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In this article, the finite-time stabilization problem is solved for a linear time-varying system with unknown control direction by exploiting a modified version of the classical extremum-seeking algorithm. We propose to use a suitable oscillatory input to modify the system dynamics, at least in an average sense, so as to satisfy a differential linear matrix inequality condition, which in turn guarantees that the system’s state remains inside a prescribed time-varying hyperellipsoid in the state space. The finite-time stability (FTS) of the averaged dynamics implies the FTS of the original system, as the distance between the original and the averaged dynamics can be made arbitrarily small by choosing a sufficiently high value of the dithering frequency used by the extremum-seeking algorithm. The main advantage of the proposed approach resides in its capability of dealing with systems with unknown control direction, and/or with a control direction that changes over time. Being FTS a quantitative approach, this article also gives an estimate of the necessary minimum dithering/mixing frequency provided, and the effectiveness of the proposed finite-time stabilization approach is analyzed by means of numerical examples. [ABSTRACT FROM AUTHOR]

This article aims to reveal an interesting phenomenon about the time delays (TDs) in control systems: stability preservation with over large intermittently or unbounded variable TDs. That is, given a system asymptotically stable for bounded TDs, it could remain to be stable when the TDs are enhanced to have peak values over large intermittently or unbounded. The investigation is carried out by theoretical analysis plus case studies with the stochastic system models. The stability of systems is the core issue; the explicit involvement of the TDs in the stability criteria is the premise for them being concerned. To these ends, a series of lemmas are established preliminarily, a new type stability theorem with its practical version is established in succession, and a stability criterion with the TDs as explicit parameters is induced. The Lyapunov function method is applied for less restriction and conservativeness. The mechanism and the approach for the emergence of the phenomenon under probe are analyzed. Implementation is considered for the linear system model. Typical variable TDs are constructed for illustrations theoretically. Finally, the obtained results are verified with numerical experiments. [ABSTRACT FROM AUTHOR]

For a linear time varying (LTV) system containing stable and unstable modes, this article gives the necessary and sufficient conditions for asymptotic decoupling of stable modes. Establishment of the conditions relies on newly developed techniques for analyzing matrix exponential and state transition matrix. The research exploits fundamental properties of LTV systems and advances the knowledge base of linear control theory. The results can find a direct application in the autonomous synchronization problem for heterogeneous multiagent systems. The research methodology is based on rigorous theoretical proofs and numerical verification. [ABSTRACT FROM AUTHOR]

In this article, positivity and global exponential stability for a class of timescale-type differential-difference equations with bounded time-varying delay are considered. At first, a sufficient and necessary condition for the positivity of matrix-valued timescale-type exponential function is obtained. Based on this, an explicit criterion for the positivity of coupled timescale-type differential-difference equations with delays bounded above is acquired by timescale-type induction method. Besides, a sufficient and necessary condition of its positivity is obtained when the delay is both bounded above and below by two positive reals. Then, a concise criterion that guarantees the global exponential stability of the considered system is derived based upon analytical techniques of time scales and positivity constraint. On top of that, the theoretical results are applied to investigate the positivity and stability for a class of timescale-type neutral linear systems with bounded time-varying delays. Two numerical examples are given to illustrate the validity of the results. [ABSTRACT FROM AUTHOR]

The Lyapunov–Krasovskii functional (LKF) method is an effective tool to obtain delay-dependent stability criteria of time-varying delay systems. However, in the method, the requirement that there must exist an appropriate LKF for a system with a delay varying in a delay interval is too strict and may lead to considerable conservativeness. In order to get less conservative stability criteria, a novel stability analysis method called a delay-mode-based LKF (DMBLKF) method is first presented for nonlinear delay systems. Compared with the LKF method, the main advantage of the DMBLKF method lies in relaxing the aforementioned requirement and leading to a less conservative criterion in stability analysis. Then, with the help of the novel method, new stability criteria of linear delay systems are derived. Finally, the effectiveness of the proposed method and the obtained criteria is illustrated through two numerical examples. [ABSTRACT FROM AUTHOR]

This article addresses stability and stabilization of infinite delay systems, which are more general but also more challenging to deal with than bounded delay systems. Based on a general model of infinite delay systems and a newly approved key technical lemma, several new Lyapunov theorems for uniform asymptotical stability and exponential stability, respectively, are established. The stability results are more general than existing stability results, and the corresponding conditions are more easily satisfied than existing ones. Therefore, the new stability results are expected to be more widely applicable. These new Lyapunov theorems are then applied to the problems of stabilizing both time-invariant and time-varying linear systems with distributed infinite input delays, and the corresponding stabilizing controllers are developed. Key to obtaining the stabilizing controllers is to construct appropriate Lyapunov functionals for these systems via introduction of a regulation term into the Lyapunov functional. A distinctive advantage of the Lyapunov-based time-domain method proposed in this article over the existing frequency domain method is that the former can be adopted to deal with more general systems, such as time-varying linear systems or even nonlinear systems. Several examples are provided to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

Inspired by the idea of multiple Lyapunov functions ($\mathtt {MLFs}$), we use piecewise continuous scalar functions to investigate the stabilizability of time-varying switched systems. Starting with time-varying switched linear systems, we first combine the idea of $\mathtt {MLFs}$ with the existence of asymptotically (exponentially, uniformly exponentially) stable functions to provide necessary and sufficient conditions for their asymptotic (exponential, uniform exponential) stabilizability. Compared to traditional differential Lyapunov inequalities, we release the requirement on negative definiteness of the derivatives of $\mathtt {MLFs}$. Successively, the above results are extended to time-varying switched nonlinear systems. Then, two illustrative examples are given to show the applicability of our theoretical results. In the end, we consider the computation issue of our current results for a special class of nonautonomous switched systems, i.e., rational nonautonomous switched systems. [ABSTRACT FROM AUTHOR]

