Adaptive Robust Gain Scheduled Control of Vehicle Semi-active Suspension for Improved Ride Comfort and Road Handling Mahmoud Ahmed and Ferdinand Svaricek Department of Aerospace Engineering Group of Control Engineering University of the Federal Armed Forces Munich 85577 Neubiberg, Germany
[email protected];
[email protected] Abstract-An adaptive robust controller is presented and applied on a vehicle semi-active suspension system. Due to the variation of the chassis mass, a robust controller was designed, such that two linear H-infinity controllers are used for both the minimum and maximum limits of the mass variation. The outputs of the two controllers are interpolated to obtain a desired control force signal that corresponds to the variation of the estimated chassis mass. To apply the control algorithm, one acceleration sensor is used. The performance of the proposed controller is compared with the Skyhook, SH-ADD, and Groundhook algorithms. The simulation and experimental results have shown a higher performance regarding ride comfort and road handling.
I.
INTRODUCTION
Vehicle suspension is one of the main elements inside a vehicle, which mostly affect its entire dynamic behavior. It establishes the link between the road and the vehicle body affecting both safety and driving comfort. Semi-active suspension systems were firstly controlled using the Skyhook approach proposed by Karnopp et al. in [1]. Many studies have been carried out on the skyhook control strategy, as it represents a simple way to achieve a good comfort requirement, e.g. the non-jerk version by Ahmadian et al. in [2]. Some extended versions of the Skyhook control was developed, such as the adaptive one by Song et al., the gain scheduled one by Hong et al., and the SH-Linear by Sammier et al. and Sohn et al. in [3]-[6]. Another Acceleration Driven Damper (ADD) Control Strategy based on the chassis acceleration and suspension deflection speed was presented by Savaresi et al. in [7]. A Mixed Skyhook-ADD (SH-ADD), which combines the best performance of both the SH and ADD algorithms, was presented also by Savaresi et al. in [8]. Another 1-Sensor-Mix algorithm that applies the idea of the frequency-range selector was proposed by Savaresi et al. in [9]-[11], such that only one sensor is used to measure the body acceleration. A Power Driven Damper (PDD) control strategy, which avoids the chattering effect, was proposed by Morselli and Zanasi using the port Hamiltonian techniques in [12]. Very few studies have been devoted for the improvement of road-holding such as the Groundhook approach, which aims at reducing the road-tire forces, as illustrated by Valášek and Kortüm in [13].
978-1-4673-1388-9/13/$31.00 ©2013 IEEE
Many studies have concerned the application of the H∞ control approach for the suspension system. However, most of the results were obtained for active suspensions, as by Zin et al. in [14]. An application of the H∞ optimal control theory to the design of a fully active suspension system was described by Palmeri et al. in [15]. Several H∞ controllers based on the state space optimization techniques were proposed for the fully active case by Rossi and Lucente in [16]. These controllers are then adapted to the semi-active suspension using the clipped approach. A H∞ controller is proposed by Sammier et al. in [17], and by Fialho and Balas in [18], such that only the suspension deflection signal was used as a measured variable because of its low cost and simplicity in the industry. A robust linear controller was presented by Lauwery et al. in [19] based on a linear black box model. A linear parameter varying approach, was presented by Poussot-Vassal et al. in [20]. A robust H∞ controller based on the combination of a linear matrix inequality solver and a genetic algorithm was designed by Fallah et al. in [21] to optimize the control gains. In the field of industry, there have been many published patents concerning the control of the semi-active suspension. Many sensor signals e.g. the longitudinal and lateral accelerations are used and combined together to indicate the whole vehicle dynamics and accordingly adjust the dampers with respect to the driving requirements as given by Volkswagen AG and Audi AG in [22],[23]. In this paper, a robust H∞ controller is designed taking into consideration the body acceleration and tire deflection based on a linear time invariant quarter-car model. The desired current for adjusting the damping force is calculated by an inverse damper model using the clipped approach. In order to compensate the mass variation due to the change of loading conditions, the outputs of two linear H∞ controllers are interpolated according to the estimated mass to generate the desired force. In Section II, the nonlinear control oriented model for both the semi-active damper and the quarter-car model is presented. The adaptive robust H∞ control algorithm and the design requirements are illustrated in Section III. Simulation and experimental results of applying the control algorithm on a quarter car test rig are discussed in Section IV and Section V. Finally, conclusions are shown in Section VI.
376
A general semi-active model that focuses on the vehicle vertical dynamics is presented in Fig. 1. To apply the model for a controlled semi-active suspension system, the controlled damper has been replaced by a force actuator and a passive damper with a fixed parameter .
M Fk ,nl
c (u (t ))
k
m Fkt , nl
M
z (t )
⇔
Fk ,nl
z (t )
c0
k
3000
3000 Total Force Linear Component Nonlinear Component
2000 1000
-1000 -2000
m Fkt , nl
kt
zt (t )
kt
zr (t )
zr (t )
Fig. 1. The quarter-car model: The semi-active model (left), the control oriented model (right)
-3000
-4000
-4000 -0.1
+
,
,
+
−
−
− ,
=−
−
−
+ + , ,
+
,
,
,
−
,
(1)
The identification of the suspension and tire stiffness is carried out on an Audi A4 quarter-car test rig. This test rig is coupled with a virtual quarter car model to simulate the corresponding chassis acceleration based on a virtual mass and the measured force between the frame and the suspension, see Fig. 2.
ztr
zr
∫
&z&tr
1 M
Fch,tr
−
z r ,tr
4000
4000
3000
3000
2000
0.015
2000 1000
0
0
-1000
-1000
-2000
(3)
-1
-0.5
0 Velocity [m/s]
0.5
1
-2000
-1
-0.5
0 Velocity [m/s]
0.5
1
Fig. 4. The semi-active damper characteristic curve (left) and modified semi-active damper characteristic curve for the simulation model (right)
In Table I, the parameters of the quarter-car model are shown. TABLE I THE QUARTER-CAR MODEL PARAMETERS Symbol M m k kt c0 ß
B. Parameter Identification
z&tr
0.01
Linear Component Nonlinear Component at 0 Amp. Noninear Component at 1.8 Amp.
