IMECE ACTIVE VIBRATION CONTROL OF ACTIVE FUEL MANAGEMENT ENGINES USING ACTIVE ENGINE MOUNTS

ACTIVE VIBRATION CONTROL OF ACTIVE FUEL MANAGEMENT ENGINES USING ACTIVE

ENGINE MOUNTS

Kwang-Keun Shin / General Motors R&D

ABSTRACT

Given the realities of today’s world, the goal of achieving vehicular fuel economy is of paramount importance. One cost effective solution to improve fuel economy without major modification of engines is using Active Fuel Management (AFM), which refers to on-demand cylinder activation and deactivation. One general characteristic of AFM engines is higher level of ignition force resulting in higher torque variation. Consequently the noise and vibration (N&V) performance of a vehicle with an AFM engine can reach an un-acceptable level with aggressive cylinder deactivation. One solution to improve fuel economy without degrading N&V performance is the use of Active Engine Mount (AEM). This paper studies the control methods for active engine mount. Both open-loop and closed-loop control are developed based upon single-tone adaptive feed-forward control framework. The details of the algorithm are discussed and the stability and the robustness are examined. Integrated open-loop and closed-loop control is proposed to ensure fast response, enhance performance and robustness. A series of simulations are performed to demonstrate the control algorithm. It is shown that the integrated open-loop and closed-loop control algorithm yields the most promising results.

INTRODUCTION

Active Fuel Management (AFM), which refers to on-demand cylinder activation, is a cost effective means of enhancing fuel economy of automobiles without major modification of conventional engines. An AFM engine operates in full-cylinder mode when regular power is required and in half-cylinder mode when power requirement is minimal. To generate the same level of driving torque with the half-cylinder mode, the higher level of firing force is required in each active cylinder. This higher firing force induces higher torque variations which, in turn, induce the higher level of structural vibrations. Also, the firing frequency of the half-cylinder reduces to the half of the full-cylinder mode firing frequency, resulting in more excitation to structurally sensitive frequency range. Therefore the conventional passive approach of vibration suppression can hardly meet noise and vibration requirements for both half and full-cylinder modes. The degradation of noise

and vibration performance is one of the major obstacles of AFM engines.

Engines are usually supported by conventional engine mounts which has damping and spring element that are tuned to reduce vibration sensitivity. Tuning the conventional mount to reduce the vibration of AFM engine is very limited because of the higher firing force and changing frequency of the AFM engines. One solution of resolving the vibration issue of AFM engines is using Active Engine Mount (AEM) [1-3] which is usually an electro-hydraulic device designed to isolate body structure from engine motion.

The details of the design depend on the AEM manufacturer, and this paper does not study a specific design. However, a simplified model that represents the concept of AEM is shown in Figure 1.

Figure 1 Schematic of Active Engine Mount As shown in the figure, an active engine mount has a force generator inside, usually an electromagnetic device such as solenoid or voice coil. When passive, the force generator does not produce force and the mount behaves as a usual passive mount consisting of spring and damper that support static and dynamic load. When active, however, the dynamic force of the force generator excites AEM and resultantly isolates the force transmitted from the engine to the vehicle structure. The bandwidth of the AEM varies from one manufacturer to another but it is in between 60 to 100 hz.

This paper presents the control structure and method of active engine mount for AFM application. Open-loop control is

Force Generator Active Engine Mount Engine Vehicle Structure Mount Force Mount Force Proceedings of IMECE2007 2007 ASME International Mechanical Engineering Congress and Exposition November 11-15, 2007, Seattle, Washington, USA

designed based upon crank speed, manifold absolute pressure (MAP), and closed-loop is designed based upon single-tone adaptive feed-forward control framework. This paper also presents simulation results to discuss the effectiveness of open-loop and closed-open-loop control.

CONTROL SYSTEM DESCRIPTION

Figure 2 Physical Configuration of AEM Control System Figure 2 shows an example of the physical configuration of an AEM control system. The engine is mounted by four mounts: two AEMs on the front and rear; and two passive mounts on the right and left. In this particular example, two passive mounts are located on the neutral torque axis and does not transmit vibrations. As engine operates, therefore, most of the engine vibrations are transmitted to the body through the two AEMs. The controller receives manifold absolute pressure (MAP) and crank angle from the engine and it also receives the two force signals from the force sensors that measure the force between the AEMs and the body. The controller calculates appropriate control inputs to drive the power amplifiers that supply powered voltage to the AEMs, to cancel the force between the AEMs and the body.

