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 - https://doi.org/10.1109/CDC.2013.6760293
DO - https://doi.org/10.1109/CDC.2013.6760293
M3 - منشور من مؤتمر
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 -