2009 International Conference on Advances in Recent Technologies in Communication and Computing
A Non-Iterative Controller Design for a BLDC Drive System C.Ganesh
S.Jeba
R.Saranya
S.Geethu
S.K. Patnaik
Department of Electrical and Electronics Engineering Sri Ramakrishna Institute of Technology Coimbatore, India
[email protected]
Department of Electrical and Electronics Engineering Anna University Chennai, India
3. Iron losses are negligible. 4. The motor is unsaturated. Based on the above assumptions, the three-phase input voltages are expressed as follows.
Abstract— Brushless DC (BLDC) Drive systems find immense applications in control systems. One such application is position control that requires very good stability limits, steady state and transient responses. Hence brushless dc drive systems should respond to meet these requirements. In this paper, few BLDC motor drive systems are considered. It is found that such systems cannot yield desired performance when conventional controller design procedure is employed. Hence, a modified design methodology for controller design of such systems is suggested. Simulation results of responses of BLDC drive systems are obtained for various desired specifications. Experimental results of responses of a BLDC drive system are obtained for various desired specifications using the proposed and conventional design methodologies to justify the simulation results. It is concluded that proposed design technique is more suitable for the controller design of such BLDC drive systems. Moreover, it can be understood that the controller design technique is non-iterative.
di va = Ria + L a + ea dt
di vb = Rib + L b + eb dt dic vc = Ric + L + ec dt The Electromagnetic torque is expressed as 1 Te = (eaia + ebib + ecic ) ω
INTRODUCTION
The controllers such as PID controller and compensating networks are used in real time control system applications for improving performance to the expected levels [1]. It is a common practice to design compensators for such applications using simple lag, lead networks, and obtain the desired transient and steady responses with safe stability margins. Controller for BLDC motor drive systems using lag and lead compensators is designed for obtaining the desired specifications [2]. It is found that such compensators are more suited for the systems with only real poles. However, there are number of systems such as BLDC systems that contain real as well as complex poles. These compensators are found to be unsuitable for such systems, since the desired specifications cannot be met. Hence, a modified design methodology for obtaining the desired performance of BLDC systems with real as well as complex poles is presented in this paper. II.
(3)
(4)
TL(s)
Ea(s)
+ -
1 Ls + R
Kt
+ +
1 Js + B
ω(s) 1
MODELLING
The BLDC motor drive system is modelled based on the following assumptions [3] [4]. 1. All the stator phase windings have equal resistance per phase and constant self and mutual inductances. 2. Power semiconductor devices are ideal. 978-0-7695-3845-7/09 $26.00 $25.00 © 2009 IEEE DOI 10.1109/ARTCom.2009.86
(2)
The electromagnetic torque can also be expressed as 2 EI (5) Te = ω The electromagnetic torque can be expressed in terms of mechanical parameters as dω Te = TL + J + Bω (6) dt where va, vb and vc are the stator input voltages of phase a, b and c respectively; ea, eb and ec are the back–emfs of phase a, b and c respectively; ia, ib and ic are the phase currents of phase a, b and c respectively, R and L are per phase resistance and inductance of each stator winding, TL is the load torque, J is inertia, ω is angular speed, B is viscous damping coefficient, Kb is back–emf constant, Kt is torque constant, Ea(s) is armature voltage and θ(s) is position. E = ea = eb = ec (7) I = ia = ib = ic (8) Block diagram of the BLDC motor [3] is shown in figure 1.
Keywords- controller; settling time; overshoot; phase margin; pwm
I.
(1)
Kb
Figure 1. Block diagram of BLDC motor
141
s
θ(s)
In this paper, various parameters of BLDC motors are listed in Table I [5]. Table II lists their respective transfer functions of BLDC motors [3].
Step Response 1.4
1.2
1
BLDC Motor 1
BLDC Motor 2
No. of Poles
8
4
Rated Voltage (Volts)
24
36
Rated Speed (rpm)
4000
4000
Rated Torque (Nm)
Parameter
Output
PARAMETERS OF BLDC MOTORS
0.8
0.6
0.4
Rise Time: 3.19msecs Settling Time: 66msecs Overshoot: 39%
0.2
0
0
0.02
0.04
0.06
0.08
0.1
0.12
Time (sec)
0.125
0.11
Torque Constant (Nm/A)
0.036
0.063
Resistance per Phase (Ohms) Inductance per Phase (H) Rotor Inertia (gm-cm2)
1.08 0.0016 48
2.25 0.0067 7.5
Figure 3.
