Optimal Fuzzy Controller Design: Local Concept
Approach
Shinq-Jen Wu and Chin-Teng Lin, Senior Member, IEEE
Abstract—In this paper, we present a global optimal and stable fuzzy controller design method for both continuous- and discrete-time fuzzy systems under both finite and infinite horizons. First, a sufficient condition is proposed which indicates that the global optimal effect can be achieved by the fuzzily combined local optimal controllers. Based on this sufficient condition, we derive a local concept approach to designing the optimal fuzzy controller by applying traditional linear optimal control theory. The stability of the entire closed-loop continuous fuzzy system can be ensured by the designed optimal fuzzy controller. The optimal feedback continuous fuzzy system can not only be guaranteed to be exponentially stable, but also be stabilized to any desired degree. Also, the total energy of system output is absolutely finite. Moreover, the resultant feedback continuous fuzzy system possesses an infinite gain margin; that is, its stability is guaranteed no matter how large the feedback gain becomes. Two examples are given to illustrate the proposed optimal fuzzy controller design approach and to demonstrate the proved stability properties.
Index Terms—Converse theorem, degree of stability, exponen-tially stable, finite energy, gain margin, global optimal, Riccati equation, T-S type fuzzy model.
I. INTRODUCTION
N
ONLINEARITY and uncertainty are always bothersome in controlling a real system, since a physical system is usually partly known and difficult to describe, has few mea-surements available, or is highly nonlinear. Fuzzy modeling can mimic a real system well, fuzzy control can support more robust control than linear control does, and, moreover, optimal control can provide the best possible system. Hence, an analytic design scheme of the optimal fuzzy controller for a fuzzy system (i.e., the system described by a fuzzy model) is of theoretical and practical interest. Although the research in fuzzy modeling and fuzzy control has been quite matured [1]–[11], it seems that the field of optimal fuzzy control is nearly open [12]. The goal of this work is to propose a scheme for designing a global optimal fuzzy controller to control and stabilize a continuous- or dis-crete-time fuzzy system in finite or infinite horizon (time) con-sideration. A simple stability criterion is proposed and the gain margin of the resultant closed-loop fuzzy system is discussed.Stability and optimality are the most important requirements
for any control system. Most of the existed works are based on Takagi–Sugeno (T-S) type fuzzy model combined with parallel distribution compensation (PDC) concept [1] and apply Lya-punov's method to do stability analysis. Tanaka and coworkers
Manuscript received January 11, 1999; revised December 10, 1999. The authors are with the Department of Electrical and Control Engineering, National Chiao-Tung University, Hsinchu, Taiwan, R.O.C.
Publisher Item Identifier S 1063-6706(00)03204-5.
reduced the stability analysis and control design problems to linear matrix inequality (LMI) problems [2], [4]. Furthermore, they relaxed the stability condition, and then, derived a fuzzy controller based on the relaxed LMI stability condition [13], [14]. Moreover, they also dealt with uncertainty issue [3]. This approach had been applied to several control problems such as control of chaos [4], of an articulated vehicle [5], of a mobile robot with multiple trailers [7], and of a modal car [15]. A fre-quency shaping method to achieve systematic design of fuzzy controllers was also performed by them [16]. Sun and coworkers developed a separation scheme to design a fuzzy observer and a fuzzy controller independently [9]. 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 fuzzy discrete-time control system and an equivalent prin-ciple to do stability analysis [11]. Even with the aforementioned research results on the theoretic aspect of fuzzy control, Tanaka and others' work mentioned in the above always treat the sta-bility of general linear feedback fuzzy controllers.
On the issue of optimal fuzzy control, Wang developed an
optimal controller to stabilize a linear time-invariant system via
Pontryagin maximum principle [12]. However, although fuzzy control of linear systems could be a good starting point for better understanding of some issues in fuzzy control synthesis, it does not have much practical implications since using the fuzzy con-troller designed for a linear system directly as the concon-troller may not be a good choice [12]. Moreover, the cited stability cri-teria 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 min-imize the upper bound of the quadratic performance function by linear-matrix-inequality (LMI) approach based on the
as-sumption of local-linear-feedback-gain control structure.
Nev-ertheless, no theoretical analysis on this design scheme of op-timal-fuzzy-control structure was proposed.
In this work, a global optimal fuzzy controller design method for a fuzzy system is achieved from a local viewpoint and the properties of the constructed optimal fuzzy controller are ex-posed based on the linear optimal control theory. The derived control law is demonstrated to be the best for the entire system to reach the optimal performance index. Moreover, the optimal feedback continuous fuzzy system can not only be guaranteed to be exponentially stable, but also be stabilized to any desired degree. Furthermore, we elicit that this kind of fuzzy controller can stabilize a continuous fuzzy system to any prescribed de-gree of stability, and the corresponding closed-loop continuous 1063–6706/00$10.00 © 2000 IEEE
fuzzy system possesses an infinite gain margin. Moreover, the total energy of the system output of the feedback continuous fuzzy system is absolutely finite.
