  This work provides a framework for data-driven control of discrete time
systems with unknown input-output dynamics and outputs controllable by the
inputs. This framework leads to stable and robust real-time control of the
system such that a feasible output trajectory can be tracked. This is made
possible by rapid real-time stable learning of the unknown dynamics using
H\"{o}lder-continuous learning schemes that are designed as discrete-time
stable disturbance observers. This observer learns from prior input-output
history and it ensures finite-time stable convergence of model estimation
errors to a bounded neighborhood of the zero vector if the system is known to
be Lipschitz-continuous with respect to outputs, inputs, internal parameters
and states, and time. In combination with nonlinearly stable controller
designs, this makes the proposed framework nonlinearly stable and robust to
disturbances, model uncertainties, and unknown measurement noise. Nonlinear
stability and robustness analyses of the observer and controller designs are
carried out using discrete Lyapunov analysis. H\"{o}lder-continuous Finite-time
stable observer and controller designs in this framework help to prove
robustness of these schemes and guaranteed convergence of outputs to a bounded
neighborhood of the desired output trajectory. A numerical experiment on a
nonlinear second-order system demonstrates the performance of this discrete
nonlinear model-free control framework.
