Analytical approximation method for the center manifold in the nonlinear output regulation problem

Abstract

In nonlinear output regulation problems, it is necessary to solve the so-called regulator equations consisting of a partial differential equation and an algebraic equation. It is known that, for the hyperbolic zero dynamics case, solving the regulator equations is equivalent to calculating a center manifold for zero dynamics of the system. The present paper proposes a successive approximation method for obtaining center manifolds and shows its effectiveness by applying it for an inverted pendulum example.

Periodic and quasi-periodic orbit design based on the center manifold theory

“…To compute the center manifold in the CRTBP, this step is based on the center manifold theorem [33,34] and the successive approximation method [31].…”

“…Independently, Nagata et al proposed the center manifold method for bounded orbits [30]. Both methods above relied on the successive approximation method to calculate center manifolds proposed by Suzuki et al [31]. Comparing other proposed techniques to obtain bounded orbits, the main advantage of the center manifold method is that all the bounded orbits including quasi-periodic orbits lying on center manifolds are parametrized by a single parameter vector.…”

Explicit Form of Station-Keeping and Formation Flying Controller for Libration Point Orbits

“…Using the nonlinear output regulation theory [14,15], a general form of controller is analytically derived to achieve station-keeping and formation flying of periodic and quasi-periodic (invariant tori) orbits embedded on a four-dimensional center manifold in the CRTBP.…”

“…Based on the output regulation theory of a linear system [11], a control law to realize the formation flying and station-keeping was proposed by Bando and Ichikawa [12]. The periodic Riccati differential equations were used by Peng et al for the station-keeping and formation flying based on the linear periodic time-varying equation of the relative motion around a libration point orbit [13].In this paper, station-keeping and formation flying along unstable libration point orbits are considered.Using the nonlinear output regulation theory [14,15], a general form of controller is analytically derived to achieve station-keeping and formation flying of periodic and quasi-periodic (invariant tori) orbits embedded on a four-dimensional center manifold in the CRTBP.A standard approach to derive such a controller is to linearize the system along a reference orbit and then stabilize the linearized system [6,16]. In our approach, the problem is solved as a nonlinear output regulation problem which allows us to derive a general form for station-keeping and formation flying controllers without the linearization assumption.…”

Trajectory Design Using the Center Manifold Theory in the Circular Restricted Three-Body Problem

“…Especially, a new semianalytical theory to approximate a solution to ordinal differential equations on center manifold 11) is applied to design libration point orbits around the Lagrangian points. Typical methods to calculate a center manifold based on the Taylor series approximation give rise to several difficulties regarding the operational complexity.…”