H∞ Control of Automotive Semi-Active Suspensions

Copyright © IFAC Advances in Automotive Control Salemo, Italy, 2004

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H oc CONTROL OF AUTOMOTIVE SEMI-ACTIVE SUSPENSIONS Carlo Rossi • , Gianluca Lucente •

• CASY - Center for Research on Complex AutomaJed Systems " G. E vangelisti " DEIS - University of Bologna. Viale Risorgimel1lo n.2. 40136 Bologna ITALY Tel. +39051 2093020. Fax. +39051 2093073 E-mail: {crossi. glucente}@deis.unibo.it

Abstract: This paper deals with the control of semi-active automotive suspensions using H oc state-space optimization techniques. After the derivation of two standard models of the system, quarter car and half car model, three Hoc controllers are developed in the fully active case and then adapted to the non-linear real system using a sort of clipped control. The first controller is focused on ride comfort optimization., the second on both comfort and handling improvement, the third is a global controller derived from half car model. Performance indices related to ride comfort and drive safety are introduced to evaluate the proposed controllers. Simulations are performed to show that semi-active controlled suspensions succeed in achieving higher levels of ride comfort and drive safety with respect to a passive setting. Copyright © 2004 [FAC Keywords: Semi-active suspensions, Automotive control, Hoc control, Disturbance rejection, Modelling

I. INTRODUCTION

Semi-active solutions are an useful trade-off between design constraints and desired performances. In the considered case, the control action consists in the variation of the suspension damping coefficient: the results obtained by this approach are near to those of active suspensions, but the complexity of the system is smaller.

The suspension system is the main tool to achieve ride comfort and drive safety for a vehicle. The design of passive suspensions has always been focused on obtaining a good compromise between these two targets, but structural limitations prevent passive solutions from achieving the best performances for both goals.

The research on controlled suspension systems is well developed since the first works on active and semiactive damping (Karnopp, 1983), (Butsuen, 1989) were published, following two main approaches. The first one follows the "skyhook" philosophy, that is the idea of designing a controller such that the passengers would perceive the sensation of a car body hangedup to the sky, totally decoupled from disturbances induced by road irregularities. Skyhook philosophy leads to a reference model design, in which the damping force is proportional to the vertical velocity of the rody of the vehicle (figure 1). This ideal damping force is then used as a force set-point for vari-

Active and semi-active suspension systems can improve driver and passengers comfort perceptions during various driving conditions, achieving a better compromise than the passive suspensions. The key feature of these systems is the generation of a control force in the active case, the variation of one or more parameters of the main suspension in the semi-active setting. A fully active control for suspensions requires the presence of a force generator between the body of the vehicle and the wheel, and this requirement increases costs, weights and complexity of the project.

559

r1 ,

Z

K

.r-----------~~~

i

Sprung mass

t

'----s-.----------'

SuspenslOf'

50

.

> ".

Unsprung mass

'

Tire

,.

..... ~(b)

Figure 1. Quarter car model able structure controllers applied to active and semiactive suspensions control, in which actuator dynamics are taken into account (Alleyne, Hedrick. 1995), (Mohan, Phadke, 1996), (Yokoyama et aI., 2001). The other approach is based on ride comfort performances optimization, which is achieved using different techniques. The problem is solved using LQG and minimum variance principles (Gordon, Best, 1994), (Fischer et a1. , 2002), but also H2 and Hoo optimization (yamashita et aI., 1990), (Son et aI., 2001), (Ohsaku et aI., 1999), (Wang et a1. , 2001).

