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Stability conditions of fuzzy systems and its application to structural and mechanical systems
Advances in Engineering Software 37 (2006) 624–629 www.elsevier.com/locate/advengsoft
Stability conditions of fuzzy systems and its application to structural and mechanical systems Cheng-Wu Chen Department of Logistics Management, Shu-Te University, 59 Hun Shan Rd., Yen Chau Kaohsiung County, Taiwan 82445, ROC Received 1 December 2003; accepted 6 December 2005 Available online 9 March 2006
Abstract This paper provided the stability conditions and controller design for a class of structural and mechanical systems represented by Takagi–Sugeno (T–S) fuzzy models. In the design procedure of controller, parallel-distributed compensation (PDC) scheme was utilized to construct a global fuzzy logic controller by blending all local state feedback controllers. A stability analysis was carried out not only for the fuzzy model but also for a real mechanical system. Furthermore, this control problem can be reduced to linear matrix inequalities (LMI) problems by the Schur complements and efficient interior-point algorithms are now available in Matlab toolbox to solve this problem. A simulation example was given to show the feasibility of the proposed fuzzy controller design method. 2006 Elsevier Ltd. All rights reserved. Keywords: Structural control; Linear matrix inequality; Fuzzy model
1. Introduction Since Zadeh [1] proposed a linguistic approach to simulate the thought process and judgement of human beings, many successful works on industrial and academic fields had been seeking (see [2–5], for example). However, all of them, there is few mathematical theory and systematic design. In 1985, Takagi and Sugeno [6] proposed a new concept of fuzzy inference system, called the Takagi– Sugeno (T–S) fuzzy model. This kind fuzzy model can combine the flexibility of fuzzy logic theory and the rigorous mathematical analysis tools into a unified framework. For the reason that it employs linear models in the consequent parts, using conventional linear system theory for analysis becomes simple. Since then, various kinds of T–S fuzzy model based controllers have been suggested [7–10]. In this type fuzzy model, local dynamics in different state space regions are represented by a set of linear sub-models. The overall model of the system is then a fuzzy ‘‘blending’’ of these linear sub-models.
E-mail address: [email protected] 0965-9978/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2005.12.002
The LMI theory is a new and rapidly growing field that provides a valuable alternative to the analytical method (see [11,12] for more details). A variety of problems that arise in system and control theory can be reduced to a few standard convex or quasiconvex optimization problems involving the LMI. These resulting optimization problems can be easily solved by numerical computation, so LMI techniques are very efficient and practical tools for solving complex control problems [13]. That is to say, the fuzzy controller design problem was formulated into solving LMI problems and the common P matrix of the stability conditions was obtained by using LMI optimization algorithms to guarantee the stability of the structural systems. Recently, with the increasing research activities in the field of structural control, many control methods have been proposed. These methods are fuzzy control, optimal control, pole placement, sliding mode control, etc. (see [14–16] and the references therein). For examples, Casciati proposed the stability checks to be conducted in the design of a fuzzy controller for nonlinear structures in 1997. In 2002, Wang and Lee developed a fuzzy sliding mode control method to guarantee the stability and robustness of
C.-W. Chen / Advances in Engineering Software 37 (2006) 624–629
the fuzzy control system based on the genetic algorithms. However, as far as we know, the analysis of stability and stabilization problem of structural systems remains an open area. For this reason, we proposed here a fuzzy control technique as well as the T–S fuzzy model to deal with structural control problems. This study is organized as follows. First, the equation of motion of structural systems is constructed and the T–S type fuzzy model is briefly presented to model the structural systems. Then, based on Lyapunov’s approach, a stability criterion is derived to guarantee the stability of fuzzy systems via the linear matrix inequality (LMI) technique. Finally, a numerical example of mechanical system with simulations is given to demonstrate the results, and the conclusions are drawn. 1.1. Equations of motion of structural systems Assume that the equation of motion for a shear-typebuilding modeled by an n-degrees-of-freedom system controlled by actuators and subjected to external force /(t) can be characterized by the following differential equation: M X€ ðtÞ þ C X_ ðtÞ þ KX ðtÞ ¼ BU ðtÞ Mr/ðtÞ;
mn c1 þ c2
c2
For controller design, the standard first-order state equation corresponding to Eq. (1) is obtained by X_ ðtÞ ¼ AX ðtÞ þ BU ðtÞ þ E/ðtÞ;
ð2Þ
where xðtÞ 0 X ðtÞ ¼ ; A¼ x_ ðtÞ M 1 K 0 0 ; E¼ ; B¼ r M 1 B
I M 1 C
;
in which X(t) is a 2n state vector; A is a 2n · 2n system matrix. For some reasons (check controllable or use pole placement, for example), the constant matrix A could be reformulated into canonical control form as follow: 3 2 0 1 0 ... 0 7 6 1 ... 0 7 60 0 7 6 .. 7 .. .. .. 6 . ð3Þ A ¼ 6 .. 7. . . . . 7 6 7 6 0 ... 1 5 40 0 a1
a2
a2n1
a2n
ð1Þ
where X ðtÞ ¼ ½x1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞ 2 Rn is an n-vector; X€ ðtÞ, X_ ðtÞ, X ðtÞ are acceleration, velocity, and displacement vectors; matrices M, C, and K are (n · n) mass, damping, and stiffness matrices, respectively; r is an n-vector denoting the influence of the external force; B is a (n · m) matrix denoting the locations of m control forces; /(t) is the excitation with a upper bound /up(t) P k/(t)k; U(t) corresponds to the actuator forces (generated via active tendon system or an active mass damper, for example); this is only a static model. The mass, stiffness and damping matrix can be expressed as 3 2 m1 7 6 m2 0 7 6 7 6 7 6 .. M ¼6 7; . 7 6 7 6 0 mn1 5 4 2
