| Publications about 'control-Lyapunov functions' |
| Articles in journal or book chapters |
| 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. |
| We discuss several issues related to the stabilizability of nonlinear systems. First, for continuously stabilizable systems, we review constructions of feedbacks that render the system input-to-state stable with respect to actuator errors. Then, we discuss a recent paper which provides a new feedback design that makes globally asymptotically controllable systems input-to-state stable to actuator errors and small observation noise. We illustrate our constructions using the nonholonomic integrator, and discuss a related feedback design for systems with disturbances. |
| The main problem addressed in this paper is the design of feedbacks for globally asymptotically controllable (GAC) control affine systems that render the closed loop systems input to state stable with respect to actuator errors. Extensions for fully nonlinear GAC systems with actuator errors are also discussed. Our controllers have the property that they tolerate small observation noise as well. |
| 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. |
| The main objective of control is to modify the behavior of a dynamical system, typically with the purpose of regulating certain variables or of tracking desired signals. Often, either stability of the closed-loop system is an explicit requirement, or else the problem can be recast in a form that involves stabilization (e.g., of an error signal). For linear systems, the associated problems can now be treated fairly satisfactorily, but in the nonlinear case the area is still far from being settled. Both of the late 1980s reports [9] and [18], with dealt with challenges and future directions for research in control theory, identified the problem of stabilization of finite-dimensional deterministic systems as one of the most important open problems in nonlinear control. We discuss some questions in this area. |
| In this expository paper, we deal with several questions related to stability and stabilization of nonlinear finite-dimensional continuous-time systems. We review the basic problem of feedback stabilization, placing an emphasis upon relatively new areas of research which concern stability with respect to "noise" (such as errors introduced by actuators or sensors). The table of contents is as follows: Review of Stability and Asymptotic Controllability, The Problem of Stabilization, Obstructions to Continuous Stabilization, Control-Lyapunov Functions and Artstein's Theorem, Discontinuous Feedback, Nonsmooth CLF's, Insensitivity to Small Measurement and Actuator Errors, Effect of Large Disturbances: Input-to-State Stability, Comments on Notions Related to ISS. |
| This paper shows that, for time varying systems, global asymptotic controllability to a given closed subset of the state space is equivalent to the existence of a continuous control-Lyapunov function with respect to the set. |
| Shorter and more expository version of "Nonsmooth control-Lyapunov functions" |
| 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. |
| We provide a formula for a stabilizing feedback law using a bounded control, under the assumption that an appropriate control-Lyapunov function is known. Such a feedback, smooth away from the origin and continuous everywhere, is known to exist via Artstein's Theorem. As in the unbounded-control case treated in a previous note, we provide an explicit and ``universal'' formula given by an algebraic function of Lie derivatives. In particular, we extend to the bounded case the result that the feedback can be chosen analytic if the Lyapunov function and the vector fields defining the system are analytic. |
| This note presents an explicit proof of the theorem - due to Artstein - which states that the existence of a smooth control-Lyapunov function implies smooth stabilizability. Moreover, the result is extended to the real-analytic and rational cases as well. The proof uses a "universal" formula given by an algebraic function of Lie derivatives; this formula originates in the solution of a simple Riccati equation. |
| It is shown that a control system in Rn is asymptotically controllable to the origin if and only if there exists a positive definite continuous functional of the states whose derivative can be made negative by appropriate choices of controls. |
| This paper was a conference version of the SIAM paper that introduced the idea of control-Lyapunov functions for arbitrary nonlinear systems. (The journal paper was submitted in 1981 but only published in 1983.) |
| Conference articles |
| This paper continues the investigation of the recently introduced integral version of input-to-state stability (iISS). We study the problem of designing control laws that achieve iISS disturbance attenuation. The main contribution is an appropriate concept of control Lyapunov function (iISS-CLF), whose existence leads to an explicit construction of such a control law. The results are compared and contrasted with the ones available for the ISS case. |
| It is shown that the existence of a continuous control-Lyapunov function (CLF) is necessary and sufficient for null asymptotic controllability of nonlinear finite-dimensional control systems. The CLF condition is expressed in terms of a concept of generalized derivative (upper contingent derivative). This result generalizes to the non-smooth case the theorem of Artstein relating closed-loop feedback stabilization to smooth CLF's. It relies on viability theory as well as optimal control techniques. A "non-strict" version of the results, analogous to the LaSalle Invariance Principle, is also provided. |
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