This paper introduces the internally positive representation (IPR) of linear coupled differential-difference systems with time-varying delays. Then, a stability analysis is conducted on this class of systems: Relying on results established under positivity assumptions, the IPR technique leads to a novel sufficient condition for the delay-independent asymptotic stability of coupled differential-difference systems that holds for not necessarily positive systems. As a consequence, an analogous stability condition for neutral-type equations with time-varying delays is also obtained. The results are discussed and favorably compared to similar conditions proposed in the literature, and finally validated by means of numerical examples. [ABSTRACT FROM AUTHOR]

The relationships between attractivity and asymptotic stability are fleshed out for homogeneous linear ordinary differential equations with time-varying coefficients. [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 technical note is concerned with the decay rate constrained stability analysis for linear delay systems that possess cone-invariant property. In order to capture the decay rate of such systems, we introduce a nondecreasing positive function whose reciprocal represents the decay rate. Under mild assumptions on the growth rate of this function, an explicit condition is given to ensure that a cone-preserving linear system with unbounded time-varying delays is asymptotically stable with a given decay rate. As typical cases, necessary and sufficient conditions are given to characterize the decay rate when the delay is restricted by a linear, sublinear or logarithmic growth rate. Finally, some numerical examples are given to illustrate the effectiveness of the theoretical results. [ABSTRACT FROM PUBLISHER]

We propose a prediction based stabilization approach for a general class of nonlinear time-varying systems with pointwise delay in the input. It is based on a recent new prediction strategy, which makes it possible to circumvent the problem of constructing and estimating distributed terms in the expression for the stabilizing control laws. Our result applies in cases where other recent results do not. [ABSTRACT FROM AUTHOR]

This article investigates the stability of positive switched linear systems. We start from motivating examples and focus on the case when each switched subsystem is marginally stable (in the sense that all the eigenvalues of the subsystem matrix are in the closed left-half plane with those on the imaginary axis simple) instead of asymptotically stable. A weak excitation condition is first proposed such that the considered positive switched linear system is exponentially stable. An extension to the case without dwell time assumption is also presented. Then, we study the influence of time-varying delay on the stability of the considered positive switched linear system. We show that the proposed weak excitation condition for the delay-free case is also sufficient for the asymptotic stability of the positive switched linear system under unbounded time-varying delay. In addition, it is shown that the convergence rate is exponential if there exists an upper bound for the delay, irrespective of the magnitude of this bound. The motivating examples are revisited to illustrate the theoretical results. [ABSTRACT FROM PUBLISHER]

The uniform stability notion for a class of nonlinear time-varying systems is studied using the homogeneity framework. It is assumed that the system is weighted homogeneous considering the time variable as a constant parameter, then several conditions of uniform stability for such a class of systems are formulated. The results are applied to the problem of adaptive estimation for a linear system. [ABSTRACT FROM AUTHOR]

We prove stability of time-varying systems with delays, using linear Lyapunov functionals and positive systems, and we provide robustness of the stability with respect to multiplicative uncertainty in the vector fields. We allow cases where the delay may be unknown, and where the vector fields defining the systems are not necessarily bounded. We illustrate our work using a chain of integrators and other examples. [ABSTRACT FROM AUTHOR]

In this note, the stabilization problem for a class of discrete-time switched linear systems with additive disturbances is investigated. The considered switching signals are of mode-dependent persistent dwell-time (MPDT) property and the disturbances are assumed to be amplitude-bounded. By constructing a quasi-time-varying (QTV) Lyapunov function, a QTV stabilizing controller is designed for the nominal system such that the resulting closed-loop system is globally uniformly asymptotically stable. In the presence of bounded additive disturbances, a MPDT robust positive invariant set is determined for the error system between the nominal system and disturbed system. A concept of generalized robust positive invariant (GRPI) set under admissible MPDT switching is further proposed for the error system. It is demonstrated that the disturbed system is also asymptotically stable in the sense of converging to the MPDT GRPI set that can be regarded as the cross section of a uniform tube of the disturbed system. A numerical example is provided to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

This technical note studies the stability problem of homogeneous positive systems of any degree with time-varying delays. Delay-independent conditions are derived for asymptotic and finite-time stability. Estimates on the decay rates, which reveal how the system delays affect the rates of convergence, are obtained. More precisely, this technical note features three contributions. First, we derive a necessary and sufficient condition for global polynomial stability of continuous-time homogeneous cooperative systems with time-varying delays when the degree of homogeneity is greater than one. Second, we characterize finite-time stability of continuous-time homogeneous cooperative delay-free systems of degree smaller than one. Finally, for discrete-time positive systems with time-varying delays, a local exponential stability criterion is established when the vector fields are order-preserving and homogeneous of degree greater than one. An illustrative example is given to show the effectiveness of our results. [ABSTRACT FROM AUTHOR]

This note considers the work entitled “Robust Stability and Stabilization of Fractional-Order Interval Systems with the Fractional Order $\alpha$: The $0<\alpha<1$ Case” by Lu and Chen, where the authors meant to provide sufficient and necessary conditions for stability of fractional-order linear-time-invariant interval systems. As this note proves, conditions are only sufficient, since necessity relies unsuitably on the certain case. An example is provided. [ABSTRACT FROM AUTHOR]