Joint Actuator-Sensor Design for Stochastic Linear Systems

We investigate here the joint actuator-sensor design problem for stochastic linear control systems. Specifically, we address the problem of identifying a pair of sensor and actuator which gives rise to the minimum expected value of a time-averaged quadratic cost. It is well known that for the linear-quadratic-Gaussian (LQG) control problem, the optimal feedback control law can be obtained via the celebrated separation principle. Moreover, if the system is stabilizable and detectable, then the infinite-horizon time-averaged cost exists. But such a cost depends on the placements of the sensor and the actuator. We formulate in the paper the optimization problem about minimizing the time-averaged cost over admissible pairs of actuator and sensor under the constraint that their Euclidean norms are fixed. The problem is non-convex and is in general difficult to solve. We obtain in the paper a gradient descent algorithm (over the set of admissible pairs) which minimizes the time-averaged cost. Moreover, we show that the algorithm can lead to a unique local (and hence global) minimum point under certain special conditions.

[1]  Alexander Olshevsky,et al.  Minimal Controllability Problems , 2013, IEEE Transactions on Control of Network Systems.

[2]  Lei Zhang,et al.  Communication and control co-design for networked control systems , 2006, Autom..

[3]  G. F. Wredenhagen,et al.  Curvature properties of the algebraic Riccati equation , 1993 .

[4]  Tongwen Chen,et al.  Optimal periodic scheduling of sensor networks: A branch and bound approach , 2013, Syst. Control. Lett..

[5]  John Lygeros,et al.  On Submodularity and Controllability in Complex Dynamical Networks , 2014, IEEE Transactions on Control of Network Systems.

[6]  Francesco Bullo,et al.  Controllability Metrics, Limitations and Algorithms for Complex Networks , 2013, IEEE Transactions on Control of Network Systems.

[7]  Edwin K. P. Chong,et al.  Sensor scheduling for target tracking: A Monte Carlo sampling approach , 2006, Digit. Signal Process..

[8]  Singiresu S Rao,et al.  Optimal placement of actuators in actively controlled structures using genetic algorithms , 1991 .

[9]  Mohamed-Ali Belabbas,et al.  Geometric methods for optimal sensor design , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[10]  Ling Shi,et al.  Optimal sensor scheduling for multiple linear dynamical systems , 2017, Autom..

[11]  S. Schechter,et al.  On the Inversion of Certain Matrices , 2018 .

[12]  Tamer Basar,et al.  Optimal capacity allocation for sampled networked systems , 2016, Autom..

[13]  Roger W. Brockett,et al.  Stabilization of motor networks , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[14]  George J. Pappas,et al.  Sensor placement for optimal Kalman filtering: Fundamental limits, submodularity, and algorithms , 2015, 2016 American Control Conference (ACC).

[15]  R. E. Kalman,et al.  New Results in Linear Filtering and Prediction Theory , 1961 .

[16]  Mohamed-Ali Belabbas,et al.  S Y ] 2 0 Ju l 2 01 7 Optimal actuator design for minimizing the worst-case control energy , 2017 .

[17]  G. Obinata,et al.  Optimal Sensor/actuator Placement for Active Vibration Control Using Explicit Solution of Algebraic Riccati Equation , 2000 .

[18]  Shreyas Sundaram,et al.  Sensor selection for Kalman filtering of linear dynamical systems: Complexity, limitations and greedy algorithms , 2017, Autom..

[19]  Clarence W. Rowley,et al.  Fluid flow control applications of ℋ2 optimal actuator and sensor placement , 2014, 2014 American Control Conference.