Lyapunov stability theory of nonsmooth systems

This paper develops nonsmooth Lyapunov stability theory and LaSalle's invariance principle for a class of nonsmooth Lipschitz continuous Lyapunov functions and absolutely continuous state trajectories. Computable tests based on Filipov's differential inclusion and Clarke's generalized gradient are derived. The primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right-hand side such as in nonsmooth dynamic systems or variable structure control     

Non-conservative matrix inequality conditions for stability/stabilizability of linear differential inclusions

This paper shows that the matrix inequality conditions for stability/stabilizability of linear differential inclusions derived from two classes of composite quadratic functions are not conservative. It is established that the existing stability/stabilizability conditions by means of polyhedral functions and based on matrix equalities are equivalent to the matrix inequality conditions. This implies that the composite quadratic functions are universal for robust, possibly constrained, stabilization problems of linear differential inclusions. In particular, a linear differential inclusion is stable (stabilizable with/without constraints) iff it admits a Lyapunov (control Lyapunov) function in these classes. Examples demonstrate that the polyhedral functions can be much more complex than the composite quadratic functions, to confirm the stability/stabilizability of the same system.     

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.     

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.     

一类非光滑优化及其在控制系统稳定化中的应用

研究一类来自控制系统稳定化中的非光滑优化问题．考虑Lyapunov函数是非光滑的,特别是有限个光滑函数的极大值函数．建立了相应的非光滑优化模型,进一步导出了这类非光滑优化的KKT系统,然后基于非线性互补函数将KKT系统转化成一个非光滑方程组,最后分别用广义牛顿法和光滑化牛顿法求解此非光滑方程组。使得此类稳定化设计可以具体实现．     

一类积分加权时滞型非线性微分包含的稳定性

采用多面体Lyapunov函数分析微分包含的稳定性，即通过建立一类内积型Lyapunov—Krasovskii泛函，分析一类积分加权时滞型非线性微分包含的零解渐近稳定性。得到了一些关于零解渐近稳定性和B—鲁棒渐近稳定性的充分条件，这些充分条件在计算上是简洁实用的。     

Stable Flocking Motion of Mobile Agents Following a Leader in Fixed and Switching Networks

Multiple mobile agents with double integrator dynamics, following a leader to achieve a flocking motion formation, are studied in this paper. A class of local control laws for a group of mobile agents is proposed. Prom a theoretical proof, the following conclusions are reached: (i) agents globally align their velocity vectors with a leader, (ii) they converge their velocities to the leaders velocity, (iii) collisions among interconnected agents are avoided, and (iv) agent's artificial potential functions are minimized. We model the interaction and/or communication relationship between agents by algebraic graph theory. Stability analysis is achieved by using classical Lyapunov theory in a fixed network topology, and differential inclusions and nonsmooth analysis in a switching network topology respectively. Simulation examples are provided.     

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.     

Almost periodic dynamics of the delayed complex-valued recurrent neural networks with discontinuous activation functions

The target of this article is to study almost periodic dynamical behaviors for complex-valued recurrent neural networks with discontinuous activation functions and time-varying delays. We construct an equivalent discontinuous right-hand equation by decomposing real and imaginary parts of complex-valued neural networks. Based on differential inclusions theory, diagonal dominant principle and nonsmooth analysis theory of generalized Lyapunov function method, we achieve the existence, uniqueness and global stability of almost periodic solution for the equivalent delayed differential network. In particular, we derive a series of results on the equivalent neural networks with discontinuous activation functions, constant coefficients as well as periodic coefficients, respectively. Finally, we give a numerical example to demonstrate the effectiveness and feasibility of the derived theoretical results.

     

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.     

Stability results for the solutions of certain fourth-order autonomous differential equations

Lyapunov functions have been derived for various authors to study the stability of solutions for certain fourth-order non-linear autonomous differential equations. Cartwright (1956) has studied (1) and (2) applying the method of analogy with linear systems, Ku (1964) has proposed a matrix method to study (3) and Skidmore (1966) has derived another Lyapunov function for (1) applying an extension to the method proposed by Leighton (1963). The intrinsic method proposed by Chin (1988 a, b) is applied to derive the Lyapunov functions obtained by Cartwright, Ku and Skidmore. Further, new Lyapunov functions are derived for (1), (2) and (3) applying the intrinsic method that are shown to be far less restrictive than those obtained by the various authors in the literature.     

