Sliding Mode PWM-Direct Torque Controlled Induction Motor Drive with Kalman Filtration of Estimated Load

The paper presents an application of sliding mode controller and Kalman Filter (KFSMC) in speed control of pulse-width-modulation direct torque controlled induction motor drive. Performance of the direct torque control (DTC) is degraded by uncertainty of stator resistance. In order to increase robustness of controlled system to the uncertainty, sliding mode controller (SMC) is utilized to replace proportional-integral (PI) speed controller in conventional DTC drive structure. Computation of SMC requires estimation of load, and Kalman Filter is integrated to reduce noise in load estimation and chatteringphenomenon in speed response. Simulations are carried out at di erent reference speeds in widerange noises of stator resistance. Indices including ITAE, settling time, overshoot and undershoot are employed to compare performance of drive structures. Results con rmed the desired characteristics of the proposed drive structure.

Abstract. The paper presents an application of sliding mode controller and Kalman Filter (KF-SMC) in speed control of pulse-width-modulation direct torque controlled induction motor drive. Performance of the direct torque control (DTC) is degraded by uncertainty of stator resistance. In order to increase robustness of controlled system to the uncertainty, sliding mode controller (SMC) is utilized to replace proportional-integral (PI) speed controller in conventional DTC drive structure. Computation of SMC requires estimation of load, and Kalman Filter is integrated to reduce noise in load estimation and chatteringphenomenon in speed response. Simulations are carried out at dierent reference speeds in widerange noises of stator resistance. Indices including ITAE, settling time, overshoot and undershoot are employed to compare performance of drive structures. Results conrmed the desired characteristics of the proposed drive structure.

Keywords
Induction motor drive, direct torque control, sliding mode control, load estimation, Kalman lter.  [3] can be replaced with pulse-width-modulation (PWM) integrated direct torque control (DTC) strategy [4]- [6]. Parameter tuning, stability analysis and design of proportional-integral (PI) or proportional-integral-derivative (PID) controllers require complicated techniques [7]- [10]. PI and PID controller tuning rules in tabular form are employed in various process models [7]. A set theory-based technique is utilized to obtain robust stability range of PI controller in in-terval systems [8]. Parameters of a fractional order PID controller are designed to reduce harmonics and noises in IM [9]. In order to obtain good response, and appropriate load disturbance rejection, parameters tuning of the controller is computed by fractional calculus [10].
Several methods for tuning PI or PID controllers are based on intelligent or bio-inspired techniques [11]- [14]. Whale optimization algorithm brings high performance for optimization of PID controller in a DC-DC buck converter [11]. Particle swarm optimization (PSO) provides shorter settling time and lower peak values in PID controlled liquid level system [12]. Combination of neural network and fuzzy logic reduces the time of accommodation, the steadystate error and simultaneously compensate uncertainty in system parameters [13]. Two metaheuristics method-genetic algorithm and PSO are utilized to adjust coecients of a Gaussian adaptive PID controller in a DC-DC buck converter.
In order to deal with parameter uncertainty and system disturbances in IM drives, robust control techniques can be selected [15]- [18]. In [15], the optimal preview control theory is utilized, and the robustness is tested via changes of rotor resistance and load torque. The state variable aggregation method is employed in design of VC strategy for the nonlinear model of IM [16]. In order to obtain robustness to the changes of stator and rotor resistances, an inputoutput feedback linearization control technique is applied [17]. Linear matrix inequality-based state feedback design is carried out in case of system disturbances and parameter uncertainties [18]. One particular method to robust controller design is the sliding mode control (SMC) technique [19].
Variations of SMC are applied in scientic and engineering problems [20]- [25]. Stability analysis and control synthesis of crowd dynamics models are implemented by nonlinear SMC [20]. Double integral SMC is employed for tracking maximum power point of a photovoltaic system [21]. The SMC with a stochastic sliding mode surface is used to stabilize singular Markovian systems with Brownian motion [22]. The digital SMC is designed for the discretized model of the 266 c 2021 Journal of Advanced Engineering and Computation (JAEC) piezoelectric actuator [23]. The discrete-time two-dimensional SMC is handled for Fornasini-Marchesini systems that are distorted by exogenous nonlinear disturbances [24]. For eld of IM drives, SMC is also widely utilized [25]- [28]. A sigmoid function and an adaptive gain are used to eliminate chattering for an IM drive with VC [25]. A gain margin technique is employed to attenuate chattering for continuous SMC of IM servo system [26]. In order to achieve total robustness in VC of IM motion control system, a sliding-mode based component is added to take into account disturbances and uncertainties [27]. A second-order SMC is implemented to control the dynamics of the IM with the aim of disturbance rejection and robustness assurance [28]. For simplicity, in the paper, integral SMC technique [25] is chosen to design control signal of DTC-IM drive for robustness to high uncertainty of stator resistance. In order to ensure boundary condition of system's Lyapunov stability and reduce chattering-phenomenon, a load estimation method based on dynamic equation of rotational motion and Kalman lter [6] is utilized. In next section, SMC drive structures for DTC drive are presented. Simulation results and conclusions are respectively given in two last sections.

