LMIs in Control/D-Stability/Observer D-Stability

LMIs in Control/D-Stability/Observer D-Stability

This is a WIP Page.

The System

$\begin{array}{cc}\dot{x}(t)& =Ax(t)\\ x(0)& =x_0\end{array}$

The Data

In order to properly define the acceptable region of the poles in the complex plane, we need the following three pieces of data: rise time ($t_r$), settling time ($t_s$), and percent overshoot ($M_p$). From this, we have to then define the acceptable region in the complex plane that the poles can lie on using the following inequality constraints:

Rise Time: ${\omega }_n\le \frac{1.8}{t_r}$

Settling Time: $\sigma \le \frac{-4.6}{t_s}$

Percent Overshoot: $\sigma \le \frac{-ln(M_p)}{\pi }|{\omega }_d|$

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

$\begin{array}{cc}{{\omega }_n}^2=\Vert z{\Vert }^2& =z^{*}z\\ {\omega }_d=Imz& =\frac{(z-z^{*})}{2}\\ \sigma =Rez& =\frac{(z+z^{*})}{2}\end{array}$

This then allows us to modify our inequality constraints as:

Rise Time: $z^{*}z-\frac{1.8^2}{{t_r}^2}\le 0$

Settling Time: $\frac{(z+z^{*})}{2}+\frac{4.6}{t_s}\le 0$

Percent Overshoot: $z-z^{*}+\frac{\pi }{ln(M_p)}|z+z^{*}|\le 0$

The Optimization Problem

A description of the Problem to be solved, if appropriate.

The LMI: The Observer D-Stability

Title and mathematical description of the LMI formulation.

$\begin{array}{cc}\text{Find}X:& \\ \left[\begin{array}{c}X\end{array}\right]& >0\\ \left[\begin{array}{c}{A}^{T}X+XA\end{array}\right]& <0\end{array}$

Conclusion:

Interpretation of the results of the LMI

Implementation

A link to CodeOcean or other online implementation of the LMI

Related LMIs

• [[../Controller D-Stability/]]