Finite-time dynamic output-feedback dissipative control for singular uncertainty T–S fuzzy systems with actuator saturation and output constraints

Finite-time dynamic output-feedback dissipative control for singular uncertainty T-S fuzzy systems with actuator saturation and output constraints

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Finite-time dynamic output-feedback dissipative control for singular uncertainty T-S fuzzy systems with actuator saturation and output constraints Xiaojing Han, Yuechao Ma, Lei Fu PII: DOI: Reference:

S0016-0032(20)30082-X https://doi.org/10.1016/j.jfranklin.2020.01.048 FI 4410

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

10 October 2018 2 December 2019 28 January 2020

Please cite this article as: Xiaojing Han, Yuechao Ma, Lei Fu, Finite-time dynamic output-feedback dissipative control for singular uncertainty T-S fuzzy systems with actuator saturation and output constraints, Journal of the Franklin Institute (2020), doi: https://doi.org/10.1016/j.jfranklin.2020.01.048

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Finite-time dynamic output-feedback dissipative control for singular uncertainty T-S fuzzy systems with actuator saturation and output constraints Xiaojing Han a , Yuechao Ma a,∗ , Lei Fu a a School

of Science, Yanshan University, Qinhuangdao Hebei, P.R.China, 066004;

Abstract This paper studies the problem of finite-time dynamic output-feedback control for singular uncertainty T-S time-varying delay fuzzy systems with actuator saturation and output constraints. Through a dynamic parallel distributed compensator, the (U, V, R)−α− dissipative output-feedback fuzzy controller is designed. Furthermore, by employing reciprocally convex approach and a new lemma, the sufficient conditions are obtained to guarantee that the systems are finite-time bounded subject to (U, V, R) − α− dissipative performance level. Then, the desired fuzzy state-feedback controller and dynamic output feedback controller are derived by solving the linear matrix inequalities (LMIs). Finally, the feasibility and effectiveness of the proposed method can be validated by some simulations. Key words: Singular T-S fuzzy systems; Dynamic output-feedback; Singular finite-time dissipativity; Actuator saturation; Output constraints; LMIs.

1

Introduction

Over the past three decades, singular system [1] which is also called differential algebraic system has made considerable theoretical results and fruitful applications in aviation, aerospace, robotics, power systems, electronic, chemical, ? This work is partially supported by the National Natural Science Foundation of China No. 61273004, and the Natural Science Foundation of Hebei province No. F2018203099. ??Declarations of interest: none ∗ Corresponding author. Email address: [email protected] (Yuechao Ma).

Preprint submitted to Elsevier

3 February 2020

biological and economic network and so on [2,3,43]. At present, the research of singular systems is still a hot field of control research. It is noted that the singular systems has impulsive behaviors or non-causal behaviors, as a result, the relevant research results are more complex and full of novelty. In 1999, Taniguchi t. et al. made the normal T-S fuzzy model extend to more general situation and further proposed the singular T-S fuzzy mode [4]. It is the model which combines normal T-S fuzzy systems with singular systems solves the control problem of the global nonlinear singular systems by means of linear generalized systems analysis and control measures. Therefore, many significant studies for the analysis and synthesis of descriptor T-S fuzzy systems have been conducted and extended, such as stability and stabilization analysis [5], robust control [6], output feedback control [7,8,45], reliable control [9], sliding control [10], reliable control [11], and dissipative control [12,13]. On the other hand, actuator saturation is ubiquity, which may cause dreadful performance and instability of the closed-loop systems [14,15,44]. Consequently, some scholars devote themselves to the analysis of stability and design of controller for the system with actuator saturation [16-19] in the past decade. Recently, the problem of robust anti-windup controller design of timedelay fuzzy systems with actuator saturations was presented in [16]. In [18], the problem of memory feedback H∞ control of uncertain singular T-S fuzzy time-delay system under actuator saturation was demonstrated. In [19], the problem of dissipative control for T-S fuzzy descriptor systems with actuator saturation and disturbances was discussed. It is commonly known that the classical Lyapunov stability pours more attention to the asymptotic closed-loop systems at an infinite-time interval. Nevertheless, the finite-time control has practical significance in many practical applications [20,21,46]. The concepts of the finite-time stability were first introduced in the 1960s [22,23]. Afterwards, finite-time boundedness as an extension of finite-time stability appeared in [24]. In recent years, numbers of results on finite-time stability and stabilization were gained for linear systems [25,26], nonlinear systems [27,28], Markovian jump systems [29], fuzzy systems [30,31] and so on. The results which were gained by employing the linear matrix inequality (LMI). For example, the output feedback finite-time stability of continuous linear systems was investigated in [26]. Next, the problem of the output feedback finite-time stability of continuous nonlinear systems was demonstrated in [28]. What’s more, to ensure the closed-loop system’s finite-time boundedness and decrease the effect of disturbance input on the controlled output, the problem of finite-time H∞ filtering for discrete-time Markovian jump systems has been investigated in [29]. Furthermore, [31] emphasized the problem of finite-time reliable L2 − L∞ /H∞ control for T-S fuzzy systems with actuator faults. In addition, compared with H∞ and passivity performance, dissipative perfor2

mance which was first introduced by Willems [32,33], is a unified result on the phenomenon of the robust control. Moreover, the dissipative theory has also been applied in many fields, such as control engineering, network, and circuit [19,34-37,47]. However, it is not easy to consider the problem of dissipative control, especially the problem of finite-time dissipative control for singular T-S fuzzy systems. For example, the problem of finite-time dissipative control for singular T-S fuzzy Markovian jump systems under actuator saturation with partly unknown transition rates has been demonstrated in [38]. Recently, the design problem of finite-time dissipative fault-tolerant controller of T-S fuzzy systems in a network environment has been investigated in [39]. In this paper, we apply a scheme of a dynamic parallel distributed compensator (DPDC) to design a dynamic output-feedback controller to ensure the finite time boundedness and finite time dissipative for the singular uncertain T-S time-varying delay fuzzy systems subject to actuator saturation and output constraints. The main contributions and novelty of this paper are as follows: (1) The design problem of finite-time dynamic output-feedback dissipative controller for singular uncertain T-S time-varying delay fuzzy systems with actuator saturation and output constrains is studied for the first time. (2) Based on dynamic parallel distributed compensator (DPDC), the finite-time dissipative desired controller gains and dynamic output-feedback controller gains can be tackled in terms of the linear matrix inequality by using the reciprocally convex approach and a new lemma. (3) The dissipative performance index α and finite-time index c2 are derived by a LMI optimization problem and the estimation of the largest domain is estimated. (4) The simulations demonstrate the feasibility and effectiveness of the proposed method. The remaining parts of this paper are organized as follows. The system description and problem formulation are shown in Section 2. The main results on dynamic output feedback finite-time dissipative control for singular uncertain T-S fuzzy systems with actuator saturation and output constraints are drawn in section 3. Two examples are presented in Section 4 to demonstrate the effectiveness and feasibility of the obtained results. Finally, conclusions and the direction of future research are made in Section 5. Notation: In this paper, Rn and Rn×m denote the real space with dimension n and n × m real matrices, respectively. The notation P > 0 represents that P is a symmetric and positive-definite matrix. X T and X −1 stand for the transpose and the inverse of X, respectively. The symbol * represents a matrix which can be inferred by symmetry and sym (A) denotes A + AT for simplicity. Diag {· · · } represents a block-diagonal matrix. λmax (Y ) and λmin (Y ) denote the maximum and minimum eigenvalue of matrix Y. The dimension of matrices is assumed to be compatible if not explicitly stated. 3

2

Description of the problem and main results

Consider the following singular uncertain T-S fuzzy time-varying delay system with actuator saturation and input delay, which is described by Plant Rule i : IF ε1 (t) is Mi1 , . . . , εp (t) is Mip , then   E x˙ (t)          

= (Ai + ∆Ai ) x (t) + (Aid + ∆Aid ) x (t − d (t)) + (Bωi + ∆Bωi ) ω (t) + Bi sat (u (t)) +Biτ sat (u (t − τ (t))) z1 (t) = (C1i + ∆C1i ) x (t) + (C1id + ∆C1id ) x (t − d (t)) + (Dωi + ∆Dωi ) ω (t)  z2 (t) =C2i x (t)       y (t) =C3i x (t)     x (t) =φ (t) , t ∈ [−d2 , 0] i = 1, 2, · · · , r (1) where ε1 (t) , ε2 (t) , · · · , εp (t) are the premise variables; i = 1, 2, · · · , r and r is the number of IF-THEN rules; Mil (l = 1, 2, · · · , p) are the fuzzy sets. x (t) ∈ Rn is the state vector; y (t) ∈ Rm , z1 (t) ∈ Rp , z2 (t) ∈ Rq are the control output vectors; d (t) is the time-varying continuous function that satisfies 0 ≤ d1 ≤ d (t) ≤ d2 , d˙ (t) ≤ h1 ; τ (t) is the time-varying delay of the control input satisfying 0 ≤ τ (t) ≤ τ , τ˙ (t) ≤ h2 ; d1 , d2 , τ and h1 , h2 are known scalars. Ai , Aid , Bωi , Bi , Biτ , C1i , C1id , Dωi , C2i , C3i are the given constant matrices of the system. ω (t) ∈ Rk is the disturbance input vector satisfying: RT 0

ω T (t)ω (t) dt ≤ η 2 , η ≥ 0.

