Abstract
This paper presents a construction of the optimal regulator system of a nonlinear controlled process.
Here the process is described in a power series of Kronecker products of the state vector. The nonlinear description is a natural extension of the usual linear matrix notation. All the mathematical manipulation is conducted in the tensor products. The method is straight forward to obtain the unique solution without any arbitrariness in the expression and the manipulation.
The steps towards the solution are as follows:
(1) To build the Hamiltonian, and to get the canonical equation and the maximizing equation of the Hamiltonian with respect to the control (u).
(2) To expand the adjoint vector in a series of Kronecker powers of the state vector, and to substitute it into the canonical equation.
(3) To equate the coefficients of the same Kronecker powers in the both sides of the canonical equation, and to obtain a set of equations which must be fulfilled by the optimal feedback coefficient tensors (Kn, n=1, 2, …).
The equation for K1 coincides with the Riccati equation of the linear optimal regulator problem for the linear part of the controlled object. The stationary Kn(n≥2) can be obtained from well-posed linear matrix equations successively for n=2, 3, ….
By induction, Kn's are proved to be (1, n) type complete symmetric tensors.
When the process is odd nonlinear and the performance function is even nonlinear, the optimal feedback becomes odd nonlinear.
An example with a third-power-nonlinear process, a forth-power-nonlinear performance function, and a third-power optimal feedback, showed much improved responses over the linearized optimal feedkack, especially for the large initial states.
