# Stabilization of a Class of Continuous-Time Switched Systems with State Constraints via a Mode-Dependent Switching Method

Authors: Qingyu Su; Peipei Wang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

The stability and the stabilization problems for a class of continuous-time switched systems with state constraints via a mode-dependent switching method are investigated. The paper presents an improved average dwell time method, which considers different decay rates of a Lyapunov function related to each of the active subsystems according to whether the saturations occur or not, respectively. It is shown that the improved average dwell time method is less conservative than the common average dwell time method. Based on the improved average dwell time method, the sufficient conditions and state feedback controllers for stabilization of the switched system are derived. A numerical example is given to illustrate the proposed approach.


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## Body

## 1. Introduction

A switched system is a special class of hybrid systems that consists of a finite number of subsystems and a logical rule that orchestrates switching between these subsystems [1–3]. In the last few years, due to their success in practical applications and importance in theory development, switched systems have received much attention [4–6]. Switched systems arise in many engineering applications, such as aeronautics and astronautics systems [7, 8], mechanical systems [9], and networked control systems [10]. Owing to some safety consideration or inherent limits of devices, the majority of the practical systems have states constraints.In the study of switched systems, one basic research topic is the issue of stability which has attracted most of the attention [11–14]. Many techniques have been studied in the study of switched systems, for example, common Lyapunov function, multiple Lyapunov functions, and switched Lyapunov functions [15–17]. However, the average dwell time method is generally recognized to have more flexibility in stabilization for switched systems [18–23].Up to now, there exist many literatures related to stabilization of switched systems with state constraints [24–27]. However, [28] studies the stability and stabilization of switched linear systems with mode-dependent average dwell time. Reference [29] investigates the stabilization of a class of switched systems with state constraints. It is noteworthy that both of the above literatures did not discuss the stability and stabilization of switched systems with state constraints based on MDADT. In short, according to the author, the problems of stability and stabilization for continuous-time switched systems with state constraints based on mode-independent average dwell time have not been addressed in the existing literatures.All the above observations give rise to the question of how to design the MDADT switching to stabilize the continuous-time switched systems with state constraints. This inspires us for this study.Thus, it is necessary to investigate the stabilization problem for a class of continuous-time switched systems with state constraints based on MDADT, which is an important property for a switched system. The main contributions of this paper are given as follows: (i) an improved average dwell time method is proposed, which is less conservative than the MDADT [28] and the ADT [29]; (ii) the sufficient conditions and state feedback controllers for stabilization of continuous-time switched systems with state constraints under MDADT switching are derived.The remainder of this paper is organized as follows. Section2 is the problem formulation and preliminaries. Main results are given in Section 3, including sufficient condition concerning controller design and an iterative algorithm. Finally, a numerical example is given in Section 4. Concluding remarks are drawn in Section 5.Notations. In this paper, the notation used is standard. Z+ denotes the set of nonnegative integers, Rn represents the n-dimensional space, · stands for the Euclidean vector norm, and I is the identity matrix. Symbol ∗ denotes the symmetry elements in symmetric matrices. C1 means the space of continuously differentiable functions.