625
0
3
7 6 c2 þ c3 c3 7 6 c2 7 6 7 6 .. C¼6 7; . cn1 c3 7 6 7 6 cn1 cn1 þ cn cn 5 4 0 cn cn 3 2 k1 þ k2 k 2 0 7 6 k 2 þ k 3 k 3 7 6 k 2 7 6 7 6 .. K¼6 7. . k n1 k 3 7 6 7 6 k n1 k n1 þ k n k n 5 4 0 k n kn
2. Fuzzy modeling of structural system Since control design of nonlinear systems is a difficult process and plants always have nonlinear practical control systems, many nonlinear control methods have been proposed to overcome the difficulty in controller design for real systems. However, the control schemes for nonlinear systems are so complicated that they are not suitable for practical application [17]. Due to the complexity of designing a general control scheme for a real structural and mechanical system, we proposed here a simplified model. In the past few years, fuzzy-rule-based modeling has become an active research field because of its unique merits in solving complex nonlinear system identification and control problems. Unlike traditional modeling, fuzzy rulebased modeling is essentially a multimodel approach in which individual rules are combined to describe the global behavior of the system [18]. Therefore, the fuzzy modeling is employed to represent a structural or mechanical system to simplify the controller design problem. To ensure the stability of the structural system, Takagi– Sugeno (T–S) fuzzy models and the stability analysis are recalled. Fig. 1 is a construction procedure of nonlinear systems. According to the literature [19], there are two approaches for constructing fuzzy models: 1. Identification (fuzzy modeling) using input–output data and 2. derivation from given nonlinear interconnected system equations. There has been an extensive literature on fuzzy modeling using input–output data following Takagi’s, Sugeno’s, and
626
C.-W. Chen / Advances in Engineering Software 37 (2006) 624–629
Nonlinear System
Identification using Input-Output Data
Physical Model
T-S Fuzzy Model
Stability Criterion
nonlinear in general, is again a fuzzy ‘‘blending’’ of each individual linear controller. In brief, to design fuzzy controllers, structure systems are represented by Takagi–Sugeno fuzzy models. The concept of parallel-distributed compensation (PDC) is employed to determine structures of fuzzy controllers from the T–S fuzzy models in this section. At first, the structural system (2) can be approximated by the T–S fuzzy model, which combines the fuzzy inference rule and the local linear state model. The ith rule of the T–S fuzzy model, representing the structural system (2) is the following: Rule i:
Fig. 1. Introduction of the complete design procedure.
Kang’s outstanding work (see [19] and the references therein). The procedure mainly is composed of two parts: structure identification and parameter identification. The identification approach is appropriate for plants that are unable or too difficult to be represented by analytical models. In some cases, because nonlinear dynamic models including algebraic equations for mechanical systems can be readily obtained, the second approach to derive a fuzzy model is more suitable. This paper focuses on the second approach to construct fuzzy models. Based on the T–S type fuzzy model, the linear feedback control techniques can be utilized as in the case of feedback stabilization. The procedure is as follows: First, the nonlinear plant is represented by a Takagi–Sugeno-type fuzzy model. In this type of fuzzy model, local dynamic s in different state-space regions are represented by linear models. The overall model of the system is achieved by fuzzy ‘‘blending’’ of these linear models through nonlinear fuzzy membership functions. The control design is carried out on the basis of the fuzzy model via the so-called PDC scheme and Fig. 2 shows the concept of PDC design. The idea of PDC is to derive each control rule so as to compensate each rule of a fuzzy system. The resulting overall controller, which is
Fuzzy model
Fuzzy controller
Rule r
Rule r
IF x1 ðtÞ is M i1 and and xp ðtÞ is M ip THEN X_ ðtÞ ¼ Ai X ðtÞ þ Bi U ðtÞ þ Ei /ðtÞ;
ð4Þ
where i = 1, 2, . . . , r and r is the rule number; X(t) is the state vector; Mip (p = 1, 2, . . . , g) are the fuzzy sets and x1(t), . . . , xp(t) are the premise variables. By using the fuzzy inference method with a singleton fuzzifier, product inference, and center average defuzzifier, the dynamic fuzzy model (4) can be expressed as follows: Pr wi ðtÞ½Ai X ðtÞ þ Bi U ðtÞ þ Ei /ðtÞ _ Pr X ðtÞ ¼ i¼1 i¼1 wi ðtÞ r X ¼ hi ðtÞðAi X ðtÞ þ Bi U ðtÞÞ þ Ei /ðtÞ; ð5Þ i¼1
with wi ðtÞ ¼
g Y
M ip ðxp ðtÞÞ;
p¼1
wi ðtÞ . hi ðtÞ ¼ Pr i¼1 wi ðtÞ
ð6Þ
Mip(xp(t)) is the grade of membership of xp(t) in Mip. It is assumed that wi ðtÞ P 0; i ¼ 1; 2; . . . ; r; r X wi ðtÞ > 0
ð7Þ
i¼1
P for all t. Therefore, hi(t) P 0 and ri¼1 hi ðtÞ ¼ 1 for all t. In order to design a global controller for the T–S fuzzy model (4), the PDC technique is adopted in this paper. Using the same premise as (4), the ith rule of the fuzzy logic controller (FLC) can be obtained as follows: Controller Rule i: IF x1 ðtÞ is M i1 and and xg ðtÞ is M ig
Rule 2 Rule 1
Rule 2 Rule 1
Linear controller design technique Fig. 2. Parallel-distributed-compensation (PDC) design.
THEN U ðtÞ ¼ K i X ðtÞ;
i ¼ 1; 2; . . . ; r;
ð8Þ
where Ki is the local feedback gain vector in the ith subspace. The final model-based fuzzy controller is analytically represented by Pr r X w ðtÞK i X ðtÞ Pr i ¼ U ðtÞ ¼ i¼1 hi ðtÞK i X ðtÞ. ð9Þ i¼1 wi ðtÞ i¼1
C.-W. Chen / Advances in Engineering Software 37 (2006) 624–629
The overall closed-loop fuzzy system obtained by combining (5) and (9) as follow: X_ ðtÞ ¼
r X r X i¼1
hi ðtÞhl ðtÞ½ðAi Bi K l ÞX ðtÞ þ Ei /ðtÞ.
