Analysis And Control Of Stochastic Nonlinear Systems With Low-order Powers

Compared with linear system,nonlinear system can better describe the essence of the practical system,most control systems involved in many fields of contemporary science and technology are nonlinear.Considering the influence of controlled object,external environment and control strategy,these control systems are usually modeled as nonlinear systems with multiple uncertainties.Moreover,in practical engineering system,stochastic noise is almost everywhere,but there is no feasible general method applicable to stochastic nonlinear control systems except for a very few cases.Furthermore,the research on the control of stochastic nonlinear systems with power greater than or equal to one has been widely developed.However,due to the inherent continuous non-smoothness nature of the low-order system with power less than one,its control results are still scarce.Therefore,the research of stochastic nonlinear control system with low-orders has been a hot and difficult topic in the field of control theory and engineering.In this paper,for the sake of simplicity,stochastic nonlinear system with power less than one is straightforwardly called stochastic low-order system.In addition,based on the fact that the law of change of things usually depends on the current state and the state of the past period of time,it can be concluded that most actual systems have time delay.Hence,the control of stochastic nonlinear time delay systems has received much attention in nonlinear control.On the other hand,compared with stability and asymptotic stability,finite-time stability control has faster convergence,and more robustness to uncertainty.Therefore,it is important to further investigate the finite-time stability of stochastic nonlinear systems.In view of this,by using a combination of tools such as adding a power integrator approach,dynamic gain control,homogeneous domination method,stochastic finite-time stability theory,observer design and Lyapunov function construction,the controller design and theoretical analysis are carried out for stochastic low-order nonlinear systems with various uncertainties such as unknown control coefficients,time-varying delay,and stochastic inverse dynamics.The main contents include:1.The global stabilisation is studied for a class of continuous but nonsmooth stochastic low-order nonlinear systems with time-varying delays.A concise controller is firstly designed for the nominal system by combining the traditional backstepping control with homogenous domination approach,which overcomes the problem of calculating explosion produced by the diffusion and Hessian terms.Then,by means of an appropriate Lyapunov–Krasovskii functional and a design parameter,the negative effects of time delays and nonlinear terms generated in the controller design process are dominated;the global asymptotic stability of the closed-loop system can be ensured by the simple but effective controller,which provides significant cost savings.Further,the global continuous control problem is addressed for a class of stochastic low-order cascade nonlinear systems with time-varying delay and stochastic inverse dynamics.The requirement of nonlinear functions is relaxed from smooth to continuous,and all the traditional growth conditions on unknown drift and diffusion nonlinearities are removed,which largely extends the scope of application.A continuous control scheme consisting of a delay-independent partial state feedback controller and a serial of dynamic update laws is proposed to guarantee the globally asymptotical stability of the closed-loop system.2.The global stabilization issue is discussed for a class of stochastic low-order nonlinear time-delay systems suffered from unknown control directions and disturbance.Instead of designing the adaptive law online,by utilizing the homogenous domination approach and sign function,a control gain is invoked to adapt for the formidable scene of merely continuous but nonsmooth caused by the low-order,and to compensate the negative effects of disturbance and multiple unknown nonlinear functions where both input and state time delays are taken into account.Moreover,the global stabilization is investigated for a class of stochastic continuous nonlinear time-delay systems involved with unknown control coefficients and stochastic-input-to-state-stable-like conditions.A dominate gain is adopted to deal with uncertain nonlinearities,which not only include unmeasurable state but also involve input and state time delays.On account of homogeneous domination manner and stochastic stability theory,a delay-independent controller is developed to guarantee that the closed-loop system is globally asymptotically stable in probability.3.The finite-time control problem is considered for a class of stochastic low-order nonlinear systems with lower-triangular form.Firstly,a key lemma is introduced to remove the traditional assumptions of nonlinear terms,and a state feedback controller is designed to ensure the global finite-time stability of the closed-loop system by using backstepping method.Then,the finite-time output feedback control of stochastic nonlinear systems is further studied because the states of many systems are unmeasurable in practical engineering.When the system states are unmeasurable,a finite-time output feedback controller is designed by constructing a reduced order observer and control parameters to ensure the finite-time stability in probability of the closed-loop system.4.The problem of finite-time stabilization is addressed for a class of stochastic low-order upper-triangular nonlinear systems corrupted by unknown control coefficients.Unlike the traditional adaptive compensation schemes,the control strategy draws into a dominate gain to cope with the deteriorative effects of both uncertain nonlinearities and unknown control coefficients.Then,a state feedback controller is constructed by the adding a power integrator method and modified homogeneous domination approach,to ensure the finite-time stability of the closed-loop system.Furthermore,considering the influence of stochastic inverse dynamics,the finite-time control problem is considered for a class of stochastic low-order nonlinear systems suffered from unknown control coefficients and unmeasured stochastic inverse dynamics,in which the negative effects generated by unknown control coefficients and nonlinear functions are dominated by a control gain.Finally,the finite-time stability of the closed-loop system is analysed by means of finitetime stochastic input-to-state stability and small gain lemma.