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In this technical note, a necessary and sufficient stability criterion for switched linear systems under dwell-time constraint is proposed by employing a class of time-scheduled homogeneous polynomial Lyapunov functions with a sufficiently large degree. The key feature of this nonconservative condition lies in its convexity in the system matrices, which explicitly facilitates its further extension to uncertain systems. Then, in order to obtain numerically testable condition, a family of LMI conditions are presented with the aid of the idea of dividing the dwell-time interval into a finite number of segments. It is proved that the non-conservativeness can be maintained with a sufficiently large interval dividing parameter. In the end, the result is straightforwardly extended to the uncertain case in virtue of the convexity in the system matrices. Numerical examples are presented to illustrate our findings. [ABSTRACT FROM PUBLISHER]

A moment-matching method for model reduction of linear switched systems (LSSs) is presented. The method can be seen as a non-trivial extension of the Krylov subspace methods for linear time-invariant (LTI) systems. The procedure is based on the so called “nice selections,” which represent a choice of basis in the reachability or observability space of the LSS. The framework can also be used for exact matching of the input-output behavior of an LSS with a reduced order LSS for a specific switching sequence. Conditions for applicability of the method for model reduction are stated and finally the results are illustrated on numerical examples. [ABSTRACT FROM PUBLISHER]

This technical note studies switching stabilization problems for continuous-time switched linear systems. We consider four types of switching stabilizability defined under different assumptions on the switching control input. The most general switching stabilizability is defined as the existence of a measurable switching signal under which the resulting time-varying system is asymptotically stable. Discrete switching stabilizability is defined similarly but requires the switching signal to be piecewise constant on intervals of uniform length. In addition, we define feedback stabilizability in Filippov sense (respectively, sample-and-hold sense) as the existence of a feedback law under which closed-loop Filippov solution (respectively, sample-and-hold solution) is asymptotically stable. It is proved that the four switching stabilizability notions are equivalent and their sufficient and necessary condition is the existence of a piecewise quadratic control-Lyapunov function that can be expressed as the pointwise minimum of a finite number of quadratic functions. [ABSTRACT FROM PUBLISHER]

A sufficient condition for the stability of arbitrary switching linear systems (SLSs) without control inputs is that the individual subsystems are stable and their evolution matrices are simultaneously triangularizable (ST). This sufficient condition for stability is known to be extremely restrictive and not robust, and therefore of very limited applicability. The situation can be radically different when control inputs are present. Indeed, previous results have established that, depending on the number of states, inputs and subsystems, the existence of feedback matrices for each subsystem so that the corresponding closed-loop matrices are stable and ST can become a generic property, i.e., a property valid for almost every set of system parameters. This note provides novel contributions along two lines. First, we give sufficient conditions for the genericity of the property of existence of feedback matrices so that the subsystem closed-loop matrices are ST (not necessarily stable). Second, we give conditions for the genericity of the property of existence of feedback matrices that, in addition to achieving ST, enable arbitrary eigenvalue selection for each subsystem's closed-loop matrix. The latter conditions are less stringent than existing ones, and the approach employed in their derivation can be interpreted as an extension to SLSs of specific aspects of the notion of eigenvalue controllability for (non-switching) linear systems. [ABSTRACT FROM AUTHOR]

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]

We consider discrete-time linear time-invariant (LTI) systems with state feedback, where the actuation signal is subject to switching propagation delays, due to, e.g., routing in a multi-hop communication network. We show how to model these systems as pure switching linear systems and provide an algorithm for robust stability analysis. Next, we turn to the design problem and provide an algorithm that computes in a finite number of steps the look-ahead knowledge of the delays (i.e., routing) necessary to achieve controllability and stabilizability. This generalizes well known controllability criteria for LTI systems. [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]

A state-dependent switching law that obeys a dwell time constraint and guarantees the stability of a switched linear system is designed. Sufficient conditions are obtained for the stability of the switched systems when the switching law is applied in presence of polytopic type parameter uncertainty. A Lyapunov function, in quadratic form, is assigned to each subsystem such that it is non-increasing at the switching instants. During the dwell time, this function varies piecewise linearly in time. After the dwell, the system switches if the switching results in a decrease in the value of the LF. The method proposed is also applicable to robust stabilization via state-feedback. It is further extended to guarantee a bound on the \cal L2-gain of the switching system; it is also used in deriving state-feedback control law that robustly achieves a prescribed \cal L2-gain bound. [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]

