Controller design for systems on manifolds in Euclidean space

Chang, Dong Eui

doi:10.1109/cdc.2017.8264580

Cited by 9 publications

(18 citation statements)

References 18 publications

(52 reference statements)

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“…We here review the thechnique of embedding and transversal stabilization from [2]. Let M be a regular manifold in some R n .…”

Section: Embedding and Transversal Stabilizationmentioning

confidence: 99%

“…Ambient Euclidean Space We review from [2] the technique of tracking controller design via linearization in ambient Euclidean space after embedding. Consider a reference trajectory x 0 : [0, ∞) → M for the system Σ M on M driven by a control signal u 0 : [0, ∞) → R k , so thaṫ x 0 (t) = X(x 0 (t), u 0 (t)) ∀t ≥ 0.…”

Section: Tracking Controller Design Via Linearization Inmentioning

confidence: 99%

“…Assume that (R 0 (t), Ω 0 (t)) and u 0 (t) are bounded over the time interval [0, ∞). The paper [2] provides several tracking controllers for this type of reference trajectories. Let…”

Section: Sym(a) =mentioning

confidence: 99%

“…We have studied in [2] about tracking controller synthesis for systems defined on manifolds via embedding into Euclidean space, transversal stabilization, and linearization. The main idea of this method is as follows.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the linearization of the system dynamics along a reference trajectory on M can be carried out globally, without relying on multiple local charts, in a Cartesian coordinate system in R n to easily design tracking controllers for the system on M . This methodology for controller synthesis is well illustrated with the fully actuated rigid body system in [2]. We refer the reader to [1,3] for other applications of the embedding technique in the context of optimal control and geometric numerical integrations.…”

Section: Introductionmentioning

confidence: 99%

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Observer-Based Controller Design for Systems on Manifolds in Euclidean Space

Chang

2018

2018 57th Annual Conference of the Society of Instrument and Control Engineers of Japan (SICE)

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A method of designing observers and observer-based tracking controllers is proposed for nonlinear systems on manifolds via embedding into Euclidean space and transversal stabilization. Given a system on a manifold, we first embed the manifold and the system into Euclidean space and extend the system dynamics to the ambient Euclidean space in such a way that the manifold becomes an invariant attractor of the extended system, thus securing the transversal stability of the manifold in the extended dynamics. After the embedding, we design state observers and observer-based controllers for the extended system in one single global coordinate system in the ambient Euclidean space, and then restrict them to the original state-space manifold to produce observers and observer-based controllers for the original system on the manifold. This procedure has the merit that any existing control method that has been developed in Euclidean space can be applied globally to systems defined on nonlinear manifolds, thus making nonlinear controller design on manifolds easier. The detail of the method is demonstrated on the fully actuated rigid body system.

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“…We here review the thechnique of embedding and transversal stabilization from [2]. Let M be a regular manifold in some R n .…”

Section: Embedding and Transversal Stabilizationmentioning

confidence: 99%

Section: Tracking Controller Design Via Linearization Inmentioning

confidence: 99%

“…Assume that (R 0 (t), Ω 0 (t)) and u 0 (t) are bounded over the time interval [0, ∞). The paper [2] provides several tracking controllers for this type of reference trajectories. Let…”

Section: Sym(a) =mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Observer-Based Controller Design for Systems on Manifolds in Euclidean Space

Chang

2018

2018 57th Annual Conference of the Society of Instrument and Control Engineers of Japan (SICE)

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Path Following for a Class of Underactuated Systems Using Global Parameterization

2020

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A large number of both aerial and underwater mobile robots fall in the category of underactuated systems that are defined on a manifold, which is not isomorphic to Euclidean space. Traditional approaches to designing controllers for such systems include geometric approaches and local coordinatebased representations. In this paper, we propose a global parameterization of the special orthogonal group, denoted by SO(3), to design path-following controllers for underactuated systems. In particular, we present a nine-dimensional representation of SO(3) that leads to controllers achieving path-invariance for a large class of both closed and non-closed embedded curves. On the one hand, this over-parameterization leads to a simple set of differential equations and provides a global non-ambiguous representation of systems as compared to other local or minimal parametric approaches. On the other hand, this over-parameterization also leads to uncontrolled internal dynamics, which we prove to be bounded and stable. The proposed controller, when applied to a quadrotor system, is capable of recovering the system from challenging situations such as initial upside-down orientation and also capable of performing multiple flips. INDEX TERMS Feedback linearization, nonlinear control, path following, quadrotor, underactuated system.

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Feedback Gradient Descent: Efficient and Stable Optimization with Orthogonality for DNNs

Chang

2022

AAAI

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The optimization with orthogonality has been shown useful in training deep neural networks (DNNs). To impose orthogonality on DNNs, both computational efficiency and stability are important. However, existing methods utilizing Riemannian optimization or hard constraints can only ensure stability while those using soft constraints can only improve efficiency. In this paper, we propose a novel method, named Feedback Gradient Descent (FGD), to our knowledge, the first work showing high efficiency and stability simultaneously. FGD induces orthogonality based on the simple yet indispensable Euler discretization of a continuous-time dynamical system on the tangent bundle of the Stiefel manifold. In particular, inspired by a numerical integration method on manifolds called Feedback Integrators, we propose to instantiate it on the tangent bundle of the Stiefel manifold for the first time. In our extensive image classification experiments, FGD comprehensively outperforms the existing state-of-the-art methods in terms of accuracy, efficiency, and stability.

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Controller design for systems on manifolds in Euclidean space

Cited by 9 publications

References 18 publications

Observer-Based Controller Design for Systems on Manifolds in Euclidean Space

Observer-Based Controller Design for Systems on Manifolds in Euclidean Space

Path Following for a Class of Underactuated Systems Using Global Parameterization

Feedback Gradient Descent: Efficient and Stable Optimization with Orthogonality for DNNs

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