Semiglobal optimal feedback stabilization of autonomous systems via deep neural network approximation

Abstract

A learning approach for optimal feedback gains for nonlinear continuous time control systems is proposed and analyzed . The goal is to establish a rigorous framework for computing approximating optimal feedback gains using neural networks. The approach rests on two main ingredients. First, an optimal control formulation involving an ensemble of trajectories with 'control' variables given by the feedback gain functions. Second, an approximation to the feedback functions via realizations of neural networks. Based on universal approximation properties we prove the existence and convergence of optimal stabilizing neural network feedback controllers.

Neural network optimal feedback control with enhanced closed loop stability

“…One of the main advantages of these architectures is that they automatically make the goal state x f an equilibrium. This is achieved by the [−N (x f ; θ)] term in (3.2) and (3.3), which is also suggested in [19]. This property is formalized in the following proposition, whose proof is straightforward.…”

“…Many NN-based methods attempt to solve the HJB PDE in the least-squares sense by minimizing the residual of the PDE and boundary conditions at randomly sampled collocation points [1,34,32]. Lastly, [10,27,19] propose self-supervised learning approaches to solve the HJB equation along its characteristics without generating any data. Of course, many other methods have been proposed for solving HJB equations and designing feedback controllers, but in the present work we focus on supervised learning approaches as these explicitly quantify error with respect to the optimal control.…”

Neural Network Based Nonlinear Observers

“…This network will be trained by using (P t ), and information on the value function will be recovered on the basis of (7). A related network based approach for stabilization of nonlinear systems was recently proposed in [14].…”

“…In this respect we can refer to a rather detailed description in [14] on the treatment of the optimization problem which describes the network learning step. Assuming the existence of a minimizer (θ * , x * θ,j ), j = 1, .…”

Tensor Decomposition Methods for High-dimensional Hamilton--Jacobi--Bellman Equations

“…For example, it was verified in [15] that the cost functional for the bilinear optimal control problem for the Fokker-Planck equation we present in our numerical results belongs to C ∞ in a neighborhood of the origin. In [44] sufficient conditions for C 1 -regularity of the value function on bounded sets are given.…”