State feedback Φ∞ synthesis of multivariable linear systems

Abstract

In this paper, we formulate a ${\Phi }_{\infty }$ synthesis problem for multivariable linear time-invariant (LTI) systems, i.e., controller synthesis with closed-loop phase constraints, which complements the $H_{\infty }$ synthesis problem. A direct imitation of the solution of $H_{\infty }$ synthesis would generally lead to complex-valued controllers for ${\Phi }_{\infty }$ synthesis, which are not physically realizable. We derive a necessary and sufficient condition for solvability of the state feedback ${\Phi }_{\infty }$ synthesis, formulated through a linear matrix inequality (LMI) subject to a rank condition. An algorithm is proposed to handle the difficulty caused by the rank condition. To overcome the limitation of the algorithm in lacking global convergence, we further derive a sufficient solvability condition with a clear geometric interpretation, expressed in terms of a real-coefficient LMI that ensures the resulting controller is real-valued.