Sensorless Scheme for Permanent-Magnet Synchronous Motors Susceptible to Time-Varying Load Torques

Abstract

This paper is devoted to designing a sensorless high-speed tracking control for surface-mount permanent-magnet synchronous motors, considering a time-varying load torque. This proposal consists of an extended-state observer interconnected with a PI-compensated controller, considering only the measurement of electrical variables for feedback. First, to design the extended-state observer, a rotary coordinate model of the motor is extended in one state to estimate the load torque and the rotor’s position and speed. Later, the estimations are fedback to a PI-compensated controller to attenuate the time-varying load torques. Our proposed methodology aims to overcome a restriction regarding the solution of the Riccati equation respecting the Lipschitz condition for observer stability analysis. Therefore, a PI-compensated controller described as a closed-loop provides a sensorless scheme. Lyapunov stability analysis is applied to determine sufficient conditions to ensure that the states of the closed-loop system are ultimately bounded, which is one of our main contributions. The proposed observer-based controller scheme deals with unmeasured load torque fluctuations. Furthermore, we carry out high-precision emulations to provide testing scenarios of the permanent-magnet synchronous motor with some challenging load torque magnitudes and behaviors. Finally, we conduct experiments on the Technosoft${}^{\circledR }$ development platform to corroborate the feasibility of the proposed control scheme in a real-world scenario.

1. Introduction

Electrical motors are the main actuator in domestic and industrial applications. They can be classified based on specific features, such as the power source type, construction, or electrical operation. In this context, permanent-magnet synchronous motors (PMSMs) offer a fast dynamic response, high efficiency, and a high torque-to-inertia ratio for highly efficient and precise tasks. These features of PMSMs offer advantages over DC and induction motors in variable speed applications, such as robots, machinery, and propulsion systems in electric, aquatic, and aerial vehicles [1]. Generally, for robust and high-precision control of a PMSM, the angular position or rotor speed is measured with mechanical or optical sensors. However, these sensors increase the size and cost of the PMSM. Furthermore, the sensors are exposed to malfunctions, reducing the reliability of the motor [2,3].

Sensorless control techniques for PMSMs have been developed in the last decades to address the abovementioned problems. In these techniques, only the electrical variables are available for feedback (stator currents and voltages) [4]. Sensorless techniques can be classified into two main categories: model-based and high-frequency injection (see Figure 1). The first method entails estimation of the rotor position from the back electromotive force (back-EMF), and is frequently used for high-speed applications [5]. The second method involves injection of a high-frequency signal to detect the rotor position, and is widely used for low-speed and zero-speed applications [6,7].

Model-based sensorless methods are helpful for feedback control, the main idea being to implement an observer-controller scheme. It is quite common to find contributions on the topic of back electromotive force (back-EMF) observers using the phase-locked loop (PLL) method to estimate the position and speed of the PMSM [8,9]. The general idea of the PLL estimator is to extract the frequency and electrical position of the EMF signals from the motor through the phase difference between the EMF voltage input signals estimated by the observer and the frequency output signal of the PLL. Then, it is necessary to synchronize the output signal in phase and frequency with the input signal. Another way to estimate the mechanical variables from the PMSM is to use a derivative approach, calculating the rotor position by applying a trigonometric function to the estimated EMF voltage. In this approach, a pass-low filter is usually necessary to avoid the dirty derivative’s effects [10,11].

Model-based methods depend on knowledge of the PMSM mathematical model and its parameters. Therefore, the performance and accuracy of the sensors are crucial. The aforementioned conventional EMF observers could compromise the performance of the PMSM controllers. For this reason, some proposals further exploit the computing resources of the microprocessors, such as sliding mode observers (SMO) [12,13,14,15], disturbance observers [16,17], the extended Kalman filter (EKF) [18,19,20], robust control [21,22]; and intelligent control [23,24]. The main drawback of the abovementioned methodologies are the parametric and load torque variations. In this connection, there exists significant research on the topic. For instance, the paper [25] proposed a model reference adaptive system (MRAS) for a PMSM. The MRAS performs multiparameter estimation combined with filters to achieve an effective control strategy that deals with load torque variations. There are some sensorless techniques described in the following papers [19,20,26] using estimations provided by EKFs, considering parametric variations. In the paper [27], a neuronal network observer is proposed to estimate the position and rotor speed, considering a partially unknown model of the PMSM. On the other hand, the paper [21] describes the development of robust observer-based controllers against load torque variations, where an SMO estimates the angular position and rotor speed, including an ${\mathcal{H}}_{\infty }$ controller, attenuating the effects of uniformly bounded load torque variations.

Regarding estimation methods, the extended state observer (ESO) is an approach recognized as an effective methodology with the advantage of estimating at the same time the state vector and external disturbances [28,29]. This kind of observer considers the disturbance as an extended state variable. There are papers which consider estimations provided by ESOs for the PMSM. The article [30] proposes an ESO to estimate the back-EMF; thus, improving the robustness of sensorless control. However, it does not consider the load torque variations. Looking for a way to deal with load torques, a sensorless control system based on two ESOs is proposed in the paper [31]. Here, the first observer estimates the load torque variations from the mechanical equation, and the second estimates the back-EMF. However, this paper only considers the robustness of a sensorless control system during transient-state operation. On the other hand, the paper [32] develops a third-order nonlinear ESO to estimate uncertainties caused by both parametric and load torque variations; in ref. [33], a third-order super-twisting ESO estimates the rotor speed and load torque variations from the rotor position error, and in ref. [34], an extended sliding mode observer is designed to estimate the stator current and lumped disturbances. Nonetheless, in the later contributions mentioned, the closed-loop system’s stability is limited by applying the separation principle, which is not trivial in nonlinear systems.

The presented sensorless control methodologies and their exposed disadvantages highlight the need for a robust algorithm that ensures stability throughout all time in a closed-loop scheme, maintaining as far as possible a similar complexity and easy implementation, such as back-EMF observers. Therefore, the main objective of this paper is to solve the speed tracking control problem for surface-mount PMSMs despite time-varying load torque variations, considering only the measurements of electrical variables (stator currents and voltages). The proposed scheme consists of an ESO interconnected with a PI-compensated controller. Here, the observer estimates the mechanical variables and load torque variations, and these estimations provide feedback to the system. Hence, our proposal consists of a stabilizing PI-compensated controller such that the solutions of PMSM are uniformly stable around the desired speed, while attenuating the effects of unknown load torque variations. In contrast to the works mentioned above, our sensorless scheme allows the trajectories of the motor to be bounded from the initial time, i.e., $t\ge 0$, involving feedback of the estimations provided by the ESO during the transient response of the observer’s convergence. This fact allows us to avoid the separation principle. Due to the structure of the scheme proposed in this work, it is not necessary to modify the rotating model ($d,q$) to estimate the back-EMF voltage and design a PLL to calculate the position and speed of the rotor [8], nor to apply a function trigonometric to the back-EMF [10], thus simplifying the complexity of our algorithm for subsequent implementation. Unlike more sophisticated algorithms, like the Kalman filter and intelligent control [18,24], the low complexity of our algorithm could help engineers speed up early-stage industrial projects.

The main contributions of this paper include the following: (i) a proposed sensorless controller to force the rotor speed to track a desired high-speed, despite time-varying load torques; (ii) sufficient conditions are defined to ensure uniformly bounded trajectories of the closed-loop for all time. Moreover, the controller is exponentially stable without load torque variations. To our knowledge, there is no research concerning the closed-loop analysis of an extended state observer combined with a PI controller for the PMSM.

The remainder of this paper is organized as follows: Section 2 provides the background and mathematical preliminaries. Section 3 describes the dynamical model of the PMSM, including the paper’s problem statement. Section 4 is devoted to ESO design for mechanical variables estimation, including stability analysis. Section 5 presents the sensorless controller strategy, considering a closed-loop stability analysis avoiding the separation principle. Section 6 provides high-precision emulations and experimental results. Finally, in Section 7, the paper’s conclusions are given.

2. Mathematical Preliminaries

In this paper, we express the set of real numbers with $\mathbb{R}$. The notation ${\lambda }_{min}\left\{\mathit{A}\right\}\left({\lambda }_{max}\left\{\mathit{A}\right\}\right)$ declares the minimum (maximum) eigenvalue of a matrix $\mathit{A}\in {\mathbb{R}}^{n\times n}$. The absolute value for any real number x is denoted by $\left|x\right|$, and we use $\parallel \mathit{x}\parallel$ as the Euclidean norm for a vector $\mathit{x}\in {\mathbb{R}}^n$. The symbol ${\mathit{A}}^T$ represents the transpose matrix of $\mathit{A}$. On the other hand, we use the following Definition and Lemmas.

Definition 1

([35], §4.4). A continuous function $\alpha :[0,a)\to {\mathbb{R}}^{+}$ is said to be in the class $\mathcal{K}$ if (i) $\alpha \left(0\right)=0$ and (ii) it is strictly increasing. Additionally, α is said to be in class ${\mathcal{K}}_{\infty }$ if, for all $r\ge 0$, we have that $\alpha :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ where $\alpha \left(r\right)\to \infty$ as $r\to \infty$.

In the following sections, $B_r$ denotes the set

$$B_r=\{\mathit{x}\in {\mathbb{R}}^n:\parallel \mathit{x}\parallel \le r\},$$

(1)

where r is a positive number.

Lemma 1

([35], §4.4). $V:D\to \mathbb{R}$ is positive definite if, and only if, there exists class $\mathcal{K}$ functions ${\alpha }_1$ and ${\alpha }_2$ such that

$${\alpha }_1(\parallel \mathit{x}\parallel )\le V\left(\mathit{x}\right)\le {\alpha }_2(\parallel \mathit{x}\parallel )\forall \mathit{x}\in B_r\subset D.$$

(2)

Moreover, if $D={\mathbb{R}}^n$ and $V(·)$ is radially unbounded, then ${\alpha }_1$ and ${\alpha }_2$ can be chosen in the class ${\mathcal{K}}_{\infty }$.

Lemma 2

([36]). Let $\mathit{X}$ and $\mathbf{Y}$ matrices with compatible dimensions. Then, the following inequality holds

$$2{\mathit{X}}^T\mathit{Y}\le {\mathit{X}}^T{\mathsf{\Lambda }}^{-1}\mathit{X}+{\mathit{Y}}^T\mathsf{\Lambda }\mathit{Y},$$

(3)

where Λ is a symmetric positive definite matrix with a suitable dimension.

3. Surface-Mount Permanent-Magnet Synchronous Motor Dynamical Model

The dynamical model in rotating ($d,q$) coordinates of the surface-mount permanent magnet synchronous motor, making the usual assumptions as the symmetry between phases, inductances, and negligible magnetic hysteresis, is given by [21]

$$\begin{array}{cc}\hfill \frac{\mathrm{d}}{\mathrm{d}t}{i}_d& =-\frac{R}{L}{i}_d+p{i}_q\omega +\frac{v_d}{L},\hfill \end{array}$$

(4a)

$$\begin{array}{cc}\hfill \frac{\mathrm{d}}{\mathrm{d}t}{i}_q& =-\frac{R}{L}{i}_q-p{i}_d\omega -\frac{\psi }{L}\omega +\frac{v_q}{L},\hfill \end{array}$$

(4b)

$$\begin{array}{cc}\hfill \frac{\mathrm{d}}{\mathrm{d}t}\omega & =\frac{K_M}{J}{i}_q-\frac{F}{J}\omega -\frac{T_l}{J},\hfill \end{array}$$

(4c)

$$\begin{array}{cc}\hfill \mathbf{y}& =\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right],\hfill \end{array}$$

(4d)

where the stator currents $i_d\left(t\right)\in \mathbb{R}$ and $i_q\left(t\right)\in \mathbb{R}$ are the measured outputs of the system, the stator voltages $v_d\left(t\right)\in \mathbb{R}$ and $v_q\left(t\right)\in \mathbb{R}$ are the control inputs, and $\omega \left(t\right)\in \mathbb{R}$ is the speed rotor. The parameter $R\in {\mathbb{R}}^{+}$ is the stator resistance, $L\in {\mathbb{R}}^{+}$ is the stator inductance, $K_M\in {\mathbb{R}}^{+}$ is the rotor torque constant, $\psi \in {\mathbb{R}}^{+}$ is the flux linkage of the rotor magnet, $J\in {\mathbb{R}}^{+}$ is the rotor inertia, $p\in \mathbb{N}$ is the number of pole pairs, $F\in {\mathbb{R}}_0^{+}$ is the viscous friction constant, and $T_l\left(t\right)\in \mathbb{R}$ is the load torque. Here, abrupt load torque variations, unknown in practical situations, are the main cause of disturbances. Furthermore, the PMSM physical operation domain $\mathcal{D}$ is defined by the following set:

$$\mathcal{D}=\left\{{[i_d,i_q,\omega ,T_l]}^T:\left|i_d\right|\le I_d^{max},\left|i_q\right|\le I_q^{max},\left|\omega \right|\le {\mathsf{\Omega }}^{max},\left|T_l\right|\le T_L^{max}\right\},$$

(5)

where $I_d^{max},I_q^{max},{\mathsf{\Omega }}^{max}$ and $T_L^{max}$ are the maximum values for the stator currents, rotor speed, and load torque that the PMSM is capable of withstanding, respectively. The bounds given in (5) can be calculated analytically by applying the methodology presented by [37]. On the other hand, for the purpose of engineering applications, these bounds can be taken from the PMSM manufacturer’s datasheet.

The field-oriented control (FOC) scheme is commonly applied to drive the PMSM (see Figure 2). The FOC requires a control action to set in zero the $i_d\left(t\right)$ that allows a decoupling of the system (4) and diminish the electromotive force; then, the PMSM is controlled in a similar way to a DC motor. Thereby, the control of both the rotor speed and electromagnetic torque is feasible, forcing the current $i_q\left(t\right)$ to track a stator current reference $i^{*}\left(t\right)$, which is considered as a virtual control input [38].

For technical reasons, we make the following assumptions throughout the whole paper:

(A1) 

We consider that $T_l\left(t\right)$ (load time-varying torque) is an unknown-bounded function with an a priori known upper bound, meaning that there prevails a positive constant $T_L^{max}$ such that

$$|T_l\left(t\right)|\le T_L^{max}.$$

(6)

(A2) 

The PMSM operates in the high-speed range, i.e., $\left|\omega \right(t\left)\right|\gg 0$.

The assumption A1 ensures that the solution of (4) remains uniformly bounded despite load torque variations. Assumption A2 is a supposition when we have issues estimating mechanical variables at low speed using model-based techniques because model (4) is not observable at zero speed.

The control objective of this work is to design a sensorless speed tracking control for the surface-mount PMSM, given by (4), forcing the speed $\omega \left(t\right)$ to track a desired speed ${\omega }^{*}\left(t\right)$ and stabilize the current $i_d\left(t\right)$ to zero; that is,

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim}\parallel \omega \left(t\right)-{\omega }^{*}\left(t\right)\parallel & =0,\hfill \end{array}$$

(7a)

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim}\parallel i_d\left(t\right)\parallel & =0,\hfill \end{array}$$

(7b)

despite unknown time-varying load torque variations. Since mechanical variables are not available for feedback, the stator’s signal currents and voltages are the only measurements for feedback.

