arXiv subjects

Prashant G. Mehta

Publications and source records attributed to Prashant G. Mehta.

At least 37 records · Page 2Linked to original sources

arXiv2023

An Optimal Control Derivation of Nonlinear Smoothing Equations

The purpose of this paper is to review and highlight some connections between the problem of nonlinear smoothing and optimal control of the Liouville equation. The latter has been an active area of recent research interest owing to work in mean-field games and optimal transportation theory. The nonlinear smoothing problem is considered here for continuous-time Markov processes. The observation process is modeled as a nonlinear function of a hidden state with an additive Gaussian measurement noise. A variational formulation is described based upon the relative entropy formula introduced by Newton and Mitter. The resulting optimal control problem is formulated on the space of probability distributions. The Hamilton's equation of the optimal control are related to the Zakai equation of nonlinear smoothing via the log transformation. The overall procedure is shown to generalize the classical Mortensen's minimum energy estimator for the linear Gaussian problem.

math.OC↗

arXiv2023

A Survey of Feedback Particle Filter and related Controlled Interacting Particle Systems (CIPS)

In this survey, we describe controlled interacting particle systems (CIPS) to approximate the solution of the optimal filtering and the optimal control problems. Part I of the survey is focussed on the feedback particle filter (FPF) algorithm, its derivation based on optimal transportation theory, and its relationship to the ensemble Kalman filter (EnKF) and the conventional sequential importance sampling-resampling (SIR) particle filters. The central numerical problem of FPF -- to approximate the solution of the Poisson equation -- is described together with the main solution approaches. An analytical and numerical comparison with the SIR particle filter is given to illustrate the advantages of the CIPS approach. Part II of the survey is focussed on adapting these algorithms for the problem of reinforcement learning. The survey includes several remarks that describe extensions as well as open problems in this subject.

eess.SY↗

arXiv2023

Hierarchical control and learning of a foraging CyberOctopus

Inspired by the unique neurophysiology of the octopus, we propose a hierarchical framework that simplifies the coordination of multiple soft arms by decomposing control into high-level decision making, low-level motor activation, and local reflexive behaviors via sensory feedback. When evaluated in the illustrative problem of a model octopus foraging for food, this hierarchical decomposition results in significant improvements relative to end-to-end methods. Performance is achieved through a mixed-modes approach, whereby qualitatively different tasks are addressed via complementary control schemes. Here, model-free reinforcement learning is employed for high-level decision-making, while model-based energy shaping takes care of arm-level motor execution. To render the pairing computationally tenable, a novel neural-network energy shaping (NN-ES) controller is developed, achieving accurate motions with time-to-solutions 200 times faster than previous attempts. Our hierarchical framework is then successfully deployed in increasingly challenging foraging scenarios, including an arena littered with obstacles in 3D space, demonstrating the viability of our approach.

cs.RO↗

arXiv2022

Modeling the Neuromuscular Control System of an Octopus Arm

The octopus arm is a neuromechanical system that involves a complex interplay between peripheral nervous system (PNS) and arm musculature. This makes the arm capable of carrying out rich maneuvers. In this paper, we build a model for the PNS and integrate it with a muscular soft octopus arm. The proposed neuromuscular architecture is used to qualitatively reproduce several biophysical observations in real octopuses, including curled rest shapes and target-directed arm reaching motions. Two control laws are proposed for target-oriented arm motions, and their performance is compared against a benchmark. Several analytical results, including rest-state characterization and stability properties of the proposed control laws, are provided.

eess.SY↗

arXiv2022

Energy Shaping Control of a Muscular Octopus Arm Moving in Three Dimensions

Flexible octopus arms exhibit an exceptional ability to coordinate large numbers of degrees of freedom and perform complex manipulation tasks. As a consequence, these systems continue to attract the attention of biologists and roboticists alike. In this paper, we develop a three-dimensional model of a soft octopus arm, equipped with biomechanically realistic muscle actuation. Internal forces and couples exerted by all major muscle groups are considered. An energy shaping control method is described to coordinate muscle activity so as to grasp and reach in 3D space. Key contributions of this paper are: (i) modeling of major muscle groups to elicit three-dimensional movements; (ii) a mathematical formulation for muscle activations based on a stored energy function; and (iii) a computationally efficient procedure to design task-specific equilibrium configurations, obtained by solving an optimization problem in the Special Euclidean group SE(3). Muscle controls are then iteratively computed based on the co-state variable arising from the solution of the optimization problem. The approach is numerically demonstrated in the physically accurate software environment Elastica. Results of numerical experiments mimicking observed octopus behaviors are reported.

