Input/output-to-state Stability Research Articles

Improved multiple Lyapunov functions of input–output-to-state stability for nonlinear switched systems

This study is concerned with the Lyapunov characterization of input–output-to-state stability (IOSS) for time-varying nonlinear switched systems. Considering the restriction in current methods that each subsystem must be IOSS, and the Lyapunov function of each subsystem is strictly descended in the definition domain, in this work, by extending multiple Lyapunov functions to an indefinite form, the restriction on the negativity of the time derivative of Lyapunov functions is relaxed; additionally, all subsystems are permitted to not be IOSS in the state space. The results obtained are then extended to verifying the integral input–output-to-state stability (iIOSS) and output-to-state stability (OSS); and a state-norm estimator is proposed based on the indefinite Lyapunov function as a by-product. Finally, a numerical example illustrates the effectiveness of the results.

Generalised existence condition and design method for switched nonlinear state-norm estimators

The state-norm estimation problem for the switched nonlinear system under a more generalized type of switching signals is investigated in this paper. The component subsystems consist of input/output-to-state stability (IOSS) and non-IOSS continuous systems simultaneously, whose activation time is lower and upper bounded by mode-dependent time-varying functions, respectively. The considered switching signals satisfy mode-dependent time-varying constraints, whose switching number can grow faster than those with constant average dwell time condition. Then based on multiple IOSS Lyapunov functions method, the existence condition and design method for the switched state-norm estimator are proposed. The obtained results present extensions compared with the existing results, which are applicable for the more general situations. Finally, the effectiveness and improvement of the proposed approach are illustrated by a numerical example.

New results for the realization of interfered direct-form digital filters with single two's complement nonlinearity

PurposeThis paper deals with the problem of input/output-to-state stability (IOSS) of direct-form digital filters, which simultaneously contain external disturbances and two's complement nonlinearity. The nonlinearity under consideration is confined to the sector [–1, 1], which contains saturation, zeroing, two's complement and triangular.Design/methodology/approachThe proposed condition is based on IOSS approach, which is capable of providing a framework for checking and analysing the stability of nonlinear system based on input as well as output information.FindingsA linear matrix inequality (LMI)-based new sufficient criterion for the IOSS of the suggested system is obtained. The obtained criterion is capable of detecting the output-to-state stability (OSS) and asymptotic stability of direct-form digital filters with zero external disturbances. In addition, state-norm estimator for the filter under consideration is constructed by adopting an exponential-decay IOSS criterion. Several examples are provided to illustrate the usefulness of the proposed criteria.Originality/valueThe result of the paper is introduced for the first time, and it is suitable for stability analysis of interfered direct-form digital filter with two's complement overflow using IOSS approach.

Input/output‐to‐state stability of nonlinear impulsive delay systems based on a new impulsive inequality

SummaryIn this paper, input/output‐to‐state stability (IOSS) and integral IOSS (iIOSS) are investigated for nonlinear impulsive systems with delay. Based on a new impulsive inequality, we propose some sufficient criteria for IOSS and iIOSS of impulsive delay systems. It is shown that the obtained results for IOSS and iIOSS are regardless of the length of the impulsive interval and the size of time delay if the impulsive gain satisfies a given condition. In addition, based on the average impulsive interval method, some more useful sufficient conditions are derived for IOSS and iIOSS of impulsive delay systems with persistent large‐scale destabilizing impulses. Furthermore, a relationship is established among the average impulsive interval, impulses, time delay, and the decay of the system without impulses such that the impulsive delay system is input/output‐to‐state stable and integral input/output‐to‐state stable, respectively. Two examples are given to show the validity of the obtained results.

Input/output‐to‐state stability of impulsive switched delay systems

SummaryThis paper investigates the input/output‐to‐state stability (IOSS) and integral IOSS (iIOSS) of nonlinear impulsive switched delay systems where the switching moments and impulsive moments do not necessarily coincide with each other. Some Razumikhin‐type criteria are presented to guarantee the IOSS and iIOSS of the systems, where both destabilizing and stabilizing effects of switching behavior and impulses are considered simultaneously. The counterpart results for impulsive switched systems without delay can be naturally obtained. Several examples are provided to verify the effectiveness and superiority of the proposed results.

