STABILIZING AND ROBUST CONTROL

Abstract

Stability and stabilization of linear systems have been the most fundamental problems in control theory. There are different kinds of stability problems such as Lyapunov stability, bounded input-bounded output stability and internal stability. The system is robustly stable if the controlled system remains stable despite the uncertain property of the plant. The stabilization means to design a controller for a given plant such that the resultant closed loop system is stable. Some problems which remain open up to date are what the minimal order of all stabilizers for a given plant is, whether or not an uncertain plant can be stabilized robustly, when there exists an H? optimal controller for a plant with delayed state, whether or not a plant is concurrently stabilized by a decentralized controller, and how linear quadratic regulation (LQR) approach can be applied to the systems with time delay. These problems are addressed in the thesis.

Recall (Morari et.al., 1989) that a system is internally stable if and only if all transfer functions between any two points in the system is stable. For single loop feedback systems, the internal stability problem has been solved by Rosenbrock (1974). Rosenbrock's result is here extended to complex linear feedback systems. It is shown that an interconnected linear system is internally stable if and only if all the roots of a specific polynomial, which can be found easily from the structure of the system, lie in the open left half of the complex plane. This greatly simplifies stability test.

It is well known that a linear plant is stabilizable by output feedback if and only if it is controllable and observable. However, for practical implementation, a low-order stabilizer is usually preferred to a full order of an observer-based stabilizer. In the thesis a new lower bound for orders of stabilizers and a sufficient condition for the existence of a low-order stabilizer are derived for all-pole plants. And a multi-stage approach is also presented to construct a low-order stabilizer. In the case of general SISO plants, a new algorithm is developed for the low-order stabilization. H? optimal control is one of the main approaches in robust control theory. In the thesis, the dynamic differential game theory is applied to the design of an H? controller for the systems with delayed state. A delay-independent and a delay-dependent sufficient condition are established for the existence of such a controller.

It is well known that a multivariable system is decentralized stabilizable if and only if all its "fixed modes" lie in the left half of the complex plane. In the thesis, the "fixed modes" are further characterized as the common roots of a set of polynomials and these polynomials are explicitly given. If the stabilizability condition is satisfied, then a novel procedure is presented to construct a stabilizing decentralized controller. The decentralized concurrent stabilization problem is also addressed and a new algorithm is given for its solution.

Robust stabilization of uncertain linear plants has received considerable attentions. In the thesis, the problem is approached using the extension property of Hurwitz polynomials. A method for the controller synthesis is then given in a simple way for a class of uncertain linear plants and the simultaneous stabilizability of any set of minimum-phase plants thus becomes its easy corollary. The simultaneous stabilization problem is a special kind of robust stabilization problem. The simultaneous stabilization of three plants is equivalent to a special transcendental problem and a new constructive procedure is presented for its solution.

The LQR technique is one of the most popular control designs. In the thesis, it is extended to the systems with time delay. The stability and robustness of the resultant LQR system are analyzed. This LQR solution is also applied to tune PI/PID controllers for typical industrial plants. In order to ensure closed-loop response performance, the selection of Q and R is related to the natural frequency and damping ratio of the desired closed-loop system. It is shown by simulation that the LQR tuned PI/PID control systems exhibit superior closed-loop performance and robustness due to the prediction capability and time varying nature in the LQR control law for the plants with time delay.