Finite-time dissipative control for singular discrete-time Markovian jump systems with actuator saturation and partly unknown transition rates

Accepted Manuscript

Finite-time dissipative control for singular discrete-time Markovian jump systems with actuator saturation and partly unknown transition rates Yuechao Ma, Xiaorui Jia, Deyou Liu PII: DOI: Reference:

S0307-904X(17)30473-0 10.1016/j.apm.2017.07.035 APM 11885

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

17 March 2016 27 May 2017 17 July 2017

Please cite this article as: Yuechao Ma, Xiaorui Jia, Deyou Liu, Finite-time dissipative control for singular discrete-time Markovian jump systems with actuator saturation and partly unknown transition rates, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.07.035

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ACCEPTED MANUSCRIPT Highlights

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• A new control problem for discrete singular Markov system is investigated. • Lyapunov function method, LMI technique and convex optimization are used. • Both actuator saturation and partly unknown transition rates are considered. • The derived conditions are less conservative have wider application range.

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Finite-time dissipative control for singular discrete-time Markovian jump systems with actuator saturation and partly unknown transition rates

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Yuechao Ma a , Xiaorui Jia a,∗ , Deyou Liu a

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

Abstract

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In this work, the finite-time dissipative control problem is considered for singular discrete-time Markovian jumping systems with actuator saturation and partly unknown transition rates. By constructing a proper Lyapunov-Krasonski functional and the method of linear matrix inequalities (LMIs), sufficient conditions that ensure the systems singular stochastic finite-time stability and singular stochastic finitetime dissipative are obtained. Then, the state feedback controllers are designed, and in order to get the optimal values of the dissipative level, the results are extended to LMI convex optimization problems. Finally, numerical examples are given to illustrate the validity of the proposed methods.

Introduction

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Key words: finite-time dissipative control; discrete-time singular system; actuator saturation; partly unknown transition rates; linear matrix inequality.

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Singular systems, which also referred as implicit systems, descriptor systems, generalized systems, or differential systems, have increasingly attracted researchers for the fact that singular systems have been applied in many scientific areas. For instance, the power systems, the mechanical systems, economics, networks and so on [1-3]. In recent years, many research results are obtained, ? Project supported by National Science Foundation of China (No. 61273004), and the Natural Science Foundation of Hebei province (No. F2014203085) ∗ Corresponding author. Email address: [email protected] (Xiaorui Jia).

Preprint submitted to Elsevier

31 August 2017

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such as H∞ control problems [4-7], the problems of stabilization and stability [8-10], observer design problems [11] and so forth. Markovian jumping systems, which are subjected to random abrupt changes in system parameters caused by some inner discrete events in the system, are also have been extensively studied. For instance, Senthikumar etc. investigated the stability analysis problem for Markov jump system in [12]. Asynchronous filtering for Markov jump system was studied in [13,14]. Zhang etc. researched the finite-time control for discrete-time Markov system in [15], and so on. It should be pointed out that, there are also many results about singular Markovian jump systems. Ma and Zhang studied the robust stability and H∞ control for singular Markovian jumping systems in [16]. [17] studied the state feedback stability for singular Markovian jumping systems. The problem of exponential mean-square stability of time-delay singular systems with Markovian switching was researched in [18]. Long, etc, [19] considered the delay-dependent stochastic admissibility for singular Markovian jump time delay systems. See many other results in [20-22] and the references therein.

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When faced with the Markovain jumping systems, the transition probabilities of the jumping process are critical factors which determine the behavior of the systems. However, in many practical systems, the transition probabilities are always partly unknown. Therefore, it is significant and necessary to study the Markovain jumping systems with partly unknown transition rates. Up to now, there have been some results for this kind of system, such as stability and stabilization in [23], state feedback controller design in [24], robust non-fragile H∞ in [25], and many other results in [26-29] and the reference therein.

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During the past several decades, the problem of finite-time stability or boundedness has drawn increasing attention as people more and more prefer to consider the systems in a finite interval. Finite-time stability describes the system state does not exceed a certain bound during a fixed finite time interval and in recent years, many results of finite-time stability or boundedness are obtained. For example, Zhang etc. [30] studied the observer designing problem for singular stochastic systems. In [31], the problems of stochastic finite-time boundedness for Markovian jumping neural network were considered. The problem of finite-time control for singular Markovian jump systems with partly unknown transition rates was investigated in [32], For more results, readers can consult [33-37] and the references therein. Compared with passivity and H∞ performance, dissipativity is a more general criterion, which was put forward by Willems in [38]. Over the past decades, dissipativity theory turned out to be a very useful concept in systems and control theory. And there are many literatures about dissipativity control and analysis. Such as, the problem of finite-time non-fragile dissipative control for uncertain T-S system was considered in [39]. In [40], the observer-based dissipative control for networked control systems was investigated. Dissipative 3

ACCEPTED MANUSCRIPT control for singular T-S fuzzy systems was researched in [41] and [42]. [43] studied delay-dependent dissipative control for singular stochastic singular systems. Dissipative control for linear time-delay system in [44], α-dissipativity analysis for singular time-delay systems in [45] and so on.

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On the other hand, in practical applications, nearly all systems are subject to saturation constraints which may cause instability and bad performance of systems. Consequently, the controller design and stability analysis for systems with saturations have drawn much attentions of researchers during the past decades. For instance, Ma etc. studied the problem of H∞ control for discretetime singular Makovian jump systems subject to saturation in [46]. And the observer-based H∞ control for uncertain singular time-delay systems with actuator saturation was investigated in [47]. Finite-time H∞ control for singular saturated switched system in [48], L2 /L1 control for saturated singular systems in [49], control and and analysis for time delay saturation system in [50,51], observer-based stabilization for saturation system in [52] and so on. However, to the best knowledge of our authors, there are few results about finite-time dissipative control for discrete-time singular Markovian jump systems with actuator saturation and partly unknown transition probabilities.

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Motivated by the above discussion, in this paper, we consider the problem of finite-time dissipative control for discrete-time singular Markovian jump system with partly unknown transition probabilities and actuator saturation. The main contributions of this paper lie in the following aspects: Firstly, a novelty and multiple Lyapunov-Krasonski functional that contains the time-varying delay and its lower and upper bounds has been constructed, which includes more information and will lead to a less conservative result. Secondly, with some differential inequalities and the method of LMI, sufficient conditions that ensure the systems finite-time stable and finite-time dissipative are obtained. And the state feedback controllers are designed. Thirdly, the results are extended to LMI optimization problem so that the optimal values of maximum estimation of the attraction domain and the dissipative level can be calculated by the Matlab LMI toolbox directly. Finally, some numerical examples are presented to illustrate the correctness and effectiveness of the proposed methods. 4

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Problem formulation

Consider the singular discrete-time Markovian jump systems with time delay and input saturation as follows:    Ex (k       

+ 1) = A (rk ) x (k) + Ad (rk ) x (k − d (k)) + B (rk ) sat (u (k)) + Bω (rk ) ω (k) +M (rk ) F (rk , x (k) , x (k − d (k))) , 2

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   Z (k) = C (rk ) x (k) + Cd (rk ) x (k − d (k)) + D (rk ) sat (u (k)) + Dω (rk ) ω (k) ,       x (k) = ϕ (k) , k ∈ {−d , −d + 1, · · · , 0} , 2

(1) where x (k) ∈ Rn is the state vector, u (k) ∈ Rp is the control input, z (k) ∈ Rq is the control output vector and sat: Rp → Rp is the standard function defined as follows:

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sat (u (k)) = [sat (u1 (k)) , sat (u2 (k)) , · · · , sat (up (k))]T ,

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here without loss of generality, sat (ui (k)) = sign (ui (k)) min {1, |ui (k)|}, ω (k) ∈ Rq is the disturbance and F (rk , x (k) , x (k − d (k))) is a nonlinear vector function and stands for the unknown time-varying pertubations, d (k) is the time-varying delay satisfies d1 ≤ d (k) ≤ d2 , A (rk ) , Ad (rk ) , B (rk ) , Bω (rk ) , M (rk ) , C (rk ) , Cd (rk ) , D (rk ) , Dω (rk ) are known mode-dependent constant matrices with appropriate dimensions, E ∈ Rn×n may be singular and we assume rank (E) = r ≤ n. {rk , k ∈ Z} is a Markovian chain taking values in finite space S = {1, 2, · · · , s} with transition probability matrix Q = {πij } , (i, j ∈ S), Pr {rk+1 = j |rk = i } = πij ,

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s P

i, j ∈ S,

where 0 ≤ πij ≤ 1,

j=1

πij = 1, and in this paper, the transition rates of the

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jumping process are considered to be partly accessible. For example, for s = 4, the transition rates matrix of the system may be expressed as the follows:

Y





ˆ12 π ˆ13 π14   π11 π    ˆ21 π ˆ22 π23 π24  π 

= 

π  ˆ31 

,  π ˆ32 π33 π ˆ34   

π41 π42 π43 π44

with π ˆij represents the unavailable elements. For notational clarity, for any i ∈ S , the set Si denotes: Si = Sik ∪ Siuk , 5

ACCEPTED MANUSCRIPT where Sik = {j : if πij is known }

Siuk = {j : if πij is unknown} . ∆

In addition, if Sik 6= ∅, Sik and Siuk are further described as Sik = {k1 , k2 , · · · , kmi } , mi ∈ {1, 2, · · · , s − 2} , where kj ∈ N + , j ∈ {1, 2, · · · , mi } , represents the index of Q the j th known element of the i th row of the matrix .

