Asymptotic tracking for linear and nonlinear systems: A two-point boundary value formulation

In this paper, the output tracking problem for linear and nonlinear systems in normal form is revisited in the case of non-minimum phase dynamics and in the absence of any model for the exosystem generating the periodic reference signal. The asymptotic output tracking is obtained by replacing the classical stability condition for the zero dynamics driven by the reference signal with the existence of a solution to a two-point boundary value problem. The latter solution allows to generate the ideal steady-state evolution of the zero dynamics. This strategy allows to retain two interesting features of standard tracking and regulation approaches, namely, i) the use of a simple linear feedback structure and without the need of solving any partial differential equation (as in tracking) and ii) avoiding minimum phase requirements (as in output regulation). Since the same kind of two-point boundary value problem usually arises in the application of the Pontryagin Minimum Principle in optimal control, a natural question to ask is whether there is an underlying connection; hence, in the linear case, further insights are achieved by comparison with an alternative solution based on optimal control problem formulated for an auxiliary affine system.