**5.3. Generalized multi-scalar control of induction machine supplied by CSI or VSI**

A cage induction machine fed by the CSI may be controlled in the same way as with the voltage source inverter (VSI). The generalized control is provided by an IM multi-scalar model formulated for the VSI machine control [Krzeminski Z., 1987]. The (114) - (117) multiscalar variables and additional u1 and u2 variables are used

$$
\mu\_1 = \psi\_{ra}\mu\_{s\beta} - \psi\_{r\beta}\mu\_{sa\ \prime} \tag{131}
$$

$$
\mu\_2 = \boldsymbol{\upmu}\_{r\boldsymbol{\alpha}} \boldsymbol{\upmu}\_{s\boldsymbol{\alpha}} + \boldsymbol{\upmu}\_{r\boldsymbol{\beta}} \boldsymbol{\upmu}\_{s\boldsymbol{\beta}} \,\,\,\,\,\,\tag{132}
$$

which are a scalar and vector product of the stator voltage and rotor flux vectors.

The multi-scalar model feedback linearization leads to defining the nonlinear decouplings [Krzeminski Z., 1987]:

$$
\Delta U\_1^\* = \frac{w\_\sigma}{L\_r} \left[ x\_{11} (\mathbf{x}\_{22} + \frac{L\_m}{w\_\sigma} \mathbf{x}\_{21}) + \frac{1}{T\_v} m\_1 \right] \tag{133}
$$

$$\mathbf{U}L\_{2}^{\*} = \frac{\varpi\_{\sigma}}{L\_{r}}\mathbf{I} - \mathbf{x}\_{11}\mathbf{x}\_{12} - \frac{R\_{r}L\_{m}}{L\_{r}}\mathbf{i}\_{s}^{2} - \frac{R\_{r}L\_{m}}{L\_{r}w\_{\sigma}}\mathbf{x}\_{21} + \frac{1}{T\_{v}}m\_{2}\Big].\tag{134}$$

The control variables for an IM supplied by the VSI have the form [Krzeminski Z., 1987]:

$$
\mu\_{s\alpha}^\* = \frac{\wp\_{r\alpha} \mathcal{U}\_2^\* - \wp\_{r\beta} \mathcal{U}\_1^\*}{\wp\_{21}},
\tag{135}
$$

$$
\mu\_{s\beta}^\* = \frac{\wp\_{ra} \text{U}\_1^\* + \wp\_{r\beta} \text{U}\_2^\*}{\wp\_{21}}.\tag{136}
$$

The controls (135) - (136) are reference variables treated as input to space vector modulator when the IM is supplied by the VSI.

On the other side, when the IM is fed by the CSI, calculation of the derivatives of (131) - (132) multi-scalar variables yields the following relations:

$$\frac{du\_1}{d\tau} = -\frac{R\_r}{L\_r}u\_1 - \mathbf{x}\_{11}u\_2 + \frac{R\_r L\_m}{L\_r}q\_s - \frac{1}{C\_M}\mathbf{x}\_{12} + \upsilon\_{11}\,\prime\tag{137}$$

$$\frac{du\_2}{d\tau} = -\frac{R\_r}{L\_r}u\_2 + \mathbf{x}\_{11}u\_1 + \frac{R\_r L\_m}{L\_r}p\_s + \frac{1}{C\_M}\mathbf{x}\_{22} + \mathbf{v}\_{22} \tag{138}$$

Sensorless Control of Induction Motor Supplied by Current Source Inverter 453

 

*i*

 

 

, (145)

. (146)

 

, (147)

. (148)

, (149)

, (150)

, (152)

. (153)

(154)

, (151)

and

 

variables have

, (144)

15 4 6

<sup>ˆ</sup> <sup>ˆ</sup> ˆ ˆˆ *r r r m*

<sup>ˆ</sup> <sup>ˆ</sup> ˆ ˆˆ *<sup>r</sup> <sup>r</sup> r m*

 

In accordance to the backstepping method, the virtual control must be determined together

been introduced and linked with the stator current estimation deviations (the integral

 

 

 , 1*c* 

> 

 

1

*z c*

 *a s <sup>d</sup> <sup>i</sup> d*

*b s <sup>d</sup> <sup>i</sup> d*

The stator current vector component deviations are treated as the subsystem control variables [Payam A. F. & Dehkordi B. M. 2006, Krstic M.; Kanellakopoulos I.; & Kokotovic P.

