Positive Realness in Discrete-Time Adaptive Control Systems

Recently, it was shown for continuous-time systems, that the positive realness conditions can be satisfied without the need for the prior knowledge about the order and the pole-excess of the controlled plant. Thes results are extended here to discrete-time systems. It is shown that the positive realness conditions can be satisfied in "almost stable" discrete-time multivariable linear systems, namely, systems that can be stabilized via static or dynamic output feedback, including non-minimum phase systems, at the price of bounded rather than vanishing output tracking errors. Satisfaction of the positivity conditions may facilitate implementation of simple adaptive control procedures that maintain robustness in the presence of parasitic dynamics and disturbances.