Figure 3 Structure of AEM Controller

Fig. 3 illustrates the structure of active engine mount control system. The crank angle of the engine is denoted as θ,

and crank speed is denoted as ω, which is time derivative of the crank angle. The firing frequency is p times faster than crank speed and the number p is called order. For example, the firing frequency of a six cylinder engine in half cylinder mode is 1.5ω and is called 1.5th order. The engine torque variation is denoted as x(t) and, it is a sinusoid of firing frequency and its amplitude T(MAP, ω) is the function of both manifold absolute pressure (MAP) and the crank speed ω. The torque variation x(t) excites the structure resulting in the disturbance vibration vector d(t) of the sensor points. The transfer function matrix from x(t) to d(t) is denoted as H and is called primary path transfer function.

The controller drawn in the dash-lined block inputs MAP and θ from the engine and outputs the vector of control input signal u(t) that drives the amplifiers, moves the active mounts, excites the structure, and generates the control response vector y(t) of the sensor locations. The transfer function matrix G from u(t) to y(t) is called secondary path transfer function and it is the lumped dynamics of the amplifiers, the mounts, and the structure. According to the principle of superposition, the sum of d(t) and y(t) becomes residual vector r(t) that is the actual measurement from the sensors. The control objective is to minimize the firing frequency (pth order) content of the residual vector r(t) by manipulating control response vector y(t).

The controller first generates cos(pθ) and sin(pθ) which are the basis of the firing frequency content, and it also calculates firing frequency pω from the crank angle θ. The controller consists of two main algorithms: open-loop and closed-loop control. The open-loop control depends only on the information from the engine while the closed-loop control also utilizes the sensor information. The open-loop control generates the control input uo(t) of the form:

) sin( ) cos( ) (t _o p

θ

_o p

θ

o α β u = + , (1)

and the coefficient vectors αo and βo are scheduled through a

two dimensional look-up table based on the MAP and crank speed ω. The closed-loop control generates the control input vector uc(t) of the form:

) sin( ) cos( ) (t _c p

θ

_c p

θ

c α β u = + , (2)

where the coefficients αc and βc are modified by the adaptation

algorithm driven by the residual vibration r(t). The closed-loop control is intended to control pth order and ignores other frequencies which are not of interest. Therefore, if the control is applied, the disturbances of the other frequencies remain the same. The resultant control input u(t) is sum of uo(t) and uc(t).

FREQUENCY DOMAIN DESCRIPTION

Since the frequency of interest is a single frequency of pth order, it is more convenient to design the controller in frequency domain. To continue the discussion of the frequency domain design, all the signals and systems involved in Fig. 3 need to be described in frequency domain. The Fourier coefficient of a signal r(t) associated with pth order is written as:

∫

− − = t T t t jp p p dt e t T ω ) ( 2 ~ _r r , (3) Controller Power Amp. force #2 force #1 crank angle. Force Sensors voltage #1 Engine Body voltage #2 : Active Mount MAP (Manifold Absolute Pressure)

r(t) d(t) y(t) u(t) cos(pθ) sin(pθ) αο + + H G Look-Up Table + αc βc + Adaptation Algorithm Controller βο θ x(t) = T(MAP,ω)cos(pθ + φ) uo(t) uc(t) pω cos(pθ) sin(pθ) pω R e fer enc e Gener a tion Engine MAP Closed-Loop Control Open-Loop Control

where Tp is the period of s(t), and ‘~’ denotes a complex vector (number). For example, the frequency domain version of open-loop control input uo(t) is:

o o t T t t jp o p o t e dt j T p β α u u =

∫

= − − − ω ) ( 2 ~ . (4)

Therefore, the system in Figure 3 can be re-drawn in frequency domain description in Figure 4.