This paper is organized as follows. The sufficient condition of global optimum is proposed in Section II, which indicates that fuzzily “blending” the local optimal fuzzy controllers can achieve global optimal effect. The global optimal fuzzy con-trol laws for both continuous- and discrete-time fuzzy systems during both finite and infinite horizons are derived theoretically in Section III. Several properties such as stability criteria and gain margin of the resultant closed-loop fuzzy system are dis-cussed in Section IV. The design methodology is illustrated by two examples in Section V. Section VI gives the concluding re-marks. The related linear optimal theory applied in this paper is summarized in the Appendix
.
II. SYSTEMREPRESENTATION ANDPROBLEMSTATEMENT We consider a given nonlinear plant described by the so-called T-S type fuzzy model
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; the state vector , the
system output vector , and is the system
input (i.e., control output); and ) and are,
respec-tively, and matrices whose elements are
known to be piecewise-continuous (PC) and real-valued func-tions defined on positive real space, ; in other words, they are matrix-valued functions on of class PC. We then assume the desired controller is a rule-based fuzzy controller in the 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 quadratic optimal fuzzy control problem is described as follows:
PROBLEM 1. Given the rule-based fuzzy system in (1) with and a rule-based fuzzy controller in (2), find the individual optimal control law, such that the composed optimal controller can minimize the quadratic cost functional over all possible inputs of class PC
(continuous) (3)
(discrete) (4)
where and
are state-trajectory penalties with both and belonging to symmetric positive semidefinite matrices, and
is fuel consumption.
The grounding on distributed fuzzy subsystems and rule-based fuzzy controller forces the researchers to find the controller , which can achieve global minimum effect under quadratic performance consideration defined on the
entire fuzzy system and fuzzy controller. Thus this issue has
not been attacked directly even though the T-S type fuzzy model has been available for many years. Wang [12] tried to break the deadlock by considering a linear system (instead of a fuzzy system) combined with a fuzzy controller. Tanaka and coworkers [21], [22] developed the LMI-based fuzzy control by assuming a local-linear-feedback-gain control structure. However, the quadratic optimal fuzzy control issue, in fact, remains fully open.
In the remainder of this section, the discrete-time case will be adopted for developing the local-concept-based optimization technology. From the essence of the dynamic programming for-malism, the operation of minimizing in (4) can be de-composed as follows:
(5)
where we use the lower index to denote time dependence for no-tation simplification, i.e., for . Hence, The quadratic op-timization problem is, in fact, a successively ongoing dynamic problem with regard to the state resulting from the previous de-cision, i.e., the initial state (at time step ) . More-over, according to the signal flow of a fuzzy inference system [23], we know, at any time step , the overall behavior of the fuzzy system can be captured by fuzzily blending all the fuzzy subsystems; in other words, the entire T-S type fuzzy system in (1) can be represented as
(6) with
and , where and denote,
re-spectively, the normalized firing strength of the th rule of the fuzzy model and of the th fuzzy control rule; i.e.,
where is the membership function of fuzzy term ,
and with
where is the membership function of fuzzy term . Therefore, the optimization dynamic issue is on successively finding the optimal global decision (optimal controller) for minimizing the cost functional
(7) and estimating with regard to the initial state ; and then, with the new initial state , resolving to minimize . In other words, the quadratic optimal fuzzy control problem in Problem 1 can be restated as the following dynamic problem:
PROBLEM 2. Given the fuzzy system in (6) with
successively find the optimal global decision, , for mini-mizing the quadratic cost functional in (7), where the initial state is the optimal state resulting from the previous
decision, i.e., and .
As we know, the energy of the entire fuzzy system is the sum-matin of the energy of each fuzzy subsystem. Hence, based on the additive property of energy, we know that, at any time step , if we can find the optimal local decision (optimal control law) for minimizing in (7) with regard to the fuzzy subsystem (8) then their composed global decision can be a global minimizer of the total cost, , with regard to the fuzzy system in (6). For clarity, since is only a variable to be solved irrespective of the aforementioned local optimization problem or of the global optimization issue in Problem 2, we can use to denote the op-timal local decision of the th fuzzy subsystem. Hence, based on the local viewpoint of the global optimal fuzzy control, we know that solving the quadratic optimal control problem in Problem 2 is to find only one corresponding optimal solution of the fuzzy controller for each rule of the fuzzy model. Thereupon, both the fuzzy model and admissible fuzzy controller have, more pre-cisely, the same input variables and same input space partition, and there exists only one optimal fuzzy control rule for each fuzzy subsystem described by a fuzzy rule in the fuzzy model; that is
(plant) If is is
then
(controller) If is is
then (9)
and a fuzzy subsystem and fuzzy control rule have a one-to-one correspondence ( th-rule-to- th-rule). Therefore, the optimal
global decisions in Problem 1 can be regarded as a series of optimal global decision based on the following successively ongoing local quadratic optimal issue with the initial state resulting from the previous decision.
