A robust-adaptive linearizing control method for sensorless high precision control of induction motor

Abstract

Even if there exists remarkable applications of induction machines in variable speed drives and also in speed sensorless control in the low–high speed region, open/closed loop estimators in the literature utilized on induction machine sensorless position control vary regarding to their accuracies, sensitivity, and robustness with respect to the variation of model parameter. The deterioration of dynamic performance depends on the lack of estimation techniques which provide trustable information on the flux or speed/position over a wide speed range. An effective estimator should handle the high number of parameter and model uncertainties inherent to induction machines and also torque ripple, the compensation of which is crucial for a satisfactory decoupling and linearizing control to provide the accuracy and precision requirements of demanding motion control in the field of robotics/unmanned vehicle. In this study, to address all of the above-mentioned problems, robust-adaptive linearizing schemes for the sensorless position control of induction machines based on high-order sliding modes and robust differentiators to improve performance were designed. The control schemes based on direct vector control and direct torque control are capable of torque ripple attenuation taking both space and current harmonics into account. The simulation results comprise both the estimation and sensorless speed control of induction machines over a wide operation range, especially at low and zero speed, all of which are promising and indicate significant superiority over existing solutions in the literature for the high precision, direct-drive, speed/position sensorless control of squirrel-cage induction machines.

Introduction

Since induction machines (IM) have highly nonlinear and coupled fifth-order dynamics under strong parameter and model uncertainties and make it possible to measure only three state variables, linearizing and decoupling control of IMs is a competitive problem in nonlinear control field. The breakthrough in the control of IMs has been achieved after the invention of field-oriented control (FOC) or vector control (VC). ¹ By applying VC method to the IM, speed and torque control become easier. To develop such kind of scheme, stator or rotor flux magnitude and their angles relative to the stationary stator should be accessible through a sensor measurement or an observer estimation. A large number of researches have conducted for implementing FOC approach, which transforms the IM’s control to that of a separately excited direct-current (DC) motor by setting the rotor flux constant on the direct axis. Nonlinear into linear system conversion is possible with this approach, which helps to decouple the flux and speed (or position). Unfortunately, the assumption of a constant rotor flux gives rise to an only “asymptotically” decoupled or input–output (I-O) linearized system. To cope with this kind of problem, I-O linearization methods have been studied. These methods utilize the flux dynamics,^2–5 or dynamic I-O linearization with the addition of an integrator.^6,7 In Khorrami et al., ⁸ nonsingularity conditions on the direct-quadrature (d-q) model, when the position is also added as output, have been investigated. The feedback linearization (FL) performance is influenced by the system parameters, which vary with the operating conditions. Besides the load changing from low to high, inductances are likely to saturate at high flux levels and resistances are inclined to increase as a result of temperature which is particularly effective for the stator resistance and frequency, showing itself mostly on the rotor resistances as skin effect. The parameter uncertainties which are dealt by some studies^9–11 are as follows: addressing and modeling magnetic saturation effects,^12,13 taking an adaptive approach for the compensation of the rotor resistance and external load,^14,15 developing an adaptive nonlinear feedback for torque control, ¹⁶ estimating and compensating the saturation effect, ¹⁷ proposing a robust dynamic I-O linearization scheme for rotor resistance variations, ¹⁸ developing an adaptive method against rotor resistance/load variations, and, finally, ¹⁹ developing a parallel model reference adaptive scheme (MRAS) to compensate for the variations of the rotor and stator resistances. Since the overall performance of the VC depends on the uncertainties of unpredictable parameter variations of the system, external load disturbances, unmodeled and nonlinear plant dynamics, all the above-mentioned methods require flux observers, the performance of which also depends on how well the system parameters are accessible. An observer-based solution coping with the problem of parameter variations could be given as follows: in Alagna et al., ²⁰ a sliding mode flux observer was designed to estimate the rotor flux vector for improving the performance of the sliding mode–based torque control while sustaining robustness against change in load torque and parameter variations; in Stumper and Kennel, ²¹ a nonlinear observer structure, which operates at any speed, in extremely low and zero speed up to half nominal torque, with direct FOC, was proposed to estimate the motor speed and the rotor flux; in Luo and Ke, ²² a full-order flux-observer-based speed estimation approach with an MRAS identifying the stator resistance parameter was applied to realize a rotor flux-based VC of an IM drive. Model uncertainties and nonlinearities inherent to IMs are well suited for the stochastic nature of extended Kalman filters (EKFs). ²³ Through the help of this method, online estimation of states can be made while identifying parameters in a relatively short time interval by taking system/process and measurement errors (noises) directly into account.^24–26 Passivity-based approaches,^27–29 which also depend on accurate knowledge of parameters, robust control, ³⁰ and intelligent control approaches ³¹ against parameter uncertainties were also utilized for IM control. Both FOC and DTC (direct torque control) drives necessitate robust (to parameter and model uncertainties) and speed sensorless solutions. In industrial applications, low-cost and precise speed sensorless systems able to effectively drive IMs at low and zero speed are preferable. ³² If one can eliminate the speed sensor at machine shaft without deteriorating the dynamic performance of the drive control system, the following benefits are provided: economy on both cost and size, elimination of additional wiring, drift and noisy measurements, and reduced hardware complexity. To eliminate speed or position sensors, an observer, which estimates position or speed values from measured stator voltages and currents, can be used. This procedure leads to speed/position estimation schemes, which deal with the effect of parameterization errors and model/load uncertainties. Speed/position sensorless control of IMs constitutes a problem of specific nature at very low speeds and specifically at zero speed. As the electromagnetic field maintains a fixed position in space at this operating point, the rotor-induced voltages in the stator windings have zero values. This gives rise to problems in the spatial orientation estimation of the rotor field based on measurable quantities, such as currents and voltages on the stator side. ³³ To obtain successful results, not only robustness against parameter and model uncertainties should be satisfied but also solutions must be sought in the very low-speed region, specifically at zero speed under full load and no load (at zero stator frequency) in the designed observer methods. ³⁴ Much research are titled “model-based” in that they use two-axes machine model to predict the speed and a flux angle necessitating for a high-performance vector drive.^35–38 The speed estimation below 1 Hz machine frequencies, model-based approaches do not provide remarkable performance and improvements in the online parameter estimation are not satisfactory. Tracking saliencies within an IM based on high-frequency (hf) signal injection is another method realized in the literature. This method can yield either the rotor position or an orientation flux angle without needing to a machine model^39–43 and thereby demands to solve the zero- and low-speed cases. These methods comprise two categories. The first one is related to the natural saturation saliency, which yields an orientation flux angle . ⁴⁴ Unfortunately, phases among the saturation vector and either flux vector (stator or rotor) are affected from the leakage saturation and slot design.^45,46 This kind of dependency cannot be acquired from the two-axes machine model knowledge; therefore, modeling based on finite-element method or experimental research is necessitated to quantify it. Saturation saliency tracking can be applied to a torque-controlled drive; however, a model-based speed estimator should be used for speed control. A rotor saliency arising from rotor slotting or a saliency involving circumferential variation of surface rotor resistance ⁴⁷ or leakage inductance ⁴⁸ is tracked in the second category. ⁴⁹ Rotor position is estimated to be used for sensorless position control and field orientation through VC techniques. The saturation saliency modulation of the hf stator currents interferes with the rotor saliency modulation in a fully fluxed machine and machine under load. Owing to the variable magnetic coupling among the discrete rotor bars and stator windings of an IM, a quasi-continuous rotor position signal is obtained measuring three-phase stator leakage inductance. The sampling process is conducted synchronously through the regular commutations of the pulse width modulation (PWM); thus, the injection of additional hf carriers is unnecessary. The acquired signals exhibit high spatial resolution and dynamic bandwidth. The repetitive transient excitation through the test signals was made to identify the spatial orientation of anisotropies that indicate the rotor position. The position information is contained in the inverter output voltages. These are measured against the neutral point of the star-connected stator winding. The disturbance for position identification is constituted due to the effect of the magnetic saturation upon the total leakage inductances. Two different approaches relying on anisotropy effects, the frequency-domain approach^50,51 and the time-domain approach, ⁵² were applied to estimate the rotor position angle. A transient condition by continuously injecting an hf carrier signal into the stator windings is established by the frequency-domain approach. These anisotropies are linked to spatial orientation of angular position of the rotating fundamental field. The directions in space of such anisotropies are detected from the machine response to perpetuate transient excitation through stator windings. The position information, which is acquired by phase-locked loop (PLL) technique, is mirrored in the negative sequence response signal component. The inverter dead-time phenomena, parameter dependence, low signal-to-noise ratio (SNR), the dynamic delay of the PLL, and magnetic saturation effects are unwanted effects in the estimation technique for fully loaded closed-loop position control system. ⁵³ In the time-domain approach, instantaneous samples are collected from the stator phase voltages, immediately after the inverter commutations at PWM operation. Inverters, which are switching higher transient excitation than a continuous carrier injection give rise to a high SNR. The zero sequence voltage component, which is computed using the sampled stator phase voltage, demonstrates an imbalance at the sampling instant among total leakage inductances of three phases. Such imbalance originates from the variation of the magnetic coupling among the rotor bars and stator windings, magnetic coupling degree.

To improve the parameter sensitivity, open-loop estimators or closed-loop observers have been constructed for the estimation of velocity and flux. A variety of approaches have been taken for this purpose, some performing state estimation only and some also performing load and/or parameter estimation. Various problems have been encountered in the process, that is, simultaneous estimation of the rotor and stator resistances gives rise to instability ⁵⁴ and inaccurate rotor and stator resistance estimations. ⁵⁵ Model reference adaptive laws ⁵⁶ estimate the flux and speed; however, the system response is not tested for load torque variations. Rodič et al.’s study ⁵⁷ uses sliding mode observers (SMOs), while Mehta et al.’s study ⁵⁸ develops Gopinath’s reduced-order observer. Zaky ⁵⁹ develops an adaptive flux observer (AFO) using the Lyapunov and Popov criteria, respectively, and requires adjustments to the observer gain matrix with changing operating conditions to suppress the oscillations on the estimated velocity. In the speed sensorless study, ⁶⁰ in which the stator resistance has been estimated based on a two-timescale approach, it has been stated that even a very small offset in either the voltage or the current measurement causes instability of the drive at zero stator frequency and rated load. SMOs, as in Umida and Guidi ⁶¹ and Edwards et al., ⁶² have also been designed with the error correction term reflected to the observer as a multiplicative term. Theoretically, the switching frequency should be infinite to avoid chattering on the sliding surface, ⁶³ which poses problems in some applications. In Lascu et al.’s study, ⁶⁴ a DTC-based on an SMO using a dual reference frame motor model was achieved at very-low-speed sensorless operation. The performance was obtained without using hf signal injection or saliency effects for the flux and speed estimation. Gaballah et al. ⁶⁵ present a chattering-free SMO, whose switching function is replaced by a rule-based fuzzy logic system, for the velocity with high-accuracy tracking performance. Unlike adaptive methods, which require persistently exciting input signals, EKF uses the above-mentioned system noise to fulfill the hf signal requirement for accurate and fast convergence. This feature also makes EKF an attractive option in the sensorless control of IMs, despite of its computational complexity. Among studies related to the EKF estimation for IMs, Mannan et al. ⁶⁶ and Hinkkanen ⁶⁷ estimate the velocity and flux with full-order observers, while Zaky ⁶⁸ uses an adaptive flux-observer insensitive to stator resistance variations. In addition, in sensorless control, the velocity is estimated as a constant parameter, with the assumption of slow variation. However, the estimation performance at very low- and zero-speed region is not satisfactory. A Luenberger observer in Jouili et al. ⁶⁹ was designed for velocity estimation, but very low- and zero-speed operation has not been addressed. For improved performance against the problems of sensorless control in the very low- or zero-speed region, methods have been proposed that make use of magnetic or electrical anisotropies of the rotor for the determination of the rotor position. ⁷⁰