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Remark 1 For the case mi = s − 1 , which means we have only one unknown element, one can calculate it from the known elements by the property of the transition rate matrix. In this paper, we design the discrete-time state feedback controller as the following form:

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u (k) = K (rk ) x (k) ,

(2)

where K (rk ) is the controller gains to be determined. Then the discrete-time singular stochastic system (1) with the controller (2) can be written in the form of the control system as follows: + 1) = A (rk ) x (k) + Ad (rk ) x (k − d (k)) + B (rk ) sat (K (rk ) x (k)) + Bω (rk ) ω (k) +M (rk ) F (rk , x (k) , x (k − d (k))) ,

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   Ex (k       

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   Z (k) = C (rk ) x (k) + Cd (rk ) x (k − d (k)) + D (rk ) sat (K (rk ) x (k)) + Dω (rk ) ω (k) ,       x (k) = ϕ (k) , k ∈ {−d , −d + 1, · · · , 0} . 2

2

(3)

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To simplify the notational presentation of this paper, for every possible rk ∈ S, a matrix A (rk ) will be represented by Ai , such as, B (rk ) will be represented by Bi , Bω (rk ) will be Bωi and so on, in addition, F (rk , x (k) , x (k − d (k))) will be represented by Fi (x (k) , x (k − d (k))).

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Definition 1 [46] (1) The matrix pair (E, Ai ) is said to be regular if, for each i ∈ S, the characteristic polynomial det (sE − Ai ) is not identically zero; (2) The matrix pair (E, Ai ) is said to be causal if, for each i ∈ S, deg (det (sE − Ai )) = rank (E) . Definition 2 (singular stochastic finite-time stability (SSFTS) ). The closedloop singular stochastic system (3) with ω (k) = 0 is said to be SSFTS with respect to (c1 , c2 , N, Ri ). With 0 < c1 < c2 , and Ri > 0 , if the system is 6

ACCEPTED MANUSCRIPT regular and causal in time ∀k ∈ {1, 2, · · · , N } and satisfies n

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n

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{ E xT (k1 ) E T Ri Ex (k1 ) , kEk2 xT (k1 ) Ri x (k1 ) ≤ c21 ⇒ E xT (k2 ) E T Ri Ex (k2 ) ≤ c22 (4) with k1 ∈ {−d2 , −d2 + 1, · · · , 0} , k2 ∈ {1, 2, · · · , N }.

τ P

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Definition 3 The closed-loop singular stochastic system (3) is said to be finite time (Q, S, R) − γ-dissipative, if the system is finite-time stability and for any i ∈ S and some γ > 0, the following condition is satisfied and with zero initial state E {G (ω, z, τ )} ≥ γE {hω, ωiτ } , ∀τ ≥ 0 (5) where G (ω, z, τ ) = hz, Qziτ + 2hz, Sωiτ + hω, Rωiτ , G (ω, z, τ ) is the energy supply function of system (3); Q, S and R are real matrices of appropriate dimensions with Q and R symmetric, τ is an integer and ha, biτ = aT (k)b (k) . Without loss of generality, it is assumed that Q ≤ 0, and de-

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noted that −Q = QT − Q− .

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Remark 2 From the definition, the finite-time (Q, S, R) − γ-dissipative includes H∞ performance as a special case, that is ,if Q = −I, S = 0, R = (γ + γ 2 ) I, Eq.(5) corresponds to an H∞ performance requirement. Assumption 1 The nonlinear function Fi (x (k) , x (k − d (k))) satisfies:

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Fi T (x (k) , x (k − d (k))) Fi (x (k) , x (k − d (k))) ≤ xT (k) G2i x (k)+xT (k − d (k)) L2i x (k − d (k)) where Gi and Li are known mode-dependent real matrices.

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For a matrix Fi ∈ Sp×n , donate the l th row of fil , and define L (Fi ) as

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L (Fi ) = {x (k) ∈ Rn : |fil x (k)| ≤ 1, l = 1, 2, · · · , p} .

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Let P ∈ Rn×n be a symmetric matrix and E T P E ≥ 0, α > 0 be a scalar, and  T donate by Ω E P E, α the following set: 



n

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Ω E T P E, α = x (k) ∈ Rn : xT (k) E T P Ex (k) ≤ α .

Denoting the state trajectory of system (1) with initial condition x (k) = ϕ (k) ∈ Cn,d2 [−d2 , 0] by x (k, ϕ), then the domain of attraction of the origin is:   ∆ Γ = ϕ (k) , k = −d2 , −d2 + 1, · · · , 0 : lim x (k, ϕ) = 0 . k→∞

7

ACCEPTED MANUSCRIPT It should be pointed out that in practical it is impossible to determinate the exact domain of attraction. An estimate Bδ ⊂ Γ of the domain of attraction is given by (

)

Bδ = ϕ (k) ∈ Cn,d2 [−d2 , 0] : max kϕ (k)k ≤ δ . [−d2 ,0]

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Let D be the set of p × p diagonal matrices whose diagonal elements are either 1or 0. Suppose each element of D is labeled as Dj , j = 1, 2, · · · , p, and denote Dj− = I − Dj . Clearly, if Dj ∈ D, then Dj− ∈ D. Lemma 1 [52] Let Ki , Hi ∈ Rn×n . Then for any x (k) ∈ L (Hi ), n

o

or, equivalently,

sat (Ki x (k)) =

2p P

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sat (Ki x (k)) ∈ co Dl Ki x (k) + Dl− Hi x (k) , l = 1, 2, · · · , 2p ,





αl (k) Dl Ki + Dl− Hi x (k) .

(6)

where co stands for the convex hull, αl (k) for l = 1, 2, · · · , 2p are some modedependent scalars which satisfy 0 < αl (k) < 1 and

2p P

l=1

αl (k) = 1.

+ 1) =

 z (k) =

l=1

l=1

αl (k) Ail x (k) + Adi x (k − d (k)) + Bωi ω (k) + Mi Fi (x (k) , x (k − d (k))) ,

αl (k) Cil x (k) + Cdi x (k − d (k)) + Dωi ω (k) ,

x (k) = ϕ (k) , k = −d2 , · · · , 0,

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     

2p P

2p P

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   Ex (k     

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Applying the above lemma to system (3), then we have the following system:



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with Ail = Ai + Bi Dl Ki +

Dl− Hi





, Cil = Ci + Di Dl Ki +

Dl− Hi



(7) .

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Lemma 2 [46] Given matrices X, Y, Z with appropriate dimensions, and Y is symmetric positive definite. Then the following inequality holds: −Z T Y Z ≤ X T Z + Z T X + X T Y −1 X.



Z



M

 ≥ 0, integers d1 , d2 and d (k) satis∗ Z fying d1 ≤ d (k) ≤ d2 , and vector function x (k + ·) : N [−d2 , −d1 ] → Rn , such

Lemma 3 [22] For any matrix 

8

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where ς (k) =



α=k−d2

ς T (α)Zς (α) ≤ ς T (k) Ως (k) ,

x (k − d (k)) x (k − d1 ) x (k − d2 )

x (α + 1) − x (α), sym (M ) = M + M T , and

T



+ sym (M ) Z − M Z − M  ∗

−Z

MT

∗

∗

−Z

    

Main results

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3

 −2Z     

, d12 = d2 − d1 , ς (α) =

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Ω=

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T

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−d12

k−d 1 −1 X

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In this section, the finite-time dissipative control for the discrete-time singular stochastic system (7) with partly unknown transition rates will be investigated. First, the finite-time stability criteria for the system will be given, then will be the finite-time dissipative, finally the controller gains will be obtained and the results will be transformed into LMI so that can solve them by LMI toolbox of Matlab.

