ECE 8830 Electric Drives

1
ECE 8830 - Electric Drives
Topic 18 Sensorless and Adaptive Vector
Control of Induction Motor Drives
Spring 2004
2
Introduction

• Position encoders/resolvers are expensive and
  introduce reliability concerns for vector
  controlled ac motor drives. It is therefore
  desirable to have a vector control scheme that
  does not require this type of sensor.
• The concept of sensorless control is to use
  estimation techniques to estimate the position of
  the rotor from motor terminal voltage and current
  signals. These signal processing methods are then
  implemented into ac motor drives using DSP chips.

3
Introduction (contd)

• For drives where only moderate dynamic
  performance is required, three types of open loop
  control approaches may be used
• Back emf-based estimation
• Constant V/Hz control
• Space harmonics-based speed estimation
• These are discussed in the Holtz paper in some
  detail.

4
Introduction (contd)

• For high performance drives, vector control
  based systems can be used. These methods include
• Rotor field orientation
• Model reference adaptive systems
• Feedforward control of stator voltages
• Stator flux orientation
• Estimation of rotor flux and torque current

5
Introduction (contd)

• As the rotor speed drops, the open loop
  estimation models lose accuracy. At lower speeds,
  closed loop approaches provide improved
  performance. Also, adaptive/self-tuning
  approaches are useful when machine parameters are
  not fully known. We will consider various
  adaptive approaches in this discussion. Finally,
  rotor speed estimation is not possible at motor
  start up and so special techniques to start the
  motor must be used. These are not described here.

6
Rotor Speed Estimation Methods

• Rotor speed estimation methods for an ac
  induction motor may be classified as follows
• Slip calculation
• Direct synthesis from state equations
• Model referencing adaptive system
• Speed adaptive flux observer
• Extended Kalman filtering
• Slot harmonics
• Injection of auxiliary signal on salient rotor

7
Slip Calculation

• If we know the slip frequency ?sl then we can
  calculate the rotor speed from the relation, ?r
  ?e- ?sl. How can we determine ?sl and ?e ?
• where and .

8
Slip Calculation (contd)

• Accurate calculation of ?sl is difficult for
  high efficiency motors, especially near
  synchronous speed because the signal amplitude is
  small and strongly dependent on motor parameters.
  Also, at low speeds, direct integration of the
  motor terminal voltages is problematic to obtain
  ?sl and ?e.

9
Direct Synthesis from State Equations

• The state equations in the ds-qs reference
  frame can be manipulated to yield the rotor
  speed.
• The stator voltage in the ds-qs reference
  frame is given by
• But
• ? where

10
Direct Synthesis from State Equations (contd)

• This equation can be rewritten as
• A similar expression can be derived for ?qrs as
• The rotor flux equations in a stationary ds-qs
• reference frame can be written as
• and

11
Direct Synthesis from State Equations (contd)

• The angle ?e between de and ds is given by
• But
• Ignoring higher order terms, we can write

?qrs
qs
?e
?drs
ds
de
12
Direct Synthesis from State Equations (contd)

• ?
• Combining these equations and some
• algebra gives

13
Direct Synthesis from State Equations (contd)

• A block diagram of this method is shown
  below
• Note This approach is highly sensitive to
  motor parameter values.

14
Model Reference Adaptive System (MRAS)

• In the model reference adaptive system (MRAS)
  approach, the output of a reference model is
  compared to the output of an adjustable/adaptive
  model until the errors between the two models
  converge. The reference model is based on stator
  equations and the adaptive model is based on the
  rotor equations. A figure showing speed
  estimation using the MRAS scheme is shown on the
  next slide.

15
Model Reference Adaptive System (MRAS) (contd)

16
Model Reference Adaptive System (MRAS) (contd)

• The stator-side equations are given by
• where vdss and idss are the stator-side d-axis
  voltage and currents in the stationary reference
  frame and vqss and iqss are the stator-side
  q-axis voltage and currents in the stationary
  reference frame.

17
Model Reference Adaptive System (MRAS) (contd)

• Thus the rotor fluxes and can be
  obtained by integration of these equations.
• The adaptive model is developed from the
  rotor-side current flux equations given by

18
Model Reference Adaptive System (MRAS) (contd)

• With the correct value of rotor speed, the
  fluxes determined from the two models should
  match. An adaptation algorithm with P-I control
  can be used to tune the speed value until the two
  flux values match.
• Three issues are important regarding this
  approach
• 1) Stability of the adaptation control loop
• 2) Convergence of the adaptation algorithm.
• 3) Integrator drift/inaccuracy

19
Model Reference Adaptive System (MRAS) (contd)

• The overall stability of the system can be
  achieved using Popovs criteria for
  hyperstability (see Bose text).
• Accuracy and drift problems inherent to the
  integration process in the reference model at low
  speed are alleviated by using a delay element
  (low pass filter) instead of an integrator in the
  stator model (see Holtzs paper for details).

20
Slot Harmonics

• This is one of the simplest methods for rotor
  speed estimation. The slots on the surface of the
  rotor of an induction motor provide reluctance
  modulation which creates harmonics in the airgap
  flux. These in turn modulate the stator flux
  linkage with a frequency proportional to the
  rotor speed. Thus, induced stator voltage waves
  will contain a ripple voltage component whose
  frequency and magnitude are proportional to the
  rotor speed.

21
Slot Harmonics (contd)

• If the number of rotor slots is not a multiple
  of 3, the desired slot harmonic signals can be
  separated from the much larger fundamental emf by
  taking the sum of the 3 phase voltages in a wye
  connected winding. This eliminates all
  nontriplen harmonic voltage components (including
  the fundamental) and the slot harmonic voltages
  add up. Their frequency is proportional to the
  rotor speed.

22
Slot Harmonics (contd)

• The slot harmonic frequency is locked onto
  using a PLL while other harmonics are filtered
  using an adaptive BPF. The output of the PLL
  gives the rotor speed.
• Because of the low number of rotor slots, the
  speed resolution of this approach is poor at low
  speeds. Nevertheless, it is a useful method for
  high speed sensorless drive applications.

23
Adaptive Observers

• The accuracy of open loop estimation methods
  decreases as the rotor speed decreases. The
  performance of these techniques depends on how
  closely the machine parameters match those used
  in the models.
• The robustness of sensorless control to
  parameter mismatch and noise can be improved
  using closed loop estimation methods. Such closed
  loop estimators are referred to as observers (as
  opposed to estimators) and are described next.

24
Full Order Nonlinear Observers

• A signal flow diagram for a full order
  nonlinear observer is shown below

25
Full Order Nonlinear Observers (contd)

• The addition of the error compensator makes it
  an observer. The error between the model current
  and the motor current is . The error signal is
  used to correct the inputs to the dynamic
  subsystems of the stator and rotor. These
  corrections are based on the following observer
  equations (derived from the motor model - see
  Holtzs paper)

26
Full Order Nonlinear Observers (contd)

• The complex gain factors G(?) are selected so
  as to ensure good dynamic response of the control
  system. It should also be recognized that the
  gain factors themselves are dependent on the
  estimated angular mechanical rotor speed ? since
  the system is nonlinear.
• This type of full order nonlinear observer has
  demonstrated performance down to speeds as low as
  0.034 p.u. or 50 rpm.

27
Sliding Mode Control

• The effective gains of the error compensator
  can be increased by using a sliding mode
  controller to tune the observer for both speed
  adaptation and for rotor flux estimation. It
  provides robust performance for a drive with
  respect to variations in motor parameters as well
  as rapid changes in load torque.
• This control approach is nonlinear where the
  drive response is forced to slide along a
  predefined trajectory in a phase plane by a
  switching algorithm despite parameter variations
  or load disturbances.

28
Sliding Mode Control (contd)

• The control DSP detects any deviation from the
  predefined trajectory and changes the switching
  strategy to get the system back on track.
• The general principle of sliding mode control
  will be reviewed and then its application to
  vector control of induction motors studied.

29
Sliding Mode Control (contd)

• Consider a sliding mode controller (SMC) for a
  simple second-order undamped linear system with a
  variable plant gain, K. The SMC controller
  comprises two switches with the option of
  positive or negative feedback as shown in the
  figure below.

30
Sliding Mode Control (contd)

• In either the positive or negative feedback
  case, the system can be shown to be unstable.
  However, when switched between the two states,
  not only can stability be achieved but the system
  can be made robust against variations in K.

31
Sliding Mode Control (contd)

• Consider first the case of negative feedback,
  i.e. switch 1 closed. In this case,
• X1R-C
• or R-X1C where X1loop error
• Differentiating this expression gives
• or

32
Sliding Mode Control (contd)

• To satisfy the loop relation, we can also
  write
• Combining these equations gives
• The general solution to this equation is

33
Sliding Mode Control (contd)

• Combining these equations gives
• This is the equation of an ellipse as shown
  below

34
Sliding Mode Control (contd)

• Similarly, in the positive feedback mode,
  (switch 2 closed) the equations become
• and
• Combining these equations gives

35
Sliding Mode Control (contd)

• The general solution to this equation is
• Squaring and combining these equations gives
• This equation describes a set of hyperbolas as
  shown on the next slide.

36
Sliding Mode Control (contd)

• The straight line asymptote equations are
  obtained by setting B1B20 which gives
• gt

37
Sliding Mode Control (contd)

• The system can be switched back and forth
  between these two modes. The superposition of the
  two phase plane diagrams results in the figure
  shown below
• The hyperbolic asymptote line is described by
• where ?0 is on
  the line.

38
Sliding Mode Control (contd)

• Assume that system at t0 is in -ve feedback
  mode at pt. X10. It moves along the ellipse until
  the ve feedback mode is invoked at pt. B. It
  will then (ideally) move along B0 to settle at 0
  at steady state, where X1 and X1 are zero. Let us
  define a straight line reference trajectory by
  the equation
• where Clt so that the line slope is lower than
  and beyond the range of the variation in K.

39
Sliding Mode Control (contd)

• Notice that the ve and -ve feedback ellipses
  and hyperbolas cross the reference trajectory in
  opposite directions. This results in a zigzag
  variation about the reference trajectory until
  steady state is reached (as the operating
  condition is switched back and forth between ve
  and -ve feedback).

40
Sliding Mode Control (contd)

• The time domain response is given by
• where t0 is the time to reach the sliding
  line. This equation represents a deceleration to
  the steady state point with an exponential decay
  of X1.

41
Sliding Mode Control (contd)

• Note the polarities of ?, ?X1 and ?X2 above
  and below the sliding line shown in the previous
  figure. The strategy of switching control is
  defined by these polarities.
• In order to ensure that the reference
  trajectory is crossed on each switching action, a
  reaching equation must be satisfied as given
  below
• for ? 0

42
Sliding Mode Control (contd)

• We now consider how to apply SMC to a
  vector-controlled induction motor drive. A block
  diagram of such a drive is shown below

43
Sliding Mode Control (contd)

• We want to make the drive response robust to
  variations in the following parameters
• Torque constant, Kt, Moment of inertia, J
• Friction damping coeff., B and load torque
• disturbance, TL.
• If we have a step command of ?r, we can
  write down the following equations
• 
  .

44
Sliding Mode Control (contd)

• The second-order plant model can be expressed
  in state-space form in terms of the state
  variables X1 and X2 as follows
• ?
• where bB/J, aKtK1/J, and d1/J.

45
Sliding Mode Control (contd)

• The proposed sliding mode control is shown in
  detail in the figure below

46
Sliding Mode Control (contd)

• Three main loops in this control system
• 1) Primary loop receives position loop error X1
  and generates U1with gains ?i and ?i.
• 2) Secondary loop takes X2 input (from the speed
  input ?m) and generates U2 with gains ?i and ?i.
• 3) An auxiliary loop injects a constant signal A
  and generates output U0 to eliminate steady state
  error due to coulomb friction and load torque TL.
• All the loops contribute to the resultant
  signal U which is the sum of U0, U1 and U2.

47
Sliding Mode Control (contd)

• The sliding trajectory during acceleration,
  deceleration and constant speed is shown below

48
Sliding Mode Control (contd)

• The sliding trajectory may be defined as
  follows for the three segments
• Acceleration segment
• where X10 initial position error
• Constant speed segment
• where -X20 max. ve speed

49
Sliding Mode Control (contd)

• Deceleration segment
• Note In all cases, ?0 gt reference
  trajectory.
• Only the primary loop is required but the
  secondary loop improves system performance.
• The control parameters for each of the loops
  is derived for each segment in the Bose text. The
  resulting control rules are presented in the next
  three slides.

