The Kalman–Yakubovich–Popov inequality for differential-algebraic systems

In this paper we revisit the Kalman–Yakubovich–Popov lemma for differential-algebraic control systems. This lemma relates the positive semi-definiteness of the Popov function on the imaginary axis to the solvability of a linear matrix inequality on a certain subspace. Further emphasis is placed on the Lur'e equation, whose solution set consists, loosely speaking, of the rank-minimizing solutions of the Kalman–Yakubovich–Popov inequality. We show that there is a correspondence between the solution set of the Lur'e equation and the deflating subspaces of certain even matrix pencils. Finally, we show that under certain conditions the Lur'e equation admits stabilizing, anti-stabilizing, and extremal solutions. We note that, for our results, we neither assume impulse controllability nor make any assumptions on the index of the system.