On The Stability Issues Of Linear Takagi~sugeno Fuzzy Models ...
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 6, NO. 3, AUGUST 1998
On the Stability Issues of Linear Takagi–Sugeno Fuzzy Models Joongseon Joh, Ye-Haw Chen, Member, IEEE, and Reza Langari
Abstract—Stability issues of linear Takagi–Sugeno (T–S) fuzzy models are investigated. We first propose a systematic way of searching for a common matrix, which, in turn, is related to stability for subsystems that are under a pairwise commutative assumption. The robustness issue under uncertainty in each subsystem is then considered. We then show that the pairwise commutative assumption can, in fact, be relaxed by a similar approach as that for uncertainty. The result is applicable to a rather broad class of T–S models, which are non-Hurwitz and/or nonpairwise commutative.
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Index Terms— Fuzzy systems, robustness, stability, uncertain systems, uncertainty.
I. INTRODUCTION
F
UZZY logic has been considered as an efficient and effective tool in managing uncertainties of systems since Zadeh’s seminal paper [1]. Among many applications of fuzzy logic, control design appears to be one that has attracted a large amount of attention in the past two decades. In general, fuzzy control system can be classified as Mamdani type and Takagi–Sugeno (T–S) type. The Mamdani-type fuzzy control system is well recognized and received by the society. The T–S-type fuzzy system mainly focuses on the modeling aspect (e.g., [2]). There are some works in literature that are mainly concerned with the stability analysis of T–S fuzzy model. The existence of a proper T–S fuzzy model is first assumed. Tanaka and Sugeno [3] showed that the stability of a T–S fuzzy model could be shown by finding a common symmetric positive definite matrix for subsystems. This has been considered a very important result and some refining efforts have been pursued thereafter. There has not been, however, a systematic matrix in a general framework. way to find the common Kawamoto et al. [4] only considered a second-order system. Tanaka [5] suggested the idea of using linear matrix inequality matrix. Xia and Chai (LMI) for finding the common [6] proposed a stability condition that is based on ad hoc membership values. Zhao et al. [7] extended some past work to consider uncertainty. The current work endeavors to tackle the stability issue of the T–S fuzzy model by first considering the problem as related Manuscript received August 26, 1996; revised March 17, 1997. J. Joh is with the Department of Control and Instrumentation Engineering, Changwon National University, Changwon City, Kyungnam, Korea. Y.-H. Chen is with the The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA. R. Langari is with the Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843 USA. Publisher Item Identifier S 1063-6706(98)05548-9.
to the switching system [8]–[10]. In a sense, the central issue lies in the search for a common positive definite matrix for multiple matrix equations. We then introduce uncertainty into the setting and robustness analysis for stability is performed. It is shown that a somewhat stringent assumption for the can, in fact, be relaxed by following a common matrix similar procedure for robustness analysis. The main contributions of the paper are threefold. First, an iterative algorithm for the choice of a common matrix is proposed. It is shown that the algorithm can indeed reach a solution for systems under a structural assumption. Second, modeling uncertainty is introduced. A robustness analysis on the influence of uncertainty toward stability is also suggested. The analysis is nonconservative and computationally straightforward. Third, we show that the structural assumption can, in fact, be relaxed by following the similar robustness analysis procedure. This, in turn, means our algorithm is applicable to both the uncertainty and nonstructural case. II. BACKGROUND MATERIALS A. Takagi and Sugeno’s Fuzzy Model Takagi and Sugeno [2] proposed an effective way to represent a fuzzy model of a dynamical system. It uses a linear input–output relation as its consequence of individual plant plant rules that rules.1 A T–S fuzzy model is composed of can be represented as
(1) where discrete time index; th state (or linguistic) variable; a fuzzy term of Mj selected for Rule i; fuzzy term set2 of ; state vector and
1 Each plant rule produces the next behavior of the system from current system states and inputs. 2 See Driankov et al. [11, pp. 103–115] for the description of fuzzy term set.
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JOH et al.: ON THE STABILITY ISSUES OF LINEAR TAKAGI–SUGENO FUZZY MODELS
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Theorem 2: Suppose tative, i.e., Lyapunov equations:
input vector and
; . For any current state vector T–S fuzzy model infers model as follows:
and are Schur and commu. Consider the following two
(8) (9)
and input vector , the as the output of the fuzzy
and and are the unique positive definite where solutions of (8) and (9), respectively. Then we always have (2) (10)
where (3)
. Proof: Substituting the
in (9) into (8) yields
and
(11) (4) Since , (2) can be written as
For a free system (i.e.,
and are commutative, we have and, hence, . Therefore, (11)
becomes (5)
From now on, we shall assume a proper T–S fuzzy model in the form of (5) is available. B. Formulation of the Stability Problem Tanaka and Sugeno [3] suggested an important criterion for the stability of the T–S fuzzy model. Theorem 1 (Tanaka and Sugeno [3]): The equilibrium ) is globally state of the fuzzy system (5) (namely, asymptotically stable if there exists a common positive definite such that matrix for all
(12) where (13) and is Schur, the “solution” and is Since unique. (End of Proof). By choosing the Lyapunov function candidate
(6)
So far, however, no systematic way of finding the common exists. We suggest to translate the stability problem of T–S fuzzy model to be one for the following -simultaneous linear systems:
(14) for both systems, for the
system we have
(7) where the system matrix as (1).
(15)
for the th plant rule is the same since
from (8). In addition,
III. STABILITY ANALYSIS OF LINEAR T–S FUZZY MODEL Narendra and Balakrishnan [10] suggested a systematic matrix of -simultaneous way of finding the common continuous-time linear systems under a pairwise commutativity assumption. The results are now extended to discrete-time systems. for simplicity. Let a T–S fuzzy We first consider and . model be with two plant rules, i.e.,
(16) system since . Equations (15) and (16) show for the for and, thus, can be used as that the common matrix of the T–S fuzzy model by Theorem 1. The technique can be extended to a T–S fuzzy model with plant rules, i.e., .
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Theorem 3: Suppose that is Schur for all and ’s are pairwise commutative, i.e.,
TABLE I COMMON MATRIX
P3
(17) Consider the following
Lyapunov equations:
.. . (18) and is the unique positive where definite symmetric solution of each equation. Then we always have
TABLE II
COMMON MATRIX
P3 WITH PERTURBATIONS ON A MATRICES
(19) Proof: The proof follows the procedure similar to that of Theorem 2. (End of Proof). is Remark 1: Apparently the choice of the ordering of in the nonunique. One may, in practice, attempt to label most appropriate way in order that the pairwise commutativity is achieved. The following example illustrates the use of Theorem 3. Example 1: Consider a T–S fuzzy model with three plant rules and two state variables. Let the corresponding ’s be
and
IV. QUADRATIC STABILITY ANALYSIS OF LINEAR T–S FUZZY MODEL
It can be easily seen that they are all Schur. Furthermore, they are pairwise commutative since
and
Let
an arbitrary symmetric . The resulting is indeed a common symmetric positive definite matrix. The symmetric is obtained under the assumption positive definite matrix of Hurwitz and pairwise commutative system matrices.
