Dual-lattice theorems in the geometric approach

A one-to-one correspondence is shown to exist between the lattice of all self-bounded (A, ℬ)-controlled invariants contained in ℒ and the lattice of all self-hidden (A, ℒ)-conditioned invariants containing ℬ. This correspondence, stated herein as the main dual-lattice theorem, allows a straightforward derivation of the universal bounds of the lattices, particularly when additional constraints are imposed, such as to contain a given subspace ℘ for the elements of the former lattice and to be contained in a given subspace ℒ for the elements of the latter. Then, two further minor dual-lattice theorems, dual to each other, are presented, and some connections and applications of the new theory to standard control and observation problems are briefly discussed.