ð10Þ
l¼1
Therefore, Eq. (10) is the controlled system of the structural system (1) via the T–S fuzzy representation and the technique of PDC control. Definition 1 ([20]). LMI formulation of the design specifications. The linear matrix inequality (LMI) is any constraint of the form m X F ðvÞ ¼ F 0 þ vi F i > 0; ð11Þ i¼1
where v = [v1, v2, . . . , vm] 2 Rm is the variable vector, and the symmetric matrices F i ¼ F Ti 2 Rnn , i = 0, . . . , m, are given. It can be shown that the solution set {v j F(v) > 0} may be empty, but it is always convex. Thus, although (11) has no analytic solution in general, it can be solved numerically by efficient numerical algorithms. Many control problems can be reformulated into LMI’s and solved efficiently by recently developed interior-point methods [11]. According to Eq. (2), the Lyapunov function of the structural system without the input of the PDC fuzzy control is a linear system. Moreover, Eq. (2) is represented by the T–S fuzzy model with PDC fuzzy control in Eq. (10). The resulting controlled system, which is a fuzzy ‘‘blending’’ of each individual linear controlled system, is nonlinearity. In the following, we have to reformulate the controller design problem into solving an LMI problem based on Lyapunov direct method and Schur complement representation. A typical stability condition for fuzzy system (10) is proposed here as follows: Theorem 1. The equilibrium point of fuzzy control system (10) is stable in the large if there exist a common positive definite matrix P and feedback gains K such that the following two inequalities are satisfied: T
ðAi Bi K i Þ P þ P ðAi Bi K i Þ þ
1 PEi ETi P < 0 g2
and T ðAi Bi K l Þ þ ðAl Bl K i Þ P 2 ðAi Bi K l Þ þ ðAl Bl K i Þ 1 þP þ 2 PEi ETi P < 0 2 g
The time derivative of V is V_ ¼ X_ T ðtÞPX ðtÞ þ X T ðtÞP X_ ðtÞ ( )T r X r X ¼ hi ðtÞhl ðtÞ½ðAi Bi K l ÞX ðtÞ þ Ei /ðtÞ PX ðtÞ i¼1 l¼1( ) r X r X T hi ðtÞhl ðtÞ½ðAi Bi K l ÞX ðtÞ þ Ei /ðtÞ þ X ðtÞP i¼1
l¼1
ðA2Þ ¼
r X r X i¼1
T
hi ðtÞhl ðtÞX T ðtÞ½ðAi Bi K l Þ P
l¼1
þ P ðAi Bi K l ÞX ðtÞ þ /T ðtÞETi PX ðtÞ þ X T ðtÞPEi /ðtÞ 1 T 2 T T g / ðtÞ/ðtÞ þ 2 X ðtÞPEi Ei PX ðtÞ g 1 2 T þ g / ðtÞ/ðtÞ þ 2 X T ðtÞPEi ETi PX ðtÞ ðA3Þ g r h X T 6 h2i ðtÞX T ðtÞ ðAi Bi K i Þ P þ P ðAi Bi K i Þ i¼l¼1
i 1 T PE E P X ðtÞ i i g2 " T r X ðAi Bi K l Þ þ ðAl Bl K i Þ T þ2 hi ðtÞhl ðtÞX ðtÞ P 2 i
2
þ g2 k/up ðtÞk . (The * could be represented as T 1 ðPEi Þ X ðtÞ g/ðtÞ < 0.) g Based on Theorem completed. h
1,
T 1 ðPEi Þ X ðtÞ g
the
proof
ðA4Þ T g/ðtÞ is
thereby
Lemma 1 ([21,22]: Schur complements). The LMI QðxÞ SðxÞ > 0; ð14Þ SðxÞ RðxÞ where Q(x) = QT(x), R(x) = RT(x) and S(x) depends on x is equivalent to RðxÞ > 0; QðxÞ SðxÞR1 ðxÞS T ðxÞ > 0. ð15Þ In other words, the set of nonlinear inequalities (15) can be represented as the LMI (14).
ð12Þ
ð13Þ
with P = PT > 0, for i < l 6 r and i = 1,2, . . . ,r. Proof. Using the Lyapunov function candidate for the fuzzy system (10) V ¼ X T ðtÞPX ðtÞ.
627
ðA1Þ
Remark 1. Theorem 1 can be reformulated into the linear matrix inequality (LMI) problem and efficient interiorpoint algorithms are now available in Matlab toolbox to solve this problem. Therefore, Eqs. (12) and (13) are transformed to the LMI by the following procedure. By introducing new variables Hi = Ai BiKi, H il ¼ ðAi Bi K l ÞþðAl Bl K i Þ , 2
W = P1, Yi = HiW, and Yil = HilW. Eqs. (12) and (13) can be rewritten as follows: 1 Y i þ Y Ti þ 2 Ei ETi < 0 ð16Þ g and 1 Y il þ Y Til þ 2 Ei ETi < 0. ð17Þ g
C.-W. Chen / Advances in Engineering Software 37 (2006) 624–629
Remark 2. Theorem 1 states that the stability of a T–S fuzzy model can be achieved by finding a common symmetric positive definite matrix P for r subsystems. The stability analysis is reduced to solving an eigenvalue problem using the interior-point method associated with linear matrix inequality (LMI) techniques. This stability condition can be reduced to that of linear systems when r = 1. 3. A simulation example In this section, we apply the proposed method to the control of the mechanical system. Its equation of motion can be derived as follow [23]: 8 x_ 1 ¼ x2 ; > > > > < x_ ¼ Mgl sin x k ðx x Þ; 2 1 1 3 I I ð18Þ > > _ ¼ x ; x 3 4 > > : x_ 4 ¼ Jk ðx1 x3 Þ þ J1 u; where x1 = q1, x2 ¼ q_ 1 , x3 = q2 and x4 ¼ q_ 2 ; and M = 1 kg and I = 1 kg m2 are the mass and the inertia of the link, L = 1 m is the length of the link, k = 1 N/m is the elasticity constant of the spring which represents the elastic coupling with the joint, J = 1 kg m2 is the inertia of the actuator shaft, and g = 9.8 m/s2 is the gravity constant. We can transform the mechanic system to the normal form below [24]: 8 z_ 1 ¼ z2 ; > > > < z_ ¼ z ; 2 3 ð19Þ > z _ ¼ z 3 4; > > : z_ 4 ¼ aðzÞ þ bðzÞu; where a(z) = 9.8 sin(z1)(z2 + 9.8 cos(z1) + 1) + (z3 9.8 sin(z1))(2 + 9.8 cos(z1)) and b(z) = 1. To design the fuzzy controller, we must have a fuzzy model, which represents the dynamics of the control system. Therefore, we represent the system (19) by a T–S fuzzy model. Here, we approximate the system by the following four-rule fuzzy model. Plant rules Rule 1: IF z1 is about 0 and z2 is about 0 THEN z_ 1 ¼ A1 z þ B1 u; Rule 2: IF z1 is about 0 and z2 is about p THEN z_ 2 ¼ A2 z þ B2 u; Rule 3: IF z1 is about p and z2 is about 0 THEN z_ 3 ¼ A3 z þ B3 u; Rule 4: IF z1 is about p and z2 is about p THEN z_ 4 ¼ A4 z þ B4 u;
where 2 A1
A2
A3
A1
B1
3 0 1 0 0 6 0 0 1 07 6 7 ¼6 7; 4 0 0 0 15 9:8 0 11:8 0 2 3 0 1 0 0 6 0 0 1 07 6 7 ¼6 7; 4 0 0 0 15 86:9221 0 11:8 0 2 3 0 1 0 0 6 0 0 1 07 6 7 ¼6 7; 4 0 0 0 15 9:8 0 7:8 0 2 3 0 1 0 0 6 0 0 1 07 6 7 ¼6 7; 4 0 0 0 15 86:9221 0 7:8 0 2 3 0 607 6 7 ¼ B 2 ¼ B3 ¼ B4 ¼ 6 7 . 405 1
ð20Þ
The membership functions for z1 and z2 are chosen as in Fig. 3. Based on Eq. (20) and Theorem 1, the following common positive definite matrix P and K gains are obtained by LMI optimization algorithm in Matlab toolbox. The simulation result is shown in Fig. 4: ¼ 103 ½ 9:9902 4:0000 0:5882 0:0400 ; ¼ 104 ½ 1:0087 0:4000 0:0588 0:0040 ; ¼ 104 ½ 1:0010 0:4000 0:0608 0:0040 ; ¼ 103 ½ 9:9131 4:0000 0:6078 0:0400 ; 2 3 89:4008 31:3796 2:7444 0:0003 6 31:3796 16:0866 1:2989 0:0088 7 7. P ¼6 4 2:7444 1:2989 0:3075 0:0032 5 0:0003 0:0088 0:0032 0:0003
k1 k2 k3 k4
1
MF2 (about π)
MF1 (about 0)
0.9 0.8 0.7 0.6
h(x1)
628
0.5 0.4 0.3 0.2 0.1 0
0
0.5
1
1.5
2
2.5
Fig. 3. Membership functions of state z1 and z2.
3
C.-W. Chen / Advances in Engineering Software 37 (2006) 624–629 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5
0
1
2
3
4
5 Time (sec)
6
7
8
9
10
Fig. 4. Closed-loop controlled system trajectory of q1.
4. Conclusions This paper is concerned with the stability problem of the structural systems represented by T–S type fuzzy model. In the design procedure, we represent the fuzzy system as a family of local state space models, and construct a global fuzzy logic controller by blending all such local state feedback controllers. A stability criterion in terms of Lyapunov’s direct method is proposed to guarantee the stability of the fuzzy system. Based on this criterion, the fuzzy controller design via LMI technique can stabilize the proposed fuzzy systems. Acknowledgements The authors wish to express sincere gratitude to Prof. B.H.V. Topping for his help and the anonymous reviewers for their constructive suggestions which led to substantial improvements of this paper. We also appreciated the National Science Council of Republic of China for granting the financial support to the work under the Contract No. NSC 93-2211-E-008-004. References [1] Zadeh LA. Fuzzy sets. Inform Control 1968;8:338–53. [2] Chang SSL, Zadeh LA. On fuzzy mapping and control. IEEE Trans Syst Man Cybern 1972;2:30–4. [3] Zadeh L. Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans Syst Man Cybern 1973; 3:28–44.
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[4] Kickert WJM, Mamdani EH. Analysis of a fuzzy logic controller. Fuzzy Sets Syst 1978;1:29–44. [5] Braae M, Rutherford DA. Theoretical and linguistic aspects of the fuzzy logic controller. Automatica 1979;15:553–77. [6] Takagi T, Sugeno M. Fuzzy identification of systems and its applications to modeling and control. IEEE Trans Syst Man Cybern 1985;15:116–32. [7] Tanaka K, Sano M. A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer. IEEE Trans Fuzzy Syst 1994;2:119–34. [8] Tanaka K, Kosaki T. Design of a stable fuzzy controller for an articulated vehicle. IEEE Trans Syst Man Cybern 1997;27:552–8. [9] Cao SG, Rees NW, Feng G. Analysis and design for a class of complex control systems Part II: Fuzzy controller design. Automatica 1997;33:1029–39. [10] Hsiao FH, Chiang WL, Chen CW. Application and fuzzy H1 control via T–S fuzzy models for nonlinear time-delay systems. Int J Artif Intel Tools 2003;12:117–37. [11] Boyd S, El Ghaoui L, Feron E, Balakrishnan V. Linear matrix inequalities in system and control theory. Philadelphia (PA): SIAM; 1994. [12] Nesterov Y, Nemirovsky A. Interior-point polynomial methods in convex programming. Philadelphia (PA): SIAM; 1994. [13] Park CW, Kang HJ, Yee YH, Park M. Numerical robust stability analysis of fuzzy feedback linearisation regulator based on linear matrix inequality approach. IEE Proc—Contr Theory Appl 2002;149: 1–10. [14] Casciati F, Faravelli L, Yao T. Control of nonlinear structures using the fuzzy control approach. Nonlinear Dynam 1996;11:171–87. [15] Casciati F. Checking the stability of a fuzzy controller for nonlinear structures. Comp Aid Civil Infra Eng 1997;12:205–15. [16] Wang AP, Lee CD. Fuzzy sliding mode control for a building structure based on genetic algorithms. Earth Eng Struct Dynam 2002; 31:881–95. [17] Chen BS, Tseng CS, Uang HJ. Robustness design of nonlinear dynamic systems via fuzzy linear control. IEEE Trans Fuzzy Syst 1999;7:571–85. [18] Chiang WL, Chen CW, Hsiao FH. Stability analysis of nonlinear interconnected systems via T–S fuzzy models. Int J Comp Intel App 2004;4:41–55. [19] Tanaka K, Wang HO. Fuzzy control systems design and analysis. New York: John Wiley & Sons. Inc.; 2001. [20] Lu LT, Chiang WL, Tang JP. Application of model reduction and LQG/LTR robust control methodology in active structure control. J Eng Mech, ASCE 1998;124:446–54. [21] Horn R, Johnson C. Topics in matrix analysis. Cambridge: Cambridge University Press; 1991. [22] Lu LT, Chiang WL, Tang JP, Liu MY, Chen CW. Active control for a benchmark building under wind excitations. J Wind Eng Ind Aerodynam 2003;91:469–93. [23] Slotine JJE, Li W. Applied nonlinear control. Englewood Cliffs (NJ): Prentice-Hall; 1991. [24] Chang W, Joo YH, Park JB, Chen G. Robust fuzzy-model-based controller for uncertain systems. In: 1999 IEEE int fuzzy conf proc, 1999, I-486-491, Korea.
Stability conditions of fuzzy systems and its application to structural and mechanical systems Cheng-Wu Chen Department of Logistics Management, Shu-Te University, 59 Hun Shan Rd., Yen Chau Kaohsiung County, Taiwan 82445, ROC Received 1 December 2003; accepted 6 December 2005 Available online 9 March 2006
Abstract This paper provided the stability conditions and controller design for a class of structural and mechanical systems represented by Takagi–Sugeno (T–S) fuzzy models. In the design procedure of controller, parallel-distributed compensation (PDC) scheme was utilized to construct a global fuzzy logic controller by blending all local state feedback controllers. A stability analysis was carried out not only for the fuzzy model but also for a real mechanical system. Furthermore, this control problem can be reduced to linear matrix inequalities (LMI) problems by the Schur complements and efficient interior-point algorithms are now available in Matlab toolbox to solve this problem. A simulation example was given to show the feasibility of the proposed fuzzy controller design method. 2006 Elsevier Ltd. All rights reserved. Keywords: Structural control; Linear matrix inequality; Fuzzy model
1. Introduction Since Zadeh [1] proposed a linguistic approach to simulate the thought process and judgement of human beings, many successful works on industrial and academic fields had been seeking (see [2–5], for example). However, all of them, there is few mathematical theory and systematic design. In 1985, Takagi and Sugeno [6] proposed a new concept of fuzzy inference system, called the Takagi– Sugeno (T–S) fuzzy model. This kind fuzzy model can combine the flexibility of fuzzy logic theory and the rigorous mathematical analysis tools into a unified framework. For the reason that it employs linear models in the consequent parts, using conventional linear system theory for analysis becomes simple. Since then, various kinds of T–S fuzzy model based controllers have been suggested [7–10]. In this type fuzzy model, local dynamics in different state space regions are represented by a set of linear sub-models. The overall model of the system is then a fuzzy ‘‘blending’’ of these linear sub-models.
E-mail address: [email protected] 0965-9978/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2005.12.002
The LMI theory is a new and rapidly growing field that provides a valuable alternative to the analytical method (see [11,12] for more details). A variety of problems that arise in system and control theory can be reduced to a few standard convex or quasiconvex optimization problems involving the LMI. These resulting optimization problems can be easily solved by numerical computation, so LMI techniques are very efficient and practical tools for solving complex control problems [13]. That is to say, the fuzzy controller design problem was formulated into solving LMI problems and the common P matrix of the stability conditions was obtained by using LMI optimization algorithms to guarantee the stability of the structural systems. Recently, with the increasing research activities in the field of structural control, many control methods have been proposed. These methods are fuzzy control, optimal control, pole placement, sliding mode control, etc. (see [14–16] and the references therein). For examples, Casciati proposed the stability checks to be conducted in the design of a fuzzy controller for nonlinear structures in 1997. In 2002, Wang and Lee developed a fuzzy sliding mode control method to guarantee the stability and robustness of
C.-W. Chen / Advances in Engineering Software 37 (2006) 624–629
