Control of chaotic vibration in automotive wiper systems

Chaos, Solitons and Fractals 39 (2009) 168–181 www.elsevier.com/locate/chaos

Control of chaotic vibration in automotive wiper systems Zheng Wang *, K.T. Chau Department of Electrical and Electronic Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China Accepted 2 January 2007

Abstract Chaotic vibration has been identified in the automotive wiper system at certain wiping speeds. This irregular vibration not only decreases the wiping efficiency, but also degrades the driving comfort. The purpose of this paper is to propose a new approach to stabilize the chaotic vibration in the wiper system. The key is to employ the extended time-delay feedback control in such a way that the applied voltage of the wiper motor is online adjusted according to its armature current feedback. Based on a practical wiper system, it is verified that the proposed approach can successfully stabilize the chaotic vibration, and provide a wide range of wiping speeds.  2009 Published by Elsevier Ltd.

1. Introduction It has been identified that chaotic behaviors occur in mechanical systems with dry friction or rattling vibration [1,2]. As an important device in automobiles, the wiper system also suffers from chaotic vibration. Some researchers have experimentally detected chaotic or chatter vibration in the wiper system at certain wiping speeds [3]. This irregular vibration not only decreases the wiping efficiency, but also degrades the driving comfort. Also, the disturbance creates a safety hazard. In order to suppress this chaotic vibration, the linear state feedback control method has been proposed for stabilization [4]. However, this control method requires online measurement of the angular speeds of the wiper arms, which is impractical for realization. The purpose of this paper is to propose a new approach to stabilize the chaotic vibration in the automotive wiper system. The proposed method will offer two distinct merits, namely the high practicality and the high effectiveness. The former one can enable practical implementation based on a reasonable cost, whereas the latter one can ensure effective stabilization at various conditions. In order to achieve high practicality, the feedback parameters should be easily measurable. Thus, the electrical parameters such as the voltage and current of the wiper motor are preferred to the mechanical parameters such as the angular deflection and angular speed of the wiper arms. Since the armature current is directly proportional to the generated torque of the wiper motor, which is actually a permanent magnet DC (PMDC) motor, it is selected as the measurable feedback parameter to stabilize the chaotic vibration in the wiper system. In fact, the armature current has played an important role in identification and control of chaotic motion in DC drive systems [5–7].

*

Corresponding author.

0960-0779/$ - see front matter  2009 Published by Elsevier Ltd. doi:10.1016/j.chaos.2007.01.118

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In order to achieve high effectiveness, the control method should not involve complicated computation for implementation or impose unnecessary perturbation onto the system dynamics. Many methods have been proposed to stabilize chaos, such as the OGY method [8,9], the time-delay feedback method [10], the non-feedback method [11,12], the proportional feedback method [13,14], the nonlinear control method [15,16], the adaptive control method [17,18], the neutral networks method [19,20] and the fuzzy control method [21]. Among them, the time-delay feedback method takes some definite advantages: it does not desire priori analytical knowledge of the system dynamics; it does not require the reference signal corresponding to the desired unstable periodic orbit (UPO); and it does not need fast sampling or computer analysis of the state of the system. Also, the corresponding perturbation is small when the delayed time is close to the period of the desired UPO [22]. The extended time-delay auto-synchronization (ETDAS) method is an attractive extended version of the time-delay feedback method, since it can significantly extend the parametric domain of effective control [23,24]. Thus, it is anticipated that the ETDAS method can be utilized to stabilize chaotic vibration of wipers over a wide range of wiping speeds. In Section 2, the dynamic model of the automotive wiper system will be formulated. In Section 3, the corresponding chaotic vibration will be analyzed. Then, the use of ETDAS to control the chaotic vibration will be discussed and implemented in Section 4. In Section 5, the stabilization of chaotic vibration at different wiping speeds will be given. Finally, conclusions will be drawn in Section 6. 2. Dynamic model An automotive wiper system is composed of three main parts: an electric motor, two wipers, and a mechanical linkage. Each wiper consists of an arm and a blade, which are moveable. As shown in Fig. 1, the electric motor provides the torque for the mechanical linkage which in turn generates the desired motion for the wiper arms and blades on the driver’s side and passenger’s side. In order to analyze the chaotic motion of the above automotive wiper system, the corresponding dynamic model needs to be formulated. Some researchers utilized the Langrange equations to model the mechanical linkage, and adopted the Hamilton principle to simulate the motion of the wipers [25]. In order to reduce the modeling complexity, the mechanical linkage was described by stiffness and damping, and the frictional force on the wiper blades was approximated by a cubic polynomial [3]. Based on these two works, the dynamic model of the automotive wiper system used in this paper is given by: 8 ð1Þ ð1Þ > < Ri  Di  M i ðni Þ ðni –0Þ ð2Þ ð1Þ ð1Þ I i ðli hi  nli x Þ ¼ Ri  Di  M i ðni Þ ðnð1Þ ð1Þ i ¼ 0; jRi j P N i li l0 Þ > : ð1Þ 0 ðni ¼ 0; jRi j < N i li l0 Þ J m xð1Þ ¼ K T ia  Bx  nRM