The standard approach to the problem of controlling linear systems with large parameter uncertainty is to seek a controller that stabilizes the system and achieves a required performance over the whole polytope of uncertainty. In the case where the latter polytope is large, the design may become very conservative. We present an alternative approach where the uncertainty polytope is divided into overlapping smaller regions and where each of these regions is assigned to a separate subsystem. Assuming that there is online information on which of the regions the parameters of the system move to, a recently developed method for H\infty design of switched system with dwell time is applied. A Lyapunov Function (LF) in a quadratic form, which is non-increasing at the switching instants, is assigned to each subsystem. This function is used to determine the stability and to find a bound on the \cal L2 -gain of the switched system. The obtained results are used to solve the corresponding robust H\infty state-feedback and static output-feedback control problems. [ABSTRACT FROM AUTHOR]

The technical note studies a class of linear systems whose piecewise-constant dynamic matrix is subject to both stochastic jumps, governed by a Markov chain, and deterministic switches. These systems will be dubbed switching dynamics Markov jump linear systems (SD-MJLS). Sufficient conditions for exponential almost sure stability (EAS-stability) are established under either hard or average constraints on the dwell-time between switching instants. The proof relies on easy-to-check norm contractivity conditions and the ergodic law of large numbers. [ABSTRACT FROM AUTHOR]

This paper deals with model reduction by balanced truncation of switched linear systems (SLS). We consider switched linear systems whose dynamics, depending on the switching signal, switches between finitely many linear systems with a common state space. These linear systems are called the modes of the SLS. The idea is to seek for conditions under which there exists a single state space transformation that brings all modes of the SLS in balanced coordinates. As a measure of reachability and observability of the state components of the SLS, we take the average of the diagonal gramians. We then perform balanced truncation by discarding the state components corresponding to the smallest diagonal elements of this average balanced gramian. In order to carry out this program, we derive necessary and sufficient conditions under which a finite collection of linear systems with common state space can be balanced by a single state space transformation. Among other things, we derive sufficient conditions under which global uniform exponential stability of the SLS is preserved under simultaneous balanced truncation. Likewise, we derive conditions for preservation of positive realness or bounded realness of the SLS. Finally, in case that the conditions for simultaneous balancing do not hold, or we simply do not want to check these conditions, we propose to compute a suitable state space transformation on the basis of minimization of an overall cost function associated with the modes of the SLS. We show that in case our conditions do hold, this transformation is in fact simultaneously balancing, bringing us back to the original method described in this paper. [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]

This technical note is concerned with stability analysis and stabilization of Itô stochastic systems with Markovian switching. A couple of eigenvalue sets for some positive operator associated with the stochastic system under study are defined to characterize its stability in the mean square sense. Properties for these eigenvalue sets are established based on which we show that the spectral abscissa of these eigenvalues sets are the same and thus these eigenvalue sets are equivalent in the sense of characterizing the stability of the system. Also, it is shown that the guaranteed convergence rate of the Markovian jump Itô stochastic systems can be determined by some eigenvalue set. Finally, a linear matrix inequality based approach is proposed to design controllers such that the closed-loop system has guaranteed convergence rate. Some numerical examples are carried out to illustrate the effectiveness of the proposed approach. The research in this technical note opens several perspectives for future work stated as some open problems. [ABSTRACT FROM AUTHOR]

In this paper, a unified framework is proposed to study the exponential stability of discrete-time switched linear systems and, more generally, the exponential growth rates of their trajectories under three types of switching rules: arbitrary switching, optimal switching, and random switching. It is shown that the maximum exponential growth rates of system trajectories over all initial states under these three switching rules are completely characterized by the radii of convergence of three suitably defined families of functions called the strong, the weak, and the mean generating functions, respectively. In particular, necessary and sufficient conditions for the exponential stability of the switched linear systems are derived based on these radii of convergence. Various properties of the generating functions are established, and their relations are discussed. Algorithms for computing the generating functions and their radii of convergence are also developed and illustrated through examples. [ABSTRACT FROM AUTHOR]

Two state observers are designed for some classes of switched linear systems with unknown inputs. The design of the proposed observers assumes that all switching subsystems fulfill a property of “strong detectability” refid="ref4"/ that allows to implement suitable reduced-order unknown-input switched observers. The synthesis of the observers is based on the feasibility of a certain system of LMIs. Two main schemes are presented. For the case when the set of possible unknown-input distribution matrices are linearly dependent, an observer is suggested that guarantees the asymptotic state reconstruction without imposing any slow-switching dwell-time constraint about the sequence of the switching times. For the general case, the existence of a minimal average dwell-time for every switching sequence is assumed. By appropriate Lyapunov analysis, the convergence of the state estimate is proven to be exponential in both cases. Simulation results confirm the predicted performance. [ABSTRACT FROM AUTHOR]

