On Non-quadratic Local Static Output Feedback

Proceedings of the 30th Chinese Control Conference July 22-24, 2011, Yantai, China

On Non-quadratic Local Static Output Feedback Controller for Continuous-time Takagi-Sugeno models PAN Juntao1,2 , FEI Shumin1 , GUERRA Thierry Marie2 , XUE Mingxiang1 1. Key Laboratory of Measurement and Control of CSE of Ministry of Educatio, Southeast University, Nanjing 210096, P. R. China E-mail: [email protected], [email protected] 2. LAMIH, UMR CNRS8503, University of Valenciennes et Hainaut-Cambresis, ´ Le Mont Houy, 59313 Valenciennes Cedex 9, France E-mail: [email protected] Abstract: This paper focuses on the problem of static output feedback controller synthesis for continuous-time Takagi-Sugeno (T-S) fuzzy system. The new perspective introduced in this paper lies on the observation that most of the previous results found in the literature intended to establish global conditions while employing non-quadratic Lyapunov functions, which can not be reached for many nonlinear systems. The well known problem of handing time-derivatives of membership function (MFs) is overcome by reducing global goals to the estimation of region of attraction. It is shown that the derived local conditions actually leads to reasonable advantages over the existing quadratic approach as well as some previous non-quadratic attempts. Moreover, conditions for the solvability of the static output feedback controller design given here are written in the form of linear matrix inequalities which can be efficiently solved by convex optimization techniques. Simulation example is given to demonstrate the validity and applicability of the proposed approaches. Key Words: Non-quadratic Lyapuonv function, Local stabilization condition, Continuous-time Takagi-Sugeno fuzzy models, Linear matrix inequality

1 Introduction Nonlinear control systems based on the Takagi-Sugeno (T-S) fuzzy model have received a great deal of attention over the last decade [1]. The stability analysis and controller synthesis issues in T-S fuzzy systems are commonly investigated based on a quadratic Lyapunov function (QLF) [2]. All those conditions are written in the form of linear matrix inequalities (LMIs) that can be efficiently solved by convex optimization techniques. Unfortunately, the LMI-based conditions derived by QLF are found to be very conservative when applied to fuzzy systems [3]. Conservativeness comes from different sources: the type of T-S fuzzy model [4, 5], the way the membership functions are dropped off to obtain LMIs expressions [6, 24], the integration of membership functions (MFs) information [8, 9], the choice of Lyapunov function [3, 10–12]. This work is concerned with a relaxation in the latter sense which demands a change of perspective from global to local conditions. Several alternative classes of Lyapunov function candidates have been proposed in the literture. It was shown in [10] that piecewise Lyapunov function (PWLF) have been straightforwardly applied to those T-S models that induce state-space partitions form the fact that not all their linear components are simultaneously activated. However, the T-S fuzzy model constructed via the sector nonlinearity approach lack this property. Another alternative Lyapunov function candidate has been developed for continuous-time T-S fuzzy systems in [3], and discrete-time T-S fuzzy systems in [13], namely non-quadratic Lyapunov functions (NQLF). The approaches proposed by [3, 13] are employed to intensively reduce the conservativeness given by QLF, though the results in continuous-time domain have not been as powerful as those corresponding to the discrete case. This asymmetry is explained by the fact that most of these Lyapunov funcThis work is supported by the State Key Program of National Natural Science Foundation of China under Grant 60835001.

