Robust Combined Adaptive Passivity-Based Control for Induction Motors

Abstract

The need for industrial and commercial machinery to maintain high torque while accurately following a variable angular speed is increasing. To meet this demand, induction motors (IMs) are commonly used with variable speed drives (VSDs) that employ a field-oriented control (FOC) scheme. Over the last thirty years, IMs have been replacing independent connection direct current motors due to their cost-effectiveness, reduced maintenance needs, and increased efficiency. However, IMs and VSDs exhibit nonlinear behavior, uncertainties, and disturbances. This paper proposes a robust combined adaptive passivity-based control (CAPBC) for this class of nonlinear systems that applies to angular rotor speed and stator current regulation inside an FOC scheme for IMs’ VSDs. It uses general Lyapunov-based design energy functions and adaptive laws with $\sigma$-modification to assure robustness after combining control and monitoring variables. Lyapunov’s second method and the Barbalat Lemma prove that the control and identification error tends to be zero over time. Moreover, comparative experimental results with a standard proportional–integral controller (PIC) and direct APBC show the proposed CAPBC’s effectiveness and robustness under normal and changing conditions.

1. Introduction

Since the late 1800s, machines used in industry and commerce have relied on direct current (DC) motors. Since the 20th century, DC variable speed drives (VSDs) have been used based on Thyristor rectifiers for high starting torque and variable speed accuracy. However, these DC motors tend to spark and are susceptible to threading, grooving, and flashover, as noted in [1]. As a result, induction motors (IMs), particularly the squirrel cage type, have gradually replaced them over the past three decades. IMs are more cost-effective and efficient and require less maintenance, as stated in [2]. Consequently, the sales of IMs have increased by 85%, accounting for 60% of the total electricity consumption in the industrial sector [2].

However, due to their nonlinear characteristics, alternating current (AC) VSDs are more complex than DC VSDs [2]. They require IGBT-based inverters to regulate the stator voltage and frequency. AC VSDs use three main control schemes: scalar control [3], direct torque control (DTC) [4,5], and field-oriented control (FOC) [6,7]. Nevertheless, the indirect FOC (IFOC) scheme [7] delivers higher output torque, higher stationary speed accuracy, and fast and nonoscillatory transient behavior. It performs more closely to DC’s VSDs for the machinery under study in this manuscript.

The IFOC method simplifies the mathematical model of an IM by choosing a specific electrical angular slip. This simplification allows for the independent control of the electromagnetic torque and the rotor magnetic flux [8]. The basic IFOC relies on knowledge of the rotor time constant (${\tau }_r$) and uses proportional–integral controllers (PICs). These PICs assume constant angular rotor speed operation and neglect disturbances such as load torque and the inverter model uncertainty to deal with simple linear dynamical systems (LDSs) [9]. Adjusting them also requires information on all the motor load parameters, which can be obtained from diverse methods [10], such as offline algorithms [11,12], offline tests [13,14], and self-commissioning tests [15,16].

Improving controller performance for FOC of IM has been a hot topic for the last decades and today [17,18,19,20,21,22,23,24,25]. The work in [20] proposes substituting the PICs with parallel PICs, improving robustness and performance indexes. However, [20] (Section 4) recognizes tuning difficulties with a doubled number of parameters, using genetic algorithms and particle swarm optimization (PSO) algorithms. The studies in [21,23] propose using a fractional-order PIC for the speed control outer loop, again with tuning issues for the fractional order value. The proposals in [24,25] design sliding mode controllers (SMCs), which are also nonadaptive depending on the parameter’s knowledge. The review in [22] discusses this SMC issue and compares it with DMRAC, fuzzy, and artificial neural network (ANN) controllers. These last two need optimization techniques for minimal error, error adjustment, and membership function, with a more complex design and adjustment, as described in [22,26]. The proposed DAPBC [17] has a more straightforward design but still has tuning difficulties, depending on trial and error. The work in [27] enhances [17] by providing a specific formula depending on the operational range and not on the plant parameters for the DAPBC controller parameters tuning; however, it is not applied to FOC controllers.

It is clear that robust adaptive controllers ensure robustness under parameter variations without relying on their explicit knowledge [28,29,30]. There are three approaches to adaptive control—direct (D), indirect (I), and combined (C) [30]. The direct method is the most widely used. It has been applied to various applications such as self-piloted crafts [31,32,33], robotics [34,35,36], power systems [37,38], including induction motors [17,39,40]. However, the C method proposed in the work in [41] aims to improve the transient performance beyond the direct and indirect dynamic methods. Nevertheless, all these techniques have tuning issues. Therefore, this manuscript proposes a combined approach with a tuning method by expanding [27] with a tuning method.

In particular, the work in [41] introduced the C approach for the model reference adaptive control (MRAC) technique for scalar LDS. Later, ref. [42] applied CMRAC to pH control for a chemical reactor, outperforming a PID controller and DMRAC. The method was then extended to single-input and single-output (SISO) LDS by controlling longitudinal airplane movement [43]. These studies consider unknown plant parameters with the known sign of the input parameter b, referred to as the known control direction (KCD). The KCD assumes that the input parameter equals its unknown modulus multiplied by its known sign. This is valid for SISO plants and multiple-input multiple-output (MIMO) systems with a diagonal input matrix B, where $B=\left|B\right|sign\left(B\right)$, such as IMs.

