A time-periodic Lyapunov approach for motion planning of controllable driftless systems on SU(n)

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.

Explicit control laws for the periodic motion planning of controllable driftless systems on SU(n)

“…Although (3) is not needed in the proof of Theorem 2, (8) and (10) are fundamental. Of crucial importance is also the more general version of Barbalat's Lemma given below.…”

“…Suppose that (1) is p-controllable. Following [3], it is straightforward to verify from (1) and (4) that the timedependent change of coordinates…”

“…. , u r m (t), t ∈ R + ) of (1) with period T and initial condition X r (0) = X ∞ , and developing Lyapunovlike convergence results, the authors formulated in [3] an algorithm that determined, in a finite number of steps, piecewise-continuous open-loop controls u k (t) that assure the validity of (2) in case (1) is regular (see [3, Algorithm 1 and Theorem 2]). The controls u k : R + → R are obtained by numerical integration of (1) and they depend on the period T and on the desired goal state X ∞ .…”

“…The mathematical proof of the solution presented is given in Section III. In Section IV, the control strategy is applied to the same quantum system (n = 4) that was considered in [3]. Once again, the aim is the generation of the C-NOT (Controlled-NOT) quantum logic gate.…”

“…In a previous work ( [3]), the authors have introduced and treated the periodic motion planning problem for rightinvariant driftless systems the forṁ…”