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In this paper, we introduce a method to obtain necessary and sufficient stability conditions for systems governed by one-dimensional nonlinear hyperbolic partial-differential equations with closed-loop integral controllers, when the linear frequency analysis cannot be used anymore. We study the stability of a general nonlinear transport equation where the control input and the measured output are both located on the boundaries. The principle of the method is to extract the limiting part of the stability from the solution using a projector on a finite-dimensional space and then use a Lyapunov approach. This paper improves a result of Trinh, Andrieu, and Xu, and gives an optimal condition for the design of the controller. The results are illustrated with numerical simulations where the predicted stable and unstable regions can be clearly identified. [ABSTRACT FROM AUTHOR]

This paper proposes a less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions by constructing a novel Lyapunov functional. Compared with the Lyapunov functional in the literature, this paper fully utilizes the sector and slope restrictions in our Lyapunov functional, which includes integral terms involved with the sector restriction of the nonlinearities $\phi (y_{i})$ and $\phi (y_{i+1})$ , integral terms involved with the slope restriction between $\phi (y_{i})$ and $\phi (y_{i+1})$ , and a quadratic term with an augmented vector related to available vectors for representing upper and lower bounds of all integral terms. The positive definiteness of a matrix appearing in the quadratic term can be relaxed by utilizing the lower bounds of all integral terms. Based on the Lyapunov functional, an improved stability criterion is derived in terms of linear matrix inequalities. Numerical examples show the effectiveness of the proposed criterion. [ABSTRACT FROM AUTHOR]

A popular design framework for networked control systems (NCSs) is the emulation-based approach combined with Lyapunov-based analysis techniques for hybrid systems. In virtually all papers that use this framework, bounds in terms of the maximal allowable transmission interval (MATI) are provided to guarantee stability, and performance properties of the NCS. However, having only such a MATI condition is rather restrictive, and unrealistic in practice due to various network effects such as packet losses, leading to conservative bounds. In this article, we therefore consider an alternative condition on the communication instants to better capture the time-varying properties of the transmission intervals. In particular, we propose, in addition to the existence of a MATI, to also impose a bound on the average allowable transmission interval, expressed in terms of a reverse average dwell-time condition on the transmission intervals. We demonstrate by means of a novel Lyapunov-based analysis that stability of the NCS can still be guaranteed under this different condition on the transmission intervals, which can, in addition, lead to a significant improvement of the MATI. The strengths of these new results will be illustrated on a numerical example. [ABSTRACT FROM AUTHOR]

It is proved in this paper that the existence of a delay-dependent suitable Lyapunov function is a necessary and sufficient condition for a discrete-time fully nonlinear time-delay system, with given delays 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 delays digraph, the less is the number of inequalities that are involved in the provided necessary and sufficient Lyapunov conditions. The case of arbitrary time-varying time delays, with no constraints as long as bounded, is covered as a special case. [ABSTRACT FROM AUTHOR]

Periodic event-triggered control (PETC) is an appealing paradigm for the implementation of controllers on platforms with limited communication resources, a typical example being networked control systems. In PETC, transmissions over the communication channel are triggered by an event generator, which depends solely on the available plant and controller data and is only evaluated at given sampling instants to enable its digital implementation. In this paper, we consider the general scenario, where the controller communicates with the plant via multiple decoupled networks. Each network may contain multiple nodes, in which case a dedicated protocol is used to schedule transmissions among these nodes. The transmission instants over the networks are asynchronous and generated by local event generators. At given sampling instants, the local event generator evaluates a rule, which only involves the measurements and the control inputs available locally, to decide whether a transmission is needed over the considered network. Following the emulation approach, we show how to design local triggering generators to ensure input-to-state stability and $\mathcal {L}_p$ stability for the overall system based on a continuous-time output-feedback controller that robustly stabilizes the network-free system. The method is applied to a class of Lipschitz nonlinear systems, for which we formulate the design conditions as linear matrix inequalities. The effectiveness of the scheme is illustrated via simulations of a nonlinear example. [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 paper deals with stabilization of feedforward multiple cascade dynamics under sampling. It is shown that u-average passivity concepts and Lyapunov methods can be profitably exploited to provide a systematic sampled-data design procedure. The proposed methodology recalls the continuous-time feedforwarding steps and can be applied under the same assumptions as those set over the continuous-time cascade dynamics. The final sampled feedback is carried out through a three-steps procedure that involves iterative passivation and stabilization in the u-average sense. Constructive aspects are developed to compute approximate solutions which are indeed implemented in practice. An example is worked out with comparative simulations with respect to usual sampled-and-hold implementations. [ABSTRACT FROM AUTHOR]

