Juan C Tudon-Martinez

Rubén Morales-Menéndez

Ricardo A Ramirez-Mendoza

Olivier Sename
email: olivier.sename@gipsa-lab.grenoble-inp.fr

Luc Dugard
email: luc.dugard@gipsa-lab.grenoble-inp.fr

Juan C Tudón-Martínez

Ruben Morales-Menendez

Ricardo Ramírez-Mendoza

A Comparison between a Model-free and Model-based Controller of an Automotive Semi-active Suspension System : Independent Wheel-stations

Keywords: Semi-active Suspension Control, Magneto-Rheological Damper, Linear Parameter Varying Control, Heuristic Control

published or not. The documents may come

INTRODUCTION

Recently, suspension systems play a key role in vehicle design. In the standard ISO 2631, it has been established that the human body could suffer different health damages if this is exposed to constant vibrations, specially at frequencies between 0.5 -20 Hz. Thus, the design in automotive suspension systems is focused to ensure not only the wheel-road contact (road holding) but also to improve the passengers comfort by isolating the road irregularities of the vehicle body. In contrast with passive suspension systems, which are designed mechanically and have constant damping ratio, the semi-active and active suspension systems can regulate the vehicle motions by using a control strategy over the damping force.

By good performance, low price, low energy consumption and easy implementation, the semi-active suspension systems are widely used in automobile industry nowadays. Commercially, there are different technologies of semi-active dampers, the main advantages of a ⋆ Authors thank to Tecnológico de Monterrey (Autotronics research chair) and CONACyT (PCP 03/2010) for their partial support.

Magneto-Rheological (MR) damper versus others semiactive dampers are: (a) fast time response (20-40 ms), (b) large force range and, (c) long bandwidth of control.

An MR damper contains metallic particles into the oil that modify the rheological properties of the fluid when a magnetic field is manipulated by an electric current signal; the flow resistance variation causes a highly nonlinear behavior in the damping ratio. The MR damper in an Automotive Suspension Control System (ASCS ) is used to dissipate the energy for reducing the motion in the sprung mass and maintaining the wheel-road contact.

The semi-active suspension control design has been tackled with many approaches based on different frameworks during last decades. For its facility of design and widely experimental evaluation, the classical Sky-Hook (SH ) controller proposed by [START_REF] Karnopp | Vibration Control using Semi-active Force Generators[END_REF] and its improvements such as the modified SH [START_REF] Hong | Modified Skyhook Control of Semi-Active Suspensions: A New Model, Gain Scheduling, and Hardware-in-the-Loop Tuning[END_REF]], mixed SH-ADD [START_REF] Savaresi | Mixed Sky-hook and ADD: Approaching the Filtering Limits of a Semiactive Suspension[END_REF]], Mix-1-sensor [START_REF] Spelta | Experimental Analysis of a Motorcycle Semi-active Rear Suspension[END_REF]] and others, offer a good alternative to improve comfort; however, such control strategies neglect the road holding objective into the control law. In [START_REF] Sung | Road Test Evaluation of Vibration Control Performance of Vehicle Suspension Featuring Electrorheological Shock Absorbers[END_REF], [START_REF] Sankaranarayanan | Semiactive Suspension Control of a Light Commercial Vehicle[END_REF] 2012) is showed the performance of some of these controllers by using a full vehicle semi-active suspension system with independent wheel-stations.

On the contrary, the Ground-Hook (GH ) controller [START_REF] Valasek | Extended Ground-hook -New Concept of Semi-Active Control of Truck's Suspension[END_REF]], designed in a similar way to SH, improves only the road holding. By combining the SH and GH techniques, the hybrid controller proposed in [START_REF] Ahmadian | A Hybrid Semiactive Control for Secondary Suspension Applications[END_REF] allows to improve both control objectives; however, a comparison between the hybrid controller and an heuristic control strategy, named Frequency Estimation-Based (FEB ) controller proposed by [START_REF] Lozoya-Santos | A LPV Quarter of Car with Semi-active Suspension Model including Dynamic Input Saturation[END_REF], shows that the heuristic approach has better results of comfort and handling in a full vehicle semi-active suspension system with softer changes in the manipulation of the semi-active dampers. Other heuristic controllers [START_REF] Ikenaga | Active Suspension Control of Ground Vehicle Based on a Full-vehicle Model[END_REF], [START_REF] Swevers | A Model-Free Control Structure for the On-line Tuning of the Semi-Active Suspension of a Passenger Car[END_REF], [START_REF] Dong | Neural Network Compensation of Semi-active Control for a Magneto-rheological Suspension with Time Delay Uncertainty[END_REF]], considered as free of model, have been designed to improve comfort in a vehicle, but they use a master controller to include the coupling features among the four wheel-stations and have been analyzed for small or medium-size vehicles and not pick-up trucks.

On the other hand, the nonlinear control techniques such as Linear Parameter-Varying (LPV ) control [START_REF] Poussot-Vassal | A New Semi-active Suspension Control Strategy through LPV Technique[END_REF], [START_REF] Do | An LPV Control Approach for Semi-Active Suspension Control with Actuator Constraints[END_REF]] and Sliding Mode Control (SMC ) [START_REF] Assadsangabi | Hybrid Sliding Mode Control of Semi-Active Suspension Systems[END_REF]] offer interesting simulating results in a Quarter of Vehicle (QoV ) design, where the objectives of comfort and road holding are considered into the control law by including the properties of hysteresis, saturation and semi-activity of the damper, e.g. the LPV approach presented in [START_REF] Do | An LPV Control Approach for Semi-Active Suspension Control with Actuator Constraints[END_REF]. These nonlinear techniques also are evaluated in full-vehicle models [START_REF] Chamseddine | Sliding Mode Control Applied to Active Suspension using Nonlinear Full Vehicle and Actuator Dynamics[END_REF], [START_REF] Yoon | Design and Evaluation of a Unified Chassis Control System for Rollover Prevention and Vehicle Stability Improvement on a Virtual Test Track[END_REF][START_REF] Poussot-Vassal | Attitude and Handling Improvements Through Gain-scheduled Suspensions and Brakes Control[END_REF]] with good results; however, their design is considerable complex since two levels of control are demanded or different control modes are defined.

