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This paper addresses the stability issue of a class of delayed switched cascade nonlinear systems consisting of separate subsystems and coupling terms between them. Some global and local asymptotic stability sufficient conditions are proposed, drawing stability conclusion of the overall cascade system from those of separate systems. These results essentially rely on the following observation: For a general delayed switched nonlinear system being asymptotically stable, the trajectories of the perturbed system asymptotically approach zero if so does the perturbation. This observation is one of the main results in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we propose new algebraic Gramians for continuous-time switched linear systems, which satisfy generalized Lyapunov equations. The main contribution of this paper is twofold. First, we show that the ranges of those Gramians encode the reachability and observability spaces of a switched linear system. As a consequence, a simple Gramian-based criterion for reachability and observability is established. Second, a balancing-based model order reduction technique is proposed and, under some sufficient conditions, stability preservation and an error bound are shown. Finally, the efficiency of the proposed method is illustrated by means of numerical examples. [ABSTRACT FROM AUTHOR]

This paper addresses robustness to external disturbances of switched discrete and continuous systems with multiple equilibria. First, we prove that if each subsystem of the switched system is input-to-state stable (ISS), then under switching signals that satisfy an average dwell-time bound, the solutions are ultimately bounded within a compact set. The size of this set varies monotonically with the supremum norm of the disturbance signal. These results generalize existing ones in the common equilibrium case to accommodate multiple equilibria. Then, we relax the (global) ISS conditions to consider equilibria that are locally exponentially stable (LES), and we establish practical stability for such switched systems under disturbances. Our motivation for studying this class of switched systems arises from certain motion planning problems in robotics, where primitive movements, each corresponding to an equilibrium point of a dynamical system, must be composed to obtain more complex motions. As a concrete example, we consider the problem of realizing safe adaptive locomotion of a three-dimensional biped under persistent external force by switching among motion primitives characterized by LES limit cycles. The results of this paper, however, are relevant to a much broader class of applications, in which composition of different modes of behavior is required. [ABSTRACT FROM AUTHOR]

This paper addresses the problems of determining the \mathcal H_2 norm and the root mean square (RMS) gain of continuous-time switched linear systems. A novel class of Lyapunov functions is proposed for reaching this goal, called homogeneous rational Lyapunov functions (HRLFs). It is shown that sufficient conditions for establishing upper bounds of the sought performance indexes in the case of arbitrary switching can be given in terms of linear matrix inequality (LMI) feasibility tests by searching for an HRLF of chosen degree. Moreover, it is shown that these conditions are also necessary by searching for an HRLF of degree sufficiently large. It is worth mentioning that necessary and sufficient LMI conditions have not been proposed yet in the literature for the considered problems. Hence, the paper continues by considering the case of switching with dwell time constraints, showing that analogous LMI conditions can be obtained for this case by searching for a family of HRLFs mutually constrained by the dwell time specification. Some numerical examples illustrate the proposed methodology and highlight the advantages with respect to the existing works. [ABSTRACT FROM AUTHOR]

This paper addresses the event-triggered dynamic output feedback control for switched linear systems with frequent asynchronism. Different from existing work, which limits at most once switching during an interevent interval, we adopt the average dwell time approach without limiting the minimum dwell time of each subsystem, and thus frequent switching is allowed to happen in an interevent interval. Since the difficulty in acquiring the full information of system states, the dynamic output feedback controller is taken into account to stabilize the switched system. By employing a controller-mode-dependent Lyapunov functional, stability criterion is proposed for the resulting closed-loop system, based on which the dynamic output feedback controller together with the mode-dependent event-triggered mechanism is codesigned. Besides, the existence of the lower bound on interevent intervals is attentively discussed, which gets rid of the Zeno behavior. Finally, the effectiveness of the proposed method is illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

This paper employs detectability ideas to decide uniform global asymptotic stability (UGAS) of the trivial solution for a class of switched nonlinear time-varying systems when the trivial solution is uniformly globally stable. Using the notion of limiting behaviors of the state, output, and switching signals, the concept of a limiting zeroing-output solution is introduced. This leads to a definition of weak zero-state detectability (WZSD) that can be used to check UGAS, (uniformly for a set of switched signals). En route to establish this, a number of new stability results are derived. For example, under appropriate conditions, it is feasible to decide UGAS even when the switching signal does not satisfy an averaged dwell-time condition. It is also shown that WZSD of the original switched system can be verified by detectability conditions of much simpler auxiliary systems. Moreover, UGAS can be guaranteed without requiring that in each allowable system (without switching), the trivial solution is attractive. The effectiveness of the proposed concept is illustrated by a few examples including a switched semi-quasi-Z-source inverter. [ABSTRACT FROM AUTHOR]

