Chapter 8

Application of Chaos Stabilization

The stabilization of chaos in electric drive systems has been investigated in Chapter 6. This can regulate the chaotic motion in such a way that any harmful vibration can be avoided. It has also been discovered that chaotic behaviors exist in many applications. In particular, some electromechanical systems exhibit chaotic behaviors due to their special configurations or operating characteristics. Such chaotic behaviors not only deflect the system performance but also harm the safety of human beings. Because of the high controllability and flexibility of electric drives, they offer the ability to stabilize those undesirable chaotic behaviors.

In this chapter, the stabilization of chaos in three different applications – namely the wiper system in automobiles, the centrifugal governor system for internal combustion engines, and the rate gyro system in space vehicles – are investigated. The corresponding modeling, analysis, and stabilization are discussed.

8.1 Chaos Stabilization in Automotive Wiper Systems

Various harmful vibrations have been identified in automotive wiper systems (Grenouillat and Leblanc, 2002). As there is a trend for cars to become increasingly quieter, these vibrations not only decrease the wiping efficiency, but also degrade the driving comfort. Also, the disturbance creates a safety hazard. The vibrations are classified into three main groups: squeal noise with a frequency above 1 kHz, reversal noise with a frequency of approximately 500 Hz, and chattering with a frequency below 100 Hz (Goto, Takahashi, and Oya, 2001). Chattering is mainly caused by the stick-slip motion of the rubber blades which leave a stripped pattern on the windshield. There exists a critical wiping speed beyond which chattering will disappear. Actually, chattering is a common phenomenon of mechatronic systems with stick-slip friction (Owen and Croft, 2003). Also, chaotic behaviors are detected under certain wiping speeds in the chattering region (Suzuki and Yasuda, 1998).

Many control methods, including feedforward schemes and feedback schemes, have been put forward to suppress vibrations in various mechatronic systems (Park et al., 2006). For automotive wiper systems, an attempt has been made to reduce chattering by adjusting the attack angle between the blades and the windshield (Grenouillat and Leblanc, 2002). However, the control of attack angles is impractical and may not be implemented. Recently, a linear state feedback control method has also been proposed to stabilize chattering (Chang and Lin, 2004), but this control method requires the online measurement of the angular deflection of the wiper arms, which again is impractical to realize.

Electric motors are considered to be the heart of many mechatronic systems (Straete et al., 1998). Rather than using mechanical means which involve the design of wipers and feedback of wiper motion, the use of electrical means can directly stabilize the driving device – that is, the electric motor – in such a way that the chaotic chattering in wipers can be suppressed. The key is to apply a proper control method to regulate the electric motor based on an electrical parameter feedback rather than a mechanical parameter feedback. Hence, it offers two distinct advantages: high practicality and high effectiveness. The former can enable practical implementation based on a reasonable cost, while the latter can ensure effective stabilization in various conditions.

In order to achieve high practicality, the feedback parameters should be easily measurable. Thus, electrical parameters, such as the voltage and current of the wiper motor, are preferred to 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 chaotic vibration in the wiper system.

8.1.1 Modeling

An automotive wiper system is composed of three main parts: an electric motor, a mechanical linkage assembly, and two wipers (one on the driver's side and one on the passenger's side). Each wiper consists of an arm and a rubber blade. As shown in Figure 8.1, the electric motor provides the torque for the mechanical linkage which, in turn, generates the desired motion for the wiper arms and blades. It has been identified that the mechanical linkage between the two wipers can be described by stiffness and damping, and the frictional force on the windshield can be approximated by a cubic polynomial (Suzuki and Yasuda, 1998). Also, the electric motor in this wiper system can be represented by a second-order dynamical equation (Hsu and Ling, 1990). Moreover, the linkage between the wipers and the motor can be described by stiffness (Lévine, 2004). Based on these three works, the dynamical model of the whole wiper system can be formulated as:

(8.1)

(8.2)

(8.3)

(8.4)

where indicate the driver's side and the passenger's side respectively, are the moments of inertia of the wiper arms, are the lengths of the wiper arms, are the angular deflections of the wiper arms with respect to their positions when no deflections occur, is the speed reduction ratio between the mechanical linkage and the motor, is the motor speed, are the relative velocities of the wiper blades with respect to the screen; with respect to the motor, is the input voltage, is the armature current, is the back EMF constant, is the torque constant, is the viscous damping, is the armature resistance, is the armature inductance, and is the moment of inertia. The corresponding equivalent model is shown in Figure 8.2.

