# Patent

## Title
Method in connection with sensorless induction motors

## Abstract

A method for the stabilization of full-order flux observers for speed-sensorless induction motors in the regenerative mode. The method comprises determining the current vector of the induction motor, determining the stator voltage vector of the induction motor, forming a full-order flux observer having a system matrix (A) and a gain matrix (L), the state-variable observer being augmented with a speed adaptation loop, and producing an estimated rotor flux linkage vector and an estimated stator current vector, determining an estimation error of the stator current vector, defining a correction angle, and forming a speed adapt-tion law based on the cross product of the estimation error of the stator current vector and the estimated rotor flux linkage vector, where the correction angle is used to turn the rotor flux linkage vector or the estimation error of the stator current vector in order to keep the observer stable.

## Background
FIELD OF THE INVENTION
The present invention relates to use a of full-order flux observers, and particularly to stabilization of the full-order flux observers for speed-sensorless induction motors in the regenerative mode.
BACKGROUND OF THE INVENTION
Speed-sensorless induction motor drives have developed significantly during the past few years. Speed-adaptive full-order flux observers are promising flux estimators for induction motor drives. The speed-adaptive observer consists of a state-variable observer augmented with a speed-adaptation loop. The observer gain and the speed-adaptation law determine the observer dynamics.
The conventional speed-adaptation law was originally derived using the Lyapunov stability theorem or the Popov hyperstability theorem. However, the stability of the adaptation law is not guaranteed since, controversial assumptions regarding nonmeasurable states have been used in and the positive-realness condition is not satisfied in. An unstable region encountered in the regenerating mode at low speeds is well known. The regenerating-mode low-speed operation is problematic also for the estimators based on the voltage model as shown in.
In the case of the speed-adaptive full-order flux observer, the size of the unstable region could be reduced or, in theory, even removed by choosing the observer gain suitably. However, based on the simulations carried out, the methods and are sensitive to very small errors in the motor parameters. Furthermore, a seamless transition from the regenerating-mode low-speed operation to higher-speed operation or motoring-mode operation may be problematic.
Another approach to remedy the instability is to modify the speed-adaptation law. This approach seems to be almost unexplored. In changing the direction of the error projection of the adaptation law was discussed (but not Studied) for a filtered back-emf-based observer. In the rotor flux estimate included in the adaptation law was replaced with the stator flux estimate, but this does not remove the unstable region.
BRIEF DESCRIPTION OF THE INVENTION
An object of the present invention is to provide a method so as to solve the above problem. The object of the invention is achieved by a method, which is characterized by what is stated in the independent claim. The preferred embodiments of the invention are disclosed in the dependent claims.
The method of the invention is based on a modified speed-adaptation law where the direction of the error projection is changed in the regenerating-mode low-speed operation. Thus instead of using only the current estimation error perpendicular to the estimated flux, the parallel component is also exploited in the regenerating mode.
An advantage of the method of the invention is that the control of sensorless induction motor will be stabile in all operating points including low-speed regeneration. The control of an induction motor based on the method of the invention is fast to implement and reliable.



BRIEF DESCRIPTION OF THE DRAWINGS
In the following the invention will be described in greater detail by means of preferred embodiments with reference to the attached drawings, in which
FIGS. 1 and 2 illustrate loci of the current estimation error;
FIG. 3a illustrates the linearized model of the observer using the prior art speed-adaptation law;
FIG. 3b illustrates the linearized model of the observer using the speed adaptation law according to the invention;
FIG. 4a shows a part of the root loci of the prior art speed adaptation law;
FIG. 4b shows a part of the root loci of the inventive speed adaptation law;
FIG. 5 illustrates the experimental setup;
FIG. 6 illustrates a rotor-flux-oriented controller;
FIGS. 7a-b and 8 illustrate experimental results in the regenerating mode;
FIG. 9a illustrates simulation results; and
FIG. 9b illustrates experimental results.



DETAILED DESCRIPTION OF THE INVENTION
In the description the induction motor model and the speed-adaptive flux observer are first defined. Then, steady-state analysis is used to clarify the problem underlying the invention and its solution according to the invention. The stability is also studied by using root loci of the linearized system. Finally, after describing a control system based on the rotor flux orientation, simulation and experimental results are presented.
Induction Motor Model
The parameters of the dynamic Γ-equivalent circuit of an induction motor are the stator resistance Rs, the rotor resistance RR, the stator transient inductance L′s, and the magnetizing inductance LM. The angular speed of the rotor is denoted by ωm, the angular speed of the reference frame ωk, the stator current space vector is is, and the stator voltage us. When the stator flux ψs and the rotor flux ψR are chosen as state variables, the state-space representation of the induction motor becomes 



  
    
      
        
          
            X
            .
          
