Welcome to the Laboratory for Uncertainty Quantification at Texas A&M University. We focus on understanding the influence of uncertainty in engineering systems and developing algorithms to control them in the presence of such uncertainties. We consider uncertainty in dynamics, communication and computation in our research. Fundamentals from optimization theory, approximation theory, control & estimation theory, and information theory are applied to develop new modeling, analysis and synthesis tools for designing robust engineering systems. The following are highlights from our research.

## Real-Time Predictive Analytics for Space Situational Awareness

Space situational awareness is concerned with tracking of space objects and classifying it with respect to certain characteristics. In this research, we are developing novel algorithms for uncertainty propagation and state estimation. Challenges include non Gaussian uncertainty supported on cylindrical coordinate systems $\mathbb{R}^5\times\mathbb{S}$, sparse sensing and unknown sensor characteristics. These algorithms are used for conjunction analysis, which is the process of predicting upcoming object encounters in an effort to notify satellite operators and avoid high risk encounters.

Selected Papers

1. N. Das, R. Bhattacharya, Sparse Sensing Architecture for Kalman Filtering with Guaranteed Error Bound, 1st IAA Conference on Space Situational Awareness (ICSSA), Orlando, FL, USA, 2017.
2. N. Das, R. P. Ghosh, N. Guha, R. Bhattacharya, B. K. Mallick, Optimal Transport Based Tracking of Space Objects in Cylindrical Manifolds (under review)

## Asynchronous Numerical Algorithms for Exascale Computing

Future exascale machines are expected to have $10^5–10^6$ processors, providing a deep hierarchy of systems and resources. However, many challenges exist, which must be overcome before exascale systems can be utilized as an effective tool to further understanding critical scientific inquires. Among the main obstacle to scale code to exascale levels, is the communication necessary in tightly coupled problems, for example in uncertainty propagation, turbulence flow simulations at high Reynolds numbers, and large-scale convex optimization. The synchronization across processors can cause 50-80% processor idle time. In our research, we focus on asynchronous numerical algorithms that do not wait for data to be synchronized. Communication between processor is modeled as a stochastic channel, and the behaviour of the numerical algorithm is analyzed in a stochastic jump dynamical system framework.

Selected Papers

1. K. Lee, R. Bhattacharya, J. Dass, V. Sakuru, and R. Mahapatra, A Relaxed Synchronization Approach for Solving Parallel Quadratic Programming Problems with Guaranteed Convergence, IPDPS, Chicago,2016.
2. K. Lee and R. Bhattacharya, On the Relaxed Synchronization for Massively Parallel Numerical Algorithms, American Control Conference, 2016.
3. K. Lee, R. Bhattacharya, and V. Gupta, A Switched Dynamical System Framework for Analysis of Massively Parallel Asynchronous Numerical Algorithms, ACC, 2015.

## Robust Tensegrity Robotics

A tensegrity system is an arrangement of axially-loaded elements (no element bends, even though the overall structure bends), that we loosely characterize as a network of bars and cables. The bars take compressive axial load and the cables take tensile load. Tensegrity structures can be designed to be extremely light, for a given stiffness. In our research, we focus on efficient multi-body modeling and control of tensegrity systems. Applications considered include dexterous robotics, biomimetic air vehicles, all terrain ground robots, prosthetics, and exoskeleton.

## Uncertainty Quantification in Cyber Physical Systems

Cyber physical systems have strong coupling between physics, communication and computation. In our research, we develop algorithms for quantifying uncertainty in system behaviour due uncertainties in the physics (unmodelled dynamics, process and sensor noise), communication (irregular channels, packet loss, etc), computation (jitter in real-time tasks, CPU transients, etc). The system level behaviour is modeled as a stochastic jump system and new uncertainty propagation algorithms for such jump systems are developed. New stochastic scheduling algorithms are developed that switch between computational tasks to ensure system-level robustness.

Selected Papers

1. K. Lee, R. Bhattacharya, Design of Resource-Optimal Switching for Resource-Constrained Dynamical Systems, International Journal of Control, Automation, and Systems, 2018.
2. K. Lee, R. Bhattacharya, Stability Analysis of Large-Scale Distributed Networked Control Systems with Random Communication Delays: A Switched System Approach, System & Control Letters, 2015.
3. K. Lee, A. Halder, R. Bhattacharya, Probabilistic Robustness Analysis of Stochastic Jump Linear Systems, ACC, 2014.
4. P. Dutta, A. Halder, R. Bhattacharya, Uncertainty Quantification for Stochastic Nonlinear Systems using Perron-Frobenius Operator and Karhunen-Loeve Expansion, IEEE Multi-Conference on Systems and Control, Dubrovnik, Oct 2012.
5. R. Bhattacharya, G. J. Balas, Control in Computationally Constrained Environments, IEEE Control Systems Technology, Volume 17, Issue 3, 2009.
6. R. Bhattacharya, G. J. Balas, Anytime Control Algorithm: Model Reduction Approach, Journal of Guidance, Control, and Dynamics, 2004, Vol. 27, No.5, pp. 767-776, 2004.

