Nonfragile observer-based guaranteed cost finite-time control of discrete-time positive impulsive switched systems

Abstract This paper considers the nonfragile observer-based guaranteed cost finite-time control of discrete-time positive impulsive switched systems(DPISS). Firstly, the positive observer and nonfragile positive observer are designed to estimate the actual state of the underlying systems, respectively. Secondly, by using the average dwell time(ADT) approach and multiple linear co-positive Lyapunov function (MLCLF), two guaranteed cost finite-time controller are designed and sufficient conditions are obtained to guarantee the corresponding closed-loop systems are guaranteed cost finite-time stability(GCFTS). Such conditions can be solved by linear programming. Finally, a numerical example is provided to show the effectiveness of the proposed method.


Introduction
The switched system is a type of hybrid systems. It comprises a set of a di erential or di erence equations and a switched controller, which designate the switching between subsystems at a speci c interval of time. It has been studied very well, see [1][2][3][4]. As a special kind of switched systems, the positive switched systems whose output and state are non-negative have been paid much attention and adopted to many practical applications, such as communication networks [5], viral mutation [6], formation ying [7], and so on. There have been many available results about discrete-time positive switched systems [8][9][10]. Because of sudden changes in the state of the system at certain instants of switching, many practical switched systems exhibit impulsive dynamical behavior, these systems are usually called impulsive switched systems [11]. For discretetime positive impulsive switched system, a few results have been obtained, see [12,13]. In [12], the exponential stability for a class of discrete-time positive impulsive switched linear systems was studied. In [13], the nitetime stability for a class of discrete-time positive impulsive switched time-delay systems under asynchronous switching was discussed. However, [12] and [13] are based on the assumption of the known state.
Moreover, it is necessary to design a state observer, because the states of the systems are not all measurable in practice [14]. Recently, the exponential stability property of the proposed switching observer was discussed, and an LMI-based algorithm was given for discrete-time impulsive switched nonlinear systems with time-varying delays in [15]. The problem of state observation for continuous-time and discrete-time impulsive switched systems was investigated in [16]. Furthermore, when the state observer gain variations could not be avoided, a kind of nonfragile state observers for discrete-time switched nonlinear systems was proposed in [17]. However, [15][16][17] were involved in non-positive systems and the design of nite time controller was not considered.
On the other hand, in most practical applications, the researchers are more interested in designing the control system, which is not only nite-time stable but also guarantees an adequate level of performance. One method to this problem is so-called guaranteed cost nite time control. Some remarkable results have been presented, see [18][19][20][21][22][23]. These results mainly focus on non-positive systems. Very recently, In [24], guaranteed cost nite-time control was extended to fractional-order positive switched systems and a cost function for fractional-order positive systems (or fractional-order positive switched systems) was proposed. In [25], the problem of guaranteed cost nite-time control for positive switched linear systems with time-varying delays was considered and a cost function of positive systems (or positive switched systems) was also presented. It is worth noting that [24] and [25] are involved in continuous-time positive switched systems with known states. To the best of our knowledge, the problem of observer-based guaranteed cost nite-time control of discretetime impulsive switched systems has not been fully investigated, especially for DPISS, which motivates us for this study.
In this paper, the problem of nonfragile observer-based guaranteed cost nite-time control of DPISS is considered. The co-positive type Lyapunov function with average dwell time (ADT) technique is constructed. The main contributions lie in two aspects: 1) For DPISS, the design methods of positive observer and nonfragile positive observer are rstly proposed, respectively. 2) Two types of guaranteed cost nite-time controller are designed to guarantee the corresponding closed-loop systems are GCFTS, the obtained conditions can be easily solved by linear programming. The rest of the paper is organised as follows: Section 2 gives some necessary preliminaries and problem statements. In Section 3, the main results are given. In Section 4, a numerical example is provided. Section 5 concludes the paper.

