State Estimation For Markovian Random Delayed Systems

It is well known that time-delays are frequently encountered in various prac-tical control systems, such as manufacturing systems, power systems, networked control systems, and etc. The aforementioned delays usually degrade the system performance, and are the source of potential instability, and even lead to the oc-currence of chaos phenomenon. So study on filtering problems with time delays is of great theoretical and practical significance. The state estimation for sys-tems with time delays can be traced back to the 1960's, while most of the existed works focused on the constant delayed systems. In fact, the time-delays occur in a random way, rather than a deterministic way, for a number of engineering applications, such as wireless sensor network systems, and etc. In such a random case, the overall system is no longer a deterministic model, and thus the filter design for the systems with random delays becomes more complicated.This thesis is devoted to the study of state estimation problems for several kinds of random delayed systems. It is assumed that the random delay is either modeled by a finite state Markov chain, or characterized by a multiple Bernoulli distribution model. Also, we will give the filtering design method for the Markov jump linear systems with constant observation delays. The key techniques applied for treating the delay terms are the observation reconstructed method, partial difference or partial differential method, and then different kinds of estimators are obtained by using the complete square method, the stochastic approximation method and the innovation analysis method. The main contents of this paper are list as follows:Firstly, we study the linear minimum mean square error estimation for Marko-vian jump linear systems with delayed measurements. For the discrete-time case, we first reconstruct a new delay free observation sequence which contains the same information as the original one. Then, we introduce a new state variable which is concerned on the original state and the Markov jumping parameters simulta-neously. The optimal filter on the new state variable is derived based on the innovation analysis method in the Hilbert space, and its analytical solution is ob-tained in terms of two Riccati difference equations. Finally, the estimator of the original state is obtained directly from the new state estimator via the Markov jumping properties. Also, the estimation problems for the continuous-time case are considered. Following the similar design procedure as in the discrete-time case, the optimal filter is obtained by using the innovation analysis method and the Ito differential rule, while its solution is given in terms of two differential Riccati equations.Secondly, we consider the optimal estimation problems for the dynamic sys-tems with random observation delays, where the delay process is modeled as a finite state Markov chain. On the estimation of discrete-time systems, two situ-ations are considered:the first one deals with the class of optimal Markov jump filters where the jump delay is assumed accessible; while in the second situation the jump delay is not accessible, and we derive the minimum linear mean square error filter. For both of these situations, the single random delayed measure-ment is firstly rewritten as the two channel constant delayed measurements with multiplicative noises, and further transformed into the delay-free ones via the ob-servation reconstruction technique. Then a set of new Markov chains is defined according to the new observations, which representing the same jumping proper-ties as the random delays. As a result, a standard Markov jump linear system is obtained. Then for the case that the jumping parameter is known, an opti-mal Markov jump filter is obtained by using the complete square method and the stochastic approximation method. The filter depends just on the present value of the Markov parameter, rather on the entire past history of the modes, and the so-lution to the filter is given by solving a set of coupled Riccati difference equations, which have the same dimension as the original systems. In the situation that the Markov chain is not observed, a linear minimum mean square error estimator is derived by using the innovation analysis method and an analytical solution to this estimator is presented via two generalized Riccati difference equations. On the estimation of continuous-time systems, also two situations are considered:For the case that the random delay is known on-line, an optimal Markov jump filter is developed, and for the case that the random delay is unknown, a linear minimum mean square error estimator is developed. But distinguishing from the discrete-time cases, the Ito differential rule needs to be employed in the continuous-time domain, and at last the solutions are given in terms of two kinds of differential Riccati equations.Finally, We investigate the estimation problems for systems with multiple random delays. For the discrete-time case, the optimal estimation for more general systems with random state and measurement delays is considered. The aim is to present a partial difference equation approach to the optimal estimation. By introducing a set of binary random variables, the system is firstly converted into the one with both multiplicative noises and constant delays. Then an estimator which includes the case of smoothing and filtering is derived via the projection formula, and the solution is given in terms of a partial difference Riccati equation with Lyapunov equations. Also, a predictor for such systems is presented based on the proposed filter and smoother. It can be found that the estimators have the same dimensions as the original system. For the continuous-time case, the optimal estimators including filter, predictor and smoother are developed in the linear minimum variance sense, and the solution is given in terms of partial differential Riccati equations. It should be pointed out that the partial differential equations are difficult to solve, and an analytical solution to these equations might not be possible. In general, the solution can be given in a numerical form by the approximation method.