Outer Loop Controller Design of a Switched Reluctance Motor Driven System
Syed A. Hossain
Member, IEEE Globe Motors Dayton, OH 454041249 USA.U
Iqbal Husain
Senior Member, IEEE Department of Electrical Engineering The University of Akron Akron, OH 443253904 Email: ihusain@uakron.edu
Abstract The development of an outer loop PID controller, feedforward controller and the combination of both PID and feedforward controller for a switched reluctance motor (SRM) drive is presented in this paper. A small signal model of the SRM is derived to linearize the overall control system. The PID and feedforward controllers are developed from the linear controls theory. The PID controlled SRM drive causes the motor to dither during position hold due to the strong nonlinear properties of the system. A new controller described in this paper combines the use of PID and feedforward controls. The use of the feedforward path helps avoid the dithering and reduce the acoustic noise in the SRM.
I. INTRODUCTION A precise control of speed and position is essential for servotype applications. In such applications, a feedback control is generally used. The output variables, such as position or speed are sensed and fed back to be compared with the desired values. The error between the reference and the actual values are utilized to control the drive system to minimize the error. A properly designed feedback controller makes the system insensitive to disturbances and system parameter variations. The motion control systems often must respond to large changes in the desired values of the torque, speed and position. They must reject large, unexpected load disturbances. For large disturbances, the overall system is very nonlinear. This nonlinearity comes about because the SRM is inherently nonlinear. Additional nonlinearity is introduced from the mechanical load. The nonlinearity of the overall SRM drive significantly increases the
complexity of designing the outer loop controller. A Fuzzy controller works well for nonlinear drives, but it increases the execution time and processor memory requirement. The SRM drive needs a cascade control structure that is required in any precise control of speed and position [12]. The cascade structure is commonly used because of its flexibility. The controller has two parts: an outer loop controller and an inner loop controller. The innermost loop is the current control loop, which is necessary for precise torque control or to limit the motor phase current within a specified bound. The position loop is the outermost loop with the speed loop being an intermediate loop. The outer loop generates the reference torque or reference speed from the position or speed error. Cascade control requires that the bandwidth (speed of response) be increased towards the inner loop, with the current control being the fastest and the position or speed loop being the slowest. The control system shown in Fig. 1 is shown simplified in Fig. 2, where Gp(s) is the Laplace domain transfer function of the plant consisting of the powerprocessing unit, the motor, and the mechanical load. Gc(s) is the outer loop controller transfer function. In response to a desired input U*(s), the output of the system is Y(s). The controller Gc(s) is designed with the following objectives: o a minimum steady state error o a good dynamic response o dither free operation of the system.
ω /θ
*
*
+

Outer Loop Controller
Tref
Torque Controller
i* ph
Current Controller
Gate Signals Converter
SRM
ω/θ
Operating quadrant determination
Angle Calculator
θon θoff
Electronic Commutator
i ph
θ
Fig. 1: Cascade control structure of SRM drive system.
0780378830/03/$17.00 ? 2003 IEEE
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Controller Gc(s)
Plant Gp(s)
ω
*
ω
Fig. 2: Simplified control system representation.
U (s) = θ
*
*
PID, Gc(s)
i* 1
i
Inner loop
1 dL * i0 Js dθ
Gp(s)
1 s
Y(s) = θ
Fig. 3: Simplified SRM drive. II. OUTER LOOP CONTROLLER DESIGN The outer loop controller is designed with specific objectives. The first step is to perform a small signal analysis of the system. The small signal analysis is accomplished by considering linearity around a steady state operating point, which will allow the use of basic concepts of linear control theory. The outer loop controller can be designed based on the linear control theory. The entire system can be simulated under a large signal condition to evaluate the adequacy of the controller. 2.1 Small Signal Analysis The first step to develop a small signal model is to assume that around the steady state operating point, the input reference changes and the load disturbances are small. In such a smallsignal analysis, the overall system can be assumed to be linear around the steady state operating point. The mechanical dynamic of the system is where i0 is the steady state current value. The bandwidth of the outer speed loop can be assumed to be at least one order of magnitude smaller than that of the current loop. Therefore, the inner currentcontrol loop can be assumed to be ideal for design purposes and represented by a unity gain transfer function, as shown in Fig. 3. 2.2 PID Controller Design The PID controller is a good approach for costeffective solutions in many applications. The small signal analysis of the SRM drive is used during designing the PID controller. The PID controller of the following form is considered for the outer loop.
Gc ( s ) = k p +
ki + kd s . s
The open loop transfer function, G(s) (= Gc(s)* Gp(s)) can be approximated as,
2 i0 dL k d s + k p s + k i G (s) = * * . J dθ s3
& =? ω+ ω
B J
N ? 1 ? ph ?∑ T j ? TL ? J ? j =1 ?
(3)
(1)
Expanding the nonlinear state Eq. (1) into a Taylor series about a steady state operating point and neglecting all the higherorder terms yields
& = f (i0 ) + ω
where
?f ?i
i0
(i ? i0 )
f (i ) = ?
B 1 ? 1 dL ? ? TL ? . ω + ? i2 J J ? 2 dθ ?
(2)
This open loop transfer function is necessary to design the outer loop controller. The frequency at which the gain equals unity is defined as the crossover frequency. At the crossover frequency, the phase delay introduced by the open loop transfer function must be less that 1800 in order for the closed loop feedback system to be stable. Choosing a reasonable value of crossover frequency ω0 and phase margin φ0 , Eq. (3) can be written as
2 i0 dL k d s + k p s + k i * * J dθ s3
Therefore,
=1
s = jω 0
and
?f &= ?ω ?i
1 dL * i0 ?i i0 (i ? i0 ) = J dθ
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2 i0 dL k d s + k p s + k i ∠ * * J dθ s3
= ?180 0 + ? 0
s = jω 0
. For design simplification, we assume that
(4)
ki = 0.1 . kp
Solving these three equations gives the gain constants of the PID controller. Here kp is responsible for process stability and ki is responsible for driving the error to zero. Systems in which process dynamics will change in predictable ways during operation often benefit from gain scheduling. Some derivative action is added through a derivative kd coefficient term to improve system response and to reduce speed oscillations. Too low a value of kp can cause drift, and too high a value will cause oscillations. A very high value of ki will also invoke oscillations or instability. When the command changes, the system needs a large control input for fast response. To produce a large control input, the PID controller should have a large kp, small ki, and large kd. When the motor reaches the command value, we need a small kp to avoid overshoot and a large ki to reduce steady state error. Thus, the PID parameters can be developed as a function of the position or speed error as follows:
Fig. 4 shows the block diagram of a position control system utilizing feedforward controller. The PID controller works with the error of the position control system. If the system involves large time lags, it will take some time before any corrective action takes place. This drawback can be overcome by the feed forward controller. As soon as a change in the position command occurs, a corrective measure will be taken simultaneously. The overall transfer function of the control system is
C ( s ) G p ( s )[G f ( s ) + Gc ( s )] = . R( s) 1 + Gc ( s )G p ( s )
(6)
Feed forward control can minimize the transient error, but since the feed forward control is openloop control, there are limitations to its functional accuracy. Again, the control input to the system, Tref is limited by the motor design. Therefore, it is difficult to get an advantage from the feed forward controller in terms of improving response time. 2.4 Overall Outer Loop Controller Design A high value of proportional gain of the PID controller is chosen to achieve quick response time of the SRM driven actuator system. The combination of a PID controller and a feed forward controller as mentioned in section 2.3 does not perform adequately for a nonlinear system, since the reference input torque is limited by the motor mechanical constraints. Fig. 5 shows the position command, reference torque from a PID controller and desired reference torque to meet the position requirement. The torque command generated from a PID controller allows the SRM to dither around a constant position. This dithering forces the SRM to use more than one phase, which is beneficial for uniform heat distribution among the phases and minimizes resistance (a parameter) variation. On the other hand, the dithering excites various modes of the system, which increases the acoustic noise level of the system. Vibration and the accompanying acoustic noise are undesirable in many applications using SRM drives. To operate the SRM drive at a fixed position against a certain load, a constant but appropriate torque command must be generated from the outer loop controller. Feed forward controller will be useful for openloop control of the system
k p = k p ,min + k i = k i ,max ?
k p ,max ? k p ,min emax ? ki ,min
*e
(5)
ki ,max
emax
*e
where the gain constants vary as an instantaneous function of error within specified limits. 2.3 Feed Forward Controller Design If disturbances are measurable, feed forward control is a useful method of canceling their effects on the system output [3]. It controls the undesirable effects of measurable disturbances by approximating, and then compensating for them before they materialize. This is advantageous, because in a usual feedback control system the corrective action starts only after the output has been affected. Feed Forward Controller, Gf(s) + PID Controller, Gc(s) θ +
θ*
R(s)
+
Tref
Process, Gp(s)
θ
C(s)
Fig. 4: Block diagram of PID and feedforward controlled SRM drive system.
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in a steady state case if the system dynamics are known. The load torque at a fixed position against a spring is a function of spring compliance, k. In a steady state case, it is considered as
Tl = k * θ *
Position command
where Tl is the load torque and θ* is the position command. Since the dynamics of the nonlinear system is not exactly known, the PID controller is used in the dynamic operation. Whenever the position output reaches the position command, the system is operated using the feed forward controller in an open loop fashion. Fig. 6 shows the use of both a PID and a feed forward controller independently as the outer loop controller. The controller select block choose the either the output of the PID controller or the feedforward controller based on the values of the position error, command changes and position changes. The reference torque from feed forward controller is
Tref = k *θ * + Tloss , prediction
(a) Time
(7)
where Tloss, prediction is determined from the overall system model. III. EXPERIMENTAL RESULTS To test the outer loop controller, a SRM driven hydraulic system, as shown in Fig. 7, consisting of a SRM driving a hydraulic load via a gear and ballscrew assembly is considered. The position sensor output of the SRM is translated into the linear displacement of the piston by using the gear and ballscrew ratio. The SRM driven hydraulic system was operated with two different outer loop controllers: one is a PID controller and the other is a combination of a PID and a feedforward controller. Representative figures for each case will show the linear displacement vs. time, motor speed vs. time, and phase currents vs. time to maintain a constant position. 3.1 Results with PID Controller The developed PID controller was used in the linear displacement control loop of the hydraulic drive system as shown in Fig. 7 where the force acting on the piston is a linear displacement dependent term. The PID controller developed in section II is used in a linear displacement control loop. Fig. 8 shows experimental
Ref. Torque (PID)
Time
(b) Ref. Torque (Desirable)
(c)
Time
Fig. 5: Tentative plot of (a) position command vs. time (b) reference torque from PID controller vs. time, and (c) desired reference torque vs. time.
