Hybrid Optimal Control for Semi-active Suspension Systems L. Flores1, A. Drivet1, R. A. Ramírez-Mendoza** O. Sename2, C. Poussot Vassal2, L. Dugard2 1 Mechatronics Department, Tecnológico de Monterrey, Campus Monterrey, Avenida Eugenio Garza Sada 2501 Sur, Monterrey, N.L., 64849, México email:{leonardo.flores ,drivet} @itesm.mx 2 Laboratoire d’Automatique de Grenoble,UMR CNRS-INPG-UJF, ENSIEG-BP 46, 38402 Saint Martin d’Heres Cedex, France, email:{olivier.sename,charles.poussot, luc.dugard }@lag.ensieg.inpg.fr ** Corresponding author:
[email protected]
ABSTRACT In this paper we compare optimal skyhook, optimal ground hook and optimal hybrid control laws for the stabilization of the semi-active suspension in a quarter of a Renault Scenic 2001. A Magneto-Rheological (MR) damper will be used in the implementation of the system. This damper is modeled using a grey box structure based on a Bouc-Wen model and a Non linear Autoregressive with eXogenous variable, (E. Niño et al. 2006). The paper is divided in four parts. The first one concerns with the modeling of the suspension using the quarter of vehicle model, the second part shows different control techniques that are applied to suspensions, the third part introduces optimal hybrid control and finally in the fourth part some simulation results are shown. Keywords: Vehicle Dynamic Modeling, Semiactive Suspension, Hybrid Control, Optimal Control
1. INTRODUCTION The vehicle suspension system is located between the vehicle body and the vehicle wheels. Its main function is to improve comfort and road holding by isolating the vehicle body from road disturbances. Passive, active and semi-active are the strategies that are extensively used in order to modify the dynamic response of vehicle suspensions. A passive suspension system consists of conventional springs and dampers, once the tuned parameters they can not be modified in order to adapt the suspension system to the road conditions.These kind of suspension systems show a compromise between ride quality and handling performance, but they have the advantage of low cost and hight reliability. Active systems supply energy to the system in order to generate the control force and require the presence of a force generator between the vehicle body and the vehicle wheels (D.Hrovat 1997). Althought these systems can be adapted to a very different road conditions improving both the confort and the road holding at the same time, they are very expensive. The semiactive control systems in other hand combine the best features of the active and passive approaches (C. Rossi et al 2004). They have the reliability of the passive systems, and the flexibility of the active ones. The semiactive suspension systems modify one or more suspension parameters in order to adapt the system to different road conditions. The most extensive semi-active control strategy is the variation of the suspension damping coefficient obtaining a
performance close to the active systems but with less complexity and relatively small cost. These semi-active systems use controllable fluids which can change their rheological properties in milliseconds when they are exposed to an electric or a magnetic field. MR dampers are semi-active control devices that use MR fluids to provide controllable force outputs, these devices can be viewed as fail-safe devices because they become passive dampers when the control hardware malfunctions. MR dampers are used to reduce the vibrations of mechanical systems; the advantage of these smart devices is the low power input requirements and the high force output.
2. QUARTER VEHICLE MODEL Ms
Zs
Sprung Mass
Ks
U
Cs Mu Unsprung Mass
Kt
Zu
Ct
Zr
Figure 1: Quarter vehicle model
In order to test the different control techniques the quarter of a vehicle model is used, this model is shown in figure 1. In this model we use the following parameters : Ms is the sprung mass, Mu is the unsprung mass, Ks is the spring stiffness, Cs is the damping coefficient of the damper, U is the force applied by the MR damper, Kt is the stiffness of the tire , Ct is the damping coefficient of the tire, Zs is the displacement of the sprung mass, Zu is the displacement of the unsprung mass and Zr is the uneveness of the road..(A. Drivet et al 2006) Applying Newton’s second law to the model we get: MsZs = − Ks ( Zs − Zu ) − Cs ( Zs − Zu ) + U MuZu = Ks ( Zs − Zu ) + Cs ( Zs − Zu ) − Kt ( Zu − Zr ) − Ct ( Zu − Zr ) − U
Considering the following state variables
X 1 = Zs ; X 2 = Zs ; X 3 = Zu and X 4 = Zu We get the following matrix equation
X = AX + BW + VZr
Where: 0 1 0 0 ⎡ ⎤ ⎢ ⎥ Ks Ms Cs Ms − Ks Ms − Cs Ms ⎢ ⎥ A=⎢ ⎥ 0 0 0 1 ⎢ ⎥ ⎢⎣ Ks Mu Cs Mu − ( Ks + Kt ) Mu − ( Cs + Ct ) Mu ⎥⎦ ⎡ X1 ⎤ 0 ⎤ ⎡ 0 ⎡ 0 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢X ⎥ 0 1 Ms ⎥ 0 ⎥ ⎡ Zr ⎤ 2 ⎢ ⎢ W =⎢ ⎥ V =⎢ B=⎢ X =⎢ ⎥ U⎦ 0 0 ⎥ 0 ⎥ ⎢X ⎥ ⎣ 3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 Ct Mu Kt Mu Mu − ⎣ ⎦ ⎣ ⎦⎥ ⎢X ⎥ ⎣ 4⎦
We define the following output variables: C1 : Sprung mass acceleration. C3 : Sprung mass displacement.. C5 : Unsprung mass velocity. C7 : Damper deflection velocity. C9 : Tire deformation.
