Field of the invention
This invention relates to a speed sensorless control method
for an induction motor that is supplied by a PWM inverter through an output LC filter.

Background of the invention
A voltage source PWM inverter enables stepless speed and
torque control of AC motors, allowing reduced energy consumption and increased control
performance. The use of a PWM inverter, however, not only brings advantages but
also causes unwanted effects in the motor. The output voltage of the inverter consists
of sharp-edged voltage pulses, producing bearing currents and high voltage stresses
in motor insulations [1], [2]. The oscillation at the switching frequency causes
additional losses and acoustic noise. These phenomena can be eliminated by adding
an LC filter to the output of the PWM inverter. In addition, the EMI shielding of
the motor cable may be avoided if the voltage is nearly sinusoidal.

Adding an LC filter to a variable speed drive makes the
motor control more difficult. Usually, a simple volts-per-hertz control method is
chosen. Better control performance is achieved by using vector control, i.e. field
oriented control. However, there are only few publications that deal with the vector
control of a motor fed through an LC filter [3]-[5]. In these papers, an extra current
or voltage measurement was necessary, and a speed encoder was used. In order to
obtain cost savings and reliability improvements, a full-order observer was proposed
in [6], making additional current or voltage measurements unnecessary.

Recently, the speed sensorless control of ac motors has
become popular. Promising estimation methods for speed sensorless induction motor
drives are speed-adaptive full -order observers [7] combined with improvements in
regeneration mode operation [8], [9]. However, a speed sensorless control methods
for induction motor supplied through an LC filter are not yet published.

The problem associated with the prior art control systems
with output filter is the need for either the measurements of current or voltage
from motor or the use of speed encoders. Both of the above solutions increase costs
both for installation and maintenance on the system.

Brief description [disclosure] 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 the use of an adaptive
full-order observer, and no additional voltage, current or speed measurements are
needed for the vector control of the motor. The rotor speed adaptation is based
on the estimation error of the inverter current. The rotor speed adaptation can
be based on the measured inverter current due to the surprisingly noticed fact,
that the quadrature components of the inverter current and the stator current are
almost identical, i.e. the applied LC filter does not distort the q component of
the current.

An advantage of the method of the invention is that the
induction motor can be controlled without any additional measurements even when
LC filter is used in the inverter output. With the method of the invention the benefits
achieved with the inverter output LC filter can be utilized in connection with a
drive that uses no additional measurement or feedback signals.

This is achieved by using a speed-adaptive observer, which
is extended for the induction motor drive equipped with an LC filter, resulting
in a drive where only the inverter output current and dc-link voltage are measured.
A simple observer gain is used, and a speed-adaptation law basing on the estimation
error of the inverter current is employed. The regeneration mode operation at low
speeds is further stabilized by modifying the speed adaptation law. The vector control
of the motor described in the specification is based on nested control loops.

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

Figure 1 shows the principle of the control system,

Figure 2 is the signal flow diagram of the cascade control,

Figures 3a, 3b and 4 show loci of current estimation error,

Figure 5 is the signal flow diagram of linearized model,

Figure 6a and 6b shows observer poles,

Figure 7 represents simulation result,

Figure 8 shows voltage and current waveforms from the simulation of Figure 7,

Figure 9 shows the experimental setup, and

Figures 10 and 11 represent experimental results.

Detailed description of the invention
In the following first the system model and control are
explained. Then attention is given to dynamic analysis of the system and simulation
and experimental results are also described.

SYSTEM MODEL AND CONTROL
The principle of the control system is shown in Fig. 1.
The inverter output voltage *
*__u__
_{
A
} is filtered by an LC filter, and the induction motor (IM) is fed by the filtered
voltage *
*__u__
_{
s
}
*.* The inverter output current *
*__i__
_{
A
} and the dc link voltage *
*__u__
_{
dc
} are the only measured quantities, whereas the stator voltage *
*__u__
_{
s
}
*,* stator current *
*__i__
_{
s
}
*,* and the electrical angular speed *&ohgr;*
_{
m
} of the rotor are estimated by an observer (the estimated quantities being
marked by '^'). The system is controlled by nested control loops in estimated rotor
flux reference frame. It should be noted that the control arrangement presented
is only one example of suitable control.