50
Sliding Mode Control (contd)

• Acceleration Segment
• In the primary loop, ?1lt0 and ?1gt0.
• In the secondary loop,

51
Sliding Mode Control (contd)

• Deceleration Segment
• In the primary loop, ?3gt0 and ?3lt0.
• In the secondary loop,

52
Sliding Mode Control (contd)

• Constant Speed Segment
• In the primary loop, ?2gt0 and ?2lt0.
• In the secondary loop,

53
Sliding Mode Control (contd)

• Practical implementation of sliding mode
  control requires a fast signal processor and
  Holtz reports running such a controller down to
  0.036 p.u. minimum speed.
• Recently, SMCs combined with fuzzy logic have
  been reported in the literature.

54
Extended Kalman Filters

• Extended Kalman filters (EKFs) can also be
  used for rotor speed estimation and motor
  control. The EKF is a full-order stochastic
  observer that allows estimation of a nonlinear
  dynamic system corrupted by noise (due to
  measurement and modeling inaccuracies).
• A block diagram of the EKF algorithm is shown
  on the next slide.

55
Extended Kalman Filters (contd)

56
Extended Kalman Filters (contd)

• The EKF algorithm uses the full motor dynamic
  model. The augmented motor model can be expressed
  as
• where , ,
• and

57
Extended Kalman Filters (contd)
and
58
Extended Kalman Filters (contd)

• These equations are of 5th order. Assuming a
  constant rotor speed, the motor model is linear.
  For DSP implementation of the EKF algorithm, the
  model must be expressed in a discrete form as
• X(k1)AdX(k)BdU(k)V(k)
• Y(k) CdX(k)W(k)
• where V(k) and W(k) are zero mean, Gaussian
  white noise vectors of X(k) and Y(k).

59
Extended Kalman Filters (contd)

• The statistical variations due to noise and
  errors in measurements are incorporated into
  three covariance matrices, expressed as Q, R and
  P. Q is a 5x5 covariance matrix that is
  associated with system noise, R is a 2x2
  covariant matrix associated with fluctuations in
  the measurements, and P is a system state vector
  covariance matrix that is also 5x5.

60
Extended Kalman Filters (contd)

• A detailed flow diagram is shown below

61
Extended Kalman Filters (contd)

• There are two main stages - a prediction stage
  and a filtering stage.
• In the prediction stage, the next predicted
  values of states X(k1) are calculated by the
  motor model and the previous estimated values.
  Also, the predicted P(k1) is calculated using
  the covariance vector Q.

62
Extended Kalman Filters (contd)

• In the filtering stage, the next estimated
  states
• (k1) are obtained from X(k1) by adding a
  correction term eK, where eY(k1) -Y(k1) and
  KKalman filter gain. The EKF computations are
  performed iteratively until e approaches 0.
• This is a rather complex approach and is
  rather slow because of the extensive computation
  required. It is therefore not suitable for high
  speed applications. Lower order models (3rd and
  4th order) requiring less extensive computation
  have been demonstrated (see Holtzs paper).

63
Fuzzy Logic Control

• Fuzzy logic control emulates fuzzy human
  thinking. It is one of a set of techniques
  referred to as intelligent control which also
  include expert systems and neural networks.
• Fuzzy logic control is a powerful technique
  for modeling complex, nonlinear systems. The
  dynamic d-q model of an ac motor is an example of
  a multivariable, complex, nonlinear system that
  is well suited to fuzzy logic control.

64
Fuzzy Logic vs. Aristotelian Logic

• In Aristotelian logic, a quantity is either a
  member of a set or is not a member of a set. The
  set has sharp (or crisp) boundaries.
• In fuzzy logic, a quantity may be a member of
  a set to some degree or not be a member of a set
  to some degree. The boundaries of the set are
  fuzzy rather than crisp.

