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Comparison of Multisine Peak Factor Minimization Algorithms for Aircraft System Identification

Justin J. Matt

arXiv:2609.28827v1eess.SYeess.SPmath.OC

TL;DR

Flight-test system identification needs efficient data collection, and phase-optimized multisines are a common excitation signal. The paper evaluates clipping and infinity norm minimization against simplex results, finding comparable or lower peak factors for the new methods, with clipping much faster.

  • Problem

    Flight-test modeling is necessary but expensive, making efficient system-identification experiments important; phase-optimized multisines are a common flight-test excitation signal.

    Source: §1 · p. 9

  • Method

    The paper evaluates a peak-clipping algorithm and an algorithm that minimizes an approximation of the infinity norm, comparing both with simplex minimization across multisine design parameters.

    Source: §1 · p. 9, §3.1.1 · p. 11

  • Results

    Clipping and infinity norm algorithms produced comparable peak factors lower than simplex, while clipping ran orders of magnitude faster and multiple initializations further lowered its peak factors.

    Source: §5 · p. 20, §6 · p. 21

  • Takeaways & Limitations

    The clipping algorithm’s rapid execution makes it an attractive choice for rapid multisine design and may support aeroelastic modeling applications or new flight-test techniques.

    Source: §5 · p. 20, §6 · p. 21

    Snow harmonics add higher-frequency content and rates that may increase actuator wear, hit physical limits, or exceed actuator bandwidth, limiting their practical use in flight testing.

    Source: §3.2.1 · p. 14

Abstract

from arXiv · show

Two phase-optimized multisine peak factor minimization algorithms are presented and evaluated. The first algorithm minimizes peak factor by iteratively clipping the peaks of generated multisine signals. The second algorithm optimizes peak factor indirectly through minimization of an approximation of the infinity norm of the multisine. Algorithm performance was evaluated as a function of different signal properties, including the number of harmonics, harmonic spacing, and number of snow harmonics (extra harmonics included for further reduction of the peak factor). The two algorithms are compared against results obtained by minimizing peak factor directly using a simplex algorithm, which has been a common approach when designing phase-optimized multisines for system identification flight tests. Sample results show that the clipping and infinity norm algorithms produced multisine signals with comparable peak factors that were lower than that of the simplex algorithm. However, the clipping algorithm runs an order of magnitude faster than the other two algorithms, which also makes it practical to repeat the algorithm multiple times to achieve even lower peak factors.

NASA/TM–20240009677

The report evaluates two phase-optimized multisine peak-factor minimization algorithms against the simplex approach commonly used for flight-test system identification. It frames this comparison around efficient flight-test data collection and the modeling benefits of low-peak-factor inputs.

Nomenclature

  • The report’s nomenclature includes DFT, FFT, IFFT, PF, RMS, and RPF, alongside symbols for signal amplitude, frequency, harmonics, and phase.Source: Nomenclature · p. 8

1 Introduction

  • Phase-optimized multisines provide broadband content and a user-designed power spectrum while limiting peak-to-peak vehicle response for aircraft system identification.Orthogonal multisines can support simultaneous multi-input and multi-axis maneuvers.Source: §1 · p. 9
  • Minimum peak factor permits more signal power without large departures from the reference flight condition, improving signal-to-noise ratio and ultimately model accuracy.Source: §1 · p. 9
  • Earlier flight-test multisines typically used simplex minimization, while clipping and sequential p-norm approaches had also been developed outside aerospace.Source: §1 · p. 9
  • The report evaluates clipping and infinity norm algorithms against simplex minimization across multisine design parameters, including modified algorithms intended to improve performance and add capability.Source: §1 · p. 9

2 Multisine Signals

A multisine combines sinusoids at discrete, harmonically related frequencies with specified power. The section explains its frequency-domain representation and the peak-factor definition used to motivate signal optimization.

