Study On Optimal Estimation And Control For Stochastic Systems With Time Delay And Multiplicative Noises

This paper is concerned with the optimal estimation,control and its applications in the networked control of the stochastic system with time delay and multiplicative noise.It is mainly on the linear optimal estimation of the system with random delay,networked control system with time delay and packet dropout,the linear quadratic Gaussian(LQG)control problem of the stochastic system with multiplicative noise and time delay,the control problem of the system with packet dropout and multiple controllers and the linear quadratic(LQ)problem of the system with multiple delays and multiplicative noise.The main contributions and innovations are as:Firstly,we consider the linearoptimal estimation problem of the networked control system with both random delay and packet dropout.Without time-stamping,the linear optimal estimation is derived for the first time.By virtue of a new observation equation and the transformation of the ARMA model to state-space model,we transform the measurement equa-tion with random delay into the standard system with merely additive noise.Using Kalman filtering,the linear optimal estimator is obtained.Secondly,we consider the optimal control and stabilization problem of the networked control system with both time delay and packet dropout.The explicit solutions of the optimal estimation and controller are obtained for the finite-horizon case.The necessary and sufficient condition of the stabilization is derived for the infinite-horizon case.In addition,the eigenvalues of the system matrix and packet dropout determine the stabilization condition,which is uncorrelated with input delay.Thirdly,we consider the LQG control problem of the system with input delay and multiplicative noise.When the state variables can be observed exactly,the explicit solution of the LQG controller is obtained for the first time.The necessary and sufficient condition for the existence of the optimal controller is derived.When the state variables are partially observed,the suboptimal linear state estimate feedback controller is obtained.Fourthly,we con-sider the optimal control and stabilization problem of the system with packet dropout and multiple controllers.Applying the maximum principle,the necessary and suf-ficient condition for the existence of the optimal controller is derived.It is the first time to show the necessary and sufficient condition for the stabilization problem.In the research of the stabilization problem,a new method in this paper is provided for the study of other stabilization problem.Fifthly,we consider the LQ problem of the system with multiple state delays or input delays and multiplicative noise.Different from the previous references,the control weight matrix can be positive semi-definite and we obtain the explicit solution of the optimal controller.The main contents are listed as follows in the order of chapters:1.We consider the linear estimation problems for networked control system-s where the measurements are subject to random time delay and packet dropout.Without time-stamping,a new observation is introduced by the summation of all the measurements received in the same time.Then the random time delay measurement system is converted into a constant-delay measurement system with multiplicative noises where the noise is binary distributed random variables with known distribu-tions.Finally,the linear optimal estimator is derived by using the transformation of ARMA model to state-space model and the standard Kalman filtering.The conver-gence and stability of the filter are also analyzed.2.We study the control problem for discrete-time networked control systems where packet dropout and input delay occur simultaneously.Firstly,we obtain the optimal LQR controller which is a linear function of the optimal state estimator,with the feedback gain based on a standard difference Riccati equation.It is noted that the state estimator and the controller can be calculated separately.Secondly,a necessary and sufficient condition for the mean-square stabilization is derived.It should be stressed that the eigenvalues of the system matrix and the packet dropout probability determine the stabilizing condition,which is uncorrelated with input delay.3.We investigate the LQG control problem for discrete-time systems with both multiplicative noises and input delay.Firstly,when the state variables can be ob-tained exactly,the optimal LQG controller is derived which consists of the feedback form of the conditional expectation of the state and one additive deterministic term based on the coupled Riccati equations.Through the completing square approach,the optimal controller is obtained based on the complete solution to the forward and backward stochastic difference equations.Secondly,when the state variables are par-tially observed,we derive a suboptimal linear state estimate feedback controller by linearizing the linear optimal estimator and using the obtained results of the optimal LQG control.4.We consider the problems of optimal control and stabilization for networked control systems(NCSs),where the remote controller and the local controller operate the linear plant simultaneously.Firstly,a necessary and sufficient condition for the finite horizon optimal control problem is given in terms of the two Riccati equations.Secondly,it is shown that the system without the additive noise is stabilizable in the mean square sense if and only if the two algebraic Riccati equations admit the unique solutions,and a sufficient condition is given for the boundedness in the mean square sense of the system with the additive noise.5.We stduy the LQ control problem for Ito stochastic systems with state delays or input delays.The main contribution is to give the explicit optimal LQ controllers for Ito stochastic systems with input delay and with state delay respectively.For the case of state delays,the optimal LQ controller is a linear function of the state at the current time and the past time.For the case of input delays,the optimal LQ controller is a linear function of the state and the past inputs.The key technology is to define the value function and complete the square based on the coupled differential equations.