Poles

For a Transfer Functions $P\left(s\right)$ or $G\left[z\right]$, $\lambda \in \mathbb{C}$ is a:

• Pole if $P\left(\lambda \right)=\infty$

Effect of Pole Placement

There are some choices of poles that are better than others (optimal choices of poles for solving the Input-Output Parameterization equations) Arbitrarily…

• Poles close to the origin = fast settling time
• Poles need to be $\in \mathbb{D}\text{ or }{\mathbb{C}}^{-}$ for stability
• Imaginary poles add a damping effect

Note

The best choice of poles are poles that are optimal and solve In Controls and could lie anywhere in the open unit disk. We do not know this choice in advance.

One choice is to uniformly distribute the poles over the unit disk to increase the chance that a pole is close to the position of the unknown “best poles”.

Summary

An exactly uniform selection of poles over the unit disk is actually difficult to obtain, instead we approximate a uniform selection of poles using a type of spiral (Archimedes Spiral), where we also need to include the complex conjugates of the selected poles.

Notation Wise

• $m$ is the number of approximating poles in $W[z]$
• $n$ is the number of plant poles
• $\hat{n}$ is the number of stable plant poles You can choose poles $p_1\dots p_m$ such that…
• $p=r{e}^{j\theta }⟹p^k=r^k{e}^{jk\theta }$
• Where here, the magnitude controls the rate of decay, and the exponential term controls the rate of oscillation.
• Here, $r_{max}<1$
• Initial choose 20-30 poles $m=20-30$ to see if you can find a solution, then you can work on refining the amount of poles you have.
• To get poles to exist in the first place, we need to choose $m\ge 2(n-\hat{n})$ where if this equation is exactly equal, we will have one unique pole. We don’t want this since then there is no optimization problem, which means we want m to be larger to increase the design space we can optimize over.

Refining Poles

• Look for regions where your high magnitude (fast) poles are and try a tighter spiral in that localized area (set an $r_{max}$)
• This makes it easier to recover the controller after solving the optimization problem.