Adaptive Pole Positioning in MIMO Linear Systems by Periodic Multirate-Input Controllers

Abstract

In this paper, the certainty equivalence principle is used to combine the identification method with a control structure derived from the pole placement problem, which rely on periodic multirate-input controllers. The proposed adaptive pole placers, contain a sampling mechanism with different sampling period to each system input and rely on a periodically varying controller which suitably modulates the sampled outputs and reference signals of the plant under control. Such a control strategy allows us to arbitrarily assign the poles of the sampled closed-loop system in desired locations and does not make assumptions on the plant other than controllability and observability of the continuous and the sampled system, and the knowledge of a set of structural indices, namely the locally minimum controllability indices of the continuous-time plant. An indirect adaptive control scheme is derived, which estimates the unknown plant parameters (and consequently the controller 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 T0. Using the proposed algorithm, the controller determination is based on the transformation of the discrete analogous of the system under control to a phase-variable canonical form, prior to the application of the control design procedure. The solution of the problem can, then, be obtained by a quite simple utilization of the concept of state similarity transformation. 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 unstable with unstable compensators because their outputs are unbounded. The proposed control strategy avoids these problems, since here gain controllers are essentially needed to be designed. Moreover, persistency of excitation and, therefore, parameter convergence, of the continuous-time plant is provided without making any assumption either the richness of the reference signals or on the existence of specific convex sets in which the estimated parameters belong or, finally, on the coprimeness of the polynomials describing the ARMA model, as in known adaptive pole placement schemes.