eess.SY↗

arXiv2022

Duality for Nonlinear Filtering I: Observability

This paper is concerned with the development and use of duality theory for a hidden Markov model (HMM) with white noise observations. The main contribution of this work is to introduce a backward stochastic differential equation (BSDE) as a dual control system. A key outcome is that stochastic observability (resp. detectability) of the HMM is expressed in dual terms: as controllability (resp. stabilizability) of the dual control system. All aspects of controllability, namely, definition of controllable space and controllability gramian, along with their properties and explicit formulae, are discussed. The proposed duality is shown to be an exact extension of the classical duality in linear systems theory. One can then relate and compare the linear and the nonlinear systems. A side-by-side summary of this relationship is given in a tabular form (Table~II).

math.OC↗

arXiv2022

Duality for Nonlinear Filtering II: Optimal Control

This paper is concerned with the development and use of duality theory for a nonlinear filtering model with white noise observations. The main contribution of this paper is to introduce a stochastic optimal control problem as a dual to the nonlinear filtering problem. The mathematical statement of the dual relationship between the two problems is given in the form of a duality principle. The constraint for the optimal control problem is the backward stochastic differential equation (BSDE) introduced in the companion paper. The optimal control solution is obtained from an application of the maximum principle, and subsequently used to derive the equation of the nonlinear filter. The proposed duality is shown to be an exact extension of the classical Kalman-Bucy duality, and different from other types of optimal control and variational formulations given in literature.

math.OC↗

arXiv2022

Controlled Interacting Particle Algorithms for Simulation-based Reinforcement Learning

This paper is concerned with optimal control problems for control systems in continuous time, and interacting particle system methods designed to construct approximate control solutions. Particular attention is given to the linear quadratic (LQ) control problem. There is a growing interest in re-visiting this classical problem, in part due to the successes of reinforcement learning (RL). The main question of this body of research (and also of our paper) is to approximate the optimal control law {\em without} explicitly solving the Riccati equation. A novel simulation-based algorithm, namely a dual ensemble Kalman filter (EnKF), is introduced. The algorithm is used to obtain formulae for optimal control, expressed entirely in terms of the EnKF particles. An extension to the nonlinear case is also presented. The theoretical results and algorithms are illustrated with numerical experiments.

eess.SY↗

arXiv2022

How does a Rational Agent Act in an Epidemic?

Evolution of disease in a large population is a function of the top-down policy measures from a centralized planner, as well as the self-interested decisions (to be socially active) of individual agents in a large heterogeneous population. This paper is concerned with understanding the latter based on a mean-field type optimal control model. Specifically, the model is used to investigate the role of partial information on an agent's decision-making, and study the impact of such decisions by a large number of agents on the spread of the virus in the population. The motivation comes from the presymptomatic and asymptomatic spread of the COVID-19 virus where an agent unwittingly spreads the virus. We show that even in a setting with fully rational agents, limited information on the viral state can result in an epidemic growth.

math.OC↗

arXiv2022

A Sensory Feedback Control Law for Octopus Arm Movements

The main contribution of this paper is a novel sensory feedback control law for an octopus arm. The control law is inspired by, and helps integrate, several observations made by biologists. The proposed control law is distinct from prior work which has mainly focused on open-loop control strategies. Several analytical results are described including characterization of the equilibrium and its stability analysis. Numerical simulations demonstrate life-like motion of the soft octopus arm, qualitatively matching behavioral experiments. Quantitative comparison with bend propagation experiments helps provide the first explanation of such canonical motion using a sensory feedback control law. Several remarks are included that help draw parallels with natural pursuit strategies such as motion camouflage or classical pursuit.

eess.SY↗

arXiv2022

Optimality vs Stability Trade-off in Ensemble Kalman Filters

This paper is concerned with optimality and stability analysis of a family of ensemble Kalman filter (EnKF) algorithms. EnKF is commonly used as an alternative to the Kalman filter for high-dimensional problems, where storing the covariance matrix is computationally expensive. The algorithm consists of an ensemble of interacting particles driven by a feedback control law. The control law is designed such that, in the linear Gaussian setting and asymptotic limit of infinitely many particles, the mean and covariance of the particles follow the exact mean and covariance of the Kalman filter. The problem of finding a control law that is exact does not have a unique solution, reminiscent of the problem of finding a transport map between two distributions. A unique control law can be identified by introducing control cost functions, that are motivated by the optimal transportation problem or Schrödinger bridge problem. The objective of this paper is to study the relationship between optimality and long-term stability of a family of exact control laws. Remarkably, the control law that is optimal in the optimal transportation sense leads to an EnKF algorithm that is not stable.

