Spline Based Pseudo-Inversion of Sampled Data Non-Minimum Phase Systems for an Almost Exact Output Tracking

This paper considers the problem of achieving a very accurate tracking of a pre-specified desired output trajectory (y) over tilde (k), k is an element of Z(+), for linear, multiple input multiple output, non-minimum phase and/or non hyperbolic, sampled data, and closed loop control systems. The proposed approach is situated in the general framework of model stable inversion and introduces significant novelties with the purpose of reducing some theoretical and numerical limitations inherent in the methods usually proposed. In particular, the new method does not require either a preactuation or null initial conditions of the system. The desired (y) over tilde (k) and the corresponding sought input are partitioned in a transient component ((y) over tilde (t)(k) and u(t)(k), respectively) and steady-state ((y) over tilde (s)(k) and u(s)(k), respectively). The desired transient component (y) over tilde (t)(k) is freely assigned without requiring it to be null over an initial time interval. This drastically reduces the total settling time. The structure of u(t)(k) is a priori assumed to be given by a sampled smoothing spline function. The spline coefficients are determined as the least-squares solution of the over-determined system of linear equations obtained imposing that the sampled spline function assumed as reference input yield the desired output over a properly defined transient interval. The steady-state input u(s)(k) is directly analytically computed exploiting the steady-state output response expressions for inputs belonging to the same set of (y) over tilde (s)(k).