**2.1. The iteration algorithm selection of the inductor and input-output capacitors**

The inductance in dc-link Ld may be calculated as a function of the integral versus time of the difference of input voltage ed and output voltage ud in dc-link [Glab (Morawiec) M. et. al., 2005, Klonne A. & Fuchs W.F., 2003, 2004]. Calculating an inductance from [Glab (Morawiec) M. et. al., 2005, Klonne A. & Fuchs W.F., 2003] may be not enough because of the resonance problem. The parameters could be determined by simple algorithm.

Two criteria are taken into account:


The first criteria can be defined as:

$$
\Delta w\_i = \frac{\Delta i\_{d\max}}{i\_d} \tag{1}
$$

Sensorless Control of Induction Motor Supplied by Current Source Inverter 431

Δidpmax – maximum value for wi coefficient, Δidpmin – minimum value for wi coefficient,

algorithm are shown in Fig. 4 and 5 or Table 1.

ΔLd – interval of optimal value Ld, ΔCM – interval of optimal value CM.

THDipmax, THDupmax – maximum predetermined value of THD for range (ΔLd, ΔCM),

For optimal quality of stator current and voltage in a drive system THDi ought to be about 1%, THDu<2% and wi<15% in numerical process. Estimated CSC parameters by the iteration

THDipmin, THDupmin – minimum predetermined value of THD (ΔLd, ΔCM),

**Figure 3.** The iteration algorithm for selection of the inductor Ld an capacitors CM

where

Δidmax is max(idmax(t1) – idmin(t2)),

id is average value of dc-link current in one period.

The current ripples in dc-link has influence on output currents and commutation process. According to this, the wi factor, THDi stator and THDu stator must be taken into account. Optimal value of inductance Ld and output CM ensure performance of the drive system with sinusoidal output current and small THD. In Fig. 3 the iteration algorithm for choosing the inductor and capacitor is shown. In every step of iteration new values wi, THDi, THDu are received. In every of these steps new values are compared with predetermined value wip, THDip, THDup and:

N is number of iteration,

THDi – stator current total harmonic distortion,

THDu – stator voltage total harmonic distortion. Number of iteration is set for user.

In START the initial parameters are loaded. In block Set Ld inductance of the inductor is set. In Numerical process block the simulation is started. In next steps THDi, THDu and wi coefficient are calculated. THDi, THDu and wi coefficient are compared with predetermined value. If YES then CM is setting, if NO the new value of Ld must be set. Comparison with predetermined value is specified as below:

$$\begin{cases} \text{THD}\_{ip\,\text{max}} \le \text{THD}\_i(\mathbf{i}) \le \text{THD}\_{ip\,\text{min}}\\ \text{THD}\_{up\,\text{max}} \le \text{THD}\_u(\mathbf{i}) \le \text{THD}\_{up\,\text{min}}\\ \Delta i\_{dp\,\text{max}} \le \Delta i\_d(\mathbf{i}) \le \Delta i\_{dp\,\text{min}} \end{cases} \tag{2}$$

where

Δidpmax – maximum value for wi coefficient,

Δidpmin – minimum value for wi coefficient,

THDipmax, THDupmax – maximum predetermined value of THD for range (ΔLd, ΔCM),

THDipmin, THDupmin – minimum predetermined value of THD (ΔLd, ΔCM),

ΔLd – interval of optimal value Ld,

430 Induction Motors – Modelling and Control

Two criteria are taken into account:

Minimization of size and weight.

The first criteria can be defined as:

Δidmax is max(idmax(t1) – idmin(t2)),

THDip, THDup and:

N is number of iteration,

where

where

Minimization of currents ripples in the system

id is average value of dc-link current in one period.

THDi – stator current total harmonic distortion,

predetermined value is specified as below:

**2.1. The iteration algorithm selection of the inductor and input-output capacitors** 

The inductance in dc-link Ld may be calculated as a function of the integral versus time of the difference of input voltage ed and output voltage ud in dc-link [Glab (Morawiec) M. et. al., 2005, Klonne A. & Fuchs W.F., 2003, 2004]. Calculating an inductance from [Glab (Morawiec) M. et. al., 2005, Klonne A. & Fuchs W.F., 2003] may be not enough because of the

max , *<sup>d</sup>*

(1)

(2)

*d i*

*i*

The current ripples in dc-link has influence on output currents and commutation process. According to this, the wi factor, THDi stator and THDu stator must be taken into account. Optimal value of inductance Ld and output CM ensure performance of the drive system with sinusoidal output current and small THD. In Fig. 3 the iteration algorithm for choosing the inductor and capacitor is shown. In every step of iteration new values wi, THDi, THDu are received. In every of these steps new values are compared with predetermined value wip,

resonance problem. The parameters could be determined by simple algorithm.

*i*

THDu – stator voltage total harmonic distortion. Number of iteration is set for user.

In START the initial parameters are loaded. In block Set Ld inductance of the inductor is set. In Numerical process block the simulation is started. In next steps THDi, THDu and wi coefficient are calculated. THDi, THDu and wi coefficient are compared with predetermined value. If YES then CM is setting, if NO the new value of Ld must be set. Comparison with

> max min max min

*ip i ip up u up*

( ) ( )

max min

*dp d dp*

*i ii i*

( )

*THD THD i THD THD THD i THD*

*w*

ΔCM – interval of optimal value CM.

For optimal quality of stator current and voltage in a drive system THDi ought to be about 1%, THDu<2% and wi<15% in numerical process. Estimated CSC parameters by the iteration algorithm are shown in Fig. 4 and 5 or Table 1.

**Figure 3.** The iteration algorithm for selection of the inductor Ld an capacitors CM

The value of AC side capacitors CL ought to be about 25% higher than for CM because of

1,25 *C C L M* . (3)

*d*

, (4)

*d*

*d dd d d di e iR L u*

where: ud is the inverter input voltage, Rd is the inductor resistance, Ld is the inductance, ed

Equation (4) is used together with differential equation for the induction motor to derive the

The model of a squirrel-cage induction motor expressed as a set of differential equations for the stator-current and rotor-ux vector components presented in αβ stationary coordinate

, *s sr r m rm m r*

 

, *<sup>s</sup> sr r m rm m r*

 

, *r r r m r rr s r r*

, *<sup>r</sup> <sup>r</sup> r m r rr s r r*

 

 

*di R L R L R L L L i u d Lw Lw w w*

*di RL RL RL L L i u d Lw Lw w w*

*d R R L*

*dL L*

*d R R L*

*dL L*

*s rr r s*

*s rr r s*

*i*

*i*

 

 

(6)

(5)

(7)

(8)

**Pn** [kW] **Ld** [mH] **CM** [μF] **CL** [μF] **Pn** [kW] **Ld** [mH] **CM** [μF] **CL** [μF] 1,5 13,2 10 10 15 7,6 30 35 2,2 12,5 12 12 22 6,2 50 60 4 11,6 20 20 30 5,5 60 70 5,5 10,5 20 20 45 4,5 80 90 7,5 9,4 22 22 55 3 120 150 11 8,3 22 25 75 2 150 200

higher harmonics in supply network voltage:

**Table 1.** Estimated a CSC parameters

**3.1. Introduction to mathematical model** 

models of induction motor fed by the CSI.

system is as follows [Krzeminski Z., 1987]:

Differential equation for the dc-link is as follows

is the control voltage in dc-link, id is the current in dc-link.

2 2

2 2

*r r*

*r r*

**3. The mathematical model of IM supplied by CSC** 

**Figure 4.** Inductor inductance from iteration algorithm, where Pn [kW] is nominal machine power for different transistors switching frequency [kHz]

**Figure 5.** Capacitor capacitance from iteration algorithm where Pn [kW] is nominal machine power for different transistors switching frequency [kHz]

The value of AC side capacitors CL ought to be about 25% higher than for CM because of higher harmonics in supply network voltage:

$$\mathbf{C}\_{L} \approx \mathbf{1}, \mathbf{25} \cdot \mathbf{C}\_{M} \,. \tag{3}$$


**Table 1.** Estimated a CSC parameters

432 Induction Motors – Modelling and Control

different transistors switching frequency [kHz]

100kHz 50kHz

0 2

50kHz 100kHz

20kHz

12 14

L [mH]

6.6kHz

different transistors switching frequency [kHz]

**Figure 4.** Inductor inductance from iteration algorithm, where Pn [kW] is nominal machine power for

1.5 2.2 4 5.5 7.5 11 15 22 30 45 55 75 Pn [kW]

20kHz

6.6kHz

**Figure 5.** Capacitor capacitance from iteration algorithm where Pn [kW] is nominal machine power for

## **3. The mathematical model of IM supplied by CSC**

### **3.1. Introduction to mathematical model**

Differential equation for the dc-link is as follows

$$\mathbf{e}\_d = \mathbf{i}\_d \mathbf{R}\_d + \mathbf{L}\_d \frac{d\mathbf{i}\_d}{d\tau} + \mathbf{u}\_{d\tau} \tag{4}$$

where: ud is the inverter input voltage, Rd is the inductor resistance, Ld is the inductance, ed is the control voltage in dc-link, id is the current in dc-link.

Equation (4) is used together with differential equation for the induction motor to derive the models of induction motor fed by the CSI.

The model of a squirrel-cage induction motor expressed as a set of differential equations for the stator-current and rotor-ux vector components presented in αβ stationary coordinate system is as follows [Krzeminski Z., 1987]:

$$\frac{d\dot{u}\_{s\alpha}}{d\tau} = -\frac{R\_s L\_r^2 + R\_r L\_m^2}{L\_r w\_\sigma} \dot{i}\_{s\alpha} + \frac{R\_r L\_m}{L\_r w\_\sigma} \nu\_{r\alpha} + \alpha\_r \frac{L\_m}{w\_\sigma} \nu\_{r\beta} + \frac{L\_r}{w\_\sigma} u\_{s\alpha'} \tag{5}$$

$$\frac{d\dot{\mathbf{u}}\_{s\beta}}{d\tau} = -\frac{\mathcal{R}\_s \mathcal{L}\_r^2 + \mathcal{R}\_r \mathcal{L}\_m^2}{\mathcal{L}\_r w\_\sigma} \dot{\mathbf{i}}\_{s\beta} + \frac{\mathcal{R}\_r \mathcal{L}\_m}{\mathcal{L}\_r w\_\sigma} \nu\_{r\beta} - \alpha\_r \frac{\mathcal{L}\_m}{w\_\sigma} \nu\_{r\alpha} + \frac{\mathcal{L}\_r}{w\_\sigma} u\_{s\beta} \tag{6}$$

$$\frac{d\boldsymbol{\psi}\_{r\alpha}}{d\boldsymbol{\tau}} = -\frac{R\_r}{L\_r}\boldsymbol{\psi}\_{r\alpha} - \alpha\_r \boldsymbol{\psi}\_{r\beta} + \frac{R\_r L\_m}{L\_r}\dot{\mathbf{i}}\_{s\alpha'} \tag{7}$$

$$\frac{d\boldsymbol{\psi}\_{r\boldsymbol{\beta}}}{d\boldsymbol{\tau}} = -\frac{R\_r}{L\_r}\boldsymbol{\psi}\_{r\boldsymbol{\beta}} + \alpha\_r \boldsymbol{\psi}\_{r\boldsymbol{\alpha}} + \frac{R\_r L\_m}{L\_r}\dot{\mathbf{i}}\_{s\boldsymbol{\beta}\prime} \tag{8}$$

$$\frac{d\rho\_r}{d\tau} = \frac{L\_m}{\iint L\_r} (\boldsymbol{\psi}\_{r\alpha}\dot{\mathbf{i}}\_{s\beta} - \boldsymbol{\psi}\_{r\beta}\dot{\mathbf{i}}\_{s\alpha}) - \frac{1}{J}\boldsymbol{m}\_{0\prime} \tag{9}$$

, (12)

*DClink AC motor side*

The full model of the drive system in rotating reference frame xy with x axis oriented with

, *sx s r r m r m m r*

, *sy sr rm rm m r*

() , *rx r r m rx i r ry sx r r*

() , *ry <sup>r</sup> r m*

<sup>1</sup> ( ), *r m rx sy ry sx*

> , *d d d sx d dd d*

*di e R u <sup>i</sup> d LL L*

<sup>1</sup> ( ) *sx*

*M du ii u*

*sy* 1

*d C*

*M*

**4. The nonlinear multi-scalar voltage control of IM with PI controllers** 

The Nonlinear multi-scalar control was presented by authors [Krzeminski Z., 1987, Glab (Morawiec) M. et. al., 2005, 2007]. This control in classical form based on PI controllers. The simplify multi-scalar control of IM supplied by CSC for different vector components ( , *r s*

) was presented in [Glab (Morawiec) M. et. al., 2005, 2007]. These multi-scalar

 

*d L i im d JL J*

*ry i r rx sy r r*

*fx sx i sy*

*sy i sx*

*i u*

*di R L R L R L L L ii u*

 

*di RL RL RL L L*

 

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

 

*d R R L*

*d L L*

 

*sx rx i sy r ry sx*

*sy ry i sx r rx sy*

*w w*

 

(14)

*ii u*

*w w*

 

(13)

(15)

*i*

(17)

(18)

, (19)

. (20)

 *i* ),

, isx, isy are the capacitors currents.

