LMIs in Control/Stability Analysis/Discrete Time/DiscreteTimeStrongStabilizability

The System

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

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

where $A_d\in {ℝ}^{n\times n}$, $B_d\in {ℝ}^{n\times m}$, $C_d\in {ℝ}^{p\times n}$, 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 $P\in {𝕊}^n$, $Z\in {ℝ}^{n\times p}$, and $\gamma \in {ℝ}_{>0}$, where $P>0$, such that

$\begin{array}{c}\begin{array}{c}\left[\begin{array}{ccc}{A}_d^{T}P{A}_d-P-A_d^{T}Z{C}_d-C_d^T{Z}^T{A}_d& & C_d^T{Z}^T\\ *& & -P\\ *& & *& & -\gamma I\end{array}\right]<0\end{array}\\ \begin{array}{c}\left[\begin{array}{ccccccc}{N}_{11}& & (A_d+B_{d}F{)}^{T}Z& & X{B}_d& & C_d^T{Z}^T\\ *& & -\gamma I& & 0& & Z^T\\ *& & *& & -\gamma I& & 0\\ *& & *& & *& & -P\end{array}\right]<0\end{array}\\ \end{array}$

Conclusion:

where $N_{11}=(A_d+B_{d}F{)}^{T}P(A_d+B_{d}F)-P+(A_d+B_{d}F{)}^{T}Z{C}_d^T{Z}^T(A_d+B_{d}F),F=-B_d^{T}X,X=Y$ and $X\in S_n$ , $X\ge 0$ is the solution to the discrete-time Lyapunov equation given by

$\begin{array}{c}{A}_{d}X{A}_d^T+X-B_d{B}_d^{T}X=0\end{array}$

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

$\begin{array}{c}{\dot{x}}_{c,k+1}=(A_d+B_{d}F+P^{-1}Z{C}_d)x_k-P^{-1}Z{u}_k\\ y_{c,k}=-B_d^{T}X{x}_k\end{array}$

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.