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Robust control design of fuzzy dynamical systems
Applied Mathematics and Computation 164 (2005) 555–572 www.elsevier.com/locate/amc
Robust control design of fuzzy dynamical systems Taih-Shyun Lee a, Ye-Hwa Chen a
b,*
, Jason C.-H. Chuang
c
BRGA, Aerospace Electronics Systems, Honeywell International Inc., 5353 W. Bell Road Glendale, AZ 85308, USA b The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA c Marshall Space Flight Center, NASA, Huntsville, AL 35812, USA
Abstract This paper presents stability analysis and robust control design for Takagi and Sugeno (T–S) fuzzy dynamical systems. A systematic approach is proposed to check the stability of the T–S fuzzy system. We then extend the consideration to the uncertainty case, which can be nonlinear and (possibly fast) time-varying. Only the possible bound of the uncertainty is needed. If the uncertainty is matched, a robust control scheme is proposed, which renders the fuzzy system practically stable. If the uncertainty is mismatched, we show that a mismatched threshold is needed to ensure stability. 2004 Elsevier Inc. All rights reserved. Keywords: Fuzzy systems; Robust control; Stability
*
Corresponding author. E-mail address: [email protected] (Y.-H. Chen).
0096-3003/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.06.045
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1. Introduction The basic framework of fuzzy controllers was originally established and applied to control a steam engine in [1]. The applications of the Mamdani-type fuzzy control architecture to numerous industrial processes have been generally recognized as a success. The new control architecture has provided control engineers an alternative to cope with complicated control problems. A number of industrial applications by fuzzy control have been developed [2–4]. In addition to this, fuzzy logic theory has been recognized as a means to describe uncertainty. This is especially important if one wishes to view the uncertainty in terms of the extent of occurrence of an event. The Takagi and Sugeno (T–S) fuzzy model was proposed to serve this purpose by Takagi and Sugeno [5]. Stability is generally recognized one vital issue in designing a control system. It is important for a control system to maintain stability as well as to achieve good performance. The perspective and advancement of fuzzy control systems heavily rely on the development of stability analysis. There have been a number of papers presented to analyze the stability issue of fuzzy control systems based on the T–S fuzzy model. A stability condition for T–S fuzzy model was proposed in [6]. To meet the stability condition, a common positive definite matrix is needed for a given T–S fuzzy system. We propose to study the stability issue for control system analysis and design when uncertainty appears. The uncertainty is nonlinear and (possibly fast) time-varying. Only the possible bound of the uncertainty is needed. The resulting system performance (e.g., practical stability) can be prescribed and guaranteed. The main contributions of the paper are threefold. First, we propose a systematic approach to find a common positive definite matrix. This in turn helps to determine the stability of an uncertainty-free T–S fuzzy system. We then extend to the consideration of an uncertainty case by adopting the means for quadratic stability. Second, we suggest a robust control design for the T–S fuzzy system by utilizing the Lyapunov minimax approach. Under the matching condition, the controlled T–S fuzzy system is guaranteed to be (globally) practically stable. This is regardless of the realization of the uncertainty. Third, we further investigate the mismatched uncertainty case. The practical stability performance is still guaranteed if the uncertainty bound is within a prescribed threshold.
2. T–S fuzzy model A T–S fuzzy model, originally proposed by Takagi and Sugeno in [5], is described by a number of fuzzy IF–THEN production rules. Let x =
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557
[x1, x2, . . ., xn]T be the state variable and u = [u1, u2, . . ., um]T be the input vector. The ith rule of a continuous T–S fuzzy model is expressed as the following: Ri : If x1 ðtÞ is M ix1 and . . . and xn ðtÞ is M ixn and u1 ðtÞ is M iu1 and . . . and um ðtÞ is M ium ; Then
x_ i ðtÞ ¼ Ai xðtÞ þ Bi uðtÞ;
i ¼ 1; 2; . . . ; N ;
ð2:1Þ
where Ri denotes the ith implication, M ixp and M iuq are the fuzzy sets of xp and uq in the ith implication, p = 1, 2, . . ., n, q = 1, 2, . . ., m, respectively, xi(t) denotes the value of state variable obtained from the ith implication, Ai 2 Rn·n, Bi 2 Rn·m, and N is the number of fuzzy implications (i.e., rules). For a given z(t) = [x(t)T, u(t)T]T, the output of the fuzzy model is inferred as follows: PN N wi ðzðtÞÞ½Ai xðtÞ þ Bi uðtÞ X ki ðzðtÞÞ½Ai xðtÞ þ Bi uðtÞ; ð2:2Þ x_ ðtÞ ¼ i¼1 PN ¼ i¼1 i¼1 wi ðzðtÞÞ where wi(z(t)) is the overall truth value (i.e., weight) of the premise of the ith implication. It is calculated as wi ðzðtÞÞ ¼
n Y p¼1
M ixp ðtÞ
m Y
M iuq ðtÞ ;
ð2:3Þ
q¼1
wi ðzðtÞÞ ki ðzðtÞÞ ¼ PN : i¼1 wi ðzðtÞÞ
ð2:4Þ
PN It is reasonable to assume that P wi(z(t)) P 0 and i¼1 wi ðzðtÞÞ > 0. ConseN quently, we have ki(z(t)) P 0 and i¼1 ki ðzðtÞÞ ¼ 1. It has been shown that a set of fuzzy implications described in (2.1) can express a highly nonlinear function in spite of a small number of fuzzy implications [7]. Tanaka pointed out that the dynamics of linear models and neural network models can be perfectly represented by the T–S fuzzy models [8]. Furthermore, it has been shown that the standard fuzzy systems and multilayered feedforward neural networks are special cases of the T–S fuzzy models. This occurs as the output consequence of each implication is expressed as a linear combination of Lipschitz continuous functions [9].
3. Basic stability condition for T–S fuzzy models Stability is one of the most important and fundamental issues in control system design and analysis. We first propose an alternative approach to analyze the stability of an open-loop (i.e., uncontrolled) T–S fuzzy model.
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Consider the following open-loop system: PN N wi ðzðtÞÞAi xðtÞ X ki ðzðtÞÞAi xðtÞ: x_ ðtÞ ¼ i¼1 ¼ PN i¼1 i¼1 wi ðzðtÞÞ
ð3:1Þ
A stability condition for (3.1) is given below [6]. Theorem 1. The equilibrium of (3.1) is globally asymptotically stable if there exists a common positive definite matrix P such that ATi P þ PAi < 0
ð3:2Þ
for i = 1,2, . . ., N. Remark 1. If we consider a discrete T–S fuzzy model whose open-loop system is described as xðk þ 1Þ ¼
N X
ki ðzðkÞÞAi xðkÞ:
ð3:3Þ
i¼1
A stability condition for (3.3) is then the existence of P > 0 such that ATi PAi P < 0
ð3:4Þ
for i = 1,2, . . ., N. The existence of such a common positive definite matrix P described in (3.2) or (3.4) is a key issue to check the stability of a T–S fuzzy model. However, there has been no systematic way of finding such a P in a general case. To develop a more specific way of finding P, the following assumption is made. Assumption 1. For all i = 1,2, . . ., N, Ai is Hurwitz and A i Aj ¼ Aj Ai ;
j ¼ 1; 2; . . . ; N :
ð3:5Þ
The following theorem was proposed in [10]. Theorem 2. For a given positive definite matrix Q, let P1, P2, . . ., PN > 0 be the unique solutions of the following Lyapunov equations: AT1 P 1 þ P 1 A1 ¼ Q; ATi P i þ P i Ai ¼ P i1 ;
i ¼ 2; . . . ; N :
ð3:6Þ
If Assumption 1 holds, then the function V(x) = xTPNx is a common Lyapunov function for each of the individual systems x_ ¼ Ai x, i = 1, 2, . . ., N.
