BIBO stable <=> Stable

IOP equations:

$$A\left[\begin{array}{c}w\\ x\\ \hat{x}\end{array}\right]=b$$

This represents a set of linear equations. A solution exists if $A$ is full rank and ($\#$ columns of $A$) $\ge$ ($\#$ rows of A).

• ($\#$ rows of $A$) = $m+n+(n-\hat{n})$
• ($\#$ columns of $A$) = $2m+\hat{n}$

Then, we need:

$$\begin{array}{rl}2m+\hat{n}& \ge m+n+(n-\hat{n})\\ ⟹m& \ge 2(n-\hat{n})\end{array}$$

We want greater than, not equal to, so that we have more flexibility when designing the controller.

Theorem: BIBO stability $\iff$ Stable

Suppose $G[z]$ is real, rational, and proper. Then $G[z]$ is stable if and only if $G[z]$ is BIBO stable.

We will prove this for $G[z]$ strictly proper.

First, we prove Stable $⟹$ BIBO stable.

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

Proof:

• Consider a bounded input signal $u[k]$ which implies that $\exists \overline{u}>0$ such that $u[k]\le 0\forall k\in {\mathbb{Z}}_{\ge 0}$ . [definition of bounded]
• Let $y[m]=(g\ast u)[m]$. Then:

$$\begin{array}{rlr}|y[m]|& =|\sum _{\kappa =0}^{m}g[m-\kappa ]u[\kappa ]|\le \sum _{\kappa =0}^m|g[m-\kappa ]u[\kappa ]|& [\text{triangle inequality}]\\ & =\sum _{\kappa =0}^m|g[m-\kappa ]||u[\kappa ]|& [\text{property of }|\cdot |]\\ & \le \sum _{\kappa =0}^m|g[m-\kappa ]|\overline{u}& \\ & =\overline{u}\sum _{\kappa =0}^m|g[m-\kappa ]|& [\text{linearity of sums}]\\ & =\overline{u}\sum _{k=0}^m|g[k]|& [\text{change of variables: }k=m-\kappa ]\\ & =\overline{u}\sum _{k=0}^m\left|\sum _{i=1}^n\sum _{j=1}^{n_i}{c}_{i,j}\right(\genfrac{}{}{0ex}{}{k-1}{j-1}\left)p_i^{k-j}\right|& [\text{expression for }g[k]\text{ from class}]\\ & & \\ & =& \end{array}$$

Then, we prove BIBO $⟹$ stable.

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