Chapter 6
Chaos is observed as an unpredictable phenomenon due to its sensitivity to initial states. It is a kind of steady-state but locally unstable behavior, and exhibits irregular properties. Such random-like phenomenon is normally regarded as unstable operation which results in additional loss, and therefore is a harmful behavior. Various control methods have been proposed to stabilize the chaotic behavior, such as the Ott–Grebogi–Yorke (OGY) method (Ott, Grebogi, and Yorke, 1990; Hunt, 1991), the time-delay feedback method (Pyragas, 1992), the non-feedback method (Rajasekar, Murali, and Lakshmanan, 1997; Ramesh and Narayanan, 1999), the proportional feedback method (Jackson and Grosu, 1995; Casas and Grebogi, 1997), the nonlinear control method (Khovanov et al., 2000; Tian, 1999), the adaptive control method (Boccaletti, Farini, and Arecchi, 1997; Liao and Lin, 1999), the neutral networks method (Hirasawa et al., 2000; Poznyak, Yu, and Sanchez, 1999), and the fuzzy control method (Tanaka, Ikeda, and Wang, 1998). Some of them have also been proposed to stabilize the chaotic behavior in electric drive systems.
In this chapter, various control approaches, including the time-delay feedback control, the nonlinear feedback control, the backstepping control, the dynamic surface control and the sliding mode control, are introduced to stabilize the chaos that occurs in both DC and AC drive systems.
6.1 Stabilization of Chaos in DC Drive System
6.1.1 Modeling
As shown in Figure 6.1, the time-delay feedback control method is used to stabilize chaos in a voltage-controlled DC drive system. Time-delay feedback control has some definite advantages: it does not desire a priori analytical knowledge of the system dynamics; it does not require a reference signal corresponding to the desired unstable periodic orbit (UPO); and it does not need fast sampling or a 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 (Kittel, Parisi, and Pyragas, 1995).
Figure 6.1 Voltage-controlled DC drive system with time-delay feedback control
The speed control of the DC drive is implemented by constant-frequency pulse width modulation (PWM). The ramp voltage signal 
, which functions to generate the PWM signal, is represented by:
(6.1) 
where 
, 
, and T are the lower limit, upper limit and period of the sawtooth wave. On the other hand, the speed error signal 
is given by:
(6.2) 
where g is the speed feedback gain, which is actually the overall gain of the speed encoder, frequency-to-voltage (F/V) converter, and operational amplifier 1 (OA1), ω is the actual speed, and 
is the reference speed. This speed error is chosen as the feedback variable. By using the bucket-brigade delay (BBD) line, the corresponding delayed signal 
for the time-delay feedback control is described as:
(6.3) 
where τ is the time delay. When the desired orbit is the embedded unstable period-p orbit, 
is normally chosen to be pT. Thus, the feedback control signal 
can be represented as:
(6.4) 
where η is the delayed feedback gain, which is actually the gain in the operational amplifier 2 (OA2). The operational amplifier 3 (OA3) simply adopts a unity gain. Then, 
and 
are fed to the comparator (CM), and thus generate the PWM signal to turn on and off the power switch S. When 
is larger than 
, S is turned off and the diode D conducts. Otherwise, S is turned on and D is turned off. The inductor L is connected in series with the DC motor to ensure a continuous conduction mode of operation. So, the voltage-controlled DC drive system with a time-delay feedback control can be modeled as:
It should be noted that 
comprises a delayed component, and the dynamical equation in (6.5) is actually a time-delayed differential equation. The delayed feedback only exists in the switching condition 
. As discussed in Chapter 3, the corresponding Poincaré map with multiple switching pulses can be represented by:
The switching points 
can be determined by using the switching condition described in (6.5):
Then, the switching points between the interval 
and 
are given by:
(6.8) 
The corresponding system states in (6.7) are given by:
Thus, the switching points can be calculated by using (6.6), (6.7), and (6.9), which yields:
Given the values of 
, the values of 
can be calculated by using (6.10).
