**Robust Active Suspension Control for Vibration Reduction of Passenger's Body**

Takuma Suzuki and Masaki Takahashi *Keio University Japan* 

## **1. Introduction**

20 Robust Control

92 Challenges and Paradigms in Applied Robust Control

Levant, A. (1998). Robust exact differentiation via sliding mode technique, *Automatica,* Vol.

Ljung, L. (1999). *System identification - theory for the user*, Prentice Hall, Upper Saddle River, NJ,

Ljung, L. (2006). *System identification toolbox for use with Matlab*, The Mathworks Inc., Natick,

Perruquetti, W. & Barbot, J. (2002). *Sliding mode control in engineering*, Marcel Dekker Inc. New

Schopp, G., Burkhardt, T., Dingl, J., Schwarz, R. & Eisath, C. (2010). Funktionsentwicklung

Utkin, V. (1977). Variable structure systems with sliding modes, *IEEE Transactions on Automatic*

Utkin, V., Guldner, J. & Shi, J. (2009). *Sliding mode control in electromechanical systems*, CRC

Young, K., Utkin, V. & Özgüner, . (1999). A control engineer's guide to sliding mode control, *IEEE Transactions on Control Systems Technology,* Vol. 7, No. 3: pp. 328 – 342.

und Kalibration für aufgeladene Motoren - Modellbasiert vom Konzept bis zur Serie, *in* R. Isermann (ed.), *Elektronisches Management motorischer Antriebe*, Vieweg

Reif, K. (2007). *Automobilelektronik*, Vieweg & Teubner, Wiesbaden, Germany.

34: 379 – 384.

MA, USA.

York, Basel.

& Teubner, Wiesbaden, Germany.

Press, Boca Raton, London, New York.

*Control,* Vol. 22: 212 – 222.

USA.

An automotive performance has improved from the demand of ride comfort and driving stability. Many research have proposed various control system design methods for active and semi-active suspension systems. These studies evaluated the amount of reduced vibration in the vehicle body, i.e., the vertical acceleration in the center-of-gravity (CoG) of the vehicle's body (Ikeda *et al.,* 1999; Kosemura *et al.,* 2008; Itagaki *et al.,* 2008). However, any passengers always do not sit in the CoG of the vehicle body. In the seated position that is not the CoG of the vehicle body, vertical acceleration is caused by vertical, roll and pitch motion of the vehicle. In nearly the resonance frequency of the seated human, the passenger's vibration becomes larger than the seated position's vibration of the vehicle body due to the seated human dynamics.

The seated human dynamics and human sensibility of vibration are cleared by many researchers. So far some human dynamics model has been proposed (Tamaoki *et al.,* 1996, 1998, 2002; Koizumi *et al.,* 2000). Moreover, some of them are standardized in ISO (ISO-2631- 1*,* 1997; ISO-5982, 2001). At the research as for automotive comfort with the passengervehicle system, M.Oya et al. proposed the suspension control method considering the passenger seated position in the half vehicle model (Oya *et al.,* 2008). G.J. Stein et al. evaluated passenger's head acceleration at some vehicle velocities and some road profiles (Guglielmino *et al.,* 2008). There are few active suspension control design methods which are positively based on a passenger's dynamics and the seating position. These methods can be expected to improve the control performance.

In this paper, new active suspension control method is developed to reduce the passenger's vibration. Firstly, a vehicle and passenger model including those dynamics at seated position is constructed. Next, a generalized plant that uses the vertical acceleration of the passenger's head as one of the controlled output is constructed to design the linear *H*<sup>∞</sup> controller. In this paper, this proposed method defines as "Passenger Control". "Passenger Control" means passenger's vibration control. Moreover, in an active suspension control, it is very important to reduce the vibration at the condition of the limited actuating force. Then, we design two methods which are "Vehicle CoG Control", and "Seat Position Control", and compare the proposed method with two methods. "Vehicle CoG Control" means vibration control of vehicle. "Seat Position Control" means vibration control of seat position. Finally, several simulations are carried out by using a full vehicle model which has active suspension system. From the result, it was confirmed that in nearly the resonance

Robust Active Suspension Control for Vibration Reduction of Passenger's Body 95

S4

*py*3

S3

*py*4

P3

P4

*Lf Lr Tr*

Symbol Value Symbol Value *Mb* 1900 kg *Tf* 1.53 m *Mt* 50 kg *Tr* 1.50 m *Ir* 600 kgm2 *Hr* 0.45 m *Ip* 3000 kgm2 *Hp* 0.53 m *Kf* 33×103 N/m *px*1,2 0.04 m *Kr* 31×103 N/m *py1* 0.4 m *Kt* 260×103 N/m *py2* -0.4 m *Lf* 1.34 m *pz*1,2 -0.045 m

degree of freedom (DOF) which is longitudinal, lateral, and vertical motions. The head has 3 DOF. First, the head moves up and down to the body parts. Second, the head rotates around the point, Pp, at the pitch direction. Third, the head rotates around the point, Pr, at the roll direction. Thus the passenger model has a total of 6 DOF. Between the each part, it has a

spring and a damper. The equation of motion of the passenger model is as follows.

Pitch CTR Roll CTR

Side view Rear view

*Mt Mt Mt Mt*

*q f*

*F*3 *F*3

*K*r*Cr*

*zr*<sup>3</sup> *zr*<sup>3</sup> *zr*<sup>4</sup> *Kt Kt Kt*

*zu*<sup>3</sup> *zu4*

*K*r

*F*3 *F*3

*Cr*

*F*4 *F*4

*Tf Tr*

*px*<sup>1</sup> *px*<sup>3</sup>

*px*4

*zcg*

*<sup>p</sup> pz*3,4 *<sup>z</sup>*1,2

*K*r*Cr*

*Lr* 1.46 m

P1

*py*1

S1

S2

*py*2

Plane view

*zu*<sup>1</sup> *zu3*

*F*1 *F*1 P2

*px*2

*zr*1

*Kt*

Fig. 1. Full vehicle model

Table 1. Specification of vehicle mode

*Kf Cf*

*Mb , Ir , Ip*

Forward

frequency of a passenger's head in the vertical direction, "Passenger Control" is effective in reducing a passenger's vibration better than "Vehicle CoG Control" and "Seat Position Control". The numerical simulation results show that the proposed method has the highest control performance which is vibration reduction of the passenger's head per generated force by the active suspension. Moreover, the results show that the proposed method has robustness for the difference in passenger's vibration characteristic.

## **2. Modeling**

#### **2.1 Modeling of the vehicle**

Figure 1 shows a full vehicle model which is equipped with an active suspension between each wheel and the vehicle body. The weight of the vehicle body is supported by the spring. We assume that a vehicle model is a generic sedan car as shown in Table 1. The equations of motion which are, bounce, roll, pitch and each unsprung motion are as follows:

$$\mathcal{M}\_b \ddot{\mathbf{z}}\_{cg} = \sum\_{i=1}^{4} f\_{si} \,\prime \,\prime \,\tag{1}$$

$$M\_r \ddot{\phi} = \frac{T\_f}{2} (f\_{s1} - f\_{s2}) + \frac{T\_r}{2} (f\_{s3} - f\_{s4}) + M\_b g H\_r \phi \, , \tag{2}$$

$$L\_p \ddot{\theta} = -L\_f \left( f\_{s1} + f\_{s2} \right) + L\_r \left( f\_{s3} + f\_{s4} \right) + M\_b g H\_p \theta \, , \tag{3}$$

$$M\_{\rm ti} \ddot{z}\_{\rm ui} = -F\_{\rm di} + K\_t z\_{\rm ti} \ \left(\mathbf{i} = \mathbf{1}, \cdots, \mathbf{4}\right). \tag{4}$$

where *Hr* is the distance from a roll center to the CoG of the vehicle body, and *Hp* is the distance from a pitch center to the CoG of the vehicle body. These parameters are constant. The spring coefficients of each wheel are different from each other, and were set to *K*1, 2 = *Kf* , *K*3, 4 = *K*r, *zsi* means a suspension stroke of each wheel, *zti* means deformation of the each tire.

