Kybernetika 36 no. 5, 497-529, 2000

A new indirect adaptive pole placer for possibly non-minimum phase MIMO linear systems

K. G. Arvanitis, G. Kalogeropoulos and I. K. Kookos


The use of generalized sampled-data hold functions, in order to synthesize adaptive pole placers for linear multiple-input, multiple-output systems with unknown parameters, is investigated in this paper, for the first time. Such a control scheme relies on a periodically varying controller, which suitably modulates the sampled outputs of the controlled plant. The proposed control strategy allows us to assign the poles of the sampled closed-loop system arbitrarily 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. The indirect adaptive control scheme presented here, estimates the unknown plant parameters (and hence the parameters of the desired modulating matrix function) 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 $T_0$. The controller determination is based on the transformation of the discrete analogue 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, whereas known indirect adaptive pole placement techniques require the solution of matrix polynomial Diophantine equations. 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 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 assumptions 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.