By defining as the torques produced by the elastic forces, as the torques produced by the damping forces, as the restoring torque produced by the motor, and as the torques produced by the frictional forces between the wiper blades and screen, they can be expressed as:

(8.5)

(8.6)

(8.7)

(8.8)

(8.9)

(8.10)

(8.11)

where and are the self-stiffness of the wipers, which are represented by

respectively; , , and are the mutual stiffness among the wipers and the motor which are described as

respectively; and are the self-damping of the wipers, which are represented as and , respectively; is the mutual damping between the wipers, which equals ; , , and are the stiffness coefficients on the driver's side, the passenger's side and the motor's side, respectively; and are the damping coefficients on the passenger's side and the mutual damping coefficients between the driver's side and the passenger's side, respectively; are the forces pressed on the screen by the blades; is the coefficient of dry friction between the wiper blades and the screen; and , , and are constants.

8.1.2 Analysis

As listed in Table 8.1, the parameters of a practical automotive wiper system are adopted for exemplification. Since the electric motor is a PMDC motor, the wiping behaviors under different are analyzed. By using the mathematical model described by (8.1) –(8.11), the bifurcation diagram of versus is shown in Figure 8.3. The corresponding points represent the locally maximum and minimum values of at each . It can be seen that the bifurcation diagram is divided into five regions. In region V, where V_in > 14.5 V, is a fixed value so that chattering does not occur. On the other hand, when V_in ≤ 14.5 V, oscillates between the maximum and minimum values, indicating that chattering occurs.

Table 8.1 Parameters of an automotive wiper system.

Inertia of the wiper arm at driver's side ID	1.91 × 10−2 kgm2
Inertia of the wiper arm at passenger's side IP	1.65 × 10−2 kgm2
Length of the wiper arm at driver's side lD	4.70 × 10−1 m
Length of the wiper arm at passenger's side lP	4.50 × 10−1 m
Force pressed by blade at driver's side ND	7.35 N
Force pressed by blade at passenger's side NP	5.98 N
Dry friction coefficient component μ0	1.18
Dry friction coefficient component μ1	−9.84 × 10−1
Dry friction coefficient component μ2	4.74 × 10−1
Stiffness coefficient on driver's side KD	7.20 × 102 Nm/rad
Stiffness coefficient on passenger's side KP	7.51 × 102 Nm/rad
Stiffness coefficient on motor's side KM	3.53 × 102 Nm/rad
Damping coefficient on passenger's side CP	1.00 × 10−2 Nm/rad/s
Mutual damping coefficient CDP	1.00 × 10−2 Nm/rad/s
Speed reduction ratio n	1.59 × 10−2
Torque constant of motor KT	1.36 × 10−1 Nm/A
Back EMF constant of motor KE	1.36 × 10−1 V/rad/s
Viscous damping of motor B	1.91 × 10−5 Nm/rad/s
Inertia of motor Jm	2.30 × 10−5 kgm2
Armature resistance of motor Ra	9.00 × 10−1 Ω
Armature inductance of motor La	3.00 × 10−3 H

As shown in Figure 8.3, the chattering region can be further divided into regions I to IV. For each region, a typical is chosen to observe the motion. Namely, Figure 8.4 plots the trajectories of versus under V_in = 4 V, V_in = 8 V, V_in = 10.8 V, and V_in = 12 V. It can be seen that the trajectories display period-1 vibration under V_in = 8 V in region II and under V_in = 12 V in region IV; the trajectory shows subharmonic motion under V_in = 10.8 V in region III; whereas the trajectory exhibits an irregular but bounded behavior under V_in = 4 V in region I. Although an irregular but bounded behavior is an important property of chaos, it is not mathematically sufficient to confirm that the chattering in region I is chaotic. Therefore, the maximum Lyapunov exponent needs to be calculated. When V_in = 4 V, is computed to be 0.346. A positive value of mathematically confirms that there is chaotic chattering in region I.

