Lyapunov stability and generalized invariance principle for nonconvex differential inclusions

This paper studies the system stability problems of a class of nonconvex differential inclusions. At first, a basic stability result is obtained by virtue of locally Lipschitz continuous Lyapunov functions. Moreover, a generalized invariance principle and related attraction conditions are proposed and proved to overcome the technical difficulties due to the absence of convexity. In the technical analysis, a novel set-valued derivative is proposed to deal with nonsmooth systems and nonsmooth Lyapunov functions. Additionally, the obtained results are consistent with the existing ones in the case of convex differential inclusions with regular Lyapunov functions. Finally, illustrative examples are given to show the effectiveness of the methods.     

基于非光滑线性Lipschitz连续平面的滑模控制设计

利用Ｆｉｌｉｐｐｏｖ解、Ｃｌａｒｋｅ广义梯度和非光滑Ｌｙａｐｕｎｏｖ稳定理论,对一类滑模面设计为非光滑线性Ｌｉｐｓｃｈｉｔｚ连续平面的二阶系统滑模控制问题进行深入讨论．首先设计控制律,使闭环系统在有限时间内能够到达所设计的滑模面;然后证明系统在滑模面上的运动是渐近稳定的．放宽了对滑模控制中滑模面设计的要求,提高了所提出设计方法的灵活性,有利于改善系统性能．仿真结果验证了所提出设计方法的正确性和有效性．     

一类非线性系统全局有限时间观测器设计

本文首先讨论了非线性系统的有限时间稳定, 并给出了其全局有限时间稳定的一个充分条件. 然后, 利用几何齐次理论、Lyapunov稳定性理论, 并通过构造一个增益适应律, 对一类具有下三角结构的非线性系统, 讨论其全局有限时间稳定状态观测器的设计问题, 所设计的观测器是连续非光滑的, 能够在有限时间段内实现状态的精确重构.     

Global stabilization of cascade systems by C/sup 0/ partial-state feedback

This paper shows how the nonsmooth but continuous feedback design approach developed recently for global stabilization of nonlinear systems with uncontrollable unstable linearization, and the notion and properties of the input-to-state stability Lyapunov function can be effectively coupled, resulting in globally stabilizing C/sup 0/ partial-state feedback controllers for a class of cascade systems which may not be smoothly stabilizable, even locally.     

Global finite-time stabilization of a class of uncertain nonlinear systems

This paper studies the problem of finite-time stabilization for nonlinear systems. We prove that global finite-time stabilizability of uncertain nonlinear systems that are dominated by a lower-triangular system can be achieved by Hölder continuous state feedback. The proof is based on the finite-time Lyapunov stability theorem and the nonsmooth feedback design method developed recently for the control of inherently nonlinear systems that cannot be dealt with by any smooth feedback. A recursive design algorithm is developed for the construction of a Hölder continuous, global finite-time stabilizer as well as a C¹ positive definite and proper Lyapunov function that guarantees finite-time stability.     

Nonpathological Lyapunov functions and discontinuous Carathéodory systems

Differential equations with discontinuous right-hand side and solutions intended in Carathéodory sense are considered. For these equations, sufficient conditions which guarantee both Lyapunov stability and asymptotic stability in terms of nonsmooth Lyapunov functions are given. An invariance principle is also proven.     

The Exponential Stability for a Class of Hybrid Systems

The exponential stability with a nonsmooth Lyapunov function for a class of hybrid systems is studied in this paper. First, a sufficient condition is derived that has to be satisfied by the feedback control for the hybrid systems. Then, for the special case where the Lyapunov function involved is a kind of nonsmooth function, the maximum of finitely many smooth functions (for short, max‐type function), the stability of a hybrid system is considered, and a convenient criterion to determine the stability of the system is established. Finally, a numerical method of determining the control input value is developed.     

