Abstract: In this paper, the control of an active suspension system using a quarter car model has been investigated. Due to the presence of non-linearities such ...
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A sliding mode controller for vehicle active suspension systems with non-linearities C Kim and P I Ro Department of Mechanical and Aerospace Engineering, North Carolina State University, USA Abstract: In this paper, the control of an active suspension system using a quarter car model has been investigated. Due to the presence of non-linearities such as a hardening spring, a quadratic damping force and the `tyre lift-off' phenomenon in a real suspension system, it is very difficult to achieve desired performance using linear control techniques. To ensure robustness for a wide range of operating conditions, a sliding mode controller has been designed and compared with an existing non-linear adaptive control scheme in the literature. The sliding mode scheme utilizes a variant of a sky-hook damper system as a reference model which does not require real-time measurement of road input. The robustness of the scheme is investigated through computer simulation, and the efficacy of the scheme is shown both in time and frequency domains. In particular, when the vertical load to the sprung mass is changed, the sliding mode control resumes normal operation faster than the non-linear self-tuning control and the passive system by factors of 3 and 6, respectively, and suspension deflection is kept to a minimum. Other results showed advantages of the sliding mode control scheme in a quarter car system with realistic non-linearities. Keywords: sliding mode, active suspension, non-linearities NOTATION ci csh csm fd fs ft g ki k sm K ÄL ms msm mu u xi xr xs xu Äx
coefficients of non-linear damping force damping constant of sky-hook system damping constant of reference model damping force spring force tyre force acceleration of gravity coefficients of non-linear spring force spring constant of reference model switching gain lateral load transfer sprung mass sprung mass of reference model unsprung mass control force input states of suspension system road disturbance variation sprung mass displacement unsprung mass displacement suspension deflection
The MS was received on 17 February 1997 and was accepted for publication on 18 July 1997. D00897 # IMechE 1998
1
INTRODUCTION
The suspension system of an automobile has two major tasks. The first is to isolate the car body with its passenger from external disturbance inputs which mainly come from road irregularities. The second is to maintain a firm contact between the road and the tyres to provide guidance along the track. In a conventional suspension system which comprises only passive components, the task of providing both ride comfort and good handling calls for conflicting requirements. To support the weight and to follow the track a stiff suspension is needed on one hand, but, to isolate the disturbance from the road a soft suspension is required on the other. Many kinds of active suspensions have been developed to improve both ride quality and handling performance. For more than 20 years, research has shown that a linear optimal control scheme provides a good way to design an active suspension system which can improve the vehicle ride and handling performance at the same time. This is based on the assumption that there exists a perfect (broadbandwidth) actuator, which can generate the required force fast enough and the system can be linearized within some operating region. In most cases, linear optimal control theory has been applied using a two degree-of-freedom model, with full state or partial state feedbacks (1±5). A control scheme, so-called `sky-hook damping', in which the absolute Proc Instn Mech Engrs Vol 212 Part D
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velocity of sprung mass is required, has also been applied to commercial systems (6). Up to this point most researchers have dealt with a linear model in developing control laws. However, since a real vehicle suspension has inherent non-linearities and uncertainties it is not sufficient to represent the real system with a linear model (7, 8). In the early 1990s many studies began to consider non-linearities, uncertainties and unmodelled parts of a real suspension system, which required the use of a non-linear model and some adaptive or robust form of control scheme. A classical form of adaptive scheme for a vehicle active suspension system was introduced in the late 1980s by Hac (9). During driving, the interaction between the road and vehicle changes drastically depending on the road surface and vehicle speed. This change must be taken into account to ensure optimal operation of the system in all driving conditions. This optimal operation