Theorem 1 For given scalars α1 , α3 > 0, the closed-loop system (7) for ev-

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ery initial condition belongs to

s T

i=1





Ω E T Pi E, γ 2 µ is SSFTS with respect to

(c1 , c2 , N, Ri ), if there exist µ ≥ 1, β, ε, θ1 , θ2 , θ3 , θ4 > 0, symmetric positive definite matrices Pi , Q1 , Q2 , Q3 , Z, Si and matrices M, and nonsingular matrices Ni such that the following conditions holds for all: i, j = 1, 2, · · · , s, l = 1, 2, · · · , 2p 



Z M  

∗ Z

9



≥0

(8)

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ψ1 =

 Σ11    ∗    ∗     ∗     ∗    ∗     ∗  

∗

Σ12 0

0

Σ15

Σ22 Σ23

Σ24

AT il

d12



AT il

−E

d12 AT di

Σ25 AT di

∗ Σ33 E T M E 0

0

0

∗

∗

Σ44

0

0

0

∗

∗

∗

Σ55

0

0

∗

∗

∗

∗ −Pˆi−1

∗

∗

∗

∗

∗

−Z −1

∗

∗

∗

∗

∗

∗

T



d12 Mi

NiT



   α1 I    0     0     0    α3 I     0   

<0

(9)

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

−Si

1 0 < Qj < θj Ri , (j = 1, 2, 3) , Ri < Pi < Ri , 0 < Z < θ4 Ri ε  2 1/ε + (d1 α1 + d2 α2 + d2 α3 ) /kEk + d12 (d1 + d2 − 1) α3 /2kEk2 

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+2d212 (d1 + d2 + 1) ε2 c21 ≤ µ−N c22

(10a)

(10b)

where P T 2 T T T 11 = Q1 + Q2 + (1 + d 12 ) Q3 − µE Pi E + βGi + Ail R Ni + Ni RAil , Ail = Ai + Bi Dl Ki + Dl− Hi P

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P

12

T T = α1 A T il R + Ni RAdi ,

22

= −Q3 + sym E T (M − Z) E + α1 RAdi + βL2i ,



T T = α3 A T il R + Ni RMi ,

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P

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T Σ23 = E T (Z − M ) E, Σ24 = E T Z − M T E, Σ25 = α1 RMi + α3 AT di R ,

Σ44 = −Q2 − E T ZE, Σ55 = −βI + sym (α3 RMi ) ,

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Σ33 = −Q1 − E T ZE ,

P P Pj , Pˆi = πij Pj + (1 − πia )

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j∈Sik

j∈Sik

πia =

P

j∈Sik

πij , 



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R ∈ Rn×n is any constant matrix satisfying: E T R = 0, and Ω E T Pi E, γ 2 µ ∈ L (Hi ) . Moreover, the domain is presented by δmax =

s

γ 2 µ1−N Υ

(11)

with

n



Υ = sup λmax E T Pi E i∈S

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+ d1 λmax (Q1 ) + d2 λmax (Q2 ) + d3 λmax (Q3 ) + 



+2d212 (d1 + d2 + 1) λmax E T ZE . 10

d12 (d1 +d2 −1) λmax 2

(Q3 )

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Proof: First, we should prove that the system is regular and causal. As 2p P

l=1

αl (k) = 1, then p

2 X

αl (k) Ψ1 < 0,

(12)

l=1

with Schur complement, Eq.(12) is equivalent to: T T T −1 Tˆ T ψ10 + ΛT 1 R Λ2 + Λ2 RΛ1 + Λ2 Si Λ2 + Λ1 Pi Λ1 + Λ3 ZΛ3 < 0,

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where

ψ10

0  Σ 11

=

           



0



T

0 T

∗ −Q3 + sym E (M − Z) E E (Z − M ) E E ∗

∗

∗

∗

∗

∗

−Q1 − E T ZE

T



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

(13)

0

Z −M

T

E TM E



E



0 

0

0

∗

−Q2 − E T ZE

0

∗

∗

−βI

           

 l=1

l=1 

Λ3 = d12

 2p P

l=1

l=1

2p P

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Λ2 = Ni α1 I 0 0 α3 I ,

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Σ011 = Q1 + Q2 + (1 + d12 ) Q3 − µE T Pi E + βG2i  2p 2p 2p P P P Λ1 = αl (k) Ail αl (k) Adi 0 0 αl (k) Mi , αl (k) (Ail − E)

l=1

2p P

l=1



αl (k) Mi ,

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from lemma 3,

αl (k) Adi 0 0

T T T −1 T T ΛT 1 R Λ2 + Λ2 RΛ1 + Λ2 Si Λ2 ≥ −Λ1 R Si RΛ1 ,

(14)

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then it follows (13) and (14) that: T Tˆ T ψ10 − ΛT 1 R Si RΛ1 + Λ1 Pi Λ1 + Λ3 ZΛ3 < 0,

(15)

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then we have

Q1 + Q2 + (1 + d12 ) Q3 − µE T Pi E + βG2i + +

2p

P

l=1

2p

2p

l=1

l=1

2p P

l=1

αl (k) (Ail − E)T Z 2p

2p P

l=1

ˆ P αl (k) Ail − P αl (k) AT RT Si R P αl (k) Ail < 0. αl (k) AT il Pi il

αl (k) (Ail − E)

l=1

(16)

For rank (E) = r ≤ n, there exist two nonsingular U, V ∈ Rn×n matrices such 11

ACCEPTED MANUSCRIPT that





Ir 0  U EV =  ,  0 0 

U

l=1

 Pi1

V −T Pi V −1 = 

2p P



αl (k)Ail V = 



Ail3 Ail4



Pi2 

Pi3 Pi4



 Ail1 Ail2  

,

RV −1 = R1 R2 ,

,



? 

?

? T AT il4 R2 Si R2 Ail4

  

< 0,

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considering that Q1 , Q2 , Q3 , Pi and Z are positive definite matrices and E T R = 0. Pre-multiplying and post-multiplying (16) by U T and U , we have

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where ? stands for the matrix block that we do not need. Then we have T AT il4 R2 Si R2 Ail4 < 0, that is Ail4 is nonsingular and the system is regular and casual. Now, it is time to prove the system (7) is SSFTS with respect to (c1 , c2 , N, Ri ). Define that y (k) = x (k + 1) − x (k) , and choose the Lyapunov-Krasovski functional as follows: V (rk , x (k)) = V1 (rk , x (k)) + V2 (rk , x (k)) + V3 (rk , x (k)) + V4 (rk , x (k)) ,

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where

k−1 P

s=k−d1

xT (s) Q1 x (s) +

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V2 (rk , x (k)) =

ED

V1 (rk , x (k)) = xT (k) E T P (rk ) Ex (k) ,

k−1 P

s=k−d(k)

CE

V3 (rk , x (k)) =

AC

V4 (rk , x (k)) = d12 then

k−1 P

s=k−d2

xT (s) Q3 x (s) +

−dP 1 −1 k−1 P

xT (s) Q2 x (s),

−d P1

k−1 P

θ=−d2 +1 s=k+θ

xT (s) Q3 x (s),

y T (s) E T ZEy (s),

θ=−d2 s=k+θ

∆V1 (rk , x (k)) = E (V1 (rk+1 , x (k + 1))) − V1 (rk , x (k)) = xT (k + 1) Q

s P

j=1

πij Pj x (k + 1) − xT (k) Pi x (k

since the transition probability matrix is partly accessible, we should bound s P the combination matrix πij Pj , with the property of transition probability j=1

matrix and consult the methods in L. Zhang’s 2010 TAC work, we have the following: 12

ACCEPTED MANUSCRIPT s P

j=1

πij Pj =

(1 − πia )

P

P

j∈Sik

j∈Sik

P

πij Pj +

j∈Siuk

Pj

π ˆij Pj =

P

j∈Sik

πij Pj +(1 − πia )

P

j∈Siuk

π ˆij P 1−πia j

≤

P

j∈ik

πij Pj +

follwing that X + 1) E T 

j∈Sik

πij Pj + (1 − πia )



X

j∈Sik



Pj  Ex (k + 1) − xT (k) E T Pi Ex (k)

