Covariance control with observed-state feedback gains for continuous nonlinear systems using T-S fuzzy models

ISA TRANSACTIONS® ISA Transactions 43 共2004兲 389–398

Covariance control with observed-state feedback gains for continuous nonlinear systems using T-S fuzzy models Wen-Jer Chang,a,* Yi-Lin Yeh,a Kuo-Hui Tsaib,† a

Department of Marine Engineering, National Taiwan Ocean University, Keelung 202, Taiwan, R.O.C. b Department of Computer Science, National Taiwan Ocean University, Keelung 202, Taiwan, R.O.C.

共Received 27 January 2003; accepted 28 February 2004兲

Abstract The design problem of state variance constrained control for stochastic systems has received rather extensive attention in recent years. This paper solves the state variance constrained controller design problem by using the covariance control theory with observed-state feedback gains for continuous Takagi-Sugeno 共TS兲 fuzzy models. By incorporating the technique of state estimation into the practical covariance control theory, a variance constrained control methodology is developed for the continuous TS fuzzy models. Finally, a numerical example is shown to demonstrate the efficiency and applicability of the proposed approach. © 2004 ISA—The Instrumentation, Systems, and Automation Society. Keywords: Continuous Takagi-Sugeno fuzzy models; Observed-state feedback gains; Parallel distributed compensation and covariance control

1. Introduction During the last decade, many efforts have been devoted to both theoretical research and implementation techniques for fuzzy logic controllers. Fuzzy logic control is one of the most useful approaches for utilizing the qualitative knowledge of a system to design a controller. In this paper, the Takagi-Sugeno 共TS兲 type fuzzy model 关1–5兴 is used to represent a nonlinear plant. The TS fuzzy model is described by a set of fuzzy ‘‘IF-THEN’’ rules with fuzzy sets in the antecedents and dynamics systems in the consequent. In this type of fuzzy model, local dynamics in different statespace regions are represented by linear models. The controller design is carried out on the basis of *Corresponding author. Tel: ⫹886-2-24622192, Ext. 7110. E-mail address: [email protected] † Tel: ⫹886-2-24622192, Ext. 6634. E-mail address: [email protected]

the fuzzy model via the so-called parallel distributed compensation 共PDC兲 scheme 关1,5兴. The PDCbased fuzzy controller, which is nonlinear in general, is reprocessed by the fuzzy ‘‘blending’’ of each individual linear controller. Stability analysis and systematic controller design are certainly among the most important issues 关3–9兴 for fuzzy control models. For the stability analysis, it is necessary to find a common positive definite matrix P to satisfy Lyapunov inequalities 关1,4,5兴. This problem can be solved numerically through convex programming algorithms involving the linear matrix inequality 共LMI兲 method 关9,10兴. In practical applications, the states of a system are often not readily available. Under such circumstances, some papers 关11–13兴 dealt with the fuzzy observer design problem for the continuous nonlinear systems using TS fuzzy models. It is known that the quadratic optimization has been the most popular controller design method.

0019-0578/2004/$ - see front matter © 2004 ISA—The Instrumentation, Systems, and Automation Society.

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Chang, Yeh, Tsai / ISA Transactions 43 (2004) 389–398

However, it can guarantee only that the control system state vector as a whole functions well. A straightforward controller design methodology, which is called as covariance control theory 关14 – 19兴, was developed for achieving the individual state variance constraints. It allows the designers to assign state covariance matrix by solving the inverse solution of Lyapunov equation for the closed-loop stochastic systems. This paper considers the covariance control problem of continuous TS fuzzy models in which the complete state vector cannot be measured. According to the fuzzy observer design method and covariance control theory, an observed-state feedback fuzzy controller design methodology is developed to achieve a specified state covariance for the TS fuzzy models. This problem is labeled as ‘‘observed-state feedback covariance control problem 共OSFCCP兲.’’ The main task of this paper is to find the stability conditions and the solutions of OSFCCP for continuous nonlinear systems by using TS fuzzy models. The advantage of this approach is that the covariance matrices of true states and estimation errors may be separately assigned in the design of observed-state feedback controllers and optimal filter gains. The organization of this paper is as follows: Section 2 introduces the properties of the observed-state feedback TS fuzzy control models. It describes the control problem statements and discusses the optimal state estimation for continuous TS fuzzy models. Section 3 finds the conditions and solutions for the OSFCCP. In Section 4, a numerical simulation is presented to demonstrate the feasibility and applicability of this approach. Finally, conclusions are drawn in Section 5. 2. The properties of observed-state feedback TS fuzzy models The fuzzy inference engine uses the fuzzy IFTHEN rules to perform a mapping from an input linguistic vector x⫽ 关 x 1 x 2 ¯ x n x 兴 T苸Rn x to an output variable y苸Rn y . The model is described by fuzzy IF-THEN rules, which represent local linear input-output relations of nonlinear systems. In this section, a TS-type fuzzy model is used to construct a nonlinear stochastic system as follows. Plant rule i:

IF x 1 共 t 兲 is M il ¯

and

x n x 共 t 兲 is M inx ,

THEN x˙ 共 t 兲 ⫽Ai x 共 t 兲 ⫹Bi u 共 t 兲 ⫹Di ␯ 共 t 兲 , 共1兲 y 共 t 兲 ⫽Ci x 共 t 兲 ⫹Ei ␮ 共 t 兲 ,

i⫽1,2,...,r,

where x ( t ) 苸Rn x is the state vector; u ( t ) 苸Rn u is the control input vector; and y ( t ) 苸Rn y is the output vector in the ith rule. The ␯ ( t ) 苸Rn v and ␮ ( t ) 苸Rn ␮ are stationary zero-mean mutually independent white noise processes with covariance V⬎0 and ⍀⬎0, respectively. The matrices, Ai 苸Rn x ⫻n x , Bi 苸Rn x ⫻n u , Ci 苸Rn y ⫻n x , Di n x ⫻n v n y ⫻n ␮ 苸R , and Ei 苸R are constant; r is the number of IF-THEN rules. The M ij are fuzzy sets and it is assumed that Bi is full-column rank. Besides, the pairs ( Ai ,Bi ) and ( Ai ,Ci ) are controllable and observable, respectively. The state and output equations for the model can be represented in term of the rules of Eq. 共1兲 as r

r

i⫽1

i⫽1

x˙ 共 t 兲 ⫽ 兺 h i 共 t 兲 Ai x 共 t 兲 ⫹ 兺 h i 共 t 兲 Bi u 共 t 兲 r

⫹ 兺 h i 共 t 兲 Di ␯ 共 t 兲 , i⫽1

r

r

i⫽1

i⫽1

共2兲

y 共 t 兲 ⫽ 兺 h i 共 t 兲 Ci x 共 t 兲 ⫹ 兺 h i 共 t 兲 Ei ␮ 共 t 兲 , 共3兲 r where h i ( t ) ⫽ ␻ i ( t ) / 兺 i⫽1 ␻ i( t ) , ␻ i( t ) nx ⫽⌸ j⫽1 M ij关 x j ( t ) 兴 , and M ij关 x j ( t ) 兴 is the grade of membership of x j ( t ) in M ij ; ␻ i ( t ) is the weight of the ith rule. In some nonlinear systems, the system states usually cannot be completely measured. Therefore, the designers need to design the fuzzy observers to estimate the states for the fuzzy model in order to implement the controller design. In Ref. 关20兴, the authors consider the so-called separation property for a controller and an observer for the linear stochastic systems. The observers require to satisfy the condition x ( t ) ⫺xˆ ( t ) →0 when t→⬁, where xˆ ( t ) denotes the estimated state vector of the observers. In this paper, the fuzzy observer is described as follows: Observer rule i:

IF x 1 共 t 兲 is M il ¯

and x n x 共 t 兲 is M inx ,

THEN ˙xˆ 共 t 兲 ⫽Ai xˆ 共 t 兲 ⫹Bi u 共 t 兲 ⫹Ki 关 y 共 t 兲 ⫺yˆ 共 t 兲兴 , 共4兲 yˆ 共 t 兲 ⫽Ci xˆ 共 t 兲 , i⫽1,2,...,r,

Chang, Yeh, Tsai / ISA Transactions 43 (2004) 389–398

where Ki 苸Rn x ⫻n y is the observer gain matrix and xˆ ( t ) 苸Rn x is the state vector of observer. The y ( t ) and yˆ ( t ) are the outputs of the fuzzy model and the fuzzy observer, respectively. Then, the final estimated state and output of the fuzzy observer are characterized as follows: r

Eqs. 共2兲 and 共5兲, state and observer equations of the fuzzy model can be described as follows: r

x˙ 共 t 兲 ⫽ 兺 h i 共 t 兲 Ai x 共 t 兲 i⫽1

r

⫹兺

r

˙xˆ 共 t 兲 ⫽ 兺 h i 共 t 兲 Ai xˆ 共 t 兲 ⫹ 兺 h i 共 t 兲 Bi u 共 t 兲 i⫽1

⫹兺 ⫹兺

r

兺

i⫽1 j⫽1

h i 共 t 兲 h j 共 t 兲 Ki E j ␮ 共 t 兲 ,

共5兲

yˆ 共 t 兲 ⫽ 兺 h i 共 t 兲 Ci xˆ 共 t 兲 . i⫽1

共6兲

and x n x 共 t 兲 is M inx i⫽1,2,...,r,

r

r

˙xˆ 共 t 兲 ⫽ 兺

兺

i⫽1 j⫽1

共7兲

r

兺

i⫽1 j⫽1

⫹兺

h i 共 t 兲 h j 共 t 兲 Ki C j 关 x 共 t 兲 ⫺xˆ 共 t 兲兴

r

兺

i⫽1 j⫽1

h i 共 t 兲 h j 共 t 兲 Ki E j ␮ 共 t 兲 .