(2)

u (t) ∈ Rl is the control input. Rl → Rl is the standard saturation function defined as follows: sat (u (t)) = [sat (u1 (t)) , sat (u2 (t)) , · · · , sat (ul (t))]T , where sat (ui (t)) = sign (ui (t)) min {1, kui (t)k} . The matrix E ∈ Rn×n is singular and it is assumed that rank (E) = r ≤ n; ∆Ai , ∆Aid , ∆Bωi , ∆C1i , ∆C1id , ∆Dωi are unknown matrices denoting norm-bounded parameter uncertainties and have the following forms: 



 ∆Ai ∆Aid ∆Bωi  

∆C1i ∆C1id ∆Dωi







 H1i 

=

H2i

F





(t) E1i E2i E3i ,

where H1i , H2i E1i , E2i , E3i are known real constant matrices with appropriate dimensions and F (t) is unknown time-varying matrices satisfying F T (t) F (t) ≤ I. 4

By fuzzy blending, the overall fuzzy model can be described as follows:

    E x˙ (t)                  z (t)   1                      

=

r X

λi (ε (t)) {(Ai + ∆Ai ) x (t) + (Aid + ∆Aid ) x (t − d (t)) + (Bωi + ∆Bωi ) ω (t) + Bi sat (u (

r X

λi (ε (t)) {(C1i + ∆C1i ) x (t) + (C1id + ∆C1id ) x (t − d (t)) + (Dωi + ∆Dωi ) ω (t)}

i=1

+Biτ sat (u (t − τ (t)))}

=

z2 (t) = y (t) =

i=1 r X

i=1 r X

λi (ε (t)) C2i x (t) λi (ε (t)) C3i x (t)

i=1

x (t) =φ (t) , t ∈ [−d2 , 0] 

T

(3) Q

p ε1 (t) ε2 (t) · · · εP (t) , λi (ε (t)) = j=1 Λij (εj (t)) are the membership functions , of the system with respect to the ith pant rules. Letting

where ε (t) =

λi (ε (t)) = hi (ε (t))

l P

i=1

hi (ε (t)). We have that λi (ε (t)) ≥ 0 and

1. In the following, we denote λi (ε (t)) by λi .

r P

i=1

λi (ε (t)) =

For the desired dissipative output-feedback fuzzy control, now we consider a DPDC structure as follows:     E xˆ˙ (t)       

=

u (t) =

r X

i=1 r X

λi (ε (t)) {Ali xˆ (t) + Alid xˆ (t − d (t)) + Hli y (t)}

(4)

λi (ε (t)) Ki xˆ (t)

i=1

where xˆ (t) is the state vector of the controller, Ali , Alid , Hli , Ki are the controller gain matrices to be obtained. n×n Let ρ be a scalar, P ∈ Rn be a symmetric matrix and EoT P E ≥ 0. Then, T denote = E P E, ρ = x (t) ∈ Rn : xT (t) E T P Ex (t) ≤ ρ . For a matrix, denote the kth row of the matrix Fi as fik , we define

` (Fi ) = {x (t) ∈ Rn : |fik x (t)| ≤ 1, k ∈ [1, l]} .





Accordingly, = E T P E, ρ is an ellipsoid and ` (Fi ) is a polyhedral consisting of states for which the saturation dose not occur. Let ℵ be the set of l×l diagonal matrices in which diagonal elements are either 1 or 0. We can work on the assumption that each element of ℵ is marked as Es , s = 1, 2, · · · and denote Es− = I − Es . Thus, if Es ∈ ℵ, then Es− ∈ ℵ. 5

Denoting the state trajectory of system (1) with initial condition x0 = φ ∈ ∆ C  [−d2 , 0] by x (t, φ). Thenthe attraction domain of the origin is set ∂ = φ ∈ Cn,d2 : lim x (t, φ) = 0 . The determination of the exact domain of att→∞

traction is practically impossible [42]. An estimate χ∂ ⊂ ∂ of the domain of attraction is given by

n

o

χ∂ = φ ∈ Cn,d2 : max[−d2 ,0] |φ| ≤ ∂1 , max[−d2 ,0] φ˙ ≤ ∂2 . 

T

T

T

Defining x¯ (t) = x (t) xˆ (t) and combined with the fuzzy control law (4) and Lemma 1, the closed-loop fuzzy system is obtained as follows:    ¯ x¯˙ (t) E                   

=

z1 (t) = z2 (t) =

r X r X

i=1 j=1 r X i=1 r X

λi λj {Aijs x¯ (t) + Aids x¯ (t − d (t)) + Aiτ s x¯ (t − τ (t)) + Bωis ω (t)}

λi {C1is x¯ (t) + C1ids x¯ (t − d (t)) + Dωis ω (t)} λi C2is x¯ (t) ,

i=1

(5)

where 











− E 0   Ai + ∆Ai Bi (Es Kj + Es Fj )   Aid + ∆Aid 0  ¯ E=  , Aijs =   , Aids =   0 E Hlj C3i Alj 0 Aljd

Aiτ s =



0 

Biτ (Es Kj +

0

0 



Es− Fj ) 

 Bωis

=





 Bωi 



+ ∆Bωi  0

 C1is



= C1i + ∆C1i 0 





C1ids = C1id + ∆C1id 0 , Dωis = [Dωi + ∆Dωi ] , C2is = C2i 0 . Lemma 1 ([14]) Let K, F ∈ Rl×n , then, for any x (t) ∈ ` (F ) , we have n

sat (Kx (t)) ∈ co Es Kx (t) + Es− F x (t) , s = 1, 2, · · · , π or sat (Kx (t)) =

π X

s=1



o



αs Es K + Es− F x (t) ,

where co {} stands for the convex hull, αs for s = 1, 2, · · · , π are some scalars

which satisfy 0 ≤ αs ≤ 1 and

π P

s=1

αs = 1.

Lemma 2 ([41]) For any constant matrices Y ∈ R, Y = Y T , constant d1 > 0 and vector function x˙ : [−r, 0] → Rn such that the following integration is well 6

defined, it holds that

−d1

Z t

t−d1

 

x˙ T (θ)Y x˙ T (θ) dθ ≤ 

x (t) x (t − d1 )

T 

  −Y  



Y 

Y −Y



x (t) x (t − d1 )



 .

Lemma 3 ([8]) Let ∆ g1 , g2 , · · · , gN : Rm → R have positive values in an open subset ∆ of Rm . The reciprocally convex combination of gi over ∆ satisfies min P

{ρi |ρi >0,

i

X

ρi =1 } i

X X 1 g i (t) = g i (t) + max fij (t) gij(t) ρi i i6=j 



gi (t) fij (t)  subject to fij : Rm → R, fji (t) = fij (t) and   ≥ 0.  fij (t) gi (t) Lemma 4 ([40]) Let matrices M1 , M2 and F (t) are real matrices of appropriate dimensions with F (t) satisfying F T (t) F (t) ≤ I, then the following inequality holds for any ε > 0: M1 F (t) M2 + M2T F (t) M1T ≤ εM1 M1T + ε−1 M2T M2 . Lemma 5 For any constant matrices N1 , N2 ∈ Rn×n , L ∈ Rn×p , positive definite symmetric matrix Z ∈ Rn×n and time-varying delay d(t), then: −

Z t

t−τ (t)

n

o

T x¯˙ (s) E¯ T Z E¯ x¯˙ (s) ds ≤ ξ T (t) Γ + τ (t) T T Z −1 T ξ (t) ,

where





(6)

T = N1 0 0 0 N2 L , 

T

T

T



T

ξ (t) = x (t) 0 0 0 x (t − τ (t)) ω (t) , 

Γ=



N T E¯ + E¯ T N1 0 0 0 −N1T E¯ + E¯ T N2 E¯ T L   1    ∗ 000 0 0               



1

∗

∗00

∗

∗∗0

∗

∗∗∗

∗

∗∗∗ 1



    .  0 0    T ¯ T T  ¯ ¯ −N2 E − E N2 −E L  

0

0

∗

0





− T  Z2 Z 2T   Z Proof: Let D =  , then we have D T D =   ≥ 0. It 0 0 T T T T ZT

7

follows that: Z t

t−τ (t)



T 

¯ ¯˙ (s)  Ex   Z 

ξ (t)

Rt

 

T

T T T T ZT





¯ ¯˙ (s)   Ex  

ξ (t)

 ds

≥ 0.



(7) 

And by combining 2 t−τ (t) ξ (t)T E¯ x¯˙ (s) ds = 2ξ T (t) T T E¯ −E¯ 0 ξ (t), then we can get (6). T

T

n×n Lemma 6 ([38]) For given , Y, I ∈   matrices E, X > 0, Y and X, J ∈ R n×(n−r) T T R , if E X + Y Θ is nonsingular, then there exist matrices J > 0, I,



−1

such that EJ + IK T = E T X + Y ΘT , where Θ, K ∈ Rn×(n−r) are any matrices with full column rank satisfying E T Θ = 0, EK = 0. Definition 1 ([12])A pair (E,A) is said to be regular, if det (sE − A) 6= 0 for some complex number s; A regular pair (E,A) is said to be impulsive-free, if deg(det(E,A))=rank(E). Definition 2 ([30]) The regular singular system (5) is called singular finitetime bounded (SFTB) with respect to (c21 , c22 , η 2 , Tc , Rc ), where Rc is a symmetric positive definite matrix and some positive constants c1 , Tc , η, if there exist scalar c1 < c2 ,such that: sup −d2 ≤θ≤0

n

o

T x¯T (θ) E¯ T Rc E¯ x¯ (θ) , x¯˙ (θ) E¯ T Rc E¯ x¯˙ (θ) ≤ c21

⇒ x¯T (t) E¯ T Rc E¯ x¯ (t) ≤ c22 , ∀t ∈ [0, T ] , where x¯ (t) and E¯ are defined in (5).