CR IP T

∆V1 (rk , x (k)) ≤ xT (k



∆V2 (rk , x (k)) = E (V2 (rk+1 , x (k + 1))) − V2 (rk , x (k)) k X

=

s=k+1−d1 k−1 X

−

xT (s) Q2 x (s)

s=k+1−d2

xT (s) Q1 x (s) +

s=k−d1 T

k−1 X

s=k−d2 T



xT (s) Q2 x (s)

AN US



k X

xT (s) Q1 x (s) +

= x (k) (Q1 + Q2 ) x (k) − x (k − d1 ) Q1 x (k − d1 ) − xT (k − d2 ) Q2 x (k − d2 ) ,

∆V3 (rk , x (k)) = E (V3 (rk+1 , x (k + 1))) − V3 (rk , x (k)) k X

M

=

T

x (s) Q3 x (s) +

s=k+1−d(k+1)



T

x (s) Q3 x (s) +

ED

−

k−1 X

−d X1

k X

θ=−d2 +1 s=k+1+θ −d X1

k−1 X

θ=−d2 +1 s=k+θ T

s=k−d(k) T

xT (s) Q3 x (s)

T



x (s) Q3 x (s)

PT

≤ x (k) (1 + d12 ) Q3 x (k) − x (k − d (k)) Q3 x (k − d (k)) ,

CE

∆V4 (rk , x (k)) = E (V4 (rk+1 , x (k + 1))) − V4 (rk , x (k))

AC

=

−d 1 −1 X

k X

θ=−d2 s=k+1+θ

y T (s) E T ZEy (s) −

= d212 y T (k) E T ZEy (k) − d12

k−d 1 −1 X

−d 1 −1 k−1 X X

y T (s) E T ZEy (s)

θ=−d2 s=k+θ

y T (s) E T ZEy (s),

s=k−d2

applying lemma 3, it follows that

∆V4 (rk , x (k)) ≤ d212 y T (k) E T ZEy (k) + η T (k) Γη (k) , 13

ACCEPTED MANUSCRIPT where 



η (k) = x (k − d (k)) x (k − d1 ) x (k − d2 ) , 

 sym     

Γ=





T

T

E (M − Z) E E (Z − M ) E E ∗

T

−E T ZE

∗



Z −M

T

E TM E



E

−E T ZE

∗



  ,  

from the above, we obtain that X + 1) E T 

j∈Sik



πij Pj + (1 − πia )



CR IP T

∆V (rk , x (k)) ≤ xT (k



X

j∈Sik



Pj  Ex (k + 1) + d12 y T (k) E T ZEy (k) 

+ xT (k) Q1 + Q2 + (1 + d12 ) Q3 − E T Pi E x (k) − xT (k − d (k)) Q3 x (k − d (k))

AN US

− xT (k − d1 ) Q1 x (k − d1 ) − xT (k − d2 ) Q2 x (k − d2 ) + η T (k) Γη (k) . (17) T As E R = 0 and the following equation is true: −xT (k + 1) E T REx (k + 1) = 0,

then

ED

M

∆V (rk , x (k)) = ∆V (rk , x (k)) − xT (k + 1) E T REx (k + 1) + βFiT (x (k) , x (k − d (k))) Fi (x (k) , x (k − d (k))) − βFiT (x (k) , x (k − d (k))) Fi (x (k) , x (k − d (k))) ≤ ∆V (rk , x (k)) − xT (k + 1) E T REx (k + 1) + βxT (k) G2i x (k) + βxT (k − d (k)) L2i x (k − d (k)) − βFiT (x (k) , x (k − d (k))) Fi (x (k) , x (k − d (k))) . 



PT

Let ζ (t) = x (k) x (k − d (k)) x (k − d1 ) x (k − d2 ) Fi (x (k) , x (k − d (k))) , then we have 



CE

T Tˆ T ∆V (rk , x (k)) ≤ ζ T (k) ψ˜1 − ΛT 1 R Si RΛ1 + Λ1 Pi Λ1 + Λ3 ZΛ3 ζ (k) , (18)

AC

where



ψ˜1 =

00  Σ 11

           

∗

∗



0



T

0 T

−Q3 + sym E (M − Z) E E (Z − M ) E E ∗

−Q1 − E T ZE

T



0 Z −M

T

E TM E



0 E

0 0

∗

∗

∗

−Q2 − E T ZE

0

∗

∗

∗

∗

−βI,

14

             

ACCEPTED MANUSCRIPT

Σ0011 = Q1 + Q2 + (1 + d12 ) Q3 − E T Pi E + βG2i , then from (13) , (14) and (18),we have

CR IP T

E {V (rk+1 , x (k + 1))} − V (rk , x (k)) < (µ − 1) xT (k + 1) E T Pi Ex (k + 1) ≤ (µ − 1) V (rk , x (k)) (19) that is E {V (rk+1 , x (k + 1))} ≤ µE {V (rk , x (k))} , (20) notice that µ ≥ 1, then it follows that:

E {V (rk , x (k))} ≤ µk E {V (0)} . n

o

E {V (0)} = E xT (0) E T Pi Ex (0)



xT (s) Q1 x (s) + −1 X

T

 

 

θ=−d2 +1 s=θ



−1 X

xT (s) Q3 x (s)



−d −1 1 −1 X X

y T (s) E T ZEy (s) . (22)

PT



o

1



1

n



CE

E xT (0) E T Pi Ex (0) ≤ E xT (0) E T Ri2 P¯i Ri2 Ex (0) ≤ sup λmax P¯i

AC

 −1  X

E



xT (s) Q1 x (s) +

s=−d1

≤E

 −1  X 

1 2

i∈S

xT (s) Q2 x (s) +



2

kEk

o

1 2

−1 X

n

¯ 2i d2 sup λmax Q +

i∈S

kEk

15

2

o

c21 ,

xT (s) Q3 x (s)

1 2



o

 

1 2

¯ 2i Ri x (s) + xT (s) Ri Q

s=−d2

c21

−1 X

i∈S

s=−d(0)

¯ 1i Ri x (s) + xT (s) Ri Q

¯ 1i d1 sup λmax Q ≤

−1 X

s=−d2

s=−d1

n



θ=−d2 s=θ

−1 −1 −1 −1 for l = 1, 2, 3 and P¯i = Ri 2 Pi Ri 2 , oZ¯i = Ri 2 ZRi 2 , (k) E T Ri Ex (k) , kEk2 xT (k) Ri x (k) ≤ c21 for all k ∈

¯ li = Letting Q and noticing that E x {−d2 , −d2 + 1, · · · , 0} then we have

 

  

s=−d(0)

x (s) Q3 x (s) + E d12

−1 −1 Ri 2 QinRi 2 T

n

xT (s) Q2 x (s) +

s=−d2

M

+E

 s=−d1   −d X1

−1 X

ED

+E

 −1  X

AN US

Considering that

(21)

n



−1 X

¯ 3i d2 sup λmax Q

c21 +

i∈S

kEk2

1 2

 

¯ 3i Ri x (s) xT (s) Ri Q

s=−d2



1 2

o



c21 ,

ACCEPTED MANUSCRIPT

−d X1



−1 X

 

xT (s) Q3 x (s)



θ=−d2 +1 s=θ

=E

E d12

  

i∈S



θ=−d2 s=θ



≤ 2d12 sup λmax Z¯i i∈

o



(d1 + d2 − 1) (d2 − d1 )

2kEk2

,



 

xT (s + 1) E T ZEx (s + 1) + xT (s) E T ZEx (s) 

o

(d2 + d1 + 1) (d2 − d1 ) c21 .

