(186a) Regulating and Trapping an Ensemble of Brownian Particles by Broadcast the Stochastic Receding Horizon Control Policy

In the last decade, the attention of many control engineers has been focused in developing control strategies for regulating trajectories of micro and nano level objects in a specific medium. Emerging applications in this direction include but not limited to 1) the understanding of dynamical behaviors of biological objects and systems such as cellular interactions, DNA etc. in their natural environments for facilitating biomedical applications, 2) manipulation and regulation of nano and molecular structures for applications in drug delivery systems and laboratory on a chip technologies. The key challenge in facilitating these applications lies in the control over trajectories of these submicroscopic objects which are under continuous random fluctuations, also called Brownian fluctuations, induced by interactions of these objects with the medium in which they are placed. Thus the main question arises here is the design of optimal control strategies for suppressing or canceling these Brownian effects while achieving the desired performance of the system. Motived by this, in this work, we develop a “Broadcast Stochastic Receding Horizon Control (BSRHC)” strategy for regulating and trapping an ensemble of Brownian particles. The central idea of our BSRHC strategy lies in the integration of existing finite horizon based receding horizon control strategy with probabilistic tools such as the theory of supermartingale and the idea of broadcast of the same information to all particles in the ensemble. Further, the key feature of this new control architecture is the utilization of the minimum possible feedback information in designing optimal control inputs.

In the AICHE 2009 annual meeting, we presented our work on the broadcast model predictive control for multi-cellular systems using the aggregate system behavior as the measured feedback information [1]. We extended this work by developing a mathematical theory for regulating the behaviors of Brownian particles in one-dimensional space which was presented in the AICHE 2010 annual meeting [2]. Building on the same principles, here, we develop a novel “Broadcast Stochastic Receding Horizon Control” strategy to show the regulation and trapping of an ensemble of Brownian particles in one, two and three dimensional space by utilizing the measured position of a single particle in the ensemble as the only available feedback information. 

We consider a system consisting of n particles. We assume that all particles in the ensemble make independent decisions i.e. the interactions among particles and the hydrodynamic effects of the medium on particles are neglected. The dynamical behaviors of particles are modeled using well known discrete time random walk models. Using the Donsker's theorem, first we show that the model is a reasonable approximation of continuous time Brownian model which shows the time-wise homogeneous phenomenon in a homogeneous medium. With this, we design the broadcast stochastic receding horizon control framework. The controller objective within this framework is to drive all particles towards a minimum trapping region and keep all particles in this region forever. For this, we formulate a finite horizon based stochastic receding horizon controller by developing the conditional expectation based non-homogeneous Markov model of the system. The transition probabilities of the model serve as the manipulated or control inputs. In the experimental designs, these probabilities are typically manipulated by applying external forces such as laser induced forces, magnetic fields, electro-kinetic fields etc. on particles placed in a given medium. At a given time step, these transition probabilities are designed over a control horizon by minimizing a conditional expectation based cost function over the finite prediction horizon under the system constraints. The first optimally computed transition probabilities are then broadcast among all particles. At each time step, the position of a particle closest to the origin which is outside the trapping region is measured. This information is used as the only available feedback in designing the next control moves.

In the presentation, we show the trapping of 100 Brownian particles simultaneously in one, two and three dimensional space by implementing the BSRHC strategy in a simulation environment. We show the trapping of all particles in the ensemble for an extended period of time. Finally, we guarantee convergence and stability of the controller with probability 1.

1. G. Kumar and M. V. Kothare. Broadcast Model Predictive Control of Multi-Cellular System, CAST Plenary Session, AICHE 2009.
2. G. Kumar and M. V. Kothare. A Mathematical Theory of Manipulating Suspended Multiple Brownian Particles Simultaneously in a Solution, AICHE 2010.