In the previous article, we introduced a neural network framework based on symmetric differential equations. This novel framework exhibits complete symmetry, endowing it with perfect mathematical properties. While we have examined some of the system's mathematical characteristics, a detailed discussion of the network training methodology has not yet been presented. Drawing on the principles of the traditional backpropagation algorithm, this study proposes an alternative training approach that utilizes differential equation signal propagation instead of chain rule derivation. This approach not only preserves the effectiveness of training but also offers enhanced biological interpretability. The foundation of this methodology lies in the system's reversibility, which stems from its inherent symmetry,a key aspect of our research. However, this method alone is insufficient for effective neural network training. To address this, we further introduce a distributed Proportional-Integral-Derivative (PID) control approach, emphasizing its implementation within a closed system. By incorporating this method, we achieved both faster training speeds and improved accuracy. This approach not only offers novel insights into neural network training but also extends the scope of research into control methodologies. To validate its effectiveness, we apply this method to the MNIST dataset, demonstrating its practical utility.

This document presents the study of the problem of location and trajectory that a robot must follow. It focuses on applying the Kalman filter to achieve location and trajectory estimation in an autonomous mobile differential robot. The experimental data was carried out through tests obtained with the help of two incremental encoders that are part of the construction of the differential robot. The data transmission is carried out from a PC where the control is carried out with the Matlab/Simulink software. The results are expressed in graphs showing the path followed by the robot using PI control, the estimator of the Kalman filter in a real system.

An omnidirectional multirotor has the advantageous maneuverability of decoupled translational and rotational motions, drastically superseding the traditional multirotors' motion capability. Such maneuverability requires an omnidirectional multirotor to frequently alter the thrust amplitude and even direction, which is prone to the rotors' settling time induced from the rotors' own dynamics. Furthermore, the omnidirectional multirotor's stability for tracking control in the presence of rotor dynamics has not yet been addressed. To resolve this issue, we propose a geometric tracking controller that takes the rotor dynamics into account. We show that the proposed controller yields the zero equilibrium of the error dynamics almost globally exponentially stable. The controller's tracking performance and stability are verified in simulations. Furthermore, the single-axis force experiment with the omnidirectional multirotor has been performed to confirm the proposed controller's performance in mitigating the rotors' settling time in the real world.

This is the tenth part of "An Outsider's Tour of Reinforcement Learning." Though I've spent the last few posts casting shade at model-free methods for reinforcement learning, I am not blindly against the model-free paradigm. In fact, the most popular methods in core control systems are model free! The most ubiquitous control scheme out there is PID control, and PID has only three parameters. I'd like to use this post to briefly describe PID control, explain how it is closely connected to many of the most popular methods in machine learning, and then turn to explain what PID brings to the table over the model-free methods that drive contemporary RL research.