The stability analysis in the Section III is based on the matrices. The assumption may, pairwise commutativity of however, be restrictive in certain applications. Therefore, it is interesting to further investigate whether the assumption . Consider the following is stringent to the existence of example. Example 2: Perturbed A matrices: Consider Example 1 with some perturbations on matrices. Let the perturbed matrices be
; we then have the following:
The is symmetric and positive definite since the corresponding eigenvalues are 21.8702 and 2.2157. Now, let us check the to be the common matrix. legitimacy of is indeed a common matrix of the Table I shows that T–S fuzzy model. In summary, a common symmetric positive definite matrix, which guarantees the stability of a T–S fuzzy model with plant rules can be found systematically from Theorem 3. It consists of solving Lyapunov equations iteratively from
The stability of the perturbed system can be checked easily using (6). Table II shows that the T–S fuzzy model is still stable and obtained earlier can indeed tolerate the common matrix certain perturbations of matrices. Remark 2: The perturbed matrices in Example 2 are and Schur but nonpairwise commutative since . However, the result shows that the common can be still used for nonpairwise commutative matrix system.
JOH et al.: ON THE STABILITY ISSUES OF LINEAR TAKAGI–SUGENO FUZZY MODELS
Example 2 shows that the common matrix , which can be determined systematically under a pairwise commutativity assumption, may be obtained without actually referring to the assumption. The main assertion in Example 2 can be more formally stated as a problem of checking the stability of the following system: (20) ’s are Hurwitz and pairwise commutative as in where ’s are (possibly) time-varying perturSection III and bations. Equation (20) can be interpreted as a free system of uncertain plant rules as a T–S fuzzy model composed of follows: Rule i: if and then
is
and
and
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Fig. 1. Convex sets.
is unknown. In Section III, a common matrix (i.e., ) is chosen systematically by way of Theorem 3 under the pairwise commutativity assumption. Furthermore, Example 2 suggests can tolerate some perturbations of matrices. that the can be used as a reasonable choice Motivated by these, the of the common in (24). The following proposition is made. Proposition 1: The T–S fuzzy model (21) is quadratically stable if there exists a positive constant such that
is (25)
(21)
. It is interesting to view that (20) is a collection linear systems with time-varying uncertainties where of ’s are Schur and pairwise commutative; that is, ’s are ’s the nominal system matrices and the corresponding are time-varying uncertainties. For systems under (possibly) time-varying uncertainties, quadratic stability performance can be used as a basis for stability study. Definition 1: Consider the following uncertain system
’s are compact sets) and . Here, ’s are Schur and pairwise commutative are chosen from the corresponding nominal system and by way of Theorem 3. Proof: Direct application of Definition 2 and [3]. (End of Proof). Corollary 1: A T–S fuzzy model in (21) is quadratically stable if
for all
(22)
(26)
and is a known compact set. The uncerwhere tain system is quadratically stable if there exists a symmetric positive definite matrix and a positive constant such that
is where ’s are Hurwitz and pairwise commutative and a common symmetric positive definite matrix. Here, denotes the maximum eigenvalue of the designated symmetric matrix. Proof: Direct application of Definiton 2 and [14]. (End of Proof). Even though (26) is more explicit than (25), it is still difficult to apply it in practice. Gu et al. [12] added an additional be convex. A convex set is defined as assumption that follows (see Rao [13, pp. 87–90]). Definition 3: Consider any two points in a set. The set is convex if the line segment joining them is also in the set. Some examples of convex sets in two dimensions are shown shaded in Fig. 1. Other useful definitions for this paper are convex hyperpolyhedron sets and protruded points. They are defined as follows. Definition 4: A set is a convex hyperpolyhedron if points in the set are common to one or more half spaces. The convex hyperpolyhedron is reduced to be a convex polygon in the two-dimensional case [Fig. 2(a)] and a convex polyhedron in the three=dimensional case [Fig. 2(b)]. A protruded point can be defined in the context of convex set. Definition 5: A protruded point is a point in a convex set which does not lie on a line segment joining two other points of the set. From Definition 5, protruded points can be considered as vertices of convex hyperpolyhedron set. It is also worth noting
(23) and . for all The quadratic stability performance can be readily applied to the systems in (20). The following definition is proposed. Definition 2: Consider the following system with uncertain plant rules: (20) Revisited and are a compact set. where The uncertain system is quadratically stable if there exists a common symmetric positive definite matrix and a common positive constant such that (24) and for all . for all ’s are not necessarily pairwise It is worth noting that commutative and are more general as in Section III. On the other hand, checking quadratic stability of a T–S fuzzy model using the Definition 2 directly may be sometimes difficult, especially for high-dimensional case. Therefore, a more realistic way to use Definition 2 is required. One may only expect to use the nominal system since the uncertainty
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 6, NO. 3, AUGUST 1998
TABLE III FIRST PLANT RULE
A1
where
and and , uncertainty sets
are uncertainty parameters satisfying with . Then, the , and become
(28) (a)
(b)
are convex and there Equation (28) shows that , , and are four protruded points in each uncertainty sets. They are
Fig. 2. Convex polygon and convex polyhedron.
that protruded points are finite for the convex hyperpolyhedron set. It is shown in Gu et al. [12] under the convexity assumption, can be reached by one of the the maximum in (26) for protruded points (i.e., vertices) of the set . The additional does not degrade seriously the convexity assumption of in can often generality of uncertainty since elements of where be represented by scalar bounds, i.e., denotes the element of and denotes a possibly time-varying scalar bound. Therefore, a similar theorem can be made for a T–S fuzzy model with uncertain plant rules using the convexity assumption as follows. Theorem 4: Consider the maximization problem (27) is a fixed symmetric positive definite matrix and where ’s are compact and convex sets for the th plant rule. The maximum for each can be reached by one of the protruded points of each set . Therefore, only protruded points are needed to be checked for quadratic stability. The following example illustrates the use of Corollary 1 and Theorem 4. Example 3: Consider a T–S fuzzy model in Example 1. The system matrices are
(29) Quadratic stability of the system can be checked using Corollary 1 and Theorem 4. As in Example 1, the common symis metric positive definite matrix
by . Corollary 1 yields the first, second, and third plant rule, as seen in Tables III–V. Tables III–V show that the uncertain system is quadratically stable by Proposition 1. Remark 3: We have shown that the quadratic stability in Proposition 1 can be checked by using Theorem 4 under the assumption of compact and convex uncertainty set. It means that in Proposition 1 can be determined from the result of Theorem 4 as follows: (30) where
’s are determined from (31)
and
They are Schur and pairwise commutative, as shown in Example 1. For simplicity, let the uncertainty sets for each plant rules be given as
In summary, the symmetric positive definite common matrix can allow some perturbation of Hurwitz and pairwise matrices. It leads to the quadratic stability commutative uncertain subsystems that have Hurwitz and pairwise on commutative system matrices. With the assumption of compact and convex hyperpolyhedron uncertainty set, the quadratic stability of T–S fuzzy model with uncertainties can be checked can indeed play easily. Therefore, the common matrix more general role than Narendra and Balakrishnan [10].
