Homogeneous Lyapunov Function Research Articles

Polynomial Fuzzy Observed-State Feedback Stabilization via Homogeneous Lyapunov Methods

A homogeneously state-dependent polynomial Lyapunov function for an observed-state feedback polynomial fuzzy control system is proposed. From which an advanced sufficient sum of squares (SOS) condition formulated in terms of state-dependent matrix inequalities to achieve fuzzy stabilization is derived. Three main features, compared with existing nonhomogeneously polynomial methods, are as follows: 1) homogeneous Lyapunov function and its time derivative are nicely linked via the Hessian matrix; 2) removing the nonconvex obstacles on $\dot{P}(x)$ when deriving a nonhomogeneous Lyapunov stabilization condition $\dot{V}(x)$ ; 3) separation principle holds; since common $P$ falls out as a special case, the proposed homogeneous SOS approach is placed ahead of the existing SOS-based methods found in polynomial fuzzy stabilization controls. To verify the problem-solving theories of the homogeneous polynomial method announced, four examples are demonstrated to show the promising features of the approach adduced. Finally, an Appendix illustrating the solving procedure is added for clarification on how the examples are solved via SOS.

Output feedback Continuous Twisting Algorithm

Abstract Two output feedback controllers based on the Continuous Twisting Algorithm are provided. In those controllers, the state observers are based on the first and the second order Robust Exact Differentiators. The stability of the closed loops is proven through input-to-state stability properties. In the case of the second order differentiator, the conservation of homogeneity allows the output feedback scheme to preserve the robustness and accuracy properties of the state feedback Continuous Twisting Algorithm. In the same case, a smooth homogeneous Lyapunov function is constructed for the closed loop. A separation principle in the design of the controller and the observers is established. A qualitative analysis of the performance of the controllers in the presence of noise in the measurement is carried out. One of the schemes is used for output feedback control of a class of nonlinear systems.

Homogeneous Lyapunov Functions: From Converse Design to Numerical Implementation

The problem of the synthesis of a homogeneous Lyapunov function for an asymptotically stable homogeneous system is studied. First, for systems with nonnegative degree of homogeneity, several expressions of homogeneous Lyapunov functions are derived, which depend explicitly on the supremum or the integral (over finite or infinite intervals of time) of the system solutions. Second, a numeric procedure is proposed, which ensures the construction of a homogeneous Lyapunov function. The analytical results are illustrated by simulations.

Finite-time stability theorems of homogeneous stochastic nonlinear systems

The purpose of this paper is to study finite-time stability of a class of homogeneous stochastic nonlinear systems modeled by stochastic differential equations. An existence result of weak solutions for stochastic differential equations with continuous coefficients is derived as a preparation for discussing stochastic nonlinear systems. Then a generalization of finite-time stochastic stability theorem is given. By using some properties of homogeneous functions and homogeneous vector fields, it is proved that a homogeneous stochastic nonlinear system is finite-time stable if its coefficients have negative degrees of homogeneity, and there exists a sufficiently smooth and homogeneous Lyapunov function such that the infinitesimal generator of the stochastic system acting on it is negative definite. In the case when the drift coefficient of a stochastic system is homogeneous and has a negative degree of homogeneity, it can be shown that the stochastic system is also finite-time stable under appropriate conditions. Two examples are provided as illustrations.

Continuous-time model predictive control of under-actuated spacecraft with bounded control torques

The stabilization problem of rigid spacecraft is essential for space explorations and operations. This paper studies the model predictive stabilization problem of a class of underactuated rigid spacecrafts with two bounded control torques. A novel model predictive control (MPC) algorithm is designed by making use of the homogeneity of the system dynamics. In addition, a local homogeneous Lyapunov function is constructed based on which the approach to designing the terminal set and other parameters are developed. Finally, the conditions for ensuring algorithm feasibility and closed-loop stability are provided. We show that under the given conditions, the designed MPC algorithm is feasible, and the closed-loop system is asymptotically stable. Simulation and comparison studies verify that the developed results are effective and valid, and the designed controller fulfills the constraint satisfaction and achieves much faster convergence rate in comparison with conventional continuous time-varying controllers.

Stability analysis of discrete-time switched systems: a switched homogeneous Lyapunov function method

ABSTRACTThis paper addresses the stability issue of discrete-time switched systems with guaranteed dwell-time. The approach of switched homogeneous Lyapunov function of higher order is formally proposed. By means of this approach, a necessary and sufficient condition is established to check the exponential stability of the considered system. With the observation that switching signal is actually arbitrary if the dwell time is one sample time, a necessary and sufficient condition is also presented to verify the exponential stability of switched systems under arbitrary switching signals. Using the augmented argument, a necessary and sufficient exponential stability criterion is given for discrete-time switched systems with delays. A numerical example is provided to show the advantages of the theoretical results.

