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Online Identification of Active Absorbers in Automotive Vibration Control
Copyright @ IFAC Advances in Automotive Control, Karlsruhe, Gennany, 2001
ONLINE IDENTIFICATION OF ACTIVE ABSORBERS IN AUT0MOTIVE VIBRATION CONTROL Maik Buttelmann· Boris Lohmann· Peter Marienfeld ••• Ferdinand Svaricek·· Marat Vinogradski· Nikola Nedeljkovic·
• Institut filr A utomatisierungstechnik, Universitiit Bremen, Germany •• Continental A G, Hannover, Germany ••• ContiTech Vibration Control GmbH, Hannover, Germany
Abstract: In the past , engine-related noise and vibration in the vehicle cabin was exclusively reduced by passive absorption. Today, modern actuators and control systems make an active noise reduction possible by introducing counteracting vibration at 180 degrees phase lag. Within a cooperation of the Institute of Automation Systems and Continental AG , an approach using active absorbers at the engine mounts is investigated. As the dynamic behaviour of the active absorbers and other elements in the secondary path are time-variant (depending on temperature , age and other factors) , an online identification is carried out. By this, the implemented feedforward control strategy is supported on a precise and frequently updated model of the secondary path. The chosen approaches to online and omine identification are presented together with first results achieved in online identification and with the overall control system. Copyright ~20011FAC 1. INTRODUCTION
control (Karkosch et al. (1999); Eberhard et al. (1999); Kuo and Morgan (1996)) . The basic principle of this technique is to introduce artificially produced noise with the same intensity as the original noise but with a phase lag of 180 degrees into the target system. The target system can be the passenger cabin (acoustic noise) or the vehicle body (vibration). The objective in both situations is to reduce noise/vibration noticed by the passengers. This principle - applied to the particular problem of reducing engine-related vibration in a vehicle - is explained in Figure 1: From the measured vibration signal (primary noise signal) at the engine, x(t), the block "controller" generates a suitable control input signal y(t) to the amplifier and active absorber, producing compensating vibration y' (the signal path from y to e is called the secondary path). The error signal e indicates the effectiveness of noise compensation and makes corrections - so-called adaptation - of the control
During the past years, a trend towards smaller and lighter vehicles can be observed. The obvious advantages of small vehicles are less fuel consumption, lower price, and better suitability for heavy urban traffic (Eberhard et al. (1999)). On the other hand, use of lightweight constructions and constrained space for engine and other functional parts introduce new challenges in decreasing noise and vibration impact on passenger comfort. In parallel, resulting from increased consumer awareness of long-term health impacts from exposure to high noise and vibration levels, the demands shift towards vehicles with better sound quality and smoother ride. The trends towards smaller and lighter cars with less engine related noise in the vehicle cabin include physically opposite demands. A possible solution is the use of active noise and vibration 187
m
_
_
___ o m
rvvv·
_X_('... } _~
primary path t-_d_(t_}_ _ _ _
_.-< +
p
Sl'Condary path S
y"
identifiution
of 5 a.bptation
Fig. 1. The principal components of the active vibration control system
Fig. 2. Feedforward control structure
algorithm possible. However , adaptation requires a precise knowledge of the dynamic behaviour of the secondary path. Gaining this knowledge is the focus of this paper: As the dynamics in the signal path from y to e are slowly time-variant, an online identification of the corresponding dynamic system is required.
so called primary path P, as shown in figure 2. As x(t) is available by measurement, a filter W can be used to generate a suitable control input signal y(t) to the secondary path S, producing the compensating noise y' . The primary path P can be considered linear but slowly time-variant and therefore only partly known (this time-variance mainly results from rubber parts with stiffness depending on temperature and other influences) . It is therefore necessary to adapt and thereby optimize the filter W frequently. This can be done by incorporating the error signal e(t) into the control structure. The adaptation makes use of the error signal e(t) measured at the chassis close to the active absorber. Adaptation algorithms for online applications were proposed by many authors; an excellent overview with focus on active noise control is given by Kuo and :\lorgan (1996).
