- ▪E.D. Sontag, "Remarks on input to state stability of perturbed gradient flows, motivated by model-free feedback control learning", Systems and Control Letters, vol. 161, pp. 105138, 2022. pdfImportant: there is an error in the paper. For the LQR application, the paper only shows iISS, not ISS. See the paper "Small-disturbance input-to-state stability of perturbed gradient flows: Applications to LQR problem" for details.gradient dominance · iss · input to state stability · data-driven control · gradient systems · steepest descent · model-free control · gradient dynamics · gradient descent · gradient systems · gradient descent · numerical methods · dynamics of algorithms
Abstract
Recent work on data-driven control and reinforcement learning has renewed interest in a relatively old field in control theory: model-free optimal control approaches which work directly with a cost function and do not rely upon perfect knowledge of a system model. Instead, an "oracle" returns an estimate of the cost associated to, for example, a proposed linear feedback law to solve a linear-quadratic regulator problem. This estimate, and an estimate of the gradient of the cost, might be obtained by performing experiments on the physical system being controlled. This motivates in turn the analysis of steepest descent algorithms and their associated gradient differential equations. This paper studies the effect of errors in the estimation of the gradient, framed in the language of input to state stability, where the input represents a perturbation from the true gradient. Since one needs to study systems evolving on proper open subsets of Euclidean space, a self-contained review of input to state stability definitions and theorems for systems that evolve on such sets is included. The results are then applied to the study of noisy gradient systems, as well as the associated steepest descent algorithms.