$$\begin{aligned} z\_{s1} &= z\_{cg} + T\_f \sqrt{2\phi - L\_f \theta} - z\_{u1} \\ z\_{s2} &= z\_{cg} - T\_f \sqrt{2\phi - L\_f \theta} - z\_{u2} \\ z\_{s3} &= z\_{cg} + T\_r \sqrt{2\phi + L\_r \theta} - z\_{u3} \\ z\_{s4} &= z\_{cg} - T\_r \sqrt{2\phi + L\_r \theta} - z\_{u4} \\ z\_{ti} &= z\_{ri} - z\_{ui} \quad (i = 1, \cdots, 4) \end{aligned} \tag{5}$$

The spring and damping forces which act between the wheels and the vehicle body, are given by the following equation.

$$F\_{di} = -K\_i z\_{si} - C\_i(t) \dot{z}\_{si} \quad \text{(i = 1, \dots, 4)}\tag{6}$$

#### **2.2 Modeling of the passenger**

Various models of a seated human have been proposed so far. In this paper, the passenger's motion is expressed to the seated human model shown in Fig. 2. Therefore, it is easy to understand the passenger's motion. To the seated position, Ps, the body part has three

Fig. 1. Full vehicle model

94 Challenges and Paradigms in Applied Robust Control

frequency of a passenger's head in the vertical direction, "Passenger Control" is effective in reducing a passenger's vibration better than "Vehicle CoG Control" and "Seat Position Control". The numerical simulation results show that the proposed method has the highest control performance which is vibration reduction of the passenger's head per generated force by the active suspension. Moreover, the results show that the proposed method has

Figure 1 shows a full vehicle model which is equipped with an active suspension between each wheel and the vehicle body. The weight of the vehicle body is supported by the spring. We assume that a vehicle model is a generic sedan car as shown in Table 1. The equations of

4

1 *b c <sup>g</sup> si i*

12 34 2 2

*<sup>f</sup> <sup>r</sup> r s s s s br*

where *Hr* is the distance from a roll center to the CoG of the vehicle body, and *Hp* is the distance from a pitch center to the CoG of the vehicle body. These parameters are constant. The spring coefficients of each wheel are different from each other, and were set to *K*1, 2 = *Kf* , *K*3, 4 = *K*r, *zsi* means a suspension stroke of each wheel, *zti* means deformation of the each tire.

> 1 1 2 2 3 3 4 4

> > ( 1, , 4)

 

 

*z zT L z z zT L z z zT L z z zT L z*

*s cgf f u s cgf f u s cg r r u s cg r r u*

 

 

The spring and damping forces which act between the wheels and the vehicle body, are

Various models of a seated human have been proposed so far. In this paper, the passenger's motion is expressed to the seated human model shown in Fig. 2. Therefore, it is easy to understand the passenger's motion. To the seated position, Ps, the body part has three

*ti ri ui*

*zzzi*

12 34

*M z f* , (1)

(5)

*<sup>T</sup> <sup>T</sup> I f f f f M gH* , (2)

*p f s s rs s b <sup>p</sup> I L f f L f f M gH* , (3)

*M z F Kz i ti ui di t ti* ( 1, ,4) . (4)

*F Kz C tz i di i si i si* ( ) ( 1, , 4) (6)

motion which are, bounce, roll, pitch and each unsprung motion are as follows:

robustness for the difference in passenger's vibration characteristic.

**2. Modeling** 

**2.1 Modeling of the vehicle** 

given by the following equation.

**2.2 Modeling of the passenger** 


Table 1. Specification of vehicle mode

degree of freedom (DOF) which is longitudinal, lateral, and vertical motions. The head has 3 DOF. First, the head moves up and down to the body parts. Second, the head rotates around the point, Pp, at the pitch direction. Third, the head rotates around the point, Pr, at the roll direction. Thus the passenger model has a total of 6 DOF. Between the each part, it has a spring and a damper. The equation of motion of the passenger model is as follows.

Robust Active Suspension Control for Vibration Reduction of Passenger's Body 97

*I c k kzz czzr*

*hp h p h p h p b h p b h*

*mr k y y c y y ckr*

*hbh p p b p p b p h p h*

22 1 1 3

*hr h p h p h*

 

*f ff*

6 6

Where,

this passenger model.

**2.3 Vehicle-passenger model** 

and the motion of the vehicle have the following relation;

*I ck*

*b*

*m m* .

*b h m*

(13)

 

2 4 <sup>4</sup> 2 24 2 2

2 1

*mr k x x c x x c k r*

*hbh p p b p p b p h p h*

,

Each parameter of the passenger model is set to *mb* = 45 kg, *mh* = 7.5 kg, *Ihr* = 8.3×10-2 kgm2, *Ihp* = 5.0 kgm2, and *Ihp* = 5.5×10-2 kgm2 based on the adult male's height and weight data. In addition, the acceleration of the passenger's head is derived from a geometric relation.

> *h bh h bh xx r y y r*

As shown in Table 2, the spring, the damper, and length were adjusted to conform the passenger model and an experimental data which was reported in previous research shown in Figs. 3 and 4 (Tamaoki *et al.,* 1996, 1998). The results shown in Figs. 3(c) and 4(c) demonstrate that the gain characteristics of the model were nearly equal to the experimental ones. However, as shown in Figs. 3(b) and 4(b), there were some differences in the high-frequency band for the phase properties. To reduce these differences, the passenger model must be made more complex, but this necessitates the use of a higher order control system. Because the purpose of our controller is to reduce the vertical vibration of the passengers in comparison with the lateral vibration, we designed it using

In this section, the passenger for the vehicle model was assumed to sit in the front-left seat in designing the control system to reduce passenger vibration and motion. The vehiclepassenger model is shown in Fig. 5. The passenger model is set to the vehicle model in a front-left seat to design the controller. The translational motion of the position of the seat

> 

 1 2 () () () *pp p x t Axt B wt B ut* (15)

(14)

1 1 1 1 1 11

The equation of state of the vehicle-passenger model is defined as the following equation.

*p pz p rz p cg y x*

 

*x Hp y Hp z zp p*

 

*bh b h hbh*

*m mm m*

*f f* (11)

(12)

1 5 <sup>5</sup> 6 65 2 2

Fig. 2. Passenger model


Table 2. Specification of passenger model (\*:Nm/rad, \*\*:Nm/rad/s)

$$2\sigma\_{bh}\ddot{\mathbf{x}}\_b = 2k\_{p4}\left(\mathbf{x}\_p - \mathbf{x}\_b\right) + 2\mathbf{c}\_{p4}\left(\dot{\mathbf{x}}\_p - \dot{\mathbf{x}}\_b\right) + \left(-\mathbf{c}\_{p2}\dot{\theta}\_h - k\_{p2}\theta\_h\right)\Big|\mathbf{r}\_4\tag{7}$$

$$2m\_{bh}\ddot{y}\_b = 2k\_{p5}\left(y\_p - y\_b\right) + 2c\_{p5}\left(\dot{y}\_p - \dot{y}\_b\right) - \left(-c\_{p6}\dot{\phi}\_h - k\_{p6}\phi\_h\right)\Big|r\_5\tag{8}$$

$$\begin{split} m\_b \ddot{z}\_b &= k\_{p3} \left( z\_p - z\_b \right) + c\_{p3} \left( \dot{z}\_p - \dot{z}\_b \right) - \left[ k\_{p1} \left( z\_b - z\_h \right) + c\_{p1} \left( \dot{z}\_b - \dot{z}\_h \right) \right] \\ &+ \left( -c\_{p2} \dot{\theta}\_h - k\_{p2} \theta\_h \right) \Big/ r\_3 \end{split} \tag{9}$$

$$m\_h \ddot{z}\_h = k\_{p1} \left( z\_b - z\_h \right) + c\_{p1} \left( \dot{z}\_b - \dot{z}\_h \right) - \left( -c\_{p2} \dot{\theta}\_h - k\_{p2} \theta\_h \right) \Big| r\_3 \tag{10}$$

$$\begin{aligned} \mathbf{I}\_{\text{hr}} \ddot{\boldsymbol{\phi}}\_{\text{h}} &= -\mathbf{c}\_{p6} \dot{\boldsymbol{\phi}}\_{\text{h}} - \mathbf{k}\_{p6} \boldsymbol{\phi}\_{\text{h}} \\ &+ m\_{\text{hph}} r\_1 \Big[ 2\mathbf{k}\_{p5} \Big( \mathbf{y}\_p - \mathbf{y}\_b \Big) + 2\mathbf{c}\_{p5} \Big( \dot{\boldsymbol{y}}\_p - \dot{\boldsymbol{y}}\_b \Big) - \Big( -\mathbf{c}\_{p6} \dot{\boldsymbol{\phi}}\_{\text{h}} - \mathbf{k}\_{p6} \boldsymbol{\phi}\_{\text{h}} \Big) \Big/ r\_5 \Big] \end{aligned} \tag{11}$$

