# LMIs in Control/pages/Switched Systems Pole Placement

Pole Placement for Switched Systems

This LMI lets you provide specifications of the switched system closed loop poles. Note that arbitrarily switching between stable systems can lead to instability whilst switching can be done between individually unstable systems to achieve stability.

## The System

Suppose we were given the switched system such that

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=A_{i}x(t)+B_{i}u(t)\\y(t)&=C_{i}x(t)+D_{i}u(t)\end{aligned}}}

where ${\displaystyle A_{i}\in \mathbb {R} ^{mxm}}$ , ${\displaystyle B_{i}\in \mathbb {R} ^{mxn}}$ , ${\displaystyle C_{i}\in \mathbb {R} ^{pxm}}$ , and ${\displaystyle D_{i}\in \mathbb {R} ^{qxn}}$  for any ${\displaystyle t\in \mathbb {R} }$ .

${\displaystyle i\in 1,...,k}$  modes of operation

## The Data

In order to properly define the acceptable region of the poles in the complex plane, we need the following pieces of data:

• matrices ${\displaystyle A_{i}}$ , ${\displaystyle B_{i}}$
• desired closed loop pole bound (${\displaystyle \alpha }$ )

Having these pieces of information will now help us in formulating the optimization problem.

## The Optimization Problem

Using the data given above, we can now define our optimization problem. We first have to define the acceptable region in the complex plane that the poles can lie on. Assume that ${\displaystyle z}$  is the complex pole location, then:

${\displaystyle Re(z)\leq \alpha }$

## The LMI: An LMI for Pole Placement

Suppose there exists ${\displaystyle P>0}$  and ${\displaystyle Z}$  such that

{\displaystyle {\begin{aligned}A_{i}P+B_{i}Z+(A_{i}P+B_{i}Z)^{T}+2\alpha P&<0,\end{aligned}}}

for ${\displaystyle i=1,...,k}$

## Conclusion:

The resulting controller can be recovered by

${\displaystyle K=ZP^{-1}}$ .

## Implementation

The implementation of this LMI requires Yalmip and Sedumi /MOSEK [1]