Global optimal fuzzy tracker design based on local concept approach

(1)

Global Optimal Fuzzy Tracker Design Based on

Local Concept Approach

Shinq-Jen Wu and Chin-Teng Lin, Senior Member, IEEE

Abstract—In this paper, we propose a global optimal fuzzy tracking controller, implemented by fuzzily blending the indi-vidual local fuzzy tracking laws, for continuous and discrete-time fuzzy systems with the aim of solving, respectively, the continuous and discrete-time quadratic tracking problems with moving or model-following targets under finite or infinite horizon (time). The differential or recursive Riccati equations, and more, the differential or difference equations in tracing the variation of the target are derived. Moreover, in the case of time-invariant fuzzy tracking systems, we show that the optimal tracking controller can be obtained by just solving algebraic Riccati equations and algebraic matrix equations. Grounding on this, several fascinating characteristics of the resultant closed-loop continuous or discrete time-invariant fuzzy tracking systems can be elicited easily. The stability of both closed-loop fuzzy tracking systems can be ensured by the designed optimal fuzzy tracking controllers. The optimal closed-loop fuzzy tracking systems cannot only be guaranteed to be exponentially stable, but also be stabilized to any desired degree. Moreover, the resulting closed-loop fuzzy tracking systems possess infinite gain margin; that is, their stability is guaranteed no matter how large the feedback gain becomes. Two examples are given to illustrate the performance of the proposed optimal fuzzy tracker design schemes and to demonstrate the proved stability properties.

Index Terms—Degree of stability, exponentially stable, gain margin, global minimum, model-following, moving target, Riccati equation.

I. INTRODUCTION

A

LTHOUGH the work in fuzzy modeling and fuzzy control has been quite matured [1]–[8], the field of optimal fuzzy control is nearly open [9]. In particular, although fuzzy logic concept has been introduced into tracking control [10]–[15], the field of theoretical approach of optimal fuzzy tracking control is fully open. The goal of this work is to propose a design scheme of the global optimal fuzzy tracking controller to control and stabilize a discrete-time or continuous fuzzy system in solving, respectively, the discrete-time or continuous quadratic tracking problems with moving or model-following targets under finite or infinite horizon.

To date, the fuzzy tracking controller has been used in ceptual design only, and has always been grounded on a con-ventional tracker. For example, Ott, et al. [13] included fuzzy

Manuscript received November 20, 2000; revised July 10, 2001 and August 27, 2001. This work was supported by the R.O.C. National Science Council under Grant number NSC 90-2213-E-009-103.

S.-J. Wu is with the Department of Electrical Engineering, Da-Yeh University, Chang-Hwa, Taiwan, R.O.C.

C.-T. Lin is with the Department of Electrical and Control Engineering, Na-tional Chiao-Tung University, Hsinchu, Taiwan, R.O.C.

Publisher Item Identifier S 1063-6706(02)02962-4.

logic into an – tracker algorithm; Lea, et al. [11] used fuzzy concept to develop the software algorithm of a camera tracking system. No theoretical demonstration has been developed for fuzzy tracking controller design in the literature.

Stability and optimality are the most important requirements

for any control system. Most of the existed works on the sta-bility analysis of fuzzy control are based on Takagi–Sugeno (T–S)-type fuzzy model combined with the parallel distribu-tion compensadistribu-tion (PDC) concept [1], and apply Lyapunov’s method to do stability analysis. Tanaka and coworkers reduced the stability analysis and control design problems to linear ma-trix inequality (LMI) problems [2], [4]. They also dealt with uncertainty issue [3]. This approach had been applied to sev-eral control problems such as control of chaos [4] and of artic-ulated vehicle [5]. A frequency shaping method for systematic design of fuzzy controllers was also done by them [16]. Sun, et

al. developed a separation scheme to design fuzzy observer and

fuzzy controller independently [6]. Methods based on grid-point approach [17] and circle criteria [18], [19] were introduced to do stability analysis of fuzzy control, as well. Wang adopted a supervisory controller and introduced stability and robustness measures [20]. Cao proposed a decomposition principle to de-sign a discrete-time fuzzy control system and an equivalent prin-ciple to do stability analysis [8]. On the issue of optimal fuzzy control, Wang developed an optimal fuzzy controller to stabilize a linear continuous time-invariant system via the Pontryagin minimum principle [9]. Although fuzzy control of linear sys-tems could be a good starting point for a better understanding of some issues in fuzzy control synthesis, it does not have many practical implications since using the fuzzy controller designed for a linear system directly as the controller may not be a good choice [9]. Moreover, the cited stability criteria may be simple, but rough to do systematic analysis and also may result in a controller with less flexibility. Tanaka and coworkers [21], [22] tried to obtain a fuzzy controller to minimize the upper bound of the quadratic performance function by LMI approach based on the assumption of local-linear-feedback-gain control structure. Nevertheless, no theoretical analysis on this design scheme of optimal-fuzzy-control structure was proposed.

