An algorithm for adaptive pole placement control of linear systems based on generalized sampled-data hold functions

Abstract

The use of generalized sampled-data hold functions for adaptive pole placement control of linear systems with unknown parameters, is investigated. The particular control structure used relies on a periodic controller, which suitably modulates the sampled plant output by a multirate periodically time-varying function. Such a control strategy, allows us to assign the eigenvalues of the closed-loop monodromy matrix to the desired prespecified values and does not make assumptions on the plant other than controllability, observability and known order. The proposed indirect adaptive scheme estimates the unknown plant 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. On the basis of the proposed algorithm, the adaptive pole placement problem is reduced to a controller determination based on the well-known Ackermanns’ formula. Known adaptive control schemes usually resort to the computation of dynamic controllers through the solution of polynomial Diophantine equations, thus introducing high-order exogenous dynamics in the control loop. Moreover, in many cases, this solution might yield an unstable controller, and the overall adaptive scheme is then unstable with unstable compensators because their outputs are unbounded. The proposed control strategy avoids these problems, since here gain controllers essentially need 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.