Optimization Problems

The Idea

There is some cost or objective function, and we are trying to minimize that cost or objective by varying variable that are available, while satisfying constraints.

In Controls

For IOP With SPA we are minimizing $w$, $x$, $\hat{W}$, while being subject to constraints of $\ast$ also from IOP With SPA.

We will numerically solve these problems using:

• YALMIP
• MOSEK

Steps in solving the problem:

1. Choose Poles
2. Calculate $\alpha ,\beta ,\gamma ,\hat{\gamma }$, A, B, $\text{steady state},ste{p}_{ru},ste{p}_{ry}$ matrices in Simple Pole Approximation
3. Define objective and constraints
4. Solve optimization problem using MOSEK.
5. Recover an explicit formula for the controller $D[z]=\frac{W[z]}{X[z]}$ after solving where each of these are defined in Simple Pole Approximation

Recovering the Controller

• $\frac{zeros(w)}{zeros(x)}\frac{poles(x)}{poles(w)}$
• We want to cancel out common poles and common zeroes to make our recovery easier.
• Using $zpk(w)$ and $zpk(x)$ gives the gain of w and x, where the gain of $D$ is $\frac{gain(w)}{gain(x)}$.
• Make sure to use format long so that we can recover our poles and zeroes with more accuracy.

This takes time

See Engineering Design