2 Multisine Signals

  • A multisine is a sum of sinusoids with discrete harmonic frequencies and specified power.Source: §2 · p. 10
  • Its harmonic frequencies lie on a grid with resolution f0 = 1/T, where T is the signal period, and its power is concentrated at those harmonics.Source: §2 · p. 10
  • Because multisines are periodic, their DFT spectra represent them exactly, and inverse DFT can transform the spectra back to time-domain signals for computation.Source: §2 · p. 10
  • Peak factor is the ratio of maximum absolute signal value to RMS, and the work uses a definition that generally yields equal maximum and minimum magnitudes.Source: §2 · p. 10
  • For a fixed peak value, minimizing peak factor is equivalent to maximizing signal power; if input and output peak factors are correlated, output power also increases.Higher output power is desirable for model identification.Source: §2 · p. 11

3 Multisine Design and Optimization

Section 3 presents clipping and infinity-norm algorithms, then considers snow harmonics, joint input-output optimization, and sampling rate in multisine design.

3.1.1 Clipping Algorithm

  • Clipping iteratively trims peaks, restores the designed spectrum while retaining phases, and can restart with random phases to select the best result.The clipping threshold increases with iteration number to improve convergence.Source: §3.1.1 · pp. 11–12

3.1.2 Infinity Norm Algorithm

  • The infinity-norm method minimizes a sequence of p-norm approximations to the peak, warm-starting each solve from the previous phase solution.Under regularity conditions, this sequence can converge to the optimal solution if each approximate solution is best.Source: §3.1.2 · pp. 12–13

3.2.1 Snowing

  • Snow harmonics add algorithm-chosen power for compression and are excluded from RMS and modeling; their higher frequencies may increase actuator wear or exceed actuator limits or bandwidth.The infinity-norm formulation includes snow-harmonic powers among its optimization variables, increasing problem dimensionality and potentially run-time.Source: §3.2.1 · pp. 13–14
  • Snowing reduced the example RPF from 1.44 to 1.27; at the same peak value, useful power was 14% higher.Figure 3 compares optimized signals with and without snow and their power spectra.Source: §3.2.1 · pp. 13–14, Figure 3 · p. 14

3.2.2 Input-Output Peak Factor Minimization

  • Weighted infinity-norm optimization extends to SISO and MIMO signals; inverse-RMS weights approximately equalize input and output peak factors, while a zero weight selects one signal.The formulation can minimize peak factors for any number of input and output signals.Source: §3.2.2 · pp. 14–15
  • With 40 harmonics, joint input/time-derivative optimization yielded RPF 1.14 for both, versus 0.98 input and 1.47 output under input-only optimization.The derivative is the output of G(s) = s.Source: §3.2.2 · p. 15

3.2.3 Sampling Rate

  • Sampling at N/kmax = 16 was selected because the 95% confidence interval for relative peak-factor error falls below 1% there.The analysis compared sampled peak factors with those computed at N/kmax = 100.Source: §3.2.3 · p. 16

4 Sample Results

Across tests varying harmonic count, harmonic spacing, and added snow harmonics, clipping and infinity-norm methods generally achieved lower peak factors than simplex; clipping was much faster and benefited from repeated starts.

4 Sample Results

  • The infinity-norm algorithm had the lowest median peak factors across harmonic counts, clipping was slightly higher, and simplex was noticeably higher; clipping ran an order of magnitude faster.Clipping run-times remained below 0.1 seconds across the tested harmonic counts.Source: Figure 6 · p. 17, §4 · p. 17
  • Across harmonic spacings, median relative peak factor was smallest at spacings of one or two, while clipping's maximum run-time was 0.4 seconds versus 200 seconds for infinity norm and 50 seconds for simplex.Larger spacing increased run-time for every algorithm under the sampling-rate constraint.Source: Figure 7 · p. 18, §4 · p. 18
  • With snow harmonics, clipping produced the lowest median relative peak factor in almost all cases, reaching 0.84 as their number increased.The 0.84 minimum was appreciably lower than any result achieved without snow harmonics; the infinity-norm algorithm sometimes failed to converge.Source: Figure 8 · p. 18, §4 · p. 19
  • With 30 repetitions, clipping achieved lower peak factors and lower run-times than the infinity-norm algorithm in all tested cases.Clipping run-time increased linearly with repetitions, yet its maximum median run-time at 30 repetitions was three seconds.Source: Figure 9 · p. 19, §4 · p. 19

5 Discussion and Applications

The clipping and infinity norm algorithms produced comparable peak factors below the simplex algorithm, while clipping was much faster. The discussion highlights applications for rapid multisine design, including possible in-flight optimization.