8.1.3 Stabilization

A time-delay feedback control enables the stabilization of chaos without a prior analytical knowledge of the system dynamics or the desired reference signals (Chen et al., 2000). Since the current of the motor can readily be measured, the delayed feedback of current is used to derive for the motor. The corresponding control strategy is given by:

(8.12)

(8.13)

where is the time delay and is the feedback gain. It has been proved that chaos can be stabilized into a newly produced periodic orbit even if does not equal the period of the embedded unstable periodic orbit (UPO) and the control signal does not become zero (Franceschini, Bose and Schöll, 1999). For the sake of simplicity, is chosen to be 0.033 s, which is the period of wiper motion under V_in = 8 V and is also less than the period of wiper motion under V_in = 12 V. Consequently, the bifurcation diagram of locally maximum and minimum values of against under V_in = 12 V is plotted as shown in Figure 8.5. It can be seen that chattering can be suppressed within the range of K∈[1.5 V/A, 10.2 V/A]. However, such range of K is relatively narrow and is insufficient to stabilize chattering over the whole range of V_in.

The extended time-delay autosynchronization (ETDAS) control incorporates the advantages of time-delay feedback control while offering a wide operating range of (Pyragas, 1995). The corresponding control strategy is given by:

(8.14)

(8.15)

where is a regressive parameter. By defining and , this yields:

(8.16)

After taking , (8.16) can be rewritten as:

(8.17)

Thus, (8.14) and (8.15) can be rewritten as:

(8.18)

(8.19)

The block diagram of an ETDAS control is shown in Figure 8.6. The key to realize (8.19) is to determine proper values of R, τ, and K. Firstly, the value of R is chosen. To examine the sensitivity of feedback perturbation, a transfer function is introduced where and are the Fourier transformation of the feedback perturbation and , respectively. So, for the ETDAS control (Pyragas, 1995), is given by:

(8.20)

When , this gives for all frequencies except those narrow windows around . So, by increasing , the feedback perturbation becomes more sensitive and offers a wider range of for stabilization. Therefore, is chosen to be 0.95. A corresponding bifurcation diagram of with respect to under V_in = 12 V is plotted, as shown in Figure 8.7. It can be seen that this method offers a wide operating range of K [3.3 V/A, 19.2 V/A]. The bifurcation diagrams of with respect to K/L_a at four representative values of are plotted, as shown in Figure 8.8. It can be seen that when K/L_a is chosen to be 2167 V/AH, a period-1 motion can be attained throughout all regions. Since L_a is equal to 0.003 H, K is chosen to be 6.5 V/A.

After substituting R = 0.95, τ = 0.033 s, and K = 6.5 V/A into (8.18) and (8.19), the time-domain waveforms of under four typical values of are shown in Figure 8.9. It should be noted that the ETDAS control is applied at the instant t = 5 s, and all stabilization processes can be successfully completed within 0.4 s. Figure 8.10 shows the corresponding trajectories versus . It can be seen that the behavior of the wiper can be stabilized into period-1 motion under all conditions. So, with respect to using the ETDAS control with K = 6.5 V/A and τ = 0.033 s, the bifurcation diagram of is plotted in Figure 8.11. Compared with the bifurcation diagram without control, as shown in Figure 8.3, it can be seen that chattering can be stabilized over the whole range of .

Experiments are developed to examine the stabilization performance of the ETDAS control. The experimental setup of an automotive wiper assembly is taken from a commercial automobile. It should be pointed out that the modeling of frictional forces between the wiper blades and the screen is based on a flat windshield, which is actually an assumption that is commonly used in the available literature. However, the experimental example is an actual automotive windshield which has a curved surface. The control method is digitally implemented by a TMS320F240 DSP microcontroller which is equipped with an A/D conversion and PWM module. The sampling rate is 10 kHz, and the motor current is sampled once in every switching interval. Since the electrical time constant is much larger than the sampling interval, the motor current can be constructed using this sampling rate. The microcontroller generates proper switching pulses for a switched mode power supply which, in turn, provides a controllable input voltage of the motor. The analog accelerometer ADXL311 is mounted on the wiper to record its acceleration for the purpose of display.

