LMIs in Control/Stability Analysis/Discrete Time/DiscreteTimeStrongStabilizability

The System

Consider the continous-time LTI system, $G:L_{2e}\rightarrow L_{2e}$ with state-space realization ($A_{d},B_{d},C_{d},0$)

${\begin{aligned}{\dot {x}}(k)&=A_{d}x(k)+B_{d}u(k),\\y(k)&=C_{d}x(k),\end{aligned}}$

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

The Data

The matrices $A_{d},B_{d},C_{d}$.

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

${\begin{aligned}{\displaystyle {\begin{aligned}{\begin{bmatrix}A_{d}^{T}PA_{d}-P-A_{d}^{T}ZC_{d}-C_{d}^{T}Z^{T}A_{d}&&C_{d}^{T}Z^{T}\\*&&-P\\*&&*&&-\gamma I\end{bmatrix}}<0\end{aligned}}}\\{\displaystyle {\begin{aligned}{\begin{bmatrix}N_{11}&&(A_{d}+B_{d}F)^{T}Z&&XB_{d}&&C_{d}^{T}Z^{T}\\*&&-\gamma I&&0&&Z^{T}\\*&&*&&-\gamma I&&0\\*&&*&&*&&-P\end{bmatrix}}<0\end{aligned}}}\\\end{aligned}}$

Conclusion:

where $N_{11}=(A_{d}+B_{d}F)^{T}P(A_{d}+B_{d}F)-P+(A_{d}+B_{d}F)^{T}ZC_{d}^{T}Z^{T}(A_{d}+B_{d}F),F=-B_{d}^{T}X,X=Y$ and $X\in S_{n}$ , $X\geq 0$ is the solution to the discrete-time Lyapunov equation given by

${\begin{aligned}A_{d}XA_{d}^{T}+X-B_{d}B_{d}^{T}X=0\end{aligned}}$

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

${\begin{aligned}{\dot {x}}_{c,k+1}=(A_{d}+B_{d}F+P^{-1}ZC_{d})x_{k}-P^{-1}Zu_{k}\\y_{c,k}=-B_{d}^{T}Xx_{k}\end{aligned}}$

Implementation

• [1]-example code

Related LMIs

• https://en.wikibooks.org/wiki/LMIsinControl/StabilityAnalysis/ContinuousTime/StrongStabilizability - Continuous 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.