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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 15, NO. 3, JUNE 2007
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A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions Kazuo Tanaka, Member, IEEE, Hiroshi Ohtake, Member, IEEE, and Hua O. Wang, Senior Member, IEEE
Abstract—There has been a flurry of research activities in the analysis and design of fuzzy control systems based on linear matrix inequalities (LMIs). This paper presents a descriptor system approach to fuzzy control system design using fuzzy Lyapunov functions. The design conditions are still cast in terms of LMIs but the proposed approach takes advantage of the redundancy of descriptor systems to reduce the number of LMI conditions which leads to less computational requirement. To obtain relaxed LMI conditions, new types of fuzzy controller and fuzzy Lyapunov function are proposed. A salient feature of the LMI conditions derived in this paper is to relate the feasibility of the LMIs to the switching speed of each linear subsystem (to be exact, to the lower bounds of time derivatives of membership functions). To illustrate the validity and applicability of the proposed approach, two design examples are provided. The first example shows that the LMI conditions based on the fuzzy Lyapunov function are less conservative than those based on a common (standard) Lyapunov function. The second example illustrates the utility of the fuzzy Lyapunov function approach in comparison with a piecewise Lyapunov function approach. Index Terms—Descriptor representation, fuzzy control, fuzzy Lyapunov function, redundancy.
I. INTRODUCTION
ONLINEAR control systems based on the Takagi–Sugeno (T–S) fuzzy model [1] have received a great deal of attention over the last decade (e.g., see [2]–[15]). The main advantage of such fuzzy model-based control methodology [16] is that it provides a natural, simple and effective design approach to complement other nonlinear control techniques (e.g., [17]) that require special and rather involved knowledge. Moreover, there is no loss of generality in adopting the T–S fuzzy model based control design framework as it has been established that any smooth nonlinear control systems can be approximated by the T–S fuzzy models (with linear rule consequence) [18]. Within the general framework of T–S fuzzy model-based control systems, there has been, in particular, a flurry of research activities in the analysis and design of fuzzy control systems based on linear matrix inequalities (LMIs) (e.g., [16]).
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Manuscript received March 22, 2005; revised November 6, 2005 and January 10, 2006. This work was supported in part by a Grant-in-Aid for Scientific Research (C) 15560217 from the Ministry of Education, Science, and Culture of Japan. K. Tanaka and H. Ohtake are with the Department of Mechanical Engineering and Intelligent Systems, The University of Electro-Communications, Tokyo 182-8585, Japan (e-mail: [email protected]; [email protected]). H. O. Wang is with the Department of Aerospace and Mechanical Engineering, Boston University, Boston, MA, 02215 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2006.880005
In this paper, we present a descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions. The design conditions are still cast in terms of LMIs but the proposed approach takes advantage of the redundancy of descriptor systems to reduce the number of LMI conditions which leads to less computational requirement. It is well known that linear descriptor systems [19] describe a larger class of systems than conventional linear state-space models. A descriptor system is also much tighter than a state-space expression for representing independent parametric perturbations. Analysis and design of the linear descriptor systems have been extensively discussed in the literature. A fuzzy descriptor representation that is a kind of nonlinear descriptor system was first stated in [20], [21]. A number of papers (e.g., [22]) extended the results in [20] and [21]. However, these papers did not fully take advantage of the redundancy of descriptor representation in control system design. Furthermore, these papers only addressed the common Lyapunov function approach which typically led to conservative results. In this paper, we propose an approach with new types of fuzzy controller and fuzzy Lyapunov function to take full advantage of the redundancy of fuzzy descriptor systems to reduce the number of LMI conditions and to render less conservative stability and stabilization results. With regard to relax the conservativeness of stability and stabilization problems using Lyapunov approach, recently piecewise or switched Lyapunov function approaches [23]–[26] have received increasing attention. However, stabilization conditions for fuzzy Lyapunov functions [27] and piecewise Lyapunov functions [28], [29] are in terms of bilinear matrix inequalities (BMIs) in general. In [27], BMI conditions have been converted into LMI conditions by way of the well-known completing square technique. In general such a conversion leads to conservative results. To overcome the conservativeness, in this paper, we directly obtain LMI design conditions for stabilizing fuzzy controllers (without using the completing square technique) by introducing new types of fuzzy controller and fuzzy Lyapunov function. A salient feature of the LMI conditions derived in this paper is to relate the feasibility of the LMIs to the switching speed of each linear subsystems (exactly speaking, to the lower bounds of time derivatives of membership functions). The details are illustrated through the design examples in Section V-B. The rest of the paper is organized as follows. Section II recalls the T–S fuzzy model and control. Section III presents a design method based on common Lyapunov functions within fuzzy descriptor systems. Section IV introduces new types of fuzzy controller and fuzzy Lyapunov functions to take advantage of the redundancy of descriptor systems. Section V entails two design
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examples to demonstrate the utility of the proposed approach. The first example shows that the LMI conditions based on the fuzzy Lyapunov function are less conservative than those based on a common (standard) Lyapunov function. The second example illustrates the utility of the fuzzy Lyapunov function approach in comparison with a piecewise Lyapunov function approach [33].
The parallel distributed compensation (PDC) offers a procedure to design a fuzzy controller from the given T–S fuzzy model (2). Control Rule : If
is
and
and
is
then (4)
The overall fuzzy controller can be calculated by II. T–S FUZZY MODEL AND CONTROL Section II summarizes T–S fuzzy model and control [16]. The T–S fuzzy model is described by fuzzy IF–THEN rules which represent local linear input–output relations of a nonlinear system. The main feature of this model is to express the local dynamics of each fuzzy implication (rule) by a linear system model. The overall fuzzy model of the system is achieved by fuzzy blending of the linear system models. Consider the following nonlinear system:
(5) A sufficient condition for the stability of the feedback system consisting of (3) and (5) is given as follows: (6) (7) (8)
(1) where is a function. is the state vector and is the input vector. Based on the sector nonlinearity concept [16], we can exactly represent (1) with the T–S fuzzy model (2) (globally or at least semi-globally). Model Rule : If
is
and
and
is
where
.
III. DESIGN VIA COMMON LYAPUNOV FUNCTION In this section, we consider control design via a common Lyapunov function. To cast the system into a descriptor form, the fuzzy controller (5) is converted into (9).
then (9)
(2) is the premise variable. The where membership function associated with the th Model Rule . and th premise variable component is denoted by denotes the number of Model Rules. Each is a measurable time-varying quantity that may be states, measurable external variables and/or time. The defuzzification process of the model (2) can be represented as
denotes the matrix whose elements are zero. where From (3) and (9), we have the following descriptor representation.
(10) where
(3) and . It should be noted from the properties of membership functions that the following relations hold. where
Theorem 1 represents a sufficient stability condition for (10). satisfying (11) and Theorem 1: If there exists a matrix (12), the control system (10) is stable. (11) (12) Proof: Consider a candidate Lyapunov function
Hence (13) where . Note that satisfying (12).