necessitated a form of controller, the gain of which can be adjusted to the type of road surface and vehicle speed. This is the starting point of the adaptive control scheme, in which a set of feedback gains is varied by the change of power spectral density of terrain roughness obtained by processing the measurement data. A comparison of adaptive LQG and non-linear controllers for active suspensions can be found in reference (10). A model reference adaptive control scheme was proposed in reference (6) which resulted in better performance than active suspension systems with non-adaptive controllers and passive suspension systems. In the paper, 10 to 30 per cent variances of suspension components and sprung mass were examined to check the adaptation capability using a single degree-of-freedom model. Also, an explicit adaptive control for the active suspension system was introduced in reference (11), which is based on a selftuning controller design. It consisted of on-line low-order recursive parameter estimation, closed-form algebraic gain computation and manipulation for the control parameter. The self-tuning based adaptive control scheme can provide considerable adaptability to variances in the suspension characteristics and changes in the sprung load (payload change, brake, cornering). However, if the system is highly non-linear over the range of operation, the adaptive schemes may show severe limitations. For example, if the wheel stroke is so increased that the stiffness of the suspension is beyond the linear range, it might be difficult to identify parameters through ordinary identification. Gordon et al. (12) proposed a control scheme using a realistic non-linear suspension model, but their study of robustness was still limited to a linear region. The goal of the current study is to develop a robust control scheme for an active suspension system which maintains good performance in the presence of severe nonlinearities or uncertainties (unknown parameters and structures). This study begins by developing a realistic nonlinear suspension model which is composed of non-linear Proc Instn Mech Engrs Vol 212 Part D
spring, damper and tyre `lift-off'. By comparing the simulation results of the linear and non-linear models, it will be shown that the non-linearities of a suspension system should be considered for developing an active suspension system. Next, the non-linear self-tuning scheme (13) is reviewed to verify whether the existing non-linear control scheme does work in a realistic non-linear suspension system with highly non-linear characteristics and=or uncertainties. Then its robustness is investigated, which shows a limitation of adaptive schemes with respect to the fast-varying uncertainties. To improve the robustness, a sliding mode control scheme is developed in this study which is unique to the active suspension application. A modified sky-hook damper system which eliminates the necessity of a road signal is used as the reference model in this study. This makes the whole process simple and results in better ride quality and handling performance over the non-linear self-tuning control scheme.
2
MODELLING OF THE SUSPENSION SYSTEM
In this section, both linear and non-linear models of the quarter-car suspension system are introduced and time responses of two models are compared to demonstrate the limitation of the linear suspension model for developing an active suspension system. Figure 1 illustrates a model for the suspension system. The equations of motion for body bouncing motion are ms xs
fs
fd
mu xu f s f d
ms g ft
(1) mu g
(2)
Fig. 1 Modelling of quarter-car suspension system D00897 # IMechE 1998
A SLIDING MODE CONTROLLER
where f s and f d are spring force and damping force, respectively, between the sprung mass and the unsprung mass and f t is the tyre force between the unsprung mass and the road. The linear model assumes that these forces are linear. In view of handling characteristics, the vehicle is considered to be within the linear region when its lateral acceleration is less than 0.4 g, which represents the normal driving condition unless it experiences severe cornering. But, in view of ride-related movement, the linear region is relatively small. For example, the permissible rattle space of a passenger car is less than 150 mm due to the limitation of the body-mounting point. But the suspension load versus wheel displacement curve shows that the linearity is less than 50 mm mainly during rebound movement. Therefore, it is very important to include non-linearities to account for a more realistic operation of the vehicle. In this section, the connecting forces (e.g. spring force, damping force) are modelled as non-linear functions using measured data. In the linear model, these connecting forces were described as linear functions of the states of system: f s kx or f d c x_ . Figure 2 shows major non-linearities in real suspension systems. The non-linear spring property is mainly due to two parts. One is the bump stop which restricts the wheel travel within a given range and prevents the tyre from contacting the vehicle body. The other is the strut bushing which connects the strut with the body structure and reduces the harshness from the road input. These two non-linear effects can be included in the spring force f s with non-linear characteristic versus suspension rattle space (xs xu ). From the measured data (SPMD: suspension parameter measurement device) shown in Fig. 3, the spring force f s is modelled as a third-order polynomial function
