arXiv subjects

S. Shankar Sastry

Publications and source records attributed to S. Shankar Sastry.

88 records · Page 5Linked to original sources

arXiv2013

Nonlinear Compressive Particle Filtering

Many systems for which compressive sensing is used today are dynamical. The common approach is to neglect the dynamics and see the problem as a sequence of independent problems. This approach has two disadvantages. Firstly, the temporal dependency in the state could be used to improve the accuracy of the state estimates. Secondly, having an estimate for the state and its support could be used to reduce the computational load of the subsequent step. In the linear Gaussian setting, compressive sensing was recently combined with the Kalman filter to mitigate above disadvantages. In the nonlinear dynamical case, compressive sensing can not be used and, if the state dimension is high, the particle filter would perform poorly. In this paper we combine one of the most novel developments in compressive sensing, nonlinear compressive sensing, with the particle filter. We show that the marriage of the two is essential and that neither the particle filter or nonlinear compressive sensing alone gives a satisfying solution.

eess.SY↗

arXiv2013

Compressive Shift Retrieval

The classical shift retrieval problem considers two signals in vector form that are related by a shift. The problem is of great importance in many applications and is typically solved by maximizing the cross-correlation between the two signals. Inspired by compressive sensing, in this paper, we seek to estimate the shift directly from compressed signals. We show that under certain conditions, the shift can be recovered using fewer samples and less computation compared to the classical setup. Of particular interest is shift estimation from Fourier coefficients. We show that under rather mild conditions only one Fourier coefficient suffices to recover the true shift.

eess.SY↗

arXiv2013

Energy Disaggregation via Adaptive Filtering

The energy disaggregation problem is recovering device level power consumption signals from the aggregate power consumption signal for a building. We show in this paper how the disaggregation problem can be reformulated as an adaptive filtering problem. This gives both a novel disaggregation algorithm and a better theoretical understanding for disaggregation. In particular, we show how the disaggregation problem can be solved online using a filter bank and discuss its optimality.

stat.AP↗

arXiv2013

Incentive Mechanisms for Internet Congestion Management: Fixed-Budget Rebate versus Time-of-Day Pricing

Mobile data traffic has been steadily rising in the past years. This has generated a significant interest in the deployment of incentive mechanisms to reduce peak-time congestion. Typically, the design of these mechanisms requires information about user demand and sensitivity to prices. Such information is naturally imperfect. In this paper, we propose a \emph{fixed-budget rebate mechanism} that gives each user a reward proportional to his percentage contribution to the aggregate reduction in peak time demand. For comparison, we also study a time-of-day pricing mechanism that gives each user a fixed reward per unit reduction of his peak-time demand. To evaluate the two mechanisms, we introduce a game-theoretic model that captures the \emph{public good} nature of decongestion. For each mechanism, we demonstrate that the socially optimal level of decongestion is achievable for a specific choice of the mechanism's parameter. We then investigate how imperfect information about user demand affects the mechanisms' effectiveness. From our results, the fixed-budget rebate pricing is more robust when the users' sensitivity to congestion is "sufficiently" convex. This feature of the fixed-budget rebate mechanism is attractive for many situations of interest and is driven by its closed-loop property, i.e., the unit reward decreases as the peak-time demand decreases.

cs.NI↗

arXiv2013

Nonlinear Basis Pursuit

In compressive sensing, the basis pursuit algorithm aims to find the sparsest solution to an underdetermined linear equation system. In this paper, we generalize basis pursuit to finding the sparsest solution to higher order nonlinear systems of equations, called nonlinear basis pursuit. In contrast to the existing nonlinear compressive sensing methods, the new algorithm that solves the nonlinear basis pursuit problem is convex and not greedy. The novel algorithm enables the compressive sensing approach to be used for a broader range of applications where there are nonlinear relationships between the measurements and the unknowns.

cs.IT↗

arXiv2013

A Dynamical Systems Approach to Energy Disaggregation

Energy disaggregation, also known as non-intrusive load monitoring (NILM), is the task of separating aggregate energy data for a whole building into the energy data for individual appliances. Studies have shown that simply providing disaggregated data to the consumer improves energy consumption behavior. However, placing individual sensors on every device in a home is not presently a practical solution. Disaggregation provides a feasible method for providing energy usage behavior data to the consumer which utilizes currently existing infrastructure. In this paper, we present a novel framework to perform the energy disaggregation task. We model each individual device as a single-input, single-output system, where the output is the power consumed by the device and the input is the device usage. In this framework, the task of disaggregation translates into finding inputs for each device that generates our observed power consumption. We describe an implementation of this framework, and show its results on simulated data as well as data from a small-scale experiment.

