**1. Introduction**

For synchronous machine (SM) with damper winding and separate excitation winding, it is not unusual to operate as an AC drive system.

In hydropower generation, sometimes, there is demand for SM to work in compensating or pumping operation mode. Then, at least motor starting of the SM has to be assured. The most sophisticated starting process is synchronous starting also called variable speed operation. It is obtained by frequency converter, whether by current source inverter (CSI) or voltage source inverter (VSI). In wind power generation, SM could be also used. Then, it is also used for variable speed operation.

Except from SM used in power generation, SM could be also used as AC drive systems in industrial applications with high power demand such as coal mines, metal and cement industries. It is also used for ship propulsion.

AC drive system for SM is traditionally done by CSI topology with thyristors. Although CSI has some advantages, VSI topology has been also used lately. It is mainly due to development of fully controllable switches (IGBT, GTO, etc.) that are nowadays also used for high power demands. Due to its controllability, PWM could be easily applied on VSI topology.

Because of the salient poles, a large number of coupled variables and high nonlinearity, the SM is a complex dynamic system with a number of unknown state variables. To obtain its control, classical system uses PI controllers for stator *dq* current components control. But due to SM's complexity, it is not possible to obtain fully decoupled torque and flux control. Namely, change of any current component necessary changes both; torque and flux. Another difficulty is unknown damper winding current.

\_*id* \_*if* \_*iD* \_*iq* \_*iQ ω*\_

\_ *i* b*d* \_ *i* b *f* c\_ *iD* \_ *i* b*q* c\_ *iQ* \_ *ω*b

¼

þ

\_ *i* b*d* \_ *i* b *f* c\_ *iD* \_ *i* b*q* c\_ *iQ* \_ *ω*b

¼

**147**

¼

*a*1 0 *id* þ *a*<sup>2</sup> 0 *iqω* þ *a*<sup>3</sup> 0

*DOI: http://dx.doi.org/10.5772/intechopen.89420*

*b*1 0 *id* þ *b*<sup>2</sup> 0 *iqω* þ *b*<sup>3</sup> 0

*c*1 0 *id* þ *c*<sup>2</sup> 0 *iqω* þ *c*<sup>3</sup> 0 *iQ ω* þ *c*<sup>4</sup> 0 *if* þ *c*<sup>5</sup> 0 *iD* þ *c*<sup>6</sup> 0 *ud* þ *c*<sup>7</sup> 0 *uf*

*d*1 0 *iq* þ *d*<sup>2</sup> 0 *idω* þ *d*<sup>3</sup> 0 *ifω* þ *d*<sup>4</sup> 0 *iDω* þ *d*<sup>5</sup> 0 *iQ* þ *d*<sup>6</sup> 0 *uq*

*f* 1 0 *iq* þ *f* <sup>2</sup> 0 *idω* þ *f* <sup>3</sup> 0 *ifω* þ *f* <sup>4</sup> 0 *iDω* þ *f* <sup>5</sup> 0 *iQ* þ *f* <sup>6</sup> 0 *uq*

observer with damper winding currents.

*a*1 0 *id* þ *a*<sup>2</sup> 0 *iqω* þ *a*<sup>3</sup> 0 *i* c*<sup>Q</sup> ω* þ *a*<sup>4</sup> 0 *if* þ *a*<sup>5</sup>

*b*1 0 *id* þ *b*<sup>2</sup> 0 *iqω* þ *b*<sup>3</sup> 0 *i* c*<sup>Q</sup> ω* þ *b*<sup>4</sup> 0 *if* þ *b*<sup>5</sup>

*c*1 0 *id* þ *c*<sup>2</sup> 0 *iqω* þ *c*<sup>3</sup> 0 *i* c*<sup>Q</sup> ω* þ *c*<sup>4</sup> 0 *if* þ *c*<sup>5</sup>

*d*1 0 *iq* þ *d*<sup>2</sup> 0 *idω* þ *d*<sup>3</sup> 0 *ifω* þ *d*<sup>4</sup>

*f* 1 0 *iq* þ *f* <sup>2</sup> 0 *idω* þ *f* <sup>3</sup> 0 *ifω* þ *f* <sup>4</sup>

*k*11<sup>0</sup>

*k*21<sup>0</sup>

*k*31<sup>0</sup>

*k*41<sup>0</sup>

*k*51<sup>0</sup>

*k*61<sup>0</sup>

damper current observer Eq. (3) is obtained.

*a*1 0 *i* b *<sup>d</sup>* þ *a*<sup>2</sup> 0 *i* <sup>b</sup>*qω*<sup>b</sup> <sup>þ</sup> *<sup>a</sup>*<sup>3</sup> 0 *i* <sup>c</sup>*<sup>Q</sup> <sup>ω</sup>*<sup>b</sup> <sup>þ</sup> *<sup>a</sup>*<sup>4</sup>

*b*1 0 *i* b*<sup>d</sup>* þ *b*<sup>2</sup> 0 *i* <sup>b</sup>*qω*<sup>b</sup> <sup>þ</sup> *<sup>b</sup>*<sup>3</sup> 0 *i* <sup>c</sup>*<sup>Q</sup> <sup>ω</sup>*<sup>b</sup> <sup>þ</sup> *<sup>b</sup>*<sup>4</sup>

*c*1 0 *i* b*<sup>d</sup>* þ *c*<sup>2</sup> 0 *i* <sup>b</sup>*qω*<sup>b</sup> <sup>þ</sup> *<sup>c</sup>*<sup>3</sup> 0 *i* <sup>c</sup>*<sup>Q</sup> <sup>ω</sup>*<sup>b</sup> <sup>þ</sup> *<sup>c</sup>*<sup>4</sup> 0 *i* b *<sup>f</sup>* þ *c*<sup>5</sup>

*d*1 0 *i* b*<sup>q</sup>* þ *d*<sup>2</sup> 0 *i* <sup>b</sup>*dω*<sup>b</sup> <sup>þ</sup> *<sup>d</sup>*<sup>3</sup> 0 *i* b *<sup>f</sup>ω*<sup>b</sup> <sup>þ</sup> *<sup>d</sup>*<sup>4</sup>

*f* 1 0 *i* b*<sup>q</sup>* þ *f* <sup>2</sup> 0 *i* b *<sup>d</sup>ω*<sup>b</sup> <sup>þ</sup> *<sup>f</sup>* <sup>3</sup> 0 *i* b *<sup>f</sup>ω*<sup>b</sup> <sup>þ</sup> *<sup>f</sup>* <sup>4</sup>

> *j* 1 0 *i* b*di* b*<sup>q</sup>* þ *j* 2 0 *i* b *f i* b*<sup>q</sup>* þ *j* 3 0 *i* b*q*c*iD* þ *j* 4 0 *i* b*di* c*<sup>Q</sup>* þ *j* 5 0 *TL*

*k*11<sup>0</sup>

*k*21<sup>0</sup>

*k*31<sup>0</sup>

*k*41<sup>0</sup>

*k*51<sup>0</sup>

*k*61<sup>0</sup>

*e*<sup>1</sup> þ *k*12<sup>0</sup>

*e*<sup>1</sup> þ *k*22<sup>0</sup>

*e*<sup>1</sup> þ *k*32<sup>0</sup>

*e*<sup>1</sup> þ *k*42<sup>0</sup>

*e*<sup>1</sup> þ *k*52<sup>0</sup>

*e*<sup>1</sup> þ *k*62<sup>0</sup>

þ

*j* 1 0 *idiq* þ *j* 2 0 *if iq* þ *j* 3 0 *iq*c*iD* þ *j* 4 0 *idi* c*<sup>Q</sup>* þ *j* 5 0 *TL*

*e*<sup>1</sup> þ *k*12<sup>0</sup>

*e*<sup>1</sup> þ *k*22<sup>0</sup>

*e*<sup>1</sup> þ *k*32<sup>0</sup>

*e*<sup>1</sup> þ *k*42<sup>0</sup>

*e*<sup>1</sup> þ *k*52<sup>0</sup>

*e*<sup>1</sup> þ *k*62<sup>0</sup>

*e*<sup>2</sup> þ *k*14<sup>0</sup>

*e*<sup>2</sup> þ *k*24<sup>0</sup>

*e*<sup>2</sup> þ *k*34<sup>0</sup>

*e*<sup>2</sup> þ *k*44<sup>0</sup>

*e*<sup>2</sup> þ *k*54<sup>0</sup>

*e*<sup>2</sup> þ *k*64<sup>0</sup>

*j* 1 0 *idiq* þ *j* 2 0 *if iq* þ *j* 3 0 *iqiD* þ *j* 4 0 *idiQ* þ *j* 5 0 *TL*

*iQ ω* þ *a*<sup>4</sup>

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization…*

*iQ ω* þ *b*<sup>4</sup>

0 *if* þ *a*<sup>5</sup> 0 *iD* þ *a*<sup>6</sup> 0 *ud* þ *a*<sup>7</sup> 0 *uf*

(1)

(2)

þ

(3)

0 *if* þ *b*<sup>5</sup> 0 *iD* þ *b*<sup>6</sup> 0 *ud* þ *b*<sup>7</sup> 0 *uf*

To obtain high performance drive all SM states should be known. Since damper

*e*<sup>4</sup> þ *k*16<sup>0</sup>

*e*<sup>4</sup> þ *k*26<sup>0</sup>

*e*<sup>4</sup> þ *k*36<sup>0</sup>

*e*<sup>4</sup> þ *k*46<sup>0</sup>

*e*<sup>4</sup> þ *k*56<sup>0</sup>

*e*<sup>4</sup> þ *k*66<sup>0</sup>

Observed values are noted with "b"; *ex* are errors, differences between measured and observed value; while *kxy* are adaptive coefficients used to obtain the convergence. If the observer in Eq. (2) is made only with observed values and errors [7],

<sup>0</sup>c*iD* þ *a*<sup>6</sup> 0 *ud* þ *a*<sup>7</sup> 0 *uf*

<sup>0</sup>c*iD* þ *b*<sup>6</sup> 0 *ud* þ *b*<sup>7</sup> 0 *uf*

<sup>0</sup>c*iD* þ *c*<sup>6</sup> 0 *ud* þ *c*<sup>7</sup> 0 *uf*

> 0 *i* c*<sup>Q</sup>* þ *d*<sup>6</sup> 0 *uq*

0 *i* c*<sup>Q</sup>* þ *f* <sup>6</sup> 0 *uq*

<sup>0</sup>c*iD* þ *a*<sup>6</sup> 0 *ud* þ *a*<sup>7</sup> 0 *uf*

<sup>0</sup>c*iD* þ *b*<sup>6</sup> 0 *ud* þ *b*<sup>7</sup> 0 *uf*

<sup>0</sup>c*iD* þ *c*<sup>6</sup> 0 *ud* þ *c*<sup>7</sup> 0 *uf*

*e*6

*e*6

*e*6

*e*6

*e*6

*e*6

0 *i* c*<sup>Q</sup>* þ *d*<sup>6</sup> 0 *uq*

0 *i* c*<sup>Q</sup>* þ *f* <sup>6</sup> 0 *uq*

<sup>0</sup>c*iDω*<sup>b</sup> <sup>þ</sup> *<sup>d</sup>*<sup>5</sup>

<sup>0</sup>c*iDω*<sup>b</sup> <sup>þ</sup> *<sup>f</sup>* <sup>5</sup>

*e*<sup>4</sup> þ *k*16<sup>0</sup>

*e*<sup>4</sup> þ *k*26<sup>0</sup>

*e*<sup>4</sup> þ *k*36<sup>0</sup>

*e*<sup>4</sup> þ *k*46<sup>0</sup>

*e*<sup>4</sup> þ *k*56<sup>0</sup>

*e*<sup>4</sup> þ *k*66<sup>0</sup>

<sup>0</sup>c*iDω* þ *d*<sup>5</sup>

<sup>0</sup>c*iDω* þ *f* <sup>5</sup>

*e*6

*e*6

*e*6

*e*6

*e*6

*e*6

0 *i* b *<sup>f</sup>* þ *a*<sup>5</sup>

0 *i* b *<sup>f</sup>* þ *b*<sup>5</sup>

*e*<sup>2</sup> þ *k*14<sup>0</sup>

*e*<sup>2</sup> þ *k*24<sup>0</sup>

*e*<sup>2</sup> þ *k*34<sup>0</sup>

*e*<sup>2</sup> þ *k*44<sup>0</sup>

*e*<sup>2</sup> þ *k*54<sup>0</sup>

*e*<sup>2</sup> þ *k*64<sup>0</sup>

winding currents are normally not measured, to make all states available, an observer has to be made. In Eq. (2) is an expression of the SM deterministic

This work examines a novel control method for variable speed operation of a SM. To overcome mentioned obstacles arisen from SM complexity, novel control will be nonlinear. VSI topology is suitable to be used with this novel control. The goal of the control system is to obtain high performance speed tracking system. To achieve this, it is necessary to have an adequate observer for damper winding states, as is similarly done in induction motor drive system [1].

There are not many studies regarding SMs AC drive system; whether with linear or nonlinear control. Classical vector control is rotor field oriented control used with the following assumption: if the flux is constant, the q-current component can control electromagnetic torque. For induction motor drives this assumption holds, but if this method is used for SM control, the q-current component will essentially change the flux [2]. It is said that control is coupled and this is why SM vector control is not efficient enough. There are few ideas on how to solve this problem. In [3] stator flux orientation control is used. With this orientation, through excitation current compensation, better flux control is obtained. Unfortunately, a control system with many calculations (coordinate transformations, PI controllers, and other) has to be used. Also, damper winding current affect has not been taken into account.

Regarding nonlinear control SM applications, a few methods are used: backstepping [4], passivity [5] and adaptive Lyapunov based [6]. The passive method [5] fails to give better results and the backstepping [4] method fails to take damper windings into consideration. In [6] new algorithms are proposed, but besides of their complexity, a control in excitation system also has to be used.

The aim of this work is to find deterministic observer for a SM and to use it by nonlinear control law. Parameter adaptivity and load torque estimation is also considered. Finally, high performance VSI drive system without excitation system control is thus obtained.

### **2. Observers**

In this section observers for SM are presented. Starting from the SM dynamic system, damper winding deterministic observers are made. At first, an observer with damper winding currents is given. Then, full order and reduced order observers for damper winding fluxes are presented. Observability analysis for the full order observer is given. Stability is approved with Lyapunov stability theory.

Finally, load torque estimation system is presented. Observability of the expanded system is analyzed and the model reference adaptive system is given.

#### **2.1 Damper winding current observers**

Synchronous machine can be described as a dynamic system of six state variables. If five of them are set to be SM currents and the sixth is rotor speed, SM dynamic system is:

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization… DOI: http://dx.doi.org/10.5772/intechopen.89420*

$$\begin{bmatrix} \dot{i}\_d\\ \dot{i}\_f\\ \dot{i}\_D\\ \dot{i}\_Q\\ \dot{i}\_q\\ \dot{i}\_Q\\ \dot{i}\_Q \end{bmatrix} = \begin{bmatrix} a\_1'i\_d + a\_2'i\_q a + a\_3'i\_Q a + a\_4'i\_f + a\_5'i\_D + a\_6'u\_d + a\_7'u\_f\\ b\_1'i\_d + b\_2'i\_q a + b\_3'i\_Q a + b\_4'i\_f + b\_5'i\_D + b\_6'u\_d + b\_7'u\_f\\ c\_1'i\_d + c\_2'i\_q a + c\_3'i\_Q a + c\_4'i\_f + c\_5'i\_D + c\_6'u\_d + c\_7'u\_f\\ d\_1'i\_q + d\_2'i\_d a + d\_3'i\_f a + d\_4'i\_D a + d\_5'i\_Q + d\_6'u\_q\\ f\_1'i\_q + f\_2'i\_d a + f\_3'i\_f a + f\_4'i\_D a + f\_5'i\_Q + f\_6'u\_q\\ j\_1'i\_dq + j\_2'i\_f i\_q + j\_3'i\_qi\_D + j\_4'i\_dq + j\_5'T\_L \end{bmatrix} \tag{1}$$

To obtain high performance drive all SM states should be known. Since damper winding currents are normally not measured, to make all states available, an observer has to be made. In Eq. (2) is an expression of the SM deterministic observer with damper winding currents.

\_ *i* b*d* \_ *i* b *f* c\_ *iD* \_ *i* b*q* c\_ *iQ* \_ *ω*b 2 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 5 ¼ *a*1 0 *id* þ *a*<sup>2</sup> 0 *iqω* þ *a*<sup>3</sup> 0 *i* c*<sup>Q</sup> ω* þ *a*<sup>4</sup> 0 *if* þ *a*<sup>5</sup> <sup>0</sup>c*iD* þ *a*<sup>6</sup> 0 *ud* þ *a*<sup>7</sup> 0 *uf b*1 0 *id* þ *b*<sup>2</sup> 0 *iqω* þ *b*<sup>3</sup> 0 *i* c*<sup>Q</sup> ω* þ *b*<sup>4</sup> 0 *if* þ *b*<sup>5</sup> <sup>0</sup>c*iD* þ *b*<sup>6</sup> 0 *ud* þ *b*<sup>7</sup> 0 *uf c*1 0 *id* þ *c*<sup>2</sup> 0 *iqω* þ *c*<sup>3</sup> 0 *i* c*<sup>Q</sup> ω* þ *c*<sup>4</sup> 0 *if* þ *c*<sup>5</sup> <sup>0</sup>c*iD* þ *c*<sup>6</sup> 0 *ud* þ *c*<sup>7</sup> 0 *uf d*1 0 *iq* þ *d*<sup>2</sup> 0 *idω* þ *d*<sup>3</sup> 0 *ifω* þ *d*<sup>4</sup> <sup>0</sup>c*iDω* þ *d*<sup>5</sup> 0 *i* c*<sup>Q</sup>* þ *d*<sup>6</sup> 0 *uq f* 1 0 *iq* þ *f* <sup>2</sup> 0 *idω* þ *f* <sup>3</sup> 0 *ifω* þ *f* <sup>4</sup> <sup>0</sup>c*iDω* þ *f* <sup>5</sup> 0 *i* c*<sup>Q</sup>* þ *f* <sup>6</sup> 0 *uq j* 1 0 *idiq* þ *j* 2 0 *if iq* þ *j* 3 0 *iq*c*iD* þ *j* 4 0 *idi* c*<sup>Q</sup>* þ *j* 5 0 *TL* 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 þ *k*11<sup>0</sup> *e*<sup>1</sup> þ *k*12<sup>0</sup> *e*<sup>2</sup> þ *k*14<sup>0</sup> *e*<sup>4</sup> þ *k*16<sup>0</sup> *e*6 *k*21<sup>0</sup> *e*<sup>1</sup> þ *k*22<sup>0</sup> *e*<sup>2</sup> þ *k*24<sup>0</sup> *e*<sup>4</sup> þ *k*26<sup>0</sup> *e*6 *k*31<sup>0</sup> *e*<sup>1</sup> þ *k*32<sup>0</sup> *e*<sup>2</sup> þ *k*34<sup>0</sup> *e*<sup>4</sup> þ *k*36<sup>0</sup> *e*6 *k*41<sup>0</sup> *e*<sup>1</sup> þ *k*42<sup>0</sup> *e*<sup>2</sup> þ *k*44<sup>0</sup> *e*<sup>4</sup> þ *k*46<sup>0</sup> *e*6 *k*51<sup>0</sup> *e*<sup>1</sup> þ *k*52<sup>0</sup> *e*<sup>2</sup> þ *k*54<sup>0</sup> *e*<sup>4</sup> þ *k*56<sup>0</sup> *e*6 *k*61<sup>0</sup> *e*<sup>1</sup> þ *k*62<sup>0</sup> *e*<sup>2</sup> þ *k*64<sup>0</sup> *e*<sup>4</sup> þ *k*66<sup>0</sup> *e*6 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 (2)

Observed values are noted with "b"; *ex* are errors, differences between measured and observed value; while *kxy* are adaptive coefficients used to obtain the convergence.