On the other hand, refs. [44,45] propose a CMRAC for MIMO LDS. However, they assume a known input matrix B, substantially simplifying the adaptive control problem. Meanwhile, ref. [46] considered an adaptive control law with a known input matrix substituted by its estimate. Lastly, refs. [47,48] neglected the estimation error and considered an unknown input matrix to control uncrewed underwater and air vehicles, respectively. Hence, this manuscript uses the ideas originally proposed by CMRAC [41,42,43] to extend the D adaptive passivity-based control (APBC) technique [27], which ensures faster results than MRAC for the IFOC scheme.

As a contribution, this paper proposes a new control technique called CAPBC for a broader class of MIMO nonlinear linear dynamical systems (NLDSs). The proposed technique can handle systems with unmodeled dynamics and bounded external disturbance, including the case of IMs. The CAPBC considers the parameters closed-loop estimation error, first introduced by Duarte et al. [41]. To ensure robustness, the CAPBC incorporates a MIMO sigma-modification [28,29,30]. The proposed technique was applied to the outer and inner controllers of an IFOC scheme for IMs and tested in a laboratory. The contributions of this proposal are detailed as follows:

• Proposing a novel CAPBC. This paper proposes a novel CAPBC technique that extends the existing DAPBC scheme from [27]. Compared with previous works [41,42,43], the CAPBC can handle a wider range of MIMO NLDS with bounded external disturbance and unmodeled dynamic. In contrast to [44,45,46,48], the proposal considers the estimation error originally proposed by [41,42,43].
• Implementing a SISO CAPBC angular speed control. The proposed technique is applied to the outer loop of an IFOC for IMs, where it controls the angular speed. Implementing the CAPBC is more complex than the DAPBC from [27] but improves performance by incorporating online parameter adaptive estimation. The controller does not require knowledge of the motor load mechanical parameters, unlike the PIC.
• Implementing a MIMO CAPBC d-q axis current control. The proposed technique is also applied to the inner loop of an IFOC for IMs, where it controls the stator current vector components. In contrast to previous works [44,45,46,48], the CAPBC can handle systems with an utterly unknown B with a known control direction (KCD), which is the case for IMs.

Moreover, this paper presents experimental results that compare the proposed CAPBC, DAPBC, and PIC techniques in an IFOC scheme for IMs. These tests include more changes than the ones considered in previous studies [27,39,40]. Specifically, the tests consider changes in angular speed reference, parameters that affect field orientation, and load torque. The results demonstrate that the proposed technique is effective and outperforms DAPBC and PIC techniques.

This paper has five sections. The first section is the introduction. Section 2 explains the IM dynamical model and the IFOC control scheme for IMs. This section also provides detailed information on the PIC adjustments and CMRAC basis. Section 3 proposes the CAPBC method for a specific type of nonlinear system that includes IMs. In addition, the authors provide theoretical proof of the proposed method. Section 4 depicts comparative experimental results with PIC and DAPBC, showing the effectiveness and robustness of CMRAC. This section also includes a discussion of the results. Lastly, Section 5 concludes the findings of this paper.

2. Preliminaries

2.1. d-q IM Dynamic Model and IFOC Diagram

The IM model considers a two-pole machine whose results can be expanded for more poles. It is assumed that the rotor and stator windings are distributed symmetrically, the signals are sinusoidal (neglecting the harmonic effects), and that hysteresis, iron losses, and saturation are negligible. The machine operates within the linear zone of the magnetic field, and all motor parameters are constant and referred to the stator. Moreover, a quadrature-phase machine with a smooth air gap is considered [49] (Section 2.1.5). Kirchhoff laws for the stator and rotor circuit are applied [49], and the Park transformation [50] is used to shift the electrical equations to a rotating synchronous reference frame. The vectors are then split into real and imaginary parts, and the IM d-q model used by the FOC scheme is obtained. This is combined with the motion equation obtained from the second Newton law for rotational motion, resulting in the following [49] (Section 2.8):

$$\begin{array}{c}\hfill {\dot{I}}_{sd}=-\frac{R_s^{\prime }}{\sigma L_s}{I}_{sd}+{\omega }_e{I}_{sq}+\frac{R_r{L}_m}{\sigma L_s{L}_r^2}{\Psi }_{rd}-\frac{L_m}{\sigma L_s{L}_r}\frac{p}{2}{\omega }_r{\Psi }_{rq}+\frac{1}{\sigma L_s}{V}_{sd},\\ \hfill {\dot{I}}_{sq}=-\frac{R_s^{\prime }}{\sigma L_s}{I}_{sq}-{\omega }_e{I}_{sd}+\frac{R_r{L}_m}{\sigma L_s{L}_r^2}{\Psi }_{rq}+\frac{L_m}{\sigma L_s{L}_r}\frac{p}{2}{\omega }_r{\Psi }_{rd}+\frac{1}{\sigma L_s}{V}_{sq},\\ \hfill {\dot{\Psi }}_{rd}=-\frac{R_r}{L_r}{\Psi }_{rd}+\left({\omega }_e-\frac{p}{2}{\omega }_r\right){\Psi }_{rq}+\frac{R_r{L}_m}{L_r}{I}_{sd},\\ \hfill {\dot{\Psi }}_{rq}=-\frac{R_r}{L_r}{\Psi }_{rq}-\left({\omega }_e-\frac{p}{2}{\omega }_r\right){\Psi }_{rd}+\frac{R_r{L}_m}{L_r}{I}_{sq},\\ \hfill \dot{{\omega }_r}=-\frac{B_p}{J}{\omega }_r-\frac{1}{J}(T_e-T_l),\mathrm{with}{T}_e=\frac{3}{2}\frac{p}{2}\frac{L_m}{L_r}\left({\Psi }_{rd}{I}_{sq}-{\Psi }_{rq}{I}_{sd}\right).\end{array}$$