In our previous paper [23], we proposed a global optimal and stable fuzzy controller design method for both continuous and discrete-time fuzzy systems under both finite and infinite hori-zons. Several fascinating characteristics, exponential stability, finite energy, any prescribed degree of stability and infinite gain margin, have been shown to exist in the closed-loop fuzzy sys-tems for the infinite-horizon optimal control problem [23], [24]. In this paper, we shall develop the relative theories and tech-niques for the fuzzy tracking problems.

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Technical contributions of this paper can be described as fol-lows. Based on the local concept approach, the global optimal fuzzy controllers with the aim of tracking moving or model-fol-lowing targets under finite or infinite horizon (time) for both continuous and discrete-time fuzzy systems are theoretically derived. The optimal closed-loop time-invariant fuzzy tracking systems are guaranteed to be exponentially stable. Furthermore, we elicit that the proposed fuzzy tracking controllers can stabi-lize fuzzy tracking systems to any prescribed degree of stability, and the corresponding closed-loop fuzzy tracking systems pos-sess infinite gain margin. The design methodologies are illus-trated by examples.

II. SYSTEMREPRESENTATION ANDPROBLEMSTATEMENT We adopt the following T–S type fuzzy model as the fuzzy tracking system describing the given nonlinear plant:

If is is then

(1) where denotes the th rule of the fuzzy model;

are system states; are the input fuzzy terms in the

th rule; denotes for continuous case and

for discrete case; is the state

vector, is the system output vector,

and is the system input (i.e., control output); and

and are, respectively, and

matrices. The desired tracking controller is then assumed to be in a rule-based nonlinear fuzzy inference form of

If is is

then (2)

where are the elements of output vector

are the input fuzzy terms in the th control rule, and the plant input (i.e., control output) vector

or is in space.

Then, the optimal fuzzy tracker design scheme is to control the

fuzzy tracking system in such a way to push the output close to any desired target without excessive control-energy consumption. We describe the quadratic optimal fuzzy tracking

control problem as follows.

Problem 1: Given the rule-based fuzzy tracking system in

(1) with and a rule-based fuzzy tracking

controller in (2), find the individual optimal tracking law, , such that the composed optimal tracking controller, , can minimize the quadratic cost functional [25], , over all possible inputs of class piece-wise-continuous (PC)

(3)

(4)

for discrete-time and continuous systems, respectively, where

(5)

and are, respectively, and

nonnegative symmetric matrices; is the

state-trajectory penalty to produce smooth response;

is fuel consumption; and the last term in is related to error cost. Moreover, the performance index in (3) and (4) with

in (5) can be, respectively, rewritten as [25]

(6)

(7)

where and the desired

trajec-tory .

Adopting the same local-concept-based optimization tech-nique in our previous papers [23], [24], we know the optimal global decisions for the quadratic fuzzy tracking problem can be regarded as a series of optimal global decisions based on the following successively on-going local quadratic optimal fuzzy tracking issue with the initial state resulting from the previous decision. The time dependence is denoted by lower index for notation simplification.

Problem 2: Given the fuzzy tracking subsystem

(8) with the initial state resulting from the previous decision, i.e., 1) find the optimal local decision at time , for

mini-mizing the cost functional

(9)

(10) 2) obtain the optimal global decision at time , for

minimizing the cost functional in (9) or

(10) by fuzzily blending each local decision, i.e., .

Notice that the next-decision initial state is instead of

, since there exists the one-to-one correspondence relationship between each fuzzy tracking subsystem and the corresponding fuzzy tracking law.

III. OPTIMALFUZZYTRACKERDESIGN

We shall design the optimal trackers for the discrete-time systems in Section III-A and for continuous systems in Sec-tion III-B.

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A. Optimal Fuzzy Tracker for Discrete-Time Fuzzy System

We are going to design the optimal fuzzy tracking controllers for the discrete-time fuzzy tracking systems with moving target in Section III-A.1 and for that with model-following target in Section III-A.2.

1) Moving-Target Tracking Problem: In this subsection, we

shall discuss the finite-horizon tracking problem first, and then generalize the results into infinite-horizon tracking solutions. The local quadratic optimization problem is obviously the same as the general linear quadratic optimal tracking control problem. Therefore, it is reasonable that solving the optimal tracking problem for each fuzzy subsystem can be achieved in the way of the conventional approach. We can use cal-culus-of-variations method combined with Lagrange-multiplier method to derive the local optimal fuzzy tracking law, and then,

fuzzily blend these tracking laws to achieve the global optimal

fuzzy tracking controller.

Theorem 1 (Time-Varying Finite-Horizon Case): For the

discrete-time fuzzy tracking system and discrete-time fuzzy tracking controller represented, respectively, by (1) and (2), let be given matrices, then the follwoing hold.