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f s k 0 k 1 Äx k 2 Äx 2 k 3 Äx 3
(3)
where the coefficients are obtained from fitting the experimental data, which resulted in k 3 3170 400, k 2 73 696, k 1 12 394, k 0 2316:4. (The SPMD data from the 1992 model Hyundai Elantra front suspension were used.) Generally, the damping force is asymmetric with respect to the speed of the rattle space; damping force during bump is bigger than that during rebound in order to reduce the harshness from the road during bump while dissipating sufficient energy of oscillation during rebound at the same time. Figure 4 shows the measured data for the damper force versus relative velocity of upper and lower struts, that shows the asymmetric property. From the measured data the damper force f d is modelled as a second-order polynomial function f d c1 Ä x_ c2 Ä x_ 2
(4)
where coefficients are obtained from fitting the experimental data, which resulted in c2 524:28, c1 1 385:4. The vertical stiffness of a tyre is highly non-linear, especially when the load condition changes very severely. Also, the vertical tyre force becomes zero when the tyre loses contact with the road. To make the suspension model more realistic, this `lift-off' is modelled in this study. The tyre force is calculated as f t k t (xr
xu )
ft 0
when (xr
xu ) . 0
(5a)
when (xr
xu ) < 0
(5b)
Also, the `road-holding' characteristic of the tyre plays a
Fig. 2 Non-linear properties of suspension system D00897 # IMechE 1998
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C KIM AND P I RO
Fig. 3
Modelling of non-linear spring force [wheel stroke (m) versus suspension force (N)]
Fig. 4 Asymmetric damping property of actual suspension system [damper speed (m=s) versus damping force (N)]
very important role in handling performance. During cornering on a rough road, tyre lift-off may cause a very dangerous situation which reduces the lateral grip of the tyre and makes the vehicle unstable or uncontrollable (not able to steer). In order to show the effect of this asymmetric tyre Proc Instn Mech Engrs Vol 212 Part D
stiffness on the response of the quarter-car model, sinusoidal responses of the passive suspension system using both linear and non-linear models are examined. The linear model is derived by linearizing the non-linear property about an operation point. Two different amplitudes of road input are given to the system in order to investigate the D00897 # IMechE 1998
A SLIDING MODE CONTROLLER
effect of road input on the non-linearity. The excitation frequency (about 1.5 Hz) is chosen to be close to the natural frequency of the sprung mass bouncing mode. Figure 5 shows the time response of the sprung mass displacement for the two models. In the figure the difference between non-linear I and II is the addition of the tyre lift-off effect in the latter. When the road input is small there is no big difference between the two models, since the linear model data are approximated by constants (k s , cs ). However, as the excitation amplitude increases the responses become quite different from each other. From the result, it is clear that vehicle non-linearities should be considered in developing a more accurate system model, from which a more reliable control algorithm can be developed.
3
SLIDING MODE CONTROL SCHEME
Since the variable structure control (VSC) theory was introduced in the USSR in the early 1950s it has been applied to many fields in robotics, aerospace and automotive systems (14, 15). VSC has been used to develop a general controller for a variety of systems including singleinput single-output, multiple-input multiple-output, linear, non-linear, discrete-time and stochastic systems (16, 17). In the application for automotive systems, sliding mode control is applied to advanced braking systems (ABS) wheel-slip control due to the non-linearity of the vehicle traction system, where the results indicate that it provides
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tight wheel slip control on both dry and ice-like surfaces (18). Alleyne and Hedrick (19) applied the sliding control scheme to an electrohydraulic actuator of an active suspension system, in which the control object was to match a desired control force (`inner loop' control). 3.1
