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# Stability of a class of switched linear systems with uncertainties and average dwell time switching

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## Abstract

In this paper, the problems of stability for continuous and discrete-time polytopic uncertain switched linear systems with average dwell time switching are investigated. Firstly, the exponential stability result of general continuous-time switched systems using a discontinuous piecewise Lyapunov function approach is revisited, and the discretetime counterpart is then presented by the following similar lines in continuous-time case. Based on these and by further constructing a discontinuous piecewise parameterdependent Lyapunov function, the exponential stability criteria for both uncertain continuous and discrete-time polytopic uncertain switched linear systems with average dwell time switching are derived and formulated in terms of a set of linear matrix inequalities, respectively. Furthermore, the minimal average dwell time is obtained from the corresponding stability conditions for a given decay degree, and the admissible switching signals are consequently found such that the underlying system is robustly exponentially stable. Numerical examples are included to demonstrate the effectiveness and less conservativeness of the developed theoretical results.

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... There have been a lot of methodologies and techniques for the study of switched systems, such as the common Lyapunov function method, the switched Lyapunov function method and the average dwell time approach. Numerous researches have focused on the stability analysis and controller design of switched systems [8][9][10][11][12][13][14][15]. ...
... It is well known that time delay appears frequently in practical engineering and may deteriorate the stability and performance of the system. Thus, time delay systems have received more and more attention, and many results have been reported [12][13][14][15][16][17][18][19][20][21][22][23][24]. Some of them are concerned with switched systems with time-varying delay [12][13][14][15][16]. ...
... Thus, time delay systems have received more and more attention, and many results have been reported [12][13][14][15][16][17][18][19][20][21][22][23][24]. Some of them are concerned with switched systems with time-varying delay [12][13][14][15][16]. For example, [16] dealt with the problem of robust state-feedback control for uncertain discrete-time switched systems with mode-dependent time-varying delays. ...
Article
This paper is concerned with the problems of disturbance tolerance and rejection of discrete switched systems with time-varying delay and saturating actuator. Using the switched Lyapunov function approach, a sufficient condition for the existence of a state feedback controller is proposed such that the disturbance tolerance capability of the closed-loop system is ensured. By solving a convex optimization problem with linear matrix inequality (LMI) constraints, the maximal disturbance tolerance is estimated. In addition, the problem of disturbance rejection of the closed-loop system is solved. Two examples are given to illustrate the effectiveness of the proposed method.
... In [19], newly Lyapunov functional is proposed with piecewise-constant transition probabilities. It should be noted that average dwell time switching is very important in dynamic systems [20][21][22][23]. In [20], the average dwell time switching and uncertainties are considered. ...
... It should be noted that average dwell time switching is very important in dynamic systems [20][21][22][23]. In [20], the average dwell time switching and uncertainties are considered. Correspondingly, 2 Mathematical Problems in Engineering a dependent average dwell time approach is proposed in [21]. ...
Article
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This paper first investigates the problem of finite-time boundedness of Markovian jump system with piecewise-constant transition probabilities via dynamic output feedback control, which leads to both stochastic jumps and deterministic switches. Based on stochastic Lyapunov functional, the concept of finite-time boundedness, average dwell time, and the coupling relationship among time delays, several sufficient conditions are established for finite-time boundedness and H ∞ filtering finite-time boundedness. The system trajectory stays within a prescribed bound. Finally, an example is given to illustrate the efficiency of the proposed method.
... The stability for switched systems with time-varying delay is studied in [14]. The problem of stability for polytopic uncertain switched systems with average dwell time is studied in [15]. The stability analysis of continuous-time switched systems with a random switching signal is investigated in [16]. ...
... The stability of the periodic piecewise systems is guaranteed by (10) which gives a relationship of dwell time and λ ⋄ i for Hurwitz subsystems and non-Hurwitz subsystems over a period. From (15), it can be seen that −λ * is the exponential order of system (1). It indicates that a greater λ * can guarantee higher convergence rate of this system. ...
