My research interest is in the broad areas of data-driven learning and control of dynamical systems, cyber physical system, and distributed optimization. The research problems are guided by applications in autonomy and robotics, network power systems, and building systems.

**Data-Driven Analysis and Control of Dynamical Systems**

We have introduced novel operator theoretical methods for stability analysis and optimal control design for dynamical systems. The transformative idea we proposed is to shift the focus from the point-wise nonlinear evolution of dynamical systems on the finite-dimensional state space to ensemble linear evolution of functions on infinite dimensional space. With every nonlinear dynamical system, one can associate two linear transfer operators called as transfer Perron-Frobenius (P-F) and Koopman operators. Both these operators provide for the linear description of nonlinear dynamics in the space of densities. This linear description of nonlinear dynamics can be effectively used for the analysis and control of a nonlinear system.

In particular, using a linear transfer P-F operator we introduce the Lyapunov measure as a new tool for verifying weaker set-theoretic notion of almost everywhere stability. The Lyapunov measure is shown to be dual to the Lyapunov function and, unlike the Lyapunov function, systematic linear programming-based computational methods are proposed for the computation of the Lyapunov measure.

This duality in stability theory between Lyapunov function and Lyapunov measure also extend to the optimal control of deterministic and stochastic system. This duality leads to a convex formulation of optimal control problem in the dual space of densities. These duality results combined with the data-driven methods discovered for the finite dimensional approximation of linear operators are employed for the design of data-driven optimal control of nonlinear systems. The goal of this proposed research is to discover a comprehensive analytical and computational framework for the data-driven control of dynamical system that account explicitly for the finite amount of data available for control.

**Selected publications**

- U. Vaidya and P. G. Mehta, Lyapunov measure for almost everywhere stability, IEEE Transactions on Automatic control, Vol 53, pp. 307-323, 2008.
- U. Vaidya, P. G. Mehta and U. Shanbhag, Nonlinear stabilization via control Lyapunov measure, IEEE Transactions on Automatic Control, Vol 55, pp. 1314-1328, 2010.
- A. Raghunathan and U. Vaidya, Optimal stabilization using Lyapunov measures, IEEE Transactions on Automatic Control, 2014.
- U. Vaidya, Stochastic stability analysis of discrete time syste using Lyapunov measure, American Control Conference, 2015.
- B. Huang and U. Vaidya, Data-driven feedback stabilization using Koopman operator, Book Chapter: Koopman Operator for Control, 2019.
- R. Rajaram,U. Vaidya, M. Fardad, and B. Ganapathaysubramanian, Almost Everywhere stability: Linear transfer operator approach,Journal of Mathematical analysis and applications, Vol 368, pp. 144-156, 2010.
- B. Huang and U. Vaidya, Convex Approach to data-driven optimal control via Perron-Frobenius and Koopman operator, Preprint.

**Cyber Physical Systems**

We have discovered systematic framework for the analysis and design of network controlled system in the presence of uncertainty. This includes discovering fundamental limitation results for the estimation and stabilization of nonlinear systems over uncertain communication channels. The main contribution of this research work was to connect fundamental limitations for nonlinear stabilization and estimation with the measure-theoretic entropy of steady-state invariant measure of the open loop system. For a system with non-equilibrium open loop dynamics such as chaotic attractor or unstable limit cycles, the measure-theoretic entropy is captured by the positive Lyapunov exponents. This was the first systematic result to highlight the critical role played by non-equilibrium dynamics in nonlinear stabilization and estimation over uncertain channels and differ from the existing results based on equilibrium dynamics of local linearization of a nonlinear system. Our results generalize existing results known in the case of linear systems where Lyapunov exponents emerge as a natural generalization of eigenvalues from linear systems to nonlinear systems.

We also discovered systematic framework based on mean square notion of stability for the analysis and synthesis of nonlinear and linear network system with stochastic uncertainty. These results are used to understand fundamental trade-offs between network topology, internal agent dynamics, and uncertainty characteristics in the synchronization of large scale networks.

###### Selected publications

- A. Diwadkar and U. Vaidya, Limitations and tradeoffs in synchronization of large scale networks with uncertain links, Scientific Reports, Nature Publication Group, 2016.
- S. Pushpak, A. Diwadkar, and U. Vaidya, Mean Square-Based Stochastic Stability Analysis of Continuous time Linear Network System, IEEE Transactions of Automatic Control, Volume: 63, Issue: 12, Dec. 2018.
- A. Diwadkar and U. Vaidya, Limitations for Nonlinear observation over erasure channels, IEEE Transactions on Automatic Control, Vol. 58, Issue 2, pp-454-459, 2013.
- U.Vaidya and N. Elia, Limitations for nonlinear stabilization over erasure channel: Scalar Case, Systems and Control Letters, Volume 61, Issue 9, Pages 959-966, 2012.
- A. Diwadakar and U. Vaidya, Control of LTV system with intermittent communication, International Journal of Robust and Nonlinear Control, 2013.
- A. Diwadkar and U. Vaidya, Synchronization in large scale nonlinear network with uncertain links, Automatica, Volume 100, February 2019, Pages 194-199.
- P. G. Mehta, U. Vaidya and A. Banaszuk, Markov chains, entropy and fundamental limitations in nonlinear stabilization, IEEE Transactions on Automatic control, Vol 53, pp. 784-791,2008.

#### Information Flow in Dynamical Systems

Information flow and causality are two of the most fundamental concepts important for the analysis and design of variety of systems in engineering and natural sciences. A mathematically precise definition of information flow developed with dynamics in mind is essential for the rigorous formulation of autonomy in network dynamical system. Degree of interaction as measure by information flow between network of autonomous agents and its environment can be used to characterize degree of autonomy in network dynamical system. We have developed novel axiom-based formalism for information flow in network dynamical system using methods from ergodic theory and stochastic dynamics. The proposed formalism can be viewed as natural extension of directed information from information theory to dynamical system and is used to precisely characterize flow of information and influence structure in network dynamical system. The problem of distributed control and estimation in large scale dynamical network system are intimately connected with the flow of information among network components. We are investigating the application of information transfer for reduced order modeling of dynamical system and for inferring causal structure from brain data.

**Selected publications**

- S. Sinha and U. Vaidya, Formalism for information flow in network dynamical system, IEEE Control and Decision Conference, 2015 Osaka, Japan.
- U. Vaidya and S. Sinha, Information-based causal measure for influence characterization in network dynamical system with applications, American Control Conference, 2016.
- S. Sinha and U. Vaidya, On Data-Driven Computation of Information Transfer for Causal Inference in Discrete-time Dynamical Systems, Journal of Nonlinear Science, Volume 30, pages1651–1676(2020).
- S. Sinha, P. Sharma, U. Vaidya, and V. Ajjarapu, On Information Transfer Based Characterization of Power System Stability, IEEE Transactions of Power Systems, 2019.

**R**obust Learning via Robust Optimization

In this research we are discovering dynamical system based algorithms for solving robust optimization problem. The algorithm can be viewed as a dynamical system where the system converges asymptotically to the optimal solution of the robust optimization problem. We call this system as saddle-point dynamical system. There are several advantages of the proposed saddle-point dynamical system for solving robust optimization problem. One of the main advantage of the proposed saddle- point dynamical system is that the natural distributed structure of these dynamics are used to solve the robust optimization problem in a distributed fashion.

This novel dynamical system based algorithm for robust optimization is employed for robust learning in the presence of adversaries and robust control.

**Selected publications**

- K. Ebrahimi, N. Elia, and U. Vaidya, A continuous time dynamical system approach for solving robust optimization, European Control Conference, 2019.
- K. Ebrahimi, U. Vaidya, and N. Elia, Robust Optimization Via discrete-time saddle point algorithm, IEEE Control and Decision Conference, 2019.
- Yasaman Esfandiari, Keivan Ebrahimi, Aditya Balu, Umesh Vaidya, Nicola Elia and Soumik Sarkar, A Saddle-Point Dynamical System Approach for Robust Deep Learning, SafeAI, 2020.

**Network Power Systems**

In application involving network power system the research work is focused on couple of different problems which include real-time stability monitoring, stochastic stability and performance analysis of power system in the presence of uncertain renewable, robust distributed optimization of distributed system, data-driven analytics involving linear operator theoretic methods for reduced order modeling, and cyber security of power grid.