PROBLEM 3. Given the fuzzy subsystem
(10) with the initial state resulting from the previous decision, i.e., 1) find the optimal local decision at time-step , for
min-imizing the cost functional
(11) 2) obtain the optimal global decision at time-step , for minimizing the cost functional in (7) by fuzzily blending each local decision, i.e.,
Notice that the next-decision initial state is
instead of in (6), since there exists the one-to-one rela-tionship between each fuzzy subsystem and the corresponding fuzzy controller.
III. OPTIMALFUZZYCONTROLLERDESIGN
We shall design the optimal controllers for the contin-uous-time systems in Section III-A and for discrete-time systems in Section III-B.
A. Optimal Fuzzy Controller for Continuous-Time Fuzzy System
Since the local fuzzy system (i.e., fuzzy subsystem) is linear, its quadratic optimization problem is the same as the general linear quadratic (LQ) issue [24]. Therefore, solving the optimal control problem for fuzzy subsystem can be achieved by simply generalizing the classical theorem in Proposition 4 in the Ap-pendix from the deterministic case to fuzzy case. We summarize this generalization result here.
Theorem 1 (Solution of the Standard Fuzzy LQ Prob-lem): For the fuzzy system in (1) and fuzzy controller in (2),
let , be
given matrices. If there exists on an symmetric positive semidefinite solution to the matrix Riccati differential equation
(12) where the final value of the dependent variable , is equal to the final state penalty index , and , then there exists a local optimal fuzzy control law
where is the corresponding optimal state trajectory. And, the corresponding global minimizer is
(14) which minimizes in (3). The resulting optimal closed-loop system dynamics is described by
(15)
with .
Proof: This theorem obviously holds with Proposition 4
in the Appendix.
The above theorem considers that the horizon is fixed and is arbitrary. Does the controller exist when the horizon goes to infinity? For the general LQ problem, the an-swer is positive if the system is time-invariant and well-behaved, i.e., completely controllable and completely observable. Now, we assume our fuzzy subsystem is linear time-invariant and well-behaved. In this case, the results below for each fuzzy
sub-system are similar to those for a deterministic sub-system described
by Propositions 5 and 6 in the Appendix.
Theorem 2: For the fuzzy system in (1) and fuzzy controller
in (2), let be given constant matrices and . If is completely controllable (c.c.) and is completely observable (c.o.) for , then
1) there exists a unique symmetric positive semidef-inite solution, , of the steady-state Riccati equation (S.S.R.E.)
(16) 2) the asymptotically local optimal fuzzy control law is
(17) and their “blending” global minimizer in (14) min-imizes
(18) 3) and the optimal local feedback fuzzy subsystem
(19) is asymptotically and exponentially stable.
Proof: This theorem obviously holds with Propositions 5
and 6 in the Appendix.
B. Optimal Fuzzy Controller for Discrete-Time Fuzzy System
In the discrete-time system, the generalization of optimal con-trol theory from general “deterministic” system to “fuzzy” sub-system is also practicable. The following theorem converts the result of the general LQ problem in the Appendix into its fuzzy optimal counterpart.
Theorem 3: For the fuzzy system in (1) and fuzzy controller
in (2), let
be given matrices. If there exists a symmetric positive semidef-inite solution to the following matrix Riccati difference equation:
(20) with and the identity matrix of dimension , then there exists a local optimal fuzzy control law
(21) and the resultant global controller is
(22) which minimizes in (4). Moreover, the optimal trajec-tory is
(23) Notice that, using standard matrix manipulations, the matrix Riccati difference equation in (20) can be rewritten as
(24)
Proof: This theorem obviously holds with Proposition 8
in the Appendix.
Similarly, if the fuzzy subsystem is linear time-invariant and
well-behaved, meaning that it is stabilizable and detectable, we
still can obtain infinite-horizon optimal controller as described in the following theorem.
Theorem 4: For the fuzzy system in (1) and fuzzy controller
in (2), let be given constant matrices and . If is stabilizable and is detectable for
then
1) there exists a unique symmetric positive semidefinite so-lution of the following S.S.R.E.:
(25)
(26) 2) the asymptotically local optimal fuzzy control law is
(27) and the resultant global controller in (22) minimizes (28)
3) moreover, the optimal local feedback fuzzy subsystem (29) is asymptotically and exponentially stable.
Proof: This theorem obviously holds with Proposition 9
in the Appendix.
IV. STABILITY ANDGAINMARGIN
In this section, we are concerned with the stability of the global closed-loop system with the optimal fuzzy controller designed in the preceding section. We shall show that the controller resulting from the infinite-horizon optimal control problem gives not merely an asymptotically stable closed-loop system, but one with a prescribed degree of stability. Further-more, we also define the term gain margin to discuss what range of the feedback gain we can enlarge under the stability consideration.
A. Global Stability
The entire feedback fuzzy system is nonlinear, even though the subsystem is linear. We can thus apply the so-called converse
theorem of Lyapunov stability theory in the nonlinear system
[25] to our fuzzy system. This theorem is given in the following proposition.
Proposition 1: Consider the system
where is a function, i.e., an ordered -tuple complex-valued function, and . Define
then is an exponentially stable equilibrium of the system if and only if the linearized system
is (globally) exponentially stable. Additionally, if is defined as above and if all the eigenvalues of have negative real parts, then is an exponentially stable equilibrium.
Via the converse theorem, the stability analysis of a nonlinear system is coincidental with that of the linearized system. For the T-S type fuzzy system, a locally linearized system from the global system in (6), we know that the linearized matrix in the above proposition at some point is
Hence, the term fully handles the stability
of the fuzzy system.
From Theorems 2 and 4, we know that in the infinite-horizon optimal control problem, if the local fuzzy system is time-in-variant and well-behaved, the local feedback fuzzy system is asymptotically and exponentially stable no matter whether the system is continuous or discrete. Now our strategy is to ground on this nice local feature and step for the global system by the spectral mapping theorem in [26], which says that the spectrum of a analytic function of an operator is the analytic function of the spectrum of the operator. This will be exposed on the fol-lowing theorem.
Theorem 5: For the time-invariant fuzzy system in (1) and
fuzzy controller in (2), if is c.c. and is c.o. for then
1) the optimal feedback fuzzy system
(30) is exponentially stable;
2) the total energy of system output is finite
Proof:
1) Via the converse theorem, we know the stability of the resultant feedback fuzzy system in (30) concurs with that of the linearized fuzzy system (with respect to )
(31) For clarity, we introduce the notation to denote the local feedback system matrix. Then, we know, via Theorem 2, that each feedback fuzzy subsystem is exponentially stable, which means the spectrum of denoted by , is located in the open left-half plane of the complex space , i.e.,
. Accordingly, we have
via the spectral mapping theorem and
for all . Hence, the zero solution of
on is exponentially stable; in other words, there exists constants and such that for all
Then, the state transition matrix of the linearized fuzzy system in (31) is
where and . Therefore,
the linearized fuzzy system in (31) and also the feedback fuzzy system in (30) are exponentially stable. Hence, we can conclude that the entire continuous fuzzy system is
exponentially stable if each continuous fuzzy subsystem is exponentially stable.
2) From the above proof, we know the entire feedback fuzzy system in (30) is exponential stable and, hence, also en-sure to be uniformly asymptotically stable. Therefore, for
all and , the state satis-fies a) and b) in the following.
a) The range of mapping from to is bounded on uniformly, i.e.,
s.t.
b) The range of mapping from to tends to zero as uniformly, i.e.,
s.t. Therefore, with
From a) and b), we know that these two integrates are both finite, and accordingly we have
However, we cannot yet demonstrate that there exists such close stability relationship between the entire closed-loop system and the local feedback system for the discrete-time case. Therefore, we can only use Lyapunov's direct method or linear matrix inequality method [8] to analyze the stability of the overall feedback discrete-time fuzzy system.
B. Stabilization to Any Desired Degree
So far, we have examined the stability of the closed-loop system. We now attempt to show that the constructed optimal fuzzy controller can stabilize the entire fuzzy system to any desired degree. That is, for some prescribed constant , the state approaches zero at least by the rate of Of course, the larger the desired degree of stability, the more stable the closed-loop system. However, a high degree of closed-loop stability may only be achieved at excessive control energy consumption. Before showing this, we need the following two lemmas.
Lemma 1: For a system described by
where and are and matrices (i.e.,
). c.c. is equivalent to c.c.
for any complex value , i.e.,
c.c. c.c.
Proof:
1) is c.c. if and only if .
