Optimal Control via Neural Networks: A Convex Approach

In this article, we propose a new type of neural network called Element-Learning (EL) models that can handle nonlinear systems with multiple equilibria. Unlike traditional neural networks, EL models can learn a wider range of dynamical systems by using element-wise bijective mappings. This allows the model to represent systems with multiple equilibria and nonlinear dynamics more accurately than previous models.
The proposed EL model has several superior properties that make it an effective tool for feedback control design. Firstly, the EL model can be considered linear because of the dynamic feedback, making it easier to analyze and optimize. Secondly, the reference for y can be replaced with that for x due to the bijectivity of Φ, which makes the admissible set of u corresponding to the input constraint convex. Finally, the admissible set of x and u corresponding to the output constraint is also convex because of the convexity of Ξ.
The learning process of the EL model is similar to that of the S–HW model, except that output y is fed back into (10) to use the same procedure for learning the model. The predicted outputs are expressed with a hat to distinguish them from real data. From (10)-(13), we obtain A(d)Φ(y, d) + B(d)Ψ−1(v, y, d) + c(d) − ∂Φ/∂y and ∂Φ/∂d.
In summary, the EL model is a powerful tool for handling nonlinear systems with multiple equilibria in feedback control design. Its ability to learn a wider range of dynamical systems makes it an attractive choice for controlling complex systems. By using element-wise bijective mappings, the EL model can represent systems with multiple equilibria more accurately than previous models.