Almost perfect tracking through mixed numerical-analytical stable pseudo-inversion of non minimum phase plants

This 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 ỹ(t). The main purpose of the new approach proposed here is to alleviate some limitations 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 ỹ_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 ỹt(t) is specified to obtain a fast and smooth transition towards the steady-state trajectory ỹ_s(t), without under and/or overshoot in the case of a set point reset. The transient control input u_t(t) is “a priori” assumed to be given by a piecewise polynomial function. Once ỹ(t) has been specified, this allows the computation of the unknown u_t(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 u_s(t) is analytically computed exploiting the steady-state output response expressions for inputs belonging to the same set of ỹ_s(t).