Showing papers by "Ian R. Petersen published in 2009"

Sliding mode control of quantum systems

Abstract: This paper proposes a new robust control method for quantum systems with uncertainties involving sliding mode control (SMC). SMC is a widely used approach in classical control theory and industrial applications. We show that SMC is also a useful method for robust control of quantum systems. In this paper, we define two specific classes of sliding modes (i.e. eigenstates and state subspaces) and propose two novel methods combining unitary control and periodic projective measurements for the design of quantum SMC systems. Two examples including a two-level system and a three-level system are presented to demonstrate the proposed SMC method. One of the main features of the proposed method is that the designed control laws can guarantee the desired control performance in the presence of uncertainties in the system Hamiltonian. This SMC approach provides a useful control theoretic tool for robust quantum information processing with uncertainties.

Coherent H ∞ control for a class of linear complex quantum systems

Abstract: This paper considers a coherent H∞ control problem for a class of linear quantum systems which can be defined by complex quantum stochastic differential equations in terms of annihilation operators only. For this class of quantum systems, a solution to the H∞ control problem can be obtained in terms of a pair of complex Riccati equations. The paper also considers complex versions of the Bounded Real Lemma, the Strict Bounded Real Lemma and the Lossless Bounded Real Lemma. For the class of quantum systems under consideration, the question of physical realizability is related to the Bounded Real and Lossless Bounded Real properties.

A Discrete-Time Robust Extended Kalman Filter for Uncertain Systems With Sum Quadratic Constraints

Abstract: This technical note outlines the formulation of a novel discrete-time robust extended Kalman filter for uncertain systems with uncertainties described in terms of sum quadratic constraints. The robust filter is an approximate set-valued state estimator which is robust in the sense that it can handle modeling uncertainties in addition to exogenous noise. Riccati and filter difference equations are obtained as an approximate solution to a reverse-time optimal control problem defining the set-valued state estimator. In order to obtain a solution to the set-valued state estimation problem, the discrete-time system dynamics are modeled backwards in time.

Frequency locking of an optical cavity using linear–quadratic Gaussian integral control

Abstract: We show that a systematic modern control technique such as linear–quadratic Gaussian (LQG) control can be applied to a problem in experimental quantum optics which has previously been addressed using traditional approaches to controller design. An LQG controller which includes integral action is synthesized to stabilize the frequency of the cavity to the laser frequency and to reject low frequency noise. The controller is successfully implemented in the laboratory using a dSpace digital signal processing board. One important advantage of the LQG technique is that it can be extended in a straightforward way to control systems with multiple measurements and multiple feedback loops. This work is expected to pave the way for extremely stable lasers with fluctuations approaching the quantum noise limit and which could be potentially used in a wide range of applications.

Local Mode Dependent Decentralized Stabilization of Uncertain Markovian Jump Large-Scale Systems

Abstract: This technical note is concerned with the robust stabilization of a class of stochastic large-scale systems governed by a finite state Markov process. Using the method of integral quadratic constraints, a sufficient condition is developed to design decentralized stabilizing controllers which use local system states and local system operation modes to produce local control inputs. The condition is given in terms of a set of rank constrained linear matrix inequalities. The theory is illustrated by an example.

Robust minimax optimal control of nonlinear uncertain systems using feedback linearization with application to hypersonic flight vehicles

Abstract: For a class of multi input and multi output nonlinear uncertain systems, a feedback linearization method combined with minimax linear quadratic regulator (LQR) control is proposed. The nominal nonlinear model is assumed to be feedback linearizable with full relative degree. The uncertainties are assumed to be parametric or parameterized and satisfy a certain integral quadratic constraint condition. The proposed method systematically incorporates the parametric uncertainty bound in the design procedure and gives a locally stabilized closed loop system. This procedure consists of feedback linearization of the nominal model and linearization of the remaining nonlinear uncertain terms with respect to each individual uncertainty at a local operating points. This two-stage linearization process, followed by a robust minimax LQR control design, provides a robustly stable closed loop system. To demonstrate the effectiveness of the proposed approach, an application study is provided on longitudinal stabilization of an air-breathing hypersonic flight vehicle in the presence of uncertainties.

