Closed loop stability ⟺W,X,V stable ⟹ already guaranteed by choosing {pi}i=1m⊂D and satisfying Equations (∗) and (∗∗) from Simple Pole Approximation.
Steady-state error
(ess) is given by
ess=Tre[1]=X[1][IOP theorem part b]=1+i=1∑m1−pixi+k=1∑n^1−qkx^k
The pi terms are fixed because we choose them in advance and qk are fixed because they are from the plant.
Case 1: We want the steady state error to be zero such that ess=0
ess=1+i=1∑m1−pixi+1−qk∑k=1n^x^k=0
Case 2: We want the steady-state error to be bounded such that ∣ess∣≤C
We are consider the time horizon where k≥0. There are infinitely many such time steps. This makes the computation hard to do on a computer. At the same time, for a practical stable system, the step response will settle down to a steady-state and we don’t really need to keep enforcing this limit. Thus, we set some practical limit K>0 to give us a finite number of timesteps.