This paper presents a systematical reduced-order observer design method for a class of switched descriptor systems containing unknown inputs (UIs) in both the dynamic and the output equations. Generally speaking, when the output of the system contains UIs, the reduced-order observer design will become much more challenging. In order to overcome the difficulty brought by the UIs in the output, first, by introducing a new UI vector, a new equivalent system is obtained, which does not contain UIs in the corresponding output any more. Then, for the purpose of reduced-order observer design, the observer matching condition (OMC) and the minimal phase condition (MPC) are discussed, and it is shown that the new general system maintains both the OMC and the MPC. Subsequently, based on the discussions on the existence conditions, a reduced-order unknown input observer is developed to asymptotically estimate the state of the original system. Finally, an electronic circuit example is given to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, input-to-state stability (ISS) for discrete-time switched nonlinear time-varying (SNTV) systems is investigated. Starting with discrete-time nonlinear time-varying (NTV) systems, some improved sufficient conditions are proposed to verify the ISS of systems by using the weak implication-form ISS (WI-ISS) Lyapunov function, weak dissipative-form ISS (WD-ISS) Lyapunov function, and interval descent technique. Then, the results obtained are extended to study the ISS of discrete-time SNTV systems, several relaxed conditions are given by using piecewise WI-ISS and WD-ISS Lyapunov functions, minimum dwell time, and infinite switching methods, respectively. Comparing with the existing results, the obtained conditions release the requirement on negative definiteness of the differences of (piecewise) Lyapunov functions, moreover, all subsystems are allowed to be unstable in the case of infinite switching. Finally, a numerical example is given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. A popular method used for the stability analysis of these systems searches for a Lyapunov function with convex optimization tools. We analyze the accuracy of this method for constrained switched systems, a class of systems that has attracted increasing attention recently. We provide a new guarantee for the upper bound provided by the sum of squares implementation of the method. This guarantee relies on the $p$ -radius of the system and the entropy of the language of allowed switching sequences. We end this paper with a method to reduce the computation of the JSR of low-rank matrices to the computation of the constrained JSR of matrices of small dimension. [ABSTRACT FROM AUTHOR]

This paper studies the stability for impulsive switched time-delay systems with state-dependent impulses. Since the impulses and the switches are not necessarily synchronous, we start from a stability analysis of impulsive switched time-delay systems with time-dependent impulses. Sufficient conditions are derived to guarantee the stability property, which extends the previous results for the synchronous switch and impulse case. For the state-dependent impulse case, using the B-equivalent method, impulsive switched time-delay systems with state-dependent impulses are transformed into impulsive switched time-delay systems with time-dependent impulses. The equivalence between the original system and the transformed system is established, and stability conditions are obtained for impulsive switched time-delay systems with state-dependent impulses. Finally, a numerical example is given to demonstrate the obtained results. [ABSTRACT FROM AUTHOR]

This paper deals with the dead-beat stabilizability of autonomous discrete-time switched linear systems. Based on a constructive necessary and sufficient condition for dead-beat stabilizability, we propose two algorithms. The first one is concerned with the problem of testing dead-beat stabilizability and computing the shorter stabilizing mode sequence, whenever it exists. The other one implements a method to construct a switched system whose shorter dead-beat stabilizing sequence has a prescribed length. Then, we present numerical assessments and possible applications. [ABSTRACT FROM AUTHOR]

In this paper we study the stability of nonlinear and time-varying switched systems under restricted switching. We approach the problem by decomposing the system dynamics into a nominal-like part and a perturbation-like one. Most stability results for perturbed systems are based on the use of strong Lyapunov functions, i.e. functions of time and state whose total time derivative along the nominal system trajectories is bounded by a negative definite function of the state. However, switched systems under restricted switching may not admit strong Lyapunov functions, even when asymptotic stability is uniform over the set of switching signals considered. The main contribution of the current paper consists in providing stability results that are based on the stability of the nominal-like part of the system and require only a weak Lyapunov function. These results may have wider applicability than results based on strong Lyapunov functions. The results provided follow two lines. First, we give very general global uniform asymptotic stability results under reasonable boundedness conditions on the functions that define the dynamics of the nominal-like and the perturbation-like parts of the system. Second, we provide input-to-state stability (ISS) results for the case when the nominal-like part is switched linear-time-varying. We provide two types of ISS results: standard ISS that involves the essential supremum norm of the input and a modified ISS that involves a power-type norm. [ABSTRACT FROM AUTHOR]