Definition 3 ([39]) The regular singular system (5) is said to be singular finite-time (U, V, R) − α dissipative with respect to (c21 , c22 , η 2 , Tc , α, Rc ), if for any t ∈ [0, T ] and a scalar α > 0, the following condition is satisfied with zero initial state: Z Q (t) ≥ α

t

0

wT (s) ω (s) ds,

where

Q (t) =

Z t 0

z1T (s) U z1 (s) ds + 2

Z t 0

z1T (s) V ω (s) ds +

Z t 0

ω T (s) Rω (s) ds.

Q (t) is the quadratic energy supply function related to the closed-loop system (5); U, R are real symmetric matrices and V is a real matrix with appropriate dimensions. Without loss of generality, U ≤ 0 and−U = U−T U− f or some U− ≥ 0. 8

Remark 1 From Definition 3, the above strict (U, V, R) − α dissipative performance which contains H∞ and passivity performance as special cases: 1. When U=-I, V=0 and R = (α + α2 ) I, the finite-time (U, V, R) − α dissipative reduces to an H∞ performance. 2. When U=0,V=I and R=0, the finite-time (U, V, R) − α dissipative corresponds to a finite-time passivity or positive realness property. ¯ 0 be a set of admissible initial conditions. The objective of Remark 2 Let X this paper is to obtain the controller gain matrix Kj and the DPDC gain matrices Alj , Aljd , Hlj , such that all trajectories of system (5) starting from ¯ 0 will remain inside it, which denotes X ¯ 0 is an invariant within X set for  T ¯ ¯ ¯ ¯ system (5). Thus, X0 is regarded as an ellipsoidal = E P E, ρ . 



¯ ρ is defined as follows: The set = E¯ T P¯ E, 



o

n

¯ ρ = x¯ (t) ∈ R2n : x¯T (t) E¯ T P¯ E¯ x¯ (t) ≤ ρ . = E¯ T P¯ E, The main objective of this paper: 1. To design a fuzzy state feedback controller of the form (4) which can ensure singular finite-time (U, V, R)−α dissipative bounded of the closed-loop system (5); 2. The following hard constraints concerning control output [8] are satisfied: {z2 (t)}ξ

≤ {z2,M }ξ , t > 0, ξ = 1, · · · q,

(8)

where z2,M = [z1,max · · · zq,max ]T , zξ max and {·}ξ denote the maximum limit of each component and the ξ−th element of a vector, respectively.

3 3.1

Main results SFTB analysis and finite-time (U, V, R) − α dissipative control

In this section, we pay attention to the problem of finite-time bounded and finite-time (U, V, R) − α dissipative performance analysis for the system (5).

Theorem 1 For given positive constants c1 , η, Tc , δ and positive definite matrix Rc , if there are scalars c2 > c1 > 0, positive definite matrixes P, Z   m , Qn, Yn, S (m = 1, 2, 3; n = 1, 2) T ˜ ¯ ¯ any matrices N1 , N2 , L with appropriate dimensions, = E P E, ρ ⊂ ` F˜i and scalars d2 > d1 ≥ 0, τ ≥ 0, h1 > 0, h2 > 0 such that the following conditions hold for i, j ∈ R Ωii < 0, (9) 9

Ωij + Ωji < 0,



(10)



 Q2 

S   ≥ 0, ∗ Q2

(11) !

d3 d3 τ2 λ2 + d1 λ3 + d2 (λ4 + λ5 ) + 1 λ6 + 12 λ7 + τ λ8 + λ9 c21 +δη 2 < λ1 e−δT c22 , 2 2 2 (12) where Γ, T are defined in Lemma 5, and 

˜T Pˆ = E¯ T P˜ + S˜R

T



P

= 1

 



1 1 1 1 0 T ¯ Pˆ = E¯ T Rc2 P¯ Rc2 E, ¯ Zi = Rc2 Z¯i Rc2 (i = 1, 2, 3) , ,E 0 P 1

1

1

¯ i Rc2 (i = 1, 2) , Yi = Rc2 Y¯i Rc2 (i = 1, 2) , Qi = Rc2 Q  

















¯1 , λ1 = λmin P¯ , λ2 = λmax P¯ , λ3 = λmax Z¯1 , λ4 = λmax Z¯2 , λ5 = λmax Z¯3 , λ6 = λmax Q 

Ωij =



Ξij =





 Ξij + Γ    ∗    ∗     ∗  

d12 ΠT1

−Q−1 1

0

0

0

∗

−Q−1 2

0

0

∗

∗

− τ1 Y2−1

0

∗

∗

∗





ΠT1

T

T

− τ1 Y2



      ,      

Pˆ T Aids

0

Pˆ T Aiτ s

ϕ22

E¯ T (−S + Q2 ) E¯

E¯ T S E¯

0

∗

∗

ϕ33

E¯ T (−S + Q2 ) E¯

0

∗

∗

∗

−Z3 − E¯ T Q2 E¯

0

0

∗

∗

∗

∗

− (1 − h2 ) Y1

0

∗

∗

∗

∗

∗

−δI

3 X i=1



Pˆ T Bωis 

∗

d12 = d2 −d1 , ϕ11 = sym(ATijs Pˆ )+ ϕ33



d1 ΠT1

∗

ϕ E¯ T Q1 E¯  11               



¯ 2 , λ8 = λmax Y¯1 , λ9 = λmax Y¯2 , λ7 = λmax Q

0 0

       ,       

¯ 1 −δ E¯ Pˆ , ϕ22 = −Z1 −E¯ T Q1 E− ¯ E¯ T Q2 E, ¯ Zi −E¯ T Q1 E+Y 







¯ = − (1 − h1 ) Z2 −2E¯ T Q2 E+sym E¯ T S E¯ , Π1 = Aijs 0 Aids 0 Aiτ s Bωis ,

˜ ∈ R2n×2(n−r) is any matrix with full column rank satisfying E¯ T R ˜ = 0. R Then the closed-loop system (5) is regular, impulse free and singular finite-  2 2 2 T ˜ ¯ ¯ time bounded subject to (c1 , c2 , η , Tc , Rc ) at the origin with = E P E, ρ 10

contained in the domain of attraction, and the estimated of the attraction domain is Γθ ≤ ρ, where !

"

∂2 d3 d3 ∂22 ∂2

+ τ λ8 1 Γθ = (λ2 + d1 λ3 + d2 (λ4 + λ5 )) 1 + 1 λ6 + 12 λ7

¯ T Rc E¯

kRc k 2 2 kRc k

E 

∂22 τ2

 c2 . + λ9

2 E¯ T Rc E¯

1

Proof: First, we proof the system (5) is regular and impulse free. Choose two nonsingular matrices M, N ∈ Rn×n such that 





¯ = M EN

0 0



 A11 A12 

 I2r 0 

 , M Aijs N

=

A21 A22

,N

T





 S1 

S˜ = 

S2

.



˜ = ˜ = 0 and R ˜ ∈ R2n×2(n−r) , we can get M −T R Noting that E¯ R

˜T 0R 22

T

.

Then, pre- and post- multiplying ϕ11 < 0 by N T and N. By combining ˜ T A22 + AT R ˜ 22 S˜2T < 0, which means A22 is nonZ1 , Z2 , Z3 > 0, we have S˜2 R 22 22   ¯ Aijs is regular and impulse singular. Then, from definition 1, the pair E, free. Next, we consider the following Lyapunov-Krasovskii function(LKF): V (t) =

4 X

Vυ (t)

(13)

υ=1

where V1 (t) = x¯T (t) E¯ T P˜ E¯ x¯ (t) ,

V2 (t) =

Z t

t−d1

V3 (t) = d1

V4 (t) =

Z t

eδ(t−s) x¯T (s) Z1 x¯ (s) ds+

Z 0 Z t

Z t

−d1

t−τ (t)

t+θ

e

t−d(t)

Z t

eδ(t−s) x¯T (s) Z2 x¯ (s) ds+

T eδ(t−s) x¯˙ (s) E¯ T Q1 E¯ x¯˙ (s) dsdθ+d12

δ(t−s) T

x¯ (s) Y1 x¯ (s) ds +

Z 0 Z t

11

−τ

t+θ

Z −d1 Z t −d2

t+θ

t−d2

eδ(t−s) x¯T (s) Z3 x¯ (s) ds,

T eδ(t−s) x¯˙ (s) E¯ T Q2 E¯ x¯˙ (s) dsdθ,

T eδ(t−s) x¯˙ (s) E¯ T Y2 E¯ x¯˙ (s) dsdθ,

Then, the time-derivative of V (t) along the system (5), we have V˙ (t) = 4 P V˙ υ (t), where υ=1

V˙ 1 (t) = 2¯ xT (t) Pˆ T E¯ x¯˙ (t) , (14) T T T V˙ 2 (t) ≤δV2 (t) + x¯ (t) (Z1 + Z2 + Z3 ) x¯ (t) − x¯ (t − d1 ) Z1 x¯ (t − d1 ) − x¯ (t − d2 ) Z3 x¯ (t − d2 ) − (1 − h1 ) x¯T (t − d (t)) Z2 x¯ (t − d (t)) , (15) Z t T T T x¯˙ (s) E¯ T Q1 E¯ x¯˙ (s) ds + d2 x¯˙ (t) E¯ T Q2 E¯ x¯˙ (t) V˙ 3 (t) ≤δV3 (t) + d2 x¯˙ (t) E¯ T Q1 E¯ x¯˙ (t) − d1 1

− d12

t−d2

12

t−d1

Z t−d1 ˙T

x¯ (s) E¯ T Q2 E¯ x¯˙ (s) ds,

(16) T T T ˙ V4 (t) ≤δV4 (t) + x¯ (t) Y1 x¯ (t) − (1 − h2 ) x¯ (t − τ (t)) Y1 x¯ (t − τ (t)) + τ x¯˙ (t) E¯ T Y2 E¯ x¯˙ (t) −

Z t

t−τ

T x¯˙ (s) E¯ T Y2 E¯ x¯˙ (s)ds.