Estimation of Attractive Stability Region via Homogeneous Parameter-Dependent Quadratic Lyapunov Function for Impulsive Switched Linear System with Saturated Control Input

A class of homogeneous polynomial functions is used to estimate the attractive stability region for an impulsive switched linear system with saturated control input. Under the arbitrary switching rule and the dwell time switching rule, local stabilization conditions are obtained in terms of linear matrix inequalities. The corresponding optimization problems are formulated to obtain a larger attractive stability region. Using the simple ellipsoid method to estimate the attractive stability region produces very conservative results, because the attractive stability region is normally irregular. To solve this problem, a polyhedron constructed using a level set of the homogeneous parameter-dependent quadratic Lyapunov function is used to estimate the attractive stability region. The polyhedron is closer to the attractive stability region than the simple ellipsoid. Finally, numerical examples are given to demonstrate the effectiveness of the proposed method.

On Homogeneous Parameter-Dependent Quadratic Lyapunov Function for RobustH∞Filtering Design in Switched Linear Discrete-Time Systems with Polytopic Uncertainties

This paper is concerned with the problem of robustH∞filter design for switched linear discrete-time systems with polytopic uncertainties. The condition of being robustly asymptotically stable for uncertain switched system and less conservativeH∞noise-attenuation level bounds are obtained by homogeneous parameter-dependent quadratic Lyapunov function. Moreover, a more feasible and effective method against the variations of uncertain parameter robust switched linear filter is designed under the given arbitrary switching signal. Lastly, simulation results are used to illustrate the effectiveness of our method.

同次制御に基づくアクティブ動吸振器の制振性能解析

同次制御に基づくアクティブ動吸振器の制振性能解析

Estimation of asymptotic stability regions via composite homogeneous polynomial Lyapunov functions

In this article, we present a new method to estimate the asymptotic stability regions for a class of nonlinear systems via composite homogeneous polynomial Lyapunov functions, where these nonlinear systems are approximated as a convex hull of some linear systems. Since level set of the composite homogeneous polynomial Lyapunov functions is a union set of several homogeneous polynomial functions, the composite homogeneous polynomial Lyapunov functions are nonconservative compared with quadratic or homogeneous polynomial Lyapunov functions. Numerical examples are used to illustrate the effectiveness of our method.

Adaptive finite-time consensus in multi-agent networks

This paper is concerned with the finite-time consensus problem of distributed agents having non-identical unknown nonlinear dynamics, to a leader agent that also has unknown nonlinear control input signal. By parameterization of unknown nonlinear dynamics, a Lyapunov technique in conjunction with homogeneity technique is presented for designing a decentralized adaptive finite-time consensus control protocol in undirected networks. Homogeneous Lyapunov functions and homogeneous vector fields are introduced in the stability analysis although the whole system is not homogeneous. Theoretical analysis shows that leader-following consensus can be achieved in finite-time, meanwhile, finite-time parameter convergence can be also guaranteed under the proposed control scheme. An example is given to validate the theoretical results.

Finite-gain [formula omitted] stability for hybrid dynamical systems

We characterize the finite-gain Lp stability properties for hybrid dynamical systems. By defining a suitable concept of the hybrid Lp norm, we introduce hybrid storage functions and provide sufficient Lyapunov conditions for the Lp stability of hybrid systems, which cover the well-known continuous-time and discrete-time Lp stability notions as special cases. We then focus on homogeneous hybrid systems and prove a result stating the equivalence among local asymptotic stability of the origin, global exponential stability, existence of a homogeneous Lyapunov function with suitable properties for the hybrid system with no inputs, and input-to-state stability, and we show how these properties all imply Lp stability. Finally, we characterize systems with direct and reverse average dwell-time properties, and establish parallel results for this class of systems. We also make several connections to the existing results on dissipativity properties of hybrid dynamical systems.

Worst-Case Mahler Measure in Polytopic Uncertain Systems

The Mahler measure provides a way to quantify the unstable and plays a key role in stabilization problems. This technical brief addresses the computation of the worst-case Mahler measure in systems depending polynomially on uncertain parameters constrained in a polytope. A sufficient condition for establishing an upper bound of the worst-case Mahler measure is provided in terms of linear matrix inequality (LMI) feasibility tests, where a homogeneous parameter-dependent quadratic Lyapunov function (HPD-QLF) is searched for. Moreover, it is shown that the best upper bound guaranteed by this condition can be obtained by solving generalized eigenvalue problems. Then, the conservatism of this methodology is investigated, showing that the upper bound is monotonically nonincreasing with the degree of the HPD-QLF, and that there exists a degree for which the upper bound is guaranteed to be tight. Some numerical examples illustrate the proposed results.