(1)

Here, the variables are the amplitude of the sinusoidal signals at the motor terminals expressed as the direct and quadrature stator current amplitudes $I_{sd}$, $I_{sq}$, the direct and quadrature stator voltage amplitudes $V_{sd}$, $V_{sq}$, and the direct and quadrature rotor flux amplitudes ${\Psi }_{rd}$ and ${\Psi }_{rq}$. ${\omega }_r$ is the rotor angular speed at the shaft, ${\omega }_e$ is the angular electrical frequency or speed of the synchronous reference frame, $T_e$ is the electromagnetic motor output torque, and $T_l$ is the load torque. The parameters $R_s$, $R_r$ are the stator and rotor resistances of a phase winding, p are the poles number, J is the motor load inertia, $B_p$ is the viscous coefficient, $L_s$, $L_r$, and $L_m$ are the stator, rotor, and magnetizing inductances, respectively, $\sigma =1-\frac{L_m^2}{L_r{L}_s}$ is the dispersion coefficient, and $R_s^{\prime }=R_s+\frac{L_m^2{R}_r}{L_r^2}$ is the stator transient resistance. Finally, the model uses the following first-time derivatives: ${\dot{I}}_{sd}=\frac{d{I}_{sd}}{dt}$, ${\dot{I}}_{sq}=\frac{d{I}_{sq}}{dt}$, ${\dot{\Psi }}_{sd}=\frac{d{\Psi }_{sd}}{dt}$, ${\dot{\Psi }}_{sq}=\frac{d{\Psi }_{sq}}{dt}$, and $\dot{{\omega }_r}=\frac{d{\omega }_r}{dt}$.

Remark 1. 

The coupling between electromagnetic torque, stator current, and rotor flux can be observed in Equation (1). As a solution, the IFOC is then achieved for the IM d-q model (1) after imposing the following electrical angular frequency (please see details in Appendix A) and operating with a fixed $I_{sd}^{*}$ [5] (Section 4.1):

$${\omega }_e=\frac{p}{2}{\omega }_r+\alpha \frac{1}{{\widehat{\tau }}_r}\frac{I_{sq}^{*}}{I_{sd}^{*}}.$$

(2)

Here, $I_{sd}^{*}$, $I_{sq}^{*}$ are the required direct and quadrature stator current amplitudes. ${\widehat{\tau }}_r$ is the estimated rotor time constant, and $\alpha =1$.

As a result of applying the electrical angular frequency (2), the quadrature component of the rotor flux tends to zero ${\Psi }_{rq}\to 0$. Thus, ${\Psi }_{rd}\to L_m{I}_{sd}$, and the electromagnetic torque $T_e\to K_{T_e}{I}_{sq}$, with a constant $K_{T_e}=\frac{3}{2}\frac{p}{2}\frac{L_m^2}{L_r}{I}_{sd}$. Therefore, the following simplified IM d-q model is obtained [49] (Section 2.8):

$$\begin{array}{c}\hfill {\dot{I}}_{sd}=-\frac{R_s^{\prime }}{\sigma L_s}{I}_{sd}+\left({\omega }_e{I}_{sq}+\frac{R_r{L}_m^2}{\sigma L_s{L}_r^2}{I}_{sd}\right)+\frac{1}{\sigma L_s}{V}_{sd},\\ \hfill {\dot{I}}_{sq}=-\frac{R_s^{\prime }}{\sigma L_s}{I}_{sq}+\left(-{\omega }_e{I}_{sd}+\frac{L_m^2}{\sigma L_s{L}_r}\frac{p}{2}{\omega }_r{I}_{sd}\right)+\frac{1}{\sigma L_s}{V}_{sq},\\ \hfill \dot{{\omega }_r}=-\frac{B_p}{J}{\omega }_r-K_{T_e}{I}_{sq}+\left(\frac{1}{J}{T}_l\right).\end{array}$$

(3)

This simplified d-q model considers $I_{sq}$ as an electromagnetic torque-producing current and $I_{sd}$ as a rotor-flux-producing current. Then, $I_{sd}$ is controlled to achieve constant rotor flux control, and $I_{sq}$ is used to control the electromagnetic torque for the load demands at different rotor speeds. This way, torque and flux can be controlled independently, similar to DC machines that are separately excited, which is the aim of the IFOC (3). Figure 1 depicts the IFOC block diagram.

Remark 2. 

The simplified d-q model (3) has the nonlinear terms ${\omega }_e{I}_{sd}$, ${\omega }_e{I}_{sq}$, and ${\omega }_r{I}_{sd}$. Moreover, the load torque term $T_l$ is often considered as a disturbance.

The following section describes the adjustment of the basic PIC considered for rotor angular speed and stator current vector of the IFOC for IMs. This controller assumes the operation at a fixed angular rotor speed. Thus, ${\omega }_e$ and ${\omega }_r$ are constant, and model (3) behaves as an LDS. Furthermore, the outer loop PIC neglects the $T_l$ disturbance and expects robustness in front of its variations.