Feedback stabilization for a class of discontinuous systems driven by integrator

This paper investigates the feedback stabilization problem for a class of discontinuous systems which is characterized by Filippov differential inclusion. Lyapunov-based backstepping design method is generalized with nonsmooth Lyapunov functions to solve the control problem. A set-valued time derivative is introduced first for nonsmooth function along discontinuous vector fields, which enables us to perform Lyapunov-based design with nondifferentiable Lyapunov function. Conditions for designing a virtual control law which is shown nondifferentiable in general in the recursive design problem are proposed. Finally, as a special case, piecewise linear system is discussed to demonstrate the application of the presented design approach.     

带干摩擦的非光滑动力系统Lyapunov指数的数值计算

对带干摩擦的非光滑动力系统,在不连续点利用闭上链(cocycle)得到一个传递矩阵,由此给出一种非光滑动力系统Lyapunov指数的数值计算方法.该方法是根据计算光滑动力系统Lyapunov指数的数值方法而提出的.通过对带干摩擦的动力系统进行Lyapunov指数的数值计算,结合时间历程图和Poincare映射对其动力学特性进行了分析.数值结果表明该方法的可行性和有效性,与一般的计算非光滑动力系统Lyapunov指数的数值算法相比,该方法更便于数值计算.     

Arc-length-based Lyapunov tests for convergence and stability with applications to systems having a continuum of equilibria

In this paper, fundamental relationships are established between convergence of solutions, stability of equilibria, and arc length of orbits. More specifically, it is shown that a system is convergent if all of its orbits have finite arc length, while an equilibrium is Lyapunov stable if the arc length (considered as a function of the initial condition) is continuous at the equilibrium, and semistable if the arc length is continuous in a neighborhood of the equilibrium. Next, arc-length-based Lyapunov tests are derived for convergence and stability. These tests do not require the Lyapunov function to be positive definite. Instead, these results involve an inequality relating the right-hand side of the differential equation and the Lyapunov function derivative. This inequality makes it possible to deduce properties of the arc length function and thus leads to sufficient conditions for convergence and stability. Finally, it is shown that the converses of all the main results hold under additional assumptions. Examples are included to illustrate how our results are particularly suited for analyzing stability of systems having a continuum of equilibria.     

Stability of the solutions of impulsive integro-differential equations in terms of two measures

In the present paper the stability of the solutions of impulsive systems of integro-differential equations of Volterra type with fixed moments of impulse effect in terms of two piecewise continuous measures is investigated. The investigations are carried out by means of piecewise continuous functions of the same type as the Lyapunov functions using differential inequalities for piecewise continuous functions.     

A unified dissipativity approach for stability analysis of piecewise smooth systems

The main objective of this paper is to present a unified dissipativity approach for stability analysis of piecewise smooth (PWS) systems with continuous and discontinuous vector fields. The Filippov definition is considered for the solution of these systems. Using the concept of generalized gradients for nonsmooth functions, sufficient conditions for the stability of a PWS system are formulated based on Lyapunov theory. The importance of the proposed approach is that it does not need any a-priori information about attractive sliding modes on switching surfaces, which is in general difficult to obtain. A section on application of the main results to piecewise affine (PWA) systems followed by a section with extensive examples clearly show the usefulness of the proposed unified methodology. In particular, we present an example with a stable sliding mode where the proposed method works and previously suggested methods fail.     

Passivity analysis for delayed discontinuous neural networks

In this paper, we have integrated the passivity of delayed neural networks with discontinuous activations which are unbounded. Based on differential inclusion theory and nonsmooth analysis, some sufficient conditions are presented by means of the generalized Lyapunov method. The results are established in form of linear matrix inequality. In addition, Some numerical examples are proposed to show the effectiveness of the developed results.     

Backstepping for nonsmooth systems

The paper presents a constructive control design for integrator backstepping in nonsmooth systems. The approach is based on non smooth analysis and Lyapunov stability for nonsmooth systems and is similar in spirit with the robust control designs that have appeared in literature, but is applicable to a larger class of systems. The backstepping controller is first applied to the case of a unicycle driven by a new discontinuous kinematic controller yielding global asymptotic convergence with bounded inputs. Then it is used to implement a sliding mode controller in a hybrid system. Simulations results not only verify the convergence properties but also reveal the ability of the new backstepping controller to suppress chattering.     