JOH et al.: ON THE STABILITY ISSUES OF LINEAR TAKAGI–SUGENO FUZZY MODELS
TABLE IV SECOND PLNAT RULE
TABLE V THIRD PLANT RULE
V. ANALYSIS OF UNCERTAINTY BOUNDS OF LINEAR T–S FUZZY MODEL Section IV considers the quadratic stability of a general T–S fuzzy model with -pairwise commutative nominal system matrices. The uncertainty sets ’s are fixed, i.e., the uncertainty bounds are specified. It is, however, also interesting to finding the maximum allowable uncertainty bounds for the as a common . This T–S fuzzy model in (21) using means, the nominal system is prespecified. We formulate the problem as follows. uncertain Problem 1: Consider a T–S fuzzy model with plant rules as
407
A2
A3
By relating to the uncertainty bound, (33) means that the uncertainty bound is represented as scalar multiple to a fixed prototype of uncertainty bound. It is solvable and quite general since many problems can be formulated in this . The way by choosing appropriate fixed uncertainty set systems by uncertainty set in (33) can be extended to defining (34) where the subscript denotes the th plant rule. Problem 1 can be restated as follows using the uncertainty set in (34). uncertain Problem 2: Consider a T–S fuzzy model with plant rules as
(20) Revisited (20) Revisited ’s are where ’s are Schur and pairwise commutative, (possibly) time-varying uncertainties. Find the maximum possuch that sible sets
(32) where is the common symmetholds for arbitrary ric positive definite matrix determined from the corresponding nominal system and is a positive constant. Gu [14] proposed an algorithm calculating the maximum allowable uncertainty bound for the system (22) under the additional assumption that has fixed shape but “adjustable” with the additional assumption size. A new uncertainty set can be stated as follows. Let (33) is a compact and convex set with fixed shape where and size and is constant. The constant is shown to be nonnegative in [14] in the case of stable nominal system.
where ’s are Schur and pairwise commutative, . Here, is defined as in (34). Find the maximum with possible sets (35) . such that (20) is quadratically stable with From the pairwise commutativity of ’s and Corollary 1, such that Problem 2 can be reduced to finding
(36) . for every Gu [14] suggested an algorithm of finding the maximum possible uncertainty bound of the system (22) by using the uncertainty set defined in (33). It can be applied to systems in Problem 2 to determine the maximum possible bound of each plant rules of T–S fuzzy model in (36). The procedure is outlined as follows.
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To determine
Equation (43) is a generalized eigenvalue problem. Therefore, (38) and (39) become
for (36), solve
(37) by treating as fixed. Equation (37) has solutions for . Choosing the minimum among solutions guarantees that all the eigenvalues of the in (37) are , i.e., nonpositive ([14]). It is represented as
for
(44) and
(45) (38) is a function of . This implies that in (35) can be found by using (38) via all Next, . The same reasoning yields (38). It can be determined as follows:
(39) for every Equation (39) means that the minimum value of is the maximum uncertainty bound for the th plant rule of a T–S fuzzy model as in (35). Despite that (39) yields the required uncertainty bounds, ’s is, in general, time consearching for the entire suming. The problem can be made more tractable and less ’s to be convex time consuming by only considering hyperpolyhedrons and replacing the reciprocals of ’s as ’s, i.e.,
Furthermore, it is sufficient to only check the protruded points . The following proposition due to the convexity of holds. unProposition 2: Consider the T–S fuzzy model with certain plant rules as
(20) Revisited where
’s are Schur and pairwise commutative, (34) Revisited
’s are fixed sets that are compact and Suppose that convex hyperpolyhedron. Then the maximum possible bounds ’s defined by of
(35) Revisited
(40) is From (40), the problem of finding the minimum of converted to the problem of finding the maximum of . Fur’s thermore, the convex hyperpolyhedron assumption of enable the searching process in (39) to be limited to only searching for the finite number of protruded points. Substituting (40) into (37) yields [14]
(44) Revisited
(41)
(45) Revisited
(42)
’s. via the finite protruded points of each Proof: Straightforward by using Proposition 1 and [14]. (End of Proof) The following example illustrates the use of Proposition 2. Example 4: We shall revisit Example 3 with some modifications. Suppose that
Equation (41) means that there exists a nontrivial such that
By isolating
can be determined by solving the following:
, (42) becomes where (43)
where zero denotes an
zero matrix.