5 Discussion and Applications

  • With multiple initial conditions, clipping outperformed the other algorithms in peak factor and run-time in all tested cases; clipping was orders of magnitude faster than the other methods.The infinity norm algorithm was typically slowest, with run-times comparable to the simplex algorithm.Source: §5 · p. 20
  • The clipping and infinity norm algorithms produced comparable peak factors that were lower than those from the simplex algorithm.The conclusion depends on the stopping criteria used, which affect relative peak factor and run-time.Source: §5 · p. 20
  • Clipping could support real-time multisine redesign during flight, including excitation design for aeroelastic aircraft modeling.The passage contrasts this potential with earlier in-flight system-identification techniques using simpler square-wave inputs.Source: §5 · p. 20

6 Conclusion

The paper evaluates clipping and infinity norm algorithms against simplex minimization for multisine peak factor design. Both new algorithms yield lower peak factors than simplex, while clipping offers a particularly fast option for flight-test input design.

6 Conclusion

  • The clipping method iteratively clips generated multisine peaks, while the infinity norm method minimizes an approximation of the signal’s infinity norm.The infinity norm method can also be used for simultaneous input-output peak factor minimization.Source: §6 · pp. 20–21
  • Both clipping and infinity norm produced comparable peak factors below simplex, while multiple initial conditions let clipping achieve still lower peak factors with substantially faster execution.The comparison covers the tested cases and algorithm settings.Source: §6 · pp. 20–21
  • The algorithms were assessed across harmonic counts, harmonic spacing, and snow harmonics, and rapid clipping may facilitate in-flight signal optimization and aeroelastic modeling applications.The paper frames rapid and effective input design as increasingly important as aircraft designs and flight-test techniques grow more complex.Source: §6 · p. 21

4. TITLE AND SUBTITLE

The supplied passage labels a field for the dates covered but provides no date values.

4. TITLE AND SUBTITLE

  • The passage identifies a “DATES COVERED (From - To)” field without giving dates.Source: §4 · p. 23

6. AUTHOR(S)

The supplied text identifies NASA Langley Research Center as the performing organization.

6. AUTHOR(S)

  • NASA Langley Research Center, Hampton, Virginia 23681-2199, is listed as the performing organization.Source: §6 · p. 23

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

The sponsoring agency is NASA in Washington, DC. Distribution is unclassified and unlimited, with availability through the NASA STI Program.

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

  • The sponsoring agency is the National Aeronautics and Space Administration, Washington, DC 20546-0001.Source: §9 · p. 23

12. DISTRIBUTION/AVAILABILITY STATEMENT

  • Distribution is unclassified and unlimited; the NASA STI Program can be reached at (757) 864-9658.Source: §12 · p. 23

5e. TASK NUMBER

The record provides a work-unit identifier and contact and form details, alongside a summary of the report’s multisine algorithm comparison.

5e. TASK NUMBER

11. SPONSOR/MONITOR’S REPORT NUMBER(S)

  • The clipping and infinity norm algorithms achieved comparable peak factors below the simplex algorithm, while clipping ran an order of magnitude faster.The evaluation varied harmonic count, harmonic spacing, and the number of snow harmonics.Source: §11 · p. 23

18. NUMBER OF PAGES

  • The record includes a security-classification field and identifies Standard Form 298 (Rev. 8/98) as prescribed by ANSI Std. Z39.18.Source: §11 · p. 23, §18 · p. 23
  • The listed telephone number is (757) 864-9658.Source: §18 · p. 23
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