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This paper considers the problem of designing adaptive feedforward control (AFC) systems for uncertain single-input single-output linear systems perturbed by multisinusoidal disturbances of known frequencies. The proposed approach removes the longstanding assumption that either the sign of the real part or the imaginary part of the transfer function of a stable plant at the frequency of excitation is known for AFC to be applicable, which is referred to in this paper as a strictly positive real (SPR)-like condition. Notable features of the solution are that persistence of excitation is not required, and stability analysis tools based on averaging are avoided; hence, the requirement of an exponentially stable equilibrium for the closed-loop system is circumvented. [ABSTRACT FROM AUTHOR]

This paper deals with the certification problem for robust quadratic stability, robust state convergence, and robust quadratic performance of linear systems that exhibit bounded rates of variation in their parameters. We consider both continuous-time (CT) and discrete-time (DT) parameter-varying systems. In this paper, we provide a uniform method for this certification problem in both cases and we show that, contrary to what was claimed previously, the DT case requires a significantly different treatment compared to the existing CT results. In the established uniform approach, quadratic Lyapunov functions, which are affine in the parameter, are used to certify robust stability, robust convergence rates, and robust performance in terms of linear matrix inequality feasibility tests. To exemplify the procedure, we solve the certification problem for $\mathscr {L}_2$ -gain performance both in the CT and the DT cases. A numerical example is given to show that the proposed approach is less conservative than a method with slack variables. [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]

In this paper, we study connections between structured storage or Lyapunov functions of a class of interconnected systems (dynamical networks) and dissipativity properties of the individual systems. We prove that if a dynamical network, composed as a set of linear time invariant systems interconnected over an acyclic graph, admits an additive quadratic Lyapunov function, then the individual systems in the network are dissipative with respect to a (nonempty) set of interconnection neutral supply functions. Each supply function from this set is defined on a single interconnection link in the network. Specific characterizations of neutral supply functions are presented, which imply robustness of network stability/dissipativity to removal of interconnection links. [ABSTRACT FROM AUTHOR]

We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function, which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sum of squares certificates and, hence, such a Lyapunov function can always be found by semidefinite programming. This generalizes the classical fact that an asymptotically stable linear system admits a quadratic Lyapunov function, which satisfies a certain linear matrix inequality. In addition to homogeneous vector fields, the result can be useful for showing local asymptotic stability of nonhomogeneous systems by proving asymptotic stability of their lowest order homogeneous component. This paper also includes some negative results: We show that in absence of homogeneity, globally asymptotically stable polynomial vector fields may fail to admit a global rational Lyapunov function, and in presence of homogeneity, the degree of the numerator of a rational Lyapunov function may need to be arbitrarily high (even for vector fields of fixed degree and dimension). On the other hand, we also give a family of homogeneous polynomial vector fields that admit a low-degree rational Lyapunov function but necessitate polynomial Lyapunov functions of arbitrarily high degree. This shows the potential benefits of working with rational Lyapunov functions, particularly as the ones whose existence we guarantee have structured denominators and are not more expensive to search for than polynomial ones. [ABSTRACT FROM AUTHOR]

In this paper, the stability and $L_2$ -gain properties of linear impulsive delay systems with delayed impulses are studied. Commonly employed techniques, in which the delayed impulses are treated using Newton–Leibniz formula, may not be applicable to $L_2$ -gain analysis, since they make the disturbance input appear in the impulse part. In order to circumvent the difficulty, we first augment the considered system to a time-delay system with switching nondelayed impulses. Due to the absence of delayed impulses, this new approach has advantages in constructing Lyapunov functions and handling the effects of impulse delays on the system performance. Switching-based time-dependent Lyapunov functions are introduced to deal with the resultant switching impulses of the augmented system. Sufficient conditions for exponential stability and $L_2$ -gain properties are derived in terms of linear matrix inequalities. Numerical examples are provided to illustrate the efficiency of the new approach. [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]

