Chapter 5

Chaos in Switched Reluctance Drive Systems

The switched reluctance (SR) drive is a specific kind of doubly-salient drive. The SR drive system has the advantages of robust structure, high reliability, simple control, wide range of constant power operation, and low manufacturing cost (Lawrenson, 1992). On the other hand, it suffers from the drawbacks of large noise, large torque ripple, special converter configuration, and high nonlinearities (Miller, 1993; Zhan, Chan and Chau, 1999). Thus, it is anticipated that the SR drive system is more prone to chaos due to the high nonlinearities.

In the SR drive, there are two driving periods, namely the commutation period and the pulse width modulation (PWM) period (Miller, 1993). The commutation period is the stroke angle determined by the phase commutation, in which the phase current always has an initial value of zero. Hence, for each phase winding, the drive system operates in a discontinuous conduction mode for the term of the commutation period. The PWM period is the period of the carrier signal for PWM regulation, which is also the switching period of power devices. This PWM period is usually short enough to force the phase current to be continuous within the commutation period. The phase current oscillation, and hence the torque oscillation, are governed by the distribution of those PWM pulses within the commutation period. At normal operation, the PWM pulses are uniformly distributed. Hence, the corresponding oscillation is regular and its oscillating frequency follows the PWM frequency. However, if some pulse widths are zero, which are caused by the skipping cycles, both the current and torque oscillations will be very severe. The commutation period will become nonperiodic and generate a low-frequency oscillation. Apart from the system nonlinearities, this low-frequency oscillation is another factor contributing to the chaotic operation of the SR drive. Furthermore, the synchronization between the commutation and PWM periods is an additional factor contributing to chaotic operation. When these two periods are synchronous, asynchronous, or incommensurate – namely the ratio of the commutation period to the PWM period is respectively an integer, a rational number, or an irrational number – the corresponding operation is respectively a fundamental, subharmonic, or quasiperiodic solution.

Instead of using voltage PWM regulation, the SR drive system can be controlled by the current hysteresis regulation. Actually, this current hysteresis regulation generally prefers a voltage PWM regulation for the speed control of the SR drive. Contrary to the voltage PWM regulation, the current hysteresis regulation can be easily implemented by comparing the reference current and actual current with a hysteresis loop. Therefore, the switching frequency of the current hysteresis regulation is variable and determined by the operating conditions of the SR drive. Since chaos has been well verified to exist in a dynamical system with a hysteresis loop (Zhusubaliyev and Mosekilde, 2003), an SR drive system using current hysteresis regulation can also exhibit chaotic behavior under certain operating conditions.

The subharmonic and chaotic behavior was first identified in a voltage-controlled closed-loop rotational SR drive system (Chen et al., 2002). By tuning the speed gain, subharmonic and chaotic behaviors occur in the drive system. Consequently, the chaotic behavior in a voltage-controlled open-loop linear SR drive system has also been investigated (De Castro, Robert and Goeldel, 2008). By tuning the step frequency, which is defined to be three times the switching frequency, bifurcation and chaos phenomenon can be observed in the drive system.

In this chapter, chaos in two major classes of SR drive systems, namely the voltage-control mode and the current-control mode, are modeled, analyzed, and validated by computer simulation and practical experimentation.

5.1 Voltage-Controlled Switched Reluctance Drive System

5.1.1 Modeling

Figure 5.1 shows a typical 3-phase voltage-controlled SR drive system. The corresponding speed control is achieved by applying PWM chopping to its motor voltages. The commutation strategy uses rotor position feedback to select the turn-on angle and turn-off angle of those lower-leg power switches (A₂, B₂, and C₂). When the phase windings are conducted in turn, the dwell interval of each phase winding is selected to be equal to the commutation angle , where m is the number of phases and is the number of rotor poles. As shown in Figure 5.2, the stator phase-A winding starts to conduct at and ends at . Subsequently, the phase-B winding conducts from to , and the phase-C winding conducts from to . For each cycle of conduction of all phase windings, the rotational angle of the rotor is a total of . Figure 5.2(a) also indicates that each phase winding conducts at the instant of decreasing magnetic reluctance between the stator and the rotor, hence producing a positive torque to drive the rotor. For synchronizing the voltage PWM regulation with the phase commutation, the ramp voltage for each phase winding is a function of the instantaneous rotor displacement θ, as given by:

(5.1)

where and are the lower and upper bounds of the ramp voltage, is its period, is an integer, and is defined as the remainder of divided by . As shown in Figure 5.1, since and are linear functions of the instantaneous speed and reference speed , respectively, the speed control signal can be expressed as:

(5.2)

where g is the overall feedback gain incorporating both the frequency-to-voltage (F/V) converter and the operational amplifier (OA). Then, both and are fed into the comparator (CM) which outputs the signal to turn on or off those upper-leg power switches (A₁, B₁, and C₁), depending on the phase commutation. When exceeds , the upper-leg switch, being the same phase of the on-state lower-leg switch, is turned off; otherwise, it is turned on. The other phase switches remain off. The corresponding waveforms of and , as well as switching signals, are shown in Figure 5.2(b).

Instead of the phase current , the phase flux linkage is chosen as the state variable so that the system differential equation does not involve the calculation of and . Since m phase windings of the SR motor are conducted in turn, only two adjacent phase windings have currents at the same time when m>2. For the sake of clarity and simplicity, m-phase windings conducted in turn are represented by only two-phase windings (namely 1 and 2) activated alternately. When the phase winding is controlled by PWM regulation, it is called the activated winding; otherwise, it is called the inactivated winding. If we consider that winding 2 lags behind winding 1 by , the system equation of the SR drive can be expressed as:

(5.3)

(5.4)

(5.5)

where is the phase voltage, is the DC supply voltage, R is the phase resistance, B is the viscous damping, J is the load inertia, is the electromagnetic torque, is the load torque, and ε is the unit step function. Because of its high nonlinearity, is approximated by a series of two-dimensional quadratic Lagrange interpolation functions of and . These two-dimensional grids are formulated by using the manufacturer's design data or by employing a finite element analysis of the SR motor. This approach can significantly reduce the complexity in the calculation of flux linkages. Moreover, can similarly be obtained by the numerical inversion of the relationship of . Since the current is more easily measured than the flux linkage, it is chosen as the output variable. By defining the state vector as and the output vector as , where , the system equation given by (5.3) can be rewritten as:

(5.6)

5.1.2 Analysis

In order to construct the Poincaré map, a hyperplane Σ ℜ³ is defined as:

(5.7)

The trajectory of X under observation repeatedly passes through Σ when θ increases monotonically. Thus, the sequence of Σ crossing defines a Poincaré map P: ℜ³ × ℜ³ as given by:

(5.8)

Actually, the solution of this map, the so-called orbit (Parker and Chua, 1989), is a sequence of samples at the turn-on angle of each phase winding. In order to avoid the calculation of Σ crossing, the rotor displacement θ, rather than time t, is selected as the independent variable of the system equation given by (5.3). The next crossing of the plane can be directly calculated by integrating from to . To make θ an independent variable, (5.3) is expressed as:

(5.9)

Since is piecewise continuous in terms of θ, it is much easier to discover the discontinuity points of in (5.9) than in (5.3). Accordingly, (5.9) can be separately solved within different continuous intervals. Hence, it is much more efficient to use (5.9) than (5.3) to calculate the Poincaré map. By redefining the state vector as and the output vector as , (5.9) can be rewritten as:

(5.10)

The Poincaré map (5.8) can also be rewritten as:

(5.11)

Although is piecewise continuous, the solution of (5.10) is continuous and hence P is also continuous. It should be noted that the map in (5.10) is a noninvertible map within the whole set of the solution. For example, if and are simultaneously nonzero, X is a multivalued function of Y. However, the map in (5.11) is homeomorphism, namely both and are continuous, because one of and of is zero. It results in a new map Q: which is defined as . Thus, the maps P and Q are topologically conjugate (Wiggins, 1990), where the corresponding orbits and have the same dynamics although they represent different physical variables of the SR drive system. For example, if is a fixed point of P, then is also a fixed point of Q. The eigenvalues of their Jacobian matrices and are identical. Since the orbit represents the system states, it is useful for the analysis of periodic and chaotic orbits. On the other hand, it is convenient to use the orbit to illustrate the trajectories and waveforms since the rotor speed and armature current are easy to observe both in simulation and in experiments.