In this paper, we consider the l\infty gain characterizations of linear switched systems (LSS) and present various relevant results on their exact computation and optimization. Depending on the role of the switching sequence, we study two broad cases: first, when the switching sequence attempts to maximize, and second, when it attempts to minimize the l\infty gain. The first, named as worst-case throughout the paper, can be related to robustness of the system to uncontrolled switching; the second relates to situations when the switching can be part to the overall decision making. Although, in general, the exact computation of l\infty gains is difficult, we provide specific classes, the input-output switching systems, for which it is shown that linear programming can be used to obtain the worst-case l\infty gain. This is a sufficiently rich class of systems as any stable LSS can be approximated by one. Certain applications to robust control design are provided where we show that a switched compensation independently of the plant has no advantage over a linear time invariant (LTI) compensation, and further, if the plant is strictly causal, even a switched compensation which has a matched switching with the plant does not provide a better performance over an LTI compensation. Also, we present a new necessary and sufficient condition to check the stability of LSS in form of a model matching problem. On the other hand, if one is interested in minimizing the l\infty gain over the switching sequences, we show that, for finite impulse response (FIR) switching systems the minimizing switching sequence can be chosen to be periodic. For input-only or output-only switching an exact, readily computable, characterization of the minimal l\infty gain is provided, and it is shown that the minimizing switching sequence is constant, which, as also shown, is not true for input-output switching. [ABSTRACT FROM AUTHOR]

This paper investigates standard $H_\infty$ control of switched systems via dwell-time switchings without posing any internal stability requirements on subsystems of the switched systems. First, a sufficient condition is formed by specifying lower and upper bounds of the dwell time, constraining upper bound of derivative of a Lyapunov function of the active subsystem, and forcing the Lyapunov function values of the overall switched system to decrease at switching times to achieve standard $H_\infty$ control of unforced switched linear systems. Then, in the same framework of the dwell time, sufficient conditions are given for that of the corresponding forced switched linear systems by further designing state feedback controllers. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

Controlled distinguishability of two dynamical systems is the property of the systems that guarantees the existence of a control input generating different outputs of the systems regardless of their initial state vectors. These inputs are referred in the literature as discerning control inputs. In this paper, a necessary and sufficient condition for controlled distinguishability of continuous-time bilinear systems is established. It generalizes the classic one provided in the literature for the class of linear systems and a method for designing discerning inputs of bilinear systems that stabilize the systems is also proposed. [ABSTRACT FROM AUTHOR]

This paper considers the optimal control problem of linear switched systems with linear quadratic (LQ) cost or multiple LQ cost. By adopting an embedding transformation, the switching design problem is relaxed and transformed into a traditional optimal control problem. The bang–bang-type solutions of the embedded optimal control problems are obtained for both the positive definite LQ cost case and the multiple LQ cost case, which are the optimal solutions to the original problems. The switching sequence of modes and the switching instants can be calculated by solving a closed-form optimal switching condition. The optimal state feedback control law is determined simultaneously. Finally, numerical results are provided to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of both, local and global, uniform asymptotic stability for switched nonlinear time-varying (NLTV) systems through the detectability of output-maps. With this aim, the notion of reduced limiting control systems for switched NLTV systems whose switchings verify time/state-dependent constraints, and the concept of weak zero-state detectability for those reduced limiting systems are introduced. Necessary and sufficient conditions for the (global)uniform asymptotic stability of families of trajectories of the switched system are obtained in terms of this detectability property. These sufficient conditions in conjunction with the existence of multiple weak Lyapunov functions yield a criterion for the (global) uniform asymptotic stability of families of trajectories of the switched system. This criterion can be seen as an extension of the classical Krasovskii-LaSalle theorem. An interesting feature of the results is that no dwell-time assumptions are made. Moreover, they can be used for establishing the global uniform asymptotic stability of the switched NLTV system under arbitrary switchings. The effectiveness of the proposed results is illustrated by means of various interesting examples, including the stability analysis of a semiquasi-Z-source inverter [ABSTRACT FROM AUTHOR]

