arXiv subjects

Abhishek Halder

Publications and source records attributed to Abhishek Halder.

72 records · Page 4Linked to original sources

arXiv2019

Finite Horizon Density Steering for Multi-input State Feedback Linearizable Systems

In this paper, we study the feedback synthesis problem for steering the joint state density or ensemble subject to multi-input state feedback linearizable dynamics. This problem is of interest to many practical applications including that of dynamically shaping a robotic swarm. Our results here show that it is possible to exploit the structural nonlinearities to derive the feedback controllers steering the joint density from a prescribed shape to another while minimizing the expected control effort to do so. The developments herein build on our previous work, and extend the theory of the Schrödinger bridge problem subject to feedback linearizable dynamics.

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arXiv2019

Proximal Recursion for the Wonham Filter

This paper contributes to the emerging viewpoint that governing equations for dynamic state estimation, conditioned on the history of noisy measurements, can be viewed as gradient flow on the manifold of joint probability density functions with respect to suitable metrics. Herein, we focus on the Wonham filter where the prior dynamics is given by a continuous time Markov chain on a finite state space; the measurement model includes noisy observation of the (possibly nonlinear function of) state. We establish that the posterior flow given by the Wonham filter can be viewed as the small time-step limit of proximal recursions of certain functionals on the probability simplex. The results of this paper extend our earlier work where similar proximal recursions were derived for the Kalman-Bucy filter.

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arXiv2019

Gradient Flow Algorithms for Density Propagation in Stochastic Systems

We develop a new computational framework to solve the partial differential equations (PDEs) governing the flow of the joint probability density functions (PDFs) in continuous-time stochastic nonlinear systems. The need for computing the transient joint PDFs subject to prior dynamics arises in uncertainty propagation, nonlinear filtering and stochastic control. Our methodology breaks away from the traditional approach of spatial discretization or function approximation -- both of which, in general, suffer from the "curse-of-dimensionality". In the proposed framework, we discretize time but not the state space. We solve infinite dimensional proximal recursions in the manifold of joint PDFs, which in the small time-step limit, is theoretically equivalent to solving the underlying transport PDEs. The resulting computation has the geometric interpretation of gradient flow of certain free energy functional with respect to the Wasserstein metric arising from the theory of optimal mass transport. We show that dualization along with an entropic regularization, leads to a cone-preserving fixed point recursion that is proved to be contractive in Thompson metric. A block co-ordinate iteration scheme is proposed to solve the resulting nonlinear recursions with guaranteed convergence. This approach enables remarkably fast computation for non-parametric transient joint PDF propagation. Numerical examples and various extensions are provided to illustrate the scope and efficacy of the proposed approach.

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arXiv2019

Optimal Control of Thermostatic Loads for Planning Aggregate Consumption: Characterization of Solution and Explicit Strategies

We consider the problem of planning the aggregate energy consumption for a set of thermostatically controlled loads for demand response, accounting price forecast trajectory and thermal comfort constraints. We address this as a continuous-time optimal control problem, and analytically characterize the structure of its solution in the general case. In the special case when the price forecast is monotone and the loads have equal dynamics, we show that it is possible to determine the solution in an explicit form. Taking this fact into account, we handle the non-monotone price case by considering several subproblems, each corresponding to a time subinterval where the price function is monotone, and then allocating to each subinterval a fraction of the total energy budget. This way, for each time subinterval, the problem reduces to a simple convex optimization problem with a scalar decision variable, for which a descent direction is also known. The price forecasts for the day-ahead energy market typically have no more than four monotone segments, so the resulting optimization problem can be solved efficiently with modest computational resources.

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arXiv2019

DeGroot-Friedkin Map in Opinion Dynamics is Mirror Descent

We provide a variational interpretation of the DeGroot-Friedkin map in opinion dynamics. Specifically, we show that the nonlinear dynamics for the DeGroot-Friedkin map can be viewed as mirror descent on the standard simplex with the associated Bregman divergence being equal to the generalized Kullback-Leibler divergence, i.e., an entropic mirror descent. Our results reveal that the DeGroot-Friedkin map elicits an individual's social power to be close to her social influence while minimizing the so called "extropy" -- the entropy of the complimentary opinion.

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arXiv2018

Proximal Recursion for Solving the Fokker-Planck Equation

We develop a new method to solve the Fokker-Planck or Kolmogorov's forward equation that governs the time evolution of the joint probability density function of a continuous-time stochastic nonlinear system. Numerical solution of this equation is fundamental for propagating the effect of initial condition, parametric and forcing uncertainties through a nonlinear dynamical system, and has applications encompassing but not limited to forecasting, risk assessment, nonlinear filtering and stochastic control. Our methodology breaks away from the traditional approach of spatial discretization for solving this second-order partial differential equation (PDE), which in general, suffers from the "curse-of-dimensionality". Instead, we numerically solve an infinite dimensional proximal recursion in the space of probability density functions, which is theoretically equivalent to solving the Fokker-Planck-Kolmogorov PDE. We show that the dual formulation along with the introduction of an entropic regularization, leads to a smooth convex optimization problem that can be implemented via suitable block co-ordinate iteration and has fast convergence due to certain contraction property that we establish. This approach enables meshless implementation leading to remarkably fast computation.