Conference Paper
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This paper investigates the stability and L2-gain problems for a class of continuous-time periodic piecewise linear systems with possibly non-Hurwitz subsystems. First, the exponential stability of periodic piecewise systems is studied by allowing the Lyapunov function to possibly non-monotonically decreasing over a period. A sufficient condition is established in terms of matrix inequalities. In light of the proposed Lyapunov function, the L2-gain criterion is derived for periodic piecewise linear systems as well.
... The switching signal specifies which subsystem will be activated along the trajectory at each instant of time. Presently, many achievements have been achieved on the control of switched system [13][14][15][16][17][18][19][20][21]. Especially, in recent years, it is a hot topic to study the problem of fault diagnosis and fault tolerant control for switched systems. ...
... R t t k 1 C k 1 e ˛.t s/ T .s/.s/ds, t k 1 C k 1 6 t < t k , k D 2, 3, : : : When t 2 OEt k , t k C k /, k D 1, 2, 3, : : :, the augmented system can be written as in (4). Consider the Lyapunov function as in (19), one has ...
Article
In this chapter, the problem of FD for continuous-time switched systems under asynchronous switching is investigated. The designed FD filter is assumed to be asynchronous with the original systems. Attention is focused on designing a FD filter such that the estimation error between the residual and the fault is minimized in the sense of H ∞ norm. By employing piecewise Lyapunov function and ADT techniques, a sufficient condition for the existence of such a filter is exploited in terms of certain LMIs. Finally, an example is provided to illustrate the effectiveness of the proposed approach.
... However, for the switched systems under controlled switching signals, the corresponding H 1 performance analysis problem is somewhat complicated in finding suitable switching signals for ensuring L 2 -gain and improving disturbance attenuation performance (Liberzon 2003;Xiang and Xiao 2011). In recent works, it has been recognised that average dwell time (ADT) switching as a class of controlled switching signals is flexible and efficient in analysis and synthesis of switched systems (Xiong, Lam, Gao, and Daniel 2005;Wang et al. 2009;Zhang, Boukas, and Shi 2009;Zhang and Jiang 2010). Due to the complexity caused by considering ADT switching, fewer results on H 1 performance analysis of switched linear systems with ADT switching have been reported. ...
Article
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This article is concerned with the disturbance attenuation properties of a class of switched linear systems by using a mode-dependent average dwell time (MDADT) approach. The proposed switching law is less strict than the average dwell time (ADT) switching in that each mode in the underlying system has its own ADT. By using the MDADT approach, a sufficient condition is obtained to guarantee the exponential stability with a weighted H∞ performance for the underlying systems. A numerical example is given to show the validity and potential of the developed results on improving the disturbance attenuation performance.
... These systems have attracted considerable attention because of their applicability and significance in various areas, such as power electronics, embedded systems, chemical processes, and computer-controlled systems [1,2]. Many works in the field of stability analysis and control synthesis for switched systems have appeared (see [3][4][5][6][7][8][9][10][11] and references cited therein). However, in the real world, they may not cover all the practical cases. ...
Article
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This paper is concerned with the problem of robust reliable control for a class of uncertain discrete impulsive switched systems with state delays, where the actuators are subjected to failures. The parameter uncertainties are assumed to be norm-bounded, and the average dwell time approach is utilized for the stability analysis and controller design. Firstly, an exponential stability criterion is established in terms of linear matrix inequalities (LMIs). Then, a state feedback controller is constructed for the underlying system such that the resulting closed-loop system is exponentially stable. A numerical example is given to illustrate the effectiveness of the proposed method.
