Research On Some Control Problems For Linear Systems With Time-Delay

This thesis is devoted to the study of the control problem with time delay. It is mainly on the linear quadratic control problem of Ito system with input delay, the consensus problem of multi-agent system with communication delay, the de-lay margin problem for consensus of multi-agent system with input delay, and the Stackelberg game with input delay.The main contributions are as:We obtain the necessary and sufficient condi-tions for both the existence and uniqueness of the optimal solution and the mean-square stabilizability of the stochastic system with input delay; We illustrate the mathematical derivation of the commonly used control protocol for the consensus of the multi-agent system, and give the maximum input delay margin for the consensus of the multi-agent system; We derive the necessary and sufficient condition for the existence of a unique Stackelberg strategy, and design the optimal controller based on three symmetric and decoupled Riccati equations which in fact greatly improve the nonsymmetric and coupled Riccati equations in the literature. We also extend this result to the game problem with time delay.The main innovations are as:It is first that the theoretical technique is proposed to solve a class of stochastic forward-backward differential equation with time delay, and then the necessary and sufficient solvability conditions for the unique solution of the stochastic control problem with time delay and equivalent conditions of the mean-square stabilizability of the stochastic system are firstly obtained based on the establishment of the explicit relationship between the optimal co-state and the state; It is first that the mathematical derivation of the consensus protocol, the convergence rate, the consensus with communication delay and the delay margin of consensus are studied for the multi-agent system by applying the stochastic approximation theory and the control theory of time-delay system; It is first that the game problem with time delay is considered by introducing new co-states and new state. The main contents, results and innovations are listed as follows in the order of chapters:1. We study the stochastic control problem with time delay. Using the vari-ational method, a novel and intuitive proof is stated for the stochastic maximum principle. Based on the maximum principle, we establish the linear and nonhomo-geneous relationship between the optimal co-state and the state, and then obtain the necessary and sufficient solvability conditions of the problem. Together with the complete square method, the explicitly optimal solution is given in terms of Riccati equation and linear partial differential equation. Furthermore, for the stochastic op-timal control problem in infinite horizon, we obtain the necessary and sufficient con-dition for the mean-square stabilizability of the stochastic system and the explicitly optimal controller. It is shown that the derived results break through the traditional control theory method. The main innovation is the theory of solution of stochastic forward and backward differential equation with time delay, i.e., the establishment of the linear and nonhomogeneous relationship between the optimal co-state and the state, which lays a theoretical foundation for the study of the stochastic control with distributed delays.2. We consider the consensus problem of multi-agent systems with communica-tion noises, with communication delay and with packetdrop respectively by applying the theory of stochastic system and the control methods with delay.●For the system with communication noises, we give the mathematical ex-planation of the commonly used control protocol for the first time based on the stochastic approximation theory. Then with the using of the probabilistic method, the sufficient conditions of strong consensus are derived. In partic-ular, it is shown that the system achieves strong consensus with correlated measurement noises by applying the technique of martingale inequality.●For the case with both noises and communication delay, we derive that the mean-square consensus problem is equivalent to mean-square stability prob- lem by making transformation to the states. With the using of the theory of time-delay system, the techniques of Schwarz inequality and mathematical induction, the sufficient conditions are derived to ensure the mean-square con-sensus. Furthermore, the method is extended to the case with packetdrop and strong consensus is derived.It is noted that the convergence rate analysis is also studied for the above systems respectively which is closely related to the step size of the algorithm. The main innovations are that the stochastic approximation theory is applied to the consensus problem and the consensus conditions are derived for the systems with correlated noises, time delay and packetdrop and the analysis of the convergence rate is further given.3. We investigate the input delay margin problem for multi-agent system. With the standard control protocol, we extend the delay margin problem for the stabil-ity of delayed system to the simultaneous stability problem of multiple single-agent systems. For the continuous-time system, we obtain the maximum delay margin of consensus; For the discrete-time system, we derive the necessary and sufficient condition for strictly nonzero maximum delay margin of consensus. The main inno-vation is to give the maximum delay margin allowing consensus. Different from the results in the literature which are mainly on the critical stable open-loop system, we firstly consider the unstable system and derive the maximum delay margin.4. We study the Stackelberg game using the theories of optimal control and time-delay system, including the games with delay free and with time delay. For the game problem with delay free, we obtain the necessary and sufficient condition for the existence of a unique solution by introducing two new co-states which capture the future information of the controllers, and derive the explicit Stackelberg solu-tion in terms of three decoupled and symmetric Riccati equations. For the game problem with time delay, by introducing a new state to capture the past information which eliminates the non-causality of the control induced by time delay, we obtain the necessary and sufficient condition for the existence and uniqueness of the so- lution and obtain the explicit strategy in terms of three decoupled and symmetric Riccati equations. The main innovation is the introducing of new co-states and new state, which implies that the control information in the equilibrium conditions are divided into past, current and future information, and thus the solvability condition is obtained. These results greatly improve the theory of Stackelberg strategy for the leader-follower game.