A periodic multirate adaptive pole placer for possibly nonminimum phase plants

A new indirect adaptive algorithm is derived for pole placement control of linear continuous-time systems with unknown parameters. The control structu re proposed relies on a periodic controller which suitably modulates the sa mpled output and discrete reference signals by a multirate periodically tim e-varying function. Such a control strategy allows us to assign the poles o f the sampled closed-loop system to desired prespecified values, and does n ot make assumptions on the plant other than controllability, observability, and known order. The proposed indirect adaptive control scheme estimates t he unknown plant parameters (and consequently the controller parameters) on -line, from sequential data of the inputs and the outputs of the plant, whi ch are recursively updated within the time limit imposed by a fundamental s ampling period T-0. On the basis of the proposed algorithm the adaptive pol e placement problem is reduced to a controller determination based on the w ell-known Ackermann's formula. Known indirect adaptive pole placement schem es usually resort to the computation of dynamic controllers through the sol ution of a polynomial Diophantine equation, thus introducing high order exo genous dynamics in the control leap. Moreover, in many cases, the solution of the Diophantine equation for a desired set of closed-loop eigenvalues mi ght yield an unstable controller, and the overall adaptive pole placement s cheme 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 excita tion 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 scheme s.