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Ian R. Petersen

Publications and source records attributed to Ian R. Petersen.

At least 235 records · Page 13Linked to original sources
arXiv2012

Robust Filtering for Adaptive Homodyne Estimation of Continuously Varying Optical Phase

Recently, it has been demonstrated experimentally that adaptive estimation of a continuously varying optical phase provides superior accuracy in the phase estimate compared to static estimation. Here, we show that the mean-square error in the adaptive phase estimate may be further reduced for the stochastic noise process considered by using an optimal Kalman filter in the feedback loop. Further, the estimation process can be made robust to fluctuations in the underlying parameters of the noise process modulating the system phase to be estimated. This has been done using a guaranteed cost robust filter.

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arXiv2012

Optimal Lyapunov-based quantum control for quantum systems

Quantum Lyapunov control was developed in order to transform a quantum system from arbitrary initial states to a target state. The idea is to find control fields that steer the Lyapunov function to zero as $t\rightarrow \infty$, meanwhile the quantum system is driven to the target state. In order to shorten the time required to reach the target state, we propose two designs to optimize Lyapunov control in this paper. The first design makes the Lyapunov function decrease as fast as possible with a constraint on the total power of control fields, and the second design has the same purpose but with a constraint on each control field. Examples of a three-level system demonstrate that the evolution time for Lyapunov control can be significantly shortened, especially when high control fidelity is required. Besides, this optimal Lyapunov-based quantum control is robust against uncertainties in the free Hamiltonian and decoherence in the system compared to conventional Lyapunov control.

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arXiv2012

Robust Stability of Quantum Systems with a Nonlinear Coupling Operator

This paper considers the problem of robust stability for a class of uncertain quantum systems subject to unknown perturbations in the system coupling operator. A general stability result is given for a class of perturbations to the system coupling operator. Then, the special case of a nominal linear quantum system is considered with non-linear perturbations to the system coupling operator. In this case, a robust stability condition is given in terms of a scaled strict bounded real condition.

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arXiv2012

Minimax Linear Quadratic Gaussian Control of Nonlinear MIMO System with Time Varying Uncertainties

In this paper, a robust nonlinear control scheme is proposed for a nonlinear multi-input multi-output (MIMO) system subject to bounded time varying uncertainty which satisfies a certain integral quadratic constraint condition. The scheme develops a robust feedback linarization approach which uses standard feedback linearization approach to linearize the nominal nonlinear dynamics of the uncertain nonlinear system and linearizes the nonlinear time varying uncertainties at an arbitrary point using the mean value theorem. This approach transforms uncertain nonlinear MIMO systems into an equivalent MIMO linear uncertain system model with unstructured uncertainty. Finally, a robust minimax linear quadratic Gaussian (LQG) control design is proposed for the linearized model. The scheme guarantees the internal stability of the closed loop system and provides robust performance. In order to illustrate the effectiveness of this approach, the proposed method is applied to a tracking control problem for an air-breathing hypersonic flight vehicle (AHFV).

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arXiv2012

Singular Perturbation Approximations for General Linear Quantum Systems

This paper considers the use of singular perturbation approximations for general linear quantum systems where the system dynamics are described in terms of both annihilation and creation operators. Results that are related to the physical realizability property of the approximate system are presented.

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arXiv2012

Minimum Relative Entropy State Transitions in Linear Stochastic Systems: the Continuous Time Case

This paper is concerned with a dissipativity theory for dynamical systems governed by linear Ito stochastic differential equations driven by random noise with an uncertain drift. The deviation of the noise from a standard Wiener process in the nominal model is quantified by relative entropy. We discuss a dissipation inequality for the noise relative entropy supply. The problem of minimizing the supply required to drive the system between given Gaussian state distributions over a specified time horizon is considered. This problem, known in the literature as the Schroedinger bridge, was treated previously in the context of reciprocal processes. A closed-form smooth solution is obtained for a Hamilton-Jacobi equation for the minimum required relative entropy supply by using nonlinear algebraic techniques.

