Control and Evaluation of Slow-Active Suspensions with Preview for a Full Car

An optimal control design method based on the use of the correlation between the front and rear wheel inputs wheelbase preview is introduced and then applied to the optimum design of a slow-active suspension system. The suspension consists of a limited bandwidth actuator in series with a passive spring, the combination being in parallel with a passive damper. A threedimensional seven degrees of freedom car riding model subjected to four correlated random road inputs is considered. The performance potential of the limited bandwidth system with wheelbase preview in comparison with the nonpreview uncorrelated inputs case is investigated.


Introduction
Electronically controlled suspension systems have the potential to overcome limitations of passive systems and to reduce the need to compromise among a variety of operating conditions and among the generally conflicting goals of the suspension designs 1-7 .Such goals include providing good vibration isolation of the vehicle body ride comfort , maintaining uninterrupted contact between the tire and the road road-holding ability for ride handling , and stabilizing the vehicle body ride safety .Researchers have studied active suspension systems using a number of different vehicle models, with optimum linear quadratic regulator control methods being widely used in the suspension design.Many investigators have utilized a two degree-of-freedom quarter car model with an active suspension 8-14 .These references have used full bandwidth suspension systems, which offer improvements in ride and handling qualities but may suffer from other problems such as complexity, high cost and power consumption and noise, vibration, and harshness problems 15,16 .Consequently, such fully active systems have been unattractive commercially for production cars.However, it has been demonstrated that limited bandwidth active slow-active systems can offer significant improvements in the control of body resonances, similar to full bandwidth systems 15, 16 .The slow-active systems may be envisioned as an actuator of limited bandwidth around 3-5 Hz .This system has been realized commercially by gas springs and electrohydraulic elements 17, 18 .This type of system is of special interest for several reasons: a with an actuator bandwidth of 3 Hz, the actuators can be used to control roll in cornering and pitch in braking and accelerating such that poor handling need not to be a consequence of low wheel rates; b beyond the bandwidth of the actuator, the system is conventional; c relatively simple actuators which may be electrical or hydraulic will probably provide the necessary performance; e power consumption should be very modest.
In recent years, there has been a great interest in the use of preview of the road input in the design of active suspension systems, and very encouraging results in terms of improved performance have been obtained 19, 20 .Since the control law needs information on the road input some distance ahead of the front wheels, practical difficulties of measuring the road surface by a body-mounted road sensor are encountered.
Practically, however, as a vehicle travels straight on hard road surfaces, the rear wheel is subject to almost the same road input as the front wheel with only a time delay.The wheelbase preview information can be incorporated in the control law designs for active vehicle suspension systems, and further performance improvements are expected.The wheelbase preview control has been studied 21-23 , and promising results have been obtained.The benefits come without additional costs for sensors.The aforementioned references have used preview information with unlimited bandwidth fully active systems.Pilbeam and Sharp 23 have used a half-car model to study the effect of the wheelbase preview on the performance capabilities of a slow-active suspension system.However, the derived preview-controlled system does not have zero steady-state axles to body deflection in response to external body forces due to changes in payload, aerodynamic forces, and forces from steering, braking, and traction.
The purpose of this study is to develop a full car model equipped with a slow-active suspension system and including integral control terms in order to control the pitch and roll modes of the car body.In addition, a control strategy especially designed to include wheelbase preview is presented in order to reduce the vibration of the car body and to evaluate the effects of this strategy on the performance of the limited bandwidth suspension systems.The process of synthesizing the control law is based on modifications of the control algorithm to account for tire road input correlation and penalizing the integral control terms in the performance index cost function .This formulation provides needed flexibility in targeting body modes pitch and roll by control.

Car Ride Model
A schematic representation of the vehicle mathematical model is shown in Figure 1.It is seen to be a three-dimensional model with seven degrees of freedom.It considers bounce, pitch, and roll modes of the sprung mass and vertical bounce motions of the four unsprung masses.A slow-active suspension system is assumed to be incorporated between the sprung mass and the car axles at each corner as indicated in the figure.The slowactive system incorporates an actuator of limited bandwidth in series with a passive spring, the combination being in parallel with a passive damper.The actuator control bandwidth embraces the normal range of body resonant frequencies in bounce, pitch, and roll, but it does not extend as far as wheel-hop frequencies.This type of actuator is treated as   a displacement producer.Its inner control loop would involve displacement feedback and the actuator demand signal will notionally be a desired displacement.The limited bandwidth nature of each actuator is represented in the vehicle model by including two similar secondorder low-pass filters in series with a perfect actuator.
The equations governing the dynamic motion of the vehicle model equipped with slow-active suspension systems are briefly described below.
From Figure 1, the following equations may be derived: The equation of heave vertical motion of the sprung mass the equation of roll motion of the sprung mass the equation of pitch motion of the sprung mass the equations of vertical motion of the unsprung masses

