Nonlinear Observer-based Control of Synchronous Machine Drive System

Abstract

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 q-current component can control electromagnetic torque. For induction motor drives this assumption holds true, but if this method is used for SM control the q-current 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 a1, a2 etc. are calculated from SM parameters:

a1=(Lf Lmd2 rD − Lmd3 rD+LD2 Lf rs−LD Lmd2 rs) / (LDA) a2=(Lf Lmd2 rD − Lmd3 rD−LD2 Lmd rf+LD Lmd2 rf) / (LD A) a3=(LD2 Lf Lmq2−LD Lmd2 Lmq2−LD2 Lf Lq LQ+LD Lmd2 Lq LQ) / (LD LQ A) a4=(−Lf Lmd LQ rD+Lmd2 LQ rD) / (LD LQ A) a5=(−LD2 Lf Lmq+LD Lmd2 Lmq) / (LD LQ A) a6=(−LD2 Lf LQ+LD Lmd2 LQ) / (LD LQ A) a7=(LD2 Lmd LQ−LD Lmd2 LQ) / (LD LQ A) b1=(Ld Lmd2 rD − Lmd3 rD−LD2 Lmd rs+LD Lmd2 rs) / (LD A) b2=(Ld Lmd2 rD − Lmd3 rD+Ld LD2 rf−LD Lmd2 rf) / (LD A) b3=(−LD2 Lmd Lmq2+LD Lmd2 Lmq2+LD2 Lmd Lq LQ−LD Lmd2 Lq LQ)/(LD LQ A) b4=(−Ld Lmd LQ rD+Lmd2 LQ rD)/ (LD LQ A) b5=(LD2 Lmd Lmq−LD Lmd2 Lmq)/ (LD LQ A) b6=(LD2 Lmd LQ−LD Lmd2 LQ)/ (LD LQ A) b7=(−Ld LD2 LQ+LD Lmd2 LQ)/ (LD LQ A) c1=Lmd rD/LD c2=Lmd rD/LD c3= −rD/LD d1=(−LD Lmq2 rQ−LD LQ2)/ (LD LQ B) d2=(−Ld LD LQ2+Lmd2 LQ2) / (LD LQ B) d3=(−LD Lmd LQ2+Lmd2 LQ2) / (LD LQ B) d4=− (Lmd LQ) / (LD B) d5=(Lmq rQ) / (LQ B) d6=LQ f1=(Lmq rQ)/LQ f2= −rQ/LQ g1= − (LD Lmd Lmq−LD Lmd LQ+Lmd2 LQ+LD Lmq LQ) /(2 H LD LQ) g2= − (−LD Lmd LQ+Lmd2 LQ)/(2 H LD LQ) g3= Lmd /(2 H LD) g4= Lmd /(2 H LQ) g5= −1/(2 H) A = −Ld LD Lf+Ld Lmd2+LD Lmd2+Lf Lmd2−2 Lmd3 B= −Lmq2+Lq LQ

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 x0ϵΏ its observability matrix is (3):

is observability criterion matrix and Lkf is the k-th order Lie derivative of the function h with respect to the vector field f. If the matrix O has full rank

rank {O}=n

than the state of the system Σ is locally weakly observable at point x0.

A number of possible submatrices can be tested, but choosing matrices given in (5) and (6) it will be easy to make a proof.

O1 determinant is:

O2 determinant is:

Considering both conditions:

Det (O1) ≠0 U Det (O2) ≠0 => rank{O}=6

Matrix O is a full-rank 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 a1, a2 etc. are very similar to the already given coefficients.

Consider this observer (8):

It is Lyapunov stable because c3 and f2 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 positive-definite.

Error dynamics is defined as (7) - (9) and is given in (11).

State errors are:

and resistance errors: ΔRs and ΔRf.

Now consider Lyapunov function differential taking into account its error dynamics (11) and the usual assumption of slow resistance change:

with convergence coefficients k11, k22, etc. given as:

k31 = a4 ; k32 = b4; k33 = d4ω; k34 = g3iq; k51 = a5ω; k52 = b5ω; k53 = d5 k54 = g4 id ; k11, k22, k43, k64 > 0

and resistance adaptive rules (12), (13):

Lyapunov function differential is obtained (14):

it is negative-definite 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:

φd , φq are stator magnetic fluxes; not to be misinterpreted as φD , φQ 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 positive-definite Lyapunov function V2 (19):

By using positive coefficients kp0, kp1 and kp2, the differential of V2 is negative-definite and global asymptotic stability of the control law is obtained according to Lyapunov direct method.

Both functions V1 and V2 are Lyapunov stable, and it is concluded that dynamics of the entire system (V1+ V2 ) 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 input-output 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 h1 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.

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 variable-step of fixed-step 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:

rf = 0,0612, Lσf = 0,18.

Damper winding (p.u.) values are:

rD = 0,159, LσD = 0,117, rQ = 0,242, LσQ = 0,162.

Filter reactor (p.u.) value is: Lreact = 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).

Fig. 2.Rotor speed

Fig. 3.Rotor speed error

Fig. 4.Square of stator flux

Fig. 5.Square of stator flux error

Fig. 6.Damper D-axis flux

Fig. 7.Damper D-axis flux error

Fig. 8.Damper Q-axis flux

Fig. 9.Damper Q-axis flux error

Fig. 10.Rotor resistance adaptation

Fig. 11.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.

Fig. 12.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.

Fig. 13.Rotor speed error

Fig. 14.Electromagnetic torque

Fig. 15.D-axis flux-comparison

Fig. 16.Q-axis flux-comparison

Fig. 17.Load torque-comparison

Fig. 18.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 d-current component and speed is controlled by q-current component.

Fig. 19.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.

Fig. 20.Rotor speed

Fig. 21.Rotor speed error

Fig. 22.D-axis damper winding current

Fig. 23.Q-axis damper winding current

Fig. 24.Electromagnetic torque

Fig. 25.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 id to zero and to control torque with iq 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 ud according to the d-current zero reference. Error convergence can be easily achieved and the equation is given in (25).

where kd is convergence constant and ed is d-current error

ed = id- idref.

To find the control law for iq current, a new variable can be defined as (26):

The differential of rotor speed error (eω ) can be written as (27):

where kω is convergence constant; eα is the difference between α and its desired value α* and eω is the difference between ω and its refference ωref .

In the case that desired value of α is α* (28);

the differential of rotor speed error (eω) would be:

and it would be convergent.

According to this analysis Lyapunov fuction can be defined as (30):

uq 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).

Fig. 26.Rotor speed

Fig. 27.Rotor speed error

Fig. 28.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.

Fig. 29.Rotor speed

Fig. 30.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.

Fig. 31.Rotor speed

Fig. 32.Electromagnetic torque

 

8. Conclusion

This paper presents simulation studies of synchronous machine observer-based 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 C-code 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 – d-axis mutual inductance Ld – stator d-axis inductance Lσ – stator leakage iductance LD – damper d-axis inductance LσD – damper d-axis leakage inductance Lf – field inductance Lσf – field leakage inductance Lmq – q-axis mutual inductance Lq – stator q-axis inductance LQ – damper q-axis inductance LσQ – damper q-axis leakage inductance rs – stator resistance rf – field resistance rD – damper d-axis resistance rQ – damper q-axis resistance H – inertia constant id – stator d-axis current if – field current iq – stator q-axis current φD – damper d-axis flux φQ – damper q-axis flux φd – stator d-axis flux φq – stator q-axis flux ω – rotor speed TL – load torque