          _
        
        =
        
          
            
              
                [
                
                  
                    
                      
                        
                          -
                          
                            1
                            
                              τ
                              s
                              ′
                            
                          
                        
                        -
                        
                          j
                          ⁢
                          
                             
                          
                          ⁢
                          
                            ω
                            k
                          
                        
                      
                    
                    
                      
                        1
                        
                          τ
                          s
                          ′
                        
                      
                    
                  
                  
                    
                      
                        
                          1
                          -
                          σ
                        
                        
                          τ
                          r
                          ′
                        
                      
                    
                    
                      
                        
                          -
                          
                            1
                            
                              τ
                              r
                              ′
                            
                          
                        
                        -
                        
                          j
                          ⁢
                          
                             
                          
                          ⁢
                          
                            (
                            
                              
                                ω
                                k
                              
                              -
                              
                                ω
                                m
                              
                            
                            )
                          
                        
                      
                    
                  
                
                ]
              
              
                }
                
                  
                    A
                    _
                  
                  ⁢
                  
                     
                  
                
              
            
            ⁢
            
              X
              _
            
          
          +
          
            
              
                [
                
                  
                    
                      1
                    
                  
                  
                    
                      0
                    
                  
                
                ]
              
              
                }
                B
              
            
            ⁢
            
              
                u
                _
              
              s
            
          
        
      
    
    
      
        (
        
          1
          ⁢
          a
        
        )
      
    
  
  
    
      
        
          
            i
            _
          
          s
        
        =
        
          
            
              [
              
                
                  
                    
                      1
                      
                        L
                        s
                        ′
                      
                    
                  
                  
                    
                      -
                      
                        1
                        
                          L
                          s
                          ′
                        
                      
                    
                  
                
              
              ]
            
            
              }
              C
            
          
          ⁢
          
            X
            _
          
        
      
    
    
      
        (
        
          1
          ⁢
          b
        
        )
      
    
  




where the state vector is X=[ψs ψR]T, and the parameters expressed in terms of the Γ-equivalent circuit parameters are σ=L′s/(LM+L′s), τ′s=L′s/Rs, and τ′r=σLM/RR. The electromagnetic torque is 



  
    
      
        
          T
          e
        
        =
        
          
            
              3
              2
            
            ⁢
            p
            ⁢
            
               
            
            ⁢
            Im
            ⁢
            
              {
              
                
                  
                    i
                    _
                  
                  s
                
                ⁢
                
                  
                    ψ
                    _
                  
                  R
                  *
                
              
              }
            
          
          =
          
            
              3
              2
            
            ⁢
            p
            ⁢
            
              1
              
                L
                s
                ′
              
            
            ⁢
            Im
            ⁢
            
              {
              
                
                  
                    ψ
                    _
                  
                  s
                
                ⁢
                
                  
                    ψ
                    _
                  
                  R
                  *
                
              
              }
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




where p is the number of pole pairs and the complex conjugates are marked by the symbol *. In the specification, the parameters of a 2.2-kW four-pole induction motor given in Table I are used. It should also be understood that these parameters are used only for explaining the invention.

The method according to the invention comprises determining the current vector of the induction motor and determining the stator voltage vector of the induction motor. The current vector is obtained, for example, by measuring the currents. In a three-phase system it is usually necessary to measure only two currents.
The voltage vector is obtained, for example, by measuring the voltage in the apparatus feeding the motor. The apparatus is usually a frequency converter with a direct voltage intermediate circuit. By measuring this voltage and combining it with state information of the output switches, the output voltage vector is achieved.
Speed-Adaptive Full-Order Flux Observer
Conventionally, the stator current and the rotor flux are used as state variables in full-order flux observers. However, choosing the stator and rotor fluxes as state variables is preferred since this allows the observer to be used with stator-flux-oriented control or direct torque control [8] as well as with rotor-flux-oriented control. Consequently, the full-order flux observer is defined by 



  
    
      
        
          
            X
            _
          
          
            ^
            .
          