. Therefore, we have (11), where is a nonsingular matrix if there exists
TANAKA et al.: A DESCRIPTOR SYSTEM APPROACH TO FUZZY CONTROL SYSTEM DESIGN VIA FUZZY LYAPUNOV FUNCTIONS
The time derivative of (13) along the trajectories of (3) is
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We define (17) is reas a candidate fuzzy Lyapunov function, where . This requirement is satisfied for fuzzy quired to be at least models constructed via a sector nonlinearity approach [16] if the . system (1) is at least Theorem 2 gives a sufficient condition based on (17). for and Theorem 2: Assume that for all . Then, the control system (10) is stable satisfying if there exists
Therefore, if
(18) (19) (14)
then
for all . We obtain (11) from and . Remark 1: The number of the conditions (6)–(8) to design the fuzzy controller (5) is . In contrast, the number of the conditions (11) and (12) to design the fuzzy controller (9) . The redundancy of the descriptor representation can is drastically reduce the number of design conditions especially for large , i.e., more complicated systems. Therefore the design conditions (11) and (12) are particularly useful for complicated systems. To cast the conditions (11) and (12) into LMIs, we define as
(15)
. For the above , the conditions (11) and (12) where are LMIs with respect to the feedback gains and the variables in . Corollary 1 collects the LMIs. , the conditions Corollary 1: If we use (15) as a common (12) and (11) amount to (16), as shown at the bottom of the page, where . The symbol “ ” denotes and the transposed elements (matrices) for symmetric positions.
(20) Proof: Consider a candidate fuzzy Lyapunov function (21) where
(22) is a nonsingular As in Theorem 1, note that matrix. The condition (22) holds if the following conditions [i.e., (18)] are satisfied:
The time derivative of (21) along the trajectories of (3) is obtained as follows:
IV. DESIGN VIA FUZZY LYAPUNOV FUNCTION In this section, we present two designs based on fuzzy Lyapunov functions. The first method addresses the design of a PDC fuzzy controller via a fuzzy Lyapunov function. This is followed by Section IV-B which proposes a new fuzzy controller to obtain more relaxed stability results.
(16)
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Note that it satisfies
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From this fact, we obtain the following condition:
Multiplying the inequality on the left and right by and , respectively, we have Therefore
(23) From the properties of membership functions, we have
Therefore
Inequality (23) can be rewritten as
(24)
TANAKA et al.: A DESCRIPTOR SYSTEM APPROACH TO FUZZY CONTROL SYSTEM DESIGN VIA FUZZY LYAPUNOV FUNCTIONS
From
and
,
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B. LMIs Based on New Fuzzy Lyapunov Function and New Fuzzy Controller
for all if
We propose a new fuzzy controller (28) instead of the PDC fuzzy controller (5)
(28) Remark 2: The condition (20) reduces to the condition (12) for all . The condition (19) always holds when when for all . Therefore, Theorem 2 is always less conservative than Theorem 1. . Remark 3: In Theorem 2, we assume that However, in practical applications, it may not be easy to select to satisfy the assumption. Techniques for selecting to satisfy the assumption will be discussed in Section V.
is the feedback gain depending on and where is an LMI variable to design the controller. From (3) and (28), we have the following descriptor representation:
(29) where
A. LMIs Based on Fuzzy Lyapunov Function The conditions in Theorem 2 can be converted into the LMIs if we choose
(25)
as
We derive stability conditions for (29) using the fuzzy Lyapunov function (17). The conditions in Theorem 2 can be converted into the LMIs if we use
, where . Corollary 2: If we use matrix (25), then (30)
as Therefore, the conditions (18) and (20) reduce to (26) and (27), as shown at the bottom of the page, respectively, where . Note that the condition (19) always hold if we use (25) . as instead of in (25), should be Remark 4: If we use . In this case, even if the LMIs in Corollary 2 can not be uniquely determined from are feasible, . Section IV-B provides a solution to this problem.
, where
. Then, we obtain stable feedback gains as (31)
and in Corollary 3. from the solutions Corollary 3: If we use the matrix (30), then the conditions (18)–(20) reduce to (32)–(34), as shown at the bottom of the next page, respectively. will be addressed in Section V. How to select
(26)
(27)
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Fig. 1. Feasible area for common Lyapunov function (Corollary 1).
Fig. 2. Feasible area for Lyapunov function with (25) (Corollary 2).
V. DESIGN EXAMPLES A. Design Example 1 Consider the following system (35): (35) Under , system (35) can be exactly converted into the following fuzzy model: (36) where
, , , . The membership functions are obtained as
We compare the fuzzy Lyapunov approaches (Corollaries 2 and 3) with the common Lyapunov approach (Corollary 1). First, we design stabilizing controllers for several combinations . Fig. 1 shows of and using Corollary 1, where
the feasible area for the combinations, where the dotted area denotes the feasible area. Next, we design stabilizing controllers using Corollary 2. Fig. 2 shows the feasible area for several combinations of and using Corollary 2. It can be seen from the figures that the feasible area for the fuzzy Lyapunov function with (25) is almost the same as that of the common Lyapunov function. Finally, we design stabilizing controllers using Corollary 3. and . Since In Corollary 3, we need to select the values of , it is enough to consider only . Figs. 3–5 show the feasible area for several values of . It can be seen that the feasible areas for the fuzzy Lyapunov function with (30) is larger than that of the common Lyapunov function. In fact, if for all , Theorem 2 reduces to Theorem 1. Thus, the fuzzy Lyapunov function approach (Corollary 3) leads to less conservative results. should be selected so as To solve the LMIs in Corollary 3, . In this example, the values of are to satisfy given in advance. The next example will show how to select in practice. For cases where it is difficult to select proper , one can resort to a small value for . Of course, extremely small
(32) (33) (34)
TANAKA et al.: A DESCRIPTOR SYSTEM APPROACH TO FUZZY CONTROL SYSTEM DESIGN VIA FUZZY LYAPUNOV FUNCTIONS
Fig. 3. Feasible area for Lyapunov function with (30) (Corollary 3 and = 1:0 10 ).
0 2
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Fig. 5. Feasible area for Lyapunov function with (30) (Corollary 3 and =
01:0 2 10 ).
We can exactly represent the nonlinear dynamics with the following fuzzy model: (38) where
The membership functions are obtained as
The common Lyapunov approach guarantees the stability for . The piecewise Lyapunov functions used in this example are Fig. 4. Feasible area for Lyapunov function with (30) (Corollary 3 and = 1:0 10 ).
0 2
(39) and
values can lead to conservative results. However, the conditions in Corollary 3 always guarantee less conservative results than those in Corollaries 1 and 2. B. Design Example 2 This example shows comparative results between the proposed Lyapunov function with (30) and a couple of well-know piecewise Lyapunov functions. Consider the following nonlinear system:
(40) The stability conditions [33] for (39) and (40) are obtained for the system (37). The condition for (39) guarantees the stability . The condition for (40) guarantees the stability for for . Next, we show stability result using Corollary 3. The time derivative of the membership functions are
(41) (37) where
is at least a
function.