On the asymptotic stability of switched homogeneous systems

The stability of switched systems generated by the family of autonomous subsystems with homogeneous right-hand sides is investigated. It is assumed that for each subsystem the proper homogeneous Lyapunov function is constructed. The sufficient conditions of the existence of the common Lyapunov function providing global asymptotic stability of the zero solution for any admissible switching law are obtained. In the case where we can not guarantee the existence of a common Lyapunov function, the classes of switching signals are determined under which the zero solution is locally or globally asymptotically stable. It is proved that, for any given neighborhood of the origin, one can choose a number L > 0 (dwell time) such that if intervals between consecutive switching times are not smaller than L then any solution of the considered system enters this neighborhood in finite time and remains within it thereafter.

Robust stability of uncertain piecewise-linear systems: LMI approach

In this paper we propose sufficient conditions for the robust stability of time-invariant uncertain piecewise-linear systems using homogeneous polynomial Lyapunov functions. The proposed conditions are expressed in terms of linear matrix inequalities, which can be numerically determined. We solve the stabilization of piecewise uncertain linear control systems by using state piecewise-linear feedback. We propose an illustrative example to show the efficiency of the proposed approach.

Global stabilization of partially linear composite systems using homogeneous method

A linear feedback control scheme to globally stabilize a class of partially linear composite systems is proposed from the point view of homogeneity. Assume that the global stability of the zero dynamics of the nonlinear subsystem can be tested by using a homogeneous Lyapunov function. It is shown that the stabilization of the linear controllable subsystem from its own states equals to the stabilization of the whole systems if the nonlinearities satisfy a homogeneous inequality condition. Then we assume that the states are not measurable and also extend the method developed for state-feedback control to the output-feedback case. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

On the Minimum Stable Commutation Time for Switching Nonlinear Systems

Real systems are often driven by switching reference signals which affect dynamics and/or equilibrium points. This technical note addresses the computation of upper bounds of the minimum commutation time ensuring stability for switching nonlinear systems. Specifically, we consider the cases of constant and variable equilibrium point of interest, for polynomial systems and for a class of non-polynomial systems. We hence propose upper bounds of the sought minimum commutation time by adopting homogeneous polynomial Lyapunov functions for the former case and polynomial Lyapunov functions for the latter one, which can be computed via linear matrix inequaltiy optimizations for given Lyapunov functions.

Establishing Convexity of Polynomial Lyapunov Functions and Their Sublevel Sets

There has been renewed interest in the use of polynomial and homogeneous polynomial Lyapunov functions for addressing stability and performance issues in systems and control theory. However, no condition is yet available to establish the convexity of these functions and the convexity of their sublevel sets, properties that can be useful in many applications. In this paper, new conditions for establishing these convexity properties are proposed, which can be handled through convex linear matrix inequalities optimizations by means of sum-of-squares relaxations.

STABILIZATION OF HOMOGENEOUS SYSTEMS USING IMPLICIT CONTROL LYAPUNOV FUNCTIONS

This paper focuses on the problem of stabilization of homogeneous systems with dilations using implicit Lyapunov functions. In this paper, we define implicit homogeneous Lyapunov functions and implicit homogeneous control Lyapunov functions. Moreover, we proposed a design method of implicit homogeneous control Lyapunov functions and a controller design method based on the functions. Finally, we confirmed the effectiveness of the proposed method by computer simulation.

Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions

This paper deals with robust stability analysis of linear state space systems affected by time-varying uncertainties with bounded variation rate. A new class of parameter-dependent Lyapunov functions is introduced, whose main feature is that the dependence on the uncertain parameters and the state variables are both expressed as polynomial homogeneous forms. This class of Lyapunov functions generalizes those successfully employed in the special cases of unbounded variation rates and time-invariant perturbations. The main result of the paper is a sufficient condition to determine the sought Lyapunov function, which amounts to solving an LMI feasibility problem, derived via a suitable parameterization of polynomial homogeneous forms. Moreover, lower bounds on the maximum variation rate for which robust stability of the system is preserved, are shown to be computable in terms of generalized eigenvalue problems. Numerical examples are provided to illustrate how the proposed approach compares with other techniques available in the literature.