2.
DTC-IM drive with sliding mode speed controller State-space model [29] of IM in coordinate system [α, β] is described by Eqs. (1)- (2): where: Rotor speed is obtained according to Eqs. (10)- (11): Computation process of Kalman Filtered stator current components is described in [6]. For drive structure with SMBSC, design process of reference torque requires load information, and Kalman Filtered load estimation block is added to reduce ripple of estimated load torque. In SMBSC design, sliding surface and its time derivative are modied from [25], and expressed by Eqs. (17)-(18): where k 1 > 0. Assume that reference torqueoutput of SMC is equal to real motor torque, time derivative of rotor speed can be computed as Eq. (19):ω Assume that reference rotor speed is constant, Eq. (18) is rewritten as follows: In order to ensure the stability of the controlled system, the Lyapunov's stability theorem is utilized. Dene the Lyapunov function candidate according to Eq. (21): Its time derivative is given by Eq. (22): The controlled system is stable if the term dV /dt is negative. Similarly to [25], we can choose the time derivative of the sliding surface as Eq. (23): where k 2 > 0. From Eqs. (20), (23), we obtain the SMBSC control law: However, information of load torque is unknown, so it is considered noise. The Eq. (24) is rewritten: The coecient k 2 is designed to maintain system stability with control law described in Eq. (25). The Eq. (20) is converted into Eq. (26) thanks to Eq. (25):Ṡ Assume that load torque is bounded, k 2 is chosen as follows: where M L > |T L |. It is easy to see that the smaller M L is, the more chattering problem is reduced. So it is necessary to estimate the load torque T L . Its simple approximation T EL is described by Eq. (28): Estimated motor torque which is computed by Signal Calculation block (see Fig. 1), can be distorted (see Eqs. (12), (13), (16)) although two stator current components were ltered. Hence, estimated load is also deformed, and it is Kalman ltered according to Eqs. (29)-(36): where: x = T EL : state scalar; w, v: zero-mean Gaussian process, measurement noise scalars with unknown variances Q, R; symbols ∧, ∼ denote estimated, predicted scalars respectively. The component ML in Eq. (27) is chosen as follows: where ε is positive and arbitrarily small.

Simulation results
In this section, parameters of simulated IM and SVPWM-DTC drive are listed in Tab. 1. Simulations are implemented at reference speeds of 500 rpm, 50 rpm, 5 rpm (see Figs. 2-4) with load jump of 8 Nm at 0.3 s (see Fig. 5). In order to verify theoretical assumptions, stator resistance R s is assumed to be distorted by zero-mean Gaussian noise with variance of added relative value σ R 2 = {0.1 2 , 0.3 2 , 0.5 2 , 0.7 2 , 0.9 2 }, and vector of stator current components is also considered to be deformed by zero-mean Gaussian noise vectors with covariances σ P 2 = σ M 2 I = 0.3 2 I (see [6]).     SMC structure in Fig. 1 without Kalman ltration in load estimation SMC one, SMC struc-ture in Fig. 1 Settling times t s1 and t s2 are respectively searched in durations 0.0 s 0.3 s (before load activation), and 0.3 s 1.0 s (after load activation). Overshoot and undershoot are also calculated in two same durations respectively.         tively. The main reason for this is that KF-   SMC structure brings much lower overshoot and undershoot than PI one: overshoots and undershoots for KFSMC are respectively 2.29%-31.05% (see Tab. 3) and 9.84%-14.24% (see Tab. 4) of ones for PI. In comparison to SMC, overshoots and undershoots for KFSMC are respectively 32.09%-100.07% (see Tab. 3) and 97.30%-106.63% (see Tab However, trend of oscillation is increasing especially after load activation at ω ref = 5 rpm (see Fig. 23). Reason of this is that high stator resistance uncertainty σ R 2 = 0.92 makes estimated values of components of stator ux vector (see Eqs. (12)     Kalman ltered (see Fig. 26). The signicant dierence between stator ux and its estimated values leads to increasing errors in motor torque and load torque estimates (see Eqs. (16) & (28)). Another method that is less dependent on stator resistance can be utilized to improve the accuracy of the estimates.

Conclusions
Drive structure using SMC and Kalman ltered load estimation was presented in the paper.
Simulations were implemented at dierent reference speeds with wide ranges of noise variances of stator resistance. Proposed drive structure guaranteed system's Lyapunov stability, provided robustness to high uncertainty of stator resistance, and reduce chattering-phenomenon. It dedicated lower ITAE performance index than both the conventional and the SMC without Kalman ltration in load estimation ones, especially at lowest reference speed with ITAE value reduced by 91.8%-97.6% compared to conventional structure. Kalman Filter in its load estimate signicantly reduced chattering problem compared to the one without Kalman Filter. The structure brought lower ripples at steady state, more robust settling times than two other ones. High-order or super-twisting or non-reaching phase, chattering-phenomenon free sliding mode control techniques can be used to obtain more robust speed controller, and approaches that are less dependent on stator resistance uncertainty can be employed to provide more accurate load torque estimates. Proposed approach can be utilized in robust sensorless control of IM drive.