## 1. Introduction

In the control systems of induction motors, the state variables of the motor need to be estimated based on methods such as vector control, direct torque control and multiscalar control. The values of state variables, such as magnetic fluxes coupled with the rotor windings, and sometimes also angular speed, are applied in the control system as feedback signals. The estimation quality has a crucial impact on the accuracy of the overall control process. The Luenberger observers [

1,

2,

3] are often applied in the state variables’ reconstruction.

In the literature on modern control systems of induction machines, the authors pay attention to the fact that the accuracy of magnetic flux estimation (both the module and the flux vector argument) has a significant impact on the quality of control in these control systems. For example, the works [

4,

5,

6,

7] provide a review of various modern control systems of induction machines used in practice: Field Oriented Control (FOC), Direct Torque Control with Voltage Space Vector Modulation (DTC-SVM), Direct Torque Control with Flux Vector Modulation (DTC-FVM), Switching Table Direct Torque Control (ST-DTC) and Direct Self-Control (DSC). In all these systems, the stator or rotor magnetic flux signal from the observer is used in the feedback paths and is fed to the input of the continuous (most often proportional-integral (PI) type) or hysteresis regulators. It is also used to calculate the electromagnetic torque value in systems with Direct Torque Control (DTC). The authors of the work [

4] write that “Implementation of any high-performance drive system requires a high accuracy estimation of the actual stator or/and rotor flux vector (magnitude and position) and electromagnetic torque”, thus emphasizing the role of the flux observer and its impact on the quality of the control system.

In the works already cited [

4,

5,

6], as well as in [

8,

9,

10], control systems are presented in which the information about the stator and/or rotor flux is used to predict the values of the stator current, rotor flux and electromagnetic torque. These are systems that use prediction based on the induction machine model (Model Predictive Control (MPC)): Predictive Torque Control (PTC) and Predictive Current Control (PCC), belonging to the Finite Set Model Predictive Control class (FS-MPC). In these systems, the quantities calculated on the basis of prediction are used in the process of minimizing the cost function [

8]. Consequently, the quality of the control system in this case also significantly depends on the performance of the stator and/or rotor flux observer. Moreover, in systems using prediction, the low computational complexity of the observer is crucial, because delays in switching the power inverter keys, resulting from the time needed to implement the control algorithm and to calculate the observer’s equations, have an extremely negative impact on the control quality [

9].

In all the above-mentioned control systems of induction machines, the magnetic flux of the stator or rotor is one of the controlled quantities. The reference value of this flux is fed to the controller input or to an algorithm that minimizes the cost function. In classic control systems, this reference flux value may be constant. It may also depend, for example, on the angular speed, by analogy to the control systems of DC commutator machines. However, in modern control systems in which the active power losses in the induction machine are additionally minimized, the flux reference value is calculated in the process of ongoing (real-time) optimization, which takes into account the current operating point parameters of the machine, including the electromagnetic torque, which is calculated on the basis of the flux being estimated by the observer. Such control systems that minimize losses in the machine have been presented, among others, in the works [

11,

12,

13]. In these systems, the importance of the flux observer and its performance increases significantly because low estimation quality adversely affects the process of flux control and the optimization process of its reference value.

The authors of the work [

4] write: “There is a strong trend to avoid mechanical motion (speed/position) sensors because it reduces cost and improves reliability and functionality of the drive system.” As a result of this trend, in recent years, the literature commonly considers sensorless control systems of induction machines, presented among others in [

14], in which the rotor speed signal is obtained in the speed reconstruction system, and various structures of which are presented in [

15,

16,

17]. Virtually any of the previously mentioned control systems of an induction machine can be implemented as sensorless. Rotor speed estimation systems are built, among others, with the use of flux observers, which are equipped with an adaptive mechanism, creating Model Reference Adaptive Systems (MRAS) [

17,

18,

19]. The adaptation mechanism reconstructs the speed based on the estimated flux signals or the fluxes and the measured stator winding currents. Accordingly, the quality of the flux estimation has a significant impact on the quality of the speed reconstruction and the performance of the sensorless control system. Additionally, in sensorless control systems attention should be paid to the problem of stability of the speed estimation system, which (due to the adaptive mechanism) is a non-linear system of higher order than the flux observer used for its construction. The issue of the stability of speed reconstruction systems based on the proportional Luenberger observer is presented, among others, in [

20].