2.2. PIC Adjustment

Appendix B describes the PIC adjustment theory in detail. As a summary, the inner current controllers’ parameters are computed as follows (based on [9]):

$$\begin{array}{c}\hfill V_{sq}^{*}=\left(K_{Pi}{e}_{I_{sq}}+K_{Ii}\int e_{I_{sq}}d\tau \right)\mathrm{and}{V}_{sd}^{*}=\left(K_{Pi}{e}_{I_{sd}}+K_{Ii}\int e_{I_{sd}}d\tau \right)\\ \hfill \mathrm{where}{K}_{Ii}=\frac{R_s^{\prime }{\tau }_i{\omega }_{ni}^2}{H_o{H}_i}\mathrm{and}{K}_{Pi}=\frac{R_s^{\prime }({\tau }_{i}2{\xi }_i{\omega }_{ni}-1)}{H_o{H}_i}.\end{array}$$

(4)

Here, $V_{sq}^{*}$ and $V_{sd}^{*}$ are the direct and quadrature stator voltage references, $K_{Pi}$ and $K_{Ii}$ are the PICs inner proportional and integral parameters, $e_{I_{sq}}$ and $e_{I_{sd}}$ are the direct and quadrature stator current errors, ${\tau }_i$ is the electrical time-constant, $H_o$ is the inverter gain, and $H_i$ is the current sensor gain.

As design criteria, a root locus method is often applied. The inner damping coefficient value ${\xi }_i$ is chosen between $0.5$ and $0.8$, with the most common value for this application considering ${\xi }_i=\frac{\sqrt{2}}{2}\approx 0.707$. The inner natural frequency equals ${\omega }_{ni}=\frac{2.3}{{\tau }_i}$ [51], which in this AC drive case should be higher than the switching frequency of the inverter´s IGBTs, having a value between 1.7 kHz and 16 kHz for powers between 1500 kW and 37 kW, respectively [52].

Moreover, the outer angular speed controllers’ parameters are computed as follows (based on [9]):

$$\begin{array}{c}\hfill I_{sq}^{*}=\left(K_{Po}{e}_{{\omega }_r}+K_{Io}\int e_{{\omega }_r}d\tau \right)K_{T_e}^{-1}\\ \hfill \mathrm{where}{K}_{Io}=\frac{H_i{B}_p{\tau }_o{\omega }_{no}^2}{H_o}\mathrm{and}{K}_{Po}=\frac{H_i{B}_p(2{\xi }_o{\omega }_{no}{\tau }_o-1)}{H_o},\end{array}$$

(5)

where $I_{sq}^{*}$ is the quadrature stator current set point, $K_{Po}$ and $K_{Io}$ are the PICs outer proportional and integral parameters, $e_{{\omega }_r}$ is the rotor angular speed error, ${\tau }_o$ is the mechanical time-constant, and $H_o$ is the speed sensor gain. Here, the squared outer natural frequency is ${\omega }_{no}^2=\left(K_{Io}{H}_o\right)/\left(H_i{B}_p{\tau }_o\right)$, which is used to obtain the fixed-gain parameter $K_{Io}$ of the controller after considering ${\omega }_{no}=\frac{{\omega }_{ni}}{15}$ [51,53]. The term depending on the outer natural frequency ${\omega }_{no}$ and the outer damping coefficient ${\xi }_o$ is $2{\xi }_o{\omega }_{no}=(H_i{B}_p+K_{Po}{H}_o)/\left(H_i{B}_p{\tau }_o\right)$, and it is used to compute the fixed-gain parameter $K_{Po}$ of the controller.

Remark 3. 

Please observe that PIC adjustment depends on the knowledge of the plant parameter values. Here, the resistances vary with the motor temperature, for instance. Therefore, using an adaptive controller would assure robustness regarding parameter variations.

The following section describes the CMRAC basis this manuscript considers when proposing CAPBC.

2.3. CMRAC Basis to Be Expanded for CAPBC

The CMRAC applies to scalar LDS, with relative degree 1 of the following form:

$$\dot{y}\left(t\right)=ay\left(t\right)+bu\left(t\right).$$

(6)

Here, the plant parameters a and b $\in \mathfrak{R}$ are constant and unknown, with known $sign\left(b\right)$. The LDS´s input and output are $u\left(t\right)$ and $y\left(t\right)$ $\in \mathfrak{R}$, respectively.

The CMRAC is associated with the following equations [41]:

$$\begin{array}{ccc}\hfill {\dot{y}}_r\left(t\right)=-k_c{y}_r\left(t\right)+b_r{y}^{*}\left(t\right),& & \mathrm{Reference}\mathrm{model}\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill \dot{\widehat{y}}\left(t\right)=-k_i{e}_i+{{\widehat{\theta }}_i}^T{\omega }_i,& & \mathrm{Identification}\mathrm{model}\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill u\left(t\right)={{\widehat{\theta }}_c}^T{\omega }_c,& & \mathrm{Control}\mathrm{law}\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill \left.\begin{array}{c}{\dot{\widehat{\theta }}}_1\left(t\right)=sign\left(b\right)\left(e_c\left(t\right)y\left(t\right)+{\epsilon }_1\left(t\right)\right),\\ {\dot{\widehat{\theta }}}_2\left(t\right)=sign\left(b\right)\left(e_c\left(t\right)y_r\left(t\right)+{\epsilon }_2\left(t\right)\right),\end{array}\right\}& & \mathrm{Control}\mathrm{adaptive}\mathrm{law}\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill \left.\begin{array}{c}{\epsilon }_1\left(t\right)=-(\widehat{a}+k_c)+\widehat{b}\left(t\right){\widehat{\theta }}_1\left(t\right),\\ {\epsilon }_2\left(t\right)=-b_r+\widehat{b}\left(t\right){\widehat{\theta }}_2\left(t\right),\end{array}\right\}& & \mathrm{Closed}-\mathrm{loop}\mathrm{estimation}\mathrm{error}\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill \left.\begin{array}{c}\dot{\widehat{a}}\left(t\right)=-\left(e_i\left(t\right)y\left(t\right)-{\epsilon }_1\left(t\right)\right),\\ \dot{\widehat{b}}\left(t\right)=-\left(e_i\left(t\right)u\left(t\right)-{\epsilon }_1\left(t\right){\widehat{\theta }}_1\left(t\right)-{\epsilon }_2\left(t\right){\widehat{\theta }}_2\left(t\right)\right).\end{array}\right\}& & \mathrm{Identification}\mathrm{adaptive}\mathrm{law}\hfill \end{array}$$

