Robust Mechanism for Speed and Position Observers of Electrical Machines

Abstract

In the sensorless control system, the rotor speed or position is not measured but reconstructed in the dedicated observer structure. The observer structure is based on the mathematical model of an electrical machine. This model is often determined in the space vector form by using the stator/rotor flux vector and stator/rotor current vector components. During the machine works, there exist working points in which the observer can be unstable or its accuracy is unsatisfactory. In order to increase the observer system stability, the Lyapunov theorem should be satisfied. Using this, the observer system’s proper stabilizing function can be determined. However, in many cases, this procedure is not sufficient and in close to an unstable region properties of the speed observer structure are very poor—the estimation errors have values exceeding 5%, which causes loss of synchronization in case of synchronous machines and errors in the values of electromagnetic torque or stator/rotor fluxes. In order to prevent this undesirable phenomenon, additional laws of estimation should be introduced to the speed or position estimation mechanism, which is proposed in this chapter. This mechanism is named in this chapter “robust” because during the machine works, it increases significantly the properties of the whole sensorless control system, minimizing the speed or position estimation errors almost to zero, close to the unstable region (small rotor speed with the regenerating machine mode or close to synchronous of rotor speed in case of the doubly fed generator). The proposed robust mechanism has been tested by using simulation and experimental investigations prepared for: the squirrel-cage induction machine, permanent magnet synchronous machine, and doubly fed induction generator.

1. Introduction

In sensorless control of an electrical machine, the rotor speed value or rotor position is not measured but reconstructed by an observer structure. In the literature, methods of reproducing the rotor speed or rotor position can be divided into three [1]: algorithmic, neural network, and physical methods. The most popular is an algorithmic method in which the observer structure is based on the mathematical model of an electrical machine. This group includes state full and reduced-order observers, [2], the adaptive full-order observer (AFO), [3], Kalman filters, [4], model reference adaptive observers MRAS, [5], sliding mode observers, [6], and backstepping, [6]. The other approach to the estimation of the state variables is to extend the model of a machine with an additional state variable—an auxiliary state, [7]. The rotor speed value in these observers can be reconstructed from the classical adaptation law by using the proportional-integral controller (PI), [1, 7, 8]. The rotor position value can be obtained by using the integration of the rotor speed value in the same integration step, [7, 8]. Other approach to the reconstruction of the rotor speed value is the non-adaptive method. The rotor speed value is obtained by using the suitable algebraic transformation of the estimated state variables, [5, 6].

The main problem in the sensorless control systems is the stability of the observer structure in the wide changes of working points of the machine, [8, 9]: from zero to nominal rotor speed, under load torque injections, and for regenerating mode. Stabilization of the observer structure under regenerating mode and low speed of the induction machine, IM, was studied in many papers, [9, 10, 11]. For this case, the frequency of stator voltage is almost zero, and there exist unstable poles of the observer system, [8, 10]. Similarly, the problem occurs for the permanent or interior permanent magnet synchronous machines (PMSM/IPMSM) during the zero rotor speed; while the electromagnetic force (EMF) is not generated, [12, 13, 14]. To overcome this problem, a different value of stator current or voltage (high [13] or low [14] frequency) is injected into the stator voltage from an inverter.

A robust mechanism for the rotor speed estimation is proposed in this chapter. The proposed approach is suitable for the speed observer structures, which are based on a mathematical model of an electrical machine (algorithmic) in the space vector form. In Section 2, the mathematical model of an electrical machine is considered in the general form for the nonlinear class of systems. In Section 3, the application to IM is shown. In Section 4, the speed observer of IPMSM with the robust mechanism is proposed. In Section 5, the robust mechanism for the rotor speed and position estimation is adapted to the observer structure of DFIG.

All the theoretical derivations are confirmed by using simulation and experimental investigations.

2. Design procedure of the speed and position observer

One of the most popular design procedures for the speed observer of an electrical machine is based on the second theorem of the Lyapunov of asymptotical stability of the system in the general form

$$\dot{\mathrm{x}}=\mathrm{Ax}+\mathrm{Bu},$$E1

$$\mathrm{y}=\mathrm{Cx},$$E2

where is assumed that A, B, and C are the matrixes that including the system parameters, x is the vector of state variables, and u is the vector of controls.