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TABLE VI FIRST PLANT RULE A1
TABLE VII SECOND PLANT RULE A2
TABLE VIII THIRD PLANT RULE A3
For simplicity, it is assumed that . The results obtained from (44) and (45) are shown in Tables VI–VIII. Tables VI–VIII show that the maximum eigenvalues for are 38.8201, 8.0488, and each system matrices , , and 14.3584, respectively. This, in turn, means that the maximum uncertainty bounds for each plant rules are their reciprocals, i.e., 0.0258, 0.1242, and 0.0696, respectively. These can be interpreted as the bounds of tolerance when the nominal system is under the pairwise commutativity assumption. Example 4 can be checked indirectly by reconsidering the results of Example 3. The uncertainty bound was found to be 0.02 in Example 3. It is smaller than the maximum allowable bound 0.0258 in Example 4. Therefore, it can be reasonably explained why the system in Example 3 is quadratically stable. and in Example The greater maximum eigenvalues for 3 can be explained by the greater maximum allowable bounds and in Example 4. of Remark 4: So far, analyses are focused on the T–S fuzzy model, which has Hurwitz and pairwise commutative system matrices with some perturbation on them. One may, however, interpret it as non-Hurwitz and/or nonpairwise commutative system matrices without perturbation on them as mentioned in Remark 2. It means that the Hurwitz and pairwise commutativity assumption in Section III can be relaxed by interpreting
the uncertainties flexibly. Consider the following system (7) Revisited where ’s are non-Hurwitz and/or pairwise noncommutative. One can consider the decomposition of ’s such as (46) where (47) The original system can be rewritten as follows: (48) where ’s are Schur and pairwise commutative. The “mis’s can be treated as uncertainties as in matched” portion Section IV. Therefore, the pairwise commutativity assumption in Section III can be relaxed. In summary, uncertainty bounds of T–S fuzzy model can be determined from the symmetric positive definite matrix , which is obtained from Schur and pairwise commutative system matrices. The uncertainty bounds are represented as a ’s. scalar multiplier of fixed prototypes of uncertainties ’s, the problem In the case of convex hyperpolyhedron is limited to the search of finite protruded points. Furthermore, the uncertainties can be interpreted more flexibly in order
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to relax the Schur and pairwise commutative -matrices , assumption in Section III. Therefore, the common matrix which is determined systematically in Section III, can be used for broader classes of T–S fuzzy models. We notice, however, in the procedure is nonunique. Presumably the choose of it will affect the tolerable bounds of uncertainty, a systematic way of choosing “the best” has yet been determined. VI. CONCLUSIONS A systematic way of finding a common symmetric positive for a linear discrete T–S fuzzy model is invesmatrix tigated. It is first chosen under the assumption of pairwise commutativity of -system matrices. The common matrix can tolerate perturbations of the assumption. The problem of quadratic stability of a linear T–S fuzzy model with uncertain plant rules where the corresponding nominal system matrices are pairwise commutative is then investigated. The setting is nonconservative. The result is believed to be the first that provides a systematic procedure for the stability issues of T–S fuzzy models. In addition, we also investigate the maximum allowable bound of the uncertainty should the nominal portion is chosen. REFERENCES [1] L. A. Zadeh, “Fuzzy sets,” Inform. Contr., vol. 8, no. 3, pp. 338–353, 1965. [2] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. SMC-15, pp. 116–132, Jan. 1985. , “Stability analysis and design of fuzzy control systems,” Fuzzy [3] Sets Syst., vol. 45, pp. 135–156, 1992. [4] S. Kawamoto, K. Tada, A. Ishigame, and T. Taniguchi, “An approach to stability analysis of second order fuzzy systems,” in 1st Proc. IEEE Int. Conf. Fuzzy Syst., San Diego, CA, Mar. 1992, pp. 1427–1434. [5] K. Tanaka, “Stability and stabilizability of fuzzy-neural-linear control systems,” IEEE Trans. Fuzzy Syst., vol. 3, pp. 438–447, Nov. 1995. [6] L. Xia and T. Chai, “Assessment on robustness properties of a class of nonlinear systems with fuzzy logic controllers,” in Proc. Int. Joint Conf. CFSA/IFIS/SOFT Fuzzy Theory Applicat., Taipei, Taiwan, Dec. 1995, pp. 271–276. [7] J. Zhao, V. Wertz, and R. Gorez, “Linear T-S fuzzy model based robust stabilizing controller design,” in Proc. 34th Conf. Decision Contr., New Orleans, LA, Dec. 1995, pp. 255–260. [8] M. Fu and B. R. Barmish, “Adaptive stabilization of linear systems via switching control,” IEEE Trans. Automat. Contr., vol. AC-31, pp. 1097–1103, Dec. 1986. [9] K. S. Narendra and J. Balakrishnan, “Improving transient response of adaptive control systems using multiple models and switching,” IEEE Trans. Automat. Contr., vol. AC-39, pp. 1861–1866, Sept. 1993. [10] , “A common lyapunov functions for stable LTI systems with commuting A-matrices,” IEEE Trans. Automat. Contr., vol. 39, pp. 2469–2471, Dec. 1994. [11] D. Driankov, H. Hellendoorn, and M. Reinfrank, An Introduction to Fuzzy Control. Berlin, Germany: Springer-Verlag, 1993, pp. 103–115. [12] K. Gu, W. J. Chai, and N. K. Loh, “Toward less conservative stability criterion for discrete-time linear uncertain systems,” in Proc. Amer. Contr. Conf., San Diego, CA, June 1990, pp. 1145–1149.
[13] S. S. Rao, Optimization: Theory and Applications. New Delhi, India: Wiley Eastern, 1979, pp. 87–90. [14] K. Gu, “Quadratic stability bound of discrete-time uncertain systems,” in Proc. Amer. Contr. Conf., Boston, MA, June 1991, pp. 1951–1955.
Joongseon Joh was born in Hong-Seong, Korea. He received the B.S. degree (mechanical engineering) from the Inha University, Korea, in 1981, the M.S. degree (mechanical design and production engineering) from the Seoul National University, Seoul, Korea, in 1983, and the Ph.D. degree (mechanical engineering) from the Georgia Institute of Technology, Atlanta, in 1991. From 1983 to 1986, he was with the Central Research Center of Daewoo Heavy Industries, Korea. From 1991 to 1993 he was with the Agency for Defense Development, Korea. Since 1993, he has been with the Department of Control and Instrumentation Engineering, Changwon National University, Changwon City, Korea, where he is now an Assistant Professor. His research interests include fuzzy logic control theory, neural networks for control, automatic control, and robotics.
Ye-Haw Chen (S’85–M’85) was born in Taiwan, R.O.C. He received the B.S. degree in chemical engineering from the National Taiwan University, Taipei, Taiwan, and the M.S. and Ph.D. degrees in mechanical engineering from the University of California, Berkeley, in 1983 and 1985, respectively. From 1986 to 1988, he was with the Department of Mechanical and Aerospace Engineering of Syracuse University. Since 1988, he has been with the School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, where he is now an Associate Professor. He is a consultant to industry in matters related to automation and is the author of more than 100 refereed research papers. His research interests include fuzzy logic theory, neural networks, automatic control, and manufacturing scheduling. Dr. Chen was the recipient of the Sigma Xi Junior Faculty Award and Sigma Xi Best Research Paper Award. He is a member of ASME.
Reza Langari was born in Iran in 1960. He received the B.S., M.S., and Ph.D. degrees in mechanical engineering from the University of California at Berkeley, CA, in 1980, 1983, and 1991, respectively. He has held positions with Measurex Corporation and Integrated Systems, Inc., CA, and is presently an Associate Professor in the Department of Mechanical Engineering and Associate Director of the Center for Fuzzy Logic, Robotics, and Intelligent Systems Research (CFLISR), both at Texas A&M University, College Station, TX. He is currently a Visiting Research Scientist at the United Technologies Research Center, CT. His interests are in the area of fuzzy information processing and control, nonlinear and adaptive control systems, and computing architecture for real-time control.
