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Discussion on the mode mixing in

wave energy control systems using the

Hilbert-Huang transform

Paula B. Garcia-Rosa, Marta Molinas, and Olav B. Fosso

Abstract—A great improvement in the absorption of

energy of a wave energy converter (WEC) is obtained

with a time-varying power take-off (PTO) damping over

a constant damping. In a passive control scheme based on

the Hilbert-Huang transform (HHT), the PTO damping is

time-varying and tuned to the instantaneous frequency of

the wave excitation force. The HHT method relies on the

use of the empirical mode decomposition (EMD) method

to decompose the wave signal into a number of compo-

nents (IMFs) from the highest to the lowest frequency

component. However, the decomposition process is not

always perfect and may result in mode mixing, where an

IMF will consist of signals of widely disparate frequency

scales, or different IMFs will consist of signals with

similar frequency scales. Mode mixing can be caused by

intermittent/noisy signals, and by speciﬁc amplitude and

frequency relations of the original modes in the signal. The

aim of this paper is to extend the studies on the use of the

HHT for WEC tuning purposes by revealing how the EMD

mode mixing problem affects the WEC performance. A

comprehensive study using ﬁrstly synthetic two-tone waves

(i.e., superposition of two sinusoidal waves) is performed.

Then, the observations from the two-tone studies are used

to further improve the energy absorbed by WECs using the

HHT in real ocean wave scenarios resembling the analytic

scenario.

Index Terms—wave energy, Hilbert-Huang transform,

mode mixing.

I. INTRODUCTION

RECENT studies have shown that tuning the power

take-off (PTO) damping of a wave energy con-

verter (WEC) to time-frequency estimations obtained

from the Hilbert-Huang transform (HHT) results in

greater energy absorption than tuning the PTO to a

constant frequency of the wave spectrum [1], or to

time-frequency estimations from the extended Kalman

ﬁlter (EKF), and frequency-locked loop (FLL) method

[2]. Both the EKF and FLL methods provide single

dominant frequency estimates, whereas the HHT pro-

vides the instantaneous wave-to-wave frequency of the

oscillation modes present in a wave proﬁle. In addition,

by adopting other methods to, e.g., estimate the on-

line dominant wave frequency [3], or determine the

optimal time-varying PTO damping [4], other studies

have also shown that continuously tuning the PTO

ID 1275 track GPC.

P. B. Garcia-Rosa and O. B. Fosso are with the Department

of Electric Power Engineering, Norwegian University of Science

Technology, Trondheim, Norway (e-mails: p.b.garcia-rosa@ieee.org,

olav.fosso@ntnu.no).

M. Molinas is with the Department of Engineering Cybernetics,

Norwegian University of Science Technology, Trondheim, Norway

(e-mail: marta.molinas@ntnu.no).

result in greater energy absorption than tuning it to

a constant frequency of the wave spectrum.

The HHT method [5] relies on the use of the em-

pirical mode decomposition (EMD) to decompose the

wave signal into a number of components, named

intrinsic mode functions (IMFs), from the highest to

the lowest frequency component. However, the decom-

position process may result in mode mixing, and an

IMF will consist of signals of widely disparate fre-

quency scales, or different IMFs will consist of signals

with similar frequency scales [6]. Mode mixing can

be caused by an intermittent/noisy signal, by spe-

ciﬁc amplitude and frequency relations of the original

modes in the signal, and by a combination of both

cases. Additionally, the mode mixing effects can be

attenuated by applying, e.g., masking signals prior to

the EMD procedure [7] [8], or by using the ensemble

EMD (EEMD), a white noise-assisted EMD method [9].

Focusing on the case when mode mixing is caused

by speciﬁc amplitude and frequency relations of the

original modes in a signal, [10] presents a rigorous

mathematical analysis that shows how the EMD sepa-

rates the original modes in signals with two frequency

components. Three different domains are identiﬁed by

the authors depending on the frequency and amplitude

ratios of the modes. After the EMD procedure, the

components can be separated and correctly identiﬁed

(domain 1), considered as a single wave-form (domain

2), or the EMD does something else (domain 3) [10].

In this paper, the focus is also on mode mixing

caused by speciﬁc amplitude/frequency relations of

original modes in a wave signal. In this framework,

the aim is to extend the studies on the use of the

HHT for WEC tuning purposes with a passive control

(PC) strategy, by revealing how the EMD mode mixing

problem affects the WEC performance. A comprehen-

sive study using initially synthetic two-tone waves

(superposition of two sinusoidal waves) is performed.

Then, the mode mixing conditions observed for the

two-tone wave studies are used to further improve

the energy absorbed by WECs using the HHT in real

ocean wave scenarios resembling the synthetic two-

tone wave scenario.

II. PASS IV E CON TR OL U SI NG T HE HHT

Here, we assume linear hydrodynamic theory, and

consider a single oscillating-body represented as a

truncated vertical cylinder constrained to move in

heave. By neglecting friction and viscous forces, the

GARCIA-ROSA et al.: DISCUSSION ON THE MODE MIXING IN WAVE ENERGY CONTROL SYSTEMS USING THE HILBERT-HUANG TRANSFORM

heave motion x(t)of the ﬂoating cylinder is described

by:

M¨x(t)+

t

Z

0

hr(t−τ) ˙x(τ)dτ +Sx(t) = fe(t)+fp(t),(1)

with the kernel of convolution term given by [11]

hr(t) = 2

π

∞

Z

0

Br(ω) cos(ωt −τ)dω , (2)

where M= [m+mr(∞)],mis the body mass, mr(∞)

is the inﬁnite-frequency added mass coefﬁcient, Sis

the buoyancy stiffness, Br(ω)is the radiation damp-

ing coefﬁcient, ω, is the wave frequency, fe(t)is the

wave excitation force, and fpis the force of the PTO

mechanism.

The excitation force is calculated as

fe(t) =

∞

Z

−∞

he(t−τ)ζ(τ)dτ , (3)

where ζis the wave elevation, and heis the inverse

Fourier transform of the excitation force transfer func-

tion He(ω),

he(t) = 1

2π

∞

Z

−∞

He(ω)eiωt dω , (4)

which has low-pass ﬁlter characteristics for ﬂoating

WECs. Equation (3) is non-causal, since in fact, the

pressure distribution is the cause of the excitation force

and not the incident waves [12].

For the passive control, the PTO force is parameter-

ized as a function of the damping Bp,

fp(t) = −Bp(t) ˙x(t),(5)

where Bp∈R+is continuously modiﬁed, and tuned to

the excitation force frequency. Here, the PTO damping

is calculated as

Bp(t) = p(Br(ˆωd))2+ (ˆωd(m+mr(ˆωd)) −S/ˆωd)2,

(6)

where ˆωd(t)is the instantaneous frequency of the domi-

nant IMF component of the excitation force [1].

Prior to calculating the instantaneous frequency, the

excitation force is decomposed into a number of IMF

components by applying the EMD. Therefore, the

dominant IMF is identiﬁed through the comparison of

the energy of each IMF signal (Eci) with the energy of

the excitation force signal (Efe) [1],

Eci=ZT

0

|ci(t)|2dt , Efe=ZT

0

|fe(t)|2dt , (7)

where ci(t)is the i-th IMF component. The dominant

component cd(t)is the IMF with the highest Eci/Efe

ratio. The Hilbert transform (HT) is then applied to

cd(t)[5]:

υd(t) = 1

πPZ∞

−∞

cd(τ)

t−τdτ , (8)

where Pindicates the Cauchy principal value. Thus,

the dominant IMF is represented as an analytic signal,

zd(t) = cd(t) + jυd(t),(9)

with amplitude ˆ

Ad, phase ˆ

φd, and instantaneous fre-

quency ˆωd, respectively estimated as

ˆ

Ad(t) = qc2

d(t) + υ2

d(t),(10)

ˆ

φd(t) = arctan υd(t)

cd(t),(11)

ˆωd(t) = ˙

φd(t).(12)

Furthermore, the extracted energy by the WEC over

a time range Tis calculated as

Ea(t) = −ZT

0

˙x(t)fp(t)dt , (13)

where ˙xis the body velocity.