The Jacobian matrix of the Poincaré map P with respect to is the solution ℜ³ × ℜ³ of the variational equation of its underlying system as given by:

(5.12)

Considering winding 1 to be inactivated, it yields:

(5.13)

where ; both and can be obtained by directly differentiating ; is the Dirac delta function that is the derivative of ε; and are the discontinuity points of and , which can be obtained by solving and , respectively.

For an arbitrary point , the corresponding initial value of the variational equation given by (5.12) is usually an identity matrix. However, since , it gives . Hence, the third column of the identity matrix should be replaced by a zero vector. Actually, the period of and of X is always rather than . In order to attain the period-1 orbit of P, winding 1 always represents the inactivated winding at each iteration of P, with the result that and must exchange their values after each iteration of P. Thus, the fixed point of P and its Jacobian matrix are defined as:

(5.14)

(5.15)

where .

The fixed point of P can also be located by using a Newton–Raphson algorithm, as given by:

(5.16)

where can be evaluated from (5.12) and (5.13). By checking the characteristic multipliers, which are the eigenvalues of the Jacobian matrix, the stable region of the period-1 orbit for fundamental operation can readily be obtained.

5.1.3 Simulation

In order to assess the modeling of chaos, a corresponding analysis is carried out based on a practical 3-phase SR drive system that has been designed for an electric vehicle (Zhan, Chan and Chau, 1999). The parameter values are V_s = 150 V, N_s = 12, N_r = 8, θ_s = 15°, θ_o = 3.75°, R = 0.15 Ω, B = 0.00 075 Nm/rad/s, J = 0.025 kgm², v_u = 5 V, v_l = 1 V, n_θ = 10, θ_T = 1.5°, ω^∗ = 50 rad/s, and T_l = 8.6 Nm.

When g = 1.3 V/rad/s, the steady-state behavior of the SR drive system is the fundamental or normal operation. The corresponding simulated waveforms of , , and I as well as the simulated phase-plane trajectory of i versus are shown in Figure 5.3, in which θ is expressed as the integer multiple of . As shown in Figure 5.3(a), there is no skipping cycle during PWM regulation, namely crosses every . The fluctuation of is also small (from 4.0 to 4.4 V), and the corresponding ω is from 53.1 to 53.4 rad/s. As shown in Figure 5.3(b), the corresponding i is periodic in term of , and its fluctuation is from 23 to 60 A. Since , i has ten peaks within each , resulting in the phase-plane trajectory of this periodic solution being a cycle with ten peaks, as shown in Figure 5.3(c).

When g = 4.8 V/rad/s, the SR drive system operates in chaos. The simulated chaotic waveforms and trajectory are shown in Figure 5.4. Contrary to the periodic solution, it has skipping cycles within each (as shown in Figure 5.4(a)) in which is higher than and no intersection occurs. Furthermore, the number of skipping cycles within each is a random-like variable, with the result that the oscillating magnitudes of (being the same shape of ω) and i are all fluctuating as shown in Figures 5.4(a) and 5.4(b), respectively. As expected, Figure 5.4(c) shows that the trajectory of i versus is a phase portrait with a random-like but bounded feature. It can be found that even though the fluctuation of ω is still acceptable (from 50.6 to 51.6 rad/s), the fluctuation of is severe (from 2.9 to 7.6 V), resulting in a fluctuation of i that is exceptionally large (from 0 to 120 A) which is highly undesirable.