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 6, NO. 3, AUGUST 1998
On the Stability Issues of Linear Takagi–Sugeno Fuzzy Models Joongseon Joh, Ye-Haw Chen, Member, IEEE, and Reza Langari
Abstract—Stability issues of linear Takagi–Sugeno (T–S) fuzzy models are investigated. We first propose a systematic way of searching for a common matrix, which, in turn, is related to stability for subsystems that are under a pairwise commutative assumption. The robustness issue under uncertainty in each subsystem is then considered. We then show that the pairwise commutative assumption can, in fact, be relaxed by a similar approach as that for uncertainty. The result is applicable to a rather broad class of T–S models, which are non-Hurwitz and/or nonpairwise commutative.
N
Index Terms— Fuzzy systems, robustness, stability, uncertain systems, uncertainty.
I. INTRODUCTION
F
UZZY logic has been considered as an efficient and effective tool in managing uncertainties of systems since Zadeh’s seminal paper [1]. Among many applications of fuzzy logic, control design appears to be one that has attracted a large amount of attention in the past two decades. In general, fuzzy control system can be classified as Mamdani type and Takagi–Sugeno (T–S) type. The Mamdani-type fuzzy control system is well recognized and received by the society. The T–S-type fuzzy system mainly focuses on the modeling aspect (e.g., [2]). There are some works in literature that are mainly concerned with the stability analysis of T–S fuzzy model. The existence of a proper T–S fuzzy model is first assumed. Tanaka and Sugeno [3] showed that the stability of a T–S fuzzy model could be shown by finding a common symmetric positive definite matrix for subsystems. This has been considered a very important result and some refining efforts have been pursued thereafter. There has not been, however, a systematic matrix in a general framework. way to find the common Kawamoto et al. [4] only considered a second-order system. Tanaka [5] suggested the idea of using linear matrix inequality matrix. Xia and Chai (LMI) for finding the common [6] proposed a stability condition that is based on ad hoc membership values. Zhao et al. [7] extended some past work to consider uncertainty. The current work endeavors to tackle the stability issue of the T–S fuzzy model by first considering the problem as related Manuscript received August 26, 1996; revised March 17, 1997. J. Joh is with the Department of Control and Instrumentation Engineering, Changwon National University, Changwon City, Kyungnam, Korea. Y.-H. Chen is with the The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA. R. Langari is with the Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843 USA. Publisher Item Identifier S 1063-6706(98)05548-9.
to the switching system [8]–[10]. In a sense, the central issue lies in the search for a common positive definite matrix for multiple matrix equations. We then introduce uncertainty into the setting and robustness analysis for stability is performed. It is shown that a somewhat stringent assumption for the can, in fact, be relaxed by following a common matrix similar procedure for robustness analysis. The main contributions of the paper are threefold. First, an iterative algorithm for the choice of a common matrix is proposed. It is shown that the algorithm can indeed reach a solution for systems under a structural assumption. Second, modeling uncertainty is introduced. A robustness analysis on the influence of uncertainty toward stability is also suggested. The analysis is nonconservative and computationally straightforward. Third, we show that the structural assumption can, in fact, be relaxed by following the similar robustness analysis procedure. This, in turn, means our algorithm is applicable to both the uncertainty and nonstructural case. II. BACKGROUND MATERIALS A. Takagi and Sugeno’s Fuzzy Model Takagi and Sugeno [2] proposed an effective way to represent a fuzzy model of a dynamical system. It uses a linear input–output relation as its consequence of individual plant plant rules that rules.1 A T–S fuzzy model is composed of can be represented as
(1) where discrete time index; th state (or linguistic) variable; a fuzzy term of Mj selected for Rule i; fuzzy term set2 of ; state vector and
1 Each plant rule produces the next behavior of the system from current system states and inputs. 2 See Driankov et al. [11, pp. 103–115] for the description of fuzzy term set.
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JOH et al.: ON THE STABILITY ISSUES OF LINEAR TAKAGI–SUGENO FUZZY MODELS
403
Theorem 2: Suppose tative, i.e., Lyapunov equations:
input vector and
; . For any current state vector T–S fuzzy model infers model as follows:
and are Schur and commu. Consider the following two
(8) (9)
and input vector , the as the output of the fuzzy
and and are the unique positive definite where solutions of (8) and (9), respectively. Then we always have (2) (10)
where (3)
. Proof: Substituting the
in (9) into (8) yields
and
(11) (4) Since , (2) can be written as
For a free system (i.e.,
and are commutative, we have and, hence, . Therefore, (11)
becomes (5)
From now on, we shall assume a proper T–S fuzzy model in the form of (5) is available. B. Formulation of the Stability Problem Tanaka and Sugeno [3] suggested an important criterion for the stability of the T–S fuzzy model. Theorem 1 (Tanaka and Sugeno [3]): The equilibrium ) is globally state of the fuzzy system (5) (namely, asymptotically stable if there exists a common positive definite such that matrix for all
(12) where (13) and is Schur, the “solution” and is Since unique. (End of Proof). By choosing the Lyapunov function candidate
(6)
So far, however, no systematic way of finding the common exists. We suggest to translate the stability problem of T–S fuzzy model to be one for the following -simultaneous linear systems:
(14) for both systems, for the
system we have
(7) where the system matrix as (1).
(15)
for the th plant rule is the same since
from (8). In addition,
III. STABILITY ANALYSIS OF LINEAR T–S FUZZY MODEL Narendra and Balakrishnan [10] suggested a systematic matrix of -simultaneous way of finding the common continuous-time linear systems under a pairwise commutativity assumption. The results are now extended to discrete-time systems. for simplicity. Let a T–S fuzzy We first consider and . model be with two plant rules, i.e.,
(16) system since . Equations (15) and (16) show for the for and, thus, can be used as that the common matrix of the T–S fuzzy model by Theorem 1. The technique can be extended to a T–S fuzzy model with plant rules, i.e., .
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Theorem 3: Suppose that is Schur for all and ’s are pairwise commutative, i.e.,
TABLE I COMMON MATRIX
P3
(17) Consider the following
Lyapunov equations:
.. . (18) and is the unique positive where definite symmetric solution of each equation. Then we always have
TABLE II
COMMON MATRIX
P3 WITH PERTURBATIONS ON A MATRICES
(19) Proof: The proof follows the procedure similar to that of Theorem 2. (End of Proof). is Remark 1: Apparently the choice of the ordering of in the nonunique. One may, in practice, attempt to label most appropriate way in order that the pairwise commutativity is achieved. The following example illustrates the use of Theorem 3. Example 1: Consider a T–S fuzzy model with three plant rules and two state variables. Let the corresponding ’s be
and
IV. QUADRATIC STABILITY ANALYSIS OF LINEAR T–S FUZZY MODEL
It can be easily seen that they are all Schur. Furthermore, they are pairwise commutative since
and
Let
an arbitrary symmetric . The resulting is indeed a common symmetric positive definite matrix. The symmetric is obtained under the assumption positive definite matrix of Hurwitz and pairwise commutative system matrices.
