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Study of observer considering damper force dynamics for semi active suspension systems.

ABSTRACT

This paper reports that estimation accuracy of suspension stroke velocity is increased by considering the damping force delay characteristics to an observer. Thereby ride comfort is improved, using the simple and low-cost semi active suspension systems that use only three vertical acceleration sensors.

CITATION: Yamamoto, A., Tanaka, W., Makino, T., Tanaka, S. et al., "Study of Observer Considering Damper Force Dynamics for Semi Active Suspension Systems," SAE Int. J. Passeng. Cars - Mech. Syst. 9(2):2016.

INTRODUCTION

In recent years, energy-saving and low-cost semi active suspension systems that provide both comfortable ride quality and steering stability in cars have become commonplace. Various studies, such as skyhook control [1], have been done on controlling the ride comfort for semi active suspension systems. However, it has become necessary to estimate the stroke velocity in semi active suspension systems that control the damping force. Various methods have been examined with regards to estimating the stroke velocity. Apart from the sprung vertical acceleration sensor used for ride comfort control, research has also been done on the estimation of stroke velocity using stroke sensor [2] or stroke sensor and unsprung vertical acceleration sensor [3]. However, in order to develop simpler and lower-cost semi active suspension systems, it is necessary to develop observers that estimate stroke velocity using sprung vertical acceleration sensors used in ride comfort control.

The purpose of this paper is to improve the estimation accuracy of the observer that estimates the stroke velocity using the sprung acceleration sensor. Based on research done to date, the effectiveness of considering the non-linear characteristics of semi active damper in the observer model and of carrying out observer gain scheduling has been confirmed [4]. As a result, estimation in the vicinity of sprung resonance (approximately 1 [Hz]) is possible. However, when travelling on a complex road surface with a variety of frequencies, at least it is necessary to estimate close to the unsprung resonance frequency (approximately 10 [Hz]) in order to control the damping force. Because semi active dampers are installed between body and wheel, and damping force is generated by the relative (stroke) velocity. So damping force has two resonance frequency points. Therefore, to control the damping force, at least it is necessary to estimate the stroke velocity beyond the higher resonance point (10 [Hz]). However, as the frequency increases, the estimation accuracy of the stroke velocity reduces. This is due to errors in the model, such as friction of suspension, bound stopper, and delay of damping force, and computation cycles. In order to resolve this issue, this paper proposes a practical observer model that has two new characteristics. Firstly, it considers the dynamic characteristics of the damping force. Secondly, it makes the dynamic characteristics to vary depending on control signals used. Moreover, we designed a highly accurate and practical observer by developing a suitable computation cycle. As a result, the estimation accuracy of the stroke velocity close to the unsprung resonance frequency and the ride comfort in an observer that used a sprung vertical acceleration sensor has been confirmed to improve.

SYSTEM OUTLINE

System Components

The system structure of the semi active suspension system used in the actual vehicle verification tests for this paper is shown in Figure 1. Three sprung vertical acceleration sensors have been mounted. Based on the signals from these sensors, the ECU (Electric Control Unit) calculates the estimated stroke velocity and target damping force and adjusts the damping force of each semi active damper.

The damping force adjuster for the semi active damper is composed of a linear solenoid and has a characteristic where the damping force increases in proportion to the control current i. Using stroke velocity v, the damping force characteristics [f.sub.c] of the semi active damper can be expressed as the below equation.

[f.sub.c] = [F.sub.d] (v, i) (1)

[F.sub.d] (v,i) is stored in the ECU as a data map. It outputs the damping force that is produced in the semi active damper when the control current i is conducted through the linear solenoid and stroke velocity v. Figure 2 shows the data map of the characteristics of the damping force [F.sub.d] (v,i) of the semi active damper.

General Control Flow

Figure 3 shows the control flow, from the measurement of the state quantity by the sprung vertical acceleration sensor to the instruction of control current to the semi active damper. First of all, from the three sprung vertical acceleration sensors, the vehicle is regarded as a rigid body to obtain the sprung vertical acceleration at the wheel position of each wheel. Next, the stroke velocity estimation is done according to the observer using the sprung vertical acceleration of each wheel converted to its wheel position and the electric current i. Then, using the stroke velocity estimated by the observer and sprung vertical acceleration, the target damping force is calculated based on the non-linear H[infinity] controller and the control current i is found.

Modeling

The single suspension model (2 degrees of freedom) this paper uses is shown in Figure 4.

