Hardware-in-the-loop based comparative analysis of speed controllers for a two-mass system using an induction motor drive with ideal stator current performance

Received Sep 5, 2020 Revised Dec 12, 2020 Accepted Jan 5, 2021 A comparative study of speed control performance of an induction motor drive system connecting to a load via a non-rigid shaft. The nonrigidity of the coupling is represented by stiffness and damping coefficients deteriorating speed regulating operations of the system and can be regarded as a two-mass system. In the paper, the ability of flatness based and backstepping controls in control the two-mass system is verified through comprehensive hardwarein-the-loop experiments and with the assumption of ideal stator current loop performance. Step-by-step control design procedures are given, in addition, system responses with classical PID control are also provided for parallel comparisons.


INTRODUCTION
Asynchronous electrical drive system is found in enormous applications and is considered as wellestablished thank to the advancement in current loop control [1]. However, facing mechanical loads, speed control loop design takes a decisive role, especially when the non-rigidity and backlash of the motor-to-load connection is taken into account [2][3][4][5]. Flexible coupling phenomenon is studied in [6], the authors develop an elegant control scheme integrated with a state observer for estimating motor and load torques. Brock et al. [7] propose a filter accompanied with a neural adaptive controller to eliminate high resonance frequencies.
The controlled system is analysed through extensive experiments proving the effectiveness of the proposed solution. For a class of multi-mass electrical drive systems where measurement is not always available, a system parameter identification method is presented in [8]. The authors employ pseudo random binary signal and chirp wave to excite necessary information. The approach might be difficult to apply in a wide range of practical systems. A 2DOF PI control is systematically designed in [9] for a two-mass drive coupled via a toothed belt. The tracking results are compared with 1DOF and 1DOF with feedforward controls confirm that the method exhibits robustness and load torque rejection. Due to flat property of the system, Thomsen et al. [10] formulate a flatness based control for a finite stiffness coupling drive system fed by an inverter with speed feedback signal. Backstepping control is developed for a multi-motor system which is vastly found in  [11] suppressing flexible coupling phenomenon in linear speed and tension control system. In order to deal with nonlinear stiffness, the authors in [12] implement adaptive backstepping to reduce connecting shaft oscillations. However, system stiffness characteristic must be known to carry out the control, this requirement might be very challenging in practice. Recently, a active disturbance rejection control (ADRC) is considered as an alternative to classical PID control, an application of ADRC to two-mass system can be found in [13]. A fuzzy based control for a three-mass system is presented in [14,15], where fuzzy term is selected to modify IPD control. Other control approaches can be found in [16][17][18][19][20][21][22].
In general, there is a lack of comprehensive study on how popular control methods react when applying to two-mass drive system operating in nominal range. In the paper, we carry out a hardware-in-theloop based comparative analysis of speed responses using backstepping and flatness-based controls in nominal and field-weakening modes. Initially, we present the two-mass drive system in a view of ideal current loop. Subsequently, PID, backstepping, and flatness control design steps are given in details. Finally, hardware-in-the-loop results of three controls are compared and some important conclusions are drawn.

TWO-MASS MODEL SYSTEM WITH IDEAL CURRENT LOOP RESPONSE
Typical configuration of a two-mass system is shown in Figure 1. When operating in field weakening region, the model of the system can be given as [2].
In which, ; are components of the stator current; , rotor and synchronous speeds, respectively; , , , are rotor flux dq components; is total leakage factor; is rotor time constant: , are stator voltage components; is stator inductance, ̇1,̇2are motor and load speeds; ̈1,̈2are motor and load angle accelerations; is rotor angle; is the coupling shaft damping coefficient; is shaft stiffness. It can be seen that the system of dynamical (1) is of 7 th order. With the assumption of t perfectly designed current controller, the system model can be reduced as: where: = . The state (2) is of 4 nd order, stator current isd is used to control the motor flux and isq is dedicated to speed control. For control design purpose (2) is rewritten in the following state-space form: where we have defined, 3. SPEED CONTROL DESIGN 3.1. PI control PI controller is designed according to the symmetric optimal standard, so we have the PI control structure shown as Figure 2. The design of PI for the system is well-established, readers can refer to [23,24] for more details.