If the observer in Eq. (2) is made only with observed values and errors [7], damper current observer Eq. (3) is obtained.

\_ *i* b*d* \_ *i* b *f* c\_ *iD* \_ *i* b*q* c\_ *iQ* \_ *ω*b 2 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 5 ¼ *a*1 0 *i* b *<sup>d</sup>* þ *a*<sup>2</sup> 0 *i* <sup>b</sup>*qω*<sup>b</sup> <sup>þ</sup> *<sup>a</sup>*<sup>3</sup> 0 *i* <sup>c</sup>*<sup>Q</sup> <sup>ω</sup>*<sup>b</sup> <sup>þ</sup> *<sup>a</sup>*<sup>4</sup> 0 *i* b *<sup>f</sup>* þ *a*<sup>5</sup> <sup>0</sup>c*iD* þ *a*<sup>6</sup> 0 *ud* þ *a*<sup>7</sup> 0 *uf b*1 0 *i* b*<sup>d</sup>* þ *b*<sup>2</sup> 0 *i* <sup>b</sup>*qω*<sup>b</sup> <sup>þ</sup> *<sup>b</sup>*<sup>3</sup> 0 *i* <sup>c</sup>*<sup>Q</sup> <sup>ω</sup>*<sup>b</sup> <sup>þ</sup> *<sup>b</sup>*<sup>4</sup> 0 *i* b *<sup>f</sup>* þ *b*<sup>5</sup> <sup>0</sup>c*iD* þ *b*<sup>6</sup> 0 *ud* þ *b*<sup>7</sup> 0 *uf c*1 0 *i* b*<sup>d</sup>* þ *c*<sup>2</sup> 0 *i* <sup>b</sup>*qω*<sup>b</sup> <sup>þ</sup> *<sup>c</sup>*<sup>3</sup> 0 *i* <sup>c</sup>*<sup>Q</sup> <sup>ω</sup>*<sup>b</sup> <sup>þ</sup> *<sup>c</sup>*<sup>4</sup> 0 *i* b *<sup>f</sup>* þ *c*<sup>5</sup> <sup>0</sup>c*iD* þ *c*<sup>6</sup> 0 *ud* þ *c*<sup>7</sup> 0 *uf d*1 0 *i* b*<sup>q</sup>* þ *d*<sup>2</sup> 0 *i* <sup>b</sup>*dω*<sup>b</sup> <sup>þ</sup> *<sup>d</sup>*<sup>3</sup> 0 *i* b *<sup>f</sup>ω*<sup>b</sup> <sup>þ</sup> *<sup>d</sup>*<sup>4</sup> <sup>0</sup>c*iDω*<sup>b</sup> <sup>þ</sup> *<sup>d</sup>*<sup>5</sup> 0 *i* c*<sup>Q</sup>* þ *d*<sup>6</sup> 0 *uq f* 1 0 *i* b*<sup>q</sup>* þ *f* <sup>2</sup> 0 *i* b *<sup>d</sup>ω*<sup>b</sup> <sup>þ</sup> *<sup>f</sup>* <sup>3</sup> 0 *i* b *<sup>f</sup>ω*<sup>b</sup> <sup>þ</sup> *<sup>f</sup>* <sup>4</sup> <sup>0</sup>c*iDω*<sup>b</sup> <sup>þ</sup> *<sup>f</sup>* <sup>5</sup> 0 *i* c*<sup>Q</sup>* þ *f* <sup>6</sup> 0 *uq j* 1 0 *i* b*di* b*<sup>q</sup>* þ *j* 2 0 *i* b *f i* b*<sup>q</sup>* þ *j* 3 0 *i* b*q*c*iD* þ *j* 4 0 *i* b*di* c*<sup>Q</sup>* þ *j* 5 0 *TL* 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 þ þ *k*11<sup>0</sup> *e*<sup>1</sup> þ *k*12<sup>0</sup> *e*<sup>2</sup> þ *k*14<sup>0</sup> *e*<sup>4</sup> þ *k*16<sup>0</sup> *e*6 *k*21<sup>0</sup> *e*<sup>1</sup> þ *k*22<sup>0</sup> *e*<sup>2</sup> þ *k*24<sup>0</sup> *e*<sup>4</sup> þ *k*26<sup>0</sup> *e*6 *k*31<sup>0</sup> *e*<sup>1</sup> þ *k*32<sup>0</sup> *e*<sup>2</sup> þ *k*34<sup>0</sup> *e*<sup>4</sup> þ *k*36<sup>0</sup> *e*6 *k*41<sup>0</sup> *e*<sup>1</sup> þ *k*42<sup>0</sup> *e*<sup>2</sup> þ *k*44<sup>0</sup> *e*<sup>4</sup> þ *k*46<sup>0</sup> *e*6 *k*51<sup>0</sup> *e*<sup>1</sup> þ *k*52<sup>0</sup> *e*<sup>2</sup> þ *k*54<sup>0</sup> *e*<sup>4</sup> þ *k*56<sup>0</sup> *e*6 *k*61<sup>0</sup> *e*<sup>1</sup> þ *k*62<sup>0</sup> *e*<sup>2</sup> þ *k*64<sup>0</sup> *e*<sup>4</sup> þ *k*66<sup>0</sup> *e*6 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 (3)

**147**

Because of the salient poles, a large number of coupled variables and high nonlinearity, the SM is a complex dynamic system with a number of unknown state variables. To obtain its control, classical system uses PI controllers for stator *dq* current components control. But due to SM's complexity, it is not possible to obtain fully decoupled torque and flux control. Namely, change of any current component necessary changes both; torque and flux. Another difficulty is unknown damper

This work examines a novel control method for variable speed operation of a SM. To overcome mentioned obstacles arisen from SM complexity, novel control will be nonlinear. VSI topology is suitable to be used with this novel control. The goal of the control system is to obtain high performance speed tracking system. To achieve this, it is necessary to have an adequate observer for damper winding states,

There are not many studies regarding SMs AC drive system; whether with linear or nonlinear control. Classical vector control is rotor field oriented control used with the following assumption: if the flux is constant, the q-current component can control electromagnetic torque. For induction motor drives this assumption holds, but if this method is used for SM control, the q-current component will essentially change the flux [2]. It is said that control is coupled and this is why SM vector control is not efficient enough. There are few ideas on how to solve this problem. In [3] stator flux orientation control is used. With this orientation, through excitation current compensation, better flux control is obtained. Unfortunately, a control system with many calculations (coordinate transformations, PI controllers, and other) has to be used. Also, damper winding current affect has not been taken into

Regarding nonlinear control SM applications, a few methods are used: backstepping [4], passivity [5] and adaptive Lyapunov based [6]. The passive method [5] fails to give better results and the backstepping [4] method fails to take damper windings into consideration. In [6] new algorithms are proposed, but besides of their complexity, a control in excitation system also has to be used.

nonlinear control law. Parameter adaptivity and load torque estimation is also considered. Finally, high performance VSI drive system without excitation system

The aim of this work is to find deterministic observer for a SM and to use it by

In this section observers for SM are presented. Starting from the SM dynamic system, damper winding deterministic observers are made. At first, an observer with damper winding currents is given. Then, full order and reduced order observers for damper winding fluxes are presented. Observability analysis for the full order observer is given. Stability is approved with Lyapunov stability

Finally, load torque estimation system is presented. Observability of the expanded system is analyzed and the model reference adaptive system is given.

Synchronous machine can be described as a dynamic system of six state variables. If five of them are set to be SM currents and the sixth is rotor speed, SM

as is similarly done in induction motor drive system [1].

winding current.

*Control Theory in Engineering*

account.

control is thus obtained.

**2.1 Damper winding current observers**

**2. Observers**

theory.

**146**

dynamic system is:

There is a way to define adaptive coefficients in each one of the observers given in Eqs. (2) and (3) to prove their stability according to Lyapunov theory. The proof is extensive and is given in [8].

After each matrix member of Eq. (6) is calculated [8], its determinant

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization…*

*∂L*<sup>0</sup> *f id ∂if*

*∂L*<sup>0</sup> *f if ∂if*

*∂L*<sup>0</sup> *f iq ∂if*

*∂L*<sup>0</sup> *f ω ∂if*

*∂L*<sup>1</sup> *f id ∂if*

*∂L*<sup>1</sup> *f iq ∂if*

After each matrix member of Eq. (8) is calculated [8], its determinant

*<sup>ω</sup>*<sup>2</sup>*Lmd* �*LD*

� *LmqRQ* �*LfLmdLQ RD* <sup>þ</sup> *Lmd*

*LDLQ LdLDLf* � *LdLmd*

*∂L*<sup>0</sup> *f id ∂id*

*∂L*<sup>0</sup> *f if ∂id*

*∂L*<sup>0</sup> *f iq ∂id*

*∂L*<sup>0</sup> *f ω ∂id*

*∂L*<sup>1</sup> *f id ∂id*

*∂L*<sup>1</sup> *f iq ∂id*

<sup>2</sup> �*LdLDLf* <sup>þ</sup> *LdLmd*

<sup>2</sup> �*LdLDLf* <sup>þ</sup> *LdLmd*

*Det* (*O*1) 6¼ 0 *U Det* (*O*2) 6¼ 0 = > *rank* {*O*} = 6.

While observing both determinants (Eqs. (7) and (9)):

*<sup>V</sup>*<sup>1</sup> <sup>¼</sup> *<sup>e</sup>*<sup>2</sup> 1 2 þ *e*2 2 2 þ *e*2 3 2 þ *e*2 4 2 þ *e*2 5 2 þ *e*2 6

The second criterion matrix *O*<sup>2</sup> is chosen:

*DOI: http://dx.doi.org/10.5772/intechopen.89420*

ω*LmdLmqRD*

*∂L*<sup>0</sup> *f id ∂φ<sup>D</sup>*

*∂L*<sup>0</sup> *f if ∂φ<sup>D</sup>*

*∂L*<sup>0</sup> *f iq ∂φ<sup>D</sup>*

*∂L*<sup>0</sup> *f ω ∂φ<sup>D</sup>*

*∂L*<sup>1</sup> *f id ∂φ<sup>D</sup>*

*∂L*<sup>1</sup> *f iq ∂φ<sup>D</sup>*

2

<sup>3</sup> � � �*Lmq*

<sup>2</sup> <sup>þ</sup> *LDLmd*

Matrix *O* is full rank matrix and it could be concluded that the system is weakly

Equation (10) is positive definite function of the error variables: *e*1, *e*2, *e*3, *e*4, *e*5,

*e*6. Error dynamic system is obtained by Eqs. (4) and (5), and the result is:

To make a proof of observer Eq. (5) stability, Lyapunov function Eq. (10) is

<sup>3</sup> � � �*Lmq*

<sup>2</sup> <sup>þ</sup> *LDLmd*

*Det O*<sup>1</sup> 6¼ 0, for *ω* 6¼ 0, while *Det O*<sup>2</sup> 6¼ 0, for *ω* = 0 it is easy to see that:

<sup>2</sup> � *LDLmd*

<sup>2</sup> � *LfLmd*

*∂L*<sup>0</sup> *f id ∂φ<sup>Q</sup>*

*∂L*<sup>0</sup> *f if ∂φ<sup>Q</sup>*

*∂L*<sup>0</sup> *f iq ∂φ<sup>Q</sup>*

*∂L*<sup>0</sup> *f ω ∂φ<sup>Q</sup>*

*∂L*<sup>1</sup> *f id ∂φ<sup>Q</sup>*

*∂L*<sup>1</sup> *f iq ∂φ<sup>Q</sup>*

2

<sup>2</sup> � <sup>2</sup>*Lmd*

2 *LQ RD*

<sup>2</sup> � <sup>2</sup>*Lmd*

<sup>2</sup> (10)

<sup>3</sup> � � (7)

*∂L*<sup>0</sup> *f id ∂iq*

*∂L*<sup>0</sup> *f if ∂iq*

*∂L*<sup>0</sup> *f iq ∂iq*

*∂L*<sup>0</sup> *f ω ∂iq*

*∂L*<sup>1</sup> *f id ∂iq*

*∂L*<sup>1</sup> *f iq ∂iq*

*LfLmq* þ *LDLmd*

*Lmq* � �

<sup>2</sup> <sup>þ</sup> *LfLmd*

� �

<sup>2</sup> <sup>þ</sup> *LfLmd*

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*Lmd*

*∂L*<sup>0</sup> *f id ∂ω*

*∂L*<sup>0</sup> *f if ∂ω*

*∂L*<sup>0</sup> *f iq ∂ω*

*∂L*<sup>0</sup> *f ω ∂ω*

(8)

*∂L*<sup>1</sup> *f id ∂ω*

*∂L*<sup>1</sup> *f iq ∂ω*

<sup>2</sup> <sup>þ</sup> *LqLQ* � � �

> <sup>2</sup> <sup>þ</sup> *LqLQ* � �

> > (9)

calculation gives:

*O*<sup>2</sup> ¼

calculation gives:

*Det O*ð Þ¼� <sup>2</sup>

locally observable.

proposed:

**149**

*Det O*ð Þ¼� <sup>1</sup>

*did*

¼

*dif*

*diq*

*dω*

*d Lf id* � �

*d Lf iq* � �

*LD*

*LDLQ*

#### **2.2 Damper winding full order flux observer**

If the SM dynamics given in Eq. (1) is changed in a way that damper currents states are replaced with damper fluxes states, its dynamic system will become:

$$\begin{bmatrix} \dot{i}\_d \\ \dot{i}\_f \\ \dot{\nu\_D} \\ \dot{i}\_q \\ \dot{\nu\_Q} \\ \dot{\nu\_D} \end{bmatrix} = \begin{bmatrix} a\_1 i\_d + a\_2 i\_f + a\_3 i\_q a + a\_4 \wp\_D + a\_5 \wp\_Q \, a + a\_6 u\_d + a\_7 u\_f \\ b\_1 i\_d + b\_2 i\_f + b\_3 i\_q a + b\_4 \wp\_D + b\_5 \wp\_Q \, a + b\_6 u\_d + b\_7 u\_f \\ c\_1 i\_d + c\_2 i\_f + c\_3 \wp\_D \\ d\_1 i\_q + d\_2 i\_d \, a + d\_4 a \wp\_D + d\_5 \wp\_Q + d\_6 u\_q \\ f\_1 i\_q + f\_2 \wp\_Q \\ g\_1 i\_d i\_q + g\_2 i\_f i\_q + g\_3 i\_q \wp\_D + g\_4 i\_d \boldsymbol{\nu}\_Q + g\_5 T\_L \end{bmatrix} \tag{4}$$

Using dynamic system given in Eq. (4) it is easier to obtain an observer. As it is shown in Eq. (5), full order observer with damper fluxes is:

\_ *i* b *d* \_ *i* b *f* c\_ *ψD* \_ *i* b*q* d\_ *ψQ* \_ *ω*b 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 ¼ *<sup>a</sup>*1*id* <sup>þ</sup> *<sup>a</sup>*2*if* <sup>þ</sup> *<sup>a</sup>*3*iq<sup>ω</sup>* <sup>þ</sup> *<sup>a</sup>*4*ψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>a</sup>*5*ψ*d*<sup>Q</sup> <sup>ω</sup>* <sup>þ</sup> *<sup>a</sup>*6*ud* <sup>þ</sup> *<sup>a</sup>*7*uf* <sup>þ</sup> *<sup>k</sup>*11*e*<sup>1</sup> *<sup>b</sup>*1*id* <sup>þ</sup> *<sup>b</sup>*2*if* <sup>þ</sup> *<sup>b</sup>*3*iq<sup>ω</sup>* <sup>þ</sup> *<sup>b</sup>*4*ψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>b</sup>*5*ψ*d*<sup>Q</sup> <sup>ω</sup>* <sup>þ</sup> *<sup>b</sup>*6*ud* <sup>þ</sup> *<sup>b</sup>*7*uf* <sup>þ</sup> *<sup>k</sup>*22*e*<sup>2</sup> *<sup>c</sup>*1*id* <sup>þ</sup> *<sup>c</sup>*2*if* <sup>þ</sup> *<sup>c</sup>*3*ψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>k</sup>*31*e*<sup>1</sup> <sup>þ</sup> *<sup>k</sup>*32*e*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*33*e*<sup>4</sup> <sup>þ</sup> *<sup>k</sup>*34*e*<sup>6</sup> *<sup>d</sup>*1*iq* <sup>þ</sup> *<sup>d</sup>*2*id<sup>ω</sup>* <sup>þ</sup> *<sup>d</sup>*3*if<sup>ω</sup>* <sup>þ</sup> *<sup>d</sup>*4*ωψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>d</sup>*5*ψ*d*<sup>Q</sup>* <sup>þ</sup> *<sup>d</sup>*6*uq* <sup>þ</sup> *<sup>k</sup>*43*e*<sup>4</sup> *<sup>f</sup>* <sup>1</sup>*iq* <sup>þ</sup> *<sup>f</sup>* <sup>2</sup>*ψ*d*<sup>Q</sup>* <sup>þ</sup> *<sup>k</sup>*51*e*<sup>1</sup> <sup>þ</sup> *<sup>k</sup>*52*e*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*53*e*<sup>4</sup> <sup>þ</sup> *<sup>k</sup>*54*e*<sup>6</sup> *<sup>g</sup>*1*idiq* <sup>þ</sup> *<sup>g</sup>*2*if iq* <sup>þ</sup> *<sup>g</sup>*3*iqψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>g</sup>*4*idψ*d*<sup>Q</sup>* <sup>þ</sup> *<sup>g</sup>*5*TL* <sup>þ</sup> *<sup>k</sup>*64*e*<sup>6</sup> 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 (5)

The analysis of the observability is based on nonlinear system weak observability concept [9]. According to reference [9], rank of the observability matrix *O* has to be checked.

Regarding the measured outputs, determinant of the arbitrarily chosen observability criterion matrices has to be calculated. The first criterion matrix *O*<sup>1</sup> is chosen:

*O*<sup>1</sup> ¼ *did dif diq dω d Lf id* � � *d Lf if* � � 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 ¼ *pt ∂L*<sup>0</sup> *f id ∂id ∂L*<sup>0</sup> *f id ∂if ∂L*<sup>0</sup> *f id ∂φ<sup>D</sup> ∂L*<sup>0</sup> *f id ∂iq ∂L*<sup>0</sup> *f id ∂φ<sup>Q</sup> ∂L*<sup>0</sup> *f id ∂ω ∂L*<sup>0</sup> *f if ∂id ∂L*<sup>0</sup> *f if ∂if ∂L*<sup>0</sup> *f if ∂φ<sup>D</sup> ∂L*<sup>0</sup> *f if ∂iq ∂L*<sup>0</sup> *f if ∂φ<sup>Q</sup> ∂L*<sup>0</sup> *f if ∂ω ∂L*<sup>0</sup> *f iq ∂id ∂L*<sup>0</sup> *f iq ∂if ∂L*<sup>0</sup> *f iq ∂φ<sup>D</sup> ∂L*<sup>0</sup> *f iq ∂iq ∂L*<sup>0</sup> *f iq ∂φ<sup>Q</sup> ∂L*<sup>0</sup> *f iq ∂ω ∂L*<sup>0</sup> *f ω ∂id ∂L*<sup>0</sup> *f ω ∂if ∂L*<sup>0</sup> *f ω ∂φ<sup>D</sup> ∂L*<sup>0</sup> *f ω ∂iq ∂L*<sup>0</sup> *f ω ∂φ<sup>Q</sup> ∂L*<sup>0</sup> *f ω ∂ω ∂L*<sup>1</sup> *f id ∂id ∂L*<sup>1</sup> *f id ∂if ∂L*<sup>1</sup> *f id ∂φ<sup>D</sup> ∂L*<sup>1</sup> *f id ∂iq ∂L*<sup>1</sup> *f id ∂φ<sup>Q</sup> ∂L*<sup>1</sup> *f id ∂ω ∂L*<sup>1</sup> *f if ∂id ∂L*<sup>1</sup> *f if ∂if ∂L*<sup>1</sup> *f if ∂φ<sup>D</sup> ∂L*<sup>1</sup> *f if ∂iq ∂L*<sup>1</sup> *f if ∂φ<sup>Q</sup> ∂L*<sup>1</sup> *f if ∂ω* 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (6)

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization… DOI: http://dx.doi.org/10.5772/intechopen.89420*

After each matrix member of Eq. (6) is calculated [8], its determinant calculation gives:

$$\text{Det}(O\_1) = -\frac{\alpha L\_{md} L\_{mq} R\_D}{L\_D L\_Q \left(L\_d L\_D L\_f - L\_d L\_{md}^2 - L\_D L\_{md}^2 - L\_f L\_{md}^2 + 2L\_{md}^3\right)}\tag{7}$$

The second criterion matrix *O*<sup>2</sup> is chosen:

There is a way to define adaptive coefficients in each one of the observers given in Eqs. (2) and (3) to prove their stability according to Lyapunov theory. The proof

If the SM dynamics given in Eq. (1) is changed in a way that damper currents states are replaced with damper fluxes states, its dynamic system will become:

> *a*1*id* þ *a*2*if* þ *a*3*iqω* þ *a*4*ψ<sup>D</sup>* þ *a*5*ψ<sup>Q</sup> ω* þ *a*6*ud* þ *a*7*uf b*1*id* þ *b*2*if* þ *b*3*iqω* þ *b*4*ψ<sup>D</sup>* þ *b*5*ψ<sup>Q</sup> ω* þ *b*6*ud* þ *b*7*uf c*1*id* þ *c*2*if* þ *c*3*ψ<sup>D</sup> d*1*iq* þ *d*2*idω* þ *d*3*ifω* þ *d*4*ωψ<sup>D</sup>* þ *d*5*ψ<sup>Q</sup>* þ *d*6*uq f* <sup>1</sup>*iq* þ *f* <sup>2</sup>*ψ<sup>Q</sup> g*1*idiq* þ *g*2*if iq* þ *g*3*iqψ<sup>D</sup>* þ *g*4*idψ<sup>Q</sup>* þ *g*5*TL*

Using dynamic system given in Eq. (4) it is easier to obtain an observer. As it is

*<sup>a</sup>*1*id* <sup>þ</sup> *<sup>a</sup>*2*if* <sup>þ</sup> *<sup>a</sup>*3*iq<sup>ω</sup>* <sup>þ</sup> *<sup>a</sup>*4*ψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>a</sup>*5*ψ*d*<sup>Q</sup> <sup>ω</sup>* <sup>þ</sup> *<sup>a</sup>*6*ud* <sup>þ</sup> *<sup>a</sup>*7*uf* <sup>þ</sup> *<sup>k</sup>*11*e*<sup>1</sup> *<sup>b</sup>*1*id* <sup>þ</sup> *<sup>b</sup>*2*if* <sup>þ</sup> *<sup>b</sup>*3*iq<sup>ω</sup>* <sup>þ</sup> *<sup>b</sup>*4*ψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>b</sup>*5*ψ*d*<sup>Q</sup> <sup>ω</sup>* <sup>þ</sup> *<sup>b</sup>*6*ud* <sup>þ</sup> *<sup>b</sup>*7*uf* <sup>þ</sup> *<sup>k</sup>*22*e*<sup>2</sup> *<sup>c</sup>*1*id* <sup>þ</sup> *<sup>c</sup>*2*if* <sup>þ</sup> *<sup>c</sup>*3*ψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>k</sup>*31*e*<sup>1</sup> <sup>þ</sup> *<sup>k</sup>*32*e*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*33*e*<sup>4</sup> <sup>þ</sup> *<sup>k</sup>*34*e*<sup>6</sup> *<sup>d</sup>*1*iq* <sup>þ</sup> *<sup>d</sup>*2*id<sup>ω</sup>* <sup>þ</sup> *<sup>d</sup>*3*if<sup>ω</sup>* <sup>þ</sup> *<sup>d</sup>*4*ωψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>d</sup>*5*ψ*d*<sup>Q</sup>* <sup>þ</sup> *<sup>d</sup>*6*uq* <sup>þ</sup> *<sup>k</sup>*43*e*<sup>4</sup> *<sup>f</sup>* <sup>1</sup>*iq* <sup>þ</sup> *<sup>f</sup>* <sup>2</sup>*ψ*d*<sup>Q</sup>* <sup>þ</sup> *<sup>k</sup>*51*e*<sup>1</sup> <sup>þ</sup> *<sup>k</sup>*52*e*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*53*e*<sup>4</sup> <sup>þ</sup> *<sup>k</sup>*54*e*<sup>6</sup> *<sup>g</sup>*1*idiq* <sup>þ</sup> *<sup>g</sup>*2*if iq* <sup>þ</sup> *<sup>g</sup>*3*iqψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>g</sup>*4*idψ*d*<sup>Q</sup>* <sup>þ</sup> *<sup>g</sup>*5*TL* <sup>þ</sup> *<sup>k</sup>*64*e*<sup>6</sup>

The analysis of the observability is based on nonlinear system weak observability concept [9]. According to reference [9], rank of the observability matrix *O* has to

Regarding the measured outputs, determinant of the arbitrarily chosen observability criterion matrices has to be calculated. The first criterion matrix *O*<sup>1</sup> is chosen:

> *∂L*<sup>0</sup> *f id ∂φ<sup>D</sup>*

*∂L*<sup>0</sup> *f if ∂φ<sup>D</sup>*

*∂L*<sup>0</sup> *f iq ∂φ<sup>D</sup>*

*∂L*<sup>0</sup> *f ω ∂φ<sup>D</sup>*

*∂L*<sup>1</sup> *f id ∂φ<sup>D</sup>*

*∂L*<sup>1</sup> *f if ∂φ<sup>D</sup>*

*∂L*<sup>0</sup> *f id ∂iq*

*∂L*<sup>0</sup> *f if ∂iq*

*∂L*<sup>0</sup> *f iq ∂iq*

*∂L*<sup>0</sup> *f ω ∂iq*

*∂L*<sup>1</sup> *f id ∂iq*

*∂L*<sup>1</sup> *f if ∂iq*

*∂L*<sup>0</sup> *f id ∂φ<sup>Q</sup>*

*∂L*<sup>0</sup> *f if ∂φ<sup>Q</sup>*

*∂L*<sup>0</sup> *f iq ∂φ<sup>Q</sup>*

*∂L*<sup>0</sup> *f ω ∂φ<sup>Q</sup>*

*∂L*<sup>1</sup> *f id ∂φ<sup>Q</sup>*

*∂L*<sup>1</sup> *f if ∂φ<sup>Q</sup>*

*∂L*<sup>0</sup> *f id ∂ω*

*∂L*<sup>0</sup> *f if ∂ω*

*∂L*<sup>0</sup> *f iq ∂ω*

*∂L*<sup>0</sup> *f ω ∂ω*

*∂L*<sup>1</sup> *f id ∂ω*

*∂L*<sup>1</sup> *f if ∂ω*

*∂L*<sup>0</sup> *f id ∂if*

*∂L*<sup>0</sup> *f if ∂if*

*∂L*<sup>0</sup> *f iq ∂if*

*∂L*<sup>0</sup> *f ω ∂if*

*∂L*<sup>1</sup> *f id ∂if*

*∂L*<sup>1</sup> *f if ∂if*

shown in Eq. (5), full order observer with damper fluxes is:

(4)

(5)

(6)

is extensive and is given in [8].

*Control Theory in Engineering*

\_*id* \_*if ψ*\_ *D* \_*iq ψ*\_ *Q ω*\_

¼

\_ *i* b *d* \_ *i* b *f* c\_ *ψD* \_ *i* b*q* d\_ *ψQ* \_ *ω*b

be checked.

*O*<sup>1</sup> ¼

**148**

*did dif diq dω d Lf id* � �

¼ *pt*

*∂L*<sup>0</sup> *f id ∂id*

*∂L*<sup>0</sup> *f if ∂id*

*∂L*<sup>0</sup> *f iq ∂id*

*∂L*<sup>0</sup> *f ω ∂id*

*∂L*<sup>1</sup> *f id ∂id*

*∂L*<sup>1</sup> *f if ∂id*

*d Lf if* � �

¼

**2.2 Damper winding full order flux observer**

*O*<sup>2</sup> ¼ *did dif diq dω d Lf id* � � *d Lf iq* � � 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ¼ *∂L*<sup>0</sup> *f id ∂id ∂L*<sup>0</sup> *f id ∂if ∂L*<sup>0</sup> *f id ∂φ<sup>D</sup> ∂L*<sup>0</sup> *f id ∂iq ∂L*<sup>0</sup> *f id ∂φ<sup>Q</sup> ∂L*<sup>0</sup> *f id ∂ω ∂L*<sup>0</sup> *f if ∂id ∂L*<sup>0</sup> *f if ∂if ∂L*<sup>0</sup> *f if ∂φ<sup>D</sup> ∂L*<sup>0</sup> *f if ∂iq ∂L*<sup>0</sup> *f if ∂φ<sup>Q</sup> ∂L*<sup>0</sup> *f if ∂ω ∂L*<sup>0</sup> *f iq ∂id ∂L*<sup>0</sup> *f iq ∂if ∂L*<sup>0</sup> *f iq ∂φ<sup>D</sup> ∂L*<sup>0</sup> *f iq ∂iq ∂L*<sup>0</sup> *f iq ∂φ<sup>Q</sup> ∂L*<sup>0</sup> *f iq ∂ω ∂L*<sup>0</sup> *f ω ∂id ∂L*<sup>0</sup> *f ω ∂if ∂L*<sup>0</sup> *f ω ∂φ<sup>D</sup> ∂L*<sup>0</sup> *f ω ∂iq ∂L*<sup>0</sup> *f ω ∂φ<sup>Q</sup> ∂L*<sup>0</sup> *f ω ∂ω ∂L*<sup>1</sup> *f id ∂id ∂L*<sup>1</sup> *f id ∂if ∂L*<sup>1</sup> *f id ∂φ<sup>D</sup> ∂L*<sup>1</sup> *f id ∂iq ∂L*<sup>1</sup> *f id ∂φ<sup>Q</sup> ∂L*<sup>1</sup> *f id ∂ω ∂L*<sup>1</sup> *f iq ∂id ∂L*<sup>1</sup> *f iq ∂if ∂L*<sup>1</sup> *f iq ∂φ<sup>D</sup> ∂L*<sup>1</sup> *f iq ∂iq ∂L*<sup>1</sup> *f iq ∂φ<sup>Q</sup> ∂L*<sup>1</sup> *f iq ∂ω* 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (8)

After each matrix member of Eq. (8) is calculated [8], its determinant calculation gives:

*Det O*ð Þ¼� <sup>2</sup> *<sup>ω</sup>*<sup>2</sup>*Lmd* �*LD* 2 *LfLmq* þ *LDLmd* 2 *Lmq* � � *LD* <sup>2</sup> �*LdLDLf* <sup>þ</sup> *LdLmd* <sup>2</sup> <sup>þ</sup> *LDLmd* <sup>2</sup> <sup>þ</sup> *LfLmd* <sup>2</sup> � <sup>2</sup>*Lmd* <sup>3</sup> � � �*Lmq* <sup>2</sup> <sup>þ</sup> *LqLQ* � � � � *LmqRQ* �*LfLmdLQ RD* <sup>þ</sup> *Lmd* 2 *LQ RD* � � *LDLQ* <sup>2</sup> �*LdLDLf* <sup>þ</sup> *LdLmd* <sup>2</sup> <sup>þ</sup> *LDLmd* <sup>2</sup> <sup>þ</sup> *LfLmd* <sup>2</sup> � <sup>2</sup>*Lmd* <sup>3</sup> � � �*Lmq* <sup>2</sup> <sup>þ</sup> *LqLQ* � � (9)

While observing both determinants (Eqs. (7) and (9)):

*Det O*<sup>1</sup> 6¼ 0, for *ω* 6¼ 0, while *Det O*<sup>2</sup> 6¼ 0, for *ω* = 0 it is easy to see that: *Det* (*O*1) 6¼ 0 *U Det* (*O*2) 6¼ 0 = > *rank* {*O*} = 6.

Matrix *O* is full rank matrix and it could be concluded that the system is weakly locally observable.

To make a proof of observer Eq. (5) stability, Lyapunov function Eq. (10) is proposed:

$$V\_1 = \frac{e\_1^2}{2} + \frac{e\_2^2}{2} + \frac{e\_3^2}{2} + \frac{e\_4^2}{2} + \frac{e\_5^2}{2} + \frac{e\_6^2}{2} \tag{10}$$

Equation (10) is positive definite function of the error variables: *e*1, *e*2, *e*3, *e*4, *e*5, *e*6. Error dynamic system is obtained by Eqs. (4) and (5), and the result is:

$$\begin{bmatrix} \dot{\mathbf{e}}\_1 \\ \dot{\mathbf{e}}\_2 \\ \dot{\mathbf{e}}\_3 \\ \dot{\mathbf{e}}\_4 \\ \dot{\mathbf{e}}\_5 \\ \dot{\mathbf{e}}\_6 \\ \dot{\mathbf{e}}\_7 \\ \dot{\mathbf{e}}\_6 \end{bmatrix} = \begin{bmatrix} a\_4e\_3 + a\_5oe\_5 - k\_{11}e\_1 \\ b\_4e\_3 + b\_5oe\_5 - k\_{22}e\_1 \\ c\_3e\_3 - k\_{31}e\_1 - k\_{32}e\_2 + k\_{33}e\_4 + k\_{34}e\_6 \\ d\_4ae\_3 + d\_5e\_5 - k\_{43}e\_4 \\ \vdots \\ f\_2e\_5 - k\_{51}e\_1 - k\_{52}e\_2 - k\_{53}e\_4 - k\_{54}e\_6 \\ g\_3i\_qe\_3 + g\_4de\_5 - k\_{64}e\_6 \end{bmatrix} \tag{11}$$

Error dynamics are obtained in similar way as for the full order observer. If the coefficients *kxy* are defined as stated: *k*<sup>31</sup> = *g*<sup>3</sup> *iq*, *k*<sup>51</sup> = *g*<sup>4</sup> *id*, *k*<sup>61</sup> > 0, derivation of the Lyapunov function is negative definite and stability of the observer is proved:

If the motion dynamics equation from the dynamic system is omitted, the

This observer includes only damper winding dynamic equations, and for its

Stability can be proved in the same way as for the previous observers. If a

*<sup>V</sup>* <sup>¼</sup> *<sup>e</sup>*<sup>2</sup> þ *e*2 

It has negative definite derivation Eq. (19) and stability is proved.

**2.4 Damper winding flux observer with adaptation of resistance**

resistances. Firstly, dynamic system Eq. (4) has to be expanded:

*<sup>V</sup>*\_ <sup>¼</sup> *<sup>c</sup>*3*<sup>e</sup>* þ *f* <sup>2</sup>*e* 

Full order observer can be also used for the adaptation of the stator and rotor

*a*1*id* þ *a*2*if* þ *a*3*iqω* þ *a*4*ψ<sup>D</sup>* þ *a*5*ψ<sup>Q</sup> ω* þ *a*6*ifRf* þ *a*7*idRS* þ *a*8*ud* þ *a*9*uf b*1*id* þ *b*2*if* þ *b*3*iqω* þ *b*4*ψ<sup>D</sup>* þ *b*5*ψ<sup>Q</sup> ω* þ *b*6*ifRf* þ *b*7*idRS* þ *b*8*ud* þ *b*9*uf c*1*id* þ *c*2*if* þ *c*3*ψ<sup>D</sup> d*1*iq* þ *d*2*idω* þ *d*3*ifω* þ *d*4*ωψ<sup>D</sup>* þ *d*5*ψ<sup>Q</sup>* þ *d*6*iqRS* þ *d*7*uq f* <sup>1</sup>*iq* þ *f* <sup>2</sup>*ψ<sup>Q</sup> g*1*idiq* þ *g*2*if iq* þ *g*3*iqψ<sup>D</sup>* þ *g*4*idψ<sup>Q</sup>* þ *g*5*MT*

In a similar way as for the full order observer Eq. (5), an observer for adaptation

*<sup>a</sup>*1*id* <sup>þ</sup> *<sup>a</sup>*2*if* <sup>þ</sup> *<sup>a</sup>*3*iq<sup>ω</sup>* <sup>þ</sup> *<sup>a</sup>*4*ψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>a</sup>*5*ψ*d*<sup>Q</sup> <sup>ω</sup>* <sup>þ</sup> *<sup>a</sup>*6*ifR*b*<sup>f</sup>* <sup>þ</sup> *<sup>a</sup>*7*idR*b*<sup>s</sup>* <sup>þ</sup> *<sup>a</sup>*8*ud* <sup>þ</sup> *<sup>a</sup>*9*uf* <sup>þ</sup> *<sup>k</sup>*11*e*<sup>1</sup> *<sup>b</sup>*1*id* <sup>þ</sup> *<sup>b</sup>*2*if* <sup>þ</sup> *<sup>b</sup>*3*iq<sup>ω</sup>* <sup>þ</sup> *<sup>b</sup>*4*ψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>b</sup>*5*ψ*d*<sup>Q</sup> <sup>ω</sup>* <sup>þ</sup> *<sup>b</sup>*6*ifR*b*<sup>f</sup>* <sup>þ</sup> *<sup>b</sup>*7*idR*b*<sup>s</sup>* <sup>þ</sup> *<sup>b</sup>*8*ud* <sup>þ</sup> *<sup>b</sup>*9*uf* <sup>þ</sup> *<sup>k</sup>*22*e*<sup>2</sup> *<sup>c</sup>*1*id* <sup>þ</sup> *<sup>c</sup>*2*if* <sup>þ</sup> *<sup>c</sup>*3*ψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>k</sup>*31*e*<sup>1</sup> <sup>þ</sup> *<sup>k</sup>*32*e*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*33*e*<sup>4</sup> <sup>þ</sup> *<sup>k</sup>*34*e*<sup>6</sup> *<sup>d</sup>*1*iq* <sup>þ</sup> *<sup>d</sup>*2*id<sup>ω</sup>* <sup>þ</sup> *<sup>d</sup>*3*if<sup>ω</sup>* <sup>þ</sup> *<sup>d</sup>*4*ωψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>d</sup>*5*ψ*d*<sup>Q</sup>* <sup>þ</sup> *<sup>d</sup>*6*iqR*b*<sup>s</sup>* <sup>þ</sup> *<sup>d</sup>*7*uq* <sup>þ</sup> *<sup>k</sup>*43*e*<sup>4</sup> *<sup>f</sup>* <sup>1</sup>*iq* <sup>þ</sup> *<sup>f</sup>* <sup>2</sup>*ψ*d*<sup>Q</sup>* <sup>þ</sup> *<sup>k</sup>*51*e*<sup>1</sup> <sup>þ</sup> *<sup>k</sup>*52*e*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*53*e*<sup>4</sup> <sup>þ</sup> *<sup>k</sup>*54*e*<sup>6</sup> *<sup>g</sup>*1*idiq* <sup>þ</sup> *<sup>g</sup>*2*if iq* <sup>þ</sup> *<sup>g</sup>*3*iqψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>g</sup>*4*idψ*d*<sup>Q</sup>* <sup>þ</sup> *<sup>g</sup>*5*MT* <sup>þ</sup> *<sup>k</sup>*64*e*<sup>6</sup>

