Research On Control Design Methods Via Nonsmooth Lipschitz Surfaces

Nonsmooth systems arise in a large number of nature and physical science. In a control science point of view, there exist large numbers of dynamical systems with nonsmooth dynamics in the fields of engineering applications and theoretical studies. No matter the system is nonsmooth itself or is drived by a nonsmooth control input, the resulting system is called nonsmooth control system. The issues have received more and more attentions recently in control science and system science. It is important and theoretically meaningful to improve and develop nonsmooth control design methods in the framework of nonsmooth systems theory. Based on the relation between the trajectories of the closed-loop system and Lipschitz surfaces in the state space, the dissertation attempts to make a discussion on the analysis and synthesis of nonsmooth dynamical systems which can be described in the sense of Filippov differential inclusion, utilizing nonsmooth analysis and nonsmooth stability analysis as the mathematical tools. Several tops, such as switched control design, variable structure control design with sliding mode and on-off control design, are discussed as follows.Firstly, the switched control design problem with nonsmooth Lipschitz surfaces is investigated for a class of linear systems with parameter uncertainties. A series of Lipschitz domains with Lipschitz surfaces as their boundaries are constructed in a recursive way using the notion of self-stable region. The equilibrium is discussed, and the convergence of the trajectories to the equilibrium in the interior of the Lipschitz domain is studied utilizing contingent cone criteria. In addition, taking the boundaries of the Lipschitz domain as switching surfaces, a switched control design method is proposed to drive the trajectories entering the interior of the Lipschitz domain in finite time and maintaining in it, such that the global stability is achieved.Secondly, the sliding mode control design problem with nonsmooth Lipschitz surfaces is investigated for a class of nonlinear systems with uncertainties and perturbations. Contingent cone criteria are used as new approaching condition and existence condition of sliding mode. According to the new conditions, a sliding mode control design method based on the global contingent cone criteria and a sliding mode control design method based on the non-global contingent cone criteria, which is considered to reduce the conservative property, are proposed to drive the trajectories approaching the Lipschitz surface in finite time and sliding on it. In order to obtain the Lipschitz switching surface of the sliding mode control, a class of linear uncertain systems is considered, and a sliding mode control design method based on self-stable region is proposed. The stability of the new switching surface is analyzed and its properties are illustrated by comparing with some existing methods.Thirdly, the on-off control design problem with nonsmooth Lipschitz surfaces is investigated for a class of linear controllable systems with on-off control input. The new nonsmooth Lipschitz on-off switching surfaces are constructed by means of combination of finite alternative smooth subsurfaces. The equilibria set of the closed-loop system is discussed. The on-off control design method is proposed and the glaoblly asymptotic stability is analyzed utilizing LaSalle's invariant principle of nonsmooth system. The switching surfaces can be designed with respect to the domains in the state space and the flexibility of the on-off control design can be improved.Finally, the on-off guidance law design problem with nonsmooth Lipschitz thresholds of divert thrusters for a certain type of exo-atmospheric interceptror is investigated. The Lipschitz threshold is constructed by means of combination of finite linear surfaces, considering the characteristics of the relative motion between the interceptror and the target. The stability of the guidance system is analyzed by utilizing contingent cone criteria. By comparing with some existing methods with smooth thresholds, the effectiveness and advantage of the proposed method are illustrated by some numerical simulations.