j ,~

· 2 F f Z(s) ( ) Zu(S)-Zr( S) (b) FIgure . req. resp. 0 Zr(S) a, Zr(S)

the wheel, the axle and other rotating elements. Ks and Cs are the spring constant and the viscous friction coefficient of the main suspension, while Kt is the constant of the spring modelling the tire. In the semi-active suspension system considered, the friction coefficient of the damper can be controlled acting on the position of a spool valve, hence it is the control input. In this paper the actuator dynamics are assumed to be much faster than the other dynamics, so they are neglected. Note that controlling the friction coefficient introduces structural limitations: the damping force is dependent upon the state of the system, since it must be always concordant with the difference of velocity across the damper. This will exploited in section 4. This simple model is very useful to study the behavior of the frequency response of the sprung mass height Z and the tire deflection Zu - z,. when a road input is applied to the system; the focus is on these two variables because of their great importance in the evaluation of ride comfort and drive safety performances. When considering a passive system, system responses can be characterized with respect to the damping coefficient Cs. Figure 2.a presents the frequency response between the road input Z r and the height of the sprung mass, while figure 2.b refers to the frequency response between the road input and the tire deflection. It is confirmed the existence of a compromise in the passive systems: figure 2.a shows that ride comfort is achieved generally with a soft suspension (Cs = 700 Ns / m), because this setting allows the highest attenuation in the range between the resonance peaks of spnmg and unsprung masses, which are placed at 1 Hz and 10 Hz respectively; drive safety, instead, requires an hard setting (C.. = 2700 Ns/m), as demonstrated by (fig. 2.b) where the resonant peaks almost disappear. In figure 2 also the responses of a fully active system based on skyhook philosophy are reported, to show improvements given by suspension control.

This paper deals with the control of a semi-active suspension system of a high class-sport car, in which the mechanical design of the suspension is mainly driven by handling performances; therefore, the main objective of control is the ride comfort, with the constraint of keeping good levels of handling and drive safety. It is shown that through the use of the H 00 design methodology, both comfort and handling performances can be improved. The paper is organized as follows: in section 2 quarter car and half car models are derived and the main features of the system are described. Section 3 focuses on the various performance indices which will be used to assess the control systems performances. In section 4 three controllers for the system, based on Hoo optimization technique, are described. Section 5 describes the simulations performed to evaluate the control systems performances according to the indices previously introduced. Conclusions end the paper.

2. SYSTEM MODELS Suspensions control is usually approached starting from the simplest model of the system, the quarter car model (fig 1); half car model is used to analyze system performances in a more comprehensive way, when the focus of the study is vehicle dynamics in general, including load transfer effects caused by acceleration and braking. Quarter car model, depicted in fig. 1, is described by the following differential equations:

Mi m~

z) + Cs (tu - i) = Ks (z - Zu) + C .. (i - tu)

= Ks (Zu -

(1) + Kt(zr - Zu) (2)

The half car model (fig. 3) is useful in the evaluation of dynamic load transfer effects and bounce movements of the car during acceleration and braking operations. The angle (J is the pitch angle, which represents the

where M, the sprung mass, models the portion of body above the suspension, m, the unsprung mass, models

560

j tf

Ride comfort:

z~'f 5

Drive safety:

F::~~n =

,

=

t:

o ( ~ ) 2 dt

t 1::0 (~;::;::) 2dt

~t

I

~

I

~ ! -I

;;

!

--

--

':.

,;:. L

I I

1 --

Figure 3. Half car model

1

1 I

orientation of the car along its headway direction, according to SAE reference system (Gillespie, 1992). Pitch movements can deteriorate passengers perceptions and road holding, so the reduction of pitch dynamics will be a target, which will be considered in the simulations performed for half car model. The model has four degrees of freedom and it is described by the following dynamic equations:

Figure 4. Perceived acceleration: frequency response

In the previous indices, the acceleration of gravity 9 and the steady-state load on the wheel F z.,. stat = (M + m )g are used to normalize the expressions. Low RMS values of the dynamic forces exchanged between wheel and road denotes good levels of road holding, so one of the objectives of control is keeping low levels of this index during the various operating conditions.

