Discrete-time Control of Nonlinear Control-Affine Systems with Uncertain Dynamics

A novel approach to data-enabled control of discrete nonlinear control-affine systems with uncertain (“gray-box”) dynamics is given here. The gray-box dynamics model accounts for known dynamics and lumps together the effects of disturbance inputs and poorly known dynamics into one time-varying unknown input, which is estimated in real time. This data-enabled approach leads to robust and stable real-time tracking control of desired output trajectories in the presence of model uncertainties. The lumped unknown input is estimated by a Hölder-continuous robustly stable learning scheme, using input-output data in discrete time. This leads to finite-time convergence of the estimation errors of the unknown dynamics to a bounded neighborhood of the zero vector, provided the system is Lipschitz-continuous with respect to states, inputs, outputs, and time. A Lyapunov analysis is carried out to show the nonlinear stability and robustness of the uncertainty observer and tracking control law. This results in simultaneous real-time uncertainty estimation and tracking control. Numerical simulation results for an inverted pendulum with imperfectly known dynamics are carried out, which agree with the theoretical results on stability and robustness.