The stability analysis in the Section III is based on the matrices. The assumption may, pairwise commutativity of however, be restrictive in certain applications. Therefore, it is interesting to further investigate whether the assumption . Consider the following is stringent to the existence of example. Example 2: Perturbed A matrices: Consider Example 1 with some perturbations on matrices. Let the perturbed matrices be
; we then have the following:
The is symmetric and positive definite since the corresponding eigenvalues are 21.8702 and 2.2157. Now, let us check the to be the common matrix. legitimacy of is indeed a common matrix of the Table I shows that T–S fuzzy model. In summary, a common symmetric positive definite matrix, which guarantees the stability of a T–S fuzzy model with plant rules can be found systematically from Theorem 3. It consists of solving Lyapunov equations iteratively from
The stability of the perturbed system can be checked easily using (6). Table II shows that the T–S fuzzy model is still stable and obtained earlier can indeed tolerate the common matrix certain perturbations of matrices. Remark 2: The perturbed matrices in Example 2 are and Schur but nonpairwise commutative since . However, the result shows that the common can be still used for nonpairwise commutative matrix system.
JOH et al.: ON THE STABILITY ISSUES OF LINEAR TAKAGI–SUGENO FUZZY MODELS
Example 2 shows that the common matrix , which can be determined systematically under a pairwise commutativity assumption, may be obtained without actually referring to the assumption. The main assertion in Example 2 can be more formally stated as a problem of checking the stability of the following system: (20) ’s are Hurwitz and pairwise commutative as in where ’s are (possibly) time-varying perturSection III and bations. Equation (20) can be interpreted as a free system of uncertain plant rules as a T–S fuzzy model composed of follows: Rule i: if and then
is
and
and
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Fig. 1. Convex sets.
is unknown. In Section III, a common matrix (i.e., ) is chosen systematically by way of Theorem 3 under the pairwise commutativity assumption. Furthermore, Example 2 suggests can tolerate some perturbations of matrices. that the can be used as a reasonable choice Motivated by these, the of the common in (24). The following proposition is made. Proposition 1: The T–S fuzzy model (21) is quadratically stable if there exists a positive constant such that
is (25)
(21)
. It is interesting to view that (20) is a collection linear systems with time-varying uncertainties where of ’s are Schur and pairwise commutative; that is, ’s are ’s the nominal system matrices and the corresponding are time-varying uncertainties. For systems under (possibly) time-varying uncertainties, quadratic stability performance can be used as a basis for stability study. Definition 1: Consider the following uncertain system
’s are compact sets) and . Here, ’s are Schur and pairwise commutative are chosen from the corresponding nominal system and by way of Theorem 3. Proof: Direct application of Definition 2 and [3]. (End of Proof). Corollary 1: A T–S fuzzy model in (21) is quadratically stable if
for all
(22)
(26)
and is a known compact set. The uncerwhere tain system is quadratically stable if there exists a symmetric positive definite matrix and a positive constant such that
is where ’s are Hurwitz and pairwise commutative and a common symmetric positive definite matrix. Here, denotes the maximum eigenvalue of the designated symmetric matrix. Proof: Direct application of Definiton 2 and [14]. (End of Proof). Even though (26) is more explicit than (25), it is still difficult to apply it in practice. Gu et al. [12] added an additional be convex. A convex set is defined as assumption that follows (see Rao [13, pp. 87–90]). Definition 3: Consider any two points in a set. The set is convex if the line segment joining them is also in the set. Some examples of convex sets in two dimensions are shown shaded in Fig. 1. Other useful definitions for this paper are convex hyperpolyhedron sets and protruded points. They are defined as follows. Definition 4: A set is a convex hyperpolyhedron if points in the set are common to one or more half spaces. The convex hyperpolyhedron is reduced to be a convex polygon in the two-dimensional case [Fig. 2(a)] and a convex polyhedron in the three=dimensional case [Fig. 2(b)]. A protruded point can be defined in the context of convex set. Definition 5: A protruded point is a point in a convex set which does not lie on a line segment joining two other points of the set. From Definition 5, protruded points can be considered as vertices of convex hyperpolyhedron set. It is also worth noting
(23) and . for all The quadratic stability performance can be readily applied to the systems in (20). The following definition is proposed. Definition 2: Consider the following system with uncertain plant rules: (20) Revisited and are a compact set. where The uncertain system is quadratically stable if there exists a common symmetric positive definite matrix and a common positive constant such that (24) and for all . for all ’s are not necessarily pairwise It is worth noting that commutative and are more general as in Section III. On the other hand, checking quadratic stability of a T–S fuzzy model using the Definition 2 directly may be sometimes difficult, especially for high-dimensional case. Therefore, a more realistic way to use Definition 2 is required. One may only expect to use the nominal system since the uncertainty
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TABLE III FIRST PLANT RULE
A1
where
and and , uncertainty sets
are uncertainty parameters satisfying with . Then, the , and become
(28) (a)
(b)
are convex and there Equation (28) shows that , , and are four protruded points in each uncertainty sets. They are
Fig. 2. Convex polygon and convex polyhedron.
that protruded points are finite for the convex hyperpolyhedron set. It is shown in Gu et al. [12] under the convexity assumption, can be reached by one of the the maximum in (26) for protruded points (i.e., vertices) of the set . The additional does not degrade seriously the convexity assumption of in can often generality of uncertainty since elements of where be represented by scalar bounds, i.e., denotes the element of and denotes a possibly time-varying scalar bound. Therefore, a similar theorem can be made for a T–S fuzzy model with uncertain plant rules using the convexity assumption as follows. Theorem 4: Consider the maximization problem (27) is a fixed symmetric positive definite matrix and where ’s are compact and convex sets for the th plant rule. The maximum for each can be reached by one of the protruded points of each set . Therefore, only protruded points are needed to be checked for quadratic stability. The following example illustrates the use of Corollary 1 and Theorem 4. Example 3: Consider a T–S fuzzy model in Example 1. The system matrices are
(29) Quadratic stability of the system can be checked using Corollary 1 and Theorem 4. As in Example 1, the common symis metric positive definite matrix
by . Corollary 1 yields the first, second, and third plant rule, as seen in Tables III–V. Tables III–V show that the uncertain system is quadratically stable by Proposition 1. Remark 3: We have shown that the quadratic stability in Proposition 1 can be checked by using Theorem 4 under the assumption of compact and convex uncertainty set. It means that in Proposition 1 can be determined from the result of Theorem 4 as follows: (30) where
’s are determined from (31)
and
They are Schur and pairwise commutative, as shown in Example 1. For simplicity, let the uncertainty sets for each plant rules be given as
In summary, the symmetric positive definite common matrix can allow some perturbation of Hurwitz and pairwise matrices. It leads to the quadratic stability commutative uncertain subsystems that have Hurwitz and pairwise on commutative system matrices. With the assumption of compact and convex hyperpolyhedron uncertainty set, the quadratic stability of T–S fuzzy model with uncertainties can be checked can indeed play easily. Therefore, the common matrix more general role than Narendra and Balakrishnan [10].
