Article

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

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

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.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

... 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
Full-text available
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
Full-text available
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
Full-text available
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
Full-text available
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
Full-text available
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
Full-text available
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
Full-text available
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. ...
Article
Full-text available
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.
Article
This article is concerned with the stability analysis for a class of switched neutral systems with mixed time-varying delays. The upper bound of derivative of the discrete time-varying delay is not restricted to one. A delay-dependent criterion of exponential stability under switching signals with an average dwell time is developed by employing free weighting matrices. The criterion is given in the form of linear matrix inequality. A state decay estimation for the switched system is explicitly given by considering the individual rate of decay of state for each subsystem. Two simulation examples are given to illustrate the effectiveness of the proposed method.
Article
In this paper, the stability and stabilization problems for a class of switched linear systems with mode-dependent average dwell time (MDADT) are investigated in both continuous-time and discrete-time contexts. The proposed switching law is more applicable in practice than the average dwell time (ADT) switching in which each mode in the underlying system has its own ADT. The stability criteria for switched systems with MDADT in nonlinear setting are firstly derived, by which the conditions for stability and stabilization for linear systems are also presented. A numerical example is given to show the validity and potential of the developed techniques.
Article
The switching signal design for global exponential stability of discrete switched systems with interval time-varying delay is considered in this paper. Some LMI conditions are proposed to design the switching signal and guarantee the global exponential stability of switched time-delay system. Some nonnegative inequalities are used to reduce the conservativeness of the systems. Finally, two numerical examples are illustrated to show the main result.
Article
The problem of H∞H∞ control for a class of discrete-time Markov jump linear systems (MJLSs) characterized by piecewise-constant transition probabilities (TPs) is investigated in the paper. The so-called piecewise-constant TPs mean that the TPs are varying but invariant within an interval. The variation of the TPs considered here is subject to a typical class of slow switching signal, the average dwell time (ADT) switching, i.e., the number of switches in a finite interval is bounded and the average time between two consecutive switchings of TP matrices is not less than a constant. In this paper, the technique is illustrated and its use is exemplified with application to the popular class of multiplier–accelerator macroeconomic model.
Article
This paper investigates the problem of sliding mode control (SMC) for uncertain switched stochastic system with time-varying delay. The system under consideration is concerned with the stochastic dynamics and deterministic switching laws. An integral sliding surface is constructed and the stable sliding mode is derived. A sufficient condition for mean-square exponential stability of the sliding mode is developed under a class of switching laws based on the average dwell time method. Variable structure controllers are designed to guarantee the existence of the sliding mode from the initial time. An illustrative example is used to demonstrate the effectiveness of the proposed scheme.
Article
In this paper, several concepts of switching frequency are introduced to analyze the properties and performance of switched systems in infinite as well as finite-time intervals. The observation is very motivating that different system properties and performances depend on different switching frequencies. Sufficient conditions ensuring asymptotic stability, ℓ2 gain performance, and state boundness are derived on the basis of the notions of switching frequency, respectively. Then, on the basis of the analysis results, the control synthesis problems are addressed. LMI-based design algorithms are proposed to meet different control synthesis requirements. Numerical design examples are provided to demonstrate our results. Copyright © 2011 John Wiley & Sons, Ltd.
Article
This paper is concerned with the problems of passivity and passification for a class of uncertain switched systems subject to stochastic disturbance and time-varying delay. The passivity property is adopted to analyze the influence of the external disturbance on such systems to achieve prescribed attenuation levels. Based on average dwell time approach, free-weighting matrix method, and Jensen's integral inequality, delay-dependent sufficient conditions are obtained in terms of linear matrix inequalities, which ensure the uncertain switched stochastic time-delay system to be robustly mean-square exponentially stable and stochastically passive. Then, the switched passive controllers are synthesized by linearization techniques. Finally, two numerical examples are given to illustrate the effectiveness of the proposed methods.
Conference Paper
There exist common problems about the high energy consumption and lower technical ability of personnel in heating system. Internet of Things technology provides possibility for enhancing the existing decentralized, independent subsystems to the open, diverse networking monitoring platform. Based on the current status of equipments and personnel, this work will achieve dynamic management of optimal parameter in the production process of heating system with the function of operations supervision, quantitative comparison and training guidance, Then the performance of control is improved by management The concepts of internet of things are reflected from data to information, from information to knowledge, from knowledge to discerning property. Simultaneously, it also provides powerful technical support for the safe and efficient energy-saving environmental protection in the production process.