*s <sup>d</sup> <sup>i</sup> d* 

*s*

*i*

1*c*

*<sup>s</sup>* <sup>1</sup> *zi c* 

*<sup>s</sup>* <sup>1</sup> *zi c* 

*d*

*d* 

 

 

*d*

*d* 

by introducing the deviation defining variable, one obtains:

Transformation of (152) - (153) leads to:

*<sup>d</sup> <sup>R</sup> R L <sup>i</sup> dL L*

  

*d R R L*

*dL L*

  

*s r rr s*

*ai a a au v*

 

*r rr s r r*

*r rr s r r*

<sup>ˆ</sup> <sup>ˆ</sup> ˆ ˆˆ *<sup>s</sup>*

backstepping structure [Krstic M.; Kanellakopoulos I.; & Kokotovic P. 1995]):

with the observer stabilizing variables. In that purpose, the new

1995]. Adding and deducting the stabilizing functions, one obtains:

where

*di*

*d* 

where ps and qs are defined in (59) - (60).

By feedback linearization of the system of equations, one obtains

$$
\upsilon v\_{11} = -\frac{R\_r}{L\_r}\upsilon\_{p1} - \frac{R\_r L\_m}{L\_r} q\_s + \frac{1}{C\_M} x\_{12} + x\_{11} u\_{22} \tag{139}
$$

$$v\_{22} = -\frac{R\_r}{L\_r}v\_{p2} - \frac{R\_r L\_m}{L\_r}p\_s + \frac{1}{C\_M}\mathbf{x}\_{22} - \mathbf{x}\_{11}\mathbf{u}\_{11'} \tag{140}$$

where

vp1 and vp2 are the output of subordinated PI controllers.

The control variables of the IM fed by the CSI have the form:

$$\dot{\mathbf{u}}\_{fa} = -\mathbf{C}\_{M} \frac{\upsilon\_{11}\nu\_{r\beta} - \upsilon\_{22}\nu\_{ra}}{\upsilon\_{21}},\tag{141}$$

$$\dot{\mathbf{u}}\_{f\beta} = \mathbf{C}\_{M} \frac{v\_{11}\boldsymbol{\nu}\_{r\alpha} + v\_{22}\boldsymbol{\nu}\_{r\beta}}{\boldsymbol{\chi}\_{21}}.\tag{142}$$

As a result, two feedback loops and linear subsystems are obtained (Fig. 15).

### **6. The speed observer backstepping**

General conception of the adaptive control with backstepping is presented in references [Payam A. F. & Dehkordi B. M. 2006, Krstic M.; Kanellakopoulos I.; & Kokotovic P. 1995]. In [Krstic M.; Kanellakopoulos I.; & Kokotovic P. 1995] the adaptive back integration observer stability is proved and the stability range is given.

Proceeding in accordance with the adaptive estimator with backstepping conception, one can derive formulae for the observer, where only the state variables will be estimated as well as the rotor angular speed as an additional estimation parameter.

Treating the stator current vector components , <sup>ˆ</sup> *s i* as the observer output variables (as in [Payam A. F. & Dehkordi B. M. 2006, Krstic M.; Kanellakopoulos I.; & Kokotovic P. 1995]) and vα,<sup>β</sup>as the new input variables, which will be determined by the backstepping method, one obtains:

$$\frac{d\hat{\mathbf{i}}\_{s\alpha}}{d\tau} = -a\_1 \hat{\mathbf{i}}\_{s\alpha} + a\_5 \hat{\mathbf{y}}\_{r\alpha} + \hat{\alpha}\_r a\_4 \hat{\mathbf{y}}\_{r\beta} + a\_6 \boldsymbol{\mu}\_{s\alpha} + \boldsymbol{\upsilon}\_{\alpha} \tag{143}$$

