Describing Function Inversion: Theory and Computational Techniques

In the last few years the study of nonlinear mechanics has received the attention of numerous investigators, either under the scope of pure mathematics or from the engineering point of view. Many of the recent developments are based on the early works of H. Poincare [1] and A. Liapunov [2] As examples can be cited the perturbation method, harmonic balance, the second method of Liapunov, etc. An approximate technique developed almost simultaneously by C. Goldfarb [3] in the USSR, A. Tustin [4] in England, R. Kochenburger [5] in the USA, W, Oppelt [6] in Germany and J. Dutilh [7] and C. Ecary [8] in France, known as the describing function technique, can be considered as the graphical solution of the first approximation of the method of the harmonic balance. The describing function technique has reached great popularity, principally because of the relative ease of computation involved and the general usefulness of the method in engineering problems. However, in the past, the describing function technique has been useful only in analysis. More exactly, it is a powerful tool for the investigation of the possible existence of limit cycles and their approximate amplitudes and frequencies. Several extensions have been developed from the original describing function technique. Among these can be cited the dual-input describing function, J. C. Douce et al. [9]; the Gaussian-input describing function, R. C, Booton [10]; and the root-mean-square describing function, J. E. Gibson and K. S. Prasanna-Kumar [11]. In a recent work which employs the describing function, C. M„ Shen [12] gives one example of stabilization of a nonlinear system by introducing a saturable feedback. However, Shen’s work cannot be qualified as a synthesis method since he fixes a priori the nonlinearity to be introduced in the feedback loop. A refinement of the same principle used by Shen has been proposed by R. Haussler [13] The goal of this new method of synthesis is to find the describing function of the element being synthesized. Therefore, for Haussler’s method to be useful, a way must be found to reconstruct the nonlinearity from its describing function. This is called the inverse-describing-function-problem and is essentially a synthesis problem. This is not the only ease in which the inverse-describing-function-problem can be useful. Sometimes, in order to find the input-output characteristic of a physical nonlinear element, a harmonic test can be easier to perform rather than a static one (which also may be insufficient). The purpose of this report is to present the results of research on a question which may then be concisely stated as; If the describing function of a nonlinear element is known, what is the nonlinearity? The question may be divided into two parts, the first part being the determination of the restrictions on the nonlinearity (or its describing function) necessary to insure that the question has an answer, and the second part the practical determination of that answer when it exists. Accordingly, the material in this report is presented in two parts. Part I is concerned with determining what types of nonlinearities are (and what types are not) uniquely determined by their conventional (fundamental) describing function. This is done by first showing the non-uniqueness in general of the describing function, and then constructing a class of null functions with respect to the describing function integral, i.e., a class of nonlinearities not identically zero whose describing functions are identically zero. The defining equations of the describing function are transformed in such a manner as to reduce the inverse describing function problem to the problem of solving a Volterra integral equation, an approach similar to that used by Zadeh [18]. The remainder of Part I presents the solution of the integral equations and studies the effect of including higher order harmonics in the description of the output ware shape. The point of interest here is that inclusion of the second harmonic may cause the describing function to become uniquely invertible in some cases. Part II presents practical numerical techniques for effecting the inversion of types of describing functions resulting from various engineering assumptions as to the probable form of the nonlinearities from which said describing functions were determined. The most general method is numerical evaluation of the solution to the Volterra integral equations developed in Part I, A second method, which is perhaps the easiest to apply, requires a least squares curve fit to the given describing function data. Then use is made of the fact that the describing function of a polynomial nonlinearity is itself a polynomial to calculate the coefficients in a polynomial approximation to the nonlinearity. This approach is indicated when one expects that the nonlinearity is a smooth curve, such as a cubic characteristic. The third method presented assumes that the nonlinearity can be approximated by a piecewise linear discontinuous function, and the slopes and y-axis intercepts of each linear segment are computed. This approach is indicated when one expects a nonlinearity with relatively sharp corners. It may toe remarked that the polynomial approximation and the piecewise linear approximation are derived independently of the material in Part I. All three methods presented in Part II are suited for use with experimental data as well as with analytic expressions for the describing functions involved. Indeed, an analytical expression must toe reduced to discrete data for the machine methods to the of use. To the best of the authors® knowledge, research in the area of describing function inversion has been nonexistent with the exception of Zadeh’s paper [18] in 1956. It seems that a larger effort in this area would toe desirable in the light of recent extensions of the describing function itself to signal stabilization of nonlinear control systems by Oldentourger and Sridhar [19] and Boyer [20] and the less restrictive study of dual-input describing functions for nonautonomous systems by Gibson and Sridhar [21]. There presently exist techniques for determining a desired describing function for use in avoiding limit cycle oscillations in an already nonlinear system (Haussler [13]), and the methods presented in this report now allow the exact synthesis of the nonlinear element from the describing function data

Identification of lenvatinib prognostic index via recursive partitioning analysis in advanced hepatocellular carcinoma

