A new multirate sampled-data technique for adaptive pole positioning in linear systems

Abstract

The adaptive pole placement problem for linear systems is solved using a new class of multirate controllers, called two-point multirate controllers. In such a type of controller, the control is constrained to a certain piecewise constant signal, while the controlled plant output is detected many times over a fundamental sampling period. On the basis of the proposed strategy, the original problem is reduced to an associate discrete pole placement problem, for which a fictitious static-state feedback controller is needed to be computed. This control strategy allows us to assign the poles of the sampled closedloop system arbitrarily in desired locations, and does not make assumptions on the plant other than controllability and observability of the continuous and the sampled system, and known order. The controller determination relies on a closed-form formula, which can be thought as the extension of the Ackermann formula for multi-input/multi-output (MIMO) systems. Known indirect adaptive pole placement techniques require the solution of matrix polynomial Diophantine equations, which, in many cases, might yield an unstable controller. Moreover, the proposed adaptive scheme is readily applicable to non-minimum phase systems, and to systems which do not possess the parity interlacing property. Finally, persistency of excitation and, therefore, parameter convergence, of the continuous-time plant is provided without making assumptions either on the existence of specific convex sets in which the estimated parameters belong or on the coprimeness of the polynomials describing the ARMA model, or finally on the richeness of the reference signals, as compared to known adaptive pole placement schemes.