Nonsmooth Control System Design Based On The Solution Of Filippov's Differential Inclusion

In the field of theoretical study and the field of engineering practice, there existlarge numbers of dynamical systems that have nonsmooth dynamics. The nonunique-ness of the trajectories of nonsmooth systems will cause that it's not sufficient to handlethe problems of nonsmooth control systems in practice only through the control theorydeveloped for smooth control systems. Moreover, the current design methods are notso efficient to overcome the problems on the development of the nonsmooth systems.Thus, it's important and meaningful to improve and develop the methods to analyze anddesign the nonsmooth control systems from the viewpoint of control theory and controlengineering.For the control problems of the nonsmooth systems, this dissertation attempts to uti-lize the nonsmooth analysis as the mathmatical tool to make a discussion on the analysisand synthesis of the nonsmooth dynamical system which can be described in Filippovdifferential inclusion. Several topics, such as the motion analysis of nonsmooth dynami-cal systems, sliding mode control design with a nonsmooth Lipschitz switching surface,backstepping design method for nonsmooth Lipschitz feedback and the motion controlof Lagrange systems with saturation constraint are discussed as follows.Firstly, in order to analyze the motion of the trajectories of the nonsmooth dynami-cal systems that can be described by the solution of Filippov differential inclusion, severalcriteria are proposed to determine the relation between the trajectories and nonsmoothLipschitz surface, which are for three typical situations: no intersection, crossing thesurface, and sliding along the surface. The nonsmooth sliding motion on the switchingsurface is also analyzed through the methods of high-dimensional cross product methodsuch that the relation between the sliding motion and the nonsmooth region can be ob-tained.Secondly, the design problems on the sliding mode control with a nonsmooth linearLipschitz switching surface are considered. The cone criteria are considered as the newapproaching condition and the existence condition of sliding mode control. Accordingto the new criteria, the design method based on the global contingent cone condition andthe design method based on the non-global contingent cone condition are proposed tostabilize a class of single-input single-output nonlinear systems. In order to obtain thelinear Lipschitz switching surface of the sliding mode control, a method based on the combined switching surface is also proposed, since the linear switching surface of thesliding mode control may not be unique.Thirdly, for the stabilization and tracking control problems of a class of parametricstrict-feedback systems with uncertainties, the nonsmooth backstepping design methodis proposed to construct nonsmooth Lipschitz feedback controller for the second-ordersystems based on the crossing condition and the discontinuous design of adaptive algo-rithm. Moreover, this design method is systematically developed for the general para-metric strict-feedback systems such that the virtual control can be designed with respectto the regions in the state space and the ?exibility of the backstepping design method isimproved.Finally, motion control design of Lagrange systems with saturation constraint isdiscussed. The nonsmooth backstepping design method is employed to design nonsoothLipschitz feedback controller according to the regions that tracking error belongs to suchthat the different feedback gain can be obtained. Moreover, a sufficient condition isproposed to design the feasible patitioned regions and feedback gain such that the outputsaturation can be overcomed. Furthermore, the design method is applied to a class of2-degree-of-freedom manipulators. The simulation results show that the controller iseffective to overcome the output saturation and also has better performance.