A geometric approach to the synthesis of failure detection filters

A geometric formulation of Beard\´s failure detection filter problem is stated using the concepts of -invariant and unobservability subspaces. The notions of output separable and mutually detectable families of subspaces introduced by Beard are also clarified. It is shown that mutual detectability is a necessary and sufficient condition for the existence of a detection filter with arbitrarily assignable spectrum. Moreover, it is shown that the failure detection falter problem has a computationally simple solution when the failure events satisfy some mild restrictions. Finally, the complete duality between a generalization of Beard\´s detection filter problem and the restricted control decoupling problem is illustrated.