Nonlinear DynDOI 10.1007/s11071-013-0875-y

O R I G I NA L PA P E R

Design of neural network-based control systems for activesteering system

İkbal Eski · Ali Temürlenk

Received: 11 January 2013 / Accepted: 20 March 2013© Springer Science+Business Media Dordrecht 2013

Abstract Nowadays, safety of road vehicles is an im-portant issue due to the increasing road vehicle acci-dents. Passive safety system of the passenger vehicleis to minimize the damage to the driver and passen-ger of a road vehicle during an accident. Whereas anactive steering system is to improve the response ofthe vehicle to the driver inputs even in adverse situa-tions and thus avoid accidents. This paper presents aneural network-based robust control system design forthe active steering system. Primarily, double-pinionsteering system used modeling of the active steeringsystem. Then four control structures are used to con-trol prescribed random trajectories of the active steer-ing system. These control structures are as classicalPID Controller, Model-Based Neural Network Con-troller, Neural Network Predictive Controller and Ro-bust Neural Network Predictive Control System. Theresults of the simulation showed that the proposedneural network-based robust control system had supe-rior performance in adapting to large random distur-bances.

İ. Eski (�)Faculty of Engineering, Mechatronics EngineeringDepartment, Erciyes University, Kayseri 38039, Turkeye-mail: ikbal@erciyes.edu.tr

A. TemürlenkFaculty of Information Technologies and Engineering,Ahmet Yesevi University, Turkestan city, Kazakhstan

Keywords Active steering system · Artificial neuralnetwork · Robust control · Random road input signal

1 Introduction

The active steering system plays a significant role inimproving vehicle handling and stability. Several pa-pers, a few of which are presented below, have beenpublished in the area of the vehicle steering controlsystem, vehicle stability and some of these papers aregiven below.

Zheng and Anwar researched a yaw stability con-trol algorithm with active front wheel steering con-trol of a vehicle [1]. The yaw stability control al-gorithm was obtained the decoupling of the lateraland yaw motion of a vehicle and the vehicle’s yawdamping simultaneously by the feedback of both yawrate and front steering angle. Also, the control sys-tem was applied on a steer-by-wire vehicle, and thebenefits of the system were illustrated experimentally.A model of active steering approach for trajectorygeneration of unmanned ground vehicles were devel-oped by Yoon et al. [2]. An optimal tracking problemwas presented in the way of cost minimization un-der constraints. Simulation results show that the mod-ified parallax method reflected the threat of the obsta-cles to the vehicle considering the dimension and statevariables of the vehicle. An integrated control strat-egy presented for optimum coordination of individual

mailto:ikbal@erciyes.edu.tr

İ. Eski, A. Temürlenk

brakes and front/rear steering subsystems [3]. A low-level slip-ratio controller has been designed to gener-ate the desired longitudinal forces at small longitudi-nal slip-ratios, while averting wheel locking at largeslip-ratios. The efficiency of the suggested approachwas demonstrated through computer simulations.

Feedback linearization scheme for the control ofvehicle’s lateral dynamics was applied by Liaw andChung [4]. Feedback linearization scheme was em-ployed to construct the stabilizing control laws for thenominal model. The stability of the overall vehicle dy-namics at the saddle node bifurcation was then guar-anteed by applying the Lyapunov stability criterion.Since the remaining term of the vehicle dynamics con-tains the steering control input, which might changesystem equilibrium except the designed one. Paramet-ric analysis of system equilibrium for an example ve-hicle model was also obtained to classify the regimeof control gains for potential behavior of vehicle’s dy-namical behavior. A comparative study of different lat-eral controllers applied to the autonomous steering ofautomobiles was presented by Sotelo et al. [5]. Thenonlinear nature of vehicle dynamics makes it a chal-lenging problem in the intelligent transportation sys-tems field, as long as a stable, accurate controller wascompulsorily needed in order to ensure safety dur-ing navigation. The problem has been tackled undertwo different approaches. The first one was based onchained systems theory, while the second controller re-lied on fuzzy logic. A comparative analysis has beencarried out based on the results achieved in practicaltrials.

A kinematics model of planetary gear set and steer-ing gear with active front steering system was installedby Gao and Wang [6]. Also, a controller of variablesteering ratio for active front steering system was de-signed, and virtual road tests was made in CarMakerdriver vehicle-road simulation environment. The re-sults of simulation tests validated the controller per-formance and the advantage of variable steering ratiofunction. In addition, driving comfort was improvedat low speed especially due to the active front steer-ing system. Chu et al. [7] proposed coordinated con-trol system to improve vehicle handling and stabilityby coordinating control of electronic stability programand active front steering. Primarily, they calculated thetarget yawing moment required to keep the vehicle sta-ble according to PID control of the yaw-rate. After-wards, they proposed a fuzzy method to control elec-

tronic stability program and active front steering. Fi-nally, they used genetic algorithm to optimize the con-trol rule to ensure the correctness and accuracy of thecontrol rule. The performance of the integrated con-trol system was evaluated by computer simulations attwo different running condition and they compared theperformance of the integrated system.

Fault detection of a steering wheel sensor signalin an active front steering system was researched byMalinen et al. [8]. A kinematics constraint, mod-eled as a dynamic system, was used to estimate thesteering wheel angle. This estimated signal was com-pared with the measured signal. Using change detec-tion algorithms typical failure patterns of the steer-ing wheel sensor were detected quite easily. The es-timated results and measurements from a prototypevehicle have been presented in this study. Escalonaand Chamorro [9] researched a method for the stabil-ity analysis of the steady curving of vehicles based onequations of motion that were obtained using multibody dynamics. Owing to this method, steady cir-cular motions could be described in terms of equi-librium points rather than periodic motions. Stabilityanalyses were thus made much simpler and compu-tationally efficient. Also, the method was applied toa simple wheeled mechanism. The numerical resultsthus obtained were consistent with those of analyticaland classical theories, which testify to the accuracy ofthe proposed method. Jinlia et al. [10] proposed inter-nal model control based on combined brake and frontwheel active steering for vehicle stability control andcompared with the four wheel steering internal modelcontrol.

Discrete neural control for flight path angle and ve-locity of a generic hypersonic flight vehicle was in-vestigated by Xu et al. [11]. Primarily, strict-feedbackform was set up for the attitude subsystem consider-ing flight path angle, pitch angle, and pitch rate byaltitude-flight path angle transformation. Secondly, thedirect neural network control was proposed for at-titude subsystem via back-stepping scheme. The di-rect design was employed for system uncertainty ap-proximation with less online tuned neural networkparameters and there was no need to know the in-formation of the upper bound of control gain dur-ing the controller design. Similar neural networkcontrol was applied on velocity subsystem. Finally,the feasibility of the proposed controller was veri-fied by a simulation example. A simple new method

Design of neural network-based control systems for active steering system

Fig. 1 Schematicrepresentation of thedouble-pinion steeringsystem [20]

for dynamics of chaotic time continuous systems isproposed by Perc [12]. Determining the flexibilityof regular and chaotic attractors is researched byMarhl and Perc [13]. They deploy a systematic ap-proach, first introducing the simplest measure givenby the local divergence of the system along the at-tractor, and then develop more rigorous mathemati-cal tools for estimating the flexibility of the system’sdynamics. A new method for controlling unstableperiodic orbits is presented [14]. The effectivenessof the proposed method is shown on two differ-ent chaotic systems. Also, the chaotic behavior of adriven resonant circuit is researched [15] and theirstudy used basic nonlinear time series analysis meth-ods.

This paper is organized as follows: In Sect. 2, ve-hicle dynamic and kinematics model is presented. Ourproposed control system is given in Sect. 3, and simu-lation results are presented in Sect. 4. Finally, conclu-sions are given Sect. 5.

2 Model of active steering system

The model of active steering system is composed of adouble pinion rack mechanism, the primary mechani-cal steering system and an electric actuator motor ad-ditionally. Also, the electric actuator is connected toa different pinion gear with the steering rack [16], asillustrated in Fig. 1. The dynamics equations of the

model are given by

Jcθ̈c + Bcθ̇c + Kc(

θc − Xrrp

)= Td (1)

Jmθ̈m + Bmθ̇m + Km(

θm − XrGrp

)= Tm (2)

MrẌr + BrẊr + KrXr= Kc

rp

(θc − Xr

rp

)+ KmG

rp

(θm − XrG

rp

)+ Ft (3)

where Jc and Jm represent the inertia moment of steer-ing column and electric actuator, respectively. Bc, Bmand Br represent the damping coefficient of steeringcolumn, electric actuator and steering rack. Kc , Kmand Kt denote the spring coefficient of steering col-umn, electric actuator and tie/rack, respectively. θmand θc represent the angle of electric actuator andsteering column, respectively. rp is the rack radius, Tdis the driver torque, Tm is the torque of electric actu-ator, Mr is the rack mass, Xr is the displacement ofrack, Ft is the lateral force from road and G is the mo-tor gear ratio. The differential equations can be writ-ten in state-space notation when the state vector x isdefined as

