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

Leipo Liu; Hao Xing; Xiangyang Cao; Xiushan Cai; Zhumu Fu

doi:10.1515/math-2019-0058

Artikel Open Access

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

Leipo Liu , Hao Xing , Xiangyang Cao , Xiushan Cai und Zhumu Fu

Veröffentlicht/Copyright: 24. Juli 2019

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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.

Keywords: Guaranteed cost finite-time control; Nonfragile positive observer design; Discrete-time positive impulsive switched systems; Average dwell time

MSC 2010: 15-xx; 65-xx; 93-xx

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 a_ij > 0 (≥ 0, < 0, ≤ 0), which is also applying to a vector. A ≻ B (A ⪰ B) means that A – B ≻ 0 (A – B ⪰ 0). R+n is the n-dimensional non-negative (positive) vector space. R^n×n denotes the space of n × n matrices with real entries. A^T denotes the transpose of matrix A. N and N⁺ are the sets of non-negative and positive integers. Z⁺ denotes the set of positive integers. Matrices are assumed to have compatible dimensions for calculating if their dimensions are not explicitly stated.

2 Preliminaries and problem statements

Consider the following DPISS:

x(k+1)=Aσ(k)x(k)+Bσ(k)u(k),k≠km−1,m∈Z+x(k+1)=Eσ(k)x(k),k=km−1,m∈Z+y(k)=Cσ(k)x(k)x(k0)=x0 (1)

where k ∈ N, x(k) ∈ Rⁿ is the system state, u(k) ∈ R^m 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, A_i, B_i, C_i, E_i are the ith subsystem if σ(k) = i ∈ I. k₀ = 0 is the initial time. k_m(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). A_p, B_p, C_p, E_p 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 A_i ⪰ 0, B_i ⪰ 0, C_i ⪰ 0, E_i ⪰ 0, where i ∈ I.

2.1 Positive observer design

We construct the following DPISS (1):

x^(k+1)=(Aσ(k)−Lσ(k)Cσ(k))x^(k)+Bσ(k)u(k)+Lσ(k)Cσ(k)x(k),k≠km−1,m∈Z+x^(k+1)=Hσ(k)x^(k),k=km−1,m∈Z+u(k)=Kσ(k)x^(k)x(k0)=x0 (2)

where x̂(k) ∈ Rⁿ 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 state x̂(k) of system (2) with u(k) ⪰ 0. To this end, it is naturally required, according to Lemma 1, A_i – L_iC_i ⪰ 0, B_i ⪰ 0, L_iC_i ⪰ 0, H_i ⪰ 0, K_i ⪰ 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 ≠ k_m – 1, we have

e(k+1)x^(k+1)=Ai−LiCi0LiCiAi+BiKie(k)x^(k) (3)

When k = k_m – 1, we get

e(k+1)x^(k+1)=EiEi−Hi0Hie(k)x^(k) (4)

For the sake of convenience, we define x̃(k + 1) = [e^T(k + 1) x̂^T(k + 1)]^T. Thus, equation (3) and (4) can be rewritten as

x~(k+1)=Ai~x~(k),k≠km−1,m∈Z+x~(k+1)=Bi~x~(k),k=km−1,m∈Z+x~(k0)=x~0 (5)

where

Ai~=Ai−LiCi0LiCiAi+BiKi,Bi~=EiEi−Hi0Hi

Remark 2

According to Lemma 1, if error dynamic system (5) is positive, then it should guarantee that Ã_i ≥ 0, B̃_i ≥ 0, ∀ i ∈ I(It means A_i – L_iC_i ⪰ 0, L_iC_i ⪰ 0, A_i + B_iK_i ⪰ 0, E_i ⪰ 0, E_i – H_i ⪰ 0, H_i ⪰ 0).

Definition 2

For any switching signal σ(k) and any k₂ ≥ k₁ ≥ 0, let N_σ(k₁, k₂) denote the switching numbers over the interval [k₁, k₂). For given τ_α > 0 and N₀ > 0, if the inequality

Nσ(k1,k2)≤N0+k2−k1τα (6)

holds, then τ_α 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 T_f and two vectors α ≻ β ≻ 0, discrete-time positive impulsive switched systems (5) with is said to be FTS with respect to (α, β, T_f, σ(k)), if

x~T(0)β≤1⇒x~T(k)α<1, ∀k∈[0,Tf] (7)

Remark 3

In (5), the state x̃^T(k) includes the e(k) and x̂(k). From Definition 3, our goal is that the weighted system x̃^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 definitions for our further study.