(16)

0

 

 

> 

 

*d d sx fx p p ui u i* 

inverter output current vector is as follows

where: ωi is angular frequency of vector *<sup>f</sup> i*

**4.1. The simplified Multi-scalar control** 

( , *s s i*

 ), ( , *m s i*

2 2

*d Lw Lw* 

2 2

*d Lw Lw* 

*r r*

*r r*

*r*

*d C*

*du*

where: ud is the input six transistors bridge voltage, usx is the stator voltage component.

where

Rr, Rs are the motor windings resistance, Ls, Lr, Lm are stator, rotor and mutual inductance, usα, usβ, isα, isβ, ψrα, ψrβ are components of stator voltage, currents and rotor flux vectors, ωr is the angular rotor velocity, J is the torque of inertia, m0 is the load torque. All variables and parameters are in p. u.

### **3.2. The mathematical model of IM contains full drive system equations**

The vector components of the rotor flux together with inverter output current are used to derive model of IM fed by the CSI. The model is developed using rotating reference frame *xy* with *x* axis orientated with output current vector. The *y* component of the output current vector is equal to zero.

The variables in the rotating reference frame are presented in Fig. 6.

The output current under assumption an ideal commutator can be expressed

$$
\dot{\mathbf{i}}\_f = \mathbf{K} \cdot \dot{\mathbf{i}}\_{d'} \tag{10}
$$

where

K is the unitary commutation function (K=1).

**Figure 6.** Variables in the rotating frame of references

If the commutation function is K=1 than

$$\left| \dot{\mathbf{r}}\_f \right| \approx \dot{\mathbf{r}}\_d. \tag{11}$$

The equation (12) results from (11) taking into account ideal commutator of the CSI, according to equation

$$\begin{aligned} \mathbf{p}\_d \mathbf{p}\_{\text{DClink}} &= \mathbf{p}\_{\text{AC motor side}}\\ \mathbf{u}\_d \dot{\mathbf{i}}\_d &= \mathbf{u}\_{\text{sx}} \dot{\mathbf{i}}\_{\text{fx}} \end{aligned} \tag{12}$$

where: ud is the input six transistors bridge voltage, usx is the stator voltage component.

434 Induction Motors – Modelling and Control

parameters are in p. u.

vector is equal to zero.

where

where

0

(9)

, *<sup>f</sup> <sup>d</sup> i Ki* (10)

. *<sup>f</sup> <sup>d</sup> i i* (11)

α

x

 fi

> ψr

> > sα i

<sup>1</sup> ( ), *r m rs rs*

Rr, Rs are the motor windings resistance, Ls, Lr, Lm are stator, rotor and mutual inductance, usα, usβ, isα, isβ, ψrα, ψrβ are components of stator voltage, currents and rotor flux vectors, ωr is the angular rotor velocity, J is the torque of inertia, m0 is the load torque. All variables and

The vector components of the rotor flux together with inverter output current are used to derive model of IM fed by the CSI. The model is developed using rotating reference frame *xy* with *x* axis orientated with output current vector. The *y* component of the output current

The equation (12) results from (11) taking into account ideal commutator of the CSI,

 

 

*d L i im d JL J* 

*r*

**3.2. The mathematical model of IM contains full drive system equations** 

The variables in the rotating reference frame are presented in Fig. 6.

K is the unitary commutation function (K=1).

y

**Figure 6.** Variables in the rotating frame of references

If the commutation function is K=1 than

according to equation

The output current under assumption an ideal commutator can be expressed

β

sβ i

ψrx

ψry

The full model of the drive system in rotating reference frame xy with x axis oriented with inverter output current vector is as follows

$$\frac{d\dot{u}\_{s\mathbf{x}}}{d\tau} = -\frac{\mathcal{R}\_{\text{s}}\mathcal{L}\_{r}^{2} + \mathcal{R}\_{r}\mathcal{L}\_{m}^{2}}{\mathcal{L}\_{r}w\_{\sigma}}\dot{\imath}\_{s\mathbf{x}} + \frac{\mathcal{R}\_{r}\mathcal{L}\_{m}}{\mathcal{L}\_{r}w\_{\sigma}}\dot{\nu}\_{r\mathbf{x}} + \alpha\_{i}\dot{\imath}\_{sy} + \alpha\_{r}\frac{\mathcal{L}\_{m}}{w\_{\sigma}}\dot{\nu}\_{ry} + \frac{\mathcal{L}\_{r}}{w\_{\sigma}}u\_{s\mathbf{x}} \tag{13}$$

$$\frac{d\dot{\mathbf{u}}\_{sy}}{d\tau} = -\frac{R\_s L\_r^2 + R\_r L\_m^2}{L\_r w\_\sigma} \dot{\mathbf{i}}\_{sy} + \frac{R\_r L\_m}{L\_r w\_\sigma} \boldsymbol{\nu}\_{ry} - \alpha \dot{\mathbf{i}}\_{sx} - \alpha \dot{\mathbf{r}}\_r \frac{L\_m}{w\_\sigma} \boldsymbol{\nu}\_{rx} + \frac{L\_r}{w\_\sigma} \boldsymbol{u}\_{sy} \tag{14}$$

$$\frac{d\boldsymbol{\upmu}\_{r\boldsymbol{\chi}}}{d\boldsymbol{\upmu}} = -\frac{R\_r}{L\_r}\boldsymbol{\upmu}\_{r\boldsymbol{\upmu}} + (\boldsymbol{\upalpha}\_i - \boldsymbol{\upalpha}\_r)\boldsymbol{\upmu}\_{r\boldsymbol{\upmu}} + \frac{R\_r L\_m}{L\_r}\boldsymbol{\upalpha}\_{s\boldsymbol{\upalpha}}\tag{15}$$

$$\frac{d\boldsymbol{\psi}\_{ry}}{d\boldsymbol{\pi}} = -\frac{R\_r}{L\_r}\boldsymbol{\psi}\_{ry} - (\boldsymbol{\alpha}\_i - \boldsymbol{\alpha}\_r)\boldsymbol{\upmu}\_{rx} + \frac{R\_r L\_m}{L\_r}\boldsymbol{i}\_{sy\text{-}\boldsymbol{\pi}}\tag{16}$$

$$\frac{d\rho\_r}{d\tau} = \frac{L\_m}{JL\_r} (\wp\_{rx}i\_{sy} - \wp\_{ry}i\_{sx}) - \frac{1}{J}m\_{0\prime} \tag{17}$$

$$\frac{d\dot{\mathbf{u}}\_d}{d\pi} = \frac{e\_d}{L\_d} - \frac{R\_d}{L\_d}\dot{\mathbf{i}}\_d - \frac{\mu\_{\rm sx}}{L\_d}\tag{18}$$

$$\frac{d\mu\_{s\mathbf{x}}}{d\tau} = \frac{1}{C\_M}(\dot{\mathbf{i}}\_{f\mathbf{x}} - \dot{\mathbf{i}}\_{s\mathbf{x}}) + \alpha\_i \mu\_{sy\text{-}\mathbf{y}} \tag{19}$$

$$\frac{d\boldsymbol{u}\_{sy}}{d\tau} = -\frac{1}{C\_M}\dot{\mathbf{i}}\_{sy} - \alpha\_i \boldsymbol{u}\_{s\mathbf{x}}\,. \tag{20}$$

where: ωi is angular frequency of vector *<sup>f</sup> i* , isx, isy are the capacitors currents.

### **4. The nonlinear multi-scalar voltage control of IM with PI controllers**

### **4.1. The simplified Multi-scalar control**

The Nonlinear multi-scalar control was presented by authors [Krzeminski Z., 1987, Glab (Morawiec) M. et. al., 2005, 2007]. This control in classical form based on PI controllers. The simplify multi-scalar control of IM supplied by CSC for different vector components ( , *r s i* ), ( , *s s i* ), ( , *m s i* ) was presented in [Glab (Morawiec) M. et. al., 2005, 2007]. These multi-scalar control structures give different dynamical and statical properties of IM supplied by CSI. In this chapter the simplified control is presented. The simplification is based on (11) and (12) equations. If the capacity CM has small values (a few μF) the mathematical equations (19) - (20) can be ommitted and the output current vector in stationary state is *<sup>f</sup> <sup>s</sup> i i* . Under this simplification, to achieve the decoupling between two control paths the multi-scalar model based control system was proposed [Krzeminski Z., 1987, Glab (Morawiec) M. et. al., 2005, 2007]. The variables for the multi-scalar model of IM are defined

$$\mathbf{x}\_{11} = a \mathbf{o}\_{r'} \tag{21}$$

Sensorless Control of Induction Motor Supplied by Current Source Inverter 437

(29)

, (30)

. (31)

, (32)

, (33)

(34)

, (35)

(36)

(37)

The compensation of nonlinearities in differential equation leads to the following expressions for control variables v1 and v2 appearing in differential equations (27) - (28):

> 1 1 <sup>1</sup> , *r m sx sy d ry r di RL u v ii <sup>m</sup> L LT*

2 2 1 *r m sx sx d rx r di*

> 1 *s d ird R R TLL*

*d*

*dx R RL*

*dL L*

*i*

*r*

*d JL J*

*i*

*i x*

*d d*

*e L*

*i*

21

22

11

12

*d T* 

*dx*

*dx L*

*d T* 

*dx*

*RL u v ii <sup>m</sup> L LT* 

where m1, 2 are the PI controllers output and

The control variables are specified

variable is voltage ed in dc-link.

electromagnetic subsystem

electromechanical subsystem

The decoupled two subsystems are obtained:

when 21 d *x* ,i 0 .

1 2 21 *ry rx*

11

*x*

*v v*

 

*x*

1 2

*v v*

 

The inverter control variables are: voltage ed and the output current vector pulsation. The multi-scalar control of IM supplied by CSI was named voltage control because the control

> <sup>21</sup> <sup>22</sup> 22 , *r rm r r*

*x x*

22 2 <sup>1</sup> ( )

12 0 <sup>1</sup> , *<sup>m</sup>*

12 1 <sup>1</sup> ( ).

*x m*

*x m*

*x m*

21 *rx ry*

$$\propto\_{12} = -\dot{\mathbf{i}}\_d \boldsymbol{\Psi}\_{ry'} \tag{22}$$

$$\propto\_{21} = \left. \boldsymbol{\nu}\_{r\boldsymbol{\chi}}^{2} + \left. \boldsymbol{\nu}\_{r\boldsymbol{\chi}}^{2} \right. \tag{23}$$

$$\mathbf{x}\_{22} = \mathbf{i}\_d \boldsymbol{\upmu}\_{r\mathbf{x}'} \tag{24}$$

where

x11 is the rotor speed, x12 is the variable proportional to electromagnetic torque, x21 is the square of rotor flux and x22 is the variable named magnetized variable [Krzeminski, 1987].