共10兲

By introducing ˜x ( t ) ⫽x ( t ) ⫺xˆ ( t ) , Rij⫽ 关 ( Ai ˜ ⫽(B G ⫹Bi G j ) ⫹ ( A j ⫹B j Gi ) 兴 /2 and R ij i j ⫹B j Gi ) /2, i⬍ j⭐r, Eq. 共9兲 can be rewritten as r

x˙ 共 t 兲 ⫽ 兺 h i 共 t 兲 h i 共 t 兲共 Ai ⫹Bi Gi 兲 x 共 t 兲 i⫽1

⫹2 兺 h i 共 t 兲 h j 共 t 兲 Rijx 共 t 兲 ⫺ i⬍ j

冋兺 r

i⫽1

r

h i共 t 兲 h i共 t 兲

˜ ˜x 共 t 兲 ⫻Bi Gi˜x 共 t 兲 ⫹2 兺 h i 共 t 兲 h j 共 t 兲 R ij i⬍l

册

r

⫹ 兺 h i 共 t 兲 Di ␯ 共 t 兲 .

共11兲

i⫽1

The observer error dynamic becomes

x8 共 t 兲 ⫽ 兺 h i 共 t 兲 h i 共 t 兲共 Ai ⫺Ki Ci 兲˜x 共 t 兲 i⫽1

⫹2 兺

i⬍ j

共8兲

This observed-state feedback fuzzy controller is nonlinear in general. By substituting Eq. 共8兲 into

冋兺 r

h i 共 t 兲 h j 共 t 兲 Hij˜x 共 t 兲 ⫺ r

r

i⫽1

⫹兺

h i 共 t 兲 h j 共 t 兲共 Ai ⫹Bi G j 兲 xˆ 共 t 兲

r

where r is the number of IF-THEN rule. The overall observed-state feedback fuzzy controller becomes

u 共 t 兲 ⫽ 兺 h i 共 t 兲 Gi xˆ 共 t 兲 .

共9兲

i⫽1

r

The same weight h i ( t ) of the ith rule of the fuzzy model 共2兲 and 共3兲 is used for the fuzzy observer 共5兲 and 共6兲. The design parameter of the fuzzy observer is the gain matrix Ki in each rule. In this paper, the method of PDC 关1,5兴 is used to synthesize fuzzy control laws of observed-state feedback stabilization for the nonlinear systems, which are represented by continuous TS type fuzzy model 共1兲. The basic idea of the PDC method is to design the feedback gains for each rule in the fuzzy models. Linear control design techniques can be used to design these linear controllers for each rule. Hence the final nonlinear controller can be blended by local linear fuzzy controllers, which share the same fuzzy sets with the continuous TS type fuzzy model 共1兲. By using the observed states from the fuzzy observer, the feedback fuzzy controller becomes the following. Observed-state feedback controller rule i:

THEN u 共 t 兲 ⫽Gi xˆ 共 t 兲 ,

h i 共 t 兲 h j 共 t 兲 Bi G j xˆ 共 t 兲

⫹ 兺 h i 共 t 兲 Di ␯ 共 t 兲 ,

r

r

IF x 1 共 t 兲 is M il ¯

兺

r

r

兺 h i共 t 兲 h j 共 t 兲 Ki Cj 关 x 共 t 兲 ⫺xˆ 共 t 兲兴 i⫽1 j⫽1 r

r

i⫽1 j⫽1

i⫽1

r

391

i⫽1

h i共 t 兲 h i共 t 兲

˜ ␮共 t 兲 ⫻Ki Ei ␮ 共 t 兲 ⫹2 兺 h i 共 t 兲 h j 共 t 兲 H ij i⬍1

册

r

⫹ 兺 h i 共 t 兲 Di ␯ 共 t 兲 , i⫽1

共12兲

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Chang, Yeh, Tsai / ISA Transactions 43 (2004) 389–398

where Hij⫽ 关 ( Ai ⫺Ki C j ) ⫹ ( A j ⫺K j Ci ) 兴 /2 and ˜ ⫽ ( K E ⫹K E ) /2. Augmenting Eqs. 共11兲 H ij i j j i and 共12兲 yields r

␹˙ 共 t 兲 ⫽ 兺

r

兺

i⫽1 k⫽1

h i 共 t 兲 h i 共 t 兲 h k共 t 兲关 Li ␹ 共 t 兲 ⫹Nik¯␯ 共 t 兲兴

r

⫹2 兺 h i 共 t 兲 h j 共 t 兲关 Lij␹ 共 t 兲 ⫹Nij¯␯ 共 t 兲兴 , 共13兲 i⬍ j

where

冋 册

冋 册

Lij⫽

冋

Li ⫽

冋

Rij

˜ ⫺R ij

0

Hij

Ai ⫹Bi Gi

⫺Bi Gi

0

Ai ⫺Ki Ci

册

Nij⫽

Nik⫽

,

冋

0 0

冋

Dk

册

0

Dk ⫺Ki Ei

册

Xi ⫽ lim E 关 ␹ 共 t 兲 ␹ 共 t 兲 T兴 .

