4.7 Hurwitz and Routh Stability Criteria

In this section we present two tests for determining if the zeros of an nth-degree polynomial are located in the open left-half complex plane. In deriving these tests, we will use the Lyapunov matrix equation given by (4.6). We consider the nth-degree polynomial

with real coefficients. Associated with the polynomial p( s) is the n n real Hurwitz matrix,

In our further discussion we assume, without loss of generality, that a _n = 1. The leading principal minors of the Hurwitz matrix are

We next define

and

Thus, for example, . Let

For example, for n = 4, we have

Let

Lemma 4.1

The matrices B and A are similar; that is, there is a nonsingular matrix such that

Proof

The similarity matrix is lower triangular and has the form To determine the nonfixed coefficients t ij of the matrix T, we represent (4.36) as and compare both sides of (4.38) to determine the coefficients t ij. We obtain, for example, t 31 = ? b n, t 42 = ? b n ? b n ?1, and so on.

Let P be an n n diagonal matrix defined as

and Q be an n n diagonal matrix defined as

We represent the matrix Q as Q = c ^T c, where

Note...