$$\frac{d\hat{\vec{l}}\_{s\beta}}{d\tau} = -a\_1 \hat{\vec{l}}\_{s\beta} + a\_5 \hat{\vec{\nu}}\_{r\beta} - \hat{\alpha}\_r a\_4 \hat{\vec{\nu}}\_{r\alpha} + a\_6 \mu\_{s\beta} + \upsilon\_{\beta\gamma} \tag{144}$$

$$\frac{d\hat{\boldsymbol{\psi}}\_{r\alpha}}{d\tau} = -\frac{R\_r}{L\_r}\hat{\boldsymbol{\psi}}\_{r\alpha} - \hat{\boldsymbol{\alpha}}\_r \hat{\boldsymbol{\psi}}\_{r\beta} + \frac{R\_r L\_m}{L\_r}\hat{\boldsymbol{i}}\_{s\alpha} \tag{145}$$

$$\frac{d\hat{\boldsymbol{\psi}}\_{r\beta}}{d\tau} = -\frac{R\_r}{L\_r}\hat{\boldsymbol{\psi}}\_{r\beta} + \hat{\boldsymbol{\alpha}}\_r \hat{\boldsymbol{\psi}}\_{r\alpha} + \frac{R\_r L\_m}{L\_r}\hat{\boldsymbol{\mathbf{i}}}\_{s\beta} \,. \tag{146}$$

In accordance to the backstepping method, the virtual control must be determined together with the observer stabilizing variables. In that purpose, the new and variables have been introduced and linked with the stator current estimation deviations (the integral backstepping structure [Krstic M.; Kanellakopoulos I.; & Kokotovic P. 1995]):

$$\frac{d\tilde{\xi}\_a}{d\tau} = \tilde{\dot{\mathfrak{i}}}\_{sca} \, \, \, \, \, \tag{147}$$

$$\frac{d\tilde{\boldsymbol{\varphi}}\_b}{d\boldsymbol{\pi}} = \tilde{\boldsymbol{\mathfrak{i}}}\_{s\boldsymbol{\beta}} \,. \tag{148}$$

The stator current vector component deviations are treated as the subsystem control variables [Payam A. F. & Dehkordi B. M. 2006, Krstic M.; Kanellakopoulos I.; & Kokotovic P. 1995]. Adding and deducting the stabilizing functions, one obtains:

$$\frac{d\tilde{\zeta}\_{\alpha}}{d\tau} = \tilde{\dot{i}}\_{s\alpha} - \sigma\_{\alpha} + \sigma\_{\beta} \, \, \, \, \, \tag{149}$$

$$\frac{d\tilde{\boldsymbol{\varphi}}\_{\beta}}{d\tau} = \tilde{\boldsymbol{i}}\_{s\beta} - \boldsymbol{\sigma}\_{\alpha} + \boldsymbol{\sigma}\_{\beta} \tag{150}$$

where

452 Induction Motors – Modelling and Control

where

one obtains:

2

vp1 and vp2 are the output of subordinated PI controllers.

**6. The speed observer backstepping** 

stability is proved and the stability range is given.

Treating the stator current vector components , <sup>ˆ</sup>

*d* 

as the rotor angular speed as an additional estimation parameter.

The control variables of the IM fed by the CSI have the form:

where ps and qs are defined in (59) - (60).

2 11 1 22 22

*s*

, (138)

, (139)

, (140)

1 *<sup>r</sup> r m*

*u xu p x v dL L C*

*du R R L*

*R RL*

*R RL*

*f M*

*f M*

As a result, two feedback loops and linear subsystems are obtained (Fig. 15).

