Input-to-state stability of infinite-dimensional control systems

Lyapunov characterization of input-to-state stability for semilinear control systems over Banach spaces

This paper presents conditions for stability of interconnected nonlinear systems consisting of integral input-to-state stable subsystems. Both the necessity and the sufficiency are investigated using a Lyapunov formulation. The ultimate result is a necessary and sufficient condition for integral input-to-state stability (iISS) of the interconnection, which is believed to be the first of its kind. This paper derives two sufficient conditions. One is the iISS-ISS small-gain condition which has been proposed earlier by one of the authors. Its sufficiency is strengthened to cover situations more general than the earlier result. The other condition is a universal representation of the nonlinear small-gain condition which has been a popular tool for dealing with input-to-state stable (ISS) subsystems. This paper proves that, for establishing iISS property of the interconnection, the universal representation can deal with iISS subsystems as much as the iISS-ISS small-gain condition if supply rate functions are smooth everywhere and analytic at zero. Furthermore, two necessary conditions this paper derives demonstrate that the nonlinear small-gain condition in the universal form is also necessary for the iISS property

This paper overviews the research investigations pertaining to stability and stabilization of control systems with time‐delays. The prime focus is the fundamental results and recent progress in theory and applications. The overview sheds light on the contemporary development on the linear matrix inequality (LMI) techniques in deriving both delay‐independent and delay‐dependent stability results for time‐delay systems. Particular emphases will be placed on issues concerned with the conservatism and the computational complexity of the results. Key technical bounding lemmas and slack variable introduction approaches will be presented. The results will be compared and connections of certain delay‐dependent stability results are also discussed.

This paper introduces appropriate concepts of input-to state stability (ISS) for nonlinear impulsive hybrid systems(NIHS). First we study the case in which one kind of dynamics system without external inputs has a stability property, then by applying the ISS-Lyapunov function theory, several sufficient conditions are derived for the ISS property of the whole NIHS with inputs. The results of ISS are used to study the robustly globally uniformly exponential stability for impulsive hybrid systems. The impulsive hybrid system can keep ISS no matter how often the impulses occur. Our proposed results are evaluated using illustrative example to show their effectiveness.

On the stability of certain systems under resonance conditions

Analysis of Global Asymptotic Stability of Interconnected Nonlinear Systems Using State-Dependent Scaling

In this paper, we present Lyapunov characterizations of the global asymptotic stability, the input-to-state stability (ISS), and the incremental input-to-state stability (δ-ISS) for discrete-time nonlinear systems with uncertain and time-varying time delays. We provide necessary and sufficient conditions for the global asymptotic stability and for the ISS and sufficient conditions for the δ-ISS, in terms of discrete-time Lyapunov-Krasovskii functionals. In particular, the Lyapunov characterization of the notion of δ-ISS is motivated by problems concerning the construction of discrete abstractions approximating discrete-time nonlinear systems. Uncertain and time-varying time delays occur very frequently, for instance, in networked control systems. Examples are included that illustrate the proposed methodologies.

This paper studies the input-to-state stability (ISS) of nonlinear systems in Takagi-Sugeno's form. Following the definitions and ISS properties proposed by Sontag and Wang [1], a condition in terms of Linear Matrix Inequalities (LMIs) and predefined norm-bounded inputs is derived. Due to the convex property of the weighted sum of input matrices, the input field is bounded and the ISS property holds for systems that are global asymptotic stable (GAS) in the absence of exogenous inputs. First, a existing ISS condition for Takagi-Sugeno (TS) systems that fulfill standard quadratic Lyapunov function is revisited. Then, a ISS condition using a non-quadratic Lyapunov function candidate is derived. To illustrate the applicability of both conditions two numerical examples are studied.

This article addresses a class of large-scale stochastic impulsive systems with time delay and time-varying input disturbance having bounded magnitude. The main interest is to develop sufficient conditions for the input-to-state stability (ISS) and stabilization in the presence of impulsive effects. The method of Razumikhin–Lyapunov function is used to develop the ISS and stabilization properties. Later, these results are applied to a class of control systems where the controller actuators are susceptible to failures. It should be noted that our results are delay independent, and the designed reliable controller is robust with respect to the actuator failures and to the system uncertainties. It is also observed that if the isolated continuous system is ISS and subjected to bounded impulsive effects, then the resulting impulsive system preserves the ISS property. Moreover, if the isolated continuous subsystems are all ISS and the interconnection amongst them is bounded from above, then the impulsive interconnected system is ISS provided that the degree of stability of each subsystem is larger than the magnitude of interconnection. If the underlying continuous system is unstable, then the input-to-state stabilization of the impulsive system is guaranteed if the stabilizing impulses are applied to the system frequently. As an implication to these results, if the input disturbance is zero, then the input-to-state stability (or stabilization) reduces to the stability (or stabilization) of the equilibrium state of the underlying disturbance-free system. A numerical example and simulations are provided to illustrate the proposed results.

In this work, we study Bishop–Phelps cones (briefly, BP cones) given by an equation in Banach spaces. Due to the special form, these cones enjoy interesting properties. We show that nontrivial BP cones have given by an equation form a ‘large family’ in some sense in any Banach space and they can be used to characterize the strict convexity of the space. We obtain conditions for a convex cone to be included in or to be itself a BP cone given by an equation. Furthermore, we give an affirmative answer to the open question of whether these cones may possess a nonempty interior in infinite-dimensional spaces and introduce Lorentz cones in some classical Banach spaces of sequences as illustrations. We also obtain explicit representations for all BP cones given by an equation in some classical Banach spaces. Finally, we present some short applications in optimal control and approximation.

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A degree of flexibility in Lyapunov inequalities for establishing input-to-state stability of interconnected systems

In this paper, input-to-state stability (ISS) of continuous-time systems is analyzed via finite-time Lyapunov functions. ISS of a continuous-time system is first proved via finite-time robust Lyapunov functions for an introduced auxiliary system of the considered system. It is then obtained that the existence of a finite-time ISS Lyapunov function implies that the continuous-time system is ISS. The converse finite-time ISS Lyapunov theorem is proposed. Furthermore, we explore the properties of finite-time ISS Lyapunov functions for the continuous-time system on a bounded and compact set without a small neighborhood of the origin. The effectiveness of our results is illustrated by four examples.

This paper continues with the development of the input‐to‐state stability (ISS)‐control Leonov function (CLeF) approach. The definitions of practical ISS and integral ISS (iISS)–CLeFs are refined, and the proposed methodology for control synthesis is improved to simplify the final control law. Then, it is shown that the existence of practical ISS‐ and iISS–CLeFs is a sufficient condition to guarantee the existence of a controller that endows multistable state periodic systems with the ISS and iISS properties, respectively. Furthermore, a methodology for the design of such a controller is provided via the well‐known Sontag's universal formula. Besides, an extension of the main result is presented to connect the ISS–CLeF approach with the standard Leonov function method such that the maximal invariant set of the closed‐loop system is compact on a manifold. Finally, the proposed approach is applied to the design of an excitation controller for a synchronous generator, which guarantees global ISS properties for the closed‐loop system, unlike the usual local results reported in the literature. The obtained control is also independent of the load angle. The effectiveness of the designed controller is demonstrated in simulations.

ISS Lyapunov functions for cascade switched systems and sampled-data control