册

,

共14兲

Let the common covariance matrix for Eq. 共13兲 be X, then

Xaa Xab XTab Xbb

册

,

i⫽1,2,...,r,

共15兲

and X⫽XT⬎0. The common covariance matrix X satisfies the following Lyapunov equation for each rule 关20–23兴:

Li X⫹XLTi ⫹Ni ⌽NTi ⫽0, where

⌽⫽

冋 册 V

0

0

⍀

共18兲

共19兲

Ai Xbb⫹Xbb共 Ai ⫺Ki Ci 兲 T⫹Di VDTi ⫽0, 共20兲

t→⬁

冋

where ␴ ␸ denotes the root-mean-squared 共RMS兲 constraint for the variances of system states. This problem is referred to as the OSFCCP. Based on the common covariance matrix defined in Eq. 共15兲, Refs. 关21–23兴 provided the conditions and solutions for the optimal filter gains Ki as follows:

⫺Xbb共 Bi Gi 兲 T⫹Di VDTi ⫽0,

,

If Li is a stable matrix, the state covariance matrix Xi of each subsystem of Eq. 共13兲 can be defined by 关20–23兴

X⫽Xi ⫽

共17兲

共 Ai ⫹Bi Gi 兲 Xaa⫹Xaa共 Ai ⫹Bi Gi 兲 T⫺Bi Gi Xbb

0 ˜ . ⫺H ij

␸ ⫽1,2,...,n x ,

t→⬁

Ki ⫽XbbCTi 共 Ei ⍀ETi 兲 ⫺1 ,

␯共 t 兲 ¯␯ 共 t 兲 ⫽ ␮共 t 兲 ,

x共 t 兲 ␹ 共 t 兲 ⫽ ˜x 共 t 兲 ,

lim E 关 x 2␸ 共 t 兲兴 ⫽ 关 Xaa兴 ␸␸ ⭐ ␴ ␸2 ,

共16兲

.

The purpose of this paper is to find the set of controllers Gi which satisfy the Lyapunov equation 共16兲 such that the covariance matrix Xaa satisfies the following variance performance objectives:

where Xaa⬎0, Xbb⬎0, and Xab⫽Xbb are defined in Eq. 共15兲. Note that the above conditions are necessary and sufficient 关22兴 for the existence of optimal estimators since they satisfy the WienerHopf equation 关23兴. From the results of Refs. 关21,22兴, it can be found that the optimal filter gain Ki ⫽XbbCiT( Ei ⍀EiT) ⫺1 leads to the fact that the steady-state error between the system state x ( t ) and the estimated state xˆ ( t ) converges to zero when t→⬁. From Eq. 共18兲, if continuous TS type fuzzy model 共1兲 is corrupted only by state noise without measurement noise 共i.e., ⍀⫽0兲, then the optimal filter gain Ki does not exist. A variance constrained design methodology for continuous TS fuzzy models, based on the theory of covariance control, has been developed in Ref. 关24兴. However, it did not consider the observed-state feedback control technique. To offer a lucid presentation of the covariance control theory for continuous TS type fuzzy model 共1兲, we recall the results of the stability of the whole system with the fuzzy observers. The stability conditions of the OSFCCP are stated by using the above optimal estimations 共18兲–共20兲 via the following theorem. Theorem 1 Consider the fuzzy model 共1兲 driven by Eqs. 共5兲 and 共8兲 with the observer gain Ki defined in Eq. 共18兲. If there exist common positive definite matrices Xaa⬎0, Xbb⬎0, Xab⫽Xbb , and ( Xaa⫺Xbb) ⬎0 关as defined in Eq. 共15兲兴 satisfying the following conditions, then the equilibrium of the

Chang, Yeh, Tsai / ISA Transactions 43 (2004) 389–398

observed-state feedback fuzzy control model 共11兲 is asymptotically stable in the large,

Ai Xbb⫹XbbATi ⫺XbbCTi 共 Ei ⍀ETi 兲 ⫺1 Ci Xbb ⫹Di VDTi ⫽0,

共21兲

393

satisfy Eqs. 共22兲 and 共23兲 so that the closed-loop fuzzy model 共11兲 is asymptotically stable. Assigning the common covariance matrix X in advance, the solutions of control feedback gains Gi can be found by using the theory of generalized inverse in next section.

共 Ai ⫹Bi Gi 兲共 Xaa⫺Xbb兲 ⫹ 共 Xaa⫺Xbb兲共 Ai ⫹Bi Gi 兲 T

⫹XbbCTi 共 Ei ⍀ETi 兲 ⫺1 Ci Xbb⫽0,

共22兲

Rij共 Xaa⫺Xbb兲 ⫹ 共 Xaa⫺Xbb兲 RTij ⬍0,

i⬍ j⭐r. 共23兲

Proof: From the previous statements, it is clear that the matrix Ki performs an optimal filter gain if and only if there exist matrices Xaa⬎0, Xbb⬎0 such that Eqs. 共18兲–共20兲 are all satisfied with Xab ⫽Xbb defined in Eq. 共15兲. Substituting Eq. 共18兲 into Eq. 共20兲 and rearranging it yields

Ai Xbb⫹XbbATi ⫹Di VDTi ⫽XbbCTi 共 Ei ⍀ETi 兲 ⫺1 Ci Xbb ,

共24兲

which is equivalent to Eq. 共21兲. From Eq. 共19兲, it is easy to obtain 共 Ai ⫹Bi Gi 兲共 Xaa⫺Xbb兲 ⫹ 共 Xaa⫺Xbb兲共 Ai ⫹Bi Gi 兲 T

⫹Ai Xbb⫹XbbATi ⫹Di VDTi ⫽0.