This technical note presents a regulator synthesis approach for switched bimodal linear systems where it is desired to achieve regulation against known disturbance or reference signals. First, a set of observer-based Q-parameterized stabilizing controllers for the switched system is constructed. Then regulation conditions for the switched system are obtained and a regulator synthesis method is derived based on iteratively solving a set of linear matrix inequalities. Finally, a numerical example is presented to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Sufficient conditions are given for the stability of linear switched systems with dwell time and with polytopic type parameter uncertainty. A Lyapunov function, in quadratic form, which is non-increasing at the switching instants is assigned to each subsystem. During the dwell time, this function varies piecewise linearly in time after switching occurs. It becomes time invariant afterwards. This function leads to asymptotic stability conditions for the nominal set of subsystems that can be readily extended to the case where these subsystems suffer from polytopic type parameter uncertainties. The method proposed is then applied to stabilization via state-feedback both for the nominal and the uncertain cases. [ABSTRACT FROM AUTHOR]

This paper investigates sufficient conditions for the convergence to zero of the trajectories of linear switched systems. We provide a collection of results that use weak dwell-time, dwell-time, strong dwell-time, permanent and persistent activation hypothesis. The obtained results are shown to be tight by counterexample. Finally, we apply our result to the three-cell converter. [ABSTRACT FROM AUTHOR]

In this paper we consider a prototypical model matching problem where the various mappings involved are systems that switch arbitrarily among n stable linear time-in-variant (LTI) systems. The interest is placed on optimizing the worst-case performance of this model matching system over all possible switchings with either ℓ∞ -induced norm or ℋ2 norm as the performance criterion. This optimization is performed over all Youla-Kucera parameters that switch causally in time among n stable LTI systems. For the particular setup at hand, it is shown that the optimal Youla-Kucera parameter need not depend on the switching trajectory in the cases of partially matched switching and unmatched switching, and that it can be obtained as an LTI solution to an associated standard ℓ1 or ℋ2 optimization. In the case of matched switching, two convergent sequences to the optimal solution from above and below are formulated in terms of linear programs and quadratic programs respectively for the ℓ∞-induced and ℋ2 norm optimizations. An approximate solution with any given precision is possible by finite truncation. Applications of these results to sensitivity minimization, linear switched parameter systems, and cooperative control are provided. [ABSTRACT FROM AUTHOR]

We consider planar bilinear control systems with measurable controls. We show that any point in the reachable set can be reached by a "nice" control, specifically, a control that is a concatenation of a bang arc with either 1) a bang-bang control that is periodic after the third switch; or 2) a piecewise constant control with no more than two discontinuities. Under the additional assumption that the bilinear system is positive (or in- variant for any proper cone), we show that the reachable set is spanned by a concatenation of a bang arc with either 1) a bang-bang control with no more than two discontinuities; or 2) a piecewise constant control with no more than two discontinuities. In particular, any point in the reachable set can be reached using a piecewise-constant control with no more than three discontinuities. Several known results on the stability of planar linear switched systems under arbitrary switching follow as corollaries of our result. We demonstrate this with an example. [ABSTRACT FROM AUTHOR]

In the paper, an extension of LaSalle's Invariance Principle to a class of switched linear systems is studied. One of the motivations is the consensus problem in multi-agent systems. Unlike most existing results in which each switching mode in the system needs to be asymptotically stable, this paper allows that the switching modes are only Lyapunov stable. Under certain ergodicity assumptions, an extension of LaSalle's Invariance Principle for global asymptotic stability is obtained. Then it is used to solve the consensus reaching problem of certain multi-agent systems in which each agent is modeled by a double integrator, and the associated interaction graph is switching and is assumed to be only jointly connected. [ABSTRACT FROM AUTHOR]

Abstract-For a group of linear systems, each under a saturated linear, not necessarily stabilizing, feedback law, we design a switching scheme such that the resulting switched system is locally asymptotically stable at the origin with a large domain of attraction. By expressing each saturated linear feedback in a convex hull of a group of auxiliary linear feedbacks, we formulate and solve the problem of designing such a switching scheme as a constrained optimization problem with the objective of maximizing an estimate of the domain of attraction. Simulation results indicate that the resulting domain of attraction extends well beyond the linear regions of the actuators. [ABSTRACT FROM AUTHOR]