## 2. Problem Formulation and Preliminaries

Consider a class of state-constrained continuous-time switched systems given by(1)x˙t=hAσtxt+Bσtuσtt,where x(t)∈Rn and uσ(t)(t), respectively, denote the state vector and the control input. The two-matrix pair (Ap,Bp), ∀σ(t)=p∈L, represents the pth subsystem or pth mode of (1); symbol h(·) denotes the saturation function. σ(t) is a piecewise constant function of time, which takes its values in the finite set L={1,2,…,N};  N>1 is the number of subsystems. In addition, for a switching sequence 0<t1<⋯<ti-1<ti<⋯, symbol ti means the moment of the ith switching. When t∈(ti-1,ti), we say the σ(ti-1)th subsystem is active.Generally speaking, the ideal state feedback is treated asuσ(t)(t)=Fσ(t)x(t), where Fσ(t) is the controller gain to be determined. Then, the resulting closed-loop system is described by(2)x˙t=Apxt,non-constrained,A¯pxt,constrained,where Ap≔Ap+BpFp, which represents the closed-loop system matrix of the pth subsystem.About the saturation functionh(·), we transform it into the vertex of a convex hull to handle the saturations [30].SymbolUn denotes the set of n×n diagonal matrices. The diagonal elements of Un are 0 or 1. Assuming that every element of Un is marked as Us, Us-=I-Us, s∈{1,2,…,2n-1,2n}. Therefore, we have (3)hAx∈coUsAx+Us-L,and the diagonal elements of row diagonally dominant matrix L are negative.Now, let us first revisit the following definition and lemmas for later development.Definition 1 (see [19]).
For a switching signalσ(t) and each t2≥t1≥0, let Nσ(t)(t2,t1) denote the number of discontinuities of σ(t) in the open interval (t1,t2). We say that σ(t) has an average dwell time τa if there exist two positive numbers N0 and τa such that (4)Nσtt2,t1≤N0+t2-t1τa,∀t2≥t1≥0.Remark 2.
Definition1 means that if there exists a positive number τa such that a switching signal has the ADT property, the ADT between any two consecutive switching signals is no smaller than a common constant τa for all subsystems.Lemma 3 (see [29]).
Consider a continuous-time state-constrained switched system:(5)x˙t=fσtxt,xt∈Ω⊂Rn,σt=p∈P,with fp(0)=0, ∀p∈P. Assume that all trajectories remain inside Ω. Let α, β, and μ be some given constants satisfying α>β>0, μ≥1. If there exist some piecewise continuous functions Vp(x(t)):Ω→R, for all p∈P, such that(6)φ1xt≤Vpxt≤φ2xt,∀xt∈Ω,(7)V˙pxt≤-αVpxt,∀t∈Tfti,ti+1,-βVpxt,∀t∈Tsti,ti+1,and ∀(σ(ti)=p, σ(ti-)=q)∈P×P, p≠q,(8)Vpxti≤μVqxti,∀xt∈Ω,where φ1 and φ2 are some class K functions, then the state-constrained switched system (5) is GUAS for any switching signal with ADT(9)τa≥τa∗=ln⁡μζ,Tft0,tTst0,t≥ζ-βα-ζ>0.Lemma 4 (see [28]).
Consider the continuous-time switched systemx˙(t)=fσ(t)(x(t)), σ(t)∈P, and let λp>0 and μp>1 be given constants. Supposing that there exist positive definite C1 functions Vσ(t):Rn→R and two class K∞ functions κ1p and κ2p, p∈S, (10)κ1pxt≤Vpxt≤κ2pxt,V˙pxt≤-λpVpxt, and ∀(σ(ti)=p,σ(ti-)=q)∈S×S, p≠q, (11)Vpxti≤μpVqxti, then the system is GUAS for any switching signal with MDADT:(12)τap≥τap∗=ln⁡μpλp.


