Raktim Bhattacharya

Linear quadratic regulation of systems with stochastic parameter uncertainties

In this paper, we develop a theoretical framework for linear quadratic regulator design for linear systems with probabilistic uncertainty in the parameters. The framework is built on the generalized polynomial chaos theory. In this framework, the stochastic dynamics is transformed into deterministic dynamics in higher dimensional state space, and the controller is designed in the expanded state space. The proposed design framework results in a family of controllers, parameterized by the associated random variables. The theoretical results are applied to a controller design problem based on stochastic linear, longitudinal F16 model. The performance of the stochastic design shows excellent consistency, in a statistical sense, with the results obtained from Monte-Carlo based designs.

Nonlinear Receding Horizon Control of an F-16 Aircraft

The application of receding horizon control (RHC) with the linear, parameter varying (LPV) design methodology to a high-fidelity, nonlinear F-16 aircraft model is demonstrated. The highlights are 1) use of RHC to improve upon the performance of a LPV regulator; 2) discussion on details of implementation such as control space formulation, tuning of RHC parameters, computation time and numerical properties of the algorithms; and 3) simulated response of nonlinear RHC and LPV regulator.

Anytime Control Algorithm: Model Reduction Approach

Recently, there has been considerable interest in anytime algorithms for real-time systems. Anytime algorithms are computational models that compromise quality of result for computational time. The tolerance to fluctuating CPU time makes anytime algorithms operationally optimal for real-time task scheduling. A methodology is presented that transforms linear control algorithms into anytime control algorithms. Implementation of a linear control algorithm involves matrix‐vector multiplications that require a fixed computational time. Such algorithms fail to compute the controller output if the alloted CPU time is less than required and cannot make use of any excess CPU time that might be available. When implemented as a real-time system, the static nature of the required computational time makes it operationally suboptimal for task scheduling. Linear control algorithms are transformed to anytime control algorithms by switching between controllers of different order. Balanced truncation and residualization are considered as model reduction tools to generate a set of reduced-order controllers, and a switching algorithm is presented that smoothly switches between them to accommodate variation in available computational time. I. Introduction I N recent times, advancement in digital technology has led to the design of complex computational systems. These systems usually interact with an environment that demands more out of some algorithms and less out of others, at different times in their operation life. Therefore, it is not feasible to perform accurate computation at all times by all of the algorithms in the system. Anytime algorithms provide a technique for allocating computational resources to the most useful algorithm, thereby enabling optimal usage of hardware resources. Anytime algorithms differ from conventional computational procedures in several ways. 1 Anytime algorithms are algorithms that compromise performance for computational time. They are capable of providing results at any point in their execution. The quality, accuracy, or performance of the algorithm improves with increased processing time. The improvement in the solution is large in the early stages of computation but diminishes over time. Anytime algorithms first emerged in the area of artificial intelligence. Early applications of such algorithms can be found in medical diagnosis and mobile robot navigation. The term anytime algorithm was coined by Dean and Boddy 2,3 in the late 1980s in the context of their work on time-dependent planning. They used this idea to solve a path-planning problem involving a robot assigned to deliver packages to a set of locations. Horvitz introduced a similar idea, called flexible computation, to solve time-critical decision problems. 4 In 1991, Liu et al. 5 introduced the concept of imprecise computation and applied it to real-time systems. They showed that imprecise computation techniques provide scheduling flexibility by trading off the quality of result to meet computational deadlines. Ever since, the concept of imprecise computation has been applied to solve several diverse problems. 6−9 The idea of anytime algorithms

Polynomial Chaos-Based Analysis of Probabilistic Uncertainty in Hypersonic Flight Dynamics

In this paper, we present a novel computational framework for analyzing the evolution of the uncertainty in state trajectories of a hypersonic air vehicle due to the uncertainty in initial conditions and other system parameters. The framework is built on the so-called generalized polynomial chaos expansions. In this framework, stochastic dynamical systems are transformed into equivalent deterministic dynamical systems in higher dimensional space. Here, the evolution of uncertainty due to initial condition, ballistic coefficient, lift over drag ratio, and atmospheric density is analyzed. The problem studied here is related to the Mars entry, descent, and landing problems. We demonstrate that the polynomial chaos framework is able to predict evolution of uncertainty, in hypersonic flight, with the same order of accuracy as the Monte-Carlo methods but with more computational efficiency.