Article
Full-text available
This paper studies the quadratic regulation problem for discrete-time switched linear systems (DSLQR problem) on an infinite time horizon. A general relaxation framework is developed to simplify the computation of the value iterations. Based on this framework, an efficient algorithm is developed to solve the infinite-horizon DSLQR problem with guaranteed closed-loop stability and suboptimal performance. Due to these guarantees, the proposed algorithm can be used as a general controller synthesis tool for switched linear systems.
Conference Paper
The main goal of this paper is to investigate the stabilization problem for a class of switched positive linear systems (SPLS) with average dwell time (ADT) switching in continuous-time context. State-feedback controllers and the corresponding switching law with ADT property are designed, which stabilize the closed-loop systems while keeping the states nonnegative. The proposed conditions, formulated as linear matrix inequalities, can be directly used for controller synthesis and switching designing. Finally, a numerical example is given to demonstrate the validity of the obtained results.
Conference Paper
In this paper, the problem of non-fragile observer-based H∞ control for discrete-time switched delay systems is investigated. Both data missing and time delays are taken into account in the links from sensors to observers and from controllers to actuators. Such problem is transformed into an H∞ control problem for stochastic switched delay systems. Average dwell time (ADT) approach is used to obtain sufficient conditions on the solvability of such problems. An example is provided to show the effectiveness of the proposed method.
Article
This paper investigates the problem of finite-time boundedness of a class of discrete-time Markovian jump systems with piecewise-constant transition probabilities subject to average dwell time switching. Another set of useful regime-switching models has been given for both fixed transition probability Markov switching models and time-varying transition probabilities. Based on the knowledge of average dwell time and multiple Lyapunov function, a novel sufficient condition for finite-time boundedness of H ∞ filtering is derived and the system trajectory stays within a prescribed bound. Finally, an example is provided to illustrate the usefulness and effectiveness of the proposed method.
Article
In this paper, the fault detection problem is investigated for a class of discrete-time switched singular systems with time-varying state delays. The residual generator is firstly constructed based on a switched filter, and the design of fault detection filter is formulated as an H ∞ filtering problem, that is, minimizing the error between residual and fault in the H ∞ sense. Then, by constructing an appropriate decay-rate-dependent piecewise Lyapunov function and using the average dwell time scheme, a sufficient condition for the residual system to be regular, causal, and exponential stable while satisfying a prescribed H ∞ performance is derived in terms of linear matrix inequalities (LMIs). The corresponding solvability condition for the desired fault detection filters is also established via LMI approach. Finally, a numerical example is presented to show the effectiveness of the developed theoretical results.Copyright © 2012 John Wiley & Sons, Ltd.
Article
In this paper, a robust exponential l2 − l∞ filtering problem is addressed for discrete-time switched systems with polytopic uncertainties. The purpose of robust exponential l2 − l∞ filtering is to design a filter such that the resulting filtering error system is robustly exponentially stable with a decay rate and a prescribed exponential l2 − l∞ performance index. The robust exponential l2 − l∞ filtering problem is solved via an average dwell time approach. Sufficient conditions in terms of strict LMI are derived for checking the robust exponential stability of a filter. An explicit expression for the desired robust exponential filter is also given. Finally, a numerical example is provided to demonstrate the potential and effectiveness of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.
Article
This study considers the problem of finite-time filtering for switched linear systems with a mode-dependent average dwell time. By introducing a newly augmented Lyapunov–Krasovskii functional and considering the relationship between time-varying delays and their upper delay bounds, sufficient conditions are derived in terms of linear matrix inequalities such that the filtering error system is finite-time bounded and a prescribed noise attenuation level is guaranteed for all non-zero noises. Thus, a finite-time filter is designed for switched linear systems with a mode-dependent average dwell time. Finally, an example is given to illustrate the efficiency of the proposed methods.
Article
This paper is concerned with the problem of delay-dependent exponential H ∞ model reduction for discrete-time switched delay systems under switching signals with average dwell time (ADT). The objective is to construct a reduced-order model, which ensures that the resulting error system under switching signal with ADT is expo-nentially stable with an H ∞ norm bound. A weighting factor α is introduced to construct a Lyapunov function for switched delay systems such that ADT approach is used with piecewise Lyapunov matrices instead of common Lyapunov function matrices. Further-more, sufficient conditions for the solvability of this problem are obtained in terms of strict linear matrix inequalities (LMIs), which lessen the computation complexity. A nu-merical example is provided to show the effectiveness of the developed method.
Article
An extension of a fixed transition probability (TP) Markovian switching model to combine time-varying TPs has offered another set of useful regime-switching models. This paper is concerned with the problem of finite-time H ∞ control for a class of discrete-time Markovian jump systems with partly unknown time-varying TPs subject to average dwell time switching. The so-called time-varying TPs mean that the TPs are varying but invariant within an interval. The variation of the TPs considered here is subject to a class of slow switching signal. Based on selecting the appropriate Lyapunov–Krasovskii functional, sufficient conditions of finite-time boundedness of Markovian jump systems are derived and the system trajectory stays within a prescribed bound. Finally, an example is given to illustrate the efficiency of the proposed method.
Article
This study addresses the robust fault detection problem for discrete-time switching systems under an arbitrary switching signal. Sufficient conditions building an observer are obtained by using multiple Lyapunov function and \({H}_{\infty }\) methods. A sufficient condition for the solvability of this problem is established in terms of linear matrix inequalities. A numerical example is provided to demonstrate the effectiveness of the proposed method.
Article
This paper studies robust stability of positive switched systems (PSSs) with polytopic uncertainties in both discrete-time and continuous-time contexts. By using multiple linear copositive Lyapunov functions, a sufficient condition for stability of PSSs with dwell time is addressed. Being different from time-invariant multiple linear copositive Lyapunov functions, the Lyapunov functions constructed in this paper are time-varying during the dwell time and time-invariant afterwards. Then, robust stability of PSSs with polytopic uncertainties is solved. All conditions are solvable via linear programming. Finally, illustrative examples are given to demonstrate the validity of the proposed results.
Conference Paper
This paper studies the exponential stability of positive switched linear systems with delays and impulses. First, based on the multiple linear co-positive Lyapunov functional approach and the average dwell time method, a delay-dependent exponential stability criterion is presented for the positive switched linear system with time-varying delays. Second, the nonlinear impulsive effect occurring at the switching times is considered. By keeping impulsive perturbation within reasonable bounds and establishing multiple co-positive Lyapunov function, a sufficient condition is obtained for globally uniformly exponentially stable of impulsive positive switched system. Finally, an illustrative example is studied to support our new results.
Conference Paper
This paper investigates the problem of robustly globally uniformly exponential stability and L 2 control for a class of switched systems with time-varying delay by using mode-dependent average dwell time (MDADT) approach. The MDADT approach is more applicable in practice than the average dwell time method in which each mode in the underlying system has its own average dwell time. Firstly, some sufficient conditions for robustly exponential stability is given by using the MDADT approach and piecewise Lyapunov function. Secondly, the L 2 -gain of the switched system with the external disturbance is analyzed. Then, the state feedback controller gain is obtained to ensure the exponential stability of the switched system. Finally, a numerical example is provided to illustrate the effectiveness of the theoretical results.
Article
Full-text available
This paper is concerned with fault detection and control problem for continuous-time switched systems with average dwell time. Attention is focused on designing a fault detection observer and controller such that the impact of the unknown inputs and the faults on the system is minimized in the sense of \(H_{\infty }\) norm. By employing multiple Lyapunov function and average dwell time techniques, a sufficient condition for the existence of such an observer and controller is exploited in terms of certain linear matrix inequalities. Finally, two illustrative examples are provided to show the effectiveness and applicability of the proposed results.
Article
This paper is concerned with the problem of observer design for switched recurrent neural networks with time-varying delay. The attention is focused on designing the full-order observers that guarantee the global exponential stability of the error dynamic system. Based on the average dwell time approach and the free-weighting matrix technique, delay-dependent sufficient conditions are developed for the solvability of such problem and formulated as linear matrix inequalities. The error-state decay estimate is also given. Then, the stability analysis problem for the switched recurrent neural networks can be covered as a special case of our results. Finally, four illustrative examples are provided to demonstrate the effectiveness and the superiority of the proposed methods.