Step response of compensated system Frequency Response
50 Magnitude (dB)
TABLE I.
0 -50 -100
Gain Margin: 5.92 dB Phase Margin: 19.8 deg.
-150 -45
System BLDC Motor 1 BLDC Motor 2 III.
Phase (deg)
-90
TABLE II.
T SYSTEM TRANSFER FUNCTIONS
Transfer Function θ(s)
Compensator
System
-225 2
10
3
4
10
5
10
10
Frequency (rad/sec)
Figure 4. Frequency response of compensated system
Failure of the conventional lead compensator design procedure is attributed to the larger phase angle reduction contribution by the complex poles at the gain cross over frequency. Moreover, sufficient phase lead cannot be provided to compensate for this large phase angle reduction. Application of conventional lag compensator design technique [9] yields phase margin of 55.4 degrees and gain margin of 10.5 dB. It seems that these results are better than that obtained by the lead compensator. The step and frequency responses of this compensated system are shown in figure 5 and figure 6.
SHORT COMINGS OF CONVENTIONAL COMPENSATOR DESIGN
+ -
-180
-270 1 10
0.036 Ea(s) 7.48*10−9s3 + 5.184*10−6s2 +1.296*10−3s θ(s) 0.063 = Ea(s) 5.062*10−8s3 +1.688*10−5s2 +3.969*10−3s =
Normally, a lead compensator is designed for a system with sluggish transient response whereas a lag compensator is designed for a system with poor steady state response [6]. Lead compensator is best suited for systems with only real poles [7]. It does not improve the transient response of systems with complex open loop poles. A typical BLDC drive system with the open loop transfer function of motor 2 in table I is considered to justify this shortcoming of conventional lead compensator design. Block diagram of a typical compensated system is shown in figure 2.
Step Response 1.4 1.2 1 Output
θR(s)
-135
θ(s)
0.8 0.6
Rise Time: 10.3msecs Settling Time: 99.3msecs Overshoot: 17.7%
0.4 0.2 0
0
0.02
0.04
0.06
0.08
0.1
0.12
Time (sec)
Figure 2. Block diagram of compensated system
Figure 5. Step response of compensated system
The system is to be compensated to meet the desired specifications: Kv≥285 sec-1, Phase margin=55 degrees and Gain margin ≥ 8 dB. Application of conventional lead compensator design technique [8] yields only phase margin of 19.8 degrees and gain margin of 5.92 dB. The step and frequency responses of this compensated system are shown in figure 3 and figure 4. These results do not match with the desired specifications.
It is found from the results that the compensated system is more oscillatory. Further, a correction value has to be adjusted on a trial and error basis to get the desired specifications. Hence, such a procedure is tedious and time consuming [10]. A new compensator design procedure is suggested in this paper to overcome the shortcomings of the conventional compensator design [11] [12].
142
are found to be 8.55dB and 54.5degrees respectively. Hence, the proposed design methodology has satisfied the desired frequency domain specifications.
Frequency Response
0
-50
Gain Margin: 10.5 dB PhaseMargin: 55.4 deg.
Step Response 1.4
-100 -90
1.2
-180
0.8
1 Output
Phase (deg)
Magnitude (dB)
50
-270 0 10
1
10
2
10
3
10
4
0.6 0.4
10
Rise Time: 8.82msecs Settling Time: 55msecs Overshoot: 18.1%
Frequency (rad/sec)
0.2
Figure 6. Frequency response of compensated system
IV.
0
(1 + T2 s )
0.08
0.1
0.12
0.14
0.16
0.18
Frequency Response
Magnitude (dB)
100 0 -100
Phase (deg)
-200 -90
(9)
-135 -180 -225 -270 -2 10
Gain Margin: 8.55 dB Phase Margin: 54.5 deg. -1
10
0
10
1
10
2
10
3
10
4
10
Frequency (rad/sec)
Figure 8. Frequency response of compensated system
Comparison of these results with the results obtained from the conventional lag and lead compensators ensures that the modified procedure could yield better responses. This is because the response obtained is much faster with fewer oscillations compared to that of conventional compensators. Moreover, no iterative procedure is required as in the case of lag compensator design. Therefore, this procedure is simple, powerful and less time consuming.