In this manner, by their good results in a QoV model, a comparative analysis between an LPV controller (modelbased approach) and the FEB controller (model-free approach) is presented in this paper by using a full semiactive suspension system of a pick-up with independent controllers in the wheel-stations. In [START_REF] Lozoya-Santos | A LPV Quarter of Car with Semi-active Suspension Model including Dynamic Input Saturation[END_REF] is presented a comparison between these controllers, but the analysis of performance does not include the nonlinear dynamics of a full-vehicle model and the QoV parameters correspond to a medium-size car. In [START_REF] Dong | Comparative Research on Semi-Active Control Strategies for Magneto-rheological Suspension[END_REF], a comparative research in semi-active control strategies for an MR suspension is presented, a nonlinear controller (SMC ) showed the best performance in comfort but an heuristic controller (fuzzy) had the best road holding; the analysis is also based on a QoV system. Similarly, a quarter car test rig is used in [START_REF] Hudha | Effects of Control Techniques and Damper Constraint on the Performance of a Semi-Active Magnetorheological Damper[END_REF] to compare the SH, GH and hybrid controller.

Based on robust control theory, the LPV controller is synthesized to improve comfort and road holding by using two scheduling parameters that include the constraints of the MR damper; thus, several experiments over a commercial MR damper are considered to model the nonlinearities of the semi-active damper dynamics. On the other hand, the FEB controller defines the electric current value to apply on the MR damper by bandwidths according to the desired objective control (comfort or road holding) and using an estimated frequency signal of motion in the suspension. Physical insights of the vehicle are used to model it in CarSim T M , which is used as Software-inthe-Loop (SiL). The Bounce Sine Sweep (BSS ) test is used to quantify the comfort and road holding performances in both controllers; the Root Mean Square RMS value over the variables of interest is used as performance index.

The outline of this paper is as follows: the formulation of the LPV controller is described in the next section. Section 3 presents the FEB controller design. Section 4 shows the simulation tests and a discussion of results is presented in 5. Finally, conclusions are presented in section 6.

LPV FORMULATION

In LPV controller design, first it is described the nonlinear behavior of the QoV system and then the gain scheduling parameters, used to achieve an LPV model, are defined.

QoV representation

An experimental MR damper model represents the suspension between both masses, Fig. 1. It is assumed that the wheel-road contact is ensured. All variables are described in Table 1. The system dynamics is given by: The MR damper model embedded in the QoV is represented by the parametric model of [START_REF] Guo | Dynamical Modeling of Magneto-rheological Damper Behaviors[END_REF], which emphasizes the hysteresis dependence on the position and velocity of the piston by using a tanh function. The characteristic of a linear elastomer is included, by representing a stiffness factor. The non-linear model is defined by:

m s zs = -k s (z s -z us ) -F M R m us zus = k s (z s -z us ) -k t (z us -z r ) + F M R (1)
F M R = f c • tanh (a 1 żdef + a 2 z def ) + b 1 • żdef + b 2 • z def (2)
where, the 5 coefficients have physical meaning, Table 1.

LPV modelling

Based on the LPV control framework proposed by [START_REF] Do | An LPV Control Approach for Semi-Active Suspension Control with Actuator Constraints[END_REF], the varying parameter ρ 1 is defined by the nonlinear term of the MR damping force of (2), such that:

ρ 1 = tanh (a 1 żdef + a 2 z def ) ∈ [-1, 1] (3)
A general state-space representation of the QoV model can be obtained, rewriting (1), ( 2) and (3) as:

ẋs = A s • x s + B s • u + B s1 • z r y = C s • x s (4)
where, u is the control input in N and,

y = z def = z s -z us , C s = [ 1 0 -1 0 ] B s =       0 -ρ 1 m s 0 ρ 1 m us       , B s1 =      0 0 0 k t m us      , x s =    z s żs z us żus    A s =        0 1 0 0 - k s + b 2 m s - b 1 m s k s + b 2 m s b 1 m s 0 0 0 1 k s + b 2 m us b 1 m us - k s + b 2 + k t m us - b 1 m us        , u = f c
In order to satisfy the dissipativity constraint in the control input, the state-space representation of (4) is rewritten in (5) and u = f c -f 0 , details in [START_REF] Do | An LPV Control Approach for Semi-Active Suspension Control with Actuator Constraints[END_REF].

ẋs = (A s + B s2 • ρ 2 • C s2 ) • x s + B s • u + B s1 • z r y = C s • x s (5) B s2 = 0 - f 0 m s 0 f 0 m s T , C s2 = [ a 2 a 1 -a 2 -a 1 ]
where, f 0 is the average of f c and ρ 2 = ρ1 Cs2xs ∈ [0, 1]. Since the controlled input matrix B s depends on ρ 1 , it is required to add a low pass filter into de QoV model for getting a proper structure for the LPV based controller synthesis, [START_REF] Poussot-Vassal | A New Semi-active Suspension Control Strategy through LPV Technique[END_REF]. Additionally, a saturation constraint (ρ sat ) is embedded into the filter for does not affect the closed-loop behavior.

