On Performance Guarantees for Systems with Conic Constraints

In this thesis, we provide a number of novel algebraic means of certifying stability and performance for linear systems constrained in various ways by cones. The purpose is mainly threefold: to provide mathematical statements with applicative potential, to unify seemingly dissimilar results in the literature and thereby increase understanding, and to advance the state of the art on dynamical systems with conic constraints, an area of control still in its infancy. The main contributions of the five papers contained in the thesis are as follows. Paper I provides an analytical upper bound on the deviation from H-infinity optimality of a certain controller class as a function of the deviation from symmetry in the state matrix. Paper II goes on to establish a diagonal solution to a Riccati inequality which certifies H-infinity optimality of a particular controller both when the open-loop state matrix is symmetric and when the closed-loop system is positive. In Paper III, a necessary and sufficient condition is given in the form of a stable coefficient matrix for a nonsymmetric Riccati equation to admit a stabilizing cone-preserving solution. This result is subsequently applied in Paper IV to obtain a nonsymmetric variant of the bounded real lemma in H-infinity control on self-dual cones. Finally, Paper V establishes an equivalence between the existence of a bounded linear functional satisfying a conic inequality and the satisfaction of certain integral linear constraints on trajectories confined to a cone. This result in turn yields a non-strict version of the Kalman-Yakubovich-Popov Lemma when the cone is taken as the positive semidefinite cone, thereby serving to further bring together linear-cone theory with the dominating linear-quadratic paradigm in control.