LMIs in Control/Stability Analysis/Continuous Time/Hurwitz Stabilizability
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]
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013