III. EMD AND MODE MIXING

The calculation of the instantaneous frequency by

applying the HT (8) directly to the excitation force

signal results in negative local frequencies, as the in-

stantaneous frequency is not well deﬁned for multi-

component signals [13], i.e. signals with more than

one local extrema for each zero crossing. Thus, fe(t)

is decomposed into monocomponent signals (IMFs)

using the EMD. An intrinsic mode function is a signal

that is symmetric with respect to the local zero mean,

and has numbers of zero crossings and extrema dif-

fering at most by one. Therefore, an IMF satisﬁes the

necessary conditions for a meaningful interpretation of

the instantaneous frequency obtained from the HT [5].

The EMD algorithm identiﬁes local maxima and

minima of fe(t), and calculates upper and lower en-

velopes for such extrema using cubic splines. The

mean values of the envelopes are used to decompose

the original signal into IMFs, in a sequence from the

highest frequency component to the lowest frequency

component. The EMD algorithm is summarized in

Table I. The steps 1 to 5 are known as sifting process.

TABLE I

EMD ALGORITHM.

Step 0: Set i=1;r(t)=fe(t);

Step 1: Identify the local maxima and minima in r(t);

Step 2: Calculate the upper envelope deﬁned by the maxima,

and the lower envelope deﬁned by the minima;

Step 3: Calculate the mean envelope m(t);

Step 4: Set h(t)=r(t)−m(t);

Step 5: If h(t)is an IMF, go to next step. Otherwise, set r(t)= h(t)

and go back to step 1;

Step 6: Set ci(t)=h(t);r(t)=r(t)−ci(t);

Step 7: If i=N, deﬁne the IMF components as c1(t),...,cN(t),

and the residue as r(t). Otherwise, set i=i+ 1 and

go back to step 1.

The decomposition process may result in mode mix-

ing, and an IMF will consist of signals of widely

disparate frequency scales, or different IMFs will con-

sist of signals with similar frequency scales. Mode

GARCIA-ROSA et al.: DISCUSSION ON THE MODE MIXING IN WAVE ENERGY CONTROL SYSTEMS USING THE HILBERT-HUANG TRANSFORM

mixing can be caused by an intermittent/noisy signal,

by speciﬁc amplitude and frequency relations of the

original modes in the signal, or by a combination of

both cases. The second case is considered in this study.

In this framework, [10] presents a mathematical

analysis that shows how EMD separates the original

modes in signals with two frequency components. Fig-

ure 1 illustrates a summary of how EMD behaves for

two-tone signals. Here, a two-tone signal represents the

normalized excitation force for two wave components,

¯

fe(t) = acos (ω1t+ϕ) + cos ω2t , (14)

a=a1|He(ω1)|(a2|He(ω2)|)−1,(15)

where ω1is the low-frequency (LF) component, ω2

is the high-frequency (HF) component, ϕ=ϕ1−ϕ2,

ϕ1=∠He(ω1),ϕ2=∠He(ω2), and a1,a2, and ϕ1,

ϕ2are, respectively, the amplitudes and phases of LF

and HF components. Following [10], the frequencies

{ω1, ω2}and phases {ϕ1, ϕ2}are much lower than the

sampling frequency and then, the EMD behaviour is

only sensitive to the relative parameters ω1/ω2and ϕ.

If the two components are correctly separated by

the EMD, then the ﬁrst IMF (c1(t)) matches the high-

frequency component of the signal (cos ω2t). Two well-

deﬁned regions (Fig. 1.a) are observed depending on

the amplitude and frequency ratios of original modes:

the white region indicates a perfect separation of the

components (p1≈0), and the darkest region indicates

the separation is not perfect (p1≈1). In such a case, the

EMD either consider the signal as a single modulated

component or it does something else. The separation of

the components is not possible if either the frequency

ratio is above the cutoff frequency ωc= 0.67 (dotted

line) or the frequency and amplitude ratios are in the

area above aω1

ω22=1 (dashed line) [10].

In addition, it is important to distinguish between

the single modulated component area and the area

which EMD does something else. Notice that if the

signal is a single modulated component, then the ﬁrst

IMF should be the original signal ( ¯

fe(t)). Figure 1.b

illustrates a comparison of the ﬁrst IMF with the orig-

inal signal for the areas ω1

ω2≥ωcand aω1

ω22≥1(cases

when the separation of components is not perfect).