Cascade cavity realization for a class of complex transfer functions arising in coherent quantum feedback control

Abstract: This paper presents a realization algorithm for a class of complex transfer functions corresponding to physically realizable complex linear quantum systems. The class of complex linear quantum systems under consideration includes interconnections of passive optical components such as cavities, beam-splitters, phase-shifters and interferometers. It is shown that for almost all quantum optical systems within this class, the corresponding transfer function can be realized as a cascade connection involving only cavities and phase-shifters.

Stability analysis of positive feedback interconnections of linear negative imaginary systems

Abstract: The paper is concerned with the stability analysis of positive feedback interconnections of negative imaginary systems. Firstly, a previously established Negative Imaginary Lemma is shown to remain true even if the transfer function has poles on the imaginary axis. This is achieved by suitably extending the definitions of negative imaginary transfer functions. Secondly, a necessary and sufficient condition is established for the internal stability of the feedback interconnection of negative imaginary systems. Moreover, some properties of negative imaginary transfer functions are developed. Finally, a numerical example is presented to illustrate the theory.

On the physical realizability of general linear Quantum Stochastic Differential Equations with complex coefficients

Abstract: This paper is concerned with the physical realizability of linear Quantum Stochastic Differential Equations (QSDEs). In a recent paper, James, Nurdin and Petersen gave necessary and sufficient conditions for the physical realizability of linear QSDEs with real coefficients. These conditions were algebraic conditions on the system state space matrices. In this paper, we first study general linear QSDEs with complex coefficients and explain how to relate them via unitary transformations to the class of linear QSDEs with real coefficients considered by James, Nurdin Petersen. We use this relation between linear QSDEs with real and complex coefficients to derive necessary and sufficient algebraic conditions for the physical realizability of the general complex linear QSDEs being considered. Then we prove the equivalence between these algebraic conditions and a condition that an associated linear system is (J, J')-unitary.

Robust $H^{\infty }$ Control of an Uncertain System Via a Stable Output Feedback Controller

Abstract: This technical note presents a new approach to the robust control of an uncertain system via a stable output feedback controller. The uncertain systems under consideration contain structured uncertainty described by integral quadratic constraints. The controller is designed to achieve absolute stabilization with a specified level of disturbance attenuation. The main result involves solving a state feedback version of the problem by solving an algebraic Riccati equation dependent on a set of scaling parameters. Then two further algebraic Riccati equations are solved, which depend on a further set of scaling parameters.

Coherent LQG control for a class of linear complex quantum systems

Abstract: This paper considers a coherent LQG control problem for a class of linear quantum systems which can be defined by complex quantum stochastic differential equations in terms of annihilation operators only. For this class of quantum systems, a method for designing an LQG controller which satisfies a physical realizability condition is developed. This result is related to a recent result on constructing strictly positive real LQG controllers. The main idea of the paper involves constructing a suitable cost function for the LQG problem such that a physically realizable controller is obtained. The resulting controller provides robust stability as well as optimal performance.

Robust Output Feedback Guaranteed Cost Control of Nonlinear Stochastic Uncertain Systems Via an IQC Approach

Abstract: This technical note presents a new approach to constructive output feedback robust nonlinear controller design based on the use of IQCs and minimax LQG control. The approach involves a class of controllers which include copies of the nonlinearities in the controller. The linear part of the controller is synthesized using minimax LQG control theory which is closely related to H infin control theory.

Stabilization of uncertain negative-imaginary systems via state-feedback control

Abstract: The controller synthesis problem of uncertain negative-imaginary systems has important engineering applications in, for example, lightly damped flexible structures with colocated position sensors and force actuators. This paper provides a systematic controller synthesis procedure via an LMI approach to construct a state-feedback internally stabilizing controller such that the nominal closed-loop system satisfies negative-imaginary properties and a DC gain condition. As a result of this, the closed-loop system can then be guaranteed to be robustly stable against uncertainties that are stable strictly negative-imaginary (e.g. unmodeled spill-over dynamics in a lightly damped flexible structure). An numerical example is given to show the usefulness of the proposed results.

