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ECE 8830 Electric Drives

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Title: 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
72
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
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