*A. Filter and Motor Models*
In a reference frame rotating at angular frequency &ohgr;_{
s
} the equations for the LC filter are
$$\frac{d{\underset{\u203e}{i}}_{A}}{dt}=-j{\mathrm{\&ohgr;}}_{s}{\underset{\u203e}{i}}_{A}-\frac{{R}_{Lf}}{{L}_{f}}{\underset{\u203e}{i}}_{A}+\frac{1}{{L}_{f}}\left({\underset{\u203e}{u}}_{A}-{\underset{\u203e}{u}}_{s}\right)$$
$$\frac{d{\underset{\u203e}{u}}_{s}}{dt}=-j{\mathrm{\&ohgr;}}_{s}{\underset{\u203e}{u}}_{s}+\frac{1}{{C}_{f}}\left({\underset{\u203e}{i}}_{A}-{\underset{\u203e}{i}}_{s}\right)$$

where *L*
_{
f
} is the inductance and *R*
_{
Lf
} the series resistance of the inductor, and *C*
_{
f
} is the capacitance of the filter.

The motor model is based on the inverse-&Ggr; model [10]
of the induction motor. The stator and rotor voltage equations are
$${\underset{\u203e}{u}}_{s}={R}_{s}{\underset{\u203e}{i}}_{s}+\frac{d{\underset{\u203e}{\mathrm{\&psgr;}}}_{s}}{dt}+j{\mathrm{\&ohgr;}}_{s}{\underset{\u203e}{\mathrm{\&psgr;}}}_{s}$$
$$0={R}_{R}{\underset{\u203e}{i}}_{R}+\frac{d{\underset{\u203e}{\mathrm{\&psgr;}}}_{R}}{dt}+j\left({\mathrm{\&ohgr;}}_{s}-{\mathrm{\&ohgr;}}_{m}\right){\underset{\u203e}{\mathrm{\&psgr;}}}_{R}$$

respectively, where *R*
_{
s
} and *R*
_{
R
} are the stator and rotor resistances, respectively, and *
*__i__
_{
R
} is the rotor current. The stator and rotor flux linkages are
$${\underset{\u203e}{\mathrm{\&psgr;}}}_{s}=\left({{L}^{\prime}}_{s}+{L}_{M}\right){\underset{\u203e}{i}}_{s}+{L}_{M}{\underset{\u203e}{i}}_{R}$$
$${\underset{\u203e}{\mathrm{\&psgr;}}}_{R}={L}_{M}\left({\underset{\u203e}{i}}_{s}+{\underset{\u203e}{i}}_{R}\right)$$

respectively, where *L'*
_{
s
} denotes the stator transient inductance and *L*
_{
M
} is the magnetizing inductance. Based on (1)-(6), the state-space representation
of the system can be written as shown in (7) and (8).
$$\underset{\u203e}{\dot{\mathrm{x}}}=\underset{\underset{\u203e}{\mathrm{A}}}{\underset{\u23df}{\left[\begin{array}{cccc}-\frac{{R}_{Lf}}{{L}_{f}}-j{\mathrm{\&ohgr;}}_{s}& -\frac{1}{{L}_{f}}& 0& 0\\ \frac{1}{{C}_{f}}& -j{\mathrm{\&ohgr;}}_{s}& -\frac{1}{{C}_{f}}& 0\\ 0& \frac{1}{{{L}^{\prime}}_{s}}& -\frac{1}{{{\mathrm{\&tgr;}}^{\prime}}_{\mathrm{\&sgr;}}}-j{\mathrm{\&ohgr;}}_{s}& \frac{1}{{{L}^{\prime}}_{s}}\left(\frac{1}{{\mathrm{\&tgr;}}_{r}}-j{\mathrm{\&ohgr;}}_{m}\right)\\ 0& 0& {R}_{R}& -\frac{1}{{\mathrm{\&tgr;}}_{r}}-j\left({\mathrm{\&ohgr;}}_{s}-{\mathrm{\&ohgr;}}_{m}\right)\end{array}\right]}}\underset{\u203e}{\mathrm{x}}+\underset{\mathrm{B}}{\underset{\u23df}{\left[\begin{array}{c}\frac{1}{{L}_{f}}\\ 0\\ 0\\ 0\end{array}\right]}}{\underset{\u203e}{u}}_{A}$$
$${\underset{\u203e}{i}}_{A}=\underset{\mathrm{C}}{\underset{\u23df}{\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]}}\underset{\u203e}{\mathrm{x}}$$