The paper is organized as follows: Section 2 gives an overview of the control strategy and highlights the necessity of an online identification of the secondary path. Section 3 introduces the basic principle of identification, as used here, together with a recursive algorithm particularly suitable for real-time implementation. Section 4 presents details of the implementation and results achieved. Section 5 draws some conclusions and gives an outlook to future work.
Figure 2 (without the block "identification" ) shows the control system structure as described so far . In order to further improve the filter W by adaptation , a precise dynamic model of the secondary path S is needed. This can be obtained by system identification, providing a model of S from an analysis of the error signal e(t) and the absorber input signal y(t). Since also the transfer function S turns out to be linear but slowly time-variant, the identification is to be repeated frequently by the hard- and software on board the vehicle. Figure 2 illustrates how the block "identification" interacts with the other functional hlocks in the final control system.
2. FEEDFORWARD CONTROL STRUCTURE WITH ADAPTATION AND IDENTIFICATION The main way of engine vibration propagation trough the chassis into the passenger compartment is through the engine mounts. Therefore , the developers of the mounts have to find a compromise between two excluding objectives: The constrained space requires stiff mounts; the acoustic comfort considerations require mounts that absorb most of the vibrations produced by the engine. As stated, in such situations it is possible (Karkosch et al. (1999); Eberhard et al. (1999) ) to improve vibration reduction by active vibration/noise control.
As an alternative to the proposed feedforward structure one might think of a pure feedback configuration , exclusively using the error signal e(t) to generate the absorber input signal y. However , in general such feedback approaches are more critical with respect to stability and performance. Particularly, if the transfer functions P and S are not precisely known, the demand for precise phase lag of 180 0 can not be fulfilled . Phase errors can even lead to unstability, while the feedforward approach without adaptation is inherently stable.
In general, when designing a disturbance attenuating control system , it is advantageous to measure and consider all available disturbances. In the particular case of engine-related noise, a sensor directly installed at the engine delivers a suitable disturbance signal, the so-called primary noise signal x(t). The vibration signal d(t) at the mount can be considered as originating from x(t) by passing through the disturbance transfer path , the 188
For this reason : feedback approaches are not considered in more detail here, although they are the only possible solution if the source of the noise is unaccessible to measurement.
Calculating the sums in the terms iJ!TiJ! and iJ!T y and inverting the matrix iJ!TiJ! are memory and time consuming processes. A reduction of the identification algorithm computing effort is necessary for its viability in an environment like a ty- :cal passenger car microcontroller where memory and computing power are limited resources. Also important are real time, or near real time characteristics. Introducing a recursive calculation method for 8 reduces the computing and memory effort essentially. The algorithm can be derived from the equations given above (Isermann (1992); Ljung (1999)). The vector 8 is then calculated in three consecutive steps:
3. IDENTIFICATION The dynamic behavior of the secondary path S (as well as the dynamics of P) partly depends on the stiffness of rubber parts in the signal path. Slowly changing factors as temperature and age influence the rubber part stiffness substantially. Nonlinear effects, however, do not have much influence, as the quality of the results will show. Therefore, the dynamic of the secondary path can be assumed linear and slowly changing in time. This allows the considered transfer function S to be described by a linear parametric model. Since the model S is part of an digital control system , it is formulated as a difference equation, (Isermann (1992); Ljung (1999)): y(k)
P(k)1jJ(k ')'(k)
1)
+ ... + bmu(k -
m),
8(k
+ 1) = 8(k) + ')'(k)[y(k + 1)
P(k
+ 1) = [I -
(3)
According to the preciding equations, the vector 0 contains the parameters (aI .. a m , bI ... bm ), and the (N, 2m)-Matrix iJ! is built with measured values of u(.) and y(.) at the corresponding sample times according to (1) . Usually, N is chosen to be substantially larger than 2m which makes the linear equations system over-determined (because of unavoidable measurement disturbances). A solution 8 of minimum norm of the error vector
e
=y -
1), ... ,y(k - m)
m)f·
The described recursive algorithm was successfully implemented in a test car and in a laboratory model. The implementation is described in the next section. Alternative identification methods considering/identifying specific disturbance characteristics were not considered in this work. This is justifiable, since only small disturbances were observed and excellent identification results were achieved with the models described above, even with relatively low system orders.
