Chapter 4
Chaos in AC Drive Systems
AC drive systems have been widely accepted for industrial applications. In general, they take the advantages of a higher power density and a higher efficiency than DC drive systems. AC drive systems are composed of two major groups, namely the induction drive systems and synchronous drive systems. Among the induction drive systems, the cage-rotor is almost exclusively used for industrial applications. Among the synchronous drive systems, the permanent magnet (PM) brushless AC drive system (usually termed PM synchronous drive system) is becoming popular, whereas the synchronous reluctance drive system is receiving attention.
In this chapter, chaos is investigated in three representative AC drive systems, namely a cage-rotor induction drive system, a PM synchronous drive system, and a synchronous reluctance (SynR) drive system.
It is well known that the motor parameter variations, especially the increase in rotor resistance due to prolonged operation, in field-oriented control (FOC) based induction drive systems may violate the necessary decoupling condition, causing an unexpected speed fluctuation. In recent years, it has been identified that the compensation of this rotor resistance variation may cause chaotic phenomena (Gao and Chau, 2003a). Therefore, in this section a practical induction drive system using FOC is first modeled, then Poincaré mapping and bifurcation analysis are conducted. Consequently, computer simulation and experimental verification are given to verify the chaotic phenomena.
4.1.1 Modeling
The general d–q model of a cage-rotor induction motor is well known (Leonhard, 1996), and can be expressed as:
(4.1) 
(4.2) 
(4.5) 
where 
, 
, and 
, 
are the stator currents and voltages, respectively; 
and 
are the rotor fluxes; 
and 
are the stator and rotor resistances, respectively; 
, 
, and 
are the stator inductance, rotor inductance, and mutual inductance, respectively; η is defined as 
in which 
is the rotor time constant; γ is defined as 
; and β is defined as 
in which 
is the coupling factor, 
is the rotor speed, 
is the speed of the reference frame, J is the rotor inertia, B is the coefficient of viscous friction, and P is the number of poles.
The control diagram of the FOC-based induction drive system for speed tracking is shown in Figure 4.1. The compensator, which functions to compensate the increase of rotor resistance due to prolonged operation, is represented by:
Figure 4.1 Control diagram of FOC-based induction drive system
where 
is the rotor speed command, 
is the gain of the speed controller, and δ is the gain of the compensator. By rearranging (4.3) and (4.4), 
and 
can be expressed in terms of 
and 
. Hence, the stator current commands are given by:
Since the rotor fluxes cannot be directly measured, they are generally estimated on the basis of (4.3) and (4.4), as given by:
(4.9) 
(4.10) 
By using these estimated rotor fluxes together with (4.6) under the normal operation of 
, the stator current commands given by (4.7) and (4.8) can be rewritten as:
(4.11) 
(4.12) 
It should be noted that δ is a control variable. Namely, when δ deviates from the condition of 
, the rotor resistance compensation may cause chaos to occur. As depicted in Figure 4.1, the stator voltages can then be expressed as:
(4.13) 
(4.14) 
where 
and 
are the gains of the current controllers. Together with the speed controller, they are governed by proportional-integral (PI) control, which has the following form in the frequency domain:
(4.15) 
where 
is the proportional constant and 
is the integral constant.
By defining ten state variables, namely 
, 
, 
, 
, 
, 
, 
, 
, 
, and 
, the closed-loop system depicted in Figure 4.1 can be represented by the following first-order differential equations:
Based on the above differential equations, the dynamical behavior of the FOC-based induction drive system can readily be analyzed. Obviously, because of its high order of dimensions, numerical analysis is employed.
