Sliding Mode Self-Sensing Control of Synchronous Machine Using Super Twisting Interconnected Observers

The aim of this study is to propose a self-sensing control of internal permanent-magnet synchronous machines (IPMSMs) based on new high order sliding mode approaches. The high order sliding mode control will be combined with the backstepping strategy to achieve global or semi global attraction and ensure finite time convergence. The proposed control strategy should be able to reject the unmatched perturbations and reject the external perturbation. On the other hand, the super-twisting algorithm will be combined with the interconnected observer methodology to propose the multi-input–multi-output observer. This observer will be used to estimate the rotor position, the rotor speed and the stator resistance. The proposed controller and observer ensure the finite-time convergence to the desired reference and measured state, respectively. The obtained results confirm the effectiveness of the suggested method in the presence of parametric uncertainties and unmeasured load torque at various speed ranges.


Introduction
Internal permanent-magnet synchronous machines (IPMSMs) have been getting progressively more popular in several industrial applications owing to high efficiency, good power density and finally the high torque/current ratio [1,2]. Despite its advantages, high-speed control remains a very significant and challenging subject of research [3,4], and the feedback information on rotor position remains of paramount importance. It is noticed that the utilization of encoders has many shortcomings, like decreased reliability of the system and the cost. As a result, self-sensing control has become attractive in many industrial applications.
To perform self-sensing control operations, various methods, like the one which uses the back electromotive force to control the IPMSM [5], have been proposed in the literature. This method offers a good performance at medium and high speeds [6]; however, it does not function at zero and very low speeds because of the integrator's drift problem, particularly in the analog realization. Moreover, this method is very sensitive to the variations of the stator resistance during operation. Alternatively, an extended electromotive force based method is suggested [7]. Moreover, to evaluate the speed and the related rotor's position, several observer-based methods are introduced in the twisting algorithm and the interconnected technique provides a novel MIMO high order sliding observer. The suggested observer is to estimate both the rotor's position and speed at the different speed rate. In addition, the resistance of the rotor is estimated, which leads to better accuracy when the operating conditions change. The closed-loop system stability is provided by means of utilizing the nonlinear separation principles theorem, rather than several proposed self-sensing control methods for IPMSM.

Problem Statement
The IPMSM is usually modeled in the rotor synchronous rotating frame as [27]: where i d and i q are stator currents, Ω and θ are the rotor mechanical speed and angular position, respectively, u d and u q are the stator voltages, R s is the stator resistance, L d and L q are the dq-axis inductances, p is the number of pole pairs, Φ f is permanent-magnet flux linkage, J is the moment of inertia, T l is the load torque, and f v is the viscous friction coefficient.

Remark 1.
The operating interval of IPMSM D om is delimited by the values defined as follows: and the present maximum values are, respectively, I max  d , I max   q for currents, Ω max for speed, R max s for stator resistance and T max l for load torque; the values are defined in specification sheet from the motor's manufacturing.
Control objective: The main control design objective is to track the desired reference (Ω * ) for speed and following the desired trajectory for i * d despite uncertainties of parameters and disturbance in torque load.
Observation objective: The aim is to reconstruct the value of the IPMSM's speed, position, and stator resistance, by unique measurement of currents and voltages.

MTPA QC HOSM Control
The developed approach, as well as the application of IPMSM, will be detailed in the following section, starting by the QC HOSM method basics.

Quasi-Continuous Higher Order Sliding Mode Controller
In this section a solution for the control problem described in previous section is proposed. This solution combines the backstepping and higher order sliding mode methodologies as recommended by [28]. Consider the following nonlinear system: where i varies from 2 to (n − 1), the state x ∈ R n , with x i ∈ R, x i = [x 1 , . . . , x i ] T , and the control input is u ∈ R.
In addition, f i (x i , t) and g i (x i , t) are the smooth function, the bounded unknown disturbance term ω i (x i , t), representing the parameter variations, [29] as well as the external perturbations, admitting n − i bounded derivatives at least.
. For simplicity, suppose the relative degree of system (2) is n. For the nonlinear system class with unmatched disturbances (2), the control design procedure is presented as follows.
Step 1: The first step consists to constructing the next virtual control x i+1 = φ i . For that, let us define x 2 = φ 1 with φ 1 is (n − 1) times differentiable functions constructed as follows: where σ 1 = y − y re f . At the present stage, the virtual control consists of two parts φ 1 and u 1,1 , respectively, providing compensation for the nominal part and, on the other hand, disturbances introduced by the n − 1 integrators. The i-th step is as follows. The constructed virtual controller will be used in the next step as a reference in order to construct a new virtual control or a final control depending on the relative degree of the studied system.
Step i: Considering the virtual control such that x i+1 = φ i , where φ i is n − i times the differentiable function given by: Finally, the robust control law u is calculated at the last step.
Step n: Taking σ n = x n − φ n−1 , the actual effective control u is provided by This controller ensures the finite-time convergence of the system states to their desired references despite the presence of the perturbations and parameter uncertainties.
In the following, this control strategy will be applied for the IPMSM case.