math.DS↗

arXiv2013

Blind Identification of ARX Models with Piecewise Constant Inputs

Blind system identification is known to be a hard ill-posed problem and without further assumptions, no unique solution is at hand. In this contribution, we are concerned with the task of identifying an ARX model from only output measurements. Driven by the task of identifying systems that are turned on and off at unknown times, we seek a piecewise constant input and a corresponding ARX model which approximates the measured outputs. We phrase this as a rank minimization problem and present a relaxed convex formulation to approximate its solution. The proposed method was developed to model power consumption of electrical appliances and is now a part of a bigger energy disaggregation framework. Code will be made available online.

eess.SY↗

arXiv2013

Quadratic Basis Pursuit

In many compressive sensing problems today, the relationship between the measurements and the unknowns could be nonlinear. Traditional treatment of such nonlinear relationships have been to approximate the nonlinearity via a linear model and the subsequent un-modeled dynamics as noise. The ability to more accurately characterize nonlinear models has the potential to improve the results in both existing compressive sensing applications and those where a linear approximation does not suffice, e.g., phase retrieval. In this paper, we extend the classical compressive sensing framework to a second-order Taylor expansion of the nonlinearity. Using a lifting technique and a method we call quadratic basis pursuit, we show that the sparse signal can be recovered exactly when the sampling rate is sufficiently high. We further present efficient numerical algorithms to recover sparse signals in second-order nonlinear systems, which are considerably more difficult to solve than their linear counterparts in sparse optimization.

cs.IT↗

arXiv2012

Fast L1-Minimization Algorithms For Robust Face Recognition

L1-minimization refers to finding the minimum L1-norm solution to an underdetermined linear system b=Ax. Under certain conditions as described in compressive sensing theory, the minimum L1-norm solution is also the sparsest solution. In this paper, our study addresses the speed and scalability of its algorithms. In particular, we focus on the numerical implementation of a sparsity-based classification framework in robust face recognition, where sparse representation is sought to recover human identities from very high-dimensional facial images that may be corrupted by illumination, facial disguise, and pose variation. Although the underlying numerical problem is a linear program, traditional algorithms are known to suffer poor scalability for large-scale applications. We investigate a new solution based on a classical convex optimization framework, known as Augmented Lagrangian Methods (ALM). The new convex solvers provide a viable solution to real-world, time-critical applications such as face recognition. We conduct extensive experiments to validate and compare the performance of the ALM algorithms against several popular L1-minimization solvers, including interior-point method, Homotopy, FISTA, SESOP-PCD, approximate message passing (AMP) and TFOCS. To aid peer evaluation, the code for all the algorithms has been made publicly available.

cs.CV↗

arXiv2012

Consistent Approximations for the Optimal Control of Constrained Switched Systems

Though switched dynamical systems have shown great utility in modeling a variety of physical phenomena, the construction of an optimal control of such systems has proven difficult since it demands some type of optimal mode scheduling. In this paper, we devise an algorithm for the computation of an optimal control of constrained nonlinear switched dynamical systems. The control parameter for such systems include a continuous-valued input and discrete-valued input, where the latter corresponds to the mode of the switched system that is active at a particular instance in time. Our approach, which we prove converges to local minimizers of the constrained optimal control problem, first relaxes the discrete-valued input, then performs traditional optimal control, and then projects the constructed relaxed discrete-valued input back to a pure discrete-valued input by employing an extension to the classical Chattering Lemma that we prove. We extend this algorithm by formulating a computationally implementable algorithm which works by discretizing the time interval over which the switched dynamical system is defined. Importantly, we prove that this implementable algorithm constructs a sequence of points by recursive application that converge to the local minimizers of the original constrained optimal control problem. Four simulation experiments are included to validate the theoretical developments.

math.OC↗

arXiv2012

Provably Safe and Robust Learning-Based Model Predictive Control

Controller design faces a trade-off between robustness and performance, and the reliability of linear controllers has caused many practitioners to focus on the former. However, there is renewed interest in improving system performance to deal with growing energy constraints. This paper describes a learning-based model predictive control (LBMPC) scheme that provides deterministic guarantees on robustness, while statistical identification tools are used to identify richer models of the system in order to improve performance; the benefits of this framework are that it handles state and input constraints, optimizes system performance with respect to a cost function, and can be designed to use a wide variety of parametric or nonparametric statistical tools. The main insight of LBMPC is that safety and performance can be decoupled under reasonable conditions in an optimization framework by maintaining two models of the system. The first is an approximate model with bounds on its uncertainty, and the second model is updated by statistical methods. LBMPC improves performance by choosing inputs that minimize a cost subject to the learned dynamics, and it ensures safety and robustness by checking whether these same inputs keep the approximate model stable when it is subject to uncertainty. Furthermore, we show that if the system is sufficiently excited, then the LBMPC control action probabilistically converges to that of an MPC computed using the true dynamics.

