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Magneto-rheological dampers—model influence on the semi-active suspension performance

Recently, automotive industry has adopted semi-active damper systems to improve handling and comfort properties of vehicles. Nowadays, Magneto-Rheological (MR) dampers are among the most effective solutions; with the control algorithm used for their operation being a key element. While basic controllers do not require mathematical damper models, improved performance can be achieved if these are available. Usually, the accuracy of a particular set of models can be assessed by evaluating standard quantitative metrics. However, two models with similar error-metrics can still have widely different qualitative properties. In this context, the main aim of this paper is to study the effects that may appear in the closed-loop performance of an automotive suspension system when the damper model is unable to represent crucial nonlinear MR phenomena. To highlight the model influence on the controller synthesis and subsequently on the suspension performance, two damper models with different accuracy levels were chosen: an Artificial Neural Networks (ANN)-based model is compared with the classical Bingham model. First, their accuracy is experimentally validated using typical error-metrics. Afterwards, the same suspension control strategy is designed using both models. Frequency-Estimation-Based control was selected because it better exploits available model data than other typical strategies such as sky-hook. The resulting performance is assessed with a software-in-the-loop approach using CarSim® and complemented with a hardware-in-the-loop implementation using a CAN-bus, both closed-loop control cases use a Simulation-Oriented ANN model as benchmark to represent the MR damper nonlinearities. Results show that although the difference in error-metrics between models can be small using typical identification methods (e.g. 16% in one scenario), suspension performance in comfort and road-holding are significantly different. Error-metrics can be deceptive for assessing the effectiveness of MR damper models during the controller design phase. Accurate qualitative modeling in the pre/post-yield regions are the main factors which determine the resulting controller performance.


Introduction
Vehicle manufacturers are constantly trying to improve security and riding comfort. One of the key components involved in this process is the vehicle suspension. The main function of dampers is to dissipate the kinetic energy, reducing the motion of the sprung mass (i.e. vehicle chassis) and improving the road-holding. While it is possible to adjust the suspension elements passively by mechanical design, it has been shown that active or semi-active systems can improve performance (Dixon 2007, Savaresi et al 2010, Fijalkowski 2011. In this regard, active elements are capable of inducing arbitrary forces to provide the best performance; however, they have higher energy requirements and a complex mechanical implementation (Fijalkowski 2011). On the other hand, semi-active dampers allow modifying the dynamical properties of the suspension system with less energy consumption and mechanical complexity. The result is that semi-active systems are less expensive, yielding a better level of performance-versus-cost ratio (Savaresi et al 2010). In this context, Magneto-Rheological (MR) dampers present several attractive properties such as: low power requirement, fast response, simple structure and a wide range of adjustable damping coefficient (Dixon 2007).
In addition to an adequate actuator, semi-active suspension systems require control algorithms capable of managing the dissipativity and saturation constraints of the damper to improve the desired handling and ride-comfort properties. Several Automotive Suspension Control Systems (ASCS) have been proposed with varying levels of success. For some of these proposals an accurate damper model is unnecessary. For example, the well known sky-hook and ground-hook controllers only require data of the suspension deflection velocity and sprung or unsprung mass velocity instead of a complete damper model. On the other hand, better knowledge of the physical system can be used to improve the performance. That is, if an accurate damper model is available then modelbased control strategies can be used to achieve better results. Examples of these control schemes can be found in (Do et al 2012, Poussot-Vassal et al 2012, where Linear Parameter-Varying (LPV ) systems are used to consider the model-based actuator constraints for a robust controller design.
In MR dampers, the rheological properties of the fluid are modified by the magnetic field generated by a coil located within the damper, according to a time-varying electrical current signal. This allows modifying the properties of the vehicle suspension on-line. However, the actual relationship between the electric current and the resulting damping force is highly nonlinear. MR dampers, unlike passive ones, present variable average damping coefficients, hysteresis loops and saturation levels (Dutta and Chakraborty 2015). Moreover, other dynamical phenomena such as friction, viscous damping and yield stress have also been detected when timevarying electric currents are considered . Indeed, in a typical closed loop control operation where the MR damper current is modified on-line, additional dynamical modes are also introduced. This discussion highlights the difficulties in obtaining proper MR damper models. For this purpose two major approaches have been proposed in the literature: structured and non-structured models.
Structured models are comprised by mathematical representations with a predefined structure. This approach allows relating particular model parameters to specific physical behaviors. For instance, the yield stress phenomenon could be related to a particular parameter. Examples of these models are the classical Bingham model (Stanway et al 1987), the Bouc-Wen models (Spencer et al 1996, Werely et al 1998, Yang et al 2002, Dutta and Chakraborty 2014, Waubke and Kasess 2016 and the algebraic models proposed in (Guo et al 2006, Kwok et al 2006, Çesmeci and Engin 2010, Singru et al 2017. The Bouc-Wen and the algebraic models aim to represent common nonlinearities with high accuracy. However, these models suffer from having too many parameters, some of them related with the internal structure of the damper. Therefore, they can be difficult to characterize experimentally. Among structured models, the Bingham model is capable of reproducing variable damping coefficient, saturation and hysteresis with a simple structure; making this model attractive for practical applications. Non-structured models are mainly data-based, yielding gray or black box-type representations. Examples of this approach, based on techniques such as Artificial Neural Networks (ANN), fuzzy logic, statistical inference and polynomial approximations, can be found in (Choi et al 2001, Chang and Zhou 2002, Hong et al 2002, Guo et al 2004, Du et al 2005, Savaresi et al 2005, Zapateiro et al 2009, Metered et al 2010, Boada et al 2011, Imaduddin et al 2017, Tang et al 2017. The main advantage of these models is the possibility of achieving very high levels of accuracy at the cost of losing the physical interpretation of the model parameters. Among these techniques, ANN-based modeling yields simpler structures because the decoupling of the jounce/rebound effects of the damper is not necessary. In addition, it has been observed that ANN models have good extrapolation properties, simpler identification algorithms and it is possible to achieve a low number of model parameters (Tudón-Martínez et al 2012) when the ANN design is based on the minimal dimensions criterion. This is relevant because simpler ANN models have low computational overhead and are suitable for practical applications. In contrast, if a very high level of accuracy is required, a complex ANN can be used by introducing recurrent architectures, redundant inputs, time delayed input vectors and/or increment of the network size at the cost of higher computational overhead. An interesting study about the trade-off between complexity and modeling accuracy using an ANN is presented in (Tudón-Martínez et al 2012).
Due to the particular characteristics of structured and non-structured models their utilization for the design of closed loop control systems differs. In particular, inverse dynamics-based control is the most representative approach used with non-structured models, particularly with ANN (Chang and Zhou 2002, Savaresi et al 2005, Zapateiro et al 2009, Metered et al 2010, Boada et al 2011. Other controloriented problems considering ANN are presented in (Guo et al 2004, Imaduddin et al 2017; usually ANN-based models are very attractive to add fault-tolerance to the suspension control systems. On the other hand, structured models are especially suitable for analytical design methods, such as those based on LPV systems (Do et al 2012, Poussot-Vassal et al 2012.
The previous discussion shows that a direct comparison between the actual effectiveness of structured and nonstructured models can be difficult since the controller design approaches used for each are different. In this regard, some research projects have presented interesting comparative studies among models (Song et al 2005, Şahin et al 2010, Sandu et al 2010, Wang and Song 2011. These reports assess the models in terms of key features such as model error, number of parameters, complexity, processing time, identification methods, etc. However, these studies only highlight the resulting model properties, but not their actual usefulness for designing effective control systems. In this sense, the main contribution of this paper is to present a qualitative and quantitative analysis of the effects that can occur in the closed-loop control performance of an automotive suspension system when the damper model, used for the controller design, is simple or has limitations to represent all nonlinear MR phenomena. To clarify this situation consider that two models, A and B, have been assessed quantitatively using the Root Mean Square (rms) error, achieving both a similar level of accuracy. However, assume that it has been observed that model A is able to predict the hysteresis properties while model B is better at predicting force saturation. If the same control design method is applied with both models it would be valid to wonder which would yield the best performance. Moreover, it is not clear which model would be better for particular control objectives, maybe model A is better for improving comfort and model B for road-holding. These aspects are important if the very best performance is required. In this regard it is important to note that in current literature it is common to propose suspension controllers without considering the actual behavior of the damper, i.e. dissipativity, saturation, friction, etc (Karnopp et al 1974, Valasek et al 1997, Hong et al 2002. In fact, it is typical to assume that an arbitrary force can be exerted by the semi-active damper, which is contrary to the dissipative nature of the device (Savaresi et al 2005).
The previous discussion highlights the necessity of studying further the relationship between particular model properties and the resulting effectiveness for control design. In this article this problem is explored by first assessing the quantitative and qualitative accuracy of two models: the well known Bingham damper model and a simple ANN model. Some researches have demonstrated that Bingham model is a simple model, which is quick and practical while ANN-based model is more precise; however, their resulting effectiveness for control design is not clear, e.g. it is unknown which model is better for designing a comfort-oriented controller. Thus, the main purpose of this paper is to qualitatively and quantitatively analyze the effects of damper model accuracy in the controller design. Particularly, to determine the impact on the suspension performance (comfort and road-holding) when the damper model used during the controller design is unable to represent the typical MR phenomena in the pre-yield and post-yield stress regions, such the stiffness, friction, nonlinear viscoelasticity and hysteresis.
The control strategy applied to both models allows extracting all the relevant frequency domain information contained in the design models, which has been shown to be the main aspect which defines the dynamics of suspension systems. The resulting controllers (one based on the Bingham model and other on ANN) are then implemented using a Software-in-the-Loop (SiL) solution with the CarSim ® simulator. To represent the MR damper dynamics in a very accurate representation of the complete vehicle suspension system, a Simulation-Oriented ANN (SO-ANN) model is used in this SiL environment. The SO-ANN model, in comparison with the Bingham and simple ANN models, has such a high level of accuracy, both quantitatively and qualitatively, that it can be used as a benchmark of the behavior of the MR damper. Thus, the SO-ANN model, which mimics with high accuracy the MR damper behavior, allows comparing the closed-loop suspension performance that is achieved using each damper model (Bingham or simple ANN) during the controller design. This comparative analysis is carried-out in time and frequency domains to yield useful conclusions.
Finally, the difficulty of implementing complex control algorithms is a limitation for practical applications. To explore this aspect, the considered controller (designed using the Bingham model and ANN model) is implemented in a Hardware-in-the-Loop (HiL) configuration using a low power micro-controller and considering a networked operation using a Controller Area Network (CAN). The most used variables in this paper are defined in table 1, and the abbreviations are described in table 2.