65
Fuzzy Systems

• A fuzzy system is a rule-based mapping of
  inputs to outputs for a system.
• It can be theoretically proven that a fuzzy
  system is a universal approximator.
• see Fuzzy Engineering by Bart Kosko

66
Membership Functions
Courtesy Jim Sibigtroth, Motorola
67
Complete Fuzzy System
Input
Output
Rules Processor
Defuzzifier
Fuzzifier
Knowledge Database
68
Fuzzy System Block Diagram
Courtesy Jim Sibigtroth, Motorola
69
Rule Activation over Control Surface
Courtesy Jim Sibigtroth, Motorola
70
3-D Control Surface
Courtesy Jim Sibigtroth, Motorola
71
Example Two input, two rule Fuzzy Model
n1
Rule1
m1
F1
A1
S1
m2
Rule2
F2
n2
A2
S2
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Techniques for Inference
Mamdani Implication F1 min (m1, n1)
F2 min (m2,
n2) Defuzzification Conversion Weighted Average
A1, A2 areas of output membership functions S1,
S2 singletons of output membership functions.
73
Mamdani Approach

• Most commonly used approach to developing fuzzy
  logic models for control applications.
• Uses expert knowledge to generate rule set.
• Uses membership functions for both input and
  output variables.
• Computationally intensive compared to Sugeno
  approach.

74
Sugeno Approach

• Output membership functions are singletons
  (zero order) or polynomials (first order).
• The rule in a first order Sugeno model may be
  expressed as
• if x is A and y is B then zpxqyr
• where p, q, and r are constants.
• Computationally efficient.
• Well suited to optimization/adaptation.

75
Supervised Learning

• In supervised learning, an initial set of
• membership functions and rules are
• generated. The model is then optimized
• using neural network algorithms (e.g. back
• propagation) to minimize the error between
• training data and model-generated data.

76
Unsupervised Learning

• In unsupervised learning, the initial set of
• membership functions and rule set are self-
• generated using clustering algorithms.
• Optimization can then be performed using
• neural network algorithms.

77
Fuzzy Logic Control of Induction Motor Drive
Example

• Consider the fuzzy speed controller shown
  below for a vector-controlled drive system. The
  controller observes the pattern of the speed loop
  error signal and updates the output DU so that
  ?r ?r.

78
Fuzzy Logic Control of Induction Motor Drive
Example (contd)

• Two input signals to the fuzzy controller -
  error E ?r- ?r and CE dE/dt. Based on the
  physical operation of the controller, we can
  write the simple control rule
• IF E is near zero (ZE) AND CE is slightly
  positive (PS) THEN the controller output DU is
  slightly negative.
• This is implemented
• as shown

79
Fuzzy Logic Control of Induction Motor Drive
Example (contd)

• A two rule control is shown below with rules
• Rule 1 IF EZE AND CENS THEN DUNS
• Rule 2 IF EPS AND CENS THEN DUZE

80
Fuzzy Logic Control of Induction Motor Drive
Example (contd)

• Now consider a more detailed fuzzy logic model
  for speed control of a vector-controlled
  induction motor drive. The membership functions
  for e, ce and du in per unit values are shown on
  the next slide with the following definitions of
  terms
• NB-ve big NM-ve medium NS-ve small
• NVS-ve very small ZZero
• PVSve very small PSve small
• PMve medium PBve big

81
Fuzzy Logic Control of Induction Motor Drive
Example (contd)

82
Fuzzy Logic Control of Induction Motor Drive
Example (contd)

• The rule matrix for fuzzy speed control is
  shown in the table below with the rules in the
  form
• IF e(pu)PS AND ce(pu)NM THEN du(pu)NS

83
Fuzzy Logic Control of Induction Motor Drive
Example (contd)

• The general design control guidelines are
• 1. If both e(pu) and ce(pu) are zero, maintain
  the present control setting du(pu)0.
• 2. If e(pu) ? 0 but is approaching zero at a
  satisfactory rate, then maintain the present
  control setting.
• 3. If e(pu) is growing the change the control
  signal du(pu) depending on the magnitude and sign
  of e(pu) and ce(pu) to force e(pu) towards zero.
• See pp. 584 Bose for detailed fuzzy control
• algorithm description.

84
Fuzzy Logic Control of Induction Motor Drive
Example (contd)

• The response of the fuzzy controller for a
  particular motor with steps of speed command and
  load torque with nominal inertia (J) is shown
  below

85
Fuzzy Logic Control of Induction Motor Drive
Example (contd)

• The response of the fuzzy controller for the
  same motor with steps of speed command and load
  torque with 4x nominal inertia (4J) is shown
  below