Article
Full-text available
This paper presents a new approach for solving optimal control problems for switched systems. We focus on problems in which a prespecified sequence of active subsystems is given. For such problems, we need to seek both the optimal switching instants and the optimal continuous inputs. In order to search for the optimal switching instants, the derivatives of the optimal cost with respect to the switching instants need to be known. The most important contribution of the paper is a method which first transcribes an optimal control problem into an equivalent problem parameterized by the switching instants and then obtains the values of the derivatives based on the solution of a two point boundary value differential algebraic equation formed by the state, costate, stationarity equations, the boundary and continuity conditions, along with their differentiations. This method is applied to general switched linear quadratic problems and an efficient method based on the solution of an initial value ordinary differential equation is developed. An extension of the method is also applied to problems with internally forced switching. Examples are shown to illustrate the results in the paper.
Article
Full-text available
The use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control. Lower bounds on the optimal control cost are obtained by semidefinite programming based on the Bellman inequality. This also gives an approximation to the optimal control law. An upper bound to the optimal cost is obtained by another convex optimization problem using the given control law. A compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy
Article
Full-text available
We introduce some analysis tools for switched and hybrid systems. We first present work on stability analysis. We introduce multiple Lyapunov functions as a tool for analyzing Lyapunov stability and use iterated function systems theory as a tool for Lagrange stability. We also discuss the case where the switched systems are indexed by an arbitrary compact set. Finally, we extend Bendixson's theorem to the case of Lipschitz continuous vector fields, allowing limit cycle analysis of a class of “continuous switched” systems
Article
Delay-dependent robust stability of cellular neural networks with time-varying discrete and distributed delays is considered in this paper. Based on Lyapunov stability theory and linear matrix inequality (LMI), stability criteria are derived in terms of LMIs avoiding bounding certain cross terms which often leads to conservatism. And the restriction of derivative of time-varying delay is removed. Numerical examples are given to indicate significant improvements over some existing results.
Article
In this paper, the problem of H∞ output feedback control for switched linear discrete-time systems with time delays is investigated. The time delay is assumed to be time-varying and bounded. By constructing a switched quadratic Lyapunov function for the underlying system, both static and dynamic H∞ output feedback controllers are designed respectively such that the corresponding closed-loop system under arbitrary switching signals is asymptotically stable and a prescribed H∞ noise-attenuation level bound is guaranteed. A cone complementary linearization algorithm is exploited to design the controllers. A numerical example is presented to show the effectiveness of the developed theoretical results.
Article
In this paper, the problem of designing H∞ state-feedback controllers for switched linear discrete-time systems with polytopic uncertainties is investigated. Two approaches on designing robust and parameter-dependent H∞ controllers are proposed and the existence conditions of the desired controllers are derived and formulated in terms of a set of linear matrix inequalities. By solving the corresponding convex optimization problem, the desired controllers are obtained, respectively, and different optimal H∞ noise-attenuation level bounds of corresponding closed-loop systems are given as well. The designed controllers have their own advantages and disadvantages regarding the conservatism and realization complexity. An illustrative example emerging in networked control systems (NCS) and numerical simulations are presented to show the applicability and effectiveness of the obtained theoretic results. Copyright © 2006 John Wiley & Sons, Ltd.
Article
In this paper, the problem of robust H∞ filtering for switched linear discrete-time systems with polytopic uncertainties is investigated. Based on the mode-switching idea and parameter-dependent stability result, a robust switched linear filter is designed such that the corresponding filtering error system achieves robust asymptotic stability and guarantees a prescribed H∞ performance index for all admissible uncertainties. The existence condition of such filter is derived and formulated in terms of a set of linear matrix inequalities (LMIs) by the introduction of slack variables to eliminate the cross coupling of system matrices and Lyapunov matrices among different subsystems. The desired filter can be constructed by solving the corresponding convex optimization problem, which also provides an optimal H∞ noise-attenuation level bound for the resultant filtering error system. A numerical example is given to show the effectiveness and the potential of the proposed techniques. Copyright © 2006 John Wiley & Sons, Ltd.