Step 5: Gain of the compensated system transfer function at the desired gain crossover frequency can be expressed as, (1 + jϖT1) Gp( jϖ ) = 1 (1 + jϖT2 )
0.06
200
Step 4: Determine the gain of the uncompensated system, at the desired gain crossover frequency, ωg. Let this gain be Gp( jϖ ) ωg .
Gpc( jϖ ) = Gc
0.04
Figure 7. Step response of compensated system
A new compensator design procedure is proposed below to meet the desired specifications such as phase margin (PM), gain margin (GM) and steady-state error [13]. Step1: Determine the required gain (Gc) of the system to meet the steady state requirement. Step2: Choose the gain crossover frequency, ωg of the compensated system such that it corresponds to the desired phase margin. Step 3: Let the transfer function of the compensator be Gcomp (s ) =
0.02
Time (sec)
PROPOSED COMPENSATOR DESIGN PROCEDURE
(1 + T1s )
0
(10)
Step 6: Choose ωT1 ≥50 to nullify the phase angle contribution by the compensator at the desired gain crossover frequency and compute T1 and T2 from equation 10. Step 7: The compensated system transfer function is determined by substituting T2, T1 and Gc as,
V.
CASE STUDY OF THE PROPOSED COMPENSATOR DESIGN PROCEDURE A case study is made for two different BLDC drive systems based on the simulation results of their compensated systems. The simulation results are tabulated in Table III from which, the following observations can be made. 1. Conventional lead compensator is no longer suitable for the BLDC motor drive systems with complex open loop poles because the desired specifications cannot be met. Conventional lag compensator is better than conventional lead compensator for the BLDC drive systems with complex open poles because the desired specifications are met but its design is tedious and time consuming, as correction factor has to be adjusted on a trial and error basis (Iterative procedure) to meet the desired specifications. 2. Proposed compensator design technique is better than that of conventional compensators, because it is possible to meet the desired specifications with greater accuracy. Hence, this technique is best suited for third order systems with complex poles.
Gpc (s) = GcGp (s)Gcomp (s )
Example: A compensator is designed for a typical BLDC drive system with open loop transfer function of motor 2 in table I to meet the specifications: Kv ≥ 285 sec-1, Phase margin=55 degrees and Gain margin ≥ 8 dB. Gc is computed to satisfy the required velocity error constant, Kv ≥ 285 sec-1 as 18. To satisfy the desired phase margin, ωg should be selected as 129 rad/sec. Step 4 leads to Gp( jϖ ) 129 = 7.26 dB. Steps 5 and 6 yield T1=0.3876 and T2=0.8942. Transfer function of the designed compensator is (11) (1 + 0 .3876 s ) Gcomp ( s ) = (1 + 0 .8942 s )
The step response of the compensated system is shown in figure 7. Rise time, peak overshoot and settling time are found to be 8.82msecs, 18.1% and 55msecs respectively. Hence the system is found to respond faster with fewer oscillations. The frequency response of the compensated system is shown in figure 8. Gain margin and Phase margin
143
TABLE III. Desired Specifications
BLDC Drive System
BLDC Motor 2
PM=50deg. GM≥11dB Kv≥277sec-1 PM=40deg. GM≥8dB Kv≥972 sec-1 PM=55deg. GM≥8dB Kv≥285sec-1
BLDC Motor 2
PM=48deg. GM≥7dB Kv≥694sec-1
BLDC Motor 1 BLDC Motor 1
GM: Gain margin ts: Settling time
VI.
SIMULATION RESULTS OF THE COMPENSATED BLDC DRIVE SYSTEMS Type of Compensator
Time Response Specifications
Frequency Response Specifications
tr (msec)
ts (msec)
% Mp
GM (dB)
PM (deg.)