This paper is concerned with the regional stabilization problem for discrete time-delay systems with actuator saturations. Different from some existing delay-independent techniques handling the saturation nonlinearity, a delay-dependent polytopic approach is first proposed in a discrete-time framework. Then, by combining with an augmented Lyapunov function, a discrete Wirtinger-based inequality and some novel analysis techniques, improved delay-dependent regional stabilization conditions are established by means of linear matrix inequalities. Finally, it is shown via two numerical examples that the obtained results are less conservative than some existing ones yet providing a larger estimate of an initial condition set. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the stabilizability of controlled discrete-time switched linear systems (CSLSs), where the discrete switching control input and the continuous control input coexist. For a given positive integer $h$ , we propose a quantitative metric of stabilizability, called $h$ -contraction rate ($h$ -CR), and prove that it serves as an indicator of the stabilizability in the following sense: the $h$ -CR is less than one for sufficiently large $h$ if and only if the switched linear system is stabilizable. In addition, we develop computational tools to estimate upper and lower bounds of the $h$ -CR. In particular, the upper bound estimation is expressed as a semidefinite programming problem. We derive its Lagrangian dual problem and prove that the dual problem with an additional rank constraint gives an exact characterization of the $h$ -CR. An algorithm is developed to solve the dual problem with the rank constraint. [ABSTRACT FROM AUTHOR]

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 studies stochastic input-to-state stability and stochastic global stability for impulsive stochastic nonlinear systems. Using fixed dwell-time condition and Lyapunov-based approach, sufficient conditions are established for the stability properties. Two cases are studied: 1) the case that the continuous dynamics is stable and 2) the case that the impulsive effects are stable. Furthermore, the relations among different dwell-time conditions are studied. Finally, two examples are used to illustrate the developed theory. [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]

Event-triggered control (ETC) is a control strategy that is especially suited for applications where communication resources are scarce. By updating and communicating sensor and actuator data only when needed for stability or performance purposes, ETC is capable of reducing the amount of communications, while still retaining a satisfactory closed-loop performance. In this paper, an ETC strategy is proposed by striking a balance between conventional periodic sampled-data control and ETC, leading to so-called periodic event-triggered control (PETC). In PETC, the event-triggering condition is verified periodically and at every sampling time it is decided whether or not to compute and to transmit new measurements and new control signals. The periodic character of the triggering conditions leads to various implementation benefits, including a minimum inter-event time of (at least) the sampling interval of the event-triggering condition. The PETC strategies developed in this paper apply to both static state-feedback and dynamical output-based controllers, as well as to both centralized and decentralized (periodic) event-triggering conditions. To analyze the stability and the \cal L2-gain properties of the resulting PETC systems, three different approaches will be presented based on 1) impulsive systems, 2) piecewise linear systems, and 3) perturbed linear systems. Moreover, the advantages and disadvantages of each of the three approaches will be discussed and the developed theory will be illustrated using a numerical example. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the exponential stabilization problem for autonomous discrete-time switched linear systems (SLSs), where only the discrete mode can be controlled. Our approach is based on periodic control Lyapunov functions whose value decreases periodically instead of at each time step as in the classical control Lyapunov functions. Using periodic control Lyapunov functions, we develop stabilizability analysis and controller synthesis conditions that are less conservative than existing results in that they apply to a larger class of SLSs. Utilizing recent results on the switched optimal control problems, a constructive way to find periodic control Lyapunov functions is presented. [ABSTRACT FROM PUBLISHER]

The design of robust controllers for linear systems with structured time-varying uncertainty is a fundamental problem in control systems. This problem is still open because the available methods for robust analysis lead to nonconvex optimization whenever design variables are present. This paper proposes a novel approach that overcomes this difficulty. Specifically, continuous-time linear systems with polytopic time-varying uncertainty are considered. The basic problem consists of designing a robust static output feedback controller ensuring robust asymptotical stability. A novel approach is proposed through the introduction of robust stabilizability functions (RSFs), i.e., functions able to establish whether a controller is robustly asymptotically stabilizing. In particular, RSFs based on homogeneous polynomial Lyapunov functions (HPLFs) are searched for. The proposed approach requires the solution of two convex optimization problems with constraints expressed as linear matrix inequalities (LMIs). For any size of the LMIs, the proposed approach provides a sufficient condition for the existence of a sought controller. Moreover, this condition is also necessary when the size of the LMIs is large enough. Several extensions of the proposed methodology are presented, which deal with the synthesis of controllers ensuring desired decay rate, fixed-order dynamic output controllers, and controllers with the minimum norm. [ABSTRACT FROM PUBLISHER]