 *<sup>c</sup>*1*id* <sup>þ</sup> *<sup>c</sup>*2*if* <sup>þ</sup> *<sup>c</sup>*3*ψ*c*<sup>D</sup> <sup>f</sup>* <sup>1</sup>*iq* <sup>þ</sup> *<sup>f</sup>* <sup>2</sup>*ψ*d*<sup>Q</sup>*

" #

(16)

(18)

(19)

(17)

(20)

(21)

*<sup>V</sup>*\_ <sup>¼</sup> *<sup>c</sup>*3*<sup>e</sup>* þ *f* <sup>2</sup>*e* � *k*61*e* 

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization…*

c\_ *ψD* d\_ *ψQ*

operation only rotor and stator current components are needed.

positive definite Lyapunov function Eq. (18) is considered:

" #

simplest observer can be defined:

*DOI: http://dx.doi.org/10.5772/intechopen.89420*

\_*id* \_*if ψ*\_ *D* \_*iq ψ*\_ *Q ω*\_

could be defined:

\_ *i* b *d* \_ *i* b *f* c\_ *ψD* \_ *i* b*q* d\_ *ψQ* \_ *ω*b

Then, derivation of the Lyapunov function Eq. (10) is done. Using substitution of the Eq. (11), the results is:

$$\mathbf{V\_1 = \mathbf{a\_4}e\_1e\_3 + \mathbf{a\_5}ue\_1e\_5 - \mathbf{k\_{11}e\_1^2} + \mathbf{b\_4}e\_2e\_3 + \mathbf{b\_5}ue\_2e\_5 - \mathbf{k\_{22}e\_2^2} + \mathbf{c}\_2$$

$$+ \mathbf{c\_3e\_3^2} - \mathbf{k\_{31}e\_1e\_3} - \mathbf{k\_{32}e\_2e\_3} - \mathbf{k\_{33}e\_3e\_4} - \mathbf{k\_{34}e\_3e\_6} + \mathbf{d\_4}u\mathbf{e\_3e\_4} + \mathbf{g}$$

$$+ \mathbf{d\_5e\_4e\_5} - \mathbf{k\_{43}e\_4^2} + + \mathbf{f\_2}e\_5^2 - \mathbf{k\_{51}e\_1e\_5} - \mathbf{k\_{52}e\_2e\_5} - \mathbf{k\_{53}e\_4e\_5}$$

$$- \mathbf{k\_{54}e\_5e\_6} + + \mathbf{g}\_3i\_qe\_3e\_6 + \mathbf{g}\_4ie\_5e\_6 - \mathbf{k\_{64}e\_6^2} \tag{12}$$

If the coefficients *kxy* are defined as stated:

$$\mathbf{k}\_{31} = \mathbf{a}\_4; \mathbf{k}\_{32} = \mathbf{b}\_4; \mathbf{k}\_{33} = \mathbf{d}\_4\\\boldsymbol{\alpha}; \mathbf{k}\_{34} = \mathbf{g}\_3\\\mathbf{i}\_{\mathbf{q}}; \mathbf{k}\_{51} = \mathbf{a}\_5\\\boldsymbol{\alpha}; \mathbf{k}\_{52} = \mathbf{b}\_5\\\boldsymbol{\alpha}; \mathbf{0}$$

$$\mathbf{k}\_{53} = \mathbf{d}\_5; \mathbf{k}\_{54} = \mathbf{g}\_4\\\mathbf{i}\_{\mathbf{d}}; \mathbf{k}\_{11}, \mathbf{k}\_{22}, \mathbf{k}\_{43}, \mathbf{k}\_{64} > \mathbf{0}$$

Derivation of the Lyapunov function becomes:

$$\dot{V}\_1 = -k\_{11}e\_1^2 - k\_{22}e\_2^2 + c\_3e\_3^2 - k\_{43}e\_4^2 + f\_2e\_5^2 - k\_{64}e\_6^2 \tag{13}$$

Due to the character of the damper winding, the parameters *c*<sup>3</sup> and *f*<sup>2</sup> are negative for each SM. That is why it is easy to make Eq. (13) to be negative definite. When *V*\_ < 0 is achieved, a global asymptotic stability of the observer is proved.

#### **2.3 Damper winding reduced order flux observer**

To obtain full order observer it is necessary for the stator and rotor voltages to be known. Knowledge of the load torque is also needed. Therefore, simpler observer has been found reference [10]. If the stator and rotor current dynamics equations from the dynamic system Eq. (4) are omitted, reduced order observer could be defined:

$$
\begin{bmatrix}
\dot{\widehat{\boldsymbol{w}\_D}} \\
\dot{\widehat{\boldsymbol{w}\_Q}} \\
\dot{\widehat{\boldsymbol{w}}}
\end{bmatrix} = \begin{bmatrix}
c\_1 \dot{\boldsymbol{i}}\_d + c\_2 \dot{\boldsymbol{i}}\_f + c\_3 \widehat{\boldsymbol{\mu}\_D} + k\_{31} \boldsymbol{e}\_6 \\
f\_1 \dot{\boldsymbol{i}}\_q + f\_2 \widehat{\boldsymbol{\mu}\_Q} + k\_{51} \boldsymbol{e}\_6 \\
g\_1 \dot{\boldsymbol{i}}\_d \dot{\boldsymbol{i}}\_q + g\_2 \dot{\boldsymbol{i}}\_f \boldsymbol{i}\_q + g\_3 \widehat{\boldsymbol{i}}\_q \widehat{\boldsymbol{\mu}\_D} + g\_4 \widehat{\boldsymbol{i}}\_d \widehat{\boldsymbol{\mu}\_Q} + \mathbf{g}\_5 \boldsymbol{T}\_L + k\_{61} \boldsymbol{e}\_6
\end{bmatrix} \tag{14}
$$

It is easy to see that to obtain an observer Eq. (14) it is not needed to know the stator and rotor voltages.

Stability can be proved by the following Lyapunov function:

$$V = \frac{\varepsilon\_3^2}{2} + \frac{\varepsilon\_5^2}{2} + \frac{\varepsilon\_6^2}{2} \tag{15}$$

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization… DOI: http://dx.doi.org/10.5772/intechopen.89420*

Error dynamics are obtained in similar way as for the full order observer. If the coefficients *kxy* are defined as stated: *k*<sup>31</sup> = *g*<sup>3</sup> *iq*, *k*<sup>51</sup> = *g*<sup>4</sup> *id*, *k*<sup>61</sup> > 0, derivation of the Lyapunov function is negative definite and stability of the observer is proved:

$$
\dot{V} = c\_{\mathcal{Y}} e\_{\mathfrak{z}}^2 + f\_{\mathfrak{z}} e\_{\mathfrak{z}}^2 - k\_{61} e\_{\mathfrak{e}}^2 \tag{16}
$$

If the motion dynamics equation from the dynamic system is omitted, the simplest observer can be defined:

$$
\begin{bmatrix}
\dot{\widehat{\boldsymbol{\mu\_{D}}}}
\end{bmatrix} = \begin{bmatrix}
c\_{1}\dot{\boldsymbol{i}\_{d}} + c\_{2}\dot{\boldsymbol{i}\_{f}} + c\_{3}\widehat{\boldsymbol{\mu\_{D}}} \\
f\_{1}\dot{\boldsymbol{i}\_{q}} + f\_{2}\widehat{\boldsymbol{\mu\_{Q}}}
\end{bmatrix} \tag{17}
$$

This observer includes only damper winding dynamic equations, and for its operation only rotor and stator current components are needed.

Stability can be proved in the same way as for the previous observers. If a positive definite Lyapunov function Eq. (18) is considered:

$$V = \frac{e\_3^2}{2} + \frac{e\_5^2}{2} \tag{18}$$

It has negative definite derivation Eq. (19) and stability is proved.

$$
\dot{V} = c\_3 e\_3^2 + f\_2 e\_5^2 \tag{19}
$$

#### **2.4 Damper winding flux observer with adaptation of resistance**

Full order observer can be also used for the adaptation of the stator and rotor resistances. Firstly, dynamic system Eq. (4) has to be expanded:

$$
\begin{bmatrix}
\dot{i}\_d\\\dot{i}\_f\\\dot{w}\_D\\\dot{i}\_q\\\dot{i}\_q\\\dot{w}\_Q
\end{bmatrix} = \begin{bmatrix}
a\_1\dot{i}\_d + a\_2\dot{i}\_f + a\_3\dot{i}\_q\boldsymbol{o} + a\_4\boldsymbol{y}\_D + a\_5\boldsymbol{y}\_Q\boldsymbol{o} + a\_6\dot{i}\_f\boldsymbol{R}\_f + a\_7\dot{i}\_d\boldsymbol{R}\_S + a\_8\boldsymbol{u}\_d + a\_9\boldsymbol{u}\_f\\\dot{b}\_1\dot{i}\_d + b\_2\dot{i}\_f + b\_3\dot{i}\_q\boldsymbol{o} + b\_4\boldsymbol{y}\_D\boldsymbol{o} + b\_5\dot{i}\_f\boldsymbol{R}\_f + b\_7\dot{i}\_d\boldsymbol{R}\_S + b\_8\boldsymbol{u}\_d + b\_9\boldsymbol{u}\_f\\\ c\_1\dot{i}\_d + c\_2\dot{i}\_f + c\_3\boldsymbol{y}\_D\\\ d\_1\dot{i}\_q + d\_2\dot{i}\_f + c\_3\boldsymbol{y}\_f\\\ \dot{f}\_1\dot{i}\_q + \dot{f}\_2\boldsymbol{w}\_Q\\\ \boldsymbol{g}\_1\dot{i}\_d\dot{i}\_q + \boldsymbol{g}\_2\dot{i}\_f\dot{i}\_q + \boldsymbol{g}\_3\dot{i}\_q\boldsymbol{w}\_D + \boldsymbol{g}\_4\boldsymbol{i}\_d\boldsymbol{w}\_Q + \boldsymbol{g}\_5\boldsymbol{M}\_T
\end{bmatrix} \tag{20}
$$

In a similar way as for the full order observer Eq. (5), an observer for adaptation could be defined:

\_ *i* b *d* \_ *i* b *f* c\_ *ψD* \_ *i* b*q* d\_ *ψQ* \_ *ω*b *<sup>a</sup>*1*id* <sup>þ</sup> *<sup>a</sup>*2*if* <sup>þ</sup> *<sup>a</sup>*3*iq<sup>ω</sup>* <sup>þ</sup> *<sup>a</sup>*4*ψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>a</sup>*5*ψ*d*<sup>Q</sup> <sup>ω</sup>* <sup>þ</sup> *<sup>a</sup>*6*ifR*b*<sup>f</sup>* <sup>þ</sup> *<sup>a</sup>*7*idR*b*<sup>s</sup>* <sup>þ</sup> *<sup>a</sup>*8*ud* <sup>þ</sup> *<sup>a</sup>*9*uf* <sup>þ</sup> *<sup>k</sup>*11*e*<sup>1</sup> *<sup>b</sup>*1*id* <sup>þ</sup> *<sup>b</sup>*2*if* <sup>þ</sup> *<sup>b</sup>*3*iq<sup>ω</sup>* <sup>þ</sup> *<sup>b</sup>*4*ψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>b</sup>*5*ψ*d*<sup>Q</sup> <sup>ω</sup>* <sup>þ</sup> *<sup>b</sup>*6*ifR*b*<sup>f</sup>* <sup>þ</sup> *<sup>b</sup>*7*idR*b*<sup>s</sup>* <sup>þ</sup> *<sup>b</sup>*8*ud* <sup>þ</sup> *<sup>b</sup>*9*uf* <sup>þ</sup> *<sup>k</sup>*22*e*<sup>2</sup> *<sup>c</sup>*1*id* <sup>þ</sup> *<sup>c</sup>*2*if* <sup>þ</sup> *<sup>c</sup>*3*ψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>k</sup>*31*e*<sup>1</sup> <sup>þ</sup> *<sup>k</sup>*32*e*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*33*e*<sup>4</sup> <sup>þ</sup> *<sup>k</sup>*34*e*<sup>6</sup> *<sup>d</sup>*1*iq* <sup>þ</sup> *<sup>d</sup>*2*id<sup>ω</sup>* <sup>þ</sup> *<sup>d</sup>*3*if<sup>ω</sup>* <sup>þ</sup> *<sup>d</sup>*4*ωψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>d</sup>*5*ψ*d*<sup>Q</sup>* <sup>þ</sup> *<sup>d</sup>*6*iqR*b*<sup>s</sup>* <sup>þ</sup> *<sup>d</sup>*7*uq* <sup>þ</sup> *<sup>k</sup>*43*e*<sup>4</sup> *<sup>f</sup>* <sup>1</sup>*iq* <sup>þ</sup> *<sup>f</sup>* <sup>2</sup>*ψ*d*<sup>Q</sup>* <sup>þ</sup> *<sup>k</sup>*51*e*<sup>1</sup> <sup>þ</sup> *<sup>k</sup>*52*e*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*53*e*<sup>4</sup> <sup>þ</sup> *<sup>k</sup>*54*e*<sup>6</sup> *<sup>g</sup>*1*idiq* <sup>þ</sup> *<sup>g</sup>*2*if iq* <sup>þ</sup> *<sup>g</sup>*3*iqψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>g</sup>*4*idψ*d*<sup>Q</sup>* <sup>þ</sup> *<sup>g</sup>*5*MT* <sup>þ</sup> *<sup>k</sup>*64*e*<sup>6</sup> (21)

e\_ e\_ e\_ e\_ e\_ e\_ 

*a*4*e*<sup>3</sup> þ *a*5*ωe*<sup>5</sup> � *k*11*e*<sup>1</sup> *b*4*e*<sup>3</sup> þ *b*5*ωe*<sup>5</sup> � *k*22*e*<sup>1</sup>

þ

(12)

(13)

 

(15)

(14)

(11)

*c*3*e*<sup>3</sup> � *k*31*e*<sup>1</sup> � *k*32*e*<sup>2</sup> þ *k*33*e*<sup>4</sup> þ *k*34*e*<sup>6</sup> *d*4*ωe*<sup>3</sup> þ *d*5*e*<sup>5</sup> � *k*43*e*<sup>4</sup>

*f* <sup>2</sup>*e*<sup>5</sup> � *k*51*e*<sup>1</sup> � *k*52*e*<sup>2</sup> � *k*53*e*<sup>4</sup> � *k*54*e*<sup>6</sup> *g*3*iqe*<sup>3</sup> þ *g*4*ide*<sup>5</sup> � *k*64*e*<sup>6</sup>

<sup>þ</sup> b4e2e3 <sup>þ</sup> b5ωe2e5 � k22e2

� k51e1e5 � k52e2e5 � k53e4e5

� k31e1e3 � k32e2e3 � k33e3e4 � k34e3e6 þ d4ωe3e4þ

Then, derivation of the Lyapunov function Eq. (10) is done. Using substitution

þ þf2e2

k31 ¼ a4;k32 ¼ b4;k33 ¼ d4ω;k34 ¼ g3iq;k51 ¼ a5ω;k52 ¼ b5ω; k53 ¼ d5;k54 ¼ g4id;k11, k22, k43, k64>0

Due to the character of the damper winding, the parameters *c*<sup>3</sup> and *f*<sup>2</sup> are negative for each SM. That is why it is easy to make Eq. (13) to be negative definite.

< 0 is achieved, a global asymptotic stability of the observer is proved.

To obtain full order observer it is necessary for the stator and rotor voltages to be known. Knowledge of the load torque is also needed. Therefore, simpler observer has been found reference [10]. If the stator and rotor current dynamics equations from the dynamic system Eq. (4) are omitted, reduced order observer could be

> *<sup>c</sup>*1*id* <sup>þ</sup> *<sup>c</sup>*2*if* <sup>þ</sup> *<sup>c</sup>*3*ψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>k</sup>*31*e*<sup>6</sup> *<sup>f</sup>* <sup>1</sup>*iq* <sup>þ</sup> *<sup>f</sup>* <sup>2</sup>*ψ*d*<sup>Q</sup>* <sup>þ</sup> *<sup>k</sup>*51*e*<sup>6</sup> *<sup>g</sup>*1*idiq* <sup>þ</sup> *<sup>g</sup>*2*if iq* <sup>þ</sup> *<sup>g</sup>*3*iqψ*c*<sup>D</sup>* <sup>þ</sup> *<sup>g</sup>*4*idψ*d*<sup>Q</sup>* <sup>þ</sup> *<sup>g</sup>*5*TL* <sup>þ</sup> *<sup>k</sup>*61*e*<sup>6</sup>

It is easy to see that to obtain an observer Eq. (14) it is not needed to know the

Stability can be proved by the following Lyapunov function:

*<sup>V</sup>* <sup>¼</sup> *<sup>e</sup>*<sup>2</sup> þ *e*2 þ *e*2 

�k54e5e6 þ þ*g*3*iqe*3*e*<sup>6</sup> þ *g*4*ide*5*e*<sup>6</sup> � *k*64*e*

<sup>¼</sup> a4e1e3 <sup>þ</sup> a5ωe1e5 � k11e2

<sup>þ</sup>d5e4e5 � k43e<sup>2</sup>

If the coefficients *kxy* are defined as stated:

Derivation of the Lyapunov function becomes:

 � *k*22*e* þ *c*3*e* � *k*43*e* þ *f* <sup>2</sup>*e* � *k*64*e* 

¼ �*k*11*e*

**2.3 Damper winding reduced order flux observer**

*Control Theory in Engineering*

of the Eq. (11), the results is:

V\_

<sup>þ</sup>c3e<sup>2</sup>

*V*\_

When *V*\_

defined:

c\_ *ψD* d\_ *ψQ* \_ *ω*b

stator and rotor voltages.

