Loop shaping in mixed H2 and H∞ optimal control

Mixed H₂ and H_∞ optimal control law design by shaping of the loop transfer function to meet engineering goals is considered. The development is closely analogous to the linear quadratic Gaussian with loop transfer recovery (LQG/LTR) procedure for H₂ optimal control law design, with an additional constraint on the H_∞ norm of a certain disturbance transfer function matrix. To facilitate the numerical computation Of the controller, the main results of mixed H₂ and H_∞ optimization are modified by assuming that the design weights are interdependent and that the disturbance inputs are uncorrelated. The condition that forms the basis for loop transfer recovery, under which the dynamic compensator recovers the properties of the state feedback, static gain controller, is derived. A mixed H₂ and H_∞ version of the Kalman equality is derived to show the relationship between the achievable loop shape and the design parameters. A design procedure that implements the equations is given, and a numerical example is shown to illustrate the design procedure.