Path-Following through Control Funnel Functions

Ravanbakhsh, Hadi; Aghli, Sina; Heckman, Christoffer; Sankaranarayanan, Sriram

doi:10.1109/iros.2018.8593637

Cited by 8 publications

(3 citation statements)

References 32 publications

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“…Although this model is by no means identical to the real system, it is hoped that the CLF and the control law remain valid despite the unmodeled dynamics. Our recent work has successfully investigated physical experiments that use control Lyapunov-like functions learned from mathematical models for path following problems on a 1 8 -scale model vehicle using accurate indoor localization to obtain full state information in realtime [76]. The broader area of iterative learning controls considers the process of learning how to control a given plant at the same time as inferring a more refined model of the plant through exploration [29].…”

Section: Formal Controller Synthesismentioning

confidence: 99%

Learning control lyapunov functions from counterexamples and demonstrations

Ravanbakhsh

Sankaranarayanan

2018

Auton Robot

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We present a technique for learning control Lyapunov-like functions, which are used in turn to synthesize controllers for nonlinear dynamical systems that can stabilize the system, or satisfy specifications such as remaining inside a safe set, or eventually reaching a target set while remaining inside a safe set. The learning framework uses a demonstrator that implements a black-box, untrusted strategy presumed to solve the problem of interest, a learner that poses finitely many queries to the demonstrator to infer a candidate function, and a verifier that checks whether the current candidate is a valid control Lyapunov function. The overall learning framework is iterative, eliminating a set of candidates on each iteration using the counterexamples discovered by the verifier and the demonstrations over these counterexamples. We prove its convergence using ellipsoidal approximation techniques from convex optimization. We also implement this scheme using nonlinear MPC controllers to serve as demonstrators for a set of state and trajectory stabilization problems for nonlinear dynamical systems. We show how the verifier can be constructed efficiently using convex relaxations of the verification problem for polynomial systems to semi-definite programming (SDP) problem instances. Our approach is able to synthesize relatively simple polynomial control Lyapunov functions, and in that process replace the MPC using a guaranteed and computationally less expensive controller.

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Section: Formal Controller Synthesismentioning

confidence: 99%

Learning control lyapunov functions from counterexamples and demonstrations

Ravanbakhsh

Sankaranarayanan

2018

Auton Robot

Self Cite

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“…Especially, numerous path planning algorithms [27,22,11] rely on this task. This involves first designing a controller that follows this trajectory [15], and then determining a neighbourhood of the trajectory (a funnel [24]) where the controller fulfils its goal of reaching a given target set [12,21]. In this paper, we present an efficient method for this second task.…”

Section: Introductionmentioning

confidence: 99%

Computing Funnels Using Numerical Optimization Based Falsifiers

Fejlek¹,

Ratschan²

2021

Preprint

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In this paper, we present an algorithm that computes funnels along trajectories of systems of ordinary differential equations. A funnel is a time-varying set of states containing the given trajectory, for which the evolution from within the set at any given time stays in the funnel. Hence it generalizes the behavior of single trajectories to sets around them, which is an important task, for example, in robot motion planning.In contrast to approaches based on sum-of-squares programming, which poorly scale to high dimensions, our approach is based on falsification and tackles the funnel computation task directly, through numerical optimization. This approach computes accurate funnel estimates far more efficiently and leaves formal verification to the end, outside all funnel size optimization loops.

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“…As it is mentioned in [22], although it is not possible to asymptotically stabilize nonholonomic systems to non-admissible curves because of the non-vanishing tracking error, the practical stabilization can be achieved. It has to be noted that such problem has been addressed only for particular classes of systems, e.g., for unicycle and car-like systems [13,22,24] This paper deals with rather general formulation of the stabilization problem with non-admissible reference curves. The main contribution of our paper is twofold.…”

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confidence: 99%

Stabilization of non-admissible curves for a class of nonholonomic systems

Grushkovskaya

Zuyev

2019

2019 18th European Control Conference (ECC)

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The problem of tracking an arbitrary curve in the state space is considered for underactuated driftless control-affine systems. This problem is formulated as the stabilization of a time-varying family of sets associated with a neighborhood of the reference curve. An explicit control design scheme is proposed for the class of controllable systems whose degree of nonholonomy is equal to 1. It is shown that the trajectories of the closed-loop system converge exponentially to any given neighborhood of the reference curve provided that the solutions are defined in the sense of sampling. This convergence property is also illustrated numerically by several examples of nonholonomic systems of degrees 1 and 2. where x = (x 1 , . . . , x n ) is the state and u = (u 1 , . . . , u m ) is the control. The stabilization of such systems has been the subject of numerous studies over the last few decades, and many important results have been obtained in this area. In particular, it follows from the famous result of R.W. Brockett [7] that the trivial equilibrium of (1) is not stabilizable by a regular time-invariant feedback law if the vectors f 1 (0), f 2 (0), ..., f m (0) are linearly independent. Despite the significant progress in the development of control algorithms to stabilize the solution x = 0 of system (1) (see, e.g., [4,6,8,17,21,23,25,26,30,31], and references therein), the stabilization of nonholonomic systems to a given curve remains a challenging problem. This issue can be formulated as the trajectory tracking problem. In many papers, this problem has been addressed under the assumption that the trajectory is admissible, i.e. satisfies the system equations with some control inputs [1-3, 10, 11, 20, 27-29]. Since the number of controls m may be significantly smaller than the dimension of the state space n, not every path in the state space is admissible for system (1). However, in many applied problems, it is important to stabilize system (1) along an arbitrary curve, which is not necessarily admissible. As it is mentioned in [22], although it is not possible to asymptotically stabilize nonholonomic systems to non-admissible curves because of the non-vanishing tracking error, the practical stabilization can be achieved. It has to be noted that such problem has been addressed only for particular classes of systems, e.g., for unicycle and car-like systems [13,22,24] This paper deals with rather general formulation of the stabilization problem with non-admissible reference curves. The main contribution of our paper is twofold. First, we introduce a class of control functions for the first degree nonholonomic systems, which allows stabilizing the system in a prescribed neighborhood of an arbitrary (not necessarily admissible) curve. We also show how the obtained results can be extended to higher degree nonholonomic systems. The proposed feedback design scheme is based on the approach introduced in [14,30,32] for the stabilization and motion planning of nonholonomic systems. However, it has to be noted that the results o...

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Path-Following through Control Funnel Functions

Cited by 8 publications

References 32 publications

Learning control lyapunov functions from counterexamples and demonstrations

Learning control lyapunov functions from counterexamples and demonstrations

Computing Funnels Using Numerical Optimization Based Falsifiers

Stabilization of non-admissible curves for a class of nonholonomic systems

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