JOH et al.: ON THE STABILITY ISSUES OF LINEAR TAKAGI–SUGENO FUZZY MODELS
TABLE IV SECOND PLNAT RULE
TABLE V THIRD PLANT RULE
V. ANALYSIS OF UNCERTAINTY BOUNDS OF LINEAR T–S FUZZY MODEL Section IV considers the quadratic stability of a general T–S fuzzy model with -pairwise commutative nominal system matrices. The uncertainty sets ’s are fixed, i.e., the uncertainty bounds are specified. It is, however, also interesting to finding the maximum allowable uncertainty bounds for the as a common . This T–S fuzzy model in (21) using means, the nominal system is prespecified. We formulate the problem as follows. uncertain Problem 1: Consider a T–S fuzzy model with plant rules as
407
A2
A3
By relating to the uncertainty bound, (33) means that the uncertainty bound is represented as scalar multiple to a fixed prototype of uncertainty bound. It is solvable and quite general since many problems can be formulated in this . The way by choosing appropriate fixed uncertainty set systems by uncertainty set in (33) can be extended to defining (34) where the subscript denotes the th plant rule. Problem 1 can be restated as follows using the uncertainty set in (34). uncertain Problem 2: Consider a T–S fuzzy model with plant rules as
(20) Revisited (20) Revisited ’s are where ’s are Schur and pairwise commutative, (possibly) time-varying uncertainties. Find the maximum possuch that sible sets
(32) where is the common symmetholds for arbitrary ric positive definite matrix determined from the corresponding nominal system and is a positive constant. Gu [14] proposed an algorithm calculating the maximum allowable uncertainty bound for the system (22) under the additional assumption that has fixed shape but “adjustable” with the additional assumption size. A new uncertainty set can be stated as follows. Let (33) is a compact and convex set with fixed shape where and size and is constant. The constant is shown to be nonnegative in [14] in the case of stable nominal system.
where ’s are Schur and pairwise commutative, . Here, is defined as in (34). Find the maximum with possible sets (35) . such that (20) is quadratically stable with From the pairwise commutativity of ’s and Corollary 1, such that Problem 2 can be reduced to finding
(36) . for every Gu [14] suggested an algorithm of finding the maximum possible uncertainty bound of the system (22) by using the uncertainty set defined in (33). It can be applied to systems in Problem 2 to determine the maximum possible bound of each plant rules of T–S fuzzy model in (36). The procedure is outlined as follows.
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To determine
Equation (43) is a generalized eigenvalue problem. Therefore, (38) and (39) become
for (36), solve
(37) by treating as fixed. Equation (37) has solutions for . Choosing the minimum among solutions guarantees that all the eigenvalues of the in (37) are , i.e., nonpositive ([14]). It is represented as
for
(44) and
(45) (38) is a function of . This implies that in (35) can be found by using (38) via all Next, . The same reasoning yields (38). It can be determined as follows:
(39) for every Equation (39) means that the minimum value of is the maximum uncertainty bound for the th plant rule of a T–S fuzzy model as in (35). Despite that (39) yields the required uncertainty bounds, ’s is, in general, time consearching for the entire suming. The problem can be made more tractable and less ’s to be convex time consuming by only considering hyperpolyhedrons and replacing the reciprocals of ’s as ’s, i.e.,
Furthermore, it is sufficient to only check the protruded points . The following proposition due to the convexity of holds. unProposition 2: Consider the T–S fuzzy model with certain plant rules as
(20) Revisited where
’s are Schur and pairwise commutative, (34) Revisited
’s are fixed sets that are compact and Suppose that convex hyperpolyhedron. Then the maximum possible bounds ’s defined by of
(35) Revisited
(40) is From (40), the problem of finding the minimum of converted to the problem of finding the maximum of . Fur’s thermore, the convex hyperpolyhedron assumption of enable the searching process in (39) to be limited to only searching for the finite number of protruded points. Substituting (40) into (37) yields [14]
(44) Revisited
(41)
(45) Revisited
(42)
’s. via the finite protruded points of each Proof: Straightforward by using Proposition 1 and [14]. (End of Proof) The following example illustrates the use of Proposition 2. Example 4: We shall revisit Example 3 with some modifications. Suppose that
Equation (41) means that there exists a nontrivial such that
By isolating
can be determined by solving the following:
, (42) becomes where (43)
where zero denotes an
zero matrix.