Assumption of such machine state variables may lead to improvement of the control system quality due to the fact that e.g. the x12 variable is directly the electromagnetic torque of the machine. In FOC control methods [Klonne A. & Fuchs W.F., 2003, 2004, Salo M. & Tuusa H. 2004] the electromagnetic torque is not directly but indirectly controlled (the isq stator current component). With the assumption of a constant rotor flux modulus, such a control conception is correct. The inaccuracy of the machine parameters, asymmetry or inadequately aligned control system may lead to couplings between control circuits.

The mathematical model for new state of variables (21) - (24) used (15) - (18) is expressed by differential equations:

$$\frac{d\mathbf{x}\_{11}}{d\tau} = \frac{L\_m}{f L\_r} (\mathbf{x}\_{12}) - \frac{1}{f} m\_{0\prime} \tag{25}$$

$$\frac{d\mathbf{x}\_{12}}{d\tau} = -\frac{1}{T\_i}\mathbf{x}\_{12} + \frac{1}{L\_d}\boldsymbol{\mu}\_{sx}\boldsymbol{\nu}\_{ry} - \frac{R\_r L\_m}{L\_r}\dot{\mathbf{i}}\_{sy}\dot{\mathbf{i}}\_d + \boldsymbol{\upsilon}\_{1\prime} \tag{26}$$

$$\frac{d\mathbf{x}\_{21}}{d\tau} = -2\frac{R\_r}{L\_r}\mathbf{x}\_{21} + 2R\_r\frac{L\_m}{L\_r}\mathbf{x}\_{22} \tag{27}$$

$$\frac{d\mathbf{x}\_{22}}{d\tau} = -\frac{1}{T\_i}\mathbf{x}\_{22} + \frac{R\_r L\_m}{L\_r}\dot{\mathbf{i}}\_{\text{sx}}\dot{\mathbf{i}}\_d + \mathbf{v}\_2\,. \tag{28}$$

The compensation of nonlinearities in differential equation leads to the following expressions for control variables v1 and v2 appearing in differential equations (27) - (28):

$$
\omega\_1 = \frac{R\_r L\_m}{L\_r} i\_{sy} i\_d - \frac{u\_{sx}}{L\_d} \cdot \nu\_{ry} + \frac{1}{T\_i} m\_{1\prime} \tag{29}
$$

$$
\omega v\_2 = -\frac{R\_r L\_m}{L\_r} \dot{\mathbf{i}}\_{sx} \dot{\mathbf{i}}\_d + \frac{\mu\_{sx}}{L\_d} \cdot \nu\_{r\infty} + \frac{1}{T\_i} m\_2 \,\mathrm{s}\tag{30}
$$

where m1, 2 are the PI controllers output and

$$\frac{1}{T\_i} = \frac{R\_s}{L\_r} + \frac{R\_d}{L\_d} \,. \tag{31}$$

The control variables are specified

$$e\_d = -L\_d \cdot \frac{\left\| \boldsymbol{\nu}\_{ry} \boldsymbol{v}\_1 - \boldsymbol{\nu}\_{rx} \boldsymbol{v}\_2 \right\|}{\boldsymbol{\chi}\_{21}} \, \tag{32}$$

$$\rho\_i = \frac{\wp\_{rx}\upsilon\_1 + \wp\_{ry}\upsilon\_2}{i\_d \cdot \chi\_{21}} + \chi\_{11'} \tag{33}$$

when 21 d *x* ,i 0 .

436 Induction Motors – Modelling and Control

where

differential equations:

control structures give different dynamical and statical properties of IM supplied by CSI. In this chapter the simplified control is presented. The simplification is based on (11) and (12) equations. If the capacity CM has small values (a few μF) the mathematical equations (19) - (20) can be ommitted and the output current vector in stationary state is *<sup>f</sup> <sup>s</sup> i i* . Under this simplification, to achieve the decoupling between two control paths the multi-scalar model based control system was proposed [Krzeminski Z., 1987, Glab (Morawiec) M. et. al.,

> <sup>11</sup> , *<sup>r</sup> x*

<sup>12</sup> , *d ry x i* 

<sup>22</sup> , *d rx x i* 

x11 is the rotor speed, x12 is the variable proportional to electromagnetic torque, x21 is the square of rotor flux and x22 is the variable named magnetized variable [Krzeminski, 1987].

Assumption of such machine state variables may lead to improvement of the control system quality due to the fact that e.g. the x12 variable is directly the electromagnetic torque of the machine. In FOC control methods [Klonne A. & Fuchs W.F., 2003, 2004, Salo M. & Tuusa H. 2004] the electromagnetic torque is not directly but indirectly controlled (the isq stator current component). With the assumption of a constant rotor flux modulus, such a control conception is correct. The inaccuracy of the machine parameters, asymmetry or

The mathematical model for new state of variables (21) - (24) used (15) - (18) is expressed by

*r*

*d JL J*

*id r dx R L x u ii v*

*dx R L*

*dL L*

1 *r m*

*i r dx R L x ii v*

12 0 <sup>1</sup> () , *<sup>m</sup>*

12 1 1 1 , *r m sx ry sy d*

> <sup>21</sup> <sup>22</sup> 22 , *r m r r r*

*x Rx*

22 2

*sx d*

(25)

(27)

. (28)

(26)

*x m*

inadequately aligned control system may lead to couplings between control circuits.

11

21

22

12

*dx L*

*dTL L*

*dT L*

2 2 <sup>21</sup> , *rx ry x* 

(21)

(22)

(23)

(24)

2005, 2007]. The variables for the multi-scalar model of IM are defined

The inverter control variables are: voltage ed and the output current vector pulsation. The multi-scalar control of IM supplied by CSI was named voltage control because the control variable is voltage ed in dc-link.

The decoupled two subsystems are obtained:

electromagnetic subsystem

$$\frac{d\mathbf{x}\_{21}}{d\tau} = -2\frac{R\_r}{L\_r}\mathbf{x}\_{21} + 2\frac{R\_r L\_m}{L\_r}\mathbf{x}\_{22'} \tag{34}$$

$$\frac{d\mathbf{x}\_{22}}{d\tau} = \frac{1}{T\_i}(-\mathbf{x}\_{22} + m\_2) \,, \tag{35}$$

electromechanical subsystem

$$\frac{d\mathbf{x}\_{11}}{d\tau} = \frac{L\_m}{f L\_r} \mathbf{x}\_{12} - \frac{1}{f} m\_{0\prime} \tag{36}$$

$$\frac{d\mathbf{x}\_{12}}{d\tau} = \frac{1}{T\_i}(-\mathbf{x}\_{12} + m\_1). \tag{37}$$

### **4.2. The multi-scalar control with inverter mathematical model**

The author in [Morawiec M., 2007] revealed stability proof of simplified multi-scalar control while the parameters of the CSI are optimal selected.

When the capacitance CM is neglected the stator current vector *<sup>s</sup> i* is about ~5% out of phase to *<sup>f</sup> i* while nominal torque is set. Then the control variables and decoupling are not obtained precisely. The error is small than 2% because PI controllers improved it.

In order to compensate these errors the capacity CM to mathematical model is applied.

From (19) - (20) in stationary state lead to dependences:

$$\dot{\mathbf{u}}\_{s\mathbf{x}} = \dot{\mathbf{u}}\_{f\mathbf{x}} + \alpha \mathbf{p}\_{i\mathbf{f}} \mathbf{C}\_{M} \mathbf{u}\_{sy\ \prime} \tag{38}$$

Sensorless Control of Induction Motor Supplied by Current Source Inverter 439

, (48)

, (49)

, (50)

, (51)

. (52)

(53)

(55)

, (57)

, (58)

*s sx sy sy sx q iu iu* , (59)

*s sx sx sy sy p ui ui* . (60)

, (61)

, (54)

, (56)

*R R L*

<sup>12</sup> *fx ry if M* <sup>32</sup> *x i Cx* 

<sup>22</sup> *fx rx if M* <sup>31</sup> *x i Cx* 

<sup>31</sup> *rx sy ry sx xuu* 

<sup>32</sup> *rx sx ry sy x uu* 

*r*

*d JL J*

The multi-scalar model for new multi-scalar variables has the form:

12

*dx*

11

*dTL*

21

*dx L*

1 1

*i d*

22 2

*dx R L*

*dTL L*

1 22 32

2 12 31

 

 

1 1 *r m*

*v e x Cx C p L LL* 

*id r*

*dx R L*

*dL L*

 

 

12 0 <sup>1</sup> , *<sup>m</sup>*

12 11 22 1

*x u xx v*

*sx ry*

<sup>21</sup> <sup>22</sup> 22 , *r m r r r*

*x Rx*

22 11 12 2

*x u i xx v*

*sx rx d*

<sup>1</sup> ( ) *<sup>d</sup> r m d ry if MM s d r*

<sup>1</sup> ( ) *<sup>d</sup> r m d rx if MM s d r <sup>R</sup> R L v e x Cx C q L LL* 

The compensation of nonlinearities in differentials equation leads to the following expressions for control variables v1 and v2 appearing in differential equations (54), (56):

1 1 11 22

*sx ry*

1 1

*T L* 

*i d v m u xx*

*x m*

and

where

2 2 <sup>21</sup> *rx ry x* 

$$\mathbf{i}\_{sy} = -\alpha \mathbf{j}\_{\rm if} \mathbb{C}\_{M} \boldsymbol{\mu}\_{\rm ss} \,. \tag{39}$$

The new mathematical model of the drive system is obtained from (38) - (39) through differentiation it and used (15) - (16) in *xy* coordinate system:

$$\frac{d\dot{\mathbf{u}}\_{\rm sx}}{d\tau} = -\frac{R\_d}{L\_d}\dot{\mathbf{i}}\_d + \frac{1}{L\_d}\mathbf{e}\_d - \frac{1}{L\_d}\mathbf{u}\_x - \alpha\_{\dot{f}}\mathbf{C}\_M\dot{\mathbf{i}}\_{\rm sy} - \alpha\_{\dot{f}}^2\mathbf{C}\_M^2\mathbf{u}\_{\rm sx'} \tag{40}$$

$$\frac{d\dot{\mathbf{i}}\_{sy}}{d\tau} = -\alpha\_{\dot{f}} \mathbf{C}\_{M} \dot{\mathbf{i}}\_{d} + \alpha\_{\dot{f}} \mathbf{C}\_{M} \dot{\mathbf{i}}\_{s\mathbf{x}} - \alpha\_{\dot{f}}^{2} \mathbf{C}\_{M}^{2} \boldsymbol{u}\_{sy} \tag{41}$$

$$\frac{d\boldsymbol{\eta}\_{r\boldsymbol{\chi}}}{d\boldsymbol{\tau}} = -\frac{R\_r}{L\_r}\boldsymbol{\eta}\_{r\boldsymbol{\chi}} + (\boldsymbol{\alpha}\_{\circ\circ} - \boldsymbol{\alpha}\_r)\boldsymbol{\nu}\_{r\boldsymbol{\chi}} + \frac{R\_r L\_m}{L\_r}\boldsymbol{i}\_{\rm s\boldsymbol{\chi}}\tag{42}$$

$$\frac{d\boldsymbol{\eta}\_{ry}}{d\tau} = -\frac{R\_r}{L\_r}\boldsymbol{\eta}\_{ry} - (\boldsymbol{\alpha}\_{\hat{\boldsymbol{y}}} - \boldsymbol{\alpha}\_r)\boldsymbol{\upmu}\_{rx} + \frac{R\_r L\_m}{L\_r}\boldsymbol{\upmu}\_{sy} \tag{43}$$

$$\frac{d\dot{\mathbf{u}}\_d}{d\tau} = -\frac{R\_d}{L\_d}\dot{\mathbf{i}}\_d + \frac{1}{L\_d}\mathbf{e}\_d - \frac{1}{L\_d}\boldsymbol{u}\_{\infty\text{-}\prime} \tag{44}$$

$$\frac{d\boldsymbol{u}\_{\rm{sx}}}{d\boldsymbol{\pi}} = \frac{1}{\boldsymbol{C}\_{\mathcal{M}}} (\dot{\mathbf{i}}\_{f\boldsymbol{x}} - \dot{\mathbf{i}}\_{\rm{sx}}) + \alpha\_{\boldsymbol{if}} \boldsymbol{u}\_{\rm{sy}}.\tag{45}$$

$$\frac{d\mu\_{sy}}{d\tau} = -\frac{1}{C\_M} \dot{\iota}\_{sy} - \alpha\_{if}\mu\_{sx} \,. \tag{46}$$

Substituting (38) - (39) to multi-scalar variables [Krzeminski Z., 1987] one obtains:

$$\propto\_{11} = \alpha\_{r\text{ }\text{ }\text{ }\text{ }} \tag{47}$$

$$\mathbf{x}\_{12} = -\mathbf{i}\_{f\mathbf{x}}\boldsymbol{\upmu}\_{ry} - \boldsymbol{\upalpha}\_{if}\boldsymbol{\upalpha}\_{M}\mathbf{x}\_{32\text{ }\prime} \tag{48}$$

$$
\lambda \mathbf{x}\_{21} = \boldsymbol{\upmu}\_{rx}^2 + \boldsymbol{\upmu}\_{ry}^2. \tag{49}
$$

$$\mathbf{x}\_{22} = \mathbf{i}\_{\text{fx}} \boldsymbol{\upmu}\_{\text{rx}} + \boldsymbol{\upmu}\_{\text{if}} \mathbf{C}\_{M} \mathbf{x}\_{31'} \tag{50}$$

and

438 Induction Motors – Modelling and Control

to *<sup>f</sup> i*

**4.2. The multi-scalar control with inverter mathematical model** 

When the capacitance CM is neglected the stator current vector *<sup>s</sup>*

while the parameters of the CSI are optimal selected.