Figure 4 Frequency Domain Description of the Control System OPEN-LOOP CONTROL

The open-loop control is designed by setting closed-loop control zero, i.e., u~_c=0. The frequency domain system equation is then: o jp u G d r ~ ~( )~ ~_{= +}

_ω

_{. (5)}

The natural control objective is to make the residual vibration zero, i.e. 0 u G d+ (jp )~o = ~ ~

_ω

. (6) The weighted least square solution of equation (6) is then,

d Q G G Q G u~ [~(_jp

ω

)* ~(_jp

ω

)]1~(_jp

ω

)* ~ o − − = . (7)

Where, Q is a positive definite matrix that penalizes the relative amplitude of the residual vibration. In fact, equation (7) is the minimizing solution of the following cost function.

r Q r ~ ~ 2 1 * = J . (8)

The disturbance

d

~

can be measured off-line and is a function of excitation frequency and the intensity of the excitation. ) , ( ) ( ~ ~

_ω

_ω

MAP T jp H d= . (8)

Therefore the open-loop coefficients are function of crank speed and MAP, and are stored in the two-dimensional look-up tables. ) , ( MAP f o = α

ω

α . (9) ) , ( MAP f o = β

ω

β . (10) CLOSED-LOOP CONTROL

If the disturbance is from the engine only and the look-up table (9) and (10) are perfect, the open-loop control alone can achieve the control objective. In practice, however, un-modeled disturbance, the variations over system and time always cause some error in open-loop control. If vibration sensors are available, therefore, a closed-loop control can be applied.

Several methods of closed-loop control for narrow band disturbance rejection are available and are well summarized in [4]. The most popular approach is Adaptive Feed-forward Control (AFC) using Least Mean Square (LMS) [5-7], and some feedback approaches are also available, for example [8]. In this paper, however, a Single-Tone Adaptive Feed-forward Control (STAFC) is used for the closed-loop control method, because of its simplicity in real-time calculation and its systematic design that enables the field engineers tune the control algorithm without knowing the details of the theory.

From Figure 4, the frequency domain system equation of the closed-loop control is written as:

c o jp jp u G u G d r ~ ~( )~ ~( )~ ~_{= +}

_ω

₊

_ω

_{. (11)}

Now the control problem is to minimize the cost function in equation (12) by recursively updating u~_c.

r Q r ~ ~ 2 1 * = J . (12)

Let us assume the control input is updated from ~u_c,old to ~u_c,new in one iteration, and the corresponding residual vibration changes from ~r_old to ~r_new. The change of the control input is then Δ~u_c =u~_c,new −~u_c,old and the change of the cost is then

old new J J J = −

Δ . Assuming small changes, the following first order approximation holds by ignoring higher order terms of Tailor series.

(

J c

)

c

J ≅ ∂ ∂u~ *Δu~

Δ . (13)

The gradient

(

∂J ∂u~_c

)

at the current iteration is in fact, r Q G r Q r u u ~ ) ( ~ ) ~ ~ ( ~ 2 1 ~ * *

ω

j J c c = ∂ ∂ = ∂ ∂ . (14)

If the update algorithm is chosen as:

old c c

jp

, *

~

)

(

~

~

_G

_Q

_r

u

=

−

γ

ω

Δ

, (15)

then equation (13) becomes:

(

∂ ∂~

) (

* ∂ ∂~

)

≤0

− ≅

ΔJ

γ

J uc J uc . (16)

Therefore, for a sufficiently small γ, the cost function decreases as iteration proceeds: + H(jpω) G(jpω) Look-Up Table + Adaptation Algorithm θ ) , ( ~_x=TMAPω pω pω Engine MAP Closed-Loop Control Open-Loop Control d~ r ~ y ~ αc - jβc αo - jβo u ~ c u ~ o u ~

old new

J

J

≤

. (17)

Consequently, the update algorithm is:

old c old c new c

jp

, * , ,

~

)

(

~

~

~

_u

_G

_Q

_r

u

=

−

γ

ω

. (18)

Robustness of the update algorithm

The update algorithm in equation (18) requires transfer function matrix G~(jp

ω

). In practice, however, the true transfer function is different from the model G~(jω) due to inevitable variations. Let us now denote the true transfer function as: )] ( )[ ( ~ ) ( ~ _{ω ω ω} j j j t G I Δ G = + , σ{Δ(jω)}<1. (19)

In equation (19) the multiplicative deviation Δ(jω) of the true system from the model is assumed to be less than 1, which allows 100 % deviation from the model. With the true system, the true residual vibration is re-written as:

old c t o t old jp (jp )~, ~ ~ ) ( ~ ~ ~ _{d G u G u} r = +

ω

+

ω

. (20)

By substituting equation (20) into equation (18), one get:

]. ~ ~ ) ( ~ [ ) ( ~ ~ )}] ( ){ ( ~ ) ( ~ { [ ] ~ ~ ) ( ~ [ ) ( ~ ~ )] ( ~ ) ( ~ [ ~ ) ( ~ ~ ~ * * * , * * , , d u G Q G u Δ I G Q G I d u G Q G u G Q G I r Q G u u + − + − = + − − = − = o t old o t old c old old c new c jp j j j j jp j j j j

ω

ω

γ

ω

ω

ω

γ

ω

ω

γ

ω

ω

γ

ω

γ

(21)

Equation (21) is nothing but a first order difference equation whose convergence is, therefore, guaranteed if

[

−

γ

~*(

ω

) ~(

ω

){ + (

ω

)}

]

≤1

σ

I G j QG j I Δ j . (22)

The inequality (22) is true if

[

~*(

ω

) ~(

ω

){ + (

ω

)}

]

≤1

σ

γ

G j QG j I Δ j . (23) Since 2 )} ( { +

ω

≤

σ

I Δ jp , (24) inequality (23) is true if 2 / 1 )} ( ~ ) ( ~ { *

ω

ω

≤

σ

γ

G j QG j . (25)

Consequently, inequality (22) is true if

)} ( ~ ) ( ~ { 2 1 *

ω

ω

σ

γ

j j QG G ≤ . (26)

Therefore the convergence of the update algorithm (18) is guaranteed if the update gain γ satisfies inequality (26) and the deviation of the true system from the model is less than 100 %. Frequency-Domain Adaptation

The adaptation algorithm (18) can be implemented in frequency-domain that involves collecting a block of time

signal r(t), calculating pth order Fourier coefficient of the data block by using either equation (3) or FFT, updating the Fourier coefficients of the control input u~_c, and waiting until the transient response disappears before starting a new collection of r(t). Depending on the dynamics of the secondary path transfer function, one iteration of the frequency-domain adaptation takes couple of or more vibration periods that results in a delay of one iteration which impairs system performance.

Time-Domain Adaptation

More efficient way of implementing the adaptation algorithm (18) is using time-domain approximation that eliminates the need for collecting and processing a block data, reduces the delay, and enables smooth control. Let us assume

c

u

~ _{is not a constant but a continuous function of time} i.e.,~ tu( ). Then for one period, the following integral relationship holds. ) ( ~ ) ( ~ ) ( ~ p t T t T t t dt dt t d p − − =

∫

− u u u . (27)

If we consider u~(t)=u~_new and ~u(t−T_p)=~u_old, then equation (27) gives old old new t T t jp dt dt t d p r Q G u u u _~ ) ( ˆ~ ~ ~ ) ( ~ *

ω

γ

− = − =

∫

− . (28)

From the definition of Fourier coefficient in equation (3), equation (18) can be approximated as,

∫

∫

− − − − = t T t t jp p t T t p p dt e t jp T dt dt t d

γ

_ω

ω ) ( ) ( ~ 2 ) ( ~ * r Q G u _{. (29)}

By taking the integrations out, the time domain approximation of equation (18) is written as t jp p e t jp T dt t d ₌₋

γ

_ω

− ω ) ( ) ( ~ 2 ) ( ~ * Qr G u . (30)

In discrete-time implementation, therefore, (30) becomes:

k t j k k k t jp t e t + )=~( )−

μ

ˆ~(

ω

) ( ) − ω ( ~ * 1 u G Qr u , (31)

where μ is a lumped gain of γ and discrete sampling time, but it is, in fact, a tunable gain. As the gain μ approaches to zero, the system is always stable but the response time becomes longer. Higher gain reduces the response time but can cause instability if it is too high. Therefore, the controller gain μ should be tuned to reduce response time without causing instability. Eq. (30) is the final version of discrete-time adaptation algorithm which is implemented in simulation studies and also in actual vehicle.

SIMULATIONS

The first set of simulation is performed to show the effects of open-loop and closed-loop controls. In this simulation, a six cylinder engine is set to half-cylinder AFM mode, and the

crank speed is set to the constant of 1000 RPM. The control is activated at 0.5 sec so that one can compare the vibration level before and after 0.5 sec.

Figure 5 shows the front mount force when only open-loop control is applied. The rear mount force is omitted here since it shows very similar results with front force. As discussed earlier, the open-loop control does not require vibration measurements. For the simulation purpose some modeling error is injected to the open-loop gain table. Otherwise open-loop control achieves perfect vibration cancellation at least in the simulation environment. As can be seen in the figure, the open-loop control immediately suppresses the force as soon as the control is activated at 0.5 sec. However, the modeling error in the open-loop table causes some amount of residual force after the control is activated. The actual amount of this residual force depends on the system variations over time and from vehicle to vehicle.

Figure 6 shows the control result when only closed-loop control is applied. As discussed earlier, the closed-loop control requires vibration measurements. For the simulation purpose some modeling error is also injected into the closed-loop gain. As shown in the figure, the closed-loop control suppresses the front force in about 0.5 sec after the control is activated. However, the response time of the closed-loop control is longer than the open-loop control case.

The open-loop control is shown to have faster response time and the closed-loop control is shown to have good final performance. By integrating the open-loop and closed-loop control, one can achieve the control algorithm with fast response, good robustness and good final performance all at the same time. Figure 7 shows the simulation result when the open-loop and closed-open-loop controls are combined. As shown in the figure, the control algorithm immediately suppresses the force to small level as soon as the control is activated and the closed-loop control asymptotically suppresses the residual force down to zero. 0 0.5 1 1.5 -100 -50 0 50 100 time (sec) fo n t m o u n t fo rc e (N ) Open-Loop Control @1000 RPM

Figure 5 Open-Loop Control

0 0.5 1 1.5 -100 -50 0 50 100 time (sec) fr o n t m o u n t fo rc e (N ) Closed-Loop Control @1000 RPM

Figure 6 Closed-Loop Control

0 0.5 1 1.5 -100 -50 0 50 100 time (sec) fr o n t m o u n t fo rc e (N )

Open-Loop + Closed-Loop Control @1000 RPM

Figure 7 Integrated Open-Loop and Closed-Loop Control The second set of simulation is performed to show the effects of RPM sweep on open-loop and closed-loop control. In this simulation, the engine RPM changes with the sweep from 600 to 3000 RPM in 5 seconds in half-cylinder mode of the six cylinder engine. Figures 8 and 9 show the pth order magnitude of seat track acceleration and steering column acceleration, and Figures 10 and 11 show the pth order magnitude of corresponding front and rear mount forces, respectively. As shown in the set of figures, the open-loop control drastically reduces the vibrations. However, some residual vibrations are inevitable in the presence of the modeling errors. When closed-loop control is added to the open-closed-loop control, the residual vibration is further reduced.

500 1000 1500 2000 2500 3000 0 0.2 0.4 0.6 0.8 1 RPM A c c e le ra ti o n ( m /s e c 2)

Seat Track Acceleration (5 sec rpm sweep) Base

Open-Loop Open+Closed-Loop

Figure 8 Seat Track Vibrations with RPM Sweep

500 1000 1500 2000 2500 3000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 RPM A c c e le ra ti o n ( m /s e c 2)

Steering Column Acceleration (5 sec rpm sweep) Base

Open-Loop Open+Closed-Loop

500 1000 1500 2000 2500 3000 0 10 20 30 40 50 60 70 80 RPM M o u n t F o rc e (N )

Front Mount Force (5 sec rpm sweep) Base Open-Loop Open+Closed Loop

Figure 10 Front Mount Force with RPM Sweep

500 1000 1500 2000 2500 3000 0 20 40 60 80 100 120 RPM M o u n t F o rc e (N )

Rear Mount Force (5 sec rpm sweep) Base Open-Loop Open+Closed Loop

Figure 11 Rear Mount Force with RPM Sweep

CONCLUSIONS

Both open-loop and closed-loop controls are developed based upon single-tone adaptive feed-forward structure, which is found to be a suitable structure for AEM/AFM application. Open-loop control has faster response time and is a cost effective solution because it does not need sensors. However, the open-loop control cannot compensate for variations, over

time and across vehicles. Closed-loop control is capable of compensating for vehicle variations up to 100%. However, the response time of the closed-loop control may not meet the desired transient performance. Both fast response time and robustness to vehicle variations can be achieved at the same time by integrating open-loop and closed-loop control.

REFERENCES

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