GA Encoding for Robust Stability Analysis

In the general case, the closed-loop characteristic polynomials is defined in parametric form as

$$P(z,{\bf q}, {\bf k}) =\sum\limits_{i=0}^{n} a_i({\bf q}, {\bf k}){z}^{{\prime}i}$$ (3)

The controller parameter vector ${\bf k}$ and the operating domain Q are given. In such a polynomial family, the coefficient $a_i({\bf q}$, ${\bf k})$include any continuous nonlinear functions, in which we are particularly interested in coefficients with nonlinear dependency. The roots of this polynomial are complex and equal to $\vert z^{\prime}\vert e^{j\theta}$ and the range of ${\bf q}$ is defined by its pre-specified minimum and maximum values $[{\bf q}^-, {\bf q}^+]$. The objective of the genetic algorithm is to find:

$$f(z, q) = Min[\vert P(z^{\prime}, {\bf Q})\vert],\:\:{q \in {\bf Q}, \: z \in \bar{U}}$$ (4)

where

$${\bf Q} \subset R^n and \;\: \bar{U} \subset R^2$$

$${\bf Q} = \{{\bf q}: {\bf q}=[q_i], q_i^- \le q_i \le q_i^+, \forall i = 1,...,n\}$$

$$\bar{U} = \{z^{\prime}: \vert z^{\prime}\vert \le 1 \}$$

We exploit the fact that for real coefficients, roots are complex conjugate pairs or real. Hence, the search space can be restricted to the upper half of the unit circle shown in Figure 1.