Regret Bounds for Adaptive Nonlinear Control

Boffi, Nicholas M.; Tu, Stephen; Slotine, Jean‐Jacques E.

doi:10.48550/arxiv.2011.13101

Cited by 2 publications

(4 citation statements)

References 22 publications

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“…In this context, we could utilize Control Contraction Metrics (CCMs) [33,[77][78][79][80][81] for extending contraction theory to the systematic design of differential feedback control δu = k(x, δx, u, t) via convex optimization, achieving greater generality at the expense of computational efficiency in obtaining u. Applications of the CCM to estimation, adaptive control, and motion planning are discussed in [82], [83][84][85], and [78,[86][87][88][89], respectively, using geodesic distances between trajectories [49]. It is also worth noting that the objective function of CV-STEM has the condition number of a positive definite matrix that defines a contraction metric as one of its arguments, rendering it applicable and effective even to machine learning-based automatic control frameworks as shall be seen in Sec.…”

Section: Construction Of Contraction Metrics (Sec 3-4)mentioning

confidence: 99%

“…It is demonstrated in [101] that the aNCM is applicable to many types of systems such as robotics systems [5, p. 392], spacecraft high-fidelity dynamics [123,124], and systems modeled by basis function approximation and DNNs [125,126]. Discrete changes could be incorporated in this framework using [85,127].…”

Section: Construction Of Contraction Metrics (Sec 3-4)mentioning

confidence: 99%

“…Sample 𝑥 in bounded error tube Although we consider continuous-time dynamical systems in this section, discrete-time changes could be incorporated in this framework using [85,127]. Also, note that the techniques in this section [101] can be used with differential state feedback frameworks [33,[77][78][79][80][81] as described in Theorem 4.6, trading off added computational cost for generality (see Table 3 and [83,84]).…”

Section: Contraction Theorymentioning

confidence: 99%

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Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview

Tsukamoto,

Chung,

Slotine

2021

Preprint

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Contraction theory is an analytical tool to study differential dynamics of a non-autonomous (i.e., time-varying) nonlinear system under a contraction metric defined with a uniformly positive definite matrix, the existence of which results in a necessary and sufficient characterization of incremental exponential stability of multiple solution trajectories with respect to each other. By using a squared differential length as a Lyapunov-like function, its nonlinear stability analysis boils down to finding a suitable contraction metric that satisfies a stability condition expressed as a linear matrix inequality, indicating that many parallels can be drawn between well-known linear systems theory and contraction theory for nonlinear systems. Furthermore, contraction theory takes advantage of a superior robustness property of exponential stability used in conjunction with the comparison lemma. This yields much-needed safety and stability guarantees for neural network-based control and estimation schemes, without resorting to a more involved method of using uniform asymptotic stability for input-to-state stability. Such distinctive features permit systematic construction of a contraction metric via convex optimization, thereby obtaining an explicit exponential bound on the distance between a time-varying target trajectory and solution trajectories perturbed externally due to disturbances and learning errors. The objective of this paper is therefore to present a tutorial overview of contraction theory and its advantages in nonlinear stability analysis of deterministic and stochastic systems, with an emphasis on deriving formal robustness and stability guarantees for various learning-based and data-driven automatic control methods. In particular, we provide a detailed review of techniques for finding contraction metrics and associated control and estimation laws using deep neural networks.

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Section: Construction Of Contraction Metrics (Sec 3-4)mentioning

confidence: 99%

Section: Construction Of Contraction Metrics (Sec 3-4)mentioning

confidence: 99%

Section: Contraction Theorymentioning

confidence: 99%

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Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview

Tsukamoto,

Chung,

Slotine

2021

Preprint

Self Cite

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“…In recent years, there has been a drive to connect adaptive control methods with techniques from reinforcement learning [7]- [10]. In parallel, methods from reinforcement learning have seen an explosion of work on linear giving precise optimality guaranatees [11]- [13].…”

Section: Introductionmentioning

confidence: 99%

Wasserstein Contraction Bounds on Closed Convex Domains with Applications to Stochastic Adaptive Control

Tyler¹,

Lamperski²

2021

Preprint

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This paper is motivated by the problem of quantitatively bounding the convergence of adaptive control methods for stochastic systems to a stationary distribution. Such bounds are useful for analyzing statistics of trajectories and determining appropriate step sizes for simulations. To this end, we extend a methodology from (unconstrained) stochastic differential equations (SDEs) which provides contractions in a specially chosen Wasserstein distance. This theory focuses on unconstrained SDEs with fairly restrictive assumptions on the drift terms. Typical adaptive control schemes place constraints on the learned parameters and their update rules violate the drift conditions. To this end, we extend the contraction theory to the case of constrained systems represented by reflected stochastic differential equations and generalize the allowable drifts. We show how the general theory can be used to derive quantitative contraction bounds on a nonlinear stochastic adaptive regulation problem.

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Regret Bounds for Adaptive Nonlinear Control

Cited by 2 publications

References 22 publications

Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview

Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview

Wasserstein Contraction Bounds on Closed Convex Domains with Applications to Stochastic Adaptive Control

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