Fig. 6 shows the maximum values of guaranteeing the feasibility of Corollary 3 for each . Fig. 6 also shows the maximum
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From the assumption in Theorem 2, we have the following relation. (45) From (45) and (44), we have . Therefore, in the and nonlinear system (43), our approach is better for the piecewise Lyapunov function is better for . Again our approach explicitly relates the feasibility of the LMIs to the switching speed between rules. VI. CONCLUSION
Fig. 6. Maximum values of k guaranteeing feasibility of Corollary 3 for each .
values of guaranteeing the feasibility of common Lyapunov and the piecewise Lyapunov function function . approach It can be concluded that the fuzzy Lyapunov function approach guarantees larger feasible (stable) area (i.e., larger ) for larger . That is, the fuzzy Lyapunov function with (30) guarantees larger feasible area than the piecewise Lyapunov function . Conversely, when , the piecewise when Lyapunov function guarantees larger feasible area than the Lyapunov function with (30). As mentioned in the discussion for Example 1, it is clear that the fuzzy Lyapunov function approach (Corollary 3) is always less conservative than the common Lyapunov function approach (Corollary 1). Thus, by using our approach, we can relate stability (feasias our approach considers the time bility) area according to derivative of membership functions, i.e., the switching speed between Rule 1 and Rule 2. The piecewise Lyapunov function and . The switches crisply between switching speed of the piecewise Lyapunov function can be regarded as infinity. In other words, the piecewise Lyapunov function guarantees the stability for the infinity switching speed. Therefore the stability conditions for the piecewise Lyapunov function are conservative if the switching speed between Rules 1 and 2 is moderate. Finally, we assume that (42) Note that (42) satisfies (37) can be rewritten as
. Then, the nonlinear system
(43) is the parameter with respect to period. The time derivative of (42) is (44)
This paper has presented a descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions. New types of fuzzy controller and fuzzy Lyapunov function have been proposed to render less conservative LMI design conditions. A salient feature of the design conditions is to relate the feasibility of the LMIs to the switching speed of each linear subsystems. The design examples have illustrated the viability and utility of the proposed approach. ACKNOWLEDGMENT The authors would like to thank Mr. T. Nebuya, UEC, Japan, for his contribution to this research. REFERENCES [1] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. SMC-15, no. 1, pp. 116–132, Jan. 1985. [2] K. Tanaka and M. Sugeno, “Stability analysis of fuzzy systems using lyapunov’s direct method,” in Proc. NAFIPS’90, Toronto, Ontario, Canada, Jun. 1990, pp. 133–136. [3] R. Langari and M. Tomizuka, “Analysis and synthesis of fuzzy linguistic control systems,” in Proc. 1990 ASME Winter Annu. Meeting, Dallas, TX, Nov. 1990, pp. 35–42. [4] S. Kitamura and T. Kurozumi, “Extended circle criterion and stability analysis of fuzzy control systems,” in Proc. Int. Fuzzy Eng. Symp., Yokohama, Japan, Nov. 1991, vol. 2, pp. 634–643. [5] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Sets Syst., vol. 45, no. 2, pp. 135–156, 1992. [6] S. S. Farinwata, “Stability analysis of the fuzzy logic controller designed by the phase portrait assignment algorithm,” in Proc. 2nd IEEE Int. Conf. Fuzzy Systems, San Francisco, CA, Apr. 1993, pp. 1377–1382. [7] M. Sugeno, “On stability of fuzzy systems expressed by fuzzy rules with singleton consequents,” IEEE Trans. Fuzzy Syst., vol. 7, no. 2, pp. 201–224, Apr. 1999. [8] S. Hong and R. Langari, “Synthesis of an LMI-based fuzzy control system with guaranteed optimal H performance,” in Proc. FUZZIEEE’98, Anchorage, AK, May 1998, pp. 422–427. [9] D. Filev, “Algebraic design of fuzzy logic controllers,” in Proc. 1996 IEEE Int. Symp. Intelligent Control, Dearborn, MI, Sep. 1996, pp. 253–258. [10] G. Chen and D. Zhang, “Back-driving a truck with suboptimal distance trajectories: A Fuzzy logic control approach,” IEEE Trans. Fuzzy Syst., vol. 3, no. 1, pp. 125–131, Feb. 1995. [11] T. Furuhashi, “Fuzzy control stability analysis using generalized fuzzy petri net model,” J. Adv. Comput. Intell., vol. 3, no. 2, pp. 99–105, 1999. [12] H. Yamamoto and T. Furuhashi, “New fuzzy inference method for symbolic stability analysis of fuzzy control systems,” in Proc. 9th IEEE Int. Conf. Fuzzy Syst., San Antonio, TX, May 2000, pp. 659–664. [13] P. Baranyi, “SVD-based complexity reduction to ts fuzzy models,” IEEE Trans. Ind. Electron. , vol. 49, no. 2, pp. 433–443, Apr. 2002. [14] P. Baranyi, “SVD-based reduction to MISO TS models,” IEEE Trans. Ind. Electron., vol. 51, no. 1, pp. 232–242, Feb. 2003.
