LMIs in Control/Click here to continue/LMIs in system and stability Theory/Observer D-stability

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The System


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

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: $$\omega_n{\leq}{1.8 \over t_r}$$

Settling Time: $$\sigma{\leq}{-4.6 \over t_s}$$

Percent Overshoot: $$\sigma{\leq}{-ln({M_p}) \over {\pi}}|{\omega_d}|$$

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

\begin{align} {\omega_n}^2=\|z\|^2&=z^{*} z\\ {\omega_d}=Im{z}&={(z-z^{*}) \over 2}\\ {\sigma}=Re{z}&={(z+z^{*}) \over 2} \end{align}$$

This then allows us to modify our inequality constraints as:

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

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

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

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{align} \text{Find} \; X:&\\ \begin{bmatrix} X\end{bmatrix}&>0\\ \begin{bmatrix} A^T X+XA\end{bmatrix}&<0 \end{align}$$

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/