Nonlinear integral extension of PID control with improved convergence of perturbed second-order dynamic systems

Nonlinear extension of the integral part of a standard proportional-integral-derivative (PID) feedback control is proposed for perturbed second-order systems. The approach is model-free and requires solely the Lipschitz boundedness of the unknown matched perturbations. For constant disturbances, the global asymptotic stability is shown based on the circle criterion. For Lipschitz perturbations, an ultimately bounded output error is provided based on the steady-state behavior in frequency domain. Also the transient response to the stepwise disturbances is analyzed for the control tuning. Based on the developed analysis, the design recommendations are formulated as a step by step procedure. It is also discussed how the proposed control is applicable to second-order systems extended by additional (parasitic) actuator dynamics with low-pass characteristics. The proposed nonlinear control is proven to outperform its linear PID counterpart during the settling phase, i.e. at convergence of the residual output error. An experimental case study of the second-order system with an additional actuator dynamics and considerable perturbations is demonstrated to confirm and benchmark the control performance.