Showing papers by "Ian R. Petersen published in 2015"

Fast machine-learning online optimization of ultra-cold-atom experiments

Abstract: Machine-designed control of complex devices or experiments can discover strategies superior to those developed via simplified models. We describe an online optimization algorithm based on Gaussian processes and apply it to optimization of the production of Bose-Einstein condensates (BEC). BEC is typically created with an exponential evaporation ramp that is approximately optimal for s-wave, ergodic dynamics with two-body interactions and no other loss rates, but likely sub-optimal for many real experiments. Machine learning using a Gaussian process, in contrast, develops a statistical model of the relationship between the parameters it controls and the quality of the BEC produced. This is an online process, and an active one, as the Gaussian process model updates on the basis of each subsequent experiment and proposes a new set of parameters as a result. We demonstrate that the Gaussian process machine learner is able to discover a ramp that produces high quality BECs in 10 times fewer iterations than a previously used online optimization technique. Furthermore, we show the internal model developed can be used to determine which parameters are essential in BEC creation and which are unimportant, providing insight into the optimization process.

Robust manipulation of superconducting qubits in the presence of fluctuations

Abstract: Superconducting quantum systems are promising candidates for quantum information processing due to their scalability and design flexibility. However, the existence of defects, fluctuations, and inaccuracies is unavoidable for practical superconducting quantum circuits. In this paper, a sampling-based learning control (SLC) method is used to guide the design of control fields for manipulating superconducting quantum systems. Numerical results for one-qubit systems and coupled two-qubit systems show that the "smart" fields learned using the SLC method can achieve robust manipulation of superconducting qubits, even in the presence of large fluctuations and inaccuracies.

Performance of Sinusoidal Scanning With MPC in AFM Imaging

Abstract: An atomic force microscope (AFM) is an extremely versatile investigative tool in the field of nanotechnology, the performance of which is significantly influenced by its conventional zig-zag raster pattern scanning method. In this paper, in order to increase its imaging speed, we consider the use of a sinusoidal scanning method, i.e., a spiral scanning method with an improved model predictive control (MPC) scheme. In this approach, spirals are generated by applying waves, each with a single frequency and slowly varying amplitude, in the X-piezo (sine wave) and Y-piezo (cosine wave) of the piezoelectric tube scanner (PTS) of the AFM. As these input signals are single frequencies, the scanning can proceed faster than traditional raster scanning, without exciting the resonant mode of the PTS. The proposed MPC controller reduces the phase error between the reference position input and measured output sinusoids and provides better tracking of the reference signal. Also, a notch filter is designed and included in the feedback loop to suppress vibrations of the PTS at the resonant frequency. The experimental results show that, using the proposed method, the AFM is able to scan a 6 μm radius image within 2.04 s with a quality better than that obtained using the conventional raster pattern scanning method.

Robust Output Feedback Consensus for Networked Negative-Imaginary Systems

Abstract: A robust output feedback consensus problem for networked homogeneous negative-imaginary (NI) systems is investigated in this technical note. By virtue of NI systems theory, a set of reasonable yet elegant conditions are derived for output consensus under ${\Fraktur{L}}_{2}$ external disturbances as well as NI model uncertainty. As a byproduct, this technical note also reaffirms a previous result by Li et al. which shows the robustness of networked systems is always worse than that of single agent systems. Furthermore, the eventual convergence sets are also characterized for several special NI systems that are commonly studied in the literature. It is shown how the results in this work embed and generalize earlier results for these classes of systems. We show that the natural convergence set boils down to the centroid of the initial pattern when the initial conditions of the controllers are zero. Numerical examples are given to showcase the main results.

Damping controller design for nanopositioners: A mixed passivity, negative-imaginary, and small-gain approach

Abstract: A design of a damping controller to damp the first resonant mode of a piezoelectric tube scanner (PTS) used in most commercial atomic force microscopes (AFMs) is proposed in this study. The design of the controller is carried out by proposing a novel analytical framework. The analytical framework examines the finite-gain stability for a positive feedback interconnection between two stable linear time-invariant systems, where one system has mixed passivity, negative-imaginary, and small-gain properties and the other system has mixed negative-imaginary, negative-passivity, and small-gain properties. Experimental results are presented to show the effectiveness of the proposed analytical framework to design the proposed controller.

