Finite-Time Control For Switching Jump Systems

he traditional Lyapunov stability theory focuses on systems’ asymptotic stability in an infinite time interval. While, in practical engineering such as aerospace systems, robot control systems which acting in a short time interval, people are more interested whether the system meets certain transient performance requirements in a finite short interval except for the asymptotic stability in an infinite time interval, which means in the practical given time, the system’s states trajectory evoluting within an expected scope. This paper on the basis of finite time stability theory and considering a class of more general hybrid systems—switching jump systems, which contain both stochastic jump modes and deterministic switches, so the effect of stochastic transition probability and deterministic average dwell time on system performance are taken into account at the same time. The finite-time control problems under complex cases such as time delay, uncertain mode transition probability and bounded disturbance frequency are studied, the details are as follow:(1) Considering the discrete-time switching jump systems with time delay, the transient feature and H∞ disturbance attenuation performance in finite time interval are researched. First, by choosing the appropriate stochastic Lyapunov function and allowing it to increase at every switching instant with a limited increase rate, sufficient results on the stochastic finite-time boundness are derived, the coupling relationship among time-delay, average dwell time and given finite-time horizon of the underlying system is also discussed. Then the finite-time H∞ controller is designed based on finite-time boundedness.(2) The finite-time stabilization problem of continuous time switching jump systems with imprecisely known transition rates between jump modes is studied. First, the uncertainty of transition rates are described by a more general way, which some elements in the transition rate matrix are completely unknown, another is some elements are not known precisely but their bounds are known. Then the finite-time boundedness results are achieved by integrating the Markov processes between two adjacent switching time and combining with the average dwell time constraint condition. Finally the state feedback controller is also designed to guarantee the state trajectory of closed-loop system controlled within the expected range.(3) The Finite-time H∞ filtering for switching jump systems in finite frequency domain is researched. Considering the traditional H∞ filtering methods require systems have the same level of noise attenuation in full frequency may lead to more conservativeness, we propose a finite time based H∞ filter design method by considering the noise in specific frequency. The full frequency filtering result is given first, then the definition of finite frequency theory is introduced, so the finite frequency and finite time filtering scheme, named two dimensional filtering techniques is provided. Finally, a numerical simulation is offered to compare the two methods, which shows the proposed finite frequency filtering method has certain advantages.