Feedback System Performance Specifications

Performance specifications (specs) provide quantitative requirements/goals for control systems.

1. Overshoot
2. Settling time
3. Rise time
4. Steady-state error
5. Gain/phase margins
6. Bandwidth
7. Limits on input/output/error
   • Saturation (physical) limits
   • Safety (engineered limits)

Final Value Theorem

Let $p(t)$/ $g[k]$ be a signal with Laplace/Z-transform that is real, rational, and proper. Then:

If all poles of $P(s)$/$G[z]$ lie in ${\mathbb{C}}^{-}$/$\mathbb{D}$,

$$\underset{t\to \infty }{\lim }p(t)=0/\underset{k\to \infty }{\lim }g[k]=0$$

If all poles of $P(s)$/$G[z]$ lie in ${\mathbb{C}}^{-}$/$\mathbb{D}$ except for exactly one pole at $0$/$1$:

$$\underset{t\to \infty }{\lim }p(t)=\underset{s\to 0}{\lim }sP(s)/\underset{k\to \infty }{\lim }g[k]=\underset{z\to 1}{\lim }(z-1)G[z]$$

Proof.