        
        =
        
          
            
              
                A
                ^
              
              ⁢
              
                X
                ^
              
            
            _
          
          +
          
            B
            ⁢
            
               
            
            ⁢
            
              
                u
                _
              
              s
            
          
          +
          
            
              L
              _
            
            ⁡
            
              (
              
                
                  
                    i
                    _
                  
                  s
                
                -
                
                  
                    
                      i
                      _
                    
                    ^
                  
                  s
                
              
              )
            
          
        
      
    
    
      
        (
        
          3
          ⁢
          a
        
        )
      
    
  




îs=C{circumflex over (X)}  (3b)

where the observer state vector is X=[{circumflex over (ψ)}s {circumflex over (ψ)}R]T, and the system matrix and the observer gain are 



  
    
      
        
          
            
              A
              ^
            
            _
          
          =
          
            [
            
              
                
                  
                    
                      -
                      
                        1
                        
                          τ
                          s
                          ′
                        
                      
                    
                    -
                    
                      j
                      ⁢
                      
                         
                      
                      ⁢
                      
                        ω
                        k
                      
                    
                  
                
                
                  
                    1
                    
                      τ
                      s
                      ′
                    
                  
                
              
              
                
                  
                    
                      1
                      -
                      σ
                    
                    
                      τ
                      r
                      ′
                    
                  
                
                
                  
                    
                      -
                      
                        1
                        
                          τ
                          r
                          ′
                        
                      
                    
                    -
                    
                      j
                      ⁢
                      
                         
                      
                      ⁢
                      
                        (
                        
                          
                            ω
                            k
                          
                          -
                          
                            
                              ω
                              ^
                            
                            m
                          
                        
                        )
                      
                    
                  
                
              
            
            ]
          
        
        ,
        
          
            L
            _
          
          =
          
            [
            
              
                
                  
                    
                      l
                      _
                    
                    s
                  
                
              
              
                
                  
                    
                      l
                      _
                    
                    r
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        
          3
          ⁢
          c
        
        )
      
    
  




respectively, where the estimates are marked by the symbol ^.

Observer Gain

The simple observer gain

ls=λ[1+jsgn({circumflex over (ω)}m)], lr=λ[−1+jsgn({circumflex over (ω)}m)]  (4a)

where 



  λ
  =
  
    {
    
      
        
          
            
              λ
              ′
            
            ⁢
            
              
                
                
                  
                    ω
                    ^
                  
                  m
                
                
              
              
                ω
                λ
              
            
          
        
        
          
            
              if
              ⁢
              
                 
              
              ⁢
              
                
                
                  
                    ω
                    ^
                  
                  m
                
                
              
            
            <
            
              ω
              λ
            
          
        
      
      
        
          
            λ
            ′
          
        
        
          
            
              if
              ⁢
              
                 
              
              ⁢
              
                
                
                  
                    ω
                    ^
                  
                  m
                
                
              
            
            ≥
            
              ω
              λ
            
          
        
      
    
  




gives satisfactory behavior from zero speed to very high speeds. Parameters λ′ and ω80 are positive constants. The parameter can be considered as an impedance, which may be helpful when choosing λ′ for different motor sizes. In the specification, the observer gain is determined by λ′=1 p.u.

Speed-Adaptation Laws

Prior Art

The conventional speed-adaptation law is

{circumflex over (ω)}m=−γpIm{(is−îs){circumflex over (ψ)}*R}−γi∫Im{(is−îs){circumflex over (ψ)}*R}dt  (5)

where γp and γi are the adaptation gains. Only the current estimation error perpendicular to the estimated rotor flux is used to estimate the speed. The adaptation law works well except in the regenerating mode at low speeds. The gains γp=10 (Nm·s)−1 and γi=10000 (Nm·s2)−1 are used in this specification.

According to the Present Invention

The speed-adaptation law according to the present invention is

{circumflex over (ω)}m=−γpIm{(is−îs){circumflex over (ψ)}*Re−jφ}−γi∫Im{(is−îs){circumflex over (ψ)}*Re−jφ}dt  (6)

where the angle φ changes the direction of the error projection. In other words, the component of the current estimation error parallel to the estimated rotor flux is also exploited when φ≠0. The change in the direction of the error projection is needed to stabilize the regenerating-mode operation at low speeds. Equation (6) is simple to calculate since Im{ab*} can be interpreted as the cross product of the vectors. In the case of (6) the cross product is calculated between stator current estimation error and estimated rotor flux.

In the speed adaptation the estimated rotor flux linkage is used. The method is also applicable for estimating stator flux linkage. This allows the method to be used in a wide variety of vector control methods.
Steady-State Analysis
Based on (1) and (3), the estimation error e=X−{circumflex over (X)} of the state vector and the stator current error are 



  
    
      
        
          
            e
            .
          
