Lyapunov Asymptotic Stability Research Articles

The stability of fixed points of discrete dynamical systems in the space conv ℝ n

Conditions for the asymptotic Lyapunov stability of the fixed points of discrete dynamical systems in the space conv ℝn are established.

Fault-Tolerant Control for T–S Fuzzy Descriptor Systems with Sensor Faults: An LMI Approach

This paper focuses on the robust sensor fault-tolerant control of T–S fuzzy descriptor systems with unmeasurable premise variables subject to external disturbances. By introducing sensor fault as an auxiliary state vector, an augmented fuzzy descriptor system is constructed. A robust fuzzy descriptor observer is investigated to achieve simultaneous reconstruction of system states and sensor faults using descriptor approach and the \(L_2\) techniques to minimize the effect of the perturbations on the state estimation error. Based on the state estimated variables and by isolating the sensor fault, a fault-tolerant control scheme is proposed using a state feedback controller approach such that the resulting closed-loop system is stable with prescribed \(H_{\infty }\) performance. Sufficient conditions for the existence of a robust fuzzy descriptor observer and a fault-tolerant controller are derived in the sense of Lyapunov asymptotic stability and are formulated in terms of a series of linear matrix inequalities without additional constraints. Finally, a numerical example was discussed to prove the performance of the proposed approach.

Reliable finite-time filtering for impulsive switched linear systems with sensor failures

This paper investigates the finite-time filter design problem for a class of switched linear systems with parameter uncertainties and impulsive effects. The sensor failures with output-measurement errors and mode-detection delays are taken into account. A filter mode-dependent Lyapunov-like function construction approach is developed which can effectively reduce the filter design difficulty induced by the mode-detection delays. Sufficient conditions on finite-time boundedness and finite-time H∞ performance are first derived for the augmented switched error systems, upon which the mode-dependent finite-time H∞ filter is designed. Performance of the potential of the developed filter is illustrated by a numerical example and an application to PWM (Pulse-Width-Modulation)-driven boost converter.

Global asymptotic stabilisation in probability of nonlinear stochastic systems via passivity

ABSTRACTThe purpose of this paper is to develop a systematic method for global asymptotic stabilisation in probability of nonlinear control stochastic systems with stable in probability unforced dynamics. The method is based on the theory of passivity for nonaffine stochastic differential systems combined with the technique of Lyapunov asymptotic stability in probability for stochastic differential equations. In particular, we prove that a nonlinear stochastic differential system whose unforced dynamics are Lyapunov stable in probability is globally asymptotically stabilisable in probability provided some rank conditions involving the affine part of the system coefficients are satisfied. In this framework, we show that a stabilising smooth state feedback law can be designed explicitly. A dynamic output feedback compensator for a class of nonaffine stochastic systems is constructed as an application of our analysis.

A NEURAL NETWORK FOR THE GENERALIZED NONLINEAR COMPLEMENTARITY PROBLEM OVER A POLYHEDRAL CONE

In this paper, we consider a neural network model for solving the generalized nonlinear complementarity problem (denoted by GNCP) over a polyhedral cone. The neural network is derived from an equivalent unconstrained minimization reformulation of the GNCP, which is based on the penalized Fischer–Burmeister function${\it\phi}_{{\it\mu}}(a,b)={\it\mu}{\it\phi}_{\mathit{FB}}(a,b)+(1-{\it\mu})a_{+}b_{+}$. We establish the existence and the convergence of the trajectory of the neural network, and study its Lyapunov stability, asymptotic stability and exponential stability. It is found that a larger${\it\mu}$leads to a better convergence rate of the trajectory. Simulation results are also reported.

Lyapunov function method for analyzing stability of quasi-Hamiltonian systems under combined Gaussian and Poisson white noise excitations

The asymptotic Lyapunov stability with probability one of multi-degree-of-freedom quasi-Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises is studied by using Lyapunov function method. According to the integrability and resonance, quasi-Hamiltonian systems can be divided into five classes, namely quasi-non-integrable, quasi-completely integrable and non-resonant, quasi-completely integrable and resonant, quasi-partially integrable and non-resonant, and quasi-partially integrable and resonant. Lyapunov functions for these five classes of systems are constructed. The derivatives for these Lyapunov functions with respect to time are obtained by using the stochastic averaging method. The approximately sufficient condition for the asymptotic Lyapunov stability with probability one of quasi-Hamiltonian system under parametric excitations of combined Gaussian and Poisson white noises is determined based on a theorem due to Khasminskii. Four examples are given to illustrate the application and efficiency of the proposed method. And the results are compared with the corresponding necessary and sufficient condition obtained by using the largest Lyapunov exponent method.

Existence and stability of periodic contrast structures in the reaction-advection-diffusion problem

A singularly perturbed periodic problem for a parabolic reaction-advection-diffusion equation at low advection is studied. The case when there is an internal transition layer under unbalanced nonlinearity is considered. An asymptotic expansion of a solution is constructed. To substantiate the asymptotics thus constructed, the asymptotic method of differential inequalities is used. The Lyapunov asymptotic stability of a periodic solution is studied; the proof uses the Krein-Rutman theorem.

