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Optimal Data Fitting on Lie Groups: a Coset Approach ; Sepulchre, Rodolphe in Diehl, M. (Ed.) Recent Advances in Optimization and its Applications in Engineering (2010) This work considers the problem of fitting data on a Lie group by a coset of a compact subgroup. This problem can be seen as an extension of the problem of fitting affine subspaces in Rn to data which can ... [more ▼] This work considers the problem of fitting data on a Lie group by a coset of a compact subgroup. This problem can be seen as an extension of the problem of fitting affine subspaces in Rn to data which can be solved using principal component analysis. We show how the fitting problem can be reduced for biinvariant distances to a generalized mean calculation on an homogeneous space. For biinvariant Riemannian distances we provide an algorithm based on the Karcher mean gradient algorithm. We illustrate our approach by some examples on SO(n). [less ▲] Detailed reference viewed: 20 (1 ULg)Refining Sparse Principal Components ; ; et al in Diehl, M. (Ed.) Recent Advances in Optimization and its Applications in Engineering (2010) In this paper, we discuss methods to refine locally optimal solutions of sparse PCA. Starting from a local solution obtained by existing algorithms, these methods take advantage of convex relaxations of ... [more ▼] In this paper, we discuss methods to refine locally optimal solutions of sparse PCA. Starting from a local solution obtained by existing algorithms, these methods take advantage of convex relaxations of the sparse PCA problem to propose a refined solution that is still locally optimal but with a higher objective value. [less ▲] Detailed reference viewed: 102 (2 ULg)Optimization on manifolds : methods and applications ; ; Sepulchre, Rodolphe in Recent Advances in Optimization and its Applications in Engineering (2010) Summary. This paper provides an introduction to the topic of optimization on manifolds. The approach taken uses the language of differential geometry, however, we choose to emphasise the intuition of the ... [more ▼] Summary. This paper provides an introduction to the topic of optimization on manifolds. The approach taken uses the language of differential geometry, however, we choose to emphasise the intuition of the concepts and the structures that are important in generating practical numerical algorithms rather than the technical details of the formulation. There are a number of algorithms that can be applied to solve such problems and we discuss the steepest descent and Newton’s method in some detail as well as referencing the more important of the other approaches. There are a wide range of potential applications that we are aware of, and we briefly discuss these applications, as well as explaining one or two in more detail. [less ▲] Detailed reference viewed: 80 (2 ULg)Controlling the phase of an oscillator: a phase response curve approach ; Sacré, Pierre ; Sepulchre, Rodolphe in Proceedings of the Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference (2009, December) The paper discusses elementary control strategies to control the phase of an oscillator. Both feedforward and feedback (P and PI) control laws are designed based on the phase response curve (PRC ... [more ▼] The paper discusses elementary control strategies to control the phase of an oscillator. Both feedforward and feedback (P and PI) control laws are designed based on the phase response curve (PRC) calculated from the linearized model. The performance is evaluated on a popular model of circadian oscillations. [less ▲] Detailed reference viewed: 101 (58 ULg)Synchronization with partial state feedback on SO(n), ; Sarlette, Alain ; Sepulchre, Rodolphe in Proceedings of the 48th IEEE Conference on Decision and Control (2009, December) This paper considers the problem of constructing a distributed feedback law to achieve synchronization for a group of k agents whose states evolve on SO(n) and which exchange only partial state ... [more ▼] This paper considers the problem of constructing a distributed feedback law to achieve synchronization for a group of k agents whose states evolve on SO(n) and which exchange only partial state information along communication links. The partial state information is given by the action of the state on reference vectors in Rn. We propose a gradient based control law which achieves exponential local