Article
In this paper we present a tutorial overview of some of the issues that arise in the design of switched linear control systems. A benchmark regulation problem is then presented. Copyright © 2003 John Wiley & Sons, Ltd.
Article
This paper studies the problem of H -control for linear systems with Markovian jumping parameters. The jumping parameters considered here are two separable continuous-time, discrete-state Markov processes, one appearing in the system matrices and one appearing in the control variable. Our attention is focused on the design of linear state feedback controllers such that both stochastic stability and a prescribed H -performance are achieved. We also deal with the robust H -control problem for linear systems with both Markovian jumping parameters and parameter uncertainties. The parameter uncertainties are assumed to be real, time-varying, norm-bounded, appearing in the state matrix. Both the finite-horizon and infinite-horizon cases are analyzed. We show that the control problems for linear Markovian jumping systems with and without parameter uncertainties can be solved in terms of the solutions to a set of coupled differential Riccati equations for the finite-horizon case or algebraic Riccati equations for the infinite-horizon case. Particularly, robust H -controllers are also designed when the jumping rates have parameter uncertainties.
Article
This paper is concerned with the problem of H∞ model reduction for switched system, which is an important class of hybrid systems frequently encountered in practical situations. Two sharply different approaches are proposed to solve this problem. The first approach casts the model reduction into a convex optimization problem, which is the first attempt to solve the model reduction problem by using linearization procedure. The second one, based on the cone complementarity linearization idea, casts the model reduction problem into a sequential minimization problem subject to linear matrix inequality constraints. Both approaches have their own advantages and disadvantages concerning conservatism and computational complexity. A numerical example illustrates the effectiveness of the proposed theories.
Article
In this paper digital optimal control of pulse-width-modulated switched linear systems is considered. The equivalent discrete-time system and its deviation around a stationary solution plays an important role. An optimal linear discrete-time deviation controller is developed. The controller is chosen to be one-step-ahead predictive, which is a natural choice for control of digital pulse-modulated systems. The controller is optimal, i.e. a cost criterion is minimized. The theory is generally applicable and gives, in principle, a tool to design a digital optimal controller for an arbitrary pulse-width-modulated switched linear system. The theory is illustrated by examples from the field of switched electrical networks.
Article
In this paper, we propose a switched H∞ gain scheduled controller for uncertain nonlinear systems with exogenous signals. Each controller is constructed at a distinct and fixed value of exogenous signals using H∞ synthesis methodology. Then the constructed controller set is switched according to the proposed switching scheme for the wide range of variation of exogenous signals. Moreover, it is shown that the switched control system guarantees stability during every switching.
Article
This work presents a hybrid nonlinear control methodology for a broad class of switched nonlinear systems with input constraints. The key feature of the proposed methodology is the integrated synthesis, via multiple Lyapunov functions, of “lower-level” bounded nonlinear feedback controllers together with “upper-level” switching laws that orchestrate the transitions between the constituent modes and their respective controllers. Both the state and output feedback control problems are addressed. Under the assumption of availability of full state measurements, a family of bounded nonlinear state feedback controllers are initially designed to enforce asymptotic stability for the individual closed-loop modes and provide an explicit characterization of the corresponding stability region for each mode. A set of switching laws are then designed to track the evolution of the state and orchestrate switching between the stability regions of the constituent modes in a way that guarantees asymptotic stability of the overall switched closed-loop system. When complete state measurements are unavailable, a family of output feedback controllers are synthesized, using a combination of bounded state feedback controllers, high-gain observers and appropriate saturation filters to enforce asymptotic stability for the individual closed-loop modes and provide an explicit characterization of the corresponding output feedback stability regions in terms of the input constraints and the observer gain. A different set of switching rules, based on the evolution of the state estimates generated by the observers, is designed to orchestrate stabilizing transitions between the output feedback stability regions of the constituent modes. The differences between the state and output feedback switching strategies, and their implications for the switching logic, are discussed and a chemical process example is used to demonstrate the proposed approach.