4. Mechanical Variables Estimation

In this section, we examine the design of an ESO. That observer considers the time-varying load torque as an extended state of (4) and estimates its variations. Additionally, the proposed observer estimates the rotor angular position and the rotor speed.

Beforehand, consider $\mathit{\xi }={\left[i_d,i_q,\omega ,T_l\right]}^T$, that now (4) can be represented by an extended-state system as

$$\begin{array}{cc}\hfill \dot{\mathit{\xi }}& =\mathit{A}\mathit{\xi }+\mathsf{\Phi }(\mathit{\xi },\mathit{v})+\mathit{D}\phi ,\hfill \end{array}$$

(8a)

$$\begin{array}{cc}\hfill \mathbf{y}& =\mathit{C}\mathit{\xi },\hfill \end{array}$$

(8b)

where $\mathit{v}\left(t\right)={\left[v_d\left(t\right),v_q\left(t\right)\right]}^T$ acts as the input that allows controlling the motor, $\phi \left(t\right)$ is an unknown function but bounded, and

$$\begin{array}{cc}\hfill \mathit{A}& =\left[\begin{array}{cccc}-\frac{R}{L}& \phantom{-}0& \phantom{-}0& \phantom{-}0\\ \phantom{-}0& -\frac{R}{L}& -\frac{\psi }{L}& \phantom{-}0\\ \phantom{-}0& \phantom{-}\frac{K_M}{J}& -\frac{F}{J}& -\frac{1}{J}\\ \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0\end{array}\right],\mathsf{\Phi }(\mathit{\xi },\mathit{v})=\left[\begin{array}{c}\phantom{-}p{i}_q\omega +\frac{1}{L}{v}_d\\ -p{i}_d\omega +\frac{1}{L}{v}_q\\ 0\\ 0\end{array}\right]\hfill \end{array}$$

(9a)

$$\begin{array}{cc}\hfill \mathit{C}& =\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right],\mathit{D}=\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right].\hfill \end{array}$$

(9b)

We propose an ESO in the sequel to overcome the mechanical variables estimation and load torque variation problem.

4.1. Extended State Observer Design

First, this section describes the ESO as follows

$$\dot{\widehat{\mathit{\xi }}}=\mathit{A}\widehat{\mathit{\xi }}+\mathsf{\Phi }\left(\widehat{\mathit{\xi }},\mathit{v}\right)+\mathit{L}\left(\mathbf{y}-\mathit{C}\widehat{\mathit{\xi }}\right),$$

(10)

where $\widehat{\mathit{\xi }}\left(t\right)={\left[{\widehat{\xi }}_1,{\widehat{\xi }}_2,{\widehat{\xi }}_3,{\widehat{\xi }}_4\right]}^T={\left[{\widehat{i}}_d,{\widehat{i}}_q,\widehat{\omega },{\widehat{T}}_l\right]}^T$ defines the estimated state vector of $\mathit{\xi }\left(t\right)$, and $\mathit{L}\in {\mathbb{R}}^{4\times 2}$ is the observer design gain matrix.

Now, let us denote the observation error as $\tilde{\mathit{\xi }}\left(t\right)=\mathit{\xi }\left(t\right)-\widehat{\mathit{\xi }}\left(t\right)$. Then, considering (8) and (10), the observer error dynamics are described as follows:

$$\dot{\tilde{\mathit{\xi }}}={\mathit{A}}_O\tilde{\mathit{\xi }}+\mathsf{\Phi }(\mathit{\xi },\mathit{v})-\mathsf{\Phi }\left(\widehat{\mathit{\xi }},\mathit{v}\right)+\mathit{D}\phi .$$

(11)

Here, assuming that the pair $(\mathit{A},\mathit{C})$ is observable, the gain observer matrix $\mathit{L}$ is selected, such that, ${\mathit{A}}_O=\mathit{A}-\mathit{L}\mathit{C}$ is a Hurwitz matrix.

Meanwhile, the following is henceforth assumed:

(A3) 

We consider ${\phi }^{+}$ as an a priori known bound for the unknown function $\phi \left(t\right)$ (scalar), with

$$\left|\phi \left(t\right)\right|\le {\phi }^{+}.$$

(12)

(A4) 

We assume that $\mathsf{\Phi }(\mathit{\xi },\mathit{v})$ (nonlinear term) in (8) is uniformly bounded in $\mathit{v}$ and considered as locally Lipschitz in $\mathcal{D}$; that means the next

$$∥\mathsf{\Phi }(\mathit{\xi },\mathit{v})-\mathsf{\Phi }\left(\widehat{\mathit{\xi }},\mathit{v}\right)∥\le \gamma ∥\mathit{\xi }-\widehat{\mathit{\xi }}∥,\mathit{\xi }\in \mathcal{D},$$

(13)

is fulfilled, where $\gamma >0$ is the Lipschitz constant.

4.2. Stability Analysis

Considering a Lyapunov candidate function $V_1\left(\tilde{\mathit{\xi }}\right)={\tilde{\mathit{\xi }}}^T{\mathit{P}}_1\tilde{\mathit{\xi }}$ for the system (11) that represents the error dynamics. Here, ${\mathit{P}}_1\in {\mathbb{R}}^{4\times 4}$ is a symmetric and positive matrix; this implies that $V_1\left(\tilde{\mathit{\xi }}\right)$ is a positive definite and radially unbounded function. Hence, by Lemma 1, there exist ${\mathcal{K}}_{\infty }$ functions ${\alpha }_1(·)$ and ${\alpha }_2(·)$ such that

$${\lambda }_{min}\left\{{\mathit{P}}_1\right\}{∥\tilde{\mathit{\xi }}∥}^2={\alpha }_1\left(∥\tilde{\mathit{\xi }}∥\right)\le V_1\left(\tilde{\mathit{\xi }}\right)\le {\alpha }_2\left(∥\tilde{\mathit{\xi }}∥\right)={\lambda }_{max}\left\{{\mathit{P}}_1\right\}{∥\tilde{\mathit{\xi }}∥}^2.$$

(14)

The time derivative of $V_1\left(\tilde{\mathit{\xi }}\right)$ along the trajectories of (11) yields the following

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(\tilde{\mathit{\xi }}\right)=& {\tilde{\mathit{\xi }}}^T\left({\mathit{P}}_1{\mathit{A}}_O+{\mathit{A}}_O^T{\mathit{P}}_1\right)\tilde{\mathit{\xi }}\hfill \\ \hfill & +2{\tilde{\mathit{\xi }}}^T{\mathit{P}}_1\left(\mathsf{\Phi }(\mathit{\xi },\mathit{v})-\mathsf{\Phi }\left(\widehat{\mathit{\xi }},\mathit{v}\right)\right)+2{\tilde{\mathit{\xi }}}^T{\mathit{P}}_1\mathit{D}\phi \left(t\right).\hfill \end{array}$$

(15)

Now, according to the Lemma 2, if $X={\mathit{P}}_1\tilde{\mathit{\xi }}$ and $Y=\mathsf{\Phi }(\mathit{\xi },\mathit{v})-\mathsf{\Phi }\left(\widehat{\mathit{\xi }},\mathit{v}\right)$, then we can state the following inequality

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(\tilde{\mathit{\xi }}\right)& \le {\tilde{\mathit{\xi }}}^T\left({\mathit{P}}_1{\mathit{A}}_O+{\mathit{A}}_O^T{\mathit{P}}_1+{\mathit{P}}_1{\mathsf{\Lambda }}^{-1}{\mathit{P}}_1\right)\tilde{\mathit{\xi }}\hfill \\ \hfill & +{\left(\mathsf{\Phi }(\mathit{\xi },\mathit{v})-\mathsf{\Phi }\left(\widehat{\mathit{\xi }},\mathit{v}\right)\right)}^T\mathsf{\Lambda }\left(\mathsf{\Phi }(\mathit{\xi },\mathit{v})-\mathsf{\Phi }\left(\widehat{\mathit{\xi }},\mathit{v}\right)\right)+2{\tilde{\mathit{\xi }}}^T{\mathit{P}}_1\mathit{D}\phi \left(t\right).\hfill \end{array}$$