ð2Þ

La ið1Þ a ¼ V in  K E x  Ra ia ð1Þ ni

ð1Þ ðhi

¼

ð1Þ lðni Þ

¼

 nxÞli ;

ð1Þ l0 sgnðni Þ

ð1Þ l1 ni

þ

RD ¼ k D hD  k DP hP ; DD ¼

ð3Þ

ð1Þ M i ðni Þ

ð1Þ cD hD



ð1Þ cDP hP ;

¼

þ

ð1Þ N i li lðni Þ

RP ¼ k P hP  k PD hD ; DP ¼

ð4Þ

ð1Þ l2 ðni Þ3

ð1Þ cP hP



ð5Þ RM ¼ k MD hD þ k MP hP

ð1Þ cPD hD

Fig. 1. Structure of automotive wiper system.

ð6Þ ð7Þ

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K D ðK P þ K M Þ K P ðK D þ K M Þ K DK P ; kP ¼ ; k PD ¼ k DP ¼ KD þ KP þ KM KD þ KP þ KM KD þ KP þ KM KMKD KMKP k MD ¼ ; k MP ¼ KD þ KP þ KM KD þ KP þ KM cD ¼ C DP ; cP ¼ C DP þ C P ; cDP ¼ cPD ¼ C DP

kD ¼

ð8Þ ð9Þ ð10Þ

where i = D,P indicate the driver’s side and the passenger’s side, respectively; Ii are the moments of inertia of the wiper arms; li are the lengths of the wiper arms; hi are the angular deflections of the wiper arms; n is the speed reduction ratio between the mechanical linkage and the motor; x is the motor speed; Ri are the torques produced by the elastic forces; Di are the torques produced by the damping forces; Mi are the torques produced by the frictional forces between the ð1Þ wiper blades and the screen; Ni are the forces pressed down on the wiper arms by the springs; ni are the relative speeds of the wiper blades with respect to the screen; Vin is the input voltage of the motor; ia is the armature current of the motor; KT is the torque constant of the motor; KE is the back EMF constant of the motor; B is the viscous damping of the motor; Ra is the armature resistance of the motor; La is the armature inductance of the motor; Jm is the moment

LD θD

LP ID

IP

θP

MD

MP

CP

CDP KD

+ ea −

KP KM B

J

ia

La Ra

+ V in

−

Fig. 2. Equivalent model of automotive wiper system.

Table 1 System parameters ID IP lD lP ND NP l0 l1 l2 KD KP KM CP CDP n KT KE B Jm Ra La

4.07 · 102 3.67 · 102 4.70 · 101 4.50 · 101 7.35 5.98 1.18 9.84 · 101 4.74 · 101 7.20 · 102 7.51 · 102 3.53 · 102 1.00 · 102 1.00 · 102 1.59 · 102 1.36 · 101 1.36 · 101 1.91 · 105 2.30 · 105 9.00 · 101 3.00 · 103

kg m2 kg m2 m m N N

Nm/rad Nm/rad Nm/rad Nm/rad s1 Nm/rad s1 Nm/A V/rad s1 Nm/rad s1 Nm/rad s2 X H

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of inertia of the motor; RM is the restoring torque of the motor; l0, l1 and l2 are the coefficients of dry frictions between the wiper blades and the screen; KD, KP and KM are the stiffness coefficients of the mechanical linkage; CP and CDP are the damping coefficients of the mechanical linkage. The equivalent circuit diagram of this dynamic model is shown in Fig. 2. It should be noted that the parameters of the wipers are based on a practical automotive wiper system [3], and the parameters of the motor are based on the Sanyo Denki M818T031 PMDC motor. All these parameters are listed in Table 1.