Definition 4

[24]. Define the cost function of DPISS (5) as follows:

J=∑s=0Tf−1(xT(s)R1+uT(s)R2) (8)

where R₁ ≻ 0 and R₂ ≻ 0 are two given vectors.

Definition 5

[24]. (GCFTS) For a given time constant T_f 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, T_f, 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

x^(k+1)=(Aσ(k)−(Lσ(k)+△Lσ(k)))x^(k)+Bσ(k)u(k)+(Lσ(k)+△Lσ(k))Cσ(k)x(k)k≠km−1,m∈Z+x^(k+1)=Hσ(k)x^(k),k=km−1,m∈Z+u(k)=Kσ(k)x^(k)x(k0)=x0 (9)

where △L_i ∈ R^n×p are uncertain real-valued matrices which satisfy △Li∈[L1′,L2′].L1′∈Rn×p,L2′∈Rn×p.

We can obtain the following error system

x~(k+1)=Ai~x~(k), k≠km−1,m∈Z+x~(k+1)=Bi~x~(k), k=km−1,m∈Z+x~(k0)=x~0 (10)

where

Ai~=Ai−(Li+△Li)Ci0(Li+△Li)CiAi+BiKi,Bi~=EiEi−Hi0Hi

Remark 5

According to Lemma 1, if observer system (9) and error dynamic system (10) are positive, then it should guarantee that Ã_i ≥ 0, B̃_i ≥ 0, ∀i ∈ I(It means A_i – (L_i + △L_i)C_i ⪰ 0, (L_i + △L_i)C_i ⪰ 0, A_i + B_iK_i ⪰ 0, B_i ⪰ 0, E_i ⪰ 0, E_i – H_i ⪰ 0, K_i ⪰ 0, H_i ⪰ 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 T_f, vectors α ≻ β ≻ 0 and R₁ ≻ 0, R₂ ≻ 0, 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 ϕ₁, ϕ₂, ξ > 1, μ > 1, and such that the following inequalities hold:

(AiT−CiTLiT)νi+R1+CiTLiTυi−ξνi≺0 (11)

(AiT+KiTBiT)υi+R1+KiTR2−ξυi≺0 (12)

EiTνj−μνi≺0 (13)

(EiT−HiT)νj+HiTυj−μυi≺0 (14)

ϕ1α≺ψi≺ϕ2β (15)

Ai−LiCi≻0 (16)

Ai+BiKi≻0 (17)

Ei−Hi≻0 (18)

ϕ1>ϕ2ξTf (19)

ψ_i = [ψ_i1, ψ_i2, …, ψ_in]^T, ψ_ir represents the ith elements of the vectors ψ_i, respectively, then under the following ADT scheme

Tα>Tα∗=Tflnμlnϕ1−lnϕ2−Tflnξ (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

J=∑s=0Tf−1(xT(s)R1+uT(s)R2)≤J∗=ξTfμTfTα∗ϕ2 (21)

Proof

Construct the following multiple linear co-positive Lyapunov function for the systems (5) as follows:

Vix~(k)=x~T(k)ψi (22)

where ψi=νiTυiTT,i∈I.

Supposing a switching sequence 0 = k₀ ≤ k₁ ≤ k_m ≤ k_m+1 ≤ … ≤ T_f. Without loss of generality, we assume that subsystem i is activated at the switching instant k_m–1 and the subsystem j is activated at the switching instant k_m.

When k ∈ [k_m–1, k_m – 1), m ∈ N, σ(k) = σ(k + 1) = i, along the trajectory of system (5), the difference of the MLCLF is

△Vi(x~(k)+xT(k)R1+xT(k)KiTR2=x~T(k+1)ψi−x~T(k)ψi+xT(k)R1+xT(k)KiTR2=x~T(k)(AiT−CiTLiT)νi+R1+CiTLiTυi(AiT+KiTBiT)υi+R1+KiTR2−x~T(k)ψi (23)

From (11) and (12), we have

△Vi(x~(k))+xT(k)R1+xT(k)KiTR2≤(ξ−1)x~T(k)ψi (24)

it implies

Vi(x~(k+1))≤ξx~T(k)ψi (25)