A filter with bandwidth of 25 Hz was used for ensuring the time response of the MR damping force (∼ 40 ms):

F : ẋf u f = A f B f C f ρ sat 0 x f u (6)
where,

ρ sat = tanh(C f x f /f0) C f x f /f0
∈ [0, 1] approximates the saturation function of u f in:

u sat = f 0 if u f >> f 0 u f if -f 0 ≤ u f ≤ f 0 -f 0 if u f << -f 0 (7)
Thus, the new structure (8), presented in Figure 2, takes into account the saturation and semi-activity of the damper into the varying parameters.

ẋsf = A(ρ * 1 , ρ * 2 ) • x sf + B • u + B s1 0 • z r y = [ C s 0 ] • x sf (8) x sf = xs x f , A(ρ * 1 , ρ * 2 ) = As + ρ * 2 B s2 C s2 ρ * 1 BsC f 0 1×4 A f , B = 0 4×1 B f where, ρ * 1 = ρ 1 ρ sat ∈ [-1, 1] and ρ * 2 = ρ 2 ∈ [0, 1].
Fig. 2. QoV model with a semi-active saturated input.

LPV controller design.

In order to bound the nonlinearities of the closed-loop system, the LPV controller computes the manipulation signal by interpolation of a polytope whose vertices are Linear Time-Invariant (LTI ) controllers. In this case, a polytope of 4 LTI controllers is designed by using the H ∞ control framework; the controller is quadratically stable for all trajectories of the scheduling parameters by solving an optimization problem with LMI techniques, [START_REF] Scherer | Multiobjective Output-feedback Control Via LMI Optimization[END_REF]. Figure 3 shows the block diagram for designing the H ∞ controller in each wheel-station. The weighting functions, based on a priori knowledge of the performance objectives, are designed as:

Wz r,i = 3 × 10 -2 , W zs,f = K s,f s 2 + 2ω s1,f ζ s1,f s + ω 2 s1,f s 2 + 2ω s2,f ζ s2,f s + ω 2 s2,f Wz us,f = K us,f s 2 + 2ω us1,f ζ us1,f s + ω 2 us1,f s 2 + 2ω us2,f ζ us2,f s + ω 2 us2,f W zs,r = Ks,rω 2 s1,r s 2 + 2ω s2,r ζ s2,r s + ω 2 s2,r
, Wz us,r = Kus,rω us1,r s + ω us2,r

where W zs,i allows to reduce the amplification of zs in each quarter of vehicle for achieving the desired comfort, W zus,i is shaped to ensure the road holding and W zr,i increases the sensitivity to the road profile. By considering the control specifications, the generalized model P for the LPV control synthesis in each wheel-station is,

ẋsf = A(ρ * 1 , ρ * 2 ) • x sf + B • u + B s1 0 zr z = C 1 (ρ * 1 , ρ * 2 ) • x sf • W zs,i 0 0 Wz us,i y = C 2 • x sf (9) where C 1 (ρ * 1 , ρ * 2 ) = [ C s1 ρ * 1 D s1 C f ] and, C 2 = C s3 0 , C s3 = 1 1 -1 -1 , D s1 = -1 ms 0 C s1 = - k s + b 2 m s - b 1 m s k s + b 2 m s b 1 m s 0 0 1 0 + - f 0 m s ρ * 2 C s2 0 1×4

FEB FORMULATION

The FEB controller is considered free of model, only measurements or estimated signals are used to monitor the excitation frequency in the suspension system in order to define the electric current value required to apply over the MR damper, and thus achieve the desired control objectives, Lozoya-Santos et al. ( 2011). This control strategy is considered multi-objective (for comfort and road holding).

Since the frequency of motion in the suspension system (f ) is an indicator of its performance, with f it is possible to assign different values of damping coefficient in the semi-active shock absorber according to the frequency responses of the control objectives. Thus, it is possible to hold a specific suspension condition based on an estimated frequency ( f ), which is used to manipulate the electric current in bandwidths. The measurements of the deflection velocity ( żdef = żs -żus ) and suspension deflection (z def = z sz us ) are used to estimate the frequency of motion and to decide which damping improves the comfort and/or road holding conditions. By using the RMS values of z def and żdef of the MR damper piston, the estimation of the frequency in each QoV is given by,

f = ż2 def1 + ż2 def2 + . . . + ż2 defn (z def1 2 + z def2 2 + . . . + z defn 2 ) • 4π 2 (10)
where, n = 256 is the number of samples to compute the RMS value with a sampling time of 1/512. Since the damping coefficient is proportional to the electric current value in an MR damper, a look-up table between the manipulation signal (electric current) and the estimated frequency of the suspension motion is defined in order to assure the desired performances for comfort and road holding. Therefore, the FEB control law is completely independent of a vehicle or semi-active damper model.

SIMULATION TESTS

The vehicle model used to evaluate the controllers was characterized from real data (camber angles, caster, damping force, stiffness, etc.) obtained by a K &C (Kinematics and Compliance) test over a commercial pickup truck.