M Zb = K sI (Zu I - zI) + CsI (ZuI - Zt ) + + Kd z u2 - Z2) + Cd Zu2 - Z2) (3) 19 0 = IIKsI (Z I - Zul ) + 11 Cs1(Zl - Zu1 ) + -12Ks2 (z2 - Zu2 ) - -12Cs2 (z2 - Zu2 ) + T (4) m 1zu1 = -Ks1(zul - Zl) - C s1(Zu I - Zl ) + + Ktl (Zrl - Zu1 ) (5) m2Zu2 = -Ks2(Zu2 - Z2) - C s2(Zu2 - Z2) + + Kdzr2 - Zu2 ) (6)

Another important parameter for comfort is the perceived acceleration on the seat, a notion we can find in (Gillespie, 1992) and in the Guide to ISO 2631 norm, in which human tolerances to continuous and shock induced disturbances in transport systems are established. According to this concept, human sensibility to vertical acceleration is maximum in the frequency range between 4 and 8 Hz, because of resonances of internal organs. This sensibility has been codified in a transfer function from seat displacement to perceived acceleration, whose frequency response is plotted in figure 4. As the input of this frequency response is the seat displacement, it is important for the controllers to ensure a smooth trend of the sprung mass height after a bump input or another persistent change in road profile. The quantitative index related to perceived acceleration is its RMS value, defined as follows :

in which the symbols keep the meaning of quarter car model, with the algebraic positions Zl = Zb lIB and Z2 = Zb + 12e, linking height of center of gravity to that of suspension attach points. The pitch inertia moment is denoted by 19, while 11 and 12 are the distances from center of gravity to front and rear axles respectively. The torque T models the effects of dynamic load transfer and is given by T = Mha, where h is the height of the center of gravity and a is the longitudinal acceleration of the car. In all the equations the indices 1 and 2 stay for front and rear.

~s = ~

iT

(g(t ) . z (t » (g(t ) . z (t » dt

t =fl

3. PERFORMANCES INDICES AND PROBLEM STATEMENT

where g(t ) is the impulse response of the transfer function showed in figure 4 and * is the convolution operator.

Ride comfort is a mix of perceptions received by the passengers, involving vibrations and noises on the seats, the pedals and the steer; therefore, it is strictly linked to the vertical acceleration of the body z. On the other hand, dri ve safety is a concept which can be described by the forces dynamically exchanged by tire and road, which are defined by Fz.,. ,dyn -= Kt (z,. zr)' According to (Fischer, Bomer, Isermann, 2002) ride comfort and drive safety are related to the following RMS values, which will be considered as performance indices for the evaluation of the controllers.

For half car model, the evaluation of pitch angle becomes important, as it denotes variations in the dynamic loads on the axles, whose effects must be avoided to ensure good levels of ride comfort. In particular, the control should ensure small variations of pitch angle during acceleration and braking, because of the high sensibility ofhumans to horizontal stresses in the frequency range around 1 Hz. The reduction· of pitch movements is related to the performance index J9 = min lBl.

561

1

1 ]T

Bu= [ 0 MO-M

[

Kt ] T OOO-M

where the state variables are heights and velocities of sprung and LmSprung masses (z , oi, Zu and z~), while the disturbances to be rejected is z,. , the road profile; y and z are informative outputs and error signals respectively; the control input is denoted by u, while w is the vector of the disturbance inputs. The informative output distribution matrix C y is the identity matrix, because all the state components are considered available; the choice of the matrices C z and Dz is a design parameter, as it depends on the error signals. Dyu and D zu are null matrices of suitable dimension. The feedback of sprung and unsprung mass height is made using relative heights with respect to the road profile, in order to obtain steady-state conditions in which the main suspension and the tire are not extended nor compressed. This implies that the matrix Dy is equal to Dyw = [-10 - 1

Figure 5. Control Scheme According to the performance indices introduced in this section, the control problem can be stated as follows : find an appropriate control law for the damper friction coefficient Cs = Cs(t, x), such that a mix of ride comfort and drive safety performance indices is optimized.

