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Publications about 'universal formulas'
Articles in journal or book chapters
  1. P. Mestres, J. Cortés, and E.D. Sontag. Neural network-based universal formulas for control. 2025. Note: Submitted. Also arXiv https://arxiv.org/abs/2505.24744. Keyword(s): control-Lyapunov functions, control barrier functions, universal formulas, neural networks.
    Abstract:
    We study the problem of designing a controller that satisfies an arbitrary number of affine inequalities at every point in the state space. This is motivated by the use of guardrails in autonomous systems. Indeed, a variety of key control objectives, such as stability, safety, and input saturation, are guaranteed by closed-loop systems whose controllers satisfy such inequalities. Many works in the literature design such controllers as the solution to a state-dependent quadratic program (QP) whose constraints are precisely the inequalities. When the input dimension and number of constraints are high, computing a solution of this QP in real time can become computationally burdensome. Additionally, the solution of such optimization problems is not smooth in general, which can degrade the performance of the system. This paper provides a novel method to design a smooth controller that satisfies an arbitrary number of affine constraints. This why we refer to it as a universal formula for control. The controller is given at every state as the minimizer of a strictly convex function. To avoid computing the minimizer of such function in real time, we introduce a method based on neural networks (NN) to approximate the controller. Remarkably, this NN can be used to solve the controller design problem for any task with less than a fixed input dimension and number of affine constraints, and is completely independent of the state dimension. Additionally, we show that the NN-based controller only needs to be trained with datapoints from a compact set in the state space, which significantly simplifies the training process. Various simulations showcase the performance of the proposed solution, and also show that the NN-based controller can be used to warmstart an optimization scheme that refines the approximation of the true controller in real time, significantly reducing the computational cost compared to a generic initialization.


  2. M. Malisoff and E.D. Sontag. Universal formulas for feedback stabilization with respect to Minkowski balls. Systems Control Lett., 40(4):247-260, 2000. [PDF] Keyword(s): nonlinear control, feedback stabilization, saturation, control-Lyapunov functions, bounded inputs.
    Abstract:
    This note provides explicit algebraic stabilizing formulas for clf's when controls are restricted to certain Minkowski balls in Euclidean space. Feedbacks of this kind are known to exist by a theorem of Artstein, but the proof of Artstein's theorem is nonconstructive. The formulas are obtained from a general feedback stabilization technique and are used to construct approximation solutions to some stabilization problems.


  3. Y. Lin and E.D. Sontag. Control-Lyapunov universal formulas for restricted inputs. Control Theory Adv. Tech., 10(4, part 5):1981-2004, 1995. [PDF] Keyword(s): control-Lyapunov functions, saturation, bounded inputs.
    Abstract:
    We deal with the question of obtaining explicit feedback control laws that stabilize a nonlinear system, under the assumption that a "control Lyapunov function" is known. In previous work, the case of unbounded controls was considered. Here we obtain results for bounded and/or positive controls. We also provide some simple preliminary remarks regarding a set stability version of the problem and a version for systems subject to disturbances.


Conference articles
  1. M. Malisoff and E.D. Sontag. Universal formulas for CLF's with respect to Minkowski balls. In Proc. American Control Conf., San Diego, June 1999, pages 3033-3037, 1999.


  2. Y. Lin and E.D. Sontag. Further universal formulas for Lyapunov approaches to nonlinear stabilization. In Proc. Conf. Inform. Sci. and Systems, John Hopkins University, March 1991, pages 541-546, 1991.



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Last modified: Wed Jun 18 11:30:02 2025
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