The model was customized from a generic full-size lightload vehicle model in CarSim T M software, Fig. 4, which is used as Software-in-the-Loop (SiL). The customization is composed by the physical dimensions (weight, long, width, height, wheel base, front and rear track and so on) and suspension system (independent wheel stations at the front side and a rear solid axle at the back). The Kerb weight is around 2000 Kg, the sprung and unsprung masses as well as the QoV model parameters described in eqn. ( 1) are presented in Table 2. For SiL simulations, the semi-active suspension system is composed by four experimental MR damper models described by (2). The characterized MR damper is not symmetric in the jounce-rebound effects and only has two levels of actuation: low damping force at 0 A and high damping force at 2.5 A; the range of force is [-6000 to 11000] N (peak to peak), the damper stroke is around 50 mm with a time constant of 12 ms. The model parameters of the shock absorber are shown in Table 3; they were identified from a chirp signal with decreasing amplitude from 10 mm to 1 mm at [0.5-8] Hz. Figure 5 shows the modeling performance of the Guo structure. Table 4 shows the parameters of the weighting functions used in the LPV-H ∞ control design, the cut frequencies have been selected according to the resonance frequencies of the sprung and unsprung masses, while the damping factors and gains are designed by loop-shaping.

The BSS test at 100 Km/h, which explores the most important range of frequencies for comfort and road holding, is used to evaluate the controller performances. The RMS value over the variables of interest is used as performance index in order to compare the controller performances respect to the passive one, according to: where v is the RMS value of the variable of interest (pitch, heave and vertical acceleration for comfort, and suspension deflection and tire compression for road holding) during the test, RM S Controller refers to the controller under analysis and passive refers to the baseline suspension system by using a set of passive dampers. When the performance criterion value is greater than the unity, the passive suspension system is better than the semiactive suspension system, i.e. the index x monitors the improvement or deterioration of the ASCS.

x = v RM S Controller v RM SP assive (11)

RESULTS AND DISCUSSION

Figure 6 shows the transient response of both controllers respect to the baseline suspension system by considering the BSS test. For comfort, the semi-active suspension systems have lower oscillations in the pitch and heave signals close t=2 s (related to 2 Hz), while at t>3 s the vertical acceleration is reduced when the controllers are used.

For road holding, the improvement is more significative in whole range of frequencies with both controllers, the damper compression in the front QoV is reduced up to 20 mm at different time instants (similar results are obtained in the rear damper), while the jounce motion of the rear solid axle is reduced up to 100 mm; by analyzing the tire compression, the improvement is presented between t=3-5 s, the reduction is up to 10 mm of displacement.

For a quantitative comparison between the FEB and LPV controllers, Table 5 shows the index of improvement (x) obtained by each controller in the BSS test. In all variables that monitor the comfort and road holding objectives, both control strategies have better performance than the passive suspension, except the vertical acceleration which in some time instants has lower amplitude in the oscillations caused by the soft suspension.

By comparing the FEB and LPV transient responses, in almost all variables the FEB controller has the best performance. For comfort, the FEB controller reduces the pitch angle 19 %; while for road holding, the front suspension deflection is reduced 23% and the rear up to 58%, similarly the tire compression is also reduced up to 10%. On the other hand, Figure 7 shows that the manipulation signal in the FEB controller is more persistent with softer changes than the LPV controller, whose manipulation depends on the time instant values of the scheduling parameters that could increase the natural waste in the actuation system. LPV FEB Fig. 7. Control input and control output in both ASCS, by analyzing a single wheel station.

CONCLUSIONS

A comparative analysis between a model-free and a modelbased controller is presented in this paper; the full semiactive suspension system considers the four independent wheel stations, each Quarter of Vehicle (QoV ) has an experimental Magneto-Rheological (MR) damper. The Frequency Estimation-Based (FEB ) controller uses measurements to monitor the frequency of suspension motion in order to apply the required electric current over the MR damper; on the other hand, the Linear Parameter-Varying (LPV ) controller requires a model of the QoV by including the semi-active damper dynamics. Two measurable scheduling parameters are included in the LPV control design to represent the nonlinear behavior (saturation and semi-activeness) of the MR damper.

Simulation results of the transient response show that both control strategies improve the comfort and road holding in comparison with the passive suspension system. By comparing the controllers, the FEB approach shows slightly the best comfort and road holding performance; the improvement of comfort respect to the passive system includes a reduction of 19% in the pitch angle; while for road holding, the front suspension deflection is reduced 23% and the rear up to 58%, i.e. the lifetime of the dampers is increased. Additionally, FEB controller reduces the tire compression up to 10% and LPV 17%.

CONFIDENTIAL. Limited circulation. For review only.

Preprint submitted to IFAC Joint conference: 5th Symposium on System Structure and Control, 11th Workshop on Time-Delay Finally, the LPV-H ∞ controller is more complex because requires a reliable model of the whole system, but by design, the asymptotic stability is ensured; on the other hand, the FEB controller is more feasible to be implemented because only requires process measurements.

Fig. 1 .

 1 Fig. 1. Model for a QoV with a semi-active damper.

Fig. 3 .

 3 Fig. 3. LPV control structure based on H ∞ framework for an LTI model of a QoV.

Fig. 4 .

 4 Fig. 4. Full-size pickup truck model in CarSim T M , by including MR dampers in the suspension system.

Fig. 5 .

 5 Fig. 5. Comparison between the MR damper model and experimental data.

Fig. 6 .

 6 Fig. 6. Performance of the controllers in the BSS test.

  and Tudón-CONFIDENTIAL. Limited circulation. For review only.

Preprint submitted to IFAC Joint conference: 5th Symposium on System Structure and Control, 11th Workshop on Time-Delay Martínez et al. (

Table 1 .