(17)

From Lemma 2, 3 and 5, we can obtain −d1



Z t

T 





¯T ¯ ¯T ¯   −E Q1 E E Q1 E   x (t)  ˙x¯T (s) E¯ T Q1 E¯ x¯˙ (s) ds ≤      , t−d1 x (t − d1 ) ∗ −E¯ T Q1 E¯ x (t − d1 ) x (t)

(18)

t−d1

T −d12 ∫ ¯˙x (s) E¯ T Q2 E¯¯˙x (s) ds t−d2

1 ¯ 1 (t) − 1 ΦT2 (t) (t) E¯ T Q2 EΦ ¯ 2 (t) 6 − ΦT1 (t) E¯ T Q2 EΦ ρ1 ρ2 ¯ 1 (t) − ΦT (t) E¯ T Q2 EΦ ¯ 2 (t) − ΦT (t) E¯ T S EΦ ¯ 2 (t) − ΦT (t) E¯ T S EΦ ¯ 1 (t) 6 −ΦT1 (t) E¯ T Q2 EΦ 2 1 2 



T 



Φ1 (t)   E¯ T Q2 E¯ E¯ T S E¯   Φ1 (t)  = −      ∗ E¯ T Q2 E¯ Φ2 (t) Φ2 (t) 

T 

 

   

¯ (t − d1 )   −E¯ T Q2 E¯  x 

 

  = x ¯ (t − d (t))  

x¯ (t − d2 ) −

where

1 ρ1

=

Z t

t−τ

∗

∗

E¯ T Q2 E¯ − E¯ T S E¯ 





¯ (t − d1 )   x 



   ¯ (t − d (t))  , −2E¯ T Q2 E¯ + sym E¯ T S E¯ −E¯ T S E¯ + E¯ T Q2 E¯   x

∗

n

−E¯ T Q2 E¯

o

T x¯˙ (s) E¯ T Y2 E¯ x¯˙ (s)ds ≤ ξ T (t) Γ + τ T T Z −1 T ξ (t) ,

d(t)−d1 1 , ρ2 d12



E¯ T S E¯





(19)

(20)

 Q2 S  = d2 d−d(t) ,   ≥ 0. 12 ∗ Q2

By combing (14)-(20) and adding −δV (t) − δω T (t) ω (t) on both sides, we can obtain: 12



x¯ (t − d2 )



V˙ (t) − δV (t) − δω T (t) ω (t) n

o

≤ζ T (t) Ξij + Γ + d21 ΠT1 Q1 π1 + d212 ΠT1 Q2 Π1 + τ ΠT1 Y2 Π1 + τ T T Y2−1 T ζ (t) , (21)   where ζ T (t) = xT (t) xT (t − d1 ) xT (t − d (t)) xT (t − d2 ) xT (t − τ (t)) ω T (t) . Then, by combining Schur complement, we can obtain that Ξij + Γ + d21 ΠT1 Q1 Π1 + d212 ΠT1 Q2 Π1 + τ ΠT1 Y2 Π1 + τ T T Y2−1 T < Ωij .

(22)

And by combining (10), it is easy to obtain that: V˙ (t) < δV (t) + δω T (t) ω (t) .

(23)

Pre- and post- multiplying (23) by e−δt and integrating it from 0 to T, we can derive "

V (T ) ≤ eδT V (0) + δ

Z T 0

#

"

e−δs ω T (s) ω (s) ds ≤ eδT V (0) + δ

Z T 0

#

ω T (s) ω (s) ds .

And by combining the following inequality V (0) =¯ x (0) E¯ T Pˆ x¯ (0) + T

+

Z 0

−d2

+ d12 +

−d1

e

−δs T

x¯ (s) Z1 x¯ (s) ds +

e−δs x¯T (s) Z3 x¯ (s) ds + d1

Z −d1 Z 0 −d2

Z 0 Z 0 −τ

Z 0

θ

θ

e

Z 0 Z 0 −d1

−δs ˙ T

θ

Z 0

−d(t)

e−δs x¯T (s) Z2 x¯ (s) ds

T e−δs x¯˙ (s) E¯ T Q1 E¯ x¯˙ (s) dsdθ

x¯ (s) E¯ T Q2 E¯ x¯˙ (s) dsdθ +

Z 0

−τ (t)

e

−δs

x¯T (s) Y1 x¯ (s) ds

−δs ˙ T

e

x¯ (s) E¯ T Y2 E¯ x¯˙ (s) dsdθ

d3 d3 ≤¯ x (0) E Rc E¯ x¯ (0) + d1 λ3 + d2 (λ4 + λ5 ) + 1 λ6 + 12 λ7 2 2 n o T T T T ¯ ¯ ¯ ¯ × sup x¯ (θ) E Rc E x¯ (θ) , x¯˙ (θ) E Rc E x¯˙ (θ) ¯T

T

−d2 ≤θ≤0

τ2 + τ λ8 + λ9 2

"

!

sup −τ ≤θ≤0

n

!

o

T

x¯T (θ) E¯ T Rc E¯ x¯ (θ) , x¯˙ (θ) E¯ T Rc E¯ x¯˙ (θ) #

d3 d3 τ2 ≤ λ2 + d1 λ3 + d2 (λ4 + λ5 ) + 1 λ6 + 12 λ7 τ λ8 + λ9 c21 , 2 2 2

(24) we can get that V (t) ≤ e

δT

("

#

)

d3 τ2 d3 λ2 + d1 λ3 + d2 (λ4 + λ5 ) + 1 λ6 + 12 λ7 + τ λ8 + λ9 c21 + δη 2 . 2 2 2 (25) 13

By combining V (t) ≥ λ1 x¯T (t) E¯ T Rc E¯ x¯ (t) and (12), we have x¯T (t) E¯ T Rc E¯ x¯ (t) < c22 . From Definition 2, the closed-loop system (5) is SFTB. Furthermore, by combining (24), we can also obtain that

V (φ (t))



 ≤ (λ2 2

+

2

kφk + d1 λ3 + d2 (λ4 + λ5 )) + kRc k

2

˙

φ

d31 2

λ6 +

d312 2

λ7



τ 2

 λ9

c . 2 E¯ T Rc E¯

1

!

2

˙

φ



¯T ¯

E Rc E

kφk2 + τ λ8 kRc k

Then, we have



!

∂2 d3 d3 ∂22

x¯T (t) E¯ T P˜ E¯ x¯ (t) ≤ V (φ (t)) ≤ (λ2 + d1 λ3 + d2 (λ4 + λ5 )) 1 + 1 λ6 + 12 λ7

¯ T Rc E¯

kRc k 2 2

E 

∂2 τ2 ∂22

 c2 ≤ ρ. +τ λ8 1 + λ9

kRc k 2 E¯ T Rc E¯

1

Accordingly, we know that x¯T (t) E¯ T P˜ E¯ x¯ (t) ≤ ρ and all the trajectories of x¯ (t)  that start  from estimate χ∂max of the domain of attraction remain within T ¯ ρ . This completes the proof. = E¯ P˜ E, Remark 3 The definition of singular finite-time stability (SFTS) may be used in many papers. However, we should pay attention that SFTB is widely used and SFTS can be seen as a particular case by letting ω (t) = 0.

Theorem 2 For given positive constants c1 , η, Tc , δ and positive definite matrix Rc , if there are scalars c2 > c1 > 0, positive definite matrixes P, Z   m , Qn, Yn, S (m = 1, 2, 3; n = 1, 2) T ¯ ρ ⊂ ` F˜i any matrices N1 , N2 , L with appropriate dimensions, = E¯ P˜ E, and scalars d2 > d1 ≥ 0, τ ≥ 0, h1 > 0, h2 > 0 such that the following conditions hold for i, j ∈ R ¯ ii < 0, Ω (26)

¯ ij + Ω ¯ ji < 0, Ω

(27)



(28)



 Q2 

S   ≥ 0, ∗ Q2 14

!

d3 d3 τ2 λ2 + d1 λ3 + d2 (λ4 + λ5 ) + 1 λ6 + 12 λ7 + τ λ8 + λ9 c21 +δη 2 < λ1 e−δT c22 , 2 2 2 (29) 

2  − {z2,M }ξ



where



∗

¯ ij = Ω



ϕ E¯ T Q1 E¯  11 ¯ ij = Ξ

              



−I

¯ + Γ d1 ΠT d12 ΠT Ξ 1 1  ij               



√ Pˆ γ {C2ij }Tξ 

< 0,

ΠT1

TT

0

0

0

0

0 0

∗

−Q−1 1

∗

∗

−Q−1 2

∗

∗

∗

− τ1 Y2−1

∗

∗

∗

∗

∗

∗

∗

∗

(30)



ΠT2   0    

0  

,

 0   

− τ1 Y2 0   ∗

−I



Pˆ T Aids

0

Pˆ T Aiτ s

T V Pˆ T Bωis − C1is

∗

ϕ22

E¯ T (−S + Q2 ) E¯

E¯ T S E¯

0

0

∗

∗

ϕ33

E¯ T (−S + Q2 ) E¯

0

∗

∗

∗

−Z3 − E¯ T Q2 E¯

T −C1ids V

0

0

∗

∗

∗

∗

− (1 − h2 ) Y1

0

∗

∗

∗

∗

∗



Π2 = U¯ C1is 0 U¯ C1ids 0 0 U¯ Dωis

T

T − (R − αI) − Dωis V − V Dωis

.

