LMIs in Control/pages/D-Stabilization of Switched Systems

D-Stability Controller for Switched Systems

This LMI lets you specify desired performance metrics like rising time, settling time and percent overshoot. 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 $$A_i\in\mathbb{R}^{mxm}$$, $$B_i\in\mathbb{R}^{mxn}$$, $$C_i\in\mathbb{R}^{pxm}$$, and $$D_i\in\mathbb{R}^{qxn}$$ 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 ($$t_r$$)
 * 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. 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: $$\omega_n{\leq}{1.8 \over t_r}$$

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: An LMI for Quadratic D-Stabilization
Suppose there exists $$ P > 0 $$ and $$ Z $$ such that



\begin{align} \begin{bmatrix} -rP && A_iP+B_iZ \\ (A_iP+B_iZ)^T && -rP \end{bmatrix} < 0 \\

A_iP+B_iZ+(A_iP+B_iZ)^T +2\alpha P&<0, and\\

\end{align}$$



\begin{align} \begin{bmatrix} A_iP+B_iZ+(A_iP+B_iZ)^T && c(A_iP+B_iZ-(A_iP+B_iZ)^T) \\ c((A_iP+B_iZ)^T -(A_iP+B_iZ)) && A_iP+B_iZ+(A_iP+B_iZ)^T\end{bmatrix} < 0 \end{align}$$

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