feedback control

Motivated by the growing use of Artificial Intelligence (AI) tools in control design, this paper takes the first steps towards bridging the gap between results from Direct Gradient methods for the Linear Quadratic Regulator (LQR), and neural networks. More specifically, it looks into the case where one wants to find a Linear Feed-Forward Neural Network (LFFNN) feedback that minimizes a LQR cost. This paper starts by computing the gradient formulas for the parameters of each layer, which are used to derive a key conservation law of the system. This conservation law is then leveraged to prove boundedness and global convergence of solutions to critical points, and invariance of the set of stabilizing networks under the training dynamics. This is followed by an analysis of the case where the LFFNN has a single hidden layer. For this case, the paper proves that the training converges not only to critical points but to the optimal feedback control law for all but a set of measure-zero of the initializations. These theoretical results are followed by an extensive analysis of a simple version of the problem (the ``vector case''), proving the theoretical properties of accelerated convergence and robustness for this simpler example. Finally, the paper presents numerical evidence of faster convergence of the training of general LFFNNs when compared to traditional direct gradient methods, showing that the acceleration of the solution is observable even when the gradient is not explicitly computed but estimated from evaluations of the cost function.

Motivated by the current interest in using Artificial intelligence (AI) tools in control design, this paper takes the first steps towards bridging results from gradient methods for solving the LQR control problem, and neural networks. More specifically, it looks into the case where one wants to find a Linear Feed-Forward Neural Network (LFFNN) that minimizes the Linear Quadratic Regulator (LQR) cost. This work develops gradient formulas that can be used to implement the training of LFFNNs to solve the LQR problem, and derives an important conservation law of the system. This conservation law is then leveraged to prove global convergence of solutions and invariance of the set of stabilizing networks under the training dynamics. These theoretical results are then followed by and extensive analysis of the simplest version of the problem (the ``scalar case'') and by numerical evidence of faster convergence of the training of general LFFNNs when compared to traditional direct gradient methods. These results not only serve as indication of the theoretical value of studying such a problem, but also of the practical value of LFFNNs as design tools for data-driven control applications.

Systems theory can play an important in unveiling fundamental limitations of learning algorithms and architectures when used to control a dynamical system, and in suggesting strategies for overcoming these limitations. As an example, a feedforward neural network cannot stabilize a double integrator using output feedback. Similarly, a recurrent NN with differentiable activation functions that stabilizes a non-strongly stabilizable system must be itself open loop unstable, a fact that has profound implications for training with noisy, finite data. A potential solution to this problem, motivated by results on stabilization with periodic control, is the use of neural nets with periodic resets, showing that indeed systems theoretic analysis is instrumental in developing architectures capable of controlling certain classes of unstable systems. This short conference paper also argues that when the goal is to learn control oriented models, the loss function should reflect closed loop, rather than open loop model performance, a fact that can be accomplished by using gap-metric motivated loss functions.