A Periodic Multirate Adaptive Pole Placer for Possibly Nonminimum Phase Plants

Abstract

A new indirect adaptive algorithm is derived for pole placement control of linear continuous-time systems with unknown parameters. The control structure proposed relies on a periodic controller, which suitably modulates the sampled output and discrete reference signals by a multirate periodically time-varying function. Such a control strategy allows us to assign the poles of the sampled closed-loop system to desired prespecified values, and does not make assumptions on the plant other than controllability, observability, and known order. The proposed indirect adaptive control scheme estimates the unknown plant parameters (and consequently the controller parameters) on-line, from sequential data of the inputs and the outputs of the plant, which are recursively updated within the time limit imposed by a fundamental sampling period T0. On the basis of the proposed algorithm, the adaptive pole placement problem is reduced to a controller determination based on the well-known Ackermann’s formula. Known indirect adaptive pole placement schemes usually resort to the computation of dynamic controllers through the solution of a polynomial Diophantine equation, thus introducing high order exogenous dynamics in the control loop. Moreover, in many cases, the solution of the Diophantine equation for a desired set of closed-loop eigenvalues might yield an unstable controller, and the overall adaptive pole placement scheme is then unstable with unstable compensators because their outputs are unbounded. The proposed control strategy avoids these problems, since here gain controllers are needed to be designed. Moreover, persistency of excitation and, therefore, parameter convergence, of the continuous-time plant is provided without making any assumption 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 richness of the reference signals, as compared to known adaptive pole placement schemes.