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arXiv2012

State distributions and minimum relative entropy noise sequences in uncertain stochastic systems: the discrete time case

The paper is concerned with a dissipativity theory and robust performance analysis of discrete-time stochastic systems driven by a statistically uncertain random noise. The uncertainty is quantified by the conditional relative entropy of the actual probability law of the noise with respect to a nominal product measure corresponding to a white noise sequence. We discuss a balance equation, dissipation inequality and superadditivity property for the corresponding conditional relative entropy supply as a function of time. The problem of minimizing the supply required to drive the system between given state distributions over a specified time horizon is considered. Such variational problems, involving entropy and probabilistic boundary conditions, are known in the literature as Schroedinger bridge problems. In application to control systems, this minimum required conditional relative entropy supply characterizes the robustness of the system with respect to an uncertain noise. We obtain a dynamic programming Bellman equation for the minimum required conditional relative entropy supply and establish a Markov property of the worst-case noise with respect to the state of the system. For multivariable linear systems with a Gaussian white noise sequence as the nominal noise model and Gaussian initial and terminal state distributions, the minimum required supply is obtained using an algebraic Riccati equation which admits a closed-form solution. We propose a computable robustness index for such systems in the framework of an entropy theoretic formulation of uncertainty and provide an example to illustrate this approach.

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arXiv2012

Hardy-Schatten Norms of Systems, Output Energy Cumulants and Linear Quadro-Quartic Gaussian Control

This paper is concerned with linear stochastic control systems in state space. The integral of the squared norm of the system output over a bounded time interval is interpreted as energy. The cumulants of the output energy in the infinite-horizon limit are related to Schatten norms of the system in the Hardy space of transfer functions and the risk-sensitive performance index. We employ a novel performance criterion which seeks to minimize a combination of the average value and the variance of the output energy of the system per unit time. The resulting linear quadro-quartic Gaussian control problem involves the H2 and H4-norms of the closed-loop system. We obtain equations for the optimal controller and outline a homotopy method which reduces the solution of the problem to the numerical integration of a differential equation initialized by the standard linear quadratic Gaussian controller.

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arXiv2012

On the Physical Realizability of a Class of Nonlinear Quantum Systems

In this paper, the physical realizability property is investigated for a class of nonlinear quantum systems. This property determines whether a given set of nonlinear quantum stochastic differential equations corresponds to a physical nonlinear quantum system satisfying the laws of quantum mechanics.

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arXiv2012

Risk-sensitive Dissipativity of Linear Quantum Stochastic Systems under Lur'e Type Perturbations of Hamiltonians

This paper is concerned with a stochastic dissipativity theory using quadratic-exponential storage functions for open quantum systems with canonically commuting dynamic variables governed by quantum stochastic differential equations. The system is linearly coupled to external boson fields and has a quadratic Hamiltonian which is perturbed by nonquadratic functions of linear combinations of system variables. Such perturbations are similar to those in the classical Lur'e systems and make the quantum dynamics nonlinear. We study their effect on the quantum expectation of the exponential of a positive definite quadratic form of the system variables. This allows conditions to be established for the risk-sensitive stochastic storage function of the quantum system to remain bounded, thus securing boundedness for the moments of system variables of arbitrary order. These results employ a noncommutative analogue of the Doleans-Dade exponential and a multivariate partial differential version of the Gronwall-Bellman lemma.

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arXiv2012

Characterization and Moment Stability Analysis of Quasilinear Quantum Stochastic Systems with Quadratic Coupling to External Fields

The paper is concerned with open quantum systems whose Heisenberg dynamics are described by quantum stochastic differential equations driven by external boson fields. The system-field coupling operators are assumed to be quadratic polynomials of the system observables, with the latter satisfying canonical commutation relations. In combination with a cubic system Hamiltonian, this leads to a class of quasilinear quantum stochastic systems which retain algebraic closedness in the evolution of mixed moments of the observables. Although such a system is nonlinear and its quantum state is no longer Gaussian, the dynamics of the moments of any order are amenable to exact analysis, including the computation of their steady-state values. In particular, a generalized criterion is developed for quadratic stability of the quasilinear systems. The results of the paper are applicable to the generation of non-Gaussian quantum states with manageable moments and an optimal design of linear quantum controllers for quasilinear quantum plants.