When k = k_m – 1, σ(k + 1) = σ(k_m) = j, σ(k) = σ(k_m – 1) = i, i ≠ j. Along the trajectory of system (5), the difference of MLCLF is

Vj(x~(k+1))−μVi(x~(k))=x~T(k+1)ψj−μx~T(k)ψi=x~T(k+1)ψj−x~(k)ψi≤x~T(k)EiTνj−μνi(EiT−HiT)νj+HiTυj−μυi (26)

From (13) and (14), we have

Vj(x~(k+1))≤μVi(x~(k)),i≠j. (27)

So, when k ∈ [k_m, k_m+1), from (27), we get

Vσ(k)(x~(k))<ξk−kmVσ(km)(x~(km))<μξk−kmVσ(km−1)(x~(km−1)) (28)

Repeating the procedure of (28) and noting ξ > 1, we obtain

Vσ(k)(x~(k))<ξk−kmVσ(km)(x~(km))<μξk−kmVσ(km−1)(x~(km−1))<μξk−kmξkm−1−km−1Vσ(km−1)(x~(km−1))<μξk−km−1Vσ(km−1)(x~(km−1)) (29)

By iterative operation, we get

Vσ(k)(x~(k))<μ2ξk−km−2Vσ(km−2)(x~(km−2))≤……≤≤μNσ(k,k0)ξk−k0Vσ(k0)(x~(k0))=μNσ(k,k0)ξk−k0Vσ(k0)(x~(k0)) (30)

From (15) and (22), we have

Vσ(k)(x~(k))=x~T(k)ψσ(k)≥ϕ1x~T(k)α (31)

Vσ(k0)(x~(k0))=x~T(0)ψσ(k0)≤ϕ2x~T(0)β (32)

From (30)-(32), we obtain

x~T(k)α≤1ϕ1μTfTαξTfϕ2xT(0)β≤1ϕ1μTfTαξTfϕ2 (33)

Substituting (20) into (33), one has

x~T(k)α<1 (34)

According to Definition 3, we conclude that the system (5) with u(k) = 0 is FTS with respect to (α, β, T_f, σ(k)).

Next, we will give the guaranteed cost value of system (5).

When k ∈ [k_m–1, k_m – 1), m ∈ N, according to (24), we know

Vi(x~(k))≤ξx~T(k)νi−xT(k)R1−xT(k)KiTR2 (35)

Similar to the proof process of (25)-(30), for any k ∈ [0, T_f] and μ > 1, we can obtain

Vσ(k)(x~(k))<μNσ(k,k0)ξk−k0Vσ(k0)(x~(k0))−∑s=k0k−1xT(s)(R1+KiTR2) (36)

Noting that V_σ(k)(x̃(k)) > 0, (36) can be rewritten as

0<μNσ(k,k0)ξk−k0Vσ(k0)(x~(k0))−∑s=0k−1x(s)T(R1+KiTR2) (37)

Letting k = T_f, we get

∑s=0Tf−1x(s)T(R1+KiTR2)<μNσ(k,k0)ξk−k0(Vσ(k0)x~(k0)) <μNσ(Tf,k0)ξTf−k0(Vσ(k0)x~(k0)) (38)

Then we can obtain

J=∑s=0Tf−1xT(s)(R1+KiTR2)≤J∗=ξTfμTfTα∗ϕ2 (39)

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 K_i and L_i. So, an algorithm is presented to obtain the feedback gain matrices K_i, L_i, i ∈ I.

Algorithm 1

By adjusting the parameters ξ, μ and Letting f_p = KpTBpTυp , then solving (13)-(19) via linear programming, positive vectors ν_p, υ_p, K_p and L_p can be obtained.
Substituting υ_p and K_p into f_p = KpTBpTυp , f̃_p can be obtained. The gain L_p, ν_p and υ_p are substituted into (12).
If f̃_p – f_p ⪯ 0 and (12) are all satisfied at the same time, then K_p and L_p are admissible. Otherwise, return to Step 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 T_f, vectors α≻β≻0,L1′,L2′ and R₁ ≻ 0, R₂ ≻ 0, 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 ϕ₁, ϕ₂, ξ > 1, μ > 1, such that (12)-(15), (17)-(19) and the following inequalities hold:

