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

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

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The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)\\x(0)&=x_{0}\end{aligned}}$

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:

${\begin{aligned}{\omega _{n}}^{2}=\|z\|^{2}&=z^{*}z\\{\omega _{d}}=Im{z}&={(z-z^{*}) \over 2}\\{\sigma }=Re{z}&={(z+z^{*}) \over 2}\end{aligned}}$

This then allows us to modify our inequality constraints as:

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

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.

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

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