Figure 8.12(a) shows the measured acceleration of the wiper without control under V_in = 12 V. It can be observed that there exists chattering vibration on the wiper which has also been predicted in Section 8.1.2. As previously mentioned, this is due to the stick-slip motion of the wipers on the surface of the windshield. Then, the ETDAS control is implemented using a DSP microcontroller to suppress this chattering vibration. In the experimentation, the wet condition of the windshield surface is kept to be almost the same for fair comparison. It should be noted that it is not really possible to match the parameters of the experimental wiper system with those for the aforementioned analysis. So, given τ = 0.05 s, which is smaller than the period of chattering, different values of K are selected for the ETDAS control until the optimal performance for the suppression of chattering is achieved. Thus, Figure 8.12(b) shows the measured acceleration of the wiper using the ETDAS control with K = 4 V/A under V_in = 12 V. It can be observed that the chattering amplitude is significantly suppressed. Also, the time-delay feedback control is implemented for comparison. Again, different values of K and are selected until optimal performance is achieved. Figure 8.12(c) shows the performance using a time-delay feedback control with K = 4 V/A and τ = 0.05 s under V_in = 12 V. It can be observed that the time-delay feedback control cannot provide the same performance as the ETDAS control, owing to the fact that its operating range of K is not wide enough to suppress the chattering of this experimental system.

8.2 Chaos Stabilization in Centrifugal Governor Systems

The centrifugal governor functions to control the speed of the engine automatically and prevent the engine from damage caused by a sudden change of load torque. Figure 8.13 shows the configuration of a hexagonal centrifugal governor coupled to an engine (Ge and Lee, 2003). The engine drives the rotational axis of the governor. There are four rods which are joined to the hinges at the two ends of the axis. The upper and lower rods are attached to two fly-balls, and a linear spring is attached to the sleeve. If the speed of the engine drops below the desired speed, the centrifugal force acting on the fly-balls will decrease, the fuel injection control valve will open wider, and, as more fuel will be supplied, the speed of the engine will increase until equilibrium is reached. On the other hand, if the engine speed increases, the fuel supply will be reduced and hence the speed will be reduced accordingly.

It has been identified that chaotic behaviors exist in the centrifugal governor system under a harmonic load torque (Ge and Lee, 2003). Various control methods have been proposed to stabilize this chaotic motion, such as the addition of constant motor torque, the addition of periodic force, the use of a periodic impulse input, a delayed feedback control, an adaptive control, a bang-bang control, an external force control, and an optimal control.

8.2.1 Modeling

The dynamical model of the centrifugal governor system can be obtained by calculating its kinetic and potential energies as follows (Ge and Lee, 2003):

(8.21)

(8.22)

where KE is the kinetic energy, PE is the potential energy, m is the mass of each fly-ball, k is the stiffness of the spring, η is the speed of the rotational axis, l is the length of each rod, r is the distance between the rotational axis and the hinge, and ϕ is the angle between the rotational axis and the rod. Thus, the Lagrange equation is given by:

(8.23)

Consequently, the dynamical equation of the governor can be derived as:

(8.24)

where is the damping coefficient.

For the rotational machine, the net torque is the difference between the engine torque and the load torque. The dynamics of the rotational machine can therefore be expressed as:

(8.25)

where ω is the speed of the engine, J is the inertia of the rotational machine, is a proportional constant, and is an equivalent torque of the load. By defining the time scale , the dynamical system represented by (8.24) and (8.25) can be rewritten as:

(8.26)

(8.27)

(8.28)

where

and n is the gear ratio between the rotational axis and the engine shaft. Hence, the dynamics of this centrifugal governor system is a three-dimensional autonomous system.

8.2.2 Analysis

When the load torque is harmonic, it can be represented by a constant term and a harmonic term , where , A, and are constants. Thus, the dynamical equations (8.26)–(8.28) can be expressed as:

(8.29)

(8.30)

(8.31)

Given the parameters , , , , , , , and ω = 1, the bifurcation diagram of the deflection angle with respect to the control parameter Q is plotted as shown in Figure 8.14. It can be observed that the centrifugal governor exhibits periodic motions when Q is small. If Q increases beyond the threshold, chaotic motion will occur in the system. With Q = 2, the trajectory of versus is plotted as shown in Figure 8.15. As can be seen, the system exhibits a period-1 motion. On the other hand, the trajectory depicted in Figure 8.16 exhibits chaotic motion under Q = 2.6. These results are in good agreement with the bifurcation diagram shown in Figure 8.14.

