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Adaptive Network Trained Controller for Automotive Steering Systems.

INTRODUCTION

The automotive industries are directing significant attention to technical progressions in the vehicle safety, dynamics and stability. In automotive business, the vehicle dynamics plays an important role in the development of a vehicle program. The security and the relief of the passengers are playing very important role in which the steering system is becoming one of the most important fragments of a vehicle development. The steering system of passenger cars are mainly classified into three groups i.e. mechanical steering system, hydraulic steering system and electric power assist steering (EPAS) system. An EPAS system is a feedback control system that electrically amplifies the driver steering torque inputs to the vehicle for improved steering comfort and performance. An EPAS system consists of a steering wheel, a column, a rack, an electric motor, a gearbox assembly, as well as pinion torque, position, and speed sensors. The essential operation of an EPAS system can be depicted in the system diagram shown in Figure 1.

Currently, all EPAS systems employ a pinion torque sensor, between the steering column and pinion, to determine the amount of the torque assist to the driver [1] - [5]. This torque assist is calculated via a tunable nonlinear boost curve [6] - [8]. Then, this signal is used as a control command to the electric motor to achieve a desirable level of assist. A control design must ensure several criteria so that the overall steering feel and performance will be similar to or better than a conventional hydraulic steering system. Control of EPAS systems is a very challenging task, due to several factors. First, road conditions and behavior of a driver are not predictable. In order to maintain consistency in the steering feel over a large range of the driver's torque, such as when the driver rotates the wheel and produces very high assist torque, the slopes of the boost curve change dramatically [6]. This will affect significantly overall system stability and performance in a fundamental manner. Second, there are several nonlinear components in the system such as friction, saturation and backlash [7]. These nonlinearities change as mechanical systems degrade and system conditions vary (lubrications, temperature, etc.). Finally, road disturbances and sensor noises are significant [10]. The system must be capable of maintaining robustness and eliminating undesirable vibrations or nibble (vertical vibrations in the steering wheel) that the driver might feel.

Analysis of stability of EPAS systems has been reported on several EPAS products. In [8], analytical expressions of transfer functions are derived to model assist dynamic, steering system compliance, and driver feel of the road for a dual-pinion system. Along a similar direction, analysis of stiffness and feel was reported in [10]. Mathematical equations, which relate rack force to the driver torque and steering angles, were developed to help EPAS gear engineers design an appropriate boost curve. These mathematical equations can assist in determining the regions of boost curves for which the steering feel varies as a function of vehicle speed. In [11], the authors studied the problem of power steering 'road feel'. The relationship between the pump pressures vs. steering torque was derived. It was argued that a gear can be designed on the basis of a set of boost curves which can be adapted as a function of vehicle speed. These boost curves can accommodate the desirable reduction of effort at parking and increased effort at straight driving as the vehicle speed increases.

Numerous feedback control techniques have been applied to EPAS systems, ranging from traditional lead lag compensation to more advanced control techniques. For example, the work presented by authors in [12] investigated control design by employing an H[infinity]-type criterion to provide an appropriate steering feel based on road information. In [6]- [7], several key points in implementing EPAS system in vehicle were described. A three-degree of- freedom (3-DOF) model of the EPAS was developed. An analysis on the closed-loop system was performed to understand compromises in system design. In [8], it was explained that increasing assist gain reduces steering torque for drivers. However, it may cause undesirable steering vibration at specified frequencies around 30 Hz. Motor angular velocity was needed for this control algorithm. An estimator design was investigated to obtain this signal without installing such a sensor. Two methods were investigated: Observer and back e.m.f.

In recent researches [1]- [15], different control methods are used to design the optimal controllers which provides improved performance and robustness of EPAS system with respect to traditional hydraulic steering system. But these controllers have showed undesired steering feel for high steering gain and acceptable performance for certain operating conditions. So, the aim of the work is to develop a controller which can provide high steering feel and increase in robustness of EPAS system. In this work, the neural networks are used in EPAS system instead of optimal controllers. A Euclidean adaptive resonance theory (EART) networks are trained according to the data collected from an H[infinity] optimal controller which represent the controller input and output signals. The said data are normalized and clustered into categories in the EART modules which are interconnected by a map field. Once the training is accomplished, the EART controller becomes ready to completely replace the optimal controller. The proposed technique is applicable to any arrangements of EPAS namely, rack, pinion and column EPAS with slice modifications in plant and feedback signal.

This work is organized in the following sections. Section II presents the model of 3-DOF nonlinear EPAS system. In section III, the formulation of the control problem is given. Section IV describes the Euclidean adaptive resonance theory (EART) based neural networks. The experimental analysis of EARTMAP neural controller is presented in section V and its performance results are shown in section VI. Finally, Section VII draws the conclusion of the work.

3-DOF NONLINEAR EPAS MODEL

A single pinion EPAS system contains several nonlinear components (Figure 1), such as friction terms in the column, rack, and motor, as well as sticking and backlash in the gearbox. These nonlinear elements can be inserted as required in the simulation environment to accurately represent their nonlinear properties. However, it is possible to approximate these terms by linear ones and treat approximation errors as model uncertainties. Consequently, it is imperative that feedback controllers that are designed on the basis of the linearized systems be sufficiently robust so that they provide stability and satisfactory performance on the original nonlinear system. Following the basic Newton's laws of motions, it can be established that a nonlinear dynamic model of the EPAS system by examining the torque acting on the steering column and the pinion drove electric motor. Finally, by considering the forces acting on the rack, the differential equations of the system can be obtained through equation 1, 2, 3 [5] - [8]. Equations (1), (2), (3) show the single pinion EPAS model having 3-DOF system that are column, rack and electric motor. The input/output relationships of each of these equations are presented by the MATLAB/Simulink model of these equations

A. Column Model

The nonlinear model of the column is described in eq. (1) is shown in Figure 2.

Driver torque and rack position are the two inputs to the model, and there are five outputs. The outputs are column force (force input to the rack model), column position and velocity, net torque to accelerate the column inertia, and sensed torque. The column is essentially a rotational spring-mass-damper system; consist of the steering column inertia, [J.sub.c], the total column damping, [b.sub.c], and the torsion spring rate, [K.sub.c]. The windup of the spring is coupled to the rack displacement through the inverse of the pinion radius. One end of the spring is positioned by the dynamics of the column inertia, and the motion of the rack positions the other end. The difference of these two angles is the net windup of the torsion spring rate of the torque sensor and all other compliance in the column, represented by [K.sub.c]. The force provided to the rack via the driver's input to the column through the pinion is [K.sub.c] times the windup, divided by the pinion radius, [r.sub.p]. The differential equations of the column model can be obtained through equation 1

[J.sub.c][[theta].sub.c] = -[b.sub.c][[theta].sub.c] - [K.sub.c][[theta].sub.c]+[K.sub.c]/[r.sub.p][X.sub.r] + [T.sub.d] + [f.sub.c]([[theta].sub.c],[[theta].sub.c]) (1)

B. Rack Model

The MATLAB/ Simulink model of the rack based on eq.. (2) is shown in Figure 3.

The inputs to the model are tie rod force, column input force and motor input force. The three outputs are the rack position, velocity and total input force. Note that the priority of the tie rod force is positive when it opposes the motion of the rack. The parameters associated with the rack model are M (inverse of the rack mass), br (viscous friction or the damping of the rack) and [G.sub.r] (the dry or static friction of the rack). The rack position output is coupled to the column and the assist actuator or electric motor model, and is the primary output of this block. The differential equations of the rack model can be obtained through equation 2

[mathematical expression not reproducible] (2)

C. Electric Motor Model

The model of the electric motor is shown in Figure 4.

Figure 4 shows that there are two main inputs to the model the assist torque Ta and the control input U. The outputs are the motor position and velocity. Note that the inner loop control of the motor is assumed to be of unity gain, implying perfect control of the electric states of the motor. The parameters associated with motor model are Motor rotational inertia [J.sub.m], motor and gearbox damping [b.sub.m], and gearbox friction gain [G.sub.Bf]. The differential equations of the column model can be obtained through equation 1.

[J.sub.m][[theta].sub.m] = -[b.sub.m][[theta].sub.m] - [K.sub.m][[theta].sub.m] +[K.sub.m]G/[r.sub.p] [X.sub.r] + [T.sub.m] + [f.sub.m] ([[theta].sub.m],[[theta].sub.m]) (3)

The nonlinear system block diagram that describes the 3-DOF EPAS system is shown in Figure 5. This nonlinear model works well for investigating system performance and stability. To investigate controls and to improve system stability this model was linearized and then analyzed with classical design techniques.

The assist torque, Ta can be calculated using the rack displacement and electric motor position as given below:

[T.sub.a] = [K.sub.m] ([[theta].sub.m]) - (G/[r.sub.p] [X.sub.r]) (4)

The motor electromagnetic torque input [T.sub.m] (Nm) is related to motor current input by an inner loop [T.sub.m] = [G.sub.in] (s) u, where u is the motor input and [G.sub.in] is the inner loop transfer function. In the frequency band of interest, [G.sub.in] = 1 (per normalization of mechanical units). As a result, one may simply view [T.sub.m] as the control input. Also, the resistance term [F.sub.t] = [K.sub.t][*X.sub.r] + [F.sub.r]. It has been noted that the resistance force on the rack is mainly [K.sub.t][*X.sub.r]. [F.sub.r] will denote the remaining disturbances from road condition changes.