(16)

Based on [39], it is possible to write the Lipschitz condition (13) as follows:

$$∥\mathsf{\Phi }(\mathit{\xi },\mathit{v})-\mathsf{\Phi }\left(\widehat{\mathit{\xi }},\mathit{v}\right)∥\le ∥\mathsf{\Gamma }\left(\mathit{\xi }-\widehat{\mathit{\xi }}\right)∥,\mathit{\xi }\in \mathcal{D},$$

(17)

where the nonlinear term $\mathsf{\Phi }(\mathit{\xi },\mathit{v})$ is locally Lipschitz in a domain $\mathcal{D}$ and uniformly bounded in v, and $\mathsf{\Gamma }=diag\{{\gamma }_1,{\gamma }_2,{\gamma }_3,{\gamma }_4\}$ is a constant matrix. Notice that the inequality (17) is a less conservative version of the Lipschitz condition (13) for some values of the matrix $\mathsf{\Gamma }$ [39,40]. If we take (17), the last inequality of ${\dot{V}}_1\left(\tilde{\mathit{\xi }}\right)$ may be expressed as

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(\tilde{\mathit{\xi }}\right)& \le {\tilde{\mathit{\xi }}}^T\left({\mathit{P}}_1{\mathit{A}}_O+{\mathit{A}}_O^T{\mathit{P}}_1+{\mathit{P}}_1{\mathsf{\Lambda }}^{-1}{\mathit{P}}_1+\mathsf{\Gamma }\mathsf{\Lambda }\mathsf{\Gamma }\right)\tilde{\mathit{\xi }}+2{\tilde{\mathit{\xi }}}^T{\mathit{P}}_1\mathit{D}\phi \left(t\right),\hfill \end{array}$$

(18)

then, we have chosen a solution ${\mathit{P}}_1$ that satisfies the following criteria: If the algebraic Riccati equation

$${\mathit{P}}_1{\mathit{A}}_O+{\mathit{A}}_O^T{\mathit{P}}_1+{\mathit{P}}_1{\mathsf{\Lambda }}^{-1}{\mathit{P}}_1+\mathsf{\Gamma }\mathsf{\Lambda }\mathsf{\Gamma }=-{\mathit{Q}}_1,$$

(19)

is satisfied for any positive definite and symmetric matrix ${\mathit{Q}}_1$, we obtain

$${\dot{V}}_1\left(\tilde{\mathit{\xi }}\right)\le -{\tilde{\mathit{\xi }}}^T{\mathit{Q}}_1\tilde{\mathit{\xi }}+2{\tilde{\mathit{\xi }}}^T{\mathit{P}}_1\mathit{D}\phi \left(t\right).$$

(20)

Now, making the assumption (A3), the later inequality of ${\dot{V}}_1\left(\tilde{\mathit{\xi }}\right)$ is given by

$${\dot{V}}_1\left(\tilde{\mathit{\xi }}\right)\le -\left({\lambda }_{min}\left\{{\mathit{Q}}_1\right\}-{\sigma }_1\right){∥\tilde{\mathit{\xi }}∥}^2-{\sigma }_1{∥\tilde{\mathit{\xi }}∥}^2+2{\phi }^{+}{\lambda }_{max}\left\{{\mathit{P}}_1\right\}∥\tilde{\mathit{\xi }}∥,$$

(21)

with $0<{\sigma }_1<{\lambda }_{min}\left\{{\mathit{Q}}_1\right\}$. Then,

$${\dot{V}}_1\left(\tilde{\mathit{\xi }}\right)\le -\left({\lambda }_{min}\left\{{\mathit{Q}}_1\right\}-{\sigma }_1\right){∥\tilde{\mathit{\xi }}∥}^2,\forall ∥\tilde{\mathit{\xi }}∥\ge {\mu }_1:=\frac{2{\phi }^{+}{\lambda }_{max}\left\{{\mathit{P}}_1\right\}}{{\sigma }_1}.$$

(22)

We conclude that the solutions are uniformly bounded ultimately. Now, from (14), we can calculate the ultimate bound, since $V_1\left(\tilde{\mathit{\xi }}\right)$ is a radially unbounded function. Thus, the ultimate bound is given by

$$b_1={\alpha }_1^{-1}\left({\alpha }_2\left({\mu }_1\right)\right)={\mu }_1\sqrt{\frac{{\lambda }_{max}\left\{{\mathit{P}}_1\right\}}{{\lambda }_{min}\left\{{\mathit{P}}_1\right\}}}.$$

(23)

Hence, ${\dot{V}}_1\left(\tilde{\mathit{\xi }}\right)$ is a negative function

$${\dot{V}}_1\left(\tilde{\mathit{\xi }}\right)\le -\left({\lambda }_{min}\left\{{\mathit{Q}}_1\right\}-{\sigma }_1\right){∥\tilde{\mathit{\xi }}∥}^2<0,\forall {\mu }_1\le ∥\tilde{\mathit{\xi }}∥\le b_1.$$

(24)

Moreover, if there are no existing load torque variations ($T_l=0\Rightarrow \mu =0$), then the origin is exponentially stable.

Finally, (10) yields the estimated rotor speed and load torque variations. We obtain the estimated rotor position by integrating the estimated rotor speed, that is,

$$\widehat{\theta }\left(t\right)={\int }_0^t\widehat{\omega }\left(\tau \right)\mathrm{d}\tau .$$

(25)

In the next section, we construct a PI controller based on the developed ESO, measuring $y\left(t\right)\in {\mathbb{R}}^2$, which means feedbacking of the estimations of the rotor speed and the time-varying load torque.

5. Design of a Sensorless Controller for the PMSM

This section introduces an observer-based compensated PI controller design for the PMSM modeled by (4) taking the FOC strategy to control the motor. First, we require some trivial but necessary assumptions for the desired speed reference ${\omega }^{*}\left(t\right)$.

(A5) 

Consider that ${\omega }^{*}\left(t\right)$ is bounded;

(A6) 

and its derivatives ${\dot{\omega }}^{*}\left(t\right),{\ddot{\omega }}^{*}\left(t\right)$ are continuously uniformly bounded and differentiable functions in t.

To close the loop of the system (4), we propose the next voltage control inputs

$$\begin{array}{cc}\hfill v_d\left(t\right)& =-L\left(\frac{k_{p_1}}{L}{i}_d+\frac{k_{i_1}}{L}{\int }_0^t{i}_d\left(\tau \right)\mathrm{d}\tau +p{i}_q\widehat{\omega }\right),\hfill \end{array}$$

(26a)

$$\begin{array}{cc}\hfill v_q\left(t\right)& =-L\left(\frac{k_{p_2}}{L}(i_q-i^{*})+\frac{k_{i_2}}{L}{\int }_0^t(i_q-i^{*})\left(\tau \right)\mathrm{d}\tau -\frac{\psi }{L}\widehat{\omega }-p{i}_d\widehat{\omega }-\frac{R}{L}{i}^{*}-{\dot{i}}^{*}\left(t\right)\right),\hfill \end{array}$$

(26b)

and, for the speed to track, the virtual control input reference $i^{*}\left(t\right)$ is given by

$$i^{*}\left(t\right)=-k_{p_3}(\widehat{\omega }-{\omega }^{*})-k_{i_3}{\int }_0^t(\widehat{\omega }-{\omega }^{*})\left(\tau \right)\mathrm{d}\tau +\frac{J}{K_M}\left(\frac{F}{J}\widehat{\omega }+\frac{{\widehat{T}}_l}{J}+{\dot{\omega }}^{*\left(t\right)}\right),$$

(27)

where $k_{p_1},k_{p_2}$ and $k_{p_3}$ are proportional gains and $k_{i_1},k_{i_2}$ and $k_{i_3}$ are integral gains.

