Choosing Poles for IOP and SPA

For control design with IOP and SPA, we want to choose poles that satisfy the equations for IOP, as well as our equations for specs for control design.

There is some “best” choice of poles for solving these equations. This best choice of poles could lie anywhere in $\mathbb{D}$ – we do not know the best choice of poles in advance.

One strategy is to choose $\{p_i{\}}_{i=1}^m$ uniformly distributed over $\mathbb{D}$ to maximize our chances of having some poles in $\{p_i{\}}_{i=1}^m$ that are close to the unknown best poles. However, an exactly uniform selection of poles over $\mathbb{D}$ is challenging to obtain. Instead, we approximate a uniform selection of poles using an Archimedes spiral, where we also include their complex conjugates as well.

In particularly, if we want $m$ total poles, where $m$ is a positive even number, then we select poles in polar coordinates by the formulas

$$\begin{array}{rl}{r}_i& =r_{\text{max}}\sqrt{\frac{i}{\frac{m}{2}}}\\ {\theta }_i& =2\sqrt{\pi i}\\ p_i& =(r_i,{\theta }_i)\\ \overline{p_i}& =(r_i,-{\theta }_i)\text{(complex conjugate)}\end{array}$$

for $i\in \left\{1,\dots ,\frac{m}{2}\right\}$ and $r_{\text{max}}\in (0,1)$ the maximum radius desired for the poles.