共25兲

Putting Eq. 共24兲 into Eq. 共25兲, Eq. 共22兲 can be obtained. Thus the conditions 共19兲 and 共20兲 can be replaced by Eqs. 共22兲 and 共21兲 with the observer gain Ki defined in Eq. 共18兲. It can be found that the optimal filter gain Ki defined in Eq. 共18兲 satisfying Eqs. 共19兲 and 共20兲 or 共21兲–共22兲 leads to the fact that the steady-state error of ˜x ( t ) approximates to zero when t→⬁ 关21–23兴. By Theorem 3 of Ref. 关9兴, one can find that if there exist a common positive definite error state covariance matrix ( Xaa⫺Xbb) satisfying Eqs. 共22兲 and 共23兲, then the equilibrium of continuous fuzzy control model 共11兲 is asymptotically stable in the large due to XbbCiT( Ei ⍀EiT) ⫺1 Ci Xbb⭓0 and ˜x ( t ) →0. Hence it can be concluded that if conditions 共21兲–共23兲 are satisfied with the observer gain Ki defined in Eq. 共18兲, then the equilibrium of the observedstate feedback fuzzy control model 共11兲 is asymptotically stable in the large. In order to achieve the stability conditions of Theorem 1, the control feedback gains Gi must

3. The design of observed-state feedback TS fuzzy controllers In this section, the results of above section are applied to develop a method for solving Gi subject to the assigned common covariance matrix X. The conditions and solutions for the existence of Gi can be found in the following theorem. Theorem 2 Consider the fuzzy model 共1兲 driven by Eqs. 共5兲 and 共8兲 with the observer gain Ki ⫽XbbCiT( Ei ⍀EiT) ⫺1 , where Xbb⬎0 satisfies Eq. 共21兲. There exist observed-state feedback gains Gi that achieve stability condition 共22兲 for a specified common positive definite error state covariance ( Xaa⫺Xbb) ⬎0 if and only if the following condition is satisfied: T T ⫹ 共 I⫺Bi B⫹ i 兲共 Ai Xaa⫹XaaAi ⫹Di VDi 兲共 I⫺Bi Bi 兲

⫽0.

共26兲

Moreover, suppose that condition 共26兲 is satisfied, and then the set of all convenient observedstate feedback gains Gi that solve OSFCCP is given by

1 Gi ⫽⫺ B⫹ 关 Ai 共 Xaa⫺Xbb兲 ⫹ 共 Xaa⫺Xbb兲 ATi 2 i ⫹XbbCTi 共 Ei ⍀ETi 兲 ⫺1 Ci Xbb⫺Jik兴 ⫻ 共 Xaa⫺Xbb兲 ⫺1 ⫹ 共 I⫺B⫹ i Bi 兲 ⌽,

共27兲

where ⌽ is arbitrary 共note ⌽⫽0 is such arbitrary兲 and the term Jik is expressed as

冋

册

¯J ¯ ikaa ⫺Ziab ˜ ˜T, Jik⬅Bi ¯ T B i 0 Ziab

共28兲

where ¯Jikaa is an arbitrary skew-symmetric matrix, ˜ is the modal matrix 关19兴 of B BT , and the eleB i i i ¯ is defined by ment Z iab

394

Chang, Yeh, Tsai / ISA Transactions 43 (2004) 389–398

冋

¯ Z ¯ ⬅ iaa Z i ¯T Z iab

¯ Z iab ¯ Z

ibb

册

冋 册

˜ T关 A 共 X ⫺X 兲 ⫹ 共 X ⫺X 兲 AT ⬅B i aa bb aa bb i i ˜ . ⫹XbbCTi 共 Ei ⍀ETi 兲 ⫺1 Ci Xbb兴 B i

共29兲

⫹XbbCTi 共 Ei ⍀ETi 兲 ⫺1 Ci Xbb .

共30兲

关 Bi Gi 共 Xaa⫺Xbb兲兴 ⫹ 关 Bi Gi 共 Xaa⫺Xbb兲兴T⫹Qi ⫽0. 共31兲

Following the results of Ref. 关25兴, Eq. 共31兲 has the solution

1 Bi Gi 共 Xaa⫺Xbb兲 ⫽⫺ 共 Qi ⫺Jik兲 , 2

共32兲

for some skew-symmetric matrix Jik . By the theory of generalized inverses 关19,26兴, it is well known that Eq. 共32兲 has the solution Gi ( Xaa ⫺Xbb) if and only if the following condition is satisfied: 共 I⫺Bi B⫹ i 兲共 Qi ⫺Jik 兲 ⫽0.

共33兲

˜ T from the left and B ˜ from the By multiplying B i i right, Eq. 共33兲 becomes

0

I

册

¯ ⫺J ¯ ¯ ¯ Z iaa ikaa Ziab⫺Jikab ⫽0, 共34兲 ¯ T ⫹J ¯T ¯ ¯ Z iab ikab Zibb⫺Jikbb

¯ ,Z ¯ , and Z ¯ 共i.e., Z ¯ ) are defined in where Z iaa iab ibb i Eq. 共29兲; and ¯Jikaa , ¯Jikab , and ¯Jikbb are defined by ¯J ⬅ ik

冋

¯J ikaa ¯ ⫺JT

ikab

I

冋 册 0

0

0

I

⫽0.