In this paper, we are concerned with the analysis of linear infinite-dimensional control systems that should be able to compensate and/or track signals that are periodic. Adopting the name given in the seminal paper by Hara et al., we call them repetitive controllers. We analyze the asymptotic stability in the Lyapunov sense of finite-dimensional positive real plants coupled with pure delays. For this class of systems, we initially prove convergence in the weak topology to later deduce convergence in the strong one. [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]

In this paper, the problem of frequency-limited \mmb H\infty model reduction for positive linear time-invariant systems is investigated. Specifically, our goal is to find a stable positive reduced-order model for a given positive system such that the \mmb H\infty norm of the error system is bounded over a frequency interval of interest. A new condition in terms of matrix inequality is developed for characterizing the frequency-limited \mmb H\infty performance. Then an equivalent parametrization of a positive reduced-order model is derived, based on which, an iterative algorithm is constructed for optimizing the reduced-order model. The algorithm utilizes coarse reduced-order models resulting from (generalized) balanced truncation as the initial value. Both continuous- and discrete-time systems are considered in the same framework. Numerical examples clearly show the effectiveness and advantages of the proposed model reduction method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of constructing a finite bisimulation quotient for a discrete-time switched linear system in a bounded subset of its state space. Given a set of observations over polytopic subsets of the state space and a switched linear system with stable subsystems, the proposed algorithm generates the bisimulation quotient in a finite number of steps with the aid of sublevel sets of a polyhedral Lyapunov function. Starting from a sublevel set that includes the origin in its interior, the proposed algorithm iteratively constructs the bisimulation quotient for the region bounded by any larger sublevel set. We show how this bisimulation quotient can be used for synthesis of switching laws and verification with respect to specifications given as syntactically co-safe Linear Temporal Logic formulae over the observed polytopic subsets. [ABSTRACT FROM AUTHOR]

This paper proposes a dynamic controller structure and a systematic design procedure for stabilizing discrete-time hybrid systems. The proposed approach is based on the concept of control Lyapunov functions (CLFs), which, when available, can be used to design a stabilizing state-feedback control law. In general, the construction of a CLF for hybrid dynamical systems involving both continuous and discrete states is extremely complicated, especially in the presence of non-trivial discrete dynamics. Therefore, we introduce the novel concept of a hybrid control Lyapunov function, which allows the compositional design of a discrete and a continuous part of the CLF, and we formally prove that the existence of a hybrid CLF guarantees the existence of a classical CLF. A constructive procedure is provided to synthesize a hybrid CLF, by expanding the dynamics of the hybrid system with a specific controller dynamics. We show that this synthesis procedure leads to a dynamic controller that can be implemented by a receding horizon control strategy, and that the associated optimization problem is numerically tractable for a fairly general class of hybrid systems, useful in real world applications. Compared to classical hybrid receding horizon control algorithms, the proposed approach typically requires a shorter prediction horizon to guarantee asymptotic stability of the closed-loop system, which yields a reduction of the computational burden, as illustrated through two examples. [ABSTRACT FROM PUBLISHER]

In this paper we consider the class of discrete-time switched systems switching between p autonomous positive subsystems. First, sufficient conditions for testing stability, based on the existence of special classes of common Lyapunov functions, are investigated, and these conditions are mutually related, thus proving that if a linear copositive common Lyapunov function can be found, then a quadratic positive definite common function can be found, too, and this latter, in turn, ensures the existence of a quadratic copositive common function. Secondly, stabilizability is introduced and characterized. It is shown that if these systems are stabilizable, they can be stabilized by means of a periodic switching sequence, which asymptotically drives to zero every positive initial state. Conditions for the existence of state-dependent stabilizing switching laws, based on the values of a copositive (linear/quadratic) Lyapunov function, are investigated and mutually related, too. [ABSTRACT FROM PUBLISHER]

The objective of the paper is to establish -stability results for the state processes of a new class of slowly time-varying hybrid continuous-time linear stochastic systems. The systems are hybrid in the sense that jumps both in the dynamics and the state may occur. In both cases a jump amounts to resetting to an initial value. Jump-times are defined by a more or less arbitrary point process. Most of the analysis is based on a carefully organized stochastic Lyapunov-function argument. We motivate our study by a brief summary of some basic problems in continuous-time recursive estimation of linear stochastic systems within a framework analogous to the one developed by Benveniste et al. [1]. [ABSTRACT FROM AUTHOR]