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Theorem 2 implies that there exists a positive definite PN such that ATi P N þ P N Ai < 0
ð3:7Þ
for i = 1, 2, . . ., N. By combining Theorem 2 and the stability condition (3.2) for the T–S fuzzy model, we obtain the following theorem. Theorem 3. For a given positive definite matrix Q, let P1, P2, . . ., PN be the unique solutions of the Lyapunov equations given by (3.6), If Assumption 1 holds, system (3.1) is asymptotically stable. Remark 2. According to Theorem 3, we can determine the asymptotic stability of system (3.1) by a systematic approach. At first, check whether Assumption 1 is satisfied. If it is, solve for P1 based on the Lyapunov Eq. (3.6) for a given positive definite matrix Q. Then solve for Pi by (3.6), i = 2, . . ., N. This carried out in a sequential manner (i.e., Q ! P1 ! ! PN). PN is the common positive matrix P mentioned in Theorem 1. We now turn to the system (2.2), which is under the u. Let the control be uðtÞ ¼ KxðtÞ;
ð3:8Þ
where K is a constant gain matrix. System (2.2) becomes x_ ðtÞ ¼
N X
ki ðzðtÞÞðAi Bi KÞxðtÞ:
ð3:9Þ
i¼1
If we can find a K such that Ai :¼ Ai Bi K
ð3:10Þ
satisfies Assumption 1, we conclude that the system (3.9) is asymptotically stable. Theorem 3 shows a systematic way to find a common positive definite matrix P in (3.4). Next, we propose to relax Assumption 1 by utilizing the technique developed in [11]. In [11], necessary and sufficient conditions of quadratic stability of uncertain linear systems are proposed. If the uncertainty region is a convex hyperpolyhedron, the corners of the hyperpolyhedron suffice to determine the quadratic stability of the uncertain system. Some of the definitions and theorem are summarized in the Appendix A. First, we decompose Ai of (3.1) as follows: Ai ¼ Ai þ DAi ;
i ¼ 1; 2; . . . ; N ;
ð3:11Þ
where Ai satisfies Assumption 1 and DAi is the extra portion. Consequently, by Theorem 3, there exists a common positive definite matrix P N such that
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A i P N þ P N Ai < 0
ð3:12Þ
for i = 1,2, . . ., N. Substituting (3.11) into (3.1), we obtain x_ ðtÞ ¼
N X
ki ðzðtÞÞðAi þ DAi ÞxðtÞ:
ð3:13Þ
i¼1
From the definition of quadratic stability given in the Appendix A, we conclude that system (3.13) is quadratically stable if there exists a scalar ai such that T
xT ½ðAi þ DAi Þ P N þ P N ðAi þ DAi Þx 6 ai kxk
2
ð3:14Þ
for all x 2 Rn. Applying Theorem A.2 in the Appendix A, we are able to check (3.14). Since DAi is a known constant matrix, (3.14) can be checked easily. Remark 3. If there is uncertainty involved in Ai, DAi represents an uncertain matrix. Let DAi 2 Ki with Ki compact and prescribed, i = 1, 2, . . ., N. We can also conclude that system (3.13) is quadratically stable if there exists a scalar ai such that xT ½ðAi þ Ei ÞT P N þ P N ðAi þ Ei Þx 6 ai kxk2
ð3:15Þ
for all x 2 Rn, where Ei 2 Ki, i = 1, 2, . . ., N. Inequality (3.15) can be checked by applying Theorem A.2. Remark 3 indicates that the stability of system (3.13) can be determined even if uncertainties exist in the system. Remark 4. A method of directly computing the uncertainty bound allowed for retaining quadratic stability is also developed in [12]. If the uncertain region Ki is not prescribed, we can apply the approach proposed in [12] to find the maximum possible set Ki.
4. Robust control design for T–S fuzzy models Most of the existing work on control design for T–S fuzzy models is developed without uncertainty. We shall extend the scope to address the modeling uncertainty issue. Based on Theorem 3, we propose a robust control, which renders the uncertain T–S fuzzy model globally practically stable by utilizing the Lyapunov minimax approach [13].
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Consider the following T–S fuzzy model x_ ðtÞ ¼
N X
ki ðzðtÞÞ½Ai ðqðtÞÞxðtÞ þ Bi ðqðtÞÞuðtÞ;
ð4:1Þ
i¼1
where q(t) 2 Rk is an unknown and possibly fast time-varying vector. In addition, qðtÞ 2 X
for all t 2 R;
ð4:2Þ
k
with X R compact and prescribed. Decompose Ai(q) and Bi(q) into Ai ðqÞ ¼ Ai þ DAi ðqÞ;
ð4:3Þ
Bi ðqÞ ¼ B þ DBi ðqÞ;
ð4:4Þ
i = 1, 2, . . ., N, where Ai satisfies Assumption 1. Therefore, there exists a common positive definite matrix PN satisfying (3.7). The following matching conditions are assumed. Assumption 2. There exist continuous functions Ei( Æ ) : X ! Rm · m and Di( Æ ):X ! Rm·n and a scalar d > 0 such that for all q 2 X, DAi ðqÞ ¼ BDi ðqÞ;
ð4:5Þ
DBi ðqÞ ¼ BEi ðqÞ;
ð4:6Þ
1 I þ ðEi ðqÞ þ ETi ðqÞÞ P dI; 2 i = 1, 2, . . ., N.
d > 0;
ð4:7Þ
Remark 5. The choice of Ai and B in (4.3) and (4.4) is not unique. For a given T–S fuzzy model, one can flexibly choose Ai and B to meet Assumptions 1 and 2. We next propose the control scheme. Choose a continuous function ^ðÞ : Rn ! Rþ such that q ^ðxÞ P q
1 sup max kDi ðqÞxk d i q2X
for all x 2 Rn :
ð4:8Þ
^ðÞ may be chosen to be qkxk=d where Remark 6. The function q q :¼ max qi ; i
qi :¼ max kDi ðqÞk: q2X
ð4:9Þ ð4:10Þ
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Define T
qðxÞ: lðxÞ :¼ B P N x^
ð4:11Þ
For any > 0, let the control scheme be uðtÞ ¼ pðxðtÞÞ; where
ð4:12Þ
8 lðxÞ > > ^ðxÞ; if klðxÞk P ; q < klðxÞk pðxÞ ¼ > > : lðxÞ q ^ðxÞ; if klðxÞk < :
ð4:13Þ
Theorem 4. If Assumption 2 holds, the control scheme given by (4.12) renders system (4.1) globally practically stable. The sizes of the uniform ultimate boundedness region and the uniform stability region can be made arbitrarily small by a suitable choice of . Proof. Choose the Lyapunov function candidate to be V ¼ xT P N x:
ð4:14Þ
The derivative of V along the trajectory of the controlled system of (4.1) is given by V_ ¼ x_ T P N x þ xT P N x_ N X T T ki f½xT ðAi þ DATi þ uT ðB þ DBTi ÞP N x ¼
ð4:15Þ
i¼1
þ xT P N ½ðAi þ DAi Þx þ ðB þ DBi Þug: Substituting (4.5) and (4.6) into (4.15) yields V_ ¼
N X
T
T
T
ki fxT ½Ai P N þ P N Ai þ ðBDi Þ P N þ P N ðBDi Þx
i¼1 T
þ uT ðI þ ETi ÞB P N x þ xT P N BðI þ Ei Þug:
ð4:16Þ
Applying the control scheme given by (4.12), we consider two cases. (i) If klk > : N X T T _ V ¼ ki xT ½Ai P N þ P N Ai þ ðBDi ÞT P N þ P N ðBDi Þx i¼1
^ T ^ T q q T l ðI þ ETi ÞB P N x x P N BðI þ Ei Þl klk klk
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¼
563
N X T T T ki xT ½Ai P N þ P N Ai þ ðBDi Þ P N þ P N ðBDi Þx i¼1
^2 T q T T x P N Bð2I þ Ei þ Ei ÞB P N x klk N X T T 6 ki ½xT ðAi P N þ P N Ai Þx þ 2dkxT P N Bk^ q 2dklk i¼1
¼
N X
T
T
ki ½xT ðAi P N þ P N Ai Þx:
ð4:17Þ
i¼1
For the sake of brevity, let T
Ai P N þ P N Ai ¼: Ri ;
ð4:18Þ
where Ri 2 Rn·n and Ri > 0, i = 1, 2, . . ., N. Substitute (4.18) into (4.17), V_ 6
N X
ki ðxT Ri xÞ 6
i¼1
N X
2
ki kmin ðRi Þkxk :
ð4:19Þ
i¼1
Let k :¼ min kmin ðRi Þ;
ð4:20Þ
i
and recall that N X
ki ¼ 1:
ð4:21Þ
i¼1
Consequently, we obtain 2 V_ 6 kkxk
N X
2
ki ¼ kkxk :
ð4:22Þ
i¼1
(ii) If klk 6 : N X T T T _ V ¼ ki xT ½Ai P N þ P N Ai þ ðBDi Þ P N þ P N ðBDi Þx i¼1
^ ^ q q T lT ðI þ ETi ÞB P N x xT P N BðI þ Ei Þl N X T T ¼ ki xT ½Ai P N þ P N Ai þ ðBDi ÞT P N þ P N ðBDi Þx i¼1
^2 T q T x P N Bð2I þ ETi þ Ei ÞB P N x
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6
N X
" ki x
T
T ðAi P N
i¼1
þ
T P N Ai Þx2dklk
klk2 2d
#
N X d T T 2 ki xT ðAi P N þ P N Ai Þx 2 ðklk klkÞ i¼1 N X d 2 1 T T T ¼ ki x ðAi P N þ P N Ai Þx 2 klk þ d 2 2 i¼1 ¼
1 2 6 kkxk þ d: 2
ð4:23Þ
Following the standard argument in [13], the controlled system is globally practically stable. The uniform boundedness region is with radius jR; if r 6 R; dðrÞ ¼ ð4:24Þ jr; if r > R; where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kmax ðP N Þ j :¼ ; kmin ðP N Þ sffiffiffiffiffi d : R :¼ 2k
ð4:25Þ
ð4:26Þ
The uniform ultimate boundedness ball is with radius dð> jRÞ and the maximum amount of time it takes to enter this ball (and remains there thereafter) is 8 0; if r 6 j1 d; > < 2 T ðd; rÞ ¼ kmax ðP N Þc2 kmin ðP N Þj2 d ð4:27Þ > ; otherwise: : 1=2 2 2 1 k j d 2 kd The uniform stability ball is with radius R. Both d and R can be made arbitrarily small by an appropriate choice of . The proof is thus completed. h The control design procedure is summarized as follows. Consider a T–S fuzzy model (4.1). 1. 2. 3. 4. 5.