On qualitative properties of differential-algebraic equations

Linear systems of ordinary differential equations with identically degenerate coefficient matrix before the derivative of the unknown vector function are considered. The structure of general solutions and the notion of singular point of such systems are discussed. From the comparison of the properties of the “perturbed” and original problems, a sufficient criterion for the Lyapunov asymptotic stability of the zero solution is obtained.

Finite-Time Vibration Control of Earthquake Excited Linear Structures with Input Time-Delay and Saturation

The problem of finite-time vibration control of earthquake excited linear structures with input time-delay and saturation is concerned in this paper. The objective of designing controllers is to guarantee the finite-time stability of closed-loop systems while attenuating earthquake-induced vibration of the structures. First, based on matrix transformation, the structural system is described as a state-space model, which contains actuator saturation and input time-delay. Then, based on a Lyapunov functional and finite-time stability analysis method, some sufficient conditions for the existence of saturation-tolerant finite-time vibration-attenuation controllers are obtained. By solving these conditions, the desired controllers can be obtained for the closed-loop system to be finite-time stable with a prescribed level of disturbance attenuation. It is shown by the simulation results that compared with some Lyapunov asymptotic stability results, finite-time stability control can result in better state responses. Furthermore, saturation-tolerant controller can result in a much lower controller gain than the ones without considering actuator saturations.

Stochastic stability of quasi-integrable and resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises

A procedure for determining the asymptotic Lyapunov stability with probability one of multi-degree-of-freedom (MDOF) quasi-integrable and resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises is proposed. For the case of resonance with α resonant relations, the averaged Ito^ stochastic differential equations (SDEs) for quasi-integrable and resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises are first derived by using the stochastic averaging method. Then, the expression for the largest Lyapunov exponent is obtained by generalizing Khasminskii׳s procedure to the averaged Ito^ SDEs and using the property of the stochastic integro-differential equations (SIDEs). Finally, the stochastic stability of the original system is determined approximately by using the largest Lyapunov exponent. An example of two non-linear damping oscillators under parametric excitations of combined Gaussian and Poisson white noises is worked out to illustrate the application of the proposed procedure. The validity of the proposed procedure is verified by the good agreement between the analytical results and those from Monte Carlo simulation.

H<sub>∞</sub> Robust Control for Networked Control System with Short Time-Delay

For a kind of networked control systems with short time-delay, establish discrete time-invariant system model. Construct the Lyapunov function based on the Lyapunov asymptotic stability principle. Using Linear Matrix Inequalities method given the sufficient condition of H∞ robust controller design method of closed-loop feedback control systems. Matlab simulation indicates the effectiveness and correctness of the controller design.

Stochastic stability of quasi-partially integrable and non-resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises

The asymptotic Lyapunov stability with probability one of multi-degree-of freedom quasi-partially integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises is studied. First, the averaged stochastic differential equations for quasi partially integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises are derived by means of the stochastic averaging method and the stochastic jump-diffusion chain rule. Then, the expression of the largest Lyapunov exponent of the averaged system is obtained by using a procedure similar to that due to Khasminskii and the properties of stochastic integro-differential equations. Finally, the stochastic stability of the original quasi-partially integrable and non-resonant Hamiltonian systems is determined approximately by using the largest Lyapunov exponent. An example is worked out in detail to illustrate the application of the proposed method. The good agreement between the analytical results and those from digital simulation show that the proposed method is effective.

Finite-Time Stability Analysis for a Class of Continuous Switched Descriptor Systems

Finite-time stability has more practical application values than the classical Lyapunov asymptotic stability over a fixed finite-time interval. The problems of finite-time stability and finite-time boundedness for a class of continuous switched descriptor systems are considered in this paper. Based on the average dwell time approach and the multiple Lyapunov functions technique, the concepts of finite-time stability and boundedness are extended to continuous switched descriptor systems. In addition, sufficient conditions for the existence of state feedback controllers in terms of linear matrix inequalities (LMIs) are obtained with arbitrary switching rules, which guarantee that the switched descriptor system is finite-time stable and finite-time bounded, respectively. Finally, two numerical examples are presented to illustrate the reasonableness and effectiveness of the proposed results.

Finite-Time Vibration-Attenuation Controller Design for Structural Systems with Parameter Uncertainties

The problem of finite-time vibration-attenuation controller design for buildings structural systems with parameter uncertainties is the concern of this paper. The objective of designing controllers is to guarantee the finite-time stability of closed-loop systems with a prescribed level of disturbance attenuation. First, based on matrix transformation, the structural system is described as state-space model, which contains parameter uncertainties. Then, based on finite-time stability analysis method, some sufficient conditions for the existence of finite-time vibration-attenuation controllers are obtained. By solving these conditions, the desired controllers can be obtained for the closed-loop system to be finite-time stable with the performance ∥ z∥2 < γ∥ω∥2. It is shown by the simulation results, that compared with some Lyapunov asymptotic stability results, finite-time stability control can obtain better state responses, especially while the system is under nonzero initial states.