Considering only (1) the model can be rewritten to the vector components form in which (αβ) is the stationary reference frame and the index k means the number of the state variables in the system defined in (1)–(2)

$${\dot{x}}_{\mathit{k\alpha }}=a_k{x}_{\mathit{k\alpha }}+\dots +b_k{u}_{\mathit{k\alpha }},$$E3

$${\dot{x}}_{\mathit{k\beta }}=a_k{x}_{\mathit{k\beta }}+\dots +b_k{u}_{\mathit{k\beta }}.$$E4

If the system (3)–(4) will be connected to the rotating reference frame, then the differential equations have the form

$${\dot{x}}_{\mathit{kd}}=a_k{x}_{\mathit{kd}}+{\omega }_{\mathit{dq}}{x}_{\mathit{kq}}+\dots +b_k{u}_{\mathit{kd}},$$E5

$${\dot{x}}_{\mathit{kq}}=a_k{x}_{\mathit{kq}}-{\omega }_{\mathit{dq}}{x}_{\mathit{kd}}+\dots +b_k{u}_{\mathit{kq}},$$E6

where ${\omega }_{\mathit{dq}}$ is the angular speed of the (d-q) reference frame, and ($u_{\mathit{kd}}$, $u_{\mathit{kq}}$) are the controls defined in (d-q).

It can be assumed that the system model belongs to the operation domain D defined by the set of values

$$\mathrm{D}=\left\{\mathrm{x}\in {\mathrm{R}}^k,\left|x_{\mathit{kd}}\right|\le x_{\mathit{kd}}^{\max },\left|x_{\mathit{kq}}\right|\le x_{\mathit{kq}}^{\max },{\omega }_{\mathit{dq}}\le {\omega }_{\mathit{dq}}^{\max }\right\},$$E7

where

$x_{\mathit{kd}}^{\max }$, $x_{\mathit{kq}}^{\max }$, ${\omega }_{\mathit{dq}}^{\max }$ are the maximum values for the state variables, and the parameters in the system a_k, b_k have known, constant, and bounded values.

Assumption 1. For the system (5)–(6) in which the ${\omega }_{\mathit{dq}}$ is treated as the parameter, it is possible to reconstruct its value by using the adaptive and non-adaptive approaches and state of variables x_k. Moreover, the controls ($u_{\mathit{kd}}$, $u_{\mathit{kq}}$) satisfied the persistent of excitation condition [14].

The first step in the procedure of design of the observer structure is to stabilize the observer for the system (5)–(6). The observer structure has the following form:

$${\dot{\widehat{x}}}_{\mathit{kd}}=a_k{\widehat{x}}_{\mathit{kd}}+{\widehat{\omega }}_{\mathit{dq}}{\widehat{x}}_{\mathit{kq}}+\dots +b_k{u}_{\mathit{kd}}+v_d,$$E8

$${\dot{\widehat{x}}}_{\mathit{kq}}=a_k{\widehat{x}}_{\mathit{kq}}-{\widehat{\omega }}_{\mathit{dq}}{\widehat{x}}_{\mathit{kd}}+\dots +b_k{u}_{\mathit{kq}}+v_q,$$E9

where the estimated values are marked by “^”, and v_d, v_q are the inputs to the observer (8)–(9), which stabilize the system.

The estimation errors between estimated (8)–(9) and real/measured values (5)–(6) are expressed by

$${\tilde{x}}_{\mathit{kd},q}={\widehat{x}}_{\mathit{kd},q}-x_{\mathit{kd},q},$$E10

$${\tilde{\omega }}_{\mathit{dq}}={\widehat{\omega }}_{\mathit{dq}}-{\omega }_{\mathit{dq}}.$$E11

For the above-defined estimation errors, it is possible to determine the model of estimation errors, which form is as follows:

$${\dot{\tilde{x}}}_{\mathit{kd}}=a_k{\tilde{x}}_{\mathit{kd}}+{\widehat{\omega }}_{\mathit{dq}}{\tilde{x}}_{\mathit{kq}}+{\tilde{\omega }}_{\mathit{dq}}{\widehat{x}}_{\mathit{kq}}-{\tilde{\omega }}_{\mathit{dq}}{\tilde{x}}_{\mathit{kq}}+v_d,$$E12

$${\dot{\tilde{x}}}_{\mathit{kq}}=a_k{\tilde{x}}_{\mathit{kq}}-{\widehat{\omega }}_{\mathit{dq}}{\tilde{x}}_{\mathit{kd}}-{\tilde{\omega }}_{\mathit{dq}}{\widehat{x}}_{\mathit{kd}}+{\tilde{\omega }}_{\mathit{dq}}{\tilde{x}}_{\mathit{kd}}+v_q.$$E13