*i C*

*i C*

By feedback linearization of the system of equations, one obtains

*r rM*

11 1 12 11 2 1 *r rm p s r rM*

22 2 22 11 1 1 *r rm p s r rM*

> 11 22 21 *r r*

, (141)

. (142)

as the observer output variables (as in

*x* 

11 22 21 *r r*

*x* 

*v v*

General conception of the adaptive control with backstepping is presented in references [Payam A. F. & Dehkordi B. M. 2006, Krstic M.; Kanellakopoulos I.; & Kokotovic P. 1995]. In [Krstic M.; Kanellakopoulos I.; & Kokotovic P. 1995] the adaptive back integration observer

Proceeding in accordance with the adaptive estimator with backstepping conception, one can derive formulae for the observer, where only the state variables will be estimated as well

[Payam A. F. & Dehkordi B. M. 2006, Krstic M.; Kanellakopoulos I.; & Kokotovic P. 1995]) and vα,<sup>β</sup>as the new input variables, which will be determined by the backstepping method,

15 4 6

 

*s r rr s di ai a a au v*

   

, (143)

<sup>ˆ</sup> <sup>ˆ</sup> ˆ ˆˆ *<sup>s</sup>*

*s i* 

*v v*

*v v p x xu L LC*

*v v q x xu L LC*

$$
\sigma\_a = -c\_1 \tilde{\mathfrak{z}}\_{a'} \; \sigma\_\beta = -c\_1 \tilde{\mathfrak{z}}\_{\beta'} \; \tag{151}
$$

by introducing the deviation defining variable, one obtains:

$$z\_{\alpha} = \tilde{\mathbf{i}}\_{s\alpha} + \mathbf{c}\_1 \tilde{\mathbf{g}}\_{\alpha} \, \, \, \, \, \tag{152}$$

$$
\omega\_{\beta} = \tilde{\mathbf{i}}\_{s\beta} + \mathbf{c}\_1 \tilde{\mathbf{c}}\_{\beta} \,. \tag{153}
$$

Transformation of (152) - (153) leads to:

$$\frac{d\tilde{\boldsymbol{\xi}}\_{\alpha}}{d\tau} = \boldsymbol{z}\_{\alpha} - \boldsymbol{c}\_{1}\tilde{\boldsymbol{\xi}}\_{\alpha} \tag{154}$$

$$\frac{d\tilde{\boldsymbol{\xi}}\_{\beta}}{d\boldsymbol{\tau}} = \boldsymbol{\z}\_{\beta} - \boldsymbol{c}\_{1}\tilde{\boldsymbol{\xi}}\_{\beta} \,. \tag{155}$$

Sensorless Control of Induction Motor Supplied by Current Source Inverter 455

, (165)

, (166)

. (167)

*r L*

.

*w*

 

deviations in

In the (160) - (163) expressions the rotor flux deviations appear, which may be neglected

(160) - (161) may be zero, thus lowering the observer order. Assuming the simplifications,

1 2 *<sup>s</sup> v ci cz*

1 2 *<sup>s</sup> v ci cz*

<sup>4</sup> ˆ ˆˆ *r rr az z*

*r m r R L*

, 6

*L w*

In Fig. 10, 11 the backstepping speed observer test is shown. When the load torque is set to ~-0.1 p.u. the rotor speed in backstepping observer is more precisely estimated than e.g.

**Figure 10.** The Speed observer test: the estimated rotor speed x11 is changed from 0.1 to -0.1 p.u., the

 

 

 

> 

 

*a*

without any change to the observer properties (143) - (146). Besides, the ,

*mL*

, 5

*a*

*w*

4

*a*

one obtains

and

where

c1, c2, γ are constant gains,

Krzeminski's speed observer.

<sup>r</sup> ˆ

<sup>r</sup> ˆ

s ˆ i

s ˆ i

rotor flux and stator current coefficients are shown

Calculation of the (152) - (153) deviation derivatives gives:

$$\dot{\boldsymbol{z}}\_{\alpha} = \boldsymbol{a}\_{5}\tilde{\boldsymbol{\nu}}\_{r\alpha} + \boldsymbol{a}\_{4} \Big[ \hat{\boldsymbol{\alpha}}\_{r}\tilde{\boldsymbol{\nu}}\_{r\beta} + \tilde{\boldsymbol{\alpha}}\_{r}(\hat{\boldsymbol{\nu}}\_{r\beta} - \tilde{\boldsymbol{\nu}}\_{r\beta}) \Big] + \boldsymbol{\upsilon}\_{\alpha} + \boldsymbol{c}\_{1}\tilde{\boldsymbol{i}}\_{s\alpha} \,\,\,\,\tag{156}$$