As n

 



xT (s + 1) − xT (s) E T ZE (x (s + 1) − x (s))

−d −1  1 −1 X X

n

 

1 2

¯ 3i Ri x (s) xT (s) Ri Q

y T (s) E T ZEy (s)

θ=−d2 s=θ

≤ E 2d12



1 2

 

−d −1  1 −1 X X

= E d12

n

¯ 3i sup λmax Q

θ=−d2 s=θ

 

−1 X

AN US



−d −1 1 −1 X X



−d X1

θ=−d2 +1 s=θ

≤

 

 

CR IP T

E

 

o



1 2



1

E {V (rk , x (k))} ≥ E x (k) E Pi Ex (k) = E x (k) E Ri P¯i Ri2 Ex (k)



n

inf λmin P¯i i∈S



o

n

M

E xT (k) ERi Ex (k)

n



o

PT

 ¯ ≤ µk  sup λmax Pi i∈S

n



CE

¯ 3i d2 µk sup λmax Q

+

i∈S

kEk2 n



i∈S

o

n

c21 +

i∈S



kEk2 n

i∈S

+

T

n





o

¯ 3i µk sup λmax Q

c21

o

i∈S

T

≥ inf λmin P¯i

¯ 1i d1 sup λmax Q

o

+ 2µk d12 sup λmax Z¯i

AC

T

ED

then we have

T

n

+

n

(23)



i∈S

kEk2

o

c21

(d1 + d2 − 1) (d2 − d1 )

2kEk2

(d2 + d1 + 1) (d2 − d1 ) c21 . (24)

Based on (10a) and (10b), we have n

o

E xT (k) ERi Ex (k) < c22 , ∀k ∈ {1, 2, · · · , N } ,

(25)

from definition 3, we know that the singular system (6) is SSFTS with respect to (c1 , c2 , N, Ri ). 16

o

E xT (k) ERi Ex (k) ,

¯ 2i d2 sup λmax Q

c21

o

o

   

ACCEPTED MANUSCRIPT On the other hand, from Eq.(21), it follows that xT (k) E T Pi Ex (k) ≤ V (k) ≤ µk V (0) ≤ γ 2 µ,

(26)

then we have

γ 2 µ1−N ∆ kϕ (k)k ≤ = = (ϕ) . Υ For any ϕ ∈ = (ϕ), it follows that = (ϕ) ≤ γ 2 µ. From (26) we can further get xT (k) E T Pi Ex (k) ≤ γ 2 µ for any ϕ ∈ = (ϕ), which means that all the trajectories of x (k) which start form the set = (ϕ) always stay in the domain  T Ω E Pi E, γ 2 µ , then the proof is completed.

CR IP T

2

Theorem 2 For given scalars α1 , α2 , α3 > 0 and matrices Q, S and R with Q and R symmetric and mathcalQ ≤ 0, the closed-loop system (6) for every initial condition belongs to



s T



Ω E T Pi E, γ 2 µ is said to be finite time

i=1

0

Σ 15 Σ

T 16 Ail Wi

0

0

T 26 Adi Wi

∗ Σ33 E T M E 0

0

M

0

0

∗

∗

0

0

0

0

∗

∗

∗

∗

∗

Σ12 0

AC

∗

0

Σ22 Σ23

Σ24

ED

Σ44

−E

∗

∗ Σ0 66 Mi Wi

d12 Mi

∗

∗

∗

∗

−P −1

∗

∗

∗

∗

∗

∗

−Z −1

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗



Dl− Hi



,

P0

15

=

T

d12 AT di

d12 Bωi

∗

where Cil = Ci +Di Dl Ki + 15 ,

P

Σ 25 Σ

d12

AT il

Σ0 55 Σ0 56 Bωi Wi

CE

ψ2 =

 Σ11    ∗    ∗     ∗     ∗    ∗     ∗    ∗     ∗  



0

PT



AN US

(Q, S, R) − γ-dissipative, if the system is SSFTS with respect to (c1 , c2 , N, Ri ) and there exist µ ≥ 1, β, ε, θ1 , θ2 , θ3 , θ4 > 0, symmetric positive definite matrices Pi , Q1 , Q2 , Q3 , Z, Si and matrices M, and nonsingular matrices Ni such that (8), (10a), (10b) and the following conditions holds for all i, j = 1, 2, · · · , s, l = 1, 2, · · · , 2p :

0



CilT

NiT



   α1 I    0 0     0 0    T Dωi α2 I     0 α3 I     0 0    0 0     Q−1 0    T Cdi

∗ −Si

T T T α2 A T il R +Ni RBωi −Cil S,

< 0,

(27) 16 =

P0

T T 0 0 Σ0 25 = α1 RBωi +α2 AT di R −Cdi S, Σ 26 = Σ25 , Σ 55 = − (R − γI)+sym (RBωi ) , T T Σ0 56 = α2 RMi + α3 Bωi R ,

Σ0 66 = −βI + sym (α3 RMi ) ,

17

ACCEPTED MANUSCRIPT Wi =

h√

i √ √ √ √ πik1 I, πik2 I, · · · , πikmi I, πia I, · · · , πia I ,

P = diag {Pk1 , Pk2 , · · · , Pkmi , Pkmi +1 , · · · , Ps } . And the other variables are the same as theorem 1.

CR IP T

Proof: When ω (t) = 0, it is easily obtained that (9) is true from (27), then the system is finite time stability. When ω (t) 6= 0, select the same Lyapunov function candidate as Theorem 1 and define the following function: J = ∆V (rk , x (k)) − (µ − 1) V (rk , x (k)) − z T (k) Qz (k) − 2ω T (k) Sz (k) − ω T (k) (R − γI) ω (k) , (28)  

let ξ (k) = x (k) x (k − d (k)) x (k − d1 ) x (k − d2 ) ω (k) Fi (x (k) , x (k − d (k))) , and use the similar handling method in theorem 1 we can obtain that 



Σ00 11 

˜1 = Λ

 2p P

l=1



∗



∗

∗ ∗

∗

αl (k) Ail



(29)

0



−Q3 + sym E T (M − Z) E E T (Z − M ) E E T Z − M T E

∗

∗



0

T

−Q1 − E ZE

M

where ψ˜2 =

              

0

∗

∗

ED



AN US

T ˜T ˜ ˜ ˜T ˆ ˜ ˜T ˜ ˜T J ≤ ξ T (k) ψ˜2 − Λ 1 R Si RΛ1 + Λ1 Pi Λ1 + Λ3 Z Λ3 − Λ4 QΛ4 ξ (k) ,

2p

P

l=1

αl (k) Adi 0 0

2p

P

αl (k) Bωi

l=1

2p

P

l=1

T

E ME

−CilT S

T S −Cdi



0   0  

0

0

∗

−Q2 − E T ZE

0

0

∗

∗

− (R − γI)

0

∗

∗

−βI

∗



αl (k) Mi , 

l=1

AC

For

l=1

αl (k) Cil

CE

Λ4 =

 2p P

PT

2p 2p 2p 2p ˜ 3 = d12 P αl (k) (Ail − E) P αl (k) Adi 0 0 P αl (k) Bωi P αl (k) Mi , Λ



2p P

l=1

l=1

l=1

αl (k) Cdi 0 0

2p P

l=1



l=1

αl (k) Dωi 0 .

T ˜ T −1 ˜ ˜ ˜T T ˜ ˜T ˜ ˜T −Λ 1 R Si RΛ1 ≤ Λ1 R Λ2 + Λ2 RΛ1 + Λ2 Si Λ2



˜2 = N α I 0 0 α I α I , where Λ i 1 2 3 considering (27) and use the Schur complement we can obtain that E {V (rk+1 , x (k + 1))} ≤ µV (rk , x (k)) + z T (k) Qz (k) + 2ω T (k) Sz (k) + ω T (k) (R − γI) ω (k) , (30) 18

    ,       

ACCEPTED MANUSCRIPT then

k

E {V (rk , x (k))} ≤ µ V (0) + +

k−1 X j=0

k−1 X

µ

k−j−1

j=0

n

o

T

E z (k) Qz (k) + 2

n

o

k−1 X j=0

n

o

µk−j−1 E ω T (k) Sz (k)

µk−j−1 E ω T (k) (R − γI) ω (k) .

k−1 X j=0

n

o

µk−j−1 E ω T (k) ω (k) ≤ +

n

k−1 X j=0

k−1 X j=0

j=0

n

o

E ω T (k) ω (k) ≤ +

τ X

j=0 τ X

j=0

o

n

o

E z T (k) Qz (k) + 2 n

M

τ X

n

k−1 X j=0

n

o

τ X

j=0

n

(32)

o

E ω T (k) Sz (k)

E ω T (k) Rω (k) .

PT

ED

(33) Then from definition 3 we know the system is finite time dissipative, the proof is complete.

