MATH319 Slides 137 Poles of the transfer function of the damped harmonic oscillator 139

T

138 Stability for systems and transfer functions

Theorem

Let $\Sigma$ be a linear system with rational transfer function $T$ . Then $\Sigma$ is BIBO stable if and only if $T$ is stable.

Proof. Suppose that the system is $B I B O$ , and that $T$ is not stable. Recall $\hat{Y}(s)=T(s)\hat{U}(s)$ . Then we can choose a bounded input $U=1$ such that $\hat{U}(s)=1/s$ . But $\Sigma$ is BIBO stable, so $Y$ is bounded, and hence $\hat{Y}(s)$ is holomorphic on $\{s:\Re s>0\}$ and $\hat{Y}(s)\rightarrow 0$ as $s\rightarrow\infty$ along $(0,\infty)$ . So $T(s)=s\hat{Y}(s)$ must be proper rational.

Suppose that $T$ has a pole at $\lambda$ . If $\Re\lambda>0$ , then $T(s)\hat{U}(s)=T(s)/s$ also has a pole at $\lambda$ . But $\hat{Y}(s)$ cannot have a pole at $s=\lambda$ by Prop 79.

Now suppose that $\Re\lambda=0$ , so $\lambda=i\nu$ for some real $\nu$ . The idea is to cause resonance, so we let $U(t)=\cos\nu t,$ which is bounded, and

\hat{U}(s)={{s}\over{s^{2}+\nu^{2}}}={{1/2}\over{s-i\nu}}+{{1/2}\over{s+i\nu}}