The next step is to determine the form of stabilizing functions, which stabilize the observer structure (8)–(9). By using the Lyapunov theorem, the observer structure will be stable if the candidate of the Lyapunov function

$$V=0.5\left({\tilde{x}}_{\mathit{kd}}^2+{\tilde{x}}_{\mathit{kq}}^2\right)+V_1\ge 0,$$E14

is positively defined, and V₁ > 0 has the form

$$V_1={\gamma }^{-1}{\tilde{\omega }}_{\mathit{dq}}^2.$$E15

Derivative of Lyapunov function (14) must be negative determined, therefore using (12)–(13), its form is determined

$$\begin{array}{rcl}\dot{V}& =& {\tilde{x}}_{\mathit{kd}}\left(a_k{\tilde{x}}_{\mathit{kd}}+{\widehat{\omega }}_{\mathit{dq}}{\tilde{x}}_{\mathit{kq}}+v_d\right)+{\tilde{x}}_{\mathit{kq}}\left(a_k{\tilde{x}}_{\mathit{kq}}-{\widehat{\omega }}_{\mathit{dq}}{\tilde{x}}_{\mathit{kd}}+v_q\right)\\ & & +{\tilde{\omega }}_{\mathit{dq}}\left({\gamma }^{-1}{\dot{\tilde{\omega }}}_{\mathit{dq}}+{\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kd}}-{\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kq}}\right)\le 0.\end{array}$$E16

The observer structure will be asymptotically stable if the Lyapunov theorem is satisfied and the stabilizing functions are chosen

$$v_d=-a_k{\tilde{x}}_{\mathit{kd}},$$E17

$$v_q=-a_k{\tilde{x}}_{\mathit{kq}},$$E18

then derivative (16) has the form

$$\dot{V}={\tilde{\omega }}_{\mathit{dq}}\left({\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kd}}-{\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kq}}\right)\le 0.$$E19

To satisfy (19), the value of the parameter ${\tilde{\omega }}_{\mathit{dq}}$ should be determined by

$${\dot{\tilde{\omega }}}_{\mathit{dq}}=-\gamma \left({\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kd}}-{\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kq}}\right),$$E20

where

γ > 0 is the tuning gain.

For (20), the derivative of the Lyapunov function is always smaller than zero $\dot{V}<0$_, and the Lyapunov condition is satisfied.

2.1 Adaptive estimation of parameter ${\omega }_{\mathit{dq}}$

The estimated value of the parameter ${\widehat{\omega }}_{\mathit{dq}}$ can be determined from (20) under assumption that the derivative of the real value is constant in time ${\dot{\omega }}_{\mathit{dq}}$ and equal to zero

$${\dot{\widehat{\omega }}}_{\mathit{dq}}=-\gamma \left({\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kd}}-{\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kq}}\right).$$E21

The above estimation law is named in the literature [15] as the classical adaptation law.

Remark 1. The assumption ${\dot{\omega }}_{\mathit{dq}}=0$ is not desirable for the nonlinear system, in which the highest accuracy of estimation is needed. For ${\dot{\omega }}_{\mathit{dq}}\ne 0$ and ${\omega }_{\mathit{dq}}={\widehat{\omega }}_{\mathit{dq}}-{\tilde{\omega }}_{\mathit{dq}}$, after substitution (17)–(18) to (16), the derivative of the Lyapunov function has the following form:

$$\dot{V}={\tilde{\omega }}_{\mathit{dq}}\left(-\frac{1}{\gamma }\left({\dot{\widehat{\omega }}}_{\mathit{dq}}-{\dot{\tilde{\omega }}}_{\mathit{dq}}\right)\right)\le 0.$$E22

After substitution (21) to (22), the update form of derivative of the Lyapunov function is achieved

$$\dot{V}=-{\tilde{\omega }}_{\mathit{dq}}\left(-\left({\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kd}}-{\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kq}}\right)+\frac{1}{\gamma }{\dot{\tilde{\omega }}}_{\mathit{dq}}\right)\le 0.$$E23

It is easy to check that in (23) the dependence in the internal bracket $\left({\widehat{\mathrm{x}}}_k\times {\tilde{\mathrm{x}}}_k\right)={\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kd}}-{\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kq}}$ means the cross-product of the pair of two vectors that occur in the observer system. The cross-product for can be determined by using Lagrange’s identity [16].

