Quantum Internal Model Principle: Decoherence Control

In this article, we study the problem of designing a Decoherence Control for quantum systems with the help of a scalable ancillary quantum control and techniques from geometric control theory, in order to successfully and completely decouple an open quantum system from its environment. We re-formulate the problem of decoherence control as a disturbance rejection scheme which also leads us to the idea of Internal Model Principle for quantum control systems which is first of its kind in the literature. It is shown that decoupling a quantum disturbance from an open quantum system, is possible only with the help of a quantum controller which takes into account the model of the environmental interaction. This is demonstrated for a simple 2-qubit system wherein the effects of decoherence are completely eliminated. The theory provides conditions to be imposed on the controller to ensure perfect decoupling. Hence the problem of decoherence control naturally gives rise to the quantum internal model principle which relates the disturbance rejecting control to the model of the environmental interaction. Classical internal model principle and disturbance decoupling focus on different aspects viz. perfect output tracking and complete decoupling of output from external disturbances respectively. However for quantum systems, the two problems come together and merge in order to produce an effective platform for decoherence control. In this article we introduce a seminal connection between disturbance decoupling and the corresponding analog for internal model principle for quantum systems.Comment: Submitted to IEEE Transactions on Automatic Control, Mar 15 2010. A basic introduction appeared in 46th IEEE CDC 2007. Acknowledgements: The authors would like to thank the Center for Quantum Information Science and Technology at Tsinghua University, R.-B. Wu, J. Zhang, J.-W. Wu, M. Jiang, C.-W. Li and G.-L. Long for their valuable comments and suggestion

Reliable H∞ filtering for discrete time-delay systems with randomly occurred nonlinearities via delay-partitioning method

The official published version can be found at the link below.In this paper, the reliable H∞ filtering problem is investigated for a class of uncertain discrete time-delay systems with randomly occurred nonlinearities (RONs) and sensor failures. RONs are introduced to model a class of sector-like nonlinearities that occur in a probabilistic way according to a Bernoulli distributed white sequence with a known conditional probability. The failures of sensors are quantified by a variable varying in a given interval. The time-varying delay is unknown with given lower and upper bounds. The aim of the addressed reliable H∞ filtering problem is to design a filter such that, for all possible sensor failures, RONs, time-delays as well as admissible parameter uncertainties, the filtering error dynamics is asymptotically mean-square stable and also achieves a prescribed H∞ performance level. Sufficient conditions for the existence of such a filter are obtained by using a new Lyapunov–Krasovskii functional and delay-partitioning technique. The filter gains are characterized in terms of the solution to a set of linear matrix inequalities (LMIs). A numerical example is given to demonstrate the effectiveness of the proposed design approach

Decoherence Control in Open Quantum System via Classical Feedback

In this work we propose a novel strategy using techniques from systems theory to completely eliminate decoherence and also provide conditions under which it can be done so. A novel construction employing an auxiliary system, the bait, which is instrumental to decoupling the system from the environment is presented. Our approach to decoherence control in contrast to other approaches in the literature involves the bilinear input affine model of quantum control system which lends itself to various techniques from classical control theory, but with non-trivial modifications to the quantum regime. The elegance of this approach yields interesting results on open loop decouplability and Decoherence Free Subspaces(DFS). Additionally, the feedback control of decoherence may be related to disturbance decoupling for classical input affine systems, which entails careful application of the methods by avoiding all the quantum mechanical pitfalls. In the process of calculating a suitable feedback the system has to be restructured due to its tensorial nature of interaction with the environment, which is unique to quantum systems. The results are qualitatively different and superior to the ones obtained via master equations. Finally, a methodology to synthesize feedback parameters itself is given, that technology permitting, could be implemented for practical 2-qubit systems to perform decoherence free Quantum Computing.Comment: 17 pages, 4 Fig

Feedback control of spin systems

The feedback stabilization problem for ensembles of coupled spin 1/2 systems is discussed from a control theoretic perspective. The noninvasive nature of the bulk measurement allows for a fully unitary and deterministic closed loop. The Lyapunov-based feedback design presented does not require spins that are selectively addressable. With this method, it is possible to obtain control inputs also for difficult tasks, like suppressing undesired couplings in identical spin systems.Comment: 16 pages, 15 figure

Fault detection filter design for linear systems

This dissertation considers residual generation for robust fault detection of linear systems with control inputs, unknown disturbances and possible faults. First, multi-objective fault detection problems such as $\mathscr{H_-}/ \mathscr{H_\infty}$, $\mathscr{H}_2/\mathscr{H_\infty}$ and $\mathscr{H_\infty}/\mathscr{H_\infty}$ have been formulated for linear continuous time-varying systems (LCTVS) in time domain for finite horizon and infinite horizon case, respectively. It is shown that under mild assumptions, the optimal solution is an observer determined by solving a standard differential Riccati equation (DRE). The solution is also extended to the case when the initial state for the system is unknown. Second, the parallel problems are also solved for linear discrete time-varying systems in time domain. The solution is again an observer whose gain is determined by solving a standard recursive difference Riccati equation (DDRE). The solution is also extended to the case when the initial state for the system is unknown. Third, for the general case in which $G_d$ (the transfer matrix from disturbance to output) may be a tall or square transfer matrix, and $D_d$ may not have full column rank for linear discrete time invariant systems (LDTIS), the common $\mathscr{H_-}/ \mathscr{H_\infty}$, $\mathscr{H}_2/\mathscr{H_\infty}$ and $\mathscr{H_\infty}/\mathscr{H_\infty}$ frameworks are not applicable. Based on several novel definitions of norms over a certain subspace, we propose a new problem formulation with both disturbance decoupling and fault sensitivity optimization. It is shown that the solution is an observer determined by a generalized Riccati equation (or Riccati system, alternatively). To be more specific, with this filter, some faults in certain subspace can be completely decoupled from the residual signal, while the others are optimized in terms of fault sensitivity. Furthermore, the completely non-decoupling and decoupling conditions are given. Disturbance rejection based on the solution is discussed. A direct procedure for deriving the fault detection filter in transfer matrix form is also proposed. Finally, some potential further research problems are outlined

H infinity control design for generalized second order systems based on acceleration sensitivity function

This article presents an Hinfinty control design method based on the Acceleration Sensitivity (AS) function. This approach can be applied to any fully actuated generalized second order system. In this framework, classical modal specifications(pulsations / damping ratios) are expressed in terms of Hinfinty templates allowing other frequency domain specifications to betaken into account. Finally, a comparison between AS with a more classical Hinfinty approach and with the Cross Standard Form(CSF) is presented. A 2 degrees of freedom spring-damper-mass academic example is used to illustrate the properties of the AS,though this method was developed and is used for atmospheric reentry control design

Robust H2/H∞-state estimation for discrete-time systems with error variance constraints

Copyright [1997] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This paper studies the problem of an H∞-norm and variance-constrained state estimator design for uncertain linear discrete-time systems. The system under consideration is subjected to time-invariant norm-bounded parameter uncertainties in both the state and measurement matrices. The problem addressed is the design of a gain-scheduled linear state estimator such that, for all admissible measurable uncertainties, the variance of the estimation error of each state is not more than the individual prespecified value, and the transfer function from disturbances to error state outputs satisfies the prespecified H∞-norm upper bound constraint, simultaneously. The conditions for the existence of desired estimators are obtained in terms of matrix inequalities, and the explicit expression of these estimators is also derived. A numerical example is provided to demonstrate various aspects of theoretical results