An explicit criterion for finite-time stability of linear nonautonomous systems with delays

In this paper, the problem of finite-time stability of linear nonautonomous systems with time-varying delays is considered. Using a novel approach based on some techniques developed for linear positive systems, we derive new explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. These conditions are shown to be relaxed for the Lyapunov asymptotic stability. A numerical example is given to illustrate the effectiveness of the obtained result.

Stochastic stability of quasi-integrable and non-resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises

A procedure for calculating the largest Lyapunov exponent and determining the asymptotic Lyapunov stability with probability one of multi-degree-of-freedom (MDOF) quasi-integrable and non-resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises is proposed. The averaged stochastic differential equations (SDEs) of quasi-integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises are first derived by using the stochastic averaging method for quasi-Hamiltonian systems and the stochastic jump-diffusion chain rule. Then, the expression for the largest Lyapunov exponent is obtained by generalizing Khasminskii's procedure to the averaged SDEs and the stochastic stability of the original systems is determined approximately. An example is given to illustrate the application of the proposed procedure and its effectiveness is verified by comparing with the results from Monte Carlo simulation.

Fuzzy Fault-Tolerant Control of Wind-Diesel Hybrid Systems Subject to Sensor Faults

A fuzzy fault-tolerant control (FFTC) framework is proposed for wind-diesel-hybrid systems (WDHS) with time-varying bounded sensor faults. The algorithm utilizes fuzzy systems based on “Takagi-Sugeno” (TS) fuzzy models to represent nonlinear systems. A fuzzy proportional-integral estimation observer (FPIEO) design is proposed to achieve fault estimation of TS models with abrupt sensor faults. Sufficient conditions are derived for robust stabilization in the sense of Lyapunov asymptotic stability and are formulated in the format of linear matrix inequalities (LMIs) to obtain controller gains and observer gains. The proposed algorithm maximizes the produced power, minimizes the voltage ripple, and is able to maintain the system's stability during the sensor faults. A physical model of the WDHS is presented and transformed into a TS model. Then, an FFTC algorithm is developed and applied to a WDHS to demonstrate the effectiveness of this method.

Simultaneous positioning and non-minimum phase vibration suppression of slewing flexible-link manipulator using only joint actuator

This paper presents a new global terminal sliding mode control (GTSMC) methodology for the simultaneous positioning and vibration suppression of slewing a flexible-link manipulator. The proposed control scheme is realized using only a joint actuator without an extra actuator (for example, the widely used piezoelectric zirconate titanate), and the non-minimum phase vibration control problem is solved. Firstly, based on the differential geometry method, the initial dynamic system is decomposed into two subsystems, namely an input-output subsystem and a zero-dynamics subsystem. Secondly, a continuous GTSMC strategy without chattering phenomenon is designed for the input-output subsystem, and the limited convergence time is deduced. Moreover, the eigenvalues of the zero-dynamics subsystem can be pointed by setting proper controller parameters for Lyapunov asymptotical stability. Finally, simulation and experimental results demonstrated the efficacy and feasibility of the proposed control methodology.

Lyapunov function construction for nonlinear stochastic dynamical systems

Though the Lyapunov function method is more efficient than the largest Lyapunov exponent method in evaluating the stochastic stability of multi-degree-of-freedom (MDOF) systems, the construction of Lyapunov function is a challenging task. In this paper, a specific linear combination of subsystems’ energies is proposed as Lyapunov function for MDOF nonlinear stochastic dynamical systems, and the corresponding sufficient condition for the asymptotic Lyapunov stability with probability one is then determined. The proposed procedure to construct Lyapunov function is illustrated and validated with several representative examples, where the influence of coupled/uncoupled dampings and excitation intensities on stochastic stability is also investigated.

Stochastic stability of quasi non-integrable Hamiltonian systems under parametric excitations of Gaussian and Poisson white noises

The asymptotic Lyapunov stability with probability one of n-degree-of-freedom (n-DOF) quasi non-integrable Hamiltonian systems subject to weakly parametric excitations of combined Gaussian and Poisson white noises is studied by using the largest Lyapunov exponent. First, an n-DOF quasi non-integrable Hamiltonian system subject to weakly parametric excitations of combined Gaussian and Poisson white noises is reduced to a one-dimensional averaged Itô stochastic differential equation (SDE) for Hamiltonian by using the stochastic averaging method for quasi non-integrable Hamiltonian systems. Then, the expression for the Lyapunov exponent of the averaged Itô SDE is derived and the approximately necessary and sufficient condition for the asymptotic Lyapunov stability with probability one of the trivial solution of the original system is obtained. Finally, one example is worked out to illustrate the proposed procedure and its effectiveness is confirmed by comparing with Monte Carlo simulation. It is found that analytical and simulation results agree well.