Input/Output-to-State Stability of Impulsive Switched Systems With Time Delays

In this paper, we study the input/output-to-state stability (IOSS) of impulsive switched systems with time delays, where the derivative of Lyapunov functions are assumed to be time-varying and mode-dependent. Based on the Lyapunov-Razumikhin method and the mode-dependent average dwell time (MDADT) condition, some Rzaumikhin-type criteria for IOSS of impulsive switched systems with the coexistence of the destabilizing and stabilizing impulses are established. The derivative of Lyapunov functions of subsystems are allowed to be indefinite, which implies that the criteria can be applied to impulsive switched systems involving the coexistence of IOSS and non-IOSS subsystems. Furthermore, under mode-dependent conditions, the dwell time of each subsystem allows to be small or large enough. Finally, two illustrative examples are provided to demonstrate the effectiveness of the proposed results.

Input/output-to-state stability of impulsive switched systems

This paper studies the input/output-to-state stability (IOSS) of impulsive switched systems. With the help of Lyapunov and average dwell-time (ADT) methods, some sufficient conditions for IOSS of impulsive switched systems are obtained, where both types of impulses, stabilizing impulses and destabilizing impulses, are considered. It is shown that when all of the modes are IOSS, a switched system under an ADT scheme is IOSS even if there exist destabilizing impulses, and when none of the modes is IOSS, IOSS can still be achieved under the designed ADT scheme coupled with stabilizing impulses. In particular, for a special case in which an impulsive switched system is composed of some IOSS modes and some non-IOSS modes, a relationship is established among the ADT scheme, impulses, and the total dwell time between non-IOSS and IOSS modes such that the impulsive switched system is IOSS. Two examples are provided to illustrate the applications of our results.

Stability analysis of digital filters subjected to interference using generalized overflow nonlinearities

In this paper, we feature a new criterion using an input–output-to-state stability (IOSS) approach for digital filters employing fixed-point arithmetic with generalized overflow nonlinearities in the presence of external disturbances. The proposed linear matrix inequality (LMI)-based criterion eliminates the presence of limit cycles and ensures not only the IOSS of digital filters but also their asymptotic stability when external interference is exhausted. Through several numerical examples, the effectiveness of the proposed criterion is illustrated, and it is shown that it covers a broad class of nonlinearities, namely saturation, zeroing, two's complement, and triangular.

Improving the performance of semiglobal output controllers for nonlinear systems

For a large class of nonlinear control systems, the main drawback of a semiglobal stabilizing output feedback controllers $(\mathcal{U}_R)_{R>0}$ with increasing regions of attraction $(\Omega_R)_{R>0}$ is that, when the region of attraction $\Omega_R$ is large, the convergence of solutions of the closed-loop system to the origin becomes slow. To improve the performance of a semiglobal controller, we look for a new feedback control law that preserves the semiglobal stability of the nonlinear system under consideration and that is equal to some fast controller $\mathcal{U}_{R_0}$ on a neighborhood of the origin. Under an input-output-to-state stability (IOSS) assumption, we propose a new semiglobal stabilizing hybrid feedback controller that unifies a slow controller that has a large region of attraction with a fast controller having a small region of attraction. This unification is inspired from the elegant hybrid unification of a local controller with a global one given in [21]. Moreover, this unification is different from the recent result [24], since in the cited paper the objective is just the stabilization; whereas in our study, the objective is the stabilization with high performance. Finally, we illustrate our main result by means of two numerical examples.

Some new results on the stability of direct-form digital filters with finite wordlength nonlinearities

This paper proposes a new criterion for the input/output-to-state stability (IOSS) of interfered direct-form digital filters with finite wordlength nonlinearities based on an augmented Lyapunov function. Without external interferences, the output-to-state stability (OSS) and asymptotic stability of direct-form digital filters are guaranteed under the proposed criterion. An improved criterion is also proposed for the IOSS and OSS of interfered direct-form digital filters, with the introduction of four slack matrices. A new state-norm estimator for direct-form digital filters is constructed based on an exponential-decay IOSS criterion. The criteria presented in this paper are expressed by linear matrix inequalities (LMIs). Two numerical examples demonstrate the effectiveness of the proposed results.