[M.sub.b] : Body Mass

[M.sub.t] : Wheel Mass

[K.sub.c] : Suspension Stiffness

[K.sub.t] : Tire Vertical Stiffness

[C.sub.0] : Linear Damping Coefficient

[C.sub.v] : Variable Damping Coefficient

[x.sub.b] : Body Displacement

[x.sub.t] : Wheel Displacement

[x.sub.r] : Road Profile

ESTIMATION OF STROKE VELOCITY

The conventional observer that estimates the stroke velocity from sprung acceleration performs the estimation only around 1 [Hz] of the sprung resonance vicinity. If the unsprung body resonates when travelling on complex road surfaces, the estimation accuracy of the stroke velocity is reduced.

Thus, this paper aims to improve the ride comfort on complex road surfaces by improving the observer estimation accuracy up to 10 [Hz] of the unsprung resonance vicinity. Therefore, it is assumed that the target estimation accuracy is 1 - 10 [Hz] and the phase difference between the true value and the estimated stroke velocity is within [+ or -]45 [deg]. The observer structure is shown in Figure 5.

Model Reduction

To increase the estimation accuracy to the unsprung resonance, it is necessary to reduce the model errors and speed up the computation cycle. However, when considering the practical applications, making the model multidimensional and speeding up the computation cycles will increase the burden of computation and make it harder to realize. Thus, this paper aims to make the linear model as low dimensional as possible so as to consider only the necessary model and speed up the computation cycles in order to increase the estimation accuracy of the unsprung resonance point. This paper will conduct the study using the simplest 1 degree of freedom model as shown in Figure 6.

By assuming the minimum and maximum values of the damping coefficient in the stroke velocity range when driving to be C[m.sub.0], C[m.sub.1] respectively, the damping force f can be expressed as the following equation:

[f.sub.c] = [C.sub.s] ([X.sub.b] - [X.sub.r]) ([C.sub.m0] [less than or equal to] [C.sub.s] [less than or equal to] [C.sub.m1]) (2)

Hence, the equation of motion in Figure 6 becomes equation (3).

[M.sub.b][X.sub.b] = - [K.sub.c] ([X.sub.b] - [X.sub.r])-[C.sub.S] ([X.sub.b] - [X.sub.r])

Assume the state quantity to be [x.sub.m] = [[[x.sub.b] - [x.sub.r] [x.sub.b] - [x.sub.r]].sup.T], the output to be [y.sub.m] = [[x.sub.b]] and the disturbance to be w = [[x.sub.r]]. In addition, the damping coefficient [C.sub.s] is approximated to be proportional to the control current i. By introducing the variable width and dimensionless parameter q, the following state equation is obtained from equation (3).

[x.sub.m] =A(q)[x.sub.m]+B[w.sub.m] [y.sub.m] = c(g)[x.sub.m] (4)

Whereby,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

In addition, [i.sub.min], [i.sub.max] represent the minimum and maximum values of the respective control currents i. When min i = [i.sub.min], i = [i.sub.max], the semi active damper will have the respective soft and hard damping characteristics. The observer gain L is found in this state equation. However, the linear observer theory cannot be applied directly as the model indicated in equations (4), (5) changes depending on the parameter q. Thus, observer gain [L.sub.m] (q) that changes in response to a variable model is introduced. For a model where q = 0 and q = 1, using gains [L.sub.m0], [L.sub.m1], designed from the respective linear observer theories,) ([L.sub.m] (q) is given as the following equation.

[L.sub.m](q) = (1-q)[L.sub.m0]+q[L.sub.m1] (6)

From the above, applying the gain scheduling of Nakai et al [4], the observer becomes the following equation.

[x.sub.m] = (A(q)-[L.sub.m](q)C(q))[x.sub.m] + [L.sub.m](q)[y.sub.m] (7)

Delay of Damping Force

By carrying out gain scheduling, it is possible to design an observer suitable for soft to hard times. However, in reality, due to the dynamic delay of damping forces, it is necessary to not only consider the non-linear characteristics but also the dynamic characteristics [5]. The damping force when the semi active damper is stroked at the speed of 10[Hz], 0.1[m/s] is shown in Figure 7.

[f.sub.c] is the damping force obtained from [F.sub.d], (v,i) where i is the control current and V is the stroke velocity, [f.sub.delay] is the measured value of the damping force that the semi active damper actually produces. From Figure 7, it is understood that [f.sub.delay] is delayed with respect to [f.sub.c] for both soft and hard. In addition, the amount of delay differs between soft and hard. Therefore, the delay of damping force needs to be variable. Here, it is assumed that the delay when soft and delay when hard are [T.sub.0], [T.sub.1], respectively. This delay, which is the time constant of the primary delay, is approximated by the following equation.

