Quantum-inspired optimal tuning of complex fractional-order PID controllers for nonlinear MIMO CSTR systems under uncertainty via Schrödinger Optimizer

This study investigates the design and optimization of a complex fractional-order proportional–integral–derivative (COPID) controller for a highly nonlinear two-tank 2 × 2 MIMO continuous stirred tank reactor (CSTR) system with strong cross-coupling effects. Although COPID controllers offer enhanced flexibility due to their extended parameter space, their tuning constitutes a challenging high-dimensional optimization problem. To address this issue, a Schrödinger Optimizer (SRA), inspired by quantum mechanical principles, is employed for optimal parameter tuning. The performance of the proposed SRA-based COPID controller is evaluated against classical PID, fractional-order PID (FOPID), and conventional COPID controllers under identical operating conditions. The results demonstrate that the proposed approach significantly improves transient response and cumulative error performance while maintaining comparable steady-state accuracy. Robustness analyses under external disturbances and parameter uncertainties further reveal that the SRA-based COPID controller provides more consistent and reliable performance than the benchmark methods. Extensive Monte Carlo simulations confirm that the proposed method achieves lower mean error values and reduced performance variability, highlighting its strong robustness against model uncertainties. Moreover, improvements in the primary output do not compromise the performance of the secondary output, preserving the overall balance of the MIMO system. Overall, the findings demonstrate that integrating quantum-inspired metaheuristic optimization with advanced controller structures provides an effective and practical framework for the robust control of nonlinear and strongly coupled MIMO processes.