Practical Finite-Time Contraction of Impulsive Systems With Application to Design of Impulsive Control for PFTC

This paper investigates practical finite-time contraction (PFTC) for impulsive systems with application to design of impulsive control for PFTC. The PFTC notions including practical finite-time incremental stability (PFTIS) and practical finite-time stability (PFTS) are defined. By using the variation method, the equivalence of PFTC and PFTIS is established for an impulsive system via PFTS of its virtual dynamics. Based on the equivalence, the PFTC criteria and the settling time estimates are obtained for impulsive systems. Then, the results are used to design impulsive control for PFTC. Three types of impulsive control: time-triggered impulsive control (T-IC), state-triggered impulsive control (S-IC), and time-state-mixed-triggered impulsive control (TS-IC), are designed, respectively, for PFTC. The theoretical results as well as numerical simulations with application to the precision control of missile dynamics are shown that TS-IC is the optimal impulsive control with the shortest settling time, i.e., the fastest entry and stay in the target area, the least number of impulsive control, the lowest impulse frequency, and strongest robustness compared to T-IC and S-IC. And TS-IC also improves the mechanism of impulsive control in the literature, which is either purely time or state triggered. Note to Practitioners—In practical applications, many systems often struggle to achieve complete consistency with the target state, and even if achievable, the conditions to be met will be very strict. The study introduces “practical finite-time contraction” (PFTC), which ensures systems quickly reach and stay near a target state without requiring exact convergence. This is useful in real-world systems where minor fluctuations are tolerable (e.g., voltage regulation in power grids). This approach is more feasible in practical implementation and holds significant applied value. This study focuses on the stability control problem of dynamical systems (e.g., power systems or robotic systems) and proposes a solution to ensure that the system state enters and remains within a predefined acceptable range within a finite time without requiring exact convergence to equilibrium. By using the Lyapunov-like function metric, the PFTC criteria and the PFTC settling time are obtained. Three control strategies are designed for finite-time stabilization: time-triggered impulsive control (T-IC), state-triggered impulsive control (S-IC), and time-state-mixed-triggered (TS-IC). Both theoretical analysis and numerical simulations with application to the precision control of missile dynamics demonstrate that TS-IC achieves faster stabilization with fewer control actions and lower impulse frequency compared to T-IC and S-IC, and also stronger robustness w.r.t. the perturbation, making it the optimal strategy for practical applications.