Two approaches are taken in the literature: frequency-domain approach based on the injection of hf signals, a major disadvantage being the demand for additional hf signals, sensitive to system parameters, the dynamic delay originated from PLL, low SNR, and magnetic saturation due to the fundamental field; and time-domain approach, making use of the inverter PWM signals, which provide a high SNR. The latter approach is quite effective for the control of IMs (in the very low- and zero-speed region); however, problems arise due to signal acquisition in the high-speed region—therefore, special measures (such as the use of a different model) must be taken when it is aimed to perform velocity control over a wide speed range. This issue was addressed by taking the angular velocity into consideration via the equation of motion in an EKF scheme that also estimates the load torque and rotor resistance. The torque term in the equation of motion helps conveying the rotor information to the stator side, which is otherwise lost at zero stator frequency. The approach has yielded successful results over a wide velocity range, also including very low and zero speed.^71–73 Finally, another factor to consider for precise position control in IMs is the torque pulsations caused by the interaction of the space harmonics of the motor and the current harmonics. Those pulsations, also known as torque ripple, affect the position output significantly and are unacceptable for high-performance motion control and robotics applications. Therefore, to establish the compatibility of IMs with high-cost, high-performance motors for those applications, solutions must also be sought for torque ripple minimization. There have been numerous studies on the modeling of torque ripple^74,75 for IMs, but studies dealing with the elimination of torque ripple are mostly related to DTC only, which inherently has the drawback of high torque ripple due to variable switching frequency, for example, Toro et al. ⁷⁶ address the problem of current harmonics only by different switching strategies for DTC, that is, by using space vector method (SVM). Studies also corresponding to the effects of space harmonics are very rare, such as Vernet et al., ⁷⁷ which uses a Kalman filter to estimate the two components of the harmonic rotor flux generated by one space harmonic for wound rotor IMs but has a speed sensor. To summarize the results of our review of existing literature on IM control, it has been noted that successful examples of sensorless IM speed control in the very low- and zero-speed range and similarly sensorless IM position control are very rare.^78–80 Those solutions mostly require extra precautions for signal acquisition and/or address low-speed operation only. However, high-performance position control also requires a good transient performance, calling for improved estimation/control methods over a wide speed range.

In this paper, computer simulations related to the robust-adaptive linearizing schemes based on higher order sliding-mode controllers (HOSMC) were conducted for DTC. It is intended to contribute significantly to the challenging area of sensorless position control of IMs with novel control and observer schemes that are specifically designed to solve the major problems encountered in the high-performance control of IMs. The outcomes will also contribute significantly to the IM position/tracking control with sensors, as well as sensorless speed control in a wide speed range, including very low/zero speed. Most studies in the literature address only certain subsets of IM control problems. However, for the IMs to be compatible with their more expensive counterparts in high-performance motion control, all of the above-mentioned problems should be solved. In this paper, the following approaches are proposed for this purpose:

• A multiple EKF-based observer approach for the online estimation of a high number of parameters and states. The proposed estimation scheme meets the parameter/state requirements of the observer scheme as well as those of the linearizing and decoupling controllers.

• HOSMC-based robust-adaptive linearization schemes for increased robustness and accuracy.

• A torque ripple cancelation approach for FOC and DTC schemes, with the consideration of both space and current harmonics. Modeling of torque ripple in IMs has been frequently addressed, but its minimization is mostly addressed for DTC and by reducing the current harmonics only.

The contributions of this paper are not limited to IM applications; for example, the proposed observer could be used with any nonlinear control system requiring the reliable estimation of a high number of state/parameters. Similarly, the proposed continuous HOSMC and adaptive torque ripple minimization schemes could address robustness and accuracy problems in a variety of electromechanical systems.

IM model

Before constructing control/observer schemes, the mathematical model of IM is derived in d-q axis, with the inclusion of the motor angular position—with the definition of $p$ : number of pole pairs; $L_{\sigma }=\sigma L_s$ : stator transient inductance; $\sigma$ : leakage or coupling factor; $L_s$ , $R_s$ : stator inductance and resistance, respectively; $L_r$ : rotor inductance; $R_r$ : rotor resistance referred to the stator side; $L_m$ : magnetizing inductance; $J_L$ and $B_L$ : total mechanical inertia and friction coefficient of the system; $i_{\mathrm{sd}}$ and $i_{\mathrm{sq}}$ : d-q axis components of stator currents; $v_{\mathrm{sd}}$ and $v_{\mathrm{sq}}$ : d-q axis components of stator voltages; ${\psi }_{\mathrm{rd}}$ and ${\psi }_{\mathrm{rq}}$ : d-q axis components of rotor fluxes; ${\omega }_s$ , ${\omega }_r$ , and ${\omega }_m$ : d-q axis angular velocity, rotor currents angular velocity, and rotor angular velocity, respectively; $t_L$ and $t_{\nu }$ : load and ripple torques, respectively; and ${\theta }_m$ : rotor angular position.

The state-space form of the IM model is given in equation (1)

$$\begin{array}{cc}\multicolumn{1}{c}{\left[\begin{array}{c}{\stackrel{·}{\theta }}_m\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\omega }}_m}\\ \multicolumn{1}{c}{{\stackrel{·}{\psi }}_{\mathrm{rd}}}\\ \multicolumn{1}{c}{{\stackrel{·}{\psi }}_{\mathrm{rq}}}\\ \multicolumn{1}{c}{{\stackrel{·}{i}}_{\mathrm{sd}}}\\ \multicolumn{1}{c}{{\stackrel{·}{i}}_{\mathrm{sq}}}\end{array}\right]}& =\underset{{\underset{¯}{A}}_e}{\underbrace{\left[\begin{array}{cccccc}0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -\frac{B_L}{J_L}\hfill & 0\hfill & 0\hfill & -\frac{1}{J_L}\frac{3}{2}p\frac{L_m}{L_r}{\psi }_{\mathrm{rq}}\hfill & \frac{1}{J_L}\frac{3}{2}p\frac{L_m}{L_r}{\psi }_{\mathrm{rd}}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -\frac{R_r}{L_r}\hfill & {\omega }_s-p{\omega }_m\hfill & \frac{R_r}{L_r}{L}_m\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -({\omega }_s-p{\omega }_m)\hfill & -\frac{R_r}{L_r}\hfill & 0\hfill & \frac{R_r}{L_r}{L}_m\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & \frac{L_m{R}_r}{L_{\sigma }{L}_r^2}\hfill & \frac{L_m}{L_{\sigma }{L}_r}p{\omega }_m\hfill & -(\frac{R_s}{L_{\sigma }}+\frac{L_m^2{R}_r}{L_{\sigma }{L}_r^2})\hfill & {\omega }_s\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -\frac{L_m}{L_{\sigma }{L}_r}p{\omega }_m\hfill & \frac{L_m{R}_r}{L_{\sigma }{L}_r^2}\hfill & -{\omega }_s\hfill & -(\frac{R_s}{L_{\sigma }}+\frac{L_m^2{R}_r}{L_{\sigma }{L}_r^2})\hfill \end{array}\right]}}\underset{{\underset{¯}{x}}_e}{\underbrace{\left[\begin{array}{c}{\theta }_m\hfill \\ \multicolumn{1}{c}{{\omega }_m}\\ \multicolumn{1}{c}{{\psi }_{\mathrm{rd}}}\\ \multicolumn{1}{c}{{\psi }_{\mathrm{rq}}}\\ \multicolumn{1}{c}{i_{\mathrm{sd}}}\\ \multicolumn{1}{c}{i_{\mathrm{sq}}}\end{array}\right]}}\hfill \\ \multicolumn{1}{c}{}& +\underset{{\underset{¯}{B}}_e}{\underbrace{\left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \\ \multicolumn{1}{c}{\frac{1}{L_{\sigma }}}& 0\hfill \\ \multicolumn{1}{c}{0}& \frac{1}{L_{\sigma }}\hfill \end{array}\right]}}\left[\begin{array}{c}{v}_{\mathrm{sd}}\hfill \\ \multicolumn{1}{c}{v_{\mathrm{sq}}}\end{array}\right]+\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{-\frac{1}{J_L}}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right]t_L+\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{\frac{1}{J_L}}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right]t_{\nu }\hfill \end{array}$$

(1)

The d-q axis angular velocity can be represented in terms of electrical parameters as in equation (2)

$${\omega }_s={\stackrel{·}{\theta }}_s=p{\omega }_m+\frac{R_r}{L_r}{L}_m\frac{i_{\mathrm{sq}}}{{\psi }_r}$$

(2)

where ${\theta }_s={\tan }^{-1}({\psi }_{r\beta }/{\psi }_{r\alpha })$ : flux angle; and ${\psi }_{r\alpha }$ and ${\psi }_{r\beta }$ : stator stationary axis components of rotor fluxes.

The magnitude of the rotor flux is defined as in equation (3)

$${\psi }_{\mathrm{r}}=\sqrt{{\psi }_{r\alpha }^2+{\psi }_{r\beta }^2}=\sqrt{{\psi }_{\mathrm{rd}}^2+{\psi }_{\mathrm{rq}}^2}={\psi }_{\mathrm{rd}}$$

(3)

For further simplification, the following definitions are made for the parameters

$$\begin{array}{c}{a}_1=\frac{3}{2}p\frac{L_m}{L_r},a_2=\frac{B_L}{J_L},a_3=p\frac{L_m}{L_r{L}_{\sigma }},a_4=\frac{R_s}{L_{\sigma }},\hfill \\ a_5=\frac{1}{L_{\sigma }},a_6=\frac{R_r}{L_r},a_7=\frac{R_r{L}_m}{L_r},a_8=\frac{R_r{L}_m}{L_{\sigma }{L}_r^2}\hfill \end{array}$$

All parameters are assumed to vary slower than the states, and hence, their time derivatives are taken as zero. Under the assumption of a good control for ${\psi }_r$ , it can be taken $\left|{\overrightarrow{\psi }}_r\right|={\psi }_{\mathrm{rd}}={\psi }_r$ and ${\psi }_{\mathrm{rq}}=0$ . Using this assumption and the parameter definitions above, equation (1) in matrix form is reorganized as in equations (4) to (8)

$$\frac{d{\theta }_m}{\mathrm{dt}}={\omega }_m$$

(4)

$$\frac{d{\omega }_m}{\mathrm{dt}}=\frac{a_1}{J_L}{\psi }_r{i}_{\mathrm{sq}}-a_2{\omega }_m-\frac{1}{J_L}{t}_L+\frac{1}{J_L}{t}_{\nu }$$

(5)

$$\frac{d{\psi }_r}{\mathrm{dt}}=-a_6{\psi }_r+a_7{i}_{\mathrm{sd}}$$

(6)

$$\frac{d{i}_{\mathrm{sd}}}{\mathrm{dt}}=p{\omega }_m{i}_{\mathrm{sq}}+a_7\frac{i_{\mathrm{sq}}^2}{{\psi }_r}-a_4{i}_{\mathrm{sd}}+a_8{\psi }_r+a_5{v}_{\mathrm{sd}}$$

(7)

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{d{i}_{\mathrm{sq}}}{\mathrm{dt}}=}& -a_3{\omega }_m{\psi }_r-p{\omega }_m{i}_{\mathrm{sd}}\hfill \\ \multicolumn{1}{c}{}& -a_7\frac{i_{\mathrm{sq}}{i}_{\mathrm{sd}}}{{\psi }_r}-a_4{i}_{\mathrm{sq}}+a_5{v}_{\mathrm{sq}}\hfill \end{array}$$

(8)

The system outputs are chosen as in equation (9)

$$y={\left[\begin{array}{cc}{\theta }_m\hfill & {\psi }_r\hfill \end{array}\right]}^T$$

(9)

In the “Control strategies” section, different control strategies are proposed to achieve I-O linearization for the VC and DTC of IMs.