(12)

Here, the variable $y_r\left(t\right)\in \mathfrak{R}$ is the reference model output and $\widehat{y}\left(t\right)\in \mathfrak{R}$ is the identification model output. The bounded reference trajectory is $y^{*}\left(t\right)\in \mathfrak{R}$. Furthermore, $e_c=(y_r\left(t\right)-y\left(t\right))\in \mathfrak{R}$ and $e_i\left(t\right)=(\widehat{y}\left(t\right)-y\left(t\right))\in \mathfrak{R}$ are the control and identification errors. The adaptive parameters for control are ${\widehat{\theta }}_c\left(t\right)$, and for identification, ${\widehat{\theta }}_i\left(t\right)\in {\mathfrak{R}}^2$. The designer chooses the model parameters $b_r,k_c\in {\mathfrak{R}}^{+}$, and $k_i\in \mathfrak{R}$, and ${\omega }_c$, ${\omega }_i\in {\mathfrak{R}}^2$ are the control and identification vector information. The estimated controller parameter ${\widehat{\theta }}_c^T=\left[\begin{array}{cc}{\widehat{\theta }}_1& {\widehat{\theta }}_2\end{array}\right]$, and the estimated plant parameter ${\widehat{\theta }}_i^T=\left[\begin{array}{cc}\widehat{a}& \widehat{b}\end{array}\right]$. Finally, the ideal controller parameters ${\theta }_1,{\theta }_2\in \mathfrak{R}$ fulfill the following condition:

$$-(a+k_c)+b{\theta }_1=0\mathrm{and}-b_r+b{\theta }_2=0.$$

(13)

Once this CMRAC is applied to system (6) and it is assumed that $b=\left|b\right|sign\left(b\right)$, the obtained closed-loop autonomous system ensures that $e_c\left(t\right)$, $e_i\left(t\right)$, ${\epsilon }_1\left(t\right)$, and ${\epsilon }_2\left(t\right)$ tend asymptotically to zero.

Figure 2 shows the CMRAC control diagram for LDS dynamical systems.

Remark 4. 

The C approach enhances the D and I approaches by taking into account the closed-loop estimation error. However, it does not apply to NLDS with disturbances and unmodeled dynamics. Furthermore, CMRAC has had tuning issues for Kc and Ki, which have recently been resolved [27].

Based on the previous background and as a solution to the described issues, the following section proposes a robust CAPBC for MIMO NLDS.

3. Proposed CAPBC

This section outlines the proposed CAPBC, which combines the D and I approaches, considering the closed-loop estimation error of the adaptive parameters based on the CMRAC ideas [41,42,43] described in Section 2.3. Additionally, the proposal is adaptive and does not rely on explicit knowledge of the plant parameters, as in the case of PIC described in Section 2.2. However, this CAPBC has specific tuning formulas similar to the PIC, but depending on the known plant operational expansion the DAPBC [27], and applies to both LDS and NLDS with disturbances and unmodeled dynamics. Consequently, the proposal solves the issues described in Remark 4 of CMRAC.

Figure 3 illustrates the proposed CAPBC diagram, which is then applied to the inner and outer control loops of the IFOC diagram described in Section 2.1 for IMs.

The proposed CAPBC shown in Figure 3 lacks of a reference model [41] and applies to MIMO dynamical systems of the following form:

$$\dot{y}\left(t\right)=A^{T}f\left(y\right)+B^{T}g{\left(y\right)}^{T}u+{\delta }^T\Delta +\zeta .$$

(14)

Here, the output $y\left(t\right)\in {\mathfrak{R}}^n$ and the input $u\left(t\right)\in {\mathfrak{R}}^n$ are accessible. The functions $g\left(y\right)\in {\mathfrak{R}}^{nxn}$, and $f\left(y\right)\in {\mathfrak{R}}^m$ are known, like the disturbance portion $\Delta \in {\mathfrak{R}}^n$. The unmodeled dynamics $\zeta \in {\mathfrak{R}}^n$ are unknown, as well as the plant parameters $A^T\in {\mathfrak{R}}^{nxm}$, $B^T,{\delta }^T\in {\mathfrak{R}}^{nxn}$.

The following theorem details the proposal for the class of NLDS (14):

Theorem 1. 

The following CAPBC assures the output $y\left(t\right)\in {\mathfrak{R}}^n$ of NLDS (14) tracks the reference $y^{*}\left(t\right)\in {\mathfrak{R}}^n$ while observing it via $\widehat{y}\left(t\right)\in {\mathfrak{R}}^n$:

$$\begin{array}{ccc}\hfill \dot{\widehat{y}}\left(t\right)=-K_i▿V_{e_i}{\left(t\right)}^T+{\widehat{\theta }}_i^T{\omega }_i,& & \mathrm{Identification}\mathrm{model}\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill u\left(t\right)=g{\left(y\right)}^{-1}{\widehat{\theta }}_c^T{\omega }_c,& & \mathrm{Control}\mathrm{law}\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill {\dot{\widehat{\theta }}}_c^T=\left(S^T▿V_{e_c}{\omega }_c^T-S^T{\epsilon }^T{\Gamma }_{\epsilon }^{-1}+{\sigma }_c{\widehat{\theta }}_c^T\right){\Gamma }_c,& & \mathrm{Control}\mathrm{adaptive}\mathrm{law}\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill \epsilon =\left[\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\\ {\epsilon }_3\end{array}\right]=\left[\begin{array}{c}{\widehat{B}}^T{\widehat{\theta }}_1^T+{\widehat{A}}^T\\ {\widehat{B}}^T{\widehat{\theta }}_2^T-I_n\\ {\widehat{B}}^T{\widehat{\theta }}_3^T+{\widehat{\delta }}^T\end{array}\right],& & \mathrm{Closed}-\mathrm{loop}\mathrm{estimation}\mathrm{error}\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill {\dot{\widehat{\theta }}}_i^T=\left({\omega }_i^T▿V_{e_i}-{\epsilon }^T{\Gamma }_{\epsilon }^{-1}(P_1^T+{\widehat{\theta }}_c{P}_2^T)+{\sigma }_i{\widehat{\theta }}_i^T\right){\Gamma }_i,& & \mathrm{Identification}\mathrm{adaptive}\mathrm{law}\hfill \end{array}$$

(19)

Here, $▿V_{e_c}\in {\mathfrak{R}}^{1xn}$ and $▿V_{e_i}\in {\mathfrak{R}}^{1xn}$ are the gradients of the design Lyapunov-type energy functions $V_{e_c}$ and $V_{e_i}$. The control and identification errors are $e_c\left(t\right)=y^{*}\left(t\right)-y\left(t\right)$ and $e_i\left(t\right)=y\left(t\right)-\widehat{y}\left(t\right)$, with $e_c\left(t\right),e_i\left(t\right)\in {\mathfrak{R}}^n$. The adaptive controller and identification parameters are ${\widehat{\theta }}_c^T$ and ${\widehat{\theta }}_i^T\left(t\right)\in {\mathfrak{R}}^{nx(2n+m)}$, depending on the control and identification adaptive law modifications ${\sigma }_c$ and ${\sigma }_i\in {\mathfrak{R}}^{nxn}$, which are positive-definite. The information vectors for control and identification are ${\omega }_c$ and ${\omega }_i\in R^{(2n+m)x1}$, where ${\omega }_c^T={\left[\begin{array}{ccc}f{\left(y\right)}^T& {(K_c{e}_c+{\dot{y}}^{*})}^T& {\Delta }^T\end{array}\right]}^T$ and ${\omega }_i^T={\left[\begin{array}{ccc}f{\left(y\right)}^T& {\left(g{\left(y\right)}^{T}u\right)}^T& {\Delta }^T\end{array}\right]}^T$. The auxiliary known parameters $P_1=\left[\begin{array}{ccc}{I}_m& 0& 0\\ 0& 0_n& 0\\ 0& 0& I_n\end{array}\right]$ and $P_2=\left[\begin{array}{ccc}{0}_m& I_n& 0_n\end{array}\right]$, where $I_m$ and $I_n$ are identity matrices of order m and n, respectively. Moreover, $0_n$ and $0_m$ are null matrices of order n and m, respectively. The estimated plant parameter ${\widehat{\theta }}_i^T=\left[\begin{array}{ccc}{\widehat{A}}^T& {\widehat{B}}^T& {\widehat{\delta }}^T\end{array}\right]$ (19) finds $\widehat{A}\in R^{nxm}$, $\widehat{B}$, and $\widehat{\delta }\in R^{nxn}$. The estimated controller parameter ${\widehat{\theta }}_c^T=\left[\begin{array}{ccc}{\widehat{\theta }}_1^T& {\widehat{\theta }}_2^T& {\widehat{\theta }}_3^T\end{array}\right]$ (17), computes ${\widehat{\theta }}_1\in R^{nxm}$, ${\widehat{\theta }}_2$, and ${\widehat{\theta }}_3\in R^{nxn}$. Later, these results allow implementing the closed-loop estimation error (18).

The designer adjusts the control and identification gains ${\Gamma }_c$, ${\Gamma }_i$, and ${\Gamma }_{\epsilon }\in {\mathfrak{R}}^{(2n+m)x(2n+m)}$, where ${\Gamma }_c={\Gamma }_{\epsilon }=\left(\frac{{\mu }_c}{1+{\omega }_{cn}^T{\omega }_{cn}}\right)$, ${\Gamma }_i=\left(\frac{{\mu }_i}{1+{\omega }_{in}^T{\omega }_{in}}\right)$, where ${\omega }_{cn}$ and ${\omega }_{in}$ are the vectors containing the upper operational range of each element of ${\omega }_c$ and ${\omega }_i$. Also, it adjusts the forgetting factors ${\mu }_c\in {\mathfrak{R}}^{+}$ and ${\mu }_i\in {\mathfrak{R}}^{+}$ [54]—(Section 4.3.6) and (Remark 4.3.7). Furthermore, the designer adjusts controller parameters $K_c$ and $K_i\in {\mathfrak{R}}^{nxn}$. The ideal control and identification parameters are ${\theta }_c^T$ and ${\theta }_i^T\in {\mathfrak{R}}^{nx(2n+m)}$, which are defined as follows:

$$\begin{array}{c}\hfill {\theta }_i^T=\left[\begin{array}{ccc}{A}^T& B^T& {\delta }^T\end{array}\right]\end{array}$$

(20)

$$\begin{array}{c}\hfill B^T{\theta }_1^T+A^T=0,B^T{\theta }_2^T-I_n=0\mathit{and}{B}^T{\theta }_3^T+{\delta }^T=0.\end{array}$$

(21)

The error of the parameters are ${\Phi }_c^T={\theta }_c^T-{\widehat{\theta }}_c^T$ and ${\Phi }_i^T={\theta }_i^T-{\widehat{\theta }}_i^T\in R^{nx(2n+m)}$.