In order to determine the boundary of the stable fundamental operation, all period-1 orbits are firstly located by using the Newton–Raphson algorithm as given by (5.16). Then, their characteristic multipliers are evaluated by computing the eigenvalues of the corresponding Jacobian matrices obtained from (5.12)–(5.15). By drawing the line where the magnitude of the characteristic multipliers is equal to unity, the stable region of versus g for the fundamental operation can be obtained, as shown in Figure 5.5. If the characteristic multipliers are complex conjugates while their magnitudes are less than unity, the system is still stable, but spirally converges to the fixed point. It should be noted that the unstable region for the fundamental operation involves both subharmonic and chaotic operations. Therefore, the system in Figure 5.5 is highly desirable for the designer of SR drive systems since the fundamental operation is preferred normally.

5.1.4 Experimentation

Based on the same SR drive system, experimentation has been conducted. Since ω has a large DC bias while its variation is relatively small, the speed variation cannot be clearly assessed based on the direct measurement of ω. In contrast, is the amplified speed error which not only exhibits a clear pattern of speed variation, but is also easily measured. Thus, is measured to represent ω. The measured trajectory and waveforms of i and when g = 1.3 V/rad/s are shown in Figure 5.6. This illustrates that the SR drive system operates in a period-1 orbit, which is actually a stable fundamental operation. It can be observed that i and are not of exact periodicity, which is due to the inevitable imperfections of the practical SR motor drive, such as the mechanical eccentricity of the drive shaft and the torsional oscillation of the coupler. Also, it can be found that i lies roughly between 20 and 60 A while lies between 4.0 and 4.5 V. By comparing these results with the waveforms and trajectory shown in Figure 5.3, the measured results and the simulation results have good agreement.

Moreover, by selecting g = 4.8 V/rad/s, the measured trajectory and waveforms of i and , shown in Figure 5.7, illustrate that the SR drive system operates in chaos. It can be found that the boundaries of i and lie roughly between 0 and 120 A and between 2 and 8 V, respectively. Contrary to the period-1 orbit in which the measured trajectory and waveforms are directly compared with simulation, the chaotic trajectory and waveforms measured in the experiment cannot be compared with the simulation results due to the random-like nature of chaos as well as its dependence on the initial conditions. Nevertheless, it can be found that the measured boundaries of the chaotic trajectory shown in Figure 5.7 resemble the simulation results in Figure 5.4, which is actually a property of chaos.

5.2 Current-Controlled Switched Reluctance Drive System

5.2.1 Modeling

Figure 5.8 shows a 3-phase SR drive system using current hysteresis regulation, the so-called current-controlled SR drive system. The commutation strategy also uses rotor position feedback to select the turn-on angle θ_o and turn-off angle θ_c of those lower-leg power switches (A₂, B₂, and C₂). In order to simplify the control circuit, the dwell angle of each phase winding is selected to be equal to the commutation angle , where m is the number of phases and is the number of rotor poles. For each conductive phase winding, the corresponding current control is achieved by applying a current hysteresis controller. The current reference is given by using a proportional speed controller. As the operational amplifier (OA) of the speed controller and the current multiplexer have gains of g and , respectively, the current reference signal , current feedback signal and current hysteresis band signal Δv can be expressed as:

(5.17)

(5.18)

(5.19)

where , , , and Δi are the speed reference, the instantaneous speed, the kth phase current, and its hysteresis band, respectively. Then, the current hysteresis controller outputs the signal to turn on or off those upper-leg power switches (A₁, B₁, and C₁), depending on the phase commutation. When , the upper-leg switch, being the same phase of the on-state lower-leg switch, is turned off until ; then it is turned on. The other phase switches remain off. The switching pattern of each phase winding is illustrated by the waveforms of and shown in Figure 5.8(c).