ẋ = Ax + BU (4)y = Cx + DU (5)where y is the output vector, U is the input vector, A isthe state matrix, B is the input matrix, C is the output

İ. Eski, A. Temürlenk

matrix, D is the feed forward matrix. We have

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 1 0 0 0 0−KcJc

−BcJc

KcJcrp

0 0 0

0 0 0 1 0 0Kc

Mrrp0 −(Kc+KmG2

Mrr2p+ Kt

Mr) −Br

Mr

KmGMrrp

0

0 0 0 0 0 1

0 0 KmGJmrp

0 −KmJm

−BmJm

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 01Jc

0

0 0

0 0

0 0

0 1Jm

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(7)

C =[0 −Km −KmGrp 0 Km 0

],

D = [0 0](8)

x =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

θc

θ̇c

θm

θ̇m

xr

ẋr

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

, y = Km(

θm − XrGrp

)= Ta (9)

U =[

Td

u

](10)

The geometric parameters of the active steering sys-tem are given in Table 1.

3 Control structures

In this study, four different control structures are usedto control the active steering system. These controlstructures are as PID Controller, Model-Based NeuralNetwork Controller, Neural Network Predictive Con-troller and Robust Neural Network Predictive ControlSystem. Also, these control structures are given in subsections.

3.1 Model Based Neural Network Controller(MBNNC)

MBNNC structure uses two neural networks: a con-troller network and a plant model network, as shown inFig. 2. The plant model network is a multi-layer neu-ral network with a back-propagation learning scheme.The controller is a recurrent learning multi-layer neu-ral network with “Plant Information” feedback alongwith other required feedback. The purpose of this con-troller is to provide the appropriate control action,given the current state of the system, in order to ob-tain a new state of the system such that the succes-sive application of the control law drives the systemtoward a desired final state without violating the con-straints imposed on the system and on the control vari-ables.

Identification of the plant can be achieved by ob-serving the input-output behavior of the plant. Theplant model is used to generate the model error. Thenthe controller network is trained in such a way thatthe system response tracks the desired response (inputto the model). Each of these networks has two layerswith delayed inputs and outputs. The number of de-lays is proportional to the order of the system. Themore the complexity of the system, the higher num-ber of delays required. There is also flexibility so as tochoose the number of hidden layers for each of these

Table 1 Active steeringsystem parameters Parameters Value Parameters Value

Bc 0.0225 N m s/rad Jm 4.52 × 10−4 kg m2Br 3920 N m s/rad Kc 172 N m/rad

Bm 3.34 × 10−3 N m s/rad Km 125 N m/radG 0.4686 Kt 23900 N m/rad

Jc 0.04 kg m2 Mr 32 kg

rp 0.0071 m

Design of neural network-based control systems for active steering system

Fig. 2 Model-based neuralnetwork controller

networks [17]. The MBNNC was designed to controlthe active steering system. It consisted of a third-orderlinear reference model and a neural network. The pro-posed neural controllers’ law for active steering sys-tem is given by

FN(t)

=5∑

j=1

((2

(1+ e−2[∑5

j=1[θa(t)w1j (t)+θc(t)w2j (t)]+1])−1

)

× wj1(t) + 1)

(11)

where FN(t) is the force of the NN controller, θa(t) isthe desired angle input of the steering system for neu-ral controller. w1j (t) is the weight matrices betweenfirst input layer neuron and in the hidden layer neu-rons. w2j (t) is the weight matrices between secondinput layer neuron and in the hidden layer neurons.θc(t) is the actual angle input of the steering systemfor neural controller.

3.2 Neural network predictive controller (NNPC)

There are typically two steps involved NNPC: systemidentification and control design. The system identifi-cation stage of NNPC is to train a neural network topresent the forward dynamics of the plant. Figure 3shows a schematic representation of the three layeredfeed forward neural network plant model. The predic-tion error between the plant output and the neural net-work output is used as the neural network training sig-nal. The process is represented by Fig. 3. The neural

network plant model uses previous plant inputs andprevious plant outputs to predict future values of theplant output [18]. In the control stage, the plant modelis used by the controller to predict future performance.The neural network model predicts the plant response.We have

FN(t) =5∑

j=1f

(θa(t + j) − θn(t + j)

)2

+ρ2∑

j=1g(F ′N(t + j − 1) − F ′N(t + j − 2)

)2

(12)

where f and g are activation functions of the hid-den layer and the output layer, respectively, as fol-lows:

f (t) = tan sig(t) = 2(1 + e−2t ) − 1, g(t) = t

The F ′N(t) variable is the tentative control signal,θa(t) is the desired output, θn(t) is the neural networkmodel output. The ρ value determines the contributionthat the sum of the squares of the control incrementsprovides for the performance index. The optimizationblock determines the control input that optimizes plantperformance over a finite time horizon.

