Adaptive MIMO fuzzy PID controller based on Runge-Kutta model

Abstract

Nonlinear multi-input multi-output (MIMO) systems are difficult to control because of input–output coupling, measurement noise, and parameter variations. This study proposes a Runge–Kutta (RK) model-based predictive adaptive MIMO fuzzy proportional–integral–derivative (PID) controller for nonlinear dynamical systems. A fourth-order Runge–Kutta (RK4) model predicts future outputs and provides the sensitivity information required to update the fuzzy PID scaling parameters online using the Levenberg–Marquardt (LM) algorithm. An additive correction mechanism provides supplementary control action during transient conditions and plant–model mismatch. Furthermore, a discrete-time Lyapunov-based stability analysis is established, guaranteeing the uniform ultimate boundedness (UUB) of the tracking errors and the boundedness of adaptive parameters under aggregate disturbances. The proposed controller is evaluated on a nonlinear three-tank system under nominal conditions, measurement noise, and time-varying parametric uncertainty. For comparative evaluation, an existing RK model-based adaptive MIMO PID controller is employed as a benchmark under the same operating scenarios. Both controllers maintain bounded tracking responses, control inputs, and adaptive parameters. The composite performance-index values obtained with the proposed fuzzy PID controller are 0.2660, 0.2701, and 1.0739, corresponding to improvements of 6.11%, 6.12%, and 3.59%, respectively, over the benchmark controller. In the parametric-uncertainty case, the RK-based parameter estimator follows the variations in the uncertain inter-tank flow coefficient and reduces the resulting plant–model mismatch. Although the benchmark adaptive PID controller has lower computational requirements, the average and maximum cycle times of both controllers remain substantially below the sampling period. The results demonstrate that the proposed adaptive MIMO fuzzy PID controller improves the combined tracking and control-variation performance while retaining computational feasibility.