Subhrajit Sinha

On information transfer in discrete dynamical systems

In this paper, we propose a new definition of information transfer in a discrete dynamical system. The information transfer is based on how much entropy (uncertainty) is transferred from state x to state y, as the dynamical system evolves in time. In our previous work, we had provided an axiomatic definition of information transfer where we had considered absolute entropy to define the transfer. Our new definition can be viewed as a natural generalization of directed information from information theory to dynamical systems and is based on transfer of conditional entropy. This new definition also satisfies the properties of a) zero transfer, b) transfer asymmetry and c) information conservation. We start with a definition of one-step information transfer and then generalize this definition to n-step information transfer and average information transfer over infinite time step. We also provide analytical expressions for information transfer for linear systems. Some basic examples are provided to understand the physical meaning of information transfer and how this can be used to infer causality and influence in dynamical system setting.

Causality preserving information transfer measure for control dynamical system

In this paper, we show through examples, how the existing definitions of information transfer, namely directed information and transfer entropy fail to capture true causal interaction between states in control dynamical system. Furthermore, existing definitions are shown to be too weak to have any implication on two of the most fundamental concepts in system theory, namely controllability and observability. We propose a new definition of information transfer, based on the ideas from dynamical system theory, and show that this new definition can not only capture true causal interaction between states, but also have implication on system controllability and observability properties. In particular, we show that non-zero transfer of information from input-to-state and state-to-output implies structural controllability and observability properties of the control dynamical system respectively. Analytical expression for information transfer between state-to-state, input-to-state, state-to-output, and input-to-output are provided for linear system. There is a natural extension of our proposed definition to define information transfer over n time steps and average information transfer over infinite time step. We show that the average information transfer in feedback control system between plant output and input is equal to the entropy of the open loop dynamics thereby re-deriving the Bode fundamental limitation results using the proposed definition of transfer.

Optimal placement of actuators and sensors for control of nonequilibrium dynamics

In this paper, we provide a systematic convex programming-based approach for the optimal locations of static actuators and sensors for the control of nonequilibrium dynamics. The problem is motivated with regard to its application for control of nonequilibrium dynamics in the form of temperature in building systems and control of oil spill in oceanographic flow. The controlled evolution of a passive scalar field, modeling the temperature distribution or the density of oil dispersant, is governed by the linear advection partial differential equation (PDE) with spatially located actuators and sensors. Spatial locations of actuators and sensors are optimized to maximize the controllability and observability of the linear advection PDE. Linear transfer Perron-Frobenius and Koopman operators, associated with the advective velocity field, are used to provide analytical characterization for the controllable and observable spaces of the advection PDE. Set-oriented numerical methods are used for the finite dimensional approximation of the transfer operators and in the formulation of the optimization problem. Application of the framework is demonstrated for the optimal placement of actuators for the release of dispersant for oil spill control.

Information-based measure for influence characterization in dynamical systems with applications

In this paper, we introduce novel measure based on information to define influence in a dynamical system. The objective is to determine how a particular state (or a linear combination of states) in dynamical system influence or participate in the dynamics of another state (or a linear combination of states). An important parameter in determining the influence is the definition of the influence. We propose information or entropy-based measure for influence characterization. In particular, state x is said to influence or participate in the dynamics of state y, if the evolution of state x results in change in entropy or information content of state y. This work builds on our prior work on formalism for information transfer in dynamical network [1]. We discuss the applications of the developed framework for influence characterization in power system and social network. For power system, the proposed influence measure is used for the computation of participation factor of individual generator to the inter-area oscillation mode of the power system. For social network application, we use a Twitter network for influence characterization. The influence measure is used to understand the distribution of influential nodes and for influence-based clustering of the Twitter network.

Formalism for information transfer in dynamical network

In this paper, we discover a novel approach for defining information transfer in a linear network dynamical system. We provide entropy based characterization of the information transfer where the information transfer from state x to state y is measured by the amount of entropy/uncertainty that is transferred from state x to y over one time step. Our proposed definition of information transfer is based on three axioms. The first axiom has to do with zero information transfer, which says that if state x is not connected (or appears) in the dynamics of y then information transfer from x → y is zero. The second axiom captures the asymmetric nature of information transfer i.e., x is not connected to the dynamics of y but y is connected to the dynamics of x then information transfer from x → y is zero but the transfer from y → x is not zero. The third axiom is on information conservation. Information conservation axiom says that if y space can be split into two subspace, y1 and y2, then the information transfer from x → y will be equal to the sum of the information transfers from x → y1 and x → y2 provided y1 and y2 are “dynamical independent”. Similar conservation property also applies for the case where x is split into two parts x1 and x2 with y intact. We provide an analytical expression for information transfer satisfying these three axioms. Preliminary results are provided for identifying information-based most influential nodes and clusters in network system with small world network topology.