The authors of many publications indicate that Luenberger observers used for magnetic flux estimation show relatively good properties and can be used in various control systems of induction machines. These observers are compared with others in terms of various criteria, including steady-state accuracy, performance in dynamic states, estimation quality at very low speed, robustness against machine model parameter variations, robustness against noises, computational complexity and real-time implementation. The conclusions are that the Luenberger observers are characterized by relatively good robustness to changes in the parameters of the machine model, easy hardware implementation and low computational complexity. Moreover, it is relatively easy to shape their dynamic properties (by locating their poles on the complex plane in the desired position), which is important in ensuring the stability of the control system. The literature on the subject shows that Luenberger observers display a slightly worse resistance to noises and interference of input signals than Kalman filters, for example, but due to other disadvantages of Kalman filters (complicated implementation, high computational complexity) and other estimation systems based on neural networks or fuzzy logic, the use of Luenberger observers seems to be the right choice. Such conclusions were drawn, among others, by the authors of [

21], in which a comparison is made between the proportional Luenberger observer, the Sliding Mode Observer (SMO) and the Extended Kalman Filter (EKF), operating in the sensorless Direct Field Oriented Control (DFOC) system. Similarly, in [

22,

23] the proportional Luenberger observer, the Kalman filter and the observer using neural networks are compared in the sensorless system. In [

24] a nonlinear observer with a structure similar to the Luenberger observer and the Kalman filter are compared in the control system with speed measurement. In each of the studies mentioned, similar conclusions are obtained, which indicate the benefits of using the Luenberger observer (or an observer with a similar structure).

Among all known types of Luenberger observers, only the proportional type is commonly applied in induction motor control systems [

1]. This observer, as well as other solutions obtained from that, for example the one described in [

25], have one great advantage—it is easy to calculate their gains. In case of the observer described in [

1], eigenvalues are proportional to the motor’s values, and only a value for the proportionality factor should be assumed. The estimation error attenuation is stronger when this factor is higher, which results in a better estimation quality. However, in sensorless control system, this dependency is preserved only when the proportionality factor is relatively low (i.e., less than about 1.75). Once this value is exceeded, the estimation quality deteriorates, as the observer tends to amplify noises. This results in lowered robustness. A similar observation is made for a wider class of observers in [

26].

In control systems, the observer operates in the presence of noises and parameter variations. In most cases, the reference voltage (sinusoidal waveform) is calculated by the control system and is passed to the observer’s input. However, the actual voltages feeding the motor are generated by a Pulse Width Modulation (PWM) inverter (square waveforms). The difference between the calculated and generated voltage waveforms results from the presence of higher voltage harmonics, non-ideal compensation of dead time and voltage drops on the inverter switches. This difference should be treated as a noise overlaying the observer’s input signal. Another set of observer’s input signals consists of measured stator winding currents. These currents contain components that result from the physical phenomena present in the motor but not taken into consideration in the motor’s mathematical model (i.e., the nonlinearity of the magnetic core and slot harmonics). These components should also be treated as noises. Moreover, the parameter values of the motor’s mathematical model used for an observer design are usually different from the real values. This results from identification errors and parameter variations due to physical phenomena, such as thermal changes of winding resistances. Therefore, it is advisable to design the observer by considering an increase in its robustness. The robustness of the observer may be ensured by either the proper observer’s gains selection or by application of the observer’s feedback different than the proportional one. This paper aims to discuss the observer design techniques based on both methods.