CE

Theorem 3 For given positive scalars α1 , α2 , α3 , ρm (m = 1, · · · , 4) , and matrices Q, S and R with Q and R symmetric and Q ≤ 0, the closed-loop   s T system (7) for every initial condition belongs to Ω E T Pi E, γ 2 µ is said to i=1

AC

be finite time (Q, S, R) − γ-dissipative, if the system is SSFTS with respect to (c1 , c2 , N, Ri ) and there exist ε, θ1 , θ2 , θ3 , θ4 > 0, symmetric positive definite matrices Yi , Q1 , Q2 , Q3 , Z, Si and matrices M, Vi1 , Vi2 , Xi such that (8)and the following conditions holds for all i, j = 1, 2, · · · , s, l = 1, 2, · · · , 2p : 

1  − η2 u



∗

XiT HijT XiT E T + EXi + Yi 19

  

≤ 0,

o

µk−j−1 E ω T (k) Sz (k)

µk−j−1 E ω T (k) Rω (k) ,

then

γ

o

µk−j−1 E z T (k) Qz (k) + 2

AN US

γ

CR IP T

(31) Noting that V (rk , x (k)) is positive and under zero initial, then it follows:

(34)

ACCEPTED MANUSCRIPT

ψ3 =

 ψ31    ∗    ∗     ∗  

T O1i

(−2ρ1 I +

∗

 

ρ21 Q1 )

1+

T d12 O1i

T βO3i

0

0

0

∗

(−2ρ2 I + ρ22 Q2 )

0

0

∗

∗

(−2ρ3 I + ρ23 Q3 )

0

∗

∗

∗

−βI

             

< 0,

(35)

0 < Qj < θj Ri , (j = 1, 2, 3) , εRi−1 < Yi < Ri−1 , 0 < Z < θ4 Ri d1 θ1 +d2 θ2 +d2 θ3 kEk2

+

d12 (d1 +d2 −1)θ3 2kEk2

∗

where ˜ ˜  Σ11 Σ12 0

AC

˜ 17 Σ

˜ 18 Σ

d12 ATdi

˜ 33 Σ ˜ 34 0 ∗ Σ

0

∗

∗

˜ 44 0 ∗ Σ

0

∗

∗

∗

T ˜ 55 Σ ˜ 56 Bωi ∗ Σ Wi

∗

∗

∗

∗

˜ 66 MiT Wi ∗ Σ

∗

∗

∗

∗

∗

∗

−Y

∗

∗

∗

∗

∗

∗

∗

(−2ρ4 I + ρ24 Z)

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗



O1i = Xi 01×9 ,

ED 

O3i = Gi Xi 01×9 

0

0

0

0

˜ 12 = α1 Ai Xi + Bi Dl Vi1 + Bi D− Vi2 Σ l 

˜ 15 = − Ci Xi + Di Dl Vi1 + Di D− Vi2 Σ l RBωi ,

T

T

(36a)

≤0

(36b)



˜ 19 Σ

I   

T Cdi α1 I  



0  

0



 0  

 T Dωi α2 I  

d12 MiT 0





0

T d12 Bωi

˜ 11 = µXiT E T +µEXi +µYi + Ai Xi + Bi Dl Vi1 + Bi D− Vi2 Σ l 

−ε

∗

CE 

˜ 15 Σ ˜ 16 0 Σ

˜ 22 Σ ˜ 23 Σ ˜ 24 Σ ˜ 25 Σ ˜ 26 ATdi Wi ∗ Σ

PT

ψ31 =

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



+ 2d212 (d1 + d2 − 1) θ4 c21 c1 

M





AN US



√

T O1i

CR IP T



  0 α3 I     0 0    0 0     −1 Q 0   

∗ −Si

T



RT + RAdi , 

S T +α2 Ai Xi + Bi Dl Vi1 + Bi Dl− Vi2

20



RT +R Ai Xi + Bi Dl Vi1 + Bi Dl− Vi2 ,

T

RT +

ACCEPTED MANUSCRIPT 

˜ 16 = α3 Ai Xi + Bi Dl Vi1 + Bi D− Vi2 Σ l 

T

RT +RMi ,

˜ 18 = d12 Ai Xi + Bi Dl Vi1 + Bi D− Vi2 − EXi Σ l 

T



˜ 17 = Ai Xi + Bi Dl Vi1 + Bi D− Vi2 Σ l 

˜ 19 = Ci Xi + Di Dl Vi1 + Di D− Vi2 Σ l

,



˜ 22 = −Q3 + sym E T (M − Z) E + α1 RAdi + βL2 , Σ ˜ 23 = E T (Z − M ) E, Σ i 



˜ 24 =E T Z − M T E, Σ ˜ 25 = α1 RBωi + α2 AT RT − C T S T , Σ di di ˜ 33 = −Q1 − E T ZE, Σ

T ˜ 26 =α1 RMi + α3 AT Σ di R ,

˜ 55 = − (R − γI) + sym (α2 RBωi − SDωi ) , Σ

˜ 66 = −βI + sym (α3 RMi ) Σ Wi =

h√

i √ √ √ √ πik1 I, πik2 I, · · · , πikmi I, πia I, · · · , πia I ,

AN US

Y −1 = P = diag {Pk1 , Pk2 , · · · , Pkmi , Pkmi +1 , · · · , Ps } ,

˜ 56 = Σ

CR IP T

˜ 44 = −Q2 − E T ZE, Σ T T α2 RMi + α3 Bωi R ,

˜ 34 = E T M E, Σ





M

R ∈ Rn×n is any constant matrix satisfying: E T R = 0, and Ω E T Pi E, η 2 u ∈ L (Hi ).   Then closed-loop system (7) for every initial condition belongs to Ω E T Pi E, γ 2 µ is said to be finite time (Q, S, R) − γ-dissipative, and the controller feedback gains are given as: Ki = Vi1 Xi−1 . n

ED

Proof: Let Xi = Ni−1 , pre- and post-multiply (27) by diag Xi−1 , I, I, I, I, I, I, I, I, I and denote Ki Xi = Vi1 , Hi Xi = Vi2 , from lemma 2

PT

−µXiT E T Pi EXi ≤ µXiT E T + µEXi + µPi−1 ,

considering Schur complement, (35) can be obtained. 



AC

CE

On the other hand, Ω E T Pi E, γ 2 µ ∈ L (Hi ) is equivalent to HijT Pi−1 Hij ≤ 1 which by Schur complement and pre- and post-multiply diag {I, Xi } is γ2u equivalent   1

 − γ2u 

∗

Hij Xi

−XiT E T Pi EXi

where Hij is the j th row of Hi , with lemma 2

 

≤0

−XiT E T Pi EXi ≤ XiT E T + EXi + Pi−1 = XiT E T + EXi + Yi , then the conclusion of the Theorem 3 is true.

21

o

T

T

,

Wi

ACCEPTED MANUSCRIPT h√ i √ √ Remark 3 Let Y −1 = P = diag {P1 , P2 , · · · , Ps } and Wi = πi1 I, πi2 I, · · · , πis I in the above Theorem, and the results become finite-time dissipative control for singular discrete-time saturated Markovian jump systems with completely known transition rates.

min κ

with

     

∗

T



E  Yi



≥ 0, ω2 I − Q1 ≥ 0, ω3 I − Q2 ≥ 0,

ω4 I − Q3 ≥ 0, ω5 I − E T ZE1 ≥ 0,

κ = ω1 + d1 ω2 + d2 ω3 + d2 ω4 +

(37)

AN US

s.t.

   ω I  1    

CR IP T

Remark 4 With all the ellipsoids satisfying the set invariance conditions of the theorems, we are interested in finding a large estimate of the domain of attraction. Based on the results obtained above, an optimization problem with LMI constraints can be formulated as follows:

d12 (d1 + d2 − 1) ω4 + 2d212 (d1 + d2 − 1) ω5 . 2

M

The positive scalars ω1 , ω2 , ω3 , ω4 , ω5 are introduced to bound the different items in (11) for getting a larger estimate of the attraction for the system. Therefore, a maximal estimate of the attraction domain can be obtained

ED

δmax =

s

γ 2 µ1−N . κmin

PT

Remark 5 The feasibility of conditions stated in Theorem 3 can be represented by the following LMIs based a parameter µ

Pi , Q1 , Q2 , Q3 , Z, Si > 0, M, Vi1 , Vi2 , Xi s.t. (7) , (9) , (31) , (32) and (34) .

AC

CE

min c22 + γ 2

4

Numerical examples

In this section, some examples are presented to demonstrate the effectiveness of the obtained results.

22

ACCEPTED MANUSCRIPT Example 1 Considering remark 2, the proposed methods can be used to resolve the finite-time H∞ control problem. In order to present our methods are better than some existing results, we can consider the same example as [15], and the coefficients are as follows: Mode 1 

 









1.5 0  0 0 1 0  A1 =   , B1 =   , Bω1 =   , Ad1 =   1 1 0 0.5 0 1 0

M1 = 



0

00

,



0

Cd1 = 



0.5 

,

0 0





 0.5  ,

D1 = 

0

Mode 2 1

A2 = 

1

01





, 



0

Ad2 = 





Pr =  





0 0

,

,



 





 0.1 



0.7 0.3  1 , E =  0.4 0.6 0



0 1

,

0

,



R= 

0



0

00

0 1 B2 =   , Bω2 =  1 0

D2 = 

M

In addition,

1 

0 −0.5

0 0  0.5 M2 =    , Cd2 =  00 0



1

Dω1 = 

AN US





, .

