Controller Design And Stability Analysis For High-order Stochastic Nonlinear Systems

The problems of control design for high-order stochastic nonlinear systems have been attracting more and more attention. From the view point of system model, high-order stochastic nonlinear systems represent a more general form of strict feedback stochastic nonlinear systems that have been extensively investigated. On the other hand, high-order stochastic nonlinear systems include a class of under-actuated, weakly coupled, mechan-ical systems, which is very important in practice. Therefore, the investigations of con-trol problems for high-order stochastic nonlinear systems have important theoretical and practical significance. However, the control problems can not be solved by feedback lin-earization method.This dissertation focuses on state-feedback and output-feedback controller design for several classes of high-order stochastic nonlinear systems. By using the method of integrator backstepping, homogeneous domination and some important inequalities, sta-bilizing state-feedback controllers and output-feedback controllers are designed respec-tively, and detailed design procedures of controllers are provided. In addition, by means of existence and uniqueness theory of stochastic differential equations and Lyapunov sta-bility theorem for stochastic nonlinear systems, the corresponding stability analysis are given. Numerical examples and simulations illustrate the advantages and effectiveness of the proposed approaches. The main contents of this dissertation are composed of the following five parts:Firstly, for a class of high-order stochastic nonlinear systems which are neither nec-essarily feedback linearizable nor affine in the control variables without the strict tri-angular conditions, the problem of state-feedback stabilization is investigated. Under some moderate assumptions, by developing a systematic design approach for stochastic nonlinear systems in the absence of strict triangular conditions, smooth state-feedback controllers are designed, which ensure that the closed-loop system has an almost surely unique solution on [0,∞), and the equilibrium at the origin of the closed-loop system is globally asymptotically stable in probability. Finally, an illustrative example is provided to demonstrate the effectiveness of the proposed control design methodology.Secondly, based on the ideas of the homogeneous system theory and integrator back-stepping technique, the problem of state-feedback stabilization is considered for a class of high-order stochastic nonlinear systems by relaxing the power order restriction com-pletely and growth condition to a more general form. A state-feedback controller is con-structed to ensure that the closed-loop system has an almost surely unique solution on [0,∞), the equilibrium at the origin of the closed-loop system is globally asymptotically stable in probability, and the problem of inverse optimal stabilization in probability is solved. The efficiency of the state-feedback controller is demonstrated by a simulation example.Thirdly, for a class of large-scale high-order stochastic nonlinear systems which are neither necessarily feedback linearizable nor affine in the control input, the problem of the decentralized state-feedback stabilization is investigated. Under some moderate assump-tions, smooth decentralized state-feedback controllers are designed, which ensure that the closed-loop system has an almost surely unique solution on [0,∞), the equilibrium at the origin of the closed-loop system is globally asymptotically stable in probability, and the problem of decentralized inverse optimal stabilization in probability is solved. The efficiency of the control scheme is demonstrated by a simulation example.Fourthly, the problem of output-feedback stabilization for a class of high-order stochastic nonlinear systems in which the diffusion terms depend on unmeasurable states besides the output is investigated. By introducing a new rescaling transformation, adopt-ing an effective observer and choosing the appropriate Lyapunov function, a smooth output-feedback controller is constructed to ensure that the equilibrium at the origin of the closed-loop system is globally asymptotically stable in probability, the output can be regulated to the origin almost surely, and the problem of output-feedback inverse optimal stabilization in probability is solved. The efficiency of the output-feedback controller is demonstrated by several numerical examples.Fifthly, Under the more general conditions on the power order and the nonlinear growth functions, the output-feedback stabilization problem for a class of more general high-order stochastic nonlinear systems is investigated. It is shown that, based on the in-tegrator backstepping design method and homogeneous domination technique, the power order restriction for high-order stochastic nonlinear systems is completely removed and the growth condition is largely relaxed. An output-feedback controller is constructed, which can ensure that the closed-loop system has an almost surely unique solution on [0,∞) and the equilibrium at the origin of the closed-loop system is globally asymptot-ically stable in probability. The efficiency of the control scheme is demonstrated by a simulation example.Finally, the results of the dissertation are summarized and further research topics are pointed out.