iJ!O
is given by (see for example (Isermann (1992))) :
8 = [iJ!TiJ!]-I
(6)
These steps are repeated N times while measurement of u(.) and y(.) is in progress. Each cycle consists of measurement of y(k) and u(k): construction of 1jJ(k + 1), calculation of P(k) and 8(k + I) , and calculation of P(k + 1) v:.hich is needed for the next cycle. Only P (k + 1), O( k + 1) and 1jJ(k + 1) have to be saved for use in the next calculation cycle, which is an fundamental decrease in memory demand compared with the classical method where all measured data have to be stored. Contrary to the classical method where 8 can be estimated at any point in time with k > 2m , the usage of recursive method requires initial values. This can be estimated (or calculated with the classical method from a small number of measured values (Isermann (1992))). In most cases the use of the simple estimation P(O) = oI where 0 is a large positive number and 8(0) = 0 is sufficient if the number of calculation cycles is large.
(2)
+ 1) ... , y(k - 1), y(k)f .
= [y(k -
I u(k -1), ... ,u(k -
with y = [y(k - N
(5)
(1) 1jJ(k)
= iJ!O
')'(k)1jJT(k
(4)
where
where y(.) and u(.) denote the sequences of sampled output and input signal values of the system, and k, k - 1, ... , k - m indicate the discrete uniformly spaced sample times. The task of parameter identification is now to find the 2m parameters (al .. a m , bl ... bm ) such that the real system behaviour is described as accurately as possible by the difference equation (1). In order to do this, equation (1) is considered at N sample times to build a system of linear equations with known values of y and u . This system of equations is easily written in vector notation: y
+ 1)8(k)] + l)]P(k) .
_1jJT(k
+ aly(k - 1) + ... + amy(k - m)
= b1u(k -
+ 1)
= 1jJT(k + l)P(k)1jJ(k + 1) + 1
. iJ!T y. 189
InputSignal(generated) 0.03 0.02 Q)
~
0.01
C
0
Cl
E-0.01 -0.02
2
time[s]
Outputsignal( measured) 0.4 Q)
r----~-r.------------,
0.2
"0
~ C
0 1--vNI,._
Cl
co
E
Fig. 3. Laboratory setup with engine mount, active absorber and error sensor. For vibration decoupling the whole equipment is mounted on an massive steel table
-0.2 -0.4
L -_ _ _ _...IL._ _ _
o
~
2
_ __
_ _---J
time[s]
Fig. 4. Test signal (Franklin chirp) and measured output signal (e in figure 2)
4. IMPLEMENTATION AND RESULTS
The combined SIMULlNK/dSPACE tools provide the user wit h an user-friendly environment . However, due to some restrictions in matrix/vector computations, the implementation of the recursive algorithms required some workarounds for online use . As a consequence, the implemented recursive algorithm showed a relatively high demand of computing power, even on a DSll04 (PowerPC) model. The use of C or C+ + programming within so-called S-functions might be a remedy.
In the scope of previous research and development at Continental AG (Karkosch et al. (1999) ; Eberhard et al. (1999)) an active absorber system was already implemented in a standard vehicle. Within a SIMULINK/dSPACEenvironment the feedforward control structure as described in section 2 was implemented as a part of the active absorber system. By this, the programming environment for the identification algorithm was predetermined ; the choice of SIMULINK should further assure easy maintenance and development of the programs. In parallel a laboratory system was built to allow for test of different configurations of active absorber and disturbance sensor positions . In both cases the active absorbers were produced by Continental AG . A laboratory setup with absorber, engine mount , and disturbance sensor is shown in figure 3.
Since no physical system model is available, the "correct" model order is unknown . Therefore, the "best" model order was determined iteratively. Identifications with different model orders were carried out and model responses to the test signal were determined. The applicability of the used model order was assessed through comparison of the model response with the measured system response. In figure 5 model response and the squared deviation from measured response are displayed for three different orders. As an extension of the approach (1) , not only the order m was iteratively changed , but also the number of non-zero coefficients bv in (1) was altered , i.e. the grades of numerator and denominator of the corresponding z-transfer-function were prescribed separately.
