LMI-Based Multivariable PID Controller Design and its Application to the Control of the Surface Shape of Magnetic Fluid Deformable Mirrors

In this paper, the design of multivariable proportional-integral-derivative (PID) controllers for discrete-time systems based on linear matrix inequalities (LMIs) is studied. First the stability problem of the closed-loop system with the multivariable PID controller is investigated. To adjust the transient response of the resulting closed-loop system, the design method is extended to account for a pole clustering constraint. PID controller design methods that yield closed-loop systems with H_∞ and H₂ performance specifications are then investigated. Algorithms based on properly formulated LMIs are developed for the above different cases. Finally, the performance of the proposed controllers is experimentally evaluated in an adaptive optics system where it is desired to control the shape of a magnetic fluid deformable mirror. The experimental results indicate the proposed controllers can successfully provide the desired shape tracking performance in the closed-loop adaptive optics system.