math.OC↗

arXiv2012

Statistical Results on Filtering and Epi-convergence for Learning-Based Model Predictive Control

Learning-based model predictive control (LBMPC) is a technique that provides deterministic guarantees on robustness, while statistical identification tools are used to identify richer models of the system in order to improve performance. This technical note provides proofs that elucidate the reasons for our choice of measurement model, as well as giving proofs concerning the stochastic convergence of LBMPC. The first part of this note discusses simultaneous state estimation and statistical identification (or learning) of unmodeled dynamics, for dynamical systems that can be described by ordinary differential equations (ODE's). The second part provides proofs concerning the epi-convergence of different statistical estimators that can be used with the learning-based model predictive control (LBMPC) technique. In particular, we prove results on the statistical properties of a nonparametric estimator that we have designed to have the correct deterministic and stochastic properties for numerical implementation when used in conjunction with LBMPC.

math.OC↗

arXiv2012

Compressive Phase Retrieval From Squared Output Measurements Via Semidefinite Programming

Given a linear system in a real or complex domain, linear regression aims to recover the model parameters from a set of observations. Recent studies in compressive sensing have successfully shown that under certain conditions, a linear program, namely, l1-minimization, guarantees recovery of sparse parameter signals even when the system is underdetermined. In this paper, we consider a more challenging problem: when the phase of the output measurements from a linear system is omitted. Using a lifting technique, we show that even though the phase information is missing, the sparse signal can be recovered exactly by solving a simple semidefinite program when the sampling rate is sufficiently high, albeit the exact solutions to both sparse signal recovery and phase retrieval are combinatorial. The results extend the type of applications that compressive sensing can be applied to those where only output magnitudes can be observed. We demonstrate the accuracy of the algorithms through theoretical analysis, extensive simulations and a practical experiment.

math.ST↗

arXiv2012

On the Lagrangian Biduality of Sparsity Minimization Problems

Recent results in Compressive Sensing have shown that, under certain conditions, the solution to an underdetermined system of linear equations with sparsity-based regularization can be accurately recovered by solving convex relaxations of the original problem. In this work, we present a novel primal-dual analysis on a class of sparsity minimization problems. We show that the Lagrangian bidual (i.e., the Lagrangian dual of the Lagrangian dual) of the sparsity minimization problems can be used to derive interesting convex relaxations: the bidual of the $\ell_0$-minimization problem is the $\ell_1$-minimization problem; and the bidual of the $\ell_{0,1}$-minimization problem for enforcing group sparsity on structured data is the $\ell_{1,\infty}$-minimization problem. The analysis provides a means to compute per-instance non-trivial lower bounds on the (group) sparsity of the desired solutions. In a real-world application, the bidual relaxation improves the performance of a sparsity-based classification framework applied to robust face recognition.

cs.CV↗

arXiv2011

Dimension Reduction Near Periodic Orbits of Hybrid Systems

When the Poincaré map associated with a periodic orbit of a hybrid dynamical system has constant-rank iterates, we demonstrate the existence of a constant-dimensional invariant subsystem near the orbit which attracts all nearby trajectories in finite time. This result shows that the long-term behavior of a hybrid model with a large number of degrees-of-freedom may be governed by a low-dimensional smooth dynamical system. The appearance of such simplified models enables the translation of analytical tools from smooth systems-such as Floquet theory-to the hybrid setting and provides a bridge between the efforts of biologists and engineers studying legged locomotion.

math.DS↗

arXiv2010

Mean-square boundedness of stochastic networked control systems with bounded control inputs

We consider the problem of controlling marginally stable linear systems using bounded control inputs for networked control settings in which the communication channel between the remote controller and the system is unreliable. We assume that the states are perfectly observed, but the control inputs are transmitted over a noisy communication channel. Under mild hypotheses on the noise introduced by the control communication channel and large enough control authority, we construct a control policy that renders the state of the closed-loop system mean-square bounded.

math.OC↗