Control strategies

Robust-adaptive linearizing controller based on HOSMC for VC

An HOSMC-based adaptive linearizing control is explained for both v_sq and v_sd. Two HOSMC approaches will be taken as follows:

1. HOSMC with discontinuous input, which depends on the artificial increase of the system order, to reduce chattering effects on the output for increased accuracy.

2. HOSMC scheme with continuous input to avoid the requirement of high-order state derivatives due to the artificial order increase.

The parameter and state estimation required for linearization and decoupling should be provided by an observer scheme. The HOSMC is designed to provide robustness against uncanceled nonlinearities as well as the torque ripple effects, which are not taken into consideration for direct cancelation in this scheme. HOSMC has proven to be very effective particularly against load uncertainties as well as low-order torque ripple terms, which tend to be the more effective in electric motors. ⁸¹

Defining the outputs as $y_1={\theta }_m$ and $y_5={\psi }_r$ , the system equations can be organized as in equations (10) to (14)

$${\stackrel{·}{y}}_1=y_2$$

(10)

$${\stackrel{·}{y}}_2=y_3$$

(11)

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{y}}_3=}& -\frac{a_1}{J_L}(a_4+a_6){\psi }_r{i}_{\mathrm{sq}}-\frac{a_1}{J_L}{a}_3{\omega }_m{\psi }_r^2\hfill \\ \multicolumn{1}{c}{}& -\frac{a_1}{J_L}p{\omega }_m{i}_{\mathrm{sd}}{\psi }_r+\frac{a_1}{J_L}{a}_5{\psi }_r{v}_{\mathrm{sq}}-a_2{\stackrel{·}{\omega }}_m\hfill \\ \multicolumn{1}{c}{}& -\frac{1}{J_L}{\stackrel{·}{t}}_L+\frac{1}{J_L}{\stackrel{·}{t}}_{\nu }\hfill \end{array}$$

(12)

$${\stackrel{·}{y}}_5=y_6$$

(13)

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{y}}_6=}& a_6{a}_6{\psi }_r-a_6{a}_7{i}_{\mathrm{sd}}+a_{7}p{\omega }_m{i}_{\mathrm{sq}}-a_7{a}_4{i}_{\mathrm{sd}}\hfill \\ \multicolumn{1}{c}{}& +\frac{a_7{a}_7{i}_{\mathrm{sq}}{i}_{\mathrm{sq}}}{{\psi }_r}+a_7{a}_8{\psi }_r+a_7{a}_5{v}_{\mathrm{sd}}\hfill \end{array}$$

(14)

The nonlinear terms given in equations (12) and (14) are simplified by defining new control inputs as in equations (15) and (16)

$$\begin{array}{cc}\multicolumn{1}{c}{u_q=}& -\frac{a_1}{J_L}(a_4+a_6){\psi }_r{i}_{\mathrm{sq}}-\frac{a_1}{J_L}{a}_3{\omega }_m{\psi }_r^2\hfill \\ \multicolumn{1}{c}{}& -\frac{a_1}{J_L}p{\omega }_m{i}_{\mathrm{sd}}{\psi }_r+\frac{a_1}{J_L}{a}_5{\psi }_r{v}_{\mathrm{sq}}\hfill \end{array}$$

(15)

$$\begin{array}{cc}\multicolumn{1}{c}{u_d=}& -a_6{a}_7{i}_{\mathrm{sd}}+a_{7}p{\omega }_m{i}_{\mathrm{sq}}-a_7{a}_4{i}_{\mathrm{sd}}\hfill \\ \multicolumn{1}{c}{}& +\frac{a_7{a}_7{i}_{\mathrm{sq}}{i}_{\mathrm{sq}}}{{\psi }_r}+a_7{a}_8{\psi }_r+a_7{a}_5{v}_{\mathrm{sd}}\hfill \end{array}$$

(16)

Using equations (15) and (16)— $y_4=u_q$ and ${\stackrel{·}{y}}_4={\stackrel{·}{u}}_q=u_1$ , $y_7=u_d$ and ${\stackrel{·}{y}}_7={\stackrel{·}{u}}_d=u_2$ —the system equations are obtained as in equation (17)

$$\begin{array}{c}{\stackrel{·}{y}}_1=y_2\hfill \\ {\stackrel{·}{y}}_2=y_3\hfill \\ {\stackrel{·}{y}}_3=-a_2{y}_3-\frac{1}{J_L}{\stackrel{·}{t}}_L+\frac{1}{J_L}{\stackrel{·}{t}}_{\nu }+y_4\hfill \end{array}$$

(17)

The control inputs applied to the motor are represented as in equations (18) and (19)

$$v_{\mathrm{sq}}=\frac{(u_q+\frac{a_1}{J_L}(a_4+a_6){\psi }_r{i}_{\mathrm{sq}}+\frac{a_1}{J_L}{a}_3{\omega }_m{\psi }_r^2+\frac{a_1}{J_L}p{\omega }_m{i}_{\mathrm{sd}}{\psi }_r)}{\frac{a_1}{J_L}{a}_5{\psi }_r}$$

(18)

$$v_{\mathrm{sd}}=\frac{(u_d+a_6{a}_7{i}_{\mathrm{sd}}-a_{7}p{\omega }_m{i}_{\mathrm{sq}}+a_7{a}_4{i}_{\mathrm{sd}}-\frac{a_7{a}_7{i}_{\mathrm{sq}}{i}_{\mathrm{sq}}}{{\psi }_r}-a_7{a}_8{\psi }_r)}{a_7{a}_5}$$

(19)

As can be easily seen from equations (18) and (19), the system is not state-input linearizable. In addition, it must be ensured that ${\psi }_r$ is nonzero. A robust control input is needed for u_q against the derivatives of the disturbance and harmonic effects. Next, u_q and u_d are derived based on HOSMC.

Derivation of u_q for position control

For position control, the relative degree is 4, and hence, a 4-HOSMC is proposed for position control^82–84 for avoiding the chattering effect from the control input. For 4-HOSMC, the error dynamics become ${\sigma }_{\theta }={\stackrel{·}{\sigma }}_{\theta }={\stackrel{··}{\sigma }}_{\theta }=0$ . Assume that ${\sigma }_{\theta },{\stackrel{·}{\sigma }}_{\theta },{\stackrel{··}{\sigma }}_{\theta }$ are differentiable functions, where ${\sigma }_{\theta }={\theta }_m-{\theta }_m^{\mathrm{ref}}=y_1-y_1^{\mathrm{ref}}$ . Thus, the control law is obtained as in equation (20)

$$u_q=\int u_1\mathrm{dt}=-A_q\int [\theta +3{({\stackrel{··}{\sigma }}_{\theta }^6+{\stackrel{·}{\sigma }}_{\theta }^4+{\left|{\sigma }_{\theta }\right|}^3)}^{\frac{1}{12}}]\mathrm{sign}({\stackrel{··}{\sigma }}_{\theta }+{({\stackrel{·}{\sigma }}_{\theta }^4+{\left|{\sigma }_{\theta }\right|}^3)}^{\frac{1}{6}}\mathrm{sign}({\stackrel{·}{\sigma }}_{\theta }+0.5{\left|{\sigma }_{\theta }\right|}^{\frac{3}{4}}\mathrm{sign}\left({\sigma }_{\theta }\right)))\mathrm{dt}$$

(20)

where $A_q$ is a constant control parameter.

Derivation of u_d for flux control

For flux control, the relative degree is 3, and hence, a 3-HOSMC is proposed considering the error dynamics as ${\sigma }_{\psi }={\stackrel{·}{\sigma }}_{\psi }={\stackrel{··}{\sigma }}_{\psi }=0$ . Assume that ${\sigma }_{\psi },{\stackrel{·}{\sigma }}_{\psi },{\stackrel{··}{\sigma }}_{\psi }$ are differentiable functions, where ${\sigma }_{\psi }={\psi }_r-{\psi }_r^{\mathrm{ref}}=y_4-y_4^{\mathrm{ref}}$ . The control law is defined as in equation (21)

$$u_d=\int u_2\mathrm{dt}=-B_d\int [{\stackrel{··}{\sigma }}_{\psi }+z{({\left|{\stackrel{·}{\sigma }}_{\psi }\right|}^3+{\sigma }_{\psi }^2)}^k]\mathrm{sign}({\stackrel{·}{\sigma }}_{\psi }+{\left|{\sigma }_{\psi }\right|}^{\frac{2}{3}}\mathrm{sign}\left({\sigma }_{\psi }\right))\mathrm{dt}$$

(21)

where $B_d$ is a constant control parameter.

Disturbance effect, unmodeled dynamics, torque ripples and remaining nonlinear terms after linearization will be eliminated by the HOSMC controllers. To apply this controller, one needs information on the following variables: ${\psi }_r,{\stackrel{·}{\psi }}_r,{\stackrel{··}{\psi }}_r,{\theta }_m,{\omega }_m,{\stackrel{·}{\omega }}_m,{\stackrel{··}{\omega }}_m$ . The EKF-based observer was designed for sensorless control to estimate ${\psi }_r$ , ${\theta }_m$ , and ${\omega }_m$ , and then robust differentiators were designed for the remaining variables. ⁸⁵

Robust-adaptive FL approach with torque ripple minimization for VC

In this section, a robust-adaptive feedback linearizing scheme was implemented for the position tracking control of IMs with torque ripple minimization via the q-axis current. The adaptive I-O linearizing inner loop was designed for the cancelation of torque ripple terms and can be augmented for the cancelation of sinusoidal or constant load terms (including constant friction). As the robust outer loop, it was designed an HOSMC scheme with continuous input to avoid the calculation of high-order derivatives. To eliminate the torque ripple, the frequencies of the most significant components were determined with a self-commissioning scheme with the unknown amplitudes calculated by parameter adaptation.

For the design of the control law, the q-axis current component, equation (5) is written as in equation (22)

$$\frac{d{\omega }_m}{\mathrm{dt}}=\frac{a_1}{J_L}{\psi }_r(i_{\mathrm{sq}}^{\mathrm{ref}}-e_q)-a_2{\omega }_m-\frac{1}{J_L}{t}_L+\frac{1}{J_L}{t}_{\nu }$$

(22)

where $e_q=i_{\mathrm{sq}}^{\mathrm{ref}}-i_{\mathrm{sq}}$ , $i_{\mathrm{sq}}^{\mathrm{ref}}=i_{\mathrm{sq}}^{\mathrm{SMC}}+i_{\mathrm{sq}}^{\mathrm{AFL}}$ , and $i_{\mathrm{sq}}^{\mathrm{SMC}}$ are robust outer loop controller and $i_{\mathrm{sq}}^{\mathrm{AFL}}$ is the adaptive linearizing control input. Define $e_{\psi }={\psi }_r^{\mathrm{ref}}-{\psi }_r$ and nonlinear disturbance term: $d=-(a_1/J_L)({\psi }_r{e}_q+e_{\psi }{i}_{\mathrm{sq}}^{\mathrm{SMC}})-a_2{\omega }_m$ ; ${\psi }_r^{\mathrm{ref}}$ : constant.