IPMSM Controller Design
In accordance with the control design procedure, a design of a quasi-continuous homogeneous control law for the synchronous motor is achieved, starting with the mathematical machine model (1), written as follows: where [x 1,1 , While the controlled outputs are [Ω, i d ] T , the relative degree of the vector is equal to [2, 1] T . Models (6) and (7) are expressed as an interconnected system; in each term the subsystems are bounded and considered as external disturbances: with ω 1,1 , ω 1,2 , and ω 2,1 being limited terms due, on the one hand, to the variable parameters and, on the other, to the terms of the connection, as well as to the external disturbances (namely, the load Sliding surfaces are chosen that guarantee the trajectories are asymptotically stable during the sliding phase (see [28] for more details). The sliding surfaces chosen are proposed to realize the control objective by considering both model uncertainties and external disturbances.
Speed control loop A homogeneous QC control is now applied to the speed loop. Consider the tracking error σ Ω : The virtual control u 1,1 can be written as: after, the control action u q = u 1 can be elaborated, considering: Next, the first subsystem control can be written as Finally, we obtain Considering the IPMSM torque as follows: the d-axis current has been subject to a control by forcing it to zero, which means i * d = 0. Yet the present approach does not make use of electromagnetic torque of an IPMSM. Despite considerable research on the current control of the axis d (i.e., i * d = 0), the electromagnetic torque of IPMSM remains inefficiently used. The Maximum-Torque-Per-Ampere control approach gives a highest torque/current proportion, wherefrom its effectiveness is improved [30]. Based on the MTPA control method below rated speed, the full use of the reluctance torque and the motor's operation with optimum performance determine the i * d . The process is achieved through the differentiation of Equation (11) maintaining the constancy of the current stators' absolute value i d at the level of its highest value I m as shown in [30]. The interconnection of the currents of the stator i d & i q , i d = f (i q ) can be written as: To implement the MTPA approach, the current i d is pushed to go with the reference calculated in (12). Consider the next sliding surface Note that r 2 = 1. Then, the control effect for the second subsystem is written as where u (2,1) = −λ 2,1 sign(σ 2,1 ). Finally, the controller is given by

The Super Twisting Algorithm
Consider the next systemẋ The system (15) is resolved by forming it in the sense of Filippov ([31]). Supposing that the Lebesgue-measurable function f (t, x 1 , x 2 , u) and the uncertainty function ζ(t, x 1 , x 2 , u) are uniformly bounded in all compact regions of the state space (x 1 , x 2 ), then the common form of the ST observer would be as follows˙x Considering the estimated states asx 1 andx 2 , and the observation gains as α 1 , α 2 .

HOSM Rotor Position Observer
To design a novel high order sliding mode observer, a combination of both the interconnected observer methodology and the super twisting method [23] is required. To evaluate the position and speed of the motor rotor as well as the stator resistance, an observer has been employed. The mathematical model of IPMSM within an α − β framework is written as: where It is essential to reformulate the model in an appropriate expression, in order to design these observers. For this reason, the following change of coordinates is considered with K s = D −1 L 0 φ f , the system dynamics are: where Since the system (19) is currently in the appropriate form, it is feasible to conceive a finite-time observer for the MIMO system which is the SISO's extension as in [23].
Then, for the previous system (19), the design of an HOSM interconnected observer is performed: where ξ i1ξi2 T andξ i1 = ξ i1 −ξ i1 are, respectively, the Σ ξ,i system's state estimation and the estimation error, for i = 1, 2; the rotor position estimation can be written, according to (20), aŝ Remark 2. For the observer implementation, a four-quadrant inverse tangent (ATAN2 when using Matlab software) was used.
With purpose of guaranteeing the existence of f i as an upper bound, in practice, the states ξ 1 and ξ 2 are supposed to operate in a limited range within D om , [23], such that holds for all ξ 1 , ξ 2 , in the domain D om , ξ i2 ≤ 2 sup ξ i2 is verified.

Assumption 1.
Assume there is an upper bound for the uncertainties ∆ζ i1 and their corresponding derivative, such that: where h i and h i are positive bounds.

Theorem 1.
Taking system (19), and admitting the verification of Assumption 1 and condition (23), in all operating ranges of D om , the initial conditions ξ i1 (0), ξ i2 (0) andξ i1 (0),ξ i2 (0) are associated to observer gain The estimate stateξ i,j converges in finite-time to the real states ξ i,j , for i, j = 1, 2, and there exists a time constant T 0 such that for all T 0 ≤ t,ξ i,j = ξ i,j , for i, j = 1, 2.

