Minimization of L2-Sensitivity for Linear Discrete-Time Systems Subject to L2-Scaling Constraints Using a Lagrange Function and a Bisection Method

In this paper, the problem of minimizing an L
₂
-sensitivity measure subject to L
₂
-norm dynamic-range scaling constraints for linear discrete-time systems is investigated. To this end, an expression for evaluating the L
₂
-sensitivity is introduced and an L
₂
-sensitivity minimization problem subject to the L
₂
-scaling constraints is formulated. An efficient iterative algorithm is then explored to solve the constraint optimization problem directly. The proposed solution method relies on neither converting the constrained optimization problem into an unconstrained one nor using a quasi-Newton algorithm, but utilizes a Lagrange function, some matrix-theoretic techniques, and a bisection method. The optimal state-space model structure with minimum L
₂
-sensitivity and no overflow oscillations is constructed by applying an appropriate coordinate-transformation matrix. A numerical example is presented to illustrate the utility of the proposed technique