Adaptive pole positioning in MIMO linear systems by periodic multirate-input controllers

In this paper, the certainty equivalence principle is used to combine the i dentification method with a control structure derived from the pole placeme nt problem, which rely on periodic multirate-input controllers. The propose d adaptive pole placers, contain a sampling mechanism with different sampli ng period to each system input and rely on a periodically varying controlle r which suitably modulates the sampled outputs and reference signals of the plant under control. Such a control strategy allows us to arbitrarily assi gn the poles of the sampled closed-loop system in desired locations and doe s not make assumptions on the plant other than controllability and observab ility of the continuous and the sampled system, and the knowledge of a set of structural indices, namely the locally minimum controllability indices o f the continuous-time plant. An indirect adaptive control scheme is derived , which estimates the unknown plant parameters land consequently the contro ller parameters) on-line, from sequential data of the input and outputs of the plant, which are recursively updated within the time limit imposed by a fundamental sampling period T-0. Using the proposed algorithm, the control ler determination is based on the transformation of the discrete analogous of the system under control to a phase-variable canonical form, prior to th e application of the control design procedure. The solution of the problem can, then, be obtained by a quite simple utilization of the concept of stat e similarity transformation. Known indirect adaptive pole placement schemes usually resort to the computation of dynamic controllers through the solut ion of a polynomial Diophantine equation, thus introducing high order exoge nous dynamics in the control loop. Moreover, in many cases, the solution of the Diophantine equation for a desired set of closed-loop eigenvalues migh t yield an unstable controller, and the overall adaptive pole placement sch eme is unstable with unstable compensators because their outputs are unboun ded. The proposed control strategy avoids these problems, since here gain c ontrollers are essentially needed to be designed. Moreover, persistency of excitation and, therefore, parameter convergence, of the continuous-time pl ant is provided without making any assumption either the richness of the re ference signals or on the existence of specific convex sets in which the es timated parameters belong or, finally, on the coprimeness of the polynomial s describing the ARMA model, as in known adaptive pole placement schemes. ( C) 1999 Academic Press.