Control and Coordination of Time-Delayed Systems

時滯系統的控制與協調

Student thesis: Doctoral Thesis

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Award date15 Aug 2018

Abstract

Time-delayed systems are often described by functional differential equations (FDEs), which are infinite dimensional, as opposed to ordinary differential equations (ODEs). In many practical systems, time delays are inherent features and are often the main cause for instability and poor performance. Control of time-delayed systems has been a topic of continuous interest in the control community. Although much progress has been made on this topic, there are still some open problems presenting numerous challenges. One of the most challenging problems is how to deal with systems with infinite
delays.

Recently, multi-agent systems with time delays are frequently considered. Multi-agent systems have been widely studied due to their broad applications in many areas including multiple spacecraft vehicles, biological systems, networked multiple mobile robots, teams of unmanned aerial vehicles, sensor networks and so on. In cooperative control of multi-agent systems, a fundamental problem is to design appropriate controllers such that a group of agents can reach an agreement on certain quantities of interest while each agent has only access to the information of its neighboring agents. Such a problem is called the consensus. The first part of this thesis addresses the consensus problem of heterogeneous linear multi-agent systems with time delays. The delays in this part are assumed to be bounded. The main results of this part can be summarized as follows.

Consensus of continuous-time heterogeneous linear multi-agent systems with communication delays is first considered. Novel distributed dynamic controllers are proposed for heterogeneous linear multi-agent systems with fixed and switching directed communication topologies respectively. It is shown that the controlled heterogeneous linear multi-agent systems can reach consensus for arbitrarily large constant, time-varying and distributed communication delays under some sufficient conditions.

Consensus of discrete-time heterogeneous linear multi-agent systems with three types of delays, that is, communication delays, input delays and output delays is then considered. A distributed predictor-based controller is proposed and it is shown that the output consensus problem can be solved via the proposed controller.

In the second part of this thesis, control and coordination of systems with infinite delays are considered. Infinite delays, also known as unbounded delays, are much more general but also much more difficult to deal with. The main results of this part can be summarized as follows.

Stabilization of linear systems with distributed infinite input delays is first considered. By introducing a stability result on systems with infinite delays, it is shown that a stabilizable linear system with distributed infinite input delays can be globally asymptotically stabilized with low gain controllers as long as the open-loop system is not exponentially unstable. This result includes constant delay and bounded distributed delay as its special cases.

Semi-global stabilization of linear systems with distributed infinite input delays
and input saturation is then considered. Based on the result on stabilization of linear systems with distributed infinite input delays, a new stabilizing controller is proposed for linear systems with distributed infinite input delays and input saturation. It is shown that a stabilizable linear system with distributed infinite input delays and input saturation can be semi-globally asymptotically stabilized with low gain controllers as long as the open-loop system is not exponentially unstable.

Consensus of multi-agent systems with distributed infinite transmission delays is further considered. A novel low gain controller is proposed and it is shown that the proposed controller can solve the consensus problem if the open loop agent dynamics have no exponentially unstable eigenvalues and the communication topology is fixed and contains a spanning tree. The realization of our controller is also discussed. Moreover, the special case when the open loop agent dynamics have all their eigenvalues at the origin is also considered and it is shown that in this case, the controller could be much simpler. One of the distinctive advantages of the proposed controller is that it does not require the knowledge of transmission delays.

Input-to-state stability (ISS) of nonlinear systems with infinite delays is finally discussed. The ISS stability for nonlinear systems with infinite delays is newly defined. An ISS Lyapunov functional is proposed for the concerned systems. Then using the proposed ISS Liapunov functional, a link between the exponential stability of an unforced system and its input-to-state stability is established. It is shown that a nonlinear system with infinite delays is ISS if its corresponding unforced version is exponentially stable, provided that some Lipschitz and/or Lipschitz-like conditions are satisfied.