High-order fully-actuated system approaches: Part IX. Generalised PID control and model reference tracking

Tracking control of general dynamical systems in high-order fully actuated (HOFA) system representation is solved. For the case of tracking a constant or slow time-varying signal in the presence of constant or slow time-varying disturbances, a generalised PID control scheme is proposed, which realises asymptotical tracking to a prescribed signal and also guarantees that the state derivatives of certain orders converge to the origin. For the case that the signal to be tracked is generated by a reference model, a model reference tracking (MRT) controller is presented, which relies on the solution to a type of generalised Sylvester matrix equations and guarantees the desired asymptotical tracking requirement. Due to the full-actuation property of the HOFA models, closed-loop systems under both control schemes are constant and linear. Furthermore, based on a general parametric solution to the type of Sylvester matrix equations and a general parametric eigenstructure assignment result, simple and complete parameterisation of the two types of control designs are provided, and feasibility conditions in terms of the system initial values for sub-fully actuated systems are also derived. An illustrative example is presented to demonstrate the application of the proposed approaches and their effects.