From (19) - (20) in stationary state lead to dependences:

differentiation it and used (15) - (16) in *xy* coordinate system:

*di*

*d*

The author in [Morawiec M., 2007] revealed stability proof of simplified multi-scalar control

while nominal torque is set. Then the control variables and decoupling are not

obtained precisely. The error is small than 2% because PI controllers improved it.

In order to compensate these errors the capacity CM to mathematical model is applied.

*sx fx if M sy i i Cu* 

*sy if M sx i Cu* 

The new mathematical model of the drive system is obtained from (38) - (39) through

*sx d* 1 1 2 2

*sy* 2 2

( ) *rx r r m*

( ) *ry <sup>r</sup> r m*

*d R R L*

*d L L*

*d R R L*

*d L L*

1 1 *d d*

<sup>1</sup> ( ) *sx*

*du ii u*

*M*

<sup>11</sup> *<sup>r</sup> x* 

*M*

*sy* 1

*d C*

Substituting (38) - (39) to multi-scalar variables [Krzeminski Z., 1987] one obtains:

*d C*

*du*

*di R ieu dLLL*

 

 

*di R i e u Ci Cu*

*ddd*

*d LLL*

*d d x if M sy if M sx*

*if M d if M sx if M sy*

*rx if r ry sx r r*

*ry if r rx sy r r*

> *d d sx ddd*

> > *fx sx if sy*

*sy if sx*

*i u*

 

*Ci Ci Cu*

 

, (41)

*i*

*i*

, (44)

, (45)

. (46)

, (47)

, (43)

, (42)

, (40)

*i* 

, (38)

. (39)

is about ~5% out of phase

$$
\Delta \mathbf{x}\_{31} = \boldsymbol{\upmu}\_{rx} \boldsymbol{\upmu}\_{sy} - \boldsymbol{\upmu}\_{ry} \boldsymbol{\upmu}\_{sx} \tag{51}
$$

$$
\Delta\Upsilon\_{32} = \mathcal{\Psi}\_{rx}\mu\_{s\chi} + \mathcal{\Psi}\_{ry}\mu\_{sy}\,. \tag{52}
$$

The multi-scalar model for new multi-scalar variables has the form:

$$\frac{d\mathbf{x}\_{11}}{d\tau} = \frac{L\_m}{\|L\_r\|} \mathbf{x}\_{12} - \frac{1}{J} m\_{0'} \tag{53}$$

$$\frac{d\mathbf{x}\_{12}}{d\tau} = -\frac{1}{T\_i}\mathbf{x}\_{12} + \frac{1}{L\_d}\boldsymbol{u}\_{s\mathbf{x}}\boldsymbol{\nu}\_{ry} - \boldsymbol{x}\_{11}\boldsymbol{x}\_{22} + \boldsymbol{v}\_{1\text{ \textquotedblleft}}\tag{54}$$

$$\frac{d\mathbf{x}\_{21}}{d\tau} = -2\frac{\mathbf{R}\_r}{L\_r}\mathbf{x}\_{21} + 2\mathbf{R}\_r\frac{L\_m}{L\_r}\mathbf{x}\_{22} \tag{55}$$

$$\frac{d\mathbf{x}\_{22}}{d\tau} = -\frac{1}{T\_i}\mathbf{x}\_{22} - \frac{1}{L\_d}\boldsymbol{\mu}\_{\rm sx}\boldsymbol{\nu}\_{\rm rx} + \frac{R\_r L\_m}{L\_r}\mathbf{i}\_d^2 + \boldsymbol{\chi}\_{11}\mathbf{x}\_{12} + \boldsymbol{\upsilon}\_{22} \tag{56}$$

where

$$\upsilon v\_1 = -\frac{1}{L\_d} \varepsilon\_d \nu\_{ry} + o\_{\circlearrowright} \{ \mathbf{x}\_{22} - \frac{R\_d}{L} \mathbf{C}\_M \mathbf{x}\_{32} - \mathbf{C}\_M \frac{R\_r L\_m}{L\_r} p\_s \} \, \, \, \tag{57}$$

$$
\sigma\_2 = \frac{1}{L\_d} e\_d \nu\_{rx} + o\_{if}(-\mathbf{x}\_{12} + \frac{R\_d}{L} \mathbf{C}\_M \mathbf{x}\_{31} + \mathbf{C}\_M \frac{R\_r L\_m}{L\_r} q\_s) \tag{58}
$$

$$\boldsymbol{\eta}\_{s} = \mathbf{i}\_{s\mathbf{x}} \boldsymbol{\mu}\_{sy} - \mathbf{i}\_{sy} \boldsymbol{\mu}\_{s\mathbf{x}} \tag{59}$$

$$\mathbf{p}\_s = \boldsymbol{\mu}\_{s\mathbf{x}} \mathbf{i}\_{s\mathbf{x}} + \boldsymbol{\mu}\_{sy} \mathbf{i}\_{sy} \,. \tag{60}$$

The compensation of nonlinearities in differentials equation leads to the following expressions for control variables v1 and v2 appearing in differential equations (54), (56):

$$
\omega v\_1 = \frac{1}{T\_i} m\_1 - \frac{1}{L\_d} \mu\_{sx} \nu\_{ry} + \mathbf{x}\_{11} \mathbf{x}\_{22} \tag{61}
$$

$$\dot{m}\_2 = \frac{1}{T\_i} m\_2 + \frac{1}{L\_d} \mu\_{sx} \nu\_{rx} - \mathbf{x}\_{11} \mathbf{x}\_{12} - \frac{R\_r L\_m}{L\_r} \mathbf{i}\_d^2 \tag{62}$$

and the control variables

$$\mathcal{e}\_d = L\_d \frac{V\_2 \mathbf{x}\_{41} - V\_1 \mathbf{x}\_{42}}{\nu\_{rx} \mathbf{x}\_{41} + \nu\_{ry} \mathbf{x}\_{42}} \, \text{} \tag{63}$$

Sensorless Control of Induction Motor Supplied by Current Source Inverter 441

*JL <sup>J</sup>* , (71)

*sx ry sy d*

*m sx rx sx d*

( )( )

*r r r m m r r r i*

3 33 4 *e ke e* , (73)

, (76)

, (77)

, (78)

. (74)

, (75)

, (79)

, (70)

\* <sup>4</sup> <sup>22</sup> <sup>22</sup> 2 2 *m m r r r r L L e Rx Rx L L*

The e4 tracking error is defined in (70), it does not influence on the control system properties

1 2 11 *m r L m e e ke*

2 1 2 1 1 0 12 11 22 1 1 1 <sup>ˆ</sup> *r r r m*

2 22 2

The Lyapunov function derivative, with (71) – (74) taken into account, may be expressed as:

*V ke ke ke ke e f v e f av*

*d ve x L* 

1

*d ve x L* 

*L JL <sup>L</sup> f it e k k e k e m x u JL L L TL*

lim ( ) 4( ) 4( ) 2

1 12 1 1 2 2 1 2 0 12

2 2

*u i xx*

2 2( ) 2

*r m rm rm sx rx d r d r r*

*RL RL RL*

*L L L L*

11 22 33 44 2 1 1 4 2 32

1 41 1

2 42

1 1 lim ( ) <sup>ˆ</sup> *m r <sup>r</sup> sx ry r m mi d*

2 2 2 2 12 3 3 3 4 4 3 4 21 22 22

*R R R L f it e k e k e k e x L x x L L L T*

*d rx if*

 

11 12

*d ry if*

 

*R R RL RL RL e ke ke x L x x u i i*

4( ) 4( ) 2 2 2( )

*L L LT LL L*

*r r r i r d r*

*r r rm rm r m*

*mm i d r JL L R L e ke k e m x u x x i i v LL TL L* 

0

, (72)

 

and is only an accepted simplification in the format of decoupling variables.

where: x11, x12, x21 and x22 are defined in (47) - (50).

Derivatives of the (67) - (70) errors take the form

2

11 12 2

*r m r m r r*

*RL RL xx v L L*

11 22

*x x*

2 2

where

4 33 34 21 22 22

2222

0 12

*<sup>e</sup> L m m ke J L*

2

1 0

<sup>ˆ</sup> ( ) *<sup>r</sup> m*

$$\alpha\_i = \frac{V\_1 \wp\_{rx} + V\_2 \wp\_{ry}}{\wp\_{rx} \wp\_{41} + \wp\_{ry} \wp\_{42}} \, ' \tag{64}$$

where

$$\mathbf{x}\_{41} = \mathbf{x}\_{22} - \frac{R\_d}{L} \mathbf{C}\_M \mathbf{x}\_{32} - \mathbf{C}\_M \frac{R\_r L\_m}{L\_r} p\_{s'} \tag{65}$$

$$\mathbf{x}\_{42} = -\mathbf{x}\_{12} + \frac{R\_d}{L} \mathbf{C}\_M \mathbf{x}\_{31} + \mathbf{C}\_M \frac{R\_r L\_m}{L\_r} \mathbf{q}\_{s'} \tag{66}$$

1 *<sup>i</sup> <sup>T</sup>* is determined in (31).

The decoupled two subsystems are obtained as in (34) - (37).

## **4.3. The multi-scalar adaptive-backstepping control of an IM supplied by the CSI**

The backstepping control can be appropriately written for an induction squirrel-cage machine supplied from a VSI. In literature the backstepping control is known for adaptation of selected machine parameters, written for an induction motor [Tan H. & Chang J., 1999, Young Ho Hwang, 2008]. In [Tan H. & Chang J., 1999, Young Ho Hwang, 2008] the authors defined the machine state variables in the dq coordinate system, oriented in accordance with the rotor flux vector (FOC). The control method presented in [Tan H. & Chang J., 1999, Young Ho Hwang, 2008] is based on control of the motor state variables: ωr – rotor angular speed, rotor flux modulus and the stator current vector components: isd and isq. Selection of the new motor state variables, as in the case of multi-scalar control with linear PI regulators, leads to a different form of expressions describing the machine control and decoupling. The following state variables have been selected for the multi-scalar backstepping control

$$\mathbf{x}\_{1} = \mathbf{x}\_{11}^{\*} - \mathbf{x}\_{11'} \tag{67}$$

$$e\_2 = \stackrel{\*}{\mathbf{x}\_{12}} - \stackrel{\*}{\mathbf{x}\_{12}} \,\prime\tag{68}$$

$$\mathbf{x}\_{3} = \mathbf{x}\_{21}^{\*} - \mathbf{x}\_{21} \,\prime \tag{69}$$

$$e\_4 = \left(2R\_r \frac{L\_m}{L\_r} \mathbf{x}\_{22}\right)^\* - 2R\_r \frac{L\_m}{L\_r} \mathbf{x}\_{22} \tag{70}$$

where: x11, x12, x21 and x22 are defined in (47) - (50).

The e4 tracking error is defined in (70), it does not influence on the control system properties and is only an accepted simplification in the format of decoupling variables.

Derivatives of the (67) - (70) errors take the form

$$
\dot{e}\_1 = \frac{L\_m}{fL\_r} e\_2 - k\_1 e\_1 - \frac{\tilde{m}\_0}{f} \tag{71}
$$

$$\dot{e}\_{2} = k\_{1}e\_{2} - k\_{1}^{2} \frac{\text{IL}\_{r}}{\text{L}\_{m}} e\_{1} + \frac{\text{L}\_{r}}{\text{L}\_{m}} \dot{\hat{m}}\_{0} + \frac{1}{T\_{i}} \mathbf{x}\_{12} - \frac{1}{\text{L}\_{d}} u\_{sx} \boldsymbol{\nu}\_{ry} + \mathbf{x}\_{11} \mathbf{x}\_{22} + \frac{R\_{r} L\_{m}}{\text{L}\_{r}} \dot{\mathbf{i}}\_{sy} \mathbf{i}\_{d} - \boldsymbol{\nu}\_{1} \tag{72}$$

$$
\dot{e}\_3 = -k\_3 e\_3 + e\_4 \,\prime \tag{73}
$$

$$\begin{split} \dot{\nu}\_{4} &= -k\_{3}^{2}e\_{3} + k\_{3}e\_{4} - 4(\frac{R\_{r}}{L\_{r}})^{2}x\_{21} + 4(\frac{R\_{r}}{L\_{r}})^{2}L\_{m}x\_{22} + 2\frac{R\_{r}L\_{m}}{L\_{r}T\_{i}}x\_{22} + 2\frac{R\_{r}L\_{m}}{L\_{r}L\_{d}}u\_{\mathrm{sr}}\nu\_{\mathrm{rx}} - 2(\frac{R\_{r}L\_{m}}{L\_{r}})^{2}i\_{\mathrm{sr}}i\_{d} + \\ &- 2\frac{R\_{r}L\_{m}}{L\_{r}}x\_{11}x\_{12} - 2\frac{R\_{r}L\_{m}}{L\_{r}}\nu\_{2} \end{split} \tag{74}$$

The Lyapunov function derivative, with (71) – (74) taken into account, may be expressed as:

$$\begin{split} \dot{V} &= -k\_1^2 e\_1 - k\_2^2 e\_2 - k\_3^2 e\_3 - k\_4^2 e\_4 + e\_2 (f\_1 - v\_1) + e\_4 (f\_2 - a\_3 v\_2) + \\ &+ \tilde{m}\_0 (-\frac{e\_1}{J} - k\_1 \frac{L\_r}{L\_m} e\_2 + \frac{\dot{\tilde{m}}\_0}{\mathcal{I}}) \end{split} \tag{75}$$

where

440 Induction Motors – Modelling and Control

and the control variables

*<sup>i</sup> <sup>T</sup>* is determined in (31).

where

1

**CSI** 

2

, (62)

, (63)

, (64)

*R L*

2 2 11 12

*d d*

*i*

1 1 *r m*

*i d r*

2 41 1 42 41 42

 *x x*

41 42 *rx ry*

*d r m M M s*

*d r m M M s*

*r <sup>R</sup> R L x x Cx C p L L* , (65)

*r*

, (66)

1 11 11 *ex x* , (67)

2 12 12 *ex x* , (68)

3 21 21 *ex x* , (69)

 

 

*rx ry*

1 2

*V V x x*

*<sup>R</sup> R L x x Cx C q L L*

**4.3. The multi-scalar adaptive-backstepping control of an IM supplied by the** 

following state variables have been selected for the multi-scalar backstepping control

\*

\*

\*

The backstepping control can be appropriately written for an induction squirrel-cage machine supplied from a VSI. In literature the backstepping control is known for adaptation of selected machine parameters, written for an induction motor [Tan H. & Chang J., 1999, Young Ho Hwang, 2008]. In [Tan H. & Chang J., 1999, Young Ho Hwang, 2008] the authors defined the machine state variables in the dq coordinate system, oriented in accordance with the rotor flux vector (FOC). The control method presented in [Tan H. & Chang J., 1999, Young Ho Hwang, 2008] is based on control of the motor state variables: ωr – rotor angular speed, rotor flux modulus and the stator current vector components: isd and isq. Selection of the new motor state variables, as in the case of multi-scalar control with linear PI regulators, leads to a different form of expressions describing the machine control and decoupling. The

41 22 32

42 12 31

The decoupled two subsystems are obtained as in (34) - (37).

*rx ry*

*v m u xx i T L L* 

> *Vx Vx e L*

*sx rx d*

$$w\_1 = -\frac{1}{L\_d} \varepsilon\_d \nu\_{ry} + \alpha\_{\hat{\mu}} \chi\_{41'} \tag{76}$$

$$w\_2 = \frac{1}{L\_d} e\_d \nu\_{rx} + o\_{if} x\_{42} \,\prime \tag{77}$$

$$\begin{split} f\_1 &= \liminf\_{l \ge 1} \mathbf{i}\_{12} \cdot \mathbf{e}\_1 (\frac{\mathbf{L}\_m}{\|\mathbf{L}\_r\|} - k\_1^2 \frac{\|\mathbf{L}\_r\|}{\mathbf{L}\_m}) + \mathbf{k}\_2 \mathbf{e}\_2 + \mathbf{k}\_1 \mathbf{e}\_2 + \frac{\mathbf{L}\_r}{\mathbf{L}\_m} \dot{\hat{\mathbf{m}}}\_0 + \frac{1}{T\_i} \mathbf{x}\_{12} - \frac{1}{\mathbf{L}\_d} \boldsymbol{\mu}\_{\text{sx}} \boldsymbol{\nu}\_{ry} + \\ &+ \mathbf{x}\_{11} \mathbf{x}\_{22} \end{split} \tag{78}$$

$$\begin{split} f\_{2} &= \text{lim}\,\dot{\boldsymbol{x}}\_{12} \cdot (\boldsymbol{e}\_{3} - \boldsymbol{k}\_{3}^{2}\boldsymbol{e}\_{3}) + \boldsymbol{k}\_{4}\,\boldsymbol{e}\_{4} + \boldsymbol{k}\_{3}\,\boldsymbol{e}\_{4} - 4(\frac{\boldsymbol{R}\_{r}}{\boldsymbol{L}\_{r}})^{2}\mathbf{x}\_{21} + 4(\frac{\boldsymbol{R}\_{r}}{\boldsymbol{L}\_{r}})^{2}\boldsymbol{L}\_{m}\mathbf{x}\_{22} + 2\frac{\boldsymbol{R}\_{r}\boldsymbol{L}\_{m}}{\boldsymbol{L}\_{r}\boldsymbol{T}\_{i}}\mathbf{x}\_{22} + \\ &+ 2\frac{\boldsymbol{R}\_{r}\boldsymbol{L}\_{m}}{\boldsymbol{L}\_{r}\boldsymbol{L}\_{d}}\boldsymbol{u}\_{\text{s}\mathbf{y}}\boldsymbol{\nu}\_{\text{rx}} - 2(\frac{\boldsymbol{R}\_{r}\boldsymbol{L}\_{m}}{\boldsymbol{L}\_{r}})^{2}\dot{\boldsymbol{i}}\_{d}^{2} - 2\frac{\boldsymbol{R}\_{r}\boldsymbol{L}\_{m}}{\boldsymbol{L}\_{r}}\mathbf{x}\_{11}\mathbf{x}\_{12} \end{split} \tag{79}$$

$$a\_3 = 2\frac{R\_r L\_m}{L\_r}.$$

limit12 – is a dynamic limitation in the motor speed control subsystem,

limit22 – is a dynamic limitation in the rotor flux control subsystem,

k1…k4 and γ are the constant gains.

The control variables take the form:

$$\sigma\_d = L\_d \frac{\mathbf{x}\_{41} f\_2 - a\_3 \mathbf{x}\_{42} f\_1}{a\_3 (\boldsymbol{\upmu}\_{rx} \mathbf{x}\_{41} + \boldsymbol{\upmu}\_{ry} \mathbf{x}\_{42})} \, ^\prime \tag{80}$$

Sensorless Control of Induction Motor Supplied by Current Source Inverter 443

<sup>22</sup> *x* variables appearing in the e2 and e4 deviations can be

12lim max 21 22 *<sup>s</sup> x Ixx* , (83)

22 lim max max 11 ( , ,) *s s x fU I x* , (84)

*<sup>L</sup>* , (85)

, (86)

, (87)

, (88)

of the set variables are not suitable for direct adaptation in the drive systems. Therefore, a solution often quoted in literature is the use of a PI or PID speed controller at the torque

dynamically limited and the dynamic limitations are defined by the expressions

2 2

2

*m*

2 12lim 12

2 12lim 12

4 22lim 22

limit 0,

ex x

limit 0,

2 21

12lim max 21 *s* 2

giving the relationship between the x21 variable, the stator current modulus Ismax, and the

For the multi-scalar backstepping control, to the f1 and f2 variables the limit12 and limit22 variables were introduced; they assume the 0 or 1 value depending on the need of limiting

Limitation of variables in the Lyapunov function-based control systems may be performed

ex x *if then*

limit 0, x < x then

*if*

else 12 limit 1 ,

ex x *if then*

\* <sup>12</sup>

\* <sup>12</sup>

\* <sup>22</sup>

12 12lim

xx

12 12lim

22 22lim

xx

*<sup>x</sup> x Ix*

2 2

control circuit input.

where

The set values of the \*

[Adamowicz M.; Guzinski J., 2005]:

x12lim – the set torque limitation, x22lim – the x22 variable limitation,

motor set torque limitation.

the set variable.

in the following way:

<sup>12</sup> *<sup>x</sup>* , \*

Ismax – maximum value of the stator current modulus, Usmax – maximum value of the stator voltage modulus.

The above given expressions may be modified to:

$$\alpha\_i = \frac{a\_3 \wp\_{rx} f\_1 + \wp\_{ry} f\_2}{a\_3 (\wp\_{rx} x\_{41} + \wp\_{ry} x\_{42})} \,. \tag{81}$$

The inverter control variables are: voltage ed and the output current vector pulsation. The two decoupled subsystems are obtained as in (34) - (37).

The load torque m0 can be estimated from the formula:

$$
\dot{\hat{m}}\_0 = \gamma (\frac{e\_1}{J} + k\_1 \frac{L\_r}{L\_m} e\_2) \,. \tag{82}
$$

### **4.4. Dynamic limitations of the reference variables**

In control systems with the conventional linear controllers of the PI or PID type, the reference (or controller output) variable dynamics are limited to a constant value or dynamically changed by (83) - (84), depending on the drive working point.

Control systems where the control variables are determined from the Lyapunov function (like in backstepping control) have no limitations in the set variable control circuits. The reference variable dynamics may be limited by means of additional first order inertia elements (e.g. on the set speed signal).

The author of this paper has not come across a solution of the problem in the most significant backstepping control literature references, e.g. [Tan H. & Chang J., 1999, Young Ho Hwang, 2008]. In the quoted reference positions, the authors propose the use of an inertia elements on the set variable signals. Such approach is an intermediate method, not giving any rational control effects. The use of an inertia element on the reference signal, e.g. of the rotor angular speed, will slow down the reference electromagnetic torque reaction in proportion to the inertia element time-constant. In effect a "slow" build-up of the motor electromagnetic torque is obtained, which may be acceptable in some applications. In practice the aim is to limit the electromagnetic torque value without an impact on the buildup dynamics. Control systems with the Lyapunov function-based control without limitation of the set variables are not suitable for direct adaptation in the drive systems. Therefore, a solution often quoted in literature is the use of a PI or PID speed controller at the torque control circuit input.