Assumption 2. Considering the pair of vectors $\left({\widehat{\mathrm{x}}}_k,{\tilde{\mathrm{x}}}_k\right)$ defined in the observer system (8)–(9) and for the estimation errors (10)–(13), there exists Lagrange’s identity [16], which has the following form:

$${\left({\widehat{\mathrm{x}}}_k\times {\tilde{\mathrm{x}}}_k\right)}^2\equiv {\left|{\widehat{\mathrm{x}}}_k\right|}^2{\left|{\tilde{\mathrm{x}}}_k\right|}^2-{\left({\widehat{\mathrm{x}}}_k\cdot {\tilde{\mathrm{x}}}_k\right)}^2.$$E24

Considering the vector components defined in (d-q) reference frame (24) can be rewritten as

$${\tilde{x}}_{\mathit{kd}}{\widehat{x}}_{\mathit{kq}}-{\tilde{x}}_{\mathit{kq}}{\widehat{x}}_{\mathit{kd}}=\sqrt{\left({\widehat{x}}_{\mathit{kd}}^2+{\widehat{x}}_{\mathit{kq}}^2\right)\left({\tilde{x}}_{\mathit{kd}}^2+{\tilde{x}}_{\mathit{kq}}^2\right)-{\left({\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kd}}+{\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kq}}\right)}^2}.$$E25

Substituting (25) to (23), the derivative of the Lyapunov function has the following form:

$$\dot{V}={\tilde{\omega }}_{\mathit{dq}}\left(\sqrt{\left({\widehat{x}}_{\mathit{kd}}^2+{\widehat{x}}_{\mathit{kq}}^2\right)\left({\tilde{x}}_{\mathit{kd}}^2+{\tilde{x}}_{\mathit{kq}}^2\right)-{\left({\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kd}}+{\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kq}}\right)}^2}-\frac{1}{\gamma }{\dot{\tilde{\omega }}}_{\mathit{dq}}\right)\le 0.$$E26

The Lyapunov theorem is satisfied if

$${\dot{\tilde{\omega }}}_{\mathit{dq}}=\gamma \sqrt{\left({\widehat{x}}_{\mathit{kd}}^2+{\widehat{x}}_{\mathit{kq}}^2\right)\left({\tilde{x}}_{\mathit{kd}}^2+{\tilde{x}}_{\mathit{kq}}^2\right)-{\left({\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kd}}+{\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kq}}\right)}^2},$$E27

where $\left(\left({\widehat{x}}_{\mathit{kd}}^2+{\widehat{x}}_{\mathit{kq}}^2\right)\left({\tilde{x}}_{\mathit{kd}}^2+{\tilde{x}}_{\mathit{kq}}^2\right)-{\left({\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kd}}+{\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kq}}\right)}^2\right)\ge 0$.

Remark 2: To satisfy the above condition, the negative sign-in (27) must be changed to positive. The form (27) is determined as follows:

$${\dot{\tilde{\omega }}}_{\mathit{dq}}=\gamma \sqrt{\left({\widehat{x}}_{\mathit{kd}}^2+{\widehat{x}}_{\mathit{kq}}^2\right)\left({\tilde{x}}_{\mathit{kd}}^2+{\tilde{x}}_{\mathit{kq}}^2\right)+{\left({\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kd}}+{\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kq}}\right)}^2}.$$E28

Dependence (28) can be used to find the updated form of the classical estimation law (21). It provides an improvement to the stability range of the observer system.

Assumption 3. The expression (21) has the form of an open integrator. There is a lack of additional stabilizing function, interconnecting the observer system. To improve the stability range of the observer system, it is proposed to introduce additional input s_ω

$${\dot{\widehat{\omega }}}_{\mathit{dq}}=-\gamma \left({\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kd}}-{\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kq}}\right)+s_{\omega }.$$E29

To stabilize the integrator (29), the stabilization function s_ω should be $s_{\omega }\equiv {\dot{\tilde{\omega }}}_{\mathit{dq}}$.

The updated estimation law has the following form:

$${\dot{\widehat{\omega }}}_{\mathit{dq}}=-\gamma \left({\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kd}}-{\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kq}}+{\gamma }_1{k}_f\sqrt{\left({\widehat{x}}_{\mathit{kd}}^2+{\widehat{x}}_{\mathit{kq}}^2\right)\left({\tilde{x}}_{\mathit{kd}}^2+{\tilde{x}}_{\mathit{kq}}^2\right)+{\left({\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kd}}+{\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kq}}\right)}^2}\right),$$E30

where γ₁ is the additional gain, and $k_f=\mathit{sign}\left({\widehat{\omega }}_{\mathit{dq}}\right)$ is the sign of the estimated parameter.

