Application of Sensorless Sliding Mode Observer in Control of Induction Motor Drive

Induction motors are widely used in an industry and it is necessary to improve control methods for induction motors to increase the efficiency of them. In this paper, sliding mode controllers are proposed instead of traditional PI controllers in vector control of induction motor drives. Moreover, rotor speed is estimated by a sliding mode observer. In addition, the robustness of control and observer algorithms are also proved by Lyapunov’s criterion. The experiments are obtained in different speed changes of an induction motor drive. These experimental results confirm the dynamic properties of a sensorless sliding mode control of an induction motor drive.


Introduction
Induction motors are a type of AC electrical drives.They are widely used because they are robust, sturdy and require low maintenance.However, control of an induction motor is very difficult because of variable frequency, complex dynamic and parameter variation, etc. [1].
Vector control is one the most popular methods for controlling induction motor drives because of efficiency in a wide speed range.However, in traditional vector control, PI controllers are used.PI controllers depend on parameters of the system and are sensitive to external noise.Sliding mode control was introduced in 1977, which overcomes disadvantages of PI control [1].However, disadvantage of sliding mode control is chattering phenomenon.In recent years, many researchers have been involved in solving problems such as fuzzy sliding mode control [4], rotor resistance estimation [3], chattering reducing [1], etc.
During last years, sensorless control of AC machine has been developed.Sensorless means that the rotor speed is estimated from an observer or an estimator such as Extended Kalman Filter (EKF), Model Reference Adaptive System (MRAS) [5], Luenberger Observer (LO), Sliding Mode Observer (SMO) [2] and [6], etc. Sensorless control increases the reliability and decreases cost and complexity of the system.Among them, MRAS and EKF are suitable for applications with a medium speed.In contrast, LO and SMO are robust to noise and parameter variation [7].
In this paper, integral sliding mode control is proposed in controlling of induction motor drives.Moreover, a sliding mode observer is used to estimate rotor speed.Firstly, a mathematical description of the induction motor is derived.Next, the sliding mode observer, sliding mode controller and the proof of their robustness are in the second part.The structure of the sensorless sliding mode control of the induction motor is introduced in the third part.A laboratory stand is built and some experimental results are presented in the fourth part.Finally, some conclusions are summarized.

Mathematical Model of Induction Motor
The state representation of an induction motor is: where T is the input vector and y is the output vector.A, B and C are the state matrix, the input matrix and the output matrix, respectively; and

Sensorless Sliding Mode Control of Induction Motor
In this section, a sliding mode observer and an integral sliding mode controller are developed for controlling induction motor drive.

Sliding Mode Observer
The sliding mode observer can be given as: where x is estimated state vector, ŷ is the estimated output vector, G is the gain matrix of the sliding mode observer, S is the sliding mode surface.G and S are defined as: The rotor speed is estimated online by a PI controller as following: where K P ω , K Iω are proportional and integral constant of a PI controller, and

Sliding Mode Control
In this section, the sliding mode control is proposed.
A sliding mode controller is designed in two steps: choosing a sliding mode surface and choosing control signals.
Integral sliding mode surfaces are chosen as: where S x and S y are sliding surfaces, i * m is the reference magnetizing current amplitude, i m is the actual magnetizing current amplitude, ω * m is the reference mechanical speed and ω m = (2/p) • ω R is the actual mechanical speed with p is the number of poles of induction motor.
In Eq. (15) and Eq. ( 16), the c x and c y are chosen to get the desired characteristics.The flux current component and torque current component are: where T R = L R /R R is the rotor time constant, K T is the proportional constant between the motor torque and the torque current component, J is the moment of inertia.
Suppose that the rotor time constant and the load torque are changed but are bounded as: The system is asymptotic stable if and only if K x and K y satisfy: Proof: The Lyapunov function is chosen as: The Eq. ( 23) is positive definite function.The first derivative of Lyapunov function is: The Eq. ( 24) is negative definite function if: where sign is the signum function: We have: Replace i Sx in Eq. ( 17) into Eq.(28), Eq. (28) becomes: The first derivative of S x has the opposite sign with S x so condition for K x is derived as in Eq. ( 21).Next, the first derivative of S y is: Similarly, the first derivative of S y has the opposite sign with S y and the condition for K y is in Eq. ( 22).
Therefore, with the control signals as in Eq. ( 17) and Eq. ( 18) and with the condition of K x and K y as in Eq. ( 21) and Eq. ( 22), the system is asymptotic stable.However, to eliminate the chattering problem, the saturation is used instead of signum function in Eq. ( 17) and Eq.(18).

