A mixed numerical-analytical stable pseudo-inversion method aimed at attaining an almost exact tracking

SummaryThis paper considers the problem of computing the input u(t) of an internally asymptotically stable, possibly non‐minimum phase, linear, continuous time system Σ yielding a very accurate tracking of a pre‐specified desired output trajectory y˜(t). The main purpose of the new approach proposed here is to alleviate some limitations that inherent the classical methods developed in the framework of the preview‐based stable inversion, which represents an important reference context for this class of control problems. In particular, the new method allows one to deal with arbitrary and possibly uncertain initial conditions and does not require a pre‐actuation. The desired output y˜s(t) to be exactly tracked in steady state is here assumed to belong to the set of polynomials, exponential, and sinusoidal time functions. The desired transient response y˜t(t) is specified to obtain a fast and smooth transition toward the steady‐state trajectory y˜s(t), without under and/or overshoot in the case of a set point reset. The transient control input ut(t) is a priori assumed to be given by a piecewise polynomial function. Once y˜(t) has been specified, this allows the computation of the unknown ut(t) as the approximate least squares solution of the Fredholm's integral equation corresponding to the explicit formula of the output forced response. The steady‐state input us(t) is analytically computed exploiting the steady‐state output response expressions for inputs belonging to the same set of y˜s(t). Copyright © 2013 John Wiley & Sons, Ltd.