Robust stabilization of the Takagi-Sugeno fuzzy model via bilinearmatrix inequalities

Quadratic stability has enabled, mainly via the linear matrix inequality framework, the analysis and design of a nonlinear control system from the local matrices of the system's Takagi-Sugeno (T-S) fuzzy model. It is well known, however, that there exist stable differential inclusions, hence T-S fuzzy models whose stability is unprovable by a globally quadratic Lyapunov function. At present, literature in the broader area of stability analysis suggests piecewise-quadratic stability as a means to avoid such conservatism. This paper generalizes the idea and proposes a framework that supports less conservative sufficient conditions for the stability of the T-S model by using piecewise-quadratic generalized Lyapunov functions. The advocated approach results in the formulation of the controller synthesis, which, herein, aims for robust stabilization, as a problem of bilinear rather than linear matrix inequalities. Simulation studies, which include an algorithm for solution of bilinear matrix inequalities, demonstrate the proposed method     

On formalism and stability of switched systems

In this paper,we formulate a uniform mathematical framework for studying switched systems with piecewise linear partitioned state space and state dependent switching.Based on known results from the the...     

灰色随机线性时滞系统的渐近稳定性

首先提出了灰色随机线性时滞系统及其渐近稳定性的概念;然后,利用矩阵理论和随机微分时滞方程解的渐近收敛定理及李雅普诺夫函数,研究了灰色随机线性时滞系统的渐近稳定性,得到了随机淅近稳定的几个充分性条件;最后,通过数值例子说明了所得结果在实际应用中的方便性和有效性.     

General Solution of Stability Problem for Plane Linear Switched Systems and Differential Inclusions

Characterization and control of stability of switched dynamical systems and differential inclusions have attracted significant attention in the recent past. The most of the current results for this problem are obtained by application of the Lyapunov function method which provides sufficient but frequently over conservative stability conditions. For planar systems, practically verifiable necessary and sufficient conditions are found only for switched systems with two subsystems. This paper provides explicit necessary and sufficient conditions for asymptotic stability of switched systems and differential inclusions with arbitrary number of subsystems; these conditions turned out to be identical for the both classes of systems. A precise upper bound for the number of switching points in a periodic solution, corresponding to the break of stability, is found. It is shown that, for a switched system, the break of stability may also occur on a solution with infinitely fast switching (chattering) between some two subsystems.     

Antiperiodic dynamical behaviors of discontinuous neutral-type Cohen-Grossberg neural networks with mixed time delays

This article presents a class of neutral-type Cohen-Grossberg neural networks with discontinuous activations and mixed time delays. Based on the functional differential inclusions theory, inequality technique, and nonsmooth analysis theory with Lyapunovlike approach, some new sufficient criteria are given to ascertain the existence, uniqueness, and globally exponential stability of the antiperiodic solution. Finally, two topical numerical examples and the corresponding computer simulations are delineated to substantiate the correctness of our theoretical predictions. The obtained results of this article are new and complement some related earlier works.     

Stability analysis of discontinuous quantum control systems with dipole and polarizability coupling

Closed quantum systems under the influence of a laser field, whose interaction is modeled by a Schrödinger equation, with a coupling control operator containing both a linear (dipole) and a quadratic (polarizability) term, are analyzed. Discontinuous feedbacks, obtained by a Lyapunov trajectory tracking procedure, have recently been proposed to control these types of system. The purpose of this paper is to study the asymptotic stability by considering the solutions in the Filippov sense. The analysis is developed by applying a variant of the LaSalle invariance principle for differential inclusions. Numerical simulations are included to illustrate the efficiency of the discontinuous control.     

DYNAMIC POSITIONING CONTROL OF SHIPS VIA RELAY OBSERVER DESIGN

A dynamically positioned ship maintains its position (fixed location or predetermined track) exclusively by means of active thrusters. In this paper, a control scheme for dynamic positioning control of ships using a relay observer design is presented. Our analysis relies on nonsmooth strict Lyapunov functions to demonstrate global asymptotic stability of the closed‐loop system equilibrium. Using simulation results we illustrate the good performance of the proposed scheme under noisy ship position measurements.     

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．     

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.