Use of moving frames in the representation of nonlinear and nonplanar state-space flows

It is shown that the velocity field in a state space associated with the dynamical behaviour of a multivariable system may be represented by a surface in a spin space. When two vector fields are combined in the state space, the corresponding combination of the surfaces in the spin space takes place according to a simple vector-addition rule. The technique may be applied to nonlinear systems. For linear systems, the surfaces involved are intimately associated with circle representations of the flow in a 2-dimensional space. They may thus be useful in developing design techniques for multivariable regulators by a partial representation of the flow in terms of sets of planar loci, and by then using the additive property of the representations.