Remark 3. Under the assumption that in (30) $\left({\widehat{x}}_{\mathit{kd}}^2+{\widehat{x}}_{\mathit{kq}}^2\right)\left({\tilde{x}}_{\mathit{kd}}^2+{\tilde{x}}_{\mathit{kq}}^2\right)\ll {\left({\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kd}}+{\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kq}}\right)}^2$ the update estimation law can be simplified to the following form

$${\dot{\widehat{\omega }}}_{\mathit{dq}}=-\gamma \left({\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kd}}-{\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kq}}+{\gamma }_1{k}_f{s}_{\mathit{\omega f}}\right),$$E31

where $s_{\omega }={\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kd}}+{\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kq}}$, and s_ωf is their filtered value by using a low-pass filter LPF (to avoid the algebraic loop).

In (31), there is the cross and scalar product $\left({\widehat{\mathrm{x}}}_k·{\tilde{\mathrm{x}}}_k\right)={\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kd}}+{\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kq}}$ of two vectors. It is worth noticing that for the perpendicular vectors, the scalar product is equal to zero; however, in other cases, it is different from zero and additionally stabilizes the estimation law.

2.2 Non-adaptive estimation of parameter ${\omega }_{\mathit{dq}}$

In the previous section, the parameter ${\omega }_{\mathit{dq}}$ was reconstructed from the adaptive law. However, this value can be estimated non-adaptively. Under the assumption of the steady-state for $a_k\approx 1$, ${\tilde{\omega }}_{\mathit{dq}}\approx 0$ and v_{d, q} = 0, from the model of estimation error, the following approximations can be achieved:

$${\tilde{x}}_{\mathit{kd}}\approx {\widehat{\omega }}_{\mathit{dq}}{\widehat{x}}_{\mathit{kq}},$$E32

$${\tilde{x}}_{\mathit{kq}}\approx -{\widehat{\omega }}_{\mathit{dq}}{\widehat{x}}_{\mathit{kd}},$$E33

for whose the following relationships are satisfied

$${\tilde{x}}_{\mathit{kd}}^2+{\tilde{x}}_{\mathit{kq}}^2={\widehat{\omega }}_{\mathit{dq}}^2\left({\widehat{x}}_{\mathit{kd}}^2+{\widehat{x}}_{\mathit{kq}}^2\right),$$E34

$${\widehat{\omega }}_{\mathit{dq}}=\frac{{\tilde{x}}_{\mathit{kd}}{\widehat{x}}_{\mathit{kq}}-{\tilde{x}}_{\mathit{kq}}{\widehat{x}}_{\mathit{kd}}}{{\widehat{x}}_{\mathit{kd}}^2+{\widehat{x}}_{\mathit{kq}}^2},$$E35

where ${\widehat{x}}_{\mathit{kd}}^2+{\widehat{x}}_{\mathit{kq}}^2\ne 0$.

Substituting (34)–(35) to (24), the following quadratic function is obtained:

$$\begin{array}{ll}f\left({\widehat{\omega }}_{\mathit{dq}}\right)=& -\left({\widehat{x}}_{\mathit{kd}}^2+{\widehat{x}}_{\mathit{kq}}^2\right)\left({\widehat{x}}_{\mathit{kd}}^2+{\widehat{x}}_{\mathit{kq}}^2\right){\widehat{\omega }}_{\mathit{dq}}^2+{\widehat{\omega }}_{\mathit{dq}}\left({\widehat{x}}_{\mathit{kd}}^2+{\widehat{x}}_{\mathit{kq}}^2\right)\left({\tilde{x}}_{\mathit{kd}}{\widehat{x}}_{\mathit{kq}}-{\tilde{x}}_{\mathit{kq}}{\widehat{x}}_{\mathit{kd}}\right)+\\ & ({\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kd}}{+{\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kq}})}^2.\end{array}$$E36

One of the roots of the function $f\left({\widehat{\omega }}_{\mathit{dq}}\right)$ can be calculated as follows:

$${\widehat{\omega }}_{\mathit{dq}}=\frac{{\tilde{x}}_{\mathit{kd}}{\widehat{x}}_{\mathit{kq}}-{\tilde{x}}_{\mathit{kq}}{\widehat{x}}_{\mathit{kd}}+k_f\sqrt{\left({\left({\tilde{x}}_{\mathit{kd}}{\widehat{x}}_{\mathit{kq}}-{\tilde{x}}_{\mathit{kq}}{\widehat{x}}_{\mathit{kd}}\right)}^2+4{\gamma }_1{\left({\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kd}}+{\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kq}}\right)}^2\right)}}{2\left({\widehat{x}}_{\mathit{kd}}^2+{\widehat{x}}_{\mathit{kq}}^2\right)},$$E37

where γ₁ is the additional tuning gain and $k_f=\mathit{sign}\left({\widehat{\omega }}_{\mathit{dq}}\right)$.