$$\begin{aligned} \mathbf{I}\_{hp}\ddot{\boldsymbol{\theta}}\_{h} &= -\mathbf{c}\_{p2}\dot{\boldsymbol{\theta}}\_{h} - \mathbf{k}\_{p2}\boldsymbol{\theta}\_{h} + \left[\mathbf{k}\_{p1}\left(\mathbf{z}\_{b} - \mathbf{z}\_{h}\right) + \mathbf{c}\_{p1}\left(\dot{\mathbf{z}}\_{b} - \dot{\mathbf{z}}\_{h}\right)\right] \mathbf{r}\_{3} \\ &- m\_{hbl}r\_{2}\Big[2\mathbf{k}\_{p4}\left(\mathbf{x}\_{p} - \mathbf{x}\_{b}\right) + 2\boldsymbol{c}\_{p4}\left(\dot{\mathbf{x}}\_{p} - \dot{\mathbf{x}}\_{b}\right) + \left(-\mathbf{c}\_{p2}\dot{\boldsymbol{\theta}}\_{h} - \mathbf{k}\_{p2}\boldsymbol{\theta}\_{h}\right)\Big|\mathbf{r}\_{4}\Big] \end{aligned} \tag{12}$$

Where,

96 Challenges and Paradigms in Applied Robust Control

*r*<sup>2</sup> *r*<sup>1</sup>

Forward

*xb*

*xp*

Pr Pp

*cp*4

*kp*4

*xh*

Front view Side view

 *kpi cpi rpi i* [N/m] [N/m/s] [m] 1 40000 2000 0.1 2 15\* 0.9\*\* 0.1 3 96000 1120 0.05 4 22500 600 0.2 5 2000 400 0.3 6 20\* 1.2\*\* 0.3

4 4 2 2 <sup>4</sup> 2 2

5 5 6 6 <sup>5</sup> 2 2

1 1 2 23

33 11

*bb p p b p p b p b h p b h*

*mz k z z c z z k z z c z z*

*cp*5

*mb*

*r*5

*<sup>p</sup>*5,*cp*<sup>5</sup>

Fig. 2. Passenger model

*zp*

*fh*

Ps

*k k <sup>p</sup>*<sup>5</sup>

*yp*

Table 2. Specification of passenger model (\*:Nm/rad, \*\*:Nm/rad/s)

*ph ph*

*ckr*

2 23

 

*yb zb*

*kp*3, *cp*<sup>3</sup>

*zh mh*

*yh*

,*Ihr*

*kp*1*,c <sup>k</sup> <sup>p</sup>*<sup>1</sup> *<sup>p</sup>*1,*cp*<sup>1</sup>

*kp*6*,cp*<sup>6</sup>

*kp*2*,cp*<sup>2</sup>

*r*4

*kp*4,*cp*<sup>4</sup>

*zh*

*h q*

*r*3

*zb*

*kp*3*,cp*<sup>3</sup>

Ps

*mz k z z c z z c k r hh p b h p b h p h p h* (10)

*mx k x x c x x c k r bh b p p b p p b p h p h* (7)

*mbh b p p b p p b p h p h y k y y c y y ckr* (8)

 

 

  (9)

*zp*

$$m\_{\text{blr}} = m\_{\text{b}} + m\_{\text{h}} \quad \quad m\_{\text{hblr}} = \frac{m\_{\text{b}}}{m\_{\text{b}} + m\_{\text{h}}} \cdot 1$$

Each parameter of the passenger model is set to *mb* = 45 kg, *mh* = 7.5 kg, *Ihr* = 8.3×10-2 kgm2, *Ihp* = 5.0 kgm2, and *Ihp* = 5.5×10-2 kgm2 based on the adult male's height and weight data. In addition, the acceleration of the passenger's head is derived from a geometric relation.

$$\begin{aligned} \ddot{\mathbf{x}}\_h &= \ddot{\mathbf{x}}\_b + \ddot{\boldsymbol{\theta}}\_h \Big/ r\_2 \\ \ddot{\mathbf{y}}\_h &= \ddot{\mathbf{y}}\_b + \ddot{\boldsymbol{\phi}}\_h \Big/ r\_1 \end{aligned} \tag{13}$$

As shown in Table 2, the spring, the damper, and length were adjusted to conform the passenger model and an experimental data which was reported in previous research shown in Figs. 3 and 4 (Tamaoki *et al.,* 1996, 1998). The results shown in Figs. 3(c) and 4(c) demonstrate that the gain characteristics of the model were nearly equal to the experimental ones. However, as shown in Figs. 3(b) and 4(b), there were some differences in the high-frequency band for the phase properties. To reduce these differences, the passenger model must be made more complex, but this necessitates the use of a higher order control system. Because the purpose of our controller is to reduce the vertical vibration of the passengers in comparison with the lateral vibration, we designed it using this passenger model.

## **2.3 Vehicle-passenger model**

In this section, the passenger for the vehicle model was assumed to sit in the front-left seat in designing the control system to reduce passenger vibration and motion. The vehiclepassenger model is shown in Fig. 5. The passenger model is set to the vehicle model in a front-left seat to design the controller. The translational motion of the position of the seat and the motion of the vehicle have the following relation;

$$\begin{aligned} x\_{p1} &= \left(H\_p + p\_{1z}\right)\theta\\ y\_{p1} &= -\left(H\_r + p\_{1z}\right)\phi\\ z\_{p1} &= z\_{c\chi} + p\_{1y}\phi - p\_{1x}\theta \end{aligned} \tag{14}$$

The equation of state of the vehicle-passenger model is defined as the following equation.

$$
\dot{\mathbf{x}}(t) = A\_p \mathbf{x}(t) + B\_{p1} w(t) + B\_{p2} \boldsymbol{\mu}(t) \tag{15}
$$

Robust Active Suspension Control for Vibration Reduction of Passenger's Body 99

*zr*1

*<sup>f</sup> <sup>q</sup>*

Front view Side view

*xt z z z z z z z z z z x y z z x y z z*

*F*1 *F*1

*<sup>z</sup> <sup>z</sup> <sup>r</sup>*<sup>3</sup> *<sup>r</sup>*<sup>2</sup> *zr*<sup>1</sup> *Kt Kt Kt*

*zu*<sup>2</sup> *zu*<sup>1</sup>

Fig. 5. Vehicle-passenger model

 

**3. Design of controller** 

*K*r

Passenger 2 Passenger 1

*zh*<sup>1</sup> *yh*<sup>1</sup>

*fh*1

*F*2

*yp*2 *zp*2

P2

*yb*2 *zb*2

Pr

*zh*<sup>2</sup> *yh*<sup>2</sup>

*fh*<sup>2</sup>

*F*2

1234

*rrrr T*

1234

**3.1 Disturbance accommodating control** 

*wt z z z z*

*ut FFFF*

*Cr*

*yp*1 *zp*1

> 

*T*

defined by ISO (ISO-8608, 1995). The filter is as follows:

*di*

0.706 *<sup>d</sup>* .

P1

*yb*1 *zb*1

Pr

*K*r*Cr*

Where,

*<sup>d</sup>* 50 2

 and 

Pitch CTR Roll CTR

*Tr Lf Lr*

12 3 4 12 3 4 1 11 111 11 11

*u u u u cg u u u u cg b b b h h h b b b h h h*

 

We found that feedforward control of disturbance information in the finite frequency range and feedback control improve performance (Okamoto *et al.,* 2000). The power spectral density of the actual velocity of disturbances had flat characteristics in a low frequency, and decreased according to frequency at a region of high frequency. We assumed that it regarded as the colored noise formed by shaping filter which has a transfer function with low-pass characteristics. This filter of the each wheel is based on the road condition which

> 2 2 2

*w s <sup>i</sup>*

*di di di di gi*

*x t Ax t Bw t <sup>Q</sup> wt Cx t i*

*i di di i d gi dd d*

*w s s s*

() () ()

( ) ( ) ( 1, ,4)

( ) ( 1, ,4) ( ) <sup>2</sup>

 

where, *wgi* is road input of the each wheel, *wi* is road input of the vehicle-passenger model of the generalized plant to design the controller as shown in Fig. 6. It was referred to as

*Kt*

*Kf Cf*

*Mb , Ir , Ip*

Forward

*<sup>M</sup> Mt Mt <sup>t</sup> Mt*

*zu*<sup>1</sup> *zu3*

*F*1 *F*1

*xp*1

P1 *zp*1

> 

*<sup>T</sup>*

*zh*1 *xh*1

*h*1 *q*

*zb*1 *xb*1

Pp

*zcg*

*K*r*Cr*

*xp*3

P3 *zp*3

Passenger 1 Passenger 3

*zh*3 *xh*3

*h*3 *q*

*zb*3 *xb*3

Pp

 

. (16)

.

