Research On Stability Analysis And Control Problems Of Random Nonlinear System

In real systems,stochastic noises are prevalent and they profoundly affect the dynamics of the system.In addition,stochastic systems can more accurately reflect the dynamic characteristics of natural,social and engineering systems.This implies that modeling based on stochastic systems can improve the accuracy of system modeling and thus further enhance the control performance of the system.Therefore,it is crucial to study stochastic systems from both theoretical and practical perspectives.Moreover,the rapid development of science and technology has placed higher demands on the performance of real systems.These systems are required to exhibit faster convergence,have excellent robustness and immunity to interference,and satisfy constraints on the output or state.These demands have motivated the emergence of constraint control,finite-time stabilization control,and fixed-time stabilization control as current research hotspots.In this thesis,we focus on stochastic systems and investigate in depth the stochastic input-to-state stability(SISS)and stochastic integral input-to-state stability(SiISS)and their associated stabilizing control problems.Our main goal is to provide more effective and innovative analytical tools and methods than those based on finite-time stochastic input-to-state stability(FT-SISS)to provide strong support for finite-time stability analysis or,more desirably,fixedtime stability analysis.Additionally,we further study the state-feedback and output-feedback control problem for stochastic high-order nonlinear systems with output constraints and the exponential stabilization problem for stochastic feedforward nonlinear systems with unknown growth rates.Through these studies,we expect to provide new perspectives and more innovative theoretical approaches to the stability analysis and stabilization problems of stochastic systems.This will help to further improve and develop the theory of stochastic systems and provide a solid theoretical foundation for engineering practice and social applications.Ⅰ.An in-depth study of finite-time stochastic input-to-state stability:concepts,methods,and applications1.In the framework of FT-SISS,the finite-time stabilization problem for stochastic nonlinear systems with stochastic inverse dynamics and unknown time-varying orders is investigated.By combining the technique of adding a power integrator and the general stochastic finite-time stability theory,a new finite-time control design and analysis method is proposed.This method guarantees that the closed-loop systems exists a continuous solution,all closed-loop signals are almost surely bounded,and the trivial solution of the closed-loop system is stochastically finitetime stable.The study results can be applied to address the finite-time stabilization problem for a class of liquid-level system.2.In the framework of FT-SISS,the finite-time state-feedback control problem for stochastic high-order nonlinear systems with stochastic inverse dynamics and output constraints is studied.In order to deal with the output constraints,we introduce a nonlinear transformation to transform the original system into an unconstrained equivalent one.On this basis,a statefeedback controller is designed by carefully analyzing the intrinsic properties of the system’s nonlinearities and combining the technique of adding a power integrator and homogeneous domination.As a result,we can prove that the closed-loop system has a continuous solution,the output constraint is not violated almost surely,and the trivial solution of the closed-loop system is stochastically finite-time stable.This research not only overcomes the limitations of traditional methods,but also provides a novel idea and scheme for solving the stabilization problem of stochastic nonlinear systems with output constraints.The study results can be applied to address the stabilization problem for a class of underactuated system with weak coupling.3.Based on FT-SISS,we establish a more comprehensive finite-time stability analytical framework,namely finite-time stochastic integral input-to-state stability(FT-SiISS).This investigation encompasses the following aspects:(ⅰ)We introduce the concept of FT-SiISS and rigorously prove that this concept includes FT-SISS.Consequently,FT-SiISS stands as the most advanced and comprehensive finite-time stability paradigm to date.Moreover,we clarify the relationship between FT-SiISS and other stochastic stability concepts,and this theoretical breakthrough lays a solid foundation for our comprehensive understanding of finite-time stability of stochastic systems.(ⅱ)We construct two FT-SiISS small-gain conditions and a novel finitetime stability theorem.These theoretical results not only address existing gaps in the current theories,but also provide important theoretical support for the finite-time stability analysis of interconnected stochastic systems exhibiting FT-SiISS properties.(ⅲ)Based on the above theoretical results,we design an unified finite-time control scheme for stochastic nonlinear systems with FT-SiISS or FT-SISS inverse dynamics.4.Based on FT-SISS,we develop a more optimal analytical framework,namely,fixed-time stochastic input-to-state stability(SFT-ISS).Our research encompasses the following aspects:(ⅰ)We introduce the concept of SFT-ISS for the first time.This new concept has the following three key properties.First,the corresponding stochastic system is stochastic fixed-time stable when the external input of the system is zero.Second,this definition provides a key robustness feature,i.e.,for any almost surely bounded input,the resulting solution of the stochastic system is almost surely bounded as well.Third,one of the significances of this definition lies in its Lyapunov characterization.Specifically,any stochastic system equipped with an SFT-ISS Lyapunov function is SFT-ISS.This property provides an effective tool and method for stochastic systems.(ⅱ)Based on the concept of SFT-ISS,we develop a series of powerful analytical tools.First,we introduce two SFT-ISS small-gain conditions and explore their relationships.Second,we propose a new fixed-time stability theorem for stochastic interconnected systems with SFT-ISS properties.(ⅲ)By applying these theoretical results,we successfully solve the fixed-time stabilization problem for strict-feedback stochastic nonlinear systems with SFT-ISS inverse dynamics.Ⅱ.Feedback control with more delicate objective for stochastic nonlinear systems5.For stochastic high-order nonlinear systems with output constraints,we further investigate the output-feedback control problem.In order to deal with the output constraint,we introduce a new coordinate transformation to transform the original system into an unconstrained equivalent system.On this basis,by using the adding a power integrator and homogeneous domination techniques,and exploiting the nonlinear properties of the system,we establish two new design and analysis frames of output feedback stabilization to achieve respectively stochastic asymptotic stability and stochastic finite-time stability of the trivial solution of the closed-loop system while guaranteeing the requirements of almost sure boundedness of the closed-loop signals and the symmetric output constraint.The study results can be applied to address the output-feedback control problem for a class of mass-spring mechanical system.6.The problem of exponential stabilization of output feedback for stochastic feedforward nonlinear systems is investigated.Essentially different from the related existing literature,the systems allow serious parametric uncertainties coupling to the unmeasured states.To this end,we design an adaptive output feedback controller by introducing an innovative dynamic gain.This controller not only handles the severe parameter uncertainties present in the system,but also ensures the exponential convergence of the system.By means of advanced stochastic analysis techniques and combined with the nonnegative seminartingale convergence theorem,we prove that the closed-loop system has a unique solution,all closed-loop signals are almost surely bounded,and the solution of the closed-loop system tends to the equilibrium position exponentially fast.Finally,we demonstrate the effectiveness of this method by applying it to a liquid level control resonant circuit system.