1) The local optimal tracking law is

(11) and their “blending” global optimal tracking controller

(12)

minimizes in (3), where , the local

feed-back gain, and , the introduced external local input, are given by

(13)

(14) where is the symmetric positive–semidefinite solution of the following recursive Riccati equation:

(15)

with , zero matrix of dimension , and

being the introduced target-dependent variable satisfying

(16) 2) The optimal feedback subsystem is

(17)

and then the global optimal closed-loop tracking system is

(18)

Proof: The proof is similar to the derivation in our

pre-vious papers [23], [24].

So far, we have solved the optimal finite-horizon fuzzy tracking problem by finding the optimal solution to the general time-varying case. We are now concerned with the infinite-horizon tracking problem, which is the case that the operating time goes to infinity or is much larger than the time-constant of the dynamic system. In other words, the performance index is

(19) We are eager to know if a time-invariant fuzzy tracking system can give rise to a time-invariant linear optimal tracking law with regard to each subsystem, and then generate a more imple-mentable and important design scheme. The following theorem demonstrates that a time-invariant fuzzy tracking system cannot give rise to the time-invariant linear optimal fuzzy tracking law except in the case of constant target.

Theorem 2 (Time-Invariant Infinite-Horizon Case):

Con-sider the discrete time-invariant fuzzy tracking system and discrete-time fuzzy tracking controller described, respectively, by (1) and (2). If is completely controllable (c.c.) and is completely observable (c.o.) for all , then the following hold.

(21) minimizes in (19), where the local constant feed-back gain, , and the external local input, , are calculated by

(22) (23)

(24) where is the unique symmetric positive semidefinite solution of the following discrete-time algebraic Riccati equation:

(4)

2) The optimal feedback subsystem is

(26) and then the global optimal closed-loop tracking system is

(27)

Proof:

1) The optimal solution indeed follows the optimal solution in Theorem 1 except that all the parameters in (1), (2), and (19) are now constant. It is easy to show that the asymp-totic solution of the recursive Riccati equation in (15) is also the steady-state solution, i.e.,

, and this solution results in the asymptotic solution of the recursive equation in (16) which is equivalent to

in (24), i.e., .

2) Moreover, according to Lemma 2 in the Appendix, we

know that being c.c. and being c.o.,

, guarantee the existence of an unique sym-metric positive semidefinite solution of the algebraic Ric-cati equation in (25). Hence, the proof is completed.

2) Model-Following Tracking Problem: In this subsection,

we are devoted to the model-following tracking problem, where the tracked target is the response of some reference model. Similar to the previous subsection, the finite-horizon tracking problem is discussed first. The derived optimal solutions can then be generalized into those for the infinite-horizon problem as we did in the last subsection. We adopt the same T-S type fuzzy tracking system as in Section II, and thereupon, the standard model-following tracking problem can be described as the following issue.

Problem 3: Given a discrete-time fuzzy tracking system and

a fuzzy tracking controller, respectively, in (1) and (2) with

and , find to minimize

in (3), where the desired output is the response of a linear system or model

(28)

with , to the command input , which is

the zero-input response of the system:

and with [25], where

and are system states;

and are matrices of

and , respectively.

Accordingly, the desired tracked system, via letting , can be rewritten as the following augmented system [25]:

(29)

We further define a new variable [25],

and then Problem 3 can be simplified as the following issue.

Problem 4: Given a fuzzy tracking system

(30)

with and

, find to minimize

(31)

where the parameters are as shown in the equation at the bottom of the page. Notice that the fuzzy tracking controller is

.

Obviously, the optimal solutions for the augmented optimal quadratic tracking problem in Problem 4 follow from Theorem 1 except that in Section III-A.1 are zero vectors now. Then, via complicated matrix manipulations, we can obtain the optimal solutions for the original optimal quadratic tracking problem in Problem 3 as follows. The identity input weighting factor is set to get more concise formula in the remainder of this section, i.e., for all time steps.

fuzzy tracking system and fuzzy tracking controller repre-sented, respectively, by (1) and (2), let the desired trajectory,

, come from , where is the output

of the tracked model in (28). Then, the following hold. 1) The local optimal tracking law is

(33)

(5)

minimizes in (3), where the feedback gain, , and the introduced matrix, , are calculated by

(34)

(35) where is the symmetric positive–semidefinite solution of the following recursive Riccati equation:

(36)

with , and satisfies

(37)

with .

(39)

Proof:

1) For notation simplification, the identity and zero matrices of any dimension will be denoted by and , respectively. We can still, based on the inference in Section II, decom-pose the quadratic tracking problem in Problem 4 into linear quadratic tracking problems as in Problem 2 ex-cept that are zero vectors now. Then, grounding on Theorem 1, we have the following local optimal solu-tion:

(40)

(41) where is the symmetric positive–semidefinite solu-tion of the the following recursive Riccati equasolu-tion:

(42) 2) Now, let

We obtain in (32) from (41) via

(43) and

, where the time-dependence is omitted for notation simplification; in (36) and in (37) are derived from (42) with the aid of ; and then we have in (38) from (40).