## Uncertainty Quantification in Planetary Entry, Descent, and Landing

Hypersonic flight leading to entry descent landing of a large spacecraft on the surface of Mars has been identified as a research area by NASA. The requirement is to land within a few kilometers of the robotic test sites. One of the major concerns of high mass entry is the mismatch between entry conditions and deceleration capabilities provided by supersonic parachute technologies. In such applications, there are uncertainties present in initial conditions and other system parameters. Estimation of parameters for these systems is a hard problem because of the nonlinearities in the system and the lack of frequent measurements. The evolution of uncertainty (as shown in the figure) is non Gaussian. In our work, we develop new algorithms for UQ, state-estimation, and guidance algorithms. The controlled descent ensures robustness with respect to system uncertainties, and guarantees landing at the desired site with high accuracy.

Selected Papers

1. A. Halder, R. Bhattacharya, Dispersion Analysis in Hypersonic Flight During Planetary Entry Using Stochastic Liouville Equation, AIAA Journal of Guidance, Control, and Dynamics,2011, 0731-5090 vol.34 no.2 (459-474).
2. P. Dutta & R. Bhattacharya, Nonlinear Estimation of Hypersonic State Trajectories in Bayesian Framework with Polynomial Chaos, Journal of Guidance, Control, and Dynamics, vol.33 no.6 (1765-1778), 2011.
3. P. Dutta & R. Bhattacharya, Hypersonic State Estimation Using Frobenius-Perron Operator, AIAA Journal of Guidance, Control, and Dynamics,Volume 34, Number 2, 2011.
4. J. Fisher, R. Bhattacharya, Linear Quadratic Regulation of Systems with Stochastic Parameter Uncertainties, Automatica, 2009.

## Integrated Design & Engineering of Aerospace Systems (IDEAS)

IDEAS is a multidisciplinary applied research initiative that integrates aerodynamics, structural design and flight control design in a single unified framework. The objective is to develop next generation tools for rapid custom design of high confidence unmanned air vehicles for various industries including defense, oil & gas, and precision agriculture. The vision is to codesign much of the system engineering aspect by integrating state-of-the-art in computational fluid dynamics, structural mechanics, robust control theory, CAD software and 3D printing. The application focus is currently on aerospace systems, but can be extended to general autonomous systems.

This also serves as a platform for research and teaching elements of flight control such aircraft modeling – disturbance models(gust models), linear parameter varying (LPV) models; choice of control variables; reduction of control variables – explicit ganging, pseudo control, daisy chain; design methodology – eigen structure assignment, dynamic inversion, $\mu$–analysis, LPV control and SDRE methods; computational framework – LMIs and convex optimization; and finally use of MATLAB and Simulink to solve the design problem and implement it in real-time embedded systems.

Selected Papers

1. S. C. Hsu, R. Bhattacharya, Design of Stochastic Collocation Based Linear Parameter Varying Quadratic Regulator, American Control Conference, 2017.
2. A. Halder, K. Lee, and R. Bhattacharya, Optimal Transport Approach for Probabilistic Robustness Analysis of F-16 Controllers, AIAA Journal of Guidance, Control, and Dynamics, 2015.
3. R. Bhattacharya, S. Mijanovic , E. Scholte , A. Ferrari , M. Huzmezan, M. Lelic, M. Atalla, Rigorous Design of Real-Time Embedded Control Systems, IEEE Advanced Process Control Applications for Industry, Vancouver, May, 2006.
4. R. Bhattacharya, G. J. Balas, Implementation of Online Control Customization within the Open Control Platform, Software-Enabled Control: Information Technologies for Dynamical Systems, A John Wiley/IEEE Press Publication, 2003.
5. R. Bhattacharya, G. J. Balas, M. Alpay Kaya, A. Packard, Nonlinear Receding Horizon Control of an F-16 Aircraft, Journal of Guidance, Control, and Dynamics, Vol. 25, No. 5, pp. 924-931, 2002.