Feedback for physicists: A tutorial essay on control
TL;DR: In this paper, a tutorial essay aims to give enough of the formal elements of control theory to satisfy the experimentalist designing or running a typical physics experiment and enough to satisfy a theorist wishing to understand its broader intellectual context.
Abstract: Feedback and control theory are important ideas that should form part of the education of a physicist but rarely do. This tutorial essay aims to give enough of the formal elements of control theory to satisfy the experimentalist designing or running a typical physics experiment and enough to satisfy the theorist wishing to understand its broader intellectual context. The level is generally simple, although more advanced methods are also introduced. Several types of applications are discussed, as the practical uses of feedback extend far beyond the simple regulation problems where it is most often employed. Sketches are then provided of some of the broader implications and applications of control theory, especially in biology, which are topics of active research.
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TL;DR: In this article, the authors characterize the relevant experimental parameters and compare to theory the observed power spectra and variance for a particle in a virtual harmonic potential, and show that deviations from the dynamics expected of a continuous potential are measured by the ratio of these small time scales to the relaxation time scale of the virtual potential.
Abstract: Feedback traps can create arbitrary virtual potentials for exploring the dynamics of small Brownian particles. In a feedback trap, the particle position is measured periodically and, after each measurement, one applies the force that would be produced by the gradient of the “virtual potential,” at the particle location. Virtual potentials differ from real ones in that the feedback loop introduces dynamical effects not present in ordinary potentials. These dynamical effects are caused by small time scales associated with the feedback, including the delay between the measurement of a particle’s position and the feedback response, the feedback response that is applied for a finite update time, and the finite camera exposure from integrating motion. Here, we characterize the relevant experimental parameters and compare to theory the observed power spectra and variance for a particle in a virtual harmonic potential. We show that deviations from the dynamics expected of a continuous potential are measured by the ratio of these small time scales to the relaxation time scale of the virtual potential.
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Posted Content•
TL;DR: In this article, a quantum sensor for measuring non-magnetic spin-dependent interactions is presented, which utilizes spin-exchange pumped comagnetometers, which are continuously polarized transverse to a pulsed bias field.
Abstract: We demonstrate a novel quantum sensor for measuring non-magnetic spin-dependent interactions. This sensor utilizes $^{131}$Xe, $^{129}$Xe, and $^{85}$Rb which are continuously polarized transverse to a pulsed bias field. The transverse geometry of this spin-exchange pumped comagnetometer suppresses longitudinal polarization, which is an important source of systematic error. Simultaneous excitation of both Xe isotopes is accomplished by frequency modulating the repetition rate of the bias field pulses at subharmonics of the Xe Larmor resonance frequencies. The area of each bias pulse causes $2\pi$ Larmor precession of the Rb. We present continuous dual-species Xe excitation and discuss a temperature-dependent wall interaction that limits the $^{129}$Xe polarization. The Rb atoms serve as an embedded magnetometer for detection of the Xe precession. We discuss Rb magnetometer phase shifts, and show that even first-order treatments of these phase shifts can result in order-of-magnitude improvements in the achieved field suppression when performing comagnetometry. The sensing bandwidth of the presented device is 1 Hz, and we demonstrate a white-noise level of 7 $\mu$Hz/$\sqrt{\text{Hz}}$ and a bias instability of $\sim1$ $\mu$Hz.
8 citations
Additional excerpts
...We stabilize the drive frequency of each isotope by servoing [37] the detected phase δ̃ ∓ ̃z....
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TL;DR: In this paper, the authors present a method to facilitate Monte Carlo simulations in the grand canonical ensemble given a target mean particle number, which imposes a fictitious dynamics on the chemical potential, to be run concurrently with the Monte Carlo sampling of the physical system.
Abstract: We present a method to facilitate Monte Carlo simulations in the grand canonical ensemble given a target mean particle number. The method imposes a fictitious dynamics on the chemical potential, to be run concurrently with the Monte Carlo sampling of the physical system. Corrections to the chemical potential are made according to time-averaged estimates of the mean and variance of the particle number, with the latter being proportional to thermodynamic compressibility. We perform a variety of tests, and in all cases find rapid convergence of the chemical potential-inexactness of the tuning algorithm contributes only a minor part of the total measurement error for realistic simulations.
8 citations
01 Jan 2017
TL;DR: Garrett et al. as mentioned in this paper used the Kelvin probe force microscopy (KPFM) to determine the e ect of patch potentials on both the sphere and the plate, but the force calculated from the patch potential was found to be much less than the measured force.
Abstract: Title of dissertation: THE EFFECTS OF GEOMETRY AND PATCH POTENTIALS ON CASIMIR FORCE MEASUREMENTS Joseph Landon Garrett, Doctor of Philosophy, 2017 Dissertation directed by: Professor Jeremy Munday Department of Electrical and Computer Engineering Electromagnetic uctuations of the quantum vacuum cause an attractive force between surfaces, called the Casimir force. In this dissertation, the rst Casimir force measurements between two gold-coated spheres are presented. The proximity force approximation (PFA) is typically used to compare experiment to theory, but it is known to deviate from the exact calculation far from the surface. Bounds are put on the size of possible deviations from the PFA by combining several sphere-sphere and sphere-plate measurements. Electrostatic patch potentials have been postulated as a possible source of error since the rst Casimir force measurements sixty years ago. Over the past decade, several theoretical models have been developed to characterize how the patch potentials contribute an additional force to the measurements. In this dissertation, Kelvin probe force microscopy (KPFM) is used to determine the e ect of patch potentials on both the sphere and the plate. Patch potentials are indeed present on both surfaces, but the force calculated from the patch potentials is found to be much less than the measured force. In order to better understand how KPFM resolves patch potentials, the artifacts and sensitivities of several di erent KPFM implementations are tested and characterized. In addition, we introduce a new technique, called tunable spatial resolution (TSR) KPFM, to control resolution by altering the power-law separation dependence of the KPFM signal. THE EFFECTS OF GEOMETRY AND PATCH POTENTIALS ON CASIMIR FORCE MEASUREMENTS
8 citations
Cites background from "Feedback for physicists: A tutorial..."
...By further optimizing the feedback loops the bandwidth might be increased [113,115,116]....
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TL;DR: From the data, it is clear that the underlying system exerts an important constraint on what are normally viewed as individual-level decisions, and calls on extending efforts to understand the role of system science in studies of child welfare systems, with a particular emphasis on therole of feedback as a causal influence.
Abstract: Although system is a word frequently invoked in discussions of foster care policy and practice, there have been few if any attempts by child welfare researchers to understand the ways in which the foster care system is a system. As a consequence, insights from system science have yet to be applied in meaningful ways to the problem of making foster care systems more effective. In this study, we draw on population biology to organize a study of admissions and discharges to foster care over a 15-year period. We are interested specifically in whether resource constraints, which are conceptualized here as the number of beds, lead to a coupling of admissions and discharges within congregate care. The results, which are descriptive in nature, are consistent with theory that ties admissions and discharges together because of a resource constraint. From the data, it is clear that the underlying system exerts an important constraint on what are normally viewed as individual-level decisions. Our discussion calls on extending efforts to understand the role of system science in studies of child welfare systems, with a particular emphasis on the role of feedback as a causal influence.
8 citations
Cites background from "Feedback for physicists: A tutorial..."
...In systems of this type, feedback plays an important part in the causal mechanisms that regulate how the system evolves over time [34]....
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References
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Book•
01 Jan 1991TL;DR: The author examines the role of entropy, inequality, and randomness in the design of codes and the construction of codes in the rapidly changing environment.