 1 Definition of Variables.

	Variable	Description
	a i	Pre-yield viscous damping coefficients
	b i	Post-yield viscous damping coefficients
	fc	Dynamic yield MR force
	F M R	MR damping force
	I	Electric current
	ks	Spring stiffness coefficient
	kt	Stiffness coefficient of the wheel tire
	ms	Sprung mass in the QoV
	mus	Unsprung mass in the QoV
	usat	Filtered and bounded controller output
	żdef	Damper piston velocity (deflection velocity)

zr Road profile (input disturbance) zs Vertical position of the mass ms żs Vertical velocity of the mass ms zs Vertical acceleration of the mass ms zus Vertical position of the mass mus żus Vertical velocity of the mass mus zus Vertical acceleration of the mass mus ρ * 1 Varying parameter for hysteresis/saturation of F M R ρ * 2 Varying parameter for dissipativity constraint of u

Table 2 .

 2 QoV model parameters.

	Front QoV	Rear QoV
	Parameter	Value	Parameter	Value
	ms	630 (Kg)	ms	387 (Kg)
	mus	81.5 (Kg)	mus	139.5 (Kg)
	ks	42,500 (N/m)	ks	37,300 (N/m)
	kt	295,200 (N/m)	kt	295,200 (N/m)

Table 3 .

 3 MR damper model coefficients.

	Coefficient	Value	Units
	a 1	21.3843	s/m
	a 2	14.8223	1/m
	b 1	4,630	s/m
	b 2	-3,948.6	1/m
	fc	[951.5 -3067]	N

Table 4 .

 4 LPV-H ∞ controller parameters.

		Front QoV
	Parameter	Value	Parameter	Value
	K s,f	1	K us,f	10
	ζ s1,f	0.3	ζ us1,f	0.3
	ω s1,f	1(rad/s)	ω us1,f	9(rad/s)
	ζ s2,f	1	ζ us2,f	1
	ω s2,f	3(rad/s)	ω us2,f	9(rad/s)
		Rear QoV
	Parameter	Value	Parameter	Value
	Ks,r	0.1	Kus,r	0.1
	ω s1,r	1(rad/s)	ω us1,r	9(rad/s)
	ζ s2,r	1	ω us2,r	9(rad/s)
	ω s2,r	3(rad/s)

Table 5 .

 5 Performance comparison respect to the passive suspension system by using the x index.

			Analysis of comfort
	Control	Heave	Pitch		Vertical
	Approach				acceleration
	FEB	0.99	0.81		1.13
	LPV	0.99	0.84		1.14
			Analysis of road holding
	Control	Damper Compression	Damper Compression	Tire Compression	Tire Compression	Jounce rear
	Approach	front QoV	rear QoV	front QoV	rear QoV	solid axle
	FEB	0.77	0.45	0.97	0.90	0.42
	LPV	0.84	0.46	0.98	0.83	0.46

Preprint submitted to IFAC Joint conference: 5th Symposium on System Structure and Control, 11th Workshop on Time-Delay


Jorge De-J Lozoya-Santos

Ruben Morales-Menendez

Ricardo A Ramirez-Mendoza

Olivier Sename
email: olivier.sename@gipsa-lab.grenoble-inp.fr

Luc Dugard
email: luc.dugard@gipsa-lab.grenoble-inp.fr

Adaptive Semi-Active Suspension Design Using Gain-Scheduling ⋆

Keywords: Semi-active suspension, adaptive controller, experimental validation, MR damper

A novel approach to design a road adaptive semi-active suspension is proposed. Two scheduling parameters are considered: the road frequency and the type of road. A single corner vehicle model with a nonlinear dynamic model of the semi-active damper is considered. The suspension deflection and its derivative are used as feedback signals. Nonlinear simulations show that an adaptive suspension controller: Frequency-Estimation-Based with road Adaptation (FEBA) provides better comfort and road holding over different types of road. The improvement of the performance ranges from 5-10 %.

INTRODUCTION

The automotive semi-active suspension control methods have been widely explored and different solutions have been proposed. There are several approaches in the control of semi-active suspensions that can be organized as (a) those with an experimental validation and (b) those with a simulation validation. In the first group, the free model controllers such as the Sky-Hook, [START_REF] Karnopp | Vibration Control Using Semi-Active Force Generators[END_REF] and Mix-1-sensor, [START_REF] Savaresi | A Single-Sensor Control Strategy for Semi-Active Suspensions[END_REF] are the most efficient for comfort, and the GroundHook, [START_REF] Valasek | Extended Ground-Hook -New Concept of Semi-Active Control of Truck Suspension[END_REF] is the best for road holding. For comfort and road-holding, some of the nonlinear control techniques are a model predictive and sliding mode control, [START_REF] Dong | Comparative Research on Semi-Active Control Strategies for Magneto-Rheological Suspension[END_REF], the human simulated intelligent controller, [START_REF] Yu | Human Simulated Intelligent Control of Vehicle Suspension System with MR Dampers[END_REF], and the model-free controller named the Hybrid policy, [START_REF] Goncalves | A Hybrid Control Policy for Semi-Active Vehicle Suspensions[END_REF]. In the second group, the H ∞ , [START_REF] Choi | Vibration Control of Magnetorheological Damper System subjected to Parameter Variations[END_REF], using a linear MR damper model and the nonlinear control based on LPV /H ∞ , [START_REF] Do | An LPV Control Approach for Semi-Active Suspension Control with Actuator Constraints[END_REF] have been validated by simulation. [START_REF] Poussot-Vassal | Survey and Performance Evaluation on Some Automotive Semi-Active Suspension Control Methods: A Comparative Study on a Single-Corner Model[END_REF] shows a complete comparison of benchmark model-free controllers versus Model Predictive Control (MPC ) and LPV approaches. However, the adaptation to the road profile type or the forward vehicle has not been widely explored.