tions depend on the same membership functions of the model, hereby taking into account structural information at the price of dealing with the time-derivative of MFs. Due to the explicit presence of the time-derivative of MFs in the stability and stabilization conditions, fewer results concerning the NQLF in continuous-time domain have been reported. In [3, 12], upper bounds for the time-derivative of the MFs must be considered in order to express the conditions in the form of LMIs, thus increasing conservativeness. A new line-integral Lyapunov function approach to analysis and design for T-S fuzzy systems are proposed in [11]. However, the line-integral is asked to be path-independent thus significantly reducing its applicability. A change of perspective for non-quadratic stability analysis of T-S fuzzy model has been proposed in [14, 15]. This approach reduce global goals to less exigent conditions, thereby showing an estimation of region of attraction can be found. All the aforementioned studies are infeasible in many practical situations due to the unmeasurable system states. Thus, output feedback control problem has attracted attention of many researchers recently [17–21]. Among output feedback control strategies, the static output feedback control (SOFC) is important in its own right, because static controller are less expensive to be implemented and more reliable in practice. However, the synthesis of a SOFC is nonconvex. In this paper, we aim to extend the previous results [14] so they can be applied to the SOFC design. A descriptor system approach [12, 19] is employed to avoid the appearance of crossing terms between the controller and system matrix. Unlike the previous studies [17, 22] on SOFC where the stability conditions are not strictly LMIs, the derived conditions for the solvability of the static output feedback controller design are written in the form of LMIs which are easier to be solved by convex optimization techniques. Notation. Throughout this paper, a real symmetric matrix P > 0 (P ≥ 0) denotes P being a positive definite (or positive semidefinite) matrix, and A > B (A ≥ B) means A − B > 0

2964

(A − B ≥ 0). I is used to denote an identity matrix with proper dimension. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. The notation “∗” represents transpose terms in a symmetric matrix. For brevity, the following notations are adopted r

Uz = ∑ hi (z(t))Ui ;

Uz−1

i=1



U˙ z =

d

∑ri=1 hi (z(t))Ui dt

r

Uzz = ∑

=



r

∑ hi (z(t))Ui

−1

i=1

 ;

U˙ z−1

=

d



∑ri=1 hi (z(t))Ui

−1 

dt

∑ hi (z(t))h j (z(t))Ui j ;

i=1 j=1

where hi (z(t)) are the scalar functions satisfying the convex sum property hi (z(t)) ≥ 0 and ∑ri=1 hi (z(t)) = 1.

2 Problem Formulation and Preliminaries Let us consider a nonlinear model, which is given by  x(t) ˙ = fx (z(t))x(t) + g(z(t))u(t) (1) y(t) = fy (z(t))x(t) with fx (·), fy (·), g(·) nonlinear functions. x(t) ∈ Rn the state vector, u(t) ∈ Rm the input vector, y(t) ∈ Rq the output vector, z(t) ∈ R p the premise vector assumed to be bounded and smooth in a compact set C of the state space including the origin. Let nlk (·) ∈ [ nlk , nlk ] be the set of bounded nonlinearities of (1) in C. Employing the sector nonlinearity approach [2] to establish a T-S fuzzy model of (1), the following weighting functions can be constructed: ⎧ nlk −nlk (·) k ⎪ ⎪ ⎨ w0 (·) = nlk −nlk , k ∈ {1, 2, . . . , p} (2) ⎪ ⎪ nl (·)−nl k k k ⎩ w (·) = In this case the weighting functions wk0 (·) and wk1 (·) share the properties: wk0 (·) ≥ 0, wk1 (·) ≥ 0, wk0 (·) + wk1 (·) = 1 With the aid of previous weights, the membership functions (MFs) can be set as p

× 21 + · · · + i

× 2 p−1

i=1

i=1

∈ {1, . . . , 2 p }

where i = 1 + i1 p 2 is the rule number, k ∈ {1, . . . , p}, and ik ∈ {0, 1} is the kth digit of the p−digit binary representation of i − 1. The number r = 2 p ∈ N will be referred in the sequel as the number of rules. Note that those MFs satisfy the convex sum property ∑ri=1 hi (·) = 1, hi (·) ≥ 0 in C. When convenient, x(t), u(t), z(t) and y(t) are denotes as x, u, z and y, respectively. Then, the following T-S fuzzy model is derived:  x˙ = ∑ri=1 hi (z)(Ai x + Bi u) = Az x + Bz u (4) y = ∑ri=1 hi (z)Ci x = Cz x

(5)

XTY X > 0

(6)