This paper studies input-to-state stability for hybrid systems with memory, which models hybrid dynamics affected by time delays. Using both Lyapunov–Razumikhin functions and Lyapunov–Krasovskii functionals, Lyapunov-based sufficient conditions are established for input-to-state stability. In addition, further extensions and relaxations are proposed for special cases, such as the stable flow/jump cases and the cases that Lyapunov functions do not decrease strictly during flow/jumps. Finally, two examples are used to illustrate the developed results [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]

Previous small-gain results for interconnected integral input-to-state (iISS) systems have been restricted to systems that are strongly iISS. This paper removes this restriction by allowing cross terms between external inputs and states in the Lyapunov decrease of component systems, and subsequently constructing a nonseparable Lyapunov function. An example demonstrates the use of this new small-gain formulation. [ABSTRACT FROM AUTHOR]

In this paper, the stability, stabilization, and $L_{2}$ -gain problems are investigated for periodic piecewise systems with time-varying subsystems. Continuous Lyapunov function with time-varying Lyapunov matrix is adopted. A condition guaranteeing the negative definiteness of a matrix polynomial, deriving from the Lyapunov derivative, is first obtained. Based on such a condition, an exponential stability condition is provided. Moreover, a state-feedback controller with time-varying gain is developed to stabilize the unstable periodic piecewise time-varying system. The $L_{2}$ -gain criterion for periodic piecewise time-varying system is also studied. Numerical examples are given to show the validity of the proposed techniques. [ABSTRACT FROM AUTHOR]

In this paper, the effects of bounded disturbances on decentralized event-triggered control systems are studied. The input-to-state (practical) stability of integral-based event-triggered control systems and dynamic event-triggered control systems is analyzed in a uniform framework by utilizing a new Lyapunov functional approach. An estimation on the upper bound of the input-to-state stability gain is given analytically. First, Zeno behavior is excluded with the time-regularized mechanisms, that is, a prespecified lower bound of inter-event times is introduced. Then, the conditions are presented under which the considered event-triggered control systems ensure Zeno-freeness without time regularization. Finally, a numerical example is given to illustrate the efficiency and feasibility of the proposed results. [ABSTRACT FROM AUTHOR]

Compositional control methods are well-suited for the design of large-scale control systems. QSR-dissipativity and passivity are closely related to stability and may be utilized to show Lyapunov and finite-gain stability for dynamical systems under certain conditions. Additionally, passivity is preserved under feedback and parallel interconnections of passive systems and may be utilized to build larger stable systems. In this paper, we present a QSR-dissipativity and passivity based analysis for event-triggered networked control systems (NCSs). We characterize QSR-dissipativity, passivity, and stability conditions for an event-triggered NCS in cases where an input–output event-triggering sampler is located on the plant's side, controller's side, or both sides leading to a considerable decrease in the communication load among subunits. We calculate passivity levels for the entire NCS. We show that the general passivity and stability conditions of the NCS depend on passivity properties of the plant and controller. Our results illustrate the tradeoffs amongst passivity levels, stability, and the rates of communication between subsystems. The motivation behind our work is that by designing and connecting individual stable passive event-triggered NCSs with desired passivity levels, one can design larger stable passive compositional networked systems that are suitable for the design and control of large-scale systems, such as cyber-physical systems. [ABSTRACT FROM AUTHOR]

Although the finite-time control theory and application have been studied very well in the literature, few results are concerned with the effect of sampled-data control. In this paper, first, the stability analysis of a chain of integrators under a discretized finite-time controller is given. Then, in the presence of disturbance, the robustness of the closed-loop system is also investigated. Based on a Lyapunov method, the ultimate bound for the steady states of the considered closed-loop system is estimated. Some simulation and examples are provided to verify the proposed theoretical results. [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]

Equilibrium-independent dissipativity (EID) is a recently introduced system property that requires a system to be dissipative with respect to any forced equilibrium configuration. This paper is a detailed examination of EID with quadratic supply rates for a common class of nonlinear control-affine systems. We provide an algebraic characterization of EID for such systems in the spirit of the Hill–Moylan lemma, where the usual stability condition is replaced by an incremental stability condition. Based on this characterization, we state results concerning internal stability, feedback stability, and absolute stability of EID systems. Finally, we study EID for discrete-time systems, providing the relevant definitions and an analogous Hill–Moylan-type characterization. Results for both continuous-time and discrete-time systems are illustrated through examples on physical systems and convex optimization algorithms. [ABSTRACT FROM AUTHOR]