For a practical voltage-controlled DC drive system, there is only a single switching pulse. So, the corresponding Poincaré map can be represented by:
When 
, the instant of switching action in (6.10) is given by:
When 
, the instant of switching action in (6.10) is given by:
Since 
can be represented by 
and 
, (6.13) is rewritten as:
It should be noted that 
is the switching conditions when 
is on the left side of 
, and 
is the switching conditions when 
is on the right side of 
. The stability of control can be described by (6.11), (6.12), and (6.14). Since τ is normally chosen to be the period T, namely p = 1, the corresponding Poincaré map can be simplified as:
(6.15) 
When 
, 
is an explicit function of 
; and when 
, 
is an implicit function of 
and 
. Let 
, the Poincaré map can be rewritten as:
where 
and 
are the Poincaré maps corresponding to the conditions that 
and 
, respectively.
6.1.2 Analysis
The Poincaré map with the time-delay feedback control is different from that of the system without control. By choosing a suitable delay feedback gain η, the chaotic behavior can be stabilized into periodic behavior. The effective range for stabilization is called as the stable domain.
6.1.2.1 Fundamental Operation
The fundamental operation of the DC drive system corresponds to the period-1 orbit, namely 
. So, the delay component in (6.16) and (6.17) become zero. Namely, 
and 
are independent of η. Therefore, the fixed point of period-1 orbit can be written as:
The characteristic multiplier of (6.20) is also independent of η. According to the implicit theorem, the Jacobian matrices of (6.18) and (6.19) are given by:
(6.21) 
(6.22) 
where
(6.27) 
Since 
, 
can be represented by:
Theorem 6.1
When the eigenvalues of a matrix 
are zero, the eigenvalues of the matrix 
are also zero.
Proof
When 
is the Jordan normalization form of 
, it results in 
. Thus, it yields 
. When the eigenvalues of 
are equal to zero, 
can be described as 
.
Hence, it deduces that 
, and the eigenvalues of 
are also zero.
Theorem 6.2
When the two eigenvalues of 
in (6.25) are zero, two of the four eigenvalues of 
are zero, and the other two eigenvalues of 
are equal to those of 
. These two eigenvalues are also equal to the eigenvalues of the matrix 
, which is expressed as:
Proof
Given 
and 
, yields 
.
From (6.23) and (6.24), it can be deduced that 
. The eigenvalues of 
are the same as those of 
. Two of the eigenvalues of 
are the eigenvalues of 
, and the other two are the eigenvalues of 
. Thus, from (6.23)–(6.26), 
can be described as:
(6.30) 
According to Theorem 1, when the eigenvalues of 
are all equal to zero, the eigenvalues of 
are zero. When 
and 
, the eigenvalues of 
are zero. From (6.23) –(6.28), 
can be described as:
(6.31) 
So, the eigenvalues of 
or 
are equal to the eigenvalues of 
in (6.29).
Because
the eigenvalues of S are equal to zero. Thus, the aforementioned theorems can be applied to a voltage-controlled DC drive system. The characteristic multipliers of a period-1 orbit can be calculated by the eigenvalues of 
in (6.29). By calculating the characteristic multipliers, the stable domain of the control parameters – namely the time-delay feedback gain versus various system parameters for effective stabilization – can be determined. It should be noted that the calculation of the eigenvalues of 
is easier than those for 
and 
. In particular, the dimension of 
is twice that of 
.
6.1.2.2 Subharmonic Operation
The stable domain for the period-p (p > 1) is illustrated by the period-2 operation. When 
, a second-order Poincaré map of the drive system can be obtained by using (6.18) and (6.19):
(6.32) 
(6.33) 
Contrary to the fixed point of the period-1 orbit, the fixed point of the period-2 orbit is dependent on η, which can be represented by:
Substituting (6.34) into (6.35), 
and 
can be deduced by solving (6.35), and the period-2 orbit 
can be obtained from (6.34). Similar to the period-1 orbit, the characteristic multipliers for the period-2 orbit are the eigenvalues of the Jacobian matrix 
which can be deduced from the equation:
(6.36) 
where
(6.37) 
(6.38) 
By calculating the characteristic multipliers, the corresponding stable domain of the control parameters can be obtained.
6.1.3 Simulation
In order to validate the aforementioned stabilization of chaos, a computer simulation is carried out. The parameters adopted for the simulation are based on a practical voltage-controlled DC drive system, namely R = 4.1 Ω, KE = 0.1356 V/rad/s, KT = 0.1324 Nm/A, J = 0.000557 kgm2, B = 0.000275 Nm/rad/s, L = 28 mH, Tl = 0.5 Nm, 
= 2.2 V, vl = 0 V, T = 6.667 ms, and 
. The delayed feedback gain η, speed feedback gain g, and supply voltage 
are the control parameters for this simulation.