OPTRAGEN: A MATLAB Toolbox for Optimal Trajectory Generation

OPTRAGEN is a MATLAB toolbox for numerically solving optimal control problems. OPTRAGEN translates optimal control problems to nonlinear programming problems. The transcription of optimal control problem (OCP) to nonlinear programming (NLP) problem is done by parameterizing trajectories as splines. The output of the transcription is a cost function and a constraint function that can be interfaced with any commercially available nonlinear programming solver. OPTRAGEN can be considered to be a parser that translates optimal control problems to nonlinear programming problems, and is not dependent on any nonlinear programming solver

Stability analysis of stochastic systems using polynomial chaos

A novel framework for stability analysis of linear and polynomial stochastic systems is presented. The framework is built on generalized polynomial chaos theory, which enables analysis of dynamical systems with probabilistic uncertainty on system parameters with various distributions. The theory allows for the transformation of stochastic problems into a higher dimensional deterministic problem, that is able to accurately approximate the evolution of uncertainty in the state trajectories due to stochastic system parameters. The developed theory is applied to analyze a linear flight control design for an F-16 aircraft model. The problem of generating stability certificates for stochastic polynomial systems is also considered.

Mathematical equations as executable models of mechanical systems

Cyber-physical systems comprise digital components that directly interact with a physical environment. Specifying the behavior desired of such systems requires analytical modeling of physical phenomena. Similarly, testing them requires simulation of continuous systems. While numerous tools support later stages of developing simulation codes, there is still a large gap between analytical modeling and building running simulators. This gap significantly impedes the ability of scientists and engineers to develop novel cyber-physical systems. We propose bridging this gap by automating the mapping from analytical models to simulation codes. Focusing on mechanical systems as an important class of physical systems, we study the form of analytical models that arise in this domain, along with the process by which domain experts map them to executable codes. We show that the key steps needed to automate this mapping are 1) a light-weight analysis to partially direct equations, 2) a binding-time analysis, and 3) symbolic differentiation. In addition to producing a prototype modeling environment, we highlight some limitations in the state of the art in tool support of simulation, and suggest ways in which some of these limitations could be overcome.

Optimal Trajectory Generation With Probabilistic System Uncertainty Using Polynomial Chaos

In this paper, we develop a framework for solving optimal trajectory generation problems with probabilistic uncertainty in system parameters. The framework is based on the generalized polynomial chaos theory. We consider both linear and nonlinear dynamics in this paper and demonstrate transformation of stochastic dynamics to equivalent deterministic dynamics in higher dimensional state space. Minimum expectation and variance cost function are shown to be equivalent to standard quadratic cost functions of the expanded state vector. Results are shown on a stochastic Van der Pol oscillator.

Nonlinear Estimation of Hypersonic State Trajectories in Bayesian Framework with Polynomial Chaos

This paper presents a nonlinear estimation algorithm to estimate state trajectories of a hypersonic vehicle with initial condition uncertainty. Polynomial chaos theory is used to predict the evolution of state uncertainty of the nonlinear system, and a Bayesian estimation algorithm is used to estimate the posterior probability density function of the nonlinear random process. The nonlinear estimation algorithm is then applied to the hypersonic reentry of a spacecraft in Martian atmosphere. Its performance is compared with estimators based on an extended Kalman filtering and unscented Kalman filtering framework. It is observed that for the particular application, the proposed estimator outperforms extended Kalman filtering and unscented Kalman filtering, highlighting its need in the current scenario.

Hypersonic State Estimation Using the Frobenius-Perron Operator

This paper presents a nonlinear state-estimation algorithm that combines the Frobenius-Perron operator theory with the Bayesian estimation theory. The Frobenius―Perron operator is used to predict evolution of uncertainty in the nonlinear system and obtain the prior probability density function in the estimation process. The Bayesian update rule is used to determine the posterior density function from the available measurements. The framework for this filter is similar to particle filters where the density function is sampled using a cloud of points and the system dynamics are integrated with these points as the initial condition. The key issue in particle filters is that the weight for the sample points are typically determined using histograms to obtain the prior density function, and thus requires many samples for acceptable accuracy. Moreover, the weights of the majority of the particles converge to zero after a few iterations, rendering them useless for state-estimation purposes. This issue can be resolved with the application of the Frobenius―Perron operator, which determines the time evolution of the weights along sample paths. This greatly simplifies the determination of the prior density function and can be achieved with fewer sample points. Consequently, the associated computational time is also greatly reduced. The presented algorithm is demonstrated on a hypersonic reentry problem with uncertain initial states, with given initial probability density functions. The performance is compared with particle filters, and it is observed that the proposed algorithm is computationally superior as expected.