This paper considers the stability analysis of nonlinear Lurie type systems where the nonlinearity is both (locally) sector and slope restricted. Convex conditions for verifying stability, computing outer estimates of reachable sets, and upper bounds on the induced $\mathcal {L}_2$ gain in a local or global domain are proposed. The conditions use a Lyapunov function that is quadratic on both the states and the nonlinearity and has an integral term on the nonlinearity. Numerical examples outline the benefits of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper presents a small-gain result for large-scale interconnected systems composed of hybrid subsystems. It is assumed that the subsystems are input-to-output practically stable (IOpS) with offset and unboundedness observability (UO), and the interconnected system can be proved to be IOpS with offset and UO as long as the proposed small-gain condition is satisfied. If, moreover, the assumption of IOpS on the subsystems is reduced to input-to-output stability (IOS), then the interconnected system is IOS. [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]

This paper aims to solve the finite-time stabilization problem for stochastic high-order nonlinear systems with stochastic inverse dynamics. By characterizing unmeasured stochastic inverse dynamics with finite-time stochastic input-to-state stability (FT-SISS), borrowing from some ideas of the Lyapunov function, sign function, backstepping and FT-SISS methods, and using stochastic finite-time stability theory, a new design and analysis method is given to guarantee global finite-time stability in probability of the closed-loop stochastic systems. [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 addresses integral input-to-state stability (iISS) for stochastic nonlinear dynamical systems and develops Lyapunov-type criteria. The developments offer an iISS small-gain framework for verifying robustness of interconnected systems with respect to disturbances. The framework not only includes input-to-state stability (ISS) as a special case, but also highlights fundamental differences between the stochastic and deterministic systems. In particular, dealing with nonlinearities in interconnection gives rise to a unique issue of degradation in the presence of stochastic noises. This paper addresses this issue by coping judiciously with the effect of the gradient of Lyapunov functions of subsystems through diffusion fields in composing a Lyapunov function of the entire system. This paper employs two types of robustness notions, and demonstrates that the influence of the stochastic degradation on one notion is considerably different from the influence on the other notion in dealing with interconnected systems. [ABSTRACT FROM PUBLISHER]

This paper presents a unification and a generalization of the small-gain theory subsuming a wide range of existing small-gain theorems. In particular, we introduce small-gain conditions that are necessary and sufficient to ensure input-to-state stability (ISS) with respect to closed sets. Toward this end, we first develop a Lyapunov characterization of $\omega$ ISS via finite-step $\omega$ ISS Lyapunov functions. Then, we provide the small-gain conditions to guarantee $\omega$ ISS of a network of systems. Finally, applications of our results to partial ISS, ISS of time-varying systems, synchronization problems, incremental stability, and distributed observers are given. [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 gives a solution to the problem of verifying stability of networks consisting of integral input-to-state stable (iISS) subsystems. The iISS small-gain theorem developed recently has been restricted to interconnections of two subsystems. For large-scale systems, stability criteria relying only on gain-type information that were previously developed address only input-to-state stable (ISS) subsystems. To address the stability problem involving iISS subsystems interconnected in general structure, this paper shows how to construct Lyapunov functions of the network by means of a sum of nonlinearly rescaled individual Lyapunov functions of subsystems under an appropriate small-gain condition. [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 stability and L\infty-gain analysis problem for continuous-time positive fractional-order delay systems with incommensurate orders between zero and one. A necessary and sufficient condition is firstly given to characterize the positivity of continuous-time fractional-order systems with bounded time-varying delays. Moreover, by exploiting the monotonic and asymptotic property of the constant delay system by virtue of the positivity, and comparing the trajectory of the time-varying delay system with that of the constant delay system, it is proved that the asymptotic stability of positive fractional-order systems is not sensitive to the magnitude of delays. In addition, it is shown that the L\infty-gain of a positive fractional-order system is independent of the magnitude of delays and fully determined by the system matrices. Finally, a numerical example is given to show the validity of the theoretical results. [ABSTRACT FROM PUBLISHER]

Event-triggered control consists of closing the feedback loop whenever a predefined state-dependent criterion is satisfied. This paradigm is especially well suited for embedded systems and networked control systems since it is able to reduce the amount of communication and computation resources needed for control, compared to the traditional periodic implementation. In this paper, we propose a framework for the event-triggered stabilization of nonlinear systems using hybrid systems tools, that is general enough to encompass most of the existing event-triggered control techniques, which we revisit and generalize. We also derive two new event-triggering conditions which may further enlarge the inter-event times compared to the available policies in the literature as illustrated by two physical examples. These novel techniques exemplify the relevance of introducing additional variables for the design of the triggering law. The proposed approach as well as the new event-triggering strategies are flexible and we believe that they can be used to address other event-based control problems. [ABSTRACT FROM AUTHOR]

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]