The set values of the \* <sup>12</sup> *<sup>x</sup>* , \* <sup>22</sup> *x* variables appearing in the e2 and e4 deviations can be dynamically limited and the dynamic limitations are defined by the expressions [Adamowicz M.; Guzinski J., 2005]:

$$\mathbf{x}\_{12\text{ lim}} = \sqrt{\mathbf{I}\_{s\text{ max}}^2 \mathbf{x}\_{21} - \mathbf{x}\_{22}^2} \; \prime \tag{83}$$

$$\propto\_{22\text{lim}} = f(\mathcal{U}\_{s\text{max}}^2, \mathcal{I}\_{s\text{max}}^2, \mathbf{x}\_{11}) \text{ \text{\textquotedblleft}} \tag{84}$$

where

442 Induction Motors – Modelling and Control

k1…k4 and γ are the constant gains. The control variables take the form: <sup>3</sup> 2 *r m r R L*

*<sup>L</sup>* .

41 2 3 42 1 3 41 42 ( ) *d d rx ry*

> 31 2 3 41 42 ( ) *rx ry*

 

 

*m*

. (82)

*af f ax x*

*rx ry*

The inverter control variables are: voltage ed and the output current vector pulsation. The

1 0 12 ˆ ( ) *<sup>r</sup>*

*<sup>e</sup> <sup>L</sup> m ke J L* 

In control systems with the conventional linear controllers of the PI or PID type, the reference (or controller output) variable dynamics are limited to a constant value or

Control systems where the control variables are determined from the Lyapunov function (like in backstepping control) have no limitations in the set variable control circuits. The reference variable dynamics may be limited by means of additional first order inertia

The author of this paper has not come across a solution of the problem in the most significant backstepping control literature references, e.g. [Tan H. & Chang J., 1999, Young Ho Hwang, 2008]. In the quoted reference positions, the authors propose the use of an inertia elements on the set variable signals. Such approach is an intermediate method, not giving any rational control effects. The use of an inertia element on the reference signal, e.g. of the rotor angular speed, will slow down the reference electromagnetic torque reaction in proportion to the inertia element time-constant. In effect a "slow" build-up of the motor electromagnetic torque is obtained, which may be acceptable in some applications. In practice the aim is to limit the electromagnetic torque value without an impact on the buildup dynamics. Control systems with the Lyapunov function-based control without limitation

  , (80)

. (81)

*x f ax f e L ax x* 

*a*

limit12 – is a dynamic limitation in the motor speed control subsystem,

*i*

dynamically changed by (83) - (84), depending on the drive working point.

two decoupled subsystems are obtained as in (34) - (37). The load torque m0 can be estimated from the formula:

**4.4. Dynamic limitations of the reference variables** 

elements (e.g. on the set speed signal).

limit22 – is a dynamic limitation in the rotor flux control subsystem,

x12lim – the set torque limitation,

x22lim – the x22 variable limitation,

Ismax – maximum value of the stator current modulus,

Usmax – maximum value of the stator voltage modulus.

The above given expressions may be modified to:

$$\chi\_{12\text{lim}} = \sqrt{I\_{s\text{max}}^2 \chi\_{21} - \frac{\chi\_{21}^2}{L\_m^2}}\,\,\,\,\tag{85}$$

giving the relationship between the x21 variable, the stator current modulus Ismax, and the motor set torque limitation.

For the multi-scalar backstepping control, to the f1 and f2 variables the limit12 and limit22 variables were introduced; they assume the 0 or 1 value depending on the need of limiting the set variable.

Limitation of variables in the Lyapunov function-based control systems may be performed in the following way:

$$\text{if } \left( \mathbf{x}\_{12}^{\*} > \mathbf{x}\_{12\text{lim}} \right) \text{ then } \begin{cases} \text{limit}\_{12} = \mathbf{0},\\ \mathbf{e}\_{2} = \mathbf{x}\_{12\text{lim}} - \mathbf{x}\_{12} \end{cases} \text{,} \tag{86}$$

$$\text{if } \left( \mathbf{x}\_{12}^{\*} < -\mathbf{x}\_{12\text{lim}} \right) \text{ then } \begin{cases} \text{limit}\_{12} = \mathbf{0},\\ \mathbf{e}\_{2} = -\mathbf{x}\_{12\text{lim}} - \mathbf{x}\_{12} \end{cases} \text{}\prime \tag{87}$$

else 12 limit 1 ,

$$\text{if } \left( \mathbf{x}\_{22}^{\*} > \mathbf{x}\_{22\text{lim}} \right) \text{ then } \begin{cases} \text{limit}\_{22} = \mathbf{0},\\ \mathbf{e}\_{4} = \mathbf{x}\_{22\text{lim}} - \mathbf{x}\_{22} \end{cases} \text{}\prime \tag{88}$$

$$\text{if } \left( \mathbf{x}\_{22}^{\*} < -\mathbf{x}\_{22\text{lim}} \right) \text{ then } \begin{cases} \text{limit}\_{22} = 0, \\ \mathbf{e}\_{4} = -\mathbf{x}\_{22\text{lim}} - \mathbf{x}\_{22} \end{cases} \text{}\prime \tag{89}$$

12 signal. In this way a system reacting to the change of machine

value 1*e* 0 and lack of full control over maintaining the rotor set angular speed. Compensation of the limit12 limitation introduced to the control system is possible by

A corrector in the form of an e1 signal integrating element was added to the set

real load torque was obtained. The introduced correction minimizes the rotor angular speed

*k*

The gain ke1 should be adjusted that the speed overregulation in the intermediate state does

For *<sup>e</sup>*1 1 *k k* the KTL signal will become an oscillation element and may lead to the control

The correction element amplification must not be greater than k1, or:

\*

12 1 1 *r*

*m JL x k e KT L*

The use of (93) in the angular speed control circuit improves the load torque estimation and

*L*

*t*

deviation and the corrector signal may be treated as the estimated load torque value.

*L e t KT k e d*

12 expression eliminates the intermediate

, (90)

1 1 0 0,1 *<sup>e</sup> k k* , (91)

*<sup>e</sup>*1 1 *k k* . (92)

, (93)

<sup>0</sup> ˆ*m KTL* . (94)

12 for a steady state gives the deviation

speed control. Omitting 0 *m*ˆ in the set torque x\*

electromagnetic torque x\*

KTL – correction element,

system loss of stability.

The x\*

where

not exceed 5%:

where

state speed over-regulation. But absence of 0 *m*ˆ in x\*

installing a corrector in the rotor angular speed control circuit.

The correction element is determined by the expression:

tk-1…tk is the e1 signal integration range,

*<sup>e</sup>*<sup>1</sup> *k* – is the correction element amplification.

The KTL signal must be limited to the x12lim value.

eliminates the steady state speed error.

12 set value expression must be modified:

$$\text{else } \liminf\_{22} = 1$$

The dynamic limitations effected in accordance with expressions (83) – (84) limit properly the value of \* <sup>12</sup> *<sup>x</sup>* and \* <sup>22</sup> *x* variables without any interference in the reference signal build-up dynamics.

Fig. 7 presents the variable simulation diagrams. The backstepping control dynamic limitations were used.

**Figure 7.** Diagrams of multi-scalar variables in the machine dynamic states, the x12ogr = 1.0 and x22ogr = 0.74 limitations were set for a drive system with an induction squirrel-cage machine supplied from a CSC-simulation diagrams, x\* 12 – diagram of the machine set electromagnetic torque (without signal limitation), x\* 22 – diagram of the x\*22 set signal (without limitation).

### **4.5. Impact of the dynamic limitation on the estimation of parameters**

The use of a variable limitation algorithm may have a negative impact on the control system estimated parameters. This has a direct connected with the limited deviation values, which are then used in an adaptive parameter estimation. Such phenomenon is presented in Fig. 7. The estimated parameter in the control system is the motor load torque 0 *m*ˆ . The set electromagnetic torque is limited to the x12lim = 1.0 value. Fig. 7 shows that the estimated load torque increases slowly in the intermediate states. Limitation of the set electromagnetic torque causes the limitation of deviation e2, which in turn causes limited increase dynamics of the estimated load torque. The 0 *m*ˆ value for limit12 = 0 in the dynamic states does not reach the real value of the load torque, which should be 0 12 *m x* ˆ . A large *m*<sup>0</sup> estimation error occurs in the intermediate states, which can be seen in Fig. 8. The estimation error in the intermediate states is 0 *m* 0 because the torque limitation, introduced to the control system, is not compensated. The simulation and experimental tests have shown that the load torque estimation error in the intermediate state has an insignificant impact on the speed control. Omitting 0 *m*ˆ in the set torque x\* 12 expression eliminates the intermediate state speed over-regulation. But absence of 0 *m*ˆ in x\* 12 for a steady state gives the deviation value 1*e* 0 and lack of full control over maintaining the rotor set angular speed. Compensation of the limit12 limitation introduced to the control system is possible by installing a corrector in the rotor angular speed control circuit.

A corrector in the form of an e1 signal integrating element was added to the set electromagnetic torque x\* 12 signal. In this way a system reacting to the change of machine real load torque was obtained. The introduced correction minimizes the rotor angular speed deviation and the corrector signal may be treated as the estimated load torque value.

The correction element is determined by the expression:

$$KT\_L = k\_{e1} \int\_{t\_{k-1}}^{t\_k} e\_1 d\tau \,\tag{90}$$

where

444 Induction Motors – Modelling and Control

<sup>12</sup> *<sup>x</sup>* and \*

the value of \*

limitations were used.

CSC-simulation diagrams, x\*

limitation), x\*

dynamics.

\* <sup>22</sup>

limit 0, x < x then ex x *if*

The dynamic limitations effected in accordance with expressions (83) – (84) limit properly

Fig. 7 presents the variable simulation diagrams. The backstepping control dynamic

**Figure 7.** Diagrams of multi-scalar variables in the machine dynamic states, the x12ogr = 1.0 and x22ogr = 0.74 limitations were set for a drive system with an induction squirrel-cage machine supplied from a

The use of a variable limitation algorithm may have a negative impact on the control system estimated parameters. This has a direct connected with the limited deviation values, which are then used in an adaptive parameter estimation. Such phenomenon is presented in Fig. 7. The estimated parameter in the control system is the motor load torque 0 *m*ˆ . The set electromagnetic torque is limited to the x12lim = 1.0 value. Fig. 7 shows that the estimated load torque increases slowly in the intermediate states. Limitation of the set electromagnetic torque causes the limitation of deviation e2, which in turn causes limited increase dynamics of the estimated load torque. The 0 *m*ˆ value for limit12 = 0 in the dynamic states does not reach the real value of the load torque, which should be 0 12 *m x* ˆ . A large *m*<sup>0</sup> estimation error occurs in the intermediate states, which can be seen in Fig. 8. The estimation error in the intermediate states is 0 *m* 0 because the torque limitation, introduced to the control system, is not compensated. The simulation and experimental tests have shown that the load torque estimation error in the intermediate state has an insignificant impact on the

22 – diagram of the x\*22 set signal (without limitation).

**4.5. Impact of the dynamic limitation on the estimation of parameters** 

4 22lim 22

<sup>22</sup> *x* variables without any interference in the reference signal build-up

12 – diagram of the machine set electromagnetic torque (without signal

, (89)

22 22lim

else 22 limit 1 .

tk-1…tk is the e1 signal integration range,

KTL – correction element,

*<sup>e</sup>*<sup>1</sup> *k* – is the correction element amplification.

The gain ke1 should be adjusted that the speed overregulation in the intermediate state does not exceed 5%:

$$0 < k\_{e1} \le 0, 1 \cdot k\_{1'} \tag{91}$$

The correction element amplification must not be greater than k1, or:

*<sup>e</sup>*1 1 *k k* . (92)

For *<sup>e</sup>*1 1 *k k* the KTL signal will become an oscillation element and may lead to the control system loss of stability.

The KTL signal must be limited to the x12lim value.

