This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON CYBERNETICS

1

Controller Design for TS Models Using Delayed Nonquadratic Lyapunov Functions Zsófia Lendek, Member, IEEE, Thierry-Marie Guerra, Member, IEEE, and Jimmy Lauber

Abstract—In the last few years, nonquadratic Lyapunov functions have been more and more frequently used in the analysis and controller design for Takagi–Sugeno fuzzy models. In this paper, we developed relaxed conditions for controller design using nonquadratic Lyapunov functions and delayed controllers and give a general framework for the use of such Lyapunov functions. The two controller design methods developed in this framework outperform and generalize current state-of-the-art methods. The proposed methods are extended to robust and H∞ control and α-sample variation. Index Terms—Controller design, discrete-time Takagi–Sugeno models, LMI, nonquadratic Lyapunov functions.

I. I NTRODUCTION AKAGI–SUGENO (TS) fuzzy systems [1] are nonlinear, convex combinations of local linear models, and have the property that they are able to exactly represent a large class of nonlinear systems [2]. In order to analyze the stability or to design controllers and observers for a TS fuzzy model, the direct Lyapunov approach has been used. Stability conditions have been derived using quadratic Lyapunov functions [3]–[5], piecewise continuous Lyapunov functions [6], [7], and more recently, to reduce the conservativeness of the conditions, nonquadratic Lyapunov functions [8]–[10]. Other works try to introduce some properties of the membership function [11], or try to reduce the complexity of the LMI conditions. The stability or design conditions are generally derived in the form of linear matrix inequalities (LMIs). Although also used for continuous-time TS models [10], [12]–[14], nonquadratic Lyapunov functions have shown a real improvement of the design conditions in the discretetime case [8], [15]–[17]. It has been proven that the solutions

T

Manuscript received October 7, 2013; revised February 3, 2014 and May 8, 2014; accepted May 20, 2014. This work was supported by the Romanian National Authority for Scientific Research, CNCS UEFISCDI, through Contract 74/05.10.2011, under Project PN-II-RU-TE2011-3-0043, by the International Campus on Safety and Intermodality in Transportation in the European Community, by the Délegation Régionale à la Recherche et à la Technologie, by the Ministére de L’Enseignement Supérieur et de la Recherche the Region Nord Pas de Calais, and by the Centre Nationale de la Recherche Scientifique. This paper was recommended by Associate Editor H. Gao. Zs. Lendek is with the Department of Automation, Technical University of Cluj-Napoca, Cluj-Napoca 400114, Romania (e-mail: [email protected]). T. M. Guerra and J. Lauber are with the University of Valenciennes and Hainaut-Cambresis, LAMIH UMR CNRS 8201, Valenciennes 59313, France (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2014.2327657

obtained by nonquadratic Lyapunov functions include and extend the set of solutions obtained using the quadratic framework. A different type of improvement in the discrete case has been developed in [9], conditions being obtained by replacing the classical one sample variation of the Lyapunov function by its variation over several samples (α-sample variation). More recently, by using Polya’s theorem [18], [19] asymptotically necessary and sufficient (ANS) LMI conditions have been obtained for stability and stabilization in the sense of a chosen quadratic or nonquadratic Lyapunov function and control law. For linear-time invariant systems [20] investigated stability based on Polya’s theorem and homogeneously polynomially parameter-dependent Lyapunov functions and ANS conditions in the sense of the membership function-dependent Lyapunov matrix have been obtained. Ding [21] extended the results of [20] to TS models and homogeneous polynomially parameter-dependent nonparallel distributed compensation law, thus giving ANS stability conditions for both membership function-dependent model and membership function-dependent Lyapunov matrix. By increasing the complexity of the homogeneously polynomially parameterdependent Lyapunov functions and the complexity of the homogeneous polynomially parameter dependent control laws, in theory any sufficiently smooth Lyapunov function and control law can be approximated. Thus, [21] represents all Lyapunov functions and all control laws that are continuous in the membership functions. This issue deservers further study, as the conditions in many cases cannot be relaxed. Being ANS may not give a solution, due to computational intractability. The number of LMIs that have to be solved increase quickly, leading to numerical intractability [22]. Although slack matrices can be introduced to relax the conditions, these also increase the computational burden. Constructing a family of Lyapunov functions that covers the whole set of continuous Lyapunov functions can be done, but deciding which one is the most likely to be used to obtain good results is still an open problem. Moreover, a common assumption in all the results enumerated above is that the scheduling variables may not depend on the control input, in order to avoid solving implicit equations. Although this assumption is highly impractical as it means that the system has to be input-affine, it is necessary for the results in the literature. With the considerations above, in this paper, we propose a general framework for using delayed nonquadratic Lyapunov functions for controller design. For discrete time TS models, in the nonquadratic framework, delayed controllers and observers have been proposed in [23]. The observer design method has

c 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2168-2267  See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