A discrete-time robust extended Kalman filter

Abstract: A discrete-time robust extended Kalman filter (REKF) formulation for uncertain systems expressed in terms of a set-valued state estimator is described in this paper. The robust filter and Riccati equations are derived as an approximate solution to a reverse-time optimal control problem defining this set-valued state estimator. As presented, the uncertainties are modeled by a sum quadratic constraint (SQC) that takes into account both modeling uncertainties as well as uncertainties introduced from exogenous noise sources.

Homodyne locking of a squeezer.

Abstract: We report on the successful implementation of an approach to locking the frequencies of an optical parametric oscillator (OPO)-based squeezed-vacuum source and its driving laser. The technique allows the simultaneous measurement of the phase shifts induced by a cavity, which may be used for the purposes of frequency locking, as well as the simultaneous measurement of the sub-quantum-noise-limited (sub-QNL) phase quadrature output of the OPO. The homodyne locking technique is cheap, easy to implement, and has the distinct advantage that subsequent homodyne measurements are automatically phase locked. The homodyne locking technique is also unique in that it is a sub-QNL frequency discriminator.

On lossless negative imaginary systems

Abstract: The paper is concerned with the notion of lossless negative imaginary systems and their stabilization using a strictly negative imaginary controller through positive feedback. Firstly, some properties of lossless negative imaginary transfer functions are studied. Secondly, a Lossless Negative Imaginary Lemma is given which establishes conditions on matrices appearing in a minimal state-space realization that are necessary and sufficient for a transfer function to be lossless negative imaginary. Thirdly, a necessary and sufficient condition is developed for the stabilization of a lossless negative imaginary system by a strictly negative imaginary controller. Finally, a numerical example is presented to illustrate the theory.

Robust H∞ control of an uncertain system via a strict bounded real output feedback controller

Abstract: This paper presents a new approach to the robust H∞ control of an uncertain system via an output feedback controller that is both stable and has an H∞ norm strictly less than a specified value. The uncertain systems under consideration contain structured uncertainty described by integral quadratic constraints. The controller is designed to achieve absolute stabilization with a specified level of disturbance attenuation. The main result involves solving a state feedback version of the problem by solving an algebraic Riccati equation dependent on a set of scaling parameters. Then two further algebraic Riccati equations are solved, which depend on a further set of scaling parameters. The required controller is constructed from the Riccati solutions. Copyright © 2008 John Wiley & Sons, Ltd.

Uncertainty Modeling for Robust Multivariable Control Synthesis of Hypersonic Flight Vehicles

Abstract: The non-standard dynamic characteristics of air-breathing hypersonic flight vehicles (AHFVs) and model uncertainties make the flight control of these vehicles highly challenging. In addition to the significant interactions between engine and structural dynamics, the operating conditions and mass distribution vary significantly over the flight envelope. Even considering only a single operating condition, one needs to describe the subsystem interactions and uncertainties and incorporate them into the flight control design. Focusing on robust control approaches, the description of the subsystems involves developing effective uncertainty models for AHFVs. The uncertainty modeling studies reported in the literature so far are based on linearized dynamics and incorporate the uncertainty of each individual parameter in fictitious ways, ignoring the physical nature of the sources of the uncertainties. The focus of this paper is to derive uncertainty models for the longitudinal motion dynamics of AHFVs based on a nonlinear nominal model incorporating norm bounded parametric uncertainties, and to analyze how the uncertainty in each parameter enters into the system dynamics and how it affects the performance of the system. The technique used in developing uncertainty model for AHFVs is demonstrated to be systematic and is presented in a form suitable for the application of robust control design procedures for the stabilization of longitudinal dynamics of AHFVs.