Hosm Speed and Resistance Observer
Start by the observation that the behavior of resistance of the stator changes very slowly in a time lapse, which allows to model it by a piecewise function such thatṘ s = 0, as a result of which the IPMSM (1) mathematical model can be decomposed into subsystems as follows.
To write subsystems (25) and (26) so that the super-torque algorithm can be implemented, the following coordinate changes are considered: Then, the representation of systems (25) and (26) can be written as: where χ i1 and χ i2 for i = 1, 2 are, respectively, the measured output and the unmeasured state that will be estimated, so T l is taken into account as a limited bound disturbance because it is not measured. The terms Γ i , H i and φ i for i = 1, 2 can be written as: where ∆ρ ij for i, j = 1, 2; are the interrelated terms.These terms are thought of as disturbances.
Based on the extension of [23], an interrelated second-order sliding mode (ISOSM) observer for subsystems (25) and (26) is designed as follows. The HOSM interconnected observer for systems (25) and (26) is considered as By considering (27), the resistance and speed estimation of the stator is given by: χ 11 , Remark 3. In the same way as that adopted in Section 4.3, the convergence of observers (29) can be proved. [4,23], the principle of separation is achieved by consolidating the observer once its temporal convergence is limited, and consolidating a controller. The controller will still not be applied in the first time interval from 0 to T, while, at this time, the state estimation converges to the state of the system.

Remark 4. Separation principle. Presented in
While state estimation occurs, the state of the system is reached in a time interval after T (i.e., t > T to ∞); with this, the tracking enforcement of a reference can be achieved by the controller.

Simulation Results
The simulation validation is given in the following to underline the effectiveness and performance of the control strategy presented. The different parts of the self-sensing control method are illustrated in Figure 1. The nominal IPMSM parameters utilized in the simulations are presented in Table 1. Using Matlab/Simulink software, the simulation was performed. The engine is tested conforming to the industrial test trajectory [32] shown in Figure 2 which presents the trajectories of the chosen benchmark.   At the beginning, the motor is operated at zero speed without load torque to test the algorithms are developed in the critical observability area. Then the speed is increased to 100 rad/s and the load torque is applied for 1 s (from 1.5 s to 2.5 s). From the 4th s until the 6th s, the speed is increased up to 314 rad/s, and then it is kept constant for 10 s. By the 7th s, a load torque is applied. Stage two allows testing the observer during a high transition of speed as it enables testing its stability at a high-speed. By the final stage, both the motor's speed and the load torque reached zero; the former progressively dragged to zero (0) by the 13th s, while the later set to zero by the 15th s. In the case of nominal parameters, results depicted in Figures 3-6 confirm the efficiency of the implemented self-sensing control for IPMSM. The evolution of both estimated and observed speed schemes is presented in Figure 3a. The speed error caused by the perturbation is minimal; it tends rapidly to zero after the transients resulting from the application of the load torque (see Figure 3b). Figure 4 gives both position estimation and its measure. It is explicit that the observed position follows the real position exactly.  In Figure 5, there is a good convergence of the estimated resistance with respect to its real value (see Figure 5b). It seems that the observer gives adequate results for these types of estimations. Figure 6 shows the input voltages and the dq currents. The effectiveness of the proposed strategy, especially in terms of chattering effect reduction, can be remarked on in this figure.  The influence of parameter variations was examined to show the robustness of the self-sensing control scheme. The Voluntary add of the variation of parameters in the controller observer scheme was introduced. Figures 7 and 8 and Figures 9 and 10 present the response of the system with, respectively, +30% and −30% of the variation in stator resistance. The effectiveness of the resistance estimator is noticed mutually at the different speed ranges.       The suggested resistance estimator makes possible the mitigation of the unwanted effects of stator resistance variations. The latest test is to introduce a fluctuation of ±20% in the stator inductances. The system's response under the control observer action is shown in Figures 11 and 12, where it can be seen that the variations in the stator inductance do not affect the system. The robustness and efficiency of the self-sensing control suggested, with variations of parameters and load torque, are apparent. The controller's gains are selected as follows: λ 1,1 = 1500, λ 1,2 = 400, λ 2,1 = 200.

Conclusions
In this work a self-sensing control of a permanent magnet magnet synchronous machine is introduced. The described strategy is based on a higher order sliding mode controller and observer. The proposed controller is a good combination between the backstepping and higher order sliding mode strategies. Therefore, the gains of sliding mode are reduced compared to the classical one. With the proposed controller the finite time convergence of the tracking errors is obtained and the chattering effect is attenuated as can be seen in the control output. Moreover, this controller allowed the rejection of the unmatched perturbations (parameter uncertainties in this case) and the rejection of the external perturbations. On the other hand, the super twisting algorithm is generalised for multi-input-multi-output systems by combining it with the interconnected observer strategy. The new observer is applied to estimate the rotor position, rotor speed and the stator resistance. The finite-time convergence of the estimated states to the measured one is proven. As the finite-time convergence of the controller and the observer are proven separately, the stability of the proposed observer-controller scheme can be achieved according to the separation principle. The proposed controller and observer are implemented in simulation to realise the self-sensing control of IPMSM. The obtained results show clearly the effectiveness of the developed strategy. The robustness tests made to show the efficiency of the proposed one, despite the presence of electrical parameter uncertainties, show its good ability.
Our future work will be on how to deal with the problem of saturation of machine inductance and, after that, to generalise the proposed strategy to all electrical machines.