 

Its error dynamics Eqs. (20) and (21) are obtained:

$$\begin{bmatrix} \dot{\mathbf{e}}\_1 \\ \dot{\mathbf{e}}\_2 \\ \dot{\mathbf{e}}\_3 \\ \dot{\mathbf{e}}\_4 \\ \dot{\mathbf{e}}\_5 \\ \dot{\mathbf{e}}\_6 \end{bmatrix} = \begin{bmatrix} a\_4e\_3 + a\_5oe\_5 + a\_6i\_f\Delta R\_f + a\_7i\_d\Delta R\_s - k\_{11}e\_1 \\ b\_4e\_3 + b\_5oe\_5 + b\_6i\_f\Delta R\_f + b\_7i\_d\Delta R\_s - k\_{22}e\_1 \\ c\_3e\_3 - k\_{31}e\_1 - k\_{32}e\_2 + k\_{33}e\_4 + k\_{34}e\_6 \\ d\_4ae\_3 + d\_5e\_5 + d\_6i\_q\Delta R\_s - k\_{43}e\_4 \\ f\_2e\_5 - k\_{51}e\_1 - k\_{52}e\_2 - k\_{53}e\_4 - k\_{54}e\_6 \\ g\_3i\_qe\_3 + g\_4i\_de\_5 - k\_{64}e\_6 \end{bmatrix} \tag{22}$$

Observability analysis of the Eq. (27) is obtained according to the nonlinear system weak observability concept [9]. Observability criterion matrix *O1* (28) has

> *∂L*<sup>0</sup> *f id ∂iq*

*∂L*<sup>0</sup> *f if ∂iq*

*∂L*<sup>0</sup> *f iq ∂iq*

*∂L*<sup>0</sup> *f γ ∂iq*

*∂L*<sup>1</sup> *f id ∂iq*

*∂L*<sup>1</sup> *f iq ∂iq*

*∂L*<sup>1</sup> *f γ ∂iq*

*∂L*<sup>2</sup> *f γ ∂iq*

After each matrix member of Eq. (28) is calculated [8], its determinant calcula-

*<sup>ω</sup>*<sup>2</sup>*LmdLQ* �*LDLf Lmq* <sup>þ</sup> *Lmd*<sup>2</sup>

*∂L*<sup>0</sup> *f id ∂iq*

*∂L*<sup>0</sup> *f if ∂iq*

*∂L*<sup>0</sup> *f iq ∂iq*

*∂L*<sup>0</sup> *f γ ∂iq*

*∂L*<sup>1</sup> *f id ∂iq*

*∂L*<sup>1</sup> *f if ∂iq*

*∂L*<sup>1</sup> *f γ ∂iq*

*∂L*<sup>2</sup> *f γ ∂iq*

*∂L*<sup>0</sup> *f id ∂φ<sup>Q</sup>*

*∂L*<sup>0</sup> *f if ∂φ<sup>Q</sup>*

*∂L*<sup>0</sup> *f iq ∂φ<sup>Q</sup>*

*∂L*<sup>0</sup> *f γ ∂φ<sup>Q</sup>*

*∂L*<sup>1</sup> *f id ∂φ<sup>Q</sup>*

*∂L*<sup>1</sup> *f iq ∂φ<sup>Q</sup>*

*∂L*<sup>1</sup> *f γ ∂φ<sup>Q</sup>*

*∂L*<sup>2</sup> *f γ ∂φ<sup>Q</sup>*

*Lmq* � �

<sup>3</sup> � � �*Lmq*<sup>2</sup> <sup>þ</sup> *LqLQ*

� �

<sup>3</sup> � � �*Lmq*<sup>2</sup> <sup>þ</sup> *LqLQ*

*∂L*<sup>0</sup> *f id ∂φ<sup>Q</sup>*

*∂L*<sup>0</sup> *f if ∂φ<sup>Q</sup>*

*∂L*<sup>0</sup> *f iq ∂φ<sup>Q</sup>*

*∂L*<sup>0</sup> *f γ ∂φ<sup>Q</sup>*

*∂L*<sup>1</sup> *f id ∂φ<sup>Q</sup>*

*∂L*<sup>1</sup> *f if ∂φ<sup>Q</sup>*

*∂L*<sup>1</sup> *f γ ∂φ<sup>Q</sup>*

*∂L*<sup>2</sup> *f γ ∂φ<sup>Q</sup>*

*∂L*<sup>0</sup> *f id ∂ω*

*∂L*<sup>0</sup> *f if ∂ω*

*∂L*<sup>0</sup> *f iq ∂ω*

*∂L*<sup>0</sup> *f γ ∂ω*

*∂L*<sup>1</sup> *f id ∂ω*

*∂L*<sup>1</sup> *f iq ∂ω*

*∂L*<sup>1</sup> *f γ ∂ω*

*∂L*<sup>2</sup> *f γ ∂ω*

*RD*

*∂L*<sup>0</sup> *f id ∂ω*

*∂L*<sup>0</sup> *f if ∂ω*

*∂L*<sup>0</sup> *f iq ∂ω*

*∂L*<sup>0</sup> *f γ ∂ω*

*∂L*<sup>1</sup> *f id ∂ω*

*∂L*<sup>1</sup> *f if ∂ω*

*∂L*<sup>1</sup> *f γ ∂ω*

*∂L*<sup>2</sup> *f γ ∂ω*

*∂L*<sup>0</sup> *f id ∂γ*

*∂L*<sup>0</sup> *f if ∂γ*

*∂L*<sup>0</sup> *f iq ∂γ*

*∂L*<sup>0</sup> *f γ ∂γ*

*∂L*<sup>1</sup> *f id ∂γ*

*∂L*<sup>0</sup> *f if ∂γ*

*∂L*<sup>0</sup> *f γ ∂γ*

*∂L*<sup>2</sup> *f γ ∂γ*

*∂L*<sup>0</sup> *f id ∂γ*

*∂L*<sup>0</sup> *f if ∂γ*

*∂L*<sup>0</sup> *f iq ∂γ*

*∂L*<sup>0</sup> *f γ ∂γ*

*∂L*<sup>1</sup> *f id ∂γ*

*∂L*<sup>0</sup> *f iq ∂γ*

*∂L*<sup>0</sup> *f γ ∂γ*

*∂L*<sup>2</sup> *f γ ∂γ*

*∂L*<sup>0</sup> *f id ∂TL*

*∂L*<sup>0</sup> *f if ∂TL*

*∂L*<sup>0</sup> *f iq ∂TL*

*∂L*<sup>0</sup> *f γ ∂TL*

*∂L*<sup>1</sup> *f id ∂TL*

(28)

*∂L*<sup>0</sup> *f iq ∂TL*

*∂L*<sup>0</sup> *f γ ∂TL*

*∂L*<sup>2</sup> *f γ ∂TL*

� � �

(29)

(30)

� �

*∂L*<sup>0</sup> *f id ∂TL*

*∂L*<sup>0</sup> *f if ∂TL*

*∂L*<sup>0</sup> *f iq ∂TL*

*∂L*<sup>0</sup> *f γ ∂TL*

*∂L*<sup>1</sup> *f id ∂TL*

*∂L*<sup>0</sup> *f if ∂TL*

*∂L*<sup>0</sup> *f γ ∂TL*

*∂L*<sup>2</sup> *f γ ∂TL*

been chosen:

*did dif diq dγ d Lf id* � � *d Lf iq* � � *d Lf γ* � � *d L*<sup>2</sup> *f γ* � �

*∂L*<sup>0</sup> *f id ∂id*

*DOI: http://dx.doi.org/10.5772/intechopen.89420*

*∂L*<sup>0</sup> *f if ∂id*

*∂L*<sup>0</sup> *f iq ∂id*

*∂L*<sup>0</sup> *f γ ∂id*

*∂L*<sup>1</sup> *f id ∂id*

*∂L*<sup>1</sup> *f iq ∂id*

*∂L*<sup>1</sup> *f γ ∂id*

*∂L*<sup>2</sup> *f γ ∂id*

*∂L*<sup>0</sup> *f id ∂if*

*∂L*<sup>0</sup> *f if ∂if*

*∂L*<sup>0</sup> *f iq ∂if*

*∂L*<sup>0</sup> *f γ ∂if*

*∂L*<sup>1</sup> *f id ∂if*

*∂L*<sup>1</sup> *f iq ∂if*

*∂L*<sup>1</sup> *f γ ∂if*

*∂L*<sup>2</sup> *f γ ∂if*

*∂L*<sup>0</sup> *f id ∂φ<sup>D</sup>*

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization…*

*∂L*<sup>0</sup> *f if ∂φ<sup>D</sup>*

*∂L*<sup>0</sup> *f iq ∂φ<sup>D</sup>*

*∂L*<sup>0</sup> *f γ ∂φ<sup>D</sup>*

*∂L*<sup>1</sup> *f id ∂φ<sup>D</sup>*

*∂L*<sup>1</sup> *f iq ∂φ<sup>D</sup>*

*∂L*<sup>1</sup> *f γ ∂φ<sup>D</sup>*

*∂L*<sup>2</sup> *f γ ∂φ<sup>D</sup>*

<sup>2</sup>*HLDLQ* �*LdLDLf* <sup>þ</sup> *LdLmd*<sup>2</sup> <sup>þ</sup> *LDLmd*<sup>2</sup> <sup>þ</sup> *Lf Lmd*<sup>2</sup> � <sup>2</sup>*Lmd*

<sup>2</sup>*HLDLQ* �*LdLDLf* <sup>þ</sup> *LdLmd*<sup>2</sup> <sup>þ</sup> *LDLmd*<sup>2</sup> <sup>þ</sup> *LfLmd*<sup>2</sup> � <sup>2</sup>*Lmd*

*∂L*<sup>0</sup> *f id ∂φ<sup>D</sup>*

*∂L*<sup>0</sup> *f if ∂φ<sup>D</sup>*

*∂L*<sup>0</sup> *f iq ∂φ<sup>D</sup>*

*∂L*<sup>0</sup> *f γ ∂φ<sup>D</sup>*

*∂L*<sup>1</sup> *f id ∂φ<sup>D</sup>*

*∂L*<sup>1</sup> *f if ∂φ<sup>D</sup>*

*∂L*<sup>1</sup> *f γ ∂φ<sup>D</sup>*

*∂L*<sup>2</sup> *f γ ∂φ<sup>D</sup>*

� *LmqRQ* �*Lf LmdRD* <sup>þ</sup> *Lmd*<sup>2</sup>

Observability criterion matrix *O*<sup>2</sup> has been chosen:

*∂L*<sup>0</sup> *f id ∂if*

*∂L*<sup>0</sup> *f if ∂if*

*∂L*<sup>0</sup> *f iq ∂if*

*∂L*<sup>0</sup> *f γ ∂if*

*∂L*<sup>1</sup> *f id ∂if*

*∂L*<sup>1</sup> *f if ∂if*

*∂L*<sup>1</sup> *f γ ∂if*

*∂L*<sup>2</sup> *f γ ∂if*

*∂L*<sup>0</sup> *f id ∂id*

*∂L*<sup>0</sup> *f if ∂id*

*∂L*<sup>0</sup> *f iq ∂id*

*∂L*<sup>0</sup> *f γ ∂id*

*∂L*<sup>1</sup> *f id ∂id*

*∂L*<sup>1</sup> *f if ∂id*

*∂L*<sup>1</sup> *f γ ∂id*

*∂L*<sup>2</sup> *f γ ∂id*

¼

*O*<sup>1</sup> ¼

tion gives:

*Det O*ð Þ¼� <sup>1</sup>

*O*<sup>2</sup> ¼

**153**

*did dif diq dγ d Lf id* � � *d Lf if* � � *d Lf γ* � � *d L*<sup>2</sup> *f γ* � �

¼

For the positive definite Lyapunov function:

$$V\_1 = \frac{e\_1^2}{2} + \frac{e\_2^2}{2} + \frac{e\_3^2}{2} + \frac{e\_4^2}{2} + \frac{e\_5^2}{2} + \frac{e\_6^2}{2} + \frac{\Delta R\_f^2}{2} + \frac{\Delta R\_s^2}{2} \tag{23}$$

under the assumption that the changes of the rotor and stator resistances are much slower than the changes of electromagnetic states, derivation of the Eq. (23) is:

V\_ <sup>1</sup> <sup>¼</sup> a4e1e3 <sup>þ</sup> a5ωe1e5 <sup>þ</sup> a6ife1*Δ*Rf <sup>þ</sup> a7ide1*Δ*Rs � k11e<sup>2</sup> <sup>1</sup> þ b4e2e3 þ b5ωe2e5þ <sup>þ</sup>*b*6*ife*2*ΔRf* <sup>þ</sup> *<sup>b</sup>*7*ide*1*ΔRs* � *<sup>k</sup>*22*e*<sup>2</sup> <sup>2</sup> <sup>þ</sup> *<sup>c</sup>*3*e*<sup>2</sup> <sup>3</sup> � *k*31*e*1*e*<sup>3</sup> � *k*32*e*2*e*<sup>3</sup> � *k*33*e*3*e*<sup>4</sup> �*k*34*e*3*e*<sup>6</sup> <sup>þ</sup> d4ωe3e4 <sup>þ</sup> d5e4e5 <sup>þ</sup> d6iqe4*Δ*Rs � k43e2 <sup>4</sup> <sup>þ</sup> f2e2 <sup>5</sup> � k51e1e5 �k52e2e5 � k53e4e5 � �*k*54*e*5*e*<sup>6</sup> þ *g*3*iqe*3*e*<sup>4</sup> þ *g*4*ide*5*e*<sup>6</sup> � *k*64*e* 2 <sup>6</sup> � *ΔRs* c\_ <sup>R</sup>*<sup>s</sup>* � *<sup>Δ</sup>Rf* <sup>c</sup>\_ R*f* (24)

If the rules for resistance adaptation are given as stated:

$$
\stackrel{\leftarrow}{\mathbf{R}\_f} = a\_6 i\_f e\_1 + b\_6 i\_f e\_2 \tag{25}
$$

$$
\dot{\hat{\mathbf{R}}\_s} = a\_7 i\_d e\_1 + b\_7 i\_d e\_2 + d\_6 i\_q e\_4 \tag{26}
$$

Derivation of the Lyapunov function in Eq. (24) becomes the same as the one given in Eq. (12), and stability of the observer Eq. (21) is proved.

#### **2.5 Load torque estimation**

To accomplish the SM speed tracking control, except from damper winding observer, load torque estimation is also necessary to be done. SM dynamic system given in Eq. (4) is expended with more state variables. One of them is rotor angle (*γ*) which is measured state variable. Another is load torque (*TL*) that is not measured. Although load torque dynamic is not known, according to reference [11] it could be added as a state variable with the first derivation equal to zero. Expended dynamic system is:

$$\begin{bmatrix} \dot{i}\_d \\ \dot{i}\_f \\ \dot{\nu}\_D \\ \dot{i}\_q \\ \dot{\nu}\_Q \\ \dot{\nu}\_Q \\ \dot{\nu}\_L \\ \dot{\nu}\_L \end{bmatrix} = \begin{bmatrix} a\_1 i\_d + a\_2 i\_f + a\_3 i\_q a + a\_4 \wp\_D + a\_5 \wp\_Q a + a\_6 \jmath\_d + a\_7 \wp\_f \\ b\_1 i\_d + b\_2 i\_f + b\_3 i\_q a + b\_4 \wp\_D + b\_5 \wp\_Q a + b\_6 \wp\_d + b\_7 \wp\_f \\ c\_1 i\_d + c\_2 i\_f + c\_3 \wp\_D \\ d\_1 i\_q + d\_2 i\_f + c\_4 \wp\_D + d\_3 \wp\_Q + d\_4 \wp\_Q + d\_5 \wp\_L \\ f\_1 i\_q + f\_2 \wp\_Q \\ g\_1 i\_d i\_q + g\_2 i\_f i\_q + g\_3 i\_q \wp\_D + g\_4 i\_d \wp\_Q + g\_5 T\_L \\ \omega \\ 0 \end{bmatrix} \tag{27}$$

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization… DOI: http://dx.doi.org/10.5772/intechopen.89420*

Observability analysis of the Eq. (27) is obtained according to the nonlinear system weak observability concept [9]. Observability criterion matrix *O1* (28) has been chosen:

*O*<sup>1</sup> ¼ *did dif diq dγ d Lf id* � � *d Lf iq* � � *d Lf γ* � � *d L*<sup>2</sup> *f γ* � � 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ¼ *∂L*<sup>0</sup> *f id ∂id ∂L*<sup>0</sup> *f id ∂if ∂L*<sup>0</sup> *f id ∂φ<sup>D</sup> ∂L*<sup>0</sup> *f id ∂iq ∂L*<sup>0</sup> *f id ∂φ<sup>Q</sup> ∂L*<sup>0</sup> *f id ∂ω ∂L*<sup>0</sup> *f id ∂γ ∂L*<sup>0</sup> *f id ∂TL ∂L*<sup>0</sup> *f if ∂id ∂L*<sup>0</sup> *f if ∂if ∂L*<sup>0</sup> *f if ∂φ<sup>D</sup> ∂L*<sup>0</sup> *f if ∂iq ∂L*<sup>0</sup> *f if ∂φ<sup>Q</sup> ∂L*<sup>0</sup> *f if ∂ω ∂L*<sup>0</sup> *f if ∂γ ∂L*<sup>0</sup> *f if ∂TL ∂L*<sup>0</sup> *f iq ∂id ∂L*<sup>0</sup> *f iq ∂if ∂L*<sup>0</sup> *f iq ∂φ<sup>D</sup> ∂L*<sup>0</sup> *f iq ∂iq ∂L*<sup>0</sup> *f iq ∂φ<sup>Q</sup> ∂L*<sup>0</sup> *f iq ∂ω ∂L*<sup>0</sup> *f iq ∂γ ∂L*<sup>0</sup> *f iq ∂TL ∂L*<sup>0</sup> *f γ ∂id ∂L*<sup>0</sup> *f γ ∂if ∂L*<sup>0</sup> *f γ ∂φ<sup>D</sup> ∂L*<sup>0</sup> *f γ ∂iq ∂L*<sup>0</sup> *f γ ∂φ<sup>Q</sup> ∂L*<sup>0</sup> *f γ ∂ω ∂L*<sup>0</sup> *f γ ∂γ ∂L*<sup>0</sup> *f γ ∂TL ∂L*<sup>1</sup> *f id ∂id ∂L*<sup>1</sup> *f id ∂if ∂L*<sup>1</sup> *f id ∂φ<sup>D</sup> ∂L*<sup>1</sup> *f id ∂iq ∂L*<sup>1</sup> *f id ∂φ<sup>Q</sup> ∂L*<sup>1</sup> *f id ∂ω ∂L*<sup>1</sup> *f id ∂γ ∂L*<sup>1</sup> *f id ∂TL ∂L*<sup>1</sup> *f iq ∂id ∂L*<sup>1</sup> *f iq ∂if ∂L*<sup>1</sup> *f iq ∂φ<sup>D</sup> ∂L*<sup>1</sup> *f iq ∂iq ∂L*<sup>1</sup> *f iq ∂φ<sup>Q</sup> ∂L*<sup>1</sup> *f iq ∂ω ∂L*<sup>0</sup> *f iq ∂γ ∂L*<sup>0</sup> *f iq ∂TL ∂L*<sup>1</sup> *f γ ∂id ∂L*<sup>1</sup> *f γ ∂if ∂L*<sup>1</sup> *f γ ∂φ<sup>D</sup> ∂L*<sup>1</sup> *f γ ∂iq ∂L*<sup>1</sup> *f γ ∂φ<sup>Q</sup> ∂L*<sup>1</sup> *f γ ∂ω ∂L*<sup>0</sup> *f γ ∂γ ∂L*<sup>0</sup> *f γ ∂TL ∂L*<sup>2</sup> *f γ ∂id ∂L*<sup>2</sup> *f γ ∂if ∂L*<sup>2</sup> *f γ ∂φ<sup>D</sup> ∂L*<sup>2</sup> *f γ ∂iq ∂L*<sup>2</sup> *f γ ∂φ<sup>Q</sup> ∂L*<sup>2</sup> *f γ ∂ω ∂L*<sup>2</sup> *f γ ∂γ ∂L*<sup>2</sup> *f γ ∂TL* 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (28)

After each matrix member of Eq. (28) is calculated [8], its determinant calculation gives:

$$\text{Det}(O\_1) = -\frac{\alpha^2 L\_{md} L\_Q \left(-L\_D L\_f L\_{mq} + L\_{md}^2 L\_{mq}\right)}{2\text{HL}\_D L\_Q \left(-L\_d L\_D L\_f + L\_d L\_{md}^2 + L\_D L\_{md}^2 + L\_f L\_{md}^2 - 2L\_{md}^3\right) \left(-L\_{mq}^2 + L\_q L\_Q\right)} - \frac{\alpha^2 L\_{md} L\_Q \left(-L\_{md} L\_Q \left(-L\_{md} L\_Q + L\_{md}^2 + L\_{md}^2 - 2L\_{md} L\_Q\right) + L\_{md} L\_Q\right)}{2\text{HL}\_D L\_Q \left(-L\_d L\_D L\_f + L\_d L\_{md}^2 + L\_D L\_{md}^2 + L\_f L\_{md}^2 + L\_f L\_{md}^2 - 2L\_{md}^3\right) \left(-L\_{mq}^2 + L\_q L\_Q\right)} \tag{29}$$

Observability criterion matrix *O*<sup>2</sup> has been chosen:

*O*<sup>2</sup> ¼ *did dif diq dγ d Lf id* � � *d Lf if* � � *d Lf γ* � � *d L*<sup>2</sup> *f γ* � � 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ¼ *∂L*<sup>0</sup> *f id ∂id ∂L*<sup>0</sup> *f id ∂if ∂L*<sup>0</sup> *f id ∂φ<sup>D</sup> ∂L*<sup>0</sup> *f id ∂iq ∂L*<sup>0</sup> *f id ∂φ<sup>Q</sup> ∂L*<sup>0</sup> *f id ∂ω ∂L*<sup>0</sup> *f id ∂γ ∂L*<sup>0</sup> *f id ∂TL ∂L*<sup>0</sup> *f if ∂id ∂L*<sup>0</sup> *f if ∂if ∂L*<sup>0</sup> *f if ∂φ<sup>D</sup> ∂L*<sup>0</sup> *f if ∂iq ∂L*<sup>0</sup> *f if ∂φ<sup>Q</sup> ∂L*<sup>0</sup> *f if ∂ω ∂L*<sup>0</sup> *f if ∂γ ∂L*<sup>0</sup> *f if ∂TL ∂L*<sup>0</sup> *f iq ∂id ∂L*<sup>0</sup> *f iq ∂if ∂L*<sup>0</sup> *f iq ∂φ<sup>D</sup> ∂L*<sup>0</sup> *f iq ∂iq ∂L*<sup>0</sup> *f iq ∂φ<sup>Q</sup> ∂L*<sup>0</sup> *f iq ∂ω ∂L*<sup>0</sup> *f iq ∂γ ∂L*<sup>0</sup> *f iq ∂TL ∂L*<sup>0</sup> *f γ ∂id ∂L*<sup>0</sup> *f γ ∂if ∂L*<sup>0</sup> *f γ ∂φ<sup>D</sup> ∂L*<sup>0</sup> *f γ ∂iq ∂L*<sup>0</sup> *f γ ∂φ<sup>Q</sup> ∂L*<sup>0</sup> *f γ ∂ω ∂L*<sup>0</sup> *f γ ∂γ ∂L*<sup>0</sup> *f γ ∂TL ∂L*<sup>1</sup> *f id ∂id ∂L*<sup>1</sup> *f id ∂if ∂L*<sup>1</sup> *f id ∂φ<sup>D</sup> ∂L*<sup>1</sup> *f id ∂iq ∂L*<sup>1</sup> *f id ∂φ<sup>Q</sup> ∂L*<sup>1</sup> *f id ∂ω ∂L*<sup>1</sup> *f id ∂γ ∂L*<sup>1</sup> *f id ∂TL ∂L*<sup>1</sup> *f if ∂id ∂L*<sup>1</sup> *f if ∂if ∂L*<sup>1</sup> *f if ∂φ<sup>D</sup> ∂L*<sup>1</sup> *f if ∂iq ∂L*<sup>1</sup> *f if ∂φ<sup>Q</sup> ∂L*<sup>1</sup> *f if ∂ω ∂L*<sup>0</sup> *f if ∂γ ∂L*<sup>0</sup> *f if ∂TL ∂L*<sup>1</sup> *f γ ∂id ∂L*<sup>1</sup> *f γ ∂if ∂L*<sup>1</sup> *f γ ∂φ<sup>D</sup> ∂L*<sup>1</sup> *f γ ∂iq ∂L*<sup>1</sup> *f γ ∂φ<sup>Q</sup> ∂L*<sup>1</sup> *f γ ∂ω ∂L*<sup>0</sup> *f γ ∂γ ∂L*<sup>0</sup> *f γ ∂TL ∂L*<sup>2</sup> *f γ ∂id ∂L*<sup>2</sup> *f γ ∂if ∂L*<sup>2</sup> *f γ ∂φ<sup>D</sup> ∂L*<sup>2</sup> *f γ ∂iq ∂L*<sup>2</sup> *f γ ∂φ<sup>Q</sup> ∂L*<sup>2</sup> *f γ ∂ω ∂L*<sup>2</sup> *f γ ∂γ ∂L*<sup>2</sup> *f γ ∂TL* 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (30)

Its error dynamics Eqs. (20) and (21) are obtained:

*a*4*e*<sup>3</sup> þ *a*5*ωe*<sup>5</sup> þ *a*6*ifΔRf* þ *a*7*idΔRs* � *k*11*e*<sup>1</sup> *b*4*e*<sup>3</sup> þ *b*5*ωe*<sup>5</sup> þ *b*6*ifΔRf* þ *b*7*idΔRs* � *k*22*e*<sup>1</sup> *c*3*e*<sup>3</sup> � *k*31*e*<sup>1</sup> � *k*32*e*<sup>2</sup> þ *k*33*e*<sup>4</sup> þ *k*34*e*<sup>6</sup> *d*4*ωe*<sup>3</sup> þ *d*5*e*<sup>5</sup> þ *d*6*iqΔRs* � *k*43*e*<sup>4</sup> *f* <sup>2</sup>*e*<sup>5</sup> � *k*51*e*<sup>1</sup> � *k*52*e*<sup>2</sup> � *k*53*e*<sup>4</sup> � *k*54*e*<sup>6</sup> *g*3*iqe*<sup>3</sup> þ *g*4*ide*<sup>5</sup> � *k*64*e*<sup>6</sup>

under the assumption that the changes of the rotor and stator resistances are much

<sup>2</sup> <sup>þ</sup> *<sup>c</sup>*3*e*<sup>2</sup>

Derivation of the Lyapunov function in Eq. (24) becomes the same as the one

To accomplish the SM speed tracking control, except from damper winding observer, load torque estimation is also necessary to be done. SM dynamic system given in Eq. (4) is expended with more state variables. One of them is rotor angle (*γ*) which is measured state variable. Another is load torque (*TL*) that is not measured. Although load torque dynamic is not known, according to reference [11] it could be added as a state variable with the first derivation equal to zero. Expended dynamic system is:

> *a*1*id* þ *a*2*if* þ *a*3*iqω* þ *a*4*ψ<sup>D</sup>* þ *a*5*ψ<sup>Q</sup> ω* þ *a*6*ud* þ *a*7*uf b*1*id* þ *b*2*if* þ *b*3*iqω* þ *b*4*ψ<sup>D</sup>* þ *b*5*ψ<sup>Q</sup> ω* þ *b*6*ud* þ *b*7*uf c*1*id* þ *c*2*if* þ *c*3*ψ<sup>D</sup> d*1*iq* þ *d*2*idω* þ *d*3*ifω* þ *d*4*ωψ<sup>D</sup>* þ *d*5*ψ<sup>Q</sup>* þ *d*6*uq f* <sup>1</sup>*iq* þ *f* <sup>2</sup>*ψ<sup>Q</sup> g*1*idiq* þ *g*2*if iq* þ *g*3*iqψ<sup>D</sup>* þ *g*4*idψ<sup>Q</sup>* þ *g*5*TL ω* 0

slower than the changes of electromagnetic states, derivation of the Eq. (23) is:

<sup>1</sup> <sup>¼</sup> a4e1e3 <sup>þ</sup> a5ωe1e5 <sup>þ</sup> a6ife1*Δ*Rf <sup>þ</sup> a7ide1*Δ*Rs � k11e<sup>2</sup>

If the rules for resistance adaptation are given as stated:

c\_

**2.5 Load torque estimation**

\_*id* \_*if ψ*\_ *D* \_*iq ψ*\_ *Q ω*\_ *γ*\_ *T*\_ *L*

¼

**152**

c\_

given in Eq. (12), and stability of the observer Eq. (21) is proved.

�*k*34*e*3*e*<sup>6</sup> <sup>þ</sup> d4ωe3e4 <sup>þ</sup> d5e4e5 <sup>þ</sup> d6iqe4*Δ*Rs � k43e2

�k52e2e5 � k53e4e5 � �*k*54*e*5*e*<sup>6</sup> þ *g*3*iqe*3*e*<sup>4</sup> þ *g*4*ide*5*e*<sup>6</sup> � *k*64*e*

*ΔR*<sup>2</sup> *f* 2 þ

*ΔR*<sup>2</sup> *s*

<sup>3</sup> � *k*31*e*1*e*<sup>3</sup> � *k*32*e*2*e*<sup>3</sup> � *k*33*e*3*e*<sup>4</sup>

R*<sup>f</sup>* ¼ *a*6*ife*<sup>1</sup> þ *b*6*ife*<sup>2</sup> (25)

R*<sup>s</sup>* ¼ *a*7*ide*<sup>1</sup> þ *b*7*ide*<sup>2</sup> þ *d*6*iqe*<sup>4</sup> (26)

<sup>4</sup> <sup>þ</sup> f2e2

2 <sup>6</sup> � *ΔRs*

<sup>2</sup> (23)

<sup>1</sup> þ b4e2e3 þ b5ωe2e5þ

<sup>5</sup> � k51e1e5

(27)

c\_ <sup>R</sup>*<sup>s</sup>* � *<sup>Δ</sup>Rf* <sup>c</sup>\_ R*f* (24)

(22)

e\_ 1 e\_ 2 e\_ 3 e\_ 4 e\_ 5 e\_ 6

¼

For the positive definite Lyapunov function:

<sup>þ</sup>*b*6*ife*2*ΔRf* <sup>þ</sup> *<sup>b</sup>*7*ide*1*ΔRs* � *<sup>k</sup>*22*e*<sup>2</sup>

*<sup>V</sup>*<sup>1</sup> <sup>¼</sup> *<sup>e</sup>*<sup>2</sup> 1 2 þ *e*2 2 2 þ *e*2 3 2 þ *e*2 4 2 þ *e*2 5 2 þ *e*2 6 2 þ

*Control Theory in Engineering*

V\_

After each matrix member of Eq. (30) is calculated [8], its determinant calculation gives:

$$\text{Det}(\text{Oz}) = \frac{\text{oL}\_{md}L\_{mq}R\_D}{2\text{HL}\_{\text{D}}L\_{\text{Q}}\left(-\text{L}\_{d}\text{L}\_{\text{D}}\text{L}\_{\text{f}} + \text{L}\_{d}\text{L}\_{md}\text{}^2 + \text{L}\_{\text{D}}\text{L}\_{md}^2 + \text{L}\_{\text{f}}\text{L}\_{md}^2 - 2\text{L}\_{md}^3\right)} \tag{31}$$

While observing both Eqs. (29) and (31):

$$\text{Let } \mathcal{O}\_1 \neq \mathbf{0} \text{, for } \boldsymbol{\alpha} = \mathbf{0} \text{, while } \text{Det } \mathcal{O}\_2 \neq \mathbf{0} \text{, for } \boldsymbol{\alpha} \neq \mathbf{0}$$

It is easy to see that: *Det* (*O*1) 6¼ 0 *U Det* (*O*2) 6¼ 0 = > *rank* {*O*} = 8.

Matrix *O* is full rank matrix and it could be concluded that the system in Eq. (27) is weakly locally observable. After it is concluded that the system is observable, a load torque estimator has to be made.

Using comparison between measured and calculated rotor speed values, a model reference adaptive system (MRAS) has been made.

Starting from the system that includes only rotor angle and rotor speed dynamics Eq. (32), the stability analysis of the proposed MRAS estimation has been made.

$$
\begin{bmatrix}
\dot{\mathcal{V}} \\
\dot{\boldsymbol{\alpha}}
\end{bmatrix} = \begin{bmatrix}
\boldsymbol{\alpha} \\
\boldsymbol{\mathcal{g}}\_{\mathfrak{F}} \boldsymbol{T}\_{L} + \frac{\mathbf{1}}{2H} \boldsymbol{T}\_{\mathfrak{e}}
\end{bmatrix} \tag{32}
$$

Expression in Eq. (36) can be noted as:

*DOI: http://dx.doi.org/10.5772/intechopen.89420*

*ε*\_ *γ ε*\_ *ω*

the condition:

satisfied if:

**3. Control law**

appeared.

where:

**155**

when *t* ≥ 0,*γ<sup>0</sup>* ≥ 0.

<sup>¼</sup> 0 1 0 0 � � *εγ*

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization…*

ð*t*

0

ð*t*

*εγεω* � �

ð*t*

0

c*L*ð Þþ 0 *kp εω*

each positive value of the proportional *kp* and integral *ki* coefficients.

0

T c*<sup>L</sup>* ¼ T

square of stator magnetic flux: *<sup>ω</sup>*c,ψb<sup>2</sup>

\_

Equation (42) could be noted as:

b

until in its expressions an input variable appears.

With further expansion of the Eq. (38), stability condition becomes:

*εωg*<sup>5</sup> *TL* � *T*

*εω*

According to the Popov stability criterion, stability will be proved by achieving

½ � *<sup>ε</sup> <sup>T</sup>*½ � <sup>W</sup> *dt* <sup>≥</sup> � *<sup>γ</sup>*<sup>2</sup>

0 *g*<sup>5</sup> *TL* � *T*

c*L* � � " #<sup>≥</sup> � *<sup>γ</sup>*<sup>2</sup>

c*L* � �<sup>≥</sup> � *<sup>γ</sup>*<sup>2</sup>

According to the literature reference [12] it is obvious that inequality Eq. (40) is

1 2*H* � �

According to [12] stability of the load torque estimation Eq. (41) is achieved for

Nonlinear control system is made by feedback linearization technique. It is not possible to obtain exact linearization for the SM system, so partial input output linearization has been applied. Using Lie algebra, the decoupled control system has been made. Control demand is to make a tracking of two outputs: rotor speed, and

> *<sup>d</sup>* <sup>þ</sup> <sup>ψ</sup>b<sup>2</sup> *q*.

*<sup>ω</sup>*<sup>b</sup> <sup>¼</sup> g1*idiq* <sup>þ</sup> *<sup>g</sup>*2*if iq* <sup>þ</sup> *<sup>g</sup>*3*iqφ*c*<sup>D</sup>* <sup>þ</sup> *<sup>g</sup>*4*idφ*c*<sup>Q</sup>* <sup>þ</sup> *<sup>g</sup>*5*TL* <sup>þ</sup> *<sup>k</sup>*64*e*<sup>6</sup> (42)

*<sup>ω</sup>*<sup>b</sup> <sup>¼</sup> <sup>b</sup>*h*<sup>11</sup> <sup>þ</sup> *<sup>g</sup>*5*TL* <sup>þ</sup> *<sup>Δ</sup>* (43)

h11 <sup>¼</sup> g1*idiq* <sup>þ</sup> *<sup>g</sup>*2*if iq* <sup>þ</sup> *<sup>g</sup>*3*iqφ*c*<sup>D</sup>* <sup>þ</sup> *<sup>g</sup>*4*idφ*c*<sup>Q</sup>* (44)

*<sup>s</sup>* <sup>¼</sup> <sup>ψ</sup>b<sup>2</sup>

\_

According to the feedback linearization technique, output should be derived

After the first derivation of the rotor speed Eq. (42), output variable has not

þ *ki* ð*t* 0 *εω* 1 2*H*

" #

� <sup>0</sup> *g*<sup>5</sup> *TL* � *T*

c*L*

� � " # (37)

<sup>0</sup> (38)

<sup>0</sup> (39)

<sup>0</sup> (40)

� �*dt* (41)

" #

where *Te* states for electromagnetic torque. Then, an observer is proposed:

$$
\begin{bmatrix}
\dot{\hat{\mathcal{V}}} \\
\dot{\hat{\mathcal{w}}}
\end{bmatrix} = \begin{bmatrix}
\hat{\mathcal{w}} \\
\mathbf{g}\_{\mathfrak{F}} \widehat{T\_{L}} + \frac{1}{2H} T\_{\mathfrak{e}}
\end{bmatrix} \tag{33}
$$

Both, reference Eq. (32) and observed Eq. (33) systems can be noted in the form of linear systems as is given respectively in Eqs. (34) and (35):

$$\mathbf{E}\left[\dot{\mathbf{X}}\right] = \mathbf{[A]}[\mathbf{X}] + \mathbf{[B]}[\mathbf{U}] + \mathbf{[D]};\tag{34}$$

$$\left[\dot{\hat{X}}\right] = \left[\mathbf{A}\right]\left[\hat{X}\right] + \left[\mathbf{B}\right]\left[U\right] + \left[\hat{\mathbf{D}}\right];\tag{35}$$

where:

$$\mathbf{A} = \begin{bmatrix} \mathbf{0} & \mathbf{1} \\ \mathbf{0} & \mathbf{0} \end{bmatrix}; \quad \mathbf{B}\mathbf{U} = \begin{bmatrix} \mathbf{0} \\ \mathbf{1} \\ \overline{2H} \end{bmatrix}; \quad D = \begin{bmatrix} \mathbf{0} \\ \mathbf{g}\_{\mathfrak{F}} T\_{L.} \end{bmatrix}$$

Error dynamics is obtained by Eqs. (34) and (35):

$$\mathbf{[\dot{e}]} = [\mathbf{A}][\mathbf{e}] - [\mathbf{W}] \tag{36}$$

where:

$$\boldsymbol{\varepsilon} = \begin{bmatrix} \boldsymbol{\varepsilon}\_{\boldsymbol{\gamma}} \\ \boldsymbol{\varepsilon}\_{\boldsymbol{\alpha}} \end{bmatrix} = \begin{bmatrix} \boldsymbol{\chi} - \widehat{\boldsymbol{\gamma}} \\ \boldsymbol{\alpha} - \widehat{\boldsymbol{\alpha}} \end{bmatrix}; \quad \boldsymbol{W} = \begin{bmatrix} \mathbf{0} \\ \mathbf{g}\_{\boldsymbol{\xi}} \end{bmatrix} \begin{pmatrix} \boldsymbol{T}\_{L} - \widehat{\boldsymbol{T}\_{L}} \end{pmatrix}.$$

**154**

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization… DOI: http://dx.doi.org/10.5772/intechopen.89420*

Expression in Eq. (36) can be noted as:

$$
\begin{bmatrix}
\boldsymbol{\varepsilon}\_{\gamma}^{\cdot} \\
\boldsymbol{\varepsilon}\_{\boldsymbol{\alpha}}^{\cdot}
\end{bmatrix} = \begin{bmatrix}
\mathbf{0} & \mathbf{1} \\
\mathbf{0} & \mathbf{0}
\end{bmatrix} \begin{bmatrix}
\boldsymbol{\varepsilon}\_{\gamma} \\
\boldsymbol{\varepsilon}\_{\boldsymbol{\alpha}}
\end{bmatrix} - \begin{bmatrix}
\mathbf{0} \\
\mathbf{g}\_{5} \left(\boldsymbol{T}\_{L} - \widehat{\boldsymbol{T}\_{L}}\right)
\end{bmatrix} \tag{37}
$$

According to the Popov stability criterion, stability will be proved by achieving the condition:

$$\int\_{0}^{t} \left[\boldsymbol{\varepsilon}\right]^{T} \left[\mathbf{W}\right] dt \geq -\chi\_{0}^{2} \tag{38}$$

when *t* ≥ 0,*γ<sup>0</sup>* ≥ 0.

After each matrix member of Eq. (30) is calculated [8], its determinant calcula-

ω*LmdLmqRD*

*Det O*<sup>1</sup> 6¼ 0, for *ω* ¼ 0, while *Det O*<sup>2</sup> 6¼ 0, for *ω* 6¼ 0

Matrix *O* is full rank matrix and it could be concluded that the system in Eq. (27) is weakly locally observable. After it is concluded that the system is observable, a

Using comparison between measured and calculated rotor speed values, a model

*ω*

*ω*b

" #

1 2*H Te*

þ ½ � B ½ �þ *U* Db

0 1 <sup>2</sup>*<sup>H</sup> Te*

; *<sup>W</sup>* <sup>¼</sup> <sup>0</sup>

*g*5 � �

" #

*<sup>X</sup>*\_ � � <sup>¼</sup> ½ � <sup>A</sup> ½ �þ *<sup>X</sup>* ½ � <sup>B</sup> ½ �þ *<sup>U</sup>* ½ � <sup>D</sup> ; (34)

h i

; *<sup>D</sup>* <sup>¼</sup> <sup>0</sup>

½ �¼ *ε*\_ ½ � A ½ �� *ε* ½ � W (36)

*TL* � *T* c*<sup>L</sup>* � �

*g*5*TL:* � �

; (35)

" #

1 2*H Te*

*g*5*TL* þ

It is easy to see that: *Det* (*O*1) 6¼ 0 *U Det* (*O*2) 6¼ 0 = > *rank* {*O*} = 8.