The x\* 12 set value expression must be modified:

$$\mathbf{x}\_{12}^{\*} = \frac{\text{J}\mathbf{L}\_r}{\text{L}\_m}\mathbf{k}\_1\mathbf{e}\_1 + \text{K}\mathbf{T}\_L\text{-}\prime\tag{93}$$

where

$$
\hat{m}\_0 \approx KT\_L \,. \tag{94}
$$

The use of (93) in the angular speed control circuit improves the load torque estimation and eliminates the steady state speed error.

Fig. 9 presents the load torque (determined in (94)) estimation as well as x12 and the limit12 limitations.

Sensorless Control of Induction Motor Supplied by Current Source Inverter 447

, (95)

*<sup>L</sup>* . (96)

, (97)

, (98)

, (99)

, (100)

, (101)

, (102)

, (103)

equation may be introduced to the induction machine mathematical model to obtain the set CSI output current component. The time constant T is equal the inductance Ld, it can be

> <sup>i</sup> <sup>1</sup> (i i ) *<sup>s</sup> f s*

> > *d d R T*

The simplified version of the CSI output current control it is assumed that the output capacitors have negligibly small capacitance, so their impact on the drive system dynamics is small. Assuming that the cartesian coordinate system, where the mathematical model variables are defined, is associated with the CSI output current vector (which with this simplification is the machine stator current) the mathematical model can be obtained (95)

> <sup>11</sup> *<sup>r</sup> x*

<sup>12</sup> *sx rx x i* 

<sup>22</sup> *sx rx x i* 

*r*

1 *i*

*d JL J*

*sx i* is treated as the output current vector component and *<sup>f</sup> <sup>s</sup> i i* .

11

*dx*

21

*dx L*

12

*d T* 

*dx R RL*

*dL L*

For those variables, the multi-scalar model has the form:

2 2 <sup>21</sup> *rx ry x* 

12 0 *<sup>m</sup>* 1

12 1

*x v*

<sup>21</sup> <sup>22</sup> 2 2 *r rm r r*

*x x*

*x m*

*dt T*

*d*

**5.1. The simplified multi-scalar current control of induction machine** 

written:

and (42) - (43) equations).

where

The multi-scalar variables have the form

**Figure 8.** Impact of the electromagnetic torque limitation x12lim on the estimated load torque 0 *m*ˆ (82).

**Figure 9.** Diagrams of the limit12 variable, *KTL* load torque and electromagnetic torque x12..

## **5. The nonlinear multi-scalar current control of induction machine**

Conception of the CSI current control is based on forced components of the CSI output current. The dc-link circuit inductor could be modeled as the first order inertia element with the time constant T of a value equal to the dc-link circuit time constant value. The dc-link equation may be introduced to the induction machine mathematical model to obtain the set CSI output current component. The time constant T is equal the inductance Ld, it can be written:

$$\frac{d\vec{\mathbf{i}}\_s}{dt} = \frac{1}{T}(\vec{\mathbf{i}}\_f - \vec{\mathbf{i}}\_s) \,\tag{95}$$

$$T = \frac{R\_d}{L\_d} \,. \tag{96}$$

### **5.1. The simplified multi-scalar current control of induction machine**

The simplified version of the CSI output current control it is assumed that the output capacitors have negligibly small capacitance, so their impact on the drive system dynamics is small. Assuming that the cartesian coordinate system, where the mathematical model variables are defined, is associated with the CSI output current vector (which with this simplification is the machine stator current) the mathematical model can be obtained (95) and (42) - (43) equations).

The multi-scalar variables have the form

$$\alpha\_{11} = \alpha\_{r\text{ \textquotedblleft}}\tag{97}$$

$$
\propto\_{12} = -i\_{\rm sx} \psi\_{\rm rx} \,\tag{98}
$$

$$\propto \mathbf{x}\_{21} = \boldsymbol{\upmu}\_{r\mathbf{x}}^{2} + \boldsymbol{\upmu}\_{r\mathbf{y}}^{2} \tag{99}$$

$$\mathbf{x}\_{22} = \mathbf{i}\_{sx} \boldsymbol{\upmu}\_{rx} \,\,\,\,\,\,\tag{100}$$

where

446 Induction Motors – Modelling and Control

0 mˆ

KTL

limitations.

Fig. 9 presents the load torque (determined in (94)) estimation as well as x12 and the limit12

0 mˆ

**Figure 8.** Impact of the electromagnetic torque limitation x12lim on the estimated load torque 0 *m*ˆ (82).

**Figure 9.** Diagrams of the limit12 variable, *KTL* load torque and electromagnetic torque x12..

**5. The nonlinear multi-scalar current control of induction machine** 

Conception of the CSI current control is based on forced components of the CSI output current. The dc-link circuit inductor could be modeled as the first order inertia element with the time constant T of a value equal to the dc-link circuit time constant value. The dc-link *sx i* is treated as the output current vector component and *<sup>f</sup> <sup>s</sup> i i* .

For those variables, the multi-scalar model has the form:

$$\frac{d\mathbf{x}\_{11}}{d\tau} = \frac{L\_{\text{gt}}}{fL\_r}\mathbf{x}\_{12} - \frac{1}{f}m\_0 \tag{101}$$

$$\frac{d\mathbf{x}\_{12}}{d\tau} = -\frac{1}{T\_i}\mathbf{x}\_{12} + \mathbf{v}\_{1\prime} \tag{102}$$

$$\frac{d\mathbf{x}\_{21}}{d\tau} = -2\frac{R\_r}{L\_r}\mathbf{x}\_{21} - 2\frac{R\_r L\_m}{L\_r}\mathbf{x}\_{22} \tag{103}$$

$$\frac{d\mathbf{x}\_{22}}{d\tau} = -\frac{1}{T\_i}\mathbf{x}\_{22} + \frac{R\_r L\_m}{L\_r}\mathbf{i}\_d + \upsilon\_2\ . \tag{104}$$

Applying the linearization method, the following relations are obtained, where m1 is the subordinated regulator output in the speed control line and m2 is the subordinated regulator output in the flux control line

$$w\_1 = \frac{1}{T\_i} m\_{1'} \tag{105}$$

Sensorless Control of Induction Motor Supplied by Current Source Inverter 449

, (111)

Deriving **u**s(k) from (38), the motor stator voltage is obtained as a function of the output

u ( ) i ( ) i ( ) u ( 1) *imp s fs s*

In the equations (5) - (9) representing the cage induction motor mathematical model the stator current vector components are appeared but the direct control variables do not. The motor stator current vector components cannot be the control variables because the multiscalar model relations are derived from them. This is a different situation than with the FOC control. The FOC control is based on the machine stator current components described in a coordinate system associated with the rotor flux and the stator current components are the control variables. Therefore, control variables must be introduced into the mathematical model (5) - (9). The control may be introduced considering the machine currents (5) - (6) and equation (95) written for the αβ components and describing the dc-link circuit dynamics. Adding the respective sides of equations (5) and (95) and equations (6) and (95), where equation (95) must be written with the (αβ) components – the mathematical model of the

> 2 2 1 2 2 2 22

> 2 2 1 2 2 2 22

, (113)

, (112)

*s rr r s f*

 

*s rr r s f*

 

 

 

, (115)

, (116)

. (117)

, (114)

 

 

*s sr rm r r m m r*

*s sr rm r r m m r*

*di R L T R L T L w R L L L <sup>i</sup> u i d Lw T Lw w w T*

*di RLT RL T Lw RL L L <sup>i</sup> u i d Lw T Lw w w T*

> <sup>11</sup> *<sup>r</sup> x*

<sup>12</sup> *rs rs xii* 

<sup>22</sup> *rs rs xii* 

Introducing the multi-scalar variables (114) - (117), the multi-scalar model of an IM fed by

 

   

2 2 <sup>21</sup> *r r x* 

> 

 

 

*r r*

*r r*

The multi-scalar variables are assumed like in [Krzeminski Z., 1987]:

  

*k kk k <sup>C</sup>*

*M*

u( ) *<sup>s</sup> <sup>k</sup>* is the stator voltage vector at the k-th moment, Timp - sampling period.

*T*

current, stator current and stator voltage, in the form:

drive system fed by the CSI is obtained:

and equations (7) - (8).

the CSI is obtained:

where

$$
\omega\_2 = \frac{1}{T\_i} m\_2 - \frac{R\_r L\_m}{L\_r} \mathbf{i}\_d \,. \tag{106}
$$

The control variables are modulus of the CSI output current and the output current vector pulsation, given by the following relations:

$$\left| \mathbf{i}\_{\mathbf{f}} \right| = T \frac{\upsilon\_{2} \nu\_{rx} - \upsilon\_{1} \nu\_{ry}}{\propto\_{21} \nu\_{ry}} \, , \tag{107}$$

$$
\omega \rho\_i = \frac{\upsilon\_1 + \upsilon\_2}{\varkappa\_{21}\dot{\iota}\_{sx}} + \varkappa\_{11} \,. \tag{108}
$$

where: Ld *–*inductance, Ti– the system time constant.

### **5.2. The multi-scalar current control of induction machine**

The current control analysis presented in the preceding sections does not take the CSI output capacitors into account. Such simplification may be applied because of the small impact of the capacitors upon the control variables (the machine stator current and voltage are measured). The capacitor model will have a positive impact on the control system dynamics.

The output capacitor relations have the form:

$$\frac{d\vec{\mathbf{u}}\_s}{dt} = \frac{1}{C\_M} (\vec{\mathbf{i}}\_f - \vec{\mathbf{i}}\_s) \tag{109}$$

where: u*<sup>s</sup>* is the capacitor voltage vector, i*<sup>f</sup>* is the current source inverter output current vector, i*<sup>s</sup>* is the stator current vector.

Using the approximation method, relation (38) may be written as follows:

$$\frac{\vec{\mathbf{u}}\_s(k) - \vec{\mathbf{u}}\_s(k-1)}{T\_{imp}} = \frac{1}{C\_M} \left[ \vec{\mathbf{i}}\_f(k) - \vec{\mathbf{i}}\_s(k) \right]. \tag{110}$$

Deriving **u**s(k) from (38), the motor stator voltage is obtained as a function of the output current, stator current and stator voltage, in the form:

$$\vec{\mathbf{u}}\_s(k) = \frac{T\_{imp}}{C\_M} \left[ \vec{\mathbf{i}}\_f(k) - \vec{\mathbf{i}}\_s(k) \right] + \vec{\mathbf{u}}\_s(k-1) \,\tag{111}$$

where

448 Induction Motors – Modelling and Control

output in the flux control line

dynamics.

where: u*<sup>s</sup>*

vector, i*<sup>s</sup>* 

pulsation, given by the following relations:

where: Ld *–*inductance, Ti– the system time constant.

The output capacitor relations have the form:

is the capacitor voltage vector, i*<sup>f</sup>*

is the stator current vector.

**5.2. The multi-scalar current control of induction machine** 

22

22 2

*x iv*

*d*

*d*

. (104)

*<sup>T</sup>* , (105)

*T L* . (106)

, (107)

. (108)

, (109)

. (110)

is the current source inverter output current

1 *r m*

*dx R L*

*dT L*

2 2

*T*

f

*i*

*i r*

Applying the linearization method, the following relations are obtained, where m1 is the subordinated regulator output in the speed control line and m2 is the subordinated regulator

> 1 1 1 *i v m*

1 *r m*

*R L*

*i r*

The control variables are modulus of the CSI output current and the output current vector

2 1

*v v*

*x* 

1 2

*sx*

The current control analysis presented in the preceding sections does not take the CSI output capacitors into account. Such simplification may be applied because of the small impact of the capacitors upon the control variables (the machine stator current and voltage are measured). The capacitor model will have a positive impact on the control system

<sup>u</sup> <sup>1</sup> (i i ) *<sup>s</sup>*

*d*

Using the approximation method, relation (38) may be written as follows:

*M*

u ( ) u ( 1) <sup>1</sup> i ( ) i( ) *s s*

*k k k k*

*imp M*

*T C*

*dt C* 

*f s*

*f s*

21

*x i*

*v v*

21 i *rx ry*

*ry*

11

*x*

*vm i*

u( ) *<sup>s</sup> <sup>k</sup>* is the stator voltage vector at the k-th moment, Timp - sampling period.

