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:
For our , what if has an unstable pole? Will that also be a pole of ? The coefficient of each unstable pole in must be zero for to be stable.
We can then re-order the poles of the plant :
Then, the poles of are contained in
because is stable.
So, we can write
where and are variable coefficients in .
Matching coefficients of the above equation with gives us:
- The final equation ensures all unstable poles have zero coefficients.
- These 3 are essentially another representation of IOP eq. (i)
IOP equation (ii):
Since only contains simple poles like , we can u se the same procedure as above for calculating to find and then .
However, as and , we do not need to satisfy our desired specs or recover our controller. We just need to ensure that is stable.
Thus, we just reuse the result from Eq. () applied to to ensure all coefficients of the unstable poles in are zero:
- ,
The first sum corresponds to the poles in , the second sum to the poles in , and the term comes from the in that is not in .