the fuzzy control system based on the genetic algorithms. However, as far as we know, the analysis of stability and stabilization problem of structural systems remains an open area. For this reason, we proposed here a fuzzy control technique as well as the T–S fuzzy model to deal with structural control problems. This study is organized as follows. First, the equation of motion of structural systems is constructed and the T–S type fuzzy model is briefly presented to model the structural systems. Then, based on Lyapunov’s approach, a stability criterion is derived to guarantee the stability of fuzzy systems via the linear matrix inequality (LMI) technique. Finally, a numerical example of mechanical system with simulations is given to demonstrate the results, and the conclusions are drawn. 1.1. Equations of motion of structural systems Assume that the equation of motion for a shear-typebuilding modeled by an n-degrees-of-freedom system controlled by actuators and subjected to external force /(t) can be characterized by the following differential equation: M X€ ðtÞ þ C X_ ðtÞ þ KX ðtÞ ¼ BU ðtÞ Mr/ðtÞ;
mn c1 þ c2
c2
For controller design, the standard first-order state equation corresponding to Eq. (1) is obtained by X_ ðtÞ ¼ AX ðtÞ þ BU ðtÞ þ E/ðtÞ;
ð2Þ
where xðtÞ 0 X ðtÞ ¼ ; A¼ x_ ðtÞ M 1 K 0 0 ; E¼ ; B¼ r M 1 B
I M 1 C
;
in which X(t) is a 2n state vector; A is a 2n · 2n system matrix. For some reasons (check controllable or use pole placement, for example), the constant matrix A could be reformulated into canonical control form as follow: 3 2 0 1 0 ... 0 7 6 1 ... 0 7 60 0 7 6 .. 7 .. .. .. 6 . ð3Þ A ¼ 6 .. 7. . . . . 7 6 7 6 0 ... 1 5 40 0 a1
a2
a2n1
a2n
ð1Þ
where X ðtÞ ¼ ½x1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞ 2 Rn is an n-vector; X€ ðtÞ, X_ ðtÞ, X ðtÞ are acceleration, velocity, and displacement vectors; matrices M, C, and K are (n · n) mass, damping, and stiffness matrices, respectively; r is an n-vector denoting the influence of the external force; B is a (n · m) matrix denoting the locations of m control forces; /(t) is the excitation with a upper bound /up(t) P k/(t)k; U(t) corresponds to the actuator forces (generated via active tendon system or an active mass damper, for example); this is only a static model. The mass, stiffness and damping matrix can be expressed as 3 2 m1 7 6 m2 0 7 6 7 6 7 6 .. M ¼6 7; . 7 6 7 6 0 mn1 5 4 2
625
0
3
7 6 c2 þ c3 c3 7 6 c2 7 6 7 6 .. C¼6 7; . cn1 c3 7 6 7 6 cn1 cn1 þ cn cn 5 4 0 cn cn 3 2 k1 þ k2 k 2 0 7 6 k 2 þ k 3 k 3 7 6 k 2 7 6 7 6 .. K¼6 7. . k n1 k 3 7 6 7 6 k n1 k n1 þ k n k n 5 4 0 k n kn
2. Fuzzy modeling of structural system Since control design of nonlinear systems is a difficult process and plants always have nonlinear practical control systems, many nonlinear control methods have been proposed to overcome the difficulty in controller design for real systems. However, the control schemes for nonlinear systems are so complicated that they are not suitable for practical application [17]. Due to the complexity of designing a general control scheme for a real structural and mechanical system, we proposed here a simplified model. In the past few years, fuzzy-rule-based modeling has become an active research field because of its unique merits in solving complex nonlinear system identification and control problems. Unlike traditional modeling, fuzzy rulebased modeling is essentially a multimodel approach in which individual rules are combined to describe the global behavior of the system [18]. Therefore, the fuzzy modeling is employed to represent a structural or mechanical system to simplify the controller design problem. To ensure the stability of the structural system, Takagi– Sugeno (T–S) fuzzy models and the stability analysis are recalled. Fig. 1 is a construction procedure of nonlinear systems. According to the literature [19], there are two approaches for constructing fuzzy models: 1. Identification (fuzzy modeling) using input–output data and 2. derivation from given nonlinear interconnected system equations. There has been an extensive literature on fuzzy modeling using input–output data following Takagi’s, Sugeno’s, and
626
C.-W. Chen / Advances in Engineering Software 37 (2006) 624–629
Nonlinear System
Identification using Input-Output Data
Physical Model
T-S Fuzzy Model
Stability Criterion
nonlinear in general, is again a fuzzy ‘‘blending’’ of each individual linear controller. In brief, to design fuzzy controllers, structure systems are represented by Takagi–Sugeno fuzzy models. The concept of parallel-distributed compensation (PDC) is employed to determine structures of fuzzy controllers from the T–S fuzzy models in this section. At first, the structural system (2) can be approximated by the T–S fuzzy model, which combines the fuzzy inference rule and the local linear state model. The ith rule of the T–S fuzzy model, representing the structural system (2) is the following: Rule i:
Fig. 1. Introduction of the complete design procedure.
Kang’s outstanding work (see [19] and the references therein). The procedure mainly is composed of two parts: structure identification and parameter identification. The identification approach is appropriate for plants that are unable or too difficult to be represented by analytical models. In some cases, because nonlinear dynamic models including algebraic equations for mechanical systems can be readily obtained, the second approach to derive a fuzzy model is more suitable. This paper focuses on the second approach to construct fuzzy models. Based on the T–S type fuzzy model, the linear feedback control techniques can be utilized as in the case of feedback stabilization. The procedure is as follows: First, the nonlinear plant is represented by a Takagi–Sugeno-type fuzzy model. In this type of fuzzy model, local dynamic s in different state-space regions are represented by linear models. The overall model of the system is achieved by fuzzy ‘‘blending’’ of these linear models through nonlinear fuzzy membership functions. The control design is carried out on the basis of the fuzzy model via the so-called PDC scheme and Fig. 2 shows the concept of PDC design. The idea of PDC is to derive each control rule so as to compensate each rule of a fuzzy system. The resulting overall controller, which is
Fuzzy model
Fuzzy controller
Rule r
Rule r
IF x1 ðtÞ is M i1 and and xp ðtÞ is M ip THEN X_ ðtÞ ¼ Ai X ðtÞ þ Bi U ðtÞ þ Ei /ðtÞ;
ð4Þ
where i = 1, 2, . . . , r and r is the rule number; X(t) is the state vector; Mip (p = 1, 2, . . . , g) are the fuzzy sets and x1(t), . . . , xp(t) are the premise variables. By using the fuzzy inference method with a singleton fuzzifier, product inference, and center average defuzzifier, the dynamic fuzzy model (4) can be expressed as follows: Pr wi ðtÞ½Ai X ðtÞ þ Bi U ðtÞ þ Ei /ðtÞ _ Pr X ðtÞ ¼ i¼1 i¼1 wi ðtÞ r X ¼ hi ðtÞðAi X ðtÞ þ Bi U ðtÞÞ þ Ei /ðtÞ; ð5Þ i¼1
with wi ðtÞ ¼
g Y
M ip ðxp ðtÞÞ;
p¼1
wi ðtÞ . hi ðtÞ ¼ Pr i¼1 wi ðtÞ
ð6Þ
Mip(xp(t)) is the grade of membership of xp(t) in Mip. It is assumed that wi ðtÞ P 0; i ¼ 1; 2; . . . ; r; r X wi ðtÞ > 0
ð7Þ
i¼1
P for all t. Therefore, hi(t) P 0 and ri¼1 hi ðtÞ ¼ 1 for all t. In order to design a global controller for the T–S fuzzy model (4), the PDC technique is adopted in this paper. Using the same premise as (4), the ith rule of the fuzzy logic controller (FLC) can be obtained as follows: Controller Rule i: IF x1 ðtÞ is M i1 and and xg ðtÞ is M ig
Rule 2 Rule 1
Rule 2 Rule 1
Linear controller design technique Fig. 2. Parallel-distributed-compensation (PDC) design.
THEN U ðtÞ ¼ K i X ðtÞ;
i ¼ 1; 2; . . . ; r;
ð8Þ
where Ki is the local feedback gain vector in the ith subspace. The final model-based fuzzy controller is analytically represented by Pr r X w ðtÞK i X ðtÞ Pr i ¼ U ðtÞ ¼ i¼1 hi ðtÞK i X ðtÞ. ð9Þ i¼1 wi ðtÞ i¼1
C.-W. Chen / Advances in Engineering Software 37 (2006) 624–629
The overall closed-loop fuzzy system obtained by combining (5) and (9) as follow: X_ ðtÞ ¼
r X r X i¼1
hi ðtÞhl ðtÞ½ðAi Bi K l ÞX ðtÞ þ Ei /ðtÞ.
ð10Þ
l¼1
Therefore, Eq. (10) is the controlled system of the structural system (1) via the T–S fuzzy representation and the technique of PDC control. Definition 1 ([20]). LMI formulation of the design specifications. The linear matrix inequality (LMI) is any constraint of the form m X F ðvÞ ¼ F 0 þ vi F i > 0; ð11Þ i¼1
where v = [v1, v2, . . . , vm] 2 Rm is the variable vector, and the symmetric matrices F i ¼ F Ti 2 Rnn , i = 0, . . . , m, are given. It can be shown that the solution set {v j F(v) > 0} may be empty, but it is always convex. Thus, although (11) has no analytic solution in general, it can be solved numerically by efficient numerical algorithms. Many control problems can be reformulated into LMI’s and solved efficiently by recently developed interior-point methods [11]. According to Eq. (2), the Lyapunov function of the structural system without the input of the PDC fuzzy control is a linear system. Moreover, Eq. (2) is represented by the T–S fuzzy model with PDC fuzzy control in Eq. (10). The resulting controlled system, which is a fuzzy ‘‘blending’’ of each individual linear controlled system, is nonlinearity. In the following, we have to reformulate the controller design problem into solving an LMI problem based on Lyapunov direct method and Schur complement representation. A typical stability condition for fuzzy system (10) is proposed here as follows: Theorem 1. The equilibrium point of fuzzy control system (10) is stable in the large if there exist a common positive definite matrix P and feedback gains K such that the following two inequalities are satisfied: T
ðAi Bi K i Þ P þ P ðAi Bi K i Þ þ
1 PEi ETi P < 0 g2
and T ðAi Bi K l Þ þ ðAl Bl K i Þ P 2 ðAi Bi K l Þ þ ðAl Bl K i Þ 1 þP þ 2 PEi ETi P < 0 2 g
The time derivative of V is V_ ¼ X_ T ðtÞPX ðtÞ þ X T ðtÞP X_ ðtÞ ( )T r X r X ¼ hi ðtÞhl ðtÞ½ðAi Bi K l ÞX ðtÞ þ Ei /ðtÞ PX ðtÞ i¼1 l¼1( ) r X r X T hi ðtÞhl ðtÞ½ðAi Bi K l ÞX ðtÞ þ Ei /ðtÞ þ X ðtÞP i¼1
l¼1
ðA2Þ ¼
r X r X i¼1
T
hi ðtÞhl ðtÞX T ðtÞ½ðAi Bi K l Þ P
l¼1
þ P ðAi Bi K l ÞX ðtÞ þ /T ðtÞETi PX ðtÞ þ X T ðtÞPEi /ðtÞ 1 T 2 T T g / ðtÞ/ðtÞ þ 2 X ðtÞPEi Ei PX ðtÞ g 1 2 T þ g / ðtÞ/ðtÞ þ 2 X T ðtÞPEi ETi PX ðtÞ ðA3Þ g r h X T 6 h2i ðtÞX T ðtÞ ðAi Bi K i Þ P þ P ðAi Bi K i Þ i¼l¼1
i 1 T PE E P X ðtÞ i i g2 " T r X ðAi Bi K l Þ þ ðAl Bl K i Þ T þ2 hi ðtÞhl ðtÞX ðtÞ P 2 i
2
þ g2 k/up ðtÞk . (The * could be represented as T 1 ðPEi Þ X ðtÞ g/ðtÞ < 0.) g Based on Theorem completed. h
1,
T 1 ðPEi Þ X ðtÞ g
the
proof
ðA4Þ T g/ðtÞ is
thereby
Lemma 1 ([21,22]: Schur complements). The LMI QðxÞ SðxÞ > 0; ð14Þ SðxÞ RðxÞ where Q(x) = QT(x), R(x) = RT(x) and S(x) depends on x is equivalent to RðxÞ > 0; QðxÞ SðxÞR1 ðxÞS T ðxÞ > 0. ð15Þ In other words, the set of nonlinear inequalities (15) can be represented as the LMI (14).
ð12Þ
ð13Þ
with P = PT > 0, for i < l 6 r and i = 1,2, . . . ,r. Proof. Using the Lyapunov function candidate for the fuzzy system (10) V ¼ X T ðtÞPX ðtÞ.
627
ðA1Þ
Remark 1. Theorem 1 can be reformulated into the linear matrix inequality (LMI) problem and efficient interiorpoint algorithms are now available in Matlab toolbox to solve this problem. Therefore, Eqs. (12) and (13) are transformed to the LMI by the following procedure. By introducing new variables Hi = Ai BiKi, H il ¼ ðAi Bi K l ÞþðAl Bl K i Þ , 2
W = P1, Yi = HiW, and Yil = HilW. Eqs. (12) and (13) can be rewritten as follows: 1 Y i þ Y Ti þ 2 Ei ETi < 0 ð16Þ g and 1 Y il þ Y Til þ 2 Ei ETi < 0. ð17Þ g
C.-W. Chen / Advances in Engineering Software 37 (2006) 624–629
Remark 2. Theorem 1 states that the stability of a T–S fuzzy model can be achieved by finding a common symmetric positive definite matrix P for r subsystems. The stability analysis is reduced to solving an eigenvalue problem using the interior-point method associated with linear matrix inequality (LMI) techniques. This stability condition can be reduced to that of linear systems when r = 1. 3. A simulation example In this section, we apply the proposed method to the control of the mechanical system. Its equation of motion can be derived as follow [23]: 8 x_ 1 ¼ x2 ; > > > > < x_ ¼ Mgl sin x k ðx x Þ; 2 1 1 3 I I ð18Þ > > _ ¼ x ; x 3 4 > > : x_ 4 ¼ Jk ðx1 x3 Þ þ J1 u; where x1 = q1, x2 ¼ q_ 1 , x3 = q2 and x4 ¼ q_ 2 ; and M = 1 kg and I = 1 kg m2 are the mass and the inertia of the link, L = 1 m is the length of the link, k = 1 N/m is the elasticity constant of the spring which represents the elastic coupling with the joint, J = 1 kg m2 is the inertia of the actuator shaft, and g = 9.8 m/s2 is the gravity constant. We can transform the mechanic system to the normal form below [24]: 8 z_ 1 ¼ z2 ; > > > < z_ ¼ z ; 2 3 ð19Þ > z _ ¼ z 3 4; > > : z_ 4 ¼ aðzÞ þ bðzÞu; where a(z) = 9.8 sin(z1)(z2 + 9.8 cos(z1) + 1) + (z3 9.8 sin(z1))(2 + 9.8 cos(z1)) and b(z) = 1. To design the fuzzy controller, we must have a fuzzy model, which represents the dynamics of the control system. Therefore, we represent the system (19) by a T–S fuzzy model. Here, we approximate the system by the following four-rule fuzzy model. Plant rules Rule 1: IF z1 is about 0 and z2 is about 0 THEN z_ 1 ¼ A1 z þ B1 u; Rule 2: IF z1 is about 0 and z2 is about p THEN z_ 2 ¼ A2 z þ B2 u; Rule 3: IF z1 is about p and z2 is about 0 THEN z_ 3 ¼ A3 z þ B3 u; Rule 4: IF z1 is about p and z2 is about p THEN z_ 4 ¼ A4 z þ B4 u;
where 2 A1
A2
A3
A1
B1
3 0 1 0 0 6 0 0 1 07 6 7 ¼6 7; 4 0 0 0 15 9:8 0 11:8 0 2 3 0 1 0 0 6 0 0 1 07 6 7 ¼6 7; 4 0 0 0 15 86:9221 0 11:8 0 2 3 0 1 0 0 6 0 0 1 07 6 7 ¼6 7; 4 0 0 0 15 9:8 0 7:8 0 2 3 0 1 0 0 6 0 0 1 07 6 7 ¼6 7; 4 0 0 0 15 86:9221 0 7:8 0 2 3 0 607 6 7 ¼ B 2 ¼ B3 ¼ B4 ¼ 6 7 . 405 1
ð20Þ
The membership functions for z1 and z2 are chosen as in Fig. 3. Based on Eq. (20) and Theorem 1, the following common positive definite matrix P and K gains are obtained by LMI optimization algorithm in Matlab toolbox. The simulation result is shown in Fig. 4: ¼ 103 ½ 9:9902 4:0000 0:5882 0:0400 ; ¼ 104 ½ 1:0087 0:4000 0:0588 0:0040 ; ¼ 104 ½ 1:0010 0:4000 0:0608 0:0040 ; ¼ 103 ½ 9:9131 4:0000 0:6078 0:0400 ; 2 3 89:4008 31:3796 2:7444 0:0003 6 31:3796 16:0866 1:2989 0:0088 7 7. P ¼6 4 2:7444 1:2989 0:3075 0:0032 5 0:0003 0:0088 0:0032 0:0003
k1 k2 k3 k4
1
MF2 (about π)
MF1 (about 0)
0.9 0.8 0.7 0.6
h(x1)
628
0.5 0.4 0.3 0.2 0.1 0
0
0.5
1
1.5
2
2.5
Fig. 3. Membership functions of state z1 and z2.
3
C.-W. Chen / Advances in Engineering Software 37 (2006) 624–629 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5
0
1
2
3
4
5 Time (sec)
6
7
8
9
10
Fig. 4. Closed-loop controlled system trajectory of q1.
4. Conclusions This paper is concerned with the stability problem of the structural systems represented by T–S type fuzzy model. In the design procedure, we represent the fuzzy system as a family of local state space models, and construct a global fuzzy logic controller by blending all such local state feedback controllers. A stability criterion in terms of Lyapunov’s direct method is proposed to guarantee the stability of the fuzzy system. Based on this criterion, the fuzzy controller design via LMI technique can stabilize the proposed fuzzy systems. Acknowledgements The authors wish to express sincere gratitude to Prof. B.H.V. Topping for his help and the anonymous reviewers for their constructive suggestions which led to substantial improvements of this paper. We also appreciated the National Science Council of Republic of China for granting the financial support to the work under the Contract No. NSC 93-2211-E-008-004. References [1] Zadeh LA. Fuzzy sets. Inform Control 1968;8:338–53. [2] Chang SSL, Zadeh LA. On fuzzy mapping and control. IEEE Trans Syst Man Cybern 1972;2:30–4. [3] Zadeh L. Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans Syst Man Cybern 1973; 3:28–44.
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