共36兲

T 共 I⫺Bi B⫹ i 兲关 Ai 共 Xaa⫺Xbb 兲 ⫹ 共 Xaa⫺Xbb 兲 Ai

共37兲

Then, a given error state covariance matrix ( Xaa ⫺Xbb) ⬎0 is assignable if there exists a Gi satisfying Eq. 共22兲, i.e.,

冋 册冋

0

˜ TQ B ˜ B i i i

⫹XbbCTi 共 Ei ⍀ETi 兲 ⫺1 Ci Xbb兴共 I⫺Bi B⫹ i 兲 ⫽0.

Qi ⬅Ai 共 Xaa⫺Xbb兲 ⫹ 共 Xaa⫺Xbb兲 ATi

0

0

This is equivalent to

Proof First, let us define

0

0

册

¯J ikab ˜ TJ B ˜ ⬅B i ik i . ¯J ikbb

共35兲

Expanding Eq. 共34兲 reveals that ¯Jikaa is an arbi¯ trary skew-symmetric matrix, ¯Jikab⫽⫺Z iab and ¯J ⫽Z ¯ . Note that ¯J is skew-symmetric ikbb ibb ikbb ¯ is symmetric. Thus ¯J ⫽Z ¯ is poswhile Z ibb ikbb ibb ¯ ¯ sible if and only if Zibb⫽ 关 0 兴 . The condition Z ibb ⫽ 关 0 兴 can be written as

Substituting Eq. 共21兲 into Eq. 共37兲, the condition equation 共26兲 will be a necessity. To prove the sufficient condition, suppose that Eq. 共37兲 is satisfied. Since Eq. 共37兲 is equivalent to ¯ ⫽ 关 0 兴 . Then, from Eqs. 共34兲 and equation Z ibb 共35兲, any choice of skew-symmetric matrix ¯Jikaa yields the skew-symmetric matrix Jik of Eq. 共28兲 that satisfies Eq. 共33兲. Via the consistency of Eq. 共33兲 and the theory of generalized inverses 关19,26兴, there exists a matrix Gi such that Eq. 共32兲 is satisfied for the given positive definite error state covariance ( Xaa⫺Xbb) ⬎0. Adding the transpose of Eq. 共32兲 to itself, one can obtain Eq. 共31兲 since Jik defined in Eq. 共28兲 is skew-symmetric. Equation 共31兲 is equivalent to Eq. 共22兲, thus the sufficiency is proved. Moreover, it is assumed that condition 共26兲 is satisfied and hence Eq. 共27兲 is true for any Jik of the form of Eq. 共28兲. Then Eq. 共27兲 is the general solution to Eq. 共32兲 关19,26兴. The fact that every Gi can be stabilized follows immediately from the Lyapunov stability theory 关19,27兴. Theorem 2 provides the conditions and solutions for the existence of observed-state feedback gains Gi such that the stability condition 共22兲 is satisfied. From the above results, the designing steps for the variance constrained design procedure of the OSFCCP can be summarized as follows. Step 1. Solve the positive definite matrix Xbb from algebraic Riccati-like equation 共21兲. Step 2. Assign the diagonal elements of matrix Xaa to satisfy 关 Xbb兴 ␸␸ ⭐ 关 Xaa兴 ␸␸ ⭐ ␴ ␸2 , ␸ ⫽1,2,...,n x , which can guarantee that the constraint 共17兲 is satisfied. Step 3. Use variance constrained design methodology 关15兴 to solve the off-diagonal elements of Xaa from Eq. 共26兲. Step 4. If the matrix Xaa solved from step 3 does not satisfy the condition ( Xaa⫺Xbb) ⬎0, then go to step 2; otherwise, continue.

Chang, Yeh, Tsai / ISA Transactions 43 (2004) 389–398

395

Step 5. Substitute Xaa and Xbb into Eqs. 共18兲 and 共27兲 to obtain optimal filter gains Ki and observed-stated feedback gains Gi , respectively. Step 6. Substitute Xaa , Xbb , and Gi into Eq. 共23兲 to check whether Eq. 共23兲 is satisfied. If Eq. 共23兲 is not satisfied, it is necessary to go to step 2 to reassign common state covariance matrix Xaa . In the following section, we will illustrate the application of the proposed variance constrained design approach for the OSFCCP with a numerical example.

4. A numerical example Applying the proposed approach to design the fuzzy controller and the fuzzy observer, it is necessary to use a TS type fuzzy model to represent the dynamics of a nonlinear stochastic plant. To minimize the design effort and complexity, we try to use as few rules as possible. In this section, a nonlinear continuous stochastic system is considered as follows:

x˙ 1 共 t 兲 ⫽⫺2x 2 共 t 兲 ⫹ ␯ 共 t 兲 ,

共38a兲

x˙ 2 共 t 兲 ⫽⫺ 关 9.598⫺2.048 cos x 1 共 t 兲兴 sin x 1 共 t 兲

1 u共 t 兲 4⫺1.44 cos x 1 共 t 兲 共38b兲

y 共 t 兲 ⫽⫺3x 1 共 t 兲 ⫹4x 2 共 t 兲 ⫹ ␮ 共 t 兲 ,

共38c兲

where the covariance matrices of zero-mean white noises ␯ ( t ) and ␮ ( t ) are V⫽10 and ⍀⫽1, respectively. It is assumed that the nonlinear state variable x 1 ( t ) 苸 ( ⫺ ␲ /2,␲ /2) . The nonlinear system 共38兲 can be represented by the following two-rule 共i.e., r⫽2) TS fuzzy model 关9兴. Plant rule 1: IF x 1 ( t ) is about 0