Equation (22) is transformed into equation (23)

$$\frac{d{\omega }_m}{\mathrm{dt}}=\frac{a_1}{J_L}{\psi }_r^{\mathrm{ref}}{i}_{\mathrm{sq}}^{\mathrm{SMC}}+\frac{a_1}{J_L}{\psi }_r{i}_{\mathrm{sq}}^{\mathrm{AFL}}-d+\frac{1}{J_L}{t}_{\nu }-\frac{1}{J_L}{t}_L$$

(23)

Taking ${\theta }_m$ and ${\psi }_r$ as outputs and $i_{\mathrm{sq}}$ and $v_{\mathrm{sd}}$ as inputs, the system equations are derived as in equation (24)

$$\begin{array}{c}{\stackrel{·}{y}}_1=y_2\hfill \\ {\stackrel{·}{y}}_2=y_3\hfill \\ {\stackrel{·}{y}}_3=\frac{a_1}{J_L}{\psi }_r^{\mathrm{ref}}{i}_{\mathrm{sq}}^{\mathrm{SMC}}+\frac{a_1}{J_L}{\psi }_r{i}_{\mathrm{sq}}^{\mathrm{AFL}}-d+\frac{1}{J_L}{t}_{\nu }-\frac{1}{J_L}{t}_L\hfill \end{array}$$

(24)

where $u_d=a_{7}p{\omega }_m{i}_{\mathrm{sq}}+a_7^2\frac{i_{\mathrm{sq}}^2}{{\psi }_r}-a_7{a}_4{i}_{\mathrm{sd}}+a_7{a}_5{v}_{\mathrm{sd}}$

An HOSMC-based robust outer loop for position control is derived as a novel chattering-free HOSMC with continuous input based on the Lyapunov approach as in equation (25)

$$\begin{array}{c}{i}_{\mathrm{sq}}^{\mathrm{SMC}}=\frac{J_L}{a_1{\psi }_r^{\mathrm{ref}}T}\hfill \\ ({\stackrel{·}{e}}_{\theta }+(D_{q1}+D_{q2})e_{\theta }+D_{q1}{D}_{q2}\int e_{\theta }\mathrm{dt})\hfill \end{array}$$

(25)

where $e_{\theta }={\theta }_m^{\mathrm{ref}}-{\theta }_m$ , ${\stackrel{·}{e}}_{\theta }={\stackrel{·}{\theta }}_m^{\mathrm{ref}}-{\stackrel{·}{\theta }}_m$ , and $D_{q1}$ and $D_{q2}$ are positive design parameters and T is sampling time. By substituting equation (25) into equation (24), the error dynamics are obtained as in equation (26)

$$\begin{array}{c}0={\stackrel{··}{e}}_{\theta }+\frac{1}{T}\hfill \\ ({\stackrel{·}{e}}_{\theta }+(D_{q1}+D_{q2})e_{\theta }+D_{q1}{D}_{q2}\int e_{\theta }\mathrm{dt})\hfill \\ +\frac{a_1}{J_L}{\psi }_r{i}_{\mathrm{sq}}^{\mathrm{AFL}}-d+\frac{1}{J_L}{t}_{\nu }-\frac{1}{J_L}{t}_L\hfill \end{array}$$

(26)

An adaptive linearization scheme was designed to suppress sinusoidal and/or constant load terms “ $t_C$ ” (i.e. Coulomb friction) as well as torque ripple terms “ $t_s$ .” General representation of those terms is given as in equation (27)

$$\begin{array}{c}{t}_L+t_{\nu }=t_C+t_s\sin \theta +a_1{\psi }_r{i}_{\mathrm{sq}}\hfill \\ \sum _{n=1}^{\infty }[k_{c6n}\cos 6\mathrm{np}\theta +k_{s6n}\sin 6\mathrm{np}\theta ]\hfill \\ =a_1{\psi }_r\left[\frac{t_C}{a_1{\psi }_r}\frac{t_s}{a_1{\psi }_r}\begin{array}{ccccc}{k}_{c6}\hfill & k_{s6}\hfill & k_{c12}\hfill & k_{s12}\hfill & \cdots \hfill \end{array}\right]\hfill \\ \left[\begin{array}{c}\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{\sin \theta }\\ \multicolumn{1}{c}{\cos 6p\theta }\end{array}\hfill \\ \multicolumn{1}{c}{\sin 6p\theta }\\ \multicolumn{1}{c}{\cos 12p\theta }\\ \multicolumn{1}{c}{\sin 12p\theta }\\ \multicolumn{1}{c}{⋮}\end{array}\right]=a_1{\psi }_r{P}^{T}W\hfill \end{array}$$

(27)

The control input, $i_{\mathrm{sq}}^{\mathrm{AFL}}$ , is derived with the parameter estimation vector, ${\hat{P}}^T$ , and regressor, $W$ , as in equation (28)

$$i_{\mathrm{sq}}^{\mathrm{AFL}}=-{\hat{P}}^{T}W$$

(28)

Substituting equations (27) and (28) into equation (26), the error dynamic becomes as in equation (29)

$$\begin{array}{c}{\tilde{P}}^{T}W={\stackrel{··}{e}}_{\theta }+\frac{1}{T}\hfill \\ ({\stackrel{·}{e}}_{\theta }+(D_{q1}+D_{q2})e_{\theta }+D_{q1}{D}_{q2}\int e_{\theta }\mathrm{dt})\hfill \end{array}$$

(29)

For the update law of parameters, a Lyapunov function is chosen as in equation (30)

$$\begin{array}{c}V=\frac{1}{2}{({\stackrel{·}{e}}_{\theta }+{\beta }_1{e}_{\theta }+{\beta }_2\int e_{\theta }\mathrm{dt})}^2\hfill \\ +\frac{1}{2}{({\beta }_3{e}_{\theta }+{\beta }_4\int e_{\theta }\mathrm{dt})}^2+\frac{1}{2}{\tilde{P}}^T{\Gamma }^{-1}\tilde{P}\hfill \end{array}$$

(30)

where ${\beta }_1,{\beta }_2,{\beta }_3,\mathrm{and}{\beta }_4$ are positive coefficients and $\Gamma$ is positive matrix. The derivative of the Lyapunov function is given as in equation (31)

$$\begin{array}{c}\stackrel{·}{V}=({\stackrel{··}{e}}_{\theta }+{\beta }_1{\stackrel{·}{e}}_{\theta }+{\beta }_2{e}_{\theta })\hfill \\ ({\stackrel{·}{e}}_{\theta }+{\beta }_1{e}_{\theta }+{\beta }_2\int e_{\theta }\mathrm{dt})+({\beta }_3{\stackrel{·}{e}}_{\theta }+{\beta }_4{e}_{\theta })\hfill \\ ({\beta }_3{e}_{\theta }+{\beta }_4\int e_{\theta }\mathrm{dt})+{\tilde{P}}^T{\Gamma }^{-1}\stackrel{·}{\stackrel{~}{P}}<0\hfill \end{array}$$

(31)

The negative definites can be ensured by fulfilling the following equalities and inequalities given in equations (32) to (34).

• Inequality conditions

$$\begin{array}{c}{\beta }_1,{\beta }_2,{\beta }_3,{\beta }_4,D_{q1},D_{q2},T>0,{\beta }_1<\frac{1}{T},\hfill \\ {\beta }_2<\frac{D_{q1}+D_{q2}}{T}\hfill \end{array}$$

(32)

• Equality conditions

$$\begin{array}{c}{\beta }_3{\beta }_4-\frac{{\beta }_2}{T}-\frac{D_{q1}{D}_{q2}}{T}=0,{\beta }_1^2+{\beta }_3^2+{\beta }_2\hfill \\ -\frac{{\beta }_1}{T}-\frac{D_{q1}+D_{q2}}{T}=0,{\beta }_2^2+{\beta }_4^2\hfill \\ -{\beta }_2\frac{D_{q1}+D_{q2}}{T}-{\beta }_1\frac{D_{q1}{D}_{q2}}{T}=0\hfill \end{array}$$

(33)

$$\begin{array}{c}\stackrel{·}{\stackrel{~}{P}}=\stackrel{·}{\hat{P}}=\Gamma W\hfill \\ ({\stackrel{·}{e}}_{\theta }+(D_{q1}+D_{q2})e_{\theta }+D_{q1}{D}_{q2}\int e_{\theta }\mathrm{dt})\hfill \end{array}$$

(34)

Equation (34) also gives the parameter adaptation rules. Now that the $i_{\mathrm{sq}}^{\mathrm{ref}}$ is derived to achieve the tracking control of the angular position (for set point or tracking control) while also minimizing torque ripple, a control must be designed to force the current, $i_{\mathrm{sq}}$ , to track the reference. For the calculation of control law, equation (8) is taken into consideration. Choosing the sliding mode as ${\sigma }_q=i_{\mathrm{sq}}^{\mathrm{ref}}-i_{\mathrm{sq}}$ and using the Lyapunov approach, the following control law is derived in discrete form as in equation (35)

$$\begin{array}{c}{v}_{\mathrm{sq}}\left(k\right)=v_{\mathrm{sq}}(k-1)+\frac{1}{a_{5}T}\hfill \\ [(1+D_{q}T){\sigma }_q\left(k\right)-{\sigma }_q(k-1)]\hfill \end{array}$$

(35)

A chattering-free 3-HOSMC with discontinuous input for flux control shares the same structure represented in equation (21). The artificial input $({\stackrel{·}{u}}_d)$ is obtained by increasing the system order to decrease the chattering effect on the output. Novel methods for the calculation of high-order derivatives were sought. To avoid the estimation of high-order derivatives and also chattering effects of the discontinuous input, a lower order HOSMC with continuous input was implemented. The control block diagram of the system is given in Figure 1.

Figure 1.

System control block scheme for robust control of IM.

Robust-adaptive FL approach with torque ripple minimization for DTC

Fast torque and flux responses can be obtained with DTC, which requires the flux magnitude, flux angle, and torque variations. Observers deliver the information required by the control system. A robust-adaptive FL scheme with constant switching frequency to avoid extra harmonic components caused by variable switching frequency was designed. The functional block diagram for the whole system is given in Figure 2.

Figure 2.

System control block scheme for robust-adaptive linearization IM.

