Practical finite-time control of strict-feedback nonlinear systems: a systematic singularity-free backstepping approach

Finite-time backstepping control for strict-feedback nonlinear systems often faces singularity problems in the derivatives of virtual control signals. These issues degrade performance as the system states approach the origin. This work addresses this critical problem by proposing a systematic singularity-free practical finite-time (PFT) backstepping framework. The motivation stems from the need to preserve the differentiability of virtual control signals while maintaining finite-time convergence properties. The proposed method is a novel structure for virtual control signals that eliminates singular terms at each stage of backstepping design. We also extend the method to include a finite-time command filter, suitable for high-order strict-feedback systems. Furthermore, the extension of the proposed method to strict-feedback systems with lumped disturbances is presented. The proposed framework introduces a design parameter that allows explicit tuning of performance and singularity avoidance. Rigorous Lyapunov-based analysis proves practical finite-time stability and derives convergence properties. Simulation studies on four strict-feedback nonlinear systems show that the proposed method achieves finite-time convergence. Tracking errors stay within the predicted ultimate bounds, and the design parameters effectively control convergence speed and error size. In all cases, derivatives of virtual controls remain well-defined even when system states approach the origin, confirming the singularity-free nature of the proposed approach.