THEN x˙ 共 t 兲 ⫽A1 x 共 t 兲 ⫹B1 u 共 t 兲 ⫹D1 ␯ 共 t 兲 , 共39a兲 y 共 t 兲 ⫽C1 x 共 t 兲 ⫹E1 ␮ 共 t 兲 ,

IF x 1 共 t 兲 is about ⫾

冉

␲ ␲ 兩 x 1兩 ⬍ 2 2

y 共 t 兲 ⫽C2 x 共 t 兲 ⫹E2 ␮ 共 t 兲 , where

A1 ⫽

冋

⫺2

0 ⫺7.55

4.3

冋

C1 ⫽ 关 ⫺3 4 兴 ,

⫹ 关 0.5⫹0.5 cos x 1 共 t 兲兴 ␯ 共 t 兲 ,

Plant rule 2:

THEN x˙ 共 t 兲 ⫽A2 x 共 t 兲 ⫹B2 u 共 t 兲 ⫹D2 ␯ 共 t 兲 , 共39b兲

册

,

册

0 B1 ⫽ 0.3906 ,

⫹ 关 5.8⫺1.5 cos x 1 共 t 兲兴 x 2 共 t 兲 ⫹

Fig. 1. The membership function of x 1 (t).

冊

A2 ⫽

冋

0

⫺2

⫺6.11

5.8

冋 册

冋 册

,

0 B2 ⫽ 0.25 ,

冋册

1 D1 ⫽ 1 ,

C2 ⫽ 关 ⫺3 4 兴 ,

1 D2 ⫽ 0.5 ,

册

E1 ⫽E2 ⫽1.

In Fig. 1, the membership functions of nonlinear state variable x 1 ( t ) are described for the TS fuzzy model 共39兲. In this numerical example, it is assumed that the constraints for state variances of the system 共38兲 are 关 Xaa兴 11⭐5,

关 Xaa兴 22⭐6.

共40兲

Following the fuzzy controller design procedure, which is developed in Section 3, the OSFCCP can be solved as follows: Step 1: Solving the algebraic Riccati-like equation 共21兲, one can obtain the following positive definite matrix Xbb :

Xbb⫽

冋

2.29

1.1329

1.1329 1.4096

册

.

共41兲

396

Chang, Yeh, Tsai / ISA Transactions 43 (2004) 389–398

Step 2: Subject to the state variance constraints 共40兲, the diagonal entries of Xaa are accordingly assigned as 关 Xaa兴 11⫽4.5 and 关 Xaa兴 22⫽5. Step 3: Applying the variance constrained design methodology 关15兴 to solve the off-diagonal elements of Xaa from Eq. 共26兲 yields

Xaa⫽

冋

4.5 2.5 2.5

5

册

共42兲

.

Putting the matrices Xaa and Xbb into Eq. 共15兲, the common covariance matrix X becomes

X⫽

冋

4.5

2.5

2.5

5

2.29

1.1329

2.29

1.1329

1.1329 1.4029 2.29

1.1329

1.1329 1.4029 1.1329 1.4029

册

. 共43兲

Step 4: Subtracting Xbb from Xaa , one can obtain

Xaa⫺Xbb⫽

冋

2.2100 1.3671 1.3671 3.5904

册

⬎0.

共44兲

Step 5: Since the conditions 共21兲, 共26兲, and Xaa ⫺Xbb⬎0 are all satisfied, thus the optimal filter gains Ki and observed-stated feedback gains Gi can be obtained by substituting Xaa and Xbb into Eqs. 共18兲 and 共27兲, respectively,

K1 ⫽ 关 ⫺2.3385 2.2396兴 T, K2 ⫽ 关 ⫺2.3385 2.2396兴 T,

共45兲

G1 ⫽ 关 39.5919 ⫺20.5119兴 , G2 ⫽ 关 56.1024 ⫺38.0499兴 .

Fig. 2. The responses of x 1 (t) for controlled nonlinear system 共38兲 and fuzzy model 共39兲.

the state responses of x ( t ) of controlled nonlinear system 共38兲 and TS fuzzy model 共39兲. Figs. 4 and 5 show the responses of true states x ( t ) and the estimated state xˆ ( t ) for the controlled TS fuzzy model 共39兲. Besides, the responses of control input are shown in Fig. 6. From these simulation results, the state variances of closed-loop nonlinear system 共38兲 are calculated as follows:

var关 x 1 共 t 兲兴 ⫽0.2055

and var关 x 2 共 t 兲兴 ⫽0.2473, 共48兲

where var关 x ᐉ ( t ) 兴 denotes the state x ᐉ ( t ) , ᐉ⫽1, 2. It can closed-loop system is stable state variance constraints 共40兲

variance of system be found that the and the individual are also achieved.

