On Brunovsky Numbers and Observability and Controllability Indices in Nonlinear MIMO Systems
Mohammad Reza Rahmati, Gerardo Flores · SIAM Journal on Control and Optimization · 2025
Abstract. When the exact linearization problem is solvable for nonlinear multi-input multi-output systems, it is possible to conduct the linearization in two standard ways. The first way employs a sequence of integrable distributions defined by the vector fields involved in the system. The second way uses the output functions and the codistributions defined as the kernels of codistributions. In both cases, one ends up with a change of coordinates, which transforms the system to canonical block forms, where the sizes of the blocks are invariants of the system. One can associate two sets of indices, canonical invariants of the system, called Brunovsky indices. This work compares these two sets of invariants obtained in a system of dimension [Formula: see text]. The indices are classically called the Brunovsky controllability indices and Brunovsky observability indices. We prove that the two sets of invariants give transpose partitions of [Formula: see text]. That is, if [Formula: see text] are the controllability indices and [Formula: see text] are observability indices of the same nonlinear system, then the two partitions [Formula: see text] are transposed to each other. In other words, the sizes of blocks that appear in the above two canonical forms are not only in general identical, but they may also have a different number of blocks. Therefore, they generally determine two different partitions of the dimension of the system. In addition, we present several conditions that characterize the Brunovsky canonical forms, in both the controllable and observable cases, and prove their mutual equivalence. We also discuss a relevant duality between the flag varieties parametrizing the Brunovsky systems of the same type, i.e., with the same Brunovsky numbers. The duality says that the parabolic algebras associated to the two flag varieties parametrizing systems with the given Brunovsky indices [Formula: see text] and [Formula: see text] are dual parabolic algebras.