This paper considers the input-to-state stability (ISS) of time-varying switched systems with time delays, where the upper bound estimation for the operator of Lyapunov function (UBEOL) is assumed to be time varying and mode dependent. The ISS and integral ISS are investigated for time-varying switched systems with time delays by using the Lyapunov–Razumikhin and comparison theorem methods. Since the coefficient in the UBEOL is time varying and takes a positive/negative value, the subsystems consist of both ISS and non-ISS subsystems, simultaneously. It is shown that our presented results have wider applications than some existing works. Two examples, including one of the consensus for time-varying multiagent systems with cooperative and competitive protocols, are presented to demonstrate the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

This paper addresses the stability problem for linear continuous-time Markovian switching singular systems. Considering the inherent state jump behavior at the switching instants, a necessary and sufficient condition of exponential stability in the mean square sense for the Markovian switching singular system is established in terms of linear matrix inequalities by means of a stochastic Lyapunov approach. Based on the obtained stability result, sufficient conditions of exponential stability in the mean square sense for the Markovian switching singular system with uncertain and partly unknown transition probability are presented. Numerical examples are presented to illustrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper studies the stabilization problem via the co-design of controllers and a switching policy for a new class of nonstrict feedback switched nonlinear systems whose subsystems consist of a chain of integrators, feedback paths, and first-order feedforward paths. Designing only smooth feedback controllers cannot deal with the unstabilizable factors caused by feedforward paths. By exploiting the single control Lyapunov function method, an effective switching policy is co-designed to compensate the controllers. In addition, we present a generalized backstepping process, based on which the solvability of virtual controllers is guaranteed, the algebraic condition for stabilizability is identified, and the transient response of the closed-loop systems is improved. Two examples demonstrate the effectiveness. [ABSTRACT FROM AUTHOR]

In this paper, we consider the event-triggered control problem for continuous-time switched linear systems. It is assumed that only the sampled information of system state and switching signal is available to the controller at each sampling instant. Based on a mode-dependent event-triggered transmission scheme, the closed-loop system is modeled as a switched system with delayed state and augmented switching signal. Then, an exponential stability condition, characterized by the dwell time and average dwell time of the switching signal, is obtained. The condition presents an extension of the multiple Lyapunov functional method based stability analysis for sampled-data control of nonswitched system. Consequently, the design methods for state-feedback controller gains and event-triggered parameters are then formulated by the properly selected quadratic Lyapunov functional. The analysis results are significant and also lead to an important step to study the event-triggered control for switched system. Finally, based on the definition of event-trigger efficiency, the effectiveness and improvement of the proposed approach are illustrated by two numerical examples. [ABSTRACT FROM AUTHOR]

This paper addresses the cooperative output regulation problem of linear multi-agent systems under switching communication topologies. A distributed event-triggered control scheme is proposed so that the cooperative output regulation problem is solved with only intermittent communication. The communication topology is not required to be connected at every time instant under the jointly connected assumption. With the proposed triggering mechanism, each agent only transmits the information to its neighbors at its own triggering times or the switching times. By introducing a fixed timer, Zeno behavior is strictly excluded for each agent. The effectiveness of the proposed control scheme is demonstrated by an example. [ABSTRACT FROM AUTHOR]

This paper generalizes the LaSalle–Yoshizawa Theorem to switched nonsmooth systems. The Filippov and Krasovskii regularizations of a switched system are shown to be contained within the convex hull of the Filippov and Krasovskii regularizations of the subsystems, respectively. A common candidate Lyapunov function that has a negative semidefinite generalized time derivative along the trajectories of the subsystems is shown to be sufficient to establish LaSalle–Yoshizawa-like results for the switched system. Of independent interest, are the results on approximate continuity and Filippov regularization of set-valued maps, reduction of differential inclusions using Lipschitz continuous regular functions, and comparative remarks on different generalizations of the time derivative along the trajectories of a nonsmooth system. [ABSTRACT FROM AUTHOR]

