This paper aims to solve the problem of tracking optimal profiles for a nonlinear multivariable fed-batch bioprocess by a simple but efficient closed-loop control technique based on a linear algebra approach. In the proposed methodology, the control actions are obtained by solving a system of linear equations without the need for state transformations. The optimal profiles to follow are directly those corresponding to output desired variables, therefore, estimation of states for nonmeasurable variables is considered by employing a neural networks method. The efficiency of the proposed controller is tested through several simulations, including process disturbances and operation under parametric uncertainty. The optimal controller parameters are selected through the Montecarlo Randomized Algorithm. In addition, proof of convergence to zero of tracking errors is analyzed and included in this article.

In this paper, a novel adaptive PID controller for trajectory-tracking tasks is proposed. It is implemented in discrete time over a hexacopter, and it takes into consideration the unmanned aerial vehicles (UAVs) nonlinear model. The PID controller is developed following an adaptive neural technique, and its stability is verified by the Lyapunov discrete theory. Besides, the neural identification of the dynamic model of the UAV is presented to backpropagate output errors to adjust PID gains with the purpose of reducing the control errors. The validation of the proposed algorithm is performed through experimental results with a hexacopter. KEYWORDSadaptive PID, discrete stability analysis, hexacopter, identification, neural networks 74 ;33:74-91. ROSALES ET AL. 75steady-state errors, and the derivative gain is used to modify any system overshoot. However, since quadrotor is a nonlinear underactuated systems, 8 it is not always possible to use PID control directly for the quadrotor system. Consequently, some authors proposed controllers designed based only on static PID for the UAV control such as the controllers presented in other works. [9][10][11] Other studies proposed the UAV control based on PID controllers with artificial intelligence and dynamic properties, such as the controllers presented in the work of Yang et al, 12 where authors defined a single-neuron PID, where the adjusting of the weights is a function of the control errors. Besides, simulations results were displayed to validate the proposal but the stability analysis is omitted. In addition, Xu and Zhou 13 use self-tuning PID controller based on adaptive pole placement to a quadrotor by using a linearized version of the dynamic model. Another class of PID controllers had the capabilities to backpropagate control errors for a better adjust of the controller gains, some of those are described in this work. Tsakalis and Dash 14 proposed an algorithm for  ∞ for online tuning of PID gains. The estimator is very complex to implement, although is robust with respect to the excitation spectrum. The algorithm is validated by using simulations results for different transfer functions. In the work of Liu et al, 15 the design of a Gaussian potential function network PID (GPFN-PID) controller based on GPFN NN was described. The proposed controller had the functions of online training, self-training, and self-adjusting, which made control of the UAV longitudinal channel more effective. However, the results obtained were based on simulations. Babu et al 16 proposed a gradient descent-based methodology for adjusting online the gains of a PID controller based on a cost function defined to penalize errors. The controller is designed only for positioning task and a commercial quadrotor is used to experiment the controller in position task. In the work of Wang et al, 17 an intelligent PID controller based on radial basis functions (RBFs) NN was designed for longitudinal attitude control of a small UAV using a nonlinear model. An RBF NN was utilized for online updatin...

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