LMIs in Control/Stability Analysis/Continuous Time/Hurwitz Stabilizability

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This section studies the stabilizability properties of the control systems.

The System[edit | edit source]

Given a state-space representation of a linear system

Where represents the differential operator ( when the system is continuous-time) or one-step forward shift operator ( Discrete-Time system). are the state, output and input vectors respectively.

The Data[edit | edit source]

are system matrices.

Definition[edit | edit source]

The system , or the matrix pair is Hurwitz Stabilizable if there exists a real matrix such that is Hurwitz Stable. The condition for Hurwitz Stabilizability of a given matrix pair (A,B) is given by the PBH criterion:






The PBH criterion shows that the system is Hurwitz stabilizable if all uncontrollable modes are Hurwitz stable.

LMI Condition[edit | edit source]

The system, or matrix pair is Hurwitz stabilizable if and only if there exists symmetric positive definite matrix and such that:






Following definition of Hurwitz Stabilizability and Lyapunov Stability theory, the PBH criterion is true if and only if , a matrix and a matrix satisfying:












Putting (4) in (3) gives us (2).

Implementation[edit | edit source]

This implementation requires Yalmip and Mosek.

Conclusion[edit | edit source]

Compared with the second rank condition, LMI has a computational advantage while also maintaining numerical reliability.

References[edit | edit source]

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