CR IP T





0





0 0

, 

1 0

Dω2 = 

00

.



0 0 , 00

PT

ED

and R1 = R2 = I2 , c1 = 1, N = 5, d1 = d2 = 1, µ = 2.1149,according to the methods in [15], the optimal values γ = 11.6285, c2 = 33.5960, while using the methods proposed in this paper, we can obtain the optimal values γ = 9.6901, c2 = 29.2745, which are much better than the results in [15], moreover the controller gains are 







CE

K1 = −0.0486 −0.9584 , K2 = −1.0327 −0.2935 .

AC

Remark 6 In the above example, we have considered the finite-time H∞ performance of the system, which is a particular situation of the finite-time dissipativity. If the attenuation level γ and the finite-time boundedness c2 are smaller, the results will be better. From the example, it is obvious that our results are much better. Example 2 The example considers hereafter address water-quality in streams of some rivers. Indeed, for all practical purposes, it is crucial to preserve the standards of water-quality in stream, which can be measured by the concentrations of some water biochemical constituents. For this problem, we construct a discretized water pollution model with multiple operating points. The model 23

ACCEPTED MANUSCRIPT

L1 =

0

0

−0.3258

0

0

0



0

  , 0  

M1 =

0



0   



   ,  



Ad1 =



 1    0.01  

 0.01    0  

0 0  



0 0   0

 , 0.01   

G1 =

0



0   

, C1 = , Cd1 = 0.15 0  0.2 0      0 0 0.15 0.1 0 0.5

 0.3    0  

0



0 

1.5 0

  ,  

D1 =



 3.1       1.1  .    

0.5

AC

CE

0.1 0.2 0.3





, B1 = 0 0    0 0 0.01

0 0 0

 2.1    1  



AN US

 0.15    0  

Dω1 =

−0.8208

0.0023

0



0

0

 0.002    0  



0

ED

Bω1 =





M

 −1.8794    −0.1773  

PT

A1 =



CR IP T

represents the concentrations per unit volume of biological oxygen demand and dissolved oxygen, respectively, at time k. Here the state x(k) is the difference of the concentrations per unit volume of biological oxygen demand and dissolved oxygen. We want to design a set of stochastic feedback controllers for this system based on the conditions in Theorem 3 and Remark 3. Markovain jumping occur between the two modes describe by the following coefficients. Mode 1

Mode 2 24





 0.3    0  

 1.5    0  

0

0





 −2.1126       −1.2176  ,    

0.2578



0   

, 0.3 0    0 0 0.3 

1.2 0   

, 1 0   0 1

ACCEPTED MANUSCRIPT

Bω2 =

L2 =

0 0.1234

0





Dω2 =

     

0



0

0.0016

0

0

0

0.0003

0



0  

0.15 0

0



−0.5596

0

 0.15    0  

0 0.15

2.3

0

0

0.6

 ,  

0

0

 0.1286     

0.2143  

  , C2   

0  0

0.25 0.01 0.01

  ,  

=

   ,   

Ad2 =

0 0   0

 , 0.22   

0 0 0



M1 =   0  0

0



B2 =

0



0  

1.02 0

0 0.15 

0

 ,  

0







 2.134       0.1154  ,    

0.5891



 , 0.03   

0

 0.01       0.01  .    



 0.3 0  



Cd2 =

 1.5    0.5  

0

0   

, 1.6 0    0 0 0.3

M

1

0

ED

1.5

10

20

30

40

50 time(k)

60

70

80

90

100

80

90

100

PT

Figure 1. The jumping modes

0.03

0.02

CE

0.01

0

x1

−0.01

AC

−0.02

−0.03

−0.04

−0.05

−0.06

0

10

20

30

40

50 time(k)

60

70

Figure 2. The trajectory of x1 (k) The nonlinear function Fi (x (k) , x (k − d (k))) = 0.2 sin (x (k)) + 0.1 sin (x (k − d (k))) , 25



0   

, G2 =   0 0.3 0    0 0 0.3

0.01

2

Jumping modes



 0.01 0.018 0   

2.5

0.5

 0.21    0.35  



 3.21    0  

D2 =



CR IP T

 0.8650    0  



AN US

A2 =



ACCEPTED MANUSCRIPT

0.02

0.01

x2

0

−0.01

−0.02

−0.03

−0.04

0

10

20

30

40

50 time(k)

60

70

80

90

100

CR IP T

Figure 3. The trajectory of x2 (k) 0.03

0.02

0.01

x3

0

−0.01

−0.02

−0.04

−0.05

0

10

20

30

40

AN US

−0.03

50 time(k)

60

70

80

90

100

90

100

Figure 4. The trajectory of x3 (k) −3

8

x 10

M

7

6

4

3

2

1

0

10

20

30

40

50 time(k)

60

70

80

PT

0

ED

xT(k)ETRiEx(k)

5

Figure 5. The trajectory of of xT (k) E T Ri Ex (k)

CE

and it is obvious that

AC

FiT (x (k) , x (k − d (k))) Fi (x (k) , x (k − d (k)))

≤ 2(0.2 sin (x (k)) + 0.1 sin (x (k − d (k))))2 T T ≤ xT (k) GT i Gi x (k) + x (k − d (k)) Li Li x (k − d (k)) .

The transition rate matrix is Y





0.3 0.7  =  , 0.6 0.4 26

ACCEPTED MANUSCRIPT

1 0.8 0.6 0.4 0.2 x2

0 −0.2 −0.4 −0.6 −0.8 −0.5

0 x1

0.5

1

CR IP T

−1 −1

Figure 3. The estimation of attraction domain in addition, we take 1   0  

E=



0 0 1

  , 0  

R=

000

 −0.01    0  

0

0

−0.01

0

0

−0.01

0

     



   ,  



0

0 0 

 , R1 = R2 = I3 , 0 0   0.25 0.13 0

0

R=



 12    0  

M

Q=





AN US





0 0 

15 0

0 0 16

  ,  

S=



 0.1    0  

0



0   

, 0.3 0    0 0 0.1

PT



ED

and ρ1 = 1.53, ρ2 = 0.01, ρ3 = 0.004, ρ4 = 1.5, α1 = 0.1, α2 = 1.1, α3 = 0.1, N = 2, β = 1.2, µ = 2.8, d1 = 1, d2 = 2, c1 = 0.001, then by Theorem 3 and Remark 2, we can obtain the optimal values γ = 0.2652, c2 = 3.1786 and the corresponding state feedback gain matrices are as follows: 



K2 = −0.2091 0.0009 0.9515 .

CE

K1 = 0.0252 0.0069 1.3078 ,



AC

Choose the jumping signal as Fig.1, and the disturbance function is assumed to be ω (k) = 0.5e−k , then the state responses are shown in Fig.2, Fig.3, Fig.4 and Fig.5. From the figures, it is obvious the system is finite-time stability and finite time dissipative. On the other hand, for given γ = 3, c2 = 5,solve the optimization problem shown in remark ,we can obtain the maximum estimate of the attraction domain δmax = 0.0532, and it is shown in Fig.6. In this example, by constructing discrete-time Markovian control system, we have studied the water pollution model. The figures show that the with the state 27

ACCEPTED MANUSCRIPT feedback controller, the concentrations per unit volume of biological oxygen demand and dissolved oxygen the biochemical constituents of the water become stable and then the water problem has been resolved.