Generically, an EPAS system has two main external inputs: The driver torque [T.sub.d] that is not measured and the assist motor input current U, which is a control signal. Due to variations in road and tire conditions, sensor inaccuracy and measurement noise, the EPAS system is subject to external disturbances [F.sub.r] (resistance force on the rack) and d (sensor noises). The dimension of d is identical to the number of sensors utilized in the system.

The key outputs, which are to be controlled, include assist torque [T.sub.a] and the pinion torque [T.sub.c]. [T.sub.a] is calculated by passing a pinion torque signal [T.sub.c] via a nonlinear tunable boost curve. On the other hand, in all current EPAS systems a torque sensor is used to measure [T.sub.c]. Other measured signals vary with hardware configurations. The methodology introduced in this paper is generic and applicable to all types of sensor configurations. Based on a production EPAS system, It is assumed that the column angle [[theta].sub.c] and assist motor angular velocity [[theta].sub.m] are measured.

CONTROL PROBLEM FORMULATION

The objective of designing an optimal feedback control system is to ensure the stability of the closed-loop system and to provide desirable steering feel for any gain value ([K.sub.a]) of boost curve ([K.sub.a] = [T.sub.d] / [T.sub.a]) as it is in the conventional hydraulic systems (100 [greater than or equal to] [K.sub.a][greater than or equal to] 0). Figure 6 depicts, without further dynamic compensation, the stability of the closed-loop system depends critically on the gain. When [K.sub.a] = 0, the system is stable. As [K.sub.a] increases, on the other hand, the poles of the system move gradually to the right half plane as shown in Figure 7 for [K.sub.a] = 4. Consequently, dynamic compensation is necessary.

The three main objectives to be fulfilled for the efficient design of EPAS are the generation of the assist torque, the driver's feel and the robustness of the design. The goals of EPAS design can be explained as follows:

1. Assist Torque Generation

The desired level of assist is defined by a boost curve, which relates the driver torque to the desired assist torque. A typical boost curve is shown in Figure 8 below.

The boost curve can be expressed as:

[T.sub.a] = B([T.sub.d], v) (5)

Due to dynamic delays of the EPAS system in responding to changes in [T.sub.d], the actual Ta generated by the EPAS system may be significantly different from the desired [T.sub.a]*, depending on control designs. The first objective of a well-designed EPAS system is to deliver Ta, that is reasonably close to [T.sub.a]*, in a frequency band [0, [W.sub.0]], where [W.sub.0] is about 40 Hz * to deliver a desirable steering feel. A main challenge of EPAS system design stems from the fact that [T.sub.d] is not directly measured. As a result, [T.sub.a]*, is not available as a reference signal.

Currently, a pinion torque sensor is installed to measure [T.sub.c]. In most EPAS systems [2]- [5], [T.sub.c] signal is used as an approximation of [T.sub.d] for boost curve input (feed forward control) and feedback control implementations. However, it will be shown that although at steady state [T.sub.c] and [T.sub.d] are essentially identical, they differ dramatically during dynamic transitions. The difference is mostly pronounced when the assist torque is significant. Without compensation of this dynamic difference, the dynamic behavior and performance of the EPAS will be severely compromised.

2. Driver's Feel

When a road condition changes, the resistance force Fr on the rack will vary. The driver feel is purely subjective determinations. Each automotive original equipment manufacturer (OEM) for different class of vehicle defined DNA of the steering feel. Therefore, these steering feel are more subjective for each OEM rather than objective. To provide a desirable level of the driver's feel of driving conditions, the driver must perceive a suitable amount of changes in the twist torque [T.sub.c]. In other words, one can define a desired mapping from [F.sub.r] to [T.sub.c] as [T.sub.c] = Gf*Fr. For instance, one may require [G.sub.f] = [K.sub.f] = 0.15 in the low frequency range. Some other techniques have been mentioned in [16] - [17].

3. Robustness

The EPAS system contains nonlinear components such as frictions and damping. Also, unit-to-unit deviations and component wearing will introduce parameter deviations from the nominal models. Furthermore, road condition changes and sensor noises introduce uncertainties during its operation. Hence, an EPAS system must be designed to provide satisfactory performance in the presence of such uncertainties. The control technique based on neural network theory can effectively fulfill this challenge.

The detailed design and analysis of [H.sub.[infinity]] optimal controller used in EPAS system is given in [12].