We have discovered data-driven methods based on the theory of dynamical system for the real-time rotor angle and voltage stability monitoring of power system using time-series data from Phasor Measurement Units (PMUs).

Our theoretical research work on characterizing fundamental limitations for the estimation and control of nonlinear systems has lead to systematic tools for the mean square stability analysis and synthesis of network power system in the presence of stochastic disturbances.

In particular, small signal stochastic stability of network power system is analyzed to understand the effects of increase in the penetration of renewable energy resources. Similarly, distributed controller are designed that are robust to stochastic fluctuations in loads in demand response control problem.

Linear operator theoretic methods involving Koopman operator are employed for the identification and reduced order modeling of power system dynamics using sensor data. The problem of uncertainty propagation and domain of attraction computation are also formulated using spectral properties of the Koopman operator.

The novel dynamical system based approach we developed for solving robust optimization problem is applied for solving robust optimal power flow (OPF) problem in the presence of renewable uncertainty. The uncertainty from renewable is modeled as an uncertain parameter in the OPF problem. The dynamical system based approach for solving RO problem is also used for solving the robust OPF problem in distributed manner and for design of distributed real-time voltage control strategies in distribution power system.

###### Selected publications

- J. Yan, C.C. Liu, and U. Vaidya, PMU based real-time monitoring of rotor angle dynamics, IEEE Transactions on Power Systems, Vol 26, No. 4, pp 212-2123, 2011.
- S. Dasgupta, M. Paramasivam, U. Vaidya, and A. Venkataramana, Real- time monitoring of short term voltage stability using PMU data, IEEE Transactions on Power Systems, Vol 28, No 4, pp 3702-3711, 2013.
- S. Dasgupta, M. Paramasivam, U. Vaidya, and A. Venkataramana, PMU-based model free appraoch for real time rotor angle monitoring. IEEE Transactions on Power Systems, Vol. 30, No. 5, September 2015
- S. Dasgupta, M. Paramasivam, U. Vaidya, and A. Venkataramana, Entropy-based metric for characterization of delayed voltage recovery, IEEE Transactions on Power Systems, Vol. 30, No. 5, 2015.
- M. Paramasivam, S. Dasgupta, A. Venkataramana, and U. Vaidya Contingency Analysis and Identification of Dynamic Voltage Control Area, IEEE Transactions on Power Systems, Vol. 30, No. 6, 2015.
- S. Sinha, P. Sharma, U. Vaidya, and V. Ajjarapu, On Information Transfer Based Characterization of Power System Stability, IEEE Transactions of Power Systems, 2019.

**Building Systems and Fluid Dynamics**

Linear operator theoretic methods involving P-F and Koopman operators are employed to develop data-driven and model-based approaches for the estimation, tracking, and localization of contaminants in building systems application. The linearity property of the operator theoretic framework is exploited to extend the notion of controllability and observability gramian from linear to nonlinear systems to propose a systematic procedure for optimal placement of sensors and actuators in buildings. Similarly, the linearity of the operator theoretic framework is exploited to solve difficult problems of estimation of contaminants and localization of contaminant source.

###### Selected publications

- A. Fontanini, U. Vaidya, and B. Ganapathysubramanian, A stochastic approach to modeling the dynamics of natural ventilation systems, Energy and Building, Vol. 63, pp 87-97, 2013
- A. Fontanini, U. Vaidya, and B. Ganapathysubramanian, Constructing Markov matrix for real-time transient contaminant transport analysis for indoor environment, Energy and Building, 2015
- A. Fontanini, U. Vaidya, and B. Ganapathysubramanian, A methodology for optimal placement of sensors in enclosed environment: Dynamical systems approach, Energy and Building, 2015
- U.Vaidya, R. Rajaram, and S. Dasgupta, Gramian based approach for sensor and actuator placement in advection PDE, Journal of Mathematical Analysis and Application Vol. 394, pp. 213-224, 2012.
- S. Sinha, U. Vaidya, and R. Rajaram, Actuator and sensor placement for control of non-equilibrium dynamics, Journal of Mathematical Analysis and Applications, 2015.

**Robotics and Autonomy**

Coming soon….