For easy insertion of the new identification block into the existing digital controller the same sample time of ts = 0.0028 was chosen. A sufficient identification of the system is possible only if the test signal can excite all vibration modes of the examined system in the considered frequen cy range. It was found in previous work that the engine vibrations are in the range of 30 to 300Hz. Thus, an accurate description of the secondary path in this frequenca range is necessary. To excite the given range in possibly short period of time a sinus sweep signal with growing frequency from 30 to 300Hz was chosen. The signal generator was implemented in the dSP ACE system as described in (Franklin (1990))(Franklin chirp) . Figure 4 shows the Franklin chirp input signal and the measured output after the signal passed the secondary path (S in the figure 2) .
The experiments with active absorber implemented in a car were carried out with the car and its engine in standstill. Both systems (laboratory and in the car implemented active absorber) were excited solely by the sinus sweep signal applied to active absorber. As shown in figure 5, models with relatively low order (with respect to to the complexity of examined mechanical system) give reasonable signal 190
30r-----------------~-~
m 20
~ 10
x10 -3
-0
.a
0
·c::J) -10 ] -20
~
-30L...----------~----------~----~
o
100
200
frequency[Hz) 0.4 ~ 0.2
.a ·c
0
Ol
O~~--------~----------~----~
-50
I-voNj._
ID III -100 ro
ro
.I::.
E -O.2
0.
-0.4 '--_ _ _~_ _ _ _~_ _ _ _ _ _____l
o
2
-150 -200L----------~------~-~
o
timers)
100
200
frequency[Hz)
a) x10 -3
..e :1··
2· .
Q)
: .
J . . . .u,. :
I •
~.,
j: . ......
Fig. 6. Bode plot of identified model with numerator~denominator grade 10 - 11 ....
responses. As a sufficient result for integration into the adaptation algorithm the system with order 10 - 11 (numerator ~ denominator orders) was chosen (model response shown in Figure 5.b). Although the maximal error is larger than with order 6 - 11 , smaller overall error justifies the choice. Increasing the model order provides better overall response, however, the increased accuracy does not justify the substantially higher demand of computing power for its identification. Figure 6 shows the Bode plot of the identified model with order 10 - 11.
1
0.4 ~ 0.2
.a ·c
01-__. _
Ol
ro E-0.2 -0.4 L..._ _ _~_ _ _ _--,-_ _ _ _ _ _-.J
o
2
timers)
b) x10 ' 3
Figure 7 shows a comparison between the measured engine produced vibrations in the chassis with and without an active vibration control applied. As displayed , with active vibration control a substantial improvement in comparison to the passive approach can be achieved.
....
1 0.4
1.4
~ 0.2
.a ·c
~--..,...--.---..,...---,.---..,...---.----....
0 1-___•
Ol
ro E-0.2 -0.4 L..._ _ _~~_ _ _--,-_ _ _ _ _ _- - i
o
2
timers)
c) Fig. 5. Calculated output signals and the squared difference between the signal and the measured output (error) for models with different numerator~denominator orders: a) 6 - 11; b) 10 - 11; and c) 33 - 34. Fig. 7. Chassis vibrations measured with and without active vibration control 191
5. SUMMARY AND OUTLOOK As already stated, the models used for identification do not take any disturbances into account. This approach is justified for the problem considered here and delivers excellent results with relatively low system order, as described in section 4. The chosen model with order 10-11 reproduces the output of the real system accurately; it is considerably simple and therefore can be easily embedded in a real-time environment to support the filter adaptation. For the disturbed system case (e.g. with engine running), extended models including modeling of the disturbances themselves can be implemented and tested in the future. Provided the disturbance x can be measured as assumed in this paper, the described feedforward approach represents a fast response and a stable solution. However, vibrations from other sources like road or drive train can not easily be measured. In such case the feedback approach is the only possible solution to the vibration attenuation problem in the car chassis. Further work considering feasibility of feedback control approaches for active vibration attenuation will be carried out in the near future.