4.1.2 Analysis
In order to analyze the chaotic behavior of FOC-based induction drive systems, Poincaré mapping and bifurcation analysis are employed. The fundamental concept of this mapping was due to Poincaré. It replaces the solution of a continuous-time dynamical system by an iterative map, the so-called Poincaré map. This map acts like a stroboscope, and produces a sequence of samples of the continuous-time solution. Thus, the steady-state behavior of the Poincaré map, termed the orbit, corresponds to the steady-state waveform of the continuous-time dynamical system. On the other hand, it is essential to know about the formation of chaos when there is a variation in the system parameters. As a parameter is varied, a bifurcation indicates an abrupt change in the steady-state behavior of the system. A plot of the steady-state orbit against a bifurcation parameter is known as a bifurcation diagram. Thus, a bifurcation analysis facilitates the appraisal of the steady-state system behavior at a glance.
For the FOC-based induction drive system, the speed command may be fixed or time-varying. When this speed command is time-varying, it can be expressed as:
where 
and 
denote the DC component and AC amplitude, respectively, and T is the period of the oscillation. With this periodic speed command, the drive is a nonautonomous system. To derive the Poincaré map of the dynamical system described by (4.16)–(4.26), a ten-dimensional surface 
is defined as:
where 
is the solution of the state vector. The trajectory of 
repeatedly passes the surface 
for every period T (namely the period of the periodic command). The sequence of surface crossing (known as the orbit) defines the Poincaré map as given by:
where 
and 
are the nth and (n + 1)th samples of 
, respectively.
By selecting δ as a bifurcation parameter, the steady-state orbits resulting from the Poincaré map (4.28) of the drive system described by (4.16)–(4.26) can be numerically determined, hence generating the desired bifurcation diagram.
4.1.3 Simulation
A practical three-phase FOC-based induction drive system is employed for exemplification. The corresponding motor parameters are 
, and 
The control parameters of the PI current controllers are 
and 
, whereas those of the PI speed controller are 
and 
. Based on these parameters, computer simulation can be performed to illustrate the occurrence of chaos in a three-phase FOC-based induction drive system.
With different values of periodic speed command, as given by (4.26), the speed bifurcation diagrams can readily be obtained from (4.16)–(4.28). Figure 4.2 shows two bifurcation diagrams of the motor speed with respect to the variation of δ under 
and 
It can be seen that the drive system becomes chaotic from its normal operation when δ is increased beyond a threshold (about 
) for both cases, with and without DC speed offset.
When 
, as can be seen in Figure 4.2, the drive system performs its normal operation. Figures 4.3 and 4.4 show the corresponding motor speed and stator current waveforms under the cases of 
and 
, respectively. It can be found that the motor speeds (marked “actual”) closely follow the speed commands (marked “reference”) during normal operation.
Figure 4.3 Simulated waveforms under normal operation without speed offset. (a) Actual motor speed. (b) Reference motor speed. (c) Reference stator current. (d) Actual stator current
Figure 4.4 Simulated waveforms under normal operation with speed offset. (a) Actual motor speed. (b) Reference motor speed. (c) Reference stator current. (d) Actual stator current
With an increase of δ, the drive system becomes chaotic. When 
, as reflected by Figure 4.2, the system performs a chaotic operation. Figures 4.5 and 4.6 show the chaotic waveforms of the motor speed and stator current under the aforementioned two speed commands. It can be observed that both speed and current exhibit a random-like, but bounded, oscillation that is actually the key nature of chaos.
Figure 4.5 Simulated waveforms under chaotic operation without speed offset. (a) Actual motor speed. (b) Reference motor speed. (c) Reference stator current. (d) Actual stator current
Figure 4.6 Simulated waveforms under chaotic operation with speed offset. (a) Actual motor speed. (b) Reference motor speed. (c) Reference stator current. (d) Actual stator current
4.1.4 Experimentation
For experimentation, the whole FOC-based drive system is implemented. Under the same operation conditions as for simulation, the measured waveforms under normal operations are shown in Figures 4.7 and 4.8. It can be observed that they are in good agreement with the simulated waveforms in Figures 4.3 and 4.4, respectively. Also, the measured waveforms under chaotic operation are shown in Figures 4.9 and 4.10. They both offer the random-like but bounded nature. It should be noted that the simulated and measured chaotic patterns cannot be compared because the chaotic pattern is aperiodic and very sensitive to initial conditions.