Decompose Ai(q) and Bi(q) into (4.3) and (4.4), respectively. For a given Q > 0, calculate PN by using (3.6). Find d based on (4.7). ^ðxÞ by (4.8). Choose q Select and construct the control u(t) based on (4.12).
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565
5. Robust control design for mismatched case In case the matching conditions (4.5) and (4.6) are not met, we need to investigate the mismatched case. Let us decompose the uncertainty in the following way: ~ i ðqÞ; DAi ðqÞ ¼ BDi ðqÞ þ DA
ð5:1Þ
~ i ðqÞ: DBi ðqÞ ¼ BEi ðqÞ þ DB
ð5:2Þ
~ i 0, DB ~ i 0, the conRemark 7. This decomposition is always feasible. If DA trol design turns into the matched case discussed in Section 4. Let ~ i ðqÞk; qDA~ :¼ sup max kDA
ð5:3Þ
~ i ðqÞk: qDB~ :¼ sup max kDB
ð5:4Þ
i
i
q2X
q2X
Define ~ i ðqÞx þ DB ~ i ðqÞpðxÞ; ~ei ðx; qÞ :¼ DA
ð5:5Þ
^ðxÞ ¼ qkxk=d. where p(x) is given by (4.13) with q By applying (5.3) and (5.4), we can obtain N X
ki ð~eTi P TN x þ xT P N ~ei Þ ¼
i¼1
N X
~ i ðqÞx þ DB ~ i ðqÞpðxÞÞT P T x ki ½ðDA N
i¼1
~ i ðqÞx þ DB ~ i ðqÞpðxÞÞ þ xT P N ðDA N X 1 2 62 ki kmax ðP N Þ qDA~ þ qDB~ q kxk d i¼1 1 2 ¼ 2kmax ðP N Þ qDA~ þ qDB~ q kxk d ¼: ckxk2 :
ð5:6Þ
Theorem 5. Consider the T–S fuzzy model (4.1) under the mismatched case ex^ðxÞ ¼ qkxk=d pressed by (5.1) and (5.2). The control scheme given by (4.12) with q renders system (4.1) globally practically stable if c < k;
ð5:7Þ
where k is given by (4.20) and c by (5.6). Furthermore, the sizes of the uniform ultimate boundedness region and the uniform stability region can be made arbitrarily small by a suitable choice of .
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Proof. Let the Lyapunov function candidate V be the same as (4.14). The derivative of V along the trajectory of the controlled system of (4.1) is V_ ¼ x_ T P N x þ xT P N x_ N n X T T T ki ½xT ðAi þ DAi Þ þ uT ðB þ DBTi ÞP N x þ xT P N ½ðAi þ DAi Þx ¼ i¼1
o þðB þ DBi Þu þ ~eTi P TN x þ xT P N ~ei :
ð5:8Þ
By the proof of Theorem 4, we have 1 V_ 6 kkxk2 þ ~ kd þ 2
N X
ki ð~eTi P TN x þ xT P N ~ei Þ:
ð5:9Þ
i¼1
Applying (5.6), 1 1 2 2 2 kd þ ckxk ¼ ðk cÞkxk þ ~kd: V_ 6 kkxk þ ~ 2 2
ð5:10Þ
Therefore, if (5.7) holds, the controlled system of (4.1) is practically stable by following the similar argument as in the proof of Theorem 4. The size of the ultimate boundedness region and the uniform stability ball can be determined subsequently. The proof is thus completed. h ~ i 0, DB ~ i 0), c = 0. Therefore, Remark 8. In the matched case (i.e., DA condition (5.7) is always satisfied. In other words, Theorem 5 renders Theorem 4 a special case.
6. Design example In this section, we will illustrate the effectiveness of the proposed robust control design. Consider a mass-spring-damper mechanical system [14], _ þ f ðyÞ ¼ /ð_yÞu; M€y þ gðy; yÞ
ð6:1Þ
where y is the displacement, y_ is the velocity, €y is the acceleration, M is the mass, gðy; y_ Þ is the damper force, f(y) is the spring force, and /ð_y Þ is coupled with the input u. The functions g( Æ ), f( Æ ), and /( Æ ) are nonlinear and possibly uncertain. We shall first consider that the functions are known. We adopt the following numerical values and function forms: M = 1.0 kg, _ ¼ y_ , f(y) = 0.01y + 0.1y3 and /ð_y Þ ¼ 1 þ 0:13_y . The mass-damper-spring gðy; yÞ system is proposed to be represented by the following T–S fuzzy model: Plant Rule 1: If y(t) is F 11 and y_ ðtÞ is F 12 , then ^ 1 xðtÞ þ B ^ 1 uðtÞ; x_ ðtÞ ¼ A
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567
Plant Rule 2: If y(t) is F 11 and y_ ðtÞ is F 22 , then ^ 2 xðtÞ þ B ^ 2 uðtÞ; x_ ðtÞ ¼ A Plant Rule 3: If y(t) is F 21 and y_ ðtÞ is F 12 , then ^ 3 xðtÞ þ B ^ 3 uðtÞ; x_ ðtÞ ¼ A Plant Rule 4: If y(t) is F 21 and y_ ðtÞ is F 22 , then ^ 4 xðtÞ þ B ^ 4 uðtÞ; x_ ðtÞ ¼ A where xðtÞ ¼ ½ yðtÞ y_ ðtÞ T , 0 1 0 ^1 ¼ ^1 ¼ A ; B ; 0:01 1:0 1:4387 0 1 0 ^2 ¼ ^2 ¼ A ; B ; 0:01 1:0 0:5613 0 1 0 ^3 ¼ ^3 ¼ A ; B ; 0:235 1:0 1:4387 0 1 0 ^4 ¼ ^4 ¼ A ; B : 0:235 1:0 0:5613 The following y_ 2 ½ 1:5 1:5 : F 11 ðyÞ ¼ 1
fuzzy
y2 ; 2:25
are
F 21 ðyÞ ¼
y2 ; 2:25
adopted,
where
y 2 ½ 1:5
1:5 ,
y_ 3 : 6:75 ^ i and B ^ i according to (4.3) and (4.4). Let At first, we have to decompose A 0 1 Ai ¼ ; ð6:2Þ 0:01 1:0 _ ¼ 0:5 þ F 12 ðyÞ
y_ 3 ; 6:75
sets
F 22 ð_y Þ ¼ 0:5
0 B¼ ; 1
ð6:3Þ
i = 1, . . ., 4. It is noted that Assumption 1 is satisfied. Consequently, we have
; 0 0 0 0 DA2 ¼ ; 0 0 DA1 ¼
0 0
DB1 ¼
0
; 0:4387 0 DB2 ¼ ; 0:4387
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DA3 ¼
0
0
; 0 0 ; 0
0:225 0 DA4 ¼ 0:225
DB3 ¼
0
; 0:4387 0 DB4 ¼ : 0:4387
It is noted that Ai and Bi, i = 1, . . ., 4, are within the range space of B. The matching condition is therefore met. Let also 1 0 Q¼ : 0 1 We can obtain 50:505 P4 ¼ 50
50 50:5
by solving the Lyapunov Eq. (3.6). Based on (4.6) and (4.7), we can choose dnom = 0.5. We choose the nominal bounding function to be: ^nom ðxÞ ¼ 0:5kxk: q
ð6:4Þ