C2 : Sprung mass velocity. C4 : Unsprung mass acceleration. C6 : Unsprung mass displacement. C8 : Damper deflection displacement. C10 : Tire deformation velocity.
These output variables are calculated using the following equations. C1 = Zs = ⎡⎣ − Ks Ms − Cs Ms Ks Ms Cs Ms ⎤⎦ X + U Ms C2 = Zs = ⎡⎣ 0 1 0 0 ⎤⎦ X
C3 = Zs = [1 0 0 0] X
C4 = Zu = ⎡ Ks Mu Cs Mu − ( Ks + Kt ) Mu −(Cs + Ct ) Mu ⎤ X − U Mu ⎣⎢ ⎦⎥ C5 = Zu = [ 0 0 0 1] X
C6 = Zu = [ 0 0 1 0] X
C7 = Z
C8 = Z
Damdef
= [ 0 1 0 −1] X
C9 = Z = 0 0 1 0 ] X − Zr Tiredef [
= 1 0 −1 0] X Damdef [ C10 = Z = 0 0 0 1] X − Zr Tiredef [
The simulations were run using the following parameters: Ms=300 Kg; Sprung Mass Mu=45 Kg; Unsprung Mass Ks=16000 N/m; Suspensión Spring Constant Cs=1100 Ns/m; Passive damper constant Kt=160000 N/m; Tire equivalent spring Ct= Neglected. Tire equivalent damper constant
3. SKY HOOK, GROUND HOOK AND HYBRID CONTROL LAWS 3.1 Skyhook Control In the ideal skyhook configuration which is shown in figure 2 a damper Csky is connected between the sprung mass and an inertial reference attached to the sky. This configuration focuses on the sprung mass, as Csky increases the sprung mass motion decreases, however the unsprung mass motion is increased. Sky Csky
Ms Sprung Mass
Zs
Ks
Mu Unsprung Mass
Zu
Ct
Kt
Zr
Figure 2: Ideal Sky-Hook
Because of the impossibility of buiding the ideal skyhook, we emulate its behaviour varying the damping force of the MR damper located between the sprung and the unsprung masses in the model shown in figure 1, according to the following control law wich is called “Sky hook control law”(M.Ahmadian et al 2004) If Zs • ( Zs − Zu ) ≥ 0 then U = G iCsky i Zs
otherwise U = 0
3.2
Ground-hook Control
In the ideal groundhook control model the damper is connected between the unsprung mass and an inertial frame attached to the earth.This configuration is shown the figure 3. This configuration focuses on the unsprung mass adding damping to it and trying to isolating it from road signals. However as the damping is removed from the sprung mass this mass will have a lot of oscillations. (M. Ahmadian et al 2005) Ms Sprung Mass
Zs
Ks
Mu
Cgnd
Unsprung Mass
Gnd
Kt
Zu
Ct
Zr
Figure 3: Ideal Ground-Hook Model
Similarly to the skyhook case , it is impossible to build the ideal ground hook, we can only emulate its behaviour varying the damping force of the MR damper located between the sprung and the unsprung masses shown in figure 1 according to the following control law wich is called “Ground hook control law” (M. Ahmadian et al 2004) If Zu • ( Zs − Zu ) ≤ 0 then U = G iCgnd i Zu
otherwise U = 0
3.3
Hybrid Control
In order to obtain the advantages of both the skyhook and the groundhook, they are combined in the so called hybrid control technique which is a linear combination of skyhook and groundhook. In this control strategy which is shown in figure 4 we use the alpha parameter to specify how much skyhook and how much groundhook will be used. If alpha has the value of one the system behaves as pure skyhook control and if alpha takes the value of zero the system behaves as pure groundhook control . The hybrid control law is the sum of both the skyhook control and the groundhook control according to the following equation. (M. Ahmadian et al 2004) If Zs • ( Zs − Zu ) ≥ 0 then U sky = α iG iCsky i Zs otherwise U sky = 0
If Zu • ( Zs − Zu ) ≤ 0 then U gnd = (1 − α )iG iCgnd i Zu
otherwise U gnd = 0 U hyb = U sky + U gnd
The user varies the alpha parameter in order to specify the ratio between skyhook and grondhook, for instance if alpha equals 0.30 its meaning is that the user wants 30% of skyhook and 70% of ground hook. Sky Ms
Csky
Sprung Mass
Zs
Ks
Mu
Cgnd
Unsprung Mass
Gnd
Kt
Zu
Ct
Zr Figure 4: Ideal Hybrid-Control