In this paper, an optimization gain selection method has been developed based on a genetic algorithm (the optimization described in this paper is not a part of an on-line control strategy as in [

4,

5,

6,

8,

9,

10], but is a tool for an off-line observer design). The novelty of the current work is its applied fitness function that takes into consideration not only the criteria based on eigenvalues but also the additional criterion, which enhances the robustness of the observer. Moreover, the proposed method enables to enforce relations between the observer’s gains to fulfill the practical requirement for electric drives, that is to provide the same dynamics independently to rotation direction. Finally, on the contrary to the method described in [

1], the proposed method can be applied to observers that have more state variables than the motor mathematical model. This is essential in cases of observers with feedbacks that are different from the proportional one.

There are many known observers with non-proportional feedbacks containing dynamical units on the contrary to the proportional one. Such structures, although described in the system and control theory, have barely been applied in control systems of an induction motor. So far only a proportional-integral (PI) observer has been applied, and to a limited extent. This includes mostly for rotor temperature estimation [

27] and fault detection [

28], and not as a source of feedback signals for a control system. An example of a full-order PI observer for magnetic fluxes reconstruction is presented in [

29]. However, this observer is based on the mathematical model of an induction motor that uses rotor current oriented d-q transform and treats the angular speed of rotor current phasor as an input quantity. The need for estimation of the rotor current phasor angular speed decreases the practical usability of this solution. All the observers proposed in this paper are described in a stationary α-β coordinate system, therefore they do not require estimation of any phasor’s speed. A reduced-order PI observer for rotor flux components reconstruction is described in [

30]. Reconstructing two of four state variables of an induction motor, this observer cannot operate with speed adaptation mechanism, on the contrary to the full-order observers proposed by the authors.

Also, it should be mentioned that other types of non-proportional observers presented in this paper are applied for the first time in the control systems of an induction motor.

Application of non-proportional observers in their basic forms may be impossible for a certain class of observed systems, because of resulting observers’ structural instability. It can be proven, that an induction motor also belongs to this class [

31]. To solve this problem, the authors proposed the modification of original observers’ structures consisting in replacement of observers’ feedback integrators with first-order inertias. Replacing an integrator with inertia has been previously applied in simple estimators of induction motor magnetic fluxes [

32]. However, it is applied in non-proportional observers for the first time in the current work. This change affects the dynamical properties of the observers, therefore it is taken into consideration in the gain selection process, by proper modification of the observers’ state matrices.

The most important contributions are as follows:

Application of a reduced order integral unit PI observer, a modified integral observer and proportional observers with additional integrators in an induction motor control system for the first time.

Proposed modification of non-proportional observers’ mathematical models that prevents them from structural instability.

Proposed new simple gain selection criterion that enhances observer’s robustness.

## 2. Methodology

In this paper, the selection of the gains of the proportional observer is described in its general form, which is also suitable for transformed non-proportional observers. The workflow of a proposed gain selection process is presented in

Figure 1. The process starts with observed system (induction motor) mathematical model analysis, which is the base for the observer’s structure design. Next, the structure of the observer is chosen. If a non-proportional observer is chosen, then its structure must be modified, to prevent instability and transformed to the form of the proportional observer. Once the observer is transformed, its gains are to be optimized with a genetic algorithm. The transformation causes that the gain optimization is performed the same way, independently of the type of the observer. The described method may be applied with a program written in the Scilab environment as attached to this paper (

Supplementary Materials, the

START genet.sce file). The program in its original form is for gain selection of a proportional observer; however, it can also be applied for non-proportional observers. To do that, the matrices in lines 70–73 in the file

START genet.sce are replaced with proper forms from lines 67–223 of the file

START sim.sce. Also, the assumed structure of the gain matrix in lines 87–94 (

START genet.sce) should be properly changed, to have the same sizes as gain matrices in lines 67–223 (

START sim.sce).