Backstepping control
For designing step, it is assumed that all feedback signals from the motor, the shaft and the load sides are available. As in Figure 2, the backstepping control is divided into two steps. In each step, a subsystem will be studied and controlled. Selecting a proper Lyapunov candidate function for each design step can guarantees the asymptotic stability of the corresponding subsystem. After completing all two steps, the asymptotic stability of the whole system can be achieved [25]. The speed control equation is given: control design, we define: where 1 and 2 are state variables; u is the control input and y is the output (the load rotating speed). Since then the system is expressed in the strict feedback form as (7): In order to implement backstepping control we define: where 1 =̇2 is the set value of the output variable and 2 =̇1 is the virtual control. Selecting the Lyapunov candidate function as (9): Taking derivative of (9) results in: Because of ̈1 = 0 so that (8) rendered as ̇1 =̈2, substitution ̇1from (5) into (10) yields: Considerinġ1 as the virtual control variable in such a way that ̇1 =̇1 so that: If ̇1 = − 1 1 2 < 0, with ∀ 1 > 0 then the stable condition is satisfied. Defining: Substitution of (13) into (12) results in: The control specified in (14) ensures that the load speed tracks the desired value. Next, the control for the motor side is designed. It is noted that 1 =̇2 is constant, so that from (5), (8), and (13) can be shortened as: Taking derivative of (16) gives: From (11) and (12), we can write as: After some basic operations, it can be shown that: where we have defined ̇1 −̇1 = 2 , so that: At this point in order to rendeṙ2 < 0, the following conditions must be fullfied: Conditions specified in (21) imply that: From (14), it indicates that:

̈1
= − 2 ( 11 +̇) and ̇−̇1 =̇1 + 2 ( 1 1 + ) Substituting (15) and (22) into (23) gives: To proceed to calculate the final expression of the control current, first of all, we substitution from the second equation of the system (5) into (24) we have: From the expression (25) and using fundamental calculation, we will get the final expression of isq is: Based on (26), the proposed control structure for the two-mass system driven by the induction motor is shown in Figure 3.  [26], it can be deduced that: Assume that the load torque is not varying, then the load observer can be developed as: where = . then from (28) and (29), it is straightforward to show that: The error model is: The characteristic equation of (32) is: By selecting 1 and 2 such that:

Design of the reference trajectories
The Based on feedforward and feedback control design and trajectories of speed and flux reference the proposed control scheme for the two-mass system driven by the induction motor with perfect stator current controller is shown in Figure 4.

RESULT AND DISCUSSION
In this section, the proposed control strategy is verified by hardware-in-the-loop (HIL) simulation. The system hardware is simulated in real time on the HIL platform with a time step of 1μs that means very close to physical models, while the pulse width modulation carrier frequency is 5 kHz. Voltage and current controllers are implemented in DSP TMS320F2808. The simulation parameters are: rated power=2.2 kW; rated phase voltage=380/220 VRMS; rated frequency=50 Hz; d=0.015 Nm/rad; c=0.28 Nm/rad, J1=7.455 x10 -5 Kgm 2 ; J2=5.5913x10 -5 Kgm 2 ; with load torque is 1.5 Nm.
Simulation scenario is: at t=0s, the magnetic current is formed. Then at t=0.5s, the motor starts to speed up to 1000 rpm. At t=2s, the reverse speed, the motor starts to reverse to -1000 rpm. In order to test the system ability, low speed range of 1 rpm is also applied in the simulation. This section provides evaluation of HIL results of speed response. The evaluation of difference speed control strategies for the two-mass system is shown in Figure 5 and Table 1.  It is found that the nonlinear control method based on the flatness-based control principle gives better performance than the PI and backstepping controls, as the referenced load speed match with motor speed after a short period of time, 0.3 seconds at (±) 1 rpm and 0.5 seconds at (±) 1000 rpm (at up and reverse speed), without overshoot. That implies the speed controller based on the flatness-based control can suppress resonance oscillation over the entire operating range of the IM motor.
From the results, it can be seen that the two-mass system with flexible couplings when operating at the speed of (±) 1000 rpm and even at the low speed range (±) 1 rpm, the nonlinear control method, based on the flatness-based control, has proved its advantage by solving the problem of resonant vibration suppression at the spindle compared to the PI controller and backstepping. Besides of at times of transition with fast setting time, without overshoot and load speed match with motor speed. In the following simulation scenario, the motor is set in field-weakening region. The magnetizing current responses are shown in Figure 6 and Figure 7. The three control methods show stable operation not only in nominal but also field weakening range. However, it can be seen that the flatness-based control gives better transient response, i.e., without overshoot, as seen in Figure 7, and shorter settling time (0.2s in comparison with 0.5s for PI control and backstepping method).

CONCLUSION
In this paper, the two-mass system with flexible couplings comprising of an induction motor and a load is considered. At the motor side, the current response is assumed to be ideal leading to a simplified system model. PI control, Flatness-based control, and backstepping control are employed to solve flux and speed control problems of the system. The HIL based test shows better control performances of flatness control compared to the other two schemes. This conclusion implies possibility of practical uses of flatness control in dealing with systems with flexible couplings. In the future research, in order to give a concrete evaluation, robustness experiments against varying system parameter and load disturbances will be carried out.