Stability analysis of a class of coupled time-varying differential-difference systems with time-delay and sector-type nonlinearities

Stability Analysis of a Class of Coupled Time-Varying Differential-Difference Systems with Delays and Sector-Type Nonlinearities

This paper presents a unified approach to absolute stability analysis for a class of time-varying switched nonlinear systems with delays and sector-type nonlinearities. Several novel sufficient conditions, formulated as linear inequalities, to ensure that the zero solution of the switched systems is both asymptotically and exponentially stable for arbitrary switching signals and any admissible sector-type nonlinearities are provided. For time-invariant positive systems, a necessary and sufficient condition for absolute stability is established. The developed approach utilises the Lyapunov-Krasovskii function method, the comparison principle and the properties of positive systems. Examples are included to illustrate the effectiveness of the proposed method.

Periodic Motions for Estimation of the Attraction Domain in the Wheeled Robot Stabilization Problem

In this paper, the advisory train cruise control stability with a driver in the loop is investigated, where the driver's state is measured by an external device. A novel stability criterion is proposed, aware of the varying behaviour of the driver in the system. For this purpose, the driver is modelled as a time varying system, and the dynamic model of the train is formed by considering rolling and aerodynamic resistance forces. In order to ascertain stability, a sampled- data based state feedback controller is considered. Converting the sampling period into a bounded time-varying delay, the addressed problem is transformed to the problem of stability analysis of time-varying delayed system. Based on Lyapunov stability theory, a novel Lyapunov Krasovskii Functional (LKF) is designed to provide sufficient conditions for the existence of L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> stability of the system in terms of solvable Linear Matrix Inequalities. A case study is given to illustrate the effectiveness of the proposed method.

Stability of neural-network based train cruise advisory control with aperiodical measurements

The problem of on-line identification of continuous time, time-varying systems is considered via the minimization of a least squares criterion with a forgetting function. The forgetting function used depends on two time-varying parameters which play crucial roles in the stability analysis of the method.

The average impulsive interval is widely used to describe the frequency of impulsive occurrence (FIO), where the occurrence number of impulses is bounded by a linear function of time interval length. However, the linear relationship may insufficiently or excessively characterize the occurrence number of impulses to stabilize impulsive time-varying systems. In this article, the impulsive density is introduced to describe a time-varying FIO, such that the occurrence number of impulses can be characterized more explicitly. Under the impulsive density, the asymptotical stability is considered for impulsive stochastic time-varying systems, where the continuous dynamics of systems, impulsive strengths, and instants are all assumed to be time-varying. In addition, the exponential stability is also investigated for impulsive stochastic time-varying systems with time-delay, which can extend some existing results. Two examples, including one example of the consensus for impulsive time-varying multiagent systems with time-delay, are presented to demonstrate the effectiveness of the proposed results.

This paper is concerned with stability analysis of nonlinear time-varying systems by using Lyapunov function based approach. The classical Lyapunov stability theorems are generalized in the sense that the time-derivative of the Lyapunov functions are allowed to be indefinite. Then, under quite general assumptions, we first present a new converse stability theorem for a large class of time-varying systems which will be used to prove certain stability properties of nonlinear systems with perturbation. Therefore, a new Lyapunov function is presented that guarantees global asymptotic stability under some restrictions on the perturbed system. Furthermore, some illustrative examples are presented.

We present new methods for proving stability of time-varying linear systems with delays. Our main tools include positive systems and linear Lyapunov functionals. Our work applies to key classes of systems that arise in numerous engineering applications, including neutral systems, and systems that are not necessarily periodic in time and not necessarily positive. We prove stability by comparing the trajectories of the original systems with trajectories of higher dimensional positive systems. One of our key results requires an upper bound on the delay, but the delay can be unknown. Our work also provides robustness of the stability with respect to uncertainties in the coefficient matrices of the system. We illustrate our work in three examples, which show how our methods can sometimes be used with backstepping and linearization to cover even more general classes of systems.