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[15] P. Baranyi, “TP model transformation as a way to LMI based controller design,” IEEE Trans. Ind. Electron., vol. 51, no. 2, pp. 387–400, Apr. 2004. [16] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analysis: An Linear Matrix Inequality Approach. New York: Wiley, 2001. [17] R. Sepulcher, M. Jankovic, and P. Kokotovic, Constructive Nonlinear Control. New York: Springer-Verlag, 1997. [18] H. O. Wang, J. Li, D. Niemann, and K. Tanaka, “T–S fuzzy model with linear rule consequence and PDC controller: A universal framework for nonlinear control systems,” in Proc. 9th IEEE Int. Conf. Fuzzy Syst., San Antonio, TX, May 2000, pp. 549–554. [19] D. G. Luenberger, “Dynamic equations in descriptor form,” IEEE Trans. Autom. Control, vol. AC-22, no. 3, pp. 312–321, Mar. 1977. [20] T. Taniguchi, “Fuzzy descriptor systems: Stability analysis and design via LMIs,” in Proc. Amer. Control Conf., San Diego, CA, Jun. 1999, pp. 1827–1831. [21] T. Taniguchi, “Fuzzy descriptor systems and nonlinear model following control,” IEEE Trans. Fuzzy Syst., vol. 8, no. 4, pp. 442–452, Aug. 2000. [22] Y. Wang, Q. L. Zhang, and W. Q. Lin, “Stability analysis and design for T–S fuzzy descriptor systems,” in Proc. 40th Conf. Decision and Control, Orlando, FL, Dec. 2001, pp. 3962–3967. [23] L. Fang, H. Lin, and P. J. Antsaklis, “Stabilization and performance analysis for a class of switched systems,” in Proc. 43rd Conf. Decision Control, Paradise Island, Bahamas, Dec. 2004, pp. 3265–3270. [24] H. Ohtake, “Switching fuzzy controller design based on switching lyapunov function for a class of nonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 36, no. 1, pp. 13–23, Feb. 2006. [25] D. Xie, L. Wang, F. Hao, and G. Xie, “Robust stability analysis and control synthesis for discrete-time uncertain switched systems,” in Proc. 42nd Conf. Decision and Control, Honolulu, HI, Dec. 2003, pp. 4812–4816. [26] G. Feng, “Stability analysis of piecewise discrete-time linear systems,” IEEE Trans. Autom. Control, vol. 47, no. 7, pp. 1108–1112, Jul. 2002. [27] K. Tanaka, T. Hori, and H. O. Wang, “A multiple Lyapunov function approach to stabilization of fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 11, no. 4, pp. 582–589, Aug. 2003. [28] T. Taniguchi and M. Sugeno, “Stabilization of nonlinear systems based on piecewise Lyapunov functions,” in Proc. 13th IEEE Int. Conf. Fuzzy Systems, Budapest, Hungary, Jul. 2004, pp. 1607–1612. [29] H. Ohtake, K. Tanaka, and H. O. Wang, “Piecewise nonlinear control,” in Proc. 42nd IEEE Conf. Decision Control, Maui, HI, Dec. 2003, pp. 4735–4740. [30] K. Tanaka, “Fuzzy control of dynamical systems with input nonlinearity,” in Proc. Joint 2nd Int. Conference on Soft Computing and Intelligent Systems and 5th Int. Symp. on Advanced Intelligent Systems, Yokohama, Japan, 2004, TUP-7-1 in CD. [31] K. Tanaka, T. Hori, and H. O. Wang, “Dynamic output feedback designs for nonlinear systems,” in Proc. IEEE Int. Conf. Systems, Man, and Cybernetics, Tokyo, Japan, Aug. 1999, vol. 3, pp. 56–61. [32] T. Taniguchi, K. Tanaka, and H. O. Wang, “Model construction, rule reduction, and robust compensation for generalized form of takagisugeno fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 8, no. 4, pp. 525–538, Aug. 2001. [33] L. Xie, S. Shishkin, and M. Fu, “Piecewise lyapunov functions for robust stability of linear time-varying systems,” Syst. Control Lett., vol. 31, pp. 165–171, 1997. [34] H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2002. [35] G. Chen and H. Shibata, “System analysis using redundancy of descriptor representation,” J. Syst., Control Inform., vol. 47, no. 5, pp. 211–216, May 2003.
Kazuo Tanaka (S’87–M’91) received the B.S. and M.S. degrees in electrical engineering from Hosei University, Tokyo, Japan, in 1985 and 1987, respectively, and the Ph.D. degree in systems science from the Tokyo Institute of Technology, Tokyo, Japan, in 1990. He is currently a Professor in the Department of Mechanical Engineering and Intelligent Systems at The University of Electro-Communications, Tokyo, Japan. He was a Visiting Scientist in Computer Science at the University of North Carolina, Chapel
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Hill, in 1992 and 1993. He is the author of two books and a coauthor of nine books. Recently, he coauthored (with H. O. Wang) the book Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach (New York: Wiley-Interscience, 2001). His research interests include intelligent systems and control, nonlinear systems control, robotics, and applications. Dr. Tanaka is currently serving on the IEEE Control Systems Society Conference Editorial Board. He received the Best Young Researchers Award from the Japan Society for Fuzzy Theory and Systems in 1990, the Outstanding Papers Award at the 1990 Annual NAFIPS Meeting in Toronto, Canada, in 1990, the Outstanding Papers Award at the Joint Hungarian-Japanese Symposium on Fuzzy Systems and Applications in Budapest, Hungary, in 1991, the Best Young Researchers Award from the Japan Society for Mechanical Engineers in 1994, the Best Book Awards from the Japan Society for Fuzzy Theory and Systems in 1995, 1999 IFAC World Congress Best Poster Paper Prize in 1999, the IEEE TRANSACTIONS ON FUZZY SYSTEMS Outstanding Paper Award in 2000, and the Best Paper Selection at 2005 American Control Conference in Portland, OR, in 2005.
Hiroshi Ohtake (S’02–M’05) received the B.S. and M.S. degrees in mechanical and control engineering from The University of Electro-Communications, Tokyo, Japan, in 2000 and 2002, respectively. He is currently an Assistant Professor in Department of Mechanical Engineering and Intelligent Systems at The University of Electro-Communications. He was a Research Fellow of the Japan Society for the Promotion of Science from 2002 to 2004. His research interests include nonlinear mechanical systems control and robotics. Mr. Ohtake received the Outstanding Student Paper Award at the Joint 9th IFSA World Congress and the 20th NAFIPS International Conference in Vancouver, BC, Canada, in 2001, the Young Investigators Award from the Japan Society for Fuzzy Theory and Intelligent Informatics in 2003, the Best Presentation Award at the FAN Symposium 2004 in Kochi, Japan, in 2004, the American Control Conference Best Paper Selection, at American Control Conference 2005 in Portland, OR, in 2005.
Hua O. Wang (M’94–SM’01) received the B.S. degree from the University of Science and Technology of China (USTC), Hefei, in 1987, the M.S. degree from the University of Kentucky, Lexington, in 1989, and the Ph.D. degree from the University of Maryland, College Park, in 1993, all in electrical engineering. Since September 2002, he has been with Boston University, Boston, MA, where he is currently an Associate Professor of Aerospace and Mechanical Engineering. He was with the United Technologies Research Center, East Hartford, CT, from 1993 to 1996, and was a Faculty Member in the Department of Electrical and Computer Engineering at Duke University, Durham, NC, from 1996 to 2002. He served as the Program Manager (IPA) for Systems and Control with the U.S. Army Research Office (ARO) from August 2000 to August 2002. During 2000–2005, he also held the position of Cheung Kong Chair Professor and Director with the Center for Nonlinear and Complex Systems at Huazhong University of Science and Technology, Wuhan, China. His research interests include control of nonlinear dynamics, intelligent systems and control, networked control systems, robotics, cooperative control, and applications. He coauthored (with K. Tanaka) the book Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach (New York: WileyInterscience, 2001). Dr. Wang is a recipient of the 1994 O. Hugo Schuck Best Paper Award of the American Automatic Control Council, the 14th IFAC World Congress Poster Paper Prize, the 2000 IEEE TRANSACTIONS ON FUZZY SYSTEMS Outstanding Paper Award. He has served as an Associate Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, and was on the IEEE Control Systems Society Conference Editorial Board. He is an Editor for the Journal of Systems Science and Complexity. He is an appointed member of the 2006 Board of Governors of the IEEE Control Systems Society.