3.3 Robust neural feedback control system (RNFCS)

A designed control system is employed to control theactive steering system. The purpose of this proposed

İ. Eski, A. Temürlenk

Fig. 3 Neural networkpredictive controller

control system is to provide the appropriate control ac-tion. The mathematical expression of the force of theRNF control system is given by

F(t) =5∑

j=1f

(θa(t + j) − θn(t + j)

)2

+ ρ2∑

j=1g(F ′N(t + j − 1) − F ′N(t + j − 2)

)2

+ σe−βt (13)where σ and β are the robust controller parame-ters and are empirically set to σ = 100 and β =0.0001. Schematic representation the proposed neuralbased control system model is shown in Fig. 4. TheLevenberg–Marquardt algorithm is used to adjust theweights of neural network.

3.4 Resilient backpropagation algorithm (RPROP)

The Resilient Backpropagation algorithm is a lo-cal adaptive learning scheme, performing supervisedbatch learning in feed-forward neural networks [19].

The basic principle of this algorithm is to eliminate theharmful influence of the size of the partial derivative’ssize on the weight step. As a consequence, only thesign of the derivative is considered to indicate the di-rection of the weight update. This algorithm typicallyuses a sigmoid function in the hidden layer and a lin-ear function in the output layer. Here, wij is the weightmatrice, �ij (t) is the update value for each weight. Asecond learning rule is introduced which determinesthe evolution of the update value �ij (t). This estima-tion is based on the observed behavior of the partialderivative during two successive weight-steps:

�ij (t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

λ+�ij (t − 1), if ∂E∂wij (t) ∂E∂wij (t − 1) > 0λ−�ij (t − 1), if ∂E∂wij (t) ∂E∂wij (t − 1) < 0�ij (t − 1), else

(14)

where

0 < λ− < 1 < λ+. (15)

Design of neural network-based control systems for active steering system

Fig. 4 Robust neuralnetwork predictive controlsystem

The adaptation rule works in the following way.Every time the partial derivative of the correspondingweight wij changes its sign, which indicates that thelast update was too big and the algorithm has jumpedover a local minimum, the update value �ij (t) is de-creased by the factor λ−. If the derivative retains itssign, the update value is slightly increased in order toaccelerate convergence in shallow regions. Once theupdate value for each weight is adapted, the weight-update itself follows a very simple rule: if the deriva-tive is positive (increasing error), the weight is de-creased by its update-value, if the derivative is nega-tive, the update value is added:

�wij (t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−�ij (t), if ∂E∂wij (t) > 0�ij (t), if ∂E∂wij (t) < 0

0, else

(16)

wij (t + 1) = wij (t) + �wij (t) (17)

However, there is one exception. If the partialderivative changes sign, i.e. the previous step is toolarge and the minimum is missed, the previous weight-update is reverted:

�wij (t) = −�wij (t − 1),

if∂E

∂wij(t)

∂E

∂wij(t − 1) < 0 (18)

Due to that ‘backtracking’ weight-step, the deriva-tive is supposed to change its sign once again in thefollowing step. In order to avoid a double punishmentof the updatevalue, there should be no adaptation ofthe update-value in the succeeding step. In practicethis can be done by setting ∂E

∂wij(t − 1) = 0 in the �ij

update rule above. The partial derivative of the totalerror is given by

∂E

∂wij(t) = 1

2

P∑p=1

∂Ep

∂wij(t) (19)

Hence, the partial derivatives of the errors must beaccumulated for all P training patterns. This meansthat the weights are updated only after the presenta-tion of all training patterns. α (weight-decay) param-eter determines the relationship of two goals, namelyto reduce the output error (the standard goal) and toreduce the size of weights (to improve generalization).The composite error function is

E = 12

P∑p=1

NO∑j=1

(dpj − epj )2 + 110α

∑i,j

w2ij (20)

Moreover, for comparison purposes, the classicalPID controller was used for trajectory control activesteering system. The PID controller was initially tunedusing the Ziegler-Nichols method, and the PID param-eters are KP = 60, KI = 7200 and KD = 1.

İ. Eski, A. Temürlenk

4 Simulation results

This section presents simulation result of the activesteering system for random input signals using the PIDcontroller, the MBNNC, the NNPC and the RNNPCapproaches. The first structure used in the control ofactive steering system is the PID controller. The simu-lation results and the values of error obtained from therandom input signal of this organ have been presentedin Fig. 5. As is seen in the figure, the PID controlleris unable to adapt itself to random input signal, whichchanges suddenly, but it can follow the input signalafter a certain period. In this case the PID controller,as such it is, is hardly adequate to run active steeringsystem (Figs. 5b and 5d).