This paper considers a singular problem in optimal control of continuous-time bimodal switched linear systems. A relaxed switched system with a continuous-valued switching signal is considered and the representations of singular control and singular arcs are derived. The similarity in the structure between the singular control and a stabilizing switching law is revealed and an approximation of the singular control by a well-defined switching signal is addressed. The results are demonstrated by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, the input-to-state stability is studied for stochastic impulsive switched time-delay systems. Using the vector Lyapunov function, average dwell time, and the properties of $M$ -matrix, different types of sufficient conditions are established. Both the case that the continuous dynamics is stable and the case that the discrete dynamics is stable are addressed, and the stability conditions are obtained. In the obtained stability conditions, different components of the vector Lyapunov function are allowed to be coupled; the information in consecutive impulsive switching intervals is also allowed to be coupled. Therefore, the magnification on the corresponding coupling items is avoided and the obtained results are more general and less conservative than the existing results. Furthermore, we investigate the relationships among the vector Lyapunov function approach, the approach based on the comparison principle and the scalar Lyapunov function approach. According to the vector Lyapunov function, the comparison system is constructed and the scalar-Lyapunov-function-based stability conditions are established. Finally, the applicability of our results is illustrated through two examples from neural systems and the synchronization problem of chaos-based secure communication systems. [ABSTRACT FROM AUTHOR]

This paper is concerned with studying Lyapunov characterization of integral input-to-state stability (iISS) for switched nonlinear time-varying systems. Sufficient conditions are given to verify iISS for switched nonlinear time-varying systems under a time-varying state-dependent switching law designed, which allow all subsystems to be not integral input-to-state stable (iISS) and the time derivative of Lyapunov functions of individual subsystems to be indefinite. An indefinite multiple Lyapunov functions (iMLFs) method for analyzing the dynamic behavior of switched nonlinear time-varying systems is provided. Also, an iMLFs-based small-gain theorem for switched interconnected nonlinear time-varying systems is presented, where each lower dimensional subsystem is allowed to be not iISS, which extends the small-gain technique from its original nonswitched nonlinear time-invariant version to a switched nonlinear time-varying version. Finally, an illustrative example is used to demonstrate the feasibility of the theoretical results. [ABSTRACT FROM AUTHOR]

The stability and stabilization problems for a class of switched discrete-time nonlinear systems are studied in this paper. Each nonlinear subsystem of the presented switched system is modeled as a piecewise affine (PWA) one by splitting the state space into polyhedron regions. With the aid of a simple searching strategy for active state transition pairs at a switching instant, i.e., the so-called $\mathbb {S}$ -arbitrary switching approach, the stability criteria are derived via the relaxed piecewise quadratic Lyapunov function technique. Then, using the descriptor system approach, a family of PWA stabilizing controllers are designed to guarantee exponential stability of the resulting closed-loop control system, and the corresponding PWA controller gains could be calculated using numerical software. The validity and potential of the developed techniques are verified through a numerical example. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability problem of randomly switched systems. By using the probability analysis method, the almost surely globally asymptotical stability and almost surely exponential stability are investigated for switched systems with semi-Markovian switching, Markovian switching, and renewal process switching signals, respectively. Two examples are presented to demonstrate the effectiveness of the proposed results, in which an example of consensus of multiagent systems with nonlinear dynamics is taken into account. [ABSTRACT FROM AUTHOR]

This paper is concerned with dynamic output feedback control for a class of switched systems with mode-dependent average dwell-time switching. By constructing a quasi-time-dependent Lyapunov function, the issues of global uniform asymptotic stability and $\ell _{2}$ -gain analysis for the switched system are addressed first. Then, a set of reduced-order output feedback controllers is designed, which is both mode-dependent and quasi-time-dependent. Compared with time-independent criteria, the new results greatly reduce the conservatism. The effectiveness and merits of the proposed method are illustrated with a numerical example. [ABSTRACT FROM AUTHOR]

This note points out certain limitations of our results from the paper mentioned in the title, and provides a modified approach to overcome these limitations. Specifically, the observer design addressed in the aforementioned paper is, in general, only applicable to switched linear systems with invertible state reset maps and this note presents a modified algorithm for state estimation that can also handle non-invertible state reset maps. In the process, we also identify some equalities from that paper which may not hold in general for arbitrary state reset maps. [ABSTRACT FROM PUBLISHER]