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arXiv2018

Stability Theory of Stochastic Models in Opinion Dynamics

We consider a certain class of nonlinear maps that preserve the probability simplex, i.e., stochastic maps, that are inspired by the DeGroot-Friedkin model of belief/opinion propagation over influence networks. The corresponding dynamical models describe the evolution of the probability distribution of interacting species. Such models where the probability transition mechanism depends nonlinearly on the current state are often referred to as {\em nonlinear Markov chains}. In this paper we develop stability results and study the behavior of representative opinion models. The stability certificates are based on the contractivity of the nonlinear evolution in the $\ell_1$-metric. We apply the theory to two types of opinion models where the adaptation of the transition probabilities to the current state is exponential and linear, respectively--both of these can display a wide range of behaviors. We discuss continuous-time and other generalizations.

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arXiv2018

Optimal Power Consumption for Demand Response of Thermostatically Controlled Loads

We consider the problem of determining the optimal aggregate power consumption of a population of thermostatically controlled loads. This is motivated by the problem of synthesizing the demand response for a load serving entity (LSE) serving a population of such customers. We show how the LSE can opportunistically design the aggregate reference consumption to minimize its energy procurement cost, given day-ahead price, load and ambient temperature forecasts, while respecting each individual load's comfort range constraints. The resulting synthesis problem is shown to be amenable to optimal control techniques, but computationally difficult otherwise. Numerical simulations elucidate how the LSE can use the optimal aggregate power consumption trajectory thus computed, for the purpose of demand response.

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arXiv2018

On the Parameterized Computation of Minimum Volume Outer Ellipsoid of Minkowski Sum of Ellipsoids

We consider the problem of computing certain parameterized minimum volume outer ellipsoidal (MVOE) approximation of the Minkowski sum of a finite number of ellipsoids. We clarify connections among several parameterizations available in the literature, obtain novel analysis results regarding the conditions of optimality, and based on the same, propose two new algorithms for computing the parameterized MVOE. Numerical results reveal faster runtime for the proposed algorithms than the state-of-the-art semidefinite programming approach of computing the same.

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arXiv2017

Gradient Flows in Filtering and Fisher-Rao Geometry

Uncertainty propagation and filtering can be interpreted as gradient flows with respect to suitable metrics in the infinite dimensional manifold of probability density functions. Such a viewpoint has been put forth in recent literature, and a systematic way to formulate and solve the same for linear Gaussian systems has appeared in our previous work where the gradient flows were realized via proximal operators with respect to Wasserstein metric arising in optimal mass transport. In this paper, we derive the evolution equations as proximal operators with respect to Fisher-Rao metric arising in information geometry. We develop the linear Gaussian case in detail and show that a template two step optimization procedure proposed earlier by the authors still applies. Our objective is to provide new geometric interpretations of known equations in filtering, and to clarify the implication of different choices of metric.

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arXiv2017

Gradient Flows in Uncertainty Propagation and Filtering of Linear Gaussian Systems

The purpose of this work is mostly expository and aims to elucidate the Jordan-Kinderlehrer-Otto (JKO) scheme for uncertainty propagation, and a variant, the Laugesen-Mehta-Meyn-Raginsky (LMMR) scheme for filtering. We point out that these variational schemes can be understood as proximal operators in the space of density functions, realizing gradient flows. These schemes hold the promise of leading to efficient ways for solving the Fokker-Planck equation as well as the equations of non-linear filtering. Our aim in this paper is to develop in detail the underlying ideas in the setting of linear stochastic systems with Gaussian noise and recover known results.

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arXiv2016

Architecture and Algorithms for Privacy Preserving Thermal Inertial Load Management by A Load Serving Entity

Motivated by the growing importance of demand response in modern power system's operations, we propose an architecture and supporting algorithms for privacy preserving thermal inertial load management as a service provided by the load serving entity (LSE). We focus on an LSE managing a population of its customers' air conditioners, and propose a contractual model where the LSE guarantees quality of service to each customer in terms of keeping their indoor temperature trajectories within respective bands around the desired individual comfort temperatures. We show how the LSE can price the contracts differentiated by the flexibility embodied by the width of the specified bands. We address architectural questions of (i) how the LSE can strategize its energy procurement based on price and ambient temperature forecasts, (ii) how an LSE can close the real time control loop at the aggregate level while providing individual comfort guarantees to loads, without ever measuring the states of an air conditioner for privacy reasons. Control algorithms to enable our proposed architecture are given, and their efficacy is demonstrated on real data.

