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IEEE Transactions on Quantum Engineering

Robust $H_{\infty }$ Uncertainties-Tolerant Observer-Based Reference Quantum Trajectory Tracking Control for Lindblad Master Equation

Shin-Yi Wen, Bor-Sen Chen, Chun-Liang Lin

Published Dec 18, 2025• 1 citation
Publisher

Abstract

Robust output feedback reference quantum trajectory tracking control design is proposed by the simultaneous continuous weak measurement of non-commuting observables. Using the robust$H_\infty$uncertainties-tolerant observer-based reference quantum trajectory tracking control (UTOBRQTTC) design strategy, the proposed method can robustly estimate the quantum trajectory and robustly track a sequence of any reference quantum states against undesired uncertainties and potential unavailable fault signals. Smoothed signal models are embedded into the augmented bilinear quantum system derived from Lindblad master equation. With the regression of unavailable system and sensor fault signal by smoothed models, the proposed$H_\infty$UTOBRQTTC design of the augmented bilinear quantum system can proactively compensate for the corruption of fault signals. Therefore, robust quantum trajectory estimation and reference quantum trajectory tracking can be achieved simultaneously via the proposed robust$H_\infty$UTOBRQTTC design strategy. Furthermore, the nonlinear Hamilton-Jacobi inequality (HJI)-constrained optimization problem of optimal robust$H_\infty$UTOBRQTTC design strategy can be treated as an LMIs-constrained optimization problem by the upper bound of spectral radius of augmented bilinear quantum system and the proposed two-step procedure, which can be efficiently solved with the help of MATLAB LMI TOOLBOX. Finally, several simulation examples of two-level bilinear quantum systems represented by the Lindblad master equation are provided to demonstrate the estimation performance of quantum trajectory and fault signals and any arbitrary signal tracking performance for more practical applications of bilinear quantum systems.

DOI
10.1109/tqe.2025.3645732

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Robust H∞H_{\infty }H∞​ Uncertainties-Tolerant Observer-Based Reference Quantum Trajectory Tracking Control for Lindblad Master Equation
Robust output feedback reference quantum trajectory tracking control design is proposed by the simultaneous continuous weak measurement of non-commuting observables. Using the robustH∞H_\inftyH∞​uncertainties-tolerant observer-based reference quantum trajectory tracking control (UTOBRQTTC) design strategy, the proposed method can robustly estimate the quantum trajectory and robustly track a sequence of any reference quantum states against undesired uncertainties and potential unavailable fault signals. Smoothed signal models are embedded into the augmented bilinear quantum system derived from Lindblad master equation. With the regression of unavailable system and sensor fault signal by smoothed models, the proposedH∞H_\inftyH∞​UTOBRQTTC design of the augmented bilinear quantum system can proactively compensate for the corruption of fault signals. Therefore, robust quantum trajectory estimation and reference quantum trajectory tracking can be achieved simultaneously via the proposed robustH∞H_\inftyH∞​UTOBRQTTC design strategy. Furthermore, the nonlinear Hamilton-Jacobi inequality (HJI)-constrained optimization problem of optimal robustH∞H_\inftyH∞​UTOBRQTTC design strategy can be treated as an LMIs-constrained optimization problem by the upper bound of spectral radius of augmented bilinear quantum system and the proposed two-step procedure, which can be efficiently solved with the help of MATLAB LMI TOOLBOX. Finally, several simulation examples of two-level bilinear quantum systems represented by the Lindblad master equation are provided to demonstrate the estimation performance of quantum trajectory and fault signals and any arbitrary signal tracking performance for more practical applications of bilinear quantum systems.
In quantum engineering, faults may occur in a quantum control system, which will cause the quantum control system unstable or deteriorate other relevant performance of the system. This note presents an estimator-based fault-tolerant control design approach for a class of linear quantum stochastic systems subject to fault signals. In this approach, the fault signals and some commutative components of the quantum system observables are estimated, and a fault-tolerant controller is designed to compensate the effect of the fault signals. Numerical procedures are developed for controller design and an example is presented to demonstrate the proposed design approach.
In quantum engineering, faults may occur in a quantum control system, which will cause the quantum control system unstable or deteriorate other relevant performance of the system. This note presents an estimator-based fault-tolerant control design approach for a class of linear quantum stochastic systems subject to fault signals. In this approach, the fault signals and some commutative components of the quantum system observables are estimated, and a fault-tolerant controller is designed to compensate the effect of the fault signals. Numerical procedures are developed for controller design and an example is presented to demonstrate the proposed design approach.
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.
In this study, in order to generate a sequence of desired quantum states (or quantum bits) for quantum communication and computation, it is more appealing to formulate a quantum control system as a bilinear state reference tracking system.An optimal tracking control is proposed to achieve the state-tracking by solving a Hamilton-Jacobi equation (HJE).In order to avoid the difficulty in solving the HJE with a closed-form solution, the technique of formal tensor power series is employed to treat with the HJE to obtain the optimal tracking control in quantum systems from the approximate design perspective.If the quantum system suffers from stochastic parameter variations, it could be modeled as state-dependent noise.In the situation, stochastic optimal tracking control design is also developed for quantum systems.Finally, several examples are given to illustrate the design procedure and to confirm the performance of the proposed tracking control method.
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.
In this paper, a nonlinear optimal feedback control based on the nonlinear state estimator for a two-level open non-Markovian stochastic quantum system is proposed. The proposed nonlinear state estimator is designed by using the state-dependent differential Riccati equation and constructed to optimally estimate the state. The estimated state is updated by the output data of continuous weak measurement of the controlled quantum system and used to design the nonlinear state feedback controller to track the output of the reference model. The output state of reference model is chosen as the desired performance. The numerical simulation results verify the achievability of the proposed state feedback control strategy, and the capability to steer the state of system from any arbitrary initial state to the final reference target state with high state transfer success rate.