MR damper modeling
The behavior of automotive dampers is typically assessed through their Force-Velocity (FV ) diagram. Figure 1 shows the typical FV diagram of an MR damper under sinusoidal There are several approaches which have been proposed to model MR dampers. The typical approach for model assessment consists in comparing the force of a given set of models with the real force obtained from an experimental setup. Using this method both quantitative and qualitative considerations can be made. Examples of this approach can be found in (Song et al 2005, Şahin et al 2010, Sandu et al 2010, Wang and Song 2011. On the other hand, the effectiveness of a given model can be defined as how useful the model in question is for a particular application, such as mechanical design, controller design, simulation accuracy, performance testing, etc (Lozoya-Santos et al 2012). In the context of ASCS, the ultimate purpose of MR damper models is to design appropriate control algorithms. In this sense, the most effective model for ASCS is that which yields the best control algorithm. This article explores the evaluation of MR damper models taking into account this notion.
The first model considered is the well-known Bingham model (Stanway et al 1987), which has a parametric structure aimed at modeling the hysteresis FV loops, especially at high frequencies. This model comprehends a Coulomb friction element placed in parallel with a viscous damper and assumes that the fluid remains rigid in the pre-yield region. In the post-yield region, it considers a linear relation between the damping and deformation rate (Stanway et al 1987). The resulting damper force considering the Bingham model is given by: where f 0 is the preloaded damping force, f c is the dynamic yield force, c 0 is the viscous damping coefficient, I is the electric current andż def is the relative velocity of extension/compression of the damper rod, i.e. the damper deflection velocity. For the comparison a simple non-structured model is proposed, in this case an ANN-based model. One complication which arises when using non-structured models is the difficulty of knowing a priori which particular configuration (i.e. number of neurones, input variables, etc) will render the best results for a particular application. For instance, if the model is aimed at simulation accuracy then more complex structures will be required at the cost of computational overhead. On the other hand, for control design a good balance of model accuracy and complexity is preferred. That is, for ASCS applications a slight loss of accuracy is permissible if a markedly simpler model is available. In most cases this balance has to be explored through direct experimentation.
In (Tudón-Martínez et al 2012), the main features of ANN-based MR damper models are studied comprehensively. This study revealed, that suspension deflection velocity and electric current are the main variables required for accurate modeling. This reduces the necessity of including redundant input variables, such as in (Chang and Zhou 2002, Savaresi et al 2005, Du et al 2006. In addition, time delayed inputs, as those proposed in (Chang and Zhou 2002, Savaresi et al 2005, Zapateiro et al 2009, are less important if an appropriate sampling time is used. Particularly, a sample time in the range of 3 ms was found to be effective. Finally, the model accuracy can be improved by introducing recurrent ANNs, as supported by the findings reported in (   found that linear accuracy improvements when using recurrent ANNs translate to exponentially increasing computing times. Therefore, it is recommended to limit the use of recurrent ANNs to applications where accuracy is more important than model complexity. In this article feedforward ANNs are preferred over other variants because their learning algorithms do not require complex computational operations and it is not necessary to know the initial values of the parameters. MR damper models have been obtained using Radial Basis-Function networks, but these networks have the drawback of requiring a pre-clustering step which complicates the training process (Du et al 2006).
The extensive study presented in (Tudón-Martínez et al 2012) shows that an ANN-based model with a simple feedforward Multi-Layer Perceptron (MLP) structure is capable of representing two different commercial MR dampers with good accuracy, while keeping the model simple enough to be used in real-time control applications. Considering an MLP architecture as (L i , L m , L o ) with L i , L m and L o neurones in the input, middle and output layers respectively, the ANN structure (2, 10, 1) has shown in (Tudón-Martínez et al 2012) to be appropriate for MR damper modeling and can be treated as the following nonlinear mapping of the damper deflection velocity˙( ) z t def and the electric current input I(t) to the resulting damper force: . 2

MR def
In this article a SiL arrangement (described in section 4.1) is used to evaluate the resulting closed loop performance through the CarSim ® simulator. This approach has the advantage of including the complete vehicle dynamics; however, it requires a highly accurate model to represent the actual MR damper. Therefore, an additional ANN model, oriented to simulation accuracy, is also introduced. In comparison to models (1) and (2), which are used for controller design and are intended to be simple, this model can be as complex as required. After extensive testing it was determined that excellent model accuracy could be achieved for the MR damper studied in this article using a recurrent ANN with a (4, 10, 1) structure, a recurrent feedback of 2 delayed samples of the damping force, a delayed sample of the inputs and a sampling time of 0.625 ms. The general mathematical representation of this model is shown next: In addition to having a more complex internal structure than model (2), this SO-ANN model (3) has a recurrent structure which allows modeling highly nonlinear dynamics. This characteristic is important because it has been observed that in practice MR dampers do have an important dynamical component (Şahin et al 2010, Sandu et al 2010. In the following section the experimental characterization of a real MR damper using the Bingham model (1), the ANN model (2) and the SO-ANN model (3) is presented.

Experimental characterization of an MR damper
In this section a commercial MR damper manufactured by Delphi MagneRide TM is characterized using the test bench shown in figure 2. This damper allows using a continuous input signal, has a stroke of 40 mm, time constant of 15 ms and force range of ±4000 N. On the other hand, the test bench comprises an hydraulic actuator of 15 kN at 2068 kPa with a stroke of 0.06 m, an MTS-407 TM controller and a NI-9172 TM DAQ which measures the relevant variables with a sampling frequency of 1650 Hz.
Appropriate excitation of the electric current I(t) and the damper deflection z def (t) are required in order to properly characterize key nonlinearities of the semi-active damper. In particular, steady state sinusoidal excitation of the damper According with this methodology, table 3 shows the Design of Experiments (DoE) used in this article. Some of the main features of the DoE are displacement and electric current ranges of ±25 mm and 0-2.5 A, respectively, with a displacement bandwidth between 0.5 and 15 Hz. This bandwidth is enough for evaluating comfort and road holding in automotive applications.

Modeling results
In order to validate the capabilities of models (1)-(3) to represent the dynamical behavior of the commercial MR damper, different replicas of the experiments described in table 3 have been used. In order to assess the resulting performance of each model the following index, based on the rms error, is used: (·) F MR represent the estimated and experimental damping force, respectively, and n is the number samples in the experiment.
The Bingham model was identified by using a nonlinear least squares method based on the L 2 -norm of the modeling error, while the ANN-based models were trained with the Levenberg-Marquardt algorithm. Figure 3 shows the setup used in the training and testing stages. Matlab ® was used to compute and execute the learning algorithms. On the other hand, the testing procedure was carried out in a National Instruments TM cRIO 9014 real time simulator using a sampling rate of 200 Hz.
Each experiment was repeated ten times to improve the validity of the results. In each experiment 60% of the data obtained was used for model identification and the remainder for evaluating the model performance. Figure 4 illustrates the flowchart of the modeling procedure in general. The model  design framework (control or simulation oriented) follows the operation dynamics of interest according to (Lozoya-Santos et al 2012); while, the architecture of the ANN-based models is based on the minimal dimensions criterion discussed in (Tudón-Martínez et al 2012). The analysis of the modeling results has been carried out in quantitative and qualitative form, using the rms error and FV curve respectively. The selection of the identified model parameters (Bingham or ANN-based model) for control purposes depend on the vehicle dynamics required to test in a closed-loop operating condition. Table 4 shows the quantitative resulting performance of each model in the five experiments. A comparison of the mean modeling error of all the experiments of table 3 reveals that the ANN-based model has an improvement of 55% compared with the Bingham model; whereas the simulationoriented model (i.e. SO-ANN) has the lowest error level, with a mean error 52% lower than the control-oriented ANN model.
On the other hand, the results show that there are important differences in modeling error depending on the experiment. For instance, in experiment E 3 an improvement of 84% can be obtained using the ANN model instead of the Bingham model; however, in experiment E 5 the difference is only 16%. In comparison, the SO-ANN model presented an improvement of 79% over the ANN model for the same experiment. Finally, the typical damper identification method is given by experiment E 1 showing a difference of 33% in the modeling error between the ANN and Bingham models, which is a moderate error level.
The previous discussion confirms the notion that model accuracy is dependent on the experimental conditions. Therefore, it is difficult to know the effectiveness of each model for a particular task by assessing the model error quantitatively. That is, an indirect evaluation of the modeleffectiveness through a quantitative error index is difficult without further knowledge. For example, intuition indicates that if the conditions of experiment E 5 are more relevant than  those of E 3 for controller design then the ANN-based model will be only marginally more effective than the Bingham model for this purpose. In this case only a much more complex representation, such as the SO-ANN model, with its recurrent structure and low error across all conditions, would introduce significant improvements. Conversely, if the conditions of experiment E 3 are more relevant than those of E 5 then ANN-based model will be more effective for a controller design in comparison to the Bingham model. In the next section a direct evaluation of the effectiveness of each model will be presented by designing and implementing modelbased closed loop controllers. A more thorough qualitative assessment of each model for one experiment is presented next. Although more involved, this kind of comparisons can reveal deeper nuances that quantitative error indexes cannot. For instance, figure 5 compares the FV diagram obtained for each model based on the same experimental data (E 3 , road profile). Clearly, the SO-ANN model shows the best modeling performance and is able to represent the effects of jounce and rebound when the electric current changes. In comparison, the control-oriented ANN model has some limitations to fit the nonlinearities at low relative velocities. Nonetheless, in general it correctly captures the dependency between viscous damping coefficient and electric current. On the other hand, some clear modeling drawbacks are exhibited with the Bingham model: (a) in the post-yield region, the viscous damping effect is linear, (b) the pre-yield region is narrower than the experimental one, inducing an overestimation of the resulting force arounḋ In summary, the qualitative model assessment revealed that the Bingham model has problems to correctly predict: (a) the hysteresis FV loops at higher frequencies observed in experiments E 1 , E 2 and E 5 , (b) the friction phenomenon produced when there is a change of sign in the position, e.g.
in experiments E 1 , E 2 and E 5 , (c) the nonlinear viscous damping in the pre and post-yield region seen in experiments, E 1 , E 3 and E 4 , and (d) the saturation effect in all experiments.
Finally, a novel complementary experiment was performed to evaluate the extrapolation properties of the models. This experiment is comprised by a displacement SFS with an exponentially decreasing amplitude (from 12.7 to 1 mm) accompanied with a SC sequence for the electric current. In this case, the Bingham model showed an error of 15.1%, the ANN-based model 2.7% and the SO-ANN model 1.5%.

Modeling effect on the suspension controller design
The main objective of this section is to analyze the modeling effects of an MR damper into the design of a semi-active suspension controller; allowing to ascertain the relationship between the model properties obtained in the last section with the actual effectiveness of each model for controller design.
In model-based control an accurate model is required if high-performance operation is desired. This follows from the fact that if a high degree of model uncertainty is expected then the robustness margins need to be increased during the design stage. This normally results in a more conservative controller, which will typically yield a lower performance ( In this context it could be argued that an optimal MR damper model (for controller design) will be that which allows designing high-performance model-based controllers, regardless of the level of model accuracy reported by any particular quantitative measure (Prabakar et al 2014). In the last section it was shown that the ANN was, by most measures, a more accurate model than the Bingham model. Using these models as case of study, high-performance semiactive suspension controllers are designed to elucidate how well model accuracy translates to the resulting controller performance.
