1. E.D. Sontag. Remarks on input to state stability of perturbed gradient flows, motivated by model-free feedback control learning. Systems and Control Letters, 161:105138, 2022. Note: Important: 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.[PDF] Keyword(s): iss, input to state stability, data-driven control, gradient systems, steepest descent, model-free control. 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.




2. E.D. Sontag. Control of systems without drift via generic loops. IEEE Trans. Automat. Control, 40(7):1210-1219, 1995. [PDF] Keyword(s): stabilization, non-holonomic systems, path-planning, systems without drift, nonlinear control, controllability, real-analytic functions. Abstract:

   This paper proposes a simple numerical technique for the steering of arbitrary analytic systems with no drift. It is based on the generation of "nonsingular loops" which allow linearized controllability along suitable trajetories. Once such loops are available, it is possible to employ standard Newton or steepest descent methods, as classically done in numerical control. The theoretical justification of the approach relies on recent results establishing the genericity of nonsingular controls, as well as a simple convergence lemma.

1. E.D. Sontag. Gradient techniques for systems with no drift: A classical idea revisited. In Proc. IEEE Conf. Decision and Control, San Antonio, Dec. 1993, IEEE Publications, 1993, pages 2706-2711, 1993. [PDF] Keyword(s): path-planning, systems without drift, nonlinear control, controllability, real-analytic functions. Abstract:

   This paper proposes a technique for the control of analytic systems with no drift. It is based on the generation of "nonsingular loops" which allow linearized controllability. Once such loops are available, it is possible to employ standard Newton or steepest descent methods. The theoretical justification of the approach relies on results on genericity of nonsingular controls as well as a simple convergence lemma.

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