The gain selection process is performed once, during observer design. Once calculated, the gains do not change during observer’s operation. The observers described in this paper may be tested with the attached simulation model (see the Scilab-Xcos file sim model.zcos). The simulation can be run with the START sim.sce file. The model is simplified, although it enables testing the observer’s performance in the presence of noises and parameter variations. It should be noted that such tests cannot be performed in the laboratory, since in a real electric drive, the values of noises and parameter variations are unknown. This is why simulations are so helpful. However, laboratory tests are the last step of the observer design process. The laboratory test results for the observers described in this paper can be found in previously published papers by the authors.

The observers’ gains optimized with the algorithm in the file START genet.sce may be tested with simulation started by the file START sim.sce. To do that, proper observer structure should be chosen in line 68 of the file START sim.sce, and new gains values should be typed in the appropriate places between lines 67–223. Both files may be used for induction motors that are different from the one used by the authors. The motor’s rated parameters and per-unit system base quantities may be changed in the file START genet.sce in lines 46–55 and in the file START sim.sce in lines 41–53. These values in both files should be the same.

## 4. Discussion

Practical application of the non-proportional observers encounters problems that need to be solved to provide proper operation. First of all, when the number of the state variables of the observer is greater than the number of the state variables of the observed system, it may be necessary to modify the observer’s structure to provide stability. Despite this problem, the application of the observer with extended non-proportional feedback may provide better estimation quality than the application of the classical proportional observer.

Another problem is the proper gain selection. Two observers of the same structure and eigenvalues may have different gains and result in different robustness. Simulation results were presented in this paper for comparison purposes. These results were obtained under identical conditions for all presented observer structures. Presented observers were also successfully tested in the laboratory and results have been presented in the cited bibliography. Therefore, all of them are applicable in control systems of an induction motor.

All discussed observers were tested using the simulation model presented in

Figure 5, in the same conditions, i.e., the introduced noises and motor’s parameter variations. The observers operated provided with a speed-adaptation mechanism [

1,

35,

40]. The motor was fed with the voltage generated according to V/Hz = constant rule by a PWM power inverter. The observer’s inputs are passed reference voltages

u_{sαref} and

u_{sβref}, generated by the control system (sine waves), whereas the motor was fed with phase voltages based on those parameters but generated with PWM method (square waves). In the model of the motor, parameter variations were introduced. All the results were obtained with the same simulation model attached to this paper (

Supplementary Materials, the

START sim.sce file). For consecutive observers only the matrices of the observer’s simulation model were replaced. This also concerned the non-proportional observers described in

Section 3.3. Therefore, the inner structure of the observer’s block was always the same. The waveforms illustrated three consecutive transient states, the start-up (0–0.5 s), the step switching on of the load torque (0.75 s) and the reversal (1–1.8 s). The load torque was active; therefore, the motor operated as a generator for negative speed values. The values of the state variables of the motor were given as dimensionless p.u. values [

36].

Simulation results of the proportional observer are presented in

Figure 6. It shows that the observer has relatively small real parts of the eigenvalues

λ, which are smaller than the non-proportional observers described in

Section 3.3. It means that the time constants of the proportional observer are shortest; nevertheless, this observer does not provide the best estimation quality. The reconstruction quality of magnetic fluxes has an impact on the operation of the speed adaptation mechanism; therefore the transient of the estimated angular speed ω (

Figure 6) is a good measure of the observer’s performance. In this case, a significant difference between the real and estimated speed values is visible during the start-up of the motor (

t = 0.3 s). Some small differences are also visible during the reversal of the motor (

t from 1.3 to 1.6 s). It should be noted that the reversal (the change of rotation direction) of the motor is the most difficult state of operation from the observer’s point of view. In this state, the variations of the equivalent circuit parameters have the greatest impact on the estimation quality (especially variations of the stator and rotor windings resistances

R_{s} and

R_{r}). Moreover, during the reversal, the angular speed ω crosses 0. From the equivalent circuit (

Figure 2) and the state matrix

**A** (3) of the system (1), it can be seen that at the moment when ω = 0, the mathematical model divides into two subsystems that are not coupled with each other. The lack of coupling between the state variables deteriorates the operation of the observer’s feedback. This is why the reversal is a good test for the observer’s performance and robustness.

