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.
1 Introduction
The switched system is a type of hybrid systems. It comprises a set of a differential or difference equations and a switched controller, which designate the switching between subsystems at a specific 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 flying [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 discrete-time 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 finite-time 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 finite 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 finite-time stable but also guarantees an adequate level of performance. One method to this problem is so-called guaranteed cost finite 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 finite-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 finite-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 finite-time control of discrete-time 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 finite-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 firstly proposed, respectively. 2) Two types of guaranteed cost finite-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.
Notations. The representation A ≻ 0 (⪰ 0, ≺ 0, ⪯ 0) means that aij > 0 (≥ 0, < 0, ≤ 0), which is also applying to a vector. A ≻ B (A ⪰ B) means that A – B ≻ 0 (A – B ⪰ 0).
2 Preliminaries and problem statements
Consider the following DPISS:
where k ∈ N, x(k) ∈ Rn is the system state, u(k) ∈ Rm represents the control input. σ(k) represents switching signal of system and takes values in a finite set I = 1, 2, …, S, S ∈ N+. In general, Ai, Bi, Ci, Ei are the ith subsystem if σ(k) = i ∈ I. k0 = 0 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 – 1) = 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 definitions and lemmas for the system (1).
Definition 1
[5]. System (1) with u(k) ⪰ 0 is positive if x(k0) ⪰ 0 and any switching signals σ(k), the corresponding trajectories x(k) ⪰ 0, y(k) ⪰ 0 hold for all k ≥ k0.
Lemma 1
System (1) is positive if and only if Ai ⪰ 0, Bi ⪰ 0, Ci ⪰ 0, Ei ⪰ 0, where i ∈ I.
2.1 Positive observer design
We construct the following DPISS (1):
where x̂(k) ∈ Rn is the estimated state vector of x(k), ŷ(k) ∈ Rp is the observer output vector. Li ∈ Rn×p is the observer gain, Hi ∈ Rn×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 state x̂(k) of system (2) with u(k) ⪰ 0. To this end, it is naturally required, according to Lemma 1, Ai – LiCi ⪰ 0, Bi ⪰ 0, LiCi ⪰ 0, Hi ⪰ 0, Ki ⪰ 0, where i ∈ I.
Let e(k) = x(k) – x̂(k) be the estimated error, then we can obtain the following error system:
When k ≠ km – 1, we have
When k = km – 1, we get
For the sake of convenience, we define x̃(k + 1) = [eT(k + 1) x̂T(k + 1)]T. Thus, equation (3) and (4) can be rewritten as
where
Remark 2
According to Lemma 1, if error dynamic system (5) is positive, then it should guarantee that Ãi ≥ 0, B̃i ≥ 0, ∀ i ∈ I(It means Ai – LiCi ⪰ 0, LiCi ⪰ 0, Ai + BiKi ⪰ 0, Ei ⪰ 0, Ei – Hi ⪰ 0, Hi ⪰ 0).
Definition 2
For any switching signal σ(k) and any k2 ≥ k1 ≥ 0, let Nσ(k1, k2) denote the switching numbers over the interval [k1, k2). For given τα > 0 and N0 > 0, if the inequality
holds, then τα is called an average dwell time, and N0 is called a chattering bound. Generally, we choose N0 = 0.
Definition 3
(Finite-Time Stability(FTS)). For a given time Tf and two vectors α ≻ β ≻ 0, discrete-time positive impulsive switched systems (5) with is said to be FTS with respect to (α, β, Tf, σ(k)), if
Remark 3
In (5), the state x̃T(k) includes the e(k) and x̂(k). From Definition 3, our goal is that the weighted system x̃T(k)α does not exceed threshold 1 in a given time interval Tf, then the estimation error e(k) might not converge to zero in a given time interval Tf. If a smaller threshold is chosen, then the estimation error will become very small.
Now we give some new definitions for our further study.
Definition 4
[24]. Define the cost function of DPISS (5) as follows:
where R1 ≻ 0 and R2 ≻ 0 are two given vectors.
Definition 5
[24]. (GCFTS) For a given time constant Tf and two vectors α ≻ β ≻ 0, 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, Tf, σ(k)) and the cost function satisfies J ≤ J*, then the closed-loop system is called GCFTS, where J* is a guaranteed cost value and u(k) is a guaranteed cost finite-time controller.
Remark 4
[14] noted that one could not stabilize any unstable positive system by using extended Luenberger type positive observers. But finite time stability means that the system state does not exceed the specified boundary within a given time interval and it is different from the asymptotic stability. So the problem of positive observer-based guaranteed cost finite-time control of DPISS is feasible.
2.2 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 △Li ∈ Rn×p are uncertain real-valued matrices which satisfy
We can obtain the following error system
where
Remark 5
According to Lemma 1, if observer system (9) and error dynamic system (10) are positive, then it should guarantee that Ãi ≥ 0, B̃i ≥ 0, ∀i ∈ I(It means Ai – (Li + △Li)Ci ⪰ 0, (Li + △Li)Ci ⪰ 0, Ai + BiKi ⪰ 0, Bi ⪰ 0, Ei ⪰ 0, Ei – Hi ⪰ 0, Ki ⪰ 0, Hi ⪰ 0).