Then the closed-loop system (5) is regular, impulse free and singular finite-time bounded subject to (c21 , c22 , η 2 , Tc , α, Rc ) and respect  to (U, V, R) − α dissipative ¯ ρ contained in the performance level α > 0 at the origin with = E¯ T P¯ E, domain of attraction, and the hard constraints concerning control output (8) are satisfied. Moreover, the estimated of the attraction domain is Γθ ≤ ρ, where 

!

∂2 d3 d3 ∂22

Γθ = (λ2 + d1 λ3 + d2 (λ4 + λ5 )) 1 + 1 λ6 + 12 λ7

¯ T Rc E¯

kRc k 2 2

E 

∂2 τ2 ∂22

 c2 . +τ λ8 1 + λ9

kRc k 2 E¯ T Rc E¯

1 15



        ,       

Proof: Now we prove the closed-loop system (5) is singular finite-time (U, V, R)− α dissipative bounded. From (27) and by using Schur complement lemma, we can obtained that V˙ (t) − δV (t) − z1T (t) U z1 (t) − 2z1T (t) V ω (t) − ω T (t) (R − αI) ω (t) n o ¯ ij + Γ + d2 ΠT Q1 Π1 + d2 ΠT Q2 Π1 + τ ΠT Y2 Π1 + τ T T Y2−1 T + ΠT Π2 ζ (t) . ≤ζ T (t) Ξ 1

1

12

1

1

2

(31)

Similar as the method in Theorem 1 and consider the zero initial condition V (0) = 0, it can be obtained that 0 ≤ e−δT V (T ) < V (0) + =

Z T 0

Z T

h

0

h

i

e−δt z1T (t) U z1 (t) + 2z1T (t) V ω (t) + ω T (t) (R − αI) ω (t) dt i

e−δt z1T (t) U z1 (t) + 2z1T (t) V ω (t) + ω T (t) (R − αI) ω (t) dt.

Then, we have α

Z T 0

Z Th

T

ω (t) ω (t) dt <

0

i

z1T (t) U z1 (t) + 2z1T (t) V ω (t) + ω T (t) Rω (t) dt. (32)

From Definition 3, we know that the closed-loop system (5) is singular finitetime (U, V, R) − α dissipative bounded. Next, the system hard constraints concerning control output (8) are satisfied. According to the LKF(13). It is clear that 

x¯ (t) Pˆ T E¯ x¯ (t) ≤ V (t) < e−δt V (0) + T

Z t 0

h

e−δs z1T (s) U z1 (s) + 2z1T (s) V ω (s) + ω T (s) (R − αI) ω (s)

Through Young’s inequality, we obtain 

x¯T (t) Pˆ T E¯ x¯ (t) < e−δt V (0) + n

Z t 0

e−δs z1T (s) (U + V ) z1 (s) ds +

Z t 0

ω T (s) (V + R − αI) ω (s)ds o

≤ e−δt V (0) + λmax (U + V ) zmax + λmax (V + R − αI) η 2 , ∆

where zmax = maxt>0

Rt T z 0

1

(s) z1 (s) ds.

Define γ = e−δt {V (0) + λmax (U + V ) zmax + λmax (V + R − αI) η 2 }. Then, it is easy to see that the following inequalities are established: 2 max {z2 (t)}ζ t≥0

< γλmax

r X i=1

!

1 1 λi Pˆ − 2 {C2is }Tζ {C2is }ζ Pˆ − 2 , ζ = 1, 2, · · · q.

16



1 1 Thus, the constraints in (8) are established if γ Pˆ − 2 {C2is }Tζ {C2is }ζ Pˆ − 2 < {z2,M }2ζ I. This completes the proof.

Finite-time (U, V, R) − α dissipative controller design

3.2

In the following, we deal with the problem of dynamic output feedback finitetime (U, V, R) − α dissipative controller design for the system (5) by using LMI and optimization approaches. Theorem 3 For given positive constants c1 , η, Tc , δ and positive definite matrix Rc = diag {Rc1 , Rc1}, if there are scalars c > c1 > 0, positive definite ma   2  ˜ ˜ ˜  Zm1 0  ˜  Qn1 0  ˜  Yn1 0  , Q = , Y = trices Z˜m =     n   (m = 1, 2, 3; n = 1, 2), n ˜ n2 0 Z˜m2 0 Q 0 Y˜n2 















˜ ˜ ˆ ˜  N11 0  ˜  N21 0  ˆ  N11 0   S11 0  ˜ S=  , N2 =   , N1 =  ,  , N1 =  ˜12 ˜22 ˆ12 0 N 0 N 0 N 0 S˜12 



ˆ  N21 0  ˆ N2 =  , nonsingular matrices X, G, Φ, and scalars d2 > d1 > 0, τ > ˆ22 0 N 0, h1 > 0, h2 > 0, ρ > 0, γ > 0, α > 0, εij > 0, χij > 0, σ > 0, σk > 0 (k = 1, 2 · · · 7) such that C3j X = GC3j , 

−1

 −ρ    ∗  

∗

f¯ik −EX ∗ r X

(33) 

−f¯ik   0

−EX

   

≤ 0,

(34)

ˆ ii < 0, Ω

(35)



(36)

i=1

r X r  X i=1 j=1

ˆ ij + Ω ˆ ji < 0 (i 6= j, ) Ω





˜ ˜  Q21 0 S11 0     ˜ 22 0 S˜12   ∗ Q      

∗ ∗

 

˜ 21 0  ∗ Q   ˜ 22 ∗ ∗ Q 17

≥ 0,

(37)



2  − {z2,M }ζ

    

σI2n <

X

√

0

γX

{C2i }Tζ

∗

− {z2,M }2ζ X

0

∗

∗

−I

    1   I2r Rc2 M −1   









      

< 0,

(38)

I2r   ˆ T  0   0 M E¯ XM  −T 12  M Rc < I2n , 0  0

Φ

(39)



ˆ > σ3−1 Rc−1 , −Q ˜ 1 + 2X ˆ > 1 σ4−1 Rc−1 , ˆ > σ2−1 Rc−1 , −Z˜3 + 2X ˆ > σ1−1 Rc−1 , −Z˜2 + 2X −Z˜1 + 2X 2 ˜ 1 + 2X ˆ > 1 σ5−1 Rc−1 , −Y˜1 + 2X ˆ > σ6−1 Rc−1 , −Y˜2 + 2X ˆ > 1 σ7−1 Rc−1 , −Q 2 2 (40)



2 2 −δT c1  δη − c2 e

                         



d 1 c1

d 2 c1

d2 c1

d1 c1

d12 c1

τ c1

τ c1   0  

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

∗

−σ

∗

∗

−d1 σ1−1

∗

∗

∗

−d2 σ2−1

∗

∗

∗

∗

−d2 σ3−1

∗

∗

∗

∗

∗

−1 −d−1 1 σ4

∗

∗

∗

∗

∗

∗

−1 −d−1 12 σ5

∗

∗

∗

∗

∗

∗

∗

−τ σ6−1

∗

∗

∗

∗

∗

∗

∗

∗







    ,       

           

    ,       

0

                      

< 0,

−σ7−1 (41)

where 



ˆ1 ˆ2  Ωij Ωij  ˆ Ωij =  , ˆ3 ∗ Ω ij



ˆ +Γ ˆ Ξ  ij ˆ1 = Ω ij

              

ˆT d1 Π 1

ˆT d12 Π 1

ˆT Π 1

TˆT

∗

˜ 1 − 2X ˆ Q

0

0

0

∗

∗

˜ 2 − 2X ˆ Q

∗

∗

∗

∗

∗

∗

∗

∗

∗

1 τ



0 ˆ Y˜2 − 2X ∗ ∗

18



0 0

ˆT Π 2   0   0

0

− τ1 Y˜2 0 ∗

−I

ˆ ij = Ξ

ϕˆ ϕˆ ϕˆ 0 ϕˆ15 ϕˆ16   11 12 13    ∗ ϕ ˆ22 ϕˆ23 ϕˆ24 0 0    ∗

∗ ϕˆ33 ϕˆ34 0

0

∗

∗

∗ ϕˆ44 0

0

∗

∗

∗

∗ ϕˆ55 0

∗

∗

∗

∗

∗ ϕˆ66



ˆ111 ϕ

ϕˆ11 = 

δEX,



ϕˆ211

 1 ˆ11 ,ϕ

∗ ϕˆ311

  3 P T ˜11 ˜ 11 E T + Y˜11 − ET + = sym Ai X + N Z˜m1 − E Q m=1

    3 T ˆT ˆ j + E − Fˆj , ϕˆ3 = sym Aˆlj + N ˜ T E T + P Z˜m2 − ϕˆ211 = C3i Hlj + Bi Es K s 11 12 

˜ 12 E T +Y˜12 −δEX2 , ϕˆ12 =  EQ 



˜T

 −N11 E  

0

˜ 12 E T EQ

∗





˜21 Biτ Es K ˆ j + Es− Fˆj + EN  , T ˜22 ˜12 ET + EN ∗ −N

T

 Bωi

ϕˆ16 = 

˜ 11 E EQ

T

∗

−X

T







˜ 11 + EL 

˜ 12 EL



ˆ22 ,ϕ

T ˜ ˜  E −S11 + Q21 E

ϕ˜23 = ϕ˜34 =  

T V C1i

∗





ϕˆ44 =  

˜ 21 E T −Z˜31 − E Q ∗



= 

0  ˆ15 = ,ϕ A˜ljd

∗

∗

0



˜ 22 E T E −S˜12 + Q 

0 ˜ 22 E T −Z˜32 − E Q

˜ ˜  − (1 − h2 ) Y11 + sym −E N21

ϕˆ55 = 

=

T T ˜ ˜ ˜  −Z11 − E Q11 E − E Q21 E

∗



m=1

 Aid X

 ˆ13 ,ϕ

T ˜ ˜ ˜ T  − (1 − h1 ) Z21 − 2E Q21 E + sym E S11 E

ϕˆ33 = 









 ˆ  , Π2







 ˜24 ,ϕ



0 ˜ 12 E T − E Q ˜ 22 E T −Z˜12 − E Q

˜ T  E S11 E

=

∗

0 E S˜12 E T 0

0



 ,





Ω2ij =  

T H1i

000

0

00000

E1i X 0 0 0 E2i X 0 0 0 0 0

T −H2i V

T d1 H1i

0

T d12 H1i

E3i

0

0

0

0

T H1i

000

0 0 000

T ¯ H2i U

0



 .