The state vector is x = [*
*__i__
_{
A
} *
*__u__
_{
s
} *
*__i__
_{
s
} *
*__&psgr;__
_{
R
}]^{
T
} , and the two time constants are defined as *&tgr;'*
_{
&sgr;
} *=L'*
_{
s
}/*(R*
_{
s
} *+R*
_{
R
}
*)* and &tgr;_{
r
} *= L*
_{
M
} /*R*
_{
R
} *.*

*B. Cascade Control*
Figure 2 illustrates the proposed cascade control of the
system in the estimated rotor flux reference frame. In the LC filter control, the
innermost control loop governs the inverter current *
*__i__
_{
A
} by means of a Pl controller, and the stator voltage *
*__u__
_{
s
} is governed by a P-type controller in the next control loop. In both control
loops, decoupling terms are used to compensate the cross-couplings caused by the
rotating reference frame.

The motor control forms the two outermost control loops.
The stator current *
*__i__
_{
s
} is controlled by a PI-type controller with cross-coupling compensation, and
the rotor speed is governed by a PI-controller. In addition, a PI-type rotor flux
controller is used. It should be noted, however, that the presented control system
is provided only as an example.

*C. Adaptive Full-Order Observer*
The system states are estimated by means of a full-order
observer. The electrical angular speed of the rotor, included in the state space
representation (7), is estimated using an adaptation mechanism. The observer is
i m-plemented in the estimated rotor flux reference frame, i.e., in a reference
frame where__&PSgr;̂__
_{
R
} =&PSgr;̂_{
R
}
*+ j*0 . The observer is given as
$$\underset{\u203e}{\dot{\widehat{\mathrm{x}}}}=\underset{\u203e}{\widehat{\mathrm{A}}\widehat{\mathrm{x}}}+\mathrm{B}{\underset{\u203e}{u}}_{A}+\underset{\u203e}{\mathrm{K}}\left({\underset{\u203e}{i}}_{A}-{\underset{\u203e}{\widehat{i}}}_{A}\right)$$

The system matrix and the observer gain vector in (9) are
$$\underset{\u203e}{\widehat{\mathrm{A}}}=\left[\begin{array}{cccc}-\frac{{R}_{Lf}}{{L}_{f}}-j{\mathrm{\&ohgr;}}_{s}& -\frac{1}{{L}_{f}}& 0& 0\\ \frac{1}{{C}_{f}}& -j{\mathrm{\&ohgr;}}_{s}& -\frac{1}{{C}_{f}}& 0\\ 0& \frac{1}{{{L}^{\prime}}_{s}}& -\frac{1}{{{\mathrm{\&tgr;}}^{\prime}}_{\mathrm{\&sgr;}}}-j{\mathrm{\&ohgr;}}_{s}& \frac{1}{{{L}^{\prime}}_{s}}\left(\frac{1}{{\mathrm{\&tgr;}}_{r}}-j{\widehat{\mathrm{\&ohgr;}}}_{m}\right)\\ 0& 0& {R}_{R}& -\frac{1}{{\mathrm{\&tgr;}}_{r}}-j\left({\mathrm{\&ohgr;}}_{s}-{\widehat{\mathrm{\&ohgr;}}}_{m}\right)\end{array}\right]$$
$$\underset{\u203e}{\mathrm{K}}={\left[\begin{array}{cccc}\underset{\u203e}{{k}_{1}}& \underset{\u203e}{{k}_{2}}& \underset{\u203e}{{k}_{3}}& \underset{\u203e}{{k}_{4}}\end{array}\right]}^{T}$$

where the estimated states are marked by the symbol '^'.

The conventional speed adaptation law for the induction
motor [7] is modified for the case where an LC filter is used. The estimation error
of the inverter current is used for the speed adaptation, instead of the estimation
error of the stator current as in the prior art systems. In order to stabilize the
regeneration mode at low speeds, the idea of a rotated current estimation error
[9], [11] is adopted.

