A family of extremum seeking laws for a unicycle model with a moving target: theoretical and experimental studies

Grushkovskaya, Victoria; Michalowsky, Simon; Zuyev, Alexander; May, Max P.; Ebenbauer, Christian

doi:10.23919/ecc.2018.8550280

Cited by 11 publications

(7 citation statements)

References 29 publications

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“…The main idea therein is that trajectories of the extremum seeking system approximate trajectories of a so-called Lie bracket system which corresponds to a gradient-like dynamics which optimizes the cost function. Based on the Lie bracket system and its corresponding extremum seeking system, a whole analysis and design framework has been established, see, for example, References 16,17,[21][22][23][24][25][26][27][28][29][30][31][32] In particular, extremum seeking for dynamic nonlinear systems using Lie bracket approximations has been addressed in Reference 25. In that article, a combination of Lie bracket approximations and singular perturbations techniques (time-scale separation) has been proposed. In general, for dynamic extremum seeking systems, a singular perturbation approach is quite common, see, for example, the articles.…”

Section: Introductionmentioning

confidence: 99%

Extremum seeking control of nonlinear dynamic systems using Lie bracket approximations

Grushkovskaya

Ebenbauer

2020

Adaptive Control & Signal

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Summary In this article, we consider extremum seeking problems for a general class of nonlinear dynamic control systems. The main result of the article is a broad family of control laws which optimize the steady‐state performance of the system. We prove practical asymptotic stability of the optimal steady‐state and, moreover, propose sufficient conditions for the asymptotic stability in the sense of Lyapunov. The results generalize and extend existing results which are based on Lie bracket approximations. In particular, our approach does not rely on singular perturbation theory, as commonly used in extremum seeking of nonlinear dynamic systems.

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Section: Introductionmentioning

confidence: 99%

Extremum seeking control of nonlinear dynamic systems using Lie bracket approximations

Grushkovskaya

Ebenbauer

2020

Adaptive Control & Signal

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“…This approach was further improved in [30], where the velocity measurements of the unicycle is no longer required [30]. Some improvements of the tracking behavior of source seeking algorithms are presented in [12]. In addition to the unicycle model, two source seeking control schemes for actuating a three dimensional autonomous vehicle are presented in [5].…”

Section: Related Workmentioning

confidence: 99%

Practical Prescribed-Time Seeking of a Repulsive Source by Unicycle Angular Velocity Tuning

Velimir¹,

Krstić²

2022

Preprint

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All the existing source seeking algorithms for unicycle models in GPS-denied settings guarantee at best an exponential rate of convergence over an infinite interval. Using the recently introduced time-varying feedback tools for prescribed-time stabilization, we achieve source seeking in prescribed time, i.e., the convergence to the source, without the measurements of the position and velocity of the unicycle, in as short a time as the user desires, starting from an arbitrary distance from the source. The convergence is established using a singularly perturbed version of the Lie bracket averaging method, combined with time dilation and time contraction operations. The algorithm is robust, provably, even to an arbitrarily strong gradient-dependent repulsive velocity drift emanating from the source.

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“…To characterize the asymptotic behavior of trajectories of system (1), we will extend the concept of stability of a family of sets to the case of π ε -solutions. This concept has been developed, e.g., in [19] for non-autonomous differential equations and applied to control problems under the classical definition of solutions in [12,13]. Let {S t } t≥0 be a one-parameter family of non-empty subsets of R n .…”

Section: Stability Of a Family Of Setsmentioning

confidence: 99%

“…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%

“…Second, we characterize stability properties of system (1) with the proposed controls in terms of families of sets. Note that the concept of stability of families of sets was used previously in [12,13,19] for non-autonomous system admitting a Lyapunov function. In the present paper, we do not assume the existence of a control Lyapunov function and define solutions of the closed-loop system in the sense of sampling.…”

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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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A family of extremum seeking laws for a unicycle model with a moving target: theoretical and experimental studies

Cited by 11 publications

References 29 publications

Extremum seeking control of nonlinear dynamic systems using Lie bracket approximations

Extremum seeking control of nonlinear dynamic systems using Lie bracket approximations

Practical Prescribed-Time Seeking of a Repulsive Source by Unicycle Angular Velocity Tuning

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

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