The Kalman-Yakubovich-Popov inequality for differential-algebraic systems
In this well-written paper, the authors revisit the Kalman-Yakubovich-Popov inequality for differential-algebraic control systems of the form \[ E\dot x(t)=Ax(t)+Bu(t),\tag{1} \] where \(E,\,A\in \mathbb K^{n\times n}\), \(B\in \mathbb K^{n\times m}\), \(\mathbb K= \mathbb C\) or \(\mathbb R\), and the pencil \(sE-A\) is supposed to be regular. Unlike previous works on this topic, the authors neither assume impulse controllability nor make any index assumption on system (1). First, the authors point out the relation between the positive semi-definiteness of the Popov function on the imaginary axis and the solvability of the Kalman-Yakubovich-Popov inequality restricted to a certain subspace associated with system (1). Then, the authors characterize the solution set of the Lur'e equation associated with system (1) and discuss also its connection to the rank-minimizing solutions of the Kalman-Yakubovich-Popov inequality. Finally, consequences of these results for the linear-quadratic optimal control problem and comparison to other approaches are discussed.
linear-quadratic optimal control problems
scientific article; zbMATH DE number 6493724
scientific article