Here ‘‘nominal’’ stands for that no uncertainty actually occurs. However, we still have DAi and DBi following the decomposition of Ai and Bi. The bounding function based on DAi and DBi in the form suggested in Remark 6 is taken. Let = 0.1. We obtain the nominal control scheme (4.13). _ ¼ ð1 þ c1 ðtÞÞy, _ Next, we consider the uncertainty case. Let gðy; yÞ f(y) = (0.01 + c2(t))y + 0.1y3 and /ð_y Þ ¼ 1 þ 0:13_y þ c3 ðtÞ, where jc1(t)j 6 0.5, jc2(t)j 6 1.0 and jc3(t)j 6 0.25 for all t P 0. Choose the same Ai and B as in (6.2) and (6.3), we have 0 0 0 DA1 ¼ ; DB1 ¼ ; c2 ðtÞ c1 ðtÞ 0:4387 þ c3 ðtÞ 0 0 0 ; DB2 ¼ ; DA2 ¼ c2 ðtÞ c1 ðtÞ 0:4387 þ c3 ðtÞ 0 0 0 ; DB3 ¼ ; DA3 ¼ 0:225 þ c2 ðtÞ c1 ðtÞ 0:4387 þ c3 ðtÞ 0 0 0 ; DB4 ¼ : DA4 ¼ 0:225 þ c2 ðtÞ c1 ðtÞ 0:4387 þ c3 ðtÞ It is noted that the matching condition is also met. Based on (4.6) and (4.7), we can choose drob = 0.4. Following Remark 6, we can choose the robust bounding function, ^rob ðxÞ ¼ 3:375kxk: q
ð6:5Þ
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^rob ðxÞ and q ^nom ðxÞ is that the former has to address The difference between q the additional bound of uncertainty, besides what has already been addressed ^nom ðxÞ. Certainly, this renders a larger magnitude in q ^rob ðxÞ. For simulation in q purpose, we used sinusoidal functions for the uncertainties: c1(t) = 0.5 sin(0.2t), c2(t) = 0.5 cos(0.5t), c3(t) = 0.25 sin(t). Let = 125. To show the effectiveness of ^rob ðxÞ, as compared with q ^nom ðxÞ, in compensating the uncertainty, we use the q
Fig. 1. Displacement under the presence of uncertainty.
Fig. 2. Velocity under the presence of uncertainty.
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Fig. 3. Control forces of nominal and robust controls.
^ðxÞ takes either one of the two forms. The controlcontrol scheme (4.13) when q led system responses are shown in Figs. 1 and 2. The dotted lines denote the ^ðxÞ ¼ q ^nom ðxÞ). The full lines responses due to the nominal control (i.e., when q ^ðxÞ ¼ q ^rob ðxÞ). It is denote the responses due to the robust control (i.e., when q noted that significant improvement in performance is obtained while the robust control is applied. This attests the importance in compensating the uncertainty. Fig. 3 shows the histories of both controls. It is interesting to note that despite ^rob ðxÞ, a smaller robust control magnitude is observed for the use of larger q t P 1 s.
7. Conclusions A systematic way of finding the common positive definite matrix P to satisfy the stability condition (3.2) is first suggested. We further adopt the means for quadratic stability to relax Assumption 1. These appear to be an effective tool for the control design issue. We then proceed with a control scheme for the T–S fuzzy system with or without uncertainty. The uncertainty may or may not meet the matching condition. The resulting controlled system performance, under the matching condition, is (global) practical stability. In the mismatched case, if the mismatched portion of the uncertainty is within a threshold, which is designated by k, as shown in Theorem 5, the same performance is guaranteed. In a nutshell, we have extended a previous framework for stability analysis to control design for a rather general situation.
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Appendix A. Quadratic stability Consider the following system, x_ ¼ ðA þ DAðtÞÞx;
ðA:1Þ
where x 2 Rn is the vector of state variables. We assume that the nominal system matrix A is asymptotically stable. The uncertainty is represented by DA(t), which is an arbitrary Lebesgue measurable matrix function satisfying DAðtÞ 2 X;
for all t P 0;
where X is a compact set and 0 2 X. Definition A.1. The system (A.1) is quadratically stable if there exists a positive definite matrix P and a scalar c > 0 such that T
2
xT ½P ðA þ EÞ þ ðA þ EÞ P x 6 ckxk ;
ðA:2Þ
holds for arbitrary x 2 Rn and E 2 X. To introduce the necessary and sufficient conditions of quadratic stability, some notations are introduced. Sn denotes the set of n · n matrices. S nþ is the set of positive definite matrices. The set of positive semidefinite matrices is represented by S n . The set of n · n asymptotically stable matrices is denoted by Hn. For A 2 Hn, a linear operator LA: Sn ! Sn is defined as LA Q ¼ P
if ðPA þ AT P Þ ¼ Q: n
ðA:3Þ n
n
For any G 2 H , the linear operator HG: S ! S is defined as H GP ¼ W
if ðPG þ GT P Þ ¼ W :
ðA:4Þ
The composition of these two operators is represented by F AG ¼ H G LA :
ðA:5Þ
Definition A.2. Let S be a set in a Banach space. A point p 2 S is defined as a nonprotruded point if there exists a linear segment L such that p 2 L S and is not an endpoint of L. A point which is not a nonprotruded point is naturally called a protruded point. Theorem A.1. Let Q 2 Sn,A 2 Hn and X be convex. Then gðQÞ ¼ min kmin ½DAAþE ðQÞ E2X
can be reached by a protruded point of X.
ðA:6Þ
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Theorem A.1 allows one to search only the protruded points for the maximum of g(Q). Theorem A.2. System (A.1) is quadratically stable if and only if max gðQÞ > 0; Q2T
ðA:7Þ
n
where T ¼ S þ \ fQ : trðQÞ ¼ 1g. Remark A.1. It has been shown in [11] that the maximum problem is convex in the sense that any local extreme is also the global extreme. Thus the search can be carried out by many standard algorithms such as the gradient search.
References [1] E.H. Mamdani, S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, International Journal of Man Machine Studies 7 (1) (1975) 1–13. [2] K. Hirota, A. Arai, S. Hachisu, Fuzzy controlled robot arm playing two-dimensional pingpong game, Fuzzy Sets and Systems 32 (3) (1989) 140–1159. [3] M. Sugeno, M. Nishida, Fuzzy control of model car, Fuzzy Sets and Systems 16 (2) (1985) 103–113. [4] S. Yasunobu, S. Miyamoto, Automatic train operation by predictive fuzzy control, in: M. Sugeno (Ed.), Industrial Application of Fuzzy control, North-Holland, Amsterdam, pp. 1–18. [5] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Transactions on Systems Man and Cybernetics 15 (1) (1985) 116–132. [6] K. Tanaka, M. Sugeno, Stability analysis and design of fuzzy control systems, Fuzzy Sets and Systems 45 (2) (1992) 135–156. [7] M. Sugeno, K. Tanaka, Successive identification of fuzzy model and its applications to prediction of complex system, Fuzzy Sets and Systems 42 (4) (1991) 315–344. [8] K. Tanaka, Stability and stabilizability of fuzzy-neural-linear control systems, IEEE Transactions on Fuzzy Systems 3 (4) (1995) 438–447. [9] J.T. Spooner, K.M. Passino, Stable indirect adaptive control using fuzzy systems and neural networks, in: Proceedings of the 34th Conference on Decision and Control, New Orleans, LA, December, 1995, pp. 249–254. [10] K.S. Narendra, J. Balakrishnan, A common Lyapunov function for stable LTI systems with commuting A-matrices, IEEE Transactions on Automatic Control AC-39 (12) (1994) 2469– 2471. [11] K. Gu, M.A. Zohdy, N.K. Loh, Necessary and sufficient conditions of quadratic stability of uncertain linear systems, IEEE Transactions on Automatic Control AC-35 (5) (1990) 601–604. [12] K. Gu, Quadratic stability bound of discrete-time uncertain systems, in: American Control Conference, Green Valley, AZ, 1991, pp. 1951–1955. [13] M.J. Corless, G. Leitmann, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Transactions on Automatic Control AC-26 (5) (1981) 1139–1144. [14] K. Tanaka, T. Ikeda, H.O. Wang, Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, H1 control theory, and linear matrix inequalities, IEEE Transactions on Fuzzy Systems 4 (1) (1995) 1–13.
Robust control design of fuzzy dynamical systems Taih-Shyun Lee a, Ye-Hwa Chen a
b,*
, Jason C.-H. Chuang
c
BRGA, Aerospace Electronics Systems, Honeywell International Inc., 5353 W. Bell Road Glendale, AZ 85308, USA b The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA c Marshall Space Flight Center, NASA, Huntsville, AL 35812, USA
Abstract This paper presents stability analysis and robust control design for Takagi and Sugeno (T–S) fuzzy dynamical systems. A systematic approach is proposed to check the stability of the T–S fuzzy system. We then extend the consideration to the uncertainty case, which can be nonlinear and (possibly fast) time-varying. Only the possible bound of the uncertainty is needed. If the uncertainty is matched, a robust control scheme is proposed, which renders the fuzzy system practically stable. If the uncertainty is mismatched, we show that a mismatched threshold is needed to ensure stability. 2004 Elsevier Inc. All rights reserved. Keywords: Fuzzy systems; Robust control; Stability
*
Corresponding author. E-mail address: [email protected] (Y.-H. Chen).
0096-3003/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.06.045
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1. Introduction The basic framework of fuzzy controllers was originally established and applied to control a steam engine in [1]. The applications of the Mamdani-type fuzzy control architecture to numerous industrial processes have been generally recognized as a success. The new control architecture has provided control engineers an alternative to cope with complicated control problems. A number of industrial applications by fuzzy control have been developed [2–4]. In addition to this, fuzzy logic theory has been recognized as a means to describe uncertainty. This is especially important if one wishes to view the uncertainty in terms of the extent of occurrence of an event. The Takagi and Sugeno (T–S) fuzzy model was proposed to serve this purpose by Takagi and Sugeno [5]. Stability is generally recognized one vital issue in designing a control system. It is important for a control system to maintain stability as well as to achieve good performance. The perspective and advancement of fuzzy control systems heavily rely on the development of stability analysis. There have been a number of papers presented to analyze the stability issue of fuzzy control systems based on the T–S fuzzy model. A stability condition for T–S fuzzy model was proposed in [6]. To meet the stability condition, a common positive definite matrix is needed for a given T–S fuzzy system. We propose to study the stability issue for control system analysis and design when uncertainty appears. The uncertainty is nonlinear and (possibly fast) time-varying. Only the possible bound of the uncertainty is needed. The resulting system performance (e.g., practical stability) can be prescribed and guaranteed. The main contributions of the paper are threefold. First, we propose a systematic approach to find a common positive definite matrix. This in turn helps to determine the stability of an uncertainty-free T–S fuzzy system. We then extend to the consideration of an uncertainty case by adopting the means for quadratic stability. Second, we suggest a robust control design for the T–S fuzzy system by utilizing the Lyapunov minimax approach. Under the matching condition, the controlled T–S fuzzy system is guaranteed to be (globally) practically stable. This is regardless of the realization of the uncertainty. Third, we further investigate the mismatched uncertainty case. The practical stability performance is still guaranteed if the uncertainty bound is within a prescribed threshold.
2. T–S fuzzy model A T–S fuzzy model, originally proposed by Takagi and Sugeno in [5], is described by a number of fuzzy IF–THEN production rules. Let x =
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557
[x1, x2, . . ., xn]T be the state variable and u = [u1, u2, . . ., um]T be the input vector. The ith rule of a continuous T–S fuzzy model is expressed as the following: Ri : If x1 ðtÞ is M ix1 and . . . and xn ðtÞ is M ixn and u1 ðtÞ is M iu1 and . . . and um ðtÞ is M ium ; Then
x_ i ðtÞ ¼ Ai xðtÞ þ Bi uðtÞ;
i ¼ 1; 2; . . . ; N ;
ð2:1Þ
where Ri denotes the ith implication, M ixp and M iuq are the fuzzy sets of xp and uq in the ith implication, p = 1, 2, . . ., n, q = 1, 2, . . ., m, respectively, xi(t) denotes the value of state variable obtained from the ith implication, Ai 2 Rn·n, Bi 2 Rn·m, and N is the number of fuzzy implications (i.e., rules). For a given z(t) = [x(t)T, u(t)T]T, the output of the fuzzy model is inferred as follows: PN N wi ðzðtÞÞ½Ai xðtÞ þ Bi uðtÞ X ki ðzðtÞÞ½Ai xðtÞ þ Bi uðtÞ; ð2:2Þ x_ ðtÞ ¼ i¼1 PN ¼ i¼1 i¼1 wi ðzðtÞÞ where wi(z(t)) is the overall truth value (i.e., weight) of the premise of the ith implication. It is calculated as wi ðzðtÞÞ ¼
n Y p¼1
M ixp ðtÞ
m Y
M iuq ðtÞ ;
ð2:3Þ
q¼1
wi ðzðtÞÞ ki ðzðtÞÞ ¼ PN : i¼1 wi ðzðtÞÞ
ð2:4Þ
PN It is reasonable to assume that P wi(z(t)) P 0 and i¼1 wi ðzðtÞÞ > 0. ConseN quently, we have ki(z(t)) P 0 and i¼1 ki ðzðtÞÞ ¼ 1. It has been shown that a set of fuzzy implications described in (2.1) can express a highly nonlinear function in spite of a small number of fuzzy implications [7]. Tanaka pointed out that the dynamics of linear models and neural network models can be perfectly represented by the T–S fuzzy models [8]. Furthermore, it has been shown that the standard fuzzy systems and multilayered feedforward neural networks are special cases of the T–S fuzzy models. This occurs as the output consequence of each implication is expressed as a linear combination of Lipschitz continuous functions [9].
3. Basic stability condition for T–S fuzzy models Stability is one of the most important and fundamental issues in control system design and analysis. We first propose an alternative approach to analyze the stability of an open-loop (i.e., uncontrolled) T–S fuzzy model.
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Consider the following open-loop system: PN N wi ðzðtÞÞAi xðtÞ X ki ðzðtÞÞAi xðtÞ: x_ ðtÞ ¼ i¼1 ¼ PN i¼1 i¼1 wi ðzðtÞÞ
ð3:1Þ
A stability condition for (3.1) is given below [6]. Theorem 1. The equilibrium of (3.1) is globally asymptotically stable if there exists a common positive definite matrix P such that ATi P þ PAi < 0
ð3:2Þ
for i = 1,2, . . ., N. Remark 1. If we consider a discrete T–S fuzzy model whose open-loop system is described as xðk þ 1Þ ¼
N X
ki ðzðkÞÞAi xðkÞ:
ð3:3Þ
i¼1
A stability condition for (3.3) is then the existence of P > 0 such that ATi PAi P < 0
ð3:4Þ
for i = 1,2, . . ., N. The existence of such a common positive definite matrix P described in (3.2) or (3.4) is a key issue to check the stability of a T–S fuzzy model. However, there has been no systematic way of finding such a P in a general case. To develop a more specific way of finding P, the following assumption is made. Assumption 1. For all i = 1,2, . . ., N, Ai is Hurwitz and A i Aj ¼ Aj Ai ;
j ¼ 1; 2; . . . ; N :
ð3:5Þ
The following theorem was proposed in [10]. Theorem 2. For a given positive definite matrix Q, let P1, P2, . . ., PN > 0 be the unique solutions of the following Lyapunov equations: AT1 P 1 þ P 1 A1 ¼ Q; ATi P i þ P i Ai ¼ P i1 ;
i ¼ 2; . . . ; N :
ð3:6Þ
If Assumption 1 holds, then the function V(x) = xTPNx is a common Lyapunov function for each of the individual systems x_ ¼ Ai x, i = 1, 2, . . ., N.
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Theorem 2 implies that there exists a positive definite PN such that ATi P N þ P N Ai < 0
ð3:7Þ
for i = 1, 2, . . ., N. By combining Theorem 2 and the stability condition (3.2) for the T–S fuzzy model, we obtain the following theorem. Theorem 3. For a given positive definite matrix Q, let P1, P2, . . ., PN be the unique solutions of the Lyapunov equations given by (3.6), If Assumption 1 holds, system (3.1) is asymptotically stable. Remark 2. According to Theorem 3, we can determine the asymptotic stability of system (3.1) by a systematic approach. At first, check whether Assumption 1 is satisfied. If it is, solve for P1 based on the Lyapunov Eq. (3.6) for a given positive definite matrix Q. Then solve for Pi by (3.6), i = 2, . . ., N. This carried out in a sequential manner (i.e., Q ! P1 ! ! PN). PN is the common positive matrix P mentioned in Theorem 1. We now turn to the system (2.2), which is under the u. Let the control be uðtÞ ¼ KxðtÞ;
ð3:8Þ
where K is a constant gain matrix. System (2.2) becomes x_ ðtÞ ¼
N X
ki ðzðtÞÞðAi Bi KÞxðtÞ:
ð3:9Þ
i¼1
If we can find a K such that Ai :¼ Ai Bi K
ð3:10Þ
satisfies Assumption 1, we conclude that the system (3.9) is asymptotically stable. Theorem 3 shows a systematic way to find a common positive definite matrix P in (3.4). Next, we propose to relax Assumption 1 by utilizing the technique developed in [11]. In [11], necessary and sufficient conditions of quadratic stability of uncertain linear systems are proposed. If the uncertainty region is a convex hyperpolyhedron, the corners of the hyperpolyhedron suffice to determine the quadratic stability of the uncertain system. Some of the definitions and theorem are summarized in the Appendix A. First, we decompose Ai of (3.1) as follows: Ai ¼ Ai þ DAi ;
i ¼ 1; 2; . . . ; N ;
ð3:11Þ
where Ai satisfies Assumption 1 and DAi is the extra portion. Consequently, by Theorem 3, there exists a common positive definite matrix P N such that
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A i P N þ P N Ai < 0
ð3:12Þ
for i = 1,2, . . ., N. Substituting (3.11) into (3.1), we obtain x_ ðtÞ ¼
N X
ki ðzðtÞÞðAi þ DAi ÞxðtÞ:
ð3:13Þ
i¼1
From the definition of quadratic stability given in the Appendix A, we conclude that system (3.13) is quadratically stable if there exists a scalar ai such that T
xT ½ðAi þ DAi Þ P N þ P N ðAi þ DAi Þx 6 ai kxk
2
ð3:14Þ
for all x 2 Rn. Applying Theorem A.2 in the Appendix A, we are able to check (3.14). Since DAi is a known constant matrix, (3.14) can be checked easily. Remark 3. If there is uncertainty involved in Ai, DAi represents an uncertain matrix. Let DAi 2 Ki with Ki compact and prescribed, i = 1, 2, . . ., N. We can also conclude that system (3.13) is quadratically stable if there exists a scalar ai such that xT ½ðAi þ Ei ÞT P N þ P N ðAi þ Ei Þx 6 ai kxk2
ð3:15Þ
for all x 2 Rn, where Ei 2 Ki, i = 1, 2, . . ., N. Inequality (3.15) can be checked by applying Theorem A.2. Remark 3 indicates that the stability of system (3.13) can be determined even if uncertainties exist in the system. Remark 4. A method of directly computing the uncertainty bound allowed for retaining quadratic stability is also developed in [12]. If the uncertain region Ki is not prescribed, we can apply the approach proposed in [12] to find the maximum possible set Ki.
4. Robust control design for T–S fuzzy models Most of the existing work on control design for T–S fuzzy models is developed without uncertainty. We shall extend the scope to address the modeling uncertainty issue. Based on Theorem 3, we propose a robust control, which renders the uncertain T–S fuzzy model globally practically stable by utilizing the Lyapunov minimax approach [13].
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561
Consider the following T–S fuzzy model x_ ðtÞ ¼
N X
ki ðzðtÞÞ½Ai ðqðtÞÞxðtÞ þ Bi ðqðtÞÞuðtÞ;
ð4:1Þ
i¼1
where q(t) 2 Rk is an unknown and possibly fast time-varying vector. In addition, qðtÞ 2 X
for all t 2 R;
ð4:2Þ
k
with X R compact and prescribed. Decompose Ai(q) and Bi(q) into Ai ðqÞ ¼ Ai þ DAi ðqÞ;
ð4:3Þ
Bi ðqÞ ¼ B þ DBi ðqÞ;
ð4:4Þ
i = 1, 2, . . ., N, where Ai satisfies Assumption 1. Therefore, there exists a common positive definite matrix PN satisfying (3.7). The following matching conditions are assumed. Assumption 2. There exist continuous functions Ei( Æ ) : X ! Rm · m and Di( Æ ):X ! Rm·n and a scalar d > 0 such that for all q 2 X, DAi ðqÞ ¼ BDi ðqÞ;
ð4:5Þ
DBi ðqÞ ¼ BEi ðqÞ;
ð4:6Þ
1 I þ ðEi ðqÞ þ ETi ðqÞÞ P dI; 2 i = 1, 2, . . ., N.
d > 0;
ð4:7Þ
Remark 5. The choice of Ai and B in (4.3) and (4.4) is not unique. For a given T–S fuzzy model, one can flexibly choose Ai and B to meet Assumptions 1 and 2. We next propose the control scheme. Choose a continuous function ^ðÞ : Rn ! Rþ such that q ^ðxÞ P q
1 sup max kDi ðqÞxk d i q2X
for all x 2 Rn :
ð4:8Þ
^ðÞ may be chosen to be qkxk=d where Remark 6. The function q q :¼ max qi ; i
qi :¼ max kDi ðqÞk: q2X
ð4:9Þ ð4:10Þ
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Define T
qðxÞ: lðxÞ :¼ B P N x^
ð4:11Þ
For any > 0, let the control scheme be uðtÞ ¼ pðxðtÞÞ; where
ð4:12Þ
8 lðxÞ > > ^ðxÞ; if klðxÞk P ; q < klðxÞk pðxÞ ¼ > > : lðxÞ q ^ðxÞ; if klðxÞk < :
ð4:13Þ
Theorem 4. If Assumption 2 holds, the control scheme given by (4.12) renders system (4.1) globally practically stable. The sizes of the uniform ultimate boundedness region and the uniform stability region can be made arbitrarily small by a suitable choice of . Proof. Choose the Lyapunov function candidate to be V ¼ xT P N x:
ð4:14Þ
The derivative of V along the trajectory of the controlled system of (4.1) is given by V_ ¼ x_ T P N x þ xT P N x_ N X T T ki f½xT ðAi þ DATi þ uT ðB þ DBTi ÞP N x ¼
ð4:15Þ
i¼1
þ xT P N ½ðAi þ DAi Þx þ ðB þ DBi Þug: Substituting (4.5) and (4.6) into (4.15) yields V_ ¼
N X
T
T
T
ki fxT ½Ai P N þ P N Ai þ ðBDi Þ P N þ P N ðBDi Þx
i¼1 T
þ uT ðI þ ETi ÞB P N x þ xT P N BðI þ Ei Þug:
ð4:16Þ
Applying the control scheme given by (4.12), we consider two cases. (i) If klk > : N X T T _ V ¼ ki xT ½Ai P N þ P N Ai þ ðBDi ÞT P N þ P N ðBDi Þx i¼1
^ T ^ T q q T l ðI þ ETi ÞB P N x x P N BðI þ Ei Þl klk klk
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¼
563
N X T T T ki xT ½Ai P N þ P N Ai þ ðBDi Þ P N þ P N ðBDi Þx i¼1
^2 T q T T x P N Bð2I þ Ei þ Ei ÞB P N x klk N X T T 6 ki ½xT ðAi P N þ P N Ai Þx þ 2dkxT P N Bk^ q 2dklk i¼1
¼
N X
T
T
ki ½xT ðAi P N þ P N Ai Þx:
ð4:17Þ
i¼1
For the sake of brevity, let T
Ai P N þ P N Ai ¼: Ri ;
ð4:18Þ
where Ri 2 Rn·n and Ri > 0, i = 1, 2, . . ., N. Substitute (4.18) into (4.17), V_ 6
N X
ki ðxT Ri xÞ 6
i¼1
N X
2
ki kmin ðRi Þkxk :
ð4:19Þ
i¼1
Let k :¼ min kmin ðRi Þ;
ð4:20Þ
i
and recall that N X
ki ¼ 1:
ð4:21Þ
i¼1
Consequently, we obtain 2 V_ 6 kkxk
N X
2
ki ¼ kkxk :
ð4:22Þ
i¼1
(ii) If klk 6 : N X T T T _ V ¼ ki xT ½Ai P N þ P N Ai þ ðBDi Þ P N þ P N ðBDi Þx i¼1
^ ^ q q T lT ðI þ ETi ÞB P N x xT P N BðI þ Ei Þl N X T T ¼ ki xT ½Ai P N þ P N Ai þ ðBDi ÞT P N þ P N ðBDi Þx i¼1
^2 T q T x P N Bð2I þ ETi þ Ei ÞB P N x
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6
N X
" ki x
T
T ðAi P N
i¼1
þ
T P N Ai Þx2dklk
klk2 2d
#
N X d T T 2 ki xT ðAi P N þ P N Ai Þx 2 ðklk klkÞ i¼1 N X d 2 1 T T T ¼ ki x ðAi P N þ P N Ai Þx 2 klk þ d 2 2 i¼1 ¼
1 2 6 kkxk þ d: 2
ð4:23Þ
Following the standard argument in [13], the controlled system is globally practically stable. The uniform boundedness region is with radius jR; if r 6 R; dðrÞ ¼ ð4:24Þ jr; if r > R; where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kmax ðP N Þ j :¼ ; kmin ðP N Þ sffiffiffiffiffi d : R :¼ 2k
ð4:25Þ
ð4:26Þ
The uniform ultimate boundedness ball is with radius dð> jRÞ and the maximum amount of time it takes to enter this ball (and remains there thereafter) is 8 0; if r 6 j1 d; > < 2 T ðd; rÞ ¼ kmax ðP N Þc2 kmin ðP N Þj2 d ð4:27Þ > ; otherwise: : 1=2 2 2 1 k j d 2 kd The uniform stability ball is with radius R. Both d and R can be made arbitrarily small by an appropriate choice of . The proof is thus completed. h The control design procedure is summarized as follows. Consider a T–S fuzzy model (4.1). 1. 2. 3. 4. 5.
Decompose Ai(q) and Bi(q) into (4.3) and (4.4), respectively. For a given Q > 0, calculate PN by using (3.6). Find d based on (4.7). ^ðxÞ by (4.8). Choose q Select and construct the control u(t) based on (4.12).
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5. Robust control design for mismatched case In case the matching conditions (4.5) and (4.6) are not met, we need to investigate the mismatched case. Let us decompose the uncertainty in the following way: ~ i ðqÞ; DAi ðqÞ ¼ BDi ðqÞ þ DA
ð5:1Þ
~ i ðqÞ: DBi ðqÞ ¼ BEi ðqÞ þ DB
ð5:2Þ
~ i 0, DB ~ i 0, the conRemark 7. This decomposition is always feasible. If DA trol design turns into the matched case discussed in Section 4. Let ~ i ðqÞk; qDA~ :¼ sup max kDA
ð5:3Þ
~ i ðqÞk: qDB~ :¼ sup max kDB
ð5:4Þ
i
i
q2X
q2X
Define ~ i ðqÞx þ DB ~ i ðqÞpðxÞ; ~ei ðx; qÞ :¼ DA
ð5:5Þ
^ðxÞ ¼ qkxk=d. where p(x) is given by (4.13) with q By applying (5.3) and (5.4), we can obtain N X
ki ð~eTi P TN x þ xT P N ~ei Þ ¼
i¼1
N X
~ i ðqÞx þ DB ~ i ðqÞpðxÞÞT P T x ki ½ðDA N
i¼1
~ i ðqÞx þ DB ~ i ðqÞpðxÞÞ þ xT P N ðDA N X 1 2 62 ki kmax ðP N Þ qDA~ þ qDB~ q kxk d i¼1 1 2 ¼ 2kmax ðP N Þ qDA~ þ qDB~ q kxk d ¼: ckxk2 :
ð5:6Þ
Theorem 5. Consider the T–S fuzzy model (4.1) under the mismatched case ex^ðxÞ ¼ qkxk=d pressed by (5.1) and (5.2). The control scheme given by (4.12) with q renders system (4.1) globally practically stable if c < k;
ð5:7Þ
where k is given by (4.20) and c by (5.6). Furthermore, the sizes of the uniform ultimate boundedness region and the uniform stability region can be made arbitrarily small by a suitable choice of .
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Proof. Let the Lyapunov function candidate V be the same as (4.14). The derivative of V along the trajectory of the controlled system of (4.1) is V_ ¼ x_ T P N x þ xT P N x_ N n X T T T ki ½xT ðAi þ DAi Þ þ uT ðB þ DBTi ÞP N x þ xT P N ½ðAi þ DAi Þx ¼ i¼1
o þðB þ DBi Þu þ ~eTi P TN x þ xT P N ~ei :
ð5:8Þ
By the proof of Theorem 4, we have 1 V_ 6 kkxk2 þ ~ kd þ 2
N X
ki ð~eTi P TN x þ xT P N ~ei Þ:
ð5:9Þ
i¼1
Applying (5.6), 1 1 2 2 2 kd þ ckxk ¼ ðk cÞkxk þ ~kd: V_ 6 kkxk þ ~ 2 2
ð5:10Þ
Therefore, if (5.7) holds, the controlled system of (4.1) is practically stable by following the similar argument as in the proof of Theorem 4. The size of the ultimate boundedness region and the uniform stability ball can be determined subsequently. The proof is thus completed. h ~ i 0, DB ~ i 0), c = 0. Therefore, Remark 8. In the matched case (i.e., DA condition (5.7) is always satisfied. In other words, Theorem 5 renders Theorem 4 a special case.