Sampling-Based Learning Control for Quantum Systems With Uncertainties

Abstract: Robust control design for quantum systems has been recognized as a key task in the development of practical quantum technology. In this paper, we present a systematic numerical methodology of sampling-based learning control (SLC) for control design of quantum systems with uncertainties. The SLC method includes two steps of training and testing. In the training step, an augmented system is constructed using artificial samples generated by sampling uncertainty parameters according to a given distribution. A gradient flow-based learning algorithm is developed to find the control for the augmented system. In the process of testing, a number of additional samples are tested to evaluate the control performance, where these samples are obtained through sampling the uncertainty parameters according to a possible distribution. The SLC method is applied to three significant examples of quantum robust control, including state preparation in a three-level quantum system, robust entanglement generation in a two-qubit superconducting circuit, and quantum entanglement control in a two-atom system interacting with a quantized field in a cavity. Numerical results demonstrate the effectiveness of the SLC approach even when uncertainties are quite large, and show its potential for robust control design of quantum systems.

Nonlinearity Effects Reduction of an AFM Piezoelectric Tube Scanner Using MIMO MPC

Abstract: The design of a controller which compensates for the effects of creep, hysteresis, vibration, and cross-coupling in a piezoelectric tube scanner (PTS) is presented in this paper. The PTS is a key nanopositioning component installed in a commercial atomic force microscope (AFM) to perform scanning. The impediments to fast scanning due to PTS dynamics are: 1) the presence of mechanical resonances; 2) nonlinearities due to the piezoelectric characteristics; and 3) the cross-coupling effect between ${x}$ - and ${y}$ -axes in the PTS. In this paper, a multi-input multi-output model predictive control (MPC) scheme is designed to counteract the effects of creep, hysteresis, vibration, and cross-coupling in a PTS. Also, a damping compensator is included to suppress the vibration effect at its resonance frequency. The proposed controller achieves a high closed-loop bandwidth and significant damping of the resonant mode. To evaluate the performance improvement using the proposed control scheme, an experimental comparison with the existing AFM proportional-integral (PI) controller and a single-input single-output (SISO) MPC is conducted. Enhancement in the scanning speed up to 125 Hz is observed with the proposed controller.

Multivariable Negative-Imaginary Controller Design for Damping and Cross Coupling Reduction of Nanopositioners: A Reference Model Matching Approach

Abstract: This paper presents the design and experimental implementation of a novel multi-input multi-output (MIMO) control structure using two negative-imaginary damping controllers to damp the first resonant mode and attenuate cross coupling effects between the axes of a piezoelectric tube scanner. The dynamics of the scanner in the lateral and longitudinal axes are identified from measured data and the design of the controller using a MIMO framework is done based on a reference model matching approach. The controller proposed in the paper is able to achieve a bandwidth near to the first resonance frequency of the scanner and the proposed controller is robust against changes in resonance frequencies that are due to load changes on the scanner. Experimental results are presented to show the effectiveness of the proposed controller.

A MIMO Double Resonant Controller Design for Nanopositioners

Abstract: A design of a double resonant controller to enhance the high-speed nanopositioning performance of a piezoelectric tube scanner (PTS) is presented in this paper. The design of the controller is demonstrated using a multi-input multi-output framework for damping, tracking, and cross coupling control in the PTS. A reference model control technique is applied to design the controller. The controller proposed in this paper achieves a bandwidth near to the first resonance frequency of the PTS. The controller is robust against changes in the resonance frequency of the PTS due to load change on the scanner. Experimental results using open-loop, closed-loop, and the built-in AFM proportional integral controller are presented to show the effectiveness of the proposed controller.

