Time-Varying And Adaptive Feedback Control Of Uncertain Nonlinear Systems

Feedback control of uncertain nonlinear systems is a core field of control the-ory research, which has received sustained attention of many scholars. On the one hand, nonlinearities and uncertainties are inevitable in the practical mathemati-cal models, which have important influences on the control effect. Hence, system nonlinearities and uncertainties must be effectively handled, which would certainly bring great challenges to the existing control theory and methods. On the other hand, with the development of science and technology, it is necessary to continu-ally improve the control accuracy and reduce the control cost. This requires that new control methods and analysis tools be developed for various uncertain non-linear systems, to provide theoretical foundations and technical supports to the achievement of new objectives. Therefore, it is of great theoretical and practical significance to investigate feedback control of uncertain nonlinear systems.In this dissertation, several effective feedback mechanisms are established by time-varying or adaptive technique, to compensate the coupled uncertainties. More general theorems on stability and convergence are presented, to develop new analysis method of stability, convergence and boundedness. Based on these, global stabilization or practical tracking via feedback is achieved for several classes of uncertain nonlinear systems. For details, the main contribution of this dissertation consists of the following two major parts:I. Feedback control with more delicate objective for uncertain non-linear systems(i) Control design with prescribed performance is considered for uncertain nonlinear systems. The systems under investigation possess unknown control di-rections and non-parametric uncertainties. Even so, the control objectives to be achieved are still refined, not only to achieve the basic performance on stabiliza-tion/tracking, but also to guarantee certain prescribed performance required in real applications. Motivated by the funnel control method, a time-varying frame-work is developed to effectively handle serious uncertainties in the systems and to successfully establish prescribed performance. Two control design schemes for the global stabilization with a prescribed convergence rate and the global practi- cal tracking with the tracking error evolving within a prescribed funnel are pro-posed for two representative classes of uncertain nonlinear systems, respectively. Moreover, by subtly choosing the funnel, the semiglobal practical tracking with prescribed maximum overshoot and the global fixed-time practical tracking can be achieved. (Chap 2 in the dissertation)(ⅱ) Global finite-time stabilization via time-varying output-feedback is in-vestigated for uncertain nonlinear systems. The systems in question possess the homogeneous in the bi-limit unmeasurable states dependent growth with the rate being an unknown continuous function of time, and hence, allow inherent non-linearities and serious parametric uncertainties/serious time-variations coupling to the unmeasurable states, which makes it impossible even to achieve asymp-totic stabilization of the systems by the existing control schemes. By time-varying technique and homogeneous domination approach, a new time-varying output-feedback strategy is established for the stabilization of the systems under inves-tigation. It is shown that, as long as the involved time-varying gain is chosen fast enough to overtake the serious parametric unknowns and the serious time-variations, the output-feedback controller designed renders the closed-loop system states globally converge to zero in finite time. (Chap 3 in the dissertation)(ⅲ) Stabilization and destabilization via time-varying noise is considered for uncertain nonlinear systems. By explicitly constructing a time-varying stochastic noise, super-exponential stabilization is achieved for nonlinear systems with serious parametric uncertainties. Then, for the more general systems simultaneously with serious time-variations, a time-varying stochastic noise, with the time-varying gain therein fast enough, is introduced to ensure that the perturbed systems are stable with a prescribed decay rate or unstable with a prescribed growth rate. These further illustrate the essential effects of stochastic noise on stability of nonlinear systems. (Chap 4 in the dissertation)Ⅱ. More general theory on stochastic stability and stochastic time-varying/adaptive output-feedback control(ⅰ) Global stabilization via time-varying output-feedback is considered for stochastic nonlinear systems. Essentially different from the related existing liter-ature, the systems allow serious parametric uncertainties coupling to the unmea-sured states. For this, a time-varying output-feedback compensation mechanism is established by time-varying technique,to effectively handle the serious para- metric uncertainties when stochastic factors exist. By proposing a time-varying high-gain K-filter, a time-varying strategy of global output-feedback stabilization is established for the systems under consideration. It is also shown that when serious time-variations exist as well, it suffices to find a fast enough time-varying gain for the control design scheme. (Chap 5 in the dissertation)(ii) Stochastic convergence and adaptive output-feedback control is investi-gated for stochastic nonlinear systems. A very general stochastic convergence the-orem is proposed for the status that stochastic universal adaptive output-feedback control keeps unsolved. The theorem doesn’t necessarily involve a positive-definite function of system states with negative-semidefinite infinitesimal, essentially dif-ferent from the existing stochastic LaSalle’s theorem, and hence provides more opportunities to achieve stochastic convergence. As a direct extension of the con-vergence theorem, a general version of stochastic Barbalat’s lemma is obtained. Sequently, by the stochastic convergence theorem proposed and the celebrated nonnegative semimartingale convergence theorem, an effective analysis method of stochastic convergence and boundedness is developed. Furthermore, the feasibility of stochastic universal adaptive output-feedback control is established, and global stabilization via adaptive output-feedback is achieved for a class of stochastic nonlinear systems with serious parametric uncertainties coupling to unmeasured states. (Chap 6 in the dissertation)(iii) Global stability and stabilization is investigated for more general stochas-tic nonlinear systems. Due to the absence of the conventional assumptions (e.g., Lipschitz condition), the stochastic nonlinear systems under investigation may have more than one weak solution. For this, the concepts of global stability in probability and global asymptotic stability in probability in the more general sense are first introduced to cover the stochastic nonlinear systems having more than one weak solution. Then, the generalized stochastic Barbashin-Krasovskii theo-rem and LaSalle theorem are established, which present the criterions of stochastic stability for more general stochastic nonlinear systems. Moreover, based on the generalized theorems, continuous output-feedback and state-feedback stabiliza-tion are accomplished separately for two classes of high-order stochastic nonlinear systems under rather weak assumptions. (Chap 7 in the dissertation)...