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A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions Kazuo Tanaka, Member, IEEE, Hiroshi Ohtake, Member, IEEE, and Hua O. Wang, Senior Member, IEEE
Abstract—There has been a flurry of research activities in the analysis and design of fuzzy control systems based on linear matrix inequalities (LMIs). This paper presents a descriptor system approach to fuzzy control system design using fuzzy Lyapunov functions. The design conditions are still cast in terms of LMIs but the proposed approach takes advantage of the redundancy of descriptor systems to reduce the number of LMI conditions which leads to less computational requirement. To obtain relaxed LMI conditions, new types of fuzzy controller and fuzzy Lyapunov function are proposed. A salient feature of the LMI conditions derived in this paper is to relate the feasibility of the LMIs to the switching speed of each linear subsystem (to be exact, to the lower bounds of time derivatives of membership functions). To illustrate the validity and applicability of the proposed approach, two design examples are provided. The first example shows that the LMI conditions based on the fuzzy Lyapunov function are less conservative than those based on a common (standard) Lyapunov function. The second example illustrates the utility of the fuzzy Lyapunov function approach in comparison with a piecewise Lyapunov function approach. Index Terms—Descriptor representation, fuzzy control, fuzzy Lyapunov function, redundancy.
I. INTRODUCTION
ONLINEAR control systems based on the Takagi–Sugeno (T–S) fuzzy model [1] have received a great deal of attention over the last decade (e.g., see [2]–[15]). The main advantage of such fuzzy model-based control methodology [16] is that it provides a natural, simple and effective design approach to complement other nonlinear control techniques (e.g., [17]) that require special and rather involved knowledge. Moreover, there is no loss of generality in adopting the T–S fuzzy model based control design framework as it has been established that any smooth nonlinear control systems can be approximated by the T–S fuzzy models (with linear rule consequence) [18]. Within the general framework of T–S fuzzy model-based control systems, there has been, in particular, a flurry of research activities in the analysis and design of fuzzy control systems based on linear matrix inequalities (LMIs) (e.g., [16]).
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Manuscript received March 22, 2005; revised November 6, 2005 and January 10, 2006. This work was supported in part by a Grant-in-Aid for Scientific Research (C) 15560217 from the Ministry of Education, Science, and Culture of Japan. K. Tanaka and H. Ohtake are with the Department of Mechanical Engineering and Intelligent Systems, The University of Electro-Communications, Tokyo 182-8585, Japan (e-mail: [email protected]; [email protected]). H. O. Wang is with the Department of Aerospace and Mechanical Engineering, Boston University, Boston, MA, 02215 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2006.880005
In this paper, we present a descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions. The design conditions are still cast in terms of LMIs but the proposed approach takes advantage of the redundancy of descriptor systems to reduce the number of LMI conditions which leads to less computational requirement. It is well known that linear descriptor systems [19] describe a larger class of systems than conventional linear state-space models. A descriptor system is also much tighter than a state-space expression for representing independent parametric perturbations. Analysis and design of the linear descriptor systems have been extensively discussed in the literature. A fuzzy descriptor representation that is a kind of nonlinear descriptor system was first stated in [20], [21]. A number of papers (e.g., [22]) extended the results in [20] and [21]. However, these papers did not fully take advantage of the redundancy of descriptor representation in control system design. Furthermore, these papers only addressed the common Lyapunov function approach which typically led to conservative results. In this paper, we propose an approach with new types of fuzzy controller and fuzzy Lyapunov function to take full advantage of the redundancy of fuzzy descriptor systems to reduce the number of LMI conditions and to render less conservative stability and stabilization results. With regard to relax the conservativeness of stability and stabilization problems using Lyapunov approach, recently piecewise or switched Lyapunov function approaches [23]–[26] have received increasing attention. However, stabilization conditions for fuzzy Lyapunov functions [27] and piecewise Lyapunov functions [28], [29] are in terms of bilinear matrix inequalities (BMIs) in general. In [27], BMI conditions have been converted into LMI conditions by way of the well-known completing square technique. In general such a conversion leads to conservative results. To overcome the conservativeness, in this paper, we directly obtain LMI design conditions for stabilizing fuzzy controllers (without using the completing square technique) by introducing new types of fuzzy controller and fuzzy Lyapunov function. A salient feature of the LMI conditions derived in this paper is to relate the feasibility of the LMIs to the switching speed of each linear subsystems (exactly speaking, to the lower bounds of time derivatives of membership functions). The details are illustrated through the design examples in Section V-B. The rest of the paper is organized as follows. Section II recalls the T–S fuzzy model and control. Section III presents a design method based on common Lyapunov functions within fuzzy descriptor systems. Section IV introduces new types of fuzzy controller and fuzzy Lyapunov functions to take advantage of the redundancy of descriptor systems. Section V entails two design
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examples to demonstrate the utility of the proposed approach. The first example shows that the LMI conditions based on the fuzzy Lyapunov function are less conservative than those based on a common (standard) Lyapunov function. The second example illustrates the utility of the fuzzy Lyapunov function approach in comparison with a piecewise Lyapunov function approach [33].
The parallel distributed compensation (PDC) offers a procedure to design a fuzzy controller from the given T–S fuzzy model (2). Control Rule : If
is
and
and
is
then (4)
The overall fuzzy controller can be calculated by II. T–S FUZZY MODEL AND CONTROL Section II summarizes T–S fuzzy model and control [16]. The T–S fuzzy model is described by fuzzy IF–THEN rules which represent local linear input–output relations of a nonlinear system. The main feature of this model is to express the local dynamics of each fuzzy implication (rule) by a linear system model. The overall fuzzy model of the system is achieved by fuzzy blending of the linear system models. Consider the following nonlinear system:
(5) A sufficient condition for the stability of the feedback system consisting of (3) and (5) is given as follows: (6) (7) (8)
(1) where is a function. is the state vector and is the input vector. Based on the sector nonlinearity concept [16], we can exactly represent (1) with the T–S fuzzy model (2) (globally or at least semi-globally). Model Rule : If
is
and
and
is
where
.
III. DESIGN VIA COMMON LYAPUNOV FUNCTION In this section, we consider control design via a common Lyapunov function. To cast the system into a descriptor form, the fuzzy controller (5) is converted into (9).
then (9)
(2) is the premise variable. The where membership function associated with the th Model Rule . and th premise variable component is denoted by denotes the number of Model Rules. Each is a measurable time-varying quantity that may be states, measurable external variables and/or time. The defuzzification process of the model (2) can be represented as
denotes the matrix whose elements are zero. where From (3) and (9), we have the following descriptor representation.
(10) where
(3) and . It should be noted from the properties of membership functions that the following relations hold. where
Theorem 1 represents a sufficient stability condition for (10). satisfying (11) and Theorem 1: If there exists a matrix (12), the control system (10) is stable. (11) (12) Proof: Consider a candidate Lyapunov function
Hence (13) where . Note that satisfying (12).