Let , then
which means is c.c. if and only if is c.c. 2) Now, consider two systems
(32) (33) If we let , via basic differential operation, it is evident that (32) and (33) are algebraically equivalent for any ; i.e., they are related by a nonsingular linear transformation Therefore, (32) c.c. is equivalent to (33) c.c., i.e.,
c.c. c.c.
From 1) and 2), we conclude that Lemma 1 holds.
Lemma 2: For a system c.o. is equiv-alent to c.o. for any complex value , i.e.,
c.o. c.o.
Proof: The proof is similar to the proof of Lemma 1.
Now, we deduce Theorem 6 using the above two lemmas.
Theorem 6: For the fuzzy system in (1) and fuzzy controller
in (2), let be given constant matrices and in (18). If is c.c. and is c.o. for
then the fuzzy system can be stabilized to any desired degree of stability; in other words, the state of the modified feedback fuzzy system
(34)
approaches at least by the rate of , where is any positive real number, is the normalized firing strength (i.e., ), and is the positive-semidefinite solution of the modified S.S.R.E.
(35) where is the dependent variable of the algebraic equation.
Proof:
1) Via the converse theorem, we know the stability of the modified feedback fuzzy system in (34) concurs with that of the linearized fuzzy system (with respect to )
(36) Hence, we shall show that all the eigenvalues of the linearized fuzzy system in the above have real part smaller than , i.e.,
We now consider the local quadratic optimal problem
w.r.t. (37)
Let , , and .
Equation (37) can be rewritten as
w.r.t. (38)
From Lemmas 1 and 2, we know that c.c. and
c.o., if and only if c.c. and
c.o., . Hence, based on the linear quadratic theory, we know the local optimal feedback system for the modified fuzzy system in (38),
(39) is exponentially stable for all . Accordingly, from part 1) of the proof in Theorem 5, we know that the fuzzily blended global feedback fuzzy system
(40) and also the corresponding linearized global fuzzy system (with respect to ) are exponentially stable, where
for all . Thereupon, we
have
with the aid of the spectral mapping theorem. This completes the proof.
C. Gain Margin
Furthermore, we shall examine another interesting property,
gain margin, of the resultant closed-loop fuzzy 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. If the loop gain can be increased without bound; that is, instability is not encountered no matter how large the loop gain becomes, then the closed-loop system is said to possess an infinite gain margin [24]. Thus far, for time-invariant well-behaved continuous fuzzy subsystem, we know the designed global optimal controller, by (17) and (14), is
In order to measure the gain margin, we think of the following control law:
Then, the gain margin of the closed-loop fuzzy system is defined as the amount by which can be increased until the system becomes unstable. In this case, the corresponding local control
law is
Notice that in the case of , this control law is no longer an optimal control law, i.e., . Now, let
and then we have
(41) We further consider
(42)
Notice that and . Comparing
(42) to (41), we find that the larger the , the smaller the , which means that when goes to zero, the gain margin of the closed-loop fuzzy system becomes infinite.
We can include into the state penalty matrix . From The-orem 2, for any , the optimal control law with respect to (42) is
where satisfies the modified S.S.R.E.
(43) where is the dependent variable of the algebraic equation. We now first cite two important results in control theory [28] and apply them to the fuzzy feedback subsystem. We shall then find the gain margin of the entire closed-loop fuzzy system. The following propositions are cited from [28].
Proposition 2: Consider the infinite-horizon optimal control
problem as follows.
Given a linear time-invariant system
find an optimal controller to minimize
with denoting the system output, where
and Now, if is c.c. and is c.o., then
1) the positive-semidefinite solution, denoted by , of the modified S.S.R.E.
(44) uniquely exists;
2) when goes to zero, i.e., when the closed-loop system possesses an infinite gain margin, the limit value of
exists. Let denote this limit value. Furthermore, is the unique positive-semidefinite solution of the modified S.S.R.E.
Proposition 3: Consider the same infinite-horizon optimal
control problem as in Proposition 2. Assume is c.c. and is c.o., then
1) we can find an optimal control law
where is the positive-semidefinite solution of the S.S.R.E.
and the closed-loop fuzzy system possesses an infinite gain margin; i.e., the modified feedback system
is always stable for any , where and is the positive-semidefinite solu-tion of the modified S.S.R.E., (44).
2) Moreover, for any fixed , the enlarged controller
can still stabilize the modified system to any de-sired degree of stability; i.e., all the eigenvalues of
have real parts smaller than where could be any positive real number and
is the positive-semidefinite solution of the modified S.S.R.E.
where is the dependent variable of the algebraic equation.
Grounding on these propositions, we elicit the fol-lowing fascinating fact.