References Eberhard G., H.-J. Karkosh, F. Svaricek, RShoureshi, J.L. Vance. (1999) Komfortverbesserung im Kfz durch Einsatz moderner Regelungs- und Steuerungstechnik. Adaptronic Congress. Potsdam, Germany. Franklin G.F. (1990) Digital control of dynamic systems. Addison-Wesley, New-York Isermann R(1992) Identifikation dynamischer Systeme 1. Springer-Verlag, Berlin Karkosh H.-J., F. Svaricek, R.A. Shoureshi, J.L. Vance. (2000) Automotive Applications of Active Vibration Control. Proceedings of the European control conference. Karlsruhe, Germany. Kuo S.M. , D.R Morgan. (1996) Active Noise Control Systems. John Wiley & Sons, New York Ljung L.(1999) System Identification - Theory For the User. Prcntice Hall , Upper Saddle River, N.J.
192
ONLINE IDENTIFICATION OF ACTIVE ABSORBERS IN AUT0MOTIVE VIBRATION CONTROL Maik Buttelmann· Boris Lohmann· Peter Marienfeld ••• Ferdinand Svaricek·· Marat Vinogradski· Nikola Nedeljkovic·
• Institut filr A utomatisierungstechnik, Universitiit Bremen, Germany •• Continental A G, Hannover, Germany ••• ContiTech Vibration Control GmbH, Hannover, Germany
Abstract: In the past , engine-related noise and vibration in the vehicle cabin was exclusively reduced by passive absorption. Today, modern actuators and control systems make an active noise reduction possible by introducing counteracting vibration at 180 degrees phase lag. Within a cooperation of the Institute of Automation Systems and Continental AG , an approach using active absorbers at the engine mounts is investigated. As the dynamic behaviour of the active absorbers and other elements in the secondary path are time-variant (depending on temperature , age and other factors) , an online identification is carried out. By this, the implemented feedforward control strategy is supported on a precise and frequently updated model of the secondary path. The chosen approaches to online and omine identification are presented together with first results achieved in online identification and with the overall control system. Copyright ~20011FAC 1. INTRODUCTION
control (Karkosch et al. (1999); Eberhard et al. (1999); Kuo and Morgan (1996)) . The basic principle of this technique is to introduce artificially produced noise with the same intensity as the original noise but with a phase lag of 180 degrees into the target system. The target system can be the passenger cabin (acoustic noise) or the vehicle body (vibration). The objective in both situations is to reduce noise/vibration noticed by the passengers. This principle - applied to the particular problem of reducing engine-related vibration in a vehicle - is explained in Figure 1: From the measured vibration signal (primary noise signal) at the engine, x(t), the block "controller" generates a suitable control input signal y(t) to the amplifier and active absorber, producing compensating vibration y' (the signal path from y to e is called the secondary path). The error signal e indicates the effectiveness of noise compensation and makes corrections - so-called adaptation - of the control
During the past years, a trend towards smaller and lighter vehicles can be observed. The obvious advantages of small vehicles are less fuel consumption, lower price, and better suitability for heavy urban traffic (Eberhard et al. (1999)). On the other hand, use of lightweight constructions and constrained space for engine and other functional parts introduce new challenges in decreasing noise and vibration impact on passenger comfort. In parallel, resulting from increased consumer awareness of long-term health impacts from exposure to high noise and vibration levels, the demands shift towards vehicles with better sound quality and smoother ride. The trends towards smaller and lighter cars with less engine related noise in the vehicle cabin include physically opposite demands. A possible solution is the use of active noise and vibration 187
m
_
_
___ o m
rvvv·
_X_('... } _~
primary path t-_d_(t_}_ _ _ _
_.-< +
p
Sl'Condary path S
y"
identifiution
of 5 a.bptation
Fig. 1. The principal components of the active vibration control system
Fig. 2. Feedforward control structure
algorithm possible. However , adaptation requires a precise knowledge of the dynamic behaviour of the secondary path. Gaining this knowledge is the focus of this paper: As the dynamics in the signal path from y to e are slowly time-variant, an online identification of the corresponding dynamic system is required.