According to the previous modeling results, experiment E 1 explores the main nonlinear MR phenomena in the frequency band (FB) of interest. Thus, the identified parameters of the Bingham and ANN model with this data-set were chosen for the controller synthesis (i.e. in the Model Application block in figure 4).
Since the control strategy used for the comparison is based on a decentralized QoV topology its corresponding model is presented next. A QoV can be represented by a sprung mass (m s ) and an unsprung mass (m us ). A linear spring with stiffness coefficient k s and an MR damper represent the suspension between both masses. The stiffness coefficient k t models the wheel tire. The vertical position of the mass m s (m us ) is defined by z s (z us ), while z r corresponds to the road surface. It is assumed that the wheel-road contact is ensured. The system dynamics is given by: where the MR damping force is soft/hard at the minimum/ maximum actuation (electric current), andf is the on-line estimated frequency of the suspension motion. In this case, the design task consists of determining the FBs FB i according to the desired suspension performance in the frequency domain (Lozoya-Santos et al 2011). Figure 6 shows a conceptual diagram of a QoV semiactive suspension control system using the FEB controllers. In oder to calculate the MR damper force either model (i.e. ANN or Bingham) can be used. These models use the suspension deflection velocity and electric current to estimate the MR damper force. On the other hand, the FEB controller uses measurements and/or estimations of the suspension motion to calculate the frequency response of the QoV suspension system and determine the optimal current level for all the possible vehicle conditions, creating a look-up table. The look-up table definition can be performed off-line or on-line if improved damper model data are available. A signal processing module is used to estimate in a short time-window the frequency content of the QoV motion as described in (Tudon-Martinez et al 2015). This data is used on-line by the controller module to select the optimal electric current for the MR damper according to the specified comfort and/or road holding performance.
The main feature of the FEB control approach is its high dependency on the MR damper model. In particular, the model is used to determine the proper damping coefficient required to reduce the chassis motion and wheel vibrations according with the main frequency content of the measured variables. During this process no model uncertainty is considered. As such, this approach is akin to a direct model cancellation, rendering it particularly sensitive to its accuracy.  Figure 6. General structure of a QoV semi-active suspension control system. In this case, the MR force is represented by an ANN-based model.
This renders the FEB control scheme a good candidate to evaluate model effectiveness for control design. Figure 7 represents the frequency responses of a frontal QoV when the ANN and Bingham models are used to represent the semi-active damping force. The sprung mass acceleration is related with the comfort performance while the tire deflection evaluates the road holding. In all cases the frequency responses are presented in two conditions, when the electric current of the MR damper is 2.5 A and 0 A, respectively. This allows assessing the bounds of the comfort and road-holding corresponding to the maximum and minimum damping force that can be exerted by the MR damper. The main differences in the frequency responses of each model are discussed as follows: • Around the resonance frequency of m us (∼9 Hz): -Note in figure 7(c) that the Bingham model shows an almost invariant sprung mass acceleration with respect to the damper current, while the ANN model predicts a much wider gain difference, figure 7(a). -In the case of the tire deflection the gain is around 200% larger in the ANN model response, figures 7(b) and (d).
This larger magnitude obtained with the ANN-based model is caused by a better semi-activeness property in the actuator model, and thanks to the good capability of the ANN to represent nonlinearities at high frequencies.
• The previous observations indicate that a comfortoriented controller designed with the Bingham model will tend to predict much lower improvements by operating the damper with a higher current level than a controller designed with the ANN model. This difference will have an important effect when deciding whether a high or low level of current is used for the damper. • Around the resonance frequency of the sprung mass (∼2 Hz): -The Bingham model shows a trade-off between comfort and road holding because the sprung mass acceleration is reduced with the maximum damping force while the tire deflection is minimized with the minimum damping force. This trade-off is not present with the ANN-based damper model.
The previous discussion highlights key differences in the behavior predicted by each model in the most relevant FBs for this application. As follows it will be shown that these differences are effectively translated into the controller tuning parameters. According with the tuning method described in (Lozoya-Santos et al 2011) FEB controllers were synthesized with each model; resulting in the look-up tables summarized in table 6. It is clear that the resulting FB configuration is different when using each model.
The process was repeated for each vehicle corner, resulting in similar differences in the look-up tables of the corresponding FEB controllers.  Finally, the main focus of this article is to present the effects of damper model choice for high-performance modelbased control, such as the FEB scheme. Nonetheless, the following paragraph presents a complementary discussion which considers the damping coefficient bounds, denoted as c min and c max . These parameters are used for controller tuning by more traditional schemes such as the sky-hook approach. In order to estimate c min and c max it is common to calculate the damper FV diagrams considering both high and low currents. Figure 8 shows the FV diagrams of both MR damper models. In figure 8 it can be observed that the pre-yield zone in the ANN-based model is wider, i.e. the friction phenomenon is present in a larger range of suspension deflection velocity. On the other hand, the viscous damping (the slope in the postyield zone) is higher in the Bingham model, e.g. at 1.2 m s −1 the force in the ANN-based model (@ 2.5 A) is around 3000 N while in the Bingham model is around 4000 N. Similarly, this inconsistence occurs in the curve at 0 A. From figure 8 it is clear that the resulting c min and c max will be markedly different with each model.