The speed-adaptation law in the estimated rotor flux reference
frame is
$$\begin{array}{l}{\widehat{\mathrm{\&ohgr;}}}_{m}=-{K}_{p}\mathrm{\hspace{1em}Im}\left\{\left({\underset{\u203e}{i}}_{A}-{\underset{\u203e}{\widehat{i}}}_{A}\right){e}^{-j\mathrm{\varphi}}\right\}\\ \mathrm{\hspace{1em}\hspace{1em}}-{K}_{i}{\displaystyle \int \mathrm{Im}\left\{\left({\underset{\u203e}{i}}_{A}-{\underset{\u203e}{\widehat{i}}}_{A}\right){e}^{-j\mathrm{\varphi}}\right\}dt}\end{array}$$

where *K*
_{
P
} and *K*
_{
i
} are real adaptation gains, and the angle &phgr; changes the direction of
the error projection. The digital implementation of the adaptive full-order observer
is based on a simple symmetric Euler method [12].

According to the method of the invention, the inverter
output current vector *
*__i__
_{
A
} and the inverter output voltage vector *
*__u__
_{
A
} are determined. These determinations are normal current and voltage measurements.
In a practice, the inverter output voltage vector *
*__u__
_{
A
} is replaced in equation (9) by its reference *
*__u__
_{
A,ref
} *·* Typically only two phase currents are measured to obtain the
current vector. The output voltage of the inverter can be determined together from
information of the states of the output switches and from voltage of the intermediate
circuit.

Method of the invention further comprises a step of forming
a full-order observer having a system matrix __Ā__ and gain vector
__K__
as explained above. The observer produces the estimated rotor flux linkage
vector __&psgr;̂__ _{R} , the estimated stator current vector
__ī__
_{s} , the estimated stator voltage vector __ū__
_{s} and the estimated inverter output current vector *
*__ī__
_{
A
}
*.* These estimates can be used in the motor control in normal manner.

Since the inverter output current is both estimated and
determined, the difference of them can be used in a speed adaptation loop, which
produces an estimate for the electrical angular speed &ohgr;̂_{m}of
the induction machine. Basically the speed adaptation law corrects the estimate
of the angular speed so that the determined and estimated current vectors are similar.

Surprisingly the *q* components of estimated stator
current vector and inverter output current vector are almost identical, which makes
it possible to use the inverter output current instead of stator current in the
speed adaptation as explained later.

With the method of the invention, all required information
is gathered in order to control the induction machine.

STEADY-STATE ANALYSIS
The dynamics of the estimation error x̃ =
x - x̂ is obtained from (7) and (9):
$$\underset{\u203e}{\dot{\tilde{\mathrm{x}}}}=\left(\underset{\u203e}{\mathrm{A}}-\underset{\u203e}{\mathrm{K}}\mathrm{C}\right)\underset{\u203e}{\tilde{\mathrm{x}}}+\left(\underset{\u203e}{\mathrm{A}}-\underset{\u203e}{\widehat{\mathrm{A}}}\right)\underset{\u203e}{\widehat{\mathrm{x}}}$$

where the difference between system matrices is
$$\underset{\u203e}{\mathrm{A}}-\underset{\u203e}{\widehat{\mathrm{A}}}=\underset{\underset{\u203e}{\mathrm{M}}}{\underset{\u23df}{\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -j\frac{1}{{{L}^{\prime}}_{s}}\\ 0& 0& 0& j\end{array}\right]}}\left({\mathrm{\&ohgr;}}_{m}-{\widehat{\mathrm{\&ohgr;}}}_{m}\right)$$

For the steady-state analysis, the derivative of estimation
error (13) is set to zero. The operation point is determined by the synchronous
angular frequency &ohgr;_{
s
}, the slip angular frequency &ohgr;_{
r
} *= &ohgr;*
_{
s
} *―&ohgr;*
_{
m
} *,* and the estimated rotor flux &psgr;̂_{
R
} . The example values of a 2.2-kW four-pole induction motor (400 V, 50 Hz),
shown in Table I, were used for the following analysis.

Figure 3 illustrates the current estimation error as the
slip is varied from the negative rated slip to the positive rated slip, for various
values of the rotor speed estimation error &ohgr;̃_{
m
} *=&ohgr;*
_{
m
} ―&ohgr;̂_{
m
} between -0.005 p.u. and 0.005 p.u. The estimated rotor flux is constant and
the synchronous frequency &ohgr;_{
s
} is 0.1 p.u. in Fig. 3a and 0.8 p.u. in Fig. 3b. The observer gain is
__K__
= [3000s^{-1} 0 0 0]^{
T
}, and the base value of the angular frequency is 2&pgr;50 rad/s, and the
rated slip frequency is *&ohgr;*
_{
rN
} = 0.05 p.u.

