Identification of Linear Systems with Multiplicative Noise from Multiple Trajectory Data

We study identification of linear systems with multiplicative noise from multiple trajectory data. A least-squares algorithm, based on exploratory inputs, is proposed to simultaneously estimate the parameters of the nominal system and the covariance matrix of the multiplicative noise. The algorithm does not need prior knowledge of the noise or stability of the system, but requires mild conditions of inputs and relatively small length for each trajectory. Identifiability of the noise covariance matrix is studied, showing that there exists an equivalent class of matrices that generate the same second-moment dynamic of system states. It is demonstrated how to obtain the equivalent class based on estimates of the noise covariance. Asymptotic consistency of the algorithm is verified under sufficiently exciting inputs and system controllability conditions. Non-asymptotic estimation performance is also analyzed under the assumption that system states and noise are bounded, providing vanishing high-probability bounds as the number of trajectories grows to infinity. The results are illustrated by numerical simulations

Long-tail Behavior in Locomotion of Caenorhabditis elegans

The locomotion of Caenorhabditis elegans exhibits complex patterns. In particular, the worm combines mildly curved runs and sharp turns to steer its course. Both runs and sharp turns of various types are important components of taxis behavior. The statistics of sharp turns have been intensively studied. However, there have been few studies on runs, except for those on klinotaxis (also called weathervane mechanism), in which the worm gradually curves toward the direction with a high concentration of chemicals; this phenomenon was discovered recently. We analyzed the data of runs by excluding sharp turns. We show that the curving rate obeys long-tail distributions, which implies that large curving rates are relatively frequent. This result holds true for locomotion in environments both with and without a gradient of NaCl concentration; it is independent of klinotaxis. We propose a phenomenological computational model on the basis of a random walk with multiplicative noise. The assumption of multiplicative noise posits that the fluctuation of the force is proportional to the force exerted. The model reproduces the long-tail property present in the experimental data.Comment: 30 pages, 11 figures, some errors were correcte

Invariant EKF Design for Scan Matching-aided Localization

Localization in indoor environments is a technique which estimates the robot's pose by fusing data from onboard motion sensors with readings of the environment, in our case obtained by scan matching point clouds captured by a low-cost Kinect depth camera. We develop both an Invariant Extended Kalman Filter (IEKF)-based and a Multiplicative Extended Kalman Filter (MEKF)-based solution to this problem. The two designs are successfully validated in experiments and demonstrate the advantage of the IEKF design