[f.sub.dday] = [1/[1 +T(q)s]] [f.sub.c] (8)

Here,

T(q) = (1-q)[T.sub.0]+q[T.sub.1] (9)

Simulation

The observer estimation accuracy designed here will be verified using simulations. It is assumed the simulation is conducted in a continuum, the road surface input [x.sub.r] is a log sweep of 0.5 - 20 [Hz] and the speed is fixed at 0.1 [m/s]. In addition, the plant model used for the simulation uses the single-suspension 2 degrees of freedom model in Figure 4. In replacement of the damping force [f.sub.c] of the semi active damper, [f.sub.delay] as expressed in equations (8) and (9) is used. The parameters for the plant model are shown in Table 1. Furthermore, Figure 8 shows the stroke velocity estimation accuracy of the 1 degree of freedom observer that has scheduled the damping force delay and gains (Gain and Delay Scheduling referred to as GDS below) and the 1 degree of freedom observer that has scheduled only the gains (Gain Scheduling referred to as GS). From Figure 8 (a), by having the observer consider the damping force delay, the phase delay at 10 [Hz] is improved from 6[deg] to 3 [deg]. In Figure 8 (b) when hard, the phase delay at 10 [Hz] improved from 37 [deg] to 10 [deg]. From these results, by considering the damping force delay and by scheduling, the improvements of estimation accuracy at high frequency for both soft and hard are confirmed. In addition, it is confirmed that achieving the target estimation accuracy is possible even for an observer that is designed in a 1 degree of freedom in a continuous state.

Discretization

When the observer is applied to the actual vehicle, discretization is required to incorporate it into the ECU. However, as the capacity of the ECU is limited, the computation cycle cannot be shortened indefinitely. Thus, an appropriate discretization period (computation cycle) from the target estimation accuracy is required. Figure 9 shows the relationship of the phase delay at 10 [Hz] to the discretization period. From Figure 9, it is understood that the phase delay at 10 [Hz] can be within [+ or -]45 [deg] if the discretization is 1 [ms] and below. In the GS up to this point, in order to carry out the estimation for the vicinity of the sprung resonance, the discretization was 10 [ms]. However, in order to carry out the estimation for the vicinity of the unsprung resonance in the GDS that considers the damping force delay proposed in this paper, the computation cycle from the performance of the ECU is assumed to be 1 [ms].

RIDE COMFORT CONTROL

In order to improve ride comfort of vehicles, many researches have been done on the vibration suppression to reduce the vibration of the vehicle body arising from uneven road surfaces. Although skyhook control is generally widely used, this paper uses the non-linear H [infinity] theory developed to improve ride comfort.

The damping force of the semi active damper has non-linear characteristics as shown in Figure 2. Therefore, [f.sub.c] is divided into a linear component C0 and a non-linear component C to get the following equation.

[f.sub.c]=([C.sub.0]+[C.sub.v])([x.sub.b]-[x.sub.1]) (10)

Therefore the equation of motion of a single suspension 2 degrees of freedom model shown in Figure 4 is given by the following equation.

[M.sub.b][X.sub.b] = -[K.sub.c] ([X.sub.b] -[X.sub.1])-([C.sub.0]+ [C.sub.v])([X.sub.b] - [X.sub.1]) [M.sub.t][x.sub.t]=[K.sub.c] ([x.sub.b]-[x.sub.t]) + ([C.sub.0] + [C.sub.v])([x.sub.b]-[x.sub.t])-[K.sub.t]([x.sub.t]-[x.sub.r]) (11)

Non-Linear H [infinity] Control

In this paper, the state quantity is assumed to be [x.sub.p] = [[[x.sub.b] [x.sub.t] [x.sub.t] - [x.sub.b] [x.sub.r] - [x.sub.t]].sup.T] the output to be [y.sub.p] = [[x.sub.b]], the disturbances to be w = [[x.sub.r]] and the control input to be u = [C.sub.v] Converting the equation of motion in equation (4) to a state space representation, the nominal plant in equation (12) is created.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Here, each of the coefficient matrices is given as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

In order to improve the ride comfort at sprung resonance frequency, the weighting function uses a function expressed in the form of equation (14) that covers the entire sprung resonance frequency band.