Modified reference model
The first step in deriving a sliding mode controller is to choose a reference model. The well-known sky-hook damping system is chosen as a reference model as in reference (7). If the system is linear and all parameters are exactly known, the simple control law Fd csh x_ s can be used to produce the desired sky-hook damping effect. However, since the actual system contains many uncertainties, including system non-linearities, a more robust control scheme is required. In this paper, a sliding mode control scheme is proposed based on the error between the skyhook damping system as a reference model and the real suspension system. In a typical implementation of the `model following' technique, system input is provided to the reference model, and the difference of responses between the system and the reference model is compensated by a controller. However, in the case of a suspension system, it is very difficult to provide the road height signal to the reference model since it is very difficult to measure. In this study, a simplified reference model is used to solve this problem, in which a road input signal is not required. As shown in Fig. 6, a modified reference model is
Fig. 5 The time responses for different models [the displacement of the sprung mass (m) versus time] D00897 # IMechE 1998
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x_ p f (xp , t) Bp u f p (xp , t) Ä f p (xp , t) Bp u Fw
(8)
where xp [xs x_ s ]T , f p represents the nominal dynamics (linear system) and Ä f p is the difference between f p and the actual plant f (xp , t). Ä f p includes non-linearities and unmodelled parts, and Fw is the disturbance term. The equation of motion of the reference model is x_ m f m
(9)
where xm [xsm x_ sm ]T
Fig. 6 The active suspension system using the modified reference model
f m x_ sm developed based on an alternative state for the road input. Since the tyre is almost ten times stiffer than the suspension spring, in general, the motion of the unsprung mass can serve as a good approximation to the road input within the normal operating frequency range of a suspension system. Using this approximation, the measured states of the unsprung mass can be used directly as input to the skyhook damping reference model. Next, a sliding mode controller is derived to force the sprung mass to track the reference sprung mass motion T
[xs x_ s xs ] ! [xsm x_ sm xsm ] 3.2
T
1 ( f sm msm
f dm
csh x_ sm )
and f sm k sm (xsm
xu ),
f dm csm ( x_ sm
(6)
e xp
xm
e_ x_ p
x_ m
fm
(10)
Bm um f p Ä f p (xp , t) Bp u Fw (11)
From the equations of motion of the plant, equations (1) and (2), the state variable form of the equations can be obtained as follows:
1 ( x_ 2 ms
(7a) fs
f d u)
g
(7b)
x_ 3 x4 x_ 4
1 ( fs fd ft mu
(7c) u)
g
(7d)
where x1 xs , x2 x_ s , x3 xu , x4 x_ u and f s , f d , f t are spring force, damping force and tyre force, respectively. Here, each of the forces contains contributions from the nominal portion (linear force) plus an additional unmodelled portion in the linear model and uncertainties of parameters. Since the motion of sprung mass is of interest (to improve the ride comfort), the plant dynamics can be expressed as Proc Instn Mech Engrs Vol 212 Part D
x_ u )
Taking the error e as the difference between the two state vectors, xp and xm, the error dynamics can be obtained as
The derivation of control law
x_ 1 x2
g
T
Next, define a time-varying surface in the state-space such that S(t) GT e 0
(12)
where G [ g1 g 2 ]T . The sliding surface can be interpreted as the surface of sprung mass state error between the plant and the reference model. Assuming the dynamics are exactly known and no disturbance affects the system (i.e. Ä f p (xp , t) Fw 0), the equivalent control is defined as S_ 0
(13)
or S_ GT e_ GT ( x_ p
x_ m ) GT ( f p Bp u
f m) 0 (14)
Using this expression, one can find the equivalent force ueq which can track the reference model in the absence of uncertainty and disturbance. That is, D00897 # IMechE 1998
A SLIDING MODE CONTROLLER
ueq (GT Bp ) 1 (GT f m
GT f p )
(15)
or S_
where GT Bp
In order to satisfy the sliding condition despite uncertainties (non-linearities in this study) in the system (i.e. Ä f p (xp , t) Fw 6 0), a term which is discontinuous across the surface S 0 can be added to ueq as (GT Bp )
1
K sgn (S)
(16)
sgn (S)
1 1
if S . 0 if S , 0
To avoid the chattering problem a saturation function is used u
ueq ueq
KS K sgn (S)
if jSj , ä if jSj . ä
(18)
Now, the range of the switching gain K to make the system stable is to be found (16). If u is replaced in equation (14) by equation (16), the Lyapunov candidate can be tested as 1 d 2 (S ) < 2 dt
çjSj
(19)
or S S_
