The IOP equations (i)-(iii) are hard to solve because lie in an infinite-dimensional vector space. To make the problem tractable, we make a finite dimensional approximation of this infinite dimensional space. In particular, we choose the simple pole approximation (SPA).

We choose as part of our control design. We approximate:

  • are variable coefficients in

Assumption: Our plant has no repeated plots. (But this doesn’t actually matter; we can have multiple poles, it just makes it more messy.)

Then, we have

  • are the plant poles and are the coefficients in . (These are given, since the plant is known.)

Then, IOP equation (i) becomes:

Then, we can write

  • Note that we can write

Substituting back:

We can then re-order the poles so that

Then,

and

where and are variable coefficients in .

Matching coefficients gives us:

  • These 3 are essentially another representation of IOP eq. (i)

IOP equation (ii):

which becomes

  • where ,