Lyapunov Functions Of Subsystems Research Articles

Event-triggered impulsive control for stability of stochastic delayed complex networks under deception attacks

In this paper, we consider the stability of stochastic complex networks with time-varying delays (SCNTD) under deception attacks, adopting the event-triggered impulsive control (ETIC). A new event-triggered mechanism (ETM) which was used to determine the impulsive instance is given using the topology of networks and the Lyapunov functions of subsystems. For stochastic complex networks, to avoid the Zeno behavior, a positive minimum inter-event time (MIET) is guaranteed. Combining the graph theory and the Lyapunov–Razumikhin method, several criteria for achieving pth moment exponential stability (pMES) of SCNTD under deception attacks which are related to the topology of networks, the event-triggered parameters and the signal sent by the enemy are obtained. Finally, the theoretical results apply to the stochastic oscillator systems, single-link robot arms systems, and Chua’s circuit systems. The stability of these systems under deception attacks is considered separately, and to show the validity of our theories, numerical simulations are provided.

Simultaneous and Sequential Control Design for Discrete-Time Switched Linear Systems Using Semi-Definite Programming

The control of switched linear discrete-time systems occurs in multiple engineering fields, where it has been used to deal with complex and non-linear systems. This letter presents two strategies to design control laws for discrete-time switched linear systems, whilst guaranteeing asymptotic stability of the closed loop. Firstly, an arbitrary switching signal is considered. In this scenario a common quadratic Lyapunov function is used for stability, but subsystem Lyapunov functions are employed to improve local subsystem performance. Secondly, a constrained switching signal, associated with subsystem lower dwell time bounds is studied. In this case, a decrease in Lyapunov cost is achieved by design, based on dwell time constraints only, thus removing the need for both a common quadratic Lyapunov function or direct stable switches. It is shown in both cases that the control design problems can be formulated as one or a sequence of semi-definite programming problems, and therefore can be solved efficiently. Finally, two examples are provided in order to illustrate the different techniques presented.

Improved Stability Conditions for Switched Positive Linear Time-Varying Systems

This note considers the problem of asymptotic stability for switched positive linear time-varying (LTV) systems. Some improved stability conditions are given for the positive LTV systems and switched positive LTV systems by using time-varying copositive Lyapunov functions (CLFs) and switched time-varying CLF, respectively. Comparing with the existing results, the conditions obtained allow the time derivative of Lyapunov functions of subsystems (or systems) to be indefinite, and all subsystems to be unstable. Furthermore, using function-dependent dwell time methods, two improved stability conditions are also proposed for switched positive LTV systems. The conditions obtained allow the average dwell time of systems to have greater range than that of the existing results. Finally, a numerical example is given to illustrate the theoretical results.

Pth moment input-to-state stability of stochastic impulsive switched delayed systems

This paper investigates the pth moment input-to-state stability (ISS) and the pth moment integral input-to-state stability (iISS) of stochastic impulsive switched delayed system with delayed impulses. By employing the method of multiple Lyapunov-Krasovskii functionals and the uniformly exponentially stable function, some relaxed Krasovskii-type sufficient conditions ensuring the pISS/piISS of the addressed systems are developed. These conditions imply the relationship among the impulsive frequency, the time delay existing in impulses, and the coefficients of the estimated upper bound for the derivative of a Lyapunov function. It is shown that if the continuous stochastic delayed dynamics is ISS, and the impulsive effects are destabilizing, then the stochastic impulsive switched delayed system is ISS with respect to the relationship. Compared with the existing results, the conditions obtained results have three relaxations, that is, the derivative of Lyapunov functions of subsystems are allowed to be sign-changing time-varying function rather than a negative definite constant, all subsystems are allowed to be unstable, and the effect of delayed impulses are considered. Finally, an example is provided to illustrate the effectiveness of the 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.

New necessary and sufficient conditions for finite‐time stability of impulsive switched linear time‐varying systems

This note considers the problem of finite-time stability (FTS) for impulsive switched linear time-varying (LTV) systems. First, two improved necessary and sufficient conditions are proposed to verify the FTS of LTV by employing an indefinite Lyapunov function. Then, the results obtained are extended to study the FTS of impulsive switched LTV systems, two new necessary and sufficient conditions are given by using common indefinite Lyapunov function and piecewise indefinite Lyapunov functions. Compared with the existing results, the conditions obtained allow the time derivative of Lyapunov functions of subsystems to be indefinite and have a tighter upper bound, moreover, all subsystems are also allowed to be not FTS. Furthermore, the construction method of switching signal is given, moreover, using function-dependent dwell time methods, three sufficient conditions of FTS are also proposed for the systems. Two numerical examples are given to illustrate the theoretical results.