. Therefore, we have (11), where is a nonsingular matrix if there exists
TANAKA et al.: A DESCRIPTOR SYSTEM APPROACH TO FUZZY CONTROL SYSTEM DESIGN VIA FUZZY LYAPUNOV FUNCTIONS
The time derivative of (13) along the trajectories of (3) is
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We define (17) is reas a candidate fuzzy Lyapunov function, where . This requirement is satisfied for fuzzy quired to be at least models constructed via a sector nonlinearity approach [16] if the . system (1) is at least Theorem 2 gives a sufficient condition based on (17). for and Theorem 2: Assume that for all . Then, the control system (10) is stable satisfying if there exists
Therefore, if
(18) (19) (14)
then
for all . We obtain (11) from and . Remark 1: The number of the conditions (6)–(8) to design the fuzzy controller (5) is . In contrast, the number of the conditions (11) and (12) to design the fuzzy controller (9) . The redundancy of the descriptor representation can is drastically reduce the number of design conditions especially for large , i.e., more complicated systems. Therefore the design conditions (11) and (12) are particularly useful for complicated systems. To cast the conditions (11) and (12) into LMIs, we define as
(15)
. For the above , the conditions (11) and (12) where are LMIs with respect to the feedback gains and the variables in . Corollary 1 collects the LMIs. , the conditions Corollary 1: If we use (15) as a common (12) and (11) amount to (16), as shown at the bottom of the page, where . The symbol “ ” denotes and the transposed elements (matrices) for symmetric positions.
(20) Proof: Consider a candidate fuzzy Lyapunov function (21) where
(22) is a nonsingular As in Theorem 1, note that matrix. The condition (22) holds if the following conditions [i.e., (18)] are satisfied:
The time derivative of (21) along the trajectories of (3) is obtained as follows:
IV. DESIGN VIA FUZZY LYAPUNOV FUNCTION In this section, we present two designs based on fuzzy Lyapunov functions. The first method addresses the design of a PDC fuzzy controller via a fuzzy Lyapunov function. This is followed by Section IV-B which proposes a new fuzzy controller to obtain more relaxed stability results.
(16)
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Note that it satisfies
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From this fact, we obtain the following condition:
Multiplying the inequality on the left and right by and , respectively, we have Therefore
(23) From the properties of membership functions, we have
Therefore
Inequality (23) can be rewritten as
(24)
TANAKA et al.: A DESCRIPTOR SYSTEM APPROACH TO FUZZY CONTROL SYSTEM DESIGN VIA FUZZY LYAPUNOV FUNCTIONS
From
and
,
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B. LMIs Based on New Fuzzy Lyapunov Function and New Fuzzy Controller
for all if
We propose a new fuzzy controller (28) instead of the PDC fuzzy controller (5)
(28) Remark 2: The condition (20) reduces to the condition (12) for all . The condition (19) always holds when when for all . Therefore, Theorem 2 is always less conservative than Theorem 1. . Remark 3: In Theorem 2, we assume that However, in practical applications, it may not be easy to select to satisfy the assumption. Techniques for selecting to satisfy the assumption will be discussed in Section V.
is the feedback gain depending on and where is an LMI variable to design the controller. From (3) and (28), we have the following descriptor representation:
(29) where
A. LMIs Based on Fuzzy Lyapunov Function The conditions in Theorem 2 can be converted into the LMIs if we choose
(25)
as
We derive stability conditions for (29) using the fuzzy Lyapunov function (17). The conditions in Theorem 2 can be converted into the LMIs if we use
, where . Corollary 2: If we use matrix (25), then (30)
as Therefore, the conditions (18) and (20) reduce to (26) and (27), as shown at the bottom of the page, respectively, where . Note that the condition (19) always hold if we use (25) . as instead of in (25), should be Remark 4: If we use . In this case, even if the LMIs in Corollary 2 can not be uniquely determined from are feasible, . Section IV-B provides a solution to this problem.
, where
. Then, we obtain stable feedback gains as (31)
and in Corollary 3. from the solutions Corollary 3: If we use the matrix (30), then the conditions (18)–(20) reduce to (32)–(34), as shown at the bottom of the next page, respectively. will be addressed in Section V. How to select
(26)
(27)
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Fig. 1. Feasible area for common Lyapunov function (Corollary 1).
Fig. 2. Feasible area for Lyapunov function with (25) (Corollary 2).
V. DESIGN EXAMPLES A. Design Example 1 Consider the following system (35): (35) Under , system (35) can be exactly converted into the following fuzzy model: (36) where
, , , . The membership functions are obtained as
We compare the fuzzy Lyapunov approaches (Corollaries 2 and 3) with the common Lyapunov approach (Corollary 1). First, we design stabilizing controllers for several combinations . Fig. 1 shows of and using Corollary 1, where
the feasible area for the combinations, where the dotted area denotes the feasible area. Next, we design stabilizing controllers using Corollary 2. Fig. 2 shows the feasible area for several combinations of and using Corollary 2. It can be seen from the figures that the feasible area for the fuzzy Lyapunov function with (25) is almost the same as that of the common Lyapunov function. Finally, we design stabilizing controllers using Corollary 3. and . Since In Corollary 3, we need to select the values of , it is enough to consider only . Figs. 3–5 show the feasible area for several values of . It can be seen that the feasible areas for the fuzzy Lyapunov function with (30) is larger than that of the common Lyapunov function. In fact, if for all , Theorem 2 reduces to Theorem 1. Thus, the fuzzy Lyapunov function approach (Corollary 3) leads to less conservative results. should be selected so as To solve the LMIs in Corollary 3, . In this example, the values of are to satisfy given in advance. The next example will show how to select in practice. For cases where it is difficult to select proper , one can resort to a small value for . Of course, extremely small
(32) (33) (34)
TANAKA et al.: A DESCRIPTOR SYSTEM APPROACH TO FUZZY CONTROL SYSTEM DESIGN VIA FUZZY LYAPUNOV FUNCTIONS
Fig. 3. Feasible area for Lyapunov function with (30) (Corollary 3 and = 1:0 10 ).
0 2
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Fig. 5. Feasible area for Lyapunov function with (30) (Corollary 3 and =
01:0 2 10 ).
We can exactly represent the nonlinear dynamics with the following fuzzy model: (38) where
The membership functions are obtained as
The common Lyapunov approach guarantees the stability for . The piecewise Lyapunov functions used in this example are Fig. 4. Feasible area for Lyapunov function with (30) (Corollary 3 and = 1:0 10 ).
0 2
(39) and
values can lead to conservative results. However, the conditions in Corollary 3 always guarantee less conservative results than those in Corollaries 1 and 2. B. Design Example 2 This example shows comparative results between the proposed Lyapunov function with (30) and a couple of well-know piecewise Lyapunov functions. Consider the following nonlinear system:
(40) The stability conditions [33] for (39) and (40) are obtained for the system (37). The condition for (39) guarantees the stability . The condition for (40) guarantees the stability for for . Next, we show stability result using Corollary 3. The time derivative of the membership functions are
(41) (37) where
is at least a
function.