Figure 4.7 Experimental waveforms under normal operation without speed offset. (a) Motor speeds. (b) Stator currents
Figure 4.8 Experimental waveforms under normal operation with speed offset. (a) Motor speeds. (b) Stator currents
Figure 4.9 Experimental waveforms under chaotic operation without speed offset. (a) Motor speeds. (b) Stator currents
Figure 4.10 Experimental waveforms under chaotic operation with speed offset. (a) Motor speeds. (b) Stator currents
From the above results, it can be confirmed that the FOC-based induction drive system exhibits chaotic behavior when δ deviates sufficiently from the condition of 
, namely the deviation from normal rotor resistance compensation.
4.2 Permanent Magnet Synchronous Drive Systems
With the advent of high-energy PM materials, PM synchronous motors can offer higher power density and higher efficiency than induction motors (Gieras and Wing, 2002), and PM synchronous drive systems are therefore becoming more attractive for modern industrial applications. However, due to the nonlinear dynamics of PM synchronous motors, some kinds of their abnormal operation may exhibit a strange behavior. Recently, chaotic behavior in PM synchronous machines has been reported by Gao and Chau (2003b) and Ye and Chau (2005).
This section analyzes the effect of PMs on an abnormal operation of PM synchronous drive systems. After modeling the system dynamics as a periodically forced nonautonomous equation, the corresponding Poincaré map and bifurcation diagram show that the design of the PMs significantly affects the behavior of operation. Namely, chaos may occur if the PMs are not properly designed. Both computer simulation and experimental results are given to illustrate the design criterion.
4.2.1 Modeling
A three-phase PM synchronous drive system can be modeled in the d−q frame, as given by Fitzgerald, Kingsley and Umans (1991):
where 
and 
are the stator currents, 
and 
are the stator voltages, 
and 
are the stator inductances, 
is the stator resistance, P is the number of poles, 
is the PM flux, 
is the motor speed, 
is the load torque, J is the rotor inertia, and B is the viscous friction coefficient.
When the PM synchronous motor is normally fed by a three-phase balanced sinusoidal supply, 
and 
are DC quantities. Under abnormal operation, these stator voltages can be time-varying and oscillating, as given by:
where 
is the voltage amplitude, f is the oscillation frequency, and θ is the spatial angle. By substituting (4.30) into (4.29), the system dynamical equation can be written in a periodically driven nonautonomous form:
(4.31) 
4.2.2 Analysis
It is well known that the periodically forced nonautonomous system is prone to subharmonics resonance and chaos (Thompson and Stewart, 2002). Closed-form analytical solutions are usually not available, whereas numerical means such as the Poincaré map and bifurcation analysis are generally employed.
Poincaré mapping functions to replace the solution of a continuous-time dynamical system by an iterative map. The steady-state behavior of the Poincaré map, termed the orbit, corresponds to the steady-state waveform of the continuous-time dynamical system. For the nonautonomous system described by (4.3.1), a natural way to construct the Poincaré map is to sample the trajectory with a frequency f. Thus, the Poincaré surface 
is defined as:
(4.32) 
where 
is the solution of the state vector. The trajectory of 
repeatedly passes the surface 
for every period T (namely, the period of the stator voltage oscillation). The sequence of surface crossings defines the Poincaré map, as given by:
(4.33) 
where 
and 
are the nth and (n + 1)th samples of 
, respectively.
A bifurcation analysis is used to measure the changes in the system's steady-state behavior with each variation in the system's parameters. As a parameter is varied, a bifurcation is an abrupt change in the steady-state behavior of the system. A plot of the steady-state orbit against a bifurcation parameter is known as a bifurcation diagram, and the bifurcation analysis facilitates the appraisal of the steady-state behavior of the system at a glance.