For the kth phase winding of the SR drive, the voltage depends on the states of its upper-leg and lower-leg power switches. During the interval of dwell angle, the lower-leg switch is always turned on. Hence, when the upper-leg switch is turned on, , where U is the DC link voltage; otherwise , which is caused by one of the freewheeling diodes. During the other interval , which is caused by two freewheeling diodes because both the upper-leg and lower-leg switches are turned off. Since the current is always unipolar for the converter topology shown in Figure 5.8, can be considered as zero when , and hence the flux linkage becomes zero in this interval. Thus, for an m-phase SR drive whose phase current is continuously conducted within l commutation periods, the number of simultaneously nonzero voltage equations can be reduced from m to l. Consequently, the dimensionality of the state equations is reduced from (m+2) to (l+2). Since the dwell angle is equal to the commutation angle, only two adjacent phase windings have currents at the same time and hence l = 2 as shown in Figure 5.8(c). For the sake of clarity and simplicity, m phase windings conducted in turn are represented by two virtual phase windings (winding 1 and winding 2) activated alternately. When the winding is conducted, it is said to be activated, otherwise it is said to be inactivated.

Rather than using , is also chosen as the state variable in the current-controlled SR drive system so that the system differential equation does not involve the calculation of and (Stephenson and Corda, 1979). When winding 2 is lagging behind winding 1 by , the system equation of the SR drive can be expressed as:

(5.20)

(5.21)

where R is the phase resistance, B is the viscous damping, J is the load inertia, is the electromagnetic torque, and is the load torque.

The nonlinear phase flux linkage of the SR motor can be approximated by a series of two-dimensional quadratic Lagrange interpolation functions. Based on a finite element analysis of the SR motor, its two-dimensional grid of tabulated values can be obtained. The flux linkage at any operating point can be numerically calculated by using the two-dimensional quadratic polynomials constructed by nine tabulated values near the operating point. These local quadratic interpolations considerably reduce the complexity of the flux model, and ensure the continuity of its first derivatives. Thus, can be obtained by using numerical inversion of . By applying numerical integration and differentiation to , can be attained by the use of (5.21).

As the phase current, rather than the flux linkage, can be directly measured in a practical SR drive, the current is chosen as the output variable. By defining the state vector as and the output vector as , where , (5.21) can be rewritten as:

(5.22)

5.2.2 Analysis

A Poincaré map is used to analyze the chaotic behavior of this current-controlled SR drive system, and a sequence of samples of the continuous-time solution is thereby produced. The steady-state behavior of the Poincaré map, known as the orbit, corresponds to the steady-state waveform of the continuous-time dynamical system.

In order to construct the Poincaré map, a hyperplane Σ ℜ³ is defined as:

(5.23)

The trajectory of X under observation repeatedly passes through Σ when θ increases monotonically. The sequence of Σ crossings defines a Poincaré map P : ℜ³ × ℜ³ as given by:

(5.24)

Actually, the orbit of this map is a sequence of samples at the beginning of the dwell angle of each phase winding. In order to avoid calculating the sequence of Σ crossing, the rotor displacement θ, rather than time t, is selected as the independent variable of the system equation given by (5.20). The crossing of the plane can be directly calculated by integrating from to . To make θ an independent variable, (5.20) is expressed as:

(5.25)

By redefining the state vector as and the output vector as , (5.25) can be rewritten as:

(5.26)

The Poincaré map (5.24) can also be rewritten as:

(5.27)

Although is piecewise continuous, the solution of (5.26) is continuous and hence P is also continuous. Similar to Section 5.1, the map in (5.27) is a noninvertible map within the whole set of the solution. For example, if and are simultaneously nonzero, X is a multivalued function of Y. Fortunately, the map in (5.27) is a homeomorphism, namely and are continuous, because one of and of is zero. Thus, the orbits and are topologically equivalent. It follows that they have the same dynamics although they represent different physical variables of the SR drive system. The orbit is used to locate the periodic and chaotic orbits, whereas the orbit is used to depict the trajectories and bifurcation diagrams.