8.2.3 Stabilization

Contrary to the previous stabilization methods (Ge and Lee, 2003), the ETDAS control is used to stabilize any chaotic motion in this centrifugal governor system. The principle of the ETDAS control for this system can be represented as:

(8.32)

(8.33)

(8.34)

where R, τ, and K are selectable parameters for ETDAS control, and is the additional torque imposed on the engine to stabilize the chaotic behaviors in the centrifugal governor system. This additional torque can be easily generated by using an electric drive that is directly coupled to the shaft of the engine. By tuning the feedback gain K in (8.34), a bifurcation diagram of the system under Q = 2.6 is obtained, as depicted in Figure 8.17. It can be observed that when K is increased beyond the low threshold, the chaotic motion of the system can be stabilized into period-1 motion. Hence, by setting K = 50, the trajectory of ϕ versus ϕ with the ETDAS control is as plotted in Figure 8.18(a). Also, the transient waveform of is plotted in Figure 8.18(b). It can be found that any chaotic motion can be stabilized effectively and promptly by the ETDAS control.

8.3 Chaos Stabilization in Rate Gyro Systems

The gyroscope is widely used in the navigation and control system of space vehicles. It functions to measure the angular velocity in spinning space vehicles. Thus, the stability of motion of the gyro is critical for the accurate measurement of angular velocity. Recently, it has been identified that chaotic motion occurs in a rate gyro with feedback control mounted on a space vehicle that is spinning with an uncertain angular velocity (Ge and Chen, 1998).

8.3.1 Modeling

The configuration of a single-rate gyro system with feedback control mounted on a space vehicle is shown in Figure 8.19. The gimbal can turn about the output X-axis with a deflection angle . The corresponding motion is damped by the damping torque , where is the damping coefficient. By using the Lagrange equation, the dynamical equation of a rate gyro system with feedback control can be derived as (Chen, 2004):

(8.35)

where is the controlled motor torque along the output X-axis to balance the gyroscopic torque; is a constant; , , and are the angular velocity components of the platform along the output X-axis, input Y-axis, and normal Z-axis, respectively; A, B = A, C and , , are the inertias of rotor and gimbals about the gimbal axes , , , respectively.

Normally, the electric motor is a permanent magnet DC (PMDC) motor. The corresponding dynamical model is given by:

(8.36)

(8.37)

where , , , , and are the armature current, armature resistance, armature inductance, back EMF constant, and torque constant of the PMDC motor, respectively, and the corresponding feed-in voltage is proportional to the difference between the desired deflection angle and the actual deflection angle , with the amplifier gain .

The desired motion of the gyro needs to be fixed at the origin, namely . In particular, when the space vehicle undergoes an uncertain angular velocity of about the spinning Z-axis, an acceleration of about the output X-axis, and a zero about the input Y-axis, the dynamical equation of the rate gyro system with feedback control can be rewritten as:

(8.38)

where , , , , , , , , , and .

8.3.2 Analysis

As is time-varying and its value is small, it can be assumed to be zero. Also, is used to represent the uncertain angular velocity with which the rate gyro in the space vehicle spins about its Z-axis. Thus, the dynamical equation of the system can be expressed as:

(8.39)

Given the parameters , , , , , , , and , the bifurcation diagram of versus the perturbation amplitude is plotted as shown in Figure 8.20. It can be observed that chaotic behaviors occur in some regions of . By setting Z_m = 0.5 rad/s and Z_m = 1.31 rad/s, the trajectories of versus of the system are plotted, as shown in Figures 8.21 and 8.22, respectively. This exhibits periodic motion under Z_m = 0.5 rad/s, but chaotic motion under Z_m = 1.31 rad/s.

8.3.3 Stabilization

Similar to the stabilization of the automotive wiper system and centrifugal governor system, the ETDAS control is applied to stabilize the chaotic motion in this rate gyro system. The principle of the ETDAS for this system can be expressed as:

(8.40)

(8.41)

(8.42)

where R, τ. and K are tunable parameters for the ETDAS control, and is the feed-in voltage of the electric motor.

By tuning the control parameter , a bifurcation diagram of the system with EDTAS control under Z_m = 1.31 rad/s is depicted in Figure 8.23. It can be observed that the chaotic motion of the system can be stabilized into periodic motion within some regions of . Namely, when is chosen to be 50, the chaotic motion under Z_m = 1.31 rad/s can be effectively stabilized into a period-2 motion, as shown in Figure 8.24.

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