Then the closed-loop system (5) is singular finite-time bounded with respect to (c21 , c22 , η 2 , Tc , α, Rc ) subject to (U, V, R) − α− dissipative performance level α > 0 and the hard constraints concerning control output (8) are satisfied. Moreover, the suitable state feedback controller gain and dynamic output feed19





= U¯ C1i X 0 0 0 U¯ C1id X 0 0 0 0 0 U¯ Dωi ,

˜11 0 0 0 0 0 0 0 N ˜21 0 L ˜ 11 n o N  −1 Tˆ =    , Ω3ij = diag −εij I, −εij I , ˜12 0 0 0 0 0 0 0 N ˜22 L ˜ 11 0 N 

 ,

˜ 22 E + sym E S˜12 E − (1 − h1 ) Z˜22 − 2E Q T

˜22 ∗ − (1 − h2 ) Y˜12 + sym −E N ϕˆ66 =− (R − αI) − sym (V Dωi ) ,      −ˆ −ˆ ˆ ˆ B F E K + E 0 0 A X 0 0 0 0 B F E K + E A X B ωi  s j id iτ s j i i s j s j ˆ1 =  Π ,  ˆ lj C3i H Aˆlj 0 0 0 Aˆljd 0 0 0 0 0 



T



 ,



 ,

back controller gains are given as follows: ˆ j X −1 , Ali = Aˆli X −1 , Alid = Aˆlid X −1 , Hli = H ˆ li G−1 . Kj = K ˜ θ ≤ ρ, where Furthermore, the estimated of the attraction domain is Γ ˜θ = Γ

" 

σ −1 + d1 σ1 + d2 (σ2 + σ3 ) 

∂22



 c2 . +τ 2 σ7

¯ T Rc E¯

1

E

  ∂2 ∂22 ∂12

+ τ σ6 1 + d31 σ4 + d312 σ5

¯ T Rc E¯

kRc k kRc k

E

(42)

˜ S˜T is nonsingular. Proof: From Theorem 2, ϕ11 < 0, we know Pˆ = P˜ E¯ + R ^ ^ ^T ^ ^ ˆ = Pˆ −1 = E¯ P + RS , and P > 0, R ∈ R2n×2(n−r) is any matrix There exists X ^ with full column rank and satisfies E¯ R = 0. Next, we can obtain ^

ˆ =X ˆ T E¯ T = E¯ P E¯ T ≥ 0. E¯ X

(43)





ˆ 11 X ˆ 12 X  ˆ T = By denoting N −1 XM  and combining (43), it is easily obtained  ˆ ˆ X21 X13 ˆ 12 that X





ˆ 11 0 X  ˆ 11 is symmetric, then we have N −1 XM ˆ T = = 0, and X , so  ˆ ˆ X21 X13

ˆ −1 N = ˆ ˆ X Accordingly, we can that M −T X  11 and X22 are nonsingular.   conclude    −1 ˆ 11 X 0  I2r   ˆ 11 are nonsingular. ¯ XM ˆ T   = X   and I2r 0 M E −1 ˆ −1 ˆ −1 ˆ ˆ −X22 X21 X11 X22 0 Then, we have 

   I2r   N −T    I2r 

0



 I2r 

=N −T  

0

ˆ X 11

−1

=N −T N T E¯ T M T ˆ −1 = E¯ T Pˆ . =E¯ T X 



X 0 







 I2r  

ˆ T 0 M E¯ XM 

0

−1 −T I2r 0 N = N



−1



ˆ −1 N N −1 M −T X 









−1 I2r 0 N



ˆ −1  X11 0  −1  N 0 0

(44)

P 0  ˆ = Now, we let X  and Pˆ =  ,X = P −1 , pre- and post- multi∗ X ∗ P

20

  

plying (34) by diag P T , · · · , P T , {z } |   

10

ˆ i = Ki X, Aˆli = I, · · · , I, P T , P T , I, I, I  and its transpose, respectively. Denoting K |

{z 7

}



ˆ li = Hli G, and considering Schur complement and Lemma Ali X, Aˆlid = Alid X, H 4, then (35) is the sufficient condition to (26). Next, pre- and post- multiplying (38) by diag {P, P, I} and its transpose respectively, then (38) is equivalent to (30). Let

P¯ =

then we have

1 2

1 2

    −1   I2r Rc 2 M T   

¯ c P¯ Rc E¯ = ER

I2r  ˆ T   0 M E¯ XM 0

1 ¯ c Rc− 2 M T N −T N T ER

= N −T



      



 

0  

Φ−1

0

1 2



−1









− 12

 M Rc  

−1

I2r  ˆ T   I2r 0 M E¯ XM 0 0

ˆ −1  X11 0  −1 ˆ −1 . = E¯ X N  0 0 1

,

(45)

 

0  

Φ−1

(46) 1

Therefore, we can derive that E¯ T Pˆ = E¯ T Rc2 P¯ Rc2 E¯ holds. From (39), we have I2n < P¯ < σ1 I2n , and then λ1 > 1, λ2 < σ1 . On the other hand, it follows from (40) that λ3 < σ1 , λ4 < σ2 , λ5 < σ3 , λ6 < 2σ4 , λ7 < 2σ5 , λ8 < σ6 , λ9 < 2σ7 , then the following inequality is a sufficient condition for (41): h

i

σ −1 + σ1 d1 + (σ2 + σ3 ) d2 + σ4 d31 + σ5 d312 + σ6 τ + σ7 τ 2 c21 + δη 2 < c22 e−δT . (47) 







¯ ρ ⊂ ` F˜i and since the system (5) is regular and Then, from = E¯ T P˜ E, ˆ ˆ ˆ ¯ˆ impulse free, there exist two other   matrices M and N such that M E N =   ˜ ˜  I2r 0  ˆ −T ˜ ˆ −1  P11 P12  ˜ ˆ ˆ −1 x¯ (t) = ˜ ˜ =   ,M PM  with Fi N = Fi1 Fi2 and N ˜ 0 0 ∗ P22 21

− 12

 M Rc  

1

¯ N −1 Rc2 EN





¯1 (t)  x .  x¯2 (t)

It follows that Fˆi2 = 0, otherwise, let x¯1 (t) = 0 and |fi2k x2 (t)| >

1 2

˜ ¯ x¯ (t) = 0, that is |fik x (t)| > ρ 12 , it contradicts that ρ , then x¯T(t) E¯ P  E ¯ ρ ⊂ ` F˜i . Then, = E¯ T P˜ E, x¯T (t) E¯ P˜ E¯ x¯ (t) = x¯T1 (t) P11 x¯1 (t) = F˜1i x¯1 (t) ≤ ρ. 







f˜i1k 0

(48)



−1 ˜T ¯ ρ ⊂ ` F˜i is obtained by f˜i1k P11 fi1k ≤ Therefore, the condition = E¯ T P˜ E, 1 , k = 1, 2, · · · , l. Considering Schur complement lemma, we have ρ



1

 −ρ 



f˜i1k 

∗ −P11



≤



1  −ρ

  0, or   

∗





     I2r 0   P11 P12   I2r 0   −     

0 0

T 

∗ P22



≤ 0, k = 1, 2, · · · , l.

0 0

(49)

where f˜i1k is the k-th row of F˜i1 . n

ˆ −T ˜TN Then, pre- and post- multiplying inequality (48) by diag 1, X transpose, we can obtain 

1  −ρ



∗



o

and its



f¯1ik −f¯1ik   ≤ 0, ˜ −E¯ X

which is equivalent to 

−1

 −ρ    ∗  

∗

f¯ik −EX ∗



−f¯ik  0

−EX

    

≤ 0,

where f¯ik is the k-th row of F¯i1 . Moreover, it follows from (40) that λ3 < σ1 , λ4 < σ2 , λ5 < σ3 , λ6 < 2σ4 , λ7 < ˜ θ ≤ ρ, 2σ5 , λ8 < σ6 , λ9 < 2σ7 , and the estimated of the attraction domain is Γ ˜ θ > Γθ . where Γ 22

Γθ <

" 



σ −1 + d1 σ1 + d2 (σ2 + σ3 ) ∂22



 c2 . +τ 2 σ7

¯ T Rc E¯

1

E

  ∂12 ∂2 ∂22

+ τ σ6 1 + d31 σ4 + d312 σ5

¯ T Rc E¯

kRc k kRc k

E

This completes the proof.

Remark 4 It may be difficult to solve the condition (33) by using LMI toolbox of Matlab. To solve this problem, we can replace the condition by the following one that may approximate this constant: [C3i X − GC3i ]T [C3i X −GC3i ] < ϑI, where ϑ is a given sufficiently small positive constant. By using Schur complement lemma, the constrained condition (33) is equivalent to the following LMI: 



T  −ϑI (C3i X − GC3i ) 



∗

−I



< 0.

(50)

Remark 5 For descriptor time-varying delay T-S fuzzy systems with actuator saturation, dynamic output feedback controller gains are derived by solving LMIs. Moreover, for the sake of the least conservative estimate of the attraction domain, we may choose the largest ellipsoid in which all the ellipsoids satisfy the set invariance condition of Theorem 3. Next, we will obtain the largest invariant ellipsoid for the system (5) through an LMIs-based optimization algorithm. For Theorem 3, an exact invariant set with least degree of conservativeness can be formulated as follows: min κ

s.t.

