A Memory-Based Simultaneous Perturbation Stochastic Approximation Method for the Optimal PID Controller Tuning in Flexible Joint Manipulators

This paper presents a memory-based Simultaneous Perturbation Stochastic Approximation (M-SPSA) method for the optimal Proportional–Integral–Derivative (PID) controller tuning in flexible joint manip ulators. The study addresses convergence instability in standard SPSA variants by introducing a memory-retention mechanism that preserves the best-performing design variable. The research contribution is the formulation of a memory-driven SPSA framework that improves convergence stability and robustness while maintaining computational efficiency. The proposed method is validated through simulation using a nonlinear flexible joint manipulator model. Its performance is compared with two established variants, namely Norm-Limited SPSA (NL-SPSA) and Normalized SPSA (N-SPSA), to evaluate convergence behavior, energy efficiency, and tracking accuracy. Results show that the proposed M-SPSA achieves consistent and stable convergence across all simulation trials. The algorithm effectively minimizes the objective-function value and reduces control energy while maintaining accurate trajectory tracking. Compared with NL-SPSA and N-SPSA, the proposed method exhibits improved convergence reliability and smoother control input behavior. These outcomes indicate that incorporating memory retention enhances stability and resilience in stochastic optimization applied to flexible robotic systems. Future work will focus on real-time validation, and extension of the approach to multi-input multi-output and adaptive control systems to strengthen its practical applicability in advanced robotics.