Optimal linear quadratic Gaussian control for discrete time‐varying system with simultaneous input delay and state/control‐dependent noises

Abstract

This article focuses on the problem of linear quadratic Gaussian (LQG) control for discrete time-varying system with input delay and state/control-dependent noises. When the state variables can be exactly observed, first, we obtain the maximum principle by applying the method of variation. Second, a nonhomogeneous relationship between the state and the costate is developed in virtue of the obtained maximum principle and the mathematical induction. It is noted that the nonhomogeneous relationship is the solution to the forward and backward stochastic difference equations (FBSDEs). Finally, based on the solution to the FBSDEs, a necessary and sufficient condition for the optimal LQG control is derived in terms of coupled Riccati equations. Moreover, the analytical expression of the optimal controller is presented. The derived results are applied in networked control systems with packet dropout. Numerical examples are shown to illustrate the effectiveness of the proposed algorithm. K E Y W O R D SLQG control, discrete time-varying system, input delay, state/control-dependent noises INTRODUCTIONThe optimal linear quadratic Gaussian (LQG) control problem has received considerable attentions. 1-4 Generally, the standard LQG model contains the additive Gaussian white noises that are not state/control dependent. 5,6 With the rapid development of networked control systems (NCSs), the optimal control has been applied in various fields, such as vehicle industry, building automation surveillance, and Internet-based teleoperation, 7-11 which makes the control system models be more complicated. Obviously, the standard LQG model has no longer been applicable to the practical applications. Hence, we focus on the study of the LQG control systems with both state/control-dependent noises and input delay. It is generally known that the packet dropout occurred in NCSs can be described by the state/control-dependent noises. On one hand, many references have been investigated for LQG control with state/control-dependent noises. [12][13][14][15][16][17][18] Joshi 12 presented a suboptimal controller by using the "enforced separation principle." For NCSs with any arbitrary packet-drop pattern, Gupta et al 13 investigated the optimal LQG control by decomposing the problem into a standard LQR state-feedback controller, along with an optimal encoder-decoder design. Sinopoli et al 14 modeled the arrival of both observations and control packets as stochastic Bernoulli processes to describe the packet dropout. Liang and Xu 15 studied 882

Identification of dual‐rate sampled errors‐in‐variables systems with time delays

“…35,36 In network control systems, due to limitations of the network carrying capacity and the communication bandwidth, time delays are often introduced in transmission of signals. 37,38 Time delays may lead to oscillation and instability of the control systems, thus which should be accurately estimated. [39][40][41] The greedy algorithm has been widely used for identification of systems with unknown time delays, which has advantages of fast convergence rate and simple structure.…”

“…For example, in chemical process systems, due to economic considerations or technological difficulties, some key variables indicating the production states can only be analyzed offline in laboratories, resulting in long sampling periods and inevitable time delays 35,36 . In network control systems, due to limitations of the network carrying capacity and the communication bandwidth, time delays are often introduced in transmission of signals 37,38 . Time delays may lead to oscillation and instability of the control systems, thus which should be accurately estimated 39‐41 .…”