3. Chaotic analysis Since Vin is generally used to perform speed control of the PMDC motor, it is selected as the bifurcation parameter for chaotic analysis. Based on the dynamic model described by (1)–(10), the bifurcation diagrams of hD and hP with respect to Vin are plotted as shown in Fig. 3. It can be seen that the system exhibits period-one motion in the regions IV and VI, whereas unstable vibration in the regions I, II, III, and V. These phenomena actually explain why the existð1Þ ing automotive wiper has predefined several wiping speeds. Moreover, the trajectories of hD versus hD at six typical values of Vin (corresponding to the six operating regions) are plotted as shown in Fig. 4. It can be found that the trajectories with Vin = 3.5 V and Vin = 10.8 V exhibit sub-harmonic motion in the regions II and V, respectively; whereas the trajectories with Vin = 2 V and Vin = 4 V exhibit chaotic behavior in the regions I and region III, respectively. Hence, it confirms that chaotic vibration of the wipers occurs at certain range of Vin or wiping speeds. To mathematically prove the existence of chaos, the maximum Lyapunov exponent kmax needs to be computed. The solution flow of the system state variables is expressed as: X ðtÞ ¼ T t X 0

ð11Þ

t

where T is the map describing the time-t evolution of X, and the solution flow of their deviation dX is given by: dX ðtÞ ¼ U tX 0 dX 0

Fig. 3. Bifurcation diagrams of hi with respect to Vin : (a) Driver’s side and (b) passenger’s side.

ð12Þ

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ð1Þ

Fig. 4. Trajectories of hD versus hD: (a) Vin = 2 V; (b) Vin = 3.5 V; (c) Vin = 4 V; (d) Vin = 8 V; (e) Vin = 10.8 V and (f) Vin = 12 V.

where U tX 0 is the map describing the time-t evolution of dX. Then, the Lyapunov exponents ki of the d-dimension system can be computed as    Dt j  h1 U X j ei  1 X ki ði ¼ 1  dÞ ¼ lim ð13Þ log  j  ei  h!1 hDt j¼0 where Dt is the evolution time, and eji is the ith base vector of the d-dimension state space at thejthstep. It should be noted that eji should be orthogonalized and normalized at each iterative step. When Dt  1 and eji   1, (13) can be approximated as [26,27]  Dt  h1 T ðX j þ eji Þ  T Dt ðX j Þ 1 X  j ki ði ¼ 1  dÞ ¼ lim log ð14Þ ei  h!1 hDt j¼0

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Fig. 5. Waveforms of hD: (a) Vin = 2 V; (b) Vin = 3.5 V; (c) Vin = 4 V; (d) Vin = 8 V; (e) Vin = 10.8 V and (f) Vin = 12 V.

Hence, the maximum value of ki can be calculated, namely kmax = 1.671 when Vin = 2 V and kmax = 0.346 when Vin = 4 V. The corresponding positive values mathematically prove that there are chaotic vibrations in the regions I and III. Moreover, the waveforms of hD are shown in Fig. 5 in which the periodic motion exhibits regular vibration whereas the chaotic motion exhibits irregular vibration. As aforementioned, this irregular vibration will be harmful to safety ð1Þ ð1Þ driving. By sampling at a fixed value of hP , the corresponding Poincare´ maps with respect to the hD  hD plate are plotted as shown in Fig. 6. It can be seen that the period-one motion exhibits a single point as shown in Fig. 6d and f; the sub-harmonic motion shows several points as shown in Fig. 6b and e; and the chaotic motion is depicted by irregularly distributed points as shown in Fig. 6a and c. 4. Control method In order to effectively stabilize the chaotic vibration in the automotive wiper system, the ETDAS method is adopted. The principle of control can be expressed as:

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ð1Þ Fig. 6. Poincare´ maps of hD versus hD: (a) Vin = 2 V; (b) Vin = 3.5 V; (c) Vin = 4 V; (d) Vin = 8 V; (e) Vin = 10.8 V and (f) Vin = 12 V.

y ð1Þ ¼ P ðy; xÞ þ F ðy; xÞ; xð1Þ ¼ Qðy; xÞ " # 1 X q1 R yðt  qsÞ  yðtÞ F ðy; xÞ ¼ K ð1  RÞ

ð15Þ ð16Þ

q¼1

where x and y are the state variables of the system, F ðy; xÞ is the perturbation s is the delayed time, K is the feedP item, q1 yðt  qs þ sÞ and r ¼ q  1, (16) can back gain, and R 2 ½0; 1Þ is the regressive parameter. By defining SðtÞ ¼ 1 q¼1 R be written as: F ðtÞ ¼ K½ð1  RÞSðt  sÞ  yðtÞ

ð17Þ

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where S(t) is given by: 1 1 1 X X X Rr yðt  rsÞ ¼ Rr yðt  rsÞ þ yðtÞ ¼ R Rr1 yðt  rsÞ þ yðtÞ SðtÞ ¼ r¼0

r¼1

ð18Þ

r¼1

After taking c ¼ r  1, (18) can be rewritten as: 1 X Rc yðt  cs  sÞ þ yðtÞ ¼ RSðt  sÞ þ yðtÞ SðtÞ ¼ R

ð19Þ

c¼0

In order to practically implement the ETDAS method, an easily measurable electrical parameter of the PMDC motor, namely the armature current ia, is used as the feedback control parameter. Thus, ia is chosen as y in (15)– ð1Þ ð1Þ (19), whereas x in (15)–(19) represents the other state variables: hD, hD , hP, hP , and x. Then, by incorporating (16) into (3), the dynamic equation can be obtained as: " # 1 X 1 1 ð1Þ q1 ia ¼ ðV in  K E x  Ra ia Þ þ F ðtÞ ¼ ðV in  K E x  Ra ia Þ þ K ð1  RÞ R ia ðt  qsÞ  ia ðtÞ ð20Þ La La q¼1 Therefore, the input voltage reference V in of the PMDC motor is modulated by the voltage perturbation DV, which are expressed as: V in ¼ V in þ DV " DV ¼ La K ð1  RÞ

1 X

ð21Þ

# Rq1 ia ðt  qsÞ  ia ðtÞ

ð22Þ

q¼1

The corresponding control block diagram is shown in Fig. 7. Hence, the whole control system can readily be implemented by a pulse-width-modulation (PWM) DC–DC converter as shown in Fig. 8, in which vc is the control signal resulting from the difference between V in and the instantaneous input voltage vin, and vst is the sawtooth signal for PWM generation.

Vin∗

+ +

ΔV

F (t ) La

Vin

K

+

+ +

Time delay

1 −R

−

R

Fig. 7. Control block diagram of ETDAS.

MOSFET

+ PMDC motor

−

Pulse

− +

vst vc

− +

vin Vin∗

ETDAS

Fig. 8. PWM DC–DC converter with ETDAS control.

ia

ia

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The key to realize (22) is to determine proper values of R, s and K. Firstly, since the value of R affects the range of K for effective stabilization [24], R is chosen as 0.85 so that the range of K is wide enough to stabilize the system at different wiping speeds. Secondly, by analyzing the time series of ia, the periods of the fundamental and sub-harmonic orbits embedded in the system can be determined [28]. Hence, s is selected as 0.0339s which is equal to the period of the fundamental orbit at a particular value of Vin [29]. Thirdly, the bifurcation diagrams of hD with respect to K at six representative values of Vin are plotted as shown in Fig. 9. It can be seen that when K is chosen as 2000, the period-one motion can be attained throughout all regions. After substituting R = 0.85 s = 0.0339 s and K = 2000 into (21) and (22), the waveforms of ia and V in of the proposed ETDAS control method are simulated as shown in Fig. 10. It can be seen that the corresponding ripples are very small, hence verifying that the proposed method imposes only very small perturbation onto the system.