Based on the above stability analysis, the stable domain of η versus g under 
= 60 V for period-1 and period-2 behaviors is depicted in Figure 6.2a, while the stable domain of η versus 
under g = 1.4 V/rad/s is depicted in Figure 6.2b. From Figure 6.2a, it can be seen that when g < 0.8 V/rad/s the stable domain of period-1 motion D1 embraces the case η = 0. This means that when g < 0.8 V/rad/s, a DC drive system without a time-delay feedback control can operate stably. Similar to the period-1 orbit, a DC drive system without a time-delay feedback control can also exhibit a stable period-2 orbit within the domain D2 when 0.8 V/rad/s < g < 1.15 V/rad/s. It also can be observed that, with the increase of g and Vs, the range of η for D1 and D2 become narrower. This feature indicates that with an increase of g and Vs, the corresponding stable domain becomes narrower and chaos is more prone to occur. The same phenomena can be observed in Figure 6.2b.
Figure 6.2 Stable domains. (a) Delayed feedback gain versus speed feedback gain. (b) Delayed feedback gain versus supply voltage
In order to illustrate the stabilization of chaos, the trajectory of armature current i versus feedback control signal 
is used for analysis. This feedback control signal is actually used to describe the motor speed. When η = 0, Vs = 60 V and g = 1.6 V/rad/s, the system operates in a chaotic mode, and the trajectory of i versus vc is as plotted in Figure 6.3a. It can be found that the trajectory exhibits a random-like but bounded behavior, with boundaries of 
[0 V, 3.3 V] and 
[1.3 A, 6.8 A]. When η is set as 0.15 and 0.11, the chaotic behavior can be stabilized into a period-1 orbit and a period-2 orbit, as shown in Figures 6.3b and 6.3c, which are in good agreement with the stable domains shown in Figure 6.2a. The corresponding boundaries of the period-1 trajectory are 
[0.66 V, 1.9 V] and 
[2.2 A, 5.8 A], and those of the period-2 are 
[−0.5 V, 3.6 V] and 
[1.4 A, 7.4 A].
6.1.4 Experimentation
According to Figure 6.1, an experimental DC drive system is prototyped. In principle, there are three main subsystems, namely a power electronic DC chopper, a motor-generator set, and an analog electronic controller. The DC chopper, consisting of a DC power supply 
, a power MOSFET switch IRFI640G, a power diode BYW29E200, and an inductor L, functions to regulate the input power flowing into the drive. The motor-generator set includes a DC motor, a DC generator, a coupler, and an electronic load, where the mechanical load torque is electronically controlled by the current sink of the electronic load. The electronic controller involves simple hardware, namely an encoder M57962L, a F/V converter LM331, three op-amps (OA1, OA2, and OA3) LM833, a bucket-brigade delay (BBD) line MN3004 and its clock MN3101, a ramp-signal generator, a comparator (CM) LM311, and a MOSFET driver DS0026.
Based on the encoder and the F/V converter, the motor speed ω is converted into an analog signal 
(with gain γ) which is then compared with the command speed signal 
to produce the error signal 
via OA1 with gain α. Hence, the speed feedback gain g equals αγ. According to the principle of delayed self-controlling feedback (Pyragas, 1992), 
and its delayed version 
are fed into OA2 with gain β to produce the perturbation signal 
which is then compared with 
to generate the desired control signal 
via OA3 with gain unity. Finally, 
is compared with the ramp signal 
(with period T and upper and lower bound voltages 
and 
) via the CM to produce the PWM switching signal for driving the power MOSFET. The core of this controller is the BBD line and the associated clock. By tuning the clock frequency via its externally connected R1–R2–C network, the BBD line can allow for a time delay varying from 2.56 to 25.6 ms. In the controller, the time delay τ is set to the switching period T which in fact corresponds to the fundamental operation (the period-1 orbit).
