Feedback schemes for radiation damping suppression in NMR: A control-theoretical perspective
Introduction
Nuclear Magnetic Resonance (NMR) has achieved an exceptional degree of control on the dynamics of quantum spin systems, through the development of techniques such as shaped pulses, composite pulses, refocusing schemes, and effective Hamiltonians (see for example [1], [2], [3] for a review of NMR and [4], [5] for some more advanced control strategies). These control techniques allow for the precise spectroscopy of complex molecules by compensating for experimental systematic errors and departures of the real systems from ideal, simplified models. Most of these control techniques are based on open loop control, contrary to the practice in most other domains where control is based on closed loop and feedback. An instance of application of feedback in NMR is given by techniques for reducing the effects of radiation damping.
Radiation damping is a phenomenon occurring in an NMR spectrometer in the presence of a collective spin measurement [6], [7], [8]. During a measurement, the nuclear spins precess about the external magnetic field, inducing an oscillating current in the detection coil that creates an electromagnetic field. This field in turn interacts with the spins in the sample, inducing a back-action on the system observed. In high field and probes of a high quality factor, radiation damping is typically an important effect only at certain frequency ranges, for example that of the abundant spin of the solvent. At these frequencies, it behaves much like a soft pulse, steering the magnetization vector back to its thermal equilibrium direction. In other situations the back-action signal is so weak that it is dominated by the relaxation effects, and hence it is negligible. In order to focus on the radiation damping effects only, in this work we assume to be dealing with one of those situations in which radiation damping is of interest, and it is comparable or dominates over relaxation effects.
A model of radiation damping has existed since the fifties [6], [7], and assumes that the back-action is conservative, i.e., it preserves the norm of the Bloch vector. Efforts to engineer the NMR receiving/transmitting system in order to reject this form of back-action have been carried out for more than a decade. Many strategies to compensate for radiation damping have been devised, such as electronic feedback [9], [10], [11], [12], rf pulse compensation [13], gradient field, Q-switches, and composite pulse sequences, see [8], [12] for more detailed surveys. We are here interested only in the first two methods.
The aim of this paper is threefold. First, we provide a rigorous convergence analysis of the behavior induced by the radiation damping effect and described qualitatively in several papers [14], [15], [8], [16], [17]. Second, we aim to give a system-theoretic interpretation of the electronic feedback and pulse compensation control designs found in the NMR literature. In the so-called “Ecole Polytechnique design” [12], the compensation scheme can be thought of as an exact feedback matching problem, i.e., a precompensator based on the knowledge of the radiation damping field deriving from the accepted model of radiation damping. In the presence of uncertainty in the radiation damping field knowledge (or in the model leading to the evaluation of its value), a high gain variant of the same control problem can be set up in order to maintain the spin in the “fully inverted” state. This scheme however only works for this particular state; for generic states, the exact cancellation of the radiation damping alone does not achieve asymptotic stabilization. However, the precompensator can still be intended as a prefeedback to which a second active field can be linearly superimposed, in order to produce desired control actions. In control terms, this design is called a 2-degrees of freedom (DOF) control design, and resembles the schemes described in [13], [12].
The third and last aim of this paper is to explore possible alternative/improved schemes inspired by control theory. In the spirit of feedback control, we show that the 2-DOF design mentioned above can be completed with an extra feedback loop, allowing to achieve closed-loop asymptotic stabilization, a more robust concept than just exact canceling by matching. We will further see that also a high gain state feedback can be designed in order to achieve tracking of a desired trajectory up to a limited steady state tracking error. Unlike the 2 DOF scheme based on exact radiation damping cancellation, this last feedback controller does not require the explicit knowledge of the radiation damping field. For “high gain” we mean a ratio of around an order of magnitude between the actuation current and the current produced by the spin precession. Hence the task of radiation damping compensation can be performed in the soft pulse regime, meaning that real-time feedback makes sense in this context even with a single coil available. When strong pulses are instead considered, the transmitter/receiver ratio is several orders of magnitude higher, hence alternative designs such as, for example, an interleaved scheme of pulsing and measuring, should be used instead.
Section snippets
The model for radiation damping
In the following, we shall consider the model of radiation damping described e.g. in [6], [7], [8], [15], [18], focusing only on the spin 1/2 case. Further details concerning the model formulation are available in the Appendix.
Disregarding relaxation effects (i.e., when the relaxation time constants [1] tend to infinity: ) and denoting with the normalized Bloch vector, ( where is the equilibrium magnetization), the nonlinear Bloch equations for radiation damping in a
Feedback control strategies
For a coil aligned for instance with the laboratory axis, the measured NMR signal is a current which is generated by the electromotive force (emf) induced in the coil by the precessing magnetization and which oscillates with the spin resonance frequency . This may be superimposed with another emf due to the external driving, i.e., to the control input (soft pulses regime only). These two oscillating emfs (or, in the AC steady state, the two corresponding oscillating currents, see Appendix
Conclusion
As for many other aspects of the NMR literature, we find that also the methods developed for the purpose of suppressing radiation damping admit nontrivial control theoretical formulations. Part of the aim of this paper is to translate this problem and its solutions into language and techniques familiar to a control audience. In particular, we obtain that feedback control strategies can be classified into two types of methods: high gain feedback and 2 DOF controllers with a prefeedback exactly
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