Computation of Kalman Decompositions of Periodic Systems

We consider the numerically reliable computation of reachability and observability Kalman decompositions of a periodic system with time-varying dimensions. These decompositons generalize the controllability/observability Kalman decompositions for standard state space systems and have immediate applications in the structural analysis of periodic systems. We propose a structure exploiting numerical algorithm to compute the periodic controllability form by employing exclusively orthogonal similarity transformations. The new algorithm is computationally efficient and strongly backward stable, thus fulfils all requirements for a satisfactory algorithm for periodic systems

Blind source separation of underdetermined mixtures of event-related sources

International audienceThis paper addresses the problem of blind source separation for underdetermined mixtures (i.e., more sources than sensors) of event-related sources that include quasi-periodic sources (e.g., electrocardiogram (ECG)), sources with synchronized trials (e.g., event-related potentials (ERP)), and amplitude-variant sources. The proposed method is based on two steps: (i) tensor decomposition for underdetermined source separation and (ii) signal extraction by Kalman filtering to recover the source dynamics. A tensor is constructed for each source by synchronizing on the ''event'' period of the corresponding signal and stacking different periods along the second dimension of the tensor. To cope with the interference from other sources that impede on the extraction of weak signals, two robust tensor decomposition methods are proposed and compared. Then, the state parameters used within a nonlinear dynamic model for the extraction of event-related sources from noisy mixtures are estimated from the loading matrices provided by the first step. The influence of different parameters on the robustness to outliers of the proposed method is examined by numerical simulations. Applied to clinical electroencephalogram (EEG), ECG and magnetocardiogram (MCG), the proposed method exhibits a significantly higher performance in terms of expected signal shape than classical source separation methods such as piCA and FastICA

Stochastic Sensor Scheduling via Distributed Convex Optimization

In this paper, we propose a stochastic scheduling strategy for estimating the states of N discrete-time linear time invariant (DTLTI) dynamic systems, where only one system can be observed by the sensor at each time instant due to practical resource constraints. The idea of our stochastic strategy is that a system is randomly selected for observation at each time instant according to a pre-assigned probability distribution. We aim to find the optimal pre-assigned probability in order to minimize the maximal estimate error covariance among dynamic systems. We first show that under mild conditions, the stochastic scheduling problem gives an upper bound on the performance of the optimal sensor selection problem, notoriously difficult to solve. We next relax the stochastic scheduling problem into a tractable suboptimal quasi-convex form. We then show that the new problem can be decomposed into coupled small convex optimization problems, and it can be solved in a distributed fashion. Finally, for scheduling implementation, we propose centralized and distributed deterministic scheduling strategies based on the optimal stochastic solution and provide simulation examples.Comment: Proof errors and typos are fixed. One section is removed from last versio

Dynamic mode decomposition with control

We develop a new method which extends Dynamic Mode Decomposition (DMD) to incorporate the effect of control to extract low-order models from high-dimensional, complex systems. DMD finds spatial-temporal coherent modes, connects local-linear analysis to nonlinear operator theory, and provides an equation-free architecture which is compatible with compressive sensing. In actuated systems, DMD is incapable of producing an input-output model; moreover, the dynamics and the modes will be corrupted by external forcing. Our new method, Dynamic Mode Decomposition with control (DMDc), capitalizes on all of the advantages of DMD and provides the additional innovation of being able to disambiguate between the underlying dynamics and the effects of actuation, resulting in accurate input-output models. The method is data-driven in that it does not require knowledge of the underlying governing equations, only snapshots of state and actuation data from historical, experimental, or black-box simulations. We demonstrate the method on high-dimensional dynamical systems, including a model with relevance to the analysis of infectious disease data with mass vaccination (actuation).Comment: 10 pages, 4 figure