Optimal Filtering And Control For Linear Systems With Time-varying Delay

In the past decades,considerable efforts have been devoted to the study of the fil-tering and control problem of dynamic systems and many excellent results have been proposed. It is noticed that the most of those works and the references therein mainly con-centrated on the delay-free or delay-constant cases.However, for a large class of practical applications, time delay cannot be neglected and may occur in a time varying way. For example,the information transmission process in a wireless sensor network. This thesis will investigate the optimal filtering and control problem for systems with time-varying delay, and aims to proposed efficient sate estimation approaches and control strategies by applying innovation analysis and estimation theory in undefinite linear space.This work mainly focus on the following four topics:1.Kalman filtering for systems with time-varying delay. Firstly, under the assump-tion that the time-varying delay is bounded, the single time-varying delayed observation is firstly redescribed as an equivalent observation with multiple constant delays by defining a binary variable to model the arrival process of the observations.Then, by constructing certain multiple channel observation sequence that contains the same amount of informa-tion as the original one,the proposed problem is converted into the projection of the state onto the linear space spanned by the observations from the multiple channels.However, duo to the transformation of the system, the covariance matrices of the innovations maybe singular, which is much different from the delay-free or delay-constant case,where the gain formula is always solvable and the solution of the filter gain is unique.The solvabil-ity of the gain formula and the uniqueness to the Riccati equations are discussed under the time-varying case.The analytical solution to the proposed problem is obtained in terms of the solutions to the singular Riccati equations.2.H∞filtering problems are investigated by applying the linear estimation theory in undefinite space and the innovation analysis approach.Analytical solutions to these probelms are obtained in terms of the Riccati and matrices differential/difference equa-tions.Moreover, a partial differential equation approach is developed to the proposed H∞filtering problem in terms of the solution of a partial differential equation with boundary conditions.The solution clearly demonstrates the infinite dimensional nature of the prob- lem. Note that the partial differential equations are nonlinear and an analytical solution to these equations may not be possible even if it exists.However, they may be solved by numerical methods such as the finite element method3. Linear quadratic regulator for systems with time-varying delays.It should be noted that control of systems with delays in multiple input channels has attracted a lot of studies in the past years.However, existing works only address systems with a single delay in each individual channel.For the case where there exist multiple delays for a single input channel, the associated control problems are challenging and have not been solved. In this thesis, under the assumption that the time-varying delay is bounded, the the system model with time-varying delay is first transformed into a special model with single input multiple delays.Then an associated backward stochastic system is introduced. By establishing a duality between the LQR problem and the estimation for an associated backward stochastic system, the proposed problem is solved by using the Kalman filtering in undefinite linear space.4.Based on the results of the LQR problem for systems with time-varying delay,the finite horizon H∞control for discrete-time systems with time-varying delay in control and exogenous input channels is investigated. Under the assumption that time-varying delay is bounded, the problem is first transformed into the LQR problem in Krein space.Then the problem is solved by using the duality principle between the H∞control problem and smoothing estimation problem for an associated system without delays.The causal and strictly causal solutions to the proposed problem are derived. The solvability condition is given in terms of the solution of one Riccati difference equation of the same order as the original system.