The feedback of the load torque estimated by the ESO will provide robustness to the controller to keep the speed error bounded despite external disturbances. The next step is to introduce the control strategy for the PMSM, considering a time-varying torque.

Sensorless Control Closed-Loop Stability Proof

The traditional FOC strategy is based on a PI controller, where the integral action (26) and (27) introduces an additional state variable. First, considering the following state vector

$$\mathit{x}=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\\ x_5\\ x_6\end{array}\right]=\left[\begin{array}{c}{\int }_0^t{i}_d\left(\tau \right)\mathrm{d}\tau \\ {\int }_0^t(i_q-i^{*})\left(\tau \right)\mathrm{d}\tau \\ {\int }_0^t(\omega -{\omega }^{*})\left(\tau \right)\mathrm{d}\tau \\ i_d\\ i_q-i^{*}\\ \omega -{\omega }^{*}\end{array}\right].$$

(28)

On the other hand, taking the ESO error dynamics (11) and the controller states (28), we can take $\widehat{\omega }-{\omega }^{*}=x_6-{\tilde{\xi }}_3$. Then, it is possible to substitute (26) and (27) in the equations of the model of the PMSM (4) obtaining the next equations for the closed loop dynamics, that include both the controller and observer dynamics

$$\begin{array}{cc}\hfill \dot{\mathit{x}}& ={\mathit{A}}_C\mathit{x}+{\mathit{B}}_1{\tilde{\xi }}_4+\left({\mathit{E}}_1\left(\mathit{x}\right)+{\mathit{E}}_2\left(t\right)\right){\tilde{\xi }}_3+\mathit{G}\left(\tilde{\xi },t\right),\hfill \end{array}$$

(29a)

$$\begin{array}{cc}\hfill \dot{\tilde{\mathit{\xi }}}& ={\mathit{A}}_O\tilde{\mathit{\xi }}+\mathsf{\Phi }(\mathit{\xi },\mathit{v})-\mathsf{\Phi }\left(\widehat{\mathit{\xi }},\mathit{v}\right)+\mathit{D}\phi .\hfill \end{array}$$

(29b)

with

$$\begin{array}{cc}\hfill {\mathit{A}}_C& =\left[\begin{array}{cccccc}0& 0& 0& \phantom{-}1& \phantom{-}0& \phantom{-}0\\ 0& 0& 0& \phantom{-}0& \phantom{-}1& \phantom{-}0\\ 0& 0& 0& \phantom{-}0& \phantom{-}0& \phantom{-}1\\ -k_{i_1}& 0& 0& -k_{p_1}-\frac{R}{L}& \phantom{-}0& \phantom{-}0\\ 0& -k_{i_2}& 0& \phantom{-}0& -k_{p_2}-\frac{R}{L}& \phantom{-}0\\ 0& 0& -k_{i_3}& \phantom{-}0& \phantom{-}\frac{K_M}{J}& -k_{p_3}\end{array}\right],\hfill \end{array}$$

(30a)

$$\begin{array}{cc}\hfill {\mathit{B}}_1& =\left[\begin{array}{c}\phantom{-}0\\ \phantom{-}0\\ \phantom{-}0\\ \phantom{-}0\\ \phantom{-}0\\ -\frac{1}{J}\end{array}\right],{\mathit{E}}_1=\left[\begin{array}{c}0\\ 0\\ 0\\ p{x}_5\\ -\frac{\psi }{L}-p{x}_4\\ k_{p_3}-\frac{F}{J}\end{array}\right],{\mathit{E}}_2\left(t\right)=\left[\begin{array}{c}0\\ 0\\ 0\\ p{i}^{*}\left(t\right)\\ 0\\ 0\end{array}\right],\hfill \end{array}$$

(30b)

$$\begin{array}{cc}\hfill \mathit{G}\left(\tilde{\mathit{\xi }},t\right)& =\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ k_{i_3}{\int }_0^t{\tilde{\xi }}_3\left(\tau \right)\mathrm{d}\tau \end{array}\right].\hfill \end{array}$$

(30c)

For that closed-loop system (29) (with no time-varying torque applied), we have the equilibrium point ${\left[\mathit{x},\tilde{\mathit{\xi }}\right]}^T=0\in {\mathbb{R}}^{10}$ (the origin). Now, we will prove that the trajectories of (29) do not diverge to infinity when $t\to \infty$. The following Theorem summarizes the design of the compensated PI controller for the PMSM.

Theorem 1.

Let system (4) fulfil the assumptions (A5) and (A6), given that it is possible to solve both of the next equations

• Given a perturbed Riccati Equation (31) with ${\mathit{P}}_1$ as the solution (positive definite matrix)

  $${\mathit{P}}_1{\mathit{A}}_O+{\mathit{A}}_O^T{\mathit{P}}_1+{\mathit{P}}_1{\mathsf{\Lambda }}^{-1}{\mathit{P}}_1+\mathsf{\Gamma }\mathsf{\Lambda }\mathsf{\Gamma }=-{\mathit{Q}}_1,$$

  (31)
  for the matrix function ${\mathit{A}}_O=\mathit{A}-\mathit{L}\mathit{C}$ given in (9), some matrix $\mathsf{\Gamma }=diag\{{\gamma }_1,{\gamma }_2,{\gamma }_3,{\gamma }_4\}$, a matrix ${\mathit{Q}}_1>0$, and some matrix $\mathsf{\Lambda }>0$.
• Given a perturbed Lyapunov Equation (32) with ${\mathit{P}}_2$ as the solution (positive definite matrix)

  $${\mathit{P}}_2{\mathit{A}}_C+{\mathit{A}}_C^T{\mathit{P}}_2=-{\mathit{Q}}_2,$$

  (32)
  for the matrix function ${\mathit{A}}_C$ given in (30), and for some ${\mathit{Q}}_2>0$.

Then, the closed loop controller constructed by (26) and (27) with ${[i_d,i_q-i^{*},\omega -{\omega }^{*}]}^T$ solve the tracking problem for the PMSM given by (4), (10), (26), and (27); subjected to time-varying torques.

Proof. 

First, consider the Lyapunov candidate function as

$$V_2\left(\mathit{x},\tilde{\mathit{\xi }}\right)=\eta {\mathit{x}}^T{\mathit{P}}_2\mathit{x}+V_1\left(\tilde{\mathit{\xi }}\right),$$

(33)

for the closed-loop system defined in (29). Here, ${\mathit{P}}_2\in {\mathbb{R}}^{6\times 6}$ is a symmetric and positive matrix, and $\eta$ is a positive constant. Hence, by Lemma 1, there exists ${\mathcal{K}}_{\infty }$ functions ${\alpha }_1(·)$ and ${\alpha }_2(·)$ such that

$${\lambda }_{min}\left\{\mathit{P}\right\}{∥\begin{array}{c}\mathit{x}\\ \tilde{\mathit{\xi }}\end{array}∥}^2={\alpha }_1\left(\parallel \mathit{x}\parallel ,\parallel \tilde{\mathit{\xi }}\parallel \right)\le V_2\left(\mathit{x},\tilde{\mathit{\xi }}\right)\le {\alpha }_2\left(\parallel \mathit{x}\parallel ,\parallel \tilde{\mathit{\xi }}\parallel \right)={\lambda }_{max}\left\{\mathit{P}\right\}{∥\begin{array}{c}\mathit{x}\\ \tilde{\mathit{\xi }}\end{array}∥}^2,$$