Lyapunov small-gain theorems for networks of not necessarily ISS hybrid systems

We prove a novel Lyapunov-based small-gain theorem for networks composed of n≥2 hybrid subsystems which are not necessarily input-to-state stable. This result unifies and extends several small-gain theorems for hybrid and impulsive systems proposed in the last few years. We also show how average dwell-time (ADT) clocks and reverse ADT clocks can be used to modify the ISS Lyapunov functions for subsystems and to enlarge the applicability of the derived small-gain theorems.

Finite‐time stability of switched nonlinear time‐varying systems via indefinite Lyapunov functions

SummaryThis note considers the problem of finite‐time stability (FTS) for switched nonlinear time‐varying systems. First, a relaxed condition is proposed to verify the FTS of nonlinear time‐varying systems by using an indefinite Lyapunov function. Then, the result obtained is extended to study the FTS of switched nonlinear time‐varying systems. Several relaxed conditions are given by using a common indefinite Lyapunov function and multiple indefinite Lyapunov functions. Moreover, the corresponding estimates on convergence regions and times of systems are also given. Comparing with the existing results, the conditions obtained allow the time derivative of Lyapunov functions of subsystems (or systems) to be indefinite and all subsystems to be not finite‐time stable or even unstable. Finally, a numerical example is given to illustrate the theoretical results.

Relaxed Conditions for the Input-to-State Stability of Switched Nonlinear Time-Varying Systems

This technical note considers the problem of input-to-state stability (ISS) for switched nonlinear time-varying systems. First, a sufficient condition is proposed to verify the ISS of nonlinear time-varying systems by using an improved Lyapunov function. Then, the results obtained are extended to study the ISS of switched nonlinear time-varying systems. Three relaxed conditions are given by using the methods of multiple Lyapunov functions, minimum dwell time, and infinite switchings, respectively. Comparing with the existing results, the conditions obtained have two relaxations, i.e., the derivative of Lyapunov functions of subsystems are allowed to be indefinite, and all subsystems are allowed to be unstable in the case of infinite switchings. All results obtained also be used to study the uniformly asymptotic stability (UAS) of systems. Finally, a numerical example is given to illustrate the theoretical results.

Stabilisation for a class of switched nonlinear systems and its application to aero‐engines

The stabilisation problem of a more general switched non-linear system originated from the aero-engine control systems is investigated under an average dwell-time approach. First, a procedure of choosing coordinates transformations for individual subsystems is presented. Through these coordinates transformations, the original subsystems are transformed into their non-linear normal forms, respectively, based on which the state feedback controllers and the Lyapunov functions for subsystems are obtained through a constructive design method. However, each subsystem and its own Lyapunov function are expressed in individual coordinates and a common coordinates transformation cannot be obtained, which makes the design of a switching law extremely difficult. The authors overcome this difficulty to construct a switching law with average dwell time which can guarantee asymptotic stability of the closed-loop switched system. Finally, the proposed switched controller is applied to a switched model of turbofan engine, and simulation results show the feasibility and validity of the developed controller.

An iISS Framework for Stochastic Robustness of Interconnected Nonlinear Systems

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.

On the absolute stability of imprecise large-scale singularly perturbed systems

UDC 531.36 We study an imprecise large-scale singularly perturbed system of differential equations. By using matrixvalued Lyapunov functions for subsystems, we construct a scalar-valued Lyapunov function that enables us to establish the absolute parametric stability of the original system. We estimate the set of values of the parameters for which the system possesses this property. 1. Statement of the Problem Consider an imprecise singularly perturbed large-scale system of differential equations of the form

Robust Stability of Networks of iISS Systems: Construction of Sum-Type Lyapunov Functions

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.

P-Stability andp-Stabilizability of Stochastic Nonlinear and Bilinear Hybrid Systems under Stabilizing Switching Rules

The problem ofpth mean exponential stability and stabilizability of a class of stochastic nonlinear and bilinear hybrid systems with unstable and stable subsystems is considered. Sufficient conditions for thepth mean exponential stability and stabilizability under a feedback control and stabilizing switching rules are derived. A method for the construction of stabilizing switching rules based on the Lyapunov technique and the knowledge of regions of decreasing the Lyapunov functions for subsystems is given. Two cases, including single Lyapunov function and a a single Lyapunov-like function, are discussed. Obtained results are illustrated by examples.