Fig. 6 shows the maximum values of guaranteeing the feasibility of Corollary 3 for each . Fig. 6 also shows the maximum
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From the assumption in Theorem 2, we have the following relation. (45) From (45) and (44), we have . Therefore, in the and nonlinear system (43), our approach is better for the piecewise Lyapunov function is better for . Again our approach explicitly relates the feasibility of the LMIs to the switching speed between rules. VI. CONCLUSION
Fig. 6. Maximum values of k guaranteeing feasibility of Corollary 3 for each .
values of guaranteeing the feasibility of common Lyapunov and the piecewise Lyapunov function function . approach It can be concluded that the fuzzy Lyapunov function approach guarantees larger feasible (stable) area (i.e., larger ) for larger . That is, the fuzzy Lyapunov function with (30) guarantees larger feasible area than the piecewise Lyapunov function . Conversely, when , the piecewise when Lyapunov function guarantees larger feasible area than the Lyapunov function with (30). As mentioned in the discussion for Example 1, it is clear that the fuzzy Lyapunov function approach (Corollary 3) is always less conservative than the common Lyapunov function approach (Corollary 1). Thus, by using our approach, we can relate stability (feasias our approach considers the time bility) area according to derivative of membership functions, i.e., the switching speed between Rule 1 and Rule 2. The piecewise Lyapunov function and . The switches crisply between switching speed of the piecewise Lyapunov function can be regarded as infinity. In other words, the piecewise Lyapunov function guarantees the stability for the infinity switching speed. Therefore the stability conditions for the piecewise Lyapunov function are conservative if the switching speed between Rules 1 and 2 is moderate. Finally, we assume that (42) Note that (42) satisfies (37) can be rewritten as
. Then, the nonlinear system
(43) is the parameter with respect to period. The time derivative of (42) is (44)
This paper has presented a descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions. New types of fuzzy controller and fuzzy Lyapunov function have been proposed to render less conservative LMI design conditions. A salient feature of the design conditions is to relate the feasibility of the LMIs to the switching speed of each linear subsystems. The design examples have illustrated the viability and utility of the proposed approach. ACKNOWLEDGMENT The authors would like to thank Mr. T. Nebuya, UEC, Japan, for his contribution to this research. REFERENCES [1] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. SMC-15, no. 1, pp. 116–132, Jan. 1985. [2] K. Tanaka and M. Sugeno, “Stability analysis of fuzzy systems using lyapunov’s direct method,” in Proc. NAFIPS’90, Toronto, Ontario, Canada, Jun. 1990, pp. 133–136. [3] R. Langari and M. Tomizuka, “Analysis and synthesis of fuzzy linguistic control systems,” in Proc. 1990 ASME Winter Annu. Meeting, Dallas, TX, Nov. 1990, pp. 35–42. [4] S. Kitamura and T. Kurozumi, “Extended circle criterion and stability analysis of fuzzy control systems,” in Proc. Int. Fuzzy Eng. Symp., Yokohama, Japan, Nov. 1991, vol. 2, pp. 634–643. [5] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Sets Syst., vol. 45, no. 2, pp. 135–156, 1992. [6] S. S. Farinwata, “Stability analysis of the fuzzy logic controller designed by the phase portrait assignment algorithm,” in Proc. 2nd IEEE Int. Conf. Fuzzy Systems, San Francisco, CA, Apr. 1993, pp. 1377–1382. [7] M. Sugeno, “On stability of fuzzy systems expressed by fuzzy rules with singleton consequents,” IEEE Trans. Fuzzy Syst., vol. 7, no. 2, pp. 201–224, Apr. 1999. [8] S. Hong and R. Langari, “Synthesis of an LMI-based fuzzy control system with guaranteed optimal H performance,” in Proc. FUZZIEEE’98, Anchorage, AK, May 1998, pp. 422–427. [9] D. Filev, “Algebraic design of fuzzy logic controllers,” in Proc. 1996 IEEE Int. Symp. Intelligent Control, Dearborn, MI, Sep. 1996, pp. 253–258. [10] G. Chen and D. Zhang, “Back-driving a truck with suboptimal distance trajectories: A Fuzzy logic control approach,” IEEE Trans. Fuzzy Syst., vol. 3, no. 1, pp. 125–131, Feb. 1995. [11] T. Furuhashi, “Fuzzy control stability analysis using generalized fuzzy petri net model,” J. Adv. Comput. Intell., vol. 3, no. 2, pp. 99–105, 1999. [12] H. Yamamoto and T. Furuhashi, “New fuzzy inference method for symbolic stability analysis of fuzzy control systems,” in Proc. 9th IEEE Int. Conf. Fuzzy Syst., San Antonio, TX, May 2000, pp. 659–664. [13] P. Baranyi, “SVD-based complexity reduction to ts fuzzy models,” IEEE Trans. Ind. Electron. , vol. 49, no. 2, pp. 433–443, Apr. 2002. [14] P. Baranyi, “SVD-based reduction to MISO TS models,” IEEE Trans. Ind. Electron., vol. 51, no. 1, pp. 232–242, Feb. 2003.