In the equations (5) - (9) representing the cage induction motor mathematical model the stator current vector components are appeared but the direct control variables do not. The motor stator current vector components cannot be the control variables because the multiscalar model relations are derived from them. This is a different situation than with the FOC control. The FOC control is based on the machine stator current components described in a coordinate system associated with the rotor flux and the stator current components are the control variables. Therefore, control variables must be introduced into the mathematical model (5) - (9). The control may be introduced considering the machine currents (5) - (6) and equation (95) written for the αβ components and describing the dc-link circuit dynamics. Adding the respective sides of equations (5) and (95) and equations (6) and (95), where equation (95) must be written with the (αβ) components – the mathematical model of the drive system fed by the CSI is obtained:

$$\frac{d\mathbf{i}\_{s\alpha}}{d\tau} = -\frac{\mathcal{R}\_s \mathcal{L}\_r^2 T + \mathcal{R}\_r \mathcal{L}\_m^2 T + \mathcal{L}\_r w\_\sigma}{2\mathcal{L}\_r w\_\sigma T} \mathbf{i}\_{s\alpha} + \frac{\mathcal{R}\_r \mathcal{L}\_m}{2\mathcal{L}\_r w\_\sigma} \boldsymbol{\nu}\_{ra} + \alpha\_r \frac{\mathcal{L}\_m}{2w\_\sigma} \boldsymbol{\nu}\_{ra} + \frac{\mathcal{L}\_r}{2w\_\sigma} \boldsymbol{u}\_{sa} + \frac{1}{2\mathcal{T}} \mathbf{i}\_{fa},\tag{112}$$

$$\frac{d\dot{\mathbf{u}}\_{s\beta}}{d\tau} = -\frac{\mathcal{R}\_s \mathcal{L}\_r^2 T + \mathcal{R}\_r \mathcal{L}\_m^2 T + \mathcal{L}\_r w\_\sigma}{2\mathcal{L}\_r w\_\sigma T} \dot{\mathbf{i}}\_{s\beta} + \frac{\mathcal{R}\_r \mathcal{L}\_m}{2\mathcal{L}\_r w\_\sigma} \boldsymbol{\nu}\_{r\beta} - \alpha\_r \frac{\mathcal{L}\_m}{2w\_\sigma} \boldsymbol{\nu}\_{r\alpha} + \frac{\mathcal{L}\_r}{2w\_\sigma} \boldsymbol{u}\_{s\beta} + \frac{1}{2\mathcal{T}} \dot{\mathbf{i}}\_{f\beta}, \quad \text{(113)}$$

and equations (7) - (8).

The multi-scalar variables are assumed like in [Krzeminski Z., 1987]:

$$
\propto\_{11} = \alpha\_{r\text{ \AA}} \tag{114}
$$

$$\mathbf{x}\_{12} = \boldsymbol{\psi}\_{r\alpha}\mathbf{i}\_{s\beta} - \boldsymbol{\psi}\_{r\beta}\mathbf{i}\_{s\alpha} \,\,\,\,\,\tag{115}$$

$$\propto\_{21} = \left. \psi \right|\_{r\alpha}^{2} + \left. \psi \right|\_{r\beta}^{2},\tag{116}$$

$$\mathbf{x}\_{22} = \boldsymbol{\psi}\_{r\alpha}\mathbf{i}\_{s\alpha} + \boldsymbol{\psi}\_{r\beta}\mathbf{i}\_{s\beta} \,. \tag{117}$$

Introducing the multi-scalar variables (114) - (117), the multi-scalar model of an IM fed by the CSI is obtained:

$$\frac{d\mathbf{x}\_{12}}{d\tau} = -\frac{1}{2T\_i}\mathbf{x}\_{12} - \mathbf{x}\_{11}\mathbf{x}\_{22} - \mathbf{x}\_{11}\mathbf{x}\_{22}\frac{L\_m}{2w\_\sigma} + \frac{L\_r}{2w\_\sigma}(\mathbf{u}\_{s\beta}\boldsymbol{\nu}\_{ra} - \mathbf{u}\_{sa}\boldsymbol{\nu}\_{r\beta}) + \boldsymbol{\upsilon}\_{1\prime} \tag{118}$$

$$\frac{d\mathbf{x}\_{22}}{d\tau} = -\frac{1}{2T\_i}\mathbf{x}\_{22} + \mathbf{x}\_{11}\mathbf{x}\_{12} + \frac{R\_r L\_m}{2L\_r w\_\sigma}\mathbf{x}\_{21} + \frac{R\_r L\_m}{2L\_r}\dot{i}\_{s\alpha}^2 + \frac{L\_r}{2w\_\sigma}(\boldsymbol{\nu}\_{r\alpha}\boldsymbol{\mu}\_{s\alpha} + \boldsymbol{\nu}\_{r\beta}\boldsymbol{\mu}\_{s\beta}) + \boldsymbol{v}\_{2,r} \tag{119}$$

where

$$
\omega\_1 = \frac{1}{2T} \boldsymbol{\nu}\_{ra} \mathbf{i}\_{f\beta} - \frac{1}{2T} \boldsymbol{\nu}\_{r\beta} \mathbf{i}\_{fa\text{ }\prime} \tag{120}
$$

Sensorless Control of Induction Motor Supplied by Current Source Inverter 451

, (131)

, (132)

, (130)

, (133)

The time constant Ti for simplified for both control method is given

*i*

scalar variables and additional u1 and u2 variables are used

\*

(132) multi-scalar variables yields the following relations:

1

\* 2

*s*

*s*

*du R R L*

*u*

*u*

**VSI** 

[Krzeminski Z., 1987]:

when the IM is supplied by the VSI.

2 2 *r*

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

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) multi-

> <sup>1</sup> *rs rs uu u*

> <sup>2</sup> *rs rs uu u*

The multi-scalar model feedback linearization leads to defining the nonlinear decouplings

1 11 22 21 1 <sup>1</sup> ( ) *<sup>m</sup> r v w L U xx x m L wT*

2 11 12 21 2 <sup>1</sup> [ ] *rm rm s r rr v w RL RL U xx i x m L L Lw T*

> \* 2 1 21 *r r*

> \* 1 2 21 *r r*

The controls (135) - (136) are reference variables treated as input to space vector modulator

On the other side, when the IM is fed by the CSI, calculation of the derivatives of (131) -

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

*u xu q x v dL L C*

*r rM*

*x* 

*x* 

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

\* \*

\* \*

 

1 11 2 12 11

*s*

, (137)

*U U*

 

 

*U U*

  . (134)

, (135)

. (136)

 

 

 

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

 

   

 

*w LT <sup>T</sup>*

*sr rm r*

*RLT RL T Lw* 

$$
\omega\_2 = \frac{1}{2T} \boldsymbol{\nu}\_{r\alpha} \mathbf{i}\_{f\alpha} + \frac{1}{2T} \boldsymbol{\nu}\_{r\beta} \mathbf{i}\_{f\beta} \,. \tag{121}
$$

Applying the linearization method to (118) - (119), the following expressions are obtained:

$$\mathbf{w}\_{1} = \frac{1}{2T\_{i}}\mathbf{m}\_{1} + \mathbf{x}\_{11}\mathbf{x}\_{22} + \frac{L\_{m}}{2\mathbf{w}\_{\sigma}}\mathbf{x}\_{11}\mathbf{x}\_{21} + \frac{L\_{r}}{2\mathbf{w}\_{\sigma}}\left[\mathbf{u}\_{s\beta}(k-1)\boldsymbol{\upnu}\_{r\alpha} - \mathbf{u}\_{s\alpha}(k-1)\boldsymbol{\upnu}\_{r\beta}\right] - a\_{1}\mathbf{x}\_{12} \tag{122}$$

$$\mathbf{w}\_{2} = \frac{1}{2T\_{i}}\mathbf{m}\_{2} - \mathbf{x}\_{11}\mathbf{x}\_{21} - \frac{\mathbf{R}\_{r}L\_{m}}{2w\_{\sigma}L\_{r}}\mathbf{x}\_{21} - \frac{\mathbf{R}\_{r}L\_{m}}{2L\_{r}}\mathbf{i}\_{s\alpha}^{2} - \frac{L\_{r}}{2w\_{\sigma}}\left[\boldsymbol{\mu}\_{s\alpha}(k-1)\boldsymbol{\nu}\_{r\alpha} + \boldsymbol{\mu}\_{s\beta}(k-1)\boldsymbol{\nu}\_{r\beta}\right] + \boldsymbol{a}\_{1}\mathbf{x}\_{22},\tag{123}$$

where

$$\mathbf{v}\_1 = a\_1 \Big[ \mathbf{i}\_{f\beta}\boldsymbol{\upmu}\_{r\alpha} - \mathbf{i}\_{f\alpha}\boldsymbol{\upmu}\_{r\beta} \Big],\tag{124}$$

$$\mathbf{v}\_2 = -a\_1 \left[ \mathbf{i}\_{f\alpha} \boldsymbol{\psi}\_{r\alpha} + \mathbf{i}\_{f\beta} \boldsymbol{\psi}\_{r\beta} \right]. \tag{125}$$

The control variables take the form

$$\dot{a}\_{f\alpha} = a\_2 \frac{\wp\_{r\alpha} v\_2 - \wp\_{r\beta} v\_1}{\wp\_{21}} \, , \tag{126}$$

$$\mathbf{i}\_{f\beta} = a\_2 \frac{\boldsymbol{\Psi}\_{r\alpha} \boldsymbol{v}\_1 + \boldsymbol{\Psi}\_{r\beta} \boldsymbol{v}\_2}{\boldsymbol{\chi}\_{21}} \,, \tag{127}$$

where

$$a\_1 = \frac{L\_r T\_{imp}}{2\pi v\_\sigma \mathcal{C}\_M} \, \tag{128}$$

$$a\_2 = a\_1 + \frac{1}{2T}.\tag{129}$$

The time constant Ti for simplified for both control method is given

450 Induction Motors – Modelling and Control

where

where

where

12

*i*

The control variables take the form

*i*

*dx L L*

22 2

*d T w w*

*i rr dx RL RL L*

1 1 2 2 *rf rf vi i T T* 

1 1 2 2 *rf rf vi i T T* 

Applying the linearization method to (118) - (119), the following expressions are obtained:

*L L v m x x x x u k u k ax*

2 2 2 11 21 21 1 22 <sup>1</sup> - ( 1) ( 1) 2 2 22 *r r r*

*m m <sup>L</sup> v m x x x i u k u k ax*

1 1 *fr fr v ai i* 

2 1 *fr fr v ai i* 

2

2

<sup>1</sup> 2

2 1

*a a*

*a*

*f*

*f*

*i a*

*i a*

 

 

1 1 11 22 11 21 1 12 <sup>1</sup> ( 1) ( 1) 2 22 *m r*

> 

 

 

 

*d T Lw L w*

1

2

*T ww*

*RL RL*

*T wL L w*

*i r r*

12 11 22 11 22 1 <sup>1</sup> ( ) <sup>2</sup> 2 2 *m r*

22 11 12 21 2 <sup>1</sup> +( ) 2 2 22 *rm rm r*

> 

 

 

 

> 

2 1

1 2

 

*v v*

   

*v v*

 

21 *r r*

21 *r r*

*x* 

> *r imp M*

> > 1 2

*T*

*w C*

*L T*

*x* 

 

 

, (119)

*x xx x i u u v*

 

, (118)

*x xx xx uu v*

*sr sr*

*s rs rs*

, (120)

. (121)

*s rs r*

 

, (124)

. (125)

, (126)

, (127)

, (128)

. (129)

 

*s s rs r*

 

 

 

> 

> >

 , (123)

 

 , (122)

 

 

$$T\_i = \frac{w\_\sigma L\_r T}{R\_s L\_r^2 T + R\_r L\_m^2 T + L\_r w\_\sigma} \, \tag{130}$$
