Robustness of Discrete-Time System

Consider a linear time invariant discrete-time system. The stability of the system can be determined by examining its characteristic equation. The characteristic equation may be written as:

$$P(z,{\bf q})=\{p(z,{\bf q})=\sum_{i=0}^{n}a_{i}({\bf q})z^i, {\bf q \in Q}\}=0$$ (1)

where $a_i({\bf q})$, i =0, 1, ..., n are functions of the uncertain plant parameter vector ${\bf q}$, and ${\bf q} \in {\bf Q} \subset {R}_{n}$. The coefficients a_i may be interval, affine, multilinear or even exponentially dependent on the uncertain plant parameter vector ${\bf q}$. When the coefficients are nonlinear, traditional methods fail, and overbounding or gridding are our only options. Unfortunately, overbounding gives conservative stability results while gridding is only feasible for a small number of parameters.

The stability boundary for Equation (1) is |z| = 1. The stability test is to determine if there exists a system pole outside the unit circle. Using the mapping $z^{\prime} \rightarrow z^{-1}$, the instability region becomes the closed unit disc $\bar{U}$, and therefore, system instability testing reduces to searching for a pole inside or on the unit circle (see Figure 1).

If the polynomial coefficients are continuous functions of the uncertain parameters and no degree dropping occurs, the system instability test can be simplified to searching for a root on the unit circle and the mapping $z^{\prime} \rightarrow z^{-1}$ need not be used. This follows from the well known boundary crossing theorem [10].

  

Figure: Roots of $P(z^{\prime },{\bf q})$ in the unit circle

Searching for a root inside or on the unit circle can be formulated as minimizing the cost function

  

Figure 2: Minimization of a cost function

$$F = \vert P(z^{\prime},{\bf q}))\vert, \;\; z^{\prime} \in \bar{U} \; and \; {\bf q} \in {\bf Q}$$ (2)

with tolerance $\epsilon$ as shown in Figure 2 and where $\bar{U}$ denotes the closed unit disc and ${\bf Q}$ is the set of feasible parameter vectors. The robust stability problem can now be stated as follows: Given a family of characteristic functions P associated with uncertain physical parameters ${\bf q}$, search the space $\bar{U} \times {\bf Q}$ for a minimal value of $\vert P(z^{\prime}, {\bf q})\vert$, $\forall$ $z^{\prime} \in \bar{U}$ and $\forall$ ${\bf q} \in {\bf Q}$. If $\exists$ a "zero" of $\vert P(z^{\prime}, {\bf q})\vert$, then the system is unstable. If the polynomial is continuous and no degree dropping occurs, we search for the roots on the unit circle. If $P(Re^{j\theta},{\bf q}) \ne 0$, $\forall$ ${\bf q} \in {\bf Q}$, $\theta \in [0, \pi]$ and R = 1, then further search inside the unit circle will be necessary. Thus there are two main possibilities:

1.

If we find a ``zero'' on or inside the unit circle, we have sufficient evidence to conclude that the system is unstable

2.

Otherwise, if we do not find a ``zero'' there are two further possibilities.

(a)

The system is unstable but the genetic algorithm cannot find a ``zero,'' or

(b)

The system is stable

We cannot distinguish between 2a and 2b and thus our approach can only provide a sufficient condition for instability, not a necessary condition. Specifically, not finding a ``zero'' of the polynomial does not provide any information about system stability.

A key issue that arises in this approach is the extremely large search space. In such a large search space, exhaustive search methods take unacceptably long while robust search algorithms provide a promising alternative. Genetic algorithms were designed to robustly and efficiently search large, nonlinear search spaces where traditional optimization techniques are not feasible. GAs are particularly attractive because they do not require the coefficients of the characteristic polynomial to be linear and we thus choose to use GAs as the search method for this paper.