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

The System

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

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

where $A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$, 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 $P\in \mathbb {S} ^{n}$, $Z\in \mathbb {R} ^{n\times p}$, and $\gamma \in \mathbb {R} _{>0}$, where $P>0$, such that

${\begin{aligned}PA+A^{T}+ZC+C^{T}Z^{T}<0\\{\displaystyle {\begin{aligned}{\begin{bmatrix}-P(A+BF)+(A+BF)^{T}P+ZC+C^{T}Z^{T}&&-Z&&-XB\\*&&-\gamma I&&0\\*&&*&&-\gamma I\end{bmatrix}}<0\end{aligned}}}\\\end{aligned}}$

Conclusion:

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

${\begin{aligned}XA+A^{T}X-XBB^{T}X=0\end{aligned}}$

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

${\begin{aligned}{\dot {x}}_{c}=(A+BF+P^{-1}ZC)x(t)-P^{-1}Zu(t)\\y_{C}=-B^{T}Xx(t)\end{aligned}}$

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.