EUCLIDEAN ADAPTIVE RESONANCE THEORY

Adaptive resonance theory (ART) [18] neural networks are known for their ability to cluster similar vector patterns into one category. Features of the said category are close to all of the clustered pattern features. The clustering process is implemented according to a certain radius of similarity called vigilance factor. Euclidean ART neural network is an example of these ART networks. The vigilance factor in EART networks is known as threshold radius [R.sub.th]. The threshold radius determines the similarity range based on which close patterns are clustered into one category [16], [19]. The EART network is considered as unsupervised network. The supervised version of EART consists of two EART networks. The first one is used to cluster input patterns, whereas the second is used to cluster the output ones. A map field unit in the middle which links the input EART clusters with their corresponding ones in the output EART. Figure 9 shows the complete structure of the supervised EART network and will be given the name EARTMAP.

Both of the EART modules are responsible for clustering their corresponding patterns. The number of resulting clusters is proportional to the threshold radius in each module. The lower the threshold radius, the higher the number of resulting clusters. The patterns to be clustered are normalized in order to prevent the proliferation of the resulting clusters [20]. Proliferation of clusters should be avoided in order to prevent flooding the system memory with redundant clusters. It is always required to keep the number of clusters in each module under control during the learning processes.

This is done by selecting proper values for threshold radius of each module. In classification phase [19], the search process would be faster if the number of clusters is smaller. This led to a compromise between the size of the resulting network and its sensitivity to the changes in the applied patterns as suggested in [16]. A smaller network with smaller number of clusters is less sensitive to the changes in the applied patterns. This may result in larger error in the classification. The algorithm for EARTMAP training is illustrated with the help of a flowchart in Figure 10.

The equations used in the flowchart are given below:

[mathematical expression not reproducible] (6)

Where,

j is the index of a category found in [EART.sub.a]

and i is the index of the current pattern presented to [EART.sub.a] module.

da(J) = min (da(j)) (7)

[mathematical expression not reproducible] (8)

Where, db(J) is the Euclidean distance between the current [I.sub.b] and the [EART.sub.b] category [w.sup.ab], index of which is J.

[mathematical expression not reproducible] (9)

Where, [mathematical expression not reproducible] and [mathematical expression not reproducible] are patterns k of cluster J in [EART.sub.a] and [EART.sub.b] modules respectively. L is the number of cluster members of [EART.sub.a] and [EART.sub.b], respectively.

HARDWARE IN THE LOOP EXPERIMENTAL SET UP

In this work, the conventional H-infinity controller is replaced with an EARTMAP neural controller that is trained based on the data collected from the original H-infinity controller [12]. The original controller receives three signals ([T.sub.c], [[theta].sub.c], and [[theta].sub.m]) and generates one output control signal (U). Both input and output vectors should be normalized prior to the training process. The said normalization prevents the uncontrolled generation of clusters. This phenomenon is known as "proliferation of clusters" [14]. Figures 11 and 12 show the normalized input and output vectors.

The normalized vectors are sent to the training algorithm to cluster them into categories. The training process results in clustering input signals in [EART.sub.a] module and output signal in [EART.sub.b] module. Both module clusters are interrelated by the map field module. Figures 13 and 14 show the clustering maps for both [EART.sub.a] and [EART.sub.b] modules respectively.

EARTMAP neural network was trained with 26512 training patterns. The training results in only 74 clusters. This means that the EARTMAP controller consists of 74 local controllers. Each local controller has its own neighborhood of influence that is represented by [R.sub.th]. The smaller the [R.sub.th] value, the bigger the number of resulting clusters. It is always desired to keep the number of clusters as low as possible. However, this would result in increasing the classification error [21]- [22]. The classification error is increased because the influence of a cluster affects patterns that are not close enough to it. However, the advantage is the need of smaller memory and faster classification.

EARTMAP CONTROLLER PERFORMANCE RESULTS

Once the training is completed, the EARTMAP controller becomes ready for classification phase. In this phase the neural controller completely replaces the original controller. The EARMAP controller samples the input vectors into [EART.sub.a] module. The sampled vector is then normalized and compared with all of the existing clusters.

The comparison is represented by calculating the Euclidean distance between the sampled vector and the cluster to be compared to. The closer cluster wins the competition. The map field, then, selects the corresponding output vector in [EART.sub.b] module and sends it to the out port of the controller. The process continues as long as the system is running.

Figure 15 show the performance of the EARMAP controller (N = 74 cluster). Figure 16 shows the performance of the same controller for N= 15 clusters only.