Based on the same conditions for computer simulation, the measured chaotic, period-1, and period-2 phase portraits are shown in Figure 6.4. It can be found that the chaotic orbit has boundaries of 
and 
, the period-1 orbit has boundaries of 
and 
, and the period-2 orbit has boundaries of 
and 
. Comparing Figures 6.4 and 6.3, the measured results are in good agreement with the simulation results. Nevertheless, the boundaries of those phase portraits still have some discrepancies, and the stabilized orbits are slightly shaking, which is due to some inevitable imperfections in the DC drive system, such as the uneven contacts of the DC commutator, the torsional oscillation of the coupler, and the phase distortion of the BBD line.
6.2 Stabilization of Chaos in AC Drive System
As discussed in Chapter 4, chaotic behavior can be observed in the permanent magnet synchronous motor (PMSM) drive system under some operating conditions. By defining
where L is the armature winding inductance, R is the armature winding resistance, B is the rotor viscous friction coefficient, J is the rotor inertia, 
is the number of PM pole-pairs, 
is the PM flux linkage, 
is the d-axis input voltage, 
is the q-axis input voltage, and 
is the load torque, the dynamical equation of the PMSM drive can be transformed into a dimensionless model which is in the form of the well-known Lorenz system (Hemati, 1994):
where 
, 
, and 
are the transformed versions of the d-axis armature current 
, q-axis armature current 
, and motor speed ω, respectively, and 
is a free parameter.
This section presents an overview of four control methods to stabilize the chaotic motion in a PMSM drive system. Namely, the nonlinear feedback control, backstepping control, dynamic surface control, and sliding mode control are employed to stabilize the chaotic motor speed to the desired value.
6.2.1 Nonlinear Feedback Control
A nonlinear feedback control has been proposed to stabilize chaos in a PMSM drive system (Ren and Liu, 2006). This nonlinear feedback control uses 
and 
as the manipulated variables. The control law is governed by:
where 
and 
are the control objectives of 
and 
respectively, while 
and 
are the corresponding controller parameters. By substituting (6.40) and (6.41) into (6.39), the dynamical equation of a PMSM drive system with a nonlinear feedback control is rewritten as:
(6.42) 
Since 
and 
are normally constant parameters, this yields 
and 
. Thus, the control law can be reduced as:
The desired speed reference 
is a constant for the PMSM drive system. Therefore, in the controller design, 
can be calculated according to the given value of 
and 
is set based on the requirement of flux. The selection of 
and 
determines the responding speed of the system. The manipulated variables 
and 
are externally accessible, while 
, 
, and 
can be measured in real time. Hence, the controller can be physically realized.
In order to assess the performance of this nonlinear feedback control, the simulation is carried out with the parameters 
= 0, 
= 5.46, 
= 20. Given 
= 5, it results in 
= 5 according to 
, and 
= 3.
The simulation is carried out with the initial values 
= 20, 
= 0.01, and 
=− 5 as well as 
. After startup, 
and 
are set to zero and remain unchanged. At the instant of t = 25 s, 
and 
are adjusted in accordance with the control law governed by (6.43) and (6.44). As shown in Figure 6.5, it can be found that the nonlinear feedback control can successfully stabilize both of the armature current components and the motor speed.
Figure 6.5 Stabilization performance of nonlinear feedback control. (a) Transformed d-axis armature current. (b) Transformed q-axis armature current. (c) Transformed motor speed
6.2.2 Backstepping Control
The backstepping control method has also been designed to stabilize the chaotic motion in a PMSM drive system (Harb, 2004). By setting 
= 0 and 
= 0, the dynamical equation (6.39) of the PMSM drive is expressed as:
The error signals of the corresponding system states are defined as:
where 
, 
, and 
are the gains of error signals. Given 
, and by the use of (6.45), the time derivative of (6.46) can be represented as:
(6.47) 
which constitutes the control law for stabilization.
Based on these error signals, a positive Lyapunov function is constructed as:
(6.48) 
By selecting the control parameters 
, 
and 
, the system stability is guaranteed, namely 
, provided that the control law is given by:
(6.49) 
Based on the same system to assess the nonlinear feedback control, the stabilization performance of this backstepping control is testified. Also, the control law is applied at the instant of t = 25 s. As shown in Figure 6.6, it can be seen that the backstepping control can successfully stabilize the system.
Figure 6.6 Stabilization performance of backstepping control. (a) Transformed d-axis armature current. (b) Transformed q-axis armature current. (c) Transformed motor speed
6.2.3 Dynamic Surface Control
The dynamic surface control has also been used to stabilize chaos in the PMSM drive system. It takes the advantage over the backstepping control because it can avoid the problem of ‘explosion of terms’ caused by the repeated differentiation of virtual input (Wei et al., 2007).
