Positive-realness in discrete-time adaptive control systems

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Abstract
Most self-tuning and adaptive control methods hinge on the satisfaction of some positive-realness conditions. Alternatively, they require prior knowledge about an upper bound on the order of the controlled plant and about its exact pole-zero excess. Recently it was shown for continuous-time systems that the positive-realness conditions can be satisfied without the need for this prior knowledge. The results are extended here to discrete-time systems. It is shown that the positive-realness conditions can be satisfied in ‘almost stable’ discrete-time multivariate 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. It is also shown that satisfaction of the positivity conditions may facilitate implementation of simple adaptive control procedures that maintain robustness in the presence of parasitic dynamics and disturbances

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