# LMIs in Control/pages/DT-SOFS

LMIs in Control/pages/DT-SOFS

The static output feedback (SOF) problem has been investigated and analyzed by many people and the literature concerning this topic is vast. In practicality, it is not always possible to have full access to the state vector and only a partial information through a measured output is available. This explains why this problem has challenged many researchers in control theory.
Here is a systematic approach for the SOF control design for discrete time linear systems.

## The System

Consider a discrete-time LTI system, with state-space realization ${\displaystyle (Ad,Bd,Cd,0)}$ ,

{\displaystyle {\begin{aligned}x_{k+1}=A_{d}x_{k}+B_{d}u_{k}\\y_{k}=C_{d}x_{k}\end{aligned}}}

${\displaystyle x_{k}\in \mathbb {R} ^{n}}$  is the state, ${\displaystyle y_{k}\in \mathbb {R} ^{p}}$  is the measured output, ${\displaystyle u_{k}\in \mathbb {R} ^{m}}$  is the control input.

## The Data

${\displaystyle A_{d}\in \mathbb {R} ^{n\times n},B_{d}\in \mathbb {R} ^{n\times m},C_{d}\in \mathbb {R} ^{p\times n}}$  are the constant matrices of appropriate dimensions.

## The Optimization Problem

The full state is not measurable and only partial information is available through ${\displaystyle y_{k}}$  which can used for control purposes.

We have to find a static output feedback gain with respect to

${\displaystyle u_{k}=-K_{d}y_{k},}$

where ${\displaystyle K_{d}\in \mathbb {R} ^{m\times p}}$  is the output feedback gain such that the final closed loop system is asymptotically stable.

## The LMI: LMI for Discrete-Time Static Output Feedback Stabilizability

The discrete time system considered is static output feedback stabilizable under any of the following equivalent necessary or sufficient conditions.

• There exists a ${\displaystyle K_{d}\in \mathbb {R} ^{m\times p}}$  and ${\displaystyle P\in \mathbb {S} ^{n}}$  where ${\displaystyle P>0}$  such that
${\displaystyle {\begin{bmatrix}-P&(A_{d}+B_{d}K_{d}C_{d})P\\((A_{d}+B_{d}K_{d}C_{d})P)'&-P\end{bmatrix}}}$ {\displaystyle {\begin{aligned}<0\end{aligned}}.}

• There exists a ${\displaystyle K_{d}\in \mathbb {R} ^{m\times p}}$  and ${\displaystyle P\in \mathbb {S} ^{n}}$  where ${\displaystyle P>0}$  such that
${\displaystyle {\begin{bmatrix}-A_{d}PPA_{d}^{T}&A_{d}P+B_{d}K_{d}C_{d}&A_{d}P\\(A_{d}P+B_{d}K_{d}C_{d})'&-1&0\\PA_{d}^{T}&0&-P\end{bmatrix}}}$ {\displaystyle {\begin{aligned}<0\end{aligned}}.}

## Conclusion

If it is feasible we obtain a output feedback gain matrix ${\displaystyle K_{d}}$  such that the closed loop system is asymptotically stable.
While implementing the optimization problem the following conditions are assumed to be satisfied

• The Transfer matrix and its inverse are both analytical a s=0
• The matrix ${\displaystyle C_{d}A_{d}^{-1}B_{d}}$  is non-singular
• The triple ${\displaystyle (A_{d},B_{d},C_{d})}$  are reachable and observable.

## Implementation

A link to the Matlab code for a simple implementation of this problem in the Github repository:

## Related LMIs

Continuous-time Static Output Feedback Stabilizability