(34)

with

$$\mathit{P}=\left[\begin{array}{cc}\eta {\mathit{P}}_2& 0\\ 0& {\mathit{P}}_1\end{array}\right]\in {\mathbb{R}}^{10\times 10}.$$

(35)

The time derivative of $V_2\left(\mathit{x},\tilde{\mathit{\xi }}\right)$ along the trajectories of (29) yields to

$$\begin{array}{cc}\hfill {\dot{V}}_2\left(\mathit{x},\tilde{\mathit{\xi }}\right)& ={\mathit{x}}^T\left({\mathit{P}}_2{\mathit{A}}_C+{\mathit{A}}_C^T{\mathit{P}}_2\right)\mathit{x}+2{\mathit{x}}^T{\mathit{P}}_2{\mathit{B}}_1{\tilde{\mathit{\xi }}}_4+2{\mathit{x}}^T{\mathit{P}}_2{\mathit{E}}_1\left(\mathit{x}\right){\tilde{\mathit{\xi }}}_3\hfill \\ \hfill & +2{\mathit{x}}^T{\mathit{P}}_2{\mathit{E}}_2\left(t\right){\tilde{\xi }}_3+2{\mathit{x}}^T{\mathit{P}}_2\mathit{Q}\left(\tilde{\mathit{\xi }},t\right)+{\dot{V}}_1\left(\tilde{\mathit{\xi }}\right).\hfill \end{array}$$

(36)

Here, if the controller gains $k_{p_j},k_{i_j}$ ($j=1,2,3.$) are selected, such that, ${\mathit{A}}_C$ is a Hurwitz matrix, and, if the solution ${\mathit{P}}_2$ is selected such that the following algebraic Lyapunov equation

$${\mathit{P}}_2{\mathit{A}}_C+{\mathit{A}}_C^T{\mathit{P}}_2=-{\mathit{Q}}_2,$$

(37)

is satisfied for any symmetric and positive definite matrix ${\mathit{Q}}_2$, we get

$$\begin{array}{cc}\hfill {\dot{V}}_2\left(\mathit{x},\tilde{\mathit{\xi }}\right)& =-{\mathit{x}}^T{\mathit{Q}}_2\mathit{x}+2{\mathit{x}}^T{\mathit{P}}_2{\mathit{B}}_1{\tilde{\xi }}_4+2{\mathit{x}}^T{\mathit{P}}_2{\mathit{E}}_1\left(\mathit{x}\right){\tilde{\xi }}_3\hfill \\ \hfill & +2{\mathit{x}}^T{\mathit{P}}_2{\mathit{E}}_2\left(t\right){\tilde{\xi }}_3+2{\mathit{x}}^T{\mathit{P}}_2\mathit{Q}\left(\tilde{\mathit{\xi }},t\right)+{\dot{V}}_1\left(\tilde{\mathit{\xi }}\right).\hfill \end{array}$$

(38)

Before continuing with the stability analysis, we define the upper bounds for some functions given in ${\dot{V}}_2$ by taking the initial conditions of the PMSM and observer (10) in the physical operation domain $\mathcal{D}$:

• We establish $|x_4|\le I_d^{max},\left|x_5\right|\le I_q^{max}$, then the function ${\mathit{E}}_1\left(\mathit{x}\right)$ can be bounded as follows:

  $${\mathit{E}}_1\left(\mathit{x}\right)\le ∥\begin{array}{c}{0}^{3\times 1}\\ p{I}_q^{max}\\ -\frac{\psi }{L}-p{I}_d^{max}\\ k_{p_3}-\frac{F}{J}\end{array}∥={\mathit{E}}_1^{max},$$

  (39)
• The virtual input $i^{*}\left(t\right)$ can be bounded by $i^{*}\le i_o$, where $i_o$ is the maximum current that the speed controller will be able to provide; hence, the function ${\mathit{E}}_2\left(t\right)$ is bounded as

  $${\mathit{E}}_2\left(t\right)\le ∥\begin{array}{c}{0}_{3\times 1}\\ p{i}_o\\ 0\\ 0\end{array}∥={\mathit{E}}_2^{max},$$

  (40)
• The function $\mathit{G}\left(\tilde{\mathit{\xi }},t\right)$ is considered as the perturbed input control of the closed-loop system and is bounded since the integral

  $${\int }_0^t{\tilde{\xi }}_3\left(\tau \right)\mathrm{d}\tau \le \underset{0\le \tau \le t}{sup}∥{\tilde{\xi }}_3\left(\tau \right)∥\le b_1<\infty$$

  (41)

  with $b_1$ defined in (23). Hence,

  $$\mathit{G}\left(\tilde{\mathit{\xi }},t\right)\le \left[\begin{array}{c}{0}^{5\times 1}\\ k_{i_3}{b}_1\end{array}\right]={\mathit{G}}^{max},$$

  (42)

Now, continuing with the stability analysis, substitute the upper bounds (39), (40), and (42) into (36), and, regrouping terms, we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_2\left(\mathit{x},\tilde{\mathit{\xi }}\right)\le & -\eta {\lambda }_{min}\left\{{\mathit{Q}}_2\right\}{\parallel \mathit{x}\parallel }^2+2\eta \left(\parallel {\mathit{P}}_2{\mathit{E}}_1^{max}\parallel +\parallel {\mathit{P}}_2{\mathit{E}}_2^{max}\parallel +\parallel {\mathit{P}}_2{\mathit{B}}_1\parallel \right)\parallel \mathit{x}\parallel ∥\tilde{\mathit{\xi }}∥\hfill \\ \hfill & -\left({\lambda }_{min}\left\{{\mathit{Q}}_1\right\}-{\sigma }_1\right){∥\tilde{\mathit{\xi }}∥}^2+2\eta \parallel {\mathit{P}}_2{\mathit{G}}^{max}\parallel \left(\parallel \mathit{x}\parallel +∥\tilde{\mathit{\xi }}∥\right).\hfill \end{array}$$

(43)

Now, we define $\mathit{z}={\left[\parallel \mathit{x}\parallel ,∥\tilde{\mathit{\xi }}∥\right]}^T$, then (43) can be rewritten as

$${\dot{V}}_2\left(\mathit{z}\right)=-{\mathit{z}}^T\mathit{Q}\mathit{z}+2\eta \mathit{x}{\mathbf{\delta }}^T\mathit{z},$$

(44)

where $\mathbf{\delta }\in {\mathbb{R}}^{1\times 2}$, with all of its elements given by $\parallel {\mathit{P}}_2{\mathit{G}}^{max}\parallel$ and

$$\begin{array}{c}\hfill \mathit{Q}=\left[\begin{array}{cc}\eta {\lambda }_{min}\left\{{\mathit{Q}}_2\right\}& -\eta \left(\parallel {\mathit{P}}_2{\mathit{E}}_1^{max}\parallel +\parallel {\mathit{P}}_2{\mathit{E}}_2^{max}\parallel +\parallel {\mathit{P}}_2{\mathit{B}}_1\parallel \right)\\ -\eta \left(\parallel {\mathit{P}}_2{\mathit{E}}_1^{max}\parallel +\parallel {\mathit{P}}_2{\mathit{E}}_2^{max}\parallel +\parallel {\mathit{P}}_2{\mathit{B}}_1\parallel \right)& {\lambda }_{min}\left\{{\mathit{Q}}_1\right\}-{\sigma }_1\end{array}\right].\end{array}$$

(45)

If the following condition is satisfied

$$\eta <\frac{{\lambda }_{min}\left\{{\mathit{Q}}_2\right\}({\lambda }_{min}\left\{{\mathit{Q}}_1\right\}-{\sigma }_1)}{{\left(\parallel {\mathit{P}}_2{\mathit{E}}_1^{max}\parallel +\parallel {\mathit{P}}_2{\mathit{E}}_2^{max}\parallel +\parallel {\mathit{P}}_2{\mathit{B}}_1\parallel \right)}^2},$$

