Juanjuan Xu

Linear Quadratic Regulation and Stabilization of Discrete-Time Systems With Delay and Multiplicative Noise

This paper is concerned with the long-standing problems of linear quadratic regulation (LQR) control and stabilization for a class of discrete-time stochastic systems involving multiplicative noises and input delay. These fundamental problems have attracted resurgent interests due to development of networked control systems. An explicit analytical expression is given for the optimal LQR controller. More specifically, the optimal LQR controller is shown to be a linear function of the conditional expectation of the state, with the feedback gain based on a Riccati-ZXL difference equation. It is also shown that the system is stabilizable in the mean-square sense if and only if an algebraic Riccati-ZXL equation has a particular solution. These results are based on a new technical tool, which establishes a non-homogeneous relationship between the state and the costate of this class of systems, and the introduction of a new Lyapunov function for the finite-horizon optimal control design.

Input delay margin for consensusability of multi-agent systems

In this paper, we are concerned with the consensus of multi-agent systems with input delay. Among all standard static protocols that achieve the consensus for the multi-agent system under no input delay, we aim to find the maximum input delay such that the system remains consensusable under the same protocols. In the case of continuous-time systems, in view of the continuity of stability with respect to the time delay, the maximum delay margin for consensusability is given for scalar systems and vector systems with a single unstable open-loop pole. For scalar discrete-time systems, we show that the maximum delay margin for consensusability is strictly greater than zero if and only if the open-loop pole of the system is located in a specified interval.

Stochastic Approximation Approach for Consensus and Convergence Rate Analysis of Multiagent Systems

In this note, we study the consensus problem for multiagent systems with measurement noises. Different from the existing approach, the consensus problem is converted to a root finding problem for which the stochastic approximation theory can be applied. By choosing an appropriate regression function, we propose a consensus algorithm which is applicable to systems with more general measurement noise processes, including stationary autoregressive and moving average (ARMA) processes and infinite moving average (MA) processes. Further, we establish a relationship between the convergence rate and the exponent of the step size of the algorithm. Particularly, strong convergence rate for systems with a leader-follower topology is studied.

Necessary and Sufficient Condition for Two-Player Stackelberg Strategy

This technical note revisits the open-loop Stackelberg strategy for a two-player game. By introducing a new costate, which captures the future information of the control input, we present a necessary and sufficient condition for the existence and uniqueness of the two-player game. The optimal strategy is designed in terms of three decoupled and symmetric Riccati equations which improves the existing results greatly on computation.

Control for Itô Stochastic Systems With Input Delay

This paper examines the long-standing problem of linear quadratic regulation and stabilization for Itô stochastic systems with input delay. This problem remains a primary challenge because the separation principle does not hold for Itô stochastic systems. This paper presents a complete solution to the problem: 1) The (sufficient and necessary) solvability condition of the optimal control and the analytical controller are given based on the modified Riccati differential equation defined herein. 2) The sufficient and necessary stabilization condition in mean square sense is explored. We show that the Itô stochastic system with input delay is stabilized if and only if the modified algebraic Riccati equation developed in this paper has a unique positive-definite solution. The essential obstacle encountered in this paper concerns a Delayed Forward-Backward Stochastic Differential Equation (D-FBSDE), which is mathematical challenging.

Stochastic control for Itô system with state transmission delay

This paper is concerned with the fundamental problem of stochastic control for Itô system with state transmission delay. The optimal controller is firstly given based on the Maximum Principle and the relationship between the state and costate. A sufficient and necessary stabilizing condition is presented for the stochastic system with delayed state. The analytical controller is given in terms of the conditional expectation of the state and the solution to a coupled nonlinear equation developed in this paper.

General linear forward and backward Stochastic difference equations with applications

Abstract In this paper, we consider a class of general linear forward and , backward stochastic difference equations (FBSDEs) which are fully coupled. The necessary and sufficient conditions for the existence of a (unique) solution to FBSDEs are given in terms of a Riccati equation. Two kinds of stochastic LQ optimal control problem are then studied as applications. First, we derive the optimal solution to the classic stochastic LQ problem by applying the solution to the associated FBSDEs. Secondly, we study a new type of LQ problem governed by a forward–backward stochastic system (FBSS). By applying the maximum principle and the solution to FBSDEs, an explicit solution is given in terms of a Riccati equation. Finally, by exploring the asymptotic behavior of the Riccati equation, we derive an equivalent condition for the mean-square stabilizability of FBSS.

Solution to stochastic LQR problem with multiple inputs

This paper considers the stochastic linear quadratic regulation (LQR) problem for Itô stochastic systems with multiple input controllers. The explicit controllers are given in terms of two Riccati equations by introducing one new costate and establishing the homogeneous relationship between the state and the new costate. More importantly, it is more computation saving for the derived Riccati equations than the one derived by augmentation technique.