This paper examines performance of the knock detection technique typically used in engine control systems, and the margin for possible improvement. We introduce a knock signal model and obtain an analytical result for the associated receiver operating characteristic of the standard knock detector. To show the improvement potential, we derive the theoretical upper bound of performance. A special case with unknown model parameters is also considered. Numerical results stimulate the research of improved detectors

Galtier, Frédéric

Vincent, François

Vingerhoeds, Rob A.

Zadnik, Martin

English

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PERFORMANCE ANALYSIS OF KNOCK DETECTORSM. Zadnik∗, F. Vincent†, F. Galtier‡, R.A. Vingerhoeds‡∗ AMPERE, 42 av. du Ge´ne´ral de Croutte, 31100 Toulouse, France, martin.zadnik@siemens.com† ENSICA, 1 pl. ´Emile Blouin, 31056 Toulouse, France, vincent@ensica.fr‡ Siemens VDO Automotive, 1 av. Paul Ourliac, 31036 Toulouse, France, {frederic.galtier, robertus.vingerhoeds}@siemens.comKeywords: engine knock, knock signal model, detection, re-ceiver operating characteristicAbstractThis paper examines performance of the knock detection tech-nique typically used in engine control systems, and the marginfor possible improvement. We introduce a knock signal modeland obtain an analytical result for the associated receiver oper-ating characteristic of the standard knock detector. To show theimprovement potential, we derive the theoretical upper boundof performance. A special case with unknown model parame-ters is also considered. Numerical results stimulate the researchof improved detectors.1 IntroductionA high compression ratio and a proper spark timing advanceare necessary to attain good efficiency of internal combustiongasoline engines. Both parameters are limited by the apparitionof knock which is a spontaneous ignition of unburned air-fuelmixture that occurs after ignition by spark plug [1]. Knockconsequences are harmful, from excessive emission of pollu-tants, efficiency decrease, to engine damage [2]. Its detectionand prevention are thus an important part of engine control.For economic reasons, the direct method of measuring knockby in-cylinder pressure transducers is usually replaced by ex-ploiting the vibration signal from one or more accelerometersmounted on engine block surface. Knock being constituted ofseveral resonances determined by cylinder geometry [3], thestandard treatment is to apply a band-pass filter followed bycomputation of signal energy in a predefined time window. Theresult is compared to base engine noise to make a decisionwhether knock is present or not. Parameters of the treatmentlike the resonance to choose, filter frequency and time windoware determined for each engine type during calibration phase.In this paper, we investigate the detection performance result-ing from the described knock signal processing scheme bymeans of detection theory. Contrary to [4] where a similaranalysis is performed by simulation on real training data, an-alytical results give access to arbitrary false alarm rates as weare not limited by sample data size. The analysis is based ona simple signal model including some unknown parameters.The detection performance is compared to that of the optimalNeyman-Pearson test which gives the theoretical upper limitfor performance of any real world detector. The gap betweenthe two curves shows how much place there is left for possibleimprovement of actual knock detection method.The paper is organised as follows. In Section 2, we intro-duce the signal model. Section 3 gives the associated optimalNeyman-Pearson detector performance. The case where somemodel parameters are unknown is discussed in Section 4. Wederive the performance of the standard knock detection systemin Section 5. A numerical example is given in Section 6, beforethe paper is concluded in Section 7.2 Knock signal modelTime-frequency analysis of knock vibrations clearly shows amulti-frequency signal structure [5]. It also reveals that res-onance frequencies decrease with time which is due to tem-perature and sound speed decrease. Theoretical analysis andexperiments show that a linear frequency modulation (i.e. a2nd order polynomial phase) is a good approximation [6]. In asimilar manner as in [7], we write the signal model as a sumof P resonances in zero mean white Gaussian additive noise ofvariance σ2:x(t) = s(t) + n(t) ==P∑p=1ap wp(t) cos(2pi(αpt2 + βpt) + φp)+ n(t). (2.1)The normalised envelope functionswp have two parameters de-scribing time scaling and instant of knock begin:wp(t) =t− t0,pτpexp(− t− t0,pτp+ 1)Θ(t− t0,p), (2.2)Θ(t) being the Heaviside unit step function. The form of en-velope functions is shown in figure 1. Each resonance thus hasa six-element parameter vector θp = [ap, αp, βp, φp, t0,p, τp]T .The sampled signal written as a vector is x = [x(0), . . . , x(N−1)]T = s+ n.3 Neyman-Pearson performance limitWe formulate the knock detection as a two hypothesis testingproblem:H0 : x = n (no knock),H1 : x = s+ n (knock is present).(3.1)If all the model parameters are known, the optimal test in thesense of maximising the probability of detection Pd for a givenArticle pre´sente´ a` ‘‘18th International Conference on Systems Engineering’’,Coventry University, Coventry, 5-7 septembre 2006-- Page 1 --100.10.20.30.40.50.60.70.80.91tw(t)t0 Figure 1: Envelope function given by equation (2.2).probability of false alarm Pfa is the Neyman-Pearson (NP)likelihood ratio test [8]:decide H1 if Lnp(x) = p(x;H1)p(x;H0) > γ (3.2)where p(x;Hi) is the probability density function (PDF) ofdata x under i-th hypothesis. For a given signal-to-noise ra-tio, the threshold γ parametrises the receiver operating char-acteristic (ROC) curve as Pfa(γ) = Pr[Lnp(x) > γ;H0],Pd(γ) = Pr[Lnp(x) > γ;H1]. These two probabilities canbe analytically calculated. The PDFs to be inserted in (3.2) arep(x;H0) = 1(2piσ2)N2exp(− xTx2σ2), (3.3)p(x;H1) = 1(2piσ2)N2exp(− (x− s)T (x− s)2σ2). (3.4)Since taking a monotonous function ofLnp(x) does not changethe test, we can use the equivalent test statisticdecide H1 if Tnp(x) = σ2 lnLnp(x) + 12sTs == xT s > γ′ (3.5)which is found to be Tnp(x) ∼ N (0, σ2E) under H0 andTnp(x) ∼ N (E, σ2E) under H1, where we defined the signalenergy E = sT s, and N (µ, σ2) denotes a normal distributionwith mean µ and variance σ2. Note that the form of the testas written in (3.5) corresponds to computing the correlation ofmeasured data with signal model. Like the standard treatmentdiscussed in section 5, it is an energy detector but here only thecontribution of the useful signal part is extracted. Finally, wefindPd(γ′) = Q(γ′ − E√σ2E), Pfa(γ′) = Q(γ′√σ2E)(3.6)where Q(x) =∫ ∞x1√2piexp(− t22)dt is the right-tail proba-bility function.In practice, signal parameters are not all known and theNeyman-Pearson test cannot directly be applied. Replacing thetrue parameter values by their estimates degrades detection per-formance and the NP detector can be seen as the upper limit ofperformance that any detector can reach.4 Case of unknown amplitudes and phasesWe now consider the situation where the signal parameters areall known except the resonance amplitudes ap and phases φp.This is equivalent to assuming that the knock signal has a fixedform but the intensity of each resonance can vary. Althoughwithout any proof of optimality similar to that of the NP test,the standard approach is to replace the unknown parameters bytheir maximum likelihood (ML) estimates, which is known asthe generalised likelihood ratio test (GLRT).We rewrite s(t) from (2.1) ass(t) =P∑p=1[Ap wp(t) cos(2pi(αpt2 + βpt))++Bp wp(t) sin(2pi(αpt2 + βpt))] (4.1)with Ap = ap cos(φp), Bp = −ap sin(φp). Introducing theunknown parameter vector a = [A1, B1, . . . , AP , BP ]T , wecan writex = Sa+ n (4.2)where the structure of the N × 2P matrix S is obvious fromequation (4.1). The ML estimate of a is aˆ = (STS)−1STx.The detection can be reformulated as testing whether a = 0(H0) or a 6= 0 (H1).Replacing s by Saˆ and proceeding as in section 3, we obtaindecide H1 if Tglrt(x) = xTS(STS)−1STx > γ′. (4.3)The test is thus a projection of measured data onto the signalsubspace. In this case, the test statistic is found to be chi-squared distributed with 2P degrees of freedom [8]. The distri-bution is central under H0 and noncentral under H1, the non-centrality parameter being λ = Eσ2. Hence, the ROC curve canbe calculated fromPd(γ′) = Qχ′22P(λ)(γ′), Pfa(γ′) = Qχ22P(γ′) (4.4)where the two Q-functions denote the corresponding right-tailprobabilities.5 Bandpass filter energy detectorAs stated before, the classical knock detector compares thebandpass filter output energy to a threshold. We suppose thatthe filter can be written in a finite impulse response form. Thefiltered knock signal isxf = Hx = nf under H0xf = Hx = sf + nf under H1(5.1)Article pre´sente´ a` ‘‘18th International Conference on Systems Engineering’’,Coventry University, Coventry, 5-7 septembre 2006-- Page 2 --2where H is the filtering matrix. Let the filtered noise co-variance matrix and its diagonalisation be C = E[nfnTf ] =UTΛU. For convenience, we now define xu = Uxf = su +nu and therefore xf = UTxu. Note that since E[nunTu ] = Λ,the samples of xu are decorrelated.The test statistic is the filtered signal energy:decide H1 if Lbpe(x) = xTf xf = xTuUUTxu == xTuxu =N∑i=1(xu)2i > γ. (5.2)The sum terms (xu)2i are independent by construction and wecan apply the central limit theorem with Lyapunov condition[9] to approximate the PDF of test statistic (5.2), supposingthat N is large enough. In this case, Lbpe(x) will be normallydistributed with mean and variance equal toN∑i=1E[(xu)2i ] andN∑i=1var[(xu)2i ] respectively. Defining the filtered signal energyEf = sTf sf , we findLbpe(x) ∼ N(Tr(C), 2Tr(C2))under H0Lbpe(x) ∼ N(Ef +Tr(C), 4sTfCsf++ 2Tr(C2))under H1.