2.3 Practical stability of the observer system

The practical stability of the observer system was proposed in [17, 18]. Based on the theorem of practical stability and considering that the system belongs to domain D defined in (7), the observer structure will be practical stable in the Lyapunov function derivative is

$$\dot{V}={\delta }_1\left|{\tilde{x}}_{\mathit{kd}}\right|+{\delta }_2\left|{\tilde{x}}_{\mathit{kq}}\right|+{\delta }_c\left|{\tilde{\omega }}_{\mathit{dq}}\right|\le -\mathit{\mu V}+\kappa ,$$E38

where (${\delta }_1$, ${\delta }_2$,${\delta }_c$) > 0 and ${\tilde{x}}_{\mathit{kd}}\le {\epsilon }_1$, ${\tilde{x}}_{\mathit{kd}}\le {\epsilon }_2$,${\tilde{\omega }}_r\le {\epsilon }_3$, ${\epsilon }_1{,}_2{,}_3\ll 1$ are sufficient small real numbers ${\epsilon }_1{,}_2{,}_3>0$ and where

$${\gamma }_1=max\left\{\frac{\left({\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kd}}-{\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kq}}\right)}{\sqrt{\left({\widehat{x}}_{\mathit{kd}}^2+{\widehat{x}}_{\mathit{kq}}^2\right)\left({\tilde{x}}_{\mathit{kd}}^2+{\tilde{x}}_{\mathit{kq}}^2\right)+{\left({\widehat{x}}_{\mathit{kd}}{\tilde{x}}_{\mathit{kd}}+{\widehat{x}}_{\mathit{kq}}{\tilde{x}}_{\mathit{kq}}\right)}^2}}+{\delta }_c\right\},$$E39

and

$$\mu =min\left({\delta }_1-\frac{1}{2{\xi }_1^2},{\delta }_2-\frac{1}{2{\xi }_2^2},\sqrt{2}{\delta }_c\right),\kappa =0.5\left({\xi }_1^2{\eta }_1^2+{\xi }_2^2{\eta }_2^2\right)\forall {\xi }_i\in \left(0,1\right),i=1,2$$E40

Hence, (38) implies the convergence of estimated vector values to their real, in finite time, noted as t₁. The reconstructed parameter ${\widehat{\omega }}_{\mathit{dq}}$ converges exponentially to real ${\omega }_{\mathit{dq}}$ in finite time t > t₂ > t₁. This condition is satisfied for ideal and constant parameters of the system (3)–(4). According to [17, 18], the tracking errors converge to the ball of radius $\kappa /\mu$. This radius can be decreased by the properly choosing tuning gains of the observer system (8)–(9).

2.4 Conclusion

Presented in Section 2 is the design of the observer structure generalized to the class of system (3)–(4) in the space vector form. The form of the system (3)–(4) was in α-β stationary reference frame. It has been appropriately transformed by using Clark’s transformation to the rotational reference frame d-q. In system (5)–(6), there exists the parameter, which is the angular speed of the reference frame in d-q. The system (5)–(6) has been properly written with a separate parameter and has a similar form to an AC electrical machines models presented in the next sections. Therefore, the proposed procedure in Section 2 for designing the observer structure can be directly adapted to the sensorless control system of an AC electrical machine. The proposed solution is based on the classical adaptation law of estimation and non-adaptive. According to the literature, [1, 2, 6, 7, 8, 9, 10, 11, 12], the rotor speed whose value is estimated only from the classical law of adaptation and used to tune the observer structure (8)–(9) can lead to instability during the regenerating mode and low speed of the electrical machine. There exist positive poles of the observer structure for which it is unstable. The problem in the classical law of adaptation is the open form of the integrator (21) from which the value of rotor speed is estimated (in the case of an electrical machine). Therefore, in Section 2, the additional stabilization function is introduced also to the classical law of estimation. The proposed stabilization function is based on Lagrange’s identity of the pair of vectors in the observer system. The form of additional stabilization law contains the scalar product and the length of the vectors. However, after the simplification shown in Remark 3, it can be assumed that the stabilization function is proportional to the scalar product of the chosen vectors that were presented in [6].

The proposed theoretical issues in Section 2 will be confirmed in the simulation and experimental results for the squirrel-cage induction machine and interior permanent magnet synchronous machine. Also, it can be extended to estimation of the state variables of the doubly fed induction generator (DFIG).