Conv. Lag Conv. Lead Proposed Conv. Lag Conv. Lead Proposed Conv. Lag Conv. Lead
7.41 3.93 6.78 6.32 1.43 5.42 10.3 3.19
54.4 34.2 33.3 56.1 29.6 42.6 99.3 66
22 25.3 19.7 37.9 52.2 32.5 17.7 39
12.8 8.34 11.4 10.7 7.3 8.94 10.5 5.92
49.7 42 49.8 40.1 18.8 39.4 55.4 19.8
Proposed
8.82
55
18.1
8.55
54.5
Conv. Lag
8.7
88.4
Conv. Lead 1.22 39.5 Proposed 8.16 79 PM: Phase margin Kv: Velocity error constant %Mp: Percentage Maximum overshoot
30.7 60.1 25.1 tr: Rise time
8.8
47.8
13.3 7.24
13.6 47.3
AND gates. Another input to the AND gates is applied from the commutation logic circuit constructed using high-speed logic gates. The hall sensor signals are applied as inputs to the commutation logic circuit to generate gating signals according to the rotor position. The output of AND gates is used to generate PWM gating signals for the IGBTs [15] in the power inverter according to the rotor position. The PWM gating signals in turn control the average voltage applied across the winding and hence the accelerating torque of the BLDC motor. When the actual shaft position is less than the desired shaft position, the error signal is positive and the IGBTs are turned ON for the duty cycle generated by the PWM generator to accelerate the rotor.
EXPERIMENTAL SETUP
The layout of experimental set up constructed using BLDC motor 1 in Table I is shown in figure 9. The set and the actual angular displacements of shaft of the BLDC motor are the inputs to the comparator that gives the difference between set and actual displacements as its output. The displacement transducer gives out an electrical signal proportional to the actual displacement. The compensating network is designed using the proposed compensator design technique. The output of the compensating network is given to the pulse width modulation (PWM) generator [14] through the impedance matching network. This PWM signal from the generator is applied as one of the inputs to the high speed
IGBT Inverter IC
DC Supply
Displacement Sensor
+ _
BLDC MOTOR θ Gating Signals 3
T6
T5
3
3
T3
T4
3
3
T2
3
T1
Hall Sensors
Error Signal
Compensating and Impedance matching networks
2
1
2
1
2
1
2
2
1
CT6
CT5
CT4
CT3
CT1
PWM Signal
Comparator
+ _
1
1
PWM Generator
CT2
Set Displacement Input
2
7408
Commutation Logic Circuit
Rotor Position Signals
Direction signal Feedback signal (Actual Displacement)
Figure 9. Hardware description of Position control system
144
Output
Output
60 degree / div
60 degree / div
100 msecs / div
time
100 msecs / div
Figure 10. Actual displacement of BLDC motor 1 with Set angular displacement 290 degrees, Kv ≥ 277 sec-1, PM = 50 degrees and GM ≥ 5 dB[tr = 6.78 msecs; % Mp = 19.7; ts = 33.3 msecs ]
Figure 11. Actual displacement of BLDC motor 1 with Set angular displacement 290 degrees, Kv ≥ 972 sec-1, PM = 40 degrees and GM ≥ 8 dB [tr = 5.42 msecs; % Mp = 32.5; ts = 42.6 msecs ] [2]
As the rotor position approaches desired position, the accelerating torque reduces due to the reduction in the duty cycle and the accelerating torque becomes zero when the rotor shaft reaches desired position. If the rotor shaft position exceeds desired position, the accelerating torque becomes negative due to the negative error and hence the rotor shaft is rotated in the opposite direction by reversing the switching sequence of the IGBTs in the power inverter. The actual displacement of the shaft is observed for two different desired specifications and shown in figures 10 and 11. These figures are in good agreement with the simulation results. From these results, it is clear that the proposed compensator design technique is very much suited for obtaining the desired response specifications. Moreover, it can be experimentally proved that the proposed compensator could yield excellent response for similar systems.
[3]
[4]
[5] [6]
[7] [8] [9]
[10]
VII. CONCLUSIONS Proposed compensator design approach is found to be more suitable for the third order control systems such as BLDC motor drive systems with complex open loop poles. The compensated system designed using this approach could yield desired specifications with greater accuracy for different BLDC drive systems. Moreover, the transient response of the compensated system is better than that obtained using conventional design approach. The technique is simple, powerful, non - iterative and consumes less time. This technique can also be employed to improve the transient response, steady state response and stability of similar systems with complex and real open loop poles. As the compensator is of first order, this compensator technique can be easily employed for complex higher order systems by reducing such systems into equivalent lower order systems.
[11]
[12]
[13]
[14]
[15]
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time
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