Effect of Improved Tracking for Atomic Force Microscope on Piezo Nonlinear Behavior

Abstract: Nanotechnology is an area of modern science which deals with the control of matter at dimensions of 100 nm or less. In recent years, of all the available microscopy techniques, atomic force microscopy (AFM) has proven to be extremely versatile as an investigative tool in this field. However the performance of AFM is significantly limited by the effects of hysteresis, creep, cross-coupling, and vibration in its scanning unit, the piezoelectric tube scanner (PTS). This article presents the design and experimental implementation of a single-input single-output (SISO) model predictive control (MPC) scheme with a vibration compensator which is based on an identified model of the PTS. The proposed controller provides an AFM with the capability to achieve improved tracking and results in compensation of the nonlinear effects. The experimental results, which compare the tracking performances of the proposed controller for different reference signals, highlight the usefulness of the proposed control scheme.

Physical interpretations of negative imaginary systems theory

Abstract: This paper presents some physical interpretations of recent stability results on the feedback interconnection of negative imaginary systems. These interpretations involve spring mass damper systems coupled together by springs or RLC electrical networks coupled together via inductors or capacitors.

A possible implementation of a direct coupling coherent quantum observer

Abstract: This paper considers the problem of implementing a previously proposed direct coupling quantum observer for a closed linear quantum system. This observer is shown to be able to estimate some but not all of the plant variables in a time averaged sense. The paper proposes a possible experimental implementation of the observer plant system using a non-degenerate parametric amplifier.

Quantum filtering for multiple diffusive and Poissonian measurements

Abstract: We provide a rigorous derivation of a quantum filter for the case of multiple measurements being made on a quantum system. We consider a class of measurement processes which are functions of bosonic field operators, including combinations of diffusive and Poissonian processes. This covers the standard cases from quantum optics, where homodyne detection may be described as a diffusive process and photon counting may be described as a Poissonian process. We obtain a necessary and sufficient condition for any pair of such measurements taken at different output channels to satisfy a commutation relationship. Then, we derive a general, multiple-measurement quantum filter as an extension of a single-measurement quantum filter. As an application we explicitly obtain the quantum filter corresponding to homodyne detection and photon counting at the output ports of a beam splitter.

Quantum filter for a class of non-Markovian quantum systems

Abstract: In this paper we present a Markovian representation approach to constructing quantum filters for a class of non-Markovian quantum systems disturbed by Lorentzian noise. An ancillary system is introduced to convert white noise into Lorentzian noise which is injected into a principal system via a direct interaction. The resulting dynamics of the principal system are non-Markovian, which are driven by the Lorentzian noise. By probing the principal system, a quantum filter for the augmented system can be derived from standard theory, where the conditional state of the principal system can be obtained by tracing out the ancillary system.

Fault Tolerant Filtering and Fault Detection for Quantum Systems Driven By Fields in Single Photon States

Abstract: The purpose of this paper is to solve a fault tolerant filtering and fault detection problem for a class of open quantum systems driven by a continuous-mode bosonic input field in single photon states when the systems are subject to stochastic faults. Optimal estimates of both the system observables and the fault process are simultaneously calculated and characterized by a set of coupled recursive quantum stochastic differential equations.

Sampling-based Learning Control for Quantum Systems with Uncertainties

Abstract: Robust control design for quantum systems has been recognized as a key task in the development of practical quantum technology. In this paper, we present a systematic numerical methodology of sampling-based learning control (SLC) for control design of quantum systems with uncertainties. The SLC method includes two steps of "training" and "testing". In the training step, an augmented system is constructed using artificial samples generated by sampling uncertainty parameters according to a given distribution. A gradient flow based learning algorithm is developed to find the control for the augmented system. In the process of testing, a number of additional samples are tested to evaluate the control performance where these samples are obtained through sampling the uncertainty parameters according to a possible distribution. The SLC method is applied to three significant examples of quantum robust control including state preparation in a three-level quantum system, robust entanglement generation in a two-qubit superconducting circuit and quantum entanglement control in a two-atom system interacting with a quantized field in a cavity. Numerical results demonstrate the effectiveness of the SLC approach even when uncertainties are quite large, and show its potential for robust control design of quantum systems.

