Patchy solution of a Francis–Byrnes–Isidori partial differential equation

Abstract

The solution to the nonlinear output regulation problem requires one to solve a first order PDE, known as the Francis-Byrnes-Isidori (FBI) equations. In this paper we propose a method to compute approximate solutions to the FBI equations when the zero dynamics of the plant are hyperbolic and the exosystem is two-dimensional. With our method we are able to produce approximations that converge uniformly to the true solution. Our method relies on the periodic nature of two-dimensional analytic center manifolds.

On the existence and uniqueness of solutions to the output regulation equations for periodic exosystems

“…In view of the current results, in a future paper we intend to extend the numerical algorithm presented in [1] for exosystems of dimension greater than two.…”

“…In this section we present two examples illustrating the applicability of Theorem 3.1 when the center manifold theorem or the results in [1] are not applicable. The first example, taken from [3,6], was a motivation for the current paper.…”

“…The purpose of this paper is to extend the results in [1] by considering exosystems generating periodic trajectories and prove the existence and uniqueness ofπ in the possibly non-hyperbolic case. A key ingredient in the proof of our main result (Theorem 2.2) is the triangular structure of (3) which simplifies the study of the flow of (3).…”

“…Consider the controlled dynamical systeṁ x = f (x, w) + g(x, w)u w = s(w) e = h(x, w) (1) where x ∈ R n is the state variable, u ∈ R m is the control variable, e ∈ R p is the output variable, and w ∈ R q is an external variable. The mappings f : R n ×R q → R n , g : R n ×R q → R n×m , s : R q → R q , and h : R n × R q → R p , defined possibly only locally about the respective origins, are assumed to be smooth.…”

“…Consequently, the eigenvalues of the matrix ∂s ∂w (0) will lie on the imaginary axis, and thus if the eigenvalues of the matrix ∂ζ ∂z (0, 0) lie off the imaginary axis, i.e., the hyperbolic case, then by the well-known center manifold theorem [10] one can deduce that a solution (not necessarily unique) to the PDE (4) exists. In the case of real-analytic data and two dimensional exosystems, applying the main result in [2], it was deduced in [1] that (4) in fact has an unique solution, is analytic and the invariant manifold is generated by a one-parameter family of periodic solutions.…”