The paper investigates equivalent relations of detectability and persistency of excitation (PE) for general nonlinear time-varying systems. Three new definitions relating to the detectability are proposed and discussed. As a preliminary result, we show that these detectability conditions are all equivalent under a mild assumption. Furthermore, two PE conditions that are a nonlinear-version extension of the PE proposed in present literature are defined. Very interestingly, it can be shown that these PE conditions are both equivalent to the proposed detectability conditions under the same assumption. To the best of our knowledge, this is a first result about the connection between detectability and PE in the area of nonlinear time-varying systems. Furthermore, we revisit a class of nonlinear time-varying interconnected systems modified from adaptive control systems. By combining a newly developed stability criterion with the proposed results, the globally uniformly asymptotic stability of the origin can be guaranteed by verifying a PE condition. From these applications, it can be seen that the uniformly asymptotic stability can be guaranteed either using the detectability conditions or by the PE conditions based on our approach.

In this paper the extremum periodic trajectory of the two-dimensional selector-linear differential inclusion (SLDI) is used to estimate boundary of the invariant set of the nonlinear time-varying system arising in the stability analysis of the wheeled robot control. The motion is supposed to be planar without a lateral slippage. The control goal is to drive the target point of the robot plaform to the specified trajectory and to stabilize the motion along it. The trajectory consists of line segments and circular arcs. The current curvature of the trajectory of the target point is taken as control. The control must satisfy two-sided constraints. Given control law, the attraction domain estimation problem is considered. The attraction domain must be inscribed into the certain parallelepiped of the 'distance to the trajectory-orientation' phase space. Time-varying curvature of the target trajectory is considered as arbitrary varying function which takes values from the specified interval. The feedback linearization scheme is used for synthesis of the control low. The 'saturation function' is then used to take into account control constraints. The closed loop system takes form of the nonlinear system with parametric disturbances. The absolute stability approach is explored for stability analysis. Some nonlinearities take values from the interval. Other nonlinearities satisfy sector constraints. Along with the nonlinear time-varying system the uncertain linear time varying system is considered. Every solution of the nonlinear system is also solution of the time varying system for certain set of time-varying disturbances. To estimate the attraction domain of the nonlinear closed loop system, the Lyapunov function for SLDI is constructed. A convex invariant function is known to exist at the boundary of the absolute stability of SLDI. The extremum trajectory, corresponding to the boundary of the absolute stability in the second order case belongs to the level set of the invariant function and is the periodic solution. The periodic solution has finite number of switches on the period. It circumscribes the boundary of the attraction domain estimate. The illustrative example is given.

Design of optimal PID controller for multivariable time-varying delay discrete-time systems using non-monotonic Lyapunov-Krasovskii approach

This work deals with the stability analysis of linear discrete-time time-variant systems, focusing on the most restrictive stability measures for free respectively forced systems, the exponential stability respectively BIBS (bounded-input bounded-state) stability. Additionally the connection of these stability concepts with the so-called Bohl exponent is in focus of this work. The Bohl exponent can be seen as a generalization of eigenvalues for the stability analysis of time-variant systems. The contribution of this work is the combination of the exponential and BIBS stability analysis with the numerical computation of the Bohl exponent, leading to a numerical stability analysis for linear discrete-time time-variant systems. This stability analysis is applied at a time-variant drivetrain control loop.

Stability analysis and control synthesis of hybrid time-varying linear systems using a discretization-based approach

ABSTRACTThis paper explicates a pointwise frequency-domain approach for stability analysis in periodically time-varying continuous systems, by employing piecewise linear time-invariant (PLTI) models defined via piecewise-constant approximation and their frequency responses. The PLTI models are piecewise LTI state-space expressions, which provide theoretical and numerical conveniences in the frequency-domain analysis and synthesis. More precisely, stability, controllability and positive realness of periodically time-varying continuous systems are examined by means of PLTI models; then their pointwise frequency responses (PFR) are connected to stability analysis. Finally, Nyquist-like and circle-like criteria are claimed in terms of PFR's for asymptotic stability, finite-gain Lp-stability and uniformly boundedness, respectively, in linear feedbacks and nonlinear Luré connections. The suggested stability conditions have explicit and direct matrix expressions, where neither Floquet factorisations of transition matrices nor open-loop unstable poles are involved, and their implementation can be graphical and numerical. Illustrative studies are sketched to show applications of the main results.