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[15] P. Baranyi, “TP model transformation as a way to LMI based controller design,” IEEE Trans. Ind. Electron., vol. 51, no. 2, pp. 387–400, Apr. 2004. [16] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analysis: An Linear Matrix Inequality Approach. New York: Wiley, 2001. [17] R. Sepulcher, M. Jankovic, and P. Kokotovic, Constructive Nonlinear Control. New York: Springer-Verlag, 1997. [18] H. O. Wang, J. Li, D. Niemann, and K. Tanaka, “T–S fuzzy model with linear rule consequence and PDC controller: A universal framework for nonlinear control systems,” in Proc. 9th IEEE Int. Conf. Fuzzy Syst., San Antonio, TX, May 2000, pp. 549–554. [19] D. G. Luenberger, “Dynamic equations in descriptor form,” IEEE Trans. Autom. Control, vol. AC-22, no. 3, pp. 312–321, Mar. 1977. [20] T. Taniguchi, “Fuzzy descriptor systems: Stability analysis and design via LMIs,” in Proc. Amer. Control Conf., San Diego, CA, Jun. 1999, pp. 1827–1831. [21] T. Taniguchi, “Fuzzy descriptor systems and nonlinear model following control,” IEEE Trans. Fuzzy Syst., vol. 8, no. 4, pp. 442–452, Aug. 2000. [22] Y. Wang, Q. L. Zhang, and W. Q. Lin, “Stability analysis and design for T–S fuzzy descriptor systems,” in Proc. 40th Conf. Decision and Control, Orlando, FL, Dec. 2001, pp. 3962–3967. [23] L. Fang, H. Lin, and P. J. Antsaklis, “Stabilization and performance analysis for a class of switched systems,” in Proc. 43rd Conf. Decision Control, Paradise Island, Bahamas, Dec. 2004, pp. 3265–3270. [24] H. Ohtake, “Switching fuzzy controller design based on switching lyapunov function for a class of nonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 36, no. 1, pp. 13–23, Feb. 2006. [25] D. Xie, L. Wang, F. Hao, and G. Xie, “Robust stability analysis and control synthesis for discrete-time uncertain switched systems,” in Proc. 42nd Conf. Decision and Control, Honolulu, HI, Dec. 2003, pp. 4812–4816. [26] G. Feng, “Stability analysis of piecewise discrete-time linear systems,” IEEE Trans. Autom. Control, vol. 47, no. 7, pp. 1108–1112, Jul. 2002. [27] K. Tanaka, T. Hori, and H. O. Wang, “A multiple Lyapunov function approach to stabilization of fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 11, no. 4, pp. 582–589, Aug. 2003. [28] T. Taniguchi and M. Sugeno, “Stabilization of nonlinear systems based on piecewise Lyapunov functions,” in Proc. 13th IEEE Int. Conf. Fuzzy Systems, Budapest, Hungary, Jul. 2004, pp. 1607–1612. [29] H. Ohtake, K. Tanaka, and H. O. Wang, “Piecewise nonlinear control,” in Proc. 42nd IEEE Conf. Decision Control, Maui, HI, Dec. 2003, pp. 4735–4740. [30] K. Tanaka, “Fuzzy control of dynamical systems with input nonlinearity,” in Proc. Joint 2nd Int. Conference on Soft Computing and Intelligent Systems and 5th Int. Symp. on Advanced Intelligent Systems, Yokohama, Japan, 2004, TUP-7-1 in CD. [31] K. Tanaka, T. Hori, and H. O. Wang, “Dynamic output feedback designs for nonlinear systems,” in Proc. IEEE Int. Conf. Systems, Man, and Cybernetics, Tokyo, Japan, Aug. 1999, vol. 3, pp. 56–61. [32] T. Taniguchi, K. Tanaka, and H. O. Wang, “Model construction, rule reduction, and robust compensation for generalized form of takagisugeno fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 8, no. 4, pp. 525–538, Aug. 2001. [33] L. Xie, S. Shishkin, and M. Fu, “Piecewise lyapunov functions for robust stability of linear time-varying systems,” Syst. Control Lett., vol. 31, pp. 165–171, 1997. [34] H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2002. [35] G. Chen and H. Shibata, “System analysis using redundancy of descriptor representation,” J. Syst., Control Inform., vol. 47, no. 5, pp. 211–216, May 2003.
Kazuo Tanaka (S’87–M’91) received the B.S. and M.S. degrees in electrical engineering from Hosei University, Tokyo, Japan, in 1985 and 1987, respectively, and the Ph.D. degree in systems science from the Tokyo Institute of Technology, Tokyo, Japan, in 1990. He is currently a Professor in the Department of Mechanical Engineering and Intelligent Systems at The University of Electro-Communications, Tokyo, Japan. He was a Visiting Scientist in Computer Science at the University of North Carolina, Chapel
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Hill, in 1992 and 1993. He is the author of two books and a coauthor of nine books. Recently, he coauthored (with H. O. Wang) the book Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach (New York: Wiley-Interscience, 2001). His research interests include intelligent systems and control, nonlinear systems control, robotics, and applications. Dr. Tanaka is currently serving on the IEEE Control Systems Society Conference Editorial Board. He received the Best Young Researchers Award from the Japan Society for Fuzzy Theory and Systems in 1990, the Outstanding Papers Award at the 1990 Annual NAFIPS Meeting in Toronto, Canada, in 1990, the Outstanding Papers Award at the Joint Hungarian-Japanese Symposium on Fuzzy Systems and Applications in Budapest, Hungary, in 1991, the Best Young Researchers Award from the Japan Society for Mechanical Engineers in 1994, the Best Book Awards from the Japan Society for Fuzzy Theory and Systems in 1995, 1999 IFAC World Congress Best Poster Paper Prize in 1999, the IEEE TRANSACTIONS ON FUZZY SYSTEMS Outstanding Paper Award in 2000, and the Best Paper Selection at 2005 American Control Conference in Portland, OR, in 2005.
Hiroshi Ohtake (S’02–M’05) received the B.S. and M.S. degrees in mechanical and control engineering from The University of Electro-Communications, Tokyo, Japan, in 2000 and 2002, respectively. He is currently an Assistant Professor in Department of Mechanical Engineering and Intelligent Systems at The University of Electro-Communications. He was a Research Fellow of the Japan Society for the Promotion of Science from 2002 to 2004. His research interests include nonlinear mechanical systems control and robotics. Mr. Ohtake received the Outstanding Student Paper Award at the Joint 9th IFSA World Congress and the 20th NAFIPS International Conference in Vancouver, BC, Canada, in 2001, the Young Investigators Award from the Japan Society for Fuzzy Theory and Intelligent Informatics in 2003, the Best Presentation Award at the FAN Symposium 2004 in Kochi, Japan, in 2004, the American Control Conference Best Paper Selection, at American Control Conference 2005 in Portland, OR, in 2005.
Hua O. Wang (M’94–SM’01) received the B.S. degree from the University of Science and Technology of China (USTC), Hefei, in 1987, the M.S. degree from the University of Kentucky, Lexington, in 1989, and the Ph.D. degree from the University of Maryland, College Park, in 1993, all in electrical engineering. Since September 2002, he has been with Boston University, Boston, MA, where he is currently an Associate Professor of Aerospace and Mechanical Engineering. He was with the United Technologies Research Center, East Hartford, CT, from 1993 to 1996, and was a Faculty Member in the Department of Electrical and Computer Engineering at Duke University, Durham, NC, from 1996 to 2002. He served as the Program Manager (IPA) for Systems and Control with the U.S. Army Research Office (ARO) from August 2000 to August 2002. During 2000–2005, he also held the position of Cheung Kong Chair Professor and Director with the Center for Nonlinear and Complex Systems at Huazhong University of Science and Technology, Wuhan, China. His research interests include control of nonlinear dynamics, intelligent systems and control, networked control systems, robotics, cooperative control, and applications. He coauthored (with K. Tanaka) the book Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach (New York: WileyInterscience, 2001). Dr. Wang is a recipient of the 1994 O. Hugo Schuck Best Paper Award of the American Automatic Control Council, the 14th IFAC World Congress Poster Paper Prize, the 2000 IEEE TRANSACTIONS ON FUZZY SYSTEMS Outstanding Paper Award. He has served as an Associate Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, and was on the IEEE Control Systems Society Conference Editorial Board. He is an Editor for the Journal of Systems Science and Complexity. He is an appointed member of the 2006 Board of Governors of the IEEE Control Systems Society.