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TABLE VI FIRST PLANT RULE A1
TABLE VII SECOND PLANT RULE A2
TABLE VIII THIRD PLANT RULE A3
For simplicity, it is assumed that . The results obtained from (44) and (45) are shown in Tables VI–VIII. Tables VI–VIII show that the maximum eigenvalues for are 38.8201, 8.0488, and each system matrices , , and 14.3584, respectively. This, in turn, means that the maximum uncertainty bounds for each plant rules are their reciprocals, i.e., 0.0258, 0.1242, and 0.0696, respectively. These can be interpreted as the bounds of tolerance when the nominal system is under the pairwise commutativity assumption. Example 4 can be checked indirectly by reconsidering the results of Example 3. The uncertainty bound was found to be 0.02 in Example 3. It is smaller than the maximum allowable bound 0.0258 in Example 4. Therefore, it can be reasonably explained why the system in Example 3 is quadratically stable. and in Example The greater maximum eigenvalues for 3 can be explained by the greater maximum allowable bounds and in Example 4. of Remark 4: So far, analyses are focused on the T–S fuzzy model, which has Hurwitz and pairwise commutative system matrices with some perturbation on them. One may, however, interpret it as non-Hurwitz and/or nonpairwise commutative system matrices without perturbation on them as mentioned in Remark 2. It means that the Hurwitz and pairwise commutativity assumption in Section III can be relaxed by interpreting
the uncertainties flexibly. Consider the following system (7) Revisited where ’s are non-Hurwitz and/or pairwise noncommutative. One can consider the decomposition of ’s such as (46) where (47) The original system can be rewritten as follows: (48) where ’s are Schur and pairwise commutative. The “mis’s can be treated as uncertainties as in matched” portion Section IV. Therefore, the pairwise commutativity assumption in Section III can be relaxed. In summary, uncertainty bounds of T–S fuzzy model can be determined from the symmetric positive definite matrix , which is obtained from Schur and pairwise commutative system matrices. The uncertainty bounds are represented as a ’s. scalar multiplier of fixed prototypes of uncertainties ’s, the problem In the case of convex hyperpolyhedron is limited to the search of finite protruded points. Furthermore, the uncertainties can be interpreted more flexibly in order
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to relax the Schur and pairwise commutative -matrices , assumption in Section III. Therefore, the common matrix which is determined systematically in Section III, can be used for broader classes of T–S fuzzy models. We notice, however, in the procedure is nonunique. Presumably the choose of it will affect the tolerable bounds of uncertainty, a systematic way of choosing “the best” has yet been determined. VI. CONCLUSIONS A systematic way of finding a common symmetric positive for a linear discrete T–S fuzzy model is invesmatrix tigated. It is first chosen under the assumption of pairwise commutativity of -system matrices. The common matrix can tolerate perturbations of the assumption. The problem of quadratic stability of a linear T–S fuzzy model with uncertain plant rules where the corresponding nominal system matrices are pairwise commutative is then investigated. The setting is nonconservative. The result is believed to be the first that provides a systematic procedure for the stability issues of T–S fuzzy models. In addition, we also investigate the maximum allowable bound of the uncertainty should the nominal portion is chosen. REFERENCES [1] L. A. Zadeh, “Fuzzy sets,” Inform. Contr., vol. 8, no. 3, pp. 338–353, 1965. [2] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. SMC-15, pp. 116–132, Jan. 1985. , “Stability analysis and design of fuzzy control systems,” Fuzzy [3] Sets Syst., vol. 45, pp. 135–156, 1992. [4] S. Kawamoto, K. Tada, A. Ishigame, and T. Taniguchi, “An approach to stability analysis of second order fuzzy systems,” in 1st Proc. IEEE Int. Conf. Fuzzy Syst., San Diego, CA, Mar. 1992, pp. 1427–1434. [5] K. Tanaka, “Stability and stabilizability of fuzzy-neural-linear control systems,” IEEE Trans. Fuzzy Syst., vol. 3, pp. 438–447, Nov. 1995. [6] L. Xia and T. Chai, “Assessment on robustness properties of a class of nonlinear systems with fuzzy logic controllers,” in Proc. Int. Joint Conf. CFSA/IFIS/SOFT Fuzzy Theory Applicat., Taipei, Taiwan, Dec. 1995, pp. 271–276. [7] J. Zhao, V. Wertz, and R. Gorez, “Linear T-S fuzzy model based robust stabilizing controller design,” in Proc. 34th Conf. Decision Contr., New Orleans, LA, Dec. 1995, pp. 255–260. [8] M. Fu and B. R. Barmish, “Adaptive stabilization of linear systems via switching control,” IEEE Trans. Automat. Contr., vol. AC-31, pp. 1097–1103, Dec. 1986. [9] K. S. Narendra and J. Balakrishnan, “Improving transient response of adaptive control systems using multiple models and switching,” IEEE Trans. Automat. Contr., vol. AC-39, pp. 1861–1866, Sept. 1993. [10] , “A common lyapunov functions for stable LTI systems with commuting A-matrices,” IEEE Trans. Automat. Contr., vol. 39, pp. 2469–2471, Dec. 1994. [11] D. Driankov, H. Hellendoorn, and M. Reinfrank, An Introduction to Fuzzy Control. Berlin, Germany: Springer-Verlag, 1993, pp. 103–115. [12] K. Gu, W. J. Chai, and N. K. Loh, “Toward less conservative stability criterion for discrete-time linear uncertain systems,” in Proc. Amer. Contr. Conf., San Diego, CA, June 1990, pp. 1145–1149.
[13] S. S. Rao, Optimization: Theory and Applications. New Delhi, India: Wiley Eastern, 1979, pp. 87–90. [14] K. Gu, “Quadratic stability bound of discrete-time uncertain systems,” in Proc. Amer. Contr. Conf., Boston, MA, June 1991, pp. 1951–1955.
Joongseon Joh was born in Hong-Seong, Korea. He received the B.S. degree (mechanical engineering) from the Inha University, Korea, in 1981, the M.S. degree (mechanical design and production engineering) from the Seoul National University, Seoul, Korea, in 1983, and the Ph.D. degree (mechanical engineering) from the Georgia Institute of Technology, Atlanta, in 1991. From 1983 to 1986, he was with the Central Research Center of Daewoo Heavy Industries, Korea. From 1991 to 1993 he was with the Agency for Defense Development, Korea. Since 1993, he has been with the Department of Control and Instrumentation Engineering, Changwon National University, Changwon City, Korea, where he is now an Assistant Professor. His research interests include fuzzy logic control theory, neural networks for control, automatic control, and robotics.
Ye-Haw Chen (S’85–M’85) was born in Taiwan, R.O.C. He received the B.S. degree in chemical engineering from the National Taiwan University, Taipei, Taiwan, and the M.S. and Ph.D. degrees in mechanical engineering from the University of California, Berkeley, in 1983 and 1985, respectively. From 1986 to 1988, he was with the Department of Mechanical and Aerospace Engineering of Syracuse University. Since 1988, he has been with the School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, where he is now an Associate Professor. He is a consultant to industry in matters related to automation and is the author of more than 100 refereed research papers. His research interests include fuzzy logic theory, neural networks, automatic control, and manufacturing scheduling. Dr. Chen was the recipient of the Sigma Xi Junior Faculty Award and Sigma Xi Best Research Paper Award. He is a member of ASME.
Reza Langari was born in Iran in 1960. He received the B.S., M.S., and Ph.D. degrees in mechanical engineering from the University of California at Berkeley, CA, in 1980, 1983, and 1991, respectively. He has held positions with Measurex Corporation and Integrated Systems, Inc., CA, and is presently an Associate Professor in the Department of Mechanical Engineering and Associate Director of the Center for Fuzzy Logic, Robotics, and Intelligent Systems Research (CFLISR), both at Texas A&M University, College Station, TX. He is currently a Visiting Research Scientist at the United Technologies Research Center, CT. His interests are in the area of fuzzy information processing and control, nonlinear and adaptive control systems, and computing architecture for real-time control.