Input-to-State Stability of Nonlinear Impulsive Systems

We prove that impulsive systems, which possess an input-to-state stable (ISS) Lyapunov function, are ISS for time sequences satisfying the fixed dwell-time condition. If an ISS Lyapunov function is the exponential one, we provide a stronger result, which guarantees uniform ISS of the whole system over sequences satisfying the generalized average dwell-time condition. Then we prove two small-gain theorems that provide a construction of an ISS Lyapunov function for an interconnection of impulsive systems if the ISS Lyapunov functions for subsystems are known. The construction of local ISS Lyapunov functions via the linearization method is provided. Relations between small-gain and dwell-time conditions as well as between different types of dwell-time conditions are also investigated. Although our results are novel already in the context of finite-dimensional systems, we prove them for systems based on differential equations in Banach spaces that makes obtained results considerably more general.

A Decomposition Technique for Nonlinear Dynamical System Analysis

A method for analyzing large-scale nonlinear dynamical systems by decomposing them into coupled lower order subsystems that are sufficiently simple for computational analysis is presented. It is shown that the decomposition approach can be used to scale the Sum of Squares programming framework for nonlinear systems analysis. The method constructs subsystem Lyapunov functions which are used to form a composite Lyapunov function for the whole system. Further computational savings are achieved if a method based on sparsity maximization is used to obtain the subsystem Lyapunov functions.

Remarks on p-stabilizability of stochastic linear hybrid systems

Problem of the p-th mean exponential stabilizability of a class of stochastic linear control hybrid systems with unstable subsystems is considered. Sufficient conditions for the p-th mean exponential stabilizability under a feedback control and a stabilizing switching signal, that is constructed based on the Lyapunov techniques, the knowledge of regions of decreasing of Lyapunov functions for subsystems and the average dwell time approach, are derived. Three cases, including the existence of a single Lyapunov function, a multiple Lyapunov function and a single Lyapunov-like function, are discussed. Obtained results are illustrated by an example.

Robust feedback design of a class of nonlinear cascaded systems with structural uncertainty

This paper is concerned with the robust stabilization and robust L 2 disturbance attenuation of a class of cascaded nonlinear systems with structural uncertainties. The design tools are construction of a Lyapunov function for robust stabilization and a storage function for robust L 2 gain performance. It is shown that, if for each subsystem there exist Lyapunov functions satisfying Hamilton-Jacobi inequalities related with nominal subsystems, then a robust stabilizing controller can be constructed such that the sum of the functions becomes a Lyapunov function for the cascaded system with uncertainties. This concept, is extended do the construction of the storage function and the controller for the robust L 2 disturbance attenuation problem by introducing an appropriate weight into the sum of the subsystem's Lyapunov functions.

Robust stability of large-scale systems which contain time-invariant lur'e-type uncertainties in subsystems and norm-bounded uncertainties in interconnections

In this paper a new frequency-domain condition for the robust stability of a class of large-scale systems is derived. It is assumed that each subsystem contains one time-invariant nonlinearity satisfying a sector condition, and that the interconnecting functions are linearly bounded by the norms of the scalar outputs of subsystems. In deriving the above such Lur'e-type Lyapunov functions of subsystems are constructed so that their weighted sum is a Lyapunov function of the overall system. This condition is potential to give a sharper result than the circle-condition-type result which was obtained previously. Furthermore, a method to estimate the domain of attraction using the above Lyapunov function is also given when the sector condition is satisfied only locally.

Local stability of composite systems--Frequency-domain condition and estimate of the domain of attraction

This paper is concerned with such composite systems whose subsystems contain one nonlinearity each and whose interconnections are functions of the scalar outputs of subsystems. A frequency-domain condition which assures local asymptotic stability is given under the assumptions that each nonlinearity satisfies a sector condition, that interconnections are linearly bounded, and that linear parts of subsystems may have unstable poles. In deriving the above result, such Lyapunov functions of subsystems are constructed so that their weighted sum is a Lyapunov function of the overall system. A method to estimate the domain Of attraction based on the above Lyapunov functions is also studied. When the bounds on nonlinearities hold true in the entire space and when the linear parts do not have unstable poles, the present condition turns out to be the same with the L 2 -stability condition which was obtained before by Araki.