For this purpose, the d-q model is organized with motor torque as a state variable in equation (36)

$$\begin{array}{c}\frac{d{t}_e}{\mathrm{dt}}=\frac{d}{\mathrm{dt}}\left(a_1{\psi }_r{i}_{\mathrm{sq}}\right)=a_1\frac{d{\psi }_r}{\mathrm{dt}}{i}_{\mathrm{sq}}+a_1\frac{d{i}_{\mathrm{sq}}}{\mathrm{dt}}{\psi }_r\hfill \\ =-(a_6+a_1{a}_6+a_4)t_e+u_q\hfill \end{array}$$

(36)

with definitions, such as

$$\begin{array}{c}\multicolumn{1}{c}{u_q=-a_1{a}_3{\omega }_m{\psi }_r^2+a_1{a}_5{\psi }_r{v}_{\mathrm{sq}},v_{\mathrm{sq}}}\\ =\frac{a_1{a}_3{\omega }_m{\psi }_r^2+u_q}{a_1{a}_5{\psi }_r},\hfill \\ e_t=t_e^{\mathrm{ref}}-t_e\hfill \end{array}$$

The following equations are obtained as in equations (37) to (39)

$$\frac{d{\theta }_m}{dt}={\omega }_m,\frac{d{\psi }_r}{dt}=-a_6{\psi }_r+a_7{i}_{sd}$$

(37)

$$\begin{array}{c}\frac{d{\omega }_m}{\mathrm{dt}}=\frac{a_1}{J_L}{t}_e^{\mathrm{ref}}-a_2{\omega }_m-\frac{a_1}{J_L}{e}_t-\frac{1}{J_L}{t}_L+\frac{1}{J_L}{t}_{\nu },\hfill \\ \frac{d{i}_{\mathrm{sd}}}{\mathrm{dt}}=a_8{\psi }_r+u_d\hfill \end{array}$$

(38)

$$\frac{d{t}_e}{\mathrm{dt}}=-(a_6+a_1{a}_6+a_4)t_e+u_q$$

(39)

Here, ${\theta }_m$ and ${\psi }_r$ are decoupled outputs, and $v_{\mathrm{sq}},v_{\mathrm{sd}}$ are inputs. Adopting the similar approach taken for the I-O linearizing scheme developed in the “Robust-adaptive FL approach with torque ripple minimization for VC” section for the current, the reference torque, $t_e^{\mathrm{ref}}=t_e^{\mathrm{SMC}}+t_e^{\mathrm{AFL}}$ , is derived, where $t_e^{\mathrm{SMC}}$ corresponds to robust outer loop control and $t_e^{\mathrm{AFL}}$ refers to adaptive linearizing control (to perform torque ripple cancelation).

$t_e^{\mathrm{SMC}}$ is designed based on 3-HOSMC continuous control, which is represented in equation (40)

$$\begin{array}{c}{t}_e^{\mathrm{SMC}}=\frac{1}{T}({\stackrel{·}{e}}_{\theta }+(D_{t1}+D_{t2})e_{\theta }+D_{t1}{D}_{t2}\int e_{\theta }\mathrm{dt})\hfill \\ (e_{\theta }={\theta }_m^{\mathrm{ref}}-{\theta }_m)\hfill \end{array}$$

(40)

The adaptive linearizing term is $t_e^{\mathrm{AFL}}=-{\hat{P}}^{T}W$ , where ${\hat{P}}^T$ and $W$ are parameter and regressor, respectively. The parameter adaptation rule is derived as in equation (34): $\stackrel{·}{\stackrel{~}{P}}=\stackrel{·}{\hat{P}}=\Gamma W({\stackrel{·}{e}}_{\theta }+(D_{t1}+D_{t2})e_{\theta }+D_{t1}{D}_{t2}\int e_{\theta }\mathrm{dt})$ .

By choosing the sliding mode as ${\sigma }_t=t_e^{\mathrm{ref}}-t_e$ and using Lyapunov approach for 1-HOSMC with continuous input, u_q is derived as in equation (41)

$$u_q\left(k\right)=u_q(k-1)+\frac{1}{T}[(1+D_{t}T){\sigma }_t\left(k\right)-{\sigma }_t(k-1)]$$

(41)

Multiple EKF-based observer

An EKF-based observer scheme was designed for the estimation of high number of parameters and states required by I-O linearization. The IM model depends on six parameters: ( ${\hat{R}}_s$ , ${\hat{R}}_r$ , ${\hat{L}}_s$ , ${\hat{L}}_r$ , ${\hat{L}}_m$ , and ${\hat{t}}_L$ ) in addition to the states ( ${\hat{\theta }}_m$ , ${\hat{\omega }}_m$ , ${\hat{\psi }}_r$ , ${\hat{\theta }}_s$ , ${\hat{i}}_{s\alpha }$ , and ${\hat{i}}_{s\beta }$ ). The estimation of these states and parameters would require several EKFs run in parallel, due to the limited number of measured output of sensorless control. Similarly, the limitations of one EKF algorithm estimating multiple states and parameters are a well-known fact. Thus, six different EKF models were designed (6: number of estimated parameters) to run consecutively and in order, at each sampling time. Each EKF passes the one parameter it has individually estimated to the others. The estimated values are also simultaneously used by the control algorithm. Successful results obtained with two alternating EKF algorithms for the estimation of R_s and R_r will be presented in the next section. The EKF algorithms use the $\alpha -\beta$ model given in equation (42), with the definition of $i_{s\alpha }$ , $i_{s\beta }$ : stator stationary axis components of stator currents; $v_{s\alpha }$ , $v_{s\beta }$ : stator stationary axis components of stator voltages; ${\psi }_{r\alpha }$ ${\psi }_{r\beta }$ : stator stationary axis components of rotor fluxes; and X symbolizes the parameter estimated by that particular EKF and is one of the following ( ${\hat{R}}_s$ , ${\hat{R}}_r$ , ${\hat{L}}_s$ , ${\hat{L}}_r$ , ${\hat{L}}_m$ , and ${\hat{t}}_L$ ). The derivation of EKF is standard and will not be given here due to space limitations

$$\begin{array}{cc}\multicolumn{1}{c}{\left[\begin{array}{c}{\stackrel{·}{i}}_{s\alpha }\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{i}}_{s\beta }}\\ \multicolumn{1}{c}{{\stackrel{·}{\psi }}_{r\alpha }}\\ \multicolumn{1}{c}{{\stackrel{·}{\psi }}_{r\beta }}\\ \multicolumn{1}{c}{{\stackrel{·}{\omega }}_m}\\ \multicolumn{1}{c}{\begin{array}{c}\stackrel{·}{\theta }\hfill \\ \multicolumn{1}{c}{\stackrel{·}{X}}\end{array}}\end{array}\right]}& =\underset{{\underset{¯}{A}}_e}{\underbrace{\left[\begin{array}{cccccc}-(\frac{R_s}{L_{\sigma }}+\frac{L_m^2{R}_r^{\prime }}{L_{\sigma }{L}_r^{\prime 2}})\hfill & 0\hfill & \frac{L_m{R}_r^{\prime }}{L_{\sigma }{L}_r^{\prime 2}}\hfill & \frac{L_m}{L_{\sigma }{L}_r^{\prime }}p{\omega }_m\hfill & 0\hfill & 0\begin{array}{c}0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{0}& -(\frac{R_s}{L_{\sigma }}+\frac{L_m^2{R}_r^{\prime }}{L_{\sigma }{L}_r^{\prime 2}})\hfill & -\frac{L_m}{L_{\sigma }{L}_r^{\prime }}p{\omega }_m\hfill & \frac{L_m{R}_r^{\prime }}{L_{\sigma }{L}_r^{\prime 2}}\hfill & 0\hfill & 0\begin{array}{c}0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\frac{R_r^{\prime }}{L_r^{\prime }}{L}_m}& 0\hfill & -\frac{R_r^{\prime }}{L_r^{\prime }}\hfill & -p{\omega }_m\hfill & 0\hfill & 0\begin{array}{c}0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{0}& \frac{R_r^{\prime }}{L_r^{\prime }}{L}_m\hfill & p{w}_m\hfill & -\frac{R_r^{\prime }}{L_r^{\prime }}\hfill & 0\hfill & 0\begin{array}{c}0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{-\frac{1}{J_L}\frac{3}{2}p\frac{L_m}{L_r^{\prime }}{\psi }_{r\beta }}& \frac{1}{J_L}\frac{3}{2}p\frac{L_m}{L_r^{\prime }}{\psi }_{r\alpha }\hfill & 0\hfill & 0\hfill & -\frac{B_L}{J_L}\hfill & 0\begin{array}{c}0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}}& \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill & \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill & \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill & \begin{array}{c}1\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill & \begin{array}{c}0\begin{array}{c}0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{0\begin{array}{c}0\hfill \end{array}}\end{array}\hfill \end{array}\right]}}\underset{{\underset{¯}{x}}_e}{\underbrace{\left[\begin{array}{c}{i}_{s\alpha }\hfill \\ \multicolumn{1}{c}{i_{s\beta }}\\ \multicolumn{1}{c}{{\psi }_{r\alpha }}\\ \multicolumn{1}{c}{{\psi }_{r\beta }}\\ \multicolumn{1}{c}{{\omega }_m}\\ \multicolumn{1}{c}{\begin{array}{c}\theta \hfill \\ \multicolumn{1}{c}{X}\end{array}}\end{array}\right]}}\hfill \\ \multicolumn{1}{c}{}& +\left[\begin{array}{cc}\frac{1}{L_{\sigma }}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \frac{1}{L_{\sigma }}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \\ \multicolumn{1}{c}{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}}& \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill \end{array}\right]\left[\begin{array}{c}{v}_{s\alpha }\hfill \\ \multicolumn{1}{c}{v_{s\beta }}\end{array}\right]+\underset{{\underset{¯}{w}}_{\mathrm{tLe}}}{\underbrace{\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{-\frac{1}{J_L}}\\ \multicolumn{1}{c}{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}}\end{array}\right]}}{t}_L\hfill \end{array}$$

(42)

The structure and block scheme of the multiple EKF algorithm were demonstrated in Figure 3(a) and (b), respectively.

Figure 3.

(a) Structure of the EKF algorithm and (b) multiple EKF algorithm block scheme.

Multiple EKF-based sensorless DTC

The sensorless DTC scheme developed for IM requires the estimation of stator flux components ( ${\psi }_{s\alpha }$ , ${\psi }_{s\beta }$ ), angular velocity $({\omega }_m)$ , and stator current components ( $i_{s\alpha }$ , $i_{s\beta }$ ), which are also measured as output. Due to the degrading effect of their unknown variations on control performance, the load torque, $t_L$ , stator resistance, $R_s$ , together with the rotor resistance, $R_r$ , are estimated simultaneously. For this purpose, two EKF models, one with $R_s$ and one with $R_r$ , are run alternatingly, at each sampling time. The resistance value estimated at each step is used to update the corresponding value in the other model. The extended models (as referred to the stator stationary frame) used by the scheme can be given in the following general form as in equations (43) and (44)

$$\begin{array}{c}{\underset{¯}{\stackrel{·}{x}}}_{\mathrm{ej}}\left(t\right)={\underset{¯}{f}}_{\mathrm{ej}}({\underset{¯}{x}}_{\mathrm{ej}}\left(t\right),{\underset{¯}{u}}_e\left(t\right))+{\underset{¯}{w}}_{j1}\left(t\right)\hfill \\ \left(\mathrm{system}\mathrm{equation}\right)\hfill \end{array}$$

(43)

$$\begin{array}{c}\underset{¯}{Z}\left(t\right)={\underset{¯}{h}}_{\mathrm{ej}}\left({\underset{¯}{x}}_{\mathrm{ej}}\left(t\right)\right)+{\underset{¯}{w}}_{j2}\left(t\right)\hfill \\ \left(\mathrm{measurement}\mathrm{equation}\right)\hfill \end{array}$$

(44)

Here, “j = r or s” (based on whether the extended model has $R_s$ or $R_r$ ); the extended state vector ${\underset{¯}{x}}_{\mathrm{ej}}$ represents the estimated states and load torque, $t_L$ , which is included in the extended state vector as a constant state with the assumption of a slow variation with time; ${\underset{¯}{f}}_{\mathrm{ej}}$ : nonlinear function of the states and inputs; ${\underset{¯}{A}}_{\mathrm{ej}}$ : system matrix; ${\underset{¯}{u}}_e$ : control input vector; ${\underset{¯}{B}}_e$ : input matrix; ${\underset{¯}{w}}_{j1}$ : process noise; ${\underset{¯}{h}}_{\mathrm{ej}}$ : function of the outputs; ${\underset{¯}{H}}_e$ : measurement matrix; and ${\underset{¯}{w}}_{j2}$ : measurement noise.