The research of [START_REF] Huisman | Preview Estimation and Control for (Semi-) Active Suspensions[END_REF] and [START_REF] Muijderman | Preview-Based Control of Suspension Systems for Commercial Vehicles[END_REF] implemented a semi-active suspension control with a bulk of on-line computing and estimated the road profile using a Kalman Filter (KF ). The results were good in simulation, but the experimental results were not. This problem resulted from the selection of several control laws on-line. [START_REF] Fialho | Road Adaptive Active Suspension Design using Linear-Parameter-Varying Gain-Scheduling[END_REF] applied this LPV con- troller with gain-scheduling in an active suspension that complied with the comfort and suspension deflection goals and adapted to the road surface. The feasibility of this solution in practice was challenged by a set of ten controllers and the use of two scheduling parameters. The LPV controller potential under this concept was successfully shown. [START_REF] Hong | Modified Skyhook Control of Semi-Active Suspensions: A New Model, Gain Scheduling, and Hardware-in-the-Loop Tuning[END_REF] estimates the road roughness using a filter in order to gain-schedule a SkyHook controller. The voltage to be applied to the semi-active damper is obtained through a lookup table. This estimation needs a 5 th order filter. A road profile estimator uses a deflection and shock absorber force as an input to the controller, [START_REF] Lu | A Frequency-Adaptive Multi-Objective Suspension Control Strategy[END_REF]. The shock absorber force must be estimated or measured. Hence, this approach is a challenging task. A fact is that research efforts in the semi-active suspension control are including the road adaptiveness using complex algorithms, i.e. companies as Peugeot, [START_REF] Poilbout | Suspension Control Device, Vehicle comprising said Device, Production Method Thereof and Associated Program[END_REF], and Honda, [START_REF] Izawa | Control System for Adjustable Damping Force Damper[END_REF]. [START_REF] Mashadi | Influence of Ride Motions on the Handling Behaviour of a Passenger Vehicle[END_REF] shows the correlation of road roughness with the standard deviation of the side-slip angle, lateral acceleration and roll angle. A proposal, that combines the Frequency-Estimation-Based (FEB ) controller with the adaptation of the manipulation limits according to the road profile, is presented [START_REF] De-J Lozoya-Santos | Control of an Automotive Semi-Active Suspension[END_REF].

This paper consists of six sections. Section 2 reviews the single corner of a vehicle, suspensions and MR damper models. The proposal is shown in Section 3. The simulation issues and performance criteria are described in Section 4. Section 5 presents and discusses the results. A conclusion is presented in Section 6.

MODELS

The lumped parameter single corner model describes the sprung mass (m s ) corresponding to the vehicle chassis and components supported by the suspension, and the unsprung mass (m us ); it only captures vertical motions (z s , z us , żs , żus , zs , zus ). The tire is modelled by a spring linked to the road (z r ) and represented with a stiffness coefficient (k t ) while the tire damping is negligible. A contact point between the road and the tire is always assumed. z = z sz us is the suspension deflection. The parameters of the lumped parameter single corner model are computed assuming a uniform distribution of the weight, two front passengers and mid-level fuel tank.

The passive suspension, typically modelled by a damper and a spring, exerts vertical resistive forces (F c , F ks ), Fig. 1a. The vertical forces are affected by the square of the motion ratio of the damper and spring, (η), [START_REF] Barak | Magic Numbers in Design of Suspension for Passenger Cars[END_REF], and reflect the projection of the force to the vertical axis in the wheel. A nonlinear single corner model with a passive damper will be used as a baseline model:

ms zs = -η 2 F ks (z) -η 2 Fc(z, ż, z) mus zus = η 2 F ks (z) + η 2 Fc(z, ż, z) -kt (zus -zr) z ∈ [z, z], ż ∈ [ ż, ż] (1)
The controlled suspension uses a semi-active damper (F M R ) instead of the passive damper (F c ), Fig. 1b. The semi-active damper varies its damping force according to a manipulation (u sa ). The single corner model with a controlled suspension is derived as follows:

ms zs = -η 2 F ks (z) -η 2 F M R (z, ż, z, usa) mus zus = η 2 F ks (z) + η 2 F M R (z, ż, z, usa) -kt (zus -zr) z ∈ [z, z], ż ∈ [ ż, ż] (2)
where u sa is the input control of the system provided by the semi-active damper. Two dampers are considered: a

F D = k p z + c p ż + m d • z + f f • tanh (v f • ż + x f • z) +c MR post-y • I • tanh (v pre-y • ż + x pre-y • z) (3)
where F D is the damping force, k p is the stiffness coefficient, c p is the viscous damping coefficient, m d is the virtual mass of the damper, f f is the friction force, v f ric and x f ric describe the damping and stiffness coefficients when friction is present, z, ż, z are the damper displacement, velocity and acceleration, c MR post-y is the gain of damping force due to the electric current I, and v pre-y and x pre-y are the damping and stiffness coefficients when the MR damper operates in a pre-yield regime.

PROPOSAL

The main idea is to adjust the limits of upper and lower level of manipulation of the semi-active damper according to the road roughness. The low level of the road roughness agrees with a soft damped suspension, and viceversa. Thus, the controller output limits are changed according to a detected road profile.