Background: After the advent of new treatment options for advanced hepatocellular carcinoma (HCC), the identification of prognostic factors is crucial for the selection of the most appropriate therapy for each patient. Patients and methods: With the aim to fill this gap, we applied recursive partitioning analysis (RPA) to a cohort of 404 patients treated with lenvatinib. Results: The application of RPA resulted in a classification based on five variables that originated a new prognostic score, the lenvatinib prognostic index (LEP) index, identifying three groups: low risk [patients with prognostic nutritional index (PNI) >43.3 and previous trans-arterial chemoembolization (TACE)]; medium risk [patients with PNI >43.3 but without previous TACE and patients with PNI <43.3, albumin-bilirubin (ALBI) grade 1 and Barcelona Clinic Liver Cancer stage B (BCLC-B)]; high risk [patients with PNI <43.3 and ALBI grade 2 and patients with PNI <43.3, albumin-bilirubin (ALBI) grade 1 and Barcelona Clinic Liver Cancer stage C (BCLC-C)]. Median overall survival was 29.8 months [95% confidence interval (CI) 22.8-29.8 months] in low risk patients (n = 128), 17.0 months (95% CI 15.0-24.0 months) in medium risk (n = 162) and 8.9 months (95% CI 8.0-10.7 months) in high risk (n = 114); low risk hazard ratio (HR) 1 (reference group), medium risk HR 1.95 (95% CI 1.38-2.74), high risk HR 4.84 (95% CI 3.16-7.43); P < 0.0001. The LEP index was validated in a cohort of 127 Italian patients treated with lenvatinib. While the same classification did not show a prognostic value in a cohort of 311 patients treated with sorafenib, we also show a possible predictive role in favor of lenvatinib in the low risk group. Conclusions: LEP index is a promising, easy-to-use tool that may be used to stratify patients undergoing systemic treatment of advanced HCC

Transcriptional dysregulation of Interferome in experimental and human Multiple Sclerosis

Recent evidence indicates that single multiple sclerosis (MS) susceptibility genes involved in interferon (IFN) signaling display altered transcript levels in peripheral blood of untreated MS subjects, suggesting that responsiveness to endogenous IFN is dysregulated during neuroinflammation. To prove this hypothesis we exploited the systematic collection of IFN regulated genes (IRG) provided by the Interferome database and mapped Interferome changes in experimental and human MS. Indeed, central nervous system tissue and encephalitogenic CD4 T cells during experimental autoimmune encephalomyelitis were characterized by massive changes in Interferome transcription. Further, the analysis of almost 500 human blood transcriptomes showed that (i) several IRG changed expression at distinct MS stages with a core of 21 transcripts concordantly dysregulated in all MS forms compared with healthy subjects; (ii) 100 differentially expressed IRG were validated in independent case-control cohorts; and (iii) 53 out of 100 dysregulated IRG were targeted by IFN-beta treatment in vivo. Finally, ex vivo and in vitro experiments established that IFN-beta administration modulated expression of two IRG, ARRB1 and CHP1, in immune cells. Our study confirms the impairment of Interferome in experimental and human MS, and describes IRG signatures at distinct disease stages which can represent novel therapeutic targets in MS

Measurements of π±\pi^\pm, K±K^\pm, KS0K^0_S, Λ\Lambda and proton production in proton-carbon interactions at 31 GeV/cc with the NA61/SHINE spectrometer at the CERN SPS

Measurements of hadron production in p+C interactions at 31 GeV/c are performed using the NA61/ SHINE spectrometer at the CERN SPS. The analysis is based on the full set of data collected in 2009 using a graphite target with a thickness of 4% of a nuclear interaction length. Inelastic and production cross sections as well as spectra of $\pi^\pm$, $K^\pm$, p, $K^0_S$ and $\Lambda$ are measured with high precision. These measurements are essential for improved calculations of the initial neutrino fluxes in the T2K long-baseline neutrino oscillation experiment in Japan. A comparison of the NA61/SHINE measurements with predictions of several hadroproduction models is presented.Comment: v1 corresponds to the preprint CERN-PH-EP-2015-278; v2 matches the final published versio

Multiplicity and transverse momentum fluctuations in inelastic proton-proton interactions at the CERN Super Proton Synchrotron

Measurements of multiplicity and transverse momentum fluctuations of charged particles were performed in inelastic p+p interactions at 20, 31, 40, 80 and 158 GeV/c beam momentum. Results for the scaled variance of the multiplicity distribution and for three strongly intensive measures of multiplicity and transverse momentum fluctuations \$\Delta[P_{T},N]\$, \$\Sigma[P_{T},N]\$ and \$\Phi_{p_T}\$ are presented. For the first time the results on fluctuations are fully corrected for experimental biases. The results on multiplicity and transverse momentum fluctuations significantly deviate from expectations for the independent particle production. They also depend on charges of selected hadrons. The string-resonance Monte Carlo models EPOS and UrQMD do not describe the data. The scaled variance of multiplicity fluctuations is significantly higher in inelastic p+p interactions than in central Pb+Pb collisions measured by NA49 at the same energy per nucleon. This is in qualitative disagreement with the predictions of the Wounded Nucleon Model. Within the statistical framework the enhanced multiplicity fluctuations in inelastic p+p interactions can be interpreted as due to event-by-event fluctuations of the fireball energy and/or volume.Comment: 18 pages, 12 figure