Due to the nonlinearity of the flux linkage, the period of the fundamental operation (period-1 orbit) is referred to as the commutation period . However, the period of and of X is always instead of . In order to attain the period-1 orbit of P, winding 1 always stands for the inactivated winding at each iteration of P, with the result that and must exchange their values after each iteration of P. Thus, the fixed point of the period-1 orbit and the cyclic points of the period-p orbit are also defined as:

(5.28)

(5.29)

where the transformation matrix C is given by:

(5.30)

5.2.3 Simulation

Based on the system equation (5.26) and the Poincaré map (5.27), the analysis of the nonlinear dynamics in the SR drive system is carried out by computer simulation. A practical SR drive, originally designed for an electric vehicle (Chan et al., 1996), is used for exemplification. The system setup is shown in Figure 5.8, and the corresponding data are listed in Table 5.1. In the following analysis, the fundamental, subharmonic and chaotic behaviors are simulated by adopting different values of g. Also, a unique property of chaotic behavior, namely a high sensitivity to the initial conditions, is demonstrated. Various bifurcation diagrams with respect to g, , and are simulated. Hence, the route to chaos and the effect of system parameters on the nonlinear dynamics are discussed.

Table 5.1 Key parameters of SR motor.

Number of phases m	3
Stator poles	12
Rotor poles	8
Dwell angle	15°
Stroke angle	15°
Turn on angle	5.5°
DC link voltage U	150 V
Phase resistance R	0.1 Ω
Viscous damping B	0.0005 Nm/rad/s
Load inertia J	0.025 kgm2
Load torque	12 Nm
Speed reference	50 rad/s
Current hysteresis band Δi	±5 A
Current feedback gain	0.02 V/A
Speed feedback gain g	10 V/rad/s
PWM frequency	1–5 kHz

Firstly, when the speed feedback gain is selected as g = 5 V/rad/s, the steady-state solution of the SR drive has a normal periodic behavior, the so-called period-1 orbit, which usually exists in the fundamental operation. The corresponding waveforms of ω, , , , and , as well as the phase-plane trajectory of i versus ω, are shown in Figure 5.9, in which θ is expressed as an integer multiple of the stroke angle . These waveforms consist of two oscillating components – one corresponds to the commutation frequency and the other to the PWM frequency . It can be found that is almost constant when the oscillating magnitude of ω is very narrow, as shown in Figure 5.9(a). On the other hand, gradually decreases with increasing inductance within the commutation period, resulting in irregular waveforms. Due to the nonlinear distribution of electromagnetic torque within the commutation period, the oscillating component of ω and dominates the component, as shown in Figures 5.9(a) and 5.9(b). The corresponding extremes of ω and are near the commutation point. Moreover, although the distribution of (or ) is uniform, as shown in Figure 5.9(c), the distribution of is nonuniform, ranging from 10 to 18 Nm, as shown in Figure 5.9(d). The phase-plane trajectory of this periodic solution is a cycle embedded with some smaller cycles as shown in Figure 5.9(e). The boundary of ω is from 49.738 to 49.786 rad/s (474.96 to 475.42 rpm), whereas the boundary of i is from 50 to 110 A. Such a high peak current is due to the overlapping of and during commutation, even though they are actually confined between 51 and 70 A as shown in Figure 5.9(c).

When the speed feedback gain is changed to g = 9 V/rad/s, the steady-state behavior of the SR drive is a period-2 subharmonic solution, as shown in Figure 5.10. It can be found that the waveforms are of period-2 behavior, the so-called period-2 orbit. Comparing Figures 5.10(a) and 5.9(a), the oscillating magnitudes of ω of period-2 and period-1 orbits are very similar. However, the oscillating magnitude of , as shown in Figure 5.10(b), is larger than that shown in Figure 5.9(B) because of the larger g. Accordingly, the oscillating magnitudes of (or ) and increase to 50–72 A and 9.5–18 Nm, as shown in Figures 5.10(c) and 5.10(d), respectively. The phase-plane trajectory of this period-2 solution is still a cycle embedded with some smaller cycles with boundaries of ω from 49.84 to 49.889 rad/s (475.94 to 476.4 rpm) and i from 50 to 113 A, as shown in Figure 5.10(e).