          _
        
        =
        
          
            
              (
              
                
                  A
                  _
                
                -
                
                  
                    L
                    _
                  
                  ⁢
                  
                     
                  
                  ⁢
                  C
                
              
              )
            
            ⁢
            
              e
              _
            
          
          +
          
            
              [
              
                
                  
                    0
                  
                
                
                  
                    
                      j
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            ψ
                            ^
                          
                          _
                        
                        R
                      
                    
                  
                
              
              ]
            
            ⁢
            
              (
              
                
                  ω
                  m
                
                -
                
                  
                    ω
                    ^
                  
                  m
                
              
              )
            
          
        
      
    
    
      
        (
        
          7
          ⁢
          a
        
        )
      
    
  
  
    
      
        
          
            
              i
              _
            
            s
          
          -
          
            
              
                i
                ^
              
              _
            
            s
          
        
        =
        
          C
          ⁢
          
             
          
          ⁢
          
            e
            _
          
        
      
    
    
      
        (
        
          7
          ⁢
          b
        
        )
      
    
  




In the following, the estimation error e is considered in the steady state and the estimated rotor flux reference frame is used, i.e., ė=0, ωk=ωs (where ωs is the angular stator frequency), and {circumflex over (ψ)}R={circumflex over (ψ)}R+j0. For a given error ωm−{circumflex over (ω)}m, and an operating point determined by the angular stator frequency ωs, the angular slip frequency ωr=ωs−ωm, and the rotor flux estimate {circumflex over (ψ)}R, a steady-state solution for (7) can be easily found.
Stable Region
FIG. 1 depicts the loci of current estimation error for two different speed estimation errors when the angular slip frequency ωr varies from the negative rated slip to the positive rated slip. The angular stator frequency is ωs=0.1 p.u. and the base value of the angular frequency is 2π50 s−1. It can be seen that the larger the speed error, the larger the current estimation error.
FIG. 1 shows the loci of the current estimation error when the angular slip frequency ωr varies from the negative rated slip to the positive rated slip (the rated slip being 0.05 p.u.). The angular stator frequency is ωs=0.1 p.u. and two different speed estimation errors (0.002 p.u. and 0.004 p.u.) are shown. The estimated rotor flux reference frame is used in FIG. 1.
In FIG. 1, ωs>0 and {circumflex over (ω)}m>ωm. If ωs<0, the loci lie in the right half-plane. If {circumflex over (ω)}m<ωm, the loci are located in the lower half-plane.
In the estimated rotor flux reference frame, the prior art adaptation law (5) reduces to

{circumflex over (ω)}m=−γp(isq−isq){circumflex over (ψ)}R−γi∫(isq−îsq){circumflex over (ψ)}Rdt  (8)

The speed estimate thus depends on the error isq−îsq. If {circumflex over (ω)}m>ωm, the condition isq−îsq>0 should hold in order the speed estimate to converge. In FIG. 1, this condition holds for all slip frequencies including the regenerating-mode operation (where ωsωr<0).
Unstable Region
FIG. 2 shows loci of the current estimation error for a lower angular stator frequency ωs=0.01 p.u. The locus consisting of the solid curve and the dashed curve shows the current estimation error. The condition isq−îsq>0 holds in the motoring-mode operation, but in the regenerating-mode operation at higher slips, it does not hold. Hence, the observer using the prior art adaptation law becomes unstable.
FIG. 2 shows loci of the current estimation error when the angular slip frequency ωr varies from the negative rated slip to the positive rated slip. The angular stator frequency is ωs=0.01 p.u. and the speed estimation error is {circumflex over (ω)}m−ωm=0.002 p.u. The dashed/solid curve shows the locus corresponding to the prior art adaptation law. The locus consisting of the solid curve and the dash-dotted curve corresponds to the adaptation law as used in connection with the present invention. In FIG. 2 the estimated rotor flux reference frame is used.
Based on FIG. 2, it can be noticed that the regenerating mode can be stabilized by changing the direction of the error projection. Consequently, the adaptation law (6) according to the method of the invention in the estimated rotor flux reference frame is considered. The current estimation error is rotated by factor exp(−jφ) in the estimated flux reference frame. Since the prior art adaptation law works well in the motoring mode, the angle φ is selected as 



  
    
      
        φ
        =
        
          {
          
            
              
                
                  
                    φ
                    max
                  
                  ⁢
                  
                    sgn
                    ⁡
                    
                      (
                      
                        ω
                        s
                      
                      )
                    
                  
                  ⁢
                  
                    (
                    
                      1
                      -
                      
                        
                          
                          
                            ω
                            s
                          
                          
                        
                        
                          ω
                          φ
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    if
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                      
                        ω
                        s
                      
                      
                    