Time averaged consensus in a direct coupled coherent quantum observer network for a single qubit finite level quantum system

Abstract: This paper considers the problem of constructing a direct coupled quantum observer network for a single qubit quantum system. The proposed observer consists of a network of quantum harmonic oscillators and it is shown that the observer network output converges to a consensus in a time averaged sense in which each component of the observer estimates a specified output of the quantum plant.

A Possible Implementation of a Direct Coupling Coherent Quantum Observer

Abstract: This paper considers the problem of implementing a previously proposed direct coupling quantum observer for a closed linear quantum system. This observer is shown to be able to estimate some but not all of the plant variables in a time averaged sense. The paper proposes a possible experimental implementation of the observer plant system using a non-degenerate parametric amplifier.

Synthesis of Linear Coherent Quantum Control Systems Using A Differential Evolution Algorithm

Abstract: We propose a new method to construct an optimal linear coherent quantum controller based on an evolutionary optimization method, namely a differential evolution algorithm. The aim is to provide a straightforward approach to deal with both nonlinear and nonconvex constraints arising in the coherent quantum controller synthesis. The solution to this control problem involves a complex algebraic Riccati equation, which corresponds to a physical realizability condition for the coherent quantum controller. The proposed method is demonstrated through an example of an entanglement control problem for a quantum network comprising two cascaded optical parametric amplifiers.

Witnessing the boundary between Markovian and non-Markovian quantum dynamics: a Green's function approach

Abstract: This paper presents a Green's function-based root locus method to investigate the boundary between Markovian and non-Markovian open quantum systems in the frequency domain. A Langevin equation for the boson-boson coupling system is derived, where we show that the structure of the Green's function dominates the system dynamics. In addition, by increasing the coupling between the system and its environment, the system dynamics are driven from Markovian to non-Markovian dynamics, which results from the redistribution in the modes of the Green's function in the frequency domain. Both a critical transition and a critical point condition under Lorentzian noise are graphically presented using a root locus method. Related results are verified using an example of a boson-boson coupling system.

Improved mirror position estimation using resonant quantum smoothing

Abstract: Quantum parameter estimation, the ability to precisely obtain a classical value in a quantum system, is very important to many key quantum technologies. Many of these technologies rely on an optical probe, either coherent or squeezed states to make a precise measurement of a parameter ultimately limited by quantum mechanics. We use this technique to theoretically model, simulate and validate by experiment the measurement and precise estimation of the position of a cavity mirror. In non-resonant systems, the achieved estimation enhancement from quantum smoothing over optimal filtering has not exceeded a factor two, even when squeezed state probes were used. Using a coherent state probe, we show that using quantum smoothing on a mechanically resonant structure driven by a resonant forcing function can result significantly greater improvement in parameter estimation than with non-resonant systems. In this work, we show that it is possible to achieve a smoothing improvement by a factor in excess of three times over optimal filtering. By using intra-cavity light as the probe we obtain finer precision than has been achieved with the equivalent quantum resources in free-space.

Quantum network reduced-state synchronization part I-convergence under directed interactions

Abstract: We consider reduced-state synchronization of qubit networks with the aim of driving the qubits' reduced states to a common trajectory. The evolution of the quantum network's state is described by a master equation, where the network Hamiltonian is either a direct sum or a tensor product of identical qubit Hamiltonians, and the coupling terms are given by a set of permutation operators over the network. The permutations introduce naturally quantum directed interactions. This part of the paper focuses on convergence conditions. We show that reduced-state synchronization is achieved if and only if the quantum permutations form a strongly connected union graph. The proof is based on an algebraic analysis making use of the Perron-Frobenius theorem for non-negative matrices. The convergence rate and the limiting orbit are explicitly characterized. Numerical examples are provided illustrating the obtained results.

