In this paper formulas are derived for the analytic center of the solution set of linear matrix inequalities (LMIs) defining passive transfer functions. The algebraic Riccati equations that are usually associated with such systems are related to boundary points of the convex set defined by the solution set of the LMI. It is shown that the analytic center is described by closely related matrix equations, and their properties are analyzed for continuous-and discrete-time systems. Numerical methods are derived to solve these equations via steepest descent and Newton methods. It is also shown that the analytic center has nice robustness properties when it is used to represent passive systems. The results are illustrated by numerical examples.
In this work we investigate explicit and implicit difference equations and the corresponding infinite time horizon linear-quadratic optimal control problem. We derive conditions for feasibility of the optimal control problem as well as existence and uniqueness of optimal controls under certain weaker assumptions compared to the standard approaches in the literature which are using algebraic Riccati equations. To this end, we introduce and analyze a discrete-time Lur'e equation and a corresponding Kalman-Yakubovich-Popov inequality. We show that solvability of the Kalman-Yakubovich-Popov inequality can be characterized via the spectral structure of a certain palindromic matrix pencil. The deflating subspaces of this pencil are finally used to construct solutions of the Lur'e equation. The results of this work are transferred from the continuous-time case. However, many additional technical difficulties arise in this context. a discrete-time version of the inequality introduced in [36], where V (E, A, B) denotes an inequality projected on a certain subspace V (E, A, B) , i. e., V * M(P )V 0 holds for any basis matrix V of V (E, A, B) . We show statements which relate the solvability of this inequality to the non-negativity of the Popov function on the unit circle, a certain rational matrix function defined bywhere G ∼ (z) := G z −1 * for a rational matrix G(z) ∈ K(z) n×n . In Sections 4 and 5 we introduce the notion of inertia for palindromic matrix pencils evaluated on the unit circle and provide spectral characterizations regarding positivity of the Popov function, similar to the characterizations which were obtained in [35] and [41] for even matrix pencils in the continuous-time case.In Section 6 we investigate the Lur'e equation for the discrete-time optimal control problem which is a generalization of the algebraic Riccati equation (2). This means that we seek solution triples (X, K, L) ∈ K n×n × K q×n × K q×m fulfilling
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