Abhyudai Singh

Mohammad Soltani, Abhyudai Singh

Stochastic dynamics of several systems can be modeled via piecewise deterministic time evolution of the state, interspersed by random discrete events. Within this general class of systems, we consider time-triggered stochastic hybrid systems (TTSHS), where the state evolves continuously according to a linear time-varying dynamical system. Discrete events occur based on an underlying renewal process (timer), and the intervals between successive events follow an arbitrary continuous probability density function. Moreover, whenever the event occurs, the state is reset based on a linear affine transformation that allows for the inclusion of state-dependent and independent noise terms. Our key contribution is derivation of necessary and sufficient conditions for the stability of statistical moments, along with exact analytical expressions for the steady-state moments. These results are illustrated on an example from cell biology, where deterministic synthesis and decay of a gene product (RNA or protein) is coupled to random timing of cell-division events. As experimentally observed, cell-division events occur based on an internal timer that measures the time elapsed since the start of cell cycle (i.e., last event). Upon division, the gene product level is halved, together with a state-dependent noise term that arises due to randomness in the partitioning of molecules between two daughter cells. We show that the TTSHS framework is conveniently suited to capture the time evolution of gene product levels, and derive unique formulas connecting its mean and variance to underlying model parameters and noise mechanisms. Systematic analysis of the formulas reveal counterintuitive insights, such as, if the partitioning noise is large then making the timing of cell division more random reduces noise in gene product levels.

Mohammad Soltani, Abhyudai Singh

We consider a class of piecewise-deterministic Markov processes where the state evolves according to a linear dynamical system. This continuous time evolution is interspersed by discrete events that occur at random times and change (reset) the state based on a linear affine map. In particular, we consider two families of discrete events, with the first family of resets occurring at exponentially-distributed times. The second family of resets is generally-distributed, in the sense that, the time intervals between events are independent and identically distributed random variables that follow an arbitrary continuous positively-valued probability density function. For this class of stochastic systems, we provide explicit conditions that lead to finite stationary moments, and the corresponding exact closed-form moment formulas. These results are illustrated on an example drawn from systems biology, where a protein is expressed in bursts at exponentially-distributed time intervals, decays within the cell-cycle, and is randomly divided among daughter cells when generally-distributed cell-division events occur. Our analysis leads to novel results for the mean and noise levels in protein copy numbers, and we decompose the noise levels into components arising from stochastic expression, random cell-cycle times, and partitioning. Interestingly, these individual noise contributions behave differently as cell division times become more random. In summary, the paper expands the class of stochastic hybrid systems for which statistical moments can be derived exactly without any approximations, and these results have applications for studying random phenomena in diverse areas.

Mohammad Soltani, Abhyudai Singh

A Network Control System (NCS) consists of control components that interact with the plant over a shared network. The system dynamics of a NCS could be subject to noise arising from randomness in the times at which the data is transmitted over the network, corruption of the transmitted data by the communication network, and external disturbances that might affect the plant. A question of interest is to understand how the statistics of the data transmission times affects the system dynamics, and under what conditions the system is stable. Another related issue is designing a controller that meets desired performance specifications (e.g., a specific mean and variance of the system state). Here, we consider a minimal NCS that consists of a plant and a controller, and it is subject to random transmission times, channel corruption and external disturbances. We derive exact dynamics of the first two moments of the system, and use them to derive the stability conditions of the system. We further design a control law that steers the system to a desired mean and variance. Finally, we demonstrate our results using different examples, and show that under some specific conditions, randomness in the data transmission times can even reduce the variability contributed from disturbance.

Cenk Demir, Abhyudai Singh

In this paper, trajectory prediction and control design for a desired hit point of a projectile is studied. Projectiles are subject to environment noise such as wind effect and measurement noise. In addition, mathematical models of projectiles contain a large number of important states that should be taken into account for having a realistic prediction. Furthermore, dynamics of projectiles contain nonlinear functions such as monomials and sine functions. To address all these issues we formulate a stochastic model for the projectile. We showed that with a set of transformations projectile dynamics only contains nonlinearities of the form of monomials. In the next step we derived approximate moment dynamics of this system using mean-field approximation. Our method still suffers from size of the system. To address this problem we selected a subset of first- and second-order statistical moments and we showed that they give reliable approximations of the mean and standard deviation of the impact point for a real projectile. Finally we used these selected moments to derive a control law that reduces error to hit a desired point.