## 3. Main Results

In this section, we first establish the stability conclusion for the continuous-time switched systems with state constraints based on MDADT in Lemma7. The following definition is important for our result.Definition 5 (see [28]).
For a switching signalσ(t) and each T≥0, let Nσp(T,0) denote the switching numbers that the pth subsystem is activated over the interval [0,T] and let Tp(T,0) denote the total running time of the pth subsystem over the interval [0,T],p∈S. We say that σ(t) has a mode-dependent average dwell time τap if there exist two positive numbers N0p (we call N0p the mode-dependent chatter bounds here) and τap such that (13)NσpT,0≤N0p+TpT,0τap,∀T≥0.Remark 6.
Definition5 constructs a novel set of switching signals with a MDADT property: it means that if there exist positive numbers τap, p∈S, such that a switching signal has the MDADT property, we only require that the average time among the intervals associated with the pth subsystem is larger than τap (note that the intervals here are not adjacent), where Nσ(T,0)=∑p=1MNσp(T,0).Next, Lemma7 presents the stability result for the continuous-time state-constrained switched systems under MDADT switching.Lemma 7.
Consider the following switched systems:(14)x˙t=fσtxt,where x(t)∈Φ⊂Rn, and let λp>γp>0 and μp≥1, p∈L, be given constants. Supposing that there exist C1 function Vσ(t)(x(t)):Φ→R and some class W functions w1p and w2p, ∀p∈L, then(15)w1pxt≤Vpxt≤w2pxt,(16)V˙pxt≤-λpVpxt,∀t∈Rpti-1,ti;V˙pxt≤-γpVpxt,∀t∈Spti-1,ti,and ∀(σ(ti)=p, σ(ti-)=q)∈L×L, p≠q, i∈Z+,(17)Vpxti≤μpVqxtithen the switched system (14) is GUAS under any MDADT switching signal(18)τap≥τap∗=ln⁡μpρ,where λpRp(t0,t)+γpSp(t0,t)≥ρTp.Proof.
From the above, we know that the system states can be divided into the saturated state and the nonsaturated state. For example, whent>ts∈Sp(ti-1,ti), that is to say, the system state is saturated, then we can conclude from that that(19)Vpxt-e-γpt-tsVpxts<0.The inequality for the nonsaturated state can be obtained using similar techniques. Therefore, when t>ti, together with (17) and (19), it implies(20)Vσtixt≤e-λσtiRσtiti,te-γσtiSσtiti,tVσtixti≤μσtie-λσtiRσtiti,te-γσtiSσtiti,tVσti-1xti-≤μσtie-λσtiRσtiti,t-λσti-1Rσti-1ti-1,tie-γσtiSσtiti,t-γσti-1Sσti-1ti-1,tiVσti-1xti-1≤μσtiμσti-1e-λσtiRσtiti,t-λσti-1Rσti-1ti-1,tie-γσtiSσtiti,t-γσti-1Sσti-1ti-1,tiVσti-2xti-1-≤⋯≤∏p=1MμpNσpe-∑p=1MλpRpt0,t+γpSpt0,tVσt0xt0.In summary, this implies (21)Vσtixt≤∏p=1MμpNσpe-ρTpVσt0xt0≤∏p=1MμpN0pμpTp/τape-ρTpVσt0xt0≤∏p=1MμpN0pμp1/τape-ρTpVσt0xt0. Therefore, supposing that the switching signal satisfies (22)τap≥τap∗=ln⁡μpρ,we have μp1/τape-ρ≤μpρ/lnμpe-ρ=1. With Vσ(ti)(x(t)) convergence to zero along with t→∞, the asymptotic stability also can be derived.Remark 8.
In Lemma7, assume that λp>γp, and symbols λp and γp respectively denote the different decay rates of a Lyapunov function related to each of the active subsystems according to whether the saturations occur or not. For convenience of understanding, Rp(ti-1,ti) and Sp(ti-1,ti) represent separately the total time of the pth subsystem during which the state is saturated or nonsaturated within [ti-1,ti). In addition, we define Rp(t0,t)=∑f=1iRp(tf-1,tf)+Rp(ti,t) and Sp(t0,t)=∑f=1iSp(tf-1,tf)+Sp(ti,t).Remark 9.
It can be seen from Lemma3 that the parameters α, β, and μ are the same for all subsystems according to whether the saturations occur or not. However, the parameters λp, γp, and μp prescribed in Lemma 7 are mode-dependent; in other words, symbol λp and γp, respectively, denote the different decay rates of a Lyapunov function related to each of the active subsystems according to whether the saturations occur or not. Therefore, we can conclude that τap∗≤τa∗, from (7)–(9) and (16)–(18), and the mode-dependent features would reduce the conservativeness existing in Lemma 3.Remark 10.
It is worth nothing that the improved ADT (18) is always smaller than ADT (9) and (12). What is more, if we choose λp=γp and λp=α, γp=β, and μp=μ then (18) will reduce to (12) and (9), respectively. Therefore, Lemmas 3 and 4 can be regarded as two special cases of Lemma 7. It is clear that Lemma 7 presents a more general stability criterion than Lemmas 3 and 4 which corresponds to the special case of λp=γp and λp=α, γp=β, μp=μ, and ∀p∈L.In total, we can infer from Remarks9 and 10 that the MDADT switching signal has great flexibility superiorities for a switched system with state constraints.Now, based on the consequences obtained above, we present the stability condition for continuous-time switched systems with state constraints based on MDADT.Theorem 11.
Consider the following switched system:(23)x˙t=hAσtxt,where σ(t)=p∈L, and let λp>γp>0 and μp≥1, p∈L, be given constants. ∀(p,q)∈L×L, p≠q and Us≠I. Supposing that there exist matrices Qp>0 and row diagonally dominant matrices Lp, where |lii|>∑j=1,j≠in‍|lij| and lii<0, ∀i∈N,(24)ApTQp+QpAp+λpQp<0,UsAp+Up-LpTQp+QpUsAp+Us-Lp+γpQp<0,Qp-μpQq<0;therefore, the continuous-time switched system with state constraints (23) is GUAS with MDADT satisfying (18).Proof.
Here, consider the following Lyapunov function:(25)Vpxt=xTtQpxt,where ∀σ(t)=p∈L. From Lemma 7, we know that the system states can be divided into the saturated state and the nonsaturated state. Therefore, when t∈Rp(ti-1,ti), in other words, the system state is nonsaturated, we can conclude from that that(26)V˙pxt+λpVpt=xTtλpQp+ApTQp+QpApxt<0,Vpxti-μpVqxti=xTtiQp-μpQqxti<0. The inequality for the saturated state can be obtained using similar techniques: (27)V˙pxt+γpVpt=xTtγpQp+A~pTQp+QpA~pxt<0,where A~p=UsAp+Us-Lp. By simplifying the above three inequalities, we can get (24). Therefore, the continuous-time switched system with state constraints (23) is GUAS with MDADT satisfying (18).Next, we introduce the condition of stabilizing controller for continuous-time switched systems with state constraints based on MDADT.Theorem 12.
Consider the same switched system (1), where σ(t)=p∈L, and let λp>γp>0 and μp≥1, p∈L, be given constants. ∀(p,q)∈L×L, p≠q and Us≠I. Suppose that there exist matrices Pp>0 and Pq>0 and row diagonally dominant matrices Lp and Tp, where |lii|>∑j=1,j≠in‍|lij| and lii<0, ∀i∈N,(28)PpApT+TpTBpT+ApPp+BpTp+λpPp<0,(29)PpApTUsT+TpTBpUsT+UpLpUs-T+UsApPp+UsBpTp+Us-LpPp+γpPp<0,(30)Pp-μpPq<0;then, the continuous-time switched system with state constraints (1) is GUAS with MDADT satisfying (18). Therefore, if (28), (29), and (30) have a solution, the controller gains can be obtained by(31)Fp=TpPp-1.Proof.
From the closed-loop system (2) and Theorem 11, we have(32)Ap+BpFpTQp+QpAp+BpFp<-λpQp,(33)UsAp+BpFp+Us-LpTQp+QpUsAp+BpFp+Us-Lp<-γpQp,(34)Qp<μpQq.For simplifying the above inequalities, we can pre- and postmultiply Qp-1 to (32) and (33). Next, we define Tp=FpQp-1 and Pp=Qp-1, such that (28), (29), and (30) are proved to be established. Therefore, if (28), (29), and (30) have a solution, the controller gains can be obtained by Fp=TpPp-1. Then, the continuous-time switched system with state constraints (1) is GUAS with MDADT satisfying (18).Remark 13.
Theorem12 is a prolongation of [29, Theorem  3], which deals with the stabilization problem of continuous-time switched systems with state constraints. From Remarks 9 and 10, using λp, γp, μp under MDADT switching to design feasible controllers has more flexibility and less conservativeness.Finally, we give an iterative LMI algorithm for verifying the sufficient conditions of theorems for the continuous-time case. Here, we briefly describe the iterative LMI algorithm: Step  1, select aEp>0 and solve Pp and Tp from the Lyapunov equation (28); Step  2, using Pp and Tp obtained previously, solve the LMI optimization problem (29) for Lp and ξ; Step  3, using Lp obtained in the previous step, solve the LMI optimization problems (28) and (29) for Pp and Tp and ξ; Step  4, if ξ≤0, system (2) is GUAS at the origin. And the current Fp is the calculated feedback again. Otherwise, no result can be obtained. A different Ep may be selected and the algorithm may be repeated from Step  1. For more details on the iterative LMI algorithm, please refer to the reference literature [30].