If the angle correction is not used in the adaptation law
(12), the rotor speed estimate is calculated according to the imaginary part of
the current estimation error. This kind of adaptation law works well in the motoring
mode (where &ohgr;_{
s
}&ohgr;_{
r
} > 0), but at low synchronous speeds in the regeneration mode
*(&ohgr;*
_{
s
}
*&ohgr;*
_{
r
} < 0), the imaginary part of the current estimation error changes its sign
at a certain slip angular frequency as can be seen in Fig. 3(a). Beyond this point,
the estimated rotor speed is corrected to the wrong direction, leading to unstable
operation. This problem is not encountered at high synchronous speeds, as can seen
in Fig. 3(b).

The unstable operation can be avoided if the real part
of the current estimation error is also taken into account in the speed adaptation.
Correspondingly, the current estimation error is rotated by a factor e^{-j&phgr;}
. The angle &phgr; is selected as [9]
$$\mathrm{\varphi}=\{\begin{array}{ll}{\mathrm{\varphi}}_{\mathit{max}}\mathrm{sign}\left({\mathrm{\&ohgr;}}_{s}\right)\left(1-\frac{\left|{\mathrm{\&ohgr;}}_{s}\right|}{{\mathrm{\&ohgr;}}_{\mathrm{\varphi}}}\right)& \mathrm{if}\mathrm{\hspace{0.17em}}\left|{\mathrm{\&ohgr;}}_{s}\right|<{\mathrm{\&ohgr;}}_{\mathrm{\varphi}}\mathrm{\hspace{0.17em}\hspace{1em}and\hspace{1em}\hspace{0.17em}}{\mathrm{\&ohgr;}}_{s}{\widehat{\mathrm{\&ohgr;}}}_{r}<\mathit{0}\\ \mathit{0}& \mathrm{otherwise}\end{array}$$

where &phgr;_{
max
} is the maximum correction angle and &ohgr;_{&phgr;} is the limit
for the synchronous angular frequency after which the correction is not used.

The influence of the angle correction is shown in Fig.
4. The parameter values *&phgr;*
_{
max
} = 0.46&pgr; and &ohgr;_{&phgr;} = 0.75 p.u. are used in this
example. In the regeneration-mode operation, the current error vector is rotated
clockwise and the estimated rotor speed is corrected to the right direction.

Figure 4 shows loci of current estimation error as slip
varies from negative rated slip to positive rated slip for various rotor speed estimation
error values between -0.005 p.u. and 0.005 p.u. The synchronous angular frequency
0.1 p.u. The current error is rotated at negative slip angular frequencies (dashed
curves).

DYNAMIC ANALYSIS
The dynamic behavior of the speed-adaptive observer can
be analyzed via linearization. The operating point is set by the equilibrium quantities:
the rotor angular speed *&ohgr;*
_{
m0
} and the synchronous angular frequency *&ohgr;*
_{
s0
} *,* and the rotor flux *&psgr;*
_{
R0
} *.* The linearized estimation error is
$$\underset{\u203e}{\dot{\tilde{\mathrm{x}}}}=\left({\underset{\u203e}{\mathrm{A}}}_{0}-\underset{\u203e}{\mathrm{K}}\mathrm{C}\right)\underset{\u203e}{\tilde{\mathrm{x}}}+\underset{\u203e}{\mathrm{M}}{\underset{\u203e}{\mathrm{x}}}_{0}\left({\mathrm{\&ohgr;}}_{m}-{\widehat{\mathrm{\&ohgr;}}}_{m}\right)$$

The transfer function from the speed estimation error to
the inverter current estimation error obtained from (16) is
$$\underset{\u203e}{G}\left(s\right)=\frac{{\underset{\u203e}{i}}_{A}\left(s\right)-{\underset{\u203e}{\widehat{i}}}_{A}\left(s\right)}{{\mathrm{\&ohgr;}}_{m}\left(s\right)-{\widehat{\mathrm{\&ohgr;}}}_{m}\left(s\right)}=\mathrm{C}{\left(s\mathrm{I}-{\underset{\u203e}{\mathrm{A}}}_{0}+\underset{\u203e}{\mathrm{K}}\mathrm{C}\right)}^{-1}\underset{\u203e}{\mathrm{M}}{\underset{\u203e}{\mathrm{x}}}_{0}$$

