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9.16. FEEDBACK LINEARIZED CONTROL

9.16. FEEDBACK LINEARIZED CONTROL. Vector control was invented to produce separate flux and torque control as it is implicitely possible with d.c. brush motors.

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9.16. FEEDBACK LINEARIZED CONTROL

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  1. 9.16. FEEDBACK LINEARIZED CONTROL Vector control was invented to produce separate flux and torque control as it is implicitely possible with d.c. brush motors. It is also known that, with constant flux, the torque is proportional to torque current and linear torque - speed characteristics may be obtained. Such an artificial decoupling and linearization of induction motor equations may, in principle, be done with some other nonlinear transformations. Feedback linearization control [5 - 7] is such a method. Still equations ((9.116) - (9.118)) - in stator coordinates - with stator current and rotor flux dq components, together with the equation of motion, are required. The new output variables are the rotor flux squared F1(x) = lr2 and the rotor speed F2(x) = wr. Electric Drives

  2. Figure 9.52. Feedback linearization control It should be noted that the computation effort is greater than for vector control or DTFC and a thorough knowledge of motor parameters is necessary. Dynamic performance quite similar to both advanced vector control or DTFC has been obtained [32]. It seems too early to forecast the industrial prospects of feedback linearization control in competition with advanced vector control or DTFC. Electric Drives

  3. 9.17. SCALAR (V1/f1) CONTROL For pump and ventilator like applications the speed control range is only from 3 to 1 up to 10 to 1. Motion (speed) sensors are avoided in such drives. Traditionally scalar, V1 / f1, open loop control has been used for such applications. In essence the voltage amplitude V1 and its frequency f1 are related by: (9.120) V0* is called voltage boost and is required to run the motor properly at low speeds. The primary frequency is ramped as desired and, based on (9.120), an open loop PWM procedure is used to control the PWM inverter (figure 9.53). Electric Drives

  4. Figure 9.53. V1 / f1 open loop (scalar) control Electric Drives

  5. Figure 9.54. Instability zones for V1 / f1 open loop scalar control a.) imr / w1 plane V1 / f1 plane In general [33] w1min is: (9.121) where tr is the rotor time constant. Electric Drives

  6. Also it has been shown [33] that instabilities occur if the figure of merit fm is: (9.122) where tm is the mechanical time constant. The slope of two asymptotes in figure 9.54.a are directly related to tr and tm. The values of fm for a few 2, 4, 6 pole motors [33] are shown in figure 9.55. Figure 9.55. The figure of merit fm (9.122) Electric Drives

  7. Compensating the slip frequency is attempted for steady state when, in fact, the rotor flux is constant in time and thus the torque Te (9.15) is: (9.124) with (9.125) The motion equation is: (9.126) Assuming that the mechanical transients are slow the rotor flux may be considered constant. The solution of wr for such slow transients is: (9.127) (9.128) Electric Drives

  8. The principle of slip frequency compensation method [34] consists of increasing the reference frequency wr* by the estimated slip frequency Sw1 (figure 9.56) to move, for given torque, from B to C and thus make wr independent of load. Figure 9.56. Principle of slip frequency compensation Electric Drives

  9. The signal flow diagram (figure 9.57) of such a scheme illustrates the estimation of the rotor flux from the voltage model. Figure 9.57. V1 / f1 scalar control with feedforward slip frequency compensation Electric Drives

  10. The voltage model is adequate only above 2Hz but this is acceptable in V1 / f1 drives. Equations (9.124), (9.126) and figure 9.57 suggest the signal flow diagram of figure 9.58. Figure 9.58. Signal flow diagram for slip frequency compensation Electric Drives

  11. The wr / wr* transfer function G0 is simply (from figure 9.58): (9.129) G0 offers a stable response only if: (9.130) On the other hand the transfer function G1 between wr and Tload is: (9.131) Equation (9.131) suggests that for steady state (s = 0 in (9.131)) Dwr for unit step load torque is: (9.132) So: (9.133) Electric Drives

  12. For full compensation and thus (point C in figure 9.56). In this case (9.130) is automatically met. In special cases, at low speeds, ( ) in order to provide a stable operation in case of random friction torque perturbations. Good performance has been demonstrated down to 100rpm [34]. Electric Drives

  13. 9.18. SELF - COMMISSIONING By self - commissioning we mean here the parameter estimation and controller closed loops calibration by the drive itself, when working on initialization mode, on the site of application, before actual operation starts. Electric Drives

  14. Figure 9.59. a.) Direct vector current control b.) current model rotor flux estimator Electric Drives

  15. However in order to calibrate the current, flux and speed controllers: • first, the stator resistance rs and the transient time constant t’ = Lsc / (rs+rr) have to be estimated and, based on these values, the current controllers are to be calibrated; • second, with the current controllers on, the rotor time constant tr is estimated and used in the rotor flux estimator (figure 9.59.b). Further, the flux controller is calibrated. • third, the reference rotor flux level and the corresponding magnetizing current are calculated; • fourth, through a no load acceleration test, the mechanical time constant tm (9.128) is calculated. • Various methods to estimate the induction motor parameters rs, Lsc, Lm, tr, J may be classified into: • step voltage response tests at standstill; • frequency response tests at standstill; • dynamic tests (non zero speed). Electric Drives

  16. To account for the inverter dead time effect, measurements are taken for two different modulation indexes md1 and md2: (9.134) Up to this point the current controllers are only grossly calibrated. The current levels are 50% and 100% of rated current and only a voltage vector is applied (V1 for example). To estimate the motor transient time constant t’, the inverter is controlled through a binary port (with the modulator inhibited) for a few microseconds noting the time tpeak and the peak current reached, ipeak (figure 9.60). Due to the short time interval the main flux does not occur and thus the motor equation is: (9.135) (9.136) Electric Drives

  17. The average vaue of t’ from a few tests is computed. Figure 9.60. Step voltage response at standstill Based on the above results, the current controllers (figure 9.61) may be callibrated. Tc is the sampling time. Figure 9.61. D.c. (synchronous) current controller Electric Drives

  18. Based on optimal gain method [36] the PI controller constants are: (9.137) Further, we have to estimate the rotor time constant. We feed the inverter at standstill with d.c. current (applying V1 with PWM) and then turn it off and record the stator voltage: (9.138) (9.139) Approximately the rotor time constant tr is: (9.140) Electric Drives

  19. To do so we use the drive in the V1 / f1 open loop mode at 10% of rated frequency and the motor no load current is transformed into synchronous coordinates. The rotor time constant tr is changed until the iq current in synchronous coordinates is zero. The d axis current will be the rated magnetizing current imo* (provided V1 / f1 ratio is the rated one), [36]. The same test is performed above rated frequency for rated voltage to get the value of imr(wr). Also Lm = lr / imr has to be found in the process. The flux controller (figure 9.59) may be calibrated (figure 9.62). Figure 9.62. Rotor flux loop controller Electric Drives

  20. The mechanical time constant tm may be calculated from constant torque (imr, iq constant) rotor field orientation acceleration tests. Noting that the speed reached wa after the time ta, the mechanical time constant is obtained from the motion equation: (9.141) (9.142) With tm from (9.125) and J from (9.142) we get: (9.143) Electric Drives

  21. The timing of self - commissioning operations may be summarized as in figure 9.63. [36] Figure 9.63. Self - commissioning timing The whole process lasts about 60 seconds. After that the drive is ready for full performance operation. For sensorless operation slightly different tehniques are required for self - commissioning. In high performance drives the motor parameters are tuned through MRAC, on - line adaptation or through on line estimators. These aspects are beyond our scope here. [37, 38] Electric Drives

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