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

The stability and the stabilization problems for a class of continuous-time switched systems with state constraints via a modedependent 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 timemethod is less conservative than the common average dwell time method. Based on the improved average dwell timemethod, 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.


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][2][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][5][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][12][13][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][16][17].However, the average dwell time method is generally recognized to have more flexibility in stabilization for switched systems [18][19][20][21][22][23].
Up to now, there exist many literatures related to stabilization of switched systems with state constraints [24][25][26][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 continuoustime 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 continuoustime switched systems with state constraints under MDADT switching are derived.
The remainder of this paper is organized as follows.Section 2 is the problem formulation and preliminaries.Main results are given in Section 3, including sufficient condition concerning controller design and an iterative algorithm.

2
Mathematical Problems in Engineering 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, R  represents the -dimensional space, ‖ ⋅ ‖ stands for the Euclidean vector norm, and  is the identity matrix.Symbol * denotes the symmetry elements in symmetric matrices.C 1 means the space of continuously differentiable functions.

Problem Formulation and Preliminaries
Consider a class of state-constrained continuous-time switched systems given by where () ∈ R  and  () (), respectively, denote the state vector and the control input.The two-matrix pair (  ,   ), ∀() =  ∈ L, represents the th subsystem or th mode of (1); symbol ℎ(⋅) denotes the saturation function.() is a piecewise constant function of time, which takes its values in the finite set L = {1, 2, . . ., };  > 1 is the number of subsystems.In addition, for a switching sequence 0 <  1 < ⋅ ⋅ ⋅ <  −1 <   < ⋅ ⋅ ⋅ , symbol   means the moment of the th switching.When  ∈ ( −1 ,   ), we say the ( −1 )th subsystem is active.Generally speaking, the ideal state feedback is treated as  () () =  () (), where  () is the controller gain to be determined.Then, the resulting closed-loop system is described by A   () , non-constrained, where A  fl   +     , which represents the closed-loop system matrix of the th subsystem.About the saturation function ℎ(⋅), we transform it into the vertex of a convex hull to handle the saturations [30].
Symbol   denotes the set of  ×  diagonal matrices.The diagonal elements of   are 0 or 1.Assuming that every element of   is marked as   ,  −  =  −   ,  ∈ {1, 2, . . ., 2 −1 , 2  }.Therefore, we have and the diagonal elements of row diagonally dominant matrix  are negative.Now, let us first revisit the following definition and lemmas for later development.
Definition 1 (see [19]).For a switching signal () and each  2 ≥  1 ≥ 0, let  () ( 2 ,  1 ) denote the number of discontinuities of () in the open interval ( 1 ,  2 ).We say that () has an average dwell time   if there exist two positive numbers  0 and   such that Remark 2. Definition 1 means that if there exists a positive number   such that a switching signal has the ADT property, the ADT between any two consecutive switching signals is no smaller than a common constant   for all subsystems.

Main Results
In this section, we first establish the stability conclusion for the continuous-time switched systems with state constraints based on MDADT in Lemma 7. The following definition is important for our result.
Definition 5 (see [28]).For a switching signal () and each  ≥ 0, let   (, 0) denote the switching numbers that the th subsystem is activated over the interval [0, ] and let   (, 0) denote the total running time of the th subsystem over the interval [0, ],  ∈ S. We say that () has a mode-dependent average dwell time   if there exist two positive numbers  0 (we call  0 the mode-dependent chatter bounds here) and   such that where () ∈ Φ ⊂ R  , and let   >   > 0 and   ≥ 1,  ∈ L, be given constants.Supposing that there exist C 1 function  () (()) : Φ → R and some class W functions  1 and  2 , ∀ ∈ L, then and then the switched system ( 14) is GUAS under any MDADT switching signal where     ( 0 , ) +     ( 0 , ) ≥   .
Remark 9.It can be seen from Lemma 3 that the parameters , , and  are the same for all subsystems according to whether the saturations occur or not.However, the parameters   ,   , and   prescribed in Lemma 7 are modedependent; in other words, symbol   and   , 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  *  ≤  *  , from ( 7)-( 9) and ( 16)-( 18), and the modedependent 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   =   and   = ,   = , and   =  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   =   and   = ,   = ,   = , and ∀ ∈ L.
In total, we can infer from Remarks 9 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.
Proof.Here, consider the following Lyapunov function: where ∀() =  ∈ L. From Lemma 7, we know that the system states can be divided into the saturated state and the nonsaturated state.Therefore, when  ∈   ( −1 ,   ), in other words, the system state is nonsaturated, we can conclude from that that The inequality for the saturated state can be obtained using similar techniques: where Ã =     +  −    .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.
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 a   > 0 and solve   and   from the Lyapunov equation ( 28); Step 2, using   and   obtained previously, solve the LMI optimization problem (29) for   and ; Step 3, using   obtained in the previous step, solve the LMI optimization problems ( 28) and ( 29) for   and   and ; Step 4, if  ≤ 0, system ( 2) is GUAS at the origin.And the current   is the calculated feedback again.Otherwise, no result can be obtained.A different   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].

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.
Figure 2 shows that despite () has occurred saturations, with the initial state  0 = [−1,1,−1]  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.

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.
Remark 6. Definition 5 constructs a novel set of switching signals with a MDADT property: it means that if there exist positive numbers   ,  ∈ S, such that a switching signal has the MDADT property, we only require that the average time among the intervals associated with the th subsystem is larger than   (note that the intervals here are not adjacent), where   (, 0) = ∑  =1   (, 0).