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

This section studies the stabilizability properties of the control systems.

The System

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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

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  are system matrices.

Definition

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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:

 

 

 

 

 

(1)

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

LMI Condition

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

 

 

 

 

 

(2)

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

 

 

 

 

 

(3)

Letting

 

 

 

 

 

(4)

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

Implementation

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This implementation requires Yalmip and Mosek.

Conclusion

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

References

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