... For recent progress, readers can refer to survey papers [1][2][3] and the references therein. As a class of time controlled switching signals, the average dwell time (ADT) and dwell time (DT) are employed to investigate the stability and stabilization problems of such systems in the literature [4][5][6][7][8][9][10][11], where some Lyapunov function and Lyapunov-like function techniques are utilized to deal with these problems [12][13][14][15]. In addition, if the switching is attached with Markov process or Markov Chain, the switching will be in stochastic sense [16]. ...
Article
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This paper is concerned with the problem of observer design for switched linear systems with time-varying delay and exogenous disturbances. The attention is focused on designing the full-order observers that guarantee the finite-time bounded and H∞H∞ finite-time stability of the dynamic augmented system. Based on linear matrix inequalities (LMIs) technology and an average dwell time (ADT) approach, sufficient conditions which ensure the observer-based finite-time bounded and H∞H∞ finite-time stability are given, respectively. By using a state observer, the memory state feedback controller is designed to finite-time stabilize a time-delay switched system, and the conditions are formulated in terms of delay-dependent LMIs. An example is given to illustrate the efficiency of the proposed methods.
... (1), f l and T Na represent the focus and dwell time [16][17][18][19][20] of the Na þ channel, respectively, L cNa represents the transmembrane length of the Na þ channel. Generally, L cNa equals to the thickness of membrane which is approximately 3-10 nm [21], P oNa represents the open probability or initialization focus coefficient of Na þ channel, b m represents the mean closing rates of the activation particle. ...
Article
This paper is concerned with the development of a novel sodium ion channel optical model by using the equivalent lens point spread function and the optical linear superposition principle. The main parameters and the corresponding equations of proposed model are studied and calculated. The relationship among the optical response index, membrane potential and time is investigated as well. Furthermore, performance evaluation by simulating and analyzing the membrane depolarization, the local potential and the dynamic spatiotemporal processes are undertaken to prove the correctness and effectiveness of the proposed model.
... On the other hand, from a practical point of view, all control systems contain uncertainties such as nonlinear and stochastic disturbances [27][28], interval and polytopic constraints [29]. For SPLSs with uncertainties, there have been a few results [24][25]. ...
Article
This paper is concerned with the stability and robust stability of switched positive linear systems whose subsystems are all unstable. By means of the mode-dependent dwell time approach and a class of discretized co-positive Lyapunov functions, some stability conditions of switched positive linear systems with all modes unstable are derived in both the continuous-time and the discrete-time cases, respectively. The co-positive Lyapunov functions constructed in this paper are time-varying during the dwell time and time-invariant afterwards. In addition, the above approach is extended to the switched interval positive systems. A numerical example is proposed to illustrate our approach.
... where f lK represents the focus length of equivalent lens, T K and L cK represent open dwell time [23][24][25][26][27] and transmembrane length of the potassium ion channel, respectively, P oK and b n are defined as the open probability of the potassium ion channel and the mean closing rates of activation particle in channel, respectively [28]. The mean closing rates of the activation particle in potassium ion channel is obtained from ...
Article
This paper presents single and multi-potassium ion channel optical models based on optical point spread function and linear superposition principle. Its novelty lies in the dynamic description method of potential diffusion process which is generated by the whole membrane and nearby ion channel while potassium ion channels open. The main parameters and the corresponding equations of proposed model are studied. The dynamic spatiotemporal spread processes of membrane potential repolarization evoked by opened potassium ion channels are investigated as well. Furthermore, performance evaluation by computer simulation and experimental results proves the correctness and effectiveness of the idea above.
... However, for the switched systems under controlled switching signals, the corresponding H ∞ performance analysis problem is somewhat complicated in finding suitable switching signals for ensuring L 2 -gain and improving disturbance attenuation performance [6,20]. In recent works, it has been recognized that average dwell time (ADT) switching as a class of controlled switching signals is flexible and efficient in analysis and synthesis of switched systems [17,22,28,29,31]. Due to the complexity caused by considering ADT switching, fewer results on H ∞ performance analysis of switched linear systems with ADT switching have been reported. ...
Article
SUMMARY This paper addresses the problem of stability for a class of switched positive linear time-delay systems. As first attempt, the Lyapunov–Krasovskii functional is extended to the multiple co-positive type Lyapunov–Krasovskii functional for the stability analysis of the switched positive linear systems with constant time delay. A sufficient stability criterion is proposed for the underlying system under average dwell time switching. Subsequently, the stability result for system under arbitrary switching is presented by reducing multiple co-positive type Lyapunov–Krasovskii functional to the common co-positive type Lyapunov–Krasovskii functional. A numerical example is given to show the potential of the proposed techniques. Copyright © 2012 John Wiley & Sons, Ltd.
... In practice, many switched systems fail to preserve stability under arbitrary switching, but may be stable under restricted switching signals (Lin & Antsaklis, 2009;Xu & Chen, 2005;Zhao & Zeng, 2010). As pointed out in Liberzon (2003), the average dwell time (ADT) switching is a class of restricted switching signals which means that the number of switches in a finite interval is bounded and the average time between consecutive switching is not less than a constant (Liberzon, 2003;Lin & Antsaklis, 2009;Zhang & Jiang, 2010). It has been also well known that the ADT scheme characterizes a larger class of stable switching signals than dwell time scheme, and its extreme case is actually the arbitrary switching. ...
Article
In this paper, the stability analysis problem for a class of switched positive linear systems (SPLSs) with average dwell time switching is investigated. A multiple linear copositive Lyapunov function (MLCLF) is first introduced, by which the sufficient stability criteria in terms of a set of linear matrix inequalities, are given for the underlying systems in both continuous-time and discrete-time contexts. The stability results for the SPLSs under arbitrary switching, which have been previously studied in the literature, can be easily obtained by reducing MLCLF to the common linear copositive Lyapunov function used for the system under arbitrary switching those systems. Finally, a numerical example is given to show the effectiveness and advantages of the proposed techniques.
... In recent years, the study of stability for switched systems has attracted considerable attention [9][10][11][12][13][14][15][16][17][18]. Switched systems are a class of hybrid systems, which consist of a finite number of subsystems (described by differential or difference equations) and an associated switching signal governing the switching among them. ...
Article
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Abstractvar REST_ID=21;This paper deals with the problem of reliable control for discrete time systems with actuator failures. The actuator is assumed to fail occasionally and can recover over a time interval. During the time of suffering failures, the considered closed‐loop system is assumed unstable. Using an average dwell time method and under the condition that the activation time ratio between the system without actuator failures and the system with actuator failures is not less than a specified constant, an observer‐based feedback controller is developed in terms of linear matrix inequalities such that the resulting closed‐loop system is exponentially stable. An example is included to demonstrate the effectiveness of the proposed approach.
... Due to their wide applications, switched systems which are an important class of hybrid systems have drawn considerable attention in the last decade [1,2]. During these years, there have been increasing research activities in the field of stability analysis for such systems (see [3][4][5][6], and the references cited therein). Recently, impulsive switched systems as a class of special switched systems have gained research attention. ...
Article
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This paper is concerned with the problem of dynamic output feedback (DOF) control for a class of uncertain discrete impulsive switched systems with state delays and missing measurements. The missing measurements are modeled as a binary switch sequence specified by a conditional probability distribution. The problem addressed is to design an output feedback controller such that for all admissible uncertainties, the closed-loop system is exponentially stable in mean square sense. By using the average dwell time approach and the piecewise Lyapunov function technique, some sufficient conditions for the existence of a desired DOF controller are derived, then an explicit expression of the desired controller is given. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.
... Such systems, typically, contain a finite number of subsystems and a switching signal governing the switching among them. In recent years, stability analysis is the fundamental problem in the study of switched systems (see [4][5][6]23,34,36,38] and references cited therein). However, in the real world, they may not cover all the practical cases. ...