Based on (12), the transfer function from the imaginary
part of the rotated inverter current error, *I*m{(__i__
_{
A
}
*-*__ī__
_{
A
})*e*
^{
-j&phgr;
}}, to the speed estimate, &ohgr;̂_{
m
}, is
$$K\left(s\right)=-{K}_{p}-\frac{{K}_{i}}{s}$$

The resulting linearized system model for dynamic analysis
is shown in Fig. 5. This model is used for investigating the pole locations of the
linearized system at different operating points.

The observer gain
__K__
affects the stability of the system. In an induction motor drive without an
LC filter, the adaptive observer is stable in the motoring mode even with zero gain
[8]. However, zero gain cannot be used when an LC filter is present. Figure 6(a)
shows the observer poles as the synchronous angular frequency &ohgr;_{
s
} is varied from -1 p.u. to 1 p.u. and the slip is rated. The adaptive observer
with zero gain is unstable in the motoring mode, corresponding to the poles in the
right half-plane.

In order to obtain a simple observer structure, the observer
gain
$$\underset{\u203e}{\mathrm{K}}={\left[\begin{array}{cccc}{k}_{1}& 0& 0& 0\end{array}\right]}^{T}$$

is proposed. The poles obtained are shown in Fig. 6(b). All poles stay in the left
half-plane in the whole inspected operation region.

SIMULATION RESULTS
The system was investigated by computer simulation with
Mat-lab/Simulink software. The data of a 2.2-kW induction motor, given in Table
I, were used for the simulations. The LC filter was designed according to the design
rules in [13], [14]. The sampling frequency was equal to the switching frequency
of 5 kHz. The bandwidths of the controllers were 500 Hz for the inverter current,
250 Hz for the stator voltage, 150 Hz for the stator current, 15 Hz for the rotor
speed, and 1.5 Hz for the rotor flux. The speed estimate was filtered using a low-pass
filter having the bandwidth of 40 Hz. The reference voltage *
*__u__
_{
A,ref
} was used in the observer instead of the actual inverter output voltage
*
*__u__
_{
A
}
*.* The observer gain was
__K__
= [3000s^{-1} 0 0 0]^{
T
}, and the adaptation gains were chosen as *K*
_{
p
} = 10 (As)^{-1} and *K*
_{
i
} = 20000 (As^{2})^{-1}.

Figure 7 shows simulation results obtained for a sequence
consisting of a fast acceleration from zero speed to 0.8 p.u., a rated load step,
a slow speed reversal, and a stepwise load reversal to the negative rated load.
The motor was in the regenerating mode between *t* = 7.6 s and *t =* 12
s. During the rest of the sequence, the motor was in the motoring mode with the
exception of a short time in the plugging mode during the speed reversal. The first
subplot shows the rotor speed (solid) and its estimate (dashed). The second subplot
shows the *q* component of the inverter current (solid) and its estimate (dashed).
The third subplot shows the *q* component of the stator current (solid) and
its estimate (dashed). It should be noted, that the solid and dashed plots are similar,
meaning that the estimates are accurate.

Slow speed reversals at load are difficult for sensorless
induction motor drives. Although the exact motor and filter parameters are used
in the simulation, small vibrations appear in *i*
_{
Aq
} and *i*
_{
sq
} at *t* = 7.8 s when the synchronous frequency is zero. More problems
are encountered if slower speed reversals are needed or parameter estimates are
inaccurate. It is to be noted that the *q* components of the inverter and stator
currents are nearly equal, which makes it possible to use *i*
_{
Aq
} in the speed adaptation law instead of *i*
_{
s
}
*q* as in the prior art systems. The voltage and current waveforms before and
after the LC filter are illustrated in detail in Fig. 8. The stator voltage and
current are nearly sinusoidal. The first subplot shows the inverter output voltage
(phase-to-phase) and the stator voltage (phase-to-phase). The second subplot shows
the inverter current and the stator current.