The aim of this paper is to design the positive observer and nonfragile positive observer based on state feedback controller, and find a class of switching signals σ(k) for systems (2) and (6) such that the corresponding closed-loop systems are GCFTS, respectively.
3 Main results
3.1 Observer-based guaranteed cost finite-time stability analysis
In this subsection, we will focus on the problem of GCFTS for DPISS (5). The following theorem gives sufficient conditions of GCFTS for system (5).
Theorem 1
Consider the system (5), for a given time constant Tf, vectors α ≻ β ≻ 0 and R1 ≻ 0, R2 ≻ 0, if there exist a set of positive vectors νi, νj, υi, υj, i ≠ j, i ∈ I, positive matrices Ki, Li and positive constants ϕ1, ϕ2, ξ > 1, μ > 1, and such that the following inequalities hold:
ψi = [ψi1, ψi2, …, ψin]T, ψir represents the ith elements of the vectors ψi, respectively, then under the following ADT scheme
the system (5) is GCFTS with respect to (α, β, Tf, 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
Supposing a switching sequence 0 = k0 ≤ k1 ≤ km ≤ km+1 ≤ … ≤ Tf. Without loss of generality, we assume that subsystem i is activated at the switching instant km–1 and the subsystem j is activated at the switching instant km.
When k ∈ [km–1, km – 1), m ∈ N, σ(k) = σ(k + 1) = i, along the trajectory of system (5), the difference of the MLCLF is
it implies
When k = km – 1, σ(k + 1) = σ(km) = j, σ(k) = σ(km – 1) = i, i ≠ j. Along the trajectory of system (5), the difference of MLCLF is
So, when k ∈ [km, km+1), from (27), we get
Repeating the procedure of (28) and noting ξ > 1, we obtain
By iterative operation, we get
Substituting (20) into (33), one has
According to Definition 3, we conclude that the system (5) with u(k) = 0 is FTS with respect to (α, β, Tf, σ(k)).
Next, we will give the guaranteed cost value of system (5).
When k ∈ [km–1, km – 1), m ∈ N, according to (24), we know
Similar to the proof process of (25)-(30), for any k ∈ [0, Tf] and μ > 1, we can obtain
Noting that Vσ(k)(x̃(k)) > 0, (36) can be rewritten as
Letting k = Tf, we get
Then we can obtain
Therefore, according to Definition 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 effective methods to solve Ki and Li. So, an algorithm is presented to obtain the feedback gain matrices Ki, Li, i ∈ I.
Algorithm 1
3.2 Nonfragile observer-based guaranteed cost finite-time stability analysis
Now we consider system (10); the following theorem gives sufficient conditions of GCFTS for system (10).
Theorem 2
Consider the system (10), for a given time constant Tf, vectors
then under the ADT scheme (20), the system (10) is GCFTS with respect to (α, β, Tf, σ(k)) and the guaranteed cost value of the system (10) is given by
Proof
Replacing
Thus, the proof is completed.
Theorem 3
Consider the system (10), for a given time constant Tf, vectors
then under the ADT scheme (20), the system (10) is GCFTS with respect to (α, β, Tf, σ(k)) and the guaranteed cost value of the system (10) is given by
Proof
If
To obtain the feedback gain matrices Ki and Li in Theorem 3, an algorithm is presented.
Algorithm 2
4 Numerical example
We present a numerical example to show the effectiveness of the proposed approach. Without loss of generality, we consider the case of nonfragile positive observer of DPISS (10) with the parameters as follows:
Choosing Tf = 5, ξ = 1.1 and μ = 1.05. Solving the inequalities in Theorem 3 by linear programming, we have
It is easy to firm that f̃p – fp ⪯ 0 and (10) are satisfied. then Kp and Lp are admissible. According to (20), we get
The simulation results are shown in Figs. 1-5, where the initial conditions of the system (10) are x̃(0) = [2, 2, 1, 1]T, which meet the condition x̃T(0)β < 1. The state trajectory of x1 and state observation trajectory x̂1 are shown in Fig. 1. The state trajectory of x2 and state observation trajectory x̂2 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 of x̃(t)α, which implies that the corresponding closed-loop system is GCFTS with respect to (α, β, Tf, σ(k)) to (α, β, Tf, σ(k)), and the cost value J∗ = 2.33, which can be obtained by (45).
The state trajectory of x1 and state observation trajectory x̂1.
The state trajectory of x2 and state observation trajectory x̂2.
State of the error dynamic system.
Switching signal of system (5) with ADT.
The evolution of x̃T(k)α of system (5).
5 Conclusions
In this paper, we have considered the issue of nonfragile observer-based guaranteed cost finite-time control for DPISS. Based on the ADT approach and co-positive type Lyapunov function technique, two types of guaranteed cost finite-time controller based on positive observer and nonfragile positive observer are designed, and sufficient conditions are obtained to guarantee the corresponding closed-loop systems are guaranteed cost finite-time stability(GCFTS), respectively. Such conditions can be solved by linear programming. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.
Acknowledgement
The authors are thankful for the supports of the National Natural Science Foundation of China under grants U1404610, 61773350 and young key teachers plan of Henan province (2016GGJS-056).
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Conflict of Interest: The authors declare that there is no conflict of interests regarding the publication of this paper.
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