The second structure employed to control the sys-tem, however, is the MBNNC approach. The study ofFigs. 6a–6d will reveal that the values of error at thepoints of sudden change resulting from the character-istic reference input signal is higher than that in thePID controller. In addition, the MBNNCA structurehas yielded the poorest results among the other neu-ral network structure employed in the control of theactive steering system. The reason for this is that thereference model error used in the MBNNCA approachis high and that the learning algorithm used in arrang-ing the weights of the control structure to reduce thiserror has remained inadequate.

The third structure employed in simulation is theNNPC structure. The results and error values for two

Fig. 5 Variations of the steering column angle using the PID Controller. (a) Random1 input signal. (b) Error of the PID Controller.(c) Random2 input signal. (d) Error of the PID Controller

Design of neural network-based control systems for active steering system

Fig. 6 Variations of the steering column angle using the MBNN Controller. (a) Random1 input signal. (b) Error of the MBNNController. (c) Random2 input signal. (d) Error of the MBNN Controller

different random input signals have been presented inFig. 7. The analysis of the graphics has revealed thatthe NNPC approach has yielded far more favorable re-sults compared to other three controllers. The NNPCapproach adapts itself to sudden changes of randominput signal far more favorably and it has been ob-served that there have been considerable reductions insteady state errors. The reason for this superior perfor-mance is that the neural network model uses optimiza-tion algorithm while being constituted. The differencebetween the output signal of neural network model andactive steering system, i.e. error is used for the adjust-ment of the parameters of optimization algorithm.

In addition, the values of error between the outputsignal of neural network model and the output signal

of the controlled system in the system identificationsection of this controller are used for the adjustmentof the weights in the learning algorithm of the NNPC.Therefore, the current NNPC approach to the controlof active steering system is more appropriate than, butinadequate compared to, other control structures. Forthis reason, a controller with a robust constitution ca-pable of both being adapted to sudden changes of therandom input signal and of eliminating the steady stateerrors constantly has been added to the NNPC struc-ture to constitute the neural network-based robust con-trol system.

The response of developed RNNP to random in-put signal and its error values have been given inFig. 8. The values of error at sudden change points for

İ. Eski, A. Temürlenk

Fig. 7 Variations of the steering column angle using the NNP Controller. (a) Random1 input signal. (b) Error of the NNP Controller.(c) Random2 input signal. (d) Error of the NNP Controller

two different random input signals are considerablylow as compared to those for the other three controlstructures. Moreover, the system does not have steadystate errors. Since the robust structure of the proposedcontrol system constantly reduces the errors exponen-tially, it has yielded more favorable results in a shorttime than the controllers. For all these reasons, the sug-gested neural network-based robust control system isthe most appropriate approach for the control of activesteering system.

5 Conclusions

In this study, the control of the active steering system isused in four different control structures. Except for the

classical PID controller, neural network is based on allthe other control structures. The reason for preferringthe neural network approach for controlling the sys-tem is its ability to learn, their high-speed performanceowing to their parallel structures, their non-linearity,and their ability to generalize. Moreover, for compari-son purposes, the classical PID controller was used forrandom trajectory control active steering system. It hasbeen observed that the PID controller is not suitablefor the control of such systems since it cannot adaptitself to sudden changes in random input signals be-cause of their control parameters being constant. Also,in the current study, in order to test performance of thecontrollers, two different random signals are used asinput signal. Within used neural network-based con-

Design of neural network-based control systems for active steering system

Fig. 8 Variations of the steering column angle using the RNNP Control System. (a) Random1 input signal. (b) Error of the RNNPControl System. (c) Random2 input signal. (d) Error of the RNNP Control System

troller, the NNPC approach has given the best result.So, this neural network structure is used for the pro-posed robust control system. From the evaluation ofthe obtained simulation results, the proposed neuralbased control system is suitable for the control of suchsystems. In addition, the developed neural based ro-bust control system, which can be applied to any vehi-cle system, will contribute to the research in this fieldin automotive manufacturing sector.

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Design of neural network-based control systems for active steering systemAbstractIntroductionModel of active steering systemControl structuresModel Based Neural Network Controller (MBNNC)Neural network predictive controller (NNPC)Robust neural feedback control system (RNFCS)Resilient backpropagation algorithm (RPROP)

Simulation resultsConclusionsReferences