By using the control variable 
and setting 
in (6.39), the corresponding dynamical equation becomes:
The control law of this dynamic surface control includes three steps. Namely, the virtual controllers are designed in the first step and the second step, and the overall control law is designed in the third step (Wei et al., 2007).
Firstly, the dynamic surface 
is designed for 
to track the reference value 
, which is expressed as:
Hence, by differentiating (6.51) and using (6.50), the dynamics of 
is given by:
Since 
is a constant value and 
, (6.52) becomes:
(6.53) 
So, the first virtual controller is to stabilize 
by choosing 
as:
(6.54) 
where 
is the parameter of the first virtual controller. It should be noted that the dynamic surface control eliminates the need for model differentiation by passing 
through a first-order filter with a positive time constant 
:
(6.55) 
where 
serves as an estimation of 
.
Secondly, by using 
to supersede 
, the second surface 
is designed as:
Hence, by differentiating (6.56) and using (6.50), the dynamics of 
is given by:
(6.57) 
So, the second virtual controller is to stabilize 
by choosing 
as:
(6.58) 
where 
is the parameter of the second virtual controller. Consequently, 
is deduced by passing 
through a first-order filter with a positive time constant 
:
(6.59) 
Thirdly, the third surface 
is designed as:
So, by differentiating (6.60) and using (6.50), the dynamics of 
is given by:
(6.61) 
In order to stabilize 
, the final control law is designed as:
(6.62) 
where 
is the parameter of the final control law.
Based on the same system as mentioned previously, the stabilization performance of this dynamic surface control is assessed. When 
is selected and the control is activated at the instant of t = 25 s, the responses are as shown in Figure 6.7. This confirms that dynamic surface control can successfully stabilize chaos in a PMSM drive system.
Figure 6.7 Stabilization performance of dynamic surface control. (a) Transformed d-axis armature current. (b) Transformed q-axis armature current. (c) Transformed motor speed
6.2.4 Sliding Mode Control
The sliding mode control has also been developed to stabilize the chaotic behavior in a Lorenz chaotic system, such as a PMSM drive system (Yau and Yan, 2004). By setting 
and 
in (6.39), the dynamical equation of a PMSM drive system can be expressed as:
where 
is a nonlinear control variable that is a continuous nonlinear function with 
and follows the relationship:
(6.64) 
where 
and 
are nonzero positive constants. A disturbance term is defined to cancel the following nonlinear term:
(6.65) 
So, the system dynamics in (6.63) can be expressed as:
(6.66) 
For analysis, a new system state vector 
is defined, where the transformation matrix is 
. The trajectory error states are defined as 
, 
, and 
, where the transformed regulation system states are 
and 
. Thus, the error state dynamical equations are given by:
where 
. So, a sliding surface suitable for the application can be defined as:
(6.68) 
where 
is the design parameter. For a sliding mode operation, the necessary and sufficient conditions are given by:
(6.70) 
Therefore, the sliding mode dynamics can be obtained as:
(6.72) 
(6.73) 
When the design parameter 
, the stability of (6.71) is guaranteed, and 
converges to zero. According to (6.69), 
is also stable and converges to zero. It has been proved that 
will converge to zero if 
and 
converge to zero (Yau and Yan, 2004). Therefore, (
) converges to the desired value (
) when 
.
The sliding mode control law is designed as:
where
(6.75) 
In this way, a reaching condition for the sliding mode 
can be guaranteed (Yau and Yan, 2004).
In order to testify the performance of the sliding mode control, the simulation is carried out with the system parameters 
, 
, and 
. The design parameter 
is selected, while the nonlinear input is defined as:
Thus, the slope of the nonlinear sector 
and 
can be obtained, leading to the selection of 
. Based on (6.67), (6.74), and (6.76), the simulated responses of 
, 
, and 
under 
are depicted in Figure 6.8 in which the control is activated at the instant of t = 25 s. This confirms that a sliding mode control can successfully stabilize chaos in a PMSM drive system.
Figure 6.8 Stabilization performance of sliding mode control. (a) Transformed d-axis armature current. (b) Transformed q-axis armature current. (c) Transformed motor speed
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