A family of implicit self-tuning regulators (STR) is presented, based on Lyapunov analysis techniques for the control of a class of multi-input and multi-output (MIMO) dynamical systems. Linearity in the parameters is assumed to hold, but the ’estimation error’ is considered to be nonzero; this allows control of a larger class of systems and also has the effect of producing a robust controller. Moreover, the certainty equivalence (CE) principle is not used in the STR design, overcoming a major problem in adaptive control. The structure of the STR is naturally derived using a tracking error/passivity approach. The role of persistency of excitation (PE) is explored, and is used to show the boundedness of parameter estimates when gradient-based parameter tuning of STR is performed in nonideal conditions. New on-line tuning algorithms for implicit STR are derived that are similar to the epsilon-modification approach for the case of continuous-time systems and that include a modification to the adaptation gain and a correction term to the standard gradient-based tuning. These improved parameter tuning algorithms guarantee tracking as well as bounded parameter estimates in nonideal situations, so that PE is not needed. The notions of a passive STR, a dissipative STR and a robust STR are introduced. Finally, this paper provides a comprehensive theory in the development of identification, prediction and adaptive control schemes for discrete-time systems based on Lyapunov analysis.