... However, for the switched systems under controlled switching signals, the corresponding H ∞ performance analysis problem is somewhat complicated in finding suitable switching signals for ensuring L 2 -gain and improving disturbance attenuation performance [6] [20]. In recent works, it has been recognized that average dwell time (ADT) switching as a class of controlled switching signals is flexible and efficient in analysis and synthesis of switched systems [17] [22] [28] [29] [31]. Due to the complexity caused by considering ADT switching, fewer results on H ∞ performance analysis of switched linear systems with ADT switching have been reported. ...
Article
In this paper, the problems of reset stabilisation for positive linear systems (PLSs) are investigated. Some properties relating to reset control of PLSs are first revealed. It is shown that these properties are different from the corresponding ones of general linear systems. Second, a class of periodic reset scheme is designed to exponentially stabilise an unstable PLS with a prescribed decay rate. Then, for a given PLS with reset control, some discussions on the upper bound of its decay rate are presented. Meanwhile, the reset stabilisation for PLSs in a special case is probed as well. Finally, two numerical examples are used to demonstrate the correctness and effectiveness of the obtained theoretical results.
... On the other hand, as for the latter category of switched systems, the switching signals are constrained with some regular properties. A typical constrained switching signals is characterised by the property that the average time between two consecutive switches is no less than a constant, see , 2009, 2011, Zhang and Jiang (2010), Zhai, Hu, Yasuda, and Michel (2001), to name a few. ...
Article
This article is concerned with the existence problem of a common linear copositive Lyapunov function (CLCLF) for switched positive linear systems with stable and pairwise commutable subsystems. Three families of such systems composed of only continuous-time subsystems, only discrete-time subsystems and mixed continuous- and discrete-time subsystems are considered, respectively. It is demonstrated that a CLCLF can always be constructed for the underlying system whenever its subsystems are continuous-time, discrete-time or the mixed type. The case when the number of subsystems is two is first considered, then the obtained result is extended to the general case. Three numerical examples are given to verify the validity of the developed results.
... The switching signal specifies which subsystem will be activated along the trajectory at each instant of time. Presently, a lot of achievements have been achieved on the control of switched system [26,54,67,68,69,70,71,72,31]. Especially for recent years, it is a hot topic to study the problem of fault diagnosis and FTC for switched systems. ...
Book
Fault Detection for Discrete-Time Switched Systems with Interval Time-Varying Delays.- Fault Detection for Discrete-Time Switched Systems with Intermittent Measurements.- Fault Detection for Continues-Time Switched Systems under Asynchronous Switching.- Sensor Fault Estimation and Accommodation for Discrete-Time Switched Systems.- Sensor Fault Estimation and Compensation for Switched Systems with State Delay.- Fault Estimation for Nonlinear Continuous-Time Switched Systems.- Actuator Fault Estimation and Accommodation for Discrete-Time Switched Systems.- Active Fault Tolerant Control for Switched Systems with Time Delay.- Fault Estimation and Accommodation for Switched Systems with Time-Varying Delay.- Observer-Based Reliable Control for Discrete Time Switched Systems.- Conclusions and Future Research Direction.
... Its motivation comes from the fact that many practical systems are inherently multimodal and the fact that some of intelligent control methods are based on the idea of switching between different controllers. Up till now, many investigations about stability of multiform switched systems have been carried out; see, for instance, [6][7][8][9][10][11][12][13][14][15][16][17][18][19] and references therein. Hence, it is our intention in this paper to tackle such an important yet challenging problem for BIBO stability analysis of delay switched systems. ...
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The problem of bounded-input bounded-output (BIBO) stability is investigated for a class of delay switched systems with mixed time-varying discrete and constant neutral delays and nonlinear perturbation. Based on the Lyapunov-Krasovskii functional theory, new BIBO stabilization criteria are established in terms of delay-dependent linear matrix inequalities. The numerical simulation is carried out to demonstrate the effectiveness of the results obtained in the paper.
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