3) We then fuzzily blend the local optimal tracking laws and optimal tracking subsystems to obtain the corre-sponding optimal tracking controller in (33) and the optimal trajectory in (39), respectively. The scheme of generalizing the optimal tracking solution from finite-horizon problem to infinite-horizon problem for model-following target is just the same as that for moving target in Section III-A1. Therefore, we only summarizes the solutions of the infinite-horizon problem with respect to the model-following tracking issue as follows.

Theorem 4 (Time-Invariant Infinite-Horizon Case): For the

time-invariant fuzzy tracking system and fuzzy tracking con-troller represented, respectively, by (1) and (2), let the desired

trajectory, , come from , where

is the output of the tracked model in (28). If is c.c. and is c.o., for all , then, the “blending” global optimal tracking controller is

(44) which minimizes in (19); and the corresponding global optimal closed-loop tracking system is

(45) where

is the symmetric positive–semidef-inite solution of the recursive Riccati equation

(46) and satisfies

(47)

Proof: The proof, grounded on Theorem 3, follows the

same generalization of the finite-horizon case to the infinite-horizon case as that in Theorem 2.

B. Optimal Fuzzy Tracker for Continuous Fuzzy System

We are now going to design the optimal fuzzy tracking con-trollers for the continuous fuzzy tracking systems with moving target in Section III-B1 and for that with model-following target in Section III-B2.

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1) Moving-Target Tracking Problem: As remarked in

Sec-tion II, we know that, via describing the fuzzy system from the local inspection, the nonlinear quadratic optimal fuzzy tracking problem in Problem 1 can be described by the linear quadratic problem in Problem 2. Hence, the continuous fuzzy tracking problem can be solved by obtaining the local optimal decision or the tracking solutions for each fuzzy subsystem by the con-ventional approach first, and then fuzzily blending the local lutions to obtain the global optimal decision or the optimal so-lutions for the entire continuous fuzzy system as follows.

continuous fuzzy tracking system and fuzzy tracking controller represented, respectively, by (1) and (2), let

be given matrices, then the following hold. 1) The local optimal tracking law is

(49)

minimizes in (4), where is the

sym-metric positive–semidefinite solution of the following dif-ferential Riccati equation:

(50)

with , zero matrix of dimension , and

is the introduced target-dependent variable satisfying

(53)

Proof: Grounding on the calculus-of-variations method,

we introduce the Lagrange multiplier and let

, whence we obtain the local optimal tracking law in (48), and the optimal feedback subsystem in (52), where and satisfy, respectively, (50) and

(51) with and . Accordingly, we

fuzzily blend the local optimal results to get the corresponding optimal tracking controller in (49) and the optimal tracking trajectory in (53).

The aforementioned theorem covers the general continuous time-varying quadratic optimal problems, where the horizon

is fixed and is arbitrary. We are now concerned with the time-invariant infinite-horizon tracking problem. In other words, the performance index is

(54) We shall demonstrate that, even in the infinite-horizon situation, a time-invariant fuzzy tracking system cannot give rise to the time-invariant linear optimal fuzzy tracking law except in the case of constant target.

Theorem 6 (Time-Invariant Infinite-Horizon Case):

Con-sider the continuous time-invariant fuzzy tracking system and fuzzy tracking controller described, respectively, by (1) and

(2). If is c.c. and is c.o. for all ,

then the following hold.

(56) minimizes in (54), where is the unique sym-metric positive–semidefinite solution of the following steady state Riccati equation (SSRE):

(57) and the target-dependent variable is

(60)

Proof: Based on the optimal solutions in Theorem 5,

we can obtain the infinite-horizon time-invariant optimal solution by letting . Moreover, we know that the asymptotic solution of the differential Riccati equation in (50) is also the solution of the algebraic Riccatic equation in

(57), i.e., , and this solution results

in the asymptotic solution of the differential equation in (51)

which is equivalent to in (58), i.e., .

Furthermore, since is c.c. and is c.o., for all , we know, via the linear quadratic theory [25], that the symmetric positive–semidefinite solution of the algebraic Riccati equation in (57) uniquely exists.

2) Model-Following Tracking Problem: We are now

con-cerned with the fuzzy tracking problem with the target from the response of some reference model, i.e., the model-following tracking problem. The optimal solutions for the finite-horizon

(7)

tracking problem are derived first, and then they are generalized into the infinite-horizon situation as we did in the last subsec-tion. The same T–S type fuzzy tracking system as in Section II is adopted, and a continuous model-following tracking problem is formulated as follows.

Problem 5: Given a continuous fuzzy tracking system and

a fuzzy tracking controller, respectively, in (1) and (2) with and , find the individual optimal

tracking law, , such that the composed

op-timal tracking controller, , can minimize in (4), where the desired output is the response of a linear system or model,

(61)

with , to the command input , which is

the zero-input response of the system: and

with [25], where

and are system states;

and are matrices of and

, respectively.

Similar to Section III-A2, we can get a more concise problem formulation.

Problem 6: Given a fuzzy tracking system

(62)

with and

, find the individual optimal tracking law, , to minimize

(63)

where the parameters are shown in the equation at the bottom of the page.