When the speed feedback gain is further increased to g = 14 V/rad/s, the SR drive operates in chaos. The chaotic waveforms and trajectory are shown in Figure 5.11. Similar to the periodic (fundamental and subharmonic) operations, the oscillating component of chaotic operation still dominates the component and the extremes of ω and are also near the commutation point, as shown in Figures 5.11(a) and 5.11(b). However, these waveforms lose the synchronization of , leading to the irregular variation of . Hence, the corresponding (or ) and exhibit random-like fluctuations with the boundaries of 48–72 A and 9–18 Nm, as shown in Figures 5.11(c) and 5.11d, respectively. As shown in Figure 5.11(e), the trajectory of i versus ω is a random-like bounded phase portrait with boundaries of ω from 49.896 to 49.938 rad/s (476.47 to 476.87 rpm) and i from 50 to 114 A. By comparing Figures 5.9–5.11, it can be found that the chaotic waveform of exhibits a random-like variation of the peaky values even though the maximum peaky value is almost identical to that of the fundamental and subharmonic waveforms. This leads to a reduction of the torque ripples that contribute to acoustic noise. It also explains why the oscillating magnitude of ω in chaotic operation is less than that in fundamental and subharmonic operations, even though the oscillating magnitude of in chaotic operation is slightly larger than that in fundamental and subharmonic operations because of the larger g. This discovery implies that the chaotic behavior of a current-controlled SR drive system may be beneficially utilized to reduce electromagnetic interference and acoustic noise.

5.2.4 Phenomena

A key property of chaotic behavior is the divergence of nearby trajectories. This property can be used to distinguish between aperiodic behavior due to chaos and that due to external noise. The divergence of nearby trajectories results in a sensitive dependence on the initial conditions. On the other hand, the chaotic behavior can be realized by its chaotic orbit of the Poincaré map. This chaotic orbit has the twin properties of stretching and folding. The stretching property means that the orbits lying initially close together in a region are mapped into separate orbits, whereas the folding property means that the orbits outside this region are returned to its interior. The above properties can be illustrated by the error of a current reference signal , which is a linear function of the speed error Δω as given by (5.17). By using (5.27), the variation of with an initial difference of 0.0028 V (equivalent to 0.0002 rad/s) is shown in Figure 5.12. It can be found that is very sensitive to its initial value of , indicating the property of divergence of nearby trajectories. Increasingly, it shows that both stretching and folding properties occur irregularly. It is the stretching property that renders the chaotic behavior long-term unpredictable, while the folding property keeps the chaotic behavior bounded. Hence, the chaotic behavior is characterized by a random-like bounded orbit.

By computing the Poincaré map in (5.27), the bifurcation diagrams of i and ω with respect to g, , and Δi are shown in Figures 5.13–5.15, respectively. These bifurcation diagrams can depict the fundamental, subharmonic and chaotic orbits at a glance, hence illustrating how to route to chaos. While calculating the bifurcation diagrams, only the bifurcation parameter varies and the other parameters are fixed.

As shown in Figure 5.13, both i and ω bifurcate from a period-1 orbit to a period-2 orbit at g = 7.5 V/rad/s, then to a period-1 orbit again at g = 7.9 V/rad/s, then back to a period-2 orbit at g = 8 V/rad/s, then to a period-4 orbit at g = 9.6 V/rad/s, and then to chaos at g = 10.2 V/rad/s. After chaos, the system comes back to a period-1 orbit at g = 14.9 V/rad/s, and then to chaos again via a similar route. It can be also found that there are period-3 orbits within the chaotic band, the so-called period-3 window. As shown in Figure 5.14, both i and ω deviate from a period-1 orbit to a period-2 orbit at T_l = 3.5 Nm, then to a period-4 orbit at T_l = 3.7 Nm, and then to chaos at T_l = 3.82 Nm. After this chaotic band, both i and ω return to a period-1 orbit again at T_l = 4.5 Nm, and then to chaos again via a period-doubling route. It is interesting to note that this route to chaos appears again and again. As shown in Figure 5.15, both i and ω with g = 14 V/rad/s also have a period-doubling route to chaos, which repeats throughout the range of Δi. It should be noted that the reduction of Δi (which is constrained by the switching frequency of power devices) can reduce the fluctuation of i and ω, but is unable to avoid the occurrence of chaos.

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