LMIs in Control/pages/Switched Systems Rise time

Rise time Specification for Switched Systems

This LMI lets you specify desired performance (rise time). 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

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

where $A_{i}\in \mathbb {R} ^{m\times m}$, $B_{i}\in \mathbb {R} ^{m\times n}$, $C_{i}\in \mathbb {R} ^{p\times m}$, and $D_{i}\in \mathbb {R} ^{q\times n}$ for any $t\in \mathbb {R}$.

$i\in 1,...,k$ 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 $A_{i}$, $B_{i}$
• rise time ($t_{r}$)

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 using the following inequality constraints:

Rise Time: $\omega _{n}{\leq }{1.8 \over t_{r}}$

Assume that $z$ is the complex pole location, then:

${\begin{aligned}{\omega _{n}}^{2}=\|z\|^{2}&=z^{*}z\\\end{aligned}}$

This then allows us to modify our inequality constraints as:

Rise Time: $z^{*}z-{1.8^{2} \over {t_{r}}^{2}}{\leq }0$

The LMI: An LMI for Rise time Specification

Suppose there exists $P>0$ and $Z$ such that

${\begin{bmatrix}-rP&&A_{i}P+B_{i}Z\\(A_{i}P+B_{i}Z)^{T}&&-rP\end{bmatrix}}<0$

for $i=1,...,k$

Conclusion:

The resulting controller can be recovered by

$K=ZP^{-1}$.

Implementation

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

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.