Theorem 7: For the time-invariant fuzzy system in (1) and
fuzzy controller in (2) with in (18), if is c.c.
and is c.o. for then
1) we can find a fuzzy control law
where is the positive-semidefinite solution of the S.S.R.E. in (16), and the resultant closed-loop fuzzy system possesses an infinite gain margin; i.e., the modi-fied closed-loop fuzzy system
(45)
is always stable for any , where and
is the positive semidefinite solution of the modi-fied S.S.R.E. in (43).
2) Moreover, for any fixed , the enlarged controller
can still stabilize the modified system to any desired de-gree of stability; in other words, the state of the modified feedback fuzzy system
(46) approaches at least by the rate of , where could be any positive real number and is the positive-semidefinite solution of the modified S.S.R.E.
Proof: For clarity, we introduce the notations and to denote, respectively, the local feedback system ma-trices in (45) and (46), i.e.,
and
1) Since is c.c. and is c.o. for ,
we know, from 1) in Proposition 3, that the modified closed-loop fuzzy subsystem is stable for any , i.e.,
for all . Accordingly, by
part 1) of the proof in Theorem 5, we know their fuzzily blended global system in (45) is exponentially stable. 2) Then fixing at any gain margin , we shall show the state
of the modified feedback fuzzy system in (46) and also that of the corresponding linearized fuzzy system (with respect to ) approaches at least by the rate of for all . In other words, we shall demonstrate
From 2) in Proposition 3, we have
due to being c.c. and being c.o. Accord-ingly, via the spectral mapping theorem, we have
for
which results in that all local modified feedback fuzzy system
(47) are exponentially stable. Moreover, we know, via part 1) of the proof in Theorem 5, that the fuzzily blended feed-back fuzzy system
(48) is exponentially stable. Furthermore, via the converse the-orem, we have
for all . Then, we obtain
for all and . Therefore, we know that the linearized feedback fuzzy system (with respect to )
has any degree of stability. Hence, the modified fuzzy system in (46) has any degree of stability as well.
V. NUMERICALSIMULATIONS
We consider a simple nonlinear mass-spring-damper mechan-ical system for continuous-time case, and an optimal backing up control of a computer simulated trunk-trailer for discrete-time case in order to illustrate the proposed optimal fuzzy control scheme and its theoretic aspect.
A. Continuous-Time System
A mass-spring-damper system can be formulated as
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. We make the same assumptions as Tanaka et
al. did in [3], and reformulate the system as
where and .
According to the study in [3], we describe this nonlinear system by the following T-S type fuzzy model:
If is and is then If is and is then If is and is then If is and is then
and the system output is with for every
rule, where
and the membership functions of the precondition parts of the fuzzy rules are
We further assume our fuzzy controller is
If is and is then
If is and is then
If is and is then
If is and is then
Accordingly, the firing-strength of each rule is
where the normalized firing-strength of the th rule is
Now, let the penalty matrices be set as and . Then, the designed finite-horizon optimal controller according to (13) and (14) is
with
where is the symmetric positive-semidefinite solu-tion of the matrix Riccati differential equasolu-tion in (12). Since always exists and the above controller stabilizes the subsystem at any , the entire feedback system is expo-nentially stable, which can be observed from the state response of the closed-loop fuzzy system at different initial conditions in Fig. 1.
Since the fuzzy subsystem is time-invariant and well-be-haved; i.e., the subsystem is c.c. and c.o. (
and for ), there exists
a unique symmetric positive semidefinite solution, , of the S.S.R.E. in (16)
and the asymptotically optimal controller is with
Fig. 1. The state responses of the continuous-time fuzzy system with the designed optimal controller in the finite-horizon quadratic optimal control problem of Section V-A at the four initial conditions:X(0) = (01; 01) ; (01; 1) ; (1; 01) ; and (1; 1) .
Fig. 2. The state responses of the continuous-time fuzzy system with the designed optimal controller in the infinite-horizon quadratic optimal control problem of Section V-A at the four initial conditions:X(0) = (01; 01) ; (01; 1) ; (1; 01) ; and (1; 1) .
The optimal feedback fuzzy system, (30), is exponentially stable, the total energy of system output is finite, and more-over, this optimal controller can stabilize the fuzzy system to any prescribed degree of stability and generates a closed-loop fuzzy system with an infinite gain margin. Fig. 2 illustrates the position and velocity responses of the closed-loop fuzzy system in different initial conditions. From the simulation results, we find the designed optimal fuzzy controller can quickly push the system from various initial states to and stay at the desired state in both cases of finite and infinite horizons.
B. Discrete-Time System
Tanaka and Sano [29] described a computer simulated truck-trailer with the mathematical model
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 truck-trailer system:
If is about
then
If is about or
then
and the system output is with
and
where
We further assume our fuzzy controller is
If is about
then
If is about or
Fig. 3. The state responses of the discrete-time fuzzy system with the designed optimal controller in the finite-horizon quadratic optimal control problem of Section V-B at the four initial conditions:X(0) = (0=2; 03=4; 010) ; (0=2; 3=4; 010) ; (=2; 03=4; 010) ; and (=2; 3=4; 010) . With the chosen membership functions, the normalized
firing-strength is
Given the penalty matrices and , the designed finite-horizon optimal controller according to (21) and (22) is
with
where is the symmetric positive-semidefinite solution of the matrix Riccati difference equation in (20) or (24).
The original subsystem is unstable, since
However, and
for all in the intersection of the spectrum of and the
com-plement of open unit disk, i.e., ;
ac-cordingly, is stabilizable and is detectable,
. Hence, all the discrete-time subsystems are still well-be-haved. The unique symmetric positive-semidefinite solution of the S.S.R.E. in (25) or (26) is
and the local optimal fuzzy control law is
with
By the way, the designed optimal controller can stabilize the local system.
However, there is no straightforward relationship between the stability of subsystems and that of the entire system for the dis-crete-time system. We may adopt Lyapunov's direct method [8] to perform the stability analysis of overall feedback system. Figs. 3 and 4 show the tracking results in various poor initial conditions for finite-horizon and infinite-horizon optimal con-trol problems, respectively. Obviously, a perfect, fast tracking is achievable even when the variation of the initial states occurs.
Fig. 4. The state responses of the discrete-time fuzzy system with the designed optimal controller in the infinite-horizon quadratic optimal control problem of Section V-B at the four initial conditions:X(0) = (0=2; 03=4; 010) ; (0=2; 3=4; 010) ; (=2; 03=4; 010) ; and (=2; 3=4; 010) .
VI. CONCLUSION
A sufficient condition for global optimal fuzzy control was proposed in this paper. This condition shows that “blending” op-timal local fuzzy controllers can achieve global opop-timal effect. Based on this observation, the design scheme of finite-horizon global optimal fuzzy controllers in continuous-time system as well as in discrete-time system were derived. In the case of time-invariant and well-behaved fuzzy systems, the design scheme of infinite-horizon global optimal fuzzy controllers for both the continuous-time and discrete-time systems were also obtained. Several fascinating characteristics have been shown to exist in the closed-loop continuous-time fuzzy system for the infinite-horizon optimal control problem. First, we have shown that the stability of the entire closed-loop fuzzy system can be guaranteed if the simple completely controllable and completely observable criteria hold for the fuzzy subsystems. Furthermore, under this situation, the closed-loop fuzzy system has freedom in the choice of the degree of stability and gain margin, meaning that the designed optimal fuzzy controller can stabilize the fuzzy system to any desired degree of stability and the resultant closed-loop fuzzy system possesses an infinite gain margin. Simulation results have manifested that all the designed optimal fuzzy controllers can effectively drive the fuzzy system to the target points in a short time.
APPENDIX
We describe here the related optimal control concepts and re-sults adopted in this paper. We start with finite-horizon optimal
control problem in continuous-time systems, and then, the infi-nite-horizon issue. Next, we tackle these two optimal control problems in discrete-time systems, since it is difficult to get complete bibliography on this issue, and even, the mathemat-ical statements for the discrete-time systems are always diverse.
A. Continuous-Time Systems
We consider here a dynamical system represented by the fol-lowing equation:
for (49)
where the state , the input , and
are matrix-valued functions on of class PC. The gen-eral LQ problem is to find an optimal control law to min-imize the performance index in (3). The results shown in [24], [27], [28], [30]–[32] are summarized as follows.
Proposition 4 [27], [31]: Let
be given matrices. Suppose the matrix Riccati equation
(50)
with has a solution on the interval
. Then, there exists a controller which minimizes in (3) for the system in (49) with . And, the optimal control law is
Furthermore, the minimum value of is .
Next, when the horizon extends to infinity, it belongs to the infinite-horizon optimal control problem. In this case, we,
moreover, assume or . In
other words, the system output is . Then, the performance index is finite if the system is completely control-lable and the stability of the feedback system is guaranteed if the system is completely observable. For the time-invariant system, a more attractive characteristic listed below is elicited. Here, we use to denote the limit value of the solution of the afore-mentioned matrix Riccati equation, i.e.,
Proposition 5 [30]: If is positive definite and is c.c., then exists and satisfies the S.S.R.E.
(52) Moreover, is positive-definite and symmetric.