3. The speed observer structures of the squirrel-cage induction machine

The AFO speed observer structure of IM is proposed in this section. The rotor speed will be estimated by using two approaches: from the adaptive estimation law and non-adaptively.

Considering the mathematical model of the induction machine presented in [5, 6], for the pair of vectors $\left({\mathrm{\psi }}_r,{\mathrm{i}}_s\right)$ according to (8)–(9), the conventional AFO observer structure can be determined in the form

$$\frac{d{\widehat{i}}_{\mathit{s\alpha }}}{\mathit{d\tau }}=a_1{\widehat{i}}_{\mathit{s\alpha }}+a_2{\widehat{\psi }}_{\mathit{r\alpha }}+a_3{\widehat{\omega }}_r{\widehat{\psi }}_{\mathit{r\beta }}+a_4{u}_{\mathit{s\alpha }}+v_{\alpha },$$E41

$$\frac{d{\widehat{i}}_{\mathit{s\beta }}}{\mathit{d\tau }}=a_1{\widehat{i}}_{\mathit{s\beta }}+a_2{\widehat{\psi }}_{\mathit{r\beta }}-a_3{\widehat{\omega }}_r{\widehat{\psi }}_{\mathit{r\alpha }}+a_4{u}_{\mathit{s\beta }}+v_{\beta },$$E42

$$\frac{d{\widehat{\psi }}_{\mathit{r\alpha }}}{\mathit{d\tau }}=a_5{\widehat{\psi }}_{\mathit{r\alpha }}-{\widehat{\omega }}_r{\widehat{\psi }}_{\mathit{r\beta }}+a_6{\widehat{i}}_{\mathit{s\alpha }}+v_{\mathit{\psi \alpha }},$$E43

$$\frac{d{\widehat{\psi }}_{\mathit{r\beta }}}{\mathit{d\tau }}=a_5{\widehat{\psi }}_{\mathit{r\beta }}+{\widehat{\omega }}_r{\widehat{\psi }}_{\mathit{r\alpha }}+a_6{\widehat{i}}_{\mathit{s\beta }}+v_{\mathit{\psi \beta }},$$E44

where the estimated values are marked by “^”.

It is assumed that the stator current vector ${\widehat{i}}_{\mathit{s\alpha }}{,}_{\beta }$, rotor flux vector ${\widehat{\psi }}_{\mathit{r\alpha }}{,}_{\beta }$ components, and rotor speed ${\widehat{\omega }}_r$ are estimated in the observer structure (41)–(44), v_α,β, and v_ψα,β are stabilizing functions introduced to the structure. The values $i_{\mathit{s\alpha }}{,}_{\beta }$ are available in measurement and $u_{\mathit{s\alpha }}{,}_{\beta }$ are treated as the known variables (from the control system structure of the machine). The machine parameters are included in

$a_1=-\frac{R_s{L}_r^2+R_r{L}_m^2}{L_r{w}_{\sigma }}$, $a_2=\frac{R_r{L}_m}{L_r{w}_{\sigma }}$, $a_3=\frac{L_m}{w_{\sigma }}$, $a_4=\frac{L_r}{w_{\sigma }}$, $a_5=-\frac{R_r}{L_r}$, $a_6=\frac{R_r{L}_m}{L_r}$,

$$w_{\sigma }=L_r{L}_s-L_m^2.$$E45

The estimation errors for the observer system (41)–(44) are defined.

$${\tilde{\omega }}_r={\widehat{\omega }}_r-{\omega }_r,,,{\tilde{\psi }}_{\mathit{r\alpha },\beta }={\widehat{\psi }}_{\mathit{r\alpha },\beta }-{\psi }_{\mathit{r\alpha },\beta }{\tilde{i}}_{\mathit{s\alpha },\beta }={\widehat{i}}_{\mathit{s\alpha },\beta }-i_{\mathit{s\alpha },\beta }$$E46

where it is assumed that components $i_{\mathit{s\alpha }}{,}_{\beta }$, ${\psi }_{\mathit{r\alpha }}{,}_{\beta }$, ${\omega }_r$ are the real values.

The rotor speed value ${\widehat{\omega }}_r$ will be reconstructed adaptively and non-adaptively by using the observer structure (41)–(44) and based on the measurements $i_{\mathit{s\alpha }}{,}_{\beta }$, and $u_{\mathit{s\alpha }}{,}_{\beta }$.

