TY - JOUR

T1 - Optimal ensemble control of stochastic time-varying linear systems

AU - Qi, Ji

AU - Zlotnik, Anatoly

AU - Li, Jr Shin

N1 - Funding Information:
This work was supported in part by the NSF under Grants CMMI-0747877 and CMMI-1301148 and the AFOSR under Grant FA9550-10-1-0146 .

PY - 2013

Y1 - 2013

N2 - We consider the optimal guidance of an ensemble of independent, structurally identical, finite-dimensional stochastic linear systems with variation in system parameters between initial and target states of interest by applying a common control function without the use of feedback. Our exploration of such ensemble control systems is motivated by practical control design problems in which variation in system parameters and stochastic effects must be compensated for when state feedback is unavailable, such as in pulse design for nuclear magnetic resonance spectroscopy and imaging. In this paper, we extend the notion of ensemble control to stochastic linear systems with additive noise and jumps, which we model using white Gaussian noise and Poisson counters, respectively, and investigate the optimal steering problem. In our main result, we prove that the minimum norm solution to a Fredholm integral equation of the first kind provides the optimal control that simultaneously minimizes the mean square error (MSE) and the error in the mean of the terminal state. The optimal controls are generated numerically for several example ensemble control problems, and Monte Carlo simulations are used to illustrate their performance. This work has immediate applications to the control of dynamical systems with parameter dispersion or uncertainty that are subject to additive noise, which are of interest in quantum control, neuroscience, and sensorless robotic manipulation.

AB - We consider the optimal guidance of an ensemble of independent, structurally identical, finite-dimensional stochastic linear systems with variation in system parameters between initial and target states of interest by applying a common control function without the use of feedback. Our exploration of such ensemble control systems is motivated by practical control design problems in which variation in system parameters and stochastic effects must be compensated for when state feedback is unavailable, such as in pulse design for nuclear magnetic resonance spectroscopy and imaging. In this paper, we extend the notion of ensemble control to stochastic linear systems with additive noise and jumps, which we model using white Gaussian noise and Poisson counters, respectively, and investigate the optimal steering problem. In our main result, we prove that the minimum norm solution to a Fredholm integral equation of the first kind provides the optimal control that simultaneously minimizes the mean square error (MSE) and the error in the mean of the terminal state. The optimal controls are generated numerically for several example ensemble control problems, and Monte Carlo simulations are used to illustrate their performance. This work has immediate applications to the control of dynamical systems with parameter dispersion or uncertainty that are subject to additive noise, which are of interest in quantum control, neuroscience, and sensorless robotic manipulation.

KW - Ensemble control

KW - Gaussian noise

KW - Mean square error

KW - Poisson counter

KW - Singular systems

KW - Stochastic systems

UR - http://www.scopus.com/inward/record.url?scp=84884755912&partnerID=8YFLogxK

U2 - 10.1016/j.sysconle.2013.07.013

DO - 10.1016/j.sysconle.2013.07.013

M3 - Article

AN - SCOPUS:84884755912

SN - 0167-6911

VL - 62

SP - 1057

EP - 1064

JO - Systems and Control Letters

JF - Systems and Control Letters

IS - 11

ER -