Lemma 2-Schur Complement: Matrices P, A, and T being of appropriate sizes, we have  T −1  T ∗ A P A−T ⇔ >0 (7) A P P>0 Lemma 3[23]: The two next problems are equivalent. (i) Find P = PT such that T + AT PA < 0

(8)

(ii) Find P = PT , L, G such that  ∗ T + AT LT + LA 0, then

nlk −nlk

hi (·) = h1+i1 ×20 +i2 ×21 +···+i p ×2 p−1 (·) = ∏ wkik (·)

r

u = −( ∑ hi Fi )( ∑ hi Pi33 )−1 y = −Fz (Pz33 )−1 y

r

1

where Ai ∈ Rn×n , Bi ∈ Rn×m , Ci ∈ Rq×n are constantmatri ces. In what follows, we will drop the argument of hi z for simplicity. In stead of parallel distributed compensation (PDC) [2], the fuzzy controller is designed in our paper using the following generalization of non-quadratic control law [13]

(10)

where 0 ∈ Rm×m denotes the matrix whose elements are zero. From (4) and (10), we have the following descriptor representation. E˜ x˙˜ = A˜ zz x˜ where ⎡

I ⎢0 E˜ =⎢ ⎣0 0  T x˜ = x

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0 0 0 0

0 0 0 0 xT

⎤ ⎡ 0 0 ⎢ I 0⎥ ⎥ ; A˜ =⎢ 0 ⎦ zz ⎣ Cz 0 0 T T T y u

(11) Az −I 0 0

0 0 −I −Fz (Pz33 )−1

⎤ Bz 0 ⎥ ⎥ 0 ⎦ −I

Obviously, system (4) is stabilized via the control law (5) if (11) is asymptotic stable. Theorem 1 represents a sufficient stability condition for (11). Theorem 1: If there exist matrices of proper size {Pi11 = T Pi11 > 0}ri=1 , {Pi21 }ri=1 , {Pi22 }ri=1 , {Pi23 }ri=1 , {Pi24 }ri=1 , T {Pi33 = Pi33 > 0}ri=1 , {Pi41 }ri=1 , {Pi42 }ri=1 , {Pi43 }ri=1 , 44 r {Pi }i=1 , {Fi }ri=1 satisfying the following conditions ∀i ∈ {1, . . . , r} (12) ϒii < 0, 2 ϒii + ϒi j + ϒ ji < 0, ∀i, j ∈ {1, . . . , r}, i=j (13) r−1 where ⎡

ϒ11 ij ⎢ ϒ21 ⎢ ij ϒi j =⎢ 31 ⎣ ϒi j ϒ41 ij

∗ T 22 −Pj − Pj22 T −Pj23 T −Pj42 − Pj24

∗ ∗ T −Pj33 − Pj33 −Fj − Pj43

∗ ∗ ∗ T 44 −Pj − Pj44

T

with Pz11 = Pz11 > 0. Then, extending (17), we have ⎡ 11 ⎤ ∗ ∗ ∗ Θzz 22 22T ⎢ Θ21 ∗ ∗ ⎥ ⎢ zz −Pz −Pz T ⎥< 0 T ⎣ Θ31 −P32 −P23 33 33 −Pz −Pz ∗ ⎦ zz z z Θ41 Θ42 Θ43 Θ44 zzz zzz z zzz (18) where 21 21T T 41 41T T ˙ 11 Θ11 zz = Az Pz + Pz Az + Bz Pz + Pz Bz − Pz T

