LMIs in Control/Stability Analysis/Continuous Time/Strong Stabilizability

The System

Consider the continous-time LTI system, $G:L_{2e}\to L_{2e}$ with state-space realization (A,B,C,0)

$\begin{array}{cc}\dot{x}(t)& =Ax(t)+Bu(t),\\ y(t)& =Cx(t),\end{array}$

where $A\in {ℝ}^{n\times n}$, $B\in {ℝ}^{n\times m}$, $C\in {ℝ}^{p\times n}$, and it and it is assumed that (A, B) is stabilizable, (A, C) is detectable, and the transfer matrix $G(s)=C(s1-A^{-1})B$ has no poles on the imaginary axis.

The Data

The matrices $A,B,C$.

The Optimization Problem

The system G is strongly stabilizable if there exist $P\in {𝕊}^n$, $Z\in {ℝ}^{n\times p}$, and $\gamma \in {ℝ}_{>0}$, where $P>0$, such that

$\begin{array}{c}PA+A^T+ZC+C^T{Z}^T<0\\ \begin{array}{c}\left[\begin{array}{ccccc}-P(A+BF)+(A+BF{)}^{T}P+ZC+C^T{Z}^T& & -Z& & -XB\\ *& & -\gamma I& & 0\\ *& & *& & -\gamma I\end{array}\right]<0\end{array}\\ \end{array}$

Conclusion:

where $F=-B^{T}X$ and $X\in S_n$ , $X\ge 0$ is the solution to the Lyapunov equation given by

$\begin{array}{c}XA+A^{T}X-XB{B}^{T}X=0\end{array}$

Moreover, a controller that strongly stabilizes G is given by the state-space realization

$\begin{array}{c}{\dot{x}}_c=(A+BF+P^{-1}ZC)x(t)-P^{-1}Zu(t)\\ y_C=-B^{T}Xx(t)\end{array}$

Implementation

• [1] Example Code

Related LMIs

• https://en.wikibooks.org/wiki/LMIs_in_Control/Stability_Analysis/Discrete_Time/DiscreteTimeStrongStabilizability - Discrete Time Strong Stabilizability

External Links

• http://control.asu.edu/MAE598_frame.htm LMI Methods in Optimal and Robust Control- A course on LMIs in Control by Matthew Peet.

• https://https://arxiv.org/abs/1903.08599/ LMI Properties and Applications in Systems, Stability, and Control Theory] - A List of LMIs by Ryan Caverly and James Forbes.

• https://web.stanford.edu/~boyd/lmibook/ LMIs in Systems and Control Theory- A downloadable book on LMIs by Stephen Boyd.