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arXiv2012

Min-Plus Techniques for Set-Valued State Estimation

This article approaches deterministic filtering via an application of the min-plus linearity of the corresponding dynamic programming operator. This filter design method yields a set-valued state estimator for discrete-time nonlinear systems (nonlinear dynamics and output functions). The energy bounds in the process and the measurement disturbances are modeled using a sum quadratic constraint. The filtering problem is recast into an optimal control problem in the form of a Hamilton-Jacobi-Bellman (HJB) equation, the solution to which is obtained by employing the min-plus linearity property of the dynamic programming operator. This approach enables the solution to the HJB equation and the design of the filter without recourse to linearization of the system dynamics/ output equation.

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arXiv2012

Robust Stability of Uncertain Quantum Systems

This paper considers the problem of robust stability for a class of uncertain quantum systems subject to unknown perturbations in the system Hamiltonian. Some general stability results are given for different classes of perturbations to the system Hamiltonian. Then, the special case of a nominal linear quantum system is considered with either quadratic or non-quadratic perturbations to the system Hamiltonian. In this case, robust stability conditions are given in terms of strict bounded real conditions.

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arXiv2012

A Mean Value Theorem Approach to Robust Control Design for Uncertain Nonlinear Systems

This paper presents a scheme to design a tracking controller for a class of uncertain nonlinear systems using a robust feedback linearization approach. The scheme is composed of two steps. In the first step, a linearized uncertainty model for the corresponding uncertain nonlinear system is developed using a robust feedback linearization approach. In this step, the standard feedback linearization approach is used to linearize the nominal nonlinear dynamics of the uncertain nonlinear system. The remaining nonlinear uncertainties are then linearized at an arbitrary point using the mean value theorem. This approach gives a multi-input multi-output (MIMO) linear uncertain system model with a structured uncertainty representation. In the second step, a minimax linear quadratic regulation (LQR) controller is designed for MIMO linearized uncertain system model. In order to demonstrate the effectiveness of the proposed method, it is applied to a velocity and altitude tracking control problem for an air-breathing hypersonic flight vehicle.

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arXiv2012

Gaussian Stochastic Linearization for Open Quantum Systems Using Quadratic Approximation of Hamiltonians

This paper extends the energy-based version of the stochastic linearization method, known for classical nonlinear systems, to open quantum systems with canonically commuting dynamic variables governed by quantum stochastic differential equations with non-quadratic Hamiltonians. The linearization proceeds by approximating the actual Hamiltonian of the quantum system by a quadratic function of its observables which corresponds to the Hamiltonian of a quantum harmonic oscillator. This approximation is carried out in a mean square optimal sense with respect to a Gaussian reference quantum state and leads to a self-consistent linearization procedure where the mean vector and quantum covariance matrix of the system observables evolve in time according to the effective linear dynamics. We demonstrate the proposed Hamiltonian-based Gaussian linearization for the quantum Duffing oscillator whose Hamiltonian is a quadro-quartic polynomial of the momentum and position operators. The results of the paper are applicable to the design of suboptimal controllers and filters for nonlinear quantum systems.

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arXiv2011

Sliding Mode Control of Two-Level Quantum Systems

This paper proposes a robust control method based on sliding mode design for two-level quantum systems with bounded uncertainties. An eigenstate of the two-level quantum system is identified as a sliding mode. The objective is to design a control law to steer the system's state into the sliding mode domain and then maintain it in that domain when bounded uncertainties exist in the system Hamiltonian. We propose a controller design method using the Lyapunov methodology and periodic projective measurements. In particular, we give conditions for designing such a control law, which can guarantee the desired robustness in the presence of the uncertainties. The sliding mode control method has potential applications to quantum information processing with uncertainties.

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arXiv2011

A Minimax Linear Quadratic Gaussian Method for Antiwindup Control Synthesis

In this paper, a dynamic antiwindup compensator design is proposed which augments the main controller and guarantees robust performance in the event of input saturation. This is a two stage process in which first a robust optimal controller is designed for an uncertain linear system which guarantees the internal stability of the closed loop system and provides robust performance in the absence of input saturation. Then a minimax linear quadratic Gaussian (LQG) compensator is designed to guarantee the performance in certain domain of attraction, in the presence of input saturation. This antiwindup augmentation only comes into action when plant is subject to input saturation. In order to illustrate the effectiveness of this approach, the proposed method is applied to a tracking control problem for an air-breathing hypersonic flight vehicle (AHFV).

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