LMIs in Control/pages/Continuous Time D-Stability Controller

Continuous-Time D-Stability Controller

In some control problems, people are still interested in designing a controller whose poles are located in specific regions of a complex plane while also simultaneously ensuring that the controller is stable. One such method with which this can be possible is called D-stability.

The System
Suppose we were given the continuous-time system



\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ y(t)&=Cx(t)+Du(t) \end{align}$$

whose stability was not known, and where $$A\in\mathbb{R}^{mxm}$$, $$B\in\mathbb{R}^{mxn}$$, $$C\in\mathbb{R}^{pxm}$$, and $$D\in\mathbb{R}^{qxn}$$ for any $$t\in\mathbb{R}$$. Then the controller that simultaneously stabilizes the above system while ensuring that the poles are at their desired location can be achieved by the controller $$u(t)=Kx(t)$$.

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 $$A$$, $$B$$, $$C$$, $$D$$
 * rise time ($$t_r$$)
 * settling time ($$t_s$$)
 * percent overshoot ($$M_p$$)

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. In order to do that, we have to first 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: $$z-z^{*}+{{\pi} \over ln({M_p})}|z+z^{*}|{\leq}0$$

which not only allows us to map the relationship between complex pole locations and inequality constraints, but it also now allows us to easily formulate our LMIs for this problem.

The LMI: The Continuous-Time Controller D-Stability
Keeping the above inequalities in mind, we observe the following:

Suppose there now exists a symmetric matrix $$P>0$$ and matrix $$Z$$, we can now determine the controller the following LMIs:



\begin{align} \begin{bmatrix} -rP&&AP+BZ\\(AP+BZ)^T&&-rP\end{bmatrix}&<0\\ AP+BZ+(AP+BZ)^T +2\alpha P&<0, and\\ \begin{bmatrix} AP+BZ+(AP+BZ)^T&&c(AP+BZ-(AP+BZ)^T)\\c((AP+BZ)^T -(AP+BZ))&&AP+BZ+(AP+BZ)^T\end{bmatrix}&<0 \end{align}$$

Conclusion:
Given the resulting controller $$K=ZP^{-1}$$, we can now determine that the pole locations $$z\in\mathbb{C}$$ of $$A+BK$$ satisfies the inequality constraints $$|x|{\leq}r$$, $$Re(x){\leq}-{\alpha}$$ and $$z+z^{*}{\leq}-c|z-z^{*}|$$.

Implementation

 * Example Code - A GitHub link that contains code (titled "ControllerDStability.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs

 * ../Continuous Time D-Stability Observer/ - Equivalent D-stability LMI for a continuous-time observer.
 * ../D stabilization/ - LMI for $$\mathbb{D}_{(q,r)}$$-stabilization.