Based on those assumptions, system model can be given as in equation (45)

$$\begin{array}{c}\left[\begin{array}{c}{\stackrel{·}{i}}_{s\alpha }\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{i}}_{s\beta }}\\ \multicolumn{1}{c}{{\stackrel{·}{\psi }}_{s\alpha }}\\ \multicolumn{1}{c}{{\stackrel{·}{\psi }}_{s\beta }}\\ \multicolumn{1}{c}{{\stackrel{·}{\omega }}_m}\\ \multicolumn{1}{c}{{\stackrel{·}{t}}_L}\\ \multicolumn{1}{c}{{\stackrel{·}{R}}_j}\end{array}\right]=\left[\begin{array}{ccccccc}-(\frac{R_s}{L_{\sigma }}+\frac{R_r^{\prime }{L}_s}{L_r^{\prime }{L}_{\sigma }})\hfill & -p_p{\omega }_m\hfill & \frac{R_r^{\prime }}{L_r^{\prime }{L}_{\sigma }}\hfill & \frac{p_p{\omega }_m}{L_{\sigma }}\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{p_p{\omega }_m}& -(\frac{R_s}{L_{\sigma }}+\frac{R_r^{\prime }{L}_s}{L_r^{\prime }{L}_{\sigma }})\hfill & -\frac{p_p{\omega }_m}{L_{\sigma }}\hfill & \frac{R_r^{\prime }}{L_r^{\prime }{L}_{\sigma }}\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-R_s}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -R_s\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-\frac{3}{2}\frac{p_p}{J_L}{\psi }_{s\beta }}& \frac{3}{2}\frac{p_p}{J_L}{\psi }_{s\alpha }\hfill & 0\hfill & 0\hfill & 0\hfill & -\frac{1}{J_L}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]\left[\begin{array}{c}{i}_{s\alpha }\hfill \\ \multicolumn{1}{c}{i_{s\beta }}\\ \multicolumn{1}{c}{{\psi }_{s\alpha }}\\ \multicolumn{1}{c}{{\psi }_{s\beta }}\\ \multicolumn{1}{c}{{\omega }_m}\\ \multicolumn{1}{c}{t_L}\\ \multicolumn{1}{c}{R_j}\end{array}\right]\hfill \\ +\left[\begin{array}{cc}\frac{1}{L_{\sigma }}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \frac{1}{L_{\sigma }}\hfill \\ \multicolumn{1}{c}{1}& 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]\left[\begin{array}{c}{v}_{s\alpha }\hfill \\ \multicolumn{1}{c}{v_{s\beta }}\end{array}\right]+{\underset{¯}{w}}_{j1}\left(t\right),\left[\begin{array}{c}{i}_{s\alpha }\hfill \\ \multicolumn{1}{c}{i_{s\beta }}\end{array}\right]=\left[\begin{array}{ccccccc}1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]\left[\begin{array}{c}{i}_{s\alpha }\hfill \\ \multicolumn{1}{c}{i_{s\beta }}\\ \multicolumn{1}{c}{{\psi }_{s\alpha }}\\ \multicolumn{1}{c}{{\psi }_{s\beta }}\\ \multicolumn{1}{c}{{\omega }_m}\\ \multicolumn{1}{c}{t_L}\\ \multicolumn{1}{c}{R_j}\end{array}\right]+{\underset{¯}{w}}_{j2}\hfill \end{array}$$

(45)

Here, ${\psi }_{s\alpha }$ and ${\psi }_{s\beta }$ represent stator stationary axis components of stator flux.

EKF-based estimator for the speed sensorless VC

EKF-based algorithms are developed for the speed sensorless FOC and DTC of IMs. The algorithms are designed aiming minimum estimation error over a wide velocity range, including very low- and long-duration zero-speed operation. A major challenge at very low and zero speed is the lost coupling effect from the rotor to the stator, which makes the rotor angular velocity unobservable on the stator side. The state-space representation of the model is derived as in equation (46)

$$\begin{array}{cc}\multicolumn{1}{c}{\underset{{\underset{¯}{\stackrel{·}{x}}}_e}{\underbrace{\left[\begin{array}{c}{\stackrel{·}{i}}_{s\alpha }\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{i}}_{s\beta }}\\ \multicolumn{1}{c}{{\stackrel{·}{\psi }}_{r\alpha }}\\ \multicolumn{1}{c}{{\stackrel{·}{\psi }}_{r\beta }}\\ \multicolumn{1}{c}{{\stackrel{·}{\omega }}_m}\\ \multicolumn{1}{c}{\stackrel{·}{\xi }}\\ \multicolumn{1}{c}{\stackrel{·}{\chi }}\end{array}\right]}}}& =\underset{{\underset{¯}{A}}_e}{\underbrace{\left[\begin{array}{ccccccc}-(\frac{R_s}{L_{\sigma }}+\frac{L_m^2{R}_{\mathrm{rN}}^{\prime }\chi }{L_{\sigma }{L}_r^{\prime 2}})\hfill & 0\hfill & \frac{L_m{R}_{\mathrm{rN}}^{\prime }\chi }{L_{\sigma }{L}_r^{\prime 2}}\hfill & \frac{L_m}{L_{\sigma }{L}_r^{\prime }}p{\omega }_m\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -(\frac{R_s}{L_{\sigma }}+\frac{L_m^2{R}_{\mathrm{rN}}^{\prime }\chi }{L_{\sigma }{L}_r^{\prime 2}})\hfill & -\frac{L_m}{L_{\sigma }{L}_r^{\prime }}{\omega }_m\hfill & \frac{L_m{R}_{\mathrm{rN}}^{\prime }\chi }{L_{\sigma }{L}_r^{\prime 2}}\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\frac{R_{\mathrm{rN}}^{\prime }\chi }{L_r^{\prime }}{L}_m}& 0\hfill & -\frac{R_{\mathrm{rN}}^{\prime }\chi }{L_r^{\prime }}\hfill & -p{\omega }_m\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \frac{R_{\mathrm{rN}}^{\prime }\chi }{L_r^{\prime }}{L}_m\hfill & p{\omega }_m\hfill & -\frac{R_{\mathrm{rN}}^{\prime }\chi }{L_r^{\prime }}\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-\frac{3}{2}\frac{p}{J_L}\frac{L_m}{L_r^{\prime }}{\psi }_{r\beta }}& \frac{3}{2}\frac{p}{J_L}\frac{L_m}{L_r^{\prime }}{\psi }_{r\alpha }\hfill & 0\hfill & 0\hfill & 0\hfill & -\frac{1}{J_L}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]}}\underset{{\underset{¯}{x}}_e}{\underbrace{\left[\begin{array}{c}{i}_{s\alpha }\hfill \\ \multicolumn{1}{c}{i_{s\beta }}\\ \multicolumn{1}{c}{{\psi }_{r\alpha }}\\ \multicolumn{1}{c}{{\psi }_{r\beta }}\\ \multicolumn{1}{c}{{\omega }_m}\\ \multicolumn{1}{c}{\xi }\\ \multicolumn{1}{c}{\chi }\end{array}\right]}}\hfill \\ \multicolumn{1}{c}{}& +\underset{{\underset{¯}{B}}_e}{\underbrace{\left[\begin{array}{cc}\frac{1}{L_{\sigma }}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \frac{1}{L_{\sigma }}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]}}\underset{{\underset{¯}{u}}_e}{\underbrace{\left[\begin{array}{c}{v}_{s\alpha }\hfill \\ \multicolumn{1}{c}{v_{s\beta }}\end{array}\right]}}+{\underset{¯}{w}}_1\left(t\right);\underset{¯}{Z}\left(t\right)=\left[\begin{array}{ccccccc}1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]\left[\begin{array}{c}{i}_{s\alpha }\hfill \\ \multicolumn{1}{c}{i_{s\beta }}\\ \multicolumn{1}{c}{{\psi }_{r\alpha }}\\ \multicolumn{1}{c}{{\psi }_{r\beta }}\\ \multicolumn{1}{c}{{\omega }_m}\\ \multicolumn{1}{c}{\xi }\\ \multicolumn{1}{c}{\chi }\end{array}\right]+{\underset{¯}{w}}_2\left(t\right)s\hfill \end{array}$$

(46)

where ${\underset{¯}{f}}_e$ : nonlinear function vector of the states and inputs; ${\underset{¯}{x}}_e$ : extended state vector; ${\underset{¯}{A}}_e$ : system matrix; ${\underset{¯}{u}}_e$ : control input vector; ${\underset{¯}{B}}_e$ : input matrix; ${\underset{¯}{h}}_e$ : function vector of the outputs; ${\underset{¯}{H}}_e$ : measurement matrix; ${\underset{¯}{w}}_1$ , ${\underset{¯}{w}}_2$ : process and measurement noise, respectively; $\xi$ : uncertainties on the mechanical side, primarily due to load torque and friction, reflected as an additive term; and $\chi$ : uncertainties on the electrical side, related primarily to temperature- and frequency-based variations of the rotor resistance.

The simultaneous estimation of the rotor angular velocity and load torque is performed with the velocity taken into consideration and not as a constant parameter. The load torque estimation is performed as a constant parameter to evaluate Coulomb and viscous friction at steady state for enhancing the performance at very low and zero speed. The algorithms based on the rotor and stator flux are conducted under variations and reversals of the velocity and load torque over a wide velocity range and at zero speed. In all scenarios, the current estimation error has stayed within a low error regime, while satisfactory and many times, lower velocity estimation errors have been acquired over a wide velocity range and at zero speed. The control algorithm of the speed sensorless DTC was given in Figure 4.

Figure 4.

Speed sensorless DTC.

There are some studies done in the literature^86–91 to design tuning rules for Q (process noise covariance matrix) and R (measurement noise covariance matrix). The sensitivity of the covariance matrix Q is bigger than R with respect to convergence and stability. If Q is increased irresponsibly, this means the system noise and parametric uncertainties are high. Therefore, the filter gain also rises and results in rapid but high response and fluctuations. If Q is decreased, this will lead to a weaker correction and state estimation. When R is increased, the measurements are sensitive to noise and should be weighted less by filter. This causes smaller filter gain and slower transient response. In this study, by considering these facts and making some trial-and-error research, the matrices were selected as

• ${\underset{¯}{Q}}_{j1}=\mathrm{diag}\left([{10}^{-6},{10}^{-6},{10}^{-6},{10}^{-6},{10}^{-4},{10}^{-5},{10}^{-7}]\right)$ ; auto-covariance of ${\underset{¯}{w}}_{j1}$

• ${\underset{¯}{R}}_{j2}=\mathrm{diag}\left([{0.001}^2,{0.001}^2]\right)$ ; auto-covariance of ${\underset{¯}{w}}_{j2}$

• “diag” refers to a diagonal matrix.