Obviously, the optimal solutions for the augmented optimal quadratic tracking problem in Problem 6 are the same as those in Theorem 5 by setting as zero vectors now. Then via fur-ther matrix manipulations, we can obtain the optimal solutions for the original optimal quadratic tracking problem in Problem 5 as follows.

continuous fuzzy tracking system and the fuzzy tracking con-troller represented, respectively, by (1) and (2), let the desired

trajectory, , come from , where

is the output of the tracked model in (61). Then, the following hold.

(65)

minimizes in (4), where satisfies

(66) and is the symmetric positive–semidefinite so-lution of the differential Riccati equation in (50) with

.

(68)

Proof: The proof is similar to that of Theorem 3.

We now let the finite horizon approach infinity in order to get the infinite-horizon optimal solutions as follows.

Theorem 8 (Time-Invariant Infinite-Horizon Case): For the

continuous time-invariant fuzzy tracking system and the fuzzy tracking controller represented, respectively, by (1) and (2), let

the desired trajectory, , come from ,

(8)

where is the output of the tracked model in (61). If

is c.c. and is c.o., for all , then

the following hold.

(70) minimizes in (54), where is the unique sym-metric positive–semidefinite solution of the SSRE in (57), and

(73)

Proof: The proof, grounded on Theorem 7, follows the

same generalization of the finite-horizon case to the infinite-horizon case as that in Theorem 6.

IV. STABILITY ANDGAINMARGIN

So far, the design scheme of the fuzzy trackers for both con-tinuous and discrete-time fuzzy systems have been developed. We are now devoted to the stability analysis of both kinds of re-sultant closed-loop fuzzy tracking systems. In this section, we shall show that the designed fuzzy tracking controllers can not only exponentially stabilize the fuzzy tracking system, but also form a closed-loop time-invariant fuzzy tracking system with any desired degree of stability. We are also concerned with the range of the feedback gain, gain margin, to which we can in-crease under the stability consideration.

In other words, for the continuous case, we discuss the sta-bility of the following two systems:

(74)

for the moving-target tracking problem, and

(75) for the model-following-target tracking issue. Since

in (74) and in (75) are both associated with

the target only, they can be regarded as external local inputs, . Therefore, we can unify these two equations into

(76) which is a nonlinear system constituted by a set of linear fuzzy subsystems. Then, based on the converse theorem of Lyapunov stability theory [28], we know the stability of the nonlinear tracking system in (76) is coincident with that of the corre-sponding linearized system (with regard to )

(77) Therefore, the stability of the fuzzy system in (76) is governed

by the term , which also

han-dles the stability of the following zero-input fuzzy system: (78) Hence, we shall focus only on discussing the stability of the zero-input fuzzy system in the above to demonstrate the stability of the resultant closed-loop fuzzy tracking system in (76) or (74) and (75).

Furthermore, we have demonstrated in our previous paper [23] that the aforementioned zero-input fuzzy system in (78) is exponentially stable, and possesses any degree of stability and infinite gain margin if each subsystem is c.c. and c.o. (well-be-haved). Therefore, we conclude that the resultant continuous closed-loop fuzzy tracking system in (76) or (74) and (75) also possess such fantastic characteristics.

We now step for analyzing the stability property of the resul-tant discrete-time closed-loop fuzzy tracking system. In other words, the stability of the following two systems are discussed first:

(79) for the moving-target tracking problem, and

(9)

for the model-following-target tracking issue. Since in (80) is associated with the target only and can be regarded as an external local input, , (79) and (80) can be unified into one equation [(79)] by setting .

Grounding on the converse theorem, we know the stability characteristics for both discrete-time issues of moving-target tracking and model-following-target tracking can be guaranteed by the stability of the following zero-input fuzzy system:

(81) which has been demonstrated to be exponentially stable and to possess any degree of stability for each well-behaved subsystem in [24].

In the remainder of this section, we shall examine another characteristic, gain margin, of the resultant discrete-time closed-loop fuzzy tracking system. Recall that the gain margin of a closed-loop system is the amount by which the loop gain can be changed until the system becomes unstable [25]. As we remarked earlier, for a time-invariant well-behaved fuzzy tracking subsystem, the designed global optimal tracking controller in (21) and (44) can be unified into

. In order to measure the gain margin, we consider a corresponding tracking controller . The gain margin

of the closed-loop fuzzy tracking system is defined as the amount by which can be increased until the system becomes unstable.

Now, let ,

where , and then we have, by setting the

input weighting factor to be one for convenience

(82)

where and . We further consider

(83)

Notice that and . Comparing

(83) to (82), we find that the larger the is, the smaller the is, which means that when goes to zero, the gain margin of the closed-loop fuzzy tracking system becomes infinite. Now, we shall show that the resulting closed-loop fuzzy tracking system possesses an infinite gain margin.

