 0:5 m is the natural length of spring,
1 ; 2 ¼ 25 are the control input gains and b ¼ 0:4 m presents distance between the pendulum hinges. We consider the desired value of the outputs be zero (yid ¼ 0 for i ¼ 1, 2). As discussed in Section 4 the following primary PID controller are obtained after some trials and errors. Z t ei d þ e_i Þ ð49Þ upid ¼ 40ðei þ 1=4 0
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Figs. 3 and 4 present the outputs of the system where only the controller defined in equation (49) is applied to the system. Obviously the primary controller by itself is not admissible. Now we applied the proposed controller defined in (24), (25). Initially the PID controller keeps the states of system xi1 ; xi2 in the range of ½1; 1; ½5; 5. Let xi ¼ ½xi1 ; xi2 T ; zi ¼ ½xi1 ; xi2 ; vi T
Fig. 2. Two inverted pendulum connected by a spring.
Fig. 3. Performance of the PID controller in first subsystem.
Fig. 4. Performance of the PID controller in second subsystem.
and vi are defined over ½45; 45. For each fuzzy system input, we define 6 membership functions over the defined sets. Consider that all of the membership functions are defined ! by the Gaussian function j ð Þ ¼ exp
ð cÞ2 , where c is center of the 22
membership function and is its variance. We assume
542
that the initial value of i1 ð0Þ; i2 ð0Þ; uir ð0Þ; uicom ð0Þ, and v^0i ð0Þ be zero. Furthermore, it has been assumed that fmin ¼ 1, 1 ¼ 10, 2 ¼ 10, ucom ¼ 5, ur ¼ 5, v^0i ¼ 5. In equation (47) and remark (1), we assume that ¼ 0:01, " ¼ 0:01. The parameters fdm ; fmin and
Fig. 5. Performance of the proposed controller in first subsystem.
Fig. 6. Performance of the proposed controller in second subsystem.
Fig. 7. Control input u1.
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the vector ki ¼ ½ki;1 ; ki;2 ; . . . ; ki;ni T has been chosen so that the lemma 2 holds. As shown in Figs. 3–6), it is obvious that the performance of the proposed controller is promising. Figs. 7 and 8 shown the total input of each subsystem.
Decentralized Fuzzy MRAC
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Fig. 8. Control input u2.
6. Conclusion In this paper, we propose a new decentralized fuzzy model reference adaptive output tracking controller for a class of large-scale nonaffine nonlinear systems. Fuzzy systems used to approximate the part of controller let us to use the knowledge of the experts in the controller design procedure. Using the Lyapunov’s stability analysis, it has been shown that the derived adaptation laws guaranty the stability of closed-loop system, and asymptotic convergence of the tracking error to a neighborhood of zero. Robustness against external disturbances and approximation errors, relaxing the conditions and using knowledge of experts are the merits of the proposed controller.
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