$$\dot{\tilde{z}}\_{\beta} = a\_5 \tilde{\nu}\_{r\beta} - a\_4 \left[ \hat{\alpha}\_r \tilde{\nu}\_{ra} + \tilde{\alpha}\_r (\hat{\nu}\_{ra} - \tilde{\nu}\_{ra}) \right] + \upsilon\_{\beta} + c\_1 \tilde{\mathfrak{i}}\_{s\beta} \,. \tag{157}$$

By selecting the following Lyapunov function

$$V = \tilde{\zeta}\_{\alpha}^{2} + \tilde{\zeta}\_{\beta}^{2} + z\_{\alpha}^{2} + z\_{\beta}^{2} + \tilde{\wp}\_{r\alpha}^{2} + \tilde{\wp}\_{r\beta}^{2} + \frac{1}{\gamma}\tilde{\phi}\_{r}^{2} \, \, \, \, \tag{158}$$

calculating the derivative and substituting the respective expressions, new correction elements can be determined, treated in the speed observer backstepping as the input variables. The Lyapunov function is determined for the dynamics of the , , , *z* variables and for the rotor flux components. Calculating the (158) derivative, one obtains:

$$\dot{V} = -c\_1 \tilde{\zeta}\_\alpha^2 - c\_1 \tilde{\zeta}\_\beta^2 - c\_2 \tilde{\zeta}\_\alpha^2 - c\_2 \tilde{\zeta}\_\beta^2 - \frac{R\_r}{L\_r} \tilde{\psi}\_{ra}^2 - \frac{R\_r}{L\_r} \tilde{\psi}\_{r\beta}^2 + z\_a (a\_5 \tilde{\psi}\_{ra} + \hat{\alpha}\_r a\_4 \tilde{\psi}\_{r\beta} + \tilde{\alpha}\_r a\_4 (\hat{\psi}\_{r\beta} - \tilde{\psi}\_{r\beta}) + \tilde{\zeta}\_\alpha^2) \tag{159}$$

$$+ \upsilon\_a + c\_1 \tilde{\zeta}\_{sa} + c\_2 z\_a + \tilde{\zeta}\_\alpha ) + z\_\beta (a\_5 \tilde{\nu}\_{r\beta} - \hat{\alpha}\_r a\_4 \tilde{\nu}\_{ra} - \tilde{\alpha}\_r a\_4 (\hat{\nu}\_{ra} - \tilde{\nu}\_{ra}) + \upsilon\_\beta + c\_1 \tilde{\zeta}\_{s\beta} + c\_2 z\_\beta + \tilde{\zeta}\_\beta) + \tag{159}$$

$$\psi\_{ra} \left( -\frac{R\_r}{L\_r} \tilde{\nu}\_{ra} - \hat{\alpha}\_r \tilde{\nu}\_{r\beta} - \hat{\alpha}\_r (\hat{\nu}\_{r\beta} - \tilde{\nu}\_{r\beta}) \right) + \tilde{\nu}\_{r\beta} (-\frac{R\_r}{L\_r} \tilde{\nu}\_{r\beta} + \hat{\alpha}\_r \tilde{\nu}\_{rx} + \hat{\alpha}\_r (\hat{\nu}\_{ra} - \tilde{\nu}\_{ra})).$$

The input variables vα,β, resulting directly from (159), should include the estimated variables and the estimation deviations:

$$
\Delta \upsilon\_{\alpha} = -a\_5 \tilde{\nu}\_{r\alpha} - \hat{\alpha}\_r a\_4 \tilde{\nu}\_{r\beta} - c\_1 \tilde{\mathfrak{i}}\_{s\alpha} - c\_2 z\_{\alpha} - \tilde{\mathfrak{z}}\_{\alpha'} \tag{160}
$$

$$
\Delta \upsilon\_{\beta} = -a\_5 \tilde{\nu}\_{r\beta} + \hat{\alpha}\_r a\_4 \tilde{\nu}\_{r\alpha} - c\_1 \tilde{\mathfrak{i}}\_{s\beta} - c\_2 z\_{\beta} - \tilde{\mathfrak{z}}\_{\beta} \ . \tag{161}
$$