The FEB controller chooses high/low damping at each sampling time to comply with comfort or road holding goals according to the suspension deflection frequency:

I =      I 1 , 0 < f < f 1 I 2 , f 1 < f < f 2 I 3 , f 2 < f < f 3 I 4 , f 3 < f < f 4 , f = ( żrms) / (2 • π • zrms) (4)
where I is the electric current to apply, I i is the constant electric current to apply in the i th bandwidth of interest, f is the estimated frequency, f i is the frequency thresholds that define bandwidth intervals, z rms and żrms are the Root Mean Square (RMS ) values of the suspension deflection and its derivative, respectively, in the last k samples; see [START_REF] De-J Lozoya-Santos | Control of an Automotive Semi-Active Suspension[END_REF] for details. A control system that meets both comfort and road holding is chosen: a hard damping in bandwidth intervals that contain the resonance frequencies (0-2 Hz and 12-14 Hz), and soft damping for the others (2-12 and 14-20 Hz). The road profiler consists in a proportional measure to the amplitude of the road profile. A quasi-linear relation can be assumed between the road profile and the suspension deflection, Lozoya-Santos et al. [2012b], in the content of frequency and variation of amplitude. Thus, the main assumption is: if the road profile roughness amplitude changes, the suspension deflection changes in the same magnitude and sense. Then, using the experimentally-validated correlation of the road roughness with the standard deviation of the road [START_REF] Izawa | Control System for Adjustable Damping Force Damper[END_REF], a road estimator can be used based on the standard deviation of the suspension deflection:

σ i = i j=i-k q 2 j k -1 - k k -1 i j=i-k q j k 2 (5a) Σ = {σ 1 , . . . , σ ∞ } (5b) Q = Z - i j=i-k z j k (5c) Z = {z i-k , . . . , z i } (5d)
where q is the suspension deflection displacement without the offset, σ i is the q standard deviation of the i th sample in the last k samples, Σ, Q, and Z are data vectors. This way, it is expected that the standard deviation of he suspension deflection when exposed to a specific road will be different in range and magnitude. The road detection consists in:

R p = 1 0 < σ i < σ smooth 2 σ smooth < σ i < σ rough 3 σ rough < σ i < ∞ (6)
where R p is the type of detected road profile. R p changes according to the thresholds of the standard deviation eqn. (5a). The thresholds are defined observing the standard deviations in the suspension deflection. A comparison of the statistic behavior of both the road profile and the suspension deflection validates the thresholds. In this case, two thresholds have been defined in order to differentiate three types of roads. Once the road is detected, the span of the FEB controller is adjusted according to where I max and I min are the electric current values to apply in the control law, eqn. (4). Smooth runways allow to use a vehicle in an sportive way; while in rough runways, the suspension must to provide comfort. This new FEB controller with road Adaptation is named : FEBA.

SIMULATION

The parameters of the vehicle single corner model correspond to the front corner of a F-Class vehicle, Table 2 following a distribution of the body weight of 60 % in the front and 40 % in the rear. Hence, a single front corner of the vehicle has a 30 % of the total body weight. Two suspensions are considered: an uncontrolled and a controlled suspension. The analysis criteria for the semi-active suspension systems in time domain consists of:

(1) A qualitative analysis based on the following plots: (2) A quantitative analysis of the controlled semi-active and passive suspension systems using the RMS criterion:

RM S(x) = 1 T T 0 x 2 dt (7)
where x is the signal of interest, and T defines the interval of time to be analyzed.

Boxplots are used to analyze statistical behaviors. They consist of five-number summaries: the smallest observation (sample minimum), lower quartile, median, upper quartile, largest observation (sample maximum), and outliers. The spacings between the different parts of the box help indicate the degree of dispersion (spread) and skewness in the data, and identify outliers.

The simulation of three road profile sequences was done according to da [START_REF] Da Silva | Dynamical Performance of Highway Bridge Decks with Irregular Pavement Surface[END_REF] using the roughness coefficients proposed by [START_REF] Wong | Theory of Ground Vehicles[END_REF] corresponding to a smooth runway, highway with gravel, and pasture. The simulations were done for the uncontrolled suspension and two controlled suspensions: FEB controller with I max = 5 A and I min = 0 A without a road adaptation mechanism, and the FEBA controller.

The uncontrolled suspension simulations have two purposes: (1) to define the thresholds in eqn. ( 6) and the electric current spans, Table 1, (b) to use as a reference for the suspension performance. Nine simulations were done.

Each one consisted of a single corner model submitted to a given road profile under a given constant electric current. Three electric current values were tested, one per simulation. Then a comparison of the controlled suspensions simulations versus the uncontrolled in three different roads validates the proposal. The simulation was assumed to be in a distance of 100 m with a longitudinal speed of 40 km/h. The sample time was 1/512 s, therefore, each road sequence consisted of 4,655 samples. For each road sequence, the standard deviation was computed according to eqn. (5a), and a vector Σ was obtained.

RESULTS

The statistic behavior of the Σ vector for each road profile shows the maximum standard deviation according to each road profile and has its own range of variability, Fig. 3. Therefore, σ i can be used as a road profile detector. The results allow to infer that the standard deviation ranges of the suspension deflection for the three tested roads can be used in a control system of a semi-active suspension. This is because the standard deviation ranges keep their upper and lower limits. According to the ob- served results for the standard deviation spans of the suspension deflection for a given road, which is independent of the applied electric current, the limit values (thresholds) were defined. The thresholds for the road detection were defined as:

0 0.05 0.1 0 A 0 A 0 A 2.5 A 2.5 A 2.5 A 5 A 5 A 5 A Standard deviation [m]
R p = 1 0 < σ i < 0.0005m 2 0.0005m < σ i < 3 0.009m < σ i < ∞ (8)
where R p = 1, R p = 2, R p = 3 corresponds to the smooth runway, highway with grave and pasture roads, respectively. Hence, the road detection will be done according to the thresholds of the eqn. (8). The detection was proven using a road profile containing the three types of roads, one behind the other. Fig. 6 shows the simulation result of the detection mechanism showing a good qualitative detection efficiency. Although an exhaustive validation of the mechanism under several roads and conditions is needed, this mechanism will be used to show its potential application.