    (a)λ2   

< σ −1 , λ3 < σ1 , λ4 < σ2 , λ5 < σ3 , λ6 < 2σ4 , λ7 < 2σ5 , λ8 < σ6 , λ9 < 2σ7 ,

(b)inequality (34)      

(c)inequalities (35 − 41, 49)

(51)

where κ=

h

σ

−1



+ d1 σ1 + d2 (σ2 + σ3 ) kRc k

−1

−1



+τ σ6 kRc k +

d31 σ4

+

d312 σ5

Meanwhile, the estimate of the attraction domain is given by Γmax =



¯T ¯

−1

E Rc E

q

ρ/κmin .

Remark 6 We find that conditions (35), (36), (41) are not strict LMIs. However, once the parameter δ is fixed, we can turn the conditions into the LMI23

+τ

2

−1 

¯T

¯ σ7 E Rc E

based feasibility problem. Thus, the feasibility of conditions stated in Theorem 3 can be turned into the following feasibility problem with a fixed parameter δ, respectively:

min c22 + α ˜ i (i = 1, 2) , Y˜i (i = 1, 2) , σ, δ X, Z˜i (i = 1, 2, 3) , Q

(52)

s.t.LMIs (34) − (41) , (49) .

Corollary 1 For positive constants c1 , η, Tc , δ and positive definite matrix Rc , ˆ m (m = 1, 2),Yˆm (m = 1, if there are scalars c2 > c1 > 0, positive definite matrixes P, Zˆm (m = 1, 2, 3),Q ˆ1 , N ˆ2 , L ˆ with any matrices N dimensions, scalars d2 > d1 ≥ 0, τ ≥  appropriate    T ˜ 0, h1 > 0, h2 > 0 and = E P E, ρ ⊂ ` Fi , such that the following conditions hold for i, j ∈ R: ¯ = GC ˆ 3i , C3i X (53) 



f¯ik

−1

 −ρ 

 

T ¯ T −E XE f¯ik

≤ 0,

(54)

_

Ωii < 0, _

(55)

_

Ωij + Ωji < 0 (i 6= j) , 

 

(56)

ˆ ˆ  Q2 S   ≥ 0,  ˆ2 ∗ Q

(57) 

2 ¯ √ ¯T T  − {z2,M }ξ X γ X {C2i }ξ 



σIn <

    1  Ir Rc2 M −1    

∗

−I 





 Ir 

ˆ EX ˜M ˆT  0 M 0

0





< 0,  

0 

ˆ Φ

(58)

1

 M −T Rc2  

< In ,

(59)

¯ >σ ¯ >σ ¯ >σ ˆ 1 + 2X ¯ > 1σ −Zˆ1 + 2X ˆ1−1 Rc−1 , −Zˆ1 + 2X ˆ2−1 Rc−1 , −Zˆ1 + 2X ˆ3−1 Rc−1 , −Q ˆ −1 Rc−1 , 2 4 ˆ 1 + 2X ¯ > 1σ ¯ >σ ¯ > 1σ ˆ −1 Rc−1 , − Yˆ1 + 2X ˆ −1 Rc−1 , −Q ˆ6−1 Rc−1 , −Yˆ2 + +2X 2 5 2 7 (60) 24



2 2 −δT c1  δη − c2 e

                         



d 1 c1

d 2 c1

d2 c1

d1 c1

d12 c1

τ c1

τ c1   0  

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

∗

−ˆ σ

∗

∗

−d1 σ ˆ1−1

∗

∗

∗

−d2 σ ˆ2−1

∗

∗

∗

∗

−d2 σ ˆ3−1

∗

∗

∗

∗

∗

−d−1 ˆ4−1 1 σ

∗

∗

∗

∗

∗

∗

−d−1 ˆ5−1 12 σ

∗

∗

∗

∗

∗

∗

∗

−τ σ ˆ6−1

∗

∗

∗

∗

∗

∗

∗

∗

0

                      

< 0,

−ˆ σ7−1 (61)

where  _1

_

Ωij =  

_

_

 Ξij + Γ _1

Ωij =



              

ˆ 1E T ϕ EQ  11 _

Ξij =

              

_T

_2



Ωij Ωij  _3  , ∗ Ωij

_T

_T

_T

d1 Π1

d12 Π1

Π1

∗

ˆ 1 − 2X ¯ Q

0

0

0

∗

∗

ˆ 2 − 2X ¯ Q

0

0

∗

∗

∗

∗

∗

∗

∗

∗

∗

_



¯ Aid X



1 τ



¯ Yˆ2 − 2X ∗ ∗

0





Π2   0    

0  

,

 0  

0



− τ1 Yˆ2 0   ∗

−I



¯ Aiτ X

¯ T CT V Bωi − X 1i

0

0

0

¯ T CT V −X 1id

0

0 0

∗

ϕ22

ˆ2 ET E −Sˆ + Q

∗

∗

ϕ33

∗

∗

∗

ˆ 2E T −Zˆ3 − E Q

∗

∗

∗

∗

− (1 − h2 ) Yˆ1

∗

∗

∗

∗

∗

_

_

ˆ E SE

_T

T

T





ˆ2 ET E −Sˆ + Q

25

T − (R − αI) − Dωi V − V Dωi



        ,       



 _

Γ=

n

ˆ T ET + EN ˆ1 0 0 0 −N ˆ T ET + EN ˆ2 E T L ˆ N 1   1    ∗ 000 0 0               



∗

∗00

∗

∗∗0

∗

∗∗∗

∗

∗∗∗

_ ¯ + Bi Es K ¯ j + Es− F¯j ϕ11 = sym Ai X

ˆ 1E T − E Q ˆ 2E T , −Zˆ1 − E Q 

_

o

∗

+

3 P

i=1

0

_ ˆ 1 E T + Y1 − δE X, ¯ ϕ Zˆi − E Q 22 =





_ ˆ 2 E T + sym E SE ˆ T , ϕ33 = − (1 − h1 ) Zˆ2 − 2E Q



¯ + Bi Es K ¯j + Π1 = Ai X _

  0 0   ,  0 0    T T T ˆ E − EN ˆ2 −E L ˆ −N  2 



Es− F¯j





¯ 0 Biτ Es K ¯j + 0 Aid X

¯ 0 U¯ C1id X ¯ 0 0 U¯ Dωi Π2 = U¯ C1i X 

T

_

3

n

Es− F¯j





Bωi , o

, Πij = diag −ε−1 ij I, −εij I , 

T T T T T T ¯  H1i 0 0 0 0 −H2i V d1 H1i d12 H1i H1i 0 H1i U  Πij =  . ¯ 0 E2i X ¯ 0 0 E3i E1i X 0 0 0 0 0 _

2

Then the T-S fuzzy system (3) is singular finite-time bounded with respect to (c21 , c22 , η 2 , Tc , α, Rc ) subject to (U, V, R)−α− dissipative performance level α > 0 and the hard constraints concerning control output (8) are satisfied. More¯jX ¯ −1 . over, the suitable state feedback controller gain are given as: Kj = K ˆ θ ≤ ρ, where Furthermore, the estimated of the attraction domain is Γ ˆθ = Γ

"



∂22



 c21 . +τ 2 σ ˆ7

T ¯ ¯

E Rc E

4



σ ˆ −1 + d1 σ ˆ1 + d2 (ˆ σ2 + σ ˆ3 )

  ∂12 ∂22 ∂2

+ τσ ˆ6 1 + d31 σ ˆ4 + d312 σ ˆ5

¯ T Rc E¯

kRc k kRc k

E

(62)

Numerical examples

In this section, we consider two examples to demonstrate the effectiveness and feasibility of the proposed methods in this paper. Example 1 26

It is more convenient to express practical model as the singular systems than ordinary ones such as a biological economic model with differential and algebraic equation established in [19]. Thus, we consider the following nonlinear descriptor system with time-varying delay:

    ς˙1 (t)          ς˙ (t)   2              

!

c αβ ηc − ς1 (t) + ας2 (t) − ς3 (t) − ης12 (t) − ς1 (t) ς3 (t) + d11 w (t) + τ11 ς1 (t − d (t)) , = − r2 p p =βς1 (t) − r2 ς2 (t) , ! αβ ηc 0 =p − r1 − β − ς1 (t) + pς1 (t) ς3 (t) + b11 u (t) , r2 p z (t) =c11 ς1 (t) , y (t) =c21 ς1 (t) + c22 ς2 (t) + c23 ς3 (t) . 

Defining x (t) = ς1 (t) ς2 (t) ς3 (t)             

E x˙ (t) = z (t) =

           

y (t) =

2 X

i=1 2 X

i=1 2 X

T

, the above equation can be rewritten as:

λi (ς1 (t)) {Ai x (t) + Aid x (t − d (t)) + Bi u (t) + Bωi ω (t)} , λi (ς1 (t)) C1i x (t) , λi (ς1 (t)) C2i x (t) ,

i=1

where

A1 =



      

p

E=



1   0  

− αβ r2

+ ηl

β αβ r2

−



0 0 1

−

ηc p

ηc p

  , A1d 0  

− r1 − β

α 

= A2d =

000

− pc

+l

−r2

0

0

−pl



 τ11    0  



0 0

0 

   , A2  

  , B1 0  

0 00





=



      

= B2 =

p

      

− αβ r2

−

ηc p

− ηl

β αβ r2

− 

0  0

b11

ηc p

− r1 − β

   , Bω1  

α 

= Bω2 =

C11 = C12 = c11 0 0 , C21 = C22 = diag {c21 , c22 , c23 } .