convergence to a synchronization configuration under a rank condition on a generalized Laplacian matrix. Furthermore, we discuss the case of time-varying reference vectors and provide a convergence result for this case. The latter helps reach synchronization, requiring less communication links and weaker conditions on the instantaneous reference vectors. Our methods are illustrated on an attitude synchronization problem where agents exchange only their relative positions observed in the respective body frames. [less ▲] Detailed reference viewed: 35 (6 ULg)Erratum: “Clustering behaviors in networks of integrate-and-fire oscillators” [Chaos 18, 037122 (2008)] Mauroy, Alexandre ; Sepulchre, Rodolphe in Chaos (2009), 19 Detailed reference viewed: 39 (12 ULg)Consensus on homogeneous manifolds Sarlette, Alain ; Sepulchre, Rodolphe in Proceedings of the 48th IEEE Conference on Decision and Control (2009, December) The present paper considers distributed consensus algorithms for agents evolving on a connected compact homogeneous (CCH) manifold. The agents track no external reference and communicate their relative ... [more ▼] The present paper considers distributed consensus algorithms for agents evolving on a connected compact homogeneous (CCH) manifold. The agents track no external reference and communicate their relative state according to an interconnection graph. The paper first formalizes the consensus problem for synchronization (i.e. maximizing the consensus) and balancing (i.e. minimizing the consensus); it thereby introduces the induced arithmetic mean, an easily computable mean position on CCH manifolds. Then it proposes and analyzes various consensus algorithms on manifolds: natural gradient algorithms which reach local consensus equilibria; an adaptation using auxiliary variables for almost-global synchronization or balancing; and a stochastic gossip setting for global synchronization. It closes by investigating the dependence of synchronization properties on the attraction function between interacting agents on the circle. The theory is also illustrated on SO(n) and on the Grassmann manifolds. [less ▲] Detailed reference viewed: 41 (7 ULg)A PDE viewpoint on basic properties of coordination algorithms with symmetries Sarlette, Alain ; Sepulchre, Rodolphe in Proceedings of the 48th IEEE Conference on Decision and Control (2009, December) Several recent control applications consider the coordination of subsystems through local interaction. Often the interaction has a symmetry in state space, e.g. invariance with respect to a uniform ... [more ▼] Several recent control applications consider the coordination of subsystems through local interaction. Often the interaction has a symmetry in state space, e.g. invariance with respect to a uniform translation of all subsystem values. The present paper shows that in presence of such symmetry, fundamental properties can be highlighted by viewing the distributed system as the discrete approximation of a partial differential equation. An important fact is that the symmetry on the state space differs from the popular spatial invariance property, which is not necessary for the present results. The relevance of the viewpoint is illustrated on two examples: (i) ill-conditioning of interaction matrices in coordination/consensus problems and (ii) the string instability issue. [less ▲] Detailed reference viewed: 22 (4 ULg)A hypothesis to explain the effect of SK channel blockade on the firing of dopaminergic neurons Drion, Guillaume ; Collard, Anne ; Seutin, Vincent et al Poster (2009, October) Detailed reference viewed: 7 (1 ULg)Bursting modeling in dopaminergic neurons Drion, Guillaume ; Seutin, Vincent ; Sepulchre, Rodolphe Conference (2009, March) Detailed reference viewed: 8 (0 ULg)On the influence of positive and negative feedback loops on the phase response curve of biological oscillators. Sacré, Pierre ; Sepulchre, Rodolphe Conference (2009, March) Detailed reference viewed: 103 (39 ULg)An application of distributed control: segmented primary mirror for the European Extremely Large Telescope Bastin, Christian ; Sarlette, Alain ; Sepulchre, Rodolphe Conference (2009, March) Detailed reference viewed: 47 (20 ULg)Autonomous rigid body attitude synchronization Sarlette, Alain ; Sepulchre, Rodolphe ; in Automatica (2009), 45(2), 572-577 Control laws to synchronize attitudes in a swarm of fully actuated rigid bodies, in the absence of a common