                  
                  <
                  
                    
                      ω
                      φ
                    
                    ⁢
                    
                       
                    
                    ⁢
                    and
                    ⁢
                    
                       
                    
                    ⁢
                    
                      ω
                      s
                    
                    ⁢
                    
                      
                        ω
                        ^
                      
                      r
                    
                  
                  <
                  0
                
              
            
            
              
                0
              
              
                otherwise
              
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




For the given motor, φmax=0.44π (i.e., 80°) and ωφ=0.4 p.u. were chosen. In FIG. 2, the current error locus resulting from (9) consists of the solid curve and the dash-dotted curve, i.e., the dashed curve was rotated 78° around the origin in order to obtain the dash-dotted curve. Now, the condition isq−îsq>0 is valid for all slip frequencies. Actually, the selection (9) stabilizes the whole regenerating region. The parameters φmax and ωφ can be substantially varied without losing the stability.
The adaptation law according to the inventive method is not restricted to the observer gain (4). Several observer gains were studied using the steady-state analysis and the linearized model. Even the same values of φmax and ωφ as with the observer gain (4) can be used in some cases, e.g., when using the observer gain proposed in or the zero observer gain.
Linearized Model
The nonlinear and complicated dynamics of the speed-adaptive observer can be studied via linearization. The key factor in the linearization is to use a synchronous reference frame in order to obtain a dc equilibrium point. In the following, the dynamics of both the motor and the observer are taken into account. Even though the stator dynamics are included in the model, the linearized model is independent of the stator voltage and, consequently, of the current controller.
Estimation Error
In the rotor flux reference frame, the linearized model of (7a) becomes [11]



  
    
      
        
          
            e
            .
          
          _
        
        =
        
          
            
              (
              
                
                  
                    A
                    _
                  
                  0
                
                -
                
                  
                    
                      L
                      _
                    
                    0
                  
                  ⁢
                  C
                
              
              )
            
            ⁢
            
              e
              _
            
          
          +
          
            
              [
              
                
                  
                    0
                  
                
                
                  
                    
                      j
                      ⁢
                      
                         
                      
                      ⁢
                      
                        ψ
                        R0
                      
                    
                  
                
              
              ]
            
            ⁢
            
              (
              
                
                  ω
                  m
                
                -
                
                  
                    ω
                    ^
                  
                  m
                
              
              )
            
          
        
      
    
    
      
        (
        
          10
          ⁢
          a
        
        )
      
    
  




Here, the equilibrium point quantities are marked by the subscript 0, and the system matrix and the observer gain are 



  
    
      
        
          
            
              A
              _
            
            0
          
          =
          
            [
            
              
                
                  
                    
                      -
                      
                        1
                        
                          τ
                          s
                          ′
                        
                      
                    
                    -
                    
                      j
                      ⁢
                      
                         
                      
                      ⁢
                      
                        ω
                        s0
                      
                    
                  
                
                
                  
                    1
                    
                      τ
                      s
                      ′
                    
                  
                
              
              
                
                  
                    
                      1
                      -
                      σ
                    
                    
                      τ
                      r
                      ′
                    
                  
                
                
                  
                    
                      -
                      
                        1
                        
                          τ
                          r
                          ′
                        
                      
                    
                    -
                    
                      j
                      ⁢
                      
                         
                      
                      ⁢
                      
                        ω
                        r0
                      
                    
                  
                
              
            
            ]
          
        
        ,
        
          
            
              L
              _
            
            0
          
          =
          
            [
            
              
                
                  
                    
                      l
                      _
                    
                    s0
                  
                
              
              
                
                  
                    
                      l
                      _
                    
                    r0
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        
          10
          ⁢
          b
        
        )
      
    
  




respectively.

The transfer function from the estimation error of the speed ωm−{circumflex over (ω)}m to the estimation error of the current is−îs is 



  
    
      
        
          
            
              
                
                  G
                  _
                
                ⁡
                
                  (
                  s
                  )
                
              
              =
              
              ⁢
              
                
                  
                    C
                    ⁡
                    
                      (
                      
                        sI
                        -
                        
                          
                            A
                            _
                          
                          0
                        
                        +
                        
                          
                            
                              L
                              _
                            
                            0
                          
                          ⁢
                          C
                        
                      
                      )
                    
                  
                  
                    -
                    1
                  
                
                ⁡
                
                  [
                  
                    
                      
                        0
                      
                    
                    
                      
                        
                          j
                          ⁢
                          
                             
                          
                          ⁢
                          
                            ψ
                            R0
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  -
                  
                    
                      