By analyzing Figures 15 and 16, it is evident that the rms value of the error signal of the controller whose number of clusters is 75 is lower than that of the controller whose number of clusters is 15. Thus, when the size of the cluster increases the errors between the measured data versus simulated data decreases and therefore, the controller can easily predict the desired responses of each of system outputs.

CONCLUSION

This work presents the analysis of neural networks based controller which is used to control electric power assist steering (EPAS) system. For better understanding the effectiveness of the proposed controller, first analysis of a conventional controller is done. The conventional controller is developed using a three-degree of- freedom (3-DOF) model of the EPAS system and then its closed-loop analysis has been performed. The conventional controller showed acceptable performance for certain operating conditions and undesired steering feel for steering gain. In order to maintain the robustness and the desired steering fell, neural networks has been proposed to replace the conventional controllers. A Euclidean adaptive resonance theory (EART) networks is trained according to the data collected from an H[infinity] optimal controller which represent the proposed controller input and output signals. The said data are normalized and clustered into categories in the EART modules which are interconnected by a map field. Once the training is accomplished, the EART controller becomes ready to completely replace the optimal controller. The proposed controller provides improved robustness and high steering feel of EPAS system by reducing the amount required for intensive calculation. Further, the rms value of the error signal with 75 number of clusters is smaller than that of 15 clusters. When the size of the cluster increases the errors between the measured data vs simulated decreases and therefore, the controller can easily predict the desired responses of each of system outputs. Hence, the selection of number of clusters is very important in EART controller based EPAS system. The proposed technique is applicable to any arrangements of EPAS namely, rack, pinion and column EPAS with slight modifications in plant and feedback signal.

REFERENCES

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ABBREVIATIONS

[T/.sub.d]                        Driver Torque
[T/.sub.m]                        Motor Torque
[K/.sub.c]                        Steering Column and Shaft Stiffness
[B/.sub.c]                        Steering Column Damping
[K/.sub.m]                        Motor and Gearbox Rotational
                                  Stiffness
[J.sub.m]                         Motor Rotational Inertia
[B.sub.m]                         Motor and Gearbox Damping
M                                 Steering Rack and Wheel Mass
[B.sub.r]                         Rack Damping
G                                 Motor Gear Ratio
[r.sub.p]                         Pinion Radius
[K.sub.t]                         Tire Spring Rate
[[theta].sub.c]                   Steering column angle
[[theta].sub.c]                   Steering column angular velocity
[X.sub.r]                         Rack position
[X.sub.r]                         Rack linear velocity
[[theta].sub.m]                   Assist motor angle
[[theta].sub.m]                   Assist Motor angular velocity
[f.sub.r], [f.sub.c] & [f.sub.m]  Nonlinear friction terms in the model
v                                 vehicle speed

[T/.sub.d]                        Nm
[T/.sub.m]                        Nm
[K/.sub.c]                        N/m
[B/.sub.c]                        Nm/(rad/s)
[K/.sub.m]                        Nm/rad
[J.sub.m]                         Kg-[m.sup.2]
[B.sub.m]                         Nm/(rad/s)
M                                 Kg
[B.sub.r]                         N/(m/s)
G                                 dimensionless
[r.sub.p]                         m
[K.sub.t]                         N/m
[[theta].sub.c]                   rad
[[theta].sub.c]                   rad/s
[X.sub.r]                         m
[X.sub.r]                         m/sec
[[theta].sub.m]                   rad
[[theta].sub.m]                   rad/s
[f.sub.r], [f.sub.c] & [f.sub.m]
v                                 mph


Training algorithm for Flowchart in Appendix given above in Figure 10 will be executed as follows:

1. Start the training

2. n-dimensional normalized patterns are obtained

3. Inputs patterns vectors and target patterns vectors are set.

4. If number of cluster determined, then previous steps patterns will be chosen as first patterns

5. Else these patterns will be calculated using equation 6

6. Calculate equation 7 and equation 8

7. If the results of equation 8 is less than or equal to [R.sub.tha] then new pattern will be determined

8. Else begin new training.

8. Calculate equation 9

9. If there are no patterns, the training will stop and desired results are achieved.

10. Else start training again till desired results are achieved.

Rakaan Chabaan

Hyundai America Technical Center

Mohammad Saad Alam

CARET, Aligarh Muslim University

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of SAE International.

Positions and opinions advanced in this paper are those of the author(s) and not necessarily those of SAE International. The author is solely responsible for the content of the paper.

doi:10.4271/2017-01-9626
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Author:Chabaan, Rakaan; Alam, Mohammad Saad
Publication:SAE International Journal of Passenger Cars - Electronic and Electrical Systems
Article Type:Technical report
Date:May 1, 2017
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