5000
1000
(2)
where, is the vertical displacement of the sprung mass, is the vertical displacement of the unsprung mass, is the chassis mass (sprung mass), is the wheel mass (unsprung mass), is tire stiffness, is the suspension spring stiffness, is the linear damping coefficient, is the damping force added by the force actuator, , is the nonlinear force component of the suspension stiffness, , is the nonlinear force component of the tire stiffness, and is the road profile.
∫
-0.01 -0.005 0 0.005 Displacement [m]
6000 Measured Force at 0 Amp Measured Force at 1.8 Amp
Force [N]
−
-5000 -0.015
0.05
To determine the characteristic curve of the semi-active damper, a harmonic signal with amplitude of 10 mm and frequency of 10 Hz was applied on the wheel. The velocity was determined by integrating the signal of the accelerometer on the wheel. In Fig. 4, the semi-active damper characteristic curve is shown.
Force [N]
=
-0.05 0 Displacement [m]
6000
The equations of the nonlinear control oriented model, which will then be used for control design, are as follows: +
-2000
-3000
5000
−
0 -1000
Fig. 3. The suspension stiffness (left) and tire stiffness (right)
zt (t )
=
1000
0
-5000
Fd (u (t ))
Total Force Linear Component Nonlinear Component
2000
Force [N]
A. Quarter-car Model
By applying a vertical force on the wheel suspension and measuring the corresponding displacements for both the tire deflection and wheel, the following characteristic curves for the suspension and tire stiffnesses in Fig. 3 are obtained.
Force [N]
II. SUSPENSION MODELING
Quantity Sprung mass Unsprung mass Suspension linearized stiffness Tire linearized stiffness Suspension linearized damping Suspension actuator bandwidth
Value
Unit
350 50 30,000 180,000 1500 50
kg kg N/m N/m N/m/s rad/s
III. CONTROLLER DESIGN The semi-active suspension system aims at operating around linear points. So, for the design and analysis of semiactive suspension control, a linear time invariant (LTI) model will be used for implementing the control algorithms. A. Adaptive Robust Control Design In Fig. 5, the block diagram of the control scheme is shown. According to the measured chassis acceleration signal and the estimated mass, two interpolated H-infinity controllers will generate the output control signal. The scheduling parameter is adjusted depending on the estimated chassis mass . The generated desired force signal
Fig. 2. The quarter-car test rig
377
minimizing weighted transfer function norms, as given in [28]. A block diagram of the H∞ control design interconnection is shown in Fig. 6. The measured output or feedback signal 1 is the chassis acceleration . This signal is used by the controller to produce the desired force that the semi-active damper should generate. The controller acts on this signal to produce the signal to the electro-magnetic valve of the semi-active damper. The block < serves to model the sensor noise. < is set to a sensor noise value of 0.01 m/s². The weight is used to scale the magnitude of the road disturbances. A maximum road disturbance of 10 cm is assumed and hence, = 0.1 .
is used to calculate the desired control current signal from the derived inverse damper model. zr ud
u Saturation
⊥ df
Inverse Damper Model
z& def
Id
I
Current Controller
u Quarter-car &z&, &z&t
Semi-active Damper
z& def
Test Rig
z& def
H-infinity Controller 1
λ
K1
&z& H-infinity Controller 2
(1 − λ )
K2
xzr
) M
zr
Wzr
&z&, &z&t
Estimation of the Chassis Mass
z
zt − zr
Fig. 5. Block diagram of the robust adaptive controller
This interpolated controller can be defined by (4), where the scheduling parameter [0,1]. This interpolation method used, has been termed a Local Controller Network (LCN) as shown in [24]-[26]. In our application, the scheduling parameter is a function of the sprung mass , such that, =ℎ and, $ = $% + 1 −
$& .
B. Estimation of the Chassis Mass
(4)
The sprung mass is estimated using the recursive least squares method as explained by Pence et al. in [27]. Based on the model equations in Section II A, neglecting the nonlinear term of , and assuming = 0, such that both the chassis mass and the damping constant are unknown, then, − + + − + − = 0. (5)
' = − . Therefore, ' + ' + ' = − . (6) To avoid the drift problem that results from integrating a noisy acceleration signal, (7) is used, such that, Let,
(
)*
+ )
, +
(
)
+ )
,
−
+ (
%
+ )
,
−
= 0,
(7)
where, - is the Laplace transformation and Λ - is a polynomial of -. For estimating the unknown parameters and , then, − (
%
+ )
,
−
=[
]/
(
)
(
+ )
)*
+ )
,
,
−
0.
u
For brevity, (8) will be written as: 1 = θ3 ϕ. Therefore, using the recursive least square algorithm the sprung mass and the damping coefficient can be estimated, such that, 56 = 7 1 − 56 8 9 9.
(9) where, 7 is a 2 × 2 symmetric covariance matrix and 56 is the least-squares estimate of the unknown parameter vector 5.
C. H-infinity Control Design
The design of linear suspension controllers may emphasize either passenger comfort or road handling. As is standard in the H∞ framework, the performance objective is achieved via
y zt − zr
Wzt − zr
y
&z& Wact
xn
Wn
yu
Fig. 6. The design configuration of the H-infinity controller
The magnitude and frequency contents of the control force are limited by the weighting function