[X.sub.w] = [A.sub.w][x.sub.w] +[B.sub.w][y.sub.p] [y.sub.w] = [c.sub.w][x.sub.w]+[D.sub.w][y.sub.p] (14)

From equations (13) and (14), the generalized plant of equation (15) is created.

x = Ax + [B.sub.1]w + [B.sub.2] (x)u y = Cx + D(x)u (15)

Here, each of the coefficient matrices is given as follows.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

In respect of this generalized plant, the non-linear H [infinity] control theory is applied to create a controller [6].

ACTUAL VEHICLE TESTS

Here, we drive at 80 [km/h] on the evaluation path P to verify the stroke velocity estimation accuracy and ride comfort performance when controlled. The evaluation path P are road surfaces that fall under ISO8608 [7] zones C - D (normal to bad roads) at 5 [Hz] and below and zones B - C (good to normal roads) at 5 [Hz] and above. In addition, the calculation of the discretization of GS is assumed to be the conventional 10 [ms] and the discretization of GDS to be 1 [ms].

Stroke Velocity Estimation Accuracy

Figures 10 (a) and (b) show the time-series waveform of the estimated stroke velocity when controlled. Figure 11 shows the estimation accuracy of the stroke velocity. From Figure 11, target estimation accuracy was obtained in the frequency range from 0.5 to 15 [Hz]. Even when travelling on actual road surfaces, the target estimation accuracy has been confirmed to be achievable.

Ride Comfort

Figure 12 shows the PSD analysis results of the sprung acceleration. Due to the improvement of the estimation accuracy of the stroke velocity, it can be confirmed that the vibrations at the vicinity of the sprung resonance point can be reduced by 1.5 [dB].

CONCLUSIONS

In stroke velocity estimation using sprung vertical acceleration sensors, the delay of damping force and gain scheduling were applied to the observer. By making the model simple, the observer is proved to be practical, being able to speed up the computation cycle and to estimate up to the unsprung resonance.

REFERENCES

[1.=] Karnopp D. C.: Active Damping in Road Vehicle Suspension System, Vehicle System Dynamics, Vol.12, p.291-311 (1983)

[2.] Hirao, R., Kasuya, K., and Ichimaru, N., "A Semi-Active Suspension System Using Ride Control Based on Bi-linear Optimal Control Theory and Handling Control Considering Roll Feeling," SAE Technical Paper 2015-01-1501, 2015, doi:10.4271/2015-01-1501.

[3.] Nyenhuis Markus, Frohlich Martin, "The adjustable shock absorber system fitted in the BMW X5," ATZ worldwide: Volume 109, Issue 3, pp. 16-19, March 2007

[4.] Nakai, H., Yoshida, K., Ohsaku, S., and Motozono, Y., "Design of Practical Observer for Semiactive Suspensions," Transactions of the Japan Society of Mechanical Engineers Series C, Vol. 63, No. 615, pp. 202-208, 1997, in Japanese

[5.] Amano, M., Fukao, T., Itagaki, N., Ichimaru, N., Kobayashi, T., and Gankai, T., "Adaptive VSS Observer for Semi-active Suspension Systems," Dynamics & Design Conference 2008, "546-1"-"546-6," 2008-09-02

[6.] Ohsaku S., Nakayama T., Kamimura I. and MotozonoY., "Nonlinear Hinf state feedback controller for semi-active controller suspension," Proceeding of International Symposium on Advanced Vehicle Control (AVEC), pp. 63-68, 1998

[7.] ISO8608: Mechanical vibration - Road surface profiles - Reporting of measured data (1995)

Akihito Yamamoto, Wataru Tanaka, Takafumi Makino, and Shunya Tanaka

Aisin Seiki Co., Ltd.

Ken Tahara

FT TECHNO Co. Ltd.

Table 1. Single suspension model parameters

Body Mass                       350 kg
Wheel Mass                       35 kg
Suspension Stiffness          20000 N/m
Tire Vertical Stiffness      190000 N/m
Minimum Damping Coefficient     100 N s/m
Maximum Damping Coefficient    4000 N s/m
To                                1 ms
[T.sub.1]                        11 ms
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Author:Yamamoto, Akihito; Tanaka, Wataru; Makino, Takafumi; Tanaka, Shunya; Tahara, Ken
Publication:SAE International Journal of Passenger Cars - Mechanical Systems
Article Type:Report
Date:Jun 1, 2016
Words:3099
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