(46)

then $\mathit{Q}$ is a positive definite matrix; then, we can bound (44) as

$${\dot{V}}_2\left(\mathit{z}\right)\le -{\lambda }_{min}{\left\{\mathit{Q}\right\}\parallel \mathit{z}\parallel }^2+2\eta \parallel \mathbf{\delta }\parallel \parallel \mathit{z}\parallel .$$

(47)

Now, the above inequality of ${\dot{V}}_2\left(\mathit{z}\right)$ is reduced to

$${\dot{V}}_2\left(\mathit{z}\right)\le -\left({\lambda }_{min}\left\{\mathit{Q}\right\}-{\sigma }_2\right){∥\mathit{z}∥}^2-{\sigma }_2{∥\mathit{z}∥}^2+2\eta \parallel \mathbf{\delta }\parallel ∥\mathit{z}∥,$$

(48)

with $0<{\sigma }_2<{\lambda }_{min}\left\{\mathit{Q}\right\}$. Then,

$${\dot{V}}_2\left(\mathit{z}\right)\le -\left({\lambda }_{min}\left\{\mathit{Q}\right\}-{\sigma }_2\right){∥\mathit{z}∥}^2,\forall ∥\mathit{z}∥\ge {\mu }_2:=\frac{2\eta \parallel \mathbf{\delta }\parallel }{{\sigma }_2}.$$

(49)

Now, we conclude that the solutions are uniformly bounded ultimately. Then, from (34), we can calculate the ultimate bound; since $V_2\left(\mathit{z}\right)$ is a radially unbounded function the ultimate bound can be calculated as follows:

$$b_2={\alpha }_1^{-1}\left({\alpha }_2\left({\mu }_2\right)\right)={\mu }_2\sqrt{\frac{{\lambda }_{max}\left\{\mathit{P}\right\}}{{\lambda }_{min}\left\{\mathit{P}\right\}}}.$$

(50)

Now, consider a ball $B_r=\{\mathit{z}\in {\mathbb{R}}^{10}:\parallel \mathit{z}\parallel <r\}$ defined on the maximum values of the physical operation domain of a PMSM such as (5). If there exists a torque variation, with $r>b_2$, this guarantees the existence of the positive invariant sets ${\mathsf{\Omega }}_c=\{\mathit{z}\in B_r:V\left(\mathit{z}\right)\le c<{\alpha }_1\left(r\right)\}$ and ${\mathsf{\Omega }}_{\epsilon }=\{\mathit{z}\in B_r:V\left(\mathit{z}\right)\le \epsilon <{\alpha }_2\left({\mu }_1\right)\}$; therefore, for every $\mathit{z}\left(t_0\right)$ starting at ${\mathsf{\Omega }}_c$, the error trajectories $z\left(t\right)$ will enter into the set ${\mathsf{\Omega }}_{\epsilon }$, since, by the inequality (49), the function ${\dot{V}}_2$ is negative definite on the set $\{\epsilon \le V(\mathit{z})\le c\}$. For more details, see Theorem 4.18 of [35]. Thus, Theorem 1 is proven. □

6. Emulations and Experimental Results

In order to apply the theoretical results from the previous section, we developed emulations and experiments using the Technosoft${}^{\circledR }$ development platform MCK28335 Kit C-Pro. We base our proposal on the FOC scheme to control the PMSM, shown in Figure 2. Our research presents an observer-based controller that modifies the FOC scheme, as seen in Figure 3.

Figure 2 shows the controller (26)–(27) and the ESO (10) in a closed-loop without the feedback of actual position measurement. However, for a comparative study, the graphic results show the position and speed measured to compare the performance of the proposed sensorless control scheme. Now, before introducing the emulation and experimental results, we describe the development platform.

6.1. Platform Description

The development platform MCK28335 Kit C-Pro Digital Control consists of a motion control kit based on the TMS320F28335 digital signal processor manufactured by Technosoft${}^{\circledR }$; Figure 4 shows the setup of the experimental platform. The development platform includes a PMSM equipped with Hall sensors and a 500-line encoder, has a PM50v3.1 power module integrated with MOSFET transistors with a switching frequency of up to 50kHz, A/D and D/A converters, and also provides the integrated development environment (IDE) and libraries needed to streamline the programming and debugging of the C code.

The nominal parameters of the PMSM are shown in

Table 1. Motor parameters and specifications MBE.300E.500.

Symbols	Parameters	Values	Units
R	Stator resistance	4.3	ohm
L	Stator inductance	$3.56\times {10}^{-4}$	H
$K_M$	Rotor torque constant	$36.8\times {10}^{-3}$	Nm/A
$\psi$	Flux linkage	$24.5\times {10}^{-3}$	Nm/A
p	Pole-pairs	1
F	Viscous friction coefficient	3 $\times {10}^{-6}$	Nms
J	Rotor inertia	1.1 $\times {10}^{-6}$	kgm${}^2$
$P_r$	Rated power	15.7	W
$V_r$	Rated voltage	36	V
${\mathsf{\Omega }}^{max}$	Max. permissible speed	210	rad/s
$T_L^{max}$	Max. cont. torque	$30\times {10}^{-3}$	Nm

.

6.2. Emulation Results

In the emulation, the sampling time for the current loop is $1\times {10}^{-4}$ s, and for the speed loop is $1\times {10}^{-3}$ s. The initial conditions are set as

$$\begin{array}{cc}\hfill {[i_d\left(0\right),i_q\left(0\right),\omega \left(0\right)]}^T& =0\in {\mathbb{R}}^3,\hfill \end{array}$$

(51a)

$$\begin{array}{cc}\hfill {\left[{\widehat{\xi }}_1\left(0\right),{\widehat{\xi }}_2\left(0\right),{\widehat{\xi }}_3\left(0\right),{\widehat{\xi }}_4\left(0\right)\right]}^T& =0\in {\mathbb{R}}^4.\hfill \end{array}$$

(51b)

The controller gains are tuned according to the controller pole ($\varpi$) placement approach. Through some simulation tests, the poles that provide good performance to the rotor and the dynamics of the current are$\varpi ={\left[-0.145,-0.145,-58,-270,-12,077,-12077\right]}^T$. Then, the gains are calculated and set as$k_{p_1}=0,k_{p_2}=0,k_{p_3}=327,k_{i_1}=1750,$$k_{i_2}=1750,$$k_{i_3}=15,627$, and the condition in (46) is satisfied with $\eta =1\times {10}^{-3}$. Taking that set of parameters, the Lyapunov Equation (37) is solved with $P_2$ given by

$${\mathit{P}}_2=\left[\begin{array}{cccccc}34\times {10}^{+04}& \phantom{-}0& \phantom{-}0& 28.5714& \phantom{-}0& \phantom{-}0\\ \phantom{-}0& 34\times {10}^{+05}& 36\times {10}^{-02}& \phantom{-}0& 28\times {10}^{-01}& 16\times {10}^{-06}\\ \phantom{-}0& 36\times {10}^{-02}& 12\times {10}^{-01}& \phantom{-}0& -1\times {10}^{-05}& 32\times {10}^{-06}\\ 28.5714& \phantom{-}0& \phantom{-}0& 43\times {10}^{-03}& \phantom{-}0& \phantom{-}0\\ \phantom{-}0& 28\times {10}^{-01}& -1\times {10}^{-05}& \phantom{-}0& 42\times {10}^{-03}& -2\times {10}^{-04}\\ \phantom{-}0& 16\times {10}^{-06}& 32\times {10}^{-06}& \phantom{-}0& -2\times {10}^{-04}& 76\times {10}^{-06}\end{array}\right].$$

(52)