Modeling effect on the suspension control performance
In the previous sections it was shown that different semiactive damper models yield distinct ASCS behaviors and accordingly different controllers when using a model-based control scheme. In this section the effects of these differences will be explored further. The objective is to extend the notion of semi-active damper model accuracy into that of model effectiveness for model-based control applications. In particular, two studies were performed: (i) A SiL simulation by using the CarSim ® software. The vehicle simulator has a suspension control system configuration of independent corners, i.e. each wheelstation has an MR damper with an independent FEB controller. In this simulation study, the most accurate SO-ANN model in equation (3) is used as benchmark to evaluate the suspension control performance obtained with the controllers designed using models (1) and (2). (ii) An experimental HiL platform based on a CAN network. This setup allows evaluating the resulting controllers in a more realistic environment subject to noise and other perturbations. The complete QoV system (considering the SO-ANN damper model) is embedded in a real-time industrial controller to mimic the suspension phenomena as close as possible, whereas the control algorithm is hosted in a low cost microcontroller. A CAN bus keeps the communication between the two devices such as in commercial cars. This process includes converting the control and actuator signals into the analog domain before digital reacquisition and CAN bus communication in order to inject more realistic noise and perturbations.

Analysis in the CarSim ® simulator
The vehicle model used to evaluate the influence of the MR damper model into the automotive suspension system was characterized from experimental data obtained from a K&C (Kinematics and Compliance) test over a commercial vehicle (pickup truck of medium payload). A CarSim ® generic pickup model was customized for the particular vehicle considered in this study using K&C data. The customization of the suspension system is mainly composed of the physical dimensions (weight, longitude, width, height, wheel base, front and rear track, etc), kinematics real data curves (camber angles, dive angles, caster angles, damping force, jounce/ rebound stops, spring and tire stiffness, etc), structural parameters (mechanical ratios, compliance coefficients, Kingpin geometry, etc) and compliance of the suspension system (independent wheel stations at the front side and a rear solid axle at the back). Figure 9 illustrates some parameters of the vehicle model, whose kerb weight is around 2000 kg, as well as some experimental nonlinear curves introduced in CarSim ® for the kinematics and compliance of the front and rear suspension system. Although each corner is independently controlled, the global load transfer is considered in the QoV suspension performance due to the weight distribution. Figure 10 shows the block diagram of the SiL simulation in a Matlab ® environment. The CarSim ® software allows defining all vehicle model parameters as well as the conditions of the simulation test (road roughness, road curvature, driving path, etc).
The FEB controller requires knowledge of the main frequency content of the QoV motion (Lozoya-Santos et al 2011). In this case this data was estimated as in (Tudon-Martinez et al 2015) using vehicle signals obtained from CarSim ® ; only a set variables which would be available in real-world applications were used.
Two sets of FEB controllers were tested, one designed using the Bingham model and the other with the ANN model. In both cases the vehicle dynamics were simulated using CarSim ® with the SO-ANN model. As shown in section 2.2 the SO-ANN model has such a high level of accuracy, both quantitatively and qualitatively, that it can be used as a benchmark of the behavior of the MR damper. Note, however, that its complexity currently does not allow controller implementation or adjustment in real-time using low-cost embedded solutions. Therefore, in this article the use of the SO-ANN was limited for real-time QoV motion simulation using high-power computing platforms.
Three simulation scenarios in CarSim ® were used to compare the suspension performances of both controllers. Each test allows the analysis of the vehicle suspension behavior at a particular road condition: (i) A Bounce Sine Sweep (BSS) test with decreasing road amplitude (from 0.10 to 0.01 m) with varying frequency Figure 9. Some parameters and nonlinear curves to characterize the kinematics and compliance of a medium-payload vehicle model customized in CarSim ® .
(0.5-12 Hz). In this case the vehicle velocity is maintained at 30 km h −1 . This test allows to evaluate the vehicle vertical dynamics at different frequencies, including the resonance frequencies of the chassis (∼2 Hz) and unsprung mass (∼9 Hz). (ii) Three bumps in series with amplitude of 0.05 m at 30 km h −1 without braking. This test quantifies the capacity of the suspension system to reduce the motion when the vehicle is suddenly excited by a bump. In order to excite the roll and pitch motions, the separation between the bumps corresponds to the wheelbase distance whereas the bumps at the right side of the vehicle are out of phase from the ones at the left. (iii) An standardized road profile (type D according to the ISO 8608) at 50 km h −1 . The vertical dynamics are analyzed when the vehicle is driven in a typical situation over a rough road. Figure 11 shows the transient response of some variables related to the chassis motion and wheel vibrations which can be used to asses the resulting comfort and road holding performances using both controllers. Figure 11 confirms that different control performances in the suspension system are obtained in both cases.
In the BSS test, figure 11(a), the vertical acceleration of the chassis is greater from t=10 to t=13 s for the controller based on the Bingham-model. This time period is associated with an excitation frequency close to the resonance frequency of the chassis, which is considered as low frequency for this application. The increased vertical acceleration could be related to the inconsistence in the pre-yield zone (associated with lower frequencies) of the FV diagram found in the Bingham model.
Similarly, figure 11(d) illustrates that the road holding performance is deteriorated with the controller based on the Bingham model when operating close to the frequency of resonance of the unsprung mass (around 9 Hz at t = 20 s). This deterioration could be related to the limited capability of the Bingham model to capture the viscous damping phenomenon and its lack of saturation at high frequencies.
