Differential geometric approach is useful for solving a special class of nonlinear control problems. The purpose of this paper is to present a linearization method of nonlinear systems by the Riemannian geometric approach, and to apply the linearization mapping to the design of nonlinear systems.
A geomeric model can be derived by replacing the orthogonal straight coordinate frame on the state space with a suitable curvilinear frame. Such a Riemannian geometric model has been proposed by the authors after the derivation of the geodesic curve on the gravitational gauge field in Einstein's principle of general relativity.
In this paper an attention is placed on a problem to decrease the dimension of the Riemannian space of the model by a proper choice of the construction of the space, which leads to decrease the computation time remarkably. For the design of control systems, a new quadratic-form performance index is introduced using Riemannian metric tensors. A nonlinear optimal regulator is constructed which is homeomorphic to the corresponding linear optimal regulator. A method to derive a curvilinear coordinates frame fitted to the nonlinear system is proposed by solving a partial differential equation with respect to the homeomorphism. A computational algorithm is proposed and numerical examples are shown.