Minimum Projection Method for nonsmooth control Lyapunov function design on general manifolds

Asymptotic stabilization on noncontractible manifolds is known as a difficult control problem. On the other hand, an important fact is every control system that is globally asymptotically stabilizable at a desired equilibrium must have nonsmooth control Lyapunov functions.This paper considers the problem of construction of nonsmooth control Lyapunov functions on general manifolds, and we propose a nonsmooth control Lyapunov function design method called the ‘Minimum Projection Method’. The proposed method considers a simple-structured smooth manifold associated with the original manifold by a surjective immersion, and then a control Lyapunov function defined on the simple-structured manifold is projected to the original manifold. A function on the original manifold is thus obtained.In this paper, we prove that the control system on another manifold associated with a surjective immersion is determined uniquely, and the resulting function by the proposed method is a nonsmooth control Lyapunov function on the original manifold. The effectiveness of the proposed method is confirmed by examples.     

Hamilton-Jacobi inequalities and the optimality conditions in the problems of control with common end constraints

The canonical theory of the necessary and sufficient conditions for global optimality based on the sets of nonsmooth solutions of the differential Hamilton-Jacobi inequalities of two classes of weakly and strongly monotone Lyapunov type functions was developed. These functions enable one to estimate from above and below the objective functional of the optimal control problem and determine the internal and external approximations of the reachability set of the controlled dynamic system.     

Global asymptotic stability of delay BAM neural networks with impulses via nonsmooth analysis

In this paper, we study global asymptotic stability of delay bi-directional associative memory (BAM) neural networks with impulses. We obtain a sufficient condition of ensuring existence and uniqueness of equilibrium point for delay BAM neural networks with impulses basing on nonsmooth analysis. And we give a criteria of global asymptotic stability of the unique equilibrium point for delay BAM neural networks with impulses using Lyapunov method. At last, we present examples to illustrate that our results are feasible.     

Adaptive control design for a class of nonsmooth nonlinear systems with matched and linearly parameterized uncertainty

This paper presents an adaptive control design for a class of nonsmooth nonlinear systems with matched uncertainty, which is linearly parameterized with a known discontinuous function. The design framework is based on the concept of the Filippov solution as the classical Lyapunov theory for smooth systems cannot be applied to establish the stability of the adaptive control system due to the presence of the discontinuity. It is proved that as an adaptive control system, the global Lyapunov stability with the convergence of the state of the controlled system to the origin can be achieved by evaluating the monotonicity of the Lyapunov function in the state space, particularly on the discontinuous surface, while the uniqueness of the solution of the closed‐loop system is not necessarily guaranteed. Some interesting numerical examples are demonstrated with simulation results. Copyright © 2008 John Wiley & Sons, Ltd.     

Continuous output-feedback stabilization for a class of stochastic high-order nonlinear systems

This paper is concerned with the global stabilization via output-feedback for a class of high-order stochastic nonlinear systems with unmeasurable states dependent growth and uncertain control coefficients. Indeed, there have been abundant deterministic results which recently inspired the intense investigation for their stochastic analogous. However, because of the possibility of non-unique solutions to the systems, there lack basic concepts and theorems for the problem under investigation. First of all, two stochastic stability concepts are generalized to allow the stochastic systems with more than one solution, and a key theorem is given to provide the sufficient conditions for the stochastic stabilities in a weaker sense. Then, by introducing the suitable reduced order observer and appropriate control Lyapunov functions, and by using the method of adding a power integrator, a continuous (nonsmooth) output-feedback controller is successfully designed, which guarantees that the closed-loop system is globally asymptotically stable in probability.     

Stability and periodicity in fuzzy differential equations

Formulations of fuzzy differential equations (DEs) in terms of the Hukuhara derivative do not reproduce the rich and varied behavior of crisp DEs. Another interpretation in terms of differential inclusions, expressed level setwise, overcomes much of this deficiency and opens up for profitable investigation such properties as stability, attraction, periodicity, and the like. This is especially important for investigating continuous systems, which are uncertain or incompletely specified. This paper studies studies Lyapunov stability of fuzzy DEs and the periodicity of the fuzzy solution set for both the time-dependent and autonomous cases