We again apply the pole ($\varpi$) placement approach to a set of gains for the observer $\mathit{L}$. The poles are selected as $\varpi ={\left[-13,000,-13,000,-1800,-30\right]}^T$. The matrix $\mathit{L}$ is set as

$$\mathit{L}=\left[\begin{array}{cc}952& \phantom{-}0\\ 0& \phantom{-}4128\\ 0& -6.2\times {10}^{+05}\\ 0& \phantom{-}784\end{array}\right],$$

(53)

and $\mathsf{\Gamma }=diag\{210,210,0,0\}$. Consequently, the solution of (19) results in

$${\mathit{P}}_1=\left[\begin{array}{cccc}36\times {10}^{+04}& 76\times {10}^{-09}& 39\times {10}^{-11}& 19\times {10}^{-09}\\ 76\times {10}^{-09}& 37\times {10}^{+04}& 11\times {10}^{+02}& 29\times {10}^{+03}\\ 39\times {10}^{-11}& 11\times {10}^{+02}& 7.3020& 593\\ 19\times {10}^{-09}& 29\times {10}^{+03}& 593& 10\times {10}^{+04}\end{array}\right].$$

(54)

We use the high-precision emulator by Technosoft${}^{\circledR }$; we prepare the emulations to track a desired variable speed considering an unknown time-varying load torque. Here, to highlight the advantages of our proposed scheme, we contrast the performance drive system under the PI controller based on the FOC scheme (see Figure 2) (using the measurements of the mechanical variables) without load torque estimation against the performance of the drive system under the proposed sensorless control scheme, i.e., considering the controller (26) and (27) with estimation values of the mechanical variables and time-varying load torque. We show in Figure 5 the tracking behavior of the current $i_d\left(t\right)$ and the speed when a time-varying torque is applied.

To compare our results, Figure 5a,c show the traditional FOC scheme with PI, whereas the Figure 5b,d illustrate our sensorless control. In both cases, the stator currents $i_d\left(t\right)$ oscillate around the origin, Figure 5a,b. Our sensorless controller lessens the impact of time-varying load torque fluctuations and enhances the accuracy of speed tracking. Particularly where the load torque shows sinusoidal behavior, the difference between both methods stands out quite markedly, as depicted in Figure 5c,d.

Figure 6a,b illustrates the current $i_q\left(t\right)$ behavior and load torque estimation. As can be seen, the current and load torque shape are alike; this is correct since the current $i_q\left(t\right)$ is the component of the current vector that generates the electromagnetic torque in the PMSM.

We can see in Figure 6b that the estimated state ${\xi }_4\left(t\right)$ converges and remains around the load torque $T_l\left(t\right)$.

The estimation of the rotor position is presented in Figure 7.

The measured and estimated rotor position is presented in Figure 7a; zoom-in on the graph is provided to visualize the position when a load torque chance from 0.008 Nm to 0.015 Nm is applied; Figure 7b displays the calculated position error. As can be seen, there is a difference in the zero average position error in the stable state; however, it remains constant and proportional to the changes in speed throughout the test.

To evaluate our proposal, we use the mean absolute error (MAE) and the mean square error (MSE) performance indexes for comparative analysis. This helps us determine how well our proposal performs.

Table 2. Performance indexes MAE and MSE.

Error	MAE		MSE
	FOC Scheme	Proposed Scheme	FOC Scheme	Proposed Scheme
$i_d\left(t\right)$	0.0042	0.0043	0.0051	0.0053
$\omega \left(t\right)-{\omega }^{*}\left(t\right)$	2.0170	1.03336	3.3270	1.4974

shows that our robust sensorless control improves the accuracy with respect to the FOC scheme. Therefore, we can conclude that the closed-loop (29) is feasible.

Finally, the proposed sensorless scheme is subjected to resistance and inductance variations with the same load torque as in Figure 6b. The electrical parametric variations of the test in Figure 8 are to prove the robustness of the sensorless controller. Figure 8a,c show the current $i_d$(t) and the speed tracking when the resistance increases up to 30% of its nominal value—as can be seen, the robustness remains. For the PMSM used, the scheme allows us to increase the inductance up to 300% of its nominal value before weakening the controller’s performance (see Figure 8b,d).

6.3. Experimental Results

The main objective of this subsection is to evaluate the performance of the proposed sensorless control scheme in a real environment. In the experiment, we consider the same motor parameter values used in the emulation results. Additionally, in the proposed control scheme, the same controller and observation gains introduced in the previous subsection are established. However, in order to not amplify the noise and parasitic currents inherent in the PMSM windings, we adjust the observer gain $L_{42}$. The motor is subjected to a time-varying frictional load torque during this test.

Figure 9 depicts the stator current $i_d\left(t\right)$ behavior and the speed tracking performance under the effect of an unknown load torque applied to the rotor at around 0.5 s from the beginning. We can see in Figure 9a that the stator current $i_d\left(t\right)$ oscillates around the origin. Additionally, the measured and estimated rotor speeds remain stable around the reference despite time-varying frictional load torque (see Figure 9b).

Figure 10a,b illustrates the magnitude and shape of the current $i_q\left(t\right)$ and unknown frictional time-varying load torque. The similarity of both signals indicates that the load torque is well-estimated. In this figure, the estimation indicates that the PMSM is working around 50% of its maximum rated load ($T_L^{max}=30\times {10}^{-3}$ Nm).

Figure 11 presents the measured and estimated rotor position. In Figure 11a, with the zoom-in on the graph, we visualize the position when the estimated load torque peaks at around 0.024 Nm, whereas, Figure 11b displays the position error.

The emulations and experiments show that the extended state observer estimates the position, speed, and load torque well. On the controller side, the PI-compensated shows a satisfactory performance at the high-speed trajectory. We enter different load torque profiles and parametric variations into the PMSM to verify the robustness of the proposed scheme. The performance achieved does not harm the dynamics of the system. However, at low speeds (<50 rad/s), the measured (encoder) and estimated speed begin to show oscillatory behavior because, at low speeds, a small back-EMF appears in the stator. The current sensor’s limited features (precision and resolution) also affect the electrical measurement performance; therefore, it cannot estimate the states adequately.

7. Conclusions

We proposed an extended state observer interconnected with a PI-compensated controller for solving the sensorless high-speed tracking problem for surface-mount PMSMs, under unknown time-varying load torques. The electrical variables were the only measurements for feedback. Here, an extended state observer estimates the mechanical variables and the time-varying load torques; then, the estimations feedback to the system instead of the mechanical sensor signals. Hence, the proposed PI-compensated controller regulates the stator current $i_d\left(t\right)$ around zero and drives the speed rotor to track a desired speed reference. Furthermore, the proposed sensorless controller attenuates the effects of unknown time-varying load torques.

Through Lyapunov stability analysis, we provided the sufficient conditions to guarantee that closed-loop trajectories are uniformly bounded solutions from the initial time ($t\ge 0$). Since rigorous closed-loop stability analyses of sensorless control with extended-state observers are still lacking in the literature, we consider that our approach represents a novel contribution. Moreover, we deal with external perturbations by selecting an appropriate Lyapunov candidate function considering the physical operation domain of the PMSM. We use the positive invariant sets of the PMSM to drive the trajectories of the closed-loop system around the origin, even when the motor is affected by load torque variations. The emulations and experiments undertaken corroborate the effectiveness of the proposed observer-based controller scheme.

Despite the difference between the emulations and experiments with respect to the speed tracking errors, the rotor speed is well-regulated along the reference trajectory. Also, the load torque and the rotor position are estimated well. The discrepancies between the emulations and the experiments may be associated with unmodeled dynamics, parametric variations, and the resolution and noise of the current sensors. In future work, we anticipate presenting different methods for gain tuning and addressing the parametric variations in experiments.