*F*3 *F*3

Fig. 3. Transfer function from seat to the head (Translational motion, Dot: Experiment (Tamaoki *et al.,* 1998), Line: model)

Fig. 4. Transfer function from seat to the head (Rotational motion, Dot: Experiment (Tamaoki *et al.,* 1996), Line: Model)

Fig. 5. Vehicle-passenger model

Where,

98 Challenges and Paradigms in Applied Robust Control

5 10 15

5 10 15

5 10 15

*p h y* 

> -300 -200 -100 0 100 200 300

Gain[rad/m]

Phase[deg]

frequency[Hz]

(a) *xp* input (b) *yp* input (c) *zp* input

Fig. 4. Transfer function from seat to the head (Rotational motion, Dot: Experiment

5 10 15

*p h y y*

> -300 -200 -100 0 100 200 300

0 0.5 1 1.5 2 2.5 3

Gain[m/m]

Phase[deg]

5 10 15

5 10 15

5 10 15

*p h z* 

frequency[Hz]

frequency[Hz]

*xh*

*xh*

5 10 15

*p h p h z z z x* ,

*zh*

*zh*

frequency[Hz]

(a) *xp* input (b) *yp* input (c) *zp* input

Fig. 3. Transfer function from seat to the head (Translational motion, Dot: Experiment

5 10 15

5 10 15

5 10 15

*p h x* 

> -300 -200 -100 0 100 200 300

Gain[rad/m]

Phase[deg]

frequency[Hz]

(Tamaoki *et al.,* 1996), Line: Model)


0 0.5 1 1.5 2 2.5 3

Gain[m/m]

Phase[deg]

frequency[Hz]

(Tamaoki *et al.,* 1998), Line: model)

*zh*

*zh*

*xh*

5 10 15

*p h p h x z x x* ,



Gain[rad/m]

Phase[deg]

0 0.5 1 1.5 2 2.5 3

Gain[m/m]

*xh*

Phase[deg]

$$\begin{aligned} \mathbf{x}(t) &= \begin{bmatrix} z\_{u1}z\_{u2}z\_{u3}z\_{u4}z\_{cg}\ \boldsymbol{\theta} \ \boldsymbol{\theta} \ \dot{z}\_{u1}\dot{z}\_{u2}z\_{u3}z\_{u4}\ \boldsymbol{z}\_{cg}\dot{\boldsymbol{\theta}}\ \dot{\theta} \ \mathbf{x}\_{b1}y\_{b1}z\_{b1}z\_{h1}\boldsymbol{z}\_{h1}\boldsymbol{\theta}\_{h1}\boldsymbol{z}\_{b1}\boldsymbol{z}\_{b1}z\_{h1}\boldsymbol{\dot{\theta}}\_{h1}\boldsymbol{\theta}\_{h1} \\\\ \mathbf{w}(t) &= \begin{bmatrix} z\_{r1}z\_{r2}z\_{r3}z\_{r4} \end{bmatrix}^{T} \\\\ \boldsymbol{w}(t) &= \begin{bmatrix} F\_{1}F\_{2}F\_{3}F\_{4} \end{bmatrix}^{T} \end{aligned}$$

## **3. Design of controller**

#### **3.1 Disturbance accommodating control**

We found that feedforward control of disturbance information in the finite frequency range and feedback control improve performance (Okamoto *et al.,* 2000). The power spectral density of the actual velocity of disturbances had flat characteristics in a low frequency, and decreased according to frequency at a region of high frequency. We assumed that it regarded as the colored noise formed by shaping filter which has a transfer function with low-pass characteristics. This filter of the each wheel is based on the road condition which defined by ISO (ISO-8608, 1995). The filter is as follows:

$$\begin{cases} \dot{\mathbf{x}}\_{di}(\mathbf{t}) = A\_{di}\mathbf{x}\_{di}(\mathbf{t}) + B\_{di}w\_{gi}(\mathbf{t})\\ w\_i(\mathbf{t}) = \mathbb{C}\_{di}\mathbf{x}\_{di}(\mathbf{t}) \qquad \text{ ( $i = 1, \dots, 4$ )}\\ \frac{w\_i(\mathbf{s})}{w\_{gi}(\mathbf{s})} = \frac{\varpi\_d^2}{s^2 + 2\xi\_d\varpi\_d\mathbf{s} + \varpi\_d^2} \qquad \text{ ( $i = 1, \dots, 4$ )} \end{cases} \tag{16}$$

where, *wgi* is road input of the each wheel, *wi* is road input of the vehicle-passenger model of the generalized plant to design the controller as shown in Fig. 6. It was referred to as *<sup>d</sup>* 50 2 and 0.706 *<sup>d</sup>* .

.

Robust Active Suspension Control for Vibration Reduction of Passenger's Body 101

We compare the proposal method and two generalized control methods to verify the control performance. As one of the generalized control methods, the controller in which the one of the controlled values is vertical acceleration of the body CoG (Vehicle CoG Control), is designed. Another is that one of the controlled values is vertical acceleration of a seated position (Seat Position Control). The design of two generalized control methods are changed the controlled value *z*1 into the vertical acceleration of CoG of the vehicle body and seated position, respectively. Frequency weights, *W*1(*s*), *W*2(*s*), *W*3(*s*), *W*4(*s*), *Kw*2 = 400, *Kw*3 = 5000, and *Kw*4 = 1.31,

Vertical vel. (sprung)

Actuator force

Tire disp.

Vertical acce. (head)

**Frequency weights for controlled values**

*W*<sup>1</sup> *W*<sup>1</sup> (*s*) (*s*) *W*<sup>2</sup> *W*<sup>2</sup> (*s*) (*s*)

*W*<sup>4</sup> *W*<sup>4</sup> (*s*) (*s*) *W*<sup>3</sup> *W*<sup>3</sup> (*s*) (*s*)

*Kw*<sup>1</sup> *Kw*<sup>2</sup>

*Kw*<sup>4</sup> *Kw*<sup>3</sup>

*z <sup>g</sup>*(*t*)

**Controlled values**

**Measured outputs (Vertical acceleration**

1

(Actuating force)

1

*W*4(*s*)

2

2

3

**of sprung)**

*yg*(*t*)

(Tire displacement)

*W*3(*s*)

use the same value also in the three methods. The following section describes *Kw*1.

Generalized plant

*Ap*

)()()(

*tyBtxAtx*

Controller

<sup>10</sup>-1 <sup>10</sup><sup>0</sup> <sup>10</sup><sup>1</sup> <sup>10</sup><sup>2</sup> <sup>0</sup>

Frequency [Hz]

)()(

*kk k kk k*

*txCtu*

(Vertical acceleration)

*s*

*<sup>I</sup> Cp*<sup>1</sup>

*Bp*<sup>2</sup>

*Ad*

**Road disturbance model**

**Inputs (Force)**

Road *wg*(*t*)

**Disturbance**

**Vehicle-passenger model**

*u*(*t*)

0.2

0.4

0.6

Gain [-]

0.8

1

1.2

Fig. 6. Generalized plant for "Passenger Control"

*W*2(*s*)

Fig. 7. Frequency weights for controlled value

(Vertical Velocity of Sprung)

*W*1(*s*)

*Bd Cd*

*s* + *I* +

*Bp*<sup>1</sup>

++ +

## **3.2 Disturbance accommodating** *H*∞ **control.**

The feedforward control of disturbances resulted in worse accuracy outside the assumed frequency (Okamoto *et al.,* 2000). Furthermore, because each resonance frequency of the vehicles, passenger, and tire differs, the control system design considering each resonance frequency is needed. Therefore, the control system was designed by using the *H*∞ method in the control theory.