Stability and robustness analysis of the system

Exponential stability of the system

Adaptive systems are not robust to modeling error and parameter uncertainties. Robust stability can be guaranteed if the persistency of excitation of regressor is satisfied. The regressor functions in the induction motor control situation are continuous, periodic, and bounded. Therefore, it is possible to determine the conditions on exogenous signals for ensuring the persistency of excitation. ⁹²

In this section, the conditions on exogenous signals that ensure the persistency of excitation originating from the specific structure of the motor dynamics are determined, although it is not possible to find these conditions for general nonlinear systems.

Guaranteed minimum speed

Given ${\stackrel{·}{\theta }}_m>0$ , there exist $T>0$ , ${\stackrel{·}{\theta }}_{m,\max }^{\mathrm{ref}}⩾{\stackrel{·}{\theta }}_{m,\min }^{\mathrm{ref}}>0$ , and ${\stackrel{·}{\theta }}_{m,\max }⩾{\stackrel{·}{\theta }}_m$ such that ${\stackrel{·}{\theta }}_{m,\max }^{\mathrm{ref}}⩾{\stackrel{·}{\theta }}_m^{\mathrm{ref}}⩾{\stackrel{·}{\theta }}_{m,\min }^{\mathrm{ref}}$ for all $t$ implies ${\stackrel{·}{\theta }}_{m,\max }⩾{\stackrel{·}{\theta }}_m⩾{\stackrel{·}{\theta }}_{m,\min }$ for all $t⩾T$ .

Motor persistency of excitation

If ${\stackrel{·}{\theta }}_{m,\max }⩾{\stackrel{·}{\theta }}_m⩾{\stackrel{·}{\theta }}_{m,\min }>0$ for all $t⩾T$ , then regressor vector, which is defined as harmonic disturbance load $(w^T=[\begin{array}{cccccccc}1\hfill & \sin (p{\theta }_s)\hfill & \cos (p{\theta }_s)\hfill & \sin (2p{\theta }_s)\hfill & \cos (2p{\theta }_s)\hfill & \dots \hfill & \sin (\mathrm{np}{\theta }_s)\hfill & \cos (\mathrm{np}{\theta }_s)\hfill \end{array})$ , is persistently exciting for all $t⩾T$ . If motor operates at a minimum speed, it can be detected the entire torque ripple spectrum. The entire torque ripple spectrum can be captured by operating the motor at a minimum speed. For the danger of parameter drifting, updating process must be terminated when the motor speed is getting too low.

A vector-valued function $\mathrm{w}:{ℝ}_{+}\to {ℝ}^n$ is said to be persistently exciting, if there exist constant $T_0,\alpha ,\beta >0$ such that ⁹³

$$\beta I_n⩾\underset{t}{\overset{t+T_0}{\int }}w\left(\tau \right)w^T\left(\tau \right)d\tau ⩾\alpha I_n>0\mathrm{for}\mathrm{all}t$$

The existence of $\alpha ,\beta >0$ and $T_0$ must be established such that

$$I⩽\underset{t}{\overset{t+T_0}{\int }}w{w}^{T}d\tau ⩽\beta I\mathrm{for}\mathrm{all}t⩾T$$

Choose $T_0=(2\pi /p{\stackrel{·}{\theta }}_{m,\min })$ . Since ${\stackrel{·}{\theta }}_m⩾{\stackrel{·}{\theta }}_{m,\min }$ , for $t⩾T$ it can be obtained

$${\theta }_m(t+T_0)-{\theta }_m\left(t\right)=\underset{t}{\overset{t+T_0}{\int }}{\stackrel{·}{\theta }}_m\mathrm{dt}⩾{\stackrel{·}{\theta }}_{m,\min }{T}_0=\frac{2\pi }{p}$$

Then

$$\underset{t}{\overset{t+T_0}{\int }}w{w}^T\mathrm{dt}=\underset{{\theta }_m(t)}{\overset{{\theta }_m(t+T_0)}{\int }}\frac{w{w}^T}{{\stackrel{·}{\theta }}_m}d{\theta }_m$$

Since ${\stackrel{·}{\theta }}_m⩽{\stackrel{·}{\theta }}_{m,\max }$ , equation (47) can be stated in inequality form as

$$\begin{array}{c}\underset{t}{\overset{t+T_0}{\int }}w{w}^T\mathrm{dt}⩾\frac{1}{{\stackrel{·}{\theta }}_{m,\max }}\underset{{\theta }_m(t)}{\overset{{\theta }_m(t+T_0)}{\int }}w{w}^{T}d{\theta }_m\hfill \\ ⩾\frac{1}{{\stackrel{·}{\theta }}_{m,\max }}\underset{{\theta }_m(t)}{\overset{{\theta }_m(t)+\frac{2\pi }{p}}{\int }}w{w}^{T}d{\theta }_m\hfill \end{array}$$

(47)

where equation (47) is based on the facts that

$$w{w}^T⩾0\mathrm{and}{\theta }_m(t+T_0)⩾{\theta }_m\left(t\right)+\frac{2\pi }{p}$$

If the regressor vector $(w)$ is substituted into equation (47), equation (47) transforms into equation (48)

$$\begin{array}{c}\underset{t}{\overset{t+T_0}{\int }}w{w}^T\mathrm{dt}⩾\hfill \\ \frac{1}{{\stackrel{·}{\theta }}_{m,\max }}d{\theta }_s\underset{0}{\overset{\frac{2\pi }{p}}{\int }}\left[\begin{array}{cccc}1\hfill & \sin \left(p{\theta }_s\right)\hfill & ...\hfill & \cos \left(\mathrm{np}{\theta }_s\right)\hfill \\ \multicolumn{1}{c}{\sin \left(p{\theta }_s\right)}& \mathrm{si}{\mathrm{n}}^2\left(p{\theta }_s\right)\hfill & ...\hfill & \sin \left(p{\theta }_s\right)\cos \left(\mathrm{np}{\theta }_s\right)\hfill \\ \multicolumn{1}{c}{\cos \left(p{\theta }_s\right)}& \sin \left(p{\theta }_s\right)\cos \left(p{\theta }_s\right)\hfill & ...\hfill & \cos \left(p{\theta }_s\right)\cos \left(\mathrm{np}{\theta }_s\right)\hfill \\ \multicolumn{1}{c}{...}& ...\hfill & ...\hfill & ...\hfill \\ \multicolumn{1}{c}{\cos \left(\mathrm{np}{\theta }_s\right)}& \sin \left(p{\theta }_s\right)\cos \left(\mathrm{np}{\theta }_s\right)\hfill & ...\hfill & \mathrm{co}{\mathrm{s}}^2\left(\mathrm{np}{\theta }_s\right)\hfill \end{array}\right]d{\theta }_s=\hfill \\ \frac{1}{{\stackrel{·}{\theta }}_{m,\max }}\left[\begin{array}{cccc}\frac{2\pi }{p}\hfill & 0\hfill & ...\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \frac{\pi }{p}\hfill & ...\hfill & 0\hfill \\ \multicolumn{1}{c}{...}& ...\hfill & ...\hfill & ...\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & ...\hfill & \frac{\pi }{p}\hfill \end{array}\right]⩾\frac{\pi }{p{\stackrel{·}{\theta }}_{m,\max }}{I}_{2n+1}\hfill \end{array}$$

(48)

By using these formulations presented above, the conditions on exogenous signals, which ensure persistency of excitation, are specified.

In addition

$$\begin{array}{c}{\theta }_m(t+T_0)={\theta }_m\left(t\right)+\underset{t}{\overset{t+T_0}{\int }}{\stackrel{·}{\theta }}_m\mathrm{dt}⩽{\theta }_m\left(t\right)+{\stackrel{·}{\theta }}_{m,\max }{T}_0={\theta }_m\left(t\right)+\frac{{\stackrel{·}{\theta }}_{m,\max }}{{\stackrel{·}{\theta }}_{m,\min }}\frac{2\pi }{p}⩽{\theta }_m\left(t\right)+\left[\frac{{\stackrel{·}{\theta }}_{m,\max }}{{\stackrel{·}{\theta }}_{m,\min }}\right]\frac{2\pi }{p}\hfill \end{array}$$

where $[{\stackrel{·}{\theta }}_{m,\max }/{\stackrel{·}{\theta }}_{m,\min }]$ indicates the least integer greater than or equal to ${\stackrel{·}{\theta }}_{m,\max }/{\stackrel{·}{\theta }}_{m,\min }$ .

For that reason,

$$\begin{array}{c}\underset{t}{\overset{t+T_0}{\int }}w{w}^{T}d\tau ⩽\underset{{\theta }_m(t)}{\overset{{\theta }_m(t)+\left[\frac{{\stackrel{·}{\theta }}_{m,\max }}{{\stackrel{·}{\theta }}_{m,\min }}\right]\frac{2\pi }{p}}{\int }}\frac{w{w}^T}{{\stackrel{·}{\theta }}_m}d{\theta }_m\hfill \\ ⩽\frac{1}{{\stackrel{·}{\theta }}_{m,\min }}{\int }^{[\begin{array}{cccc}1\hfill & \sin \left(p{\theta }_s\right)\hfill & ...\hfill & \cos \left(\mathrm{np}{\theta }_s\right)\hfill \\ \multicolumn{1}{c}{\sin \left(p{\theta }_s\right)}& \mathrm{si}{\mathrm{n}}^2\left(p{\theta }_s\right)\hfill & ...\hfill & \sin \left(p{\theta }_s\right)\cos \left(\mathrm{np}{\theta }_s\right)\hfill \\ \multicolumn{1}{c}{\cos \left(p{\theta }_s\right)}& \sin \left(p{\theta }_s\right)\cos \left(p{\theta }_s\right)\hfill & ...\hfill & \cos \left(p{\theta }_s\right)\cos \left(\mathrm{np}{\theta }_s\right)\hfill \\ \multicolumn{1}{c}{...}& ...\hfill & ...\hfill & ...\hfill \\ \multicolumn{1}{c}{\cos \left(\mathrm{np}{\theta }_s\right)}& \sin \left(p{\theta }_s\right)\cos \left(\mathrm{np}{\theta }_s\right)\hfill & ...\hfill & \mathrm{co}{\mathrm{s}}^2\left(\mathrm{np}{\theta }_s\right)\hfill \end{array}]}d{\theta }_s=\hfill \\ \frac{1}{{\stackrel{·}{\theta }}_{m,\min }}\left[\frac{{\stackrel{·}{\theta }}_{m,\max }}{{\stackrel{·}{\theta }}_{m,\min }}\right]\left[\begin{array}{cccc}\frac{2\pi }{p}\hfill & 0\hfill & ...\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \frac{\pi }{p}\hfill & ...\hfill & 0\hfill \\ \multicolumn{1}{c}{...}& ...\hfill & ...\hfill & ...\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & ...\hfill & \frac{\pi }{p}\hfill \end{array}\right]⩽\frac{2\pi }{p{\stackrel{·}{\theta }}_{m,\min }}\left[\frac{{\stackrel{·}{\theta }}_{m,\max }}{{\stackrel{·}{\theta }}_{m,\min }}\right]I_{2n+1}\hfill \end{array}$$

Therefore, $w$ is persistently exciting on $[T,\infty )$ and the proof is finished.

Exponential stability

There exists ${\stackrel{·}{\theta }}_{m,\max }⩾{\stackrel{·}{\theta }}_{m,\min }>0$ such that ${\stackrel{·}{\theta }}_{m,\max }^{\mathrm{ref}}⩾{\stackrel{·}{\theta }}_m^{\mathrm{ref}}⩾{\stackrel{·}{\theta }}_{m,\min }^{\mathrm{ref}}$ for all $t$ implies $e_m,{\stackrel{·}{e}}_m,\tilde{P}\to 0$ exponentially with convergence rate $\zeta k+O({\zeta }^2)$ for some $k>0$ , where $\zeta$ is the adaptation rate.

Let $ϵ={\stackrel{·}{e}}_m+k_{\alpha }{e}_m$ , and without loss of generality, $\Gamma =\zeta I$ . Then, the parameter update law can be constructed as

$$\stackrel{.}{\stackrel{~}{P}}=-\zeta wϵ$$

Considering $w^T\tilde{P}$ as an input in equation (29) and solving for $ϵ$ , it can be obtained

$$ϵ=H_{\mathrm{e}}\left(w^T\tilde{P}\right)$$

where

$$H_e\left(s\right)=\frac{s+K_{\alpha }}{s^2+K_{d}s+K_p}$$

For $K_{\alpha }$ satisfying $K_d>K_{\alpha }>0$ , equation (49) can be given as

$$Re\left(H_e\left(j\omega \right)\right)=\frac{K_{\alpha }{K}_p+\left(K_d-K_{\alpha }\right){\omega }^2}{{\left(K_p-{\omega }^2\right)}^2+{\omega }^2{K}_d^2}>0\mathrm{for}\mathrm{all}\omega \in ℝ$$

(49)

and $H_e$ is strictly positive real. ⁹⁴

Theorem

Consider the adaptive control system

$$\stackrel{·}{\varphi }=-\zeta \mathrm{wH}\left(w^T\varphi \right)$$

where $\zeta >0$ is the adaptation gain.

If $H(s)$ is strictly positive real and $w$ is persistently exciting, then $\varphi$ and the internal state of $H(s)$ converge to zero exponentially with convergence rate $\zeta k+O({\zeta }^2)$ for some $k>0$ .

This theorem given above can be proved if $w$ is persistently exciting. The periodicity of the motor dynamics is utilized by this theorem. This theorem is a milestone to prove the robust stability of the system.

Robustness in slow adaptation and stability of the model

With the help of the small-gain theorem, ⁹⁵ the robustness of the system is provided. While the unperturbed adaptive system is defined as open-loop forward system, the perturbation corresponds to the feedback block. The perturbed system keeps to maintain its stability when the multiplication of the I-O gain of the forward system and that of the perturbation block is less than one. The gain of the forward system is inversely proportional to the adaptation gain in slow adaptation.

Therefore, the robust stability of the system must be considered in slow adaptation process and should not be affected from the adaptation gain. By using coupled Lyapunov function dynamics, ⁹⁶ the robust stability of the system can be proved.

Theorem

Consider the following adaptive system with linear error equation, nonlinear regressor $(w)$ , and unmodeled disturbance $(\sigma )$ originating from the modeling error

$$\begin{array}{c}\stackrel{·}{x}=\mathrm{Ax}+b(\sigma +w^T\tilde{P})\hfill \\ \stackrel{.}{\stackrel{~}{P}}=-\zeta w{c}^{T}x\hfill \\ \sigma =H_{\sigma }(x,\tilde{P},r)\hfill \end{array}$$

where $A,b,c$ are constant matrices and $H_{\sigma }$ is a nonlinear operator. Suppose that $H_{\sigma }(s)=c^T(\mathrm{sI}-A{)}^{-1}b$ is strictly positive real, $w$ is persistently exciting, and there exist ${\beta }_{\sigma },K_x,K_{\tilde{P}},K_r$ such that the inequality can be stated in equation (50)

$$\Vert H_{\sigma }(x,\tilde{P},r)\Vert ⩽K_x\Vert x\Vert +K_{\tilde{P}}\Vert \tilde{P}\Vert +K_r\Vert r\Vert +{\beta }_{\sigma }$$

(50)

where $\Vert .\Vert \mathrm{corresponds}$ to the vector-2 norm.

Under these conditions, there exists ${\zeta }^{*}$ , and constants ${\rho }_1,{\rho }_2>0$ such that if $\zeta \in (0,{\zeta }^{*}),K_x,\widetilde{K_P}$ satisfy

$$p_1{K}_x+p_2{K}_{\tilde{P}}<0$$

and $r$ is bounded, then $\sigma$ , $\tilde{P}$ , and $x$ are bounded.

The perturbed system keeps its stability until the perturbation is sufficiently big with respect to the stability of the matrix $A$ and the persistency of excitation degree of $w$ .

The following conclusions can be obtained by implementing this theorem to our system.

Considering equations (26) and (29), if $w$ is persistently exciting and ${\stackrel{·}{\theta }}_{m,d}$ is bounded, then there exists ${ϵ}^{*}>0$ such that $(\Vert g(.)-{\tau }_0\Vert /{\tau }_0)<{ϵ}^{*}$ implies boundedness of $e_m$ , ${\stackrel{·}{e}}_m$ , and $\tilde{P}$ in the model. ⁹⁷

Finally, the stability of the adaptive controller with conditions on exogenous signals is derived as follows.

Theorem

For the model described in equations (26) and (29), there exist ${\stackrel{·}{\theta }}_{m,\max }^{\mathrm{ref}}⩾{\stackrel{·}{\theta }}_{m,\min }^{\mathrm{ref}}>0$ , ${ϵ}^{*}$ , and ${\zeta }^{*}$ such that ${\stackrel{·}{\theta }}_{m,\max }^{\mathrm{ref}}⩾{\stackrel{·}{\theta }}_m^{\mathrm{ref}}⩾{\stackrel{·}{\theta }}_{m,\min }^{\mathrm{ref}}$ for all $t$ ; $\zeta \in (0,{\zeta }^{*})$ and $(\Vert g(.)-{\tau }_0\Vert /{\tau }_0)<{ϵ}^{*}$ imply boundedness of $e_m$ , ${\stackrel{·}{e}}_m$ , and $\tilde{P}$ .

$e_m$ , ${\stackrel{·}{e}}_m$ , and $\tilde{P}$ approach to a vicinity of zero with radius proportional to

$$\Vert \frac{\mathrm{g}(.)-{\tau }_0}{{\tau }_0}{\stackrel{··}{\theta }}_m^{\mathrm{ref}}+f(.)-\frac{g(.)}{{\tau }_0}{w}^T{P}^{*}\Vert$$

Parameter and tracking error convergence can be obtained choosing ${\stackrel{··}{\theta }}_m^{\mathrm{ref}}=0$ , constant velocity tracking condition, and when $f(.)$ is in the range of $w^T(.)$ . A finite number of frequency components can be eliminated from torque ripple. The tracking and parameter error converge to neighborhood of zero with radius proportional to the size of the residual torque ripple. It can be concluded that if the system is stable at a certain speed with a particular controller gain and adaptation gain, it remains stable when the motor speed and controller gain are increased or the adaptation gain is decreased. ⁹⁸

Simulation results

The simulations were conducted in MATLAB/Simulink program using fixed-step ode4 (Rung–Kutta) solver with a T_s = 1e – 5 sampling time. The parameters utilized for IM model and controllers were given in Table 1, respectively. The controller parameters were adjusted by using Simulink Design Optimization toolbox, which was utilized to optimize parameters in Simulink models. The cost function was selected as the sum squared error between the reference speed as experiment data and actual speed. The optimization method is “pattern search-Latin hypercube” with a 0.001 parameter tolerance and 0.001 function tolerance. The parallel pool was used during optimization. The simulations were conducted under the variations of the rotor and stator resistances and motor load torque (see Figures 5–12).

Table 1.

Motor and controller parameters

Motor parameters	Controller parameters
$J_L$ 0.0825 kg m 2	$A_q$ 12.1133
$B_L$ 0.001 N ms/rad	$B_d$ 1.2453
$p$ 2	$D_{q1}$ 0.325, $D_{q2}$ 1.3
$L_m$ 0.22 $H$	$D_t$ 145.2646
$L_r$ 0.2311 $H$	$D_q$ 0.0526
$L_s$ 0.2311 $H$	$D_{t1}$ –0.2375, $D_{t2}$ 1112.5
$L_{\sigma }$ 0.0217 $H$

Figure 5.

Variations of reference: (a) motor speed, (b) load torque, and (c) stator resistance and rotor resistance.

Figure 6.

Variations of errors: (a) reference motor speed versus controlled motor speed and (b) controlled motor speed versus estimated motor speed.

Figure 7.

Variations of errors: (a) reference motor torque versus controlled motor torque, (b) controlled motor torque versus estimated motor torque, and (c) detailed view.

Figure 8.

Variations of observer errors for (a) α and (b) β current component.

Figure 9.

Variation errors of reference versus estimated load torques.

Figure 10.

Variation of the error of the rotor flux.

Figure 11.

Variation of the stator flux vector.

Figure 12.

Variations of the estimated and reference: (a) stator and (b) rotor resistances.

To demonstrate the multiple EKF performance, challenging variations are given for the speed reference, load, and stator/rotor resistances. In spite of doubled $R_s$ , $R_r$ values, and persistent operation at zero speed under load and no load, simulation results obtained with multiple EKF are highly promising in comparison to the case with $R_r$ or $R_s$ estimation only. Similarly, results have been obtained by switching the $R_s$ / $R_r$ algorithms not at every sample time (T_s) but at every 100 × T_s.

Conclusion

IMs are widely used in the industry, thanks to their simplicity and high-power supplement, yet are generally not preferred for industrial robotics and high-power motion control applications due to the difficulties of the sensorless operation occurred at low-speed regions. However, it is possible to utilize the advantages of the IM without sensor in the design of high-power servo systems. The outcomes of this paper will also advance the use of low-cost, low-maintenance, high-power IMs for high-performance applications in extreme conditions, against their high-cost counterparts Permanent Magnet Synchronous Motor (PMSM), Permanent Magnet Direct Current (PMDC), which might be challenged in extreme environments due to demagnetization effects. The major contribution is a more generalized approach to address the above-mentioned challenges of IM’s sensorless position control. To this purpose, novel solutions to the DVC (direct vector control) and DTC of IMs, which involve the following, are proposed:

• Robust-adaptive linearizing schemes based on HOSMC with discontinuous and continuous control inputs, in IM control, to achieve increased robustness and accuracy.

• Model-based observer implementation approach for the estimation of multiple parameters and states required by the developed control schemes. The approach aims to enhance the performance of the linearizing and decoupling control schemes in general as well as estimation performance at very low and zero speed.

• Adaptive torque ripple identification and minimization, with the consideration of the motor torque, thereby taking into account not only the current harmonics but also the space harmonics of the motor.

The proposed control, observer, and torque ripple minimization solutions will address not only problems having connection with position and speed control of IMs but also those related to high-performance motion control, IM-based Hardware in the Loop (HIL) robot control, and other mechatronics applications, such as control of traction motors in hybrid electric vehicles as well as problems related to robust and nonlinear control theory.

In future work, the proposed control schemes take into consideration a direct-drive configuration for the connection of the IM to the dynamic load. The system consists of an IM motor coupled directly to a torque-controlled DC motor on which various dynamic load effects will be generated. The performance of the proposed control and observer schemes will be tested through a hardware-in-the-loop simulator (HILS) test-bed developed for IM-based motion control.