Performance of SD Systems

Question

What is the steady-state step response of a SD system when the SD system is closed-loop stable and $T>0$ is non-pathological?

1. For $r(t)=1(t)$
2. We know the SD system is closed-loop stable, then the DT system is closed-loop stable by a theorem from Closed Loop Stability for SD Systems.
3. We can use Final Value Theorem to determine
   1. $u_{ss}={\lim }_{k\to \infty }u[k]=T_{ru[1]}$
   2. $y_{ss}={\lim }_{k\to \infty }y[k]=T_{ry[1]}$

Lemma

If $T>0$ is non-pathological and the D.T system is closed-loop stable, then $T_{rx}$ and $T_{dx}$ are BIBO stable.

Theorem

If $T>0$ and is non-pathological according to Closed Loop Stability for SD Systems, and the SD system is closed-loop stable, and if $r(t)=1(t)$, then $y_{ss}={\lim }_{t\to \infty }y(t)=T_{ry}[1]$.

Shortcut

If you want to find $G[1]={\sum }_{i=1}^n\frac{c_i}{-{\lambda }_i}=P(0)$ which will save lots of time on an exam

• State space realizations for DT feedback system