Abstract: Preface to the Second Edition. Preface to the First Edition. Acknowledgments for the Second Edition. Acknowledgments for the First Edition. 1. Introduction and Preview. 1.1 Preview of the Book. 2. Entropy, Relative Entropy, and Mutual Information. 2.1 Entropy. 2.2 Joint Entropy and Conditional Entropy. 2.3 Relative Entropy and Mutual Information. 2.4 Relationship Between Entropy and Mutual Information. 2.5 Chain Rules for Entropy, Relative Entropy, and Mutual Information. 2.6 Jensen's Inequality and Its Consequences. 2.7 Log Sum Inequality and Its Applications. 2.8 Data-Processing Inequality. 2.9 Sufficient Statistics. 2.10 Fano's Inequality. Summary. Problems. Historical Notes. 3. Asymptotic Equipartition Property. 3.1 Asymptotic Equipartition Property Theorem. 3.2 Consequences of the AEP: Data Compression. 3.3 High-Probability Sets and the Typical Set. Summary. Problems. Historical Notes. 4. Entropy Rates of a Stochastic Process. 4.1 Markov Chains. 4.2 Entropy Rate. 4.3 Example: Entropy Rate of a Random Walk on a Weighted Graph. 4.4 Second Law of Thermodynamics. 4.5 Functions of Markov Chains. Summary. Problems. Historical Notes. 5. Data Compression. 5.1 Examples of Codes. 5.2 Kraft Inequality. 5.3 Optimal Codes. 5.4 Bounds on the Optimal Code Length. 5.5 Kraft Inequality for Uniquely Decodable Codes. 5.6 Huffman Codes. 5.7 Some Comments on Huffman Codes. 5.8 Optimality of Huffman Codes. 5.9 Shannon-Fano-Elias Coding. 5.10 Competitive Optimality of the Shannon Code. 5.11 Generation of Discrete Distributions from Fair Coins. Summary. Problems. Historical Notes. 6. Gambling and Data Compression. 6.1 The Horse Race. 6.2 Gambling and Side Information. 6.3 Dependent Horse Races and Entropy Rate. 6.4 The Entropy of English. 6.5 Data Compression and Gambling. 6.6 Gambling Estimate of the Entropy of English. Summary. Problems. Historical Notes. 7. Channel Capacity. 7.1 Examples of Channel Capacity. 7.2 Symmetric Channels. 7.3 Properties of Channel Capacity. 7.4 Preview of the Channel Coding Theorem. 7.5 Definitions. 7.6 Jointly Typical Sequences. 7.7 Channel Coding Theorem. 7.8 Zero-Error Codes. 7.9 Fano's Inequality and the Converse to the Coding Theorem. 7.10 Equality in the Converse to the Channel Coding Theorem. 7.11 Hamming Codes. 7.12 Feedback Capacity. 7.13 Source-Channel Separation Theorem. Summary. Problems. Historical Notes. 8. Differential Entropy. 8.1 Definitions. 8.2 AEP for Continuous Random Variables. 8.3 Relation of Differential Entropy to Discrete Entropy. 8.4 Joint and Conditional Differential Entropy. 8.5 Relative Entropy and Mutual Information. 8.6 Properties of Differential Entropy, Relative Entropy, and Mutual Information. Summary. Problems. Historical Notes. 9. Gaussian Channel. 9.1 Gaussian Channel: Definitions. 9.2 Converse to the Coding Theorem for Gaussian Channels. 9.3 Bandlimited Channels. 9.4 Parallel Gaussian Channels. 9.5 Channels with Colored Gaussian Noise. 9.6 Gaussian Channels with Feedback. Summary. Problems. Historical Notes. 10. Rate Distortion Theory. 10.1 Quantization. 10.2 Definitions. 10.3 Calculation of the Rate Distortion Function. 10.4 Converse to the Rate Distortion Theorem. 10.5 Achievability of the Rate Distortion Function. 10.6 Strongly Typical Sequences and Rate Distortion. 10.7 Characterization of the Rate Distortion Function. 10.8 Computation of Channel Capacity and the Rate Distortion Function. Summary. Problems. Historical Notes. 11. Information Theory and Statistics. 11.1 Method of Types. 11.2 Law of Large Numbers. 11.3 Universal Source Coding. 11.4 Large Deviation Theory. 11.5 Examples of Sanov's Theorem. 11.6 Conditional Limit Theorem. 11.7 Hypothesis Testing. 11.8 Chernoff-Stein Lemma. 11.9 Chernoff Information. 11.10 Fisher Information and the Cram-er-Rao Inequality. Summary. Problems. Historical Notes. 12. Maximum Entropy. 12.1 Maximum Entropy Distributions. 12.2 Examples. 12.3 Anomalous Maximum Entropy Problem. 12.4 Spectrum Estimation. 12.5 Entropy Rates of a Gaussian Process. 12.6 Burg's Maximum Entropy Theorem. Summary. Problems. Historical Notes. 13. Universal Source Coding. 13.1 Universal Codes and Channel Capacity. 13.2 Universal Coding for Binary Sequences. 13.3 Arithmetic Coding. 13.4 Lempel-Ziv Coding. 13.5 Optimality of Lempel-Ziv Algorithms. Compression. Summary. Problems. Historical Notes. 14. Kolmogorov Complexity. 14.1 Models of Computation. 14.2 Kolmogorov Complexity: Definitions and Examples. 14.3 Kolmogorov Complexity and Entropy. 14.4 Kolmogorov Complexity of Integers. 