4.2.3 Simulation
Different parameters affect the operation of a PM synchronous drive system. The PM size, and hence the PM flux, are of particular concern. A practical three-phase 6-pole interior PM synchronous motor with parameters listed in Table 4.1 is used for exemplification.
Table 4.1 Key parameters of PM synchronous motor.
d-axis stator inductance  |
0.237 mH |
q-axis stator inductance  |
0.152 mH |
Stator resistance  |
0.327 Ω |
| Number of poles P | 6 |
PM flux  |
0.1472 Wb |
| Rotor inertia J | 5.100 × 10−4 kgm2 |
| Viscous friction coefficient B | 2.140 × 10−3 Nm/rad/s |
When selecting Vm = 30 V and Tl = 0, the bifurcation diagram of the motor speed with respect to f is shown in Figure 4.11. Obviously, it reveals that synchronism can only be achieved when f is lower than 14.06 Hz. There exists a region (
) in which the motor behavior begins to either oscillate periodically or tremble chaotically, and PM sizing provides an explanation for this loss of synchronism. In order to verify this observation, the PM synchronous drive purposely selects the conditions of Vm = 30 V, f = 15 Hz, and Tl = 0. The corresponding bifurcation diagram of motor speed with respect to 
is shown in Figure 4.12. It illustrates that there exists a critical value of 
for stable operation. Thus, the PM synchronous drive system operating at the above conditions exhibits a chaotic behavior because the corresponding PM flux (0.1472 Wb) is larger than the critical value of 0.0914 Wb. Synchronism can be guaranteed only if the PM flux is smaller than 0.03 Wb; the motor will oscillate periodically in the region of 
.
Figure 4.11 Bifurcation diagram of motor speed with respect to oscillation frequency
Figure 4.12 Bifurcation diagram of motor speed with respect to PM flux
4.2.4 Experimentation
Based on the aforementioned PM synchronous motor and operating conditions, the resulting chaotic waveforms of 
, 
, and 
are measured as shown in Figures 4.13 and 4.14, respectively. Consequently, the corresponding chaotic trajectories on the 
, 
and 
planes are measured as depicted in Figure 4.15. It can be found that the waveforms exhibit a well-known property of chaos (namely, random-like but bounded), while the trajectories resemble the well-known Rössler attractor (Thompson and Stewart, 2002), especially the one on the 
plane.
Figure 4.14 Measured chaotic motor speed waveform
Figure 4.15 Measured chaotic trajectories. (a) Plane of d-axis current versus q-axis current. (b) Plane of d-axis current versus motor speed. (c) Plane of q-axis current versus motor speed
Based on the above findings, the designer for PM synchronous drive systems should not only size the PMs for the sake of maximizing the torque, but also take into account the critical value of PM flux to avoid the formation of chaos during abnormal operation.
4.3 Synchronous Reluctance Drive Systems
The SynR drive systems have the advantages of high mechanical integrity, high-current operation ability, high-temperature working tolerance, and low material cost (Lipo, 1991). With the use of field-oriented control (FOC) or vector control, SynR drive systems can compete favorably with induction drive systems in high-performance applications (Betz et al., 1993; Sharaf-Eldin et al., 1999).
The purpose of this section is to discuss the occurrence of bifurcation and chaotic behavior in SynR drive systems. A practical SynR drive system adopting FOC is used for exemplification. Based on the derived nonlinear system equation, a bifurcation analysis shows that the system loses stability via Hopf bifurcation when the d-axis component of its three-phase motor voltages loses its control (Gao and Chau, 2004). Moreover, calculation of corresponding Lyapunov exponent further proves the existence of chaos. Finally, computer simulations and experimental results are used to support the theoretical analysis.
4.3.1 Modeling
The SynR motor is a singly salient machine. Its stator is typically equipped with three-phase sinusoidally distributed windings, which is similar to that of other AC motors such as the induction motor or the PM synchronous motor. Its rotor is purposely constructed with salient poles so as to produce the desired reluctance torque for electromechanical energy conversion. This salient rotor can be derived by using the geometrically salient-pole structure, the flux-barrier structure or the axially laminated structure. The higher the saliency ratio, the larger the reluctance torque can be produced.
