A least-squares based pseudoinversion approach for non hyperbolic linear systems

A new method to achieve an accurate output tracking for nonminimum phase linear systems with nonhyperbolic or near nonhyperbolic internal dynamics is presented. For the classical methods, developed in the framework of the preview based stable inversion, the nonhyperbolicity represents a rather serious inconvenience because the required preactuation time tends to become unacceptably large. Different stable inversion techniques, developed for SISO systems, are based on a proper redefinition of the desired output trajectory with the aim of canceling the undesired effects of unstable zeros. The main purpose of the new approach is to alleviate some theoretical and practical limitations inherent in the above methods. The desired output is here partitioned into the transient and steady-state components. The desired output to be exactly tracked in steady-state is assumed to belong to the set of polynomials, exponential and sinusoidal time functions. The transient input is “a priori” assumed to be given by a spline function. Once the desired output trajectory has been specified, this allows the computation of the unknown transient input 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 is analytically computed exploiting the steady-state output response expressions for inputs belonging to the same set of the desired steady-state output.