How do we design controllers to meet performance specifications using IOP/SPA?

Closed-loop stability

Closed loop stability stable already guaranteed by choosing and satisfying Equations () and () from Simple Pole Approximation.

Steady-state error

() is given by

The terms are fixed because we choose them in advance and are fixed because they are from the plant.

Case 1: We want the steady state error to be zero such that

Case 2: We want the steady-state error to be bounded such that

Vector Form

We have:

which we can set to , , , etc.

Limit on control effort

Note we have

Then:

Vector form

We are consider the time horizon where . 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 to give us a finite number of timesteps.

Supposing some point in time , we have

Extending this to the full time horizon :

We can then write this constraint as:

Overshoot

We want some .

Then, we have

Note that . Then, we can write the above as

Vector Form

Similar to control effort, we can write

We can then write this as:

Settling Time (within 2%)

Let be the sample time and let .

Then:

  • where

Thus, we have

and also

Vector Form

We can borrow our result from the overshoot spec: