Optimal Synthesis of State-Estimate Feedback Controllers With Minimum $l_{2}$-Sensitivity

This paper investigates the problem of synthesizing the optimal structure of a state-estimate feedback controller with minimum
l

₂
-sensitivity and no overflow. First, the
l

₂
-sensitivity of a closed-loop transfer function with respect to the coefficients of a state-estimate feedback controller is analyzed. Next, two iterative techniques for obtaining the coordinate transformation matrix which constructs the optimal structure of a state-estimate feedback controller are developed so as to minimize an
l

₂
-sensitivity measure subject to
l

₂
-scaling constraints. One technique is based on a Lagrange function, some matrix-theoretic techniques, and an efficient bisection method. Another technique converts the problem into an unconstrained optimization formulation by using linear-algebraic techniques, and optimizes it by applying an efficient quasi-Newton method with closed-form formula for gradient evaluation. A numerical example is also presented to illustrate the utility of the proposed techniques.