EXPERIMENTAL RESULTS
The experimental setup is illustrated in Fig. 9. The 2.2-kW
four-pole induction motor was fed by a frequency converter controlled by a dSPACE
DS1103 PPC/DSP board. The parameters of the experimental setup correspond to those
given in Table I. In the LC filter three 3.3-µF capacitors were used in delta
connection, giving the per-phase capacitance value of 9.9 µF. The dc-link voltage
was measured, and the reference voltage obtained from the inverter current controller
was used for the observer. The rotor speed and the shaft torque were measured only
for monitoring. A simple current feedforward compensation for dead times and power
device voltage drops was applied [15]. A permanent magnet servo motor was used as
a load machine.

Figure 10 presents the experimental results corresponding
to the simulations shown in Fig. 7. The measured performance corresponds well to
the simulation results, but the vibrations in *i*
_{
Aq
} and *i*
_{
sq
} are pronounced at zero synchronous frequency. The vibrations are due to inaccurate
motor parameter estimates and inverter nonidealities. The explanations of Figure
10 are as in Figure 7.

The vibration can be attenuated by lowering the observer
gain at low speeds. The observer gain is selected as
$${k}_{1}=\{\begin{array}{cc}{k}_{1\mathrm{l}}+\frac{{k}_{1\mathrm{h}}-{k}_{1\mathrm{l}}}{{\mathrm{\&ohgr;}}_{d}}\left|{\widehat{\mathrm{\&ohgr;}}}_{m}\right|& \mathrm{if}\hspace{0.17em}\left|{\widehat{\mathrm{\&ohgr;}}}_{m}\right|<{\mathrm{\&ohgr;}}_{d}\\ {k}_{1\mathrm{h}}& \mathrm{otherwise}\end{array}$$

where *k*
_{1l} and *k*
_{
1h} are the minimum and the maximum gains. The speed limit after which
the maximum gain is used is &ohgr;_{
d
} . The limit angular frequencies &ohgr;_{&phgr;} for the angle correction
in (15) must also be changed because of the lower gain at low speeds. In order to
avoid stepwise changes, the limit is selected as
$${\mathrm{\&ohgr;}}_{\mathrm{\&phgr;}}=\{\begin{array}{cc}{\mathrm{\&ohgr;}}_{\mathrm{\&phgr;}l}+\frac{{\mathrm{\&ohgr;}}_{\mathrm{\&phgr;}h}-{\mathrm{\&ohgr;}}_{\mathrm{\&phgr;}l}}{{\mathrm{\&ohgr;}}_{d}}\left|{\widehat{\mathrm{\&ohgr;}}}_{m}\right|& \mathrm{if}\mathrm{\hspace{0.17em}}\left|{\widehat{\mathrm{\&ohgr;}}}_{m}\right|<{\mathrm{\&ohgr;}}_{d}\\ {\mathrm{\&ohgr;}}_{\mathrm{\&phgr;}h}& \mathrm{otherwise}\end{array}$$

where &ohgr;_{&phgr;l
} and &ohgr;_{&phgr;h
} are the limit angular frequencies for the angle correction at low and high
speeds, respectively.

Experimental results obtained for the modified observer
gain and angle correction are shown in Fig. 11. The parameters in (20) and (21)
were *k*
_{
1l} = 1000 s^{-1}, *k*
_{
1h} = 3000 s^{-1}, *&ohgr;*
_{
&phgr;l
} = 0.4 p.u., *&ohgr;*
_{
&phgr;h
} = 0.75 p.u., and *&ohgr;*
_{
d
} = 0.38 p.u. The explanations of Figure 11 are as in Figure 7.

The vibrations at the zero synchronous speed are reduced
significantly, but not totally removed.

It should be noted, that the described control system is
only one possible system for controlling an induction machine based on the method
of the invention. It will be obvious to a person skilled in the art that, as the
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 MOTOR AND THE LC FILTER
Motor Parameters Stator
resistance R_{s}
3.67 &OHgr;

Rotor resistance R_{R}
1.65 &OHgr;

Stator transient inductance
L'_{s}
0.0209H

Magnetizing inductance
L_{M}
0.264 H

Total moment of inertia
J
0.0155 kgm^{2}

Rated speed n_{N}
1430 r/min

Rated (base) current I_{N}
5.0 A

Rated torque T_{N}
14.6 Nm

*LC Filter Parameters* Inductance L_{f}
8 mH

Capacitance C_{f}
9.9 µF

Series resistance R_{If}
0.1 &OHgr;

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