Transfer operator method for the control of complex dynamics


We have introduced novel operator theoretical methods for stability analysis and control of complex dynamics in nonlinear systems. The transformative idea we proposed is to shift the focus from the point-wise nonlinear behavior of nonlinear systems on the finite dimensional state space to ensemble linear behavior of densities on infinite dimensional space. With every nonlinear dynamical system one can associated two linear transfer operators called as transfer Perron-Frobenius (P-F) and Koopman (K) operators. Both these operators provides 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 {\it 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. The Lyapunov measure is also employed for the purpose of stabilizing and optimal stabilizing feedback control. While the co-design problem for finding the Lyapunov function and a controller is non-convex, we showed the corresponding co-design problem, using the Lyapunov measure, is convex. This convexity property is exploited in the development of linear programming-based computational methods for the design of feedback controllers.

Selected publications



Fundamental limitation for estimation and control over uncertain networks


We have discovered 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 unstable system. The results are similar in sprit to the Bode fundamental limitation formula connecting limitation for stabilization with unstable eigenvalues of the open loop, plant dynamics. However, the presence of uncertainty in communication channels along with the nonlinear nature of dynamics complicate the analysis of the problem. The results have several practical implications for the analysis and design of large-scale network systems. In particular, it can be used to provide controller independent limits on the maximum tolerable stochastic interaction uncertainty among network components to maintain network stability. It can be used to compare various complex network topologies for their robustness property against stochastic interaction uncertainty and for the design of robust stabilizing controllers. The problem of identifying weakest link or ranking links in the order of their criticality to maintain stability in the network can also be addressed using the developed framework. We have applied these results to understand limitations and tradeoffs that arise in the synchronization of large-scale stochastic network systems, and to electric power networks to understand vulnerability of these network systems to stochastic link failure caused by cyber-attacks or contingency.

Selected publications



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 connection between information flows from state to state in dynamical system and the underlying steady state invariant measure of the dynamical system. We have successfully established this connection for linear dynamical system. Answer to this question will allow us to design autonomous network system with information-based specification. This connection will also provide for systematic procedure for the computation of information flow and novel information-theoretic perspective for problems of distributed control and estimation in large scale networks.

Selected publications



Real-time stability monitoring and control of electric power grid


Electric power grid is one of the most critical infrastructures of the modern era. Advancement in sensing, actuation, and network-based technologies has transformed the electric power grid in the last decade. These advancements have presented us with opportunities and challenges for designing reliable and secure power networks. These opportunities are offered in the form of development of novel tools for real-time monitoring, and control techniques for reliable and uninterrupted operations of power grids. We are currently involved in projects whose goal is to develop methods for real-time monitoring and control of power systems \cite{NSFcareer, pserc1,pserc2}. The theory behind the proposed research is adapted from ergodic and geometric theories of dynamical systems. Our main contribution is in the development of a theoretical and computational framework for the finite-time transient stability monitoring of network systems operating away from equilibrium. We are also investigating dynamical system-based methods for solving stochastic optimal power flow problems that arise in the integration of the renewable, and distributed energy resources with electric power grid.

Selected publications



Cyber security of power networks


Security of electric power grids and other critical infrastructures against cyber-attacks is a problem that has received increased attention in the recent past. In our proposed research funded by NSF under the Cyber Physical Systems (CPS) program, we are developing a unified framework, based on tools adapted from systems theory, optimization, real-time systems, and stochastic dynamics for the modeling, analysis, and design of cyber-attack resilient power grids. The methods and techniques developed by this project will be applicable to a more general network system, and will help identify vulnerabilities and design of mitigation strategies against cyber-attacks in large-scale network systems. We have discovered an uncertainty-based modeling framework for the modeling of cyber attacks in power networks. Using system theory tools, we developed a vulnerability metric to capture the impact and degree of difficulty of detecting an attack. In our current research work, we are developing an optimization-based framework to help understand degradation of performance in terms of controllability, observability, and synchronization losses in network systems caused by links and node failure attacks.

Selected publications



Analysis and control of building systems


About 40\% of the US's total energy consumption is from commercial buildings. Hence, any meaningful reduction in commercial energy consumption will go a long way towards securing a sustainable, self-reliant energy vision of the United States. Some of the main challenges that arise in the analysis and design of dynamics in building systems is due to the complex nature of the underlying fluid flow vector field, and the uncertainties associated with the environment and occupancy of the building. We have developed an uncertainty-based modeling framework for the analysis of natural dynamics in building systems using the tools of stochastic dynamical systems and bifurcation theory. Furthermore, linearity of the transfer P-F and K operators is exploited to provide algorithms for optimal placement of sensors and actuators in building system. Linear nature of the transfer operators allows us to extend the notions of controllability and observability gramians from linear to nonlinear systems. Operator theoretic methods are also proposed for the purpose of computation to provide efficient numerical methods for the propagation of contaminants in building systems and closed spaces.

Selected publications



Freeplay induced flutter instability in aeroelastic system


In this Department of Defense (DoD) funded project, we are developing modeling and analysis tools for the prediction of freeplay-induced, flutter instability in a fluid structure interaction system. Flutter, an aeroelastic phenomenon, is an unstable, self-excitation of the structure, due to an unfavorable coupling of structural elasticity and aerodynamics. Flutter is very difficult to predict and its occurrence can lead to catastrophic structural failures. The aircraft wing is a typical example of a fluid-structure interaction system. We developed an analytical and computational framework by employing system-theoretic tools to predict freeplay-induced, flutter instability. One of the main contributions of this research was the application of this analytical framework to predict the stability boundary as a function of freeplay nonlinearity. Wind tunnel tests are also performed for the experimental verification of flutter instability boundaries. The comparison between the analytically predicted stability boundary and the one obtained from the experiment results shows a good agreement.

Selected publications