Nonlinearly Stable Real-Time Learning and Model-Free Control

Abstract

This work provides a framework for nonlinear model-free control of systems with unknown input-output dynamics, but outputs that can be controlled by the inputs. This framework leads to real-time control of the system such that a feasible output trajectory can be tracked by the inputs. Unlike existing model-free or data-driven control approaches, this framework guarantees nonlinear stability. The controller and observer designs in the proposed framework are nonlinearly stable and robust to the unknown dynamics as well as unknown measurement noise. For ease of computer implementation, the framework is developed in discrete time. Nonlinear stability analyses of the discrete-time observers and controllers are carried out using discrete Lyapunov analysis. The unknown input-output dynamics is learnt in real time using a nonlinearly stable observer from prior input-output history. This observer ensures finite-time stable convergence of model estimation errors to zero if the unknown model is constant, and model estimation errors converge to a bounded neighborhood of the zero vector if the model has bounded change in discrete time. Measured outputs are filtered by a finite-time stable observer before use in feedback tracking of a desired output trajectory. Finite-time stable observer design in this framework also ensures that a nonlinear separation principle is in effect for separate controller and observer design. A model-free nonlinearly stable control scheme is then designed to ensure convergence of observed outputs to a desired output trajectory. This control scheme ensures nonlinear finite-time stable convergence of tracking errors to a manifold where the tracking errors decay asymptotically. A numerical experiment on a nonlinear second-order system demonstrates the performance of this nonlinear model-free control framework.