4. OPTIMAL HYBRID CONTROL In a passive suspension the designer adjust the spring stiffness and the damping coefficient depending on “How much comfort” and “How much road holding” is covenient for the specific car. This is done taking into account that as lower damping higher comfort and as lower damping lower road holding: In a passive suspension these quantities can’t be independently varied. When a semiactive suspension is used, the designer can continuously adjust the damping coefficient between a range which is specified by the damper manufacturer. It is common to have damping coefficients between 1000 and 10000 Ns/m. In order to adjust the damping coefficient of the MR damper the designer makes the decision according to a minimizing crtiteria that calculates the best combination of skyhook and groundhook and the best damping coefficient. The minimizing criteria was taken from Optymal Skyhook (C.Pousst et al 2006) This criteria uses the following function . I 4→30 ( Zs ) I 0→5 ( Zs ) I 0→20 ( Zu ) I 0→20 ( Zdadef ) + k2 + k3 + k4 J (α , Csky ) = k1 k max I 0→5 ( Zs ) max I 0→20 ( Zu ) max I 0→20 ( Zdadef ) max I 4→30 ( Zs )
where I a →b =
b
∫ x2 ( f )df is related to the spectral density of the signal , k1 and k2
a
are weighting factors related to comfort and finally k3 and k4 are weighting factors related to road holding. 5. SIMULATION RESULTS Because of limitation of space only two figures are presented. In figure 5 the time response is plotted. It is considered that a perturbation on the road last during one second and this perturbation consist on a bump of 25 mm height.
Figure 5: Comparison between Sky,Gnd and Hybrid
In the upper section , the displacement of the sprung mass is plotted. It can be seen that the skyhook control law shows the best performance and the ground hook control law shows the worst performance. In the lower section, the displacement of the unsprung mass is plotted, and can be seen that the performance of the previous control laws is inverted, the ground hook control law is the best and the sky hook control law is the worst, In figure 6 the frequency response for the different control techniques is shown. In the upper section the acceleration of the sprung mass which is strongly related to comfort is plotted, it can be seen that the hybrid control law is the best below frequencies of 15 hertz. In the lower section the displacement of the sprung mass is plotted, but in this part of the plot the hybrid control law does not show a good response when it is compared to the others, however the displacement of the sprung mass has less effect than the acceleration of the sprung mass when the comfort is the main concern.
6. CONCLUDING REMARKS We have seen that various well-known control techniques for suspensions were presented. For instance we saw that the Sky-hook control law focuses on avoiding oscillations of the sprung mass trying to improve comfort , on the other hand we saw also that Ground-hook control focuses on avoiding oscillations of the unsprung mass in order to improve road holding.
When a semiactive suspensión is used, the hybrid control technique can be used in order to achieve that the MR damper resembles to the sky hook control law or to the ground hook control law depending on the road unevenness. We need to look for a better strategy to switch between sky-hook and ground-hook because the simulations done to find the optimal parameters for the different control laws took a lot of time. Finally we need to find a strategy to assign the constants k1 , k2 , k3 , k4 that define the ratio between comfort and road hoding “on line” depending on the road conditions. 7. ACKNOWLEDGMENTS The authors acknowledge the support received from Tecnológico de Monterrey; Mexico and The Laboratoire d’Automatique de Grenoble, ENSIEG – INPG, France to carry out the research reported in this paper. 8. REFERENCES 1.
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