LMIs in Control/Click here to continue/LMIs in system and stability Theory/Hurwitz Stabilizability

This section studies the stabilizability properties of the control systems. The System 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 are system matrices. Definition 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 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 This implementation requires Yalmip and Mosek. https://github.com/ShenoyVaradaraya/LMI--master/blob/main/Hurwitz_Stabilizability.m Conclusion Compared with the second rank condition, LMI has a computational advantage while also maintaining numerical reliability. References 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 Return to Main Page: