## 1 Introduction
Results showed that ALO algorithm outperforms the other optimization methods for a real problem, by obtaining precise values for the regulator design. Another nature-inspired technique, namely, Grey Wolf Optimizer (GWO) [22], has been used in [23] to control a quadrotor dynamic model and stability prediction of power systems after a large disturbance [24]. Whale Optimization Algorithm (WOA) [25] is another technique used to fine-tune neural networks to obtain more reliable models [26]. Another recently invented algorithm, Harris Hawks Optimization (HHO) [27], has been applied to optimize a controller to dampen power system oscillations [28]. Although there are many examples where these algorithms have been studied in the past, there are fewer studies that have tackled the problem of external disturbances and parameter uncertainty. In this paper, an optimal tracking neural control strategy with metaheuristic parameter optimization is proposed for highly nonlinear, uncertain systems subject to external disturbances. In Section 2, a neural observer based on a RHONN is proposed to estimate the unmeasured states needed to develop a state feedback control law. Section 3 describes the stabilizing control law based on the inverse optimal control law theory. In Section 4, the novelty of this research is presented, the use of the newest metaheuristic algorithms, namely: ALO, GWO, HHO, and WOA for optimal parameter estimation in order to automatically find the optimal parameters of the control law. Section 5 describes the optimal control strategy architecture. To validate the efficacy of this control strategy, Section 6 presents a highly nonlinear biological system subject to disturbances as a case study. This model is applied to describe the aerobic digestion of organic matter in a continuous stirred tank reactor (CSTR). In Section 7, simulation results of the control strategy with metaheuristic algorithms are presented. Two cases are analyzed: the system subject to load disturbances and subject to uncertainties in the composition of the incoming wastewater. The total chemical oxygen demand (TCOD) is the variable of interest to be controlled by an optimal oxygen transfer (KLA) rate. Optimal operation of the wastewater treatment plant can improve energy efficiency and increase the treatment capacity. Finally, relevant conclusions are given in Section 8.  
where \(w_{ijk}\in R^{Li}\) is the weight vector, \(\eta_{i}\) is the learning rate, \(K_{i,k}\in R^{Li\times p}\) is the Kalman gain matrix, \(P_{i,k}\in R^{L\times l\times l\times l}\) is the prediction error associated covariance matrix, \(Q_{ijk}\in R^{L\times l\times l}\) is the state noise associated covariance matrix, \(M_{i,k}\in R^{p\times p}\) is a global scaling matrix, \(R_{i,k}\in R^{p\times p}\) is the measurement noise associated covariance matrix, and \(H_{i,k}\in R^{Li\times n}\) is a matrix of the derivatives of the network outputs with respect to all trainable neural network weights as follows:  
\[H_{i,k}=\left[\frac{\partial\hat{g}_{k}}{\partial w_{ijk}}\right] \tag{7}\]  
\(L_{i}\) is the total number of neural network weights. \(P_{i}\), \(Q_{i}\), and \(R_{i}\) are initialized as diagonal matrices. During training, \(K_{ijk}\), \(P_{i,k}\), and \(H_{i,k}\) are ensured to be bounded [31].