Preliminaries and problem statements
Consider the following DPISS: where k ∈ N, x(k) ∈ R n is the system state, u(k) ∈ R m represents the control input. σ(k) represents switching signal of system and takes values in a nite set I = , , . . . , S, S ∈ N + . In general, A i , B i , C i , E i are the ith subsystem if σ(k) = i ∈ I. k = is the initial time. km(m ∈ Z + ) denotes the bth impulsive switching instant. Moreover, σ(k) = i ∈ I means that the ith subsystem is active. σ(k − ) = j and σ(k) = i(i ≠ j) indicate that k is a switching instant at which the system is switched from the jth subsystem to the ith subsystem. At switching instants, there exist impulsive jumps described by (1). Ap, Bp, Cp , Ep are constant matrices with suitable dimensions.
Next, we will give some de nitions and lemmas for the system (1).

De nition 1 [5]. System (1) with u(k)
is positive if x(k ) and any switching signals σ(k), the corresponding trajectories x(k) , y(k) hold for all k ≥ k .

. Positive observer design
We construct the following DPISS (1): wherex(k) ∈ R n is the estimated state vector of x(k),ŷ(k) ∈ R p is the observer output vector. L i ∈ R n×p is the observer gain, H i ∈ R n×n , ∀i ∈ I is the matrix to be determined. Remark 1. For DPISS (1), it not only requires that the state of the designed observer converges to that of the considered system, but also guarantees the positivity of the estimated statex(k) of system (2) with u(k) . To this end, it is naturally required, according to Lemma Let e(k) = x(k) −x(k) be the estimated error, then we can obtain the following error system: When k = km − , we get e(k + ) For the sake of convenience, we de nex(k + ) = e T (k + )x T (k + ) T . Thus, equation (3) and (4) can be rewritten as According to Lemma 1, if error dynamic system (5) is positive, then it should guarantee thatÃ i ≥ ,

De nition 2.
For any switching signal σ(k) and any k ≥ k ≥ , let Nσ(k , k ) denote the switching numbers over the interval [k , k ). For given τα > and N > , if the inequality (6) holds, then τα is called an average dwell time, and N is called a chattering bound. Generally, we choose N = .
De nition 3. (Finite-Time Stability(FTS)). For a given time T f and two vectors α β , discrete-time positive impulsive switched systems (5) with is said to be FTS with respect to (α, β, T f , σ(k)), if Remark 3. In (5), the statex T (k) includes the e(k) andx(k). From De nition 3, our goal is that the weighted systemx T (k)α does not exceed threshold 1 in a given time interval T f , then the estimation error e(k) might not converge to zero in a given time interval T f . If a smaller threshold is chosen, then the estimation error will become very small. Now we give some new de nitions for our further study. De nition 4 [24]. De ne the cost function of DPISS (5) as follows: where R and R are two given vectors. De nition 5 [24]. (GCFTS) For a given time constant T f and two vectors α β , consider DPISS (5) and cost function (8), if there exist a control law u(k) and a positive scalar J * such that the closed-loop system is FTS with respect to (α, β, T f , σ(k)) and the cost function satis es J ≤ J * , then the closed-loop system is called GCFTS, where J * is a guaranteed cost value and u(k) is a guaranteed cost nite-time controller. Remark 4. [14] noted that one could not stabilize any unstable positive system by using extended Luenberger type positive observers. But nite time stability means that the system state does not exceed the speci ed boundary within a given time interval and it is di erent from the asymptotic stability. So the problem of positive observer-based guaranteed cost nite-time control of DPISS is feasible.

. Nonfragile positive observer design
If the state observer gain variations could not be avoided, a kind of nonfragile state observer will be designed as follows where L i ∈ R n×p are uncertain real-valued matrices which satisfy L i ∈ [L , L ]. L ∈ R n×p , L ∈ R n×p .
We can obtain the following error system According to Lemma 1, if observer system (9) and error dynamic system (10) are positive, then it should guarantee thatÃ i ≥ , . The aim of this paper is to design the positive observer and nonfragile positive observer based on state feedback controller, and nd a class of switching signals σ(k) for systems (2) and (6) such that the corresponding closed-loop systems are GCFTS, respectively.

Main results . Observer-based guaranteed cost nite-time stability analysis
In this subsection, we will focus on the problem of GCFTS for DPISS (5). The following theorem gives su cient conditions of GCFTS for system (5).

