Structured controller synthesis of linear multivariable systems

Abstract

This thesis is concerned with synthesis problems for linear multivariable systems with constraints on the control structure. Three different cases of structured controllers are considered and investigated, namely, the multivariable PID controller, the multivariable lead-lag compensator and the positivity-preserving consensus controller of positive multi-agent systems.

The well-known proportional-integral-derivative (PID) controllers, which are extensively used in industrial control systems, have a fixed structure inherently. Regarding this problem, the objective is to design non-fragile $H_{\infty}$ multivariable PID controllers with derivative filters for linear systems. In order to obtain the controller gains, the original system is associated with an augmented system such that the PID controller design can be formulated as a static output-feedback (SOF) control problem. By taking the system augmentation approach, the conditions with slack matrices for solving the non-fragile $H_{\infty}$ multivariable PID controller gains are established. Based on the results, linear matrix inequality based iterative algorithms are provided to compute the controller gains.

As one type of the most popular control structures, the family of lead, lag and lead-lag compensators has been used very extensively in various engineering applications. This dissertation considers the problem of designing lead, lag and lead-lag compensators for multi-input multi-output (MIMO) linear systems under the $H_{\infty}$ performance measure. This is the first time that the lead, lag and lead-lag compensators are generalized to the MIMO cases by preserving their classical compensator structures. Theoretical results on the stability analysis and synthesis of MIMO systems under the $H_{\infty}$ control performance with the proposed lead, lag and lead-lag compensators are obtained. Then relevant algorithms for designing lead, lag and lead-lag compensators are provided to determine the compensator parameters. Differing from traditional design methods, which mostly rely on some trial-and-error and/or heuristic procedures, the proposed methods are algorithmic and the compensators can be synthesised systematically.

The research topic on multi-agent systems, especially the consensus problem, has attracted a significant amount of interest in the last decade due to both their theoretical challenges arising in the control community and various kinds of practical potential applications. A particular focus is on the positivity-preserving consensus problem of positive homogeneous multi-agent systems. The case that all agents have identical positive state-space models with multiple inputs is investigated. Using positive system theory and analyzing the property of the overall closed-loop multi-agent system, positivity-preserving consensus analysis conditions are derived. In order to preserve the non-negative property of the agents, a system augmentation approach is employed and the positivity-preserving consensus design conditions are obtained. Different from some existing works only give the sufficient conditions for solution, a necessary and sufficient condition and a sufficient condition are both provided in this work. Then the corresponding approaches and algorithms are developed for solution.