CR IP T

Example 3 Consider a four-modes singular discrete-time Markovian jump system and the coefficient matrices are as follows:





 −1.25 0.3 0.01 

0.5

0   0  



D1 =

0.01



1

  , 0  

0 0 0

  , 0  

C1 =

PT

0   0  





000 

 0.2       0.2  ,    



Ad1 =   0 

M1 =

0 0 0

0.2

AC

0

0.03 0 

CE

L1 =



 0.08



 0.01    0  

ED

Bω1 =

−1 0

0

 

  ,  



Dω1 =





 −0.5    1  



0

 1.5    2  

0

0.05 

0.01 0 

0

0

0



1

  

. 1.6 1    1 1 0

28

  , 0  

  , 1  

1.5 1.3 

Mode 2

  , 0  

G1 =

0 0

−0.2 0  −2



0

0.01 0.001 0

M



A1 =  

AN US

Mode 1

Cd1 =



 −2.26  



 B1 =   0.8  ,     −1.3



 0.1    0  

 1.4    2.8  

0





0



0   

, 0.2 0    0 0 0.1 

2.5 0.1 

3.1 0

  , 0.2   

2

ACCEPTED MANUSCRIPT

0.8 0 

−1 0

0.02 0 −1

Bω2 =

L2 =



0   0  

0.001 0

0



0

0 0







0

 0.001    0  

Ad2 =



   , 0.001   

M2 =

  , 0  

C2 =

000



 −3.2    2  

 

Bω3 =

0

0   0  

  , 0.02   

0

1.3



0.001 0.01  0.01

0

0

CE



Ad3 =

ED



−1.2

0 

PT

A3 =

0

0



0

 0.1 0 0  

0

  ,  



M3 = 

0

  , 0  

0

Cd2 =







 −2       1 ,    

1



  , 0  

G2 =

0

 0.5    1.6  





 0.1    0  



 −0.1    0  

0



0.004

0

 0.01    0  

0

0

0

0

 

0   

, 0.1 0    0 0.1

3.1 0.1   



0

  , 0  

B3 =

0



0.01 0 

0.001



, 1.5 0.2    0 0 0.1

0

 −1.2 −1.2 1  

B2 =

0

0.001

M

Mode 3 





0

0 1



 −1.3    0  

0

  , 0  

    0  , Dω2 =  1 2.6 2  . D2 =          3 3 2 0.1



0

 0.001    0  

3

1 2 1

 0.1 

0.002

0.2 0 

0







0

0

0.001



0   0  

0

  ,  



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 −1.5    0  



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A2 =



  , 0  

0

G3 = 





 −2.15       0.1  ,    



−1.2

0   0  



0 0  

, 0 0   000



 2 1.5 0  



  , Cd3 =   3 1.5 0  , 0.8 0.2        0 0.1 0.1 0 0.3

D3 =

 

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  L3 =   0 0.2 0  , C3 =  1.9       0 0 1 0.3 



 0.3       1 ,    

0.2

Dω3 =



 1.2    2  



1.3 0.1 

. 2.5 2    1 2 3

Mode 4 29

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 2.5    −0.08  

Bω4 =

L4 =



1.8 

  , 0  



0   0  

0 0





0

  , 0  

M4 =

C4 =

000

 0.01    0  

 0    0.01  

0.01 0

 −3.2    2  

0



0.2 0 

0

  , 0  

0.1



0

0

Cd4 =

0 1



 2.3 1.5 2 

 0.1 

B4 =

0 0

   , 0.001   

  , 0  

3



0.02 0 

0



0









Ad4 =

2 0.8

0 0 0

  , 0.7   

0 00





0 0.3 

1

 0.1    0  



G4 =



 0.5    1.6  

1

0   0  

0.2 0.15



3.1 0.1   

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0.1 0.05 15

10

10

8 6

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5

0

c2

4 2

µ

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Figure 7. Local optimal bound of c2 and γ

4.5

3 modes

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4

3.5

2.5

2

1.5

1

0.5

0

10

20

30

40

50 time(k)

60

70

Figure 8. The jumping modes

30

 

, 0 0   000

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0.25



0 0

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    0.3  , Dω4 =  2.6 1.6 6  . D4 =          1 2.5 1.4 0.5

γ

 −2.2       0.1  ,    

, 1.5 0.2    0 0 0.1











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A4 =



80

90

100

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0.04

0.03

0.02

x1

0.01

0

−0.01

−0.02

−0.03

−0.04

0

10

20

30

40

50 time(k)

60

70

80

90

100

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Figure 9. The trajectory of x1 (k) 0.04

0.03

0.02

x2

0.01

0

−0.02

−0.03

−0.04

0

10

20

30

40

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−0.01

50 time(k)

60

70

80

90

100

90

100

Figure 10. The trajectory of x2 (k)

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0.04

0.03

0.01

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x3

0.02

0

−0.01

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−0.02

0

10

20

30

40

50 time(k)

60

70

80

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Figure 11. The trajectory of x3 (k)

The nonlinear function Gi x (k) + e−kx(k)k Li x (k − d (k)) , 1 + kx (k − d (k))k2 (38) T note that Gi Li = 0 for all i ∈ S = {1, 2, 3, 4}. Therefore, it follows from (35) that FiT (x (k) , x (k − d (k))) Fi (x (k) , x (k − d (k))) T T ≤ xT (k) GT i Gi x (k) + x (k − d (k)) Li Li x (k − d (k))

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Fi (x (k) , x (k − d (k))) =

which satisfied the assumption. The transition probability matrix is partly 31

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0.014

0.012

xT(k)ETRiEx(k)

0.01

0.008

0.006

0.004

0.002

0

0

10

20

30

40

50 time(k)

60

70

80

90

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Figure 12. The trajectory of of xT (k) E T Ri Ex (k) 1 0.8 0.6 0.4 0.2 x2

0 −0.2

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−0.4 −0.6 −0.8 −1 −1

−0.5

0 x1

0.5

1

Figure 13. The estimation of attraction domain known and given by

PT

AC

I3 ,

CE

in addition, we take E =

Q=



 −0.01    0  

0



= 

1   0  



?       ? 0.25 0.12 ? 

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 0.3 0.3   

?

?

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Y



?

? 0.3





  , 0  

R=

000

0

0

−0.01

0

0

−0.01



   ,  

,  0.5   

0.3 ?

0 0 1

?

R=



0   0  

 14    0  



0 0  

, 0 0   212 

0 0 

16 0

0 0 25

  ,  

R1 = R2 = R3 = R4 =

S=



 0.1    0  

0



0   

0.3 0    0 0 0.3

and ρ1 = 1.05, ρ2 = 0.1, ρ3 = 0.01, ρ4 = 1.2, α1 = 0.1, α2 = 1.3, α3 = 32

ACCEPTED MANUSCRIPT 0.54, N = 2, β = 1.8 d1 = 0.001, d2 = 1.350, c1 = 0.05, by theorem 3 we know that the optimal bound with minimum of c22 + γ 2 relies on the parameter µ. We can find feasible solution when 2.78 ≤ µ ≤ 11.81. Fig.7 shows the optimal value with different µ, when µ = 3.5,we obtain the optimal values γ = 0.1160, c2 = 4.0024 and the corresponding state feedback gain matrices are as follows:



K1 = 0.4324 0.0047 0.8038 , 

K4 = −1.0146 −2.0173 0.877







K2 = −0.0208 0.0007 0.9986 ,



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Choose the jumping signal as Fig.8, and the disturbance function is assumed to be ω (k) = 0.5e−k , then the state responses are shown in Fig.9, Fig.10, Fig.11 and Fig.12. Form the state trajectories, it can be easily found that the system is finite-time stable. On the other hand, for given γ = 4, c2 = 5,solve the optimization problem shown in remark ,we can obtain the maximum estimate of the attraction domain δmax = 0.0651, and it is shown in Fig.13.

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In this example, both the partly unknown transition rates and the actuator saturation are considered. And with the criteria shown in theorem 3, we obtain the controller gains and the optimal values that guarantee the system finitetime stable and finite-time dissipative. Otherwise, the figures further illustrate that.

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CE

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Remark 7 To illustrate the validity and utility of the proposed methods, we present the above examples. With the results and figures shown in the examples, it is obvious that the Makovain systems both with completely known transition rates (shown in example 1 and example 2) and partly unknown transition rates (shown in example 3) are finite-time stability and finite-time dissipative. Moreover, example 1 has compared with some existing results in special cases and example 2 has analyzed the water pollution model, which can visually and clearly state that the proposed methods have better conservatism and can be used in some practical cases.

5

Conclusion

This paper considers the finite-time dissipative control for singular discretetime Makovian jump systems that both contain actuator saturation and partly 33



K3 = 0.0071 0.0029 1.0958 ,

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6

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unknown transition rates. With the LMI method and proper Lyaponov- Krasonski functional that includes the information of time-varying dealy, delaydependent sufficient conditions that ensure the systems finite-time stable and finite-time dissipative are obtained. Then the state feedback controllers are designed and the optimal values are obtained by extended the results into LMI optimization problems. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed approach. Moreover as the asynchronous controller design maybe conservative and the some recent results are obtained, in our future work, we maybe will focus on finite-time asynchronous dissipative control and obtain some other important results.

Acknowledgment

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This work is partially supported by the National Natural Science Foundation of China No. 61273004, and the Natural Science Foundation of Hebei province No. F2014203085. Moreover, the authors would like to thank the editor and anonymous reviewers for their many helpful comments and suggestions to improve the quality of this paper.

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