so called primary path P, as shown in figure 2. As x(t) is available by measurement, a filter W can be used to generate a suitable control input signal y(t) to the secondary path S, producing the compensating noise y' . The primary path P can be considered linear but slowly time-variant and therefore only partly known (this time-variance mainly results from rubber parts with stiffness depending on temperature and other influences) . It is therefore necessary to adapt and thereby optimize the filter W frequently. This can be done by incorporating the error signal e(t) into the control structure. The adaptation makes use of the error signal e(t) measured at the chassis close to the active absorber. Adaptation algorithms for online applications were proposed by many authors; an excellent overview with focus on active noise control is given by Kuo and :\lorgan (1996).
The paper is organized as follows: Section 2 gives an overview of the control strategy and highlights the necessity of an online identification of the secondary path. Section 3 introduces the basic principle of identification, as used here, together with a recursive algorithm particularly suitable for real-time implementation. Section 4 presents details of the implementation and results achieved. Section 5 draws some conclusions and gives an outlook to future work.
Figure 2 (without the block "identification" ) shows the control system structure as described so far . In order to further improve the filter W by adaptation , a precise dynamic model of the secondary path S is needed. This can be obtained by system identification, providing a model of S from an analysis of the error signal e(t) and the absorber input signal y(t). Since also the transfer function S turns out to be linear but slowly time-variant, the identification is to be repeated frequently by the hard- and software on board the vehicle. Figure 2 illustrates how the block "identification" interacts with the other functional hlocks in the final control system.
2. FEEDFORWARD CONTROL STRUCTURE WITH ADAPTATION AND IDENTIFICATION The main way of engine vibration propagation trough the chassis into the passenger compartment is through the engine mounts. Therefore , the developers of the mounts have to find a compromise between two excluding objectives: The constrained space requires stiff mounts; the acoustic comfort considerations require mounts that absorb most of the vibrations produced by the engine. As stated, in such situations it is possible (Karkosch et al. (1999); Eberhard et al. (1999) ) to improve vibration reduction by active vibration/noise control.
As an alternative to the proposed feedforward structure one might think of a pure feedback configuration , exclusively using the error signal e(t) to generate the absorber input signal y. However , in general such feedback approaches are more critical with respect to stability and performance. Particularly, if the transfer functions P and S are not precisely known, the demand for precise phase lag of 180 0 can not be fulfilled . Phase errors can even lead to unstability, while the feedforward approach without adaptation is inherently stable.
In general, when designing a disturbance attenuating control system , it is advantageous to measure and consider all available disturbances. In the particular case of engine-related noise, a sensor directly installed at the engine delivers a suitable disturbance signal, the so-called primary noise signal x(t). The vibration signal d(t) at the mount can be considered as originating from x(t) by passing through the disturbance transfer path , the 188
For this reason : feedback approaches are not considered in more detail here, although they are the only possible solution if the source of the noise is unaccessible to measurement.
Calculating the sums in the terms iJ!TiJ! and iJ!T y and inverting the matrix iJ!TiJ! are memory and time consuming processes. A reduction of the identification algorithm computing effort is necessary for its viability in an environment like a ty- :cal passenger car microcontroller where memory and computing power are limited resources. Also important are real time, or near real time characteristics. Introducing a recursive calculation method for 8 reduces the computing and memory effort essentially. The algorithm can be derived from the equations given above (Isermann (1992); Ljung (1999)). The vector 8 is then calculated in three consecutive steps:
3. IDENTIFICATION The dynamic behavior of the secondary path S (as well as the dynamics of P) partly depends on the stiffness of rubber parts in the signal path. Slowly changing factors as temperature and age influence the rubber part stiffness substantially. Nonlinear effects, however, do not have much influence, as the quality of the results will show. Therefore, the dynamic of the secondary path can be assumed linear and slowly changing in time. This allows the considered transfer function S to be described by a linear parametric model. Since the model S is part of an digital control system , it is formulated as a difference equation, (Isermann (1992); Ljung (1999)): y(k)
P(k)1jJ(k ')'(k)
1)
+ ... + bmu(k -
m),
8(k
+ 1) = 8(k) + ')'(k)[y(k + 1)
P(k
+ 1) = [I -
(3)
According to the preciding equations, the vector 0 contains the parameters (aI .. a m , bI ... bm ), and the (N, 2m)-Matrix iJ! is built with measured values of u(.) and y(.) at the corresponding sample times according to (1) . Usually, N is chosen to be substantially larger than 2m which makes the linear equations system over-determined (because of unavoidable measurement disturbances). A solution 8 of minimum norm of the error vector
e
=y -
1), ... ,y(k - m)
m)f·
The described recursive algorithm was successfully implemented in a test car and in a laboratory model. The implementation is described in the next section. Alternative identification methods considering/identifying specific disturbance characteristics were not considered in this work. This is justifiable, since only small disturbances were observed and excellent identification results were achieved with the models described above, even with relatively low system orders.