4. SOLUTION OF THE CONTROL PROBLEM The system considered is non linear, in the sense that the control input Cs enters the system by a multiplication for quantities depending upon the state components. The structural limitations deriving from this feature of the damper prevents the possibility of solving the control problem using linear optimization techniques, mainly because of the null control action in steady state. The problem is firstly approached modifying the system, in order to obtain a linear system, which is used to solve Hoo synthesis procedure; the resulting linear controller generates a force set-point, which is used to calculate the effective value of Cs. The feedback path (1) in figure 5 describes the ideal linear system used for the synthesis of control, while the path (2) is that of the real nonlinear control applied to the system with the semi-active damper. The nonlinear controller structure involves two stages: in the first one a damping force is computed according to Hoo control strategy, in the second one this force is converted into an admissible value for the viscous friction coefficient which will be inserted into the system by the spool valve. The aim of Hoo control is the minimization of H 00 norm between w and z for the closed loop system by a stable dynatnic state feedback

oV.

Two Hoo controllers will be presented for quarter car model, which differs for the choice of the error signals. The first controller (Quarter 1) is obtained using sprung mass vertical velocity as error signal, that is C z = [0 1 0 0]: this choice of the error output is driven only by comfort. For quarter car model, a second Hoo controller (Quarter 2) has been developed with the aim to improve both comfort and handling performances: in this case the error signals vector z contains sprung mass vertical velocity and unsprung mass vertical displacement (tire deflection) : the corresponding choice for C z is:

Cz =

[~ ~ ~ ~]

For both controllers the weighting functions are constant values; in the second case two different constants are used, in order to give more importance to the comfort variable with respect to the handling one. The H 00 norm minimization problem is converted into a suboptimal problem, which is relatively easy to solve: find a state feedback K such that the closed loop Hoo norm is less than ,; if the problem is iterated in " a suboptimal solution near to the optimal one can be fOlmded. According to (Doyle et al., 1989), two algebraic Riccati equations (ARE) have to be solved, before obtaining a sub-optimal controller K which makes the closed loop system norm from w to z to be less than , :

K (s). The starting point is a fictitious substitution of the setni-active damper of the main suspension with an ideal force actuator: this modification of the system is described by the position Cs = 0 and by the presence of a force input u . The resulting linear system is deprived of any damping action except for the control and its quarter car model has a state-space description gj.ven by:

i = AI + Buu + Bww y = CyI + Dyuu + Dyww z = CzI + Dzuu + Dzww

, Bw=

A'X"" + X""A

+

,-2 X""BwB:"Xoo +

-X""BuB~Xoo

(7)

+ C~Cz =

0

+ Y""A' + ,-2y""C~CzYoo + -y""C~CyYoo + BwB:" = 0

(8)

AYoo

(9)

Being X"" and Y00 the solutions of Riccati equations, the suboptimal controller K (s) is a purely dynamical system described by the matrices A K , BK , CK defined below:

562

Table 1. System parameters Parameter

Value

M M1 M2 m1 m2

1670 Kg 631.5 Kg 828.5 Kg 95 Kg 115Kg 21687.4 N/m 38337.6N/ m 224420 N/m 254800 N / m 1.475 m 1.125 m 0.834 m 0.88 m 24225 Kgm2 984.4 Kgm2

K~l

K~2

Ktl Kt2 /1 12 If IT 19

Figure 6. Semi-Active Damper Regions

A + ,,/-2 BwB'.uXoo -ZooLoo Foe

AK = BK = CK = Foo = Loo = Zoo =

+ BuFoo + ZooLooCy

1~

Controllers for half car model have been developed, using Hoo optimization procedure. In this case, the

-B~Xoo

-YooC'y (I - ,,/ -2Yoo X oo ) -1

state vector is x = [Zb Zb (J 0 Zul iul Zu2 ZU2] T. The disturbances input vector contains the road inputs to both front and rear axles and the torque T which appears in model equations. This torque models the effect of dynamic load transfer between front and rear axles during acceleration and braking. The pitch angle and velocity have been included in the error signals, together with height and vertical acceleration of the center of gravity, in order to achieve ride comfort. For handling performances, the inclusion of unsprung mass relative height in the errors vector is necessary. The resulting C z and D zu are:

The order of quarter car controllers is four and the scalar output of the linear controller represents the desired damping force. This force is used as a setpoint to calculate the value of the damping coefficient, together with the actual value of the difference of velocity across the damper (path 2 in fig. 5)~ the damping force must be contained within the regions of fig. 6 which define the operating space of the damper. If the required force is outside these regions, a clipped control is adopted and the value of the friction coefficient is computed in order to obtain an effective damping force C. (iu - z) (block C. in figure 5) which is the best approximation for the desired force, computed by the Hoo controller. The nonlinear control is described by the following relations, in which fma.x (iu - z) and fmin(Zu - z) are the expressions of the damper limit characteristics, reported in figure 6:

1

0

K,,1+K II2

c. _ [

M 0 0 0 0

_ Srcq

-

0

0 I,K., - 12K.2 M 1 0 0 0

0 0 0 0

D. u

c

Description Total mass Front spnmg mass Rear spnmg mass Front tlIlSpnmg mass Rear unspnmg mass Front susp. spring stiffiless Rear susp. spring stiffness Front tire stiffuess Rear tire stiffness Front axle to c.o.g Rear axle and c.o.g. Front Semi-width Rear semi-width Pitch inertia moment Roll inertia moment

=

02. 0000

If

o

0

0

0

,o K;; , K;,'

'j 01

0 0 0 0 000 0 1 000 00010 1

o

]T

[ 0-0000 1<1

Fd,u:t_pU'tnt .

Zt.

.

-z

fma.:z:(iu - z) (zu - z) C~ = fmin(zu - z) { (iu - z) CS~·CY

if C if C

~rr.q ~r<.

>

!ma.:z:(zu - z) (zu - z)

<

!min(iu - z) (iu-z)

The Hoo global controller has order eight and its two outputs represent force set-points for front and rear dampers. The effective values of the friction coefficients are computed separately as for quarter car controllers, considering the limitations imposed by the system.

(10)

otherwise

The features of the semi-active damper prevent the control from obtaining the performances which could be achieved using a fully active control: mainly, near steady-state condition, with small or any velocity difference between the two masses, there is a very small or any control force available; furthermore, the control force must be contained in the two defined regions of the figure, which are strictly dependent upon the difference of velocity between sprung and unsprung mass. The influence of the semi-active actuator on the performances is important when the desired force is discordant with the difference of velocity across the damper: the optimal damping force can not be exerted, so there is a worsening of the performances with respect to a fully active system.

5. SIMULATIONS At'ID PERFORMANCES EVALUATION The behavior of the system has been simulated for both the models presented, using MATLAB-Simulink. The values of the parameters used for the synthesis of control and simulations are given in table 1. The characteristics of front and rear dampers are different for maximal values, but very similar in the form. In figure 6 only the front damper limit characteristics are presented. kh and !l12 are the values for sprung mass

563

"""' ......

Table 2. Quarter car: Percentage reductions with respect to passive setting System

zRMS

H= quarter 1 H co quarter 2

57 42

9~- 1

,.>

FRMS

%udyn

38 27

-23

8 1.5

for front and rear axles; for quarter car model, these values must be divided by two.

i! f

The simulations focus on the performances achievable with Hoc controllers with respect to a passive setting, in which the damper has a characteristic forcevelocity intennediate between those of figure 6. In the simulation for quarter car model it is used a road profile described by the following function of time: zr(t) = 8step(t) - 4step(t - 4) + O.3sin(4t) cm, where .~tep ( t ) is the unit step function applied at t=O. The results of this simulation are presented in table 2: for each controller the percentage reductions with respect to the passive setting are given for the considered indices. The simulation underlines the better comfort performances for the controlled systems. For half car model the simulation is performed with the same road profile. The longitudinal acceleration of the vehicle is described by the time function aCt) = 7step(t) 7.~tep(t-5 ) -1.48tep(t-8 ) m/.~2. Figure 7 illustrates the evolution of the pitch angle during the experiment: the passive system is characterized by higher values of the angle. For this simulation, involving half car model, the percentage reductions of performance indices are reported in table 3.