Conference Paper
It is shown that switching among stable linear systems results in a stable system provided that switching is “slow-on-the-average”. In particular, it is proved that exponential stability is achieved when the number of switches in any finite interval grows linearly with the length of the interval, and the growth rate is sufficiently small. Moreover, the exponential stability is uniform over all switchings with the above property. For switched systems with inputs this guarantees that several input-to-state induced norms are bounded uniformly over all slow-on-the-average switchings. These results extend to classes of nonlinear switched systems that satisfy suitable uniformity assumptions. In this paper it is also shown that, in a supervisory control context, scale-independent hysteresis can produce switching that is slow-on-the-average and therefore the results mentioned above can be used to study the stability of hysteresis-based adaptive control systems
Conference Paper
A longitudinal model of a vectored-thrust Vertical and/or Short Take-Off and Landing (VSTOL) aircraft, which operates in three distinct modes, is a high-dimensional system whose performance can be enhanced with the use of switched nonlinear control. The aircraft's performance is limited by physical constraints on such parameters as ground speed, angle of attack, and thrust, which are summarized by the aerodynamic flight envelope. We specify a least-restrictive control law which will keep the aircraft within this envelope. We independently examine the aerodynamic flight envelope constraints and design the nonlinear controller for each mode using approximate feedback linearization, then combine the results to guarantee tracking within a “safe” envelope. We provide an example of switching between these modes in a short take-off maneuver based on pilot data
A robust H<sub>∞</sub> filtering technique is proposed for convex polytopic uncertain systems. By applying a bounded real lemma to the error dynamics and using the Schur complement with the appropriate change of variables, a nonlinear matrix inequality is obtained. It is then shown that the congruence transformation, with some newly defined variables, converts this nonlinear matrix inequality into the convex optimisation problem for the design of robust H<sub>∞</sub> filters, which is expressed by linear matrix inequality and can be solved very efficiently by so called interior point algorithms. The optimal tolerance level can be directly computed without the aid of the conventional bisection method, and the proposed algorithm does not require the additional search procedures needed for dealing with the norm-bounded uncertainty. Numerical examples are given to show that the proposed filter is more robust than the robust H<sub>2</sub> filter against the parameter variation, as well as the noise in the worst-case frequency range and to illustrate the advantage of describing the uncertainty as polytopic rather than norm bounded
Article
By a switched system, we mean a hybrid dynamical system consisting of a family of continuous-time subsystems and a rule that orchestrates the switching between them. The article surveys developments in three basic problems regarding stability and design of switched systems. These problems are: stability for arbitrary switching sequences, stability for certain useful classes of switching sequences, and construction of stabilizing switching sequences. We also provide motivation for studying these problems by discussing how they arise in connection with various questions of interest in control theory and applications
Article
The well-posedness problem (existence and uniqueness of solutions) of a class of multi-modal piecewise affine systems is addressed, where binary-switches individually act under autonomous switching. First, a new transition rule on the discrete state, called the switch-based transition rule, is introduced and some relations with the mode-based transition rule are discussed. Next, a sufficient condition for such a multi-modal system to be well-posed for all external inputs is derived in terms of well-posedness of its subsystems of lower complexity "bimodal systems". Finally, an easily checkable condition for the bimodal system to be well-posed for all external inputs is given, which consequently allows us to algebraically determine well-posedness of the multi-modal systems in question.
Article
This paper addresses the problem of stability analysis and control synthesis of switched systems in the discrete-time domain. The approach followed in this paper looks at the existence of a switched quadratic Lyapunov function to check asymptotic stability of the switched system under consideration. Two different linear matrix inequality-based conditions allow to check the existence of such a Lyapunov function. The first one is classical while the second is new and uses a slack variable, which makes it useful for design problems. These two conditions are proved to be equivalent for stability analysis. Investigating the static output feedback control problem, we show that the second condition is, in this case, less conservative. The reduction of the conservatism is illustrated by a numerical evaluation.
Article
This paper provides an overview of recent developments on design of hybrid controllers for continuous-time control systems that can be described by linear or nonlinear differential state equations. Hybrid controllers provide a generalization of classical feedback controllers for linear and nonlinear systems. The benefit of hybrid controllers, that they can be used to achieve closed-loop performance objectives that cannot be achieved using classical linear or nonlinear controllers, is emphasized. This paper introduces hybrid controllers in the form of a switching control architecture and provides a summary of recently developed control approaches that utilize this control architecture. We provide a conceptual framework for these results, identify limitations of the results, and discuss the current status of hybrid control design approaches