LMIs in Control/Matrix and LMI Properties and Tools/Generalized Negative Imaginary Lemma

Introduction These systems are often related to systems involving energy dissipation. But the standard Positive real theory will not be helpful in establishing closed-loop stability. However, transfer functions of systems with a degree more than one can be satisfied with the negative imaginary conditions for all frequency values and such systems are called systems with negative imaginary frequency response. The System Consider a square continuous time Linear Time invariant system, with the state space realization The Data The LMI Consider an NI transfer matrix G1(s) and an SNI transfer matrix G2(s) = C2(s1 - A2)-1 B2 + D2. The condition λ̅ (G1(0)G2(0) < 1 is satisfied if and only if ST(-C2A2-1B2 + D2)S < 1, Conclusion The above equation holds true if and only if S ST = G1(0). Implementation This can be implemented in any LMI solver such as YALMIP, using an algorithmic solver like Gurobi. Related LMIs Negative Imaginary Lemma