Theorem 1. Consider the system (5), for a given time constant T f , vectors α β
and R , R , if there exist a set of positive vectors ν i , ν j , υ i , υ j , i ≠ j, i ∈ I, positive matrices K i , L i and positive constants ϕ , ϕ , ξ > , µ > , and such that the following inequalities hold: . . , ψ in ] T , ψ ir represents the ith elements of the vectors ψ i , respectively, then under the following ADT scheme (20) the system (5) is GCFTS with respect to (α, β, T f , σ(k)) and the guaranteed cost value of system (5) is given by Proof. Construct the following multiple linear co-positive Lyapunov function for the systems (5) as follows: where ψ i = ν T i υ T i T , i ∈ I. Supposing a switching sequence = k ≤ k ≤ km ≤ k m+ ≤ . . . ≤ T f . Without loss of generality, we assume that subsystem i is activated at the switching instant k m− and the subsystem j is activated at the switching instant km.
Next, we will give the guaranteed cost value of system (5). When k ∈ [k m− , km − ), m ∈ N, according to (24), we know Similar to the proof process of (25)-(30), for any k ∈ [ , T f ] and µ > , we can obtain Noting that V σ(k) (x(k)) > , (36) can be rewritten as Letting k = T f , we get Then we can obtain Therefore, according to De nition 5, we can conclude that the system (5) is GCFTS. Thus, the proof is completed.
In Theorem 1, (11) and (12) are nonlinear matrix inequalities, there are no e ective methods to solve K i and L i . So, an algorithm is presented to obtain the feedback gain matrices K i , L i , i ∈ I. Algorithm 1 Step 1. By adjusting the parameters ξ , µ and Letting fp = K T p B T p υp, then solving (13)

. Nonfragile observer-based guaranteed cost nite-time stability analysis
Now we consider system (10); the following theorem gives su cient conditions of GCFTS for system (10). Theorem 2. Consider the system (10), for a given time constant T f , vectors α β , L , L and R , R , if there exist a set of positive vectors ν i , ν j , υ i , υ j , i ≠ j, i ∈ I, positive matrices K i , L i and positive constants ϕ , ϕ , ξ > , µ > , such that (12)-(15), (17)- (19) and the following inequalities hold: then under the ADT scheme (20), the system (10) is GCFTS with respect to (α, β, T f , σ(k)) and the guaranteed cost value of the system (10) is given by similar to the proof of Theorem 1, we easily obtain that the resulting closed-loop system (10) is GCFTS.
Thus, the proof is completed. Theorem 3. Consider the system (10), for a given time constant T f , vectors α β , L , L and R , R , if there exist a set of positive vectors ν i , ν j , υ i , υ j , i ≠ j, i ∈ I, positive matrices K i , L i and positive constants ϕ , ϕ , ξ > , µ > , such that (12)-(15), (17)- (19) and the following inequalities hold: then under the ADT scheme (20), the system (10) is GCFTS with respect to (α, β, T f , σ(k)) and the guaranteed cost value of the system (10) is given by Then, the conditions of Theorem 2 are satis ed. So the resulting closed-loop system (10) is GCFTS.
To obtain the feedback gain matrices K i and L i in Theorem 3, an algorithm is presented. Algorithm 2 Step 1. By adjusting the parameters ξ , µ and Letting fp = K T p B T p υp, then solving (13)- (15), (17)

Numerical example
We present a numerical example to show the e ectiveness of the proposed approach. Without loss of generality, we consider the case of nonfragile positive observer of DPISS (10) with the parameters as follows: . , . .
. , . x are shown in Fig. 1. The state trajectory of x and state observation trajectoryx are depicted in Fig. 2. Fig. 3 plots the state of the error dynamic system. The switching signal σ(k) is depicted in Fig. 4. Fig. 5 plots the evolution ofx(t)α, which implies that the corresponding closed-loop system is GCFTS with respect to (α, β, T f , σ(k)) to (α, β, T f , σ(k)), and the cost value J * = . , which can be obtained by (45).

Conclusions
In this paper, we have considered the issue of nonfragile observer-based guaranteed cost nite-time control for DPISS. Based on the ADT approach and co-positive type Lyapunov function technique, two types of guaranteed cost nite-time controller based on positive observer and nonfragile positive observer are designed, and su cient conditions are obtained to guarantee the corresponding closed-loop systems are guaranteed cost nite-time stability(GCFTS), respectively. Such conditions can be solved by linear programming. Finally, a numerical example is given to illustrate the e ectiveness of the proposed method.