Sensorless Sliding Mode Control Structure of Induction Motor Drive
The control structure of Induction Motor (IM) drive is shown as in Fig. 1.This figure is the basic scheme of vector control of an induction motor drive but PI controllers in speed control loop and flux control loop are replaced by Sliding Mode Controllers (SMC).A sliding mode observer is used to estimate rotor speed and a current model is used to estimate the rotor flux components.The control of induction motor is described step by step as follows: • Step 1: i Sa , i Sb are measured by current sensors.
• Step 2: i Sα , i Sβ are calculated from i Sa , i Sb .
• Step 3: The stator angle γ and rotor flux components are estimated by a current model.

Experimental Results
A laboratory stand with an induction motor drive is designed for the experimental verification of the chosen algorithms.The experimental platform is shown in Fig. 2. In the laboratory stand, a Siemens induction motor with the catalog number 1LA7106-4AA10 is used.Parameters of the induction motor are given in Tab. 1.At the end of the induction motor, an incremental sensor is mounted and its resolution is 3000 pulses per revolution.The induction motor is connected to an indirect frequency converter with the voltage DC-link, which was designed at the Department of Electrical Power Engineering, VSB-Technical University of Ostrava.The control of the voltage inverter is performed by the sinusoidal pulse width modulation with control frequency 10 kHz and control amplitude 10 V. Therefore, the transfer constant K M is 1/2.
The control system is a Digital Signal Processing (DSP) of Texas Instrument eZdsp TM F28335 because it is highly integrated and is a high performance solution for a demanding control application.The most important part of program is ADC interrupt subroutine, which is done for every 50 µs.Setting reference speed, measuring current components, calculating current and flux components as well as control signals, etc. are done in this subroutine.The load torque is from an electrodynamometer.At the speed of 100 rpm, the maximum load torque is just 1.5 Nm.In our laboratory, the data are stored in the memory of DSP and then transferred to the MATLAB environment to draw responses.The sampling time is chosen as 10 ms.Because of the limitation of memory of the DSP, the response time is 4 s and the response of speed is divided into two cases with and without load.Without load, the reference speed is 100 rpm in the forward rotation and 60 rpm in the reverse rotation.With load, the reference speed is chosen as 100 rpm and the load torque of 1.5 Nm is added at the time of 1.5 s.The response of speed is good, the percentage of overshoot is 9 %, the time response is 0.7 s and the error between the reference speed and the actual speed is just about 1 rpm.The speed decreases to 98.5 rpm at the load.After the control process, the speed re-   turns to the value corresponding to the reference speed (see Fig. 8).

Conclusion
In this paper, the sensorless sliding mode control with sliding mode observer is presented in a low speed range.The theoretical assumptions are verified through simulations in MATLAB-Simulink environment and experiments on laboratory stand.The estimated rotor speed can be used in the speed control of the induction motor drive.The absolute error between the reference speed and the actual speed in sensorless sliding mode control of the induction motor is just 1 rpm without load and 1.5 rpm with load at the reference speed of 100 rpm (percentage of error is just 1 % to 1.5 %).When comparing with PI controllers, sliding mode controllers give better results, the relative size of overshoot is smaller, settling error is smaller as well and it is more robust to external noise.When having load torque, in sliding mode control, the decreasing of rotor speed is smaller than in PI control.
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Figure 3 ,
Figure 3, Fig. 4, Fig. 5, Fig. 6 and Fig. 7 show the time response of important parameters of the induction motor drive, which were obtained by measurement from the laboratory stand in case of no load.When having load, the time response of rotor speed, torque and magnetizing current component and flux current component are in Fig. 8, Fig. 9 and Fig. 10, respectively.
a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44 Sα and i Sβ are stator current components, u Sα and u Sβ are stator current components, ψ Rα and ψ Rβ are rotor current components in the stator coordinate system [α, β] respectively; R S and R R are stator and rotor resistances, respectively; L S and L R are stator and rotor inductances; L m is the mutual inductance; σ is the leakage constant and ω R is the rotor speed.
where i Sα , u Sβ , i Sα and i Sβ are input of sliding mode observer, which estimates the rotor speed.The stator voltage components u Sα and u Sβ depends on the u * •Step 4: Calculate i Sx , i Sy from i Sα , i Sβ and γ.• Step 9: The u Tab. 1: Parameters of the induction motor.