2.4
For the system in hand, the second-order low-pass filter LPF equations are: where u i and u i are the 2nd-and 4th-order filtered control demand signals, respectively, and

2.6
With

Road Roughness Model
The input excitation to the vehicle model is assumed to be the apparent vertical roadway motion, caused by the vehicle's forward speed along a road having an irregular profile.The road surface irregularities may be treated as a stationary Gaussian random process with zero mean and characterized by a power spectral density of the form 24 where σ 2 , α, and v are the variance of road irregularities, a coefficient depending on the shape of road irregularities, and vehicle forward speed, respectively.
From the roughness representation 3.1 , the vehicle excitation model can be obtained in the time domain by using a shaping filter of the form where ξ o is a zero-mean white noise process with in which δ • denotes the Dirac delta function.The road excitation inputs to the vehicle are treated as additional states of the system.The system states are augmented with states representing the road inputs.

Uncorrelated Road Input Case
It is assumed here that the inputs at the four wheels of the vehicle are uncorrelated.In this case, the road disturbance inputs may be modeled by the following first-order differential equation: where A w is a 4 × 4 diagonal matrix with diagonal elements equal to −αv and I w is a unit matrix.
Combining roadway excitation matrix 3.4 and 2.11 , the results may be written as ẋ Ax Bu Dξ, 3.5 where 3.6

Correlated Road Input Case
In this case, the correlation between the front and rear inputs on each side of the vehicle is considered.The time delay available between the front and rear input is expected to provide an opportunity to improve the rear axle actuator control.This is a sort of preview wheelbase preview .Since all the required sensors are already available and the preview time is reasonable L/v , where L is the wheelbase, it will be investigated here.
The pure time delay between the front and rear inputs may be represented by an Nthorder Pade approximation.The transfer function of the Nth-order Pade approximation is given in the frequency domain by where z lf and z rf are the front white road noise signals, z lr and z rr are the rear white road noise signals, and τ b is the time delay between the front and rear inputs.For the second-order Pade approximation N 2 , the constants are a o 12/τ 2 b , a 1 6/τ b , and a 2 1 25 .The state space representation of the above transfer function is given by where and the matrices A η , B η , and C η are

3.11
The submatrices are

3.14
It is clear from 3.14 that the rear disturbances have disappeared and hence the problem can be treated in the same way as in the uncorrelated case.Combining 2.11 , 3.8 , and 3.14 , the results may be written as

3.16
In the case of assuming two identical front white noise signals, 3.13 may be written as

3.18
The problem formulation may be simplified by setting 3.19

Design of Optimal Controllers
Suspension system performance criteria include ride comfort, suspension travel working space , tire deflection wheel load variation , static and dynamic attitude control, actuator force levels, and power consumption.Ride comfort in a vehicle is the general feeling of well-being.It is clearly very subjective but it is generally accepted that the acceleration spectrum, particularly in the lowfrequency range 1-10 Hz , is important.Root mean square rms value of the acceleration can be used as a measure of ride discomfort.
Suspension deflection is important from a packaging point of view; the current trends toward larger tires and lower hood designs put a premium on keeping suspension deflection as small as possible.The lateral and longitudinal forces available for cornering and/or braking depend on the normal load on the tire; clearly large dynamic variations in the normal tire force would be unacceptable.RMS values of relative displacement between the wheel and the body and of the dynamic tire deflection can be used as measures of suspension travel and wheel deflection control.
In order to obtain a control law that meets the ride comfort, ride safety road-holding ability and body controllability , and working space suspension travel objectives, the following performance index is used: where E is the expectation operator.In the performance index, ρ 1 , ρ 2 , ρ 3 , ρ 4 , ρ R are weighting factors.The weighting factors reflect the preference of the vehicle designer.By changing the weighting factors, various control laws can be derived which place different emphases on individual aspects of performance.Δ sj are suspension deflections and Δ tj are tire deflections:

4.2
Assume an output vector which contains all the important quantities to be controlled, for example, dynamic tire deflections, suspension deflections, pitch and roll motions, and so forth.C is the output matrix which may be divided into three submatrices, that is, is the number of outputs, and n 1 34.
Here, the control law for the correlated road input case is considered.
The performance index can be specified in terms of the outputs as where Q n o ×n o and R m×m are the weighting matrices.Q is a positive semidefinite matrix and R is a positive definite matrix.
The control input u that minimizes J under dynamic constraints 3.15 is given by where K η is the additional part of the gain accounting for the time delay.The matrix P f can be found from the solution of the algebraic Riccati equation: The system considered here, however, is not completely controllable.This is because part of the input vector, ξ f , in 3.15 cannot be changed by applying a control force.Consequently,