(AiT−CiT(Li+△Li)Tνi+R1+CiT(Li+△Li)Tυi−ξνi≺0 (40)

Ai−(Li+△Li)Ci≻0 (41)

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

J=∑s=0Tf−1(xT(s)R1+uT(s)R2)≤J∗=ξTfμTfTα∗ϕ2 (42)

Proof

Replacing (AiT−CiTLiT)νi+R1+CiTLiTυi in (23) with (AiT−CiT(Li+△Li)Tνi+R1+CiT(Li+△Li)Tυi, 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

(AiT−CiT(Li+L1′)Tνi+R1+CiT(Li+L2′)Tυi−ξνi≺0 (43)

Ai−(Li+L1′)Ci≻0 (44)

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

J=∑s=0Tf−1(xT(s)R1+uT(s)R2)≤J∗=ξTfμTfTα∗ϕ2 (45)

Proof

If (AiT−CiT(Li+L1′)Tνi+R1+CiT(Li+L2′)Tυi−ξνi≺0 and Ai−(Li+L1′)Ci≻0, we can easily get (AiT−CiT(Li+△Li)Tνi+R1+CiT(Li+△Li)Tυi−ξνi≺0 and A_i – (L_i + △L_i)C_i ≻ 0. Then, the conditions of Theorem 2 are satisfied. 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

By adjusting the parameters ξ, μ and Letting f_p = KpTBpTυp , then solving (13)-(15), (17)-(19) and (44) via linear programming, positive vectors ν_p, υ_p, K_p and L_p can be obtained.
Substituting υ_p and K_p into f_p = KpTBpTυp , f̃_p can be obtained. The gain L_p, ν_p and υ_p are substituted into (43).
If f̃_p – f_p ⪯ 0 and (43) are all satisfied at the same time, then K_p and L_p are admissible. Otherwise, return to Step 1.

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:

A1=0.10.30.20.3,B1=0.30.10.10.2,E1=0.20.30.30.2,H1=0.10.20.10.1,C1=0.150.2,A2=0.20.150.30.2,B2=0.10.20.20.1,E2=0.30.40.20.5,H2=0.20.20.10.4,C2=0.10.3,R1=0.50.6,R2=0.10.2,L1′=0.030.02,L2′=0.040.03α=0.2,0.35,0.1,0.4T,β=0.4,0.1,0.2,0.1T,

Choosing T_f = 5, ξ = 1.1 and μ = 1.05. Solving the inequalities in Theorem 3 by linear programming, we have

ν1=0.15220.1594,ν2=0.15220.1593,υ1=0.08170.0869,υ2=0.08150.0888,K1=0.03810.14980.00520.1234,K2=0.05830.15050.02350.1244,L1=0.38290.4170,L2=0.39530.4258,f1=0.01940.0825,f2=0.02570.0929,

It is easy to firm that f̃_p – f_p ⪯ 0 and (10) are satisfied. then K_p and L_p are admissible. According to (20), we get Tα∗=1.7.

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 x̃^T(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, T_f, σ(k)) to (α, β, T_f, σ(k)), and the cost value J^∗ = 2.33, which can be obtained by (45).

$Fig. 1 The state trajectory of x1 and state observation trajectory x̂1.$

Fig. 1

The state trajectory of x₁ and state observation trajectory x̂₁.

$Fig. 2 The state trajectory of x2 and state observation trajectory x̂2.$

Fig. 2

The state trajectory of x₂ and state observation trajectory x̂₂.

$Fig. 3 State of the error dynamic system.$

Fig. 3

State of the error dynamic system.

$Fig. 4 Switching signal of system (5) with ADT.$

Fig. 4

Switching signal of system (5) with ADT.

$Fig. 5 The evolution of x̃T(k)α of system (5).$

Fig. 5

The evolution of x̃^T(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).

Conflict of Interest: The authors declare that there is no conflict of interests regarding the publication of this paper.

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Received: 2018-09-04

Accepted: 2019-05-05

Published Online: 2019-07-24

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Artikel in diesem Heft

https://doi.org/10.1515/math-2019-0058

Schlagwörter für diesen Artikel

Guaranteed cost finite-time control; Nonfragile positive observer design; Discrete-time positive impulsive switched systems; Average dwell time

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