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arXiv2014

Performance and Robustness Analysis of Stochastic Jump Linear Systems using Wasserstein metric

This paper focuses on the performance and the robustness analysis of stochastic jump linear systems. The state trajectory under stochastic jump process becomes random variables, which brings forth the probability distributions in the system state. Therefore, we need to adopt a proper metric to measure the system performance with respect to stochastic switching. In this perspective, Wasserstein metric that assesses the distance between probability density functions is applied to provide the performance and the robustness analysis. Both the transient and steady-state performance of the systems with given initial state uncertainties can be measured in this framework. Also, we prove that the convergence of this metric implies the mean square stability. Overall, this study provides a unifying framework for the performance and the robustness analysis of general stochastic jump linear systems, but not necessarily Markovian jump process that is commonly used for stochastic switching. The practical usefulness and efficiency of the proposed method are verified through numerical examples.

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arXiv2014

Mean Square Stability for Stochastic Jump Linear Systems via Optimal Transport

In this note, we provide a unified framework for the mean square stability of stochastic jump linear systems via optimal transport. The Wasserstein metric known as an optimal transport, that assesses the distance between probability density functions enables the stability analysis. Without any assumption on the underlying jump process, this Wasserstein distance guarantees the mean square stability for general stochastic jump linear systems, not necessarily for Markovian jump. The validity of the proposed methods are proved by recovering already-known stability conditions under this framework.

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arXiv2014

Probabilistic Robustness Analysis of Stochastic Jump Linear Systems

In this paper, we propose a new method to measure the probabilistic robustness of stochastic jump linear system with respect to both the initial state uncertainties and the randomness in switching. Wasserstein distance which defines a metric on the manifold of probability density functions is used as tool for the performance and the stability measures. Starting with Gaussian distribution to represent the initial state uncertainties, the probability density function of the system state evolves into mixture of Gaussian, where the number of Gaussian components grows exponentially. To cope with computational complexity caused by mixture of Gaussian, we prove that there exists an alternative probability density function that preserves exact information in the Wasserstein level. The usefulness and the efficiency of the proposed methods are demonstrated by example.

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arXiv2014

Geodesic Density Tracking with Applications to Data Driven Modeling

Many problems in dynamic data driven modeling deals with distributed rather than lumped observations. In this paper, we show that the Monge-Kantorovich optimal transport theory provides a unifying framework to tackle such problems in the systems-control parlance. Specifically, given distributional measurements at arbitrary instances of measurement availability, we show how to derive dynamical systems that interpolate the observed distributions along the geodesics. We demonstrate the framework in the context of three specific problems: (i) \emph{finding a feedback control} to track observed ensembles over finite-horizon, (ii) \emph{finding a model} whose prediction matches the observed distributional data, and (iii) \emph{refining a baseline model} that results a distribution-level prediction-observation mismatch. We emphasize how the three problems can be posed as variants of the optimal transport problem, but lead to different types of numerical methods depending on the problem context. Several examples are given to elucidate the ideas.

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arXiv2014

Probabilistic Model Validation for Uncertain Nonlinear Systems

This paper presents a probabilistic model validation methodology for nonlinear systems in time-domain. The proposed formulation is simple, intuitive, and accounts both deterministic and stochastic nonlinear systems with parametric and nonparametric uncertainties. Instead of hard invalidation methods available in the literature, a relaxed notion of validation in probability is introduced. To guarantee provably correct inference, algorithm for constructing probabilistically robust validation certificate is given along with computational complexities. Several examples are worked out to illustrate its use.

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arXiv2014

A Probabilistic Method for Nonlinear Robustness Analysis of F-16 Controllers

This paper presents a new framework for controller robustness verification with respect to F-16 aircraft's closed-loop performance in longitudinal flight. We compare the state regulation performance of a linear quadratic regulator (LQR) and a gain-scheduled linear quadratic regulator (gsLQR), applied to nonlinear open-loop dynamics of F-16, in presence of stochastic initial condition and parametric uncertainties, as well as actuator disturbance. We show that, in presence of initial condition uncertainties alone, both LQR and gsLQR have comparable immediate and asymptotic performances, but the gsLQR exhibits better transient performance at intermediate times. This remains true in the presence of additional actuator disturbance. Also, gsLQR is shown to be more robust than LQR, against parametric uncertainties. The probabilistic framework proposed here, leverages transfer operator based density computation in exact arithmetic and introduces optimal transport theoretic performance validation and verification (V&V) for nonlinear dynamical systems. Numerical results from our proposed method, are in unison with Monte Carlo simulations.

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