LMIs in Control/pages/Quadratic Schur Stabilization
LMI for Quadratic Schur Stabilization
A discrete-time system is said to be stable if all roots of its characteristic equation lie in the open unit disk. This provides a condition for the stability of discrete-time linear systems with polytopic uncertainties and a linear time-invariant system with this property is called a Schur stable system.
The System[edit | edit source]
Consider discrete time system
where , , at any .
The system consist of uncertainties of the following form
where ,, and
The Data[edit | edit source]
The matrices necessary for this LMI are , , and
The LMI:[edit | edit source]
There exists some X > 0 and Z such that
The Optimization Problem[edit | edit source]
The optimization problem is to find a matrix such that:
According to the definition of the spectral norms of matrices, this condition becomes equivalent to:
Using the Lemma 1.2 in LMI in Control Systems Analysis, Design and Applications (page 14), the aforementioned inequality can be converted into:
Conclusion:[edit | edit source]
The Controller gain matrix is extracted as
It follows that the trajectories of the closed-loop system (A+BK) are stable for any
Implementation[edit | edit source]
Related LMIs[edit | edit source]
External Links [edit | edit source]
- 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.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMI in Control Systems Analysis, Design and Applications