                        -
                        j
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        ψ
                        R0
                      
                    
                    
                      L
                      s
                      ′
                    
                  
                
                ⁢
                
                  
                    s
                    +
                    
                      j
                      ⁢
                      
                         
                      
                      ⁢
                      
                        ω
                        s0
                      
                    
                  
                  
                    
                      A
                      ⁡
                      
                        (
                        s
                        )
                      
                    
                    +
                    
                      j
                      ⁢
                      
                         
                      
                      ⁢
                      
                        B
                        ⁡
                        
                          (
                          s
                          )
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        
          11
          ⁢
          a
        
        )
      
    
  




where 



  I
  =
  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  




is the identity matrix. The polynomials in (11a) are defined as 



  
    
      
        
          A
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            s
            2
          
          +
          
            s
            ⁡
            
              (
              
                
                  1
                  
                    τ
                    s
                    ′
                  
                
                +
                
                  1
                  
                    τ
                    r
                    ′
                  
                
                +
                
                  
                    
                      l
                      
                        sd
                        ⁢
                        
                           
                        
                        ⁢
                        0
                      
                    
                    -
                    
                      l
                      
                        r
                        ⁢
                        
                           
                        
                        ⁢
                        d
                        ⁢
                        
                           
                        
                        ⁢
                        0
                      
                    
                  
                  
                    L
                    s
                    ′
                  
                
              
              )
            
          
          -
          
            
              ω
              s0
            
            ⁢
            
              ω
              r0
            
          
          +
          
            σ
            
              
                τ
                s
                ′
              
              ⁢
              
                τ
                r
                ′
              
            
          
          +
          
            
              
                
                  ω
                  
                    s
                    ⁢
                    
                       
                    
                    ⁢
                    0
                  
                
                ⁢
                
                  l
                  
                    rq
                    ⁢
                    
                       
                    
                    ⁢
                    0
                  
                
              
              -
              
                
                  ω
                  
                    r
                    ⁢
                    
                       
                    
                    ⁢
                    0
                  
                
                ⁢
                
                  l
                  
                    sq
                    ⁢
                    
                       
                    
                    ⁢
                    0
                  
                
              
            
            
              L
              s
              ′
            
          
          +
          
            
              σ
              
                lsd
                ⁢
                
                   
                
                ⁢
                0
              
            
            
              
                τ
                r
                ′
              
              ⁢
              
                L
                s
                ′
              
            
          
        
      
    
    
      
        (
        
          11
          ⁢
          b
        
        )
      
    
  
  
    
      
        
          B
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            s
            ⁡
            
              (
              
                
                  ω
                  s0
                
                +
                
                  ω
                  r0
                
                +
                
                  
                    
                      l
                      
                        sq
                        ⁢
                        
                           
                        
                        ⁢
                        0
                      
                    
                    -
                    
                      l
                      
                        r
                        ⁢
                        
                           
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        0
                      
                    
                  
                  
                    L
                    s
                    ′
                  
                
              
              )
            
          
          +
          
            
              
                
                  ω
                  s0
                
                ⁢
                
                  τ
                  s
                  ′
                
              
              +
              
                
                  ω
                  r0
                
                ⁢
                
                  τ
                  r
                  ′
                
              
            
            
              
                τ
                s
                ′
              
              ⁢
              
                τ
                r
                ′
              
            
          
          +
          
            
              
                
                  ω
                  
                    r
                    ⁢
                    
                       
                    
                    ⁢
                    0
                  
                
                ⁢
                
                  l
                  
                    sd
                    ⁢
                    
                       
                    
                    ⁢
                    0
                  
                
              
              -
              
                
                  ω
                  
                    s
                    ⁢
                    
                       
                    
                    ⁢
                    0
                  
                
                ⁢
                
                  l
                  
                    rd
                    ⁢
                    
                       
                    
                    ⁢
                    0
                  
                
              
            
            
              L
              s
              ′
            
          
          +
          
            
              σ
              ⁢
              
                 
              
              ⁢
              
                l
                
                  sq
                  ⁢
                  
                     
                  
                  ⁢
                  0
                
              
            
            
              
                τ
                r
                ′
              
              ⁢
              
                L
                s
                ′
              
            
          
        
      
    
    
      
        (
        
          11
          ⁢
          c
        
        )
      
    
  




where the entries of the observer gain are divided into real and imaginary components: ls0=lsd0+jlsq0 and lr0=lrd0+jlrq0. Since the observer gain is allowed to be a function of the estimated rotor speed, the subscript 0 is used. It is to be noted that G(s) is independent of the speed-adaptation law.