Results and Discussions
The baseline data for the example vehicle model which reflects the characteristics of a typical car are given in Table 1 18 .The improvements in vehicle ride quality through the use of optimally controlled active suspension systems are examined.The direct comparison of active and passive suspensions is difficult.While the spring and damping parameters of the conventional passive suspension remain fixed, active suspensions can be adapted to instantaneous operating conditions measured by sensors and change their characteristics to suit particular road/speed conditions or driving style.In essence, an active system can provide a soft but stable straight line ride but stiffen up for directional changes, braking, and acceleration maneuvers.To maintain fairness of comparison, the wheel-hop damping ratios for the passive system are kept the same as the ones for the active systems.It has been recognized by Elmadany 26 , for both active and passive suspension systems, that the constrained placed on the damping ratio of the wheel-hop mode limits the extent to which the sprung mass acceleration can be reduced for ride comfort.However, for very lightly damped wheel-hop mode, lower road-holding qualities will result.In this study, damping ratios of 0.2 in the wheel-hop modes will be used.The following types of suspensions have been considered in the simulations i Passive P .
ii Slow-active with integral control terms SA I .
iii Slow-active with integral control terms and wheelbase preview SA I, P .
In addition, the slow-active SA system without integral control terms is considered where necessary to show the benefits of including the integral control terms in the design of the multivariable controllers.The numerical values of the weighting factors in the objective function used in the simulation are given Table 2.In this work, the approach used to represent the wheelbase time delay is to model it with a second-order Pade approximation.This results in a continuous control strategy to be derived for full-state feedback case.Simulations are carried out at vehicle speeds between 5 and 40 m/s, corresponding to different amount of preview.It is known that the amount of preview time depends on the vehicle speed, so as the vehicle speed increases, the preview time reduces.

Response to Random Road Disturbances
The vehicle model equipped with different suspension systems has been simulated when traveling with a constant speed of 20 m/s on a randomly profiled road.The values of the road parameters are α 0.15 and σ 2 9 × 10 −6 m 2 .They correspond to a smooth asphalt road.In Table 3, the root mean square rms values of a set of selected variables are presented for the different suspensions.From Table 3, it could be noticed that the differences between the suspension systems with integral control terms and the ones without are small, due to the small weighting factors on the control terms.Therefore, only the results for systems with integral control terms are presented in the remaining portion of this section.Since the passive suspension system provides a yardstick against which other designs are evaluated, the percentage improvements in the ride quality compared with the passive system are given in Figure 8.It can be seen that the actively controlled suspensions exhibit marked improvements in ride comfort vibration isolation and suspension travel without compromising the roadholding ability.In general, the results demonstrate that it is possible to improve vehicle performance by simultaneously reducing rms sprung mass acceleration and suspension deflection without negatively influencing rms tire deflections.The slow-active system gives very good performance characteristics compared to the passive system.The improvements in the performance index, ride comfort, suspension travels packaging , and ride handling tire deflections , from the uncorrelated nonpreview case to the correlated wheelbase preview case, are shown in Figure 2. The improvements are particularly greater at lower vehicle speeds, as expected.For soft ride mode operation i.e., increasing emphasis on ride comfort , more improvements in the performance capabilities are possible.The reason behind not having a greater effect on handling performance stems from the fact that the actuator has a limited bandwidth 3 Hz , and the wheel-hop resonances around 10 Hz are out of the range of the effective control of the actuator.
Figure 3 shows the power spectra for the vehicle running at 20 m/s.Results are given for a slow-active system with and without preview and for the passive system.The trends shown in these results are substantially the same for different vehicle speeds.The spectra show that there is very little difference between the three cases considered around wheelhop resonances, due to the limited bandwidth nature of the system.The body acceleration is improved up to 6 Hz.The rear tire deflection in the preview case is improved over the nonpreview case up to about 6 Hz.There is a marked improvement in the rear suspension deflection.At very low frequency, the preview case gives similar results as the passive system, so improving the suspension travel at low frequencies and up to wheel-hop frequency.

