Optimal predefined-time observer for nonlinear multimotor servo systems with unknown states

In modern industrial applications, nonlinear multi-motor servo systems are fundamental to achieving high-dynamic and high-precision motion control. However, the performance of these systems is largely constrained by the real-time capability and accuracy of state observation. Given their high-order and nonlinear characteristics, designing a state observer that ensures convergence within a predefined time while optimizing performance metrics remains a significant and open challenge. This paper investigates the optimal predefined time observation problem for nonlinear multimotor servo systems. First, a Lyapunov function is proposed to ensure predefined-time stability/practical predefined time stability. Leveraging the duality principle between control and observability, the observation error dynamics are reformulated as a closed-loop system incorporating a virtual input designed to minimize a cost function. Then, based on this dynamic system, an optimal predefined-time observer is designed to guarantee that the observation error converges within a predefined time while also reducing computational complexity. The proposed observer involves a Lyapunov function that satisfies the Hamilton-Jacobi-Bellman equation to ensure optimality. Additionally, the Lyapunov function serves as the theoretical foundation for observer design, facilitating the reduction of steady-state error and eliminating the coupling effects between observer gains. Finally, the simulation and experimental results validate the effectiveness of the proposed observer.