Constant BIBO Stability Proof

Suppose $c\in \mathbb{R}$ and $G[z]$ is real, rational, proper, and stable. Prove that $cG[z]$ is BIBO stable.

• Given: $G[z]$ is real, rational, proper and stable
• WTS: $cG[z]$ is BIBO stable

Strategy A:

1. Show $G[z]$ is BIBO stable
2. Show $cG[z]$ is BIBO stable

Proof:

• $G[z]$ is real, rational, proper and stable
• $⟹$ $G[z]$ is BIBO stable
  • (theorem from class)
• Let $u[k]$ be bounded, such that $|u[k]|\le \hat{u}\forall k\in {\mathbb{Z}}_{\ge 0}$.
• Let $\hat{y}[k]:=(g\ast u)[k]$
• $⟹|\hat{y}[k]|\le \overline{y}\forall k\in {\mathbb{Z}}_{\ge 0}$ for some $\hat{y}>0$ (i)
  • (definition of BIBO stability, $u$ is bounded)
• Let $y[k]:=((cg)\ast u)[k]=c(g\ast u)[k]=c\hat{y}[k]$
  • (linearity of convolution operation)
• $⟹|y[k]|=|c\hat{y}[k]|\le |c|\overline{y}\forall k\in {\mathbb{Z}}_{\ge 0}$
  • (property of absolute value, (i))
• $⟹y[k]$ is bounded
  • (definition of bounded)
• $⟹cG[z]$ is BIBO stable
  • (definition of BIBO stability)

Strategy B:

1. Show $cG[z]$ is stable
2. Show $cG[z]$ is BIBO stable

Proof:

• $G[z]$ is real, rational, proper and stable
• Let $p$ be any poles of $cG[z]$
• $⟹cG[p]=\infty$ (definition of a pole)
• $⟹G[p]=\infty$
  • ($c\in \mathbb{R}$)
• $⟹p$ is a pole of $G[z]$
  • (definition of a pole)
• $⟹p\in \mathbb{D}$
  • (definition of stability, we defined $G[z]$ as stable)
• $⟹cG[z]$ is stable
  • (definition of stability)
• $cG[z]$ is real and rational
  • (properties of real and rational functions)
• $cG[z]$ is proper
  • (example from class, $cG[z]$ is proper if $G[z]$ is proper)
• $cG[z]$ is BIBO stable
  • (theorem from class)