TY - GEN

T1 - Improving the exponential decay rate by back and forth iterations of the feedback in time

AU - Natarajan, Vivek

AU - Weiss, George

PY - 2013

Y1 - 2013

N2 - We consider the control system ẋ =Ax+Bu, where A generates a strongly continuous semigroup T on the Hilbert space X and the control operator B maps into the dual of D(A*), but it is not necessarily admissible for T. We prove that if the pair (A;B) is both forward and backward optimizable (our definition of this concept is slightly more general than the one in the literature), then the system is exactly controllable. This is a generalization of a well-known result called Russell's principle. Moreover the usual stabilization by state feedback u = Fx, where F is an admissible observation operator for the closed-loop semigroup, can be replaced with a more complicated periodic (but still linear) controller. The period t of the controller has to be chosen large enough to satisfy an estimate. This controller can improve the exponential decay rate of the system to any desired value, including -∞ (deadbeat control). The corresponding control signal u, generated by alternately solving two exponentially stable homogeneous evolution equations on each interval of length t, back and forth in time, will still be in L2. The better the decay rate that we want to achieve, the more iterations the controller needs to perform, but (unless we want to achieve -∞) the number of iterations needed on each period is finite.

AB - We consider the control system ẋ =Ax+Bu, where A generates a strongly continuous semigroup T on the Hilbert space X and the control operator B maps into the dual of D(A*), but it is not necessarily admissible for T. We prove that if the pair (A;B) is both forward and backward optimizable (our definition of this concept is slightly more general than the one in the literature), then the system is exactly controllable. This is a generalization of a well-known result called Russell's principle. Moreover the usual stabilization by state feedback u = Fx, where F is an admissible observation operator for the closed-loop semigroup, can be replaced with a more complicated periodic (but still linear) controller. The period t of the controller has to be chosen large enough to satisfy an estimate. This controller can improve the exponential decay rate of the system to any desired value, including -∞ (deadbeat control). The corresponding control signal u, generated by alternately solving two exponentially stable homogeneous evolution equations on each interval of length t, back and forth in time, will still be in L2. The better the decay rate that we want to achieve, the more iterations the controller needs to perform, but (unless we want to achieve -∞) the number of iterations needed on each period is finite.

UR - http://www.scopus.com/inward/record.url?scp=84902344684&partnerID=8YFLogxK

U2 - 10.1109/CDC.2013.6760293

DO - 10.1109/CDC.2013.6760293

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AN - SCOPUS:84902344684

SN - 9781467357173

T3 - Proceedings of the IEEE Conference on Decision and Control

SP - 2715

EP - 2719

BT - 2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 52nd IEEE Conference on Decision and Control, CDC 2013

Y2 - 10 December 2013 through 13 December 2013

ER -