Proposition 6 [24], [28]: If is c.c., is c.o., and is the positive-definite solution of the S.S.R.E. in Proposi-tion 5, then
1) is the optimal control law which
minimizes in (18);
2) all the solutions of the feedback system
tend to as ; that is, it is asymptotically stable.
Proposition 7 [31], [32]: If is c.c. and is c.o., then the system can be stabilized to any desired degree; i.e., all the eigenvalues of the feedback system have real parts smaller
than , for all .
B. Discrete-Time Systems
This section discusses how to obtain the optimal control law for the discrete-time system. We shall first introduce the issue of the finite-horizon optimal control problem, and then the infinite-horizon optimal control problem. Assume our system is
(53) The finite-horizon optimal control problem is to search an op-timal control law to minimize in (4).
Proposition 8: Let
be given matrices. Suppose the matrix Riccati equation
(54)
with , has a solution ,
then there exists an optimal control law which minimizes in (4) for the system in (53) with The op-timal control law is
and the corresponding optimal trajectory is
Moreover, the minimum value of is , and,
via some standard matrix manipulations, the matrix Riccati dif-ference equation in (54) can be rewritten as
(55)
Proof: 1) Define
By the principle of optimality in dynamic programming in [33], [34], the above equation can be rewritten as
x
We notice that . Therefore,
want to look for a solution of the form
with , where is the introduced time step vari-able. Hence, we have
(56) where
Then, we perform the minimization of with respect to
Accordingly, we can obtain
(57)
(58) Substituting (57) and (58) into (56), we have
by denoting time dependence as a lower index. Furthermore, via (58), we have
Hence, (54) holds. Now, we shall show that (54) and (55) are equivalent via the matrix manipulations as below. By omitting the time-dependence index at time-step for notation simplifi-cation, we have
which is equivalent to (55) since
Then, substituting (55) into , we obtain
This completes the proof.
Now, we turn to the infinite-horizon optimal control problem with time-invariant system and use to denote the limit value of the matrix Riccati difference equation in (54) or (55),
i.e., .
Proposition 9 [24]: Let be given matrices. If is stabilizable and is detectable, then
1) there exists a unique symmetric positive semidefinite so-lution of the S.S.R.E.
(59) which can be rewritten as
(60) 2) the asymptotically optimal control law is
which minimizes in (28), and the minimum value
of is ;
3) the closed-loop feedback system is asymptotically and exponentially stable.
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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., in 1986, the M.S. degree in chemical engineering from the National Tsing-Hua University, Hsinchu, Taiwan, R.O.C., in 1989, and the M.S. degree in electrical engineering from the University of California, Los Angeles, CA, in 1994. She is currently working toward the Ph.D. degree in electrical engineering at the National Chiao-Tung University, Hsinchu, Taiwan, R.O.C.
From September 1989 to July 1990, she was with the Laboratory for Simulation and Control Technology of the Chemical Engineering Division of Industrial Technology Research Institute, Hsinchu, Taiwan, R.O.C. Since then, until 1991, she was with 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. Her research interests include thermodynamics, transport phenomena, process control and design, especially in VLSI and the petroleum industry, system and control theory, especially in optimal control, filtering theory, fuzzy system theory, and optimal fuzzy controller and tracker design.
Ms. Wu is a member of Phi-Tau-Phi scholastic honor society.
Chin-Teng Lin (S’88–M’91–SM’95) received the B.S. degree in control engineering from the National Chiao-Tung University, Hsinchu, Taiwan, R.O.C., in 1986, and the M.S.E.E. and Ph.D. degrees in electrical engineering from Purdue University, West Lafayette, IN, in 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 of Elec-trical and Control Engineering. He has also served as the Deputy Dean of the Research and Development Office of the National Chiao-Tung University since 1998. His current research interests are fuzzy systems, neural networks, intelligent control, human–machine interface, and video and audio processing. He is the coauthor of Neural Fuzzy Systems—A
Neuro-Fuzzy Synergism to Intelligent Systems (Englewood Cliffs, NJ:
Pren-tice-Hall), and the author of Neural Fuzzy Control Systems with Structure and
Parameter Learning (Singapore: World Scientific). He has published over 45
journal papers in the areas of neural networks and fuzzy systems.
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, Cybernetics Society. He has been the Executive Council Member of the Chinese Fuzzy System Association (CFSA) since 1995, and the Supervisor of the Chinese Automation Association since 1998. He was the Vice Chairman of IEEE Robotics and Automation Taipei Chapter in 1996 and 1997. He won the Outstanding Research Award granted by National Sci-ence Council (NSC), Taiwan, in 1997 and 1999, and the Outstanding Electrical Engineering Professor Award granted by the Chinese Institute of Electrical En-gineering (CIEE) in 1997.