Lemma 1: Consider a linear time-invariant dynamical fuzzy

subsystem

(84)

with known. If is c.c., is c.o., and

is the positive–semidefinite solution of the modified discrete-time algebraic Riccati equation

(85) where is the dependent variable of the algebraic equation,

then exists and is equal to , which is the

symmetric positive–semidefinite solution of the modified dis-crete-time Riccati equation

(86)

Proof: See the Appendix.

Theorem 9 (Gain Margin for Discre-Time Case): Consider

the discrete time-invariant fuzzy tracking system and fuzzy tracking controller described, respectively, by (1) and (2) with model-following target or moving target. If is c.c.,

is c.o. and ,

for all , where denotes the

spectral radius of , then the optimal fuzzy

tracking controller

(87) generates a closed-loop fuzzy tracking system in (79) with an in-finite-gain margin, where is equal to (23) for the moving-target problem, or equal to with in Theorem 4 for the model-following-target issue. That is, the modified closed-loop fuzzy tracking system

(88)

is always stable for any , where and is

the positive–semidefinite solution of the modified discrete-time Riccati equation in (85).

Proof: As we know, the stability of the modified nonlinear

fuzzy tracking system in (88) is coincident with that of the fol-lowing zero-input fuzzy system:

(89)

Let , denote the subsystem matrix of the

fuzzy system in (89), i.e.,

, and then, (89) can be rewritten as (90)

(10)

Notice that . We shall show that each fuzzy subsystem in (89) or (90) is exponentially stable for any . Furthermore, demonstrate that the entire zero-input fuzzy sys-tems in (89) or (90) are also exponentially stable for any ; and then, prove that the modified closed-loop fuzzy tracking system in (88) is always stable for any .

1) Via Lemma 2 in the Appendix and Lemma 1, we know

and is always available even in the case

of infinite gain margin. We shall show for all . Let denote the eigenpair of

, i.e., . By (85), we have

. Hence, we have

. Therefore, for all

is also an eigenvalue of , which is equivalent to .

To ensure this, commutes with

obviously, i.e.,

and then, commutes with or,

more precisely, with .

2) Accordingly, commutes

with . Recall that if and

are commutative operators, then .

Hence, we have

(91) since

. So, the spectrum of the subsystem matrix, characterizing the dy-namical behavior of each subsystem in (89) or (90), is always located in the unit disc of the complex space; in other words, each fuzzy subsystem in (89) or (90) is ex-ponentially stable for any .

3) Then, we can use the mathematical induction method to demonstrate that there exist constant and such that the states of the entire fuzzy systems in (89) or (90) satisfy

(92)

in other words, the zero-input fuzzy system in (89) is ex-ponentially stable for any . Hence, the stability of the modified nonlinear fuzzy tracking system in (88) is ensured positively for all , and accordingly, our re-sultant closed-loop fuzzy tracking systems in (79) or (80) possess infinite-gain margin.

V. NUMERICALSIMULATIONS

In this section, a simple nonlinear mass-spring-damper me-chanical system for continuous case, and an optimal backing up control of a computer simulated trunk-trailer for discrete-time case is adopted as the tracking system to illustrate the proposed optimal fuzzy tracking control scheme and its theoretic aspect.

A. Discrete-Time Tracking System

A computer simulated trunk-trailer system is used as a tracking system to track a moving target or a model-following target. The computer simulated truck-trailer physical system was described by Tanaka and Sano [29] as

where is the length of truck, is the length of trailer, is the sampling time, and is the constant speed of the backward movement. Then, they used the following fuzzy model to repre-sent the aforementioned mathematical model:

If is about

then

If is about or

then

and the system output is with

and , where

Grounding on this fuzzy system, we assume our fuzzy tracking controller as

If

is about then If

(11)

Fig. 1. Output responses (denoted by dashed line) of the discrete-time fuzzy tracking system with the designed finite-horizon optimal fuzzy tracking controllers in Section III-A for various moving targets (denoted by solid line), where (a)Y (t) being a stepwise target. (b) Y (t) = 3+2 sin 4k. (c) Y (t) = 100+k +0:7k . (d)Y (t) = 3 + 4e .

With the chosen membership functions [29], the firing strengths are

which, in this case, are also the normalized firing-strengths of the rules for the fuzzy system and controller. The performance index for the finite-horizon tracking problem is set as

(93)

and that for the infinite-horizon tracking problem is

(94)

Now, we can design the optimal fuzzy tracking controllers for the trunk-trailer tracking system in both cases of moving target and modeling-following target by the proposed design scheme in Section III-A.

Though the fuzzy subsystem is unstable (the spectrum of

system matrix ), it is

time-in-variant and well-behaved; i.e., the fuzzy subsystem is c.c. and

c.o. ( , for all

). Then, given and in (3), the unique

symmetric positive–semidefinite solution of the discrete-time algebraic Riccati equation in (25) or (46) is

For the moving-target tracking problem, we can obtain, based on Theorems 1 and 2, the optimal trajectory of the closed-loop fuzzy tracking system with the designed optimal fuzzy tracking controller. The output responses of the resultant closed-loop fuzzy tracking system for various targets are shown in Fig. 1 for the finite-horizon problem. The output responses for the infinite-horizon problem are quite similar to those shown in Fig. 1. As for the model-following-target case, since each fuzzy subsystem is well-behaved as mentioned above, the optimal fuzzy tracking controller and the corresponding tracking trajectory can be obtained according to Theorems 3 and 4. Fig. 2 shows the finite-horizon optimal output responses of the resultant closed-loop fuzzy tracking system for the targets from the tracked model in (29) with various parameters

( and ).