Most of the existing characterizations of the integral input-to-state stability (iISS) property are not valid for time-varying or switched systems in cases where converse Lyapunov theorems for stability are not available. This paper provides a characterization that is valid for switched and time-varying systems, and shows that natural extensions of some of the existing characterizations result in only sufficient but not necessary conditions. The results provided also pinpoint suitable iISS gains and relate these to supply functions and bounds on the function defining the system dynamics. [ABSTRACT FROM PUBLISHER]

This paper addresses the output reachable set estimation problem for continuous-time switched linear systems consisting of Hurwtiz stable subsystems. Based on a common Lyapunov function approach, the output reachable set is estimated by a union of bounding ellipsoids. Then, multiple Lyapunov functions with time-scheduled structure are employed to estimate the output reachable set for switched systems under dwell-time constraint. Furthermore, the safety verification problem of uncertain switched systems is investigated based on the result of output reachable set estimation. First, a sufficient condition ensuring the existence of an approximate bisimulation relation between two switched linear systems with a prescribed precision is proposed. Then, the safety verification for an uncertain switched system can be performed through an alternative safety verification for a switched system with exact parameters. Numerical examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of common zeroing-output systems (CZOS) to analyze the stability of switched systems. The concept of CZOS allows one to verify weak zero-state detectability. It characterizes a common behavior of any individual subsystem when the output signal for each subsystem is “approaching” zero. Heuristically speaking, it removes the effect of switching behavior, and thus enables one to analyze stability properties in systems with complex switching signals. With the help of CZOS, the Krasovskii–LaSalle theorem can be extended to switched nonlinear time-varying systems with both arbitrary switching and more general restricted switching cases. For switched nonlinear time-invariant systems, the needed detectability condition is further simplified, leading to several new stability results. Particularly, when a switched linear time-invariant system is considered, it is possible to generate a recursive method, which combines a Krasovskii–LaSalle result and a nested Matrosov result, to find a CZOS if it exists. The power of the proposed CZOS is demonstrated by consensus problems in literature to obtain a stronger convergence result with weaker conditions. [ABSTRACT FROM AUTHOR]

This paper investigates the state observer design of a class of unknown input switched linear systems via mode-dependent dwell time switchings. The distinguishing feature of the proposed method is that strong detectability condition of subsystems of the switched systems is unnecessarily required. First, a time-varying coordinate transformation is introduced to design a suitable reduced-order observer for each subsystem. Then, computable sufficient conditions on the synthesis of the observers are proposed in the framework of a mode-dependent dwell time technique. Since the observer of an individual subsystem cannot be designed due to unavailability of strong detectability condition of the subsystem, the state of the switched system is estimated under the condition of confining the dwell time by a pair of upper and lower bounds, restricting the growth of the Lyapunov function of the active subsystem and forcing “energy” of the overall switched system to decrease at switching instants. Finally, an example is presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Multiple Lyapunov functions (MLFs)-based small-gain theorems are presented for switched interconnected nonlinear systems with unstable subsystems, which extend the small-gain technique from its original non-switched nonlinear version to a switched nonlinear version. Each low dimensional subsystem does not necessarily have the input-to-state stability (ISS) property in the whole state space, and it only has individual ISS property in some subregions of the state space. The novelty of this paper is that integral-type MLFs and small-gain techniques are utilized to establish some MLFs-based small-gain theorems for switched interconnected nonlinear systems, which derive various stability results under some novel switching laws designed and construct integral-type MLFs. The small-gain theorems proposed cover several recent results as special cases, which also permit removal of a common restriction in which all low dimensional subsystems in switched interconnected systems are ISS or only some are ISS and others are not. Finally, two illustrative examples are presented to demonstrate the effectiveness of the results provided. [ABSTRACT FROM AUTHOR]

This paper analyzes the passivity and feedback passivity of discrete-time-switched nonlinear systems with passive and nonpassive modes that are affine in the control input. When a nonpassive mode is active, the increase in storage function is not necessarily bounded by the energy supplied to the switched system at every time step. Therefore, a switched system with at least one nonpassive mode is defined to be nonpassive in the classical passivity theory. In this paper, we propose a framework to analyze the passivity of such switched systems in a more general sense. We consider switched nonlinear systems which are affine in the control input and may consist of passive, feedback passive modes, and modes which cannot be rendered passive using feedback. In the proposed framework, we prove that a switched nonlinear system is locally feedback passive if and only if its zero dynamics are locally passive. A lower bound on the ratio of total activation time between (feedback) passive and nonfeedback passive modes is obtained to guarantee passive zero dynamics. Finally, we prove that two important properties of classical passivity still hold for the proposed passivity definition, that is: 1) output feedback control can be used to stabilize the switched system, and 2) parallel and negative feedback interconnections of two such passive systems are also passive. [ABSTRACT FROM AUTHOR]

In this paper, the problem of stability on stochastic systems with state-dependent switching is investigated. To analyze properties of the switched system by means of Itô's formula and Dynkin's formula, it is critical to show switching instants being stopping times. When the given active-region set can be replaced by its interior, the local solution of the switched system is constructed by defining a series of stopping times as switching instants, and the criteria on global existence and stability of solution are presented by Lyapunov approach. For the case where the active-region set can not be replaced by its interior, the switched systems do not necessarily have solutions, thereby quasi-solution to the underlying problem is constructed and the boundedness criterion is proposed. The significance of this paper is that all the results presented depend on some easily-verified assumptions that are as elegant as those in the deterministic case, and the proofs themselves provide design procedures for switching controls. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of designing time-varying quantizers for the stabilization of switched linear systems with quantized output and switching delays. The detection delays of switches are assumed to be time-varying but bounded, and the dwell time of the switching signal is assumed to be larger than the maximum delay. Given a switching controller, we analyze reachable sets of the closed-loop state by using a common Lyapunov function and then construct a quantizer that guarantees asymptotic stability. A sufficient condition for the existence of such a quantizer is characterized by the maximum switching delay and the dwell time. [ABSTRACT FROM AUTHOR]

We solve the problem of global uniform input-to-state stabilization of nonlinear switched systems with time-varying and periodic dynamics, with dynamic uncertainties, and with external disturbances. The switching signal is assumed to be unknown and the dynamics of the known components of the state vector is equivalent to the general triangular form (GTF) with non-invertible input-output maps. In our first and most general result, we prove that, if the dynamic uncertainty is treated as external disturbance, then the general triangular form system can be stabilized with arbitrarily small gain w.r.t. the dynamic uncertainty by means of a switching-independent, smooth and periodic feedback. Hence, using a suitable extension of the well-known small gain theorem to our case of switched systems with arbitrary switchings, we obtain the uniform input-to-state stabilization of the entire interconnected system. The second part of the paper addresses a more special case of triangular form (TF) switched systems with right-invertible input-output (I-O) maps with unknown switchings and with dynamic uncertainties. We show that the design becomes simpler and more constructive and the controllers become time-invariant if the dynamics is autonomous in this special case. Finally, we consider an example with explicit design of the stabilizing controllers. [ABSTRACT FROM PUBLISHER]

In this paper, exponential stabilizability of continuous-time positive switched systems is investigated. For two-dimensional systems, exponential stabilizability by means of a switching control law can be achieved if and only if there exists a Hurwitz convex combination of the (Metzler) system matrices. In the higher dimensional case, it is shown by means of an example that the existence of a Hurwitz convex combination is only sufficient for exponential stabilizability, and that such a combination can be found if and only if there exists a smooth, positively homogeneous and co-positive control Lyapunov function for the system. In the general case, exponential stabilizability ensures the existence of a concave, positively homogeneous and co-positive control Lyapunov function, but this is not always smooth. The results obtained in the first part of the paper are exploited to characterize exponential stabilizability of positive switched systems with delays, and to provide a description of all the “switched equilibrium points” of an affine positive switched system. [ABSTRACT FROM AUTHOR]