T

11 31 23 T 43 T Θ31 zz = Cz Pz − Pz + Pz Az + Pz Bz

⎥ ⎥ ⎥ ⎦

41 24 T 44 T 33 −1 31 Θ41 zzz = −Pz + Pz Az + Pz Bz − Fz (Pz ) Pz

T

T = Ci Pj11 + Pj23 ATi

T

T

42 24 Θ42 − Fz (Pz33 )−1 Pz32 zzz = −Pz − Pz T

43 34 Θ43 z = −Fz − Pz − Pz T

T

44 44 Θ44 − Fz (Pz33 )−1 Pz34 − Pz34 (Pz33 )−T FzT zzz = −Pz − Pz

T

11 21 22 T 42 T ϒ21 i j = Pj − Pj + Pj Ai + Pj Bi

ϒ31 ij

T

⎤

21 21T T 41 41T T ˙ 11 ϒ11 i j = Ai Pj + Pj Ai + Bi Pj + Pj Bi − Pj T

T

11 21 22 T 42 T Θ21 zz = Pz − Pz + Pz Az + Pz Bz

T + Pj43 BTi

T

T

41 24 T 44 T ϒ41 i j = −Pj + Pj Ai + Pj Bi

Then the control system (11) is asymptotic stable. Proof : Consider a non-quadratic candidate Lyapunov function

Due to the nature of the candidate Lyapunov function (14), Pz11 , Pz21 , Pz22 , Pz23 , Pz24 , Pz31 , Pz32 , Pz33 , Pz34 , Pz41 , Pz42 , Pz43 , and Pz44 are the slack matrices which can be chosen freely. For the analysis convenience, as in [19], we choose Pz31 = Pz32 = Pz34 = 0, Pz33 > 0. Thus, (18) can be rewritten as r

(14)

i=1

where E˜ T Pz−1 = Pz−T E˜ ≥ 0. Note that Pz is a nonsingular matrix. Therefore, we have ˜ z≥0 PzT E˜ T = EP

(15)

The time derivative of (14) along the trajectories of (11) is obtained as follows V˙ (x) ˜ = x˙˜T E˜ T Pz−1 x˜ + x˜T E˜ T Pz−1 x˙˜ + x˜T E˜ T P˙z−1 x˜ Takeing into account that P˙z−1 = −Pz−1 P˙z Pz−1 . It is verified that V˙ (x) ˜ = x˙˜T E˜ T Pz−1 x˜ + x˜T E˜ T Pz−1 x˙˜ − x˜T E˜ T Pz−1 P˙z Pz−1 x˜ = x˜˙T E˜ T Pz−1 x˜ + x˜T Pz−T E˜ x˙˜ − x˜T E˜ T Pz−1 P˙z Pz−1 x˜   = x˜T A˜ Tzz Pz−1 + Pz−T A˜ zz − E˜ T Pz−1 P˙z Pz−1 x˜ From this fact, we obtain the following condition A˜ Tzz Pz−1 + Pz−T A˜ zz − E˜ T Pz−1 P˙z Pz−1 < 0

(19)

From relaxation Lemma 4, conditions (12), (13) in Theorem 1 guarantee ϒzz < 0. Thus, it follows from Lyapunov stability theory that system (11) is asymptotically stable. This completes the proof of the Theorem 1. Remark 1: An important difficulty arises when employing Theorem 1 to solve SOFC problem, since the dependency of P˙z11 on the time derivatives of MFs makes this term hard to deal with. It was reported in [19] that global conditions in the form of LMIs are derived by bounding the time derivatives of MFs assuming that they do not depend on the input. Unfortunately, obtaining those LMIs conditions for global stabilization is no longer possible since the term P˙z11 = ∑ri=1 h˙ i Pi11 depend on the time derivative of MFs, which do not convey to readily available bounds. Unlike the previous works [12, 19] intended to derive global asymptotic conditions, this paper intend to overcome the difficulty mentioned in Remark 1 via a local approach. As in [14], P˙z11 can be further developed as follows:

(16)

to ensure V˙ (x) ˜ < 0, which implies the asymptotic stability of fuzzy system (11). Multiplying the inequality (16) on the left and right by PzT and Pz , respectively. Employing Lemma 1, (16) is equivalent to PzT A˜ Tzz + A˜ zz Pz − E˜ P˙z < 0

∑ h i h j ϒi j < 0

i=1 j=1

r

V (x) ˜ = x˜T E˜ T ( ∑ hi Pi )−1 x˜ = x˜T E˜ T Pz−1 x˜

r

ϒzz = ∑

(17)

