Title :
A time-periodic Lyapunov approach for motion planning of controllable driftless systems on SU(n)
Author :
Silveira, H.B. ; Silva, P. S Pereira da ; Rouchon, P.
Author_Institution :
Dept. of Telecommun. & Control Eng., Univ. of Sao Paulo, Sao Paulo, Brazil
Abstract :
For a right-invariant and controllable driftless system on SU(n), we consider a time-periodic reference trajectory along which the linearized control system generates su(n): such trajectories always exist and constitute the basic ingredient of Coron´s return method. The open-loop controls that we propose, which rely on a left-invariant tracking error dynamics and on a fidelity-like Lyapunov function, are determined from a finite number of left-translations of the tracking error and they assure global asymptotic convergence towards the periodic reference trajectory. The role of these translations is to avoid being trapped in the critical region of this Lyapunov-like function. The convergence proof relies on a periodic version of LaSalle´s invariance principle and the control values are determined by numerical integration of the dynamics of the system. Simulations illustrate the obtained controls for n = 4 and the generation of the C-NOT quantum gate.
Keywords :
Lyapunov methods; controllability; integration; invariance; linear systems; open loop systems; path planning; C-NOT quantum gate; Coron´s return method; LaSalle´s invariance principle; controllable driftless systems; fidelity-like Lyapunov function; global asymptotic convergence; left-invariant tracking error dynamics; linearized control system; motion planning; numerical integration; open-loop controls; system dynamics; time-periodic Lyapunov approach; time-periodic reference trajectory; Algebra; Automation; Control systems; Convergence of numerical methods; Error correction; Lyapunov method; Motion control; Open loop systems; Optimal control; Trajectory;
Conference_Titel :
Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on
Conference_Location :
Shanghai
Print_ISBN :
978-1-4244-3871-6
Electronic_ISBN :
0191-2216
DOI :
10.1109/CDC.2009.5399811