24.4 Adaptive Tuning

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The schematic of this method is shown in

Figure 24.6 (Wang et al., 2005). As in the other

methods, it uses a process model as the basis of

search for the appropriate blade adjustments, but

instead of using a linear model, it uses an interval

model to accommodate process nonlinearity and

measurement noise. According to this model, the

feasible region of the process is estimated first, to

include the adjustments that will result in

acceptable vibration estimates. This feasible region

is then used to search for the blade adjustments

that will minimize the modeled vibration. If the

application of these adjustments does not result

in satisfactory vibration, the interval model will

be updated to better estimate the feasible region and improve the choice of blade adjustments for the

next flight. Important parameters of adaptive tuning are summarized in Table 24.3.

24.4.1 The Interval Model

In order to account for the stochastics and nonlinearity of vibration, an interval model (Moore, 1979) is

defined to represent the range of aircraft vibration caused by blade adjustments. The interval model used

here has the form:

Dyyj ¼

Xn

i¼1

CyjiDxi; j ¼ 1; …; m ð24:11Þ

where each coefficient is defined as an interval:

Cyji ¼ ½CLji; CUji􀀉

In the above model, the variables with the two-sided arrow, $; denote intervalled variables, CLji and CUji

represent, respectively, the current values of the lower and upper bounds of the sensitivity coefficients

between each input, Dxi; and output, Dyyj: The interval Dyyj denotes the estimated range of change of the

jth output caused by the change to the current inputs, Dx1; …; Dxn:

Helicopter

Search

Learning

Algorithm

aircraft vibration

estimated aircraft

vibration

blade

modifications

Interval

Model

FIGURE 24.6 The strategy of the proposed tuning

method.

TABLE 24.3 Summary of Adaptive Tuning

In adaptive tuning, each vibration component is defined as

Dyyj ¼

Pni

¼1

Cyji Dxi ; j ¼ 1; …; m

where each coefficient is defined as an interval:

Cyji ¼ ½CLji ; CUji 􀀉

with CLji and CUji representing, respectively, the current values of the lower and upper bounds of the sensitivity coefficients

between each input, Dxi ; and output, Dyyj : The blade adjustments are then sought by minimizing the objective function:

S ¼

PNe

􀀄Q e¼1 Distanceðxc; xeÞ Ns

s¼1 Distanceðxc; xs Þ

􀀅1=Ns

where xc represents a candidate set of blade adjustments within the feasible region, xe represents any set of blade adjustments

within the selection region, xs denotes each of the previously selected blade adjustments, and Ne and Ns represent the number

of the estimated feasible blade adjustments and the previously selected blade adjustments, respectively.

24-8 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

The fit provided by the interval model for a mildly nonlinear input/output relationship is illustrated in

Figure 24.7, where the output range is estimated relative to one explored input.1 According to Equation

24.11, the estimated range of the output becomes larger, and therefore less accurate, as the potential input

is selected farther from the current input (producing a large Dxi). This potential drawback of the interval

model is considerably reduced when multiple inputs have been explored so that the interval model can

take advantage of several inputs for estimating the output range. The estimated output, yyj; at a potential

input, xi; may be computed relative to any set of previously explored inputs, yielding different estimates

of yyj (due to different values of Dxi ). In order to cope with the multiplicity of estimates, yyj is defined as the

common range among all of the yyj estimates (Yang, 2000). The estimation of yyj using this commonality

rule is illustrated in Figure 24.8, which indicates that using this estimation approach enables

representation of the system nonlinearities in a piecewise fashion. It can be shown that the lack of

commonality between the estimated ranges of output will cause a part of the input – output relationship

to not be represented by the interval model. In such cases, however, the lack of compliance between the

interval model and the input – output relationship can be corrected by adaptation of the coefficient

intervals through learning.

24.4.2 Estimation of Feasible Region

The feasible region comprises all sets of blade adjustments that will reduce the aircraft vibration within

specifications. The feasible region is estimated here by comparing the individually estimated yyj values

with their corresponding constraints, so as to decide whether the corresponding blade adjustments

belong to the feasible region. In this method, even when the interval yyj partly overlaps the vibration

constraint, the corresponding blade adjustments are included in the estimated feasible region. The above

procedure of estimating the feasible region based on individual outputs is then extended to multiple

outputs by forming the conjunction of the estimated feasible regions from each output.

24.4.3 Selection of Blade Adjustments

The blade adjustments provide the coordinates of the feasible region, therefore, they need to provide a

balanced coverage of the input space. As such, blade adjustment selection becomes synonymous with

maximizing the distance of the selected blade adjustments from the previous blade adjustments, as well as

−4 −3 −2 −1 0 3

0

0.5

1

1.5

2

Output

Input

Actual input-output relationship

Explored input-output pair

Estimated range of output

1 2 4

FIGURE 24.7 Estimated range of output by the interval model using one reference input.

1An explored input represents an input for which the exact value of the output is available. In rotor tuning, an explored

input would denote a blade adjustment that has been applied to the helicopter, and for which the corresponding vibration

changes have been measured.

Helicopter Rotor Tuning 24-9

© 2005 by Taylor & Francis Group, LLC

bringing them closer to the center of the feasible region. This objective can be pursued by minimizing the

following objective function:

S ¼

XNe

e¼1 Distanceðxc; x Y eÞ Ns

s¼1 Distanceðxc; xsÞ

􀀍 􀀎1=Ns ð24:12Þ

where xc represents a candidate set of blade adjustments within the feasible region, xe represents any set of

blade adjustments within the selection region, xs denotes each of the previously selected blade

adjustments, and Ne and Ns represent, respectively, the numbers of the estimated feasible blade

adjustments and the previously selected blade adjustments. Note that when the candidate set, xc; is close

to the previously selected blade adjustments,

􀀄 QNs

s¼1 Distanceðxc; xsÞ

􀀅1=Ns becomes small, and when the

candidate set of blade adjustments, xc; is far from the center of the feasible region, the value of

PNe

e¼1

Distanceðxc; xeÞ becomes large. By minimizing S, the candidate blade adjustments are selected such that

the above extremes are avoided.

24.4.4 Learning

Although an interval model defined according to the sensitivity coefficients may provide a suitable initial

basis for tuning, it may not be the most representative of the rotor tuning process. As such, it may not be

able to carry the search process to the end. A noted feature of the proposed method is its learning

capability, which enables it to refine its knowledge base. To this end, the coefficients of the model are

updated by considering new values for each of the upper and lower limits of individual coefficients. The

objective is to make the range of the coefficients as small as possible while making sure that the interval

model envelopes the acquired input – output data. The learning problem can be defined as

Minimize E ¼

KX21

m¼1

XK

k.m

{½yLðm; kÞ 2 yðkÞ􀀉2 þ ½yUðm; kÞ 2 yðkÞ􀀉2} ð24:13Þ

subject to

yUðm; kÞ $ yðkÞ ð24:14Þ

yLðm; kÞ # yðkÞ ð24:15Þ

CUi 2 g $ CLi ð24:16Þ

where K represents the total number of sample points collected so far, yLðm; kÞ and yUðm; kÞ represent,

respectively, the lower and upper limits of the estimated output range at the kth sample point relative to

−5 0

0

0.2

0.4

0.6

0.8

1

Input

Output

Actual input-output relationship

Explored input-output pairs

Estimated range of output

5

FIGURE 24.8 Estimated range of output by the interval model using seven reference inputs.

24-10 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

the mth sample point, yðkÞ denotes the actual output value at the kth sample point, and CUi and CLi

represent the upper and lower limits of the ith coefficient interval, respectively. The parameter g is a small

positive number to control the range of the coefficients.

Most of the approaches that can be potentially used for adapting the coefficient intervals, such as

gradient descent (Ishibuchi et al., 1993) or nonlinear programming, cannot be applied to rotor tuning

due to their demand for rich training data and their impartiality to the initial value of the coefficients

representing the a priori knowledge of the process. As an alternative, a learning algorithm is devised

here to cope with the scarcity of track and balance data while staying true to the initial values of

the coefficients. In this algorithm, the coefficients of the interval model, initially set pointwise at the

sensitivity coefficients, are adapted after each flight in two steps: enlargement and shrinkage. First, the

vibration measurements from all of the flights completed for the present tail number are matched against

the estimated output ranges from the current interval model. If any of the measurements do not fit the

upper or lower limits of the estimates, the coefficient intervals are enlarged in small steps, iteratively, and

the output ranges are re-estimated at each iteration using the updated interval model. The enlargement

of the coefficient intervals stops when the estimated output ranges include all of the measurements. At

this point, even though the updated interval model provides a fit for the input – output data, it may be

overcompensated. In order to rectify this situation, the coefficient intervals are shrunk individually by

selecting new candidates for their upper and lower limits.

The shrinkage – enlargement learning algorithm has the form:

DCLi ¼ 2hdLDxiðm; kÞ ð24:17Þ

DCUi ¼ 2hdUDxiðm; kÞ ð24:18Þ

where, during the enlargement phase, dL and dU are defined as

dL ¼

DyL If Dxiðm; kÞ . 0 and DyL . 0

DyU If Dxiðm; kÞ , 0 and DyU , 0

0 otherwise

8>><

>>:

ð24:19Þ

dU ¼

DyU If Dxiðm; kÞ . 0 and DyU , 0

DyL If Dxiðm; kÞ , 0 and DyL . 0

0 otherwise

8>><

>>:

ð24:20Þ

and during the shrinkage phase, they are defined as

dL ¼

DyL If Dxiðm; kÞ . 0 and DyL , 0

DyU If Dxiðm; kÞ , 0 and DyU . 0

0 otherwise

8>><

>>:

ð24:21Þ

dU ¼

DyU If Dxiðm; kÞ . 0 and DyU . 0

DyL If Dxiðm; kÞ , 0 and DyL , 0

0 otherwise

8>><

>>:

ð24:22Þ

with

Dxiðm; kÞ ¼ xiðkÞ 2 xiðmÞ ð24:23Þ

DyL ¼ yLðm; kÞ 2 yðkÞ ð24:24Þ

DyU ¼ yUðm; kÞ 2 yðkÞ ð24:25Þ

Helicopter Rotor Tuning 24-11

© 2005 by Taylor & Francis Group, LLC

This procedure is repeated for each coefficient interval in an iterative fashion until the objective function

E (Equation 24.13) is minimized. The minimization of E ensures limited adaptation of the coefficient

intervals within the smallest possible range.

At the beginning of tuning, the limited number of input – output data available for learning will not

provide a comprehensive representation of the process. Therefore, the coefficient intervals should not be

shrunk drastically until enough input – output data have become available. For this, the length of each

coefficient interval ½CLi; CUi􀀉 is constrained by the minimal interval length for each tuning iteration as

min L ¼ {CUið0Þ 2 CLið0Þ}ð1 2 bÞn ð24:26Þ

where b [ ½0; 1􀀉 controls the shrinkage rate of the coefficient interval, and n denotes the number of

tuning iterations. The coefficient interval cannot be shrunk when b ¼ 0 and can be shrunk without limit

when b ¼ 1: Usually, b is selected closer to 0.

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