Measurement of negatively charged pion spectra in inelastic p+p interactions at plabp_{lab} = 20, 31, 40, 80 and 158 GeV/c

We present experimental results on inclusive spectra and mean multiplicities of negatively charged pions produced in inelastic p+p interactions at incident projectile momenta of 20, 31, 40, 80 and 158 GeV/c ($\sqrt{s} =$ 6.3, 7.7, 8.8, 12.3 and 17.3 GeV, respectively). The measurements were performed using the large acceptance NA61/SHINE hadron spectrometer at the CERN Super Proton Synchrotron. Two-dimensional spectra are determined in terms of rapidity and transverse momentum. Their properties such as the width of rapidity distributions and the inverse slope parameter of transverse mass spectra are extracted and their collision energy dependences are presented. The results on inelastic p+p interactions are compared with the corresponding data on central Pb+Pb collisions measured by the NA49 experiment at the CERN SPS. The results presented in this paper are part of the NA61/SHINE ion program devoted to the study of the properties of the onset of deconfinement and search for the critical point of strongly interacting matter. They are required for interpretation of results on nucleus-nucleus and proton-nucleus collisions.Comment: Numerical results available at: https://edms.cern.ch/document/1314605 Updates in v3: Updated version, as accepted for publicatio

Measurement of Production Properties of Positively Charged Kaons in Proton-Carbon Interactions at 31 GeV/c

Spectra of positively charged kaons in p+C interactions at 31 GeV/c were measured with the NA61/SHINE spectrometer at the CERN SPS. The analysis is based on the full set of data collected in 2007 with a graphite target with a thickness of 4% of a nuclear interaction length. Interaction cross sections and charged pion spectra were already measured using the same set of data. These new measurements in combination with the published ones are required to improve predictions of the neutrino flux for the T2K long baseline neutrino oscillation experiment in Japan. In particular, the knowledge of kaon production is crucial for precisely predicting the intrinsic electron neutrino component and the high energy tail of the T2K beam. The results are presented as a function of laboratory momentum in 2 intervals of the laboratory polar angle covering the range from 20 up to 240 mrad. The kaon spectra are compared with predictions of several hadron production models. Using the published pion results and the new kaon data, the K+/\pi+ ratios are computed.Comment: 10 pages, 11 figure

NA61/SHINE facility at the CERN SPS: beams and detector system

NA61/SHINE (SPS Heavy Ion and Neutrino Experiment) is a multi-purpose experimental facility to study hadron production in hadron-proton, hadron-nucleus and nucleus-nucleus collisions at the CERN Super Proton Synchrotron. It recorded the first physics data with hadron beams in 2009 and with ion beams (secondary 7Be beams) in 2011. NA61/SHINE has greatly profited from the long development of the CERN proton and ion sources and the accelerator chain as well as the H2 beamline of the CERN North Area. The latter has recently been modified to also serve as a fragment separator as needed to produce the Be beams for NA61/SHINE. Numerous components of the NA61/SHINE set-up were inherited from its predecessors, in particular, the last one, the NA49 experiment. Important new detectors and upgrades of the legacy equipment were introduced by the NA61/SHINE Collaboration. This paper describes the state of the NA61/SHINE facility - the beams and the detector system - before the CERN Long Shutdown I, which started in March 2013

Search for short baseline nu(e) disappearance with the T2K near detector

8 pages, 6 figures, submitted to PRD rapid communication8 pages, 6 figures, submitted to PRD rapid communicationWe thank the J-PARC staff for superb accelerator performance and the CERN NA61 collaboration for providing valuable particle production data. We acknowledge the support of MEXT, Japan; NSERC, NRC and CFI, Canada; Commissariat `a l’Energie Atomique and Centre National de la Recherche Scientifique–Institut National de Physique Nucle´aire et de Physique des Particules, France; DFG, Germany; INFN, Italy; National Science Centre (NCN), Poland; Russian Science Foundation, RFBR and Ministry of Education and Science, Russia; MINECO and European Regional Development Fund, Spain; Swiss National Science Foundation and State Secretariat for Education, Research and Innovation, Switzerland; STFC, UK; and DOE, USA. We also thank CERN for the UA1/NOMAD magnet, DESY for the HERA-B magnet mover system, NII for SINET4, the WestGrid and SciNet consortia in Compute Canada, GridPP, UK. In addition participation of individual researchers and institutions has been further supported by funds from ERC (FP7), EU; JSPS, Japan; Royal Society, UK; DOE Early Career program, USA