Quantum network reduced-state synchronization part II-the missing symmetry and switching interactions

Abstract: We consider reduced-state synchronization of qubit networks with the aim of driving the qubits' reduced states to a common trajectory. The evolution of the quantum network's state is described by a master equation, where the network Hamiltonian is either a direct sum or a tensor product of identical qubit Hamiltonians, and the coupling terms are given by a set of permutation operators over the network. The permutations introduce naturally quantum directed interactions. Part I of the paper establishes synchronization conditions for fixed quantum interactions. In this part of the paper, we further investigate the missing symmetry in the reduced-state synchronization from a graphical point of view. The information-flow hierarchy in quantum permutation operators is characterized by different layers of information-induced graphs, based on which a clear bridge between quantum and classical consensus dynamics is built. We show that the quantum synchronization equation is by nature equivalent to a cut-balanced consensus process. Then a necessary and sufficient condition is obtained for reaching quantum reduced-state synchronization in light of recent work by Hendrickx and Tsitsiklis [19].

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

Abstract: This paper is concerned with dissipativity theory and robust performance analysis and design of discrete-time stochastic systems driven by 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 Schrodinger bridge problems. In application to control systems, the minimum required conditional relative entropy supply characterizes the robustness of the system with respect to a statistically uncertain random noise. We o...

A gradient descent approach to optimal coherent quantum LQG controller design

Abstract: This paper is concerned with the Coherent Quantum Linear Quadratic Gaussian (CQLQG) control problem of finding a stabilizing measurement-free quantum controller for a quantum plant so as to minimize an infinite-horizon mean square performance index for the fully quantum closed-loop system In comparison with the observation-actuation structure of classical controllers, the coherent quantum feedback is less invasive to the quantum dynamics and quantum information Both the plant and the controller are open quantum systems whose dynamic variables satisfy the canonical commutation relations (CCRs) of a quantum harmonic oscillator and are governed by linear quantum stochastic differential equations (QSDEs) In order to correspond to such oscillators, these QSDEs must satisfy physical realizability (PR) conditions, which are organised as quadratic constraints on the controller matrices and reflect the preservation of CCRs in time The CQLQG problem is a constrained optimization problem for the steady-state quantum covariance matrix of the plant-controller system satisfying an algebraic Lyapunov equation We propose a gradient descent algorithm equipped with adaptive stepsize selection for the numerical solution of the problem The algorithm finds a local minimum of the LQG cost over the parameters of the Hamiltonian and coupling operators of a stabilizing PR quantum controller, thus taking the PR constraints into account A convergence analysis of the proposed algorithm is presented

Robust adaptive quantum phase estimation

Abstract: Quantum parameter estimation is central to many fields such as quantum computation, communications and metrology. Optimal estimation theory has been instrumental in achieving the best accuracy in quantum parameter estimation, which is possible when we have very precise knowledge of and control over the model. However, uncertainties in key parameters underlying the system are unavoidable and may impact the quality of the estimate. We show here how quantum optical phase estimation of a squeezed state of light exhibits improvement when using a robust fixed-interval smoother designed with uncertainties explicitly introduced in parameters underlying the phase noise.

Time averaged consensus in a direct coupled distributed coherent quantum observer

Abstract: This paper considers the problem of constructing a distributed direct coupling quantum observer for a closed linear quantum system. The proposed distributed observer consists of a network of quantum harmonic oscillators and it is shown that the distributed observer converges to a consensus in a time averaged sense in which each component of the observer estimates the specified output of the quantum plant.

Robust feedback stability of negative imaginary systems: An integral quadratic constraint approach

Abstract: Sufficient conditions for stability of feedback interconnections of negative imaginary systems are derived via an integral quadratic constraint (IQC) approach. These extend existing results in the literature by exploiting the flexibility present at the static and infinite frequencies to reduce conservatism. Negative imaginary transfer functions with poles on the imaginary axis are accommodated using a recently generalised IQC-based robustness result. In particular, the negative imaginary property of systems is shown to give rise to IQCs on positive frequencies that are bounded away from zero and infinity. Additional quadratic conditions are introduced to take care of the IQCs near the DC and instantaneous gains of the systems.