Figure 4.16 shows a three-phase two-pole SynR motor with an axially laminated rotor structure. Since both the field winding and damper winding are absent in the structure, the corresponding system equations are very simple, as given by:
where 
, 
, 
are the stator voltages, 
, 
, 
are the stator currents, 
, 
, 
are the stator flux linkages, and 
is the stator resistance. By applying the well-known Park transformation (Fitzgerald, Kingsley and Umans, 1991; Matsuo and Lipo, 1993), the system equations given by (4.34) can be rewritten as:
where the subscripts ds and qs represent the corresponding d-axis and q-axis quantities, respectively, 
and 
are the magnetizing inductances, 
is the stator leakage inductance, and 
is the synchronous speed. Hence, the electromagnetic torque is expressed as:
(4.36) 
where P is the number of poles. In terms of 
and 
, this can be rewritten as:
Figure 4.16 Schematic diagram of three-phase SynR motor
From (4.37), it can be found that the higher the saliency (
) of the SynR motor, and the greater the difference between 
and 
, the larger becomes the value of 
. The motion equation is then given by:
where J is the moment of inertia of the drive system, B is the viscous friction coefficient, 
is the load torque and 
is the motor speed. Based on the above d–q model, the SynR drive system can employ FOC as illustrated in Figure 4.17. Hence, the dynamical model can be deduced from (4.35), (4.37), and (4.38) as given by:
Figure 4.17 Control diagram of three-phase SynR drive system
The key merit of FOC is to decouple the field component (d-axis stator current 
) and the torque component (q-axis stator current 
) for high-performance operation. In general, the SynR motor is maintained at full flux by exciting it with a constant 
while the torque is regulated by controlling 
. Based on a voltage-fed inverter, 
and 
are governed by 
and 
, respectively. The constant value of 
can be easily deduced by substituting 
into (4.39), whereas 
is regulated by the speed error using proportional-integral-derivative (PID) control. The corresponding control criteria are set below:
where 
is the reference mechanical rotor speed.
The above FOC works well provided that 
can be kept constant and under proper control. If 
loses its control, the SynR drive system may exhibit strange behavior. For the sake of simplicity, the analytical derivation is based on the assumption of 
. Thus, based on (4.39) and (4.40), the SynR drive system can be expressed as:
After rearranging 
, 
, and 
as the state variables, (4.41) can be rewritten as:
(4.42) 
The system equation can be further simplified by transforming t to 
and (
, 
, 
) to (x, y, z) as defined by:
(4.43) 
where 
, 
and 
. Hence, the SynR drive system can be expressed as:
where 
, 
, 
, 
and 
.
4.3.2 Analysis
It is essential to analyze the stability of the equilibrium point and the trajectory around the equilibrium point. The key is to derive the eigenvalues of the system at the equilibrium point.
For the special case when 
, 
, and 
, the system equation (4.44) becomes:
The equilibrium point can be deduced by setting the derivatives in (4.45) equal to zero. Obviously, the origin is a trivial equilibrium point. The nonzero equilibria can be solved by the following equations:
As 
, 
, 
and their transformed quantities x, y, z are all realistic parameters, there are three possible cases:
- If
, there is one equilibrium point (0, 0, 0). - If
, there are three equilibrium points (0, 0, 0) and (
, 
, 
). - If
, there are five equilibrium points (0, 0, 0) and (
, 
, 
).
The local stability of the equilibrium point is described by the eigenvalues of the system characteristic equation:
where λ denotes the eigenvalues, I is the identity matrix, and J is the Jacobian matrix of the transformed system evaluated at the equilibrium point (
, 
, 
). Hence, the eigenvalues can be deduced from an explicit cubic equation, as given by:
It is easy to check that the origin is a locally stable equilibrium point since the corresponding eigenvalues are all negative, that is −a, −b. and −1. For the nonzero equilibrium case, an explicit form of (4.48) can be obtained using (4.46):
Applying the Routh–Hurwitz stability criterion, as shown in Table 4.2, the local stability is guaranteed by the following condition:
Table 4.2 Routh–Hurwitz stability criterion.
 |
1 |  |
 |
 |
 |
 |
 |
0 |
 |
 |
0 |
Accordingly, the system exhibits Hopf bifurcation (Alligood, Sauer and Yorke, 1996) if there exist a pair of complex eigenvalues satisfying the following criteria:
where 
is the critical value of c at which the Hopf bifurcation occurs.
Considering that there exists a purely imaginary nonzero characteristic root 
(
) in (4.49), it yields:
(4.52) 
Separating the real part and the imaginary part results in:
(4.53) 
For positive a, b, and c, the existence of n is guaranteed by the following condition:
This condition shows that the system parameters 
, 
, J, and B of the SynR drive system have a critical effect on the existence of bifurcation. If their values are not properly set, Hopf bifurcation and chaos may occur.
It is straightforward to check that 
for positive a and c. For this reason, the equilibrium point pair (
, 
, 
), if they exist, do not meet the bifurcation criteria in (4.51). From (4.50), they are always locally unstable.
By substituting 
into (4.54), the critical value of c at which Hopf bifurcation occurs can be obtained:
(4.55) 
The corresponding critical eigenvalues are given by:
(4.56) 
Based on the stability criterion in (4.50), when c is large enough, but still smaller than the critical value 
, the equilibrium point pair (
, 
, 
) are stable. Once the value of c exceeds the critical value, they become unstable saddle points. Furthermore, they may even become chaotic attractors.
In general, the solution of system (4.44) leads to a fifth order polynomial. Consequently, the equilibrium point (
, 
, 
) has five possible roots: five real roots; three real roots and a pair of complex conjugate roots; or one real root and two pairs of complex conjugate roots. Analytical results for this case are almost impossible, nevertheless a numerical analysis can always be performed.
Under the occurrence of Hopf bifurcation, the dynamical system may demonstrate a complicated behavior – that is, chaos. To further identify the chaotic behavior, the calculation of Lyapunov exponents plays an important role. Namely, a system exhibits chaotic behavior if at least one of its Lyapunov exponents is positive (Hilborn, 1994). For a three-dimensional state-space system described by a set of three first-order differential equations, the corresponding type of attractors (fixed point, limit cycle, quasiperiod torus, and chaotic) can be directly determined by the signs of Lyapunov exponents (Parker and Chua, 1989) as listed in Table 4.3. Obviously, the SynR drive system described by (4.44) belongs to the aforementioned three-dimensional state-space system. Thus, this SynR drive system demonstrates chaotic behavior if its Lyapunov exponents are one zero, one positive, and one negative.
Table 4.3 Relationship between signs of Lyapunov exponents and type of attractors.
| Signs | Type of attractors |
| (−, −, −) | Fixed point |
| (0, −, −) | Limit cycle |
| (0, 0, −) | Quasiperiodic torus |
| (+, 0, −) | Chaotic |
The notion of a Lyapunov exponent is a generalization of the concept of an eigenvalue as a measure of the stability of a fixed point or a characteristic exponent as a measure of the stability of a periodic orbit. For a chaotic trajectory, it is not sensible to examine the instantaneous eigenvalue of a trajectory. The next best quantity, therefore, is an eigenvalue averaged over the whole trajectory. The Lyapunov exponent is best defined by measuring the evolution (under a flow) of the tangent manifold:
(4.57) 
where 
is the solution of the system characteristic equation (4.47), and l is the system dimension. The Gram–Schmidt orthonormalization algorithm can readily be used to calculate the Lyapunov exponents.
4.3.3 Simulation
To illustrate the strange behavior of an SynR drive system, computer simulations of both waveforms and trajectories are carried out. A three-phase 4-pole SynR drive system is used for exemplification, which has the parameters P = 4, Lds = 133.3 mH, Lqs = 25.1 mH, Rs = 0.029 Ω, Tl = 0 Nm, J = 1.988 × 10−3 kgm2,B = 3.513 × 10−3 Nm/rads−1, vds = 0 V, and ωref = 0 rad/s. Notice that the values of 
, 
, and 
are simply chosen for illustration purposes, and can readily be altered without affecting the final conclusion. Most importantly, the parameters of this practical drive system meet the requirements of Hopf bifurcation, as described by (4.54).
The locus of eigenvalues obtained by the equilibrium point pair (
, 
, 
) is depicted in Figure 4.18, in which the parameter c starts from 
, namely 
. As discussed above, this pair of equilibrium points lose their stability once the parameter c exceeds its critical value. Clearly, at this critical value of 
, the locus crosses the imaginary axis and Hopf bifurcation occurs.
Figure 4.19 shows the Lyapunov exponents of the SynR drive system. From these three curves, the underlying system behavior can be easily distinguished by using Table 4.3. With a small value of c, the SynR drive system has only one stable equilibrium point (fixed point). The corresponding Lyapunov exponents are all negative. Thus, five equilibria occur with an increase in c. Once c exceeds its critical value (
), a pair of stable equilibria lose their stability and become a pair of saddle points. This pair of saddle points cause the SynR drive system to demonstrate a complex behavior. There are chaotic attractors in the regions of 
and 
. On the other hand, in the regions of 
and 
, the system exhibits limit cycles. Table 4.4 lists the Lyapunov exponents under different typical values of c.
Table 4.4 Lyapunov exponents at typical values of c
| c | Lyapunov exponents | Attractor type |
| 3.00 | −0.056, −0.056, −2.598 | Fixed point |
| 10.00 | 0.336, 0.000, −3.045 | Chaotic |
| 15.52 | 0.000, −0.014, −2.678 | Limit cycle |
| 17.00 | 0.514, 0.000, −3.224 | Chaotic |
| 18.40 | 0.000, −0.012, −2.699 | Limit cycle |
| 20.74 | 0.000, −0.010, −2.700 | Limit cycle |
Figure 4.20 shows the speed bifurcation diagram via the parameter c. This diagram is obtained by plotting the successive 200 crossing points of the steady-state trajectory with a fixed Poincaré section via the parameter c. Therefore, the underlying chaotic attractor (CA) or limit cycle (LC) can be easily identified, as labeled in the figure. It can be found that this diagram matches the phenomena obtained by Figure 4.19 and Table 4.4. Namely, when 
or 
, the system exhibits limit-cycle operation; when 
or 
, the system offers chaotic operation.
Figure 4.21 shows the simulated chaotic waveforms of 
, 
, and 
of the SynR drive system when 
and the corresponding trajectories on the 
, 
and 
planes. It can be found that the waveforms offer the well-known chaotic properties, namely random-like but bounded, while the trajectories resemble a butterfly (like the well-known Lorenz attractor). On the other hand, the SynR drive system exhibits limit-cycle operation when 
. Figure 4.22 depicts the corresponding time-domain waveforms of 
, 
, and 
, and its trajectories on the 
, 
and 
planes. It can easily be observed that both the waveforms and trajectories offer a periodic property.