reference attitude or hierarchy in the swarm, are proposed in [Smith, T. R., Hanssmann, H ... [more ▼] Control laws to synchronize attitudes in a swarm of fully actuated rigid bodies, in the absence of a common reference attitude or hierarchy in the swarm, are proposed in [Smith, T. R., Hanssmann, H., & Leonard, N.E. (2001). "Orientation control of multiple underwater vehicles with symmetry-breaking potentials". In Proc. 40th IEEE conf. decision and control (pp. 4598–4603); Nair, S., Leonard, N. E. (2007). "Stable synchronization of rigid body networks". In Networks and Heterogeneous Media, 2(4), 595–624]. The present paper studies two separate extensions with the same energy shaping approach: (i) locally synchronizing the rigid bodies’ attitudes, but without restricting their final motion and (ii) relaxing the communication topology from undirected, fixed and connected to directed, varying and uniformly connected. The specific strategies that must be developed for these extensions illustrate the limitations of attitude control with reduced information. [less ▲] Detailed reference viewed: 67 (12 ULg)Consensus optimization on manifolds Sarlette, Alain ; Sepulchre, Rodolphe in SIAM Journal on Control & Optimization (2009), 48(1), 56-76 The present paper considers distributed consensus algorithms that involve N agents evolving on a connected compact homogeneous manifold. The agents track no external reference and communicate their ... [more ▼] The present paper considers distributed consensus algorithms that involve N agents evolving on a connected compact homogeneous manifold. The agents track no external reference and communicate their relative state according to a communication graph. The consensus problem is formulated in terms of the extrema of a cost function. This leads to eﬃcient gradient algorithms to synchronize (i.e., maximizing the consensus) or balance (i.e., minimizing the consensus) the agents; a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time-varying. The cost function is linked to a speciﬁc centroid deﬁnition on manifolds, introduced here as the induced arithmetic mean, that is easily computable in closed form and may be of independent interest for a number of manifolds. The special orthogonal group SO(n) and the Grassmann manifold Grass(p,n) are treated as original examples. A link is also drawn with the many existing results on the circle. [less ▲] Detailed reference viewed: 61 (8 ULg)Clustering behaviors in networks of pulse-coupled integrate-and-fire oscillators Mauroy, Alexandre ; Sepulchre, Rodolphe Conference (2009) Detailed reference viewed: 10 (0 ULg)Recent advances and open questions on Peskin model for coupled oscillators Mauroy, Alexandre ; Sepulchre, Rodolphe Conference (2009) Detailed reference viewed: 8 (0 ULg)From subspace learning to distance learning: a geometrical optimization approach Meyer, Gilles ; ; et al in Proceedings of the 2009 IEEE Workshop on Statistical Signal Processing (SSP2009) (2009) In this paper, we adopt a differential-geometry viewpoint to tackle the problem of learning a distance online. As this prob- lem can be cast into the estimation of a fixed-rank positive semidefinite (PSD ... [more ▼] In this paper, we adopt a differential-geometry viewpoint to tackle the problem of learning a distance online. As this prob- lem can be cast into the estimation of a fixed-rank positive semidefinite (PSD) matrix, we develop algorithms that ex- ploits the rich geometry structure of the set of fixed-rank PSD matrices. We propose a method which separately updates the subspace of the matrix and its projection onto that subspace. A proper weighting of the two iterations enables to continu- ously interpolate between the problem of learning a subspace and learning a distance when the subspace is fixed. [less ▲] Detailed reference viewed: 77 (27 ULg)Synchronization in Networks of Identical Linear Systems ; Sepulchre, Rodolphe in Automatica (2009), 45(11), 2557-2562 Detailed reference viewed: 77 (4 ULg)Adaptive Regulation of Vector Controlled Induction Motors ; ; et al in IEEE Transactions on Control Systems Technology (2009), 17(3), Detailed reference viewed: 42 (6 ULg)Riemannian Metric and Geometric Mean for Positive Semidefinite Matrices of Fixed Rank ; Sepulchre, Rodolphe in SIAM Journal on Matrix Analysis & Applications (2009), 31(3), 1055--1070 Detailed reference viewed: 36 (11 ULg) |
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