Now partition matrix Pz conformably with A˜ zz defined in (11). To satisfy the inequality (15), we choose ⎡ 11 ⎤ Pz 0 0 0 ⎢ Pz21 Pz22 Pz23 Pz24 ⎥ ⎥ Pz = ⎢ ⎣ Pz31 Pz32 Pz33 Pz34 ⎦ Pz41 Pz42 Pz43 Pz44

2966

p

∂ hi T 11 ∂ hi P˙z11 = ∑ h˙ i Pi11 = ∑ ( z˙ j Pi11 ) z˙Pi = ∑ ∑ ∂ z ∂ z j i=1 j=1 i=1 i=1 r

r

r

r

p

r

p

∂ p k ( ∏ wik )˙z j Pi11 ∂ z j i=1 j=1 k=1

=∑∑ =∑∑

i=1 j=1 r

=∑ r

∂zj

p ∂wj ij

∑

i=1 j=1

=∑

∂ wijj

∂zj

p ∂wj ij

∑

i=1 j=1

∂zj

p

(

∏

k=1,k= j

wkik )˙z j Pi11

p

(

∏

k=1,k= j p

wkik (wi j + 1 − wi j ))˙z j Pi11

( ∏ wkik + (1 − wi j ) k=1

p

∏

k=1,k= j

wkik )˙z j Pi11

p p k since hi = ∏k=1 wkik , ∃hl : hl = (1 − wijj ) ∏k=1,k = j w ik = j ∂ wi j

p ∏k=1 wklk for which Therefore, we have r

P˙z11 = ∑

p

=−

∂zj

j ∂ wl j

∂zj

≤ xT (Pz11 )−1 x ≤ 1

∑ hi ∂ zk (Pg111 (i,k) − Pg112 (i,k) )˙zk ∂ wk

∂ zk

∂ wk

∂ zk

=∑

∑ hi ∂ zk0 (Pg111 (i,k) − Pg112 (i,k) )( ∂ x )T x˙

=∑

∑ hi ∂ zk0 (Pg111 (i,k) − Pg112 (i,k) )( ∂ x )T (Az x + Bz u)

=∑

∑ ∑ ∑ hi ∂ xv0 (Az )vs xs (Pg111 (i,k) − Pg112 (i,k) )

i=1 k=1 r p

i=1 k=1 r p n

∂ wk

n

i=1 k=1 v=1 s=1 r p n m

+∑

∑∑

∑ hi

i=1 k=1 v=1 e=1 p n n ∂ w0k

=∑

∑∑

∑∑

k=1 v=1 e=1

∂ w0k (Bz )ve ue (Pg111 (i,k) − Pg112 (i,k) ) ∂ xv

(Az )vs xs (Pg111 (z,k) − Pg112 (z,k) )

k=1 v=1 s=1 ∂ xv p n m ∂ w0k

+∑

1 1  u 22 = 2 xT CzT (Pz33 )−1 FzT Fz (Pz33 )−1Cz x μ2 μ

with i j = 0 and l j = 1.

∂ w0k

i=1 k=1 r p

μ is implied by

∂ xv

(Bz )ve ue (Pg111 (z,k) − Pg112 (z,k) )

(20)

where g1 (i, k) = (i − 1)/2 p+1−k 2 p+1−k + 1 + (i − 1)mod2 p−k g2 (i, k) = g1 (i, k) + 2 p−k Substituting (20) into (19), gives r

Πzz = ∑

r

∑ hi h j Πi j < 0

(21)

which by Lemma1 and Lemma 3 is equivalent to   1 T 33 −1 T 33 −1 11 −1 x≤0 C (P ) F F (P ) C − (P ) xT z z z z z μ2 z z 1 ⇔ 2 CzT (Pz33 )−1 FzT Fz (Pz33 )−1Cz − (Pz11 )−1 ≤ 0 μ ⎡ 11 −1  ⎤ −(Pz ) +CzT Cz (Pz11 )−1 ∗ ⎢ ⎥ +(Pz11 )−1CzT Cz ⎢  ⎥