− pc

0

0

−pl 

 d11       0 ,    

0

For given α = 0.15, β = 0.5, r1 = 0.2, r2 = 0.2, η = 0.001, p = 1,c = 220, τ11 = 2.0034·10−5 , d11 = −4.2344×10−5 , b11 = −0.0725, c11 = 0.0317, c21 = 0.27, c22 = −0.4, c23 = −1 and l = 220. Let U = −0.04I, V = 1.5I, R = 2, δ = 0.001, d1 = 0, d2 = 2.0, T = 2, η = 2, h1 = 0.3,c1 = 0.02. The fuzzy weighting functions 27

−l

−r2 



  ,  

Fig. 1. The restrictive relationship among three variables δ, α and c2 are considered as h1 (x1 ) = 1/[1 + exp (0.5 (x1 + 1))], h2 (x2 ) = 1 − h1 (x1 ). The disturbance input is considered as w (t) = exp(−t) sin(−t). By applying MATLAB to solve the LMIs in Theorem 3, we have α = 0.0984,c2 = 6.3067. The Fig. 1 shows the restrictive relationship among three variables δ, α and c2 . We can obtain the designed controller gains and dynamic output feedback gains as follows:









K1 = 0.2827 - 0.2716 315.7468 , K2 = 0.3062 - 0.1261 206.5465 ,

Al1 =

      

Al1d =

Hl1 =

      

- 0.3917 - 0.0079 - 0.0157 - 0.3426 - 1.1419 0.3823 

 0.0010    −0.0004  

- 6.5950  - 2.3325

- 679.1151 

0.0022

- 0.0112 0.1041

1.3419

   , Al2  

−0.0004 0.0023 

  , Al2d −0.0031   

−0.0000 −0.0000 0.0000

0.2957





- 74.6244   

      

=

=



- 0.3903 - 0.0098 - 0.0122 - 0.3429 - 1.1188 0.3794

 0.0010    −0.0004  

, Hl2 = - 0.3246 - 43.6764    0.3024 - 224.9522



- 3.4207   

, - 0.4050    - 667.9229 

−0.0004 0.0023   

, 0.0022 −0.0031    −0.0000 −0.0000 0.0000

      

- 0.3365 −0.0052 - 28.5034   0.0752

0.3933

28



. - 0.3979 - 12.8546    0.0859 - 67.8812

As a result, it is easy to observe that the methods proposed in this paper feasible for solving the problem of biological economic model. The method proposed in this paper is feasible for solving the bioeconomic model problem. Example 2



Consider the following uncertainty T-S fuzzy singular time-delay system with two fuzzy rules:     E x˙ (t)                    z1 (t)                        

=

2 X

λi (ε (t)) {(Ai + ∆Ai ) x (t) + (Aid + ∆Aid ) x (t − d (t)) + (Bωi + ∆Bωi ) ω (t) + Bi sat (u (

2 X

λi (ε (t)) {(C1i + ∆C1i ) x (t) + (C1id + ∆C1id ) x (t − d (t)) + (Dωi + ∆Dωi ) ω (t)} ,

i=1

+Biτ sat (u (t − τ (t)))} ,

=

z2 (t) = y (t) =

i=1 2 X

i=1 2 X

λi (ε (t)) C2i x (t) , λi (ε (t)) C3i x (t) ,

i=1

x (t) =φ (t) , t ∈ [−d2 , 0] .

where

E=



1   0  



0 0 1

  , A1 0  

= A2 =

Bω2 =

1   0  

0

0 0 0.1

C11d =



 0.4    0  

0



0

   , B1  

1 0.1 

−2 1

= B1τ =

0 0.2

   , A1d  





 0.5       1  , B2    

= A2d =

  , C12d  

=





 0.5    0  

0

0

= B2τ =

0 0.1 

  , Dω1  



1

0.1 

1



 0.1       1  , C11    





−0.01 0



 0.3    0  

0







= Dω2 =

 0.2 0 0.3 

0 

 −0.1    0  

=



0.1 

1   0  

   , Bω1  

=





0.1

  , C12 0  

0



0  

0.1 0

0 0.5

  , C21  

=



     C22 =   0 0.1 0  , C31 = C32 =  0.1 0.1 0  , Hij =  0.1  (i, j = 1, 2) ,             0 0 0.3 0 0 1 0.1 







E11 = E21 = E31 = 0.1 0.1 0.1 , E12 = E22 = E32 = 0.1 0.1 0.1 .

Let U = diag {−0.01, −0.01, −0.01} , V = diag {0.1, 0.1, 0.1} , R = diag {2, 4, 3}. For given parameters ρ = 20, c1 = 1, η = 1, d1 = 0.1, τ = 0.8, h1 = 0.1, h2 = 1.2, ϑ = 10−9 , Tc = 1, Rc1 = I3 , γ = 3.0,ε11 = ε12 = ε21 = ε22 = 3 . The output constraint is z2,M =



z1,max z2,max z3,max

29

T

 0.1    0  

0 0

 0.1 





, z1,max = 2.0 , z2,max =



1   0  



0 0  

, 0.1 0    0 0 1

=

0 0 0

0.4



0  

0.2 0



0.1

0.2



0  

0.1 0

 0.1 0 



0 

0.1 0

 −3    0  



0.3 0 −4

000 





 0.1    0  

0



0 1   

, 1 0    0 0 0.1 

0   

, 0.2 0    0 0 0.1

1.5 , z3,max = 2.5. By solving the LMIs optimization problem (51), the estimated domain of attraction for different upper delay d2 for the closed-loop system are obtained and shown in Table 1. From the Table 1, we can know that the bigger upper delay d2 , the smaller Γmax . Table 1 Comparison of κmin and Γmax with different d2 for Example 2. d2

0.106

0.2

0.5

1.0

2.0

2.3

κmin

1.30955

1.35966

1.61357

2.35962

6.57883

Infeasible

Γmax

3.90799

3.83531

3.52064

2.91134

1.74357

Infeasible

It should be noted that the parameter δ has an important influence on the optimal bound with minimum value of c22 +α. For given upper delay d2 = 0.106 and by combined with Theorem 3, the feasible solution can be obtained when 10−5 ≤ δ ≤ 3.97. By solving the LMI optimization problem (51) and (52), we can get the results κmin = 1.30955 and δmin = 10−5 . Considering δ = 0.01, we can obtain the optimal value α = 0.0046, c2 = 3.2404. The designed controller gains and dynamic output feedback controller gains can be obtained as follows: 



K1 = −0.0118 0.0112 0.0165 , K2 =

Al1 =



 −0.1802    −0.0430  



−0.0134 −0.0012  −0.1426 0.0007

−0.0201 0.0041 −0.1382



 −0.0020   Al1d =   −0.0000 

0.0000

Hl1 =



 −43.9499    −14.9813  

9.7862

   , Al2  



0.0010 −0.0011 −0.0017

=

      



0.0107 0.0041

0.0000 0.0000



11.2103 13.1487  

−33.4751 4.4973

−2.6733 −4.6499

  ,Hl2   

=

      

 

0.0013 0.0000    0.0001   

0.0022

−0.0008 −0.0009 0.0359

 −0.0023     , A 0.0000  l2d =  −0.0001    

0.0000 0.0000 

−0.0057





−0.0001 0.0000   

−0.0064 0.0000    0.0000 0.0000 0.0000



8.2988 −1.5257 −2.4805   

5.1780 −1.3030    −1.1671 −0.4315 0.6885 4.3285

T



T

The initial states are assumed to be x0 = −0.1 0.1 0 and xˆ0 = 0.1 −0.1 0 . The fuzzy weighting function are considered as h1 (x1 ) = 1/[1 + exp (0.5 (x + 1))], h2 (x2 ) = 1 − h1 (x1 ) and the disturbance input is considered as w (t) = exp(−t) sin(t). Then we can get the following results. The state trajectories of the closed-loop system and dynamic output feedback controller under the 30

saturated control are shown in Fig. 2 and Fig. 3, respectively. The state trajectories of saturated control input and saturated control input-delay are shown in Fig. 4 and Fig. 5, respectively. Furthermore, Fig. 6 and Fig. 7 illustrate the trajectories of closed-loop system and dynamic output feedback controller starting from the initial condition under the four different kinds of disturbances and the attraction domain, which means the states are always in the ellipsoid. All of these verify the feasibility and superiority of our methods.

Fig. 2. The state trajectories of the closed-loop system

Fig. 3. The trajectories of the dynamic output feedback controller

Fig. 4. The trajectories of the saturated control input sat (u (t)) 31

Fig. 5. The trajectories of the saturated control input with time-varying delay sat (u (t − τ (t)))

Fig. 6. The trajectories of closed-loop system and the attraction domain

Fig. 7. The trajectories of dynamic output feedback controller and the attraction domain 5

Conclusion

In this paper, the problem of finite-time dynamic output-feedback dissipative control for singular uncertain T-S fuzzy system subject to actuator saturation and output constraints has been considered. By using the methods of 32

DPDC and LKF, the sufficient conditions of singular finite-time boundeness and singular finite-time strict dissipativity of the closed-loop system have been obtained. Furthermore, the dissipative desired controller gains and dynamic output-feedback controller gains have been obtained in terms of the LMIs by using the reciprocally convex approach and a new lemma. In addition, dissipative performance index α, finite-time index c2 and the attraction domain of the system are formulated and solved by LMIs optimization problem. Finally, a practical example and a numerical example are provided to validate the effectiveness and superiority of the proposed method.

6

Acknowledgement

This work is partially supported by the National Natural Science Foundation of China No. 61273004, and the Natural Science Foundation of Hebei province No. F2018203099.

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