The Stabilization And Observer Design Of Differential Inclusion Systems

By the progress of times and the development of science, the understanding of the nature isdeepened gradually, and the practical models adopted by the design of control systems are moreand more complicated. The uncertainty and discontinuity have to be considered if the accuracyis required, such as the aging of the components, the exogenous disturbance, the discontinuouschange of some activations of the systems and so on. It leads to the fact that it is very difficult todescribe this kind of systems by differential equation, to this end, we need to use differential inclu-sion to model and analyze the systems. This requirement promotes the research on the differentialinclusion systems from the view of control theory and control engineering. Thus, it is importantand meaningful to design and analyze the differential inclusion systems. This thesis focuses on thestabilization and observer design of several kinds of the differential inclusion systems. The maincontents and results obtained in this thesis are as follows:Chapter1is a survey. Firstly, it sums up the recent development of differential inclusion sys-tems and illustrates three hot topics of differential inclusion systems after the twenty-first century.Secondly, the differential inclusion systems considered and methods used in this thesis are intro-duced with specific problems. Then, the fundamental theory of differential inclusion and usefulmathematical tools are presented, which provide the basis for the further investigation. Finally, themajor work of the paper is summarized.Chapter2studies the stabilization and tracking control of the linear differential inclusion sys-tems by the convex hull Lyapunov function (CHLF). Firstly, the stabilization of a class of lineartime-variant differential inclusion systems subjected to disturbance is discussed. By using the Tay-lor expansion, the linear time-variant differential inclusion systems are transformed into the lineardifferential inclusions with affine uncertainty. Design objectives including stabilization, distur-bance rejection with bounded state and bounded L2gain are achieved. Secondly, the stabilizationof a class of the stochastic linear differential inclusion systems is studied. The Ito formula ofstochastic linear differential inclusion is presented and the feedback laws are designed to make theclosed-loop systems exponentially stable in mean square. Then, the tracking control of the lineardifferential inclusion systems is considered. The design objective is to make the tracking error sys- tems uniformly ultimately bounded and the accurate estimation of the ultimate bound is given. Atlast, the tracking control of stochastic linear differential inclusion systems is studied. The defini-tion of the uniform ultimate boundedness of stochastic linear differential inclusion in mean squareis introduced, feedback laws are designed to make the error systems uniformly ultimately boundedin mean square and the ultimate bound is also estimated accurately.Chapter3deals with the observer design for the Lur’e differential inclusion systems. Firstly,the reduced-order observers of the Lur’e differential inclusion systems are constructed by the de-composition technique. That the reduced-order observers have the same performance as the full-order observers is verified under the conditions of the existence of the full-order ones. Secondly,for a class of the Lur’e differential inclusion systems with unknown parameters, adaptive full-orderobservers are designed via the theory of the adaptive observers. Based on the full-order observers,the reduced-order ones are also presented. Finally, the linear observers for the Lur’e differentialinclusion systems are proposed. The designed observers hold evident advantage in both practicaland theoretical aspects because they do not contain any set-valued functions. Single-input-single-output systems are considered and the results are extended to multi-input-multi-output systems.For multi-input-multi-output systems, reduced-order observers are also designed.Chapter4investigates the stabilization of the discontinuous systems, the stability of theclosed-loop systems is analyzed by the Fillipov method, which can transform the discontinuoussystems into the differential inclusion systems. Firstly, based on the control Lyapunov function(CLF), the discontinuous systems are studied. After the definition of CLF, exponentially stableCLF and finite time stable CLF for discontinuous systems are introduced, the feedback laws aredesigned to make the closed-loop systems asymptotically stable, exponentially stable and finitetime stable respectively. Secondly, the stabilization of a class of the discontinuous time-delaysystems with unknown parameters is considered, the adaptive control theory is applied to the dis-continuous systems. For a special form of the systems, the adaptive feedback laws are designed tomake the closed-loop systems asymptotically stable. For a more general form of the systems, theadaptive feedback laws are also presented.Chapter5summarizes the work presented in this thesis and provides several problems whichare worth studying further.According to the specific kinds of the systems, the main contributions of this thesis can besummarized as the following three aspects:1. The stabilization and tracking control of the linear differential inclusion systems are stud-ied by the CHLF. The topics include the stabilization of the linear time-variant differential inclusionsystems, stabilization of the stochastic linear differential inclusion systems, tracking control of thelinear differential inclusion systems and tracking control of the stochastic linear differential inclu-sion systems. This work does not only extend the application of the CHLF, but also improves the theory of the linear differential inclusion systems.2. Based on the decomposition technique, the design scheme for the reduced-order ob-servers of the Lur’e differential inclusion systems is presented for the first time; By using thetheory of the adaptive observers, adaptive observers are constructed for a class of the Lur’e dif-ferential inclusion systems with unknown parameters; Based on the former work, a new observerdesign method for the Lur’e differential inclusion systems is proposed, the advantage of the ob-servers lies in that they do not include any set-valued functions and relax the restriction of theoriginal systems obviously.3. By using the CLF, the discontinuous systems are considered for the first time, and thestabilization and finite time stabilization for the discontinuous systems are solved; The adaptivecontrol method is applied to the discontinuous systems, and the stabilization of the discontinuoustime-delay systems with unknown parameters is investigated.