This paper investigates the number of switchings and design of switching sequences for controllability of switched linear systems. Two related results are established. One is a new constructive approach to designing switching sequences. The controllable state set of each switching sequence designed via the approach coincides with the controllable subspace of switched linear systems. The other is a well-estimated value for the minimum number of switchings required for controllability. Each state in the controllable subspace can be steered to origin within this value of switching times. [ABSTRACT FROM AUTHOR]

We consider products of matrix exponentials under the assumption that the matrices span a nilpotent Lie algebra. In 1995, Gurvits conjectured that nilpotency implies that these products are, in some sense, simple. More precisely, there exists a uniform bound l such that any product can be represented as a product of no more than l matrix exponentials. This conjecture has important applications in the analysis of linear switched systems, as it is closely related to the problem of reachability using a uniformly bounded number of switches. It is also closely related to the concept of nice reachability for bilinear control systems. The conjecture is trivially true for the case of first-order nilpotency. Gurvits proved the conjecture for the case of second-order nilpotency using the Baker-Campbell-Hausdorff formula. We show that the conjecture is false for the third-order nilpotent case using an explicit counterexample. Yet, the underlying philosophy behind Gurvits' conjecture is valid in the case of third-order nilpotency. Namely, such systems do satisfy the following nice reachability property: any point in the reachable set can be reached using a piecewise constant control with no more than four switches. We show that even this form of finite reachability is no longer true for the case of fifth-order nilpotency. [ABSTRACT FROM AUTHOR]

It was recently conjectured that the Hurwitz stability of the convex hull of a set of Metzler matrices is a necessary and sufficient condition for the asymptotic stability of the associated switched linear system under arbitrary switching. In this note, we show that 1) this conjecture is true for systems constructed from a pair of second-order Metzler matrices; 2) the conjecture is true for systems constructed from an arbitrary finite number of second-order Metzler matrices; and 3) the conjecture is in general false for higher order systems. The implications of our results, both for the design of switched positive linear systems, and for research directions that arise as a result of our work, are discussed toward the end of the note. [ABSTRACT FROM AUTHOR]

This paper investigates the switching stabilizability problem for a class of continuous-time switched linear systems with time-variant parametric uncertainties. First, a necessary and sufficient condition for the asymptotic stabilizability of such uncertain switched linear system is derived, under the assumption that the closed-loop switched system does not generate sliding motions. Then, an additional condition is introduced to exclude the possibility of unstable sliding motions. Finally, a necessary and sufficient for the asymptotic stabilizability of such continuous-time uncertain switched linear systems is presented. This result improves upon conditions found in the literature which are either sufficient only or necessary only. [ABSTRACT FROM AUTHOR]

We consider continuous time switched systems that are stabilized via a computer. Several factors (sampling, computer computation, communications through a network, etc.) introduce model uncertainties produced by unknown varying feedback delays. These uncertainties can lead to instability when they are not taken into account. Our goal is to construct a switched digital control for continuous time switched systems that is robust to the varying feedback delay problem. The main contribution of this note is to show that the control synthesis problem in the context of un- known time varying delays can be expressed as a problem of stabilizability for uncertain systems with polytopic uncertainties. [ABSTRACT FROM AUTHOR]

In this note, we study the stabilization problem of systems that switch among a finite set of controllable linear systems with arbitrary switching frequency. For both eases of known and unknown switching functions, feedback laws are designed to achieve exponential stability. For the later case, a method combining on-line adaptive estimation and feedback stabilization is developed in the controller design. [ABSTRACT FROM AUTHOR]

Based on the algebraic transformation of. a switched linear measurement equation into a nonlinear, yet deterministic, equation, an asymptotic state observer is constructed for discrete-time linear systems whose observations are generated according to randomly switching measurement modes. The observer, which combines the algebraic transformation with a Newton observer applied to the resulting nonlinear measurement equation, is shown to be locally exponentially convergent under arbitrary mode sequences. [ABSTRACT FROM AUTHOR]

Studies the reachability and stabilization of switched linear systems. Construction of an event-driven switching stabilization strategy; Control techniques based on switching between different controllers; State trajectories under event-driven switching strategy.

We consider n-dimensional positive linear switched systems. A necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is Hurwitz. Several researchers conjectured that for positive linear switched systems this condition is also sufficient. Recently, Gurvits, Shorten, and Mason showed that this conjecture is true for the case n = 2, but is not true in general. Their results imply that there exists some minimal integer np, such that the conjecture is true for all n < np, but is not true for n = np. We show that np = 3. [ABSTRACT FROM AUTHOR]