Example 4.1: Closed-loop stability

Consider the following standard negative feedback system:

where we have

To check closed-loop stability, we need to check the closed-loop transfer functions.

  • Roots: -0.48, 0.34, 0.77
  • Roots: -0.48, 0.34, 0.77
  • Roots: -0.48, 0.34, 0.5, 0.77

Then:

  • are real, rational and proper.
  • Since the poles of these lie in the open unit disk the TFs are stable [definition of stability].
  • These TFs are BIBO stable [theorem from class]
  • The system is closed-loop stable [definition of closed-loop stability]

IOP Theorem

Theorem IOPa.

If results in closed-loop stability, then , , , satisfying the IOP equations:

Let’s check for our example above:

Similarly, we can show that .

The equations makes sense because

Theorem IOPb.

If satisfy the IOP equations (i)-(iii) and we choose , then , , and .

Example 4.2

Given some “arbitrary” TFs, , satisfy IOP equations, Solve for , Verify that the CLTF = .

Example 4.3: SPA

Assume that our plant only has simple poles, such that:

  • Simple pole = poles that are not repeating

Then, we approximate to be

Then, solving for from gives:

  • The poles from to are the stable poles
  • The last term is the unstable poles from

We have: