Synchronous Machine Nonlinear Control System Based on Feedback Linearization and Deterministic Observers

A classical linear control system of the SM is based on PI current controllers. Due to SM nonlinearity, with such control system, it is not possible to obtain independent torque and flux control. To overcome this obstacle, a nonlinear control system can be used. Due to unknown damper winding state variables, an observer has to be made. In this work, observers for damper winding currents and damper winding fluxes are presented. Then, based on nonlinear theory, control law with feedback linearization method is obtained. Also, a comparison of the proposed and classical control system is done. For the classical control system, field-oriented control with internal model and symmetrical optimum principles is used. To verify the proposed algorithm, extensive simulation analysis of voltage source inverter drive is made. Processor in the loop testing has been also done.


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.
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.

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.

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: 2   6  6  6  6  6  6  6  6  6  4   3   7  7  7  7  7  7  7  7  ¼ 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 1 is chosen: ¼ pt

Control Theory in Engineering
After each matrix member of Eq. (6) is calculated [8], its determinant calculation gives: The second criterion matrix O 2 is chosen: After each matrix member of Eq. (8) is calculated [8], its determinant calculation gives: While observing both determinants (Eqs. (7) and (9)): Matrix O is full rank matrix and it could be concluded that the system is weakly locally observable.

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: 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: Error dynamics are obtained in similar way as for the full order observer. If the coefficients k xy are defined as stated: k 31 = g 3 i q , k 51 = g 4 i d , k 61 > 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 simplest observer can be defined: 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: It has negative definite derivation Eq. (19) and stability is proved.

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: ¼ In a similar way as for the full order observer Eq. (5), an observer for adaptation could be defined:   For the positive definite Lyapunov function: 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: Àk 52 e 2 e 5 À k 53 e 4 e 5 À Àk 54 e 5 e 6 þ g 3 i q e 3 e 4 þ g 4 i d e 5 e 6 À k 64 e 2 6 À ΔR s If the rules for resistance adaptation are given as stated: 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.

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 (T L ) 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:  Observability analysis of the Eq. (27) is obtained according to the nonlinear system weak observability concept [9]. Observability criterion matrix O 1 (28) has been chosen:  After each matrix member of Eq. (28) is calculated [8], its determinant calculation gives: Observability criterion matrix O 2 has been chosen: After each matrix member of Eq. (30) is calculated [8], its determinant calculation gives: While observing both Eqs. (29) and (31): It is easy to see that: 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.
where Te states for electromagnetic torque. Then, an observer is proposed: 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): where: ! Error dynamics is obtained by Eqs. (34) and (35): where: Expression in Eq. (36) can be noted as: According to the Popov stability criterion, stability will be proved by achieving the condition: With further expansion of the Eq. (38), stability condition becomes: According to the literature reference [12] it is obvious that inequality Eq. (40) is satisfied if: According to [12] stability of the load torque estimation Eq. (41) is achieved for each positive value of the proportional k p and integral k i coefficients.

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: 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.
Equation (42) could be noted as: 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: where If the additional variable c u d ¼ u d À e d is introduced, Eq. (63) becomes linear differential equation of the first order for the current componenti d : Similar algebra could be done with the stator q-axis equation. Using additional variable c u q ¼ u q À e q and Eq. (66) a linear differential equation of the first order for the current component i q is obtained: Components e d , e q 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: where: It is easy to see that Eq. (68) can be controlled in a closed loop by simple PI controller: Step response for the torque disturbance is given in Figure 4. It could be seen that peek response for K pω higher than 10 is acceptable.
Then, K iω is to be defined. Firstly, time constant of the inner control loop is defined as: According to the symmetrical optimum method [13] integration time constant of the outer loop circuit should be:  Finally, K iω can be defined as: Transfer function of the open loop flux control could be obtained: 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.

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.

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.

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 Step response for input disturbance. torque in linear control has some oscillations at the beginning and at reaching of the nominal speed.

Processor in the loop testing
Model based development is an approach that can handle complexities of various range of products. It is primarily used for early error detection. Using that approach, control system can be tested in phases. The first phase is called model in the loop (MiL) testing, the second one is processor in the loop (PiL) and finally there is hardware in the loop (HiL). In this work except from MiL, also PiL testing has been done. The testing equipment consists of:  • Matlab Simulink R2015a, OS Windows 7 • Code Composer Studio CCSv5 • TI C2000, C2834x control card • TMS320C2000 XDSv1 docking station 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 is expected.

Testing scheme
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 control system has been running on the processor.   To check the novel control algorithm PiL, testing of both (SM1 and SM2) machines have been done. Testing included starting process, reversing of the speed and load step changes.

PiL testing of SM1
In Figure 8, results for the starting of the SM1 have been given. Tracking of the reference speed is precise.
In Figure 9, results for the reversing of the speed of the SM1 have been given. Tracking of the reference speed is again obtained precisely.     In Figure 10, results for the load step changes of the SM1 have been given. The step change is from no load to 100% of the nominal load. Except from rotor speed and electromagnetic torque, results of damper flux observer and load torque estimation are also given.
There is an error of about 10% in observer operation, and an error in load torque estimator of about 5%. This is due to reduction in data precision during PiL testing. In spite of that, an error in speed tracking exists only during the step change and it is about 3%.

PiL testing of SM2
In Figure 11, results for the starting of the SM2 have been given. Tracking of the reference speed is precise.
In Figure 12, results for the reversing of the speed of the SM2 have been given. Tracking of the reference speed is again obtained precisely.
In Figure 13, results for the load step changes of the SM2 have been given. The step change is from no load to 100% of the nominal load. Except from rotor speed and electromagnetic torque, results of damper flux observer and load torque estimation has been also given.
There is an error of about 15% in observer operation, and an error in load torque estimator of about 3%. This is due to reduction in data precision during PiL testing. In spite of that, an error in speed tracking exist only during the step change and it is about 2%.

Conclusion
Dynamical system of SM is characterized with high nonlinearity, variable coupling and unknown damper winding variables. If the control of the SM is done by the classical linear control system, its complexity has to be simplified. Usually, dynamics of the damper winding are neglected. Besides, classical control use currents components controllers to obtain torque and flux control. Coupling in the SM dynamical system makes that change of any current component necessary changes both; torque and flux. Due to these reasons, classical system cannot provide efficient control system with good dynamic performance.
Using nonlinear techniques, fully decoupled torque and flux control could be obtained. To make it applicable, damper windings states should be known. In this work, using damper winding observers and nonlinear control law, a high performance rotor speed tracking system is obtained. Full order and reduced order deterministic observers of damper winding currents and damper winding fluxes are presented. Nonlinear control law is obtained using feedback linearization method.
A comparison between classical linear system and novel control system has been done. At the beginning of the starting as well as at reaching of the nominal speed classical control system exhibits oscillations, while the novel control keeps tracking precisely.
Processor in the loop testing of the novel control system has been also done. Except from damper winding flux observer, load torque estimation has been also used. The system performance during starting, reversing of the speed and during load step changes has been tested. Due to reduction in data precision, some error of the damper flux observer and load torque estimator appears. In spite of that, performance of the rotor speed tracking system is precise.
It could be concluded that proposed control system has advantages over classical and gives some new opportunities.