Appendix C describes the CAPBC stability proof. ⋄

The proposed CAPBC is utilized in an IFOC scheme for IMs, as depicted in Figure 4.

The design considers the simplified IM d-q model (3) and divides it into the electrical and mechanical subsystems to be controlled in cascade. The first two equations of (3) represent the inner-loop electrical subsystem, with an inner output $y_i$ of the form $y_i={\left[\begin{array}{cc}{I}_{sq}& I_{sd}\end{array}\right]}^T$ and an inner control input $u_i$ of the form $u_i={\left[\begin{array}{cc}{V}_{sq}^{*}& V_{sd}^{*}\end{array}\right]}^T$. On the other hand, the last equation of (3) represents the outer-loop mechanical subsystem, with an outer output $y_o$ of the form $y_o={\omega }_r$ and an outer control input $u_o$ of the form $u_o=I_{sq}^{*}$.

The design of the outer loop controller $CAPB{C}_o$ considers the known disturbance portion $\Delta$ to be equal to the nominal torque $T_n$, and there is no disturbance for the inner loop. The function $g\left(y\right)$ is equal to 1, and $f\left(y\right)$ is equal to $-{\omega }_r$. Moreover. the design of the inner controller $CAPB{C}_i$ assumes there is no disturbance and that there are unmodelled dynamics of the inverter. The function $g\left(y\right)$ is equal to the identity matrix of order two $I_2$, and $f\left(y\right)$ is equal to ${\left[\begin{array}{ccccc}-I_{sq}& {\omega }_e{I}_{sq}& -I_{sd}& -{\omega }_e{I}_{sd}& \frac{p}{2}{\omega }_r{I}_{sd}\end{array}\right]}^T$.

The design Lyapunov-type energy functions are $V_{e_{co}}=e_{oc}^2/2$ and $V_{e_{io}}=e_{io}^2/2$ for the outer loop, and $V_{e_{ci}}=e_{ci}^T{e}_{ci}/2$ and $V_{e_{ii}}=e_{ii}^T{e}_{ii}/2$ for the inner loop. Therefore, their respective gradients are $▿V_{e_{co}}=e_{oc}$, $▿V_{e_{io}}=e_{io}$, $▿V_{e_{ci}}=e_{ci}^T$, and $▿V_{e_{ii}}=e_{ii}^T$.

The following section discusses the obtained comparative experimental results.

4. Experimental Results

Figure 5 shows the pictures of the test bench used to validate the proposal, joined with its control diagram.

It has a real-time simulator controller OPAL-RT 4510 v2 from OPAL-RT Technologies, Montréal, QC, Canada that inherently uses a bipolar pulse width modulation (PWM), switching at 8 kHz. It commands a two-level voltage source inverter that feeds an IM-load assembly, sending the trip pulses via fiber optic (FO) cables. Simulink version 10.4 of Matlab R2021b (9.11.0.1769968) for Win64 running on a Host PC allows building the IFOC scheme of Figure 1 using PIC, DAPBC, and the proposed CAPBC and downloading them to the control platform using the software RT-LAB v2020.2.2.82. The motor data plate has 7500 kW, 380 V, 50 Hz, 1455 rpm, $fp=0.85$, and two pairs of poles $p=2$. A rotor time constant of ${\tau }_r=0.221$ was used to implement FOC (2), which is taken from previous measurements ([16], Tables IV, Motor II).

The following controllers were programmed into the IFOC scheme:

• PIC (4) and (5). These controllers were adjusted as described in Section 2.2 and using the motor load parameter values from [16] (Tables III, IM 2) that followed the IEEE standard 112A, including DC injection, locked rotor, and free load [13] (Section 5.9).
• DAPBC [27] (Theorem 1) It uses both a SISO controller $I_{sq}^{*}={\widehat{\theta }}_{oc}{\omega }_{oc}$ and a MIMO controller $\left[\begin{array}{c}V{sq}^{*}\\ V{sd}^{*}\end{array}\right]={\widehat{\theta }}_{ic}{\omega }_{ic}$, as in [27] (Equation (4)). The motor load parameter does not need to be known to adjust DAPBC.
• Proposed CAPBC from Theorem 1. It also uses both a SISO controller $I_{sq}^{*}={\widehat{\theta }}_{oc}{\omega }_{oc}$ and a MIMO controller $\left[\begin{array}{c}V{sq}^{*}\\ V{sd}^{*}\end{array}\right]={\widehat{\theta }}_{ic}{\omega }_{ic}$, as in Figure 4.

The same 10 s duration test applies to the PIC, DAPBC, and proposed CAPBC strategies. It considers that the IM starts with a $66\%$ torque load and applies a step speed command of 25 rad/s, 60 rad/s, 85 rad/s, 120, and $152.36$ rad/s, at times 2 s, $2.5$ s, 3 s, $3.5$ s, and 4 s, respectively. Later, the load decreases to $40\%$ at 5 s and increases again to $66\%$ at 6 s. Finally, the field disorientation is considered by adjusting the value of $\alpha$ from IFOC (2) to $\alpha =0.8$ at $7.5$ s and increases again to $\alpha =1.1$ at 9 s, simulating step changes for the rotor time constant.