## 4. Numerical Example

In this section, a numerical example of the continuous-time switched system with state constraints based on MDADT is presented to show the effectiveness of the developed approaches.Example.Consider the following switched systems:(35)x˙t=hAσtxt,where σ(t)∈L={1,2} and (36)A1=-2.00.1-0.5-0.4-0.90.70.3-1.0-0.4,A2=-0.7-1.50.52.0-1.01.21.81.0-1.8. We set the parameters λ1=0.3,  γ1=0.1,  and  μ1=1.4; λ2=0.2,  γ2=0.14,  and  μ2=1.5; ρ=0.15 and R1(0,t)/S1(0,t)=0.4; R2(0,t)/S2(0,t)=0.2. In Theorem 11, (37)Q1=32.1545-3.81738.9418-3.817322.8047-4.09908.9418-4.099022.1684,Q2=41.1823-4.573412.2986-4.573421.8687-4.010112.2986-4.010121.1007,L1=-3.82560.4542-1.06390.6506-3.8869-0.6986-1.47540.6764-3.6579,L2=-3.8032-0.4224-1.13580.8064-3.85590.7071-1.98840.6484-3.4116. When the MDADT switching signal satisfies τa1≥2.4549 and τa2≥2.6273, the switched system with state constraints (35) is GUAS. So, we give τa1=2.5 and τa2=2.8 in Figure 1. Figure 2 shows that despite x(t) has occurred saturations, with the initial state x0=[-1,1,-1]T and the MDADT switching signal, the continuous-time switched system with state constraints is GUAS. From the example, we demonstrate the effectiveness of our proposed method.Figure 1
τ
a
1
=
2.5 and τa2=2.8.Figure 2
Saturated trajectory of system (35).


## 5. Conclusion

The stability and the stabilization problems for a class of continuous-time switched systems with state constraints via a mode-dependent switching method have been investigated in continuous-time context. An improved average dwell time method is proposed. It is shown that the improved average dwell time method is less conservative than the common average dwell time method. Then, based on the improved average dwell time method, the sufficient conditions and state feedback controllers for stabilization of the switched system are derived. Finally, a numerical example is given to illustrate the proposed approach.



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