Finite-Time $H_{\infty}$ Filtering for Nonlinear Stochastic Systems With Multiplicative Noises via Carleman Linearization Technique

In this article, the problem of finite-time $H_\infty$ filtering is dealt with for general nonlinear stochastic systems with multiplicative noises. The nonlinear system under investigation is not only disturbed by state-dependent noises, but also corrupted by external disturbance. By utilizing the Carleman linearization technique, a linear parameter-varying system related to state estimation is obtained and a polynomial nonlinear filter is constructed for nonlinear stochastic systems. Then, a sufficient condition in terms of parameter-dependent linear matrix inequalities (PDLMIs) is established to guarantee the finite-time boundedness and certain $H_\infty$ performance of the filtering error dynamics. Moreover, the parameters of the polynomial nonlinear filter are derived by solving PDLMIs via the sum of squares decomposition approach. Finally, the effectiveness of the developed filtering scheme is demonstrated by two examples, with one concerning the rotary steerable drilling tool system.