Input-Output Parameterization

A change of variables to accomplish the following ideas

SOTA

Developed in 2019

Ideas

1. Design closed-loop transfer functions to obtain closed-loop stability and satisfy the specs
2. Recover D[z] that results in those closed-loop transfer functions

IOP equations (feasibility constraints)

• $X\left[z\right]+G\left[z\right]W\left[z\right]=1$
• $V\left[z\right]+G\left[z\right]X\left[z\right]=0$
• Where $X$, $W$, $V$ are real, rational, proper, and stable These equations are hard to solve because $X[z]$, $W[z]$, $V[z]$ lie in an infinite dimensional space (infinite choices for the poles and zeroes).
• We want to make an approximation of this space to make this problem solvable using a well known solver.
• This approximation uses the Simple Pole Approximation

Since $G\left[z\right]$ is a known constant, these equations are linear in the variables $X\left[z\right]$, $W\left[z\right]$, $V\left[z\right]$

Theorem

a. If $D\left[z\right]$ results in closed-loop stability, then:

• $X\left[z\right]=T_{re}\left[z\right]=T_{du}\left[z\right]$
• $W\left[z\right]=T_{ru}\left[z\right]$
• $V\left[z\right]=T_{de}\left[z\right]$

Then these satisfy the IOP equations

b. If $X\left[z\right]$, $W\left[z\right]$, $V\left[z\right]$ satisfy the IOP equations, and if we choose $D\left[z\right]=\frac{W\left[z\right]}{X\left[z\right]}$ then, $T_{re}=X\left[z\right]$, $T_{du}=X\left[z\right]$, $T_{ru}=W\left[z\right]$, $T_{de}=V\left[z\right]$ which is useful because now we can do 2. from Ideas.