We integrated each state variable of the road disturbance model and frequency weights for controlled values. The frequency weights are as follows:

$$\begin{cases} \dot{\mathbf{x}}\_{wi}(t) = A\_{wi} \mathbf{x}\_{wi}(t) + B\_{wi} z\_{pi}(t) \\ z\_{gi}(t) = C\_{wi} \mathbf{x}\_{wi}(t) \\ z\_{gi}(s) = K\_{wi} \mathcal{W}\_i(s) \end{cases} \quad \text{( $i = 1, \dots, 4$ )} \tag{17}$$

$$\begin{cases} \frac{z\_{gi}(s)}{z\_{pi}(s)} = K\_{wi} \mathcal{W}\_i(s) & \text{( $i = 1, \dots, 4$ )} \\ \end{cases} \tag{17}$$

where, *zpi* is controlled value of the vehicle-passenger model, *zgi* is controlled value of the generalized plant. Figure 6 shows a block diagram of the generalized plant to design the controller, and the state-space form of the generalized plant is as follows:

$$\begin{aligned} \dot{\mathbf{x}}\_{\mathcal{S}}\left(t\right) &= A\_{\mathcal{S}} \mathbf{x}\_{\mathcal{S}}\left(t\right) + B\_{\mathcal{S}1} w\_{\mathcal{S}}\left(t\right) + B\_{\mathcal{S}2} u\left(t\right) \\ \dot{\mathbf{z}}\_{\mathcal{S}}\left(t\right) &= C\_{\mathcal{S}1} \mathbf{x}\_{\mathcal{S}}\left(t\right) \\ \dot{\mathbf{y}}\_{\mathcal{S}}\left(t\right) &= C\_{\mathcal{S}2} \mathbf{x}\_{\mathcal{S}}\left(t\right) + D\_{\mathcal{S}21} w\_{\mathcal{S}}\left(t\right) \end{aligned} \tag{18}$$

*H*∞ norm of the transfer function from disturbance *wg*(*t*) to controlled value *z*(*t*) is expressed by the following equation.

$$\left\|\mathbf{G}\_{z\_{\mathcal{S}}w\_{\mathcal{S}}}\right\|\_{\infty} = \sup\_{w} \frac{\left\|z\_{\mathcal{S}}\right\|\_{2}}{\left\|w\_{\mathcal{S}}\right\|\_{2}}\tag{19}$$

$$\min\_{u} \left\| \mathbf{G}\_{z\_{\mathcal{X}} w\_{\mathcal{X}}} \right\|\_{\infty} =: \mathcal{Y}^\* \tag{20}$$

where, \* is a minimum of *H*∞ norm of the generalized plant realized with *H*∞ controller. The controller is the following equation (Glover & Doyle, 1988).

$$\begin{aligned} \dot{\mathbf{x}}\_k(t) &= A\_k \mathbf{x}\_k(t) + B\_k \mathbf{y}(t) \\ \boldsymbol{\mu}(t) &= \mathbf{C}\_k \mathbf{x}\_k(t) \end{aligned} \tag{21}$$

The measured outputs, *y*(*t*), are four vertical accelerations of the wheel position of the vehicle body. The controlled values, *z*(*t*), are vertical acceleration of the passenger's head, vertical velocity of the sprung, tire deformation, and actuating force. Frequency weight *Wi*, shown in Fig. 7 was determined by trial and error.

A bandpass filter, *W*1, that had a peak frequency equal to the resonance frequency of the passenger's head was used based on sensitivity curves (ISO-2631-1*,* 1997), such as that being standardized by ISO and shown in Fig. 8. In order to prevent the increase of response in each resonance, a low pass filter *W*2 and a bandpass filter *W*3 are used. Moreover, to prevent steady control input and minimize energy consumption, a high pass filter, *W*4, was used.

100 Challenges and Paradigms in Applied Robust Control

The feedforward control of disturbances resulted in worse accuracy outside the assumed frequency (Okamoto *et al.,* 2000). Furthermore, because each resonance frequency of the vehicles, passenger, and tire differs, the control system design considering each resonance frequency is needed. Therefore, the control system was designed by using the *H*∞ method in

We integrated each state variable of the road disturbance model and frequency weights for

() () ()

where, *zpi* is controlled value of the vehicle-passenger model, *zgi* is controlled value of the generalized plant. Figure 6 shows a block diagram of the generalized plant to design the

> 

*H*∞ norm of the transfer function from disturbance *wg*(*t*) to controlled value *z*(*t*) is expressed

sup *g g*

\* min :

*x t Ax t B y t*

The measured outputs, *y*(*t*), are four vertical accelerations of the wheel position of the vehicle body. The controlled values, *z*(*t*), are vertical acceleration of the passenger's head, vertical velocity of the sprung, tire deformation, and actuating force. Frequency weight *Wi*,

A bandpass filter, *W*1, that had a peak frequency equal to the resonance frequency of the passenger's head was used based on sensitivity curves (ISO-2631-1*,* 1997), such as that being standardized by ISO and shown in Fig. 8. In order to prevent the increase of response in each resonance, a low pass filter *W*2 and a bandpass filter *W*3 are used. Moreover, to prevent steady control input and minimize energy consumption, a high pass filter, *W*4, was used.

 *k kk k k k*

 

*x t Ax t B w t B ut z t C x t D ut*

 *g gg g g g g gg g gg g*

1 12

*y t Cx t D w t*

*z w*

*G*

The controller is the following equation (Glover & Doyle, 1988).

shown in Fig. 7 was determined by trial and error.

g2 g21

( ) ( ) ( 1, ,4) ( )

*K Ws i*

*wi wi wi wi pi*

*x t Ax t Bz t <sup>Q</sup> z t Cx t i*

*gi wi wi*

*wi i*

controller, and the state-space form of the generalized plant is as follows:

( ) ( ) ( 1, ,4)

1 2

2 2

*g*

 *g g z w <sup>u</sup>*

*z*

*w*

*w g*

\* is a minimum of *H*∞ norm of the generalized plant realized with *H*∞ controller.

. (17)

. (18)

(19)

*G* (20)

*ut Cx t* (21)

**3.2 Disturbance accommodating** *H*∞ **control.** 

controlled values. The frequency weights are as follows:

*wi*

*gi*

*z s*

*pi*

*z s*

the control theory.

by the following equation.

where,

We compare the proposal method and two generalized control methods to verify the control performance. As one of the generalized control methods, the controller in which the one of the controlled values is vertical acceleration of the body CoG (Vehicle CoG Control), is designed. Another is that one of the controlled values is vertical acceleration of a seated position (Seat Position Control). The design of two generalized control methods are changed the controlled value *z*1 into the vertical acceleration of CoG of the vehicle body and seated position, respectively. Frequency weights, *W*1(*s*), *W*2(*s*), *W*3(*s*), *W*4(*s*), *Kw*2 = 400, *Kw*3 = 5000, and *Kw*4 = 1.31, use the same value also in the three methods. The following section describes *Kw*1.

Fig. 6. Generalized plant for "Passenger Control"

Fig. 7. Frequency weights for controlled value

Robust Active Suspension Control for Vibration Reduction of Passenger's Body 103

front-right seat. In section 4.5.2, some specifications of the passenger model are different

Figure 11 shows the time histories of the vehicle and the passenger 1's vertical acceleration for 3 second. In this paper, passenger 1 sits a front-left seat, and passenger 2 sits a frontright. In the acceleration of the vehicle body, it was confirmed that there is few differences among the three methods. On the other hand, in the acceleration of the passenger's head, the proposed method is the smallest, and it was confirmed that the proposed method is effective

The actuating force of each wheel in each method is shown in Fig. 12. In the Vehicle CoG Control, the actuating force of all wheels is generated in the same direction. In the other method, the actuating force of the left/right wheel is generated in a different direction. Therefore, the vertical accelerations of the seated position and the passenger's head are

Figure 13 shows the Lissajous figure of lateral and vertical accelerations of the seated position, the passenger's body and the head part respectively. This figure is seen from the front of vehicle. In upper-right figure of Fig. 13 (c), the proposed method has control effect which vertical acceleration of the passenger 1's head is reduced in comparison with "Vehicle CoG Control". Moreover, the proposed method has not only the vibration reduction effect of the passenger 1's head, but also the vertical acceleration reduction effect of the passenger


10-2 10-1 100 101

Spatial Frequency [cycle/m]

ISO-A

ISO-G

F L F R R L R R

FL FR RL RR

**4.5.1 Comparison with the "Vehicle CoG Control" and "Seat Position Control"** 

from the generalized plant to design controller.

for the passenger's vibration reduction.

reduced by controlling the roll motion of the vehicle body.