Response to Discrete Disturbances
Two types of discrete inputs to the vehicle system are considered.These inputs are as follows.
1 A double bump, with the amplitude of the bumps on the right side of the car being twice as large as the bumps on the left side.This type of terrain excites both the pitch and roll modes of the sprung mass.The simulations are performed for the two inputs.A set of plots comparing the responses of the actively and passively suspended vehicles are presented and discussed next.
The time histories of the vehicle responses due to the double slanted bump are shown in Figures 5 and 6.From examining these figures, it may be said that the slow active suspension system exhibits smaller and better damped responses than the passive suspension.The wheel-hop damping ratio has a significant impact on the performance of the vehicle.The effect of including the integral control of the roll angle is very clear from Figure 6.The peaks of the roll angle are very much reduced compared with the ones without the integral control terms.This shows the benefit of inclusion of control terms for attitude control and ride safety.
The suspension deflection and pitch angle response time histories due to the application of a moment at the sprung mass center of gravity are shown in Figure 7.The various suspension systems have different characteristics.During accelerating the vehicle, the suspension deflection and pitch angle are not zeros for the passive system and the slowactive without the integral control of the pitch angle.These adverse effects are eliminated by introducing the integral of the pitch angle in the controller.

Summary and Conclusions
In this paper, linear optimal control theory has been applied to the design of the control laws for a slow-active suspension system based on a full vehicle model.The theoretical background for deriving the control laws has been reviewed for the case in which the correlation between the road inputs due to the wheelbase time delay is ignored nonpreview .Then, the problem formulation has been modified so that the control law can account for the wheelbase preview.A second-order Pade approximation is used to represent the preview time wheelbase time delay .The simulation results have shown that the controller based on wheelbase preview moderately improves the general road performance of the slow-active suspensions.An improvement of about 11% at a speed of 5 m/s in ride comfort is obtained by using the wheelbase preview control strategy over than without preview.The preview control for the slow-active system makes it possible to have improvement up to 29% in the suspension working space over the nonpreview control at a vehicle speed of 20 m/s 0.14 sec preview time .The slow-active suspension system has very good performance in comparison to that of a typical passive system.It provides damping of the body resonances very effectively.Damping ratio ξ o :

Equation 3 .
4 may be extended as ẇ A w w b w1 ξ f b w2 ξ r , 3.13 where b w1 , b w2 are submatrices of I w .Substituting 3.9 into 3.13 , the result may be put in the form ẇ A w w b w1 ξ f b w2 C η η b w2 ξ f A w w b w2 C η η b w1 b w2 ξ f .

Figure 2 :
Figure 2: Percentages improvements in correlated rms values for slow-active suspension system.

Figure 3 :
Figure 3: Response power spectra; a body acceleration; b rear tire deflection; c rear suspension deflection.

Figure 7 :
Figure 7: Percentage improvements in ride quality for all types of slow-active suspension systems comparing to passive system.
z c , z d : Vertical position of right front and right rear corners of sprung mass, respectively, m z lf , z lr : Height of road under left front and left rear unsprung mass, respectively, m z rf , z rr : Height of road under right front and right rear unsprung mass, respectively, m z 1 , z 2 : Vertical position of left front and left rear unsprung mass, respectively, m z 3 , z 4 : Vertical position of right front and right rear unsprung mass, respectively, m z 5 , z 6 : Displacement of left front and left rear slow active actuator, respectively, m z 7 , z 8 : Displacement of the right front and right rear slow active actuator, respectively, m Δ sj : Suspension deflection, j 1, 2, 3, 4, m Δ tj : Tire deflection, j 1, 2, 3, 4, m α: A coefficient depending on the shape of road irregularities, m −1 δ • : Dirac delta function ζ: , B p , and D p are matrices of dimensions n × n, n × m, and n × m, where n 30 and m 4.In order to control the body motion due to changes in payload forces, resulting from steering, braking, and traction, and aerodynamic forces, the system equations are augmented with the integral of the pitch and roll angles, that is, ṗ Sx, 2.10 where p is a 2 × n matrix, S 1, 3 1, S 2, 4 1, and all other elements of S are zeros.The augmented system equations then become ẋs A s x s B s u D s w, 2.11
the Reccati equation 4.6 can be solved numerically after separation of the unknown matrix P f into four submatrices, as suggested byHac 8.The closed loop state matrix equation takes the form ẋf

Table 2 :
The numerical values of weighting factors.

Table 3 :
RMS values for passive and different types of slow-active suspension systems.
: Total left rear and left front suspension force, respectively, N F c , F d : Total right rear and right front suspension force, respectively, N H: Transfer function matrix I x , I y : Sprung mass moment of inertia about longitudinal and lateral axis through sprung mass center of gravity, respectively, kg•m 2 Longitudinal distance from sprung mass center of gravity to front and rear axle center line, respectively, m c f , c r : Damping coefficient of front and rear shock absorbers, respectively, Ns/m k f , k r : Spring constant of front and rear suspension, respectively, N/m Vertical position of sprung mass center of gravity, m z a , z b : Vertical position of left front and left rear corners of sprung mass, respectively, m Mathematical Problems in Engineering b