14.5 Algorithmically Random and Incompressible Sequences. 14.6 Universal Probability. 14.7 Kolmogorov complexity. 14.9 Universal Gambling. 14.10 Occam's Razor. 14.11 Kolmogorov Complexity and Universal Probability. 14.12 Kolmogorov Sufficient Statistic. 14.13 Minimum Description Length Principle. Summary. Problems. Historical Notes. 15. Network Information Theory. 15.1 Gaussian Multiple-User Channels. 15.2 Jointly Typical Sequences. 15.3 Multiple-Access Channel. 15.4 Encoding of Correlated Sources. 15.5 Duality Between Slepian-Wolf Encoding and Multiple-Access Channels. 15.6 Broadcast Channel. 15.7 Relay Channel. 15.8 Source Coding with Side Information. 15.9 Rate Distortion with Side Information. 15.10 General Multiterminal Networks. Summary. Problems. Historical Notes. 16. Information Theory and Portfolio Theory. 16.1 The Stock Market: Some Definitions. 16.2 Kuhn-Tucker Characterization of the Log-Optimal Portfolio. 16.3 Asymptotic Optimality of the Log-Optimal Portfolio. 16.4 Side Information and the Growth Rate. 16.5 Investment in Stationary Markets. 16.6 Competitive Optimality of the Log-Optimal Portfolio. 16.7 Universal Portfolios. 16.8 Shannon-McMillan-Breiman Theorem (General AEP). Summary. Problems. Historical Notes. 17. Inequalities in Information Theory. 17.1 Basic Inequalities of Information Theory. 17.2 Differential Entropy. 17.3 Bounds on Entropy and Relative Entropy. 17.4 Inequalities for Types. 17.5 Combinatorial Bounds on Entropy. 17.6 Entropy Rates of Subsets. 17.7 Entropy and Fisher Information. 17.8 Entropy Power Inequality and Brunn-Minkowski Inequality. 17.9 Inequalities for Determinants. 17.10 Inequalities for Ratios of Determinants. Summary. Problems. Historical Notes. Bibliography. List of Symbols. Index.
45,034 citations
"Feedback for physicists: A tutorial..." refers background in this paper
...Recent papers by Touchette and Lloyd (2000, 2004) begin to explore more formally these links and derive a fundamental relationship between the amount of control achievable (“decrease of entropy” in their formulation) and the “mutual information” (Cover and Thomas, 1991) between the dynamical system and the controller created by an initial interaction....
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23,905 citations
Book•
01 Jan 1987
TL;DR: Das Buch behandelt die Systemidentifizierung in dem theoretischen Bereich, der direkte Auswirkungen auf Verstaendnis and praktische Anwendung der verschiedenen Verfahren zur IdentifIZierung hat.
Abstract: Das Buch behandelt die Systemidentifizierung in dem theoretischen Bereich, der direkte Auswirkungen auf Verstaendnis und praktische Anwendung der verschiedenen Verfahren zur Identifizierung hat. Da ...
20,436 citations
"Feedback for physicists: A tutorial..." refers methods in this paper
...For an introduction, see Dutton et al. 1997 ; for full details, see Ljung 1999 ....
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...Alternatively, there are a number of methods that avoid the transfer function completely: from a given input u(t) and measured response y(t), they directly fit to the coefficients of a time-domain model or directly give pole and zero positions (Ljung, 1999)....
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...Alternatively, there are a number of methods that avoid the transfer function completely: from a given input u t and measured response y t , they directly fit to the coefficients of a time-domain model or directly give pole and zero positions Ljung, 1999 ....
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TL;DR: In this paper, a simple model based on the power-law degree distribution of real networks was proposed, which was able to reproduce the power law degree distribution in real networks and to capture the evolution of networks, not just their static topology.
Abstract: The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them.
Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network.
The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other.
The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology.
18,415 citations
TL;DR: Developments in this field are reviewed, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
Abstract: Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
17,647 citations
"Feedback for physicists: A tutorial..." refers background in this paper
...The structure of such networks is a topic of intense current interest (Albert and Barabási, 2002; Newman, 2003)....
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