The output responses for the infinite-horizon problem under the same simulation situations are very close to those shown in Fig. 2. Our simulation results also show that the designed

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Fig. 2. Output responses (denoted by dashed line) of the discrete-time fuzzy tracking system with the designed finite-horizon optimal fuzzy tracking controllers in Section III-B for the targets (denoted by solid line) from the tracked model with four different sets of parameters:(F ; F ) = (1; 1); (1; 0:9); (0:85; 0:85) and

(0:45; 0:45).

optimal fuzzy tracking controllers can efficiently push the simulated trunk-trailer system to trace the targets as soon as possible.

B. Continuous Tracking System

In this section, we adopt the following simple nonlinear mass-spring-damper mechanical system to track a moving or a model-following target:

where is the mass and is the force; and are

the nonlinear or uncertain terms with respect to the spring and the damper, respectively; and is the nonlinear term with respect to the input term. The tracking system can be rewritten as [3]

where and . Accordingly, we

model this nonlinear system as [3]

If is and is

then

where and , and the

system output is with for every rule,

where

and the membership functions of the fuzzy terms are , and

. The firing-strengths of the rules are .

We then assume the desired fuzzy tracking controller is

If is and is

then

Also, we set the performance index for the finite-horizon tracking problem as

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Fig. 3. Output responses (denoted by dashed line) of the continuous fuzzy tracking system with the designed finite-horizon optimal fuzzy tracking controllers in Section III.1 for various moving targets (denoted by solid line), where (a)Y (t) being a stepwise target. (b) Y (t) = 3 + 2 sin 2t. (c) Y (t) = 3 + 4e . (d)

Y (t) = 0:5 + 2 log(3 + t).

and that for the infinite-horizon tracking problem as

(96)

where .

Now, we can design the optimal fuzzy tracking controllers for the mass-spring-damper tracking system to follow the desired moving target or model-following target by the proposed design scheme in Section III-B. Let (identity matrix of dimen-sion 2) and in (4). Since each fuzzy subsystem is

well-behaved ( and for

), we have the unique symmetric positive–semidef-inite solution of the algebraic Riccati equation in (57)

and

For the moving-target tracking problem, we can obtain, based on Theorems 6 and 7, the optimal trajectory of the closed-loop fuzzy tracking system with the designed optimal fuzzy tracking controller. The output responses of the resultant closed-loop fuzzy tracking system for various moving targets ( being a stepwise function,

or ) are shown in Fig. 3 for the

fi-nite-horizon problem. The corresponding output responses for

the infinite-horizon problem are very close to those shown in Fig. 3.

As for the case of model-following target, since each fuzzy subsystem is well-behaved as mentioned above, the optimal fuzzy tracking controller and the corresponding tracking trajectory can be obtained according to Theorems 7 and 8. Fig. 4 shows the finite-horizon optimal output responses of the resultant closed-loop fuzzy tracking system for the targets from the tracked model in Problem 5 with various

parame-ters ( and

). The corresponding output responses for the infi-nite-horizon problem are quite similar to those shown in Fig. 4. Our simulation results also show that the designed optimal fuzzy tracking controller can efficiently push the simulated trunk-trailer system to trace the targets in a short time.

VI. CONCLUSION

A sufficient condition for global optimization of fuzzy con-trol was adopted in this paper. Grounded on this condition, a

nonlinear global optimal quadratic tracking problem can be

de-composed into a set of linear local optimal quadratic tracking problems, and then, the local-concept-approach design scheme of global optimal fuzzy tracking controllers for both contin-uous and discrete-time fuzzy systems was derived theoretically. Grounding on this efficient design scheme, several fascinating characteristics have been shown to exist in both kinds of resul-tant closed-loop time-invariant fuzzy tracking systems.

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Simu-Fig. 4. Output responses (denoted by dashed line) of the continuous fuzzy tracking system with the designed finite-horizon optimal fuzzy tracking controllers in Section III-B for the targets (denoted by solid line) from the tracked model with four different sets of parameters:(F ; F ) = (01; 00:2); (00:2; 01);

(05; 01=30) and (01=30;05).

lation results have manifested that the designed optimal fuzzy tracking controllers can effectively drive a fuzzy system to trace the target profile in a short time. In this paper, we consider only the noise-free tracking systems. In the future work, we shall de-velop theoretically sound stochastic fuzzy estimation or fuzzy filtering techniques based on the theorems developed in this paper to deal with the practical noise-contaminated systems.