Starting from the system that includes only rotor angle and rotor speed dynamics Eq. (32), the stability analysis of the proposed MRAS estimation has

<sup>2</sup> <sup>þ</sup> *LDLmd*

<sup>3</sup> � � (31)

<sup>2</sup> <sup>þ</sup> *LfLmd*

<sup>2</sup> � <sup>2</sup>*Lmd*

(32)

(33)

2*HLDLQ* �*LdLDLf* þ *LdLmd*

While observing both Eqs. (29) and (31):

reference adaptive system (MRAS) has been made.

where *Te* states for electromagnetic torque.

*γ*\_ *ω*\_ � �

> \_ *γ*b \_ *ω*b

" #

of linear systems as is given respectively in Eqs. (34) and (35):

\_ *X*b h i

Error dynamics is obtained by Eqs. (34) and (35):

*<sup>A</sup>* <sup>¼</sup> 0 1 0 0 � �

*<sup>ε</sup>* <sup>¼</sup> *εγ εω*

" #

¼

¼

¼ ½ � A *X*b h i

; BU ¼

<sup>¼</sup> *<sup>γ</sup>* � *<sup>γ</sup>*<sup>b</sup> *<sup>ω</sup>* � *<sup>ω</sup>*<sup>b</sup> � �

*g*5*T* c*<sup>L</sup>* þ

Both, reference Eq. (32) and observed Eq. (33) systems can be noted in the form

load torque estimator has to be made.

Then, an observer is proposed:

tion gives:

been made.

where:

where:

**154**

*Det O*ð Þ¼ <sup>2</sup>

*Control Theory in Engineering*

With further expansion of the Eq. (38), stability condition becomes:

$$\int\_{0}^{t} \left[ \varepsilon\_{\gamma} \varepsilon\_{\alpha} \right] \left[ \begin{matrix} \mathbf{0} \\ \mathbf{g}\_{5} \left( T\_{L} - \widehat{T\_{L}} \right) \end{matrix} \right] \geq -\gamma\_{0}^{2} \tag{39}$$

$$\int\_{0}^{t} \varepsilon\_{a} \mathbf{g}\_{\mathfrak{F}} \left( T\_{L} - \widehat{T\_{L}} \right) \geq -\gamma\_{0}^{2} \tag{40}$$

According to the literature reference [12] it is obvious that inequality Eq. (40) is satisfied if:

$$\widehat{\mathbf{T}\_L} = \widehat{\mathbf{T}\_L}(\mathbf{0}) + k\_p \left[ \varepsilon\_o \frac{\mathbf{1}}{2H} \right] + k\_i \int\_0^t \left[ \varepsilon\_o \frac{\mathbf{1}}{2H} \right] dt \tag{41}$$

According to [12] stability of the load torque estimation Eq. (41) is achieved for each positive value of the proportional *kp* and integral *ki* coefficients.

## **3. Control law**

Nonlinear control system is made by feedback linearization technique. It is not possible to obtain exact linearization for the SM system, so partial input output linearization has been applied. Using Lie algebra, the decoupled control system has been made. Control demand is to make a tracking of two outputs: rotor speed, and square of stator magnetic flux: *<sup>ω</sup>*c,ψb<sup>2</sup> *<sup>s</sup>* <sup>¼</sup> <sup>ψ</sup>b<sup>2</sup> *<sup>d</sup>* <sup>þ</sup> <sup>ψ</sup>b<sup>2</sup> *q*.

According to the feedback linearization technique, output should be derived until in its expressions an input variable appears.

After the first derivation of the rotor speed Eq. (42), output variable has not appeared.

$$\dot{\widehat{\boldsymbol{\alpha}}} = \mathbf{g}\_1 \mathbf{i}\_d \mathbf{i}\_q + \mathbf{g}\_2 \mathbf{i}\_f \mathbf{i}\_q + \mathbf{g}\_3 \mathbf{i}\_q \widehat{\boldsymbol{\rho}\_D} + \mathbf{g}\_4 \mathbf{i}\_d \widehat{\boldsymbol{\rho}\_Q} + \mathbf{g}\_5 T\_L + k\_{64} \mathbf{e}\_6 \tag{42}$$

Equation (42) could be noted as:

$$
\dot{\hat{\phi}} = \hat{h}\_{11} + \mathbf{g}\_{\mathfrak{F}} T\_L + \Delta \tag{43}
$$

where:

$$\mathbf{h\_{11}} = \mathbf{g\_1}i\_d i\_q + \mathbf{g\_2}i\_f i\_q + \mathbf{g\_3}i\_q \widehat{\rho\_D} + \mathbf{g\_4}i\_d \widehat{\rho\_Q} \tag{44}$$

*Control Theory in Engineering*

$$
\Delta = k\_{\text{64}} \varepsilon\_4 \tag{45}
$$

c\_ h11 c\_ *ψ*2 *s*

2 4

*DOI: http://dx.doi.org/10.5772/intechopen.89420*

e\_ 7 e\_ 8 e\_ 9

expression is given:

Lyapunov function:

� \_ *h*11*ref*

� \_ *<sup>ψ</sup>*<sup>2</sup> *sref*

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization…*

3

In Eq. (55) error dynamics of *e8* and *e9* are obtained. It is left to obtain error dynamic of the *e7*. Using Eqs. (43) and (52) error dynamic of *e7* is obtained and its

Using Eqs. (55) and (57) the complete error dynamics system is obtained:

From the Eq. (58) it is easily seen that convergence of the rotor speed (electromagnetic torque) is independent of convergence of the magnetic flux. It could be

Stability of the control system can be proved by the following positive definite

*ω*\_ � *ω*\_*ref*

*<sup>V</sup>* <sup>¼</sup> *<sup>e</sup>*<sup>2</sup> 7 2 þ *e*2 8 2 þ *e*2 9

Using Eq. (58), derivation Eq. (60) could be expanded as given:

*<sup>V</sup>*\_ ¼ �*kp*0*<sup>e</sup>*

**4. Comparison of nonlinear and linear control systems**

**4.1 Control law for linear control system**

for current components control.

**157**

2 <sup>7</sup> � *kp*1*e* 2

> 2 <sup>7</sup> � *kp*1*e* 2 <sup>8</sup> � *kp*2*e* 2

If the coefficients *kp0*, *kp1* and *kp2* are positive, derivation of the Lyapunov function Eq. (60) is negative definite and stability of the control law is proved.

Linear control system is based on stator field orientation control principle. It is cascaded control system with inner and outer control loops. Outer control loops are made for rotor speed and magnetic flux control, while inner control loops are made

At first, current components control in inner loops will be defined.

*<sup>V</sup>*\_ <sup>¼</sup> *<sup>e</sup>*7*e*<sup>8</sup> � *kp*0*<sup>e</sup>*

c\_ h11 � \_ *h*11*ref*

said that completely decoupled control system is achieved.

Derivation of the Eq. (59) Lyapunov function is:

c\_ *ψ*2 *<sup>s</sup>* � \_ *<sup>ψ</sup>*<sup>2</sup> *sref*

<sup>5</sup> <sup>¼</sup> �*kp*1*e*<sup>8</sup> � *<sup>e</sup>*<sup>7</sup> �*kp*2*e*<sup>9</sup>

*ω*\_ � *ω*\_*ref* ¼ b*h*<sup>11</sup> þ *g*5*TL* þ *Δ* � *h*11*ref* � *g*5*TL* � *kp*0*e*<sup>7</sup> � *Δ* (56)

*ω*\_ � *ω*\_*ref* ¼ *e*<sup>8</sup> � *kp*0*e*<sup>7</sup> (57)

*e*<sup>8</sup> � *kp*0*e*<sup>7</sup> �*kp*1*e*<sup>8</sup> � *e*<sup>7</sup> �*kp*2*e*<sup>9</sup>

*<sup>V</sup>*\_ <sup>¼</sup> *<sup>e</sup>*7*e*\_<sup>7</sup> <sup>þ</sup> *<sup>e</sup>*8*e*\_<sup>8</sup> <sup>þ</sup> *<sup>e</sup>*9*e*\_<sup>9</sup> (60)

2

<sup>8</sup> � *e*7*e*<sup>8</sup> � *kp*2*e*

<sup>2</sup> (59)

<sup>9</sup> (61)

<sup>9</sup> (62)

(58)

� � (55)

Since the output variable has not appeared yet, derivation of the additional output variable *h*<sup>11</sup> has been done. After the derivation of *h*11, that is actually an electromagnetic torque, output variables appear. Derivation of *h*<sup>11</sup> is given in Eq. (46), and derivation of the second output variable in Eq. (47).

$$
\dot{\hat{h}\_{11}} = \mathbf{L}\_{\hat{f}} \hat{h}\_{11} + \mathbf{L}\_{\text{g1}} \hat{h}\_{11} \mathbf{u}\_d + \mathbf{L}\_{\text{g2}} \hat{h}\_{11} \mathbf{u}\_q \tag{46}
$$

$$\widehat{\boldsymbol{\psi}^{2}}\_{s} = \mathbf{L}\_{\widehat{f}} \widehat{\boldsymbol{\psi}^{2}}\_{s} + \mathbf{L}\_{\mathfrak{g}1} \widehat{\boldsymbol{\varphi^{2}\_{s}}}\_{s} \mathbf{u}\_{d} + \mathbf{L}\_{\mathfrak{g}2} \widehat{\boldsymbol{\psi^{2}\_{s}}}\_{s} \mathbf{u}\_{q} \tag{47}$$

Dynamical system of the output variables is:

$$\begin{bmatrix} \dot{\hat{\boldsymbol{\alpha}}} \\ \dot{\hat{\boldsymbol{\alpha}}} \\ \dot{\hat{\boldsymbol{\mu}}} \\ \dot{\hat{\boldsymbol{\mu}}}^{2} \end{bmatrix} = \begin{bmatrix} \hat{h}\_{11} + \mathbf{g}\_{\boldsymbol{\xi}} T\_{L} + \boldsymbol{\Delta} \\ \mathbf{L}\_{\hat{f}} \hat{h}\_{11} + \mathbf{L}\_{\mathbf{g}1} \hat{h}\_{11} \mathbf{u}\_{d} + \mathbf{L}\_{\mathbf{g}2} \hat{h}\_{11} \mathbf{u}\_{q} \\ \mathbf{L}\_{\hat{f}} \widehat{\boldsymbol{\mu}^{2}}\_{s} + \mathbf{L}\_{\mathbf{g}1} \widehat{\boldsymbol{\mu}^{2}\_{s}} \mathbf{u}\_{d} + \mathbf{L}\_{\mathbf{g}2} \widehat{\boldsymbol{\mu}^{2}\_{s}} \mathbf{u}\_{q} \end{bmatrix} \tag{48}$$

It is possible to obtain the control of the last two variables, as stated:

$$
\begin{bmatrix}
\dot{\hat{\mathbf{h}}\_{11}} \\
\dot{\hat{\mathbf{w}}\_{s}^{2}}
\end{bmatrix} = \begin{bmatrix}
\mathbf{L}\_{\hat{f}} \hat{\boldsymbol{h}}\_{11} \\
\mathbf{L}\_{\hat{f}} \widehat{\boldsymbol{\nu}^{2}\_{s}}
\end{bmatrix} + G \begin{bmatrix}
\mathbf{u}\_{d} \\
\mathbf{u}\_{q}
\end{bmatrix} \tag{49}
$$

where *G* is decoupling matrix:

$$\mathbf{G} = \begin{bmatrix} \mathbf{L\_{g1}}\widehat{h}\_{11} & \mathbf{L\_{g2}}\widehat{h}\_{11} \\ \mathbf{L\_{g1}}\widehat{\boldsymbol{\nu\_s^2}} & \mathbf{L\_{g2}}\widehat{\boldsymbol{\nu\_s^2}} \end{bmatrix} \tag{50}$$

Now it is possible to define the control law:

$$\mathbf{u}\begin{bmatrix} \mathbf{u}\_d\\\mathbf{u}\_q \end{bmatrix} = \mathbf{G}^{-1} \begin{bmatrix} -\mathbf{L}\_j \widehat{h}\_{11} - k\_{p1} \mathbf{e}\_8 + \dot{h}\_{11ref} - \mathbf{e}\_7\\ -\mathbf{L}\_j \widehat{\boldsymbol{\nu}\prime}^2 - k\_{p2} \mathbf{e}\_9 + \boldsymbol{\nu}\_{sr\ell}^2 \end{bmatrix} \tag{51}$$

where difference form the reference values are:

$$\mathbf{e}\_{\mathcal{T}} = \widehat{\mathbf{o}} - \boldsymbol{\alpha}\_{\text{ref}}; \; \mathbf{e}\_{8} = \widehat{\mathbf{h}}\_{11} - \mathbf{h}\_{11\text{ref}}; \quad \mathbf{e}\_{9} = \widehat{\boldsymbol{\psi}\_{\text{s}}^{2}} - \boldsymbol{\psi}\_{\text{sref}}^{2}$$

If *h*11*ref* is defined given:

$$h\_{11ref} = \dot{\alpha}\_{r\text{f}} - \mathbf{g}\_{\text{5}}T\_L - k\_{p0}\mathbf{c}\_{7} - \Delta \tag{52}$$

Using (51) and (52), further expansion of Eq. (49) gives:

$$
\begin{bmatrix}
\dot{\widehat{\mathbf{h}\_{11}}} \\
\dot{\widehat{\mathbf{w}\_{\mathcal{I}}^{2}}}
\end{bmatrix} = \begin{bmatrix}
\mathbf{L}\_{\hat{f}} \widehat{\boldsymbol{h}\_{11}} \\
\mathbf{L}\_{\hat{f}} \widehat{\boldsymbol{\mu}\_{\mathcal{I}}^{2}}
\end{bmatrix} + GG^{-1} \begin{bmatrix}
\end{bmatrix} \tag{53}
$$

$$
\begin{bmatrix}
\dot{\hat{\mathbf{h}}\_{11}^{\cdot}} \\
\dot{\hat{\mathbf{w}}\_{s}^{2}}
\end{bmatrix} = \begin{bmatrix}
\mathbf{L}\_{\hat{f}} \widehat{h}\_{11} - \mathbf{L}\_{\hat{f}} \widehat{h}\_{11} - k\_{p1} \mathbf{e}\_{8} + \dot{h}\_{11ref} - \mathbf{c}\_{7} \\
\mathbf{L}\_{\hat{f}} \widehat{\boldsymbol{\mu}^{2}\_{s}} - \mathbf{L}\_{\hat{f}} \widehat{\boldsymbol{\mu}^{2}\_{s}} - k\_{p2} \mathbf{e}\_{9} + \boldsymbol{\mu}^{2}\_{sref}
\end{bmatrix} \tag{54}
$$

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization… DOI: http://dx.doi.org/10.5772/intechopen.89420*

$$
\begin{bmatrix}
\dot{\hat{h}\_{11}} - \dot{h}\_{11ref} \\
\dot{\hat{\nu\_s^2}} - \dot{\nu\_{srf}^2}
\end{bmatrix} = \begin{bmatrix}
\end{bmatrix} \tag{55}
$$

In Eq. (55) error dynamics of *e8* and *e9* are obtained. It is left to obtain error dynamic of the *e7*. Using Eqs. (43) and (52) error dynamic of *e7* is obtained and its expression is given:

$$
\dot{o} - \dot{o}\_{\text{ref}} = h\_{11} + \mathbf{g}\_{\text{\textdegree}}T\_L + \Delta - h\_{11\text{ref}} - \mathbf{g}\_{\text{\textdegree}}T\_L - k\_{p0}e\_{\text{\textdegree}} - \Delta\tag{56}
$$

$$
\dot{a} - \dot{a}\_{\text{ref}} = \mathbf{e}\_8 - k\_{p0}\mathbf{e}\_7 \tag{57}
$$

Using Eqs. (55) and (57) the complete error dynamics system is obtained:

$$
\begin{bmatrix}
\dot{\mathbf{e}}\gamma\\\dot{\mathbf{e}}\_8\\\dot{\mathbf{e}}\_9
\end{bmatrix} = \begin{bmatrix}
\dot{\boldsymbol{\alpha}} - \dot{\boldsymbol{\alpha}}\_{rf} \\\
\dot{\mathbf{h}}\_{11} - \dot{\boldsymbol{h}}\_{11ref} \\\
\dot{\boldsymbol{\mu}}\_s^2 - \boldsymbol{\mu}\_{rf}^{\dot{\boldsymbol{\gamma}}}
\end{bmatrix} = \begin{bmatrix}
\boldsymbol{e}\_8 - \boldsymbol{k}\_{p1}\boldsymbol{e}\_7 \\\
\end{bmatrix} \tag{58}
$$

From the Eq. (58) it is easily seen that convergence of the rotor speed (electromagnetic torque) is independent of convergence of the magnetic flux. It could be said that completely decoupled control system is achieved.

Stability of the control system can be proved by the following positive definite Lyapunov function:

$$V = \frac{e\_7^2}{2} + \frac{e\_8^2}{2} + \frac{e\_9^2}{2} \tag{59}$$

Derivation of the Eq. (59) Lyapunov function is:

$$
\dot{V} = \mathbf{e}\_7 \dot{\mathbf{e}}\_7 + \mathbf{e}\_8 \dot{\mathbf{e}}\_8 + \mathbf{e}\_9 \dot{\mathbf{e}}\_9 \tag{60}
$$

Using Eq. (58), derivation Eq. (60) could be expanded as given:

$$
\dot{V} = e\_7 e\_8 - k\_{p0} e\_7^2 - k\_{p1} e\_8^2 - e\_7 e\_8 - k\_{p2} e\_9^2 \tag{61}
$$

$$
\dot{V} = -k\_{p0}e\_7^2 - k\_{p1}e\_8^2 - k\_{p2}e\_9^2 \tag{62}
$$

If the coefficients *kp0*, *kp1* and *kp2* are positive, derivation of the Lyapunov function Eq. (60) is negative definite and stability of the control law is proved.

### **4. Comparison of nonlinear and linear control systems**

#### **4.1 Control law for linear control system**

Linear control system is based on stator field orientation control principle. It is cascaded control system with inner and outer control loops. Outer control loops are made for rotor speed and magnetic flux control, while inner control loops are made for current components control.

At first, current components control in inner loops will be defined.

*Δ* ¼ *k*64*e*<sup>4</sup> (45)

b*h*<sup>11</sup> þ Lg1b*h*11u*<sup>d</sup>* þ Lg2b*h*11u*<sup>q</sup>* (46)

*<sup>s</sup>* u*<sup>q</sup>* (47)

(48)

(49)

(50)

c2

*<sup>s</sup>* u*<sup>d</sup>* þ Lg2*ψ*

b*h*<sup>11</sup> þ *g*5*TL* þ *Δ*

b*h*<sup>11</sup> þ Lg1b*h*11u*<sup>d</sup>* þ Lg2b*h*11u*<sup>q</sup>*

*<sup>s</sup>* u*<sup>d</sup>* þ Lg2*ψ*

u*d* u*q*

" #

c2 *<sup>s</sup> uq*

*h*11*ref* � *e*<sup>7</sup>

3

5 (51)

*sref*

c2 <sup>s</sup> � <sup>ψ</sup><sup>2</sup> sref

*h*11*ref* � *e*<sup>7</sup>

3

5 (53)

5 (54)

*sref*

3

*h*11*ref* � *e*<sup>7</sup>

*sref*

*h*11*ref* ¼ *ω*\_*ref* � *g*5*TL* � *kp*0*e*<sup>7</sup> � *Δ* (52)

*<sup>s</sup>* � *kp*2*e*<sup>9</sup> <sup>þ</sup> \_ *<sup>ψ</sup>*<sup>2</sup>

*<sup>s</sup>* � *kp*2*e*<sup>9</sup> <sup>þ</sup> \_ *<sup>ψ</sup>*<sup>2</sup>

<sup>b</sup>*h*<sup>11</sup> � *kp*1*e*<sup>8</sup> <sup>þ</sup> \_

<sup>b</sup>*h*<sup>11</sup> � *kp*1*e*<sup>8</sup> <sup>þ</sup> \_

�L^*fψ* c2

Since the output variable has not appeared yet, derivation of the additional output variable *h*<sup>11</sup> has been done. After the derivation of *h*11, that is actually an electromagnetic torque, output variables appear. Derivation of *h*<sup>11</sup> is given in

Eq. (46), and derivation of the second output variable in Eq. (47).