math.OC↗

arXiv2021

Minimum variance constrained estimator

This paper is concerned with the problem of state estimation for discrete-time linear systems in the presence of additional (equality or inequality) constraints on the state (or estimate). By use of the minimum variance duality, the estimation problem is converted into an optimal control problem. Two algorithmic solutions are described: the full information estimator (FIE) and the moving horizon estimator (MHE). The main result is to show that the proposed estimator is stable in the sense of an observer. The proposed algorithm is distinct from the standard algorithm for constrained state estimation based upon the use of the minimum energy duality. The two are compared numerically on the benchmark batch reactor process model.

math.OC↗

arXiv2021

Modeling Presymptomatic Spread in Epidemics via Mean-Field Games

This paper is concerned with developing mean-field game models for the evolution of epidemics. Specifically, an agent's decision -- to be socially active in the midst of an epidemic -- is modeled as a mean-field game with health-related costs and activity-related rewards. By considering the fully and partially observed versions of this problem, the role of information in guiding an agent's rational decision is highlighted. The main contributions of the paper are to derive the equations for the mean-field game in both fully and partially observed settings of the problem, to present a complete analysis of the fully observed case, and to present some analytical results for the partially observed case.

math.OC↗

arXiv2021

A Dynamic Programming Formulation for the Nonlinear Filter

This paper build on our recent work where we presented a dual stochastic optimal control formulation of the nonlinear filtering problem [1]. The constraint for the dual problem is a backward stochastic differential equations (BSDE). The solution is obtained via an application of the maximum principle (MP). In the present paper, a dynamic programming (DP) principle is presented for a special class of BSDE-constrained stochastic optimal control problems. The principle is applied to derive the solution of the nonlinear filtering problem.

math.OC↗

arXiv2021

Control-oriented Modeling of Bend Propagation in an Octopus Arm

Bend propagation in an octopus arm refers to a stereotypical maneuver whereby an octopus pushes a bend (localized region of large curvature) from the base to the tip of the arm. Bend propagation arises from the complex interplay between mechanics of the flexible arm, forces generated by internal muscles, and environmental effects (buoyancy and drag) from of the surrounding fluid. In part due to this complexity, much of prior modeling and analysis work has relied on the use of high dimensional computational models. The contribution of this paper is to present a control-oriented reduced order model based upon a novel parametrization of the curvature of the octopus arm. The parametrization is motivated by the experimental results. The reduced order model is related to and derived from a computational model which is also presented. The results from the two sets of models are compared using numerical simulations which is shown to lead to useful qualitative insights into bend propagation. A comparison between the reduced order model and experimental data is also reported.

physics.bio-ph↗

arXiv2021

The Conditional Poincaré Inequality for Filter Stability

This paper is concerned with the problem of nonlinear filter stability of ergodic Markov processes. The main contribution is the conditional Poincaré inequality (PI), which is shown to yield filter stability. The proof is based upon a recently discovered duality which is used to transform the nonlinear filtering problem into a stochastic optimal control problem for a backward stochastic differential equation (BSDE). Based on these dual formalisms, a comparison is drawn between the stochastic stability of a Markov process and the filter stability. The latter relies on the conditional PI described in this paper, whereas the former relies on the standard form of PI.

math.PR↗

arXiv2021

A Dual Characterization of the Stability of the Wonham Filter

This paper revisits the classical question of the stability of the nonlinear Wonham filter. The novel contributions of this paper are two-fold: (i) definition of the stabilizability for the (control-theoretic) dual to the nonlinear filter; and (ii) the use of this definition to obtain conclusions on the stability of the Wonham filter. Specifically, it is shown that the stabilizability of the dual system is necessary for filter stability and conversely stabilizability implies that the filter asymptotically detects the correct ergodic class. The formulation and the proofs are based upon a recently discovered duality result whereby the nonlinear filtering problem is cast as a stochastic optimal control problem for a backward stochastic differential equation (BSDE). The control-theoretic proof techniques and results may be viewed as a generalization of the classical work on the stability of the Kalman filter.

math.PR↗

arXiv2021

A physics-informed, vision-based method to reconstruct all deformation modes in slender bodies

This paper is concerned with the problem of estimating (interpolating and smoothing) the shape (pose and the six modes of deformation) of a slender flexible body from multiple camera measurements. This problem is important in both biology, where slender, soft, and elastic structures are ubiquitously encountered across species, and in engineering, particularly in the area of soft robotics. The proposed mathematical formulation for shape estimation is physics-informed, based on the use of the special Cosserat rod theory whose equations encode slender body mechanics in the presence of bending, shearing, twisting and stretching. The approach is used to derive numerical algorithms which are experimentally demonstrated for fiber reinforced and cable-driven soft robot arms. These experimental demonstrations show that the methodology is accurate (<5 mm error, three times less than the arm diameter) and robust to noise and uncertainties.

cs.RO↗