The PI observer (

Section 3.3.1,

Figure 7) has much longer time constants than the proportional observer described in

Section 3.2. Nevertheless, its performance is slightly better. The weaker error attenuation, resulting from longer time constants, is compensated by structurally stronger feedback. The PI observer operates much better during the startup—angular speed estimation errors are smaller, although the performance during the reversal is slightly weaker. On the other hand, the selection of the gains is much more difficult in the case of the PI observer compared to the proportional observer. The problem results from the fact that the PI observer’s matrix

**A**_{o} (33) is twice the size of matrix

**A** (3) of the proportional observer. Also, the gain matrix

**K** has twice the amount of elements.

The PIr observer (

Section 3.3.2,

Figure 8) has similar time constants as the PI observer described in

Section 3.3.1, although it provides slightly better performance, especially during the reversal. The PIr observer is also better than the observers described in

Section 3.3.3 and

Section 3.3.4. The reduction of the integral unit has been proven to be a good compromise between the classical PI observer and the proportional observer. It combines the advantages of both of them, i.e., relatively simple structure and strong, proportional-integral feedback.

The modified integral observer (

Section 3.3.3,

Figure 9) is worse in comparison to the others described in this paper. The transient of the estimated angular speed in

Figure 9 shows significant errors at all times, even in steady-state condition (

t from 0.8 to 1 s). Potential gaining offered with less noise attenuation is dampened by disadvantageous results of replacing the integrator with the inertia. The replacement is necessary to provide observer stability, although it distorts the observed system’s outputs

**y** passed to the observer. It is noticed that the outputs of the induction motor are sine-like waveforms; therefore, the impact of the inertia on them is significant. Therefore, this structure is not a good choice in cases of an induction motor, though its assets may be beneficial in other observed systems.

In the case of the observers with additional integrators (

Section 3.3.4,

Figure 10;

Figure 11), the best results are achieved for

v = 1 (

Figure 10). Its operation during the reversal is as good as the PIr observer (

Section 3.3.2), but its performance during the start-up is slightly weaker. The performance of both observers is similar, although the observer with an additional integrator has shorter time constants, resulting in stronger error attenuation. It means that the introduced additional integrator does not offer such possibilities as the proportional-integral feedback in the PIr observer. On the other hand, let us notice that the additional integrator is intended for attenuation of only one type of noise, the one overlying the angular speed.

Increasing the number of integrators up to

v = 2 does not significantly improve the observer’s performance (

Figure 11). The result for

v = 2 is similar to that obtained for

v = 1, but the observer with two additional integrators has a more complicated structure and as many state variables as the classical PI observer (

Section 3.3.1). Based on that, it can be concluded that the application of more than one additional integrator is ungrounded. As the results obtained for

v = 3 were even worse than for the previous two cases, the laboratory tests described in [

35] were only performed for

v equal to 1 and 2.

The PI and PIr observers, as well as the proportional observer with additional integrators, applied in the control system of an induction machine, provide better estimation quality at longer time constants than the proportional observer. Acceptable longer time constants mean that the displacement of the observer’s poles in the complex plane may be lesser in reference to the position of the motor’s poles. Lesser poles’ displacement results with lower gain values, making the observer more robust against noises and parameter variations.

Future research will be focused on improving the repeatability of the applied optimization method. Possible solutions are the application of other evolutionary strategies as firefly algorithms or simulated annealing as well as the application of a hybrid two-stage algorithm, where the first stage is an evolutionary one to find a global extrema and the second is a deterministic one to precisely locate the optimal solution [

41]. Another goal for future research is finding criteria that will help to locate optimal settings of observers’ modified integrators contained in the matrix

**Ω**.