Fig. 9. Bifurcation diagrams of hD with respect to K: (a) Vin = 2 V; (b) Vin = 3.5 V; (c) Vin = 4 V; (d) Vin = 8 V; (e) Vin = 10.8 V and (f) Vin = 12 V.

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Fig. 10. Waveforms of ia and V in with ETDAS control: (a) Vin = 2 V; (b) Vin = 3.5 V; (c) Vin = 4 V; (d) Vin = 8 V; (e) Vin = 10.8 V and (f) Vin = 12 V.

5. Results ð1Þ

Firstly, the trajectory of the system on the hD  hD plate is assessed when Vin = 2 V, which corresponds to the region I. Compared with the chaotic trajectory as shown in Fig. 4a, the stable trajectory as shown in Fig. 11a verifies that the system can be successfully stabilized into a period-one motion with ETDAS control. Fig. 12a shows the corresponding transient response of hD with ETDAS control applied at t = 7 s. It can be seen that the chaotic waveform of hD as shown in Fig. 5a can be quickly stabilized. Secondly, when Vin = 3.5 V which corresponds to the region II, the sub-harmonic period-two motion as depicted by Fig. 4 and 5b can be stabilized into a period-one motion with ETDAS control as illustrated by Figs. 11 and 12b. Thirdly, similar to the case in the region I, when Vin = 4 V which corresponds to the region III, the chaotic motion as depicted by Fig. 4 and 5c can be successfully stabilized into a period-one motion with ETDAS control as illustrated by Figs. 11 and 12c. Fourthly, when Vin = 8 V which corresponds to the region IV, the originally stable period-one motion as depicted by Figs. 4 and 5d does not have any changes with the use of ETDAS control as illustrated by Figs. 11 and 12d.

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ð1Þ

Fig. 11. Trajectories of hD versus hD with ETDAS control: (a) Vin = 2 V; (b) Vin = 3.5 V; (c) Vin = 4 V; (d) Vin = 8 V; (e) Vin = 10.8 V and (f) Vin = 12 V.

Fifthly, similar to the case in the region II, when Vin = 10.8 V which corresponds to the region V, the sub-harmonic period-five motion as depicted by Figs. 4 and 5e can be stabilized into a period-one motion with ETDAS control as illustrated by Figs. 11 and 12e. Finally, when Vin = 12 V which corresponds to the region VI, the originally stable period-one motion as depicted by Figs. 4 and 5f can be further improved with the use of ETDAS control as illustrated by Figs. 11 and 12f, hence achieving a smaller amplitude of vibration. It should be noted that the originally stable period-one motion in the region IV has no further improvement in the amplitude of vibration with the use of ETDAS control, whereas the one in the region VI can be further improved by using ETDAS control. It is due to the fact that the one in the region IV has already provided the minimum amplitude of vibration even without ETDAS control. To assess the effectiveness at a glance, Fig. 13 shows the bifurcation diagrams of hD and hP with respect to Vin with ETDAS control. Compared with the bifurcation diagrams without control as shown in Fig. 3, it is obvious that the

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Fig. 12. Transient responses of hD with ETDAS control: (a) Vin = 2 V; (b) Vin = 3.5 V; (c) Vin = 4 V; (d) Vin = 8 V; (e) Vin = 10.8 V and (f) Vin = 12 V.

proposed control method not only effectively stabilizes the chaotic and sub-harmonic motions, but also minimizes the amplitude of vibrations of the wipers throughout the whole operating range.

6. Conclusions In this paper, a new approach has been proposed to stabilize the chaotic vibration in the automotive wiper system. The key is to employ the ETDAS control method in such a way that the applied voltage of the PMDC motor is online adjusted according to its armature current feedback. The proposed approach takes the definite advantages of high practicality for implementation and high effectiveness for stabilization. Based on a practical wiper system, it is verified that it not only effectively stabilizes the chaotic and sub-harmonic motions, but also minimizes the amplitude of vibrations of the wipers throughout the whole operating range.

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Fig. 13. Bifurcation diagrams of hi with respect to Vinwith ETDAS control: (a) Driver’s side and (b) passenger’s side.

Acknowledgement The work was supported and funded by a grant (HKU7154/04E) from the Research Grants Council, Hong Kong Special Administrative Region, China.

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