Therefore, three domains are deﬁned [10]: (D1) the

components are separated and correctly identiﬁed (p1≈

0); (D2) the components are considered as a single

waveform (p2≈0); (D3) the EMD does something else

(the darkest areas in Fig. 1.b, i.e., p2≈1). In domain D3,

the components are either halfway between domains

D1 and D2 or contain fake oscillations, i.e., oscillations

derived from the decomposition process and not from

the original modes.

IV. EMD M OD E MI XI NG E FFE CT O N TH E EN ER GY

ABSORBED BY WECS

In this section, we investigate the effect of EMD

mode mixing on the energy absorbed by WECs using

the HHT for tuning purposes in a passive control

strategy. The analysis will initially focus on synthetic

two-tone waves and the domains to which the mode

Fig. 1. EMD behavior for two tones signal (a) performance measure

of separation: p1=||c1(t)−cos ω2t||

||acos ω1t|| , (b) distance measure of the ﬁrst

IMF to the original signal: p2=||c1(t)−¯

fe(t)||

|| cos ω2t|| . Dotted line: ωc=0.67,

dashed line: aω1

ω22= 1. (The plots are reproduced from [10] with

10 sifting iterations and sampling period of 1ms).

mixing belongs (D2 and D3), and then it is extended

to real ocean waves that exhibit similar properties as

the synthetic two-tone waves. The ﬂoating body is a

heaving cylinder with mass m= 3.2×105kg, radius of

5meters, draught of 4meters, and resonance frequency

1.2rad/s [1].

A. Two-tone waves

By using two-tone waves in domains D2, and D3, as

incident waves, Figures 2, and 3, compare the energy

absorbed by the WEC over T= 10 min for the cases

when HHT is used for tuning purposes (Ea) with the

cases when a constant dominant frequency is used

(Ea,D). The constant dominant frequency is either ω1

or ω2depending on which component has the highest

excitation force amplitude. Only domains D2 and D3

are considered, as in domain D1 the components are

well separated and identiﬁed by the EMD. Four differ-

ent frequencies are considered for the HF components:

0.75, 0.9, 1.2, and 1.5 rad/s.

The effect of the EMD mode mixing on the energy

absorbed depends on both the domain (D2 or D3) and

the frequency of the components, which is related to

the frequency response of the ﬂoating body.

GARCIA-ROSA et al.: DISCUSSION ON THE MODE MIXING IN WAVE ENERGY CONTROL SYSTEMS USING THE HILBERT-HUANG TRANSFORM

For D2 and the cases in which HF components are

below the resonance frequency of the body (1.2rad/s),

the dominant IMFs are single modulated components

resulting in greater absorption of energy than tuning

to the constant dominant frequency (Fig. 2, top). The

beneﬁt of using the IMF modulated components is

higher for the cases when the amplitudes of the ex-

citation force are close to each other (regions around

log10 a≈0). However, when the resonance frequency

is the HF or the LF component (Fig. 2, bottom), the

single modulated IMFs can either have a positive or

negative effect on the energy absorbed by the WEC.

Figure 4 shows the IMF instantaneous frequency, and

the PTO damping, for the case when ω2= 1.2rad/s

and Ea/Ea,D = 0.94. It can be observed that the IMF

frequency varies from 1.1 rad/s to 1.7 rad/s, but since

the constant dominant frequency is the resonance fre-

quency of the WEC, the energy absorption with HHT

is less than with the constant frequency. Conversely,

when the dominant frequency is 0.96 rad/s (ω1=0.8ω2

and a1|He(ω1)|>a2|He(ω2)|), the HHT is superior. Table

II summarizes the results for domain D2.

TABLE II

SUMMARY OF RESULTS FOR DOMAIN D2 (ω1/ω2=0.8).

COMPONENTS ARE DECOMPOSED AS SINGLE WAVE-FO RM S.

HF Region Dominant Comparison of

(rad/s) component Eawith Ea,D

0.75 log10 a < 0HF (ω2= 0.75 rad/s) Ea> Ea,D

log10 a > 0LF (ω1= 0.6rad/s) Ea> Ea,D

0.9 log10 a < 0HF (ω2= 0.9rad/s) Ea> Ea,D

log10 a > 0LF (ω1= 0.72 rad/s) Ea> Ea,D

1.2 log10 a < 0HF (ω2= 1.2rad/s) Ea< Ea,D

log10 a > 0LF (ω1= 0.96 rad/s) Ea> Ea,D

1.5 log10 a < 0HF (ω2= 1.5rad/s) Ea> Ea,D

log10 a > 0LF (ω1= 1.2rad/s) Ea< Ea,D

Notice that the damping tuning equation (6) is not

optimal for instantaneous frequency tuning. Such an

equation represents the optimal damping equation for

regular waves [12], i.e., sinusoidal waves with a con-

stant frequency component. Further studies are needed

to determine the best damping tuning equation when

the instantaneous frequency is used as an input. As

an example, if the PTO damping oscillations below

2×105kg/s are ﬁltered out from the damping proﬁle

in Figure 4, the energy absorbed increases 3% when

compared to the original damping obtained directly

from the HHT approach and (6).

For domain D3 (Fig. 3), only the region log10 a > 0

is considered, as indicated by Fig. 1.b. Therefore, the

LF component (ω1) is the dominant component for all

the cases, and the HHT is superior in all such cases.

Nonetheless, in this domain the dominant IMFs are

either halfway between domains D1 and D2 or contain

oscillations derived from the decomposition process.

Figure 5.a shows the instantaneous frequency of the

ﬁrst and second IMFs for the case when ω2=1.2rad/s

and Ea/Ea,D = 1.18. The dominant IMF clearly has an

oscillation around 250 s that is different from the origi-

nal modes, causing an irregularity in the PTO damping

-1 -0.5 0 0.5 1

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

-1 -0.5 0 0.5 1

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

-1 -0.5 0 0.5 1

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

-1 -0.5 0 0.5 1

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

Fig. 2. Ratios between Eaand Ea,D in domain D2.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

1.01

1.02

1.03

1.04

1.05

1.06

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

Fig. 3. Ratios between Eaand Ea,D in domain D3.

proﬁle (Fig. 5.b). Although the energy absorption is

higher than the constant dominant frequency, such

irregularity decreases the energy absorption in about

8% for the time interval 225 s to 255 s, when compared

to the case the components are a single wave-form.

Figure 6 shows the instantaneous frequency and the

PTO damping for both the dominant IMF derived

from EMD (IMF 1) and the analytic single wave-form

for the case the components are together (i.e., single

modulated components).

150 200 250 300 350

0

1

2

3

4

5105

Fig. 4. Time-series for case ω2= 1.2rad/s, Ea/Ea,D = 0.94: (domain

D2, ω1/ω2=0.8) (a) Hilbert spectrum for the ﬁrst IMF, and constant

dominant frequency; (b) PTO damping.

150 200 250 300 350

0

0.5

1

1.5

2

2.5 106

Fig. 5. Time-series for case ω2=1.2rad/s, Ea/Ea,D = 1.18 (domain

D3, ω1/ω2=0.62): (a) Hilbert spectrum for the ﬁrst and second IMFs,

and constant dominant frequency; (b) PTO damping.

B. Real ocean waves

Previous studies have shown that tuning the damp-

ing with frequency estimates from the HHT results

in greater absorption of energy than tuning with es-

timates from EKF and FLL [2]. Here, the aim is to use

the mode mixing conditions observed for the synthetic

two-tone wave studies to further improve the energy

absorbed by the HHT for real ocean waves that fulﬁll

similar conditions. Therefore, we adopt two sea states

(S1, S2) from the northeast coast of Brazil for the

simulations. The wave elevation data consist of records

of about 20 minutes (T≈20 min) sampled at 1.28 Hz

225 230 235 240 245 250 255

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

225 230 235 240 245 250 255

0

0.5

1

1.5

2

2.5 106

Fig. 6. (a) Instantaneous frequency, and (b) PTO damping for both

the dominant IMF derived from EMD (solid line) and the analytic

single wave-form (dashed line).

