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In this paper, we study attitude synchronization for elements in the unit sphere in $\mathbb {R}^{\scriptscriptstyle {\scriptscriptstyle {3}}}$ and for elements in the three-dimensional (3-D) rotation group, for a network with switching topology. The agents’ angular velocities are assumed to be the control inputs, and a switching control law for each agent is devised that guarantees synchronization, provided that all elements are initially contained in a region, which we identify later in the paper. The control law is decentralized and it does not require a common orientation frame among all agents. We refer to synchronization of unit vectors in $\mathbb {R}^{\scriptscriptstyle {3}}$ as incomplete synchronization, and of 3-D rotation matrices as complete synchronization. Our main contribution lies in showing that these two problems can be analyzed under a common framework, where all agents’ dynamics are transformed into unit vectors dynamics on a sphere of appropriate dimension. [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 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]

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]

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 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]

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]

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 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]

It is proved in this paper that the existence of suitable multiple Lyapunov functions is a necessary and sufficient condition for a discrete-time nonlinear switching system, with given switches digraph, to be globally asymptotically stable. The same result is provided for the input-to-state stability. The less is the number of edges in the switches digraph, the less is the number of inequalities that are involved in the provided necessary and sufficient Lyapunov conditions. [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 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]

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]

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]

In this paper, we study the cooperative global robust output regulation problem for nonlinear multiagent systems in output feedback form with any relative degree subject to directed switching networks. Through the augmentation by a distributed internal model and a distributed observer of the given system, we first convert the original problem into the global robust stabilization problem of the so-called extended augmented system, which is a multi-input-coupled switched system. Then, we establish a few technical lemmas to lay the foundation for constructing a switching control law and the corresponding multiple Lyapunov functions for the switched system. Then, we present the solvability conditions of our problem via the multiple Lyapunov functions and average dwell time method. The main result will be applied to the leader-following synchronization problem for a group of hyperchaotic Lorenz multiagent systems with a directed switching network. [ABSTRACT FROM PUBLISHER]

In this paper, we study geometric properties of basins of attraction of monotone systems. Our results are based on a combination of monotone systems theory and spectral operator theory. We exploit the framework of the Koopman operator, which provides a linear infinite-dimensional description of nonlinear dynamical systems and spectral operator-theoretic notions such as eigenvalues and eigenfunctions. The sublevel sets of the dominant eigenfunction form a family of nested forward-invariant sets and the basin of attraction is the largest of these sets. The boundaries of these sets, called isostables, allow studying temporal properties of the system. Our first observation is that the dominant eigenfunction is increasing in every variable in the case of monotone systems. This is a strong geometric property which simplifies the computation of isostables. We also show how variations in basins of attraction can be bounded under parametric uncertainty in the vector field of monotone systems. Finally, we study the properties of the parameter set for which a monotone system is multistable. Our results are illustrated on several systems of two to four dimensions. [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]

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]

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]

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]

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]

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]

Super-twisting algorithm is one of the versions of high-order sliding mode control. The interest to this algorithm is explained by its attractive properties: continuous control input, finite convergence time, disturbance rejection. In this paper, the upper bound of admissible unknown disturbances and low bound of the convergence time are found and shown that both the values can be achieved with any desired accuracy. [ABSTRACT FROM AUTHOR]

In this note, we present corrections to our previous paper “Switching rule design for switched dynamic systems with affine vector fields” refid="ref1"/, refid="ref2"/ . The correction is necessary to guarantee the stability of the system under sliding motion, which is not ensured from refid="ref1"/, refid="ref2"/ due to an error in the expressions (11) of the original paper refid="ref1"/. To correct the results it is necessary to add a new condition to the results of refid="ref1"/, refid="ref2"/. If there is no sliding motion or if the sliding motion dynamics, in the sense of Filippov, satisfies a quadratic stability condition, the additional condition is not necessary. The corrections in this note are restricted to the case of two operation modes. [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]

This paper considers asymptotic stability and transient performance of discrete-time piecewise affine systems. We propose a procedure to construct a nested sequence of finite-state symbolic models, each of which abstracts the original piecewise affine system and leads to linear matrix inequalities for guaranteed stability and performance levels. This sequence is in the order of decreasing conservatism, and hence gives us the option to pay more computational cost and analyze a finer symbolic model within the sequence in return for less conservative results. Moreover, in the special case where this sequence is finite, an exact analysis of stability and performance is achieved via semidefinite programming. [ABSTRACT FROM PUBLISHER]