Lyapunov-based stability analysis of quantum sliding mode controller

Abstract

This article addresses the lack of formal stability analyses in quantum sliding mode control (QSMC) by providing rigorous proof of stability and convergence based on Lyapunov theory. It also proposes an optimized QSMC approach that reduces quantum resource requirements while preserving control performance. The study begins with a re-evaluation of the conventional QSMC formulation, which implements the sign function using three qubits, along with a Hadamard gate and a CCNOT gate. A Lyapunov-based analysis is conducted to formally demonstrate the stability and convergence of the system. Based on this result, an improved QSMC scheme is introduced. The new design replaces the original structure with a quantum sign detector, a measurement process, and a rotation gate, thus reducing the implementation to two qubits. The proposed method is validated by its application to the speed control of a DC motor. The results show that it maintains efficient performance while requiring fewer computing resources. Simulation results support the feasibility of the approach. This work strengthens the theoretical foundations of QSMC and improves its applicability. The optimized version offers a more efficient and scalable solution for implementation on current quantum hardware with limited qubit availability.