(5.3)The covariance matrix of n is σ2I and thus C = σ2HHT .Finally, the ROC curve for this detector can be calculated asfollows:Pd(γ) = Q(γ − Ef − Tr(C)√4sTfCsf + 2Tr(C2)),Pfa(γ) = Q(γ − Tr(C)√2Tr(C2)).(5.4)6 Numerical exampleWe now give an example of the ROC curves discussed in thepaper. The parameters’ orders of magnitude are determinedfrom a real knock signal. The values taken are the following:P =3, θ1 = [1.0, -2.3·105 Hz/s, 7 kHz, 3.9, 0.61 ms, 0.6 ms]T ,θ2 = [0.8, -2.8·105 Hz/s, 12 kHz, 1.1, 0.64 ms, 0.56 ms]T ,θ3 = [0.5, -3.2·105 Hz/s, 17 kHz, 5.7, 0.67 ms, 0.57 ms]T ,N =167 (corresponding to 10◦–50◦ crank angle window at3000 RPM and 75 kHz sampling frequency). Figure 2 showsthe ROC curves for signal-to-noise ratio of -6 dB. The filter Hfrom (5.1) is a 50-point finite impulse response approximationof a second order infinite impulse response filter with qualityfactor 2.3, centered on β1 (the strongest resonance).As we see, the model of actual knock detection scheme is quitefar below the NP bound, showing that there is space left for pos-sible performance improvement. This can be explained by thefact that bandpass filtering is a non-coherent treatment where10−10 10−8 10−6 10−4 10−2 1000.10.20.30.40.50.60.70.80.91PfaP dNPGLRT(ap,φp)BPEFigure 2: ROC curves for SNR=-6 dB.the integration gain does not increase with sample size. To il-lustrate the results, suppose that a detection probability of 0.9is prescribed. In this case, there is a Pfa gain factor of ≈103between BPE and GLRT(ap, φp). For a 4-cylinder engine run-ning at 3000 RPM without knock, the number of false alarmspasses from 136 to 0.13 per minute. Since false alarms engagea useless spark advance correction and thus engine efficiencydecrease, this is considerable.7 ConclusionsThe present document considered the standard engine knockdetection processing system. Based on a simple signal model,analytical results for detector performance were derived interms of receiver operating characteristic. They were comparedto the theoretical upper bound given by the Neyman-Pearsontest, as well as to the ROC curve of the case with unknownamplitudes and phases. The results show how much perfor-mance improvement can be expected. To exploit this option,our further investigations will consider the usage of a prioriknowledge on unknown parameters and/or their estimation.References[1] J.B. Heywood. Internal Combustion Engine Fundamen-tals. McGraw-Hill, 1988.[2] B. Samimy and G. Rizzoni. Time-frequency analysis forimproved detection of internal combustion engine knock.In Proceedings of the IEEE-SP International Symposiumon Time-Frequency and Time-Scale Analysis, Philadelphia,October 1994.[3] S. Carstens-Behrens, M. Urlaub, J.F. Bo¨hme, J. Forster,and F. Raichle. FEM approximation of internal combus-tion chambers for knock investigations. In Proceedings ofthe World Congress & Exhibition, Detroit, March 2002.Article pre´sente´ a` ‘‘18th International Conference on Systems Engineering’’,Coventry University, Coventry, 5-7 septembre 2006-- Page 3 --3[4] B. Samimy, G. Rizzoni, A. Sayeed, and D.L. Jones. De-sign of training data-based quadratic detectors with appli-cation to mechanical systems. In Proceedings of the IEEEICASSP, Atlanta, May 1996.[5] LJ. Stankovic´ and J.F. Bo¨hme. Time-frequency analysis ofmultiple resonances in combustion engine signals. SignalProcessing, 79(1), November 1999.[6] N. Ha¨rle and J.F. Bo¨hme. Detection of knocking for sparkignition engines based on structural vibrations. In Proceed-ings of the IEEE ICASSP, Dallas, April 1987.[7] M. Urlaub and J.F. Bo¨hme. Reconstruction of pressure sig-nals on structure-borne sound for knock investigation. InProceedings of the SAE World Congress, Detroit, March2004.[8] S.M. Kay. Fundamentals of Statistical Signal Processing,volume II : Detection Theory. Prentice-Hall, 1993.[9] M.G. Kendall. The Advanced Theory of Statistics, vol-ume I. Hafner Publishing Company, New York, 1952.Article pre´sente´ a` ‘‘18th International Conference on Systems Engineering’’,Coventry University, Coventry, 5-7 septembre 2006-- Page 4 --4

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FEM approximation of internal combustion chambers for knock investigations.

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Internal Combustion Engine Fundamentals.

Reconstruction of pressure signals on structure-borne sound for knock investigation.

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Time-frequency analysis for improved detection of internal combustion engine knock.

Time-frequency analysis of multiple resonances in combustion engine signals.

Performance analysis of knock detectors

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Performance analysis of knock detectors

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