Using the design procedure presented in Section 2, the model of estimation errors is as follows:

$$\frac{d{\tilde{i}}_{\mathit{s\alpha }}}{\mathit{d\tau }}=a_1{\tilde{i}}_{\mathit{s\alpha }}+a_2{\tilde{\psi }}_{\mathit{r\alpha }}+a_3\left({\tilde{\omega }}_r{\widehat{\psi }}_{\mathit{r\beta }}+{\widehat{\omega }}_r{\tilde{\psi }}_{\mathit{r\beta }}-{\tilde{\omega }}_r{\tilde{\psi }}_{\mathit{r\beta }}\right)+v_{\alpha },$$E47

$$\frac{d{\tilde{i}}_{\mathit{s\beta }}}{\mathit{d\tau }}=a_1{\tilde{i}}_{\mathit{s\beta }}+a_2{\tilde{\psi }}_{\mathit{r\beta }}-a_3\left({\tilde{\omega }}_r{\widehat{\psi }}_{\mathit{r\alpha }}+{\widehat{\omega }}_r{\tilde{\psi }}_{\mathit{r\alpha }}-{\tilde{\omega }}_r{\tilde{\psi }}_{\mathit{r\alpha }}\right)+v_{\beta },$$E48

$$\frac{d{\tilde{\psi }}_{\mathit{r\alpha }}}{\mathit{d\tau }}=a_5{\tilde{\psi }}_{\mathit{r\alpha }}-\left({\tilde{\omega }}_r{\widehat{\psi }}_{\mathit{r\beta }}+{\widehat{\omega }}_r{\tilde{\psi }}_{\mathit{r\beta }}-{\tilde{\omega }}_r{\tilde{\psi }}_{\mathit{r\beta }}\right)+a_6{\tilde{i}}_{\mathit{s\alpha }}+v_{\mathit{\psi \alpha }},$$E49

$$\frac{d{\tilde{\psi }}_{\mathit{r\beta }}}{\mathit{d\tau }}=a_5{\tilde{\psi }}_{\mathit{r\beta }}+\left({\tilde{\omega }}_r{\widehat{\psi }}_{\mathit{r\alpha }}+{\widehat{\omega }}_r{\tilde{\psi }}_{\mathit{r\alpha }}-{\tilde{\omega }}_r{\tilde{\psi }}_{\mathit{r\alpha }}\right)+a_6{\tilde{i}}_{\mathit{s\beta }}+v_{\mathit{\psi \beta }}.$$E50

The Lyapunov function defined for the estimation errors has the form

$$V=\frac{1}{2}\left({\tilde{i}}_{\mathit{s\alpha }}^2+{\tilde{i}}_{\mathit{s\beta }}^2+{\tilde{\psi }}_{\mathit{r\alpha }}^2+{\tilde{\psi }}_{\mathit{r\beta }}^2\right)+V_1>0,$$E51

where for the non-adaptive speed estimation $V_1=0$, and in (47)–(50), ${\tilde{\omega }}_r=0$ under the assumption (32)–(33), for the case of adaptive law of estimation $V_1=\frac{1}{\gamma }{\tilde{\omega }}_r^2$.

The derivative of the Lyapunov function will be negatively determined $\dot{V}<0$ if the stabilizing functions are chosen.

$$v_{\alpha }=-c_{\alpha }{\tilde{i}}_{\mathit{s\alpha }},v_{\beta }=-c_{\alpha }{\tilde{i}}_{\mathit{s\beta }}$$E52

$$v_{\mathit{\psi \alpha }}=-c_{\psi 1}{\tilde{i}}_{\mathit{s\alpha }}+c_{\psi }{\widehat{\omega }}_r{\tilde{i}}_{\mathit{s\beta }},v_{\mathit{\psi \beta }}=-c_{\psi 1}{\tilde{i}}_{\mathit{s\beta }}-c_{\psi }{\widehat{\omega }}_r{\tilde{i}}_{\mathit{s\alpha }}$$E53

where it is assumed ($c_{\alpha }=f\left(a_1\right)>0$) as well as $c_{\psi }=f\left(a_3\right)>0$, whereas $c_{\psi 1}\ge 0$, a₅ < 0.

The rotor speed value can be estimated directly from the adaptive estimation law (20) presented in Section 2.1, considering the pair of vectors $\left({\widehat{\mathrm{x}}}_k,{\tilde{\mathrm{x}}}_k\right)\equiv \left({\widehat{\mathrm{\psi }}}_r,{\tilde{\mathrm{i}}}_s\right)$

$${\dot{\widehat{\omega }}}_r=-\gamma \left({\tilde{i}}_{\mathit{s\alpha }}{\widehat{\psