Chapter 3

Chaos in DC Drive Systems

DC drive systems have been widely used for domestic, industrial and vehicular applications because of their technological maturity and control simplicity. In general, the speed control of DC drives can be accomplished by two methods, namely armature control and field control. In the case of permanent magnet (PM) excitation, the PM field is essentially uncontrollable. During normal operation, the armature circuit is fed by a voltage source while adopting either armature voltage control or armature current control. The voltage control has the definite advantage of simplicity and low cost, whereas the current control has the merit of direct torque control.

In this chapter, chaos in both voltage-controlled DC drive systems and current-controlled DC drive systems are investigated. The corresponding modeling, analysis, simulation, and experimentation are also discussed in detail.

3.1 Voltage-Controlled DC Drive System

The investigation into chaos in power electronic circuits was launched in the late 1980s, focusing on various kinds of switching DC–DC converters (Hamill and Jefferies, 1988; Deane, 1992; Tse, Fung, and Kwan, 1996). By extending the work to DC drive systems that involve a speed-dependent load voltage, chaotic behavior in the voltage-controlled DC drive system was first investigated in 1997 (Chau et al., 1997a). Consequently, the corresponding dynamic bifurcation (Chau, Chen, and Chan, 1997b) as well as modeling of subharmonics and chaos (Chau et al., 1997c) have also been discussed.

3.1.1 Modeling

As shown in Figure 3.1, a voltage-controlled DC chopper-fed PMDC drive system operating in continuous conduction mode is used for exemplification (Chen, Chau, and Chan, 2000a a). The corresponding equivalent circuit is shown in Figure 3.2, where the motor speed ω is controlled by constant frequency pulse width modulation (PWM).

Considering that the operational amplifier A₁ has a feedback gain g, the control signal can be expressed as:

(3.1)

where and are the instantaneous and reference motor speeds, respectively. The ramp voltage is represented by:

(3.2)

where and are, respectively, the lower and upper voltages of the ramp signal, and T is its period. Then, both and are fed into the comparator A₂ which outputs the signal to turn the power switch S on or off. When the control voltage exceeds the ramp voltage, S is off and hence the diode D comes on; otherwise, S is on and D is off. Thus, the system equation can be divided into two stages as given by:

By defining the state vector X(t) and the matrices A, E₁, E₂, E₃, E₄ as:

(3.5)

(3.6)

the system equation given by (3.3) and (3.4) can be rewritten as:

(3.7)

and the switching condition can be expressed as:

(3.8)

As k changes value when while is time dependent, the system given by equation (3.7) is in fact a time-varying state equation. Thus, this DC drive system is a second-order nonautonomous dynamical system.

3.1.2 Analysis

The analysis of system chaotic behavior begins with the solution of (3.7) in a continuous-time domain, namely X(t). Then, the iterative function that maps this X(t) at t = nT to its successive one at t = (n+1)T is defined as P: :

(3.9)

which is the so-called Poincaré map. The generalized Poincaré map can fully describe the system behavior using numerical simulation, whereas the specific one can analytically describe the system behavior in terms of periodic orbits and stability.

3.1.2.1 Solution of System Equation

Given an initial value X(t₀), the continuous-time solution of the system equation given by (3.7) can be expressed as:

(3.10)

where is known as a state transition matrix. By defining the parameters α and Δ as:

(3.11)

the eigenvalues λ₁, λ₂ of A can be expressed as:

(3.12)

(3.13)

(3.14)

Hence, the corresponding can be obtained as:

(3.15)

(3.16)

(3.17)

where 1 is the identity matrix and .

3.1.2.2 Derivation of Generalized Poincaré Map

During each cyclic operation of the drive system, there are two possible situations – either a skipping cycle because of the absence of intersection between and , or an intersecting cycle in which there is at least one intersection between and .

For the skipping cycle, S does not change its state, remaining either on or off. Making use of (3.9), the corresponding Poincaré map P can be easily deduced from (3.10) as given by:

(3.18)

For the intersecting cycle, S may change m times when crosses by m times within T. This is known as multiple pulsing. Thus, the intersections occur at:

(3.19)

Hence, the corresponding Poincaré map can be expressed as the following iterative functions:

(3.20)

(3.21)

Firstly, the derivation is based on m = 1 where there is only one intersection within T. When at t = nT is greater than or equal to , the drive operates in Stage 1 from nT to nT + δ₁T, at which point the switching condition is satisfied, and then in Stage 2 to (n+1)T. Using (3.8) and (3.10), δ₁ can be determined by evaluating the solution of the following transcendental equation:

(3.22)

Hence, the corresponding Poincaré map can be written as:

(3.23)

(3.24)

On the contrary, when at t = nT is lower than or equal to , the system operates in Stage 2 first and then in Stage 1. The corresponding Poincaré map can similarly be obtained as:

(3.25)

(3.26)

(3.27)

Similar to the derivation for m = 1, the Poincaré map for m > 1, in which there are more than one intersection within T, can be determined as:

(3.28)

(3.29)

(3.30)

where , and k equals 1 or 2 depending on whether is over or not.

It should be noted that the above Poincaré map relates to generalized mapping which covers all possible solutions, such as real and complex roots due to different system parameters and conditions. Thus, the generalized Poincaré map can be considered as the mapping method for any second-order dynamical system using similar mathematical models. Moreover, the derivation can readily be extended to those higher-order dynamical systems involving power switches.

3.1.2.3 Analysis of Periodic Orbits

The generalized Poincaré map is so general that it includes cycle skipping and multiple pulsing and, owing to the presence of these cases, it is very inconvenient to analyze the steady-state periodic orbits and their stability. Also, the presence of multiple pulsing can greatly increase the switching losses, which should be avoided by using a latch or sample-and-hold. Therefore, instead of using the generalized case, the detailed analysis of periodic orbits is focused on the case in which the orbits cross the ramp signal once per cycle, namely one intersection within T. The corresponding mapping is known as the specific Poincaré map.

The steady-state periodic solution of the DC drive system can be a period-1 orbit X∗, or a period-p orbit with p > 1. It should be noted that the period-1 orbit is the fundamental orbit, whereas the period-p orbit means the (1/p)th subharmonic orbit. The corresponding specific Poincaré maps are described as:

(3.31)

(3.32)

Firstly, the period-1 orbit is analyzed. Since m = 1 and the orbit must lie between and , the corresponding Poincaré map can be deduced from (3.23) and (3.24) as given by:

(3.33)

Substituting (3.33) into the mapping given by (3.31) and taking , the period-1 orbit can be obtained as:

(3.34)

After substitution of (3.34) into (3.22), δ₁ can be determined from the corresponding transcendental equation:

(3.35)

Hence, provided that δ₁ (0, 1), X∗ can be obtained from (3.34).

Secondly, the period-p orbit is analyzed. By defining as the p duty cycles within p periods with p > 1, it indicates that crosses p voltage ramps at . Based on the specific Poincaré map derived in (3.33), the p-fold iterative mapping can be formulated as:

(3.36)

By using (3.32) and (3.33), the period-p orbit can be obtained as:

(3.37)

(3.38)

(3.39)

where and . By substituting (3.37) and (3.38) into (3.39), d can be determined. Hence, provided that (0, 1) for i = 1, , p, can then be obtained from (3.37) and (3.38).

Due to the cyclic property of the period-p orbit, {},, {} are other period-p orbits which correspond to the same subharmonic frequency. If , the period-p orbit becomes a period-1 orbit, indicating that the period-1 is a particular case of the period-p orbit.

3.1.2.4 Stability and Characteristic Multipliers

Period-1 and period-p orbits may be stable or unstable. It is known that the stability type of a fixed point of mapping corresponds to the stability type of the underlying periodic solution, and the fixed point of mapping is stable if and only if its characteristic multipliers all lie within the unit cycle in the complex plane.

For a period-1 orbit that is a fixed point of the specific Poincaré map given by (3.33), its characteristic multipliers are eigenvalues of the Jacobian matrix of that mapping, which is given by:

(3.40)

where . According to the implicit-function theorems, can be deduced from (3.22) as:

(3.41)

For a period-p orbit , is a fixed point of the p-fold iterative specific Poincaré map. Therefore, its characteristic multipliers are the eigenvalues of the Jacobian matrix of that mapping, which is given by:

(3.42)

By substituting (3.38) into (3.39)

(3.43)

Hence, according to the implicit-function theorems, the partial derivative in (3.42) can be expressed as:

(3.44)

where

(3.45)

(3.46)

Notice that in (3.45) becomes a zero matrix when .

3.1.3 Simulation

To illustrate the derived Poincaré map, computer simulations are carried out. The simulation parameters are based on the values of a practical DC chopper-fed PMDC drive system, namely T = 10 ms, g_i = 1.1 V/A, g_ω = 0.54 V/rad/s, , R = 2.9 Ω, L = 53.7 mH, , = 0.1324 Nm/A, B = 0.000275 Nm/rad/s, J = 0.000557 Nm/rad/s², = 0.39 Nm, and = 105 rad/s. The resulting eigenvalues of matrix A are and . This indicates that although this open-loop system shows no oscillating dynamics, the corresponding closed-loop system may exhibit not only oscillating dynamics but also subharmonics and even chaos.

3.1.3.1 Bifurcation Diagrams using Numerical Computation

A natural numerical tool to obtain the steady-state solution of the generalized Poincaré map is known as the brute-force method (Parker and Chua, 1987) – repeating the iteration of the map until the transient has died out or the steady state has been reached. This method has the advantage of simplicity, but may suffer from tedious simulation due to long-lived transients. If K iterations are spent for transient operation, and N points are needed to describe an orbit while M steps are used to depict a bifurcation diagram, then the total number of iterations for constructing a bifurcation diagram will be (K+N)M.

By employing a brute-force algorithm to compute the generalized Poincaré map given by, bifurcation diagrams of ω and i versus and g can be produced as shown in Figures 3.3–3.6, respectively. As can be seen in these figures, the system exhibits a typical period-doubling route to chaos, which is valid for both chaotic speed and current.

The trajectory of i versus ω for a period-1 orbit with and g = 0.25 V/rad/s is shown in Figure 3.7. Moreover, the chaotic trajectory and its Poincaré section of i versus ω with and g = 0.9 V/rad/s are shown in Figures 3.8 and 3.9, respectively. It should be noted that the chaotic trajectory represents the phase portrait of the system solution in a continuous-time domain during chaos, whereas its Poincaré section is a set of sampled points describing the chaotic solution of the Poincaré map in the phase plane. In a Poincaré section, a period-1 orbit is denoted by a single point, a period-p orbit by p points, and a chaotic orbit by an intricate pattern of points.

Moreover, if g is large enough, a drive system with a higher switching frequency and a lower armature inductance can still exhibit chaos. For example, if the switching frequency, armature inductance, and armature resistance are changed to 2.5 kHz, 3 mH, and 1 Ω respectively, the corresponding bifurcation diagram of versus g shown in Figure 3.10 indicates that the drive system will operate in chaos when g reaches 24.3 V/rad/s. Its chaotic trajectory with g = 25 V/rad/s is shown in Figure 3.11. However, because high switching frequency operation will complicate the system behavior due to parasitic reactances, and the associated high gain value will amplify the noise caused by device switching and motor commutation, a lower switching frequency operation is therefore adopted to facilitate the illustration of theoretical and experimental results.

3.1.3.2 Bifurcation Diagrams using Analytical Approach

Based on the derived relationship in (3.34) and (3.35), the evolution of δ₁ and the corresponding ω of a period-1 orbit with respect to is shown in Figures 3.12 and 3.13, respectively. Hence, by using (3.40) and (3.41), the corresponding characteristic multipliers are shown in Figure 3.14, in which one of the magnitudes is greater than unity when . This indicates that the period-1 orbit is unstable when .

Similarly, based on (3.36)–(3.39) with p = 4, the evolution of d = (δ₁, δ₂, δ₃, δ₄) and the corresponding ω of the period-4 orbit with respect to , are shown in Figures 3.15 and 3.16, respectively. It can be found that a period-2 orbit occurs when and then bifurcates to the period-4 orbit when . Since one of the duty cycles is greater than unity after , the period-4 orbit only exists between 66.5 V and 69.5 V. According to the characteristic multipliers of the period-4 orbit shown in Figure 3.17, it can be found that a period-2 orbit with 54.6 V < V_in ≤ 69.5 V is always stable, whereas a period-4 orbit with 66.5 V < V_in ≤ 69.5 V is stable only when V_in ≤ 68.4 V. When V_in ≤ 54.6 V, the corresponding characteristics in Figures 3.15 and 3.16 are identical to those in Figures 3.12 and 3.13, respectively. This indicates that the stable period-1 orbit is a subset of the period-p orbit (p > 1).

It should be noted that the above analytical results closely agree with the bifurcation diagram (Figure 3.3) resulting from numerical computation. The required computational time based on the derived analytical solution is extremely less than that required for computation using a numerical brute-force algorithm. Moreover, the analytical solution can facilitate the identification of the desired stable operating ranges for different system parameters and conditions.

3.1.4 Experimentation

In order to further verify the theoretical analysis, an experimental DC drive system is prototyped as shown in Figure 3.18. The prototype consists of two identical PMDC motors, namely M₁ and M₂, which are directly coupled together. M₁ acts as the motor that is fed by a voltage-mode DC chopper, while M₂ performs as the generator load that is controlled by an electronic load operating as a constant current sink . The built-in tachogenerator in M₁ is used to provide the speed feedback. The mechanical load torque can be represented by , where is the friction torque.

The power stage consists of a power MOSFET IRF740 driven by a driver chip DS0026, and a fast-recovery diode BYW29E-200. The electronic controller mainly consists of two operational amplifiers LM833 and a comparator LM311. Between them, an optical coupler 6N137 is adopted in order to avoid the possible coupling of switching noise. Moreover, a switching frequency as low as 100 Hz is selected to ensure that all parasitic reactances can be neglected. Since the motor has low armature inductance, an additional inductor is connected in series with the armature of M₁ to ensure that the system operates in a continuous conduction mode. All parameters of this drive system are as follows: v_l = 0 V, , T = 100 ms, g = 0.7 V/rad/s, V_in = 60 V, R = 2.8 Ω, L = 537 mH, K_E =0.1356 V/rad/s, K_T = 0.1324 Nm/A, B = 0.000275 Nm/rad/s, J = 0.000557 Nm/rad/s², T_l = 0.38 Nm, and .

For the sake of simplicity and clarity, the speed feedback control signal is measured instead of the actual motor speed. In fact, they have the same shape and obey a linear relationship, as given by (3.1). The measured trajectory and waveforms of and i with V_in = 60 V and g = 0.25 V/rad/s are shown in Figure 3.19. This illustrates that the system operates in a period-1 orbit. Also, it can be found that i roughly lies between 2 A and 4.4 A while is between 1.3 V and 1.6 V (equivalent to ω between 1005 rpm and 1016 rpm). Since the sampling is always at the transition from Stage 1 to 2, the sampled magnitude of i is the maximum value of 4.4 A and the corresponding is roughly of 1.45 V (equivalent to ω of 1010 rpm). By comparing these experimental results with the bifurcation diagrams and trajectory shown in Figures 3.5–3.7, the theoretical analysis is verified. The superimposed high-frequency components in should be due to the torsional oscillation of the shaft coupling.

Moreover, by selecting g = 0.9, the measured trajectory and waveforms of i and shown in Figure 3.20 illustrate that the drive system operates in chaos. It can be found that the boundaries of i and ω, namely between 0.7 A and 6.5 A and between 960 rpm and 993 rpm, respectively, agree with those in the theoretical chaotic trajectory shown in Figure 3.8.

3.2 Current-Controlled DC Drive System

Similar to a current-controlled switching DC–DC converter (Deane, 1992; Tse, Fung, and Kwan, 1996), a current-controlled DC drive system is more prone to chaos and instability. Unlike the switching DC–DC converter, the DC drive system preferably adopts current control since it can offer direct torque control. Thus, it is highly appropriate for the investigation, both numerically and analytically, of the chaotic behavior of a current-controlled DC drive system. By this method the stable and chaotic ranges of system parameters can be obtained, and the occurrence of chaos can be avoided.

3.2.1 Modeling

Similar to the voltage control scheme, the DC buck-chopper-fed PMDC drive system is adopted to exemplify the current control scheme (Chen, Chau, and Chan, 2000b b). The corresponding schematic and equivalent circuit is shown in Figure 3.21.

Considering that the operational amplifier A₁ and A₂ have gains and , the speed and current control signals and can be expressed as:

(3.47)

(3.48)

where , , and are armature current, rotor speed, and reference speed of the DC motor, respectively. Then, both and are fed into the comparator A₃ which outputs the pulse to the reset of an R-S latch. The power switch S is controlled by this R-S latch which is set by clock pulses of period T. Once the latch is set by the clock pulse, S is turned on and diode D is off. S then remains closed until exceeds , where the latch begins to reset. When the latch is reset, S is turned off and D is on. S then remains open until the arrival of the next clock pulse, where it closes again. If both set and reset signals occur simultaneously, the reset will dominate the set and S will remain open until the occurrence of another clock pulse. Therefore, the system equation can be divided into two stages, as given by:

where R is the armature resistance, L is the armature inductance, is the DC supply voltage, is the back-EMF constant, is the torque constant, B is the viscous damping, J is the load inertia, and is the load torque.

By defining the state vector X(t) and the matrices A, E₁, E₂, E₃, E₄ as:

(3.51)

(3.52)

the system equation given by (3.49) and (3.50) can be rewritten as:

(3.53)

By using (3.47) and (3.48), the switching condition can be expressed as:

(3.54)

It should be noted that (3.53) is a time-varying linear state equation switching between two stages. As its switching condition, given by (3.54), depends on the external speed reference and the internal state vector, the whole system described by both (3.53) and (3.54) exhibits nonlinear dynamics.

3.2.2 Analysis

The analysis of a system's chaotic behavior begins with the solution of (3.53) in a continuous-time domain, namely X(t). Then, the solution of the system is described by a sequence of samples n =0, 1, 2, . Differing from the conventional discretization of a continuous-time state equation, the successive sample may not be taken at (n+1)T. The sample occurs at (n+m)T when there is a change of switching state after m (m ≥ 1) clock pulses, because there is no intersection between and within (m − 1) clock pulses (Hamill, Deane, and Jefferies, 1992). The corresponding mapping from to its successive sample is known as a Poincaré map. Hence, a Poincaré map that maps a sample of X(t) at t = nT to its successive one at t = (n+m)T is defined as P: :

(3.55)

It should be noted that the Poincaré map given by (3.55) is a generalized case with m ≥ 1, which can fully describe the system behavior. When m > 1, the solution of (3.55) can only be solved by using numerical simulation. On the other hand, focusing on a specific case with m = 1, the solution can be analytically solved to allow the system behavior to be described in terms of periodic orbits and stability.

3.2.2.1 Solution of System Equation

Given an initial value X(t₀), the continuous-time solution of the system equation given by (3.53) can be expressed as:

(3.56)

For a practical DC drive system, the matrix A given by (3.51) always yields a positive and hence is always invertible. Moreover, even ignoring the positive BR term, can be written as where is the mechanical time constant and is the electrical time constant. As is from tens of milliseconds to several seconds and is from tens of microseconds to tens of milliseconds, and hence are seldom close to zero. Thus, the continuous-time solution given by (3.56) can be rewritten as:

(3.57)

where is the state transition matrix. By defining the parameters α and Δ as:

(3.58)

the eigenvalues λ₁, λ₂ of A can be expressed as:

(3.59)

(3.60)

(3.61)

Hence, the corresponding can be obtained as:

(3.62)

(3.63)

(3.64)

where 1 is the identity matrix and .

3.2.2.2 Derivation of Generalized Poincaré Map

Since is always sampled at the beginning of the clock pulse that changes S from off to on, the drive system always operates in Stage 1 first and then in Stage 2 for each sampling interval. By defining the intervals of Stage 1 and Stage 2 as and , respectively, the interval of the Poincaré map becomes . Thus, and can be directly deduced from (3.57) as given by:

(3.65)

(3.66)

By substituting (3.65) into (3.54), δ can be determined based on the solution of the following transcendental equation:

(3.67)

On the other hand, m can be deduced as the minimum integer that is larger than δ while fulfilling . Hence, the Poincaré map can be written as:

(3.68)

It should be note that the derived Poincaré map is a generalized mapping which covers all possible solutions such as real and complex roots due to different system parameters and operating conditions. Thus, this generalized Poincaré map can be considered as the mapping for any second-order dynamical systems using similar mathematical models, such as other DC drive systems. Moreover, the derivation can readily be extended to those higher-order dynamical systems involving power switches.

3.2.2.3 Analysis of Periodic Orbits

Based on the above generalized Poincaré map, the dynamic bifurcation of the drive system can readily be investigated by employing a brute-force method. Thus, different bifurcation diagrams with respect to different system parameters can be obtained. However, as each iterative computation of the generalized Poincaré map needs to solve the transcendental equation, the corresponding numerical simulation is usually very tedious. In order to avoid the lengthy computation and to attain an insight into the periodic solution, the analysis can be focused on the case where the interval of mapping is the same as the clock cycle, mathematically m = 1. The corresponding mapping is known as the specific Poincaré map.

The steady-state periodic solution of the drive can be a period-1 orbit X∗, or a period-p orbit with p > 1. The corresponding specific Poincaré maps are described as:

(3.69)

(3.70)

Firstly, the period-1 orbit is analyzed. Since m = 1, the corresponding Poincaré map can be obtained from (3.68) as given by:

(3.71)

Substituting (3.71) into the mapping given by (3.69) and taking , the period-1 orbit can be deduced as:

(3.72)

After substitution of (3.72) into (3.67), δ can be determined from the corresponding transcendental equation:

(3.73)

Hence, provided that δ (0,1), X∗ can be obtained from (3.72).

Secondly, the period-p orbit is analyzed. By defining as the p duty cycles within p periods of clock pulses with p > 1, the p-fold iterative mapping can be formulated from (3.71) as:

(3.74)

By using the definition in (3.70) and (3.71), the period-p orbit can be obtained as:

(3.75)

(3.76)

(3.77)

where and . By substituting (3.75) and (3.76) into (3.77), d can be determined. Hence, provided that (0, 1) for i = 1, , p, can then be obtained from (3.75) and (3.76).

Due to the cyclic property of the period-p orbit, {}, , {} are other period-p orbits which correspond to the same subharmonic frequency. If , the period-p orbit becomes a period-1 orbit, indicating that the period-1 orbit is a particular case of the period-p orbit.

3.2.2.4 Stability and Characteristic Multipliers

Both period-1 and period-p orbits may be stable or unstable. Hence, the corresponding characteristic multipliers must be further calculated in order to test the stability of the orbits.

For a period-1 orbit that is a fixed point of the specific Poincaré map given by (3.71), its characteristic multipliers are eigenvalues of the Jacobian matrix of that mapping, which is given by:

(3.78)

where . According to the implicit-function theorems, can be deduced from (3.67) as:

(3.79)

For the period-p orbit , is a fixed point of the p-fold iterative specific Poincaré map. Therefore, its characteristic multipliers are the eigenvalues of the Jacobian matrix of that mapping, which is given by:

(3.80)

By substituting (3.76) into (3.77):

(3.81)

Hence, according to the implicit-function theorems, the partial derivative in (3.80) can be expressed as:

(3.81)

where

(3.82)

(3.83)

Notice that in (3.82) becomes a zero matrix when .

3.2.3 Simulation

To illustrate the derived Poincaré map, computer simulations are carried out. The simulation parameters are based on the same PMDC drive system that was adopted for the voltage-controlled simulation. Similar to Section 3.1.3, the resulting eigenvalues of matrix A are and , indicating that although this open-loop system shows no oscillating dynamics, the corresponding closed-loop system may exhibit not only oscillating dynamics but also subharmonics, and even chaos.

3.2.3.1 Bifurcation Diagrams using Numerical Computation

By employing the brute-force method to compute the generalized Poincaré map (Parker and Chua, 1989), the bifurcation diagrams of ω and i versus and can be calculated, as shown in Figure 3.22. The corresponding δ with respect to and are also shown in Figure 3.23. As can be seen in these figures, the system exhibits a chaotic behavior, which is valid for both chaotic speed and current. It is interesting to note that the period-2 orbit of both ω and i versus bifurcates to a period-3 orbit when is reduced to 43.4 V, whereas one branch of the period-4 orbit of both ω and i versus terminates when is increased to 1.05 V/rad/s. The reason is due to the fact that δ has to be positive, resulting in the discontinuities at 43.4 V and 1.05 V/rad/s, as shown in Figure 3.23.

The bifurcation diagrams function to illustrate the occurrence of subharmonics and chaos with respect to the variation of system parameters. In order to attain an insight into the subharmonic and chaotic behaviors, both time-domain waveforms and trajectories are investigated. For the sake of clarity, the speed and current control signals, and , are used to represent the speed and current i, respectively. In fact, they simply obey linear relationships as given by (3.47) and (3.48). The simulation waveforms of , and clock pulses, as well as the trajectory of versus for the period-1 orbit with and g_ω = 0.54 V/rad/s, are shown in Figure 3.24. This illustrates that the period-1 trajectory has boundaries of from 2.2 V to 5 V and from 4.7 V to 5.2 V. When V_in = 51 V, the system operates in a period-2 orbit, as shown in Figure 3.25, in which lies between 1.8 V and 5.2 V while is between 4.5 V and 5.5 V. Moreover, when V_in = 35 V, the system is in chaos. The corresponding chaotic waveforms and trajectory are shown in Figure 3.26. To further illustrate its chaotic behavior, a system Poincaré section consisting of 4000 sampling points is also shown in Figure 3.26.

3.2.3.2 Bifurcation Diagrams using Analytical Approach

Based on the derived relationship in (3.75)–(3.77) and (3.80)–(3.83), the bifurcation diagram of i with respect to , as well as the corresponding duty cycle δ and characteristic multipliers , are shown in Figure 3.27 in which period-1 and period-2 orbits are both involved. It can be found that the system operates in period-1 when g_ω < 0.48 V/rad/s. While gradually increases to 0.48 V/rad/s, one of the magnitudes of approaches unity, and the system begins to bifurcate to a period-2 orbit. Since one of the duty cycles of the period-2 orbit is equal to unity when = 1.1 V/rad/s, the orbit only exists between 0.48 V/rad/s and 1.1 V/rad/s. According to the characteristic multipliers of the period-2 orbit, it can be found that the period-2 orbit lying 0.48 V/rad/s < g_ω ≤ 1.1 V/rad/s is stable only when g_ω ≤ 1 V/rad/s. This bifurcation diagram, resulting from analytical modeling, closely agrees with that shown in Figure 3.22d obtained by numerical computation. In fact, without duplicating the figures, other bifurcation diagrams, namely i versus as well as ω versus and , can be obtained via analytical modeling and have the same patterns as shown in Figure 3.22.

It should be noted that the required computational time based on the derived analytical solution is much less than that required for computation using the numerical algorithm. Increasingly, the analytical solution can facilitate the identification of the desired stable operating ranges for different system parameters and conditions.

3.2.3.3 Identification of Stable Operating Ranges

For a practical DC drive system, the operating point should be designed to locate on the stable period-1 orbit. The corresponding stability is governed by the period-1 orbit of the specific Poincaré map described by (3.72) and (3.73) as well as the characteristic multipliers given by (3.78) and (3.79). By substituting into (3.73), δ is simply expressed as:

(3.84)

Due to this explicit expression given by (3.84), the period-1 orbit and its characteristic multipliers can be easily calculated by using (3.72), (3.78), and (3.79). For a given set of system parameters, the period-1 orbit is stable if and only if the magnitudes of all characteristic multipliers resulting from (3.78) are less than unity.

Based on the above procedure, the stable operating regions for typical system parameters, namely , , and , are determined. As shown in Figure 3.28, it indicates that there are different relationships between and as well as and governing the system's stability. Namely, the stable range of the speed-feedback gain not only depends on the load torque but also on the input voltage. Stable ranges of other system parameters can similarly be determined by using (3.72), (3.78), (3.79), and (3.84).

3.2.4 Experimentation

The experimental set-up is based on the current-controlled DC drive system shown in Figure 3.21a. Similar to Section 3.1.4, the mechanical load is realized by another DC machine which operates in the generator mode. The armature circuit of this machine is then connected to an electronic load which serves as a controllable current sink. Thus, can be electronically controlled to keep at the desired value. As the DC motor and the mechanical load are directly coupled together by a shaft coupling unit, this two-mass mechanical system inevitably exhibits mechanical vibration, also called torsional oscillation. Based on the system parameters, the corresponding mechanical resonant frequency is found to be about 1.1 kHz (Sugiura and Hori, 1996). Since the system switching frequency is selected as 100 Hz, which is much less than the mechanical resonant frequency, the effect of torsional oscillation on the system dynamics is ignored in both theoretical analysis and numerical simulation. As shown in Figures 3.29–3.31, there are high-frequency ripples (about 1.1 kHz), due to torsional oscillation, superimposing on the measured waveforms of . The magnitude of these ripples is insignificant compared with that of the steady-state periodic solutions.

The measured trajectory and waveforms of and with V_in = 60 V and g_ω = 0.54 V/rad/s are shown in Figure 3.29. This illustrates that the system operates in a period-1 orbit in which lies between 2.7 V and 5 V while lies between 4.7 V and 5.2 V. When V_in = 51 V, the system operates in a period-2 orbit, as shown in Figure 3.30, in which lies between 2.3 V and 5.2 V while lies between 4.5 V and 5.5 V. In a comparison with those shown in Figures 3.24 and 3.25, the measured results and the theoretical prediction have a good agreement.

Moreover, by selecting V_in = 35 V, the measured trajectory and waveforms of and shown in Figure 3.31 illustrate that the drive system is in chaotic operation. For the period-1 and period-2 orbits, the measured trajectory and waveforms were directly compared with the theoretical predictions, but in the chaotic case the measured chaotic trajectory and waveforms could not be compared with the theoretical predictions because the chaotic behavior was not periodic and, therefore, the experimental measuring period could not be the same as the theoretical analyzing period. Also, its characteristics are extremely sensitive to the system initial conditions. Nevertheless, it can be found that the measured boundaries of the chaotic trajectory shown in Figure 3.31 resemble the theoretical prediction in Figure 3.26.

References

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