iJ!O
is given by (see for example (Isermann (1992))) :
8 = [iJ!TiJ!]-I
(6)
These steps are repeated N times while measurement of u(.) and y(.) is in progress. Each cycle consists of measurement of y(k) and u(k): construction of 1jJ(k + 1), calculation of P(k) and 8(k + I) , and calculation of P(k + 1) v:.hich is needed for the next cycle. Only P (k + 1), O( k + 1) and 1jJ(k + 1) have to be saved for use in the next calculation cycle, which is an fundamental decrease in memory demand compared with the classical method where all measured data have to be stored. Contrary to the classical method where 8 can be estimated at any point in time with k > 2m , the usage of recursive method requires initial values. This can be estimated (or calculated with the classical method from a small number of measured values (Isermann (1992))). In most cases the use of the simple estimation P(O) = oI where 0 is a large positive number and 8(0) = 0 is sufficient if the number of calculation cycles is large.
(2)
+ 1) ... , y(k - 1), y(k)f .
= [y(k -
I u(k -1), ... ,u(k -
with y = [y(k - N
(5)
(1) 1jJ(k)
= iJ!O
')'(k)1jJT(k
(4)
where
where y(.) and u(.) denote the sequences of sampled output and input signal values of the system, and k, k - 1, ... , k - m indicate the discrete uniformly spaced sample times. The task of parameter identification is now to find the 2m parameters (al .. a m , bl ... bm ) such that the real system behaviour is described as accurately as possible by the difference equation (1). In order to do this, equation (1) is considered at N sample times to build a system of linear equations with known values of y and u . This system of equations is easily written in vector notation: y
+ 1)8(k)] + l)]P(k) .
_1jJT(k
+ aly(k - 1) + ... + amy(k - m)
= b1u(k -
+ 1)
= 1jJT(k + l)P(k)1jJ(k + 1) + 1
. iJ!T y. 189
InputSignal(generated) 0.03 0.02 Q)
~
0.01
C
0
Cl
E-0.01 -0.02
2
time[s]
Outputsignal( measured) 0.4 Q)
r----~-r.------------,
0.2
"0
~ C
0 1--vNI,._
Cl
co
E
Fig. 3. Laboratory setup with engine mount, active absorber and error sensor. For vibration decoupling the whole equipment is mounted on an massive steel table
-0.2 -0.4
L -_ _ _ _...IL._ _ _
o
~
2
_ __
_ _---J
time[s]
Fig. 4. Test signal (Franklin chirp) and measured output signal (e in figure 2)
4. IMPLEMENTATION AND RESULTS
The combined SIMULlNK/dSPACE tools provide the user wit h an user-friendly environment . However, due to some restrictions in matrix/vector computations, the implementation of the recursive algorithms required some workarounds for online use . As a consequence, the implemented recursive algorithm showed a relatively high demand of computing power, even on a DSll04 (PowerPC) model. The use of C or C+ + programming within so-called S-functions might be a remedy.