Taking (160) - (161) into account, the deviation derivatives may be written in the form:

$$
\dot{\boldsymbol{z}}\_{\alpha} = \tilde{\alpha}\_{r} \boldsymbol{a}\_{4} (\hat{\boldsymbol{\psi}}\_{r\beta} - \tilde{\boldsymbol{\psi}}\_{r\beta}) - \boldsymbol{c}\_{2} \boldsymbol{z}\_{a} - \tilde{\boldsymbol{\xi}}\_{a}{}\_{a} \tag{162}
$$

$$
\dot{\tilde{\omega}}\_{\beta} = -\tilde{\alpha}\_r a\_4 (\hat{\psi}\_{ra} - \tilde{\psi}\_{ra}) - c\_2 z\_{\beta} - \tilde{\xi}\_{\beta} \,. \tag{163}
$$

Using (162) - (163), the Lyapunov function may be written as follows:

$$\dot{V} = -c\_1 \tilde{\boldsymbol{\xi}}\_{\alpha}{}^2 - c\_1 \tilde{\boldsymbol{\xi}}\_{\beta}{}^2 - c\_2 \boldsymbol{z}\_{\alpha}^2 - c\_2 \boldsymbol{z}\_{\beta}^2 + \tilde{\alpha}\_r \boldsymbol{a}\_4 \left[ \boldsymbol{z}\_{\alpha} (\hat{\boldsymbol{\psi}}\_{r\beta} - \tilde{\boldsymbol{\psi}}\_{r\beta}) - \boldsymbol{z}\_{\beta} (\hat{\boldsymbol{\psi}}\_{r\alpha} - \tilde{\boldsymbol{\psi}}\_{r\alpha}) + \frac{1}{\gamma} \dot{\hat{\boldsymbol{\phi}}\_r} \right]. \tag{164}$$

The observer, defined by the (143) - (146) and (154) - (155) equations, is a backstepping type estimator.

In the (160) - (163) expressions the rotor flux deviations appear, which may be neglected without any change to the observer properties (143) - (146). Besides, the , deviations in (160) - (161) may be zero, thus lowering the observer order. Assuming the simplifications, one obtains

$$
\omega v\_{\alpha} = -c\_1 \tilde{\mathbf{i}}\_{s\alpha} - c\_2 \mathbf{z}\_{\alpha} \tag{165}
$$

$$
\sigma\_{\beta} = -c\_1 \tilde{\mathfrak{i}}\_{s\beta} - c\_2 \underline{z}\_{\beta'} \tag{166}
$$

and

454 Induction Motors – Modelling and Control

1

 

 

> 

) ( ˆ ˆ ( ) )

 

 

 

 

 

, (156)

. (157)

 *rr r*

*r r r rr r r r*

 

 

 

 

 

> 

   

 

   

 , (162)

 . (163)

 

 

> 

 

*r r rr r r r*

 

 

 

. (155)

 

 

> , , , *z*

> >

    variables

(159)

 

 

 

( ˆ ˆ ( )

 

 

 

 

, (160)

. (161)

, (158)

*z c*

5 4 <sup>1</sup> ˆ ˆ ( ) *r rr r r r <sup>s</sup> za a v ci*

5 4 <sup>1</sup> ˆ ˆ ( ) *r rr r r r <sup>s</sup> za a v ci*

2222 2 2 2 <sup>1</sup> *V zz*

calculating the derivative and substituting the respective expressions, new correction elements can be determined, treated in the speed observer backstepping as the input

> 

 

 

*r R L*

 

The input variables vα,β, resulting directly from (159), should include the estimated variables

5 4 12 ˆ *r rr s v a a ci cz*

5 4 12 ˆ *r rr s v a a ci cz*

Taking (160) - (161) into account, the deviation derivatives may be written in the form:

4 2 ( ) ˆ *rr r z a c z*

4 2 ( ) ˆ *rr r z a c z*

<sup>1</sup> ( )( ) ˆ ˆˆ *V c c cz cz a z r rr rr r <sup>z</sup>*

 

The observer, defined by the (143) - (146) and (154) - (155) equations, is a backstepping type

 

. (164)

 

 

 

 

> 

> >

 

 

*d*

*d* 

 

 

Calculation of the (152) - (153) deviation derivatives gives:

2 222 2 2

 

 

 

 

 

 

 

( ( ˆ ˆ

and the estimation deviations:

 

 

 

*r r rr r r r*

*r*

 

 

> 

*R L*

estimator.

*r*

By selecting the following Lyapunov function

 

 

and for the rotor flux components. Calculating the (158) derivative, one obtains:

1 1 22 544

*r r*

*R R V c c cz cz za a a L L v ci cz z a a a v ci cz*

 

)) ( ˆ ˆ ( )). *<sup>r</sup>*

 

 

 

> >

Using (162) - (163), the Lyapunov function may be written as follows:

 

> 

2 222 1 1 22 4

 

*r r s r rr r r r s*

1 2 544 1 2

 

 

variables. The Lyapunov function is determined for the dynamics of the

$$
\dot{\hat{\alpha}}\_r = \gamma a\_4 \left( z\_\beta \hat{\nu}\_{r\alpha} - z\_\alpha \hat{\nu}\_{r\beta} \right). \tag{167}
$$

where

c1, c2, γ are constant gains,

$$a\_4 = \frac{L\_{\rm av}}{w\_{\sigma}} \; , \; a\_5 = \frac{R\_r L\_{\rm av}}{L\_r w\_{\sigma}} \; , \; a\_6 = \frac{L\_r}{w\_{\sigma}} \; .$$

In Fig. 10, 11 the backstepping speed observer test is shown. When the load torque is set to ~-0.1 p.u. the rotor speed in backstepping observer is more precisely estimated than e.g. Krzeminski's speed observer.

**Figure 10.** The Speed observer test: the estimated rotor speed x11 is changed from 0.1 to -0.1 p.u., the rotor flux and stator current coefficients are shown

**Current Source Inverter**

**~**

**~Supply**

**Current source rectifier**

**ed**

**ud**

**Current source inverter**

> **IM ~**

**id**

**Ld**

**id**

ud

if

ua ub ia

ib **IM**

**ed\_ref**

**φsie<sup>ć</sup>**

**usup<sup>α</sup> usup <sup>β</sup>**

**PLL**

ed reference

**-**

2 2 f f i i

id Ld

xy abc

**m1**

**m2**

**Speed Observer is<sup>α</sup>**

**ifα**

**f**

**iβ**

**Multi- abc scalar**

**Transformation**

s ˆ i s ˆ i

<sup>r</sup> ˆ <sup>r</sup> ˆ

**Decouplings v1 v2**

**isβ**

**xy abc**

**αβ**

**xy**

**αβ**

**us<sup>α</sup> us<sup>β</sup>**

xy

**Figure 13.** The voltage multi-scalar adaptive backstepping control system structure

**observer backstepping**

> **x12 \* -**

> **x22 \***

**- -**

**Multi- scalar variables**

**x21\***

**-**

**x11 \***

x12 x21 x22

<sup>r</sup> ˆ <sup>r</sup> ˆ <sup>r</sup> ˆ

abc **Speed** 

usa usb

ed

isb isa

**Adaptive backstepping Controller**

**Figure 14.** The current multi-scalar control system structure.

**x12 x21**

**x11**

**x22**

**variables**

**Figure 11.** The Speed observer test: the estimated rotor speed x11 in backstepping observer is changed from 0.1 to -0.1 p.u., the estimated rotor speed \_ ˆ*r K* by Krzeminski's speed observer [Krzeminski Z., 1999] and the multi-scalar variable: x12, x21, x22 are shown . The load torque m0 is set to -0.1 p.u.
