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IEEE Transactions on Automatic Control

Fault-Tolerant Coherent $H^\infty$ Control for Linear Quantum Systems

Yanan Liu, Daoyi Dong, Ian R. Petersen, Qing Zhong Gao, Steven X. Ding, Shota Yokoyama, Hidehiro Yonezawa

Published Sep 28, 2021• 17 citations
Publisher

Abstract

Robustness and reliability are two key requirements for developing practical quantum control systems. The purpose of this article is to design a coherent feedback controller for a class of linear quantum systems suffering from Markovian jumping faults so that the closed-loop quantum system has both fault tolerance and$H^\infty$disturbance attenuation performance. This article first extends the physical realization conditions from the time-invariant case to the time-varying case for linear stochastic quantum systems. By relating the fault-tolerant$H^\infty$control problem to the dissipation properties and the solutions of Riccati differential equations, an$H^\infty$controller for the quantum system is then designed by solving a set of linear matrix inequalities. In particular, an algorithm is employed to introduce additional quantum inputs and to construct the corresponding input matrices to ensure the physical realizability of the quantum controller. Also, we propose a real application of the developed fault-tolerant control strategy to quantum optical systems. A linear quantum system example from quantum optics, where the amplitude of the pumping field randomly jumps among different values due to the fault processes, can be modeled as a linear Markovian jumping system. It is demonstrated that a quantum$H^\infty$controller can be designed and implemented using some basic optical components to achieve the desired control goal for this class of systems.

DOI
10.1109/tac.2021.3115843

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Fault-Tolerant Coherent H∞H^\inftyH∞ Control for Linear Quantum Systems
Robustness and reliability are two key requirements for developing practical quantum control systems. The purpose of this article is to design a coherent feedback controller for a class of linear quantum systems suffering from Markovian jumping faults so that the closed-loop quantum system has both fault tolerance andH∞H^\inftyH∞disturbance attenuation performance. This article first extends the physical realization conditions from the time-invariant case to the time-varying case for linear stochastic quantum systems. By relating the fault-tolerantH∞H^\inftyH∞control problem to the dissipation properties and the solutions of Riccati differential equations, anH∞H^\inftyH∞controller for the quantum system is then designed by solving a set of linear matrix inequalities. In particular, an algorithm is employed to introduce additional quantum inputs and to construct the corresponding input matrices to ensure the physical realizability of the quantum controller. Also, we propose a real application of the developed fault-tolerant control strategy to quantum optical systems. A linear quantum system example from quantum optics, where the amplitude of the pumping field randomly jumps among different values due to the fault processes, can be modeled as a linear Markovian jumping system. It is demonstrated that a quantumH∞H^\inftyH∞controller can be designed and implemented using some basic optical components to achieve the desired control goal for this class of systems.
Robustness and reliability are two key requirements for developing practical quantum control systems. The purpose of this paper is to design a coherent feedback controller for a class of linear quantum systems suffering from Markovian jumping faults so that the closed-loop quantum system has both fault tolerance and H-infinity disturbance attenuation performance. This paper first extends the physical realization conditions from the time-invariant case to the time-varying case for linear stochastic quantum systems. By relating the fault tolerant H-infinity control problem to the dissipation properties and the solutions of Riccati differential equations, an H-infinity controller for the quantum system is then designed by solving a set of linear matrix inequalities (LMIs). In particular, an algorithm is employed to introduce additional noises and to construct the corresponding input matrices to ensure the physical realizability of the quantum controller. For real applications of the developed fault-tolerant control strategy, we present a linear quantum system example from quantum optics, where the amplitude of the pumping field randomly jumps among different values. It is demonstrated that a quantum H-infinity controller can be designed and implemented using some basic optical components to achieve the desired control goal.
Coherent feedback H∞H^\inftyH∞ control of quantum linear systems
The purpose of this paper is to investigate the coherent feedback H∞H^\inftyH∞ control problem for linear quantum systems. A key contribution is a simplified design methodology that guarantees closed-loop stability and a prescribed level of disturbance attenuation. It is shown that for general linear quantum systems, a physically realizable quantum controller can be obtained by solving at most four Lyapunov equations. In the passive case, a necessary and sufficient condition is provided in terms of two uncoupled pairs of Lyapunov equations. These results represent a significant simplification over the standard approach, which requires solving two coupled algebraic Riccati equations. The effectiveness of the proposed method is demonstrated through two typical quantum optical devices: an empty optical cavity and a degenerate parametric amplifier. These results provide a computationally efficient procedure for the robust and optimal control of quantum optical and optomechanical systems.
The purpose of this paper is to design a coherent feedback controller for a Markovian jump linear quantum system suffering from a fault signal. The control objective is to bound the effect of the disturbance input on the output for the time-varying quantum system. We prove the relation between the H∞ control problem, the dissipation properties, and the solutions of Riccati differential equations, by which the H∞ controller of the Markovian jump linear quantum system is given by the solutions of Linear Matrix Inequalities (LMIs).
Coherent H∞H^{\infty }H∞ Control for a Class of Annihilation Operator Linear Quantum Systems
This paper considers the coherent robust H∞control problem for a class of linear quantum stochastic systems, whose models can be described by the quantum stochastic differential equations in terms of annihilation operators only, where the unknown disturbances from environment and the model uncertainties in the Hamiltonian and coupling operators exist in the system dynamics. We provide a framework to deal with this class of systems, where the robust H∞control problem of the original system is first transformed into that of a scaled system via the quantum bounded real lemma, and then an H∞controller is designed to attenuate the influence of the disturbances on the performance output. In the design of the H∞controller, a pair of algebra Riccati equations are used to obtain a numerical solution of system matrices in the controller. Finally, the simulation experiments on a one-dimensional harmonic oscillator system coupled to three quantum fields verify the effectiveness of the proposed method.
H∞H^{\infty}H∞ Control of Linear Quantum Stochastic Systems
The purpose of this paper is to formulate and solve aHinfincontroller synthesis problem for a class of noncommutative linear stochastic systems which includes many examples of interest in quantum technology. The paper includes results on the class of such systems for which the quantum commutation relations are preserved (such a requirement must be satisfied in a physical quantum system). A quantum version of standard (classical) dissipativity results are presented and from this a quantum version of the strict bounded real lemma is derived. This enables a quantum version of the two Riccati solution to theHinfincontrol problem to be presented. This result leads to controllers which may be realized using purely quantum, purely classical or a mixture of quantum and classical elements. This issue of physical realizability of the controller is examined in detail, and necessary and sufficient conditions are given. Our results are constructive in the sense that we provide explicit formulas for the Hamiltonian function and coupling operator corresponding to the controller.