IOSS Criterion for the Absence of Limit Cycles in Interfered Digital Filters Employing Saturation Overflow Arithmetic

In this paper, a new criterion for the absence of limit cycles in fixed-point state-space digital filters with saturation overflow arithmetic and external interference is proposed via an input/output-to-state stability (IOSS) approach. The criterion ensures not only the asymptotic stability, but also the reduction of the effect of the external interference. This criterion is represented by linear matrix inequality (LMI) and, hence, is computationally tractable. Via a numerical example, we demonstrate the effectiveness of the proposed IOSS criterion.

Input/output-to-state stability and state-norm estimators for switched nonlinear systems

In this paper, the concepts of input/output-to-state stability (IOSS) and state-norm estimators are considered for switched nonlinear systems under average dwell-time switching signals. We show that when the average dwell-time is large enough, a switched system is IOSS if all of its constituent subsystems are IOSS. Moreover, under the same conditions, a non-switched state-norm estimator exists for the switched system. Furthermore, if some of the constituent subsystems are not IOSS, we show that still IOSS can be established for the switched system, if the activation time of the non-IOSS subsystems is not too big. Again, under the same conditions, a state-norm estimator exists for the switched system. However, in this case, the state-norm estimator is a switched system itself, consisting of two subsystems. We show that this state-norm estimator can be constructed such that its switching times are independent of the switching times of the switched system it is designed for.

Input–output-to-state stability for discrete-time systems

This paper presents Lyapunov characterizations of uniform output-to-state stability and uniform input–output-to-state stability (IOSS) (with respect to disturbances) for discrete-time (DT) nonlinear systems. We show the equivalence of the following three properties for DT systems: uniform IOSS, existence of a smooth Lyapunov function for uniform IOSS, and existence of a (state-)norm estimator. This equivalence result is a DT counterpart of Krichman et al. [2001. Input–output-to-state stability. SIAM Journal on Control and Optimization 39, 1874–1928. Theorem 2.4] and a generalization of Jiang and Wang [2002. A converse Lyapunov theorem for discrete-time systems with disturbances. Systems and Control Letters 45, 49–58. Theorem 1.1] and Jiand and Wang [2001. Input-to-state stability for discrete-time nonlinear systems. Automatica 37, 857–869. Theorem 1, 1 ⇔ 4 ].

STABILIZATION OF INVARIANT SETS OF SWITCHED SYSTEMS BY OUTPUT FEEDBACK

In this paper, the notion of the input/output-to-state stability (IOSS) is extended to the case of the input/output-to-V (x) stability (IOVS), which implies detectability of the zero-value set of a storage function. Based on this notion and the notion of passivity of nonlinear systems, we propose and prove sufficient conditions under which an invariant state set of a class of switched nonlinear systems can be stabilized by output feedback. Then we show that a nonlinear system is IOVS if V(x) is an IOSS-Lyapunov function of the system.

Input-Output-to-State Stability

This work explores Lyapunov characterizations of the input-output-to-state stability (IOSS) property for nonlinear systems. The notion of IOSS is a natural generalization of the standard zero-detectability property used in the linear case. The main contribution of this work is to establish a complete equivalence between the IOSS property and the existence of a certain type of smooth Lyapunov function. As corollaries, one shows the existence of "norm-estimators," and obtains characterizations of nonlinear detectability in terms of relative stability and of finite-energy estimates.

Output-to-state stability and detectability of nonlinear systems

The notion of input-to-state stability (ISS) has proved to be useful in nonlinear systems analysis. This paper discusses a dual notion, output-to-state stability (OSS). A characterization is provided in terms of a dissipation inequality involving storage (Lyapunov) functions. Combining ISS and OSS there results the notion of input/output-to-state stability (IOSS), which is also studied and related to the notion of detectability, the existence of observers, and output injection.