Feed forward Controller Controller Select

θ
*
+
SRM Controller
SRM
PID Controller
ω
θ
Fig. 6: Combined PID and feedforward controlled SRM drive.
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SR Motor Gear
Power supply with control
Position Controller
Position Command Position Sensor Piston
Ball screw
Hydraulic load
Piston returning spring
Fig. 7: Hydraulic drive system.
4 Position (rev.) Position (rev.) 3 2 1 0 1.0 1.5 2.0 2.5 Time (sec) (a) 3.0 3.5 4.0
4 3 2 1 0 1.0 2.0 3.0 Time (sec) (d) 3000 4.0 5.0
Motor Speed (rpm)
Motor Speed (rpm) 0.8 1.0 1.2 Time (sec) (b) 1.4 1.6 1.8
2000 1000 0 1000
2000 1000
0 0.6 1.0 1.4 Time (sec) (e) 1.8 2.2 2.6
12 Phase Currents (Amps) Phase Currents (Amps) 20 15 10 5 0 2 6 10 Time (msec) (c) 14 18 10 8 6 4 2 0 0.82 0.86 0.90 Time (sec) (f) 0.94 0.98
Fig. 8: Experimental results with PID controller (ac) and with new combined controller (df): (a) measured force vs. time, (b) motor speed vs. time, (c) phase currents at holding vs. time, (d) measured force vs. time, (e) motor speed vs. time, (f) phase currents at holding vs. time.
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results using the PID controller for the outer loop control. Figs. 8(a) and 8(b) are the measured linear displacement and motor speed outputs, respectively. The oscillation at constant position comes from the dithering of SRM around zero speed. The PID controller produces bidirectional reference torque due to the system nonlinearity and saturation effect. The motor operates between the first and second quadrant due to the bidirectional torque command input to the motor controller. Figs. 8(c) shows the phase currents during position hold. 3.2 Results with Combined Outer Loop Controller The simple PID controller is modified by a combination of a feedforward and a PID controller. The PID controller used in section 2.2 is used in the dynamic case and a feedforward control is used in the steady state case. It is noted that the motor operation due to a change of position command is first considered as dynamic operation and the motor operation to maintain a constant position level is considered as steady state operation. In steady state, there is no feedback control and the system works in an open loop fashion, as shown in Fig. 9. The relationship between position command and reference torque is determined through a simulation model and later it is modified by experimental procedure. The resulting equation used in the feedforward controller is
control strategies have been implemented in a digital signal processing based hardware system. It is found through the experiments that the controller is simple, easily designed, and performs satisfactorily. The feedback loop is for the purpose of obtaining the strong convergence properties of the error. This feedback loop results in motor dithering due to the strongly nonlinear properties of the system. The feedforward path is used to operate the system in an open loop fashion and to avoid the dithering. The gain of the feedforward path is determined from the system model and then modified through experiments. The open loop control may lead to a higher steady state error, but also it reduces the overall system acoustic noise significantly. REFERENCES [1] R. Krishnan, “Switched Reluctance Motor Drives: Modeling, Simulation, Analysis, Design, and Applications”, Industrial Electronics Series, CRC Press, Boca Raton, FL 2001. [2] Ned Mohan, “Electric Drives: An Integrative Approach”, MNPERE Minneapolis, MN 2000. [3] Richard C. Dorf, “Modern Control Systems”, AddisonWesley Publishing Company, 1992.
θ*
R(s)
Feed Forward Controller, Gf(s)
Tref
Process, Gp(s)
θ C(s)
Fig. 9: Block diagram of open loop control system
Tref = k ' *θ * + Tloss , prediction
? 0.000003 * 2 * 10 7 * θ * .
Fig. 8(df) shows the experimental results for the combined outer loop controller with the system operating in the positioncontrolled loop. Fig. 8(d) is the linear displacement vs. time plot. The open loop control leads to a higher steady state error. The corresponding speed vs. time is presented in Fig. 8(e). The motor speed is maintained at zero in the steady state. Fig. 8(f) shows the phase current vs. time at steady state. Since the feedforward control avoids dithering, the motor excites only one phase at steady state. This will lead to uneven heat distribution among the motor phases. This issue must be considered at the motor design stage. V. CONCLUSIONS A new combined outer loop control scheme, which consists of PID and feedforward controls, is developed in this paper. The developed method is then applied to an SRM driven highly dynamic actuator system. The overall
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