APPENDIX A

Proof of Lemma 1:

1) We consider the optimal solution for minimizing

From Lemma 3, for any , the global minimizer is

where is the symmetric positive–semidefinite solu-tion of the modified discrete-time algebraic Riccati equa-tion in (85), and the corresponding closed-loop system

(97) is exponentially stable, i.e.,

.

2) Now, we shall check if the limit value of exists and is equal to . For notation simplification, we use

and to denote and , where

is the symmetric positive–semidefinite solution of the following equation:

(98)

Define , then

(99)

Let and denote, respectively,

and ,

then

(100)

Let , then we

obtain a discrete-time Lyapunov-like equation

(101)

¿From (1), we know , and accordingly,

the unique solution is

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In other words, for all . Hence, the function is monotonic decreasing as , and bounded below by 0; i.e.,

constantly exists for all . We can pick special

values to let , i.e.,

.

Lemma 2: For the discrete time-invariant fuzzy subsystem in

(84), if is stabilizable and is detectable, then the following hold.

1) There exists an unique symmetric positive–semidef-inite solution, , of the discrete-time algebraic Riccati equation

(103) 2) The asymptotically local optimal control law is

(104) which minimizes the local quadratic functional

. 3) The optimal feedback fuzzy subsystem

(105) is asymptotically and exponentially stable.

Proof: This lemma is a counterpart of the classical

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Shinq-Jen Wu received the B.S. degree in chemical

engineering from the National Taiwan University, Taipei, Taiwan, R.O.C., the M.S. degree in chemical engineering from the National Tsing-Hua Univer-sity, Hsinchu, Taiwan, R.O.C., the M.S. degree in electrical engineering from the University of California, Los Angeles, and the Ph.D. degree in electrical engineering from the National Chiao-Tung University, Hsinchu, Taiwan, R.O.C., in 1986, 1989, 1994, and 2000, respectively.

From September 1989 to July 1990, she was with the Laboratory for Simulation and Control Technology of the Chemical Engineering Division of the Industrial Technology Research Institute, Hsinchu, Taiwan, R.O.C. In 1991, she joined the Chemical Engineering Department, Kao-Yuan Junior College of Technology and Commerce at Kaohsiung, Taiwan, R.O.C. From 1995 to 1996, she was an Engineer at the Integration Engineering Department, Macronix International Co., Ltd., Hsinchu, Taiwan, R.O.C. She is currently with the Electrical Engineering Department, Da-Yeh University, Chang-Hwa, Taiwan, R.O.C. Her research interests include thermodynamics, transport phenomena, process control and design especially in VLSI, biomed-ical and Petroleum industry, system and control theory especially in optimal control, filtering theory, fuzzy system theory, and optimal fuzzy controller and tracker design.

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Chin-Teng Lin (S’88–M’91–SM’99) received the

B.S. degree in control engineering from the National Chiao-Tung University, Hsinchu, Taiwan, R.O.C., and the M.S.E.E. and Ph.D. degrees in electrical engineering from Purdue University, West Lafayette, IN, in 1986, 1989, and 1992, respectively.

Since August 1992, he has been with the College of Electrical Engineering and Computer Science, National Chiao-Tung University, Hsinchu, Taiwan, R.O.C., where he is currently a Professor and Chairman of the Electrical and Control Engineering Department. He served as the Deputy Dean of the Research and Development Office of the National Chiao-Tung University from 1998 to 2000. His current research interests are fuzzy systems, neural networks, intelligent control, human-machine interface, image processing, pattern recognition, video and audio (speech) processing, and intelligent transportation systems (ITS). He is the coauthor of Neural Fuzzy Systems—A Neuro-Fuzzy Synergism to Intelligent

Systems (Upper Saddle River, NJ: Prentice Hall, 1996), and the author of Neural Fuzzy Control Systems with Structure and Parameter Learning (Singapore:

World Scientific, 1994). He has published over 60 journal papers in the areas of soft computing, neural networks, and fuzzy systems, including 35 IEEE TRANSACTIONSpapers.

Dr. Lin is a member of Tau Beta Pi and Eta Kappa Nu. He is also a member of the IEEE Computer Society, the IEEE Robotics and Automation Society, and the IEEE Systems, Man, and Cybernetics Society. He has been an Execu-tive Council Member of the Chinese Fuzzy System Association (CFSA) since 1995, and the Supervisor of the Chinese Automation Association since 1998. He has been the Chairman of the IEEE Robotics and Automation Society, Taipei Chapter, since 2000, and an Associate Editor of the IEEE TRANSACTIONS ON

SYSTEMS, MAN,ANDCYBERNETICS, since 2001. He won the Outstanding Re-search Award granted by the National Science Council (NSC), Taiwan, from 1997 to 2001, the Outstanding Electrical Engineering Professor Award granted by the Chinese Institute of Electrical Engineering (CIEE) in 1997, and the Out-standing Engineering Professor Award granted by the Chinese Institute of Engi-neering (CIE) in 2000. He was also elected as one of the Ten Outstanding Young Persons in Taiwan, R.O.C., in 2000.