In the bumps test (figures 11(b) and (e)), the differences between the two controllers are more difficult to distinguish. Sudden road disturbances complicate the estimation of frequency content; therefore, in this test the performance of the FEB controller is being limited by the frequency content estimator rather than by the model quality. That is, model quality has only a slight impact in this test. As expected, figures 11(b) and (e) show almost the same responses for both controllers. After the third bump an increased vertical acceleration and pitch angle are observed when using the controller based on the Bingham model. This suggests that up to that point a very similar damper configuration was selected from the lookup table of both controllers. The third bump induced an important threshold crossing for the controller based on the Bingham model, which resulted in a significant divergence on the damper configuration of both controllers. This can occur due to numerical issues in the pre-yield zone caused by the discontinuous structure of the Bingham model in a simulation environment. Thus, it is clear that the ANN model yields a more effective FEB controller, even in conditions when the performance of the frequency content estimator is limited.
In the ISO road profile test, whose frequency of excitation is around 4 Hz, the magnitude of the vertical heave motion is smaller with the controller based on the ANN model, with a reduction of up to 0.01 m, figure 11(c). On the other hand, at 4 Hz the vertical acceleration of the unsprung mass is similar with both controllers, figure 11(f). This result is consistent with the frequency response of the tire deflection shown in figure 7, where an invariant point is located around 4 Hz.
Finally, table 7 shows a quantitative assessment of the suspension performance for both controllers based on the rms of the variables plotted in figure 11. The controller based on the ANN model has the best comfort and road holding performances with a lower rms index in all cases. In particular, using the controller based on the Bingham model deteriorates the comfort measurements 11.8% in the BSS test, 11.7% in the bumps test and 10% in the road profile test whereas the road holding is deteriorated 48.2% in the BSS test, 8% in the bumps test and 0.1% in the road profile test. According to the aforementioned results, it is notable that the influence of the MR damper model used to design the closed loop controllers is easier to perceive in some tests. It is not surprising that the major differences were observed when the vehicle suspension is operating near to its resonance frequencies. In this context, it is important to remember that vehicle velocity and road roughness are the main contributors to the excitation of the suspension system. In a practical application these variables are mostly unknown and difficult to measure. This hinders the possibility of perceiving whether an appropriate MR damper model was used for the design of the suspension control system. As follows some difficulties that can occur by using an inaccurate damper model during the controller design: • Comfort deterioration around the resonance frequency of the chassis when the stiffness and friction phenomena are not well modeled in the pre-yield region of the FV map of the damper (low deflection velocity in the damper). • Road holding deterioration around the resonance frequency of the unsprung mass when an incorrect linear viscous damping is considered in the post-yield region of the FV map of the damper instead of the real nonlinear viscous damping. • An insufficient damping force when a saturation constraint is not included into the semi-active damper model. That is, the controller may select a lower damping level than the required because an unsaturated model predicts a higher damping level than that actually achieved by the damper.

Analysis in an experimental HiL platform
In this section a further comparison of the controllers designed with the Bingham and ANN models is presented using a HiL setup. This setup allows evaluating the resulting controllers in a more realistic environment subject to noise and other perturbations. This comparative analysis was  performed in the experimental system depicted in figure 12.
The suspension system is composed by a QoV model including the SO-ANN damper model that is implemented in a FPGA NI-cRIO with a sampling frequency of 200 Hz. The FEB control algorithm is implemented in a low-cost microcontroller. Both devices, FPGA and microcontroller, are communicated through a CAN bus. The cRIO chassis includes: (a) an Analog Output (AO) module NI-9263 used to send the QoV model outputs to an Axiomatic Analog Input (AI) module and (b) a Digital Input (DI) module NI-9426 which receives the PWM control signal computed by the microcontroller. The CAN network contains an AX030100 Axiomatic AI module, an AX022400 Axiomatic PWM module and the Electronic Control Unit (ECU), in this case a microcontroller. The Axiomatic AI module converts the outputs of the QoV model into CAN frames. The ECU reads the frames of the Axiomatic AI module, computes the control signal for the MR damper (SO-ANN model in FPGA) using the Bingham or ANN model based controller and transmits the control signal through the CAN bus. The Axiomatic PWM module reads the frames from the ECU and generates a PWM signal with a duty cycle proportional to the control signal. Figure 13 shows the frequency response of the closed-loop system when the controller is designed using the Bingham and ANN models. The test signal consists of a series of 20 independent sinusoidal waves of the form:

Experimental results.
As it was previously discussed in the open-loop frequency response in figure 7, the sprung mass motion performance between 2 and 6 Hz is heavily dependent on the controller action, and consequently on the damper model used for the controller design. In the closed-loop frequency domain analysis, figure 13(a) shows that the ANN model based controller yields better comfort performance mainly due to its capacity to represent the viscous damping force in the preyield region (low frequency). On the other hand, since the Bingham model has a narrower pre-yield zone, the controller based on this model predicts a higher damping force than the real one, resulting in deterioration of comfort performance. For the road holding frequency domain analysis in figure 13(b), the performance between both controllers is similar between 2 and 6 Hz.
To experimentally analyze the transient performance of both controllers, an ISO 8606 road profile type D at 60 km h −1 was considered in the HiL platform. This test is performed to study the suspension dynamics in a typical rough road. Figure 14 shows the experimental transient response of the sprung mass used to monitor the comfort while the unsprung mass displacement is used to analyze the road holding. This test provides a high level of excitation to the suspension dynamics around 4 Hz; therefore, in concordance with the closed-loop frequency response analysis in figure 13(a), the limitations of the Bingham model based controller yield a lower comfort performance compared with the ANN model based controller. The result is that the Bingham model based controller is unable to reduce the sprung mass displacement with the ISO 8608 road profile test, figure 14(a).