In the scope of previous research and development at Continental AG (Karkosch et al. (1999) ; Eberhard et al. (1999)) an active absorber system was already implemented in a standard vehicle. Within a SIMULINK/dSPACEenvironment the feedforward control structure as described in section 2 was implemented as a part of the active absorber system. By this, the programming environment for the identification algorithm was predetermined ; the choice of SIMULINK should further assure easy maintenance and development of the programs. In parallel a laboratory system was built to allow for test of different configurations of active absorber and disturbance sensor positions . In both cases the active absorbers were produced by Continental AG . A laboratory setup with absorber, engine mount , and disturbance sensor is shown in figure 3.
Since no physical system model is available, the "correct" model order is unknown . Therefore, the "best" model order was determined iteratively. Identifications with different model orders were carried out and model responses to the test signal were determined. The applicability of the used model order was assessed through comparison of the model response with the measured system response. In figure 5 model response and the squared deviation from measured response are displayed for three different orders. As an extension of the approach (1) , not only the order m was iteratively changed , but also the number of non-zero coefficients bv in (1) was altered , i.e. the grades of numerator and denominator of the corresponding z-transfer-function were prescribed separately.
For easy insertion of the new identification block into the existing digital controller the same sample time of ts = 0.0028 was chosen. A sufficient identification of the system is possible only if the test signal can excite all vibration modes of the examined system in the considered frequen cy range. It was found in previous work that the engine vibrations are in the range of 30 to 300Hz. Thus, an accurate description of the secondary path in this frequenca range is necessary. To excite the given range in possibly short period of time a sinus sweep signal with growing frequency from 30 to 300Hz was chosen. The signal generator was implemented in the dSP ACE system as described in (Franklin (1990))(Franklin chirp) . Figure 4 shows the Franklin chirp input signal and the measured output after the signal passed the secondary path (S in the figure 2) .
The experiments with active absorber implemented in a car were carried out with the car and its engine in standstill. Both systems (laboratory and in the car implemented active absorber) were excited solely by the sinus sweep signal applied to active absorber. As shown in figure 5, models with relatively low order (with respect to to the complexity of examined mechanical system) give reasonable signal 190
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responses. As a sufficient result for integration into the adaptation algorithm the system with order 10 - 11 (numerator ~ denominator orders) was chosen (model response shown in Figure 5.b). Although the maximal error is larger than with order 6 - 11 , smaller overall error justifies the choice. Increasing the model order provides better overall response, however, the increased accuracy does not justify the substantially higher demand of computing power for its identification. Figure 6 shows the Bode plot of the identified model with order 10 - 11.
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5. SUMMARY AND OUTLOOK As already stated, the models used for identification do not take any disturbances into account. This approach is justified for the problem considered here and delivers excellent results with relatively low system order, as described in section 4. The chosen model with order 10-11 reproduces the output of the real system accurately; it is considerably simple and therefore can be easily embedded in a real-time environment to support the filter adaptation. For the disturbed system case (e.g. with engine running), extended models including modeling of the disturbances themselves can be implemented and tested in the future. Provided the disturbance x can be measured as assumed in this paper, the described feedforward approach represents a fast response and a stable solution. However, vibrations from other sources like road or drive train can not easily be measured. In such case the feedback approach is the only possible solution to the vibration attenuation problem in the car chassis. Further work considering feasibility of feedback control approaches for active vibration attenuation will be carried out in the near future.
References Eberhard G., H.-J. Karkosh, F. Svaricek, RShoureshi, J.L. Vance. (1999) Komfortverbesserung im Kfz durch Einsatz moderner Regelungs- und Steuerungstechnik. Adaptronic Congress. Potsdam, Germany. Franklin G.F. (1990) Digital control of dynamic systems. Addison-Wesley, New-York Isermann R(1992) Identifikation dynamischer Systeme 1. Springer-Verlag, Berlin Karkosh H.-J., F. Svaricek, R.A. Shoureshi, J.L. Vance. (2000) Automotive Applications of Active Vibration Control. Proceedings of the European control conference. Karlsruhe, Germany. Kuo S.M. , D.R Morgan. (1996) Active Noise Control Systems. John Wiley & Sons, New York Ljung L.(1999) System Identification - Theory For the User. Prcntice Hall , Upper Saddle River, N.J.
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