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{align} \dot x(t)&=A_ix(t)+B_iu(t)\\ y(t)&=C_ix(t)+D_iu(t) \end{align}$$

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{align} {\omega_n}^2=\|z\|^2&=z^{*} z\\ \end{align}$$

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_iP+B_iZ \\ (A_iP+B_iZ)^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