Starting from a new dynamic system description novel synchronous machine deterministic observers are proposed. Reduced and full order adaptive observer variations are presented. Based on the feedback linearization control law and the use of deterministic observer a novel control system is built. It meets the requirements of high performance tracking system. Adaptivity to stator and rotor resistance and the torque sensorless application is included. The comparison of the proposed novel control with conventional linear and nonlinear control systems is discussed. The given simulational study includes complete drive system integration.
1. Introduction
This work examines a novel control method for variable speed operation of a synchronous machine. Here, it is assumed that the synchronous machine (SM) has damper windings and a separate excitation winding.
Variable speed operation of SM is used in various cases of power generation and electric drive applications. In windmill power generation and in hydro power units, variable speed operation is a design requirement. SM drive finds particular applications in: coal mines, the metal and cement industries, ship propulsion etc. 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 a high performance tracking system it is necessary to have an adequate observer for these states, as is done in similar AC drive systems
[1]
. There are not many studies, either linear (vector) or nonlinear, of the SM control system.
AC motor vector control is used with the following assumption: if the flux is constant, the qcurrent component can control electromagnetic torque. For induction motor drives this assumption holds true, but if this method is used for SM control the qcurrent component will essentially change the flux
[2]
. In this case it is said that the control is coupled and this is why SM vector control is not efficient enough. There are few ideas on how to solve this problem. One involves coordinate transformation
[3
,
4]
. Unfortunately, a control system with many calculations (coordinate transformations, PI controllers, and other) has to be used. Because of its complexity, further development of this control does not look promising. In standard SM control systems the damper winding effect is neglected. It is well known that damper windings have positive influence on SM stability at power system nominal operation. Its importance in asynchronous starting process is also known. During synchronous starting its effect has not been studied yet, but it would be of interest especially in the case of high performance drives.
Regarding nonlinear control SM applications, a few methods are used: backstepping
[5]
, passivity
[6]
and adaptive Lyapunov based
[7
,
8]
. The passive method
[6]
fails to give better results and the backstepping
[5]
method fails to take the damper windings into consideration. In
[7
,
8]
new algorithms are proposed, but their complexity seems to make implementation impractical.
In general practice, the stability in many nonlinearly controlled AC drives is proved by Lyapunov
[9]
. This is done for the observer as for the whole system (observer + controller)
[10

13]
if necessary. Except for the missing states, the control also includes parameter variation. To achieve this, many different control methods are combined with various observers. For example, forced dynamic control is combined with sliding mode observer or model reference system
[14

17]
. Stability can be achieved for winding resistance change
[18]
or change of inductance and motor inertia
[19]
. Also, sensorless control can be achieved in regard to load torque
[20

22]
or rotor speed
[23
,
24]
. The aim of this work is to use nonlinear techniques to develop a novel control system for SM. By using completely decoupled control law and by taking into consideration the effect of damper windings, the resulting control system is made more advantageous than the existing SM control systems.
This paper is organized so as to give the complete control system modeling: the SM system and its observability analysis, observer and control law definitions and internal dynamics analysis. An extensive simulation study is then presented. The results of various observer applications and also the comparisons with linear and nonlinear control systems are given.
2. Modeling
 2.1 SM system
In order to take into consideration the effect of the damper winding, the system given in (1) will be the starting point of analysis.
Coefficients a
_{1}
, a
_{2}
etc. are calculated from SM parameters:

a1=(LfLmd2rD− Lmd3rD+LD2Lfrs−LDLmd2rs) / (LDA)

a2=(LfLmd2rD− Lmd3rD−LD2Lmdrf+LDLmd2rf) / (LDA)

a3=(LD2LfLmq2−LDLmd2Lmq2−LD2LfLqLQ+LDLmd2LqLQ) / (LDLQA)

a4=(−LfLmdLQrD+Lmd2LQrD) / (LDLQA)

a5=(−LD2LfLmq+LDLmd2Lmq) / (LDLQA)

a6=(−LD2LfLQ+LDLmd2LQ) / (LDLQA)

a7=(LD2LmdLQ−LDLmd2LQ) / (LDLQA)

b1=(LdLmd2rD− Lmd3rD−LD2Lmdrs+LDLmd2rs) / (LDA)

b2=(LdLmd2rD− Lmd3rD+LdLD2rf−LDLmd2rf) / (LDA)

b3=(−LD2LmdLmq2+LDLmd2Lmq2+LD2LmdLqLQ−LDLmd2LqLQ)/(LDLQA)

b4=(−LdLmdLQrD+Lmd2LQrD)/ (LDLQA)

b5=(LD2LmdLmq−LDLmd2Lmq)/ (LDLQA)

b6=(LD2LmdLQ−LDLmd2LQ)/ (LDLQA)

b7=(−LdLD2LQ+LDLmd2LQ)/ (LDLQA)

c1=LmdrD/LD

c2=LmdrD/LD

c3= −rD/LD

d1=(−LDLmq2rQ−LDLQ2)/ (LDLQB)

d2=(−LdLDLQ2+Lmd2LQ2) / (LDLQB)

d3=(−LDLmdLQ2+Lmd2LQ2) / (LDLQB)

d4=− (LmdLQ) / (LDB)

d5=(LmqrQ) / (LQB)

d6=LQ

f1=(LmqrQ)/LQ

f2= −rQ/LQ

g1= − (LDLmdLmq−LDLmdLQ+Lmd2LQ+LDLmqLQ) /(2 H LDLQ)

g2= − (−LDLmdLQ+Lmd2LQ)/(2 H LDLQ)

g3= Lmd/(2 H LD)

g4= Lmd/(2 H LQ)

g5= −1/(2 H)

A = −LdLDLf+LdLmd2+LDLmd2+LfLmd2−2 Lmd3

B= −Lmq2+LqLQ
 2.2 Observability analysis
Observability of the given system (1) is analysed. Here, measured states are given as:

h1=id, h2=if, h3=iq, h4=ω.
The analysis is based on nonlinear system local weak observability concept
[25
,
26]
.
Assume a nonlinear dynamical system Σ (2):
In a point from its state space x
_{0}
ϵΏ its observability matrix is (3):
is observability criterion matrix and L
^{k}
_{f}
is the kth order Lie derivative of the function h with respect to the vector field f. If the matrix O has full rank
than the state of the system Σ is locally weakly observable at point x
_{0}
.
A number of possible submatrices can be tested, but choosing matrices given in (5) and (6) it will be easy to make a proof.
O
_{1}
determinant is:
O
_{2}
determinant is:
Considering both conditions:
Det (O
_{1}
) ≠0 U Det (O
_{2}
) ≠0 => rank{O}=6
Matrix O is a fullrank matrix and it can be concluded that the system is weakly locally observable at every point of state space Ώ.
 2.3 Deterministic observers
After the successful observability analysis, an observer can be constructed. Damper winding fluxes are missing states. In addition to observing the missing states, it would be preferable for the observer to be adaptive to parameter changes.
A new observer adaptive to stator and rotor winding resistance is presented.
The idea is to extend (1) in a way that stator and rotor resistances can be separately collected. The resulting system is given in (7).
Coefficients a
_{1}
, a
_{2}
etc. are very similar to the already given coefficients.
Consider this observer (8):
It is Lyapunov stable because c
_{3}
and f
_{2}
are always negative for SM.
Although the observer is simple and stable, it is of a reduced order and because of this it is not possible to prove global stability of the whole system.
Proposition
: For the SM model given in (7), a full order stable observer adaptive to stator and rotor resistance change is given in (9).
Proof
: Consider Lyapunov function given in (10). It is positivedefinite.
Error dynamics is defined as (7)  (9) and is given in (11).
State errors are:
and resistance errors: ΔR
_{s}
and ΔR
_{f}
.
Now consider Lyapunov function differential taking into account its error dynamics (11) and the usual assumption of slow resistance change:
with convergence coefficients k
_{11}
, k
_{22}
, etc. given as:
k
_{31}
=
a
_{4}
;
k
_{32}
=
b
_{4}
;
k
_{33}
=
d
_{4}
^{ω}
;
k
_{34}
=
g
_{3}
i_{q}
;
k
_{51}
=
a
_{5}
^{ω}
;
k
_{52}
=
b
_{5}
^{ω}
;
k
_{53}
=
d
_{5}
k
_{54}
=
g
_{4}
i_{d}
;
k
_{11}
,
k
_{22}
,
k
_{43}
,
k
_{64}
> 0
and resistance adaptive rules (12), (13):
Lyapunov function differential is obtained (14):
it is negativedefinite and according to Lyapunov direct method global asymptotic stability of the observer is proved.
 2.4 Control law
The aim of nonlinear control is to achieve decoupling between flux and torque controls. As already stated, the feedback linearization method is chosen.
The control demand is to make a tracking system of two outputs (15): rotor speed, and square of stator magnetic flux:
φ_{d}
,
φ_{q}
are stator magnetic fluxes; not to be misinterpreted as
φ_{D}
,
φ_{Q}
that are damper winding fluxes used as state variables.
Although it is possible to include excitation control, excitation voltage will remain constant.
It is necessary to separate the first output into two variables; so a new one
(electromagnetic torque) is introduced.
After some algebra, the system will get the form of (16):
where G is decoupling matrix:
Now, after the control law (17) is defined:
with errors:
Similarly to
[27]
, error dynamics is gained and decoupling is achieved (18):
It is easy to make the Lyapunov proof of this error dynamics as well as to prove the convergence of the whole system (observer + controller). Consider positivedefinite Lyapunov function V
_{2}
(19):
By using positive coefficients k
_{p0}
, k
_{p1}
and k
_{p2}
, the differential of V
_{2}
is negativedefinite and global asymptotic stability of the control law is obtained according to Lyapunov direct method.
Both functions V
_{1}
and V
_{2}
are Lyapunov stable, and it is concluded that dynamics of the entire system (V
_{1}
+ V
_{2}
) is stable.
 2.5 Internal dynamics
It is not possible to obtain exact linearization for the SM system as well as some similar systems such as the induction motor
[28]
. That is why partial inputoutput linearization has been applied. The relative degree of the system is lower than the system order, so it is necessary to check the system’s internal dynamics.
In the well known theorem of bounded function it is stated that the sum and product of bounded functions is also a bounded function. The reverse is also valid.
In this system, the second output (that is of course bounded by the reference) is the sum and product composition of state variables (20):
Because the first output h
_{1}
is ω, and it is also bounded by the reference, all state variables are included and it can be concluded that all internal dynamics are bounded.
To achieve global stability, decoupling matrix G has to be globally invertible. Its determinant is (21):
According to the motor parameters given in the next paragraph (21) becomes (22).
It is not possible to eliminate the first two members in (21), (22) and to theoretically claim global stability, but in all control demands described in the following paragraphs, the determinant always remains far from singularity.
3. Simulation Model
Previous considerations are outlined in the control scheme shown in
Fig. 1
.
Control scheme
At first, it is necessary to do Park transformation to current and voltage measurements. The adaptive observer then computes all observed states and parameters. Taking into consideration references and observed values, the feedback linearization control law calculates reference voltages. Signals for inverter control are then generated by modulation technique. Modulation is done by space vector pulse width modulation (SVPWM). Symmetrical pattern with a switching frequency of 3 kHz is used. The sampling time of the discretized control system is 12 kHz.
At the output of the voltage source inverter (VSI) an RLC filter is typically used. In this study some standard filter values are taken.
Simulations are done by either variablestep of fixedstep solvers. Various precision levels according to step size and tolerances can be set.
The system is usually described in Per Unit System values, and so this system will be used in this case too.
Nominal parameters of the SM are given as Per Unit values on the SM’s stator basis; it will be necessary to calculate excitation voltage and reactor inductivity on the same basis.
SM nominal values of power, voltage, frequency, pole pairs and inertia constant are:

Sn= 8,1 kVA, Un= 400 V, fn=50 Hz, p=2, H = 0,1406 s.
Stator winding (p.u.) values are:

rs= 0,082, Lσ= 0,072, Lmd= 1,728, Lmq= 0,823.
Excitation winding (p.u.) values are:
Damper winding (p.u.) values are:

rD= 0,159, LσD= 0,117, rQ= 0,242, LσQ= 0,162.
Filter reactor (p.u.) value is: L
_{react}
= 0,158.
4. Control with Full Order Observer
The results are given in the following figures. They show very accurate performance during the speed reversing process (
Fig. 2
 rotor speed,
Fig. 3
 rotor speed error). The square of stator flux is also accurately controlled (
Fig. 4
 square of stator flux,
Fig. 5
 square of stator flux error). The flux observer also gives accurate results.
Fig. 6
shows observed and ideal value of damper D axis flux and
Fig. 7
shows its observation error. In
Figs. 8
and
9
the corresponding values in Q axis are shown. Although resistance adaptability has to be checked in the experiment; in this simulation initial values are set far from the SM model values and, as expected, they approach to the constant model values (
Figs. 10
and
11
).
Rotor speed
Rotor speed error
Square of stator flux
Square of stator flux error
Damper Daxis flux
Damper Daxis flux error
Damper Qaxis flux
Damper Qaxis flux error
Rotor resistance adaptation
Stator resistance adaptation
5. Control with Reduced Order Observer
Although the results obtained by full order observer are very accurate, an observer still has certain complexity and more importantly it needs load torque knowledge to obtain high level accuracy.
If the reduced order observer is used it is possible to overcome these obstacles. The observer (8) is simple, it does not need voltage measurements and load torque estimation scheme (
Fig. 12
.) can be implemented.
Load torque estimator
The estimator contains rotor speed calculation according to the rotor speed dynamics given in (23).
To check the control’s performance, load torque step up and step down changes (after the motor starting) have been simulated. Once again, the simulation results indicate very accurate performance. In
Figs. 15
and
16
flux components are given. Estimated load torque (
Fig. 17
) and its error (
Fig. 18
) show accurate performance. As a result of the aforementioned, load torque changes have not had an impact on the tracking system;
Fig. 13
gives rotor speed error while
Fig. 14
gives electromagnetic torque.
Rotor speed error
Electromagnetic torque
Daxis fluxcomparison
Qaxis fluxcomparison
Load torquecomparison
Load torque estimation error
6. Comparison Between Linear and Novel Control
The comparison has been done under identical conditions with both systems being in the same simulation model with the same settings. Identical simulation model blocks and parameters are used in both systems. The only difference is the control law block. In the linear control system, instead of the feedback linearization control law given and explained in the previous paragraphs, linear vector control law is implemented. It contains two loops (
Fig. 19
): the first for flux control that has one PI controller, and the second for speed control that has two PI controllers (one for speed and one for toque control). As in typical vector control, flux is controlled by dcurrent component and speed is controlled by qcurrent component.
Vector control scheme
Reference voltage calculation is done according to simplified dynamics given in (24).
The results given in the following figures show degraded efficiency of vector control, as expected. In the following figures, the results of nonlinear control are blue colored solid line while the vector control results are in red colored dot line. For the same dynamics the vector control (loaded starting) lasts twice as long (
Fig. 20
), has higher errors (
Fig. 21
) and (because of a lesser degree of synchronism) has higher damper windings current oscillations in comparison with the novel control (
Fig.(s) 22
and
23
). Eliminating damper winding current oscillations by using the novel control law decreases electromagnetic torque ripple (
Fig. 24
). The current waveforms (
Fig. 25
), show less harmonic distortion in the novel application. Because of the degraded efficiency, the vector control also needs higher DC voltage then the novel control.
Rotor speed
Rotor speed error
Daxis damper winding current
Qaxis damper winding current
Electromagnetic torque
Stator winding currents
7. Comparison Between Conventional Nonlinear and Novel Control
The vector control principle has been used to form conventional nonlinear control. The normally used vector control principle is to set i
_{d}
to zero and to control torque with i
_{q}
component. It is according to this principle that backstepping and feedback linearization control laws have been designed.
To make a proper comparison, the novel control has been simulated together with each of the other controls in the same Simulink model, under the same conditions. The starting process with load torque step up at 1,3 seconds and step down at 1,8 seconds has been simulated. The results of the novel control are shown in blue colored solid line while all others are in red colored dot line.
 7.1 Backstepping design
Again, the system given in (1) has been used and damper winding has been taken into consideration.
The first step is to define u
_{d}
according to the dcurrent zero reference. Error convergence can be easily achieved and the equation is given in (25).
where k
_{d}
is convergence constant and e
_{d}
is dcurrent error
To find the control law for i
_{q}
current, a new variable can be defined as (26):
The differential of rotor speed error (e
_{ω}
) can be written as (27):
where k
_{ω}
is convergence constant; e
_{α}
is the difference between α and its desired value α
^{*}
and e
_{ω}
is the difference between ω and its refference ω
_{ref}
.
In the case that desired value of α is α* (28);
the differential of rotor speed error (e
_{ω}
) would be:
and it would be convergent.
According to this analysis Lyapunov fuction can be defined as (30):
u
_{q}
can be obtained regarding Lyapunov function differential (31):
and is given in (32).
The simulation results for the rotor speed and electromagnetic torque are given below: rotor speed (
Fig. 26
), its error (
Fig. 27
) and electromagnetic torque (
Fig. 28
).
Rotor speed
Rotor speed error
Electromagnetic torque
Due to torque and flux decoupling in the novel control law, a better use of DC voltage and consequently better performance is achieved. During the starting process, in backstepping control also the high torque ripple appears.
 7.2 Feedback linearization without torque and flux decoupling
The vector control principle given in backstepping design has been used again to test the performance of the feedback linearization without torque and flux decoupling control law. The control law for this can be derived similarly to the already given control law (17).
Simulations have been done with and without taking into consideration the damper winding effect.
The results given in
Fig. 29
and
Fig. 30
are for the case of damper winding not considered. They show, once again, the same advantages of the novel control as described before.
Rotor speed
Electromagnetic torque
The feedback linearization with damper winding effect taken into consideration is also given. The advantage of decoupling is once again obvious (
Fig. 31
) and by taking into consideration the damper winding, the electromagnetic torque ripple will be reduced during the starting process as given in
Fig. 32
.
Rotor speed
Electromagnetic torque
8. Conclusion
This paper presents simulation studies of synchronous machine observerbased control. The presented novel control is based on the electromagnetic torque and magnetic flux decoupling principle. To accomplish full decoupling, an observer for the unknown SM states has been used. Full and reduced order observers are also presented.
The method has been checked with the speed reversing tracking system. Comparison of the novel nonlinear control with linear control and also with conventional nonlinear control has been given.
Simulation results show that precise control has been achieved. It is shown that decoupling enables much better use of DC voltage, while the use of damper winding observer reduces torque ripple. These two contributions incorporated together into the control system enhance performance and operational range of SM drives.
The control system is discretized and thus the sample data system has been defined. From Simulink blocks, the control system is easily convertible to Ccode and is being prepared for DSP implementation.
The method can also be used for torque control, in both motor and generator operation regimes, and can be applied to any kind of synchronous machine including permanent magnet synchronous machines.
Synchronous Machine Parameter and State Symbols

Lmd– daxis mutual inductance

Ld– stator daxis inductance

Lσ– stator leakage iductance

LD– damper daxis inductance

LσD– damper daxis leakage inductance

Lf– field inductance

Lσf– field leakage inductance

Lmq– qaxis mutual inductance

Lq– stator qaxis inductance

LQ– damper qaxis inductance

LσQ– damper qaxis leakage inductance

rs– stator resistance

rf– field resistance

rD– damper daxis resistance

rQ– damper qaxis resistance

H – inertia constant

id– stator daxis current

if– field current

iq– stator qaxis current

φD– damper daxis flux

φQ– damper qaxis flux

φd– stator daxis flux

φq– stator qaxis flux

ω – rotor speed

TL– load torque
BIO
Marijo Šundrica graduated at the Faculty of Electrical Engineering and Computing, University of Zagreb. from 2006 to date works as a development engineer in Končar Power Plant and Electric Traction Engineering. He is pursuing his PhD degree in the field of synchronous machine dynamics, modeling and control. He authored many scientific and professional papers.
Igor Erceg received his Ph.D.E.E. in 2010 and B.S.E.E. in 2004 at the Faculty of Electrical Engineering and Computing, University of Zagreb. From 2004 he has been working as an assistant at the Faculty of Electrical Engineering and Computing. His fields of interests are power system stability analyses, power system operation and control and excitation control of synchronous generators. He authored many papers published in journals and presented at national and international conferences.
Zlatko Maljković graduated in 1974, master of science degree in 1982 and the Ph.D. degree in 1990 all at the Faculty of Electrical Engineering, University of Zagreb. Employed at Department of Electric Machines, Drives and Automation of Faculty of Electrical Engineering and Computing of University of Zagreb where worked as assistant, assistant professor, associated professor and full professor on group Electrical Machinery. Scientific and professional interests are in the following fields: synchronous machines, dynamics of electrical machines, simulation and modelling of electrical machines. Member of IEEE, CIGRE and many Croatian societies. From 2000 to 2005 observer of International Committee of CIGRE – Paris in group A1.
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