Figure 6 exhibits the controller’s comparative rotor angular speed response. It can be seen that adaptive controllers are more robust against different variations, including load torque and IFOC-impacting parameter variations. All controllers track the reference speed. However, adaptive controllers exhibit lower maximum overshoot (MO) and faster response than PIC in all scenarios. Here, the proposed CAPBC has the lowest MO and the fastest response. The effectiveness of the proposed CAPBC is superior for the different step changes in the reference speed occurring every $2.5$ s until 4 s, as well as for the torque and IFOC variations. The CAPBC enhanced the transient response of the PIC and the DAPBC, as expected.

Table 1. Comparative performance indexes for the rotor speed controllers.

	Tests
Performance Index	Time [s]	2.0	2.5	3.0	3.5	4.0	5.0	6.0	7.0	9.0
	$\alpha$	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.80	1.10
	$T_l$ [% $T_n$]	66.0	66.0	66.0	66.0	66.0	40.0	66.0	66.0	66.0
	Speed [rad/s]	25.0	60.0	85.0	120	152.36	150	152.36	152.36	152.36
Ess [%] (Ending) $\frac{({\omega }_r^{*}-{\omega }_r)}{{\omega }_r^{*}}100$	PI	0.1299	0.1212	0.1237	0.1189	0.0180	0.0307	0.0254	0.0243	0.0851
	DAPBC	0.0850	0.0302	0.0804	0.0100	0.0063	0.0079	0.0195	0.0227	0.0468
	CAPBC	0.0512	0.0269	0.0659	0.0066	0.0029	0.0068	0.0190	0.0139	0.0171
MO [%] (Starting) $\frac{({\omega }_r^{*}-{\omega }_r)}{{\omega }_r^{*}}100$	PI	5.5983	10.6929	7.1124	5.3479	4.3302	4.1276	0.2338	2.7979	3.5007
	DAPBC	3.5168	1.7019	1.1308	0.8651	0.7095	0.3595	0.2139	0.1751	0.3143
	CAPBC	3.4904	1.4038	0.8367	0.5941	0.4640	0.2938	0.1923	0.1663	0.1667
IAE [cm.s] $\int |e_{{\omega }_r}|d\tau$	PI	0.6491	0.6494	0.6501	0.6519	0.6541	0.6544	1.043	1.52	2.2654
	DAPBC	0.0152	0.0198	0.0359	0.0368	0.0378	0.0529	0.0663	0.0694	0.1507
	CAPBC	0.0099	0.0059	0.0072	0.0102	0.0223	0.0356	0.0437	0.0452	0.1425
ISI [$\times {10}^3$] $\int I_{sq}^{*2}d\tau$	PI	0.2785	0.2649	0.2656	0.2664	0.2672	0.3915	0.8797	1.1001	3.3045
	DAPBC	0.2616	0.2619	0.2623	0.2625	0.2632	0.3907	0.8752	0.9837	3.3018
	CAPBC	0.2592	0.2594	0.2610	0.2624	0.2634	0.3900	0.8689	0.9729	3.3016

shows comparative performance indexes with more detail than Figure 6.

The CMRAC exhibits the best performance in all scenarios despite having similar steady-state errors ($E_{ss}$) across the different controllers. The proposal demonstrates the lowest control effort (please see the ISI index) and better transient behavior with the lowest MO and IAE indexes. The performance of the PIC deteriorates significantly when field disorientation occurs at seven and nine seconds with $\alpha$ equal to $0.8$ and $1.1$, respectively. The proposed CAPBC and DAPBC for the inner loop ensure robustness under this situation due to their MIMO, adaptive natures, and NLDS designs.

Figure 7, Figure 8 and Figure 9 show the consumed and reference q-axis current for torque production for the PIC, DPABC, and CAPBC, respectively. All controllers track the reference current $I_{sq}^{*}$ that produces torque, with CMRAC and DAPBC having a faster response than PIC. The PIC has difficulties tracking the reference after nine seconds when field disorientation occurs, as seen in the center-right side of Figure 7. This provokes the tracking difficulties shown in the upper-right side of Figure 6 for the slower speed control outer loop.

It can be seen in Figure 7, Figure 8 and Figure 9 that the faster speed responses of adaptive controllers are achieved with a higher reference and consumed q-axis current torque producing, as expected. Moreover, the line voltages have the typical PWM wayform, with the 50 Hz frequency corresponding to the nominal rotor angular speed, as can be seen in Figure 10, Figure 11 and Figure 12.

Finally, Figure 10, Figure 11 and Figure 12 show the oscilloscope line voltage a-b around 6.5 s for the PIC, DPABC, and CAPBC, respectively. In all cases, results show the typical two-level voltage source inverter commended by a bipolar PWM.

5. Conclusions

This paper introduced a novel CAPBC tailored for a class of nonlinear systems encompassing the IMs under an IFOC scheme, including perturbances and unmodeled dynamics. It expands the DAPBC technique [27] based on the combined approach previously proposed for MRAC [41,42,43]. The theoretical underpinnings of the proposed CAPBC are detailed in Theorem 1, and the stability proof is exhibited in Appendix C.

Later, the proposal implemented a SISO CAPBC angular speed control for the outer loop of an IFOC for IMs in cascade with the inner-loop MIMO CAPBC d-q axis current control. This paper presents comparative experimental results between the proposed CAPBC, the DAPBC, and PIC techniques in an IFOC scheme for an IM of 7.5 kW. The results demonstrate that the proposed technique is effective and outperforms DAPBC and PIC techniques. Unlike traditional PICs, the CAPBC does not need knowledge of the motor load parameters.