共46兲

Step 6: Substituting Xaa , Xbb , G1 , and G2 into Eq. 共23兲 yields

R12共 Xaa⫺Xbb兲 ⫹ 共 Xaa⫺Xbb兲 RT12 ⫽

冋

⫺5.4685

6.1169

6.1169

⫺10.6966

册

⬍0.

共47兲

It can be found that the stability condition 共23兲 is also satisfied. Since conditions 共21兲–共23兲 are all satisfied, it can be concluded that the continuous TS fuzzy model 共39兲 is asymptotically stable by applying the optimal filter gains Ki of Eq. 共45兲 and the observed-stated feedback gains Gi of Eq. 共46兲. In the simulation, the initial states are given as 关 x 1 ( 0 ) x 2 ( 0 ) 兴 T⫽ 关 1 ⫺2.5兴 T. Figs. 2 and 3 show

Fig. 3. The responses of x 2 (t) for controlled nonlinear system 共38兲 and fuzzy model 共39兲.

Chang, Yeh, Tsai / ISA Transactions 43 (2004) 389–398

Fig. 4. The responses of true state x 1 (t) and estimated state xˆ 1 (t) for controlled fuzzy model 共39兲.

5. Conclusions This paper considered the synthesis of a class of nonlinear stochastic control systems, whose state variables cannot be completely measured. The nonlinear systems are modeled by the TS type fuzzy models. The optimal filtering control technique has been used to design the observers for the TS fuzzy models. Applying these optimal observers, this paper first introduced the conditions for the existence of observed-state feedback gains, and then the theory of generalized inverse was used to solve the observed-state feedback gains for the TS fuzzy controllers. Based on the observed fuzzy control technique, the proposed approach al-

Fig. 5. The responses of true state x 2 (t) and estimated state xˆ 2 (t) for controlled fuzzy model 共39兲.

397

Fig. 6. The control input u(t).

lows the designers to assign the common state covariance matrix to achieve the individual state variance constraints. Acknowledgments The authors wish to express their sincere gratitude to three anonymous reviewers who gave them some constructive comments, criticisms, and suggestions. This work was supported by the National Science Council of the Republic of China, under Contract No. NSC91-2213-E-019-003. References 关1兴 Wang, H. O., Tanaka, K., and Griffin, M. F., An approach to fuzzy control of nonlinear systems: Stability and design issues. IEEE Trans. Fuzzy Syst. 4, 14 –23 共1996兲. 关2兴 Lian, K. Y., Chiu, C. S., Chiang, T. S., and Liu, P., LMI-based fuzzy chaotic synchronization and communications. IEEE Trans. Fuzzy Syst. 9, 539–553 共2001兲. 关3兴 Cuesta, F., Gordillo, F., Aracil, J., and Ollero, A., Stability analysis of nonlinear multivariable TakagiSugeno fuzzy control systems. IEEE Trans. Fuzzy Syst. 7, 505–520 共1999兲. 关4兴 Tanaka, K., Ikeda, T., and Wang, H. O., A unified approach to controlling chaos via an LMI-based fuzzy control system design. IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 45, 1021–1040 共1998兲. 关5兴 Wang, H. O., Tanaka, K., and Griffin, M. F., Parallel distributed compensation of nonlinear systems by Takagi-Sugeno’s fuzzy model. Proceedings of the FUZZY-IEEE’95, 1995, 531–538. 关6兴 Park, C. W., Kim, J. H., Kim, S., and Park, M., LMIbased quadratic stability analysis for hierarchical fuzzy systems. IEE Proc.-D: Control Theory Appl. 148, 340–349 共2001兲.

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Wen-Jer Chang received the B.S. degree from National Taiwan Ocean University, Taiwan, R.O.C., in 1986; the M.S. degree and the Ph.D. degree from the National Central University, Taiwan, R.O.C., in 1990 and 1995, respectively. He is currently a professor and chairman with the Department of Marine Engineering of the National Taiwan Ocean University, Taiwan, R.O.C. He is now a member of the IEEE, CIEE, CACS, CSFAT, and SNAME. In 2003, Dr. Chang was listed in the Marquis Who’s Who in Science and Engineering of 2003–2004, and won the outstanding young control engineers award granted by the Chinese Automation Control Society 共CACS兲. His recent research interests are fuzzy control, robust control, performance constrained control.

Yi-Lin Yeh was born on December 24, 1978 in Taiwan, R.O.C. He received the B.S. degree in the Department of Marine Engineering of the National Taiwan Ocean University, Taiwan, R.O.C., in 2002. He is currently working toward the M.S. degree in the Department of Marine Engineering of the National Taiwan Ocean University. His research interests focus on fuzzy control and dynamic system control.

Kuo-Hui Tsai received the B.S. degree in Electrical Engineering from National Tsing Hua University, Hsin-Chu, Taiwan, in 1982. He received the M.S. degree in Computer Engineering from National Chiao Tung University, Hsin-Chu, Taiwan, in 1984. He received the Ph.D. degree in Electrical Engineering and Computer Science from Northwestern University, Evanston, Illinois, in 1991. He is currently an Associate Professor in the Department of Computer Science at the National Taiwan Ocean University. His research interests include computer networks and embedded system.