μ analysis for interconnections of systems with negative imaginary frequency response

Abstract: Many physical systems are subject to real parametric uncertainty. A recently developed method gives exact parametric stability regions for systems with so called negative imaginary frequency response. This method is contrasted in this paper with μ analysis for the same system class. The two methods are shown to yield similar results for some simple SISO cases. Limitations of both methods are pointed out through a numerical example.

A robust gyroless attitude estimation scheme for a small fixed-wing unmanned aerial vehicle

Abstract: This paper proposes a backup attitude estimation scheme for small fixed-wing unmanned aerial vehicles (UAVs) in the event of gyroscopic failure. The attitude of the system is propagated in terms of 3 degrees-of-freedom (DoF) aircraft dynamics while a combined accelerometer and global positioning system (GPS) based attitude measurement scheme is used to update attitude errors. The uncertainties in the aircraft dynamics are modeled as norm-bound uncertainties and a discrete-time robust extended Kalman filter (REKF) is implemented to estimate the attitude of the UAV.

Local mode dependent output feedback control of uncertain Markovian jump large-scale systems

Abstract: This paper is concerned with the output feedback guaranteed cost control problem for a class of uncertain stochastic large-scale systems governed by a random parameter. The uncertainties are assumed to satisfy integral quadratic constraints, and the random parameter is a Markov process. A sufficient condition is established for the design of decentralized output feedback guaranteed cost controllers which use local system states and local system operation modes to produce local control inputs, and ensure suboptimal global quadratic performance. The condition is given in terms of a set of rank constrained linear matrix inequalities. A numerical example and simulations are also provided to illustrate the theory.

Lyapunov stability for Quantum Markov Processes

Abstract: In this paper we prove some Lyapunov stability results for Quantum systems. The evolution of open quantum systems can be described using a one parameter semigroup of completely positive operators with which we can associate a minimal quantum Markov dilation. Analogous to Lyapunov stability theorems of classical Markov processes, we develop Lyapunov stability theorems for minimal Markov dilations of quantum systems. This theory depends on a quantum version of Dynkin's formula.

Sliding mode control of quantum systems

Abstract: This paper proposes a new robust control method for quantum systems with uncertainties involving sliding mode control (SMC). Sliding mode control is a widely used approach in classical control theory and industrial applications. We show that SMC is also a useful method for robust control of quantum systems. In this paper, we define two specific classes of sliding modes (i.e., eigenstates and state subspaces) and propose two novel methods combining unitary control and periodic projective measurements for the design of quantum sliding mode control systems. Two examples including a two-level system and a three-level system are presented to demonstrate the proposed SMC method. One of main features of the proposed method is that the designed control laws can guarantee desired control performance in the presence of uncertainties in the system Hamiltonian. This sliding mode control approach provides a useful control theoretic tool for robust quantum information processing with uncertainties.

Feedback interconnection of open quantum systems: A small gain theorem

Abstract: This paper examines the stability of quantum feedback networks. We introduce a novel characterization, in terms of equivalence classes of operators, that may be used to describe open quantum systems. In this characterization, equivalence classes of operators are shown to be elements of a Banach space such that the norm of an operator is analogous to the root mean square expectation value of the operator. This characterization is ideally suited to describing quantum feedback systems and enables us to prove the uniqueness and existence of solutions for feedback systems. In this context, we prove a quantum version of the small-gain theorem that is a generalisation of the classical small gain theorem.

A pseudo-H ∞ control theory and its application to the control of pendulum-like nonlinear systems

Abstract: This paper studies the problem of control of pendulum-like nonlinear systems to achieve closed-loop Lagrange stability. A state feedback pseudo-H ∞ control theory is developed. In a similar fashion to the strict bounded real lemma in H ∞ control theory, a strict pseudo-bounded real lemma is established. A sufficient condition for the synthesis of the state feedback pseudo-H ∞ control is given, which is also applied to the control of pendulum-like nonlinear systems.