Figure 7. Evolution ofB Simulations for half car model proved that quarter car controllers can improve global performances with respect to half car controllers, because of the nonlinearities of the real system. REFERENCES AlJeyne A., Hedrick J.K. (1995). Nonlinear Adaptive Control of Active Suspensions. IEEE Transaction.~ on Control System Technology 3(1), 94-101. ButsUCll, T. (1989). The Design of Semi-Active Suspensions for Automotive Vehicles. PhD thesis. MlT. Doyle J .C., Glover K., Khargonekar P.P., Francis BA. (1989). Statespace solutions to standard H2 and HooCODtrol problems. IEEE Transactions on Automatic Contm/34(8) , 831-847. Fiscber D., BOmer M,lsermann R. (2002). Control ofrnechatronic semi-active vehicle suspensions. IFAC: Symposium Mechatronic Systems pp. 225-230. Gillespie T.D. (1992). Fundamentals of Vehicle Dynamics. SAE. Gonion TJ., Best MC. (1994). Dynamic Optimization of Nonlinear Semi-Active Suspension Controllers. Control '94. Conference Publication pp. 332-337. Kamopp, D. (1983). Active damping in road vehicles suspension system. Vehicle System Dynamics 12, 291-316. Moban B., Phadke S.B. (1996). Variable Structure Active Suspension System. Industrial Electronics. Contml, and Instrumentation, Proceedil'fgS of the 1996 IEEE IECON 22nd International Conference 3, 1945-1948. Ohsaku S., Nakayama T., Kamimura L, MotozODO Y. (1999). NonIinear H 00 control for semi-active suspension. JSAE Review (20),447-452. Son HJ. et al. (200 I). A Robust Controller Design for Performance Improvement of a Semi-Active Suspension Systems. Proceedings ofIEEE 1ntenrationa/ Symposium on industrial Electronics, ISIE 20013, 1458-1461. Wang J., Wilson DA., Halikias G.D. (2001). Hoo robust performance control of decoupled active suspension .systems based on Imi method. Proceedings of the American Control Conference pp. 2658-2663. Wang Fu-Chen, Smith MC. (2003). Active and Passive Suspension Control for Vehicle Dive and Squat Nonlinear and Hybrid Systems in Aulnmotive Control, pp. 23-40. Yamashita M , Fujimori K., Uhlik C., Kawatani R., Kimura H. (1990). Hex> control of an automotive active suspension. Proceedings ofthe 29th Conference on Decision and Control

6. CONCLUSIONS

Hoc control systems for semi-active suspensions have been proposed, in order to improve both comfort and handling performances, considering quarter car and half car model. The controllers proved to be better than a passive setting as for comfort; drive safety performances can be improved including variables related to unsprung mass dynamics in the error signals vector. Table 3. Half car: Percentage reductions with respect to passive setting System

zRMS

zRMS

40 40 36

4 14 21

perc

FRMS ZI!

IdV n

3 15

25

, ..

5 Time (a)

According to these results and many other simulations, there are very little difference in performances among quarter and global controllers, when slow road inputs are applied to the system., while the global controllers are worse as for pitch angle evolution and when fast inputs are taken into account, mainly when road input contains important harmonic components near the resonance of the unsprung mass (10 Hz). Therefore, in general, the simplest controllers can be considered better than the others, for their features of disturbance rejection, robustness and simplicity.

H oo quarterl Hooquarter2 Hex> half

I

pp. 2244-2250.

FRMS

z.,,2dyn

Yokoyama M, Hedrick J.K., Toyama S. (200 I). A Model FoIJowing Sliding Mode Controller for Semi-Active Suspension System. Proceedil'fgs of the American Control Conference pp. 26522657.

3 7 7

564

"