LMIs in Control/pages/Quadratic Schur Stabilization

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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]

Schur Complement
Schur Stabilization

External Links [edit | edit source]