0 1 2 3 4

0

0.05

0.1

0.15

0 1 2 3 4

0

1

2

3

4

1010

01234

0

0.05

0.1

0.15

0.2

01234

0

1

2

3

4

1010

Fig. 7. Wave spectra from the northeast coast of Brazil (top) and

excitation force spectra (bottom).

and registered by a wave buoy in the Port of Pec´

em

area. Figure 7 shows two-peak wave spectra and the

excitation force spectra for the cylinder. The frequency

ratio for the peak frequencies (ωp1/ωp2) of the spectra

are, 0.62 and 0.5, respectively, for sea states S1 and S2.

Figure 8 shows samples of time-series simulations

for the excitation forces fe, and the instantaneous

frequencies for IMF 1 ( ˆωd, dominant IMF) and IMF 2.

It can be noted that the IMFs consist of signals with

similar frequency scales (mode mixing), and a simi-

lar behaviour between the real ocean waves and the

synthetic two-tone waves (Fig. 5.a) is observed around

690 s (Fig. 8.a bottom) and 660 s (Fig. 8.b bottom).

The mode mixing could be attenuated by applying

masking signals to the EMD, as done in, e.g., [8].

Conversely, here the mode mixing in domain D2 is

beneﬁcial to the energy absorption. However, the small

oscillations on the PTO proﬁle, caused either by the

mode mixing in domain D3, or by the nature of the

damping tuning equation (6), have a negative impact

on the energy. Therefore, such oscillations are ﬁltered

out. Figure 9 shows the evolution of the PTO damping

Bp(t), and the body motion x(t)for the cases when

650 700 750

-2

-1

0

1

2

105

650 700 750

0

1

2

3

4

650 700 750

-2

-1

0

1

2105

650 700 750

0

1

2

3

4

Fig. 8. Time-series of the excitation force, and instantaneous fre-

quency of IMF 1 (dominant IMF) and IMF 2: (a) S1; (b) S2.

650 700 750

0

0.5

1

1.5

2106

650 700 750

-0.5

0

0.5

650 700 750

0

2

4

6106

650 700 750

-0.5

0

0.5

Fig. 9. Time-series of the PTO damping, and body motion: (a) S1;

(b) S2. Solid lines: PC with HHT, dashed lines: PC with HHT and

ﬁltering of small damping oscillations.

the PTO damping is directly obtained from HHT and

(6) (solid lines), and when the small PTO damping

oscillations are ﬁltered out (dashed lines). An energy

improvement of about 2.6%, and 3.9%, is obtained

for sea states S1, and S2, when the small damping

oscillations are ﬁltered out, which agrees with the two-

tone signal studies in Section IV-A.

V. CONCLUSION

This study revealed how the EMD mode mixing

problem affects the performance of WECs with a pas-

sive control strategy that tunes the PTO damping to

frequency estimates from the HHT. The effect of mode

mixing on the energy absorbed depends both on the

type of IMF derived from the EMD procedure and the

frequencies of the components (which is related to the

frequency response of the ﬂoating body).

A comprehensive numerical study using synthetic

two-tone waves showed that for mode mixing cases

where components are decomposed as single wave-

forms (domain 2), the effect is beneﬁcial to the absorp-

tion of energy if the constant dominant frequency is not

the resonance frequency of the WEC. Additionally, the

damping tuning law might result in small oscillations

on the PTO proﬁle that will negatively impact the

energy absorption. For cases where the components

are within two IMFs, or contain oscillations derived

from the decomposition process (domain 3), the mode

mixing effects might result in irregularities in the PTO

damping proﬁle that also decrease the absorption of

energy. If the mode mixing causes a high peak in the

instantaneous frequency of an IMF, the PTO damping

will contain a small oscillation that is otherwise not

present in the PTO damping proﬁle.

The studies with synthetic two-tone waves were

used to further improve the HHT absorbed energy for

real ocean waves by ﬁltering out the small oscillations

on the PTO proﬁle. An average energy improvement

of 3.3% was obtained for two studied cases. Such

oscillations in the damping are caused either by high-

frequency oscillations derived from the EMD process

or by the nature of the damping tuning equation (that

is not optimized for the instantaneous frequency). Fur-

ther studies are needed to determine the best damping

tuning equation when the instantaneous frequency of

waves is used as an input for the PTO tuning.

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