6. Design example In this section, we will illustrate the effectiveness of the proposed robust control design. Consider a mass-spring-damper mechanical system [14], _ þ f ðyÞ ¼ /ð_yÞu; M€y þ gðy; yÞ
ð6:1Þ
where y is the displacement, y_ is the velocity, €y is the acceleration, M is the mass, gðy; y_ Þ is the damper force, f(y) is the spring force, and /ð_y Þ is coupled with the input u. The functions g( Æ ), f( Æ ), and /( Æ ) are nonlinear and possibly uncertain. We shall first consider that the functions are known. We adopt the following numerical values and function forms: M = 1.0 kg, _ ¼ y_ , f(y) = 0.01y + 0.1y3 and /ð_y Þ ¼ 1 þ 0:13_y . The mass-damper-spring gðy; yÞ system is proposed to be represented by the following T–S fuzzy model: Plant Rule 1: If y(t) is F 11 and y_ ðtÞ is F 12 , then ^ 1 xðtÞ þ B ^ 1 uðtÞ; x_ ðtÞ ¼ A
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Plant Rule 2: If y(t) is F 11 and y_ ðtÞ is F 22 , then ^ 2 xðtÞ þ B ^ 2 uðtÞ; x_ ðtÞ ¼ A Plant Rule 3: If y(t) is F 21 and y_ ðtÞ is F 12 , then ^ 3 xðtÞ þ B ^ 3 uðtÞ; x_ ðtÞ ¼ A Plant Rule 4: If y(t) is F 21 and y_ ðtÞ is F 22 , then ^ 4 xðtÞ þ B ^ 4 uðtÞ; x_ ðtÞ ¼ A where xðtÞ ¼ ½ yðtÞ y_ ðtÞ T , 0 1 0 ^1 ¼ ^1 ¼ A ; B ; 0:01 1:0 1:4387 0 1 0 ^2 ¼ ^2 ¼ A ; B ; 0:01 1:0 0:5613 0 1 0 ^3 ¼ ^3 ¼ A ; B ; 0:235 1:0 1:4387 0 1 0 ^4 ¼ ^4 ¼ A ; B : 0:235 1:0 0:5613 The following y_ 2 ½ 1:5 1:5 : F 11 ðyÞ ¼ 1
fuzzy
y2 ; 2:25
are
F 21 ðyÞ ¼
y2 ; 2:25
adopted,
where
y 2 ½ 1:5
1:5 ,
y_ 3 : 6:75 ^ i and B ^ i according to (4.3) and (4.4). Let At first, we have to decompose A 0 1 Ai ¼ ; ð6:2Þ 0:01 1:0 _ ¼ 0:5 þ F 12 ðyÞ
y_ 3 ; 6:75
sets
F 22 ð_y Þ ¼ 0:5
0 B¼ ; 1
ð6:3Þ
i = 1, . . ., 4. It is noted that Assumption 1 is satisfied. Consequently, we have
; 0 0 0 0 DA2 ¼ ; 0 0 DA1 ¼
0 0
DB1 ¼
0
; 0:4387 0 DB2 ¼ ; 0:4387
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DA3 ¼
0
0
; 0 0 ; 0
0:225 0 DA4 ¼ 0:225
DB3 ¼
0
; 0:4387 0 DB4 ¼ : 0:4387
It is noted that Ai and Bi, i = 1, . . ., 4, are within the range space of B. The matching condition is therefore met. Let also 1 0 Q¼ : 0 1 We can obtain 50:505 P4 ¼ 50
50 50:5
by solving the Lyapunov Eq. (3.6). Based on (4.6) and (4.7), we can choose dnom = 0.5. We choose the nominal bounding function to be: ^nom ðxÞ ¼ 0:5kxk: q
ð6:4Þ
Here ‘‘nominal’’ stands for that no uncertainty actually occurs. However, we still have DAi and DBi following the decomposition of Ai and Bi. The bounding function based on DAi and DBi in the form suggested in Remark 6 is taken. Let = 0.1. We obtain the nominal control scheme (4.13). _ ¼ ð1 þ c1 ðtÞÞy, _ Next, we consider the uncertainty case. Let gðy; yÞ f(y) = (0.01 + c2(t))y + 0.1y3 and /ð_y Þ ¼ 1 þ 0:13_y þ c3 ðtÞ, where jc1(t)j 6 0.5, jc2(t)j 6 1.0 and jc3(t)j 6 0.25 for all t P 0. Choose the same Ai and B as in (6.2) and (6.3), we have 0 0 0 DA1 ¼ ; DB1 ¼ ; c2 ðtÞ c1 ðtÞ 0:4387 þ c3 ðtÞ 0 0 0 ; DB2 ¼ ; DA2 ¼ c2 ðtÞ c1 ðtÞ 0:4387 þ c3 ðtÞ 0 0 0 ; DB3 ¼ ; DA3 ¼ 0:225 þ c2 ðtÞ c1 ðtÞ 0:4387 þ c3 ðtÞ 0 0 0 ; DB4 ¼ : DA4 ¼ 0:225 þ c2 ðtÞ c1 ðtÞ 0:4387 þ c3 ðtÞ It is noted that the matching condition is also met. Based on (4.6) and (4.7), we can choose drob = 0.4. Following Remark 6, we can choose the robust bounding function, ^rob ðxÞ ¼ 3:375kxk: q
ð6:5Þ
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^rob ðxÞ and q ^nom ðxÞ is that the former has to address The difference between q the additional bound of uncertainty, besides what has already been addressed ^nom ðxÞ. Certainly, this renders a larger magnitude in q ^rob ðxÞ. For simulation in q purpose, we used sinusoidal functions for the uncertainties: c1(t) = 0.5 sin(0.2t), c2(t) = 0.5 cos(0.5t), c3(t) = 0.25 sin(t). Let = 125. To show the effectiveness of ^rob ðxÞ, as compared with q ^nom ðxÞ, in compensating the uncertainty, we use the q
Fig. 1. Displacement under the presence of uncertainty.
Fig. 2. Velocity under the presence of uncertainty.
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Fig. 3. Control forces of nominal and robust controls.
^ðxÞ takes either one of the two forms. The controlcontrol scheme (4.13) when q led system responses are shown in Figs. 1 and 2. The dotted lines denote the ^ðxÞ ¼ q ^nom ðxÞ). The full lines responses due to the nominal control (i.e., when q ^ðxÞ ¼ q ^rob ðxÞ). It is denote the responses due to the robust control (i.e., when q noted that significant improvement in performance is obtained while the robust control is applied. This attests the importance in compensating the uncertainty. Fig. 3 shows the histories of both controls. It is interesting to note that despite ^rob ðxÞ, a smaller robust control magnitude is observed for the use of larger q t P 1 s.
7. Conclusions A systematic way of finding the common positive definite matrix P to satisfy the stability condition (3.2) is first suggested. We further adopt the means for quadratic stability to relax Assumption 1. These appear to be an effective tool for the control design issue. We then proceed with a control scheme for the T–S fuzzy system with or without uncertainty. The uncertainty may or may not meet the matching condition. The resulting controlled system performance, under the matching condition, is (global) practical stability. In the mismatched case, if the mismatched portion of the uncertainty is within a threshold, which is designated by k, as shown in Theorem 5, the same performance is guaranteed. In a nutshell, we have extended a previous framework for stability analysis to control design for a rather general situation.
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Appendix A. Quadratic stability Consider the following system, x_ ¼ ðA þ DAðtÞÞx;
ðA:1Þ
where x 2 Rn is the vector of state variables. We assume that the nominal system matrix A is asymptotically stable. The uncertainty is represented by DA(t), which is an arbitrary Lebesgue measurable matrix function satisfying DAðtÞ 2 X;
for all t P 0;
where X is a compact set and 0 2 X. Definition A.1. The system (A.1) is quadratically stable if there exists a positive definite matrix P and a scalar c > 0 such that T
2
xT ½P ðA þ EÞ þ ðA þ EÞ P x 6 ckxk ;
ðA:2Þ
holds for arbitrary x 2 Rn and E 2 X. To introduce the necessary and sufficient conditions of quadratic stability, some notations are introduced. Sn denotes the set of n · n matrices. S nþ is the set of positive definite matrices. The set of positive semidefinite matrices is represented by S n . The set of n · n asymptotically stable matrices is denoted by Hn. For A 2 Hn, a linear operator LA: Sn ! Sn is defined as LA Q ¼ P
if ðPA þ AT P Þ ¼ Q: n
ðA:3Þ n
n
For any G 2 H , the linear operator HG: S ! S is defined as H GP ¼ W
if ðPG þ GT P Þ ¼ W :
ðA:4Þ
The composition of these two operators is represented by F AG ¼ H G LA :
ðA:5Þ
Definition A.2. Let S be a set in a Banach space. A point p 2 S is defined as a nonprotruded point if there exists a linear segment L such that p 2 L S and is not an endpoint of L. A point which is not a nonprotruded point is naturally called a protruded point. Theorem A.1. Let Q 2 Sn,A 2 Hn and X be convex. Then gðQÞ ¼ min kmin ½DAAþE ðQÞ E2X
can be reached by a protruded point of X.
ðA:6Þ
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Theorem A.1 allows one to search only the protruded points for the maximum of g(Q). Theorem A.2. System (A.1) is quadratically stable if and only if max gðQÞ > 0; Q2T
ðA:7Þ
n
where T ¼ S þ \ fQ : trðQÞ ¼ 1g. Remark A.1. It has been shown in [11] that the maximum problem is convex in the sense that any local extreme is also the global extreme. Thus the search can be carried out by many standard algorithms such as the gradient search.
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