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 ,