In this paper, we introduce continuous-time linear compartmental switched systems and investigate their stability and stabilizability properties. By their nature, these systems are always stable. Necessary and sufficient conditions for asymptotic stability for arbitrary switching functions, and sufficient conditions for asymptotic stability under certain dwell-time conditions on the switching functions are proposed. Finally, stabilizability is thoroughly investigated and proved to be equivalent to the existence of a Hurwitz convex combination of the subsystem matrices, a condition that, for positive switched systems, is only sufficient for stabilizability. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of networked control systems (NCSs) that are subject to time-varying transmission intervals, time-varying transmission delays, and communication constraints. Communication constraints impose that, per transmission, only one node can access the network and send its information. The order in which nodes send their information is orchestrated by a network protocol, such as, the Round-Robin (RR) and the Try-Once-Discard (TOD) protocol. In this paper, we generalize the mentioned protocols to novel classes of so-called “periodic” and “quadratic” protocols. By focusing on linear plants and controllers, we present a modeling framework for NCSs based on discrete-time switched linear uncertain systems. This framework allows the controller to be given in discrete time as well as in continuous time. To analyze stability of such systems for a range of possible transmission intervals and delays, with a possible nonzero lower bound, we propose a new procedure to obtain a convex overapproximation in the form of a polytopic system with norm-bounded additive uncertainty. We show that this approximation can be made arbitrarily tight in an appropriate sense. Based on this overapproximation, we derive stability results in terms of linear matrix inequalities (LMIs). We illustrate our stability analysis on the benchmark example of a batch reactor and show how this leads to tradeoffs between different protocols, allowable ranges of transmission intervals and delays. In addition, we show that the exploitation of the linearity of the system and controller leads to a significant reduction in conservatism with respect to existing approaches in the literature. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of determining switching rules for affine switched systems such that the system state is driven to a desired point and a guaranteed cost is minimized. The switching rule is determined by solving an LMI problem and global asymptotic stability of the tracking error dynamics is guaranteed even if sliding motions occur on any switching surface of the system. The potential of the results is illustrated on a true refrigeration system, namely a domestic refrigerator, where the purpose is to control the temperature in the fresh food and the freezer compartments. [ABSTRACT FROM PUBLISHER]

In this paper we deal with the stabilizability property for discrete-time switched linear systems. A recent necessary and sufficient characterization of stabilizability, based on set theory, is considered as the reference for comparing the computation-oriented sufficient conditions. The classical BMI conditions based on Lyapunov-Metzler inequalities are considered and extended. Novel LMI conditions for stabilizability, derived from the geometric ones, are presented that permit to combine generality with computational affordability. For the different conditions, the geometrical interpretations are provided and the induced stabilizing switching laws are given. The relations and the implications between the stabilizability conditions are analyzed to infer and compare their conservatism and their complexity. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of consistency for continuous-time switched linear systems having the switching function as a primary control signal to be designed. A switching control strategy is strictly consistent whenever it improves performance compared to the ones of all isolated subsystems. Conditions under which a min-type switching strategy is strictly consistent for the classes of \cal H2 and \cal H\infty performance indexes are determined. This property makes clear the importance of switching systems control design in both theoretical and practical application frameworks. Moreover, with this property it is not necessary to assume that all the subsystems are not stable in order to make a switching strategy design problem well posed. The theory is illustrated by means of several academic examples. [ABSTRACT FROM PUBLISHER]

This paper addresses the problem of controlling an uncertain time-varying plant by means of a finite family of candidate controllers supervised by an appropriate switching logic. It is assumed that, at every time, the plant consists of an uncertain single-input/single output linear system. It is shown that stability of the switched closed-loop system can be ensured provided that 1) at every time there is at least one candidate controller capable of potentially stabilizing the current time-invariant “frozen” plant model, and 2) the plant changes are infrequent or satisfy a slow drift condition. [ABSTRACT FROM AUTHOR]

This paper presents a characterization of observability and an observer design method for switched linear systems with state jumps. A necessary and sufficient condition is presented for observability, globally in time, when the system evolves under predetermined mode transitions. Because this characterization depends upon the switching signal under consideration, the existence of singular switching signals is studied alongside developing a sufficient condition that guarantees uniform observability with respect to switching times. Furthermore, while taking state jumps into account, a relatively weaker characterization is given for determinability, the property that concerns with recovery of the original state at some time rather than at all times. Assuming determinability of the system, a hybrid observer is designed for the most general case to estimate the state of the system and it is shown that the estimation error decays exponentially. Since the individual modes of the switched system may not be observable, the proposed strategy for designing the observer is based upon a novel idea of accumulating the information from individual subsystems. Contrary to the usual approach, dwell-time between switchings is not necessary, but the proposed design does require persistent switching. For practical purposes, the calculations also take into account the time consumed in performing computations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the derivation of a model reference adaptive control (MRAC) scheme for multimodal piecewise-affine (PWA) and piecewise-linear systems. The control allows the plant to track asymptotically the states of a multimodal piecewise affine (or smooth) reference model. The reference model can be characterized by a number and geometry of phase space regions that can be entirely different from those of the plant. Numerical simulations on a set of representative examples confirm the theoretical derivation and proof of stability. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM PUBLISHER]