When the road profiler detects a given road, the limits of the FEB controller output will vary in order to assure the comfort and road holding goals regardless of the unevenness of the road. Several simulations in a closed loop using the FEB controller and the road profiles were done in order to define the limits of the electric current. Hence, observing the RMS improvement of the vertical acceleration and tire deflection controlled suspension RMS s, both relative to the same criteria from baseline suspensions, the limits of the electric currents for each detected road profile were defined as: Table 3. Lookup table values for road adaptive mechanism in FEB Adaptive controller.

Type of road Limits for I of FEB Smooth runway (Rp = 1) 0.2, 0 Highway with gravel (Rp = 2) 3, 1 Pasture (Rp = 3) 5, 2.5

Once the road profile detection and the electric currents were defined, the FEB controller was tested in the single corner model with and without the road detection. Nine simulations were done, three for the original FEB control, three for the FEB control with Adaptation to road (FEBA), and three for the baseline suspension. Each of the three simulations correspond to a given road.

The quantitative results indicate that FEBA controller maintains the comfort and road holding goals when compared with the baseline suspension, Fig. 7. Between 5-10 % is achieved in all roads. The FEB does not have a good performance in smooth roads for both goals, not even in highways with gravel road, where the road holding is key. In pasture, both controllers are better than the passive suspension, since here the road demands the comfort and road holding. The mechanisms for the controller adaptation to the road increases the efficiency of the FEB controller to a wider suspension operation.

The different electric current exerted by the FEB and FEBA controllers has the same shape, however, the limits in the FEBA controller changes due to the adaptation. This way, for smooth roads, the road holding and comfort is achieved by the softer suspension according to the FEBA, while the FEB computes an electric current to maintain a hard suspension. The difference is to allow more suspension deflection to maintain such goals, as it can be seen in the deflections of the suspension for each road, Fig. 7.

The Force-Velocity (F-V ) diagrams of MR damper model show a linear behavior for a smooth runway, a sigmoid shape in a highway with gravel, and a sigmoid shape with a high friction component, Fig. 9. The FEBA controller shows the optimum damping characteristics for each road in order to comply with the goals: soft for a smooth runway, a trade-off between soft and hard for a highway with gravel, and a harder suspension for the pasture (off road). The higher difference was in a highway with gravel road, where a trade-off between a soft and hard damping suspension is applied, while the FEB explores the limits in the damping suspension affecting the road holding but maintaining the comfort.

-0. Examples of the vertical acceleration and tire deflection for a highway with gravel are shown in Figs 10-11. The avoidance of higher values of these key variables is achieved by the FEBA controller in a better way than by the FEB.

CONCLUSIONS

A road adaptive controller for an automotive suspension with a semi-active damper has been proposed. The approach uses the suspension deflection standard deviation of a window of samples moving in time in order to detect the type of road. The detection mechanism was used to move the limits of the applied electric current in an FEB controller, hence a new FEBA controller is proposed. The results show that the detection of the road improves the RMS performance in comfort and road holding under several types of road in a 5-10 %. The proposed algorithm is simpler and allows lower computation than others presented in literature.

⋆

  Authors thank the Autotronics research chair at Tecnológico de Monterrey and CONACyT for the PCP 06/2007.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Passive and controlled suspension schemes of a quarter of vehicle.passive and MR damper. A lookup table of the a forcevelocity plot models the baseline big-car damper. It has a damping force ratio of 3 to 1 between the rebound and compression damping zones,[START_REF] Gillespie | Fundamentals of Vehicle Dynamics[END_REF], Fig.2. It presents a trade-off for road holding and comfort performances. The MR damper is modelled by a full modified

  (a) MR damper Force versus Velocity. (b) Acceleration of the sprung mass versus time. (c) Deflection of the tire versus time.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Boxplots of standard deviations of road profiles for a receding horizon of 200 samples.

Fig. 5 .

 5 Fig. 5. Boxplots of standard deviations for the suspension deflection for a receding horizon of 200 samples under several roads and electric currents.

Fig. 6 .

 6 Fig.6. Validation of the road detection mechanism when the suspension is submitted to the three roads.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Comparison of the RMS improvement of the controlled suspensions versus the passive suspension for three different roads. A positive value means that the controlled suspension has a smaller RMS than the baseline suspension, and viceversa.

Fig. 9 .

 9 Fig. 9. Comparison of force-velocity diagrams of the controlled suspensions for three different roads. The dark line is the FEBA controller and the grey one corresponds to the FEB.

Fig. 10 .FEBAFig. 11 .

 1011 Fig. 10. Chassis acceleration comparison of controlled suspensions. The dark line is the FEBA controller and the grey one corresponds to the FEB.

Table 1 :

 1 Table 1. Span of the electric current according to the type of road. Smooth runway {I SR M in , I SR M ax } Highway with gravel {I HG M in ,I HG M ax } Pasture {I P A M in ,I P A M ax }

	Type of road	Range of I

Table 2 .

 2 Single corner model parameters.

	Component	Value Units
	Sprung mass (ms)	546.5	Kg
	Unsprung mass (mus)	50	Kg
	Spring stiffness (ks)	83	kN/m
	Tire stiffness (kt)	230	kN/m
	ms resonance frequency (fm s )	1.2	Hz
	mus resonance frequency (fm us )	11.5	Hz
	Motion ratio (η)	0.614	-