Riccati equation and event-triggered control for constrained systems

Abstract

This thesis is concerned with the control of networked systems subject to such constraints as channel noise and actuator saturation. In the increasingly popular networked control systems (NCSs), the sensors, controllers, and actuators are usually geographically dispersed, and the entire system is subject to various constraints such as channel noise, limited channel capacity, measurement quantization, time delay, packet loss, limited control energy, and actuator saturation, which has motivated enormous research interest in the control of constrained systems.

The event-triggered control (ETC) is a new technology that reduces unnecessary control updates in NCSs comparing with the periodic sampled control. In periodic control, the feedback updating action is performed according to periodic time. In ETC, feedback control is updated based on the state variation. This event-trigger strategy has attracted a great deal of research interest in the last two decades.

The Riccati equation plays an important role in mathematics and engineering. A scalar Riccati equation is an ordinary differential equation (ODE) containing a quadratic term of the dependent variable and can be regarded as a quadratic polynomial system. The matrix Riccati equation is a matrix generalization of Riccati ODE. Its algebraic version, called the algebraic Riccati equation (ARE), can be utilized to resolve problems including the semi-global stabilization of systems subject to actuator saturation and the ETC for noisy systems.

In this thesis, a modified algebraic Riccati equation (MARE) is studied in the case of a zero parameter matrix. The MARE can be applied to the event-based synchronization of linear dynamical networks and the semi-global synchronization subject to actuator saturation. For the controllable single-input systems, a novel solution of the MARE is obtained in terms of the eigenvalues of the system matrix. It is an explicit analytic solution in the sense that no eigenvector of such associated parameter matrices as the Hamiltonian matrix is required, which distinguishes this new result from other results for ARE.

The thesis also addresses the problem of ETC over noisy channels. In the presence of channel noise, the error between estimated control and desirable control is non-zero at the updating instant. Nevertheless, the stabilization is achieved via an appropriate ETC law. Based on the robust control theory, a sufficient condition for closed-loop stability is established. A highlight is that the tuning rule for parameters can be identified in a straightforward manner.

The problems of event-based global and semi-global stabilization of systems subject to actuator saturation are investigated as well. When global stabilization is considered, an inherent lower bound of the inter-event time does not exist for continuous-time input-saturated systems. A minimum inter-event time is properly selected such that the event trigger is active only after the prescribed time interval. For any marginally stable systems, the semi-global stabilization is achieved via ETC law based on ARE and Riccati ODE.