An Implementation of Rotor Speed Observer for Sensorless Induction Motor Drive in Case of Machine Parameter Uncertainty

The paper describes observers using model reference adaptive system for sensorless induction motor drive with the pulse width modulator and the direct torque control under the circumstances of incorrect information of induction motor parameters. An approximation based on the definition of the Laplace transformation is used to obtain initial values of the parameters. These values are utilized to simulate sensorless control structures of the induction motor drive in Matlab-Simulink environment. Performance comparison of two typical observers is carried out at different speed areas and in presence of parameter uncertainty. A laboratory stand with the induction motor drive and load unit is set up to verify the properties of observers. Experimental results confirm the expected dynamic properties of selected observer.


Introduction
The term "sensorless" means that the electric drive does not contain the speed or position sensor in its control structure, and the concept of sensorless control is to use estimation algorithms to obtain rotor speed, ro-tor position or flux from voltages and currents of motor terminals.These signals are measured using current and voltage sensors, the important parts of the electric drive structure, and processed by suitable observers that are embedded into a real control system with Digital Signal Processor (DSP).The main techniques which are used for rotor speed estimation can be classified as follows: • Machine model-based methods that include observers: Model Reference Adaptive System (MRAS), Extended Kalman Filter (EKF), Luenberger Observer (LO), Sliding Mode Observer (SMO), etc.
• Methods without machine model that include injection methods and soft computing methods.
The scheme of machine model-based methods is simple and easy to implement into DSPs.However, there are specific demands on these methods from different drives.The popular sensorless method, reference frame or Rotor Flux MRAS (RF-MRAS) observer, is based on the rotor-flux error calculated using two different voltage and current models of the rotor flux [1].
However, the pure integration of the voltage signals particularly at low speeds, and the low accuracy because of machine parameter uncertainties make it difficult-to-implement.Combination of the method and other techniques such as Kalman filter is one of approaches for performance improvement [2].A sixthorder discrete-time model is used to analysis convergence of the EKFs for sensorless motion control systems c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING with Induction Motor (IM) [3].The Luenberger observer can be a replacement for the observers that use machine model, have accuracy problems, especially at low speeds, and for non-linear systems.The deterministic observer comprises the deterministic plant model while the stochastic type comprises other plant model representations, and the Luenberger observer is a typical deterministic observer.The LO for sensorless IM drive is an observer that uses machine model and a feedback loop with measured stator current and rotor flux.However, the gain matrix of the full-order observer must be designed according to Lyapunov's stability theory, and the observer must be faster than the observed system [4].
Another solution is utilization of the SMO.The main difference between LO and SMO is that the error between the stator current and its estimated quantity is replaced by a sliding function of the error in SMO.This sliding function is designed thanks to sliding mode theory for robustness assurance to machine parameter deviations.The sliding-mode component ensures robustness to large parameter variations and measurement noise in combined scheme of SMO and LO [5].The SMO is used in rotor flux estimation of sensorless IM drive and gives high robustness to saturation of the magnetizing inductance and variation of the rotor resistance [6].
In the stator Current-Based MRAS (CB-MRAS) observer, the estimated value of the stator current is obtained by a model.The difference between the real stator current and its estimated value is used to derive the speed-error correction signal [1].There is no design of the gain matrix in the observer.The stability analysis of the observer was tested from the point of view of the induction motor and Proportional-Integral (PI) controller parameter changes, on the basis of the observer transfer function.The RF-MRAS, Back Elec-troMotive Force MRAS (BEMF-MRAS), CB-MRAS observers were analyzed by simulations and the CB-MRAS scheme was chosen for implementation into real control system because of its good tracking capability [7].
Parameters of an induction motor vary with its working condition, and machine model-based speed observers are sensitive to variations of parameters of induction motor such as the vector control that is very sensitive to variations in the rotor time constant [8].In general, methods of stator resistance or rotor time constant estimation are based on application of observers, EKFs, MRASs, artificial intelligence [9], [10], [11], [12], [13] and [14].These methods require complicated design and computing and does not ensure applicability in real system.Therefore, in the paper, an offline identification is done for obtaining initial values of machine parameters.Then these values are used in simulations to select an observer that has both simple structure and strong robustness to machine parameter deviation.

Offline Parameter Identification
In the paper, the following symbols are used: u s , i s , ψ r -respectively stator voltage vector, stator current vector, rotor flux vector in [α, β] coordinate system; u sα , u sβ -components of the stator voltage vector; i sα , i sβ -components of the stator current vector; ψ rα , ψ rβ -components of the rotor flux vector; R s -stator resistance; L s , L sσ -stator inductance, stator leakage inductance; R r -rotor resistance; L r , L rσ -rotor inductance, rotor leakage inductance; L m -magnetizing inductance; σ -total leakage constant; T s , T r -stator, rotor time constants; ω r -real rotor angular speed; denotes the estimated values.For induction motor, equivalent circuit of one phase is shown in Fig. 1 where u(t), i(t) are input and output signals, which are the phase stator voltage and the phase stator current respectively.Stator and rotor leakage inductances are respectively defined by following equations: Assume that the circuit is initially relaxed, following equation shows the relation between Laplace transformations of the input and the output [14]: We get the following equation by inserting Eq. ( 1) and Eq. ( 2) into Eq.( 3): In case of the phase stator voltage is a constant DC signal, Laplace transformation of the output is calculated according to Eq. ( 5): where U is an input DC voltage.The output is obtained by applying the inverse Laplace transform [15] to Eq. ( 5): where: Experiments with different values of U are carried out.Many methods can be used to approximate IM parameters.For simplicity, the saved data are processed to obtain parameters thanks to the simple approximation of Laplace transform: where n is number of samples of the saved data, k is representative of the k th sample of the time or the phase stator current.Initial values of IM parameters are listed in Tab. 1. Tab.

Model Reference Adaptive System Observers
The structure of RF-MRAS observer is shown in Fig. 2(a).In the reference model or the voltage model, the stator voltage and stator current vectors are used for obtaining rotor flux vector.In the adaptive model or the current model, the estimated rotor flux vector can be gotten from stator current vector together with the estimated value of rotor speed.
The outputs of reference model and adaptive model are calculated according to Eq. ( 11) and Eq. ( 12): The relationship for the rotor speed estimation is calculated according to Eq. ( 13) and Eq. ( 14): where ξ is called the adaptive signal, K P > 0, K I > 0. Hyperstability of the RF-MRAS observer can be proved as in [16] thanks to Popov's theorem of hyperstability and asymptotic hyperstability [17], definitions of positive real and strictly positive real transfer function matrix, and strictly positive real transfer function matrix [18].The system of the observer that consists of Eq. ( 11), Eq. ( 12), Eq. ( 13) and Eq. ( 14) fulfills conditions of the lemma "Strictly positive real transfer function matrix" because rotor time constant is always larger than zero.
The structure of CB-MRAS observer for the rotor speed estimation is shown in Fig. 2(b).This MRAS observer uses stator currents as output quantities of the reference model.The CB-MRAS scheme has one reference model (stator current of induction motor) and two adaptive models (current estimator and model current).The output of the current estimator, which is the estimated stator current vector, is calculated according to Eq. ( 15), Eq. ( 16), Eq. ( 17), Eq. ( 18) and Eq. ( 19): The current model is identical to the adaptive model which is described by Eq. ( 12).The relationship for the rotor speed estimation is calculated as follows: where ξ is also called the adaptive signal, K P > 0, K I > 0. The global asymptotic stability of the CB-MRAS observer can be proved through the second Lyapunov's method with the same Lyapunov's function candidate, as in the case of LO [19], and there are no requirements in design of gain matrix of the observer.The K I component in Eq. ( 14) and Eq. ( 21) is removed whenever one of two limits of estimated rotor speed is reached because this component has a much larger influence on the observer stability than the K P one [1].

Simulation Results
For high performance applications of IM drives, Vector Control (VC) and Direct Torque Control (DTC) are used for electromagnetic torque control.They can be comparable to each other.The advantage of the DTC methods is that their structures are not as complicated as those of VC methods and therefore, they are easy to implement on DSPs.The utilization of PWM in the DTC structure is to ensure the constant switching frequency.In this section, the PWM-DTC control structure with two MRAS observers were simulated (see Fig. 3).Parameters of induction motor used in simulations were given by Tab.To evaluate the control quality of simulated sensorless IM drive it is important to evaluate the speed responses in different situations.The simulations were performed for the reference speeds which represent two  speed areas: area of low speed (1.0 ω ref ) and area of very low speed (0.1 ω ref ).Speed deviations are evaluated as follows [7]: The two abbreviations RF and CB respectively denote the RF-MRAS and the CB-MRAS observers.Figure 5, Fig. 6, Fig. 7 and Fig. 8 show real rotor speed, estimated rotor speed, speed difference between real rotor speed and estimated rotor speed at low and very low speed areas.Speed deviations for two speed areas are listed in Tab. 2. Deviations at both dynamic and steady state operations with RF-MRAS are much larger than those with CB-MRAS at two speed areas.A test at usual speed area was also simulated to compare the performance of two observers.Figure 9 claims that RF-MRAS gives speed deviations at both operations larger than CB-MRAS does.The described observers are based on IM model.Hence, it is necessary to consider the sensitivity of the observers with parameter uncertainty.Some simulations are used to check the robust stability of two MRAS observers.Reference speed and load torque at low speed area are the same as those (see Fig. 4) with time range 0 to 0.6 seconds.For simulations, two selected parameters are R s and T r (or R r ) with changes from −10 % to +10 % of their known values.In case of R s and T r decreased by 10 % of their known value, real rotor speed and speed difference are shown in Fig. 10, speed difference with CB-MRAS is significantly smaller than that with RF-MRAS.For comparsion, deviations in both dynamic and steadystate operations are listed in Tab. 3, Tab.4, Tab. 5 and Tab.6 where smaller deviations are bold.In most cases, deviations with CB-MRAS are smaller than those with RF-MRAS.Therefore, the CB-MRAS gives better tracking capability at different speed areas and stronger robustness in presence of parameter uncertainty than the RF-MRAS does.Two coefficients K 1 , K 3 are constant with the changes of R s and R r .Percentage of changes of coefficients K 2 , K 4 compared to their known values are listed in Tab. 7. It is easy to see that the deviations with CB-MRAS are more affected by coefficient K 4 than by coefficient K 2 .For CB-MRAS, the coefficient K 4 is experimentally adjusted more elaborately than three remaining in next section.

Experimental Results
A laboratory workplace with an induction motor and a load unit was used to verify performance of two MRAS observers with PWM-DTC (see Fig. 11).The load unit is an electro-dynamometer with torque range  The control system is a TMS320F28335 digital signal processor.
Figure 12 and Fig. 13 show real and estimated rotor speeds, speed difference and motor torque without load at ω ref = 100 rpm.The time courses confirm that speed deviations with CB-MRAS at both dynamic and steady state operations are significantly smaller than those with RF-MRAS.Hence, the CB-MRAS is chosen for experiments with load (see Fig. 14, Fig. 15, Fig. 16 and Fig. 17       Maximal differences after an activation and a release of a load unit are listed in Tab. 8.The differences tend to increase when the speed is lower.

Conclusions
The sensorless IM drive using MRAS speed observers with PWM-DTC method were presented in the paper.Simulations were carried out with simultaneous changes of machine parameters to select suitable MRAS observer for implementation on real control system.The IM drive with two MRAS observers gave good dynamic responses and the estimation of the mechanical speed was good in both dynamic and steadystate operations.The CB-MRAS observer gave higher accuracy at different speed areas, and stronger robustness to uncertainty of IM parameters than the RF-MRAS observer.It was the selected solution for sensorless IM drive.
The experimental results with sensorless IM drive using the MRAS observers without load confirmed the assumptions in simulations, and the CB-MRAS was chosen for experiments at many speed areas with load.The CB-MRAS gave acceptable performance at tested speeds.However, at lowest speed with load, relative speed difference is largest.More robust or sophisticated algorithms such as sliding mode theory, artificial intelligence should be developed for parameter IM identification and speed estimation at low and very low speeds.The methods for increasing the switching frequency, and dead-time compensation of power electronics devices should be utilized.The active load unit with large torque range should be constructed.

Fig. 1 :
Fig. 1: Equivalent circuit of one phase of an induction motor.

17 :
Motor speeds (upper) and stator current (lower) with CB-MRAS at ω ref = 25 rpm.c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING Tab. 2: Speed deviations at low speed and very low speed areas.
Tab. 3: Deviation |∆ωr| in dynamic operation.Deviation |∆ωr| in steady-state operation.Percentage of changes of two coefficients K 2 , K 4 compared to their known values.