Fixed-structure controller design for linear systems with performance constraints.

Abstract

The state space approach to design feedback controllers has been an active research area since the development of LQG theory. This approach involves matrix equations such as the Lyapunov equation and the Riccati equation. The Lyapunov equation is used to describe a system which is internally stable, while the Riccati equation arises from minimizing the H$\sb2$ performance functional of the closed-loop system. In this dissertation, we extend LQG synthesis techniques by considering additional performance constraints. These performance constraints are included to improve overall closed-loop system response or to yield stability constrained controllers. Specifically, for improving overall closed-loop system response, we focus on regional pole placement and L$\sb2$/L$\sb\infty$ optimal control problems. For regional pole placement, we develop techniques that constrain the closed-loop poles to lie in left hyperbolic and the left horizontal strip regions while the closed-loop system is H$\sb2$-suboptimal. We also present an augmentation technique that effectively guarantees exact H$\sb2$ optimality with an $\alpha$-shifted pole constraint. In the L$\sb2$/L$\sb\infty$ optimal control problem, we obtain pointwise-in-time constraints to bound the L$\sb\infty$-norm of the closed-loop impulse response and use dynamic compensation to perform controller synthesis. Next, to enforce controller stability, we modify LQG design techniques to yield stable controllers and positive real controllers. We show that the modifications can be performed either before (a priori approach) or after (a posteriori approach) the optimization is executed. In the a priori approach, we introduce an additional quadratic matrix function to the standard closed-loop Lyapunov stability equation to guarantee strong stabilization. In the a posteriori approach, the controller weightings and the noise intensity matrices are modified to yield stable controllers. Finally, to design positive real controllers, we show that it is possible to carefully select the controller weightings and the noise intensity matrices to yield a strictly positive real controller if the given open-loop plant is positive real. When the open-loop plant is not positive real, we then use the a priori approach with controller positivity conditions to obtain positive real controllers. The basic approach for developing the design techniques is an extension of fixed-structure controller design. Constructive sufficient conditions are derived to yield controllers satisfying these performance constraints. The results involve coupled modified Riccati matrix equations. Numerical algorithms are developed for solving these equations to yield the controller gains.