Stochastic Systems--Stability, Control, Estimation and Robustness

A large class of physical systems have variable structures subject to random changes, which may result from the abrupt phenomena such as component and interconnection failures, parameters shifting, tracking, and the time required to measure some of the variables at different stages. Systems with this character may be modelled as hybrid ones, that is, to the continuous state variable, a discrete random variable called the mode, or regime, is appended. The mode describes the random jumps of the system parameters and the occurrence of discontinuities. One of the most important hybrid systems is the so-called Markovian jumping system (MJS), in which the mode-process is a continuous-time discrete-state Markov process taking values in a finite set. On the other hand, the dynamics of many control systems are described by high-order differential equations. However, the behaviour is governed by a few dominant parameters, with a relatively minor role being played by the remaining parameters such as small time constants, masses, moments of inertia, inductances, and capacitances. The presence of these “parasitic” parameters is often the source for the increased order and the ``stiffness’' of the systems. Singularly perturbed systems (SPS) are those whose order is reduced when the parasitic parameter is neglected. Both MJS and SPS have been widely used in electric power modelling, in control of a solar thermal central receiver, in battle management command, control, and communication systems, in armature controlled DC motor, and in electronic RC circuit design.

There are subtle difficulties which need to be overcome to solve the control and estimation problems for singularly perturbed nonlinear systems with Markovian jumping parameters. In particular: 1) The time-scale structure for such systems is much richer than that of a singularly perturbed deterministic system. This is because the time-scale separation for each of the configurations of the continuous-time system may not, in general, share a common structure, i.e., the system may have different numbers of fast modes corresponding to different configurations, and the fast states may be entirely different for different configurations. 2) The analysis of asymptotic behaviour of controllers and estimators becomes significantly more complicated due to random mode changing and the modelling of uncertainty, since the convergence rate of singular perturbations may be different at each mode.

Accordingly, in this research project, we shall systematically study the problems of control and estimation of singularly perturbed nonlinear Markovian jumping systems. The linkage between general singularly perturbed systems and Markovian jumping systems will be established. A characterization of such systems will be given when singular perturbation tends to zero and time tends to infinity. In particular, for systems containing parametric uncertainty, robust controllers and estimators will be designed, which should be independent of both uncertainties and singular perturbations in certain interval.

For more information please contact Peng Shi.