Search

In controller design, it is essential to achieve stability of the closed-loop system and various performance specifications. Frequency domain criteria such as gain margin, phase margin and H8 norms of the closed-loop transfer functions as well as time domain criteria such as settling time, rise time and overshoot can be counted among important performance specifications. Most of the controllers used in the practical world are low order controllers such as P, PI and PID controllers. It is possible to see that methods for finding stabilizing low order compensators can be considered in three main categories: methods based on Nyquist theorem, methods based on a generalized version of the Hermite- Biehler theorem, and methods based on parameter space and the concept of singular frequencies. Robust stabilization of continuous time single-input single-output (SISO) linear time invariant (LTI) systems with multiplicative uncertainties is considered in this study. In particular, it has been shown that all P and PI controllers that robustly stabilize a given uncertain SISO LTI system can be found by utilizing a generalization of the Nyquist theorem and the parameter space approach, respectively. The generalization of Nyquist stability criterion suggests to determine the number of the unstable poles for gain intervals obtained by calculating the location and direction of the crossing of the Nyquist plot with the real axis. A stable characteristic polynomial, whose roots are in the left half plane, becomes unstable if and only if at least one root crosses the imaginary axis. The parameter values of the root crossing form the stability boundaries in the parameter space, which can be classified into three cases: the real root boundary, where a root crosses the imaginary axis at the origin (substitute s= jw and w = 0 in the characteristic polynomial), the infinite root boundary, where a root leaves the left half plane at infinity (for w → ∞ ) and the complex root boundary, where a pair of conjugate complex roots crosses the imaginary axes (for 0< w<∞). These stability boundaries separate regions in which the number of closed loop system unstable poles do not change in the parameter space. Sometimes, it is not possible to represent uncertainties in a system model with parametric uncertainties. Such uncertainties are usually encapsulated in a norm bounded system block that acts on a nominal system in an additive or multiplicative manner. Although it is possible to find robust controllers that can stabilize systems with such uncertainties by the help of H∞ control theory, the resulting compensators are usually of high order (at least as high as the order of the plant) and therefore impractical in many cases. Several attempts exist to put constraints on the order of H∞ controllers in the literature. However, many of these approaches suffer from computational intractability.… [ABSTRACT FROM AUTHOR]