L^*f*

c\_ h11 c\_ *ψ*2 *s*

3 5 ¼

<sup>¼</sup> *<sup>G</sup>*�<sup>1</sup> �L^*<sup>f</sup>*

2 4

2 4

Now it is possible to define the control law:

where difference form the reference values are:

e7 <sup>¼</sup> <sup>ω</sup><sup>b</sup> � <sup>ω</sup>ref; e8 <sup>¼</sup> <sup>b</sup>

3

Using (51) and (52), further expansion of Eq. (49) gives:

L^*f*

2 4

<sup>5</sup> <sup>þ</sup> *GG*�<sup>1</sup> �L^*<sup>f</sup>*

2 4

b*h*<sup>11</sup> � L^*<sup>f</sup>*

L^*fψ* c2 *<sup>s</sup>* � L^*fψ* c2

u*d* u*q* � � L^*fψ* c2 *<sup>s</sup>* þ Lg1*ψ* c2

It is possible to obtain the control of the last two variables, as stated:

2 4

L^*f* b*h*11 3 5 þ *G*

L^*fψ* c2 *s*

*<sup>G</sup>* <sup>¼</sup> Lg1b*h*<sup>11</sup> Lg2b*h*<sup>11</sup> Lg1*ψ* c2 *<sup>s</sup>* Lg2*ψ* c2 *s*

> �L^*fψ* c2

" #

<sup>b</sup>*h*<sup>11</sup> � *kp*1*e*<sup>8</sup> <sup>þ</sup> \_

h11 � h11ref; e9 ¼ ψ

*<sup>s</sup>* � *kp*2*e*<sup>9</sup> <sup>þ</sup> \_ *<sup>ψ</sup>*<sup>2</sup>

\_ b*h*<sup>11</sup> ¼ L^*<sup>f</sup>*

c\_ *ψ*2 *<sup>s</sup>* ¼ L^*fψ* c2 *<sup>s</sup>* þ Lg1*ψ* c2

Dynamical system of the output variables is:

\_ *ω*b c\_ h11 c\_ *ψ*2 *s*

*Control Theory in Engineering*

where *G* is decoupling matrix:

If *h*11*ref* is defined given:

c\_ h11 c\_ *ψ*<sup>2</sup> *s*

3 5 ¼

L^*f* b*h*11

2 4

2 4 L^*fψ* c2 *s*

> 3 5 ¼

c\_ h11 c\_ *ψ*2 *s*

2 4

**156**

If dynamics of the damper winding are neglected, equations of the SM system could be simplified. Then, the equation in the stator *d*-axis is:

$$
\mu\_d = R\_\varepsilon i\_d + \frac{di\_d}{dt} \left( L\_d - \frac{L\_{md}^2}{L\_f} \right) + \varepsilon\_d \tag{63}
$$

*CPI*ðÞ¼ *s KP* þ

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization…*

Tuning of the PI controllers is done according to Internal model control

*acc* <sup>¼</sup> ln 9ð Þ *tr*,*cc*

and *tr,cc* is stator current response time that is for most of the industrial applica-

*CPI*ð Þ*s G s*ð Þ

*acc s* þ *acc*

> *acc s* þ *acc*

1

<sup>1</sup> <sup>þ</sup> *P s*ð Þ*C s*ð Þ (76)

*Js* (75)

The transfer function of the current control closed loop *Gcc(s)* is:

*Gcc*,*cl*ðÞ¼ *s*

complete control loop for the rotor speed is given in **Figure 1**. Open loop transfer function of the rotor speed control is:

*Gω*,*ol*ðÞ¼ *s*

factor, values of *Kp<sup>ω</sup>* should not exceed 14.

almost 60 degrees for *Kp<sup>ω</sup>* higher than 10.

**Figure 1.**

**159**

*Control loop of the rotor speed.*

Load sensitivity transfer function is obtained:

*Gcc*,*cl*ðÞ¼ *s*

Outer control loops will be also controlled by PI controllers. In that case, the

*Kpω*ð Þ *Tiωs* þ 1 *Tiωs*

According to the Eq. (75), stability analysis of the SM1 speed control loop has been done. In **Figure 2**, root locus diagram is given. It shows that, due to damping

According to the Bode diagram, given in **Figure 3**, the stability phase margin is

According to **Figure 1**, torque load could be analyzed as an input disturbance.

*P s*ð Þ

*Gdy*ðÞ¼ *s*

reference [13] (IMC) method as is given:

*DOI: http://dx.doi.org/10.5772/intechopen.89420*

tions [14] set at 5 ms.

where *acc* for the first order system is defined as:

Outer loop for speed control is then analyzed.

After some algebra Eq. (73) could written as:

*KI s*

*KP* ¼ *accLcc*,*<sup>d</sup>* (70) *Ki* ¼ *accRs* (71)

<sup>1</sup> <sup>þ</sup> *CPI*ð Þ*<sup>s</sup> G s*ð Þ (73)

(69)

(72)

(74)

where

$$
\sigma\_d = \frac{L\_{md}}{L\_f} \left( -i\_f R\_f + \mu\_f \right) - \rho\_q \text{o} \tag{64}
$$

If the additional variable *<sup>u</sup>*c*<sup>d</sup>* <sup>¼</sup> *ud* � *ed* is introduced, Eq. (63) becomes linear differential equation of the first order for the current component*id*:

$$
\widehat{u\_d} = R\_i i\_d + \frac{d i\_d}{dt} \left( L\_d - \frac{L\_{md} z^2}{L\_f} \right) \tag{65}
$$

Similar algebra could be done with the stator *q*-axis equation. Using additional variable *<sup>u</sup>*c*<sup>q</sup>* <sup>¼</sup> *uq* � *eq* and Eq. (66)

$$\mathbf{e}\_q = -\frac{L\_{mq}R\_Q}{L\_Q}\mathbf{i}\_Q + \alpha \mathbf{o}\rho\_d \tag{66}$$

a linear differential equation of the first order for the current component *iq* is obtained:

$$
\widehat{u\_q} = R\_i i\_q - \frac{L\_{mq}^{-2} - L\_q L\_Q}{L\_Q} \frac{di\_q}{dt} \tag{67}
$$

Components *ed, eq* will be incorporated into the control system as decoupling.

When the Eqs. (65) and (67) are transformed into Laplace domain, the following transfer functions are obtained:

$$G(s) = \frac{I\_{dq}(s)}{U\_{dq}(s)} = \frac{\frac{1}{R\_s}}{\tau\_{cc,dq}s + 1} \tag{68}$$

where:

$$\mathbf{L}\_{\infty,\mathbf{d}} = \mathbf{L}\_{\mathbf{d}} - \frac{\mathbf{L}\_{\mathbf{m}\mathbf{d}}^2}{\mathbf{L}\_{\mathbf{f}}}$$

$$\mathbf{L}\_{\infty,\mathbf{q}} = \mathbf{L}\_{\mathbf{q}} - \frac{\mathbf{L}\_{\mathbf{m}\mathbf{q}}^2}{\mathbf{L}\_{\mathbf{Q}}}$$

$$\mathbf{\tau}\_{\infty,\mathbf{d}} = \frac{\mathbf{L}\_{\mathbf{c}\mathbf{c},\mathbf{d}}}{\mathbf{R}\_{\mathbf{s}}}$$

$$\mathbf{\tau}\_{\infty,\mathbf{q}} = \frac{\mathbf{L}\_{\mathbf{c}\mathbf{c},\mathbf{q}}}{\mathbf{R}\_{\mathbf{s}}}$$

It is easy to see that Eq. (68) can be controlled in a closed loop by simple PI controller:

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization… DOI: http://dx.doi.org/10.5772/intechopen.89420*

$$\mathbf{C}\_{PI}(\mathbf{s}) = \mathbf{K}\_P + \frac{\mathbf{K}\_I}{\mathbf{s}} \tag{69}$$

Tuning of the PI controllers is done according to Internal model control reference [13] (IMC) method as is given:

$$K\_P = \mathfrak{a}\_{cc} L\_{cc,d} \tag{70}$$

$$K\_i = \mathfrak{a}\_{\alpha} \mathcal{R}\_s \tag{71}$$

where *acc* for the first order system is defined as:

$$a\_{cc} = \frac{\ln\left(\Re\right)}{t\_{r,cc}}\tag{72}$$

and *tr,cc* is stator current response time that is for most of the industrial applications [14] set at 5 ms.

Outer loop for speed control is then analyzed.

The transfer function of the current control closed loop *Gcc(s)* is:

$$G\_{\alpha,cl}(\mathfrak{s}) = \frac{\mathbf{C}\_{PI}(\mathfrak{s})\mathbf{G}(\mathfrak{s})}{\mathbf{1} + \mathbf{C}\_{PI}(\mathfrak{s})\mathbf{G}(\mathfrak{s})} \tag{73}$$

After some algebra Eq. (73) could written as:

$$G\_{\alpha\varepsilon d}(\mathfrak{s}) = \frac{\mathfrak{a}\_{\alpha\varepsilon}}{\mathfrak{s} + \mathfrak{a}\_{\alpha\varepsilon}}\tag{74}$$

Outer control loops will be also controlled by PI controllers. In that case, the complete control loop for the rotor speed is given in **Figure 1**.

Open loop transfer function of the rotor speed control is:

$$G\_{oo,ol}(\mathfrak{s}) = \frac{K\_{po}(T\_{io}\mathfrak{s} + \mathbf{1})}{T\_{io}\mathfrak{s}} \frac{a\_{cc}}{\mathfrak{s} + a\_{cc}\mathfrak{l}\mathfrak{s}} \frac{\mathbf{1}}{\mathfrak{ls}}\tag{75}$$

According to the Eq. (75), stability analysis of the SM1 speed control loop has been done. In **Figure 2**, root locus diagram is given. It shows that, due to damping factor, values of *Kp<sup>ω</sup>* should not exceed 14.

According to the Bode diagram, given in **Figure 3**, the stability phase margin is almost 60 degrees for *Kp<sup>ω</sup>* higher than 10.

According to **Figure 1**, torque load could be analyzed as an input disturbance. Load sensitivity transfer function is obtained:

$$G\_{dy}(s) = \frac{P(s)}{1 + P(s)C(s)}\tag{76}$$

**Figure 1.** *Control loop of the rotor speed.*

If dynamics of the damper winding are neglected, equations of the SM system

*dt Ld* � *Lmd*

�*ifRf* þ *uf*

If the additional variable *<sup>u</sup>*c*<sup>d</sup>* <sup>¼</sup> *ud* � *ed* is introduced, Eq. (63) becomes linear

*did*

Similar algebra could be done with the stator *q*-axis equation. Using additional

a linear differential equation of the first order for the current component *iq* is

Components *ed, eq* will be incorporated into the control system as decoupling. When the Eqs. (65) and (67) are transformed into Laplace domain, the following

*Udq*ð Þ*<sup>s</sup>* <sup>¼</sup>

Lcc,d <sup>¼</sup> Ld � Lmd

Lcc,q <sup>¼</sup> Lq � Lmq

<sup>τ</sup>cc,d <sup>¼</sup> Lcc,d Rs

<sup>τ</sup>cc,q <sup>¼</sup> Lcc,q Rs

It is easy to see that Eq. (68) can be controlled in a closed loop by simple PI

<sup>2</sup> � *LqLQ LQ*

> 1 *Rs*

2 Lf

2 LQ

*diq*

*dt Ld* � *Lmd*

!

2 *Lf*

!

2 *Lf*

� � � *<sup>φ</sup>q*<sup>ω</sup> (64)

*iQ* þ ω*φ<sup>d</sup>* (66)

*dt* (67)

*<sup>τ</sup>cc*,*dq <sup>s</sup>* <sup>þ</sup> <sup>1</sup> (68)

þ *ed* (63)

(65)

*did*

could be simplified. Then, the equation in the stator *d*-axis is:

*ud* ¼ *Rsid* þ

*ed* <sup>¼</sup> *Lmd Lf*

differential equation of the first order for the current component*id*:

*<sup>u</sup>*c*<sup>d</sup>* <sup>¼</sup> *Rsid* <sup>þ</sup>

*eq* ¼ � *LmqRQ LQ*

*<sup>u</sup>*c*<sup>q</sup>* <sup>¼</sup> *Rsiq* � *Lmq*

*G s*ðÞ¼ *Idq*ð Þ*<sup>s</sup>*

where

*Control Theory in Engineering*

obtained:

where:

controller:

**158**

variable *<sup>u</sup>*c*<sup>q</sup>* <sup>¼</sup> *uq* � *eq* and Eq. (66)

transfer functions are obtained:

Finally, *Ki<sup>ω</sup>* can be defined as:

*DOI: http://dx.doi.org/10.5772/intechopen.89420*

**4.2 Simulation**

*4.2.1 Results for SM1*

*4.2.2 Results for SM2*

**Figure 4.**

**161**

*Step response for input disturbance.*

*Ki<sup>ω</sup>* <sup>¼</sup> *Kp<sup>ω</sup> Ti<sup>ω</sup>*

*Kp<sup>ψ</sup> Ti<sup>ψ</sup> <sup>s</sup>* <sup>þ</sup> <sup>1</sup> *Ti<sup>ψ</sup> s*

To make a comparison between nonlinear and linear control systems, simulation studies have been done. Starting process of lower power (8.1 *kVA*) SM1 and higher power (1.56 *MVA*) SM2 synchronous machines have been simulated. Simulations have been obtained in the same file under the same circumstances. Machines were controlled only through the inverter that was connected to the stator winding. On the rotor winding constant nominal voltage was applied. Nonlinear control system have used reduced order observer, while linear control system have used damper winding currents directly from the SM model. Therefore, some advantage was given to the linear control system. Parameters of the synchronous machines have been given in Appendix.

In **Figure 5**, results for the starting of the SM1 have been given. It includes rotor speed, electromagnetic torque, rotor speed error and stator flux error. It could be seen that rotor speed error is significantly higher for the linear control system.

In **Figure 6**, results for the starting of the SM2 have been given. Rotor speed error for the linear control system is again significantly higher. Electromagnetic

It could be seen that the only difference between speed Eq. (75) and flux Eq. (80) transfer functions is in the inertia factor *J*. That is why the flux control stability is analyzed in a similar way as it is done for the speed control loop.

*acc s* þ *acc* 1 *s*

Transfer function of the open loop flux control could be obtained:

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization…*

*G<sup>ψ</sup>*,*ol*ðÞ¼ *s*

(79)

(80)

**Figure 2.** *Root locus for speed control.*

**Figure 3.** *Bode diagram for speed control.*

where ðÞ¼ *<sup>s</sup>* <sup>1</sup> *Js*; *C s*ðÞ¼ *Kpω*ð Þ *Tiωs*þ<sup>1</sup> *Tiωs acc s*þ*acc*

Step response for the torque disturbance is given in **Figure 4**. It could be seen that peek response for *Kp<sup>ω</sup>* higher than 10 is acceptable.

Then, *Ki<sup>ω</sup>* is to be defined. Firstly, time constant of the inner control loop is defined as:

$$T\_{i,cc} = \frac{L\_{cc}}{R\_s} \tag{77}$$

According to the symmetrical optimum method [13] integration time constant of the outer loop circuit should be:

$$T\_{i\nu} = \mathbf{4} T\_{i,\mathcal{cc}} \tag{78}$$

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization… DOI: http://dx.doi.org/10.5772/intechopen.89420*

Finally, *Ki<sup>ω</sup>* can be defined as:

$$K\_{i\alpha} = \frac{K\_{p\alpha}}{T\_{i\alpha}}\tag{79}$$

Transfer function of the open loop flux control could be obtained:

$$G\_{\rm \psi,ol}(\mathbf{s}) = \frac{K\_{p\rm \psi} \left(T\_{i\rm \psi}\mathbf{s} + \mathbf{1}\right)}{T\_{i\rm \psi}s} \frac{a\_{\rm cc}}{s + a\_{\rm cc}s} \frac{\mathbf{1}}{s} \tag{80}$$

It could be seen that the only difference between speed Eq. (75) and flux Eq. (80) transfer functions is in the inertia factor *J*. That is why the flux control stability is analyzed in a similar way as it is done for the speed control loop.

#### **4.2 Simulation**

To make a comparison between nonlinear and linear control systems, simulation studies have been done. Starting process of lower power (8.1 *kVA*) SM1 and higher power (1.56 *MVA*) SM2 synchronous machines have been simulated. Simulations have been obtained in the same file under the same circumstances. Machines were controlled only through the inverter that was connected to the stator winding. On the rotor winding constant nominal voltage was applied. Nonlinear control system have used reduced order observer, while linear control system have used damper winding currents directly from the SM model. Therefore, some advantage was given to the linear control system. Parameters of the synchronous machines have been given in Appendix.

### *4.2.1 Results for SM1*

In **Figure 5**, results for the starting of the SM1 have been given. It includes rotor speed, electromagnetic torque, rotor speed error and stator flux error. It could be seen that rotor speed error is significantly higher for the linear control system.

#### *4.2.2 Results for SM2*

In **Figure 6**, results for the starting of the SM2 have been given. Rotor speed error for the linear control system is again significantly higher. Electromagnetic

**Figure 4.** *Step response for input disturbance.*

where ðÞ¼ *<sup>s</sup>* <sup>1</sup>

*Bode diagram for speed control.*

of the outer loop circuit should be:

defined as:

**160**

**Figure 3.**

**Figure 2.**

*Root locus for speed control.*

*Control Theory in Engineering*

*Js*; *C s*ðÞ¼ *Kpω*ð Þ *Tiωs*þ<sup>1</sup>

*Tiωs*

that peek response for *Kp<sup>ω</sup>* higher than 10 is acceptable.

*acc s*þ*acc*

Step response for the torque disturbance is given in **Figure 4**. It could be seen

Then, *Ki<sup>ω</sup>* is to be defined. Firstly, time constant of the inner control loop is

*Ti*,*cc* <sup>¼</sup> *Lcc Rs*

According to the symmetrical optimum method [13] integration time constant

*Ti<sup>ω</sup>* ¼ 4*Ti*,*cc* (78)

(77)

• Matlab Simulink R2015a, OS Windows 7

*Synchronous Machine Nonlinear Control System Based on Feedback Linearization…*

• TMS320C2000 XDSv1 docking station

control system has been running on the processor.

Data exchange between Simulink model and C2834x control card has been done in real time by serial RS232 communication. During the PiL testing, data precision has to be reduced from double to single. For this reason some error in performance

In **Figure 7**, the scheme of PiL testing system is given. In the Simulink model energetic part (SM, inverter and DC source) has been running, while the complete

• Code Composer Studio CCSv5

*DOI: http://dx.doi.org/10.5772/intechopen.89420*

• TI C2000, C2834x control card

is expected.

**Figure 7.** *PiL testing scheme.*

**Figure 8.**

**163**

*Starting of SM1-PiL.*

**5.1 Testing scheme**

**Figure 5.**

**Figure 6.** *SM2 comparison.*

torque in linear control has some oscillations at the beginning and at reaching of the nominal speed.