**4.5 Results** 

1's body.

Power Spectrical Density [m3/cycle]

10-10

Fig. 9. PSD of road surface profile

10-10

10-10

10-10

10-10

10-10

Fig. 8. Sensitivity curve of vertical vibration (ISO-2631-1*,* 1997)

## **4. Simulation**

In this section, two kinds of numerical simulations were carried out. One is to verify control performance in comparison with other methods. Another is to verify robustness for the difference in passenger's vibration characteristic.

## **4.1 Assumption**

We verified the effectiveness of the proposed method by using the vehicle-passenger model with *H*∞ controller. We used MATLAB (The Math Work Inc.) for the calculations, and the Runge-Kutta method for the differential equations. The computational step size is 1 ms. In addition, it assumes that we perform the evaluation in an ideal condition, and the model to design the controller and the model for evaluation are same models.

## **4.2 Driving condition**

It assumes that the PSD characteristic of the road surface is C class defined by ISO (ISO-8608, 1995). The vehicle speed is 16.6 m/s (60 km/h). The vehicle runs the straight for 10 seconds, and the input of the road surface to each wheel is independent. Figure 9 shows the PSD of the road disturbance. Figure 10 shows the road displacement.

## **4.3 Design of the frequency weight** *Kw***<sup>1</sup>**

In each method, if the evaluation function of acceleration is raised, it is clear that each acceleration set as the controlled value is reduced, and the actuating force increases. To set the same actuating force, frequency weight *Kw*1 of each method was adjusted so that RMS value of the actuating force of the four wheels sets to 1000 N. The each frequency weight, *Kw*1, of "Vehicle CoG Control", "Seat Position Control" and "Passenger Control" is 244, 315, and 78 respectively.

## **4.4 Difference of vehicle-passenger model**

In the numerical simulation, there are some diffidence in the vehicle-passenger model as shown in Table 3. In sections 4.5.1 and 4.5.2, passenger models sit in the front-left seat and front-right seat. In section 4.5.2, some specifications of the passenger model are different from the generalized plant to design controller.

#### **4.5 Results**

102 Challenges and Paradigms in Applied Robust Control

<sup>10</sup> -1 <sup>10</sup><sup>0</sup> <sup>10</sup><sup>1</sup> <sup>10</sup><sup>2</sup> -30

In this section, two kinds of numerical simulations were carried out. One is to verify control performance in comparison with other methods. Another is to verify robustness for the

We verified the effectiveness of the proposed method by using the vehicle-passenger model with *H*∞ controller. We used MATLAB (The Math Work Inc.) for the calculations, and the Runge-Kutta method for the differential equations. The computational step size is 1 ms. In addition, it assumes that we perform the evaluation in an ideal condition, and the model to

It assumes that the PSD characteristic of the road surface is C class defined by ISO (ISO-8608, 1995). The vehicle speed is 16.6 m/s (60 km/h). The vehicle runs the straight for 10 seconds, and the input of the road surface to each wheel is independent. Figure 9 shows the

In each method, if the evaluation function of acceleration is raised, it is clear that each acceleration set as the controlled value is reduced, and the actuating force increases. To set the same actuating force, frequency weight *Kw*1 of each method was adjusted so that RMS value of the actuating force of the four wheels sets to 1000 N. The each frequency weight, *Kw*1, of "Vehicle CoG Control", "Seat Position Control" and "Passenger Control" is 244, 315,

In the numerical simulation, there are some diffidence in the vehicle-passenger model as shown in Table 3. In sections 4.5.1 and 4.5.2, passenger models sit in the front-left seat and

Frequency [Hz]


difference in passenger's vibration characteristic.

Fig. 8. Sensitivity curve of vertical vibration (ISO-2631-1*,* 1997)

design the controller and the model for evaluation are same models.

PSD of the road disturbance. Figure 10 shows the road displacement.

Magnitude [dB]

**4. Simulation** 

**4.1 Assumption** 

**4.2 Driving condition** 

and 78 respectively.

**4.3 Design of the frequency weight** *Kw***<sup>1</sup>**

**4.4 Difference of vehicle-passenger model** 

#### **4.5.1 Comparison with the "Vehicle CoG Control" and "Seat Position Control"**

Figure 11 shows the time histories of the vehicle and the passenger 1's vertical acceleration for 3 second. In this paper, passenger 1 sits a front-left seat, and passenger 2 sits a frontright. In the acceleration of the vehicle body, it was confirmed that there is few differences among the three methods. On the other hand, in the acceleration of the passenger's head, the proposed method is the smallest, and it was confirmed that the proposed method is effective for the passenger's vibration reduction.

The actuating force of each wheel in each method is shown in Fig. 12. In the Vehicle CoG Control, the actuating force of all wheels is generated in the same direction. In the other method, the actuating force of the left/right wheel is generated in a different direction. Therefore, the vertical accelerations of the seated position and the passenger's head are reduced by controlling the roll motion of the vehicle body.

Figure 13 shows the Lissajous figure of lateral and vertical accelerations of the seated position, the passenger's body and the head part respectively. This figure is seen from the front of vehicle. In upper-right figure of Fig. 13 (c), the proposed method has control effect which vertical acceleration of the passenger 1's head is reduced in comparison with "Vehicle CoG Control". Moreover, the proposed method has not only the vibration reduction effect of the passenger 1's head, but also the vertical acceleration reduction effect of the passenger 1's body.

Fig. 9. PSD of road surface profile

Robust Active Suspension Control for Vibration Reduction of Passenger's Body 105

Vehicle CoG Control Seat Position Control Passenger Control (Propose)

3 3.5 4 4.5 5 5.5 6

3 3.5 4 4.5 5 5.5 6

3 3.5 4 4.5 5 5.5 6

Time [s]

0 1 2 3 4 5 6 7 8 9 10

3 3Front-left Front-right Rear-left Rear-right .5 4 4.5 5 5.5 6

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Time[s]

Fig. 11. Vehicle and passenger's behavior (Vertical acceleration, unit : m/s2)




1000


1000


Actuator

Fig. 12. Actuating force

Force [N] Actuator

Force [N] Actuator

 Force [N]

0

0

0

Vehicle CoG Control

Seat Position Control

Passenger Control (Propose)

1000

0

Passenger 1's head

Seat 1

Center of gravity of vehicle

1

0

1

0

0.5


Figure 14 shows power spectrum density (PSD) of the vertical acceleration of the passenger's head in each method, and actuating force. In the frequency band of 4-7 Hz with resonance of a passenger's head, although the proposed method has the vibration reduction effect better than other methods. On the other hand, PSD of the actuating force does not necessarily have the highest value in the frequency band. In this frequency band, the proposed method can reduce the passenger's vibration by the limited actuating force.

Fig. 10. Road displacement


Table 3. Vehicle-passenger model

In each frequency band, the sensitivity of the vertical acceleration for the human is defined by sensitivity curves (ISO-2631, 1997). In this paper, we estimate the root mean square (RMS) value which is added the sensitivity compensation expressed by a high order transfer function (Rimel & Mansfield, 2007). Figure 15 shows the ratio of the RMS value of each vertical acceleration to those values of "Vehicle CoG Control". In the passenger 1, it was confirmed that the proposed method can reduce the RMS value of the passenger 1's head (head 1). Moreover, in the passenger 2, it was confirmed that the RMS value of the passenger 2's head (head 2) is not increased by the proposed method, and the proposed method had the vibration reduction effect equivalent to the generalized control methods.

104 Challenges and Paradigms in Applied Robust Control

Figure 14 shows power spectrum density (PSD) of the vertical acceleration of the passenger's head in each method, and actuating force. In the frequency band of 4-7 Hz with resonance of a passenger's head, although the proposed method has the vibration reduction effect better than other methods. On the other hand, PSD of the actuating force does not necessarily have the highest value in the frequency band. In this frequency band, the proposed method can reduce the passenger's vibration by the

0 2 4 6 8 10

0 2468 10

FR

RR

FL

RL

Simulation model

Section 4.5.1 Section 4.5.2

・Front-right <sup>¬</sup>

time [s]

Generalized plant to design controller

Seated position ・Front-left ・Front-left

Vehicle Fig. 1, Table 1 ¬ ¬

Passenger Fig. 2, Table 2 ¬ Table 4

In each frequency band, the sensitivity of the vertical acceleration for the human is defined by sensitivity curves (ISO-2631, 1997). In this paper, we estimate the root mean square (RMS) value which is added the sensitivity compensation expressed by a high order transfer function (Rimel & Mansfield, 2007). Figure 15 shows the ratio of the RMS value of each vertical acceleration to those values of "Vehicle CoG Control". In the passenger 1, it was confirmed that the proposed method can reduce the RMS value of the passenger 1's head (head 1). Moreover, in the passenger 2, it was confirmed that the RMS value of the passenger 2's head (head 2) is not increased by the proposed method, and the proposed method had the vibration reduction effect equivalent to the generalized control

limited actuating force.




