# LMIs in Control/pages/CT-SOFS

LMIs in Control/pages/CT-SOFS

In view of applications, static feedback of the full state is not
feasible in general: only a few of the state variables (or a linear
combination of them,
, called the output) can be
actually measured and re-injected into the system.
**So, we are led to the notion of static output feedback**

**The System**Edit

Consider the continuous-time LTI system, with generalized state-space realization

**The Data**Edit

**The Optimization Problem**Edit

This system is static output feedback
stabilizable (SOFS) if there exists a matrix
F such that the closed-loop system

**(obtained by replacing which means applying static output feedback)**

is asymptotically stable at the origin

** The LMI: LMI for Continuous Time - Static Output Feedback Stabilizability**Edit

The system is static output feedback stabilizable if and only if it satisfies any of the following conditions:

- There exists a and , where , such that

- There exists a and , where , such that

- There exists a and , where , such that

- There exists a and , where , such that

**Conclusion**Edit

On implementation and optimization of the above LMI using YALMIP and MOSEK (or) SeDuMi we get 2 output matrices one of which is the Symmeteric matrix (or ) and

**Implementation**Edit

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

** Related LMIs**Edit

Discrete time Static Output Feedback Stabilizability

Static Feedback Stabilizability

** External Links**Edit

- [1] - LMI in Control Systems Analysis, Design and Applications
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- D. d. S. Madeira and J. Adamy, "Static output feedback: An LMI condition for stabilizability based on passivity indices," 2016 IEEE Conference on Control Applications (CCA), Buenos Aires, 2016, pp. 960-965.