IEEE TRANSACTIONS ON CYBERNETICS

been generalized further on in [24], but the controller design had the shortcoming of an increased number of LMIs. Thanks to the use of delayed Lyapunov functions and control laws, new possibilities for relaxing the derived conditions appear. While it is not our goal and we do not derive ANS conditions in this paper, using delayed Lyapunov functions and control laws, we show that the use of delay can lead to significant improvements. Finding a good structure of the control law reduces in a very important manner the conservatism of the results. The key point is finding a suitable solution that is compatible with actual solvers. Moreover, by using a delayed controller, the assumption that the scheduling variables must not depend on the current control input is no longer necessary, as solving the implicit equation is avoided. Thus, we present new possibilities for controller design based on the past states for a wider class of nonlinear systems. Furthermore, in the proposed framework, delayed systems can be easily handled. We also extend the results for robust control, H∞ control and to α-sample variation, similar to [9]. The structure of the paper is as follows. Section II presents the notations used in this paper and the general form of the TS models, and motivates our work through a simple example. Section III develops the proposed conditions for controller design. The design methods are extended to robust control, αsample variation and H∞ control in Section IV. Section V concludes the paper. II. P RELIMINARIES The discrete-time TS model considered in this paper for controller design is of the form x(k + 1) = Az x(k) + Bz u(k) (1) r where Az denotes the convex sum Az = i=1 hi (z(k))Ai , Ai and Bi , i = 1, 2, . . . , r are the local matrices, r denotes the number of rules, k is the sample, x ∈ Rnx is the state vector, u ∈ Rnu is the control input, z ∈ Rnz is the scheduling vector. It is assumed that the scheduling variables z(k) are available at the sample k. In what follows, we will make use of the following results. Lemma 1 [25]: Consider a vector x ∈ Rnx and two matrices Q = QT ∈ Rnx ×nx and R ∈ Rm×nx such that rank(R) < nx . The two following expressions are equivalent. 1) xT Qx < 0, x ∈ {x ∈ Rnx , x = 0, Rx = 0}. 2) ∃M ∈ Rm×nx such that Q + MR + RT M T < 0. Observer and controller design for TS models often lead to double-sum negativity problems of the form x

r r  

hi (z(k))hj (z(k))ij x < 0

(2)

i=1 j=1

where ij , i, j = 1, 2, . . . , r are matrices of appropriate dimensions. Lemma 2 [26]: The double-sum (2) is negative, if ii < 0 ij + ji < 0,

ii < 0 2 ii + ij + ji < 0, r−1

i, j = 1, 2, . . . , r, i = j.

Property 1 (Congruence): Given a matrix P = PT and a full column rank matrix Q it holds that P > 0 ⇒ QPQT > 0. Property 2: Let A and B be matrices of appropriate dimensions and ranks, with B = BT > 0. Then (A − B)T B−1 (A − B) ≥ 0 ⇐⇒ AT B−1 A ≥ A + AT − B. Property 3 [28] (Schur  complement): Consider a matrix M M 11 12 , with M11 and M22 being square M = MT = T M M12 22 matrices. Then  M11 < 0 M 0, for i ∈ IGP , |GP0 | = nP . The difference is

Using Lemma 1 with (12), V1 < 0, if there exists M ∈ R2nx ×nx so that ⎛ ⎞   −P−1 0 P −1 ⎝ G0 ⎠ A A − BGB FGF H H − I + (∗) < 0. + M G G0 0 0 0 0 P−1 GP

−1 = x(k + 1)T H−T H PGP H H x(k + 1)



0 P−1 GP

Choosing M =

1

leads to (12)

Using Lemma 1, V1 < 0, if there exists M ∈ R2nx ×nx so that ⎛ ⎞ −1 −H−T P 0 PH H H G G0 G0 0 ⎝ ⎠ PGP H−1 0 H−T H GH G 1 1 1   −1 A − B F H −I B F A + M G0 + (∗) < 0. G0 G0 GH 0

In order to obtain a problem with LMI constraints encompassing the classical cases, a choice is

0 M = H−T GH



HGH 0 and congruence with 0

H H −HTGH P−1 GP G 0

0

0

0

0

0 PGP



1



(∗)

0

AGA HGH − BGB FGF 0

1

⊕ GA0 ) so that

0

V1 = V(k + 1) − V(k)

G1

0

j

0

⊕ GB0 ) ∪ (GH 0

0

0

0

0

0

GP1



1

1

G1 1 G1 −T T − x(k) H H PGP H−1H x(k) G0 0 G0 ⎛  T −H−T P P H−1 G0 GH x(k) GH 0 0 ⎝

j

i P iF0 = priGF , and HiH , iH 0 = prGH , i ∈ IGV , where GV = G0 ∪

0

0

x(k + 1)

1

AGA HGH − BGB FGF −PGP

|GP0 | = nP . The difference is

=

0

0 HTGH

so that

Remark 3: GV above is simply the multiset containing all the delays in the multiple sum in (11). Proof: Consider the Lyapunov function V = −1 T > 0, for i ∈ I , x(k)T H−T P P H H x(k), with Pi = P P H G G i G G 0

Applying to (13) Property 1 with the full-rank matrix T

HGH 0

0

−PGP (∗) 0 AGA HGH − BGB FGF −HGH − HTGH + PGP 0

1

(13)

To develop the design conditions, two different Lyapunov functions will be considered. Case 1: V = x(k)T H−T P P H−1 x(k), with Pi = PTi > 0, GH G GH

(GF0

1

1

A. Design Conditions

−PGP

< 0.

1

Using Property 2, we obtain directly (14).

B. Discussion First, we illustrate the use of the conditions (11) and (14), respectively, on the following example. Consider a two-rule fuzzy system x(k + 1) =

2 

hi (z(k))(Ai x(k) + Bi u(k))

i=1

= AGA x(k) + BGB u(k) 0

0

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6

IEEE TRANSACTIONS ON CYBERNETICS

with GA0 = GB0 = {0} for which a controller has to be designed F P and let GH 0 = {0, −1}, G0 = {0, −1}, G0 = {−1, −1}, i.e., P{−1,−1} =

2 2  

hi (z(k − 1))hj (z(k − 1))Pij

i=1 j=1

H{0,−1} =

2  2 

hi (z(k))hj (z(k − 1))Hij

i=1 j=1

F{0,−1} =

2  2 

hi (z(k))hj (z(k − 1))Fij .

i=1 j=1

Then, the conditions (11) of Theorem 2 correspond to there exist Pij , Fij , Hij , i, j = 1, 2 so that   −P{−1,−1} (∗)