Fig. 10. Road displacement

Table 3. Vehicle-passenger model

methods.

0

0

0.01 0.02

0.01

0.02

路面変位[m]

Road disp. [m]

Fig. 11. Vehicle and passenger's behavior (Vertical acceleration, unit : m/s2)

Fig. 12. Actuating force

Robust Active Suspension Control for Vibration Reduction of Passenger's Body 107

Vehicle CoG Control Seat Position Control Passenger Control (Propose)

100 10<sup>1</sup>

Frequency [Hz] (a) Vertical acceleration (Center of gravity of vehicle)

10<sup>0</sup> 101

Frequency [Hz] (b) Vertical acceleration (Passenger1's head)

<sup>10</sup><sup>0</sup> <sup>10</sup><sup>1</sup> <sup>10</sup><sup>0</sup>

Frequency [Hz]

(c) Actuator force

10-4

10-4

102

104

PSD [N2/Hz]

Fig. 14. Power spectral density

106

10-3

PSD [(m/s2

)

2/Hz]

10-2

10-1

10-3

PSD [(m/s2

)

2/Hz]

10-2

10-1

Fig. 13. Lissajous figure (Lateral and vertical acceleration)

106 Challenges and Paradigms in Applied Robust Control




Passenger 1's body


Passenger 1's seat


Passenger 1's head



Lateral Acc.[m/s2 Lateral Acc.[m/s ] 2]

(c) Passenger Control (Propose)


Passenger 2's body

1.5 -1 <sup>0</sup> <sup>1</sup> -1.5

Passenger 2's head


Passenger 1's body


Passenger 1's seat


Passenger 1's head



Lateral Acc.[m/s2 Lateral Acc.[m/s ] 2]



Vertical Acc.[m/s2]

Vertical Acc.[m/s2]


Vertical Acc.[m/s2]

(a) Vehicle CoG Control (b) Seat Position Control



Passenger 1's seat



Passenger 2's seat

Passenger 2's body

1.5 -1 <sup>0</sup> <sup>1</sup> -1.5

Passenger 2's head


Vertical Acc.[m/s2]


Fig. 13. Lissajous figure (Lateral and vertical acceleration)

Vertical Acc.[m/s2]


Vertical Acc.[m/s2]

1.5 -1 <sup>0</sup> <sup>1</sup> -1.5

Passenger 1's body

1.5 -1 <sup>0</sup> <sup>1</sup> -1.5

Passenger 1's head


Lateral Acc.[m/s2 Lateral Acc.[m/s ] 2]


Passenger 2's seat

Passenger 2's body

Passenger 2's head

1.5 -1 <sup>0</sup> <sup>1</sup> -1.5

1.5 -1 <sup>0</sup> <sup>1</sup> -1.5


Vertical Acc.[m/s2]

Vertical Acc.[m/s2]



Vertical Acc.[m/s2]



Fig. 14. Power spectral density

Robust Active Suspension Control for Vibration Reduction of Passenger's Body 109

dynamics. Varterasian & Thompson reported the seated human dynamics from a large person to a small person(Varterasian & Thompson, 1977). Robust performance is verified by supposition that such person sits in the vehicle. Figure 16 shows the frequency response from vertical vibration of seat to vertical vibration of the head. Dot is 15 subjects' resonance peak. In this section, three outstanding subjects' data of their report is modeled in the vibration characteristic of vertical direction. The damper and spring were adjusted to conform the passenger model and an experimental data. The characteristic of the passenger

Nominal model 960000 1120

Subject 1 1320000 1150 Subject 2 576000 960 Subject 3 960000 2550

*kp3* [N/m]

Nominal model Subject 1 Subject 2 Subject 3

10<sup>0</sup> 101

Frequency[Hz]

The numerical simulation is carried out on the same road surface conditions as the section 4.5.1. Figure 17 shows PSD of the vertical acceleration of the passenger 1's head and Fig. 18

*cp3* [N/m/s]

model of three outstanding subjects are shown in Table 4.

Table 4. Difference of specifications

10-4

Fig. 17. PSD of vertical acceleration (Passenger 1's head)

10-3

PSD [(m/s2

)

2/Hz]

10-2

10-1

Fig. 15. RMS value of vertical acceleration

From these results, it was confirmed that the proposed method can effectively reduce passenger's vibration by using *H*∞ control which including the dynamics of human body and seated position. By means of setting the passenger' motion to one of the amounts of evaluation function, the proposed method can directly control the passenger's vibration.

### **4.5.2 Comparison with the different passenger model**

In this section, the robust performance against the difference in a passenger's vibration characteristic is verified. In previous research, there are many reports about seated human

Fig. 16. Frequency response from seat to the head (Vertical motion, dot : Experiment (Varterasian & Thompson, 1977), Line : Model)

dynamics. Varterasian & Thompson reported the seated human dynamics from a large person to a small person(Varterasian & Thompson, 1977). Robust performance is verified by supposition that such person sits in the vehicle. Figure 16 shows the frequency response from vertical vibration of seat to vertical vibration of the head. Dot is 15 subjects' resonance peak. In this section, three outstanding subjects' data of their report is modeled in the vibration characteristic of vertical direction. The damper and spring were adjusted to conform the passenger model and an experimental data. The characteristic of the passenger model of three outstanding subjects are shown in Table 4.


Table 4. Difference of specifications

108 Challenges and Paradigms in Applied Robust Control

100.0

Vehicle CoG Control Seat Position Control Passenger Control (Propose)

CoG. Seat 1 Head 1 Seat 2 Head 2

From these results, it was confirmed that the proposed method can effectively reduce passenger's vibration by using *H*∞ control which including the dynamics of human body and seated position. By means of setting the passenger' motion to one of the amounts of evaluation function, the proposed method can directly control the passenger's vibration.

In this section, the robust performance against the difference in a passenger's vibration characteristic is verified. In previous research, there are many reports about seated human

Nominal model

2 4 6 8 10 12 14

Frequency [Hz]

Fig. 16. Frequency response from seat to the head (Vertical motion, dot : Experiment

50.262.0

55.0 100.0

52.766.0

82.4

Subject 1

100.0

> 85.9

> > 66.4

100.0

0

Fig. 15. RMS value of vertical acceleration

**4.5.2 Comparison with the different passenger model** 

0

(Varterasian & Thompson, 1977), Line : Model)

0.5

1

Subject 3

Subject 2

1.5

Gain[m/m]

 [-]

Magnitude

2

2.5

3

20

40

60

80

100

RMS ratio to Vehicle CoG control[%]

120

140

160

127.4

> 101.1

100.0

Fig. 17. PSD of vertical acceleration (Passenger 1's head)

The numerical simulation is carried out on the same road surface conditions as the section 4.5.1. Figure 17 shows PSD of the vertical acceleration of the passenger 1's head and Fig. 18

Robust Active Suspension Control for Vibration Reduction of Passenger's Body 111

Kosemura, R.; Takahashi, M. & and Yoshida, K. (2008). Control Design for Vehicle Semi-

Itagaki, N.; Fukao, T.; Amano, M.; Ichimaru, N.; Kobayashi, T. & Gankai, T. (2008). Semi-

Tamaoki, G.; Yoshimura, T. & Tanimoto, Y. (1996). Dynamics and Modeling of Human Body

Tamaoki, G.; Yoshimura, T. & Suzuki, K. (1998). Dynamics and Modeling of Human Body

Tamaoki, G. & Yoshimura, T. (2002). Effect of Seat on Human Vibrational Characteristics,

Koizumi, T.; Tujiuchi, N.; Kohama, A. & Kaneda, T. (2000). A study on the evaluation of ride

Oya, M.; Tsuchida, Y. & Qiang, W. (2008). Robust Control Scheme to Design Active

Guglielmino, E.; Sireteanu, T.; Stammers, C. G.; Ghita, G. & Giuclea, M. (2008). *Semi-Active* 