Figure 4.21 Simulated chaotic waveforms and trajectories when c = 10. (a) d-Axis current. (b) q-Axis current. (c) Motor speed. (d) Plane of d-axis current versus q-axis current. (e) Plane of d-axis current versus motor speed. (f) Plane of q-axis current versus motor speed
Figure 4.22 Simulated periodic waveforms and trajectories when c = 22. (a) d-Axis current. (b) q-Axis current. (c) Motor speed. (d) Plane of d-axis current versus q-axis current. (e) Plane of d-axis current versus motor speed. (f) Plane of q-axis current versus motor speed
4.3.4 Experimentation
Based on the aforementioned SynR drive system and operating conditions, the resulting waveforms and trajectories are measured, as shown in Figures 4.23 and 4.24. Comparing the periodic waveforms and trajectories shown in Figure 4.22 with those in Figure 4.24, the measured results are in agreement with the simulation results. On the other hand, comparing the chaotic waveforms shown in Figures 4.21 and 4.23, the simulated and measured patterns cannot be matched because the chaotic pattern is aperiodic and very sensitive to the initial conditions. Nevertheless, the random-like but bounded nature can be easily observed from their trajectories. The measured boundary values are also in good agreement with the simulated values.
Figure 4.23 Measured chaotic waveforms and trajectories when c = 10. (a) d-Axis current. (b) q-Axis current. (c) Motor speed. (d) Plane of d-axis current versus q-axis current. (e) Plane of d-axis current versus motor speed. (f) Plane of q-axis current versus motor speed
Figure 4.24 Measured periodic waveforms and trajectories when c = 22. (a) d-Axis current. (b) q-Axis current. (c) Motor speed. (d) Plane of d-axis current versus q-axis current. (e) Plane of d-axis current versus motor speed. (f) Plane of q-axis current versus motor speed
It should be noted that the above analysis regulates the classical concept of SynR drive system operation. In addition to fixed-point operation, the SynR drive system can offer chaotic operation and limit-cycle operation with variations of controllable parameters. Although the chaotic operation has not yet been widely used for industrial applications, it is anticipated that the resulting chaotic motion of this SynR drive system is beneficial to some niche areas, such as industrial mixing or domestic washing.
Alligood, K.T., Sauer, T.D., and Yorke, J.A. (1996) Chaos: An Introduction to Dynamical Systems, Springer-Verlag, New York.
Betz, R.E., Lagerquist, R., Jovanovic, M. et al. (1993) Control of synchronous reluctance machines. IEEE Transactions on Industry Applications, 29, 1110–1122.
Fitzgerald, A.E., Kingsley, C. Jr., and Umans, S.D. (1991) Electric Machinery, McGraw-Hill, Singapore.
Gao, Y. and Chau, K.T. (2003a) Chaotification of induction motor drives under periodic speed command. Electric Power Components and Systems, 31, 1083–1099.
Gao, Y. and Chau, K.T. (2003b) Design of permanent magnets to avoid chaos in PM synchronous machines. IEEE Transactions on Magnetics, 39, 2995–2997.
Gao, Y. and Chau, K.T. (2004) Hopf bifurcation and chaos in synchronous reluctance motor drives. IEEE Transactions on Energy Conversion, 19, 296–302.
Gieras, J.F. and Wing, M. (2002) Permanent Magnet Motor Technology: Design and Applications, Marcel Dekker, New York.
Hilborn, R.C. (1994) Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers, Oxford University Press, Oxford.
Leonhard, W. (1996) Control of Electrical Drives, Springer-Verlag, New York.
Lipo, T.A. (1991) Synchronous reluctance machines – a viable alternative for AC drives? Electric Machines and Power Systems, 19, 659–671.
Matsuo, T. and Lipo, T.A. (1993) Field oriented control of synchronous reluctance machine. Proceedings of IEEE Power Electronics Specialists Conference, pp. 425– 431.
Parker, T.S. and Chua, L.O. (1989) Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, New York.
Sharaf-Eldin, T., Dunnigan, M.W., Fletcher, J.E., and Williams, B.W. (1999) Nonlinear robust control of a vector-controlled synchronous reluctance machine. IEEE Transactions on Power Electronics, 14, 1111–1121.
Thompson, J.M.T. and Stewart, H.B. (2002) Nonlinear Dynamics and Chaos, Wiley, New York.
Ye, S. and Chau, K.T. (2005) Design of permanent magnets to chaoize PM synchronous motors for industrial mixer. Proceedings of IEEE International Magnetics Conference, pp. 723– 724.