Optimal Data Fitting on Lie Groups: a Coset Approach

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 ▲]

Optimization on manifolds : methods and applications

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 ▲]

Synchronization with partial state feedback on SO(n),

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 ▲]

Consensus on homogeneous manifolds

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 ▲]

On the influence of positive and negative feedback loops on the phase response curve of biological oscillators.

Conference (2009, March)

Autonomous rigid body attitude synchronization

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 ▲]

From subspace learning to distance learning: a geometrical optimization approach

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 ▲]

Adaptive Regulation of Vector Controlled Induction Motors

in IEEE Transactions on Control Systems Technology (2009), 17(3),

Nonlinear Drillstring Dynamics Analysis

in SIAM Journal on Applied Dynamical Systems (2009), 8(2), 527--553

Clustering behaviors in networks of integrate-and-fire oscillators

in Chaos (2008), 18

Clustering behavior is studied in a model of integrate-and-fire oscillators with excitatory pulse coupling. When considering a population of identical oscillators, the main result is a proof of global ... [more ▼]

Clustering behavior is studied in a model of integrate-and-fire oscillators with excitatory pulse coupling. When considering a population of identical oscillators, the main result is a proof of global convergence to a phase-locked clustered behavior. The robustness of this clustering behavior is then investigated in a population of nonidentical oscillators by studying the transition from total clustering to the absence of clustering as the group coherence decreases. A robust intermediate situation of partial clustering, characterized by few oscillators traveling among nearly phase-locked clusters, is of particular interest. The analysis complements earlier studies of synchronization in a closely related model. [less ▲]