Okamoto, B. and Yoshida, K. (2000). Bilinear Disturbance-Accommodating Optimal Control

Glover, K. & Doyle, J.C. (1988). State-space Formula for All Stabilizing Controllers that

ISO-8608 (1995). Mechanical vibration -Road surface profiles - Reporting of measured data,

Rimel, A.N. & Mansfield, N.J. (2007). Design of digital filters for Frequency Weightings

*Mechanical Engineers, Series C*, Vol.66, No.650, pp. 3297-3304

*Design Conference 2008*, 547, Kanagawa, Japan, September, 2008

*1996*, 361, pp. 522-525, Fukuoka, Japan, August, 1996

*Engineers, Series C*, Vol.64, No.617, pp. 266-272

*2008*, pp.690-695, Kobe, Japan, October, 2008

ISBN- 978-1848002302, London

*and Control letters*, 11, pp.167-172

*International Organization for Standardization*

*Industrial Health*, Vol.45, No.4, pp. 512-519

October, 2008

October, 2002

*Standardization*

Active Suspension Considering Driving Condition, *Proceedings of the Dynamics and* 

Active Suspension Systems based on Nonlinear Control, *Proceedings of the 9th International Symposium on Advanced Vehicle Control 2008*, pp. 684-689, Kobe, Japan,

Considering Rotation of the Head, *Proceedings of the Dynamics and Design Conference* 

Exposed to Multidirectional Excitation (Dynamic Characteristics of Human Body Determined by Triaxial Vibration Test), *Transactions of the Japan Society of Mechanical* 

*Proceedings of the Dynamics and Design Conference 2002*, 220, Kanazawa, Japan,

comfort due to human dynamic characteristics, *Proceedings of the Dynamics and Design Conference 2000*, 703, Hiroshima, Japan, October, 2000 ISO-2631-1 (1997). Mechanical vibration and shock–Evaluation of human exposure to whole-body vibration -, *International Organization for Standardization* ISO-5982 (2001). Mechanical vibration and shock –Range of idealized value to characterize seated body biodynamic response under vertical vibration, *International Organization for* 

Suspension Achieving the Best Ride Comfort at Any Specified Location on Vehicles, *Proceedings of the 9th International Symposium on Advanced Vehicle Control* 

*Suspension Control -Improved Vehicle Ride and Road Friendliness*, Springer-Verlag,

of Semi-Active Suspension for Automobiles, *Transactions of the Japan Society of* 

Satisfy an *H*∞-norm Bound and Relations to Risk Sensitivity, *Journal of the Systems* 

Required for Risk Assessment of workers Exposed to Vibration, *Transactions of the* 

shows RMS value. In PSD of 7 Hz or more, RMS value of vertical acceleration of subject 1's head becomes higher than the nominal model. Moreover, RMS of subject 1 is the highest. On the other hand, RMS of subjects 2 and 3 is reduced in comparison with the nominal model. The physique of subject 1 differs from other subjects. When such a person sits, the specified controller should be designed. From these results, the proposed method has robustness for the passenger of the general physique.

Fig. 18. RMS value of vertical acceleration of passenger 1's head

## **5. Conclusion**

This study aims at establishing a control design method for the active suspension system in order to reduce the passenger's vibration. In the proposed method, a generalized plant that uses the vertical acceleration of the passenger's head as one of the controlled output is constructed to design the linear *H*∞ controller. In the simulation results, when the actuating force is limited, we confirmed that the proposed method can reduce the passenger's vibration better than two methods which are not include passenger's dynamics. Moreover, the proposed method has robustness for the difference in passenger's vibration characteristic.

## **6. Acknowledgment**

This work was supported in part by Grant in Aid for the Global Center of Excellence Program for "Center for Education and Research of Symbiotic, Safe and Secure System Design" from the Ministry of Education, Culture, Sport, and Technology in Japan.

## **7. References**

Ikeda, S.; Murata, M.; Oosako, S. & Tomida, K. (1999). Developing of New Damping Force Control System -Virtual Roll Damper Control and Non-liner *H*∞ Control-, *Transactions of the TOYOTA Technical Review*, Vol.49. No.2, pp.88-93

110 Challenges and Paradigms in Applied Robust Control

shows RMS value. In PSD of 7 Hz or more, RMS value of vertical acceleration of subject 1's head becomes higher than the nominal model. Moreover, RMS of subject 1 is the highest. On the other hand, RMS of subjects 2 and 3 is reduced in comparison with the nominal model. The physique of subject 1 differs from other subjects. When such a person sits, the specified controller should be designed. From these results, the proposed method has robustness for

Nominal Subject 1 Subject 2 Subject 3

100

104.6

97.7

92.5

the passenger of the general physique.

RMS ratio to Nominal controller[%]

**5. Conclusion** 

characteristic.

**7. References** 

**6. Acknowledgment** 

100

112.1

Fig. 18. RMS value of vertical acceleration of passenger 1's head

96.1

97.1

Head 1 Head 2

This study aims at establishing a control design method for the active suspension system in order to reduce the passenger's vibration. In the proposed method, a generalized plant that uses the vertical acceleration of the passenger's head as one of the controlled output is constructed to design the linear *H*∞ controller. In the simulation results, when the actuating force is limited, we confirmed that the proposed method can reduce the passenger's vibration better than two methods which are not include passenger's dynamics. Moreover, the proposed method has robustness for the difference in passenger's vibration

This work was supported in part by Grant in Aid for the Global Center of Excellence Program for "Center for Education and Research of Symbiotic, Safe and Secure System

Ikeda, S.; Murata, M.; Oosako, S. & Tomida, K. (1999). Developing of New Damping Force

Control System -Virtual Roll Damper Control and Non-liner *H*∞ Control-,

Design" from the Ministry of Education, Culture, Sport, and Technology in Japan.

*Transactions of the TOYOTA Technical Review*, Vol.49. No.2, pp.88-93


**Modelling and Nonlinear Robust Control of** 

Yang Bin1, Keqiang Li2 and Nenglian Feng1

*1Beijing University of Technology* 

*2Tsinghua University* 

*China* 

**Longitudinal Vehicle Advanced ACC Systems** 

Safety and energy are two key issues to affect the development of automotive industry. For the safety issue, the vehicle active collision avoidance system is developing gradually from a high-speed adaptive cruise control (ACC) to the current low-speed stop and go (SG), and the future research topic is the ACC system at full-speed, namely, the advanced ACC (AACC) system. The AACC system is an automatic driver assistance system, in which the driver's behavior and the complex traffic environment ranging are taken into account from high-speed to low-speed. By combining the function of the high-speed ACC and low-speed SG, the AACC system can regulate the relative distance and the relative velocity adaptively between two vehicles according to the driving condition and the external traffic environment. Therefore, not only can the driver stress and the energy consumption caused by the frequent manipulation and the traffic congestion both be reduced effectively at the urban traffic environment, but also the traffic flow and the vehicle safety will be improved

Taking the actual traffic environment into account, the velocity of vehicle changes regularly in a wide range and even frequently under SG conditions. It is also subject to various external resistances, such as the road grade, mass, as well as the corresponding impact from the rolling resistance. Therefore, the behaviors of some main components within the power transmission show strong nonlinearity, for instance, the engine operating characteristics, automatic transmission switching logic and the torque converter capacity factor. In addition, the relative distance and the relative velocity of the inter-vehicles are also interfered by the frequent acceleration/deceleration of the leading vehicle. As a result, the performance of the longitudinal vehicle full-speed cruise system (LFS) represents strong nonlinearity and coupling dynamics under the impact of the external disturbance and the internal uncertainty. For such a complex dynamic system, many effective research works have been presented. J. K. Hedrick et al. proposed an upper+lower layered control algorithm concentrating on the high-speed ACC system, which was verified through a platoon cruise control system composed of multiple vehicles [1-3]. K. Yi et al. applied some linear control methods, likes linear quadratic (LQ) and proportional–integral–derivative (PID), to design the upper and lower layer controllers independently for the high-speed ACC system [4]. In ref.[5], Omae designed the model matching control (MMC) vehicle high-speed ACC system based on the H*-infinity* (Hinf) robust control method. To achieve a tracking control between

**1. Introduction** 

on the highway.

Varterasian, H. H. & Thompson, R. R. (1977). The Dynamic Characterristics of Automobiles Seats with Human Occupants, *SAE Paper*, No. 770249 **6** 
