A new robustness for discrete time problems with bounded uncertainty

Motivated by a number of problems arising in systems and control, a Generalized Robustness Problem (GRP) is formulated in an operator theoretic setting. Subsequently, two versions of the GRP are considered: a local problem and a global problem. In the local case, the so-called target is fixed and a lemma is established which characterizes robust solutions as feasible points for a system of inequalities. The more general results on global robustness apply to systems with variable targets and lend themselves to some interesting physical interpretations. For example, when the theory is specialized to a class of robust tracking problems, the minimum singular value of a controllability-like matrix plays an important role. In addition to the necessary and sufficient conditions for solvability of the GRP, the paper also includes an illustrative example involving robust ARMA identification. This example is interesting because it shows that for certain feasible sets of observations, the "true parameters" describing the transfer function might be excluded from the set of robust solutions. Said another way, although it is highly desirable to generate the true parameters as a solution to an "ordinary" identification problem, such a solution may be not be consistent with the range of possible observations in the robustness problem.