A framework for optimal controller design with generalized ℓ2 objectives is presented. The allowable disturbances are constrained to be in a pre-specified set; the design objective is to ensure that the resulting output errors do not belong to another pre-specified set. The solution takes the form of an affine matrix inequality (AMI), which is both a necessary
and sufficient condition for the posed problem to have a solution. In the simplest case, the resulting optimization reduces to standard ℋ∞ synthesis