The road holding performance in this test, as previously observed in the SiL simulation test, is practically the same with both controllers, figure 14(b). The reason is the invariant point located around 4 Hz in the unsprung mass frequency response.

Discussion
The results of both SiL and HiL studies show that the model used during the controller design has a major effect in the resulting suspension system performance when operating near to the resonance frequency of the sprung and unsprung masses. Although the Bingham model is a simple structure capable of representing linear damped systems accurately, it was shown that using this model for semi-active suspension control design can compromise the vehicle road-holding and comfort properties. Particularly when the road profile has a high-frequency content.
The SiL and HiL studies show that when the damper model misrepresent the stiffness and friction phenomena in the pre-yield region, the comfort performance can be deteriorated. In addition, the road holding can be affected if the post-yield region is not modeled properly. Finally, the absence of a force saturation constraint at high frequencies in the model can generate inconsistent control actions in this frequency range. Table 8 summarizes the results of both models in therms of their accuracy to represent the MR damper nonlinearities and effectiveness for a control application. BSS test in SiL explores all main nonlinear MR phenomena in the whole FB of interest, showing that the Bingham model cannot represent properly the dynamics in the pre-yield and post-yield stress zone. Thus, the ANN model is more effective for a suspension controller design at these conditions. Bumps test in SiL only explores MR phenomena at very low frequencies. In this operating region both models are accurate; therefore, both models are effective for control design purposes. The differences that may appear in simulations are mostly numerical issues due to the discontinuity in the Bingham model. For the ISO road profile tests (SiL or HiL), model accuracy depends on the road roughness and vehicle velocity. For instance, in rough roads at low vehicle velocity lower operating frequencies are present. In this condition both models are accurate enough to represent the MR phenomena; therefore, both models are effective in a suspension control application at these conditions. However, at higher vehicle velocities (higher operating frequencies) the Bingham model cannot reproduce the hysteresis and saturation properly. Thus, in this case the ANN model is recommended for the suspension controller design.
As final remark is that the performance deterioration observed when using the least accurate model can be mishandled by the engineers in charge of the implementation of the suspension system. In particular, instead of improving the quality of the damper model, the following unnecessary actions are common:

Conclusions
For model-based control, model accuracy is a key element in order to achieve high-performance levels. In this context, the accuracy of a particular model can be appreciated differently depending on the application. Therefore, the concept of model effectiveness is introduced in this article; that is, a model whose properties render it useful for a particular task. In this context, MR dampers comprise several physical phenomena and interactions which can be modeled with different degrees of accuracy, each one distinctly contributing to the effectiveness for particular applications.
This article studied the relationship between specific MR damper model properties and the resulting effectiveness for control design. In order to achieve this goal three damper models where obtained from experimental data. First, for controller design purposes, an ANN based model (MLP) and, (b) the well known Bingham model were obtained. A third model, based on a more complex ANN architecture using recurrent networks, was also obtained for simulation and benchmarking purposes; this representation is denoted here as a Simulated Oriented-ANN (SO-ANN) model. In this context, the main idea of this paper is to present the effects that can occur in the closed-loop performance of an automotive suspension system when the model is unable to represent relevant nonlinear MR phenomena. For this case of study, the ANN and Bingham models were chosen because of their different nonlinear modeling features.
The differences among the models were studied both qualitatively and quantitatively. The main qualitative differences between the Bingham and simple ANN models were found to be the pre/post yield behaviors as well as the force saturation. These differences were effectively captured by some quantitative measurements; however, it should be observed that it is not easy to determine how a particular qualitative model limitation will reflect into the actual quantitative error. This is true particularly after considering that quantitative modeling error shows a large range of variation among different tests. This suggest that specific tests could be used to identify specific qualitative limitations, but this has not been currently studied for MR dampers in a control design context. Therefore, a wide range of tests is still suggested (Lozoya-Santos et al 2012).
After the model assessment, a control approach which is applicable to both control-oriented models is used to design a FEB controller. This control strategy allows extracting all the relevant frequency domain information contained in the design models, which has been shown to be the main aspect which defines the dynamics of suspension systems. The resulting controllers (one based on the Bingham model and other on ANN) are then implemented in (a) a SiL by using CarSim ® with Matlab Simulink ® and (b) a HiL by using a real-time module based on a FPGA, CAN communication and a low cost microcontroller. The first configuration allows to evaluate the effect of the full car dynamics in the resulting suspension control system while the HiL setup allows introducing noise and perturbations into the control loop as well as evaluating the feasibility of the implementation of the studied control strategy in low-cost embedded solutions, which is in line with typical automotive control applications.
The results of the study revealed that MR damper model effectiveness for controller design is affected mainly by its properties around the chassis and unsprung masses resonance frequencies. In fact, when a controller based on the Bingham model is used the suspension performance can be severely compromised if the vehicle is operated with a road profile with high frequency content around the aforementioned frequencies.
As a summary, the major factors that limit the effectiveness of an MR damper model for automotive suspension control design are: (i) When the damper model lacks the capacity to represent the stiffness and friction phenomena in the pre-yield region (low frequencies in the FV diagram of the damper). (ii) When the post-yield region is wrongly represented by a linear viscous damping function. (iii) When the damping force is not saturated at high velocities.
Each of the aforementioned cases compromises the suspension performance in a particular manner and should be avoided at the modeling stage in order to avoid unnecessary and time-consuming effort at later stages of the control design and implementation.