Closed-Loop System

Prior Art Adaptation Law

Based on the conventional adaptation law (8), the linearized transfer function from the current error isq−îsq to the speed estimate {circumflex over (ω)}m is 



  
    
      
        
          K
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            -
            
              (
              
                
                  γ
                  p0
                
                +
                
                  
                    γ
                    i0
                  
                  s
                
              
              )
            
          
          ⁢
          
            ψ
            R0
          
        
      
    
    
      
        (
        12
        )
      
    
  




where the gains can be functions of the speed estimate. Only the imaginary component isq−îsq of the estimation error of the current is of interest. Thus only the imaginary component of (11a) is used, 



  
    
      
        
          
            G
            q
          
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            Im
            ⁢
            
              {
              
                
                  G
                  _
                
                ⁡
                
                  (
                  s
                  )
                
              
              }
            
          
          =
          
            
              -
              
                
                  ψ
                  R0
                
                
                  L
                  s
                  ′
                
              
            
            ⁢
            
              
                
                  sA
                  ⁡
                  
                    (
                    s
                    )
                  
                
                +
                
                  
                    ω
                    s0
                  
                  ⁢
                  
                    B
                    ⁡
                    
                      (
                      s
                      )
                    
                  
                
              
              
                
                  
                    A
                    2
                  
                  ⁡
                  
                    (
                    s
                    )
                  
                
                +
                
                  
                    B
                    2
                  
                  ⁡
                  
                    (
                    s
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        13
        )
      
    
  




Using (12) and (13), the closed-loop system shown in FIG. 3(a) is formed. The closed-loop transfer function corresponding to any operating point can be easily calculated using suitable computer software (e.g., MATLAB Control System Toolbox).
FIG. 4(a) shows the root loci of the linearized closed-loop system corresponding to the regenerating-mode operation. The slip frequency is rated and negative. Only the dominant poles are shown. As assumed, the system is unstable at low stator frequencies (a real pole is located in the right half-plane).
Adaptation Law According to the Invention
In the estimated rotor flux reference frame, the inventive adaptation law (6) becomes

{circumflex over (ω)}m=−γp└(isq−îsq)cos(φ)−(isd−îsd)sin(φ)┘{circumflex over (ψ)}R−γi∫[(isq−îsq)cos(φ)−(isd−îsd)sin(φ)]{circumflex over (ψ)}Rdt  (14)

The linearized system is shown in FIG. 3(b), where the transfer function from the estimation error of the speed, ωm−{circumflex over (ω)}m to the estimation error of the current isd−îsd is 



  
    
      
        
          
            G
            d
          
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            Re
            ⁢
            
              {
              
                
                  G
                  _
                
                ⁡
                
                  (
                  s
                  )
                
              
              }
            
          
          =
          
            
              -
              
                
                  ψ
                  R0
                
                
                  L
                  s
                  ′
                
              
            
            ⁢
            
              
                
                  sB
                  ⁡
                  
                    (
                    s
                    )
                  
                
                -
                
                  
                    ω
                    s0
                  
                  ⁢
                  
                    A
                    ⁡
                    
                      (
                      s
                      )
                    
                  
                
              
              
                
                  
                    A
                    2
                  
                  ⁡
                  
                    (
                    s
                    )
                  
                
                +
                
                  
                    B
                    2
                  
                  ⁡
                  
                    (
                    s
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        15
        )
      
    
  




FIG. 4(b) shows the root loci of the linearized closed-loop system corresponding to the regenerating-mode operation. In this case, the system is stable also at low stator frequencies (marginally stable when the stator frequency is zero).
FIGS. 4(a) and 4(b) show part of the root loci showing the dominant poles in the regenerating mode. The slip frequency is rated and negative. Due to symmetry, only the upper half-plane is shown in the FIGS. 4(a) and 4(b).
Control System
The regenerating-mode low-speed operation of the speed-adaptive observer was investigated by means of simulations and experiments. The MATLAB/Simulink environment was used for the simulations. The experimental setup is shown in FIG. 5. The 2.2-kW four-pole induction motor (Table I) was fed by a frequency converter controlled by a dSpace DS1103 PPC/DSP board. The PM servo motor was used as the loading machine.
The control system was based on the rotor flux orientation. The simplified overall block diagram of the system is shown in FIG. 6, where the electrical variables on the left-hand side of the coordinate transformations are in the estimated flux reference frame and the variables on the right-hand side are in the stator reference frame. The digital implementation of the observer proposed in [10] was used. The flux reference was 0.9 Wb.
A PI-type synchronous-frame current controller was used. The bandwidth of the current controller was 8 p.u. The speed estimate was filtered using a first-order low-pass filter having the bandwidth of 0.8 p.u, and the speed controller was a conventional PI-controller having the bandwidth of 0.16 p.u. The flux controller was a PI-type controller having the bandwidth of 0.016 p.u.
The sampling was synchronized to the modulation and both the switching frequency and the sampling frequency were 5 kHz. The dc-link voltage was measured, and the reference voltage obtained from the current controller was used for the flux observer. A simple current feedforward compensation for dead times and power device voltage drops was applied.
It is also understood that the experimental setup is illustrated here only for an example. The control system using the method of the invention can be any known system and is not limited to the mentioned rotor-flux-oriented system.
Results
The base values used in the following figures are: current √{square root over (2)}·5.0 A and flux 1.0 Wb. Experimental results obtained using the prior art adaptation law are shown in FIG. 7(a). The speed reference was set to 0.08 p.u. and a negative rated-load torque step was applied at t=1 s. After applying the negative load, the drive should operate in the regenerating mode. However, the system becomes unstable soon after the torque step. According to the root loci of FIG. 4(a), the operating point is unstable since the stator frequency is approximately 0.05 p.u. FIG. 7(b) depicts experimental results obtained using the adaptation law according to the invention. As expected based on the root loci of FIG. 4(b), the system behaves stably.
The first subplot of FIGS. 7(a) and 7(b) shows the measured speed (solid) and the estimated speed (dotted). The second subplot shows the q component of the stator current in the estimated flux reference frame. The third subplot presents the real and imaginary components of the estimated rotor flux in the stator reference frame.
FIG. 8 shows experiment results obtained using the adaptation law according to the invention. The speed reference was now set to 0.04 p.u. and the negative rated-load torque step was applied at t=5 s. Even though the stator frequency is only about 0.0085 p.u., both the flux and speed are correctly observed. The explanation of curves are as in FIG. 7.
Simulation results showing slow speed reversals are shown in FIG. 9(a). The adaptation law according to the invention was used. A rated-load torque step was applied at t=1 s. The speed reference was slowly ramped from 0.06 p.u. (t=5 s) to −0.06 p.u. (t=20 s) and then back to 0.06 p.u. (t=35 s). The drive operates first in the motoring mode, then in the regenerating mode, and finally again in the motoring mode.
Corresponding experimental results are shown in FIG. 9(b). The noise in the current and the speed estimate originates mainly from the incomplete dead-time compensation. At a given speed, the proportional effect of the dead-time compensation is more significant in the regenerating mode than in the motoring mode since the amplitude of the stator voltage is smaller. This kind of speed reversals require a very accurate stator resistance estimate since the stator frequency remains in the vicinity of zero for a long time. If desired, the observer could be augmented with a stator resistance adaptation scheme, Experimental results in the motoring-mode operation (demonstrating e.g. zero-speed operation) of the same speed-adaptive observer can be found in. The explanations of the curves are as in FIG. 7.
It will be obvious to a person skilled in the art that, as technology advances, the inventive concept can be implemented in various ways. The invention and its embodiments are not limited to the examples described above but may vary within the scope of the claims.







TABLE I





PARAMETERS OF THE 2.2-KW FOUR-POLE 400-V 50-Hz MOTOR.



















Stator resistance Rs
3.67
Ω



Rotor resistance RR
2.10
Ω



Magnetizing inductance LM
0.224
H



Stator transient inductance L's
0.0209
H



Moment of inertia Jtot
0.0155
kgm2



Rated speed
1430
rpm



Rated current
5.0
A



Rated torque
14.6
Nm

## Claims

1. A method for the stabilization of full-order flux observers for speed-sensorless induction motors in the regenerative mode, characterized by
determining a current vector of the induction motor, 
determining a stator voltage vector of the induction motor, 
forming a full-order flux observer having a system matrix (A) and gain matrix (L), the state-variable observer being augmented with a speed adaptation loop, and producing an estimated rotor flux linkage vector and an estimated stator current vector, 
determining an estimation error of the stator current vector, 
defining a correction angle, and 
forming a speed adaptation law based on the cross product of the estimation error of the stator current vector and the estimated rotor flux linkage vector, where the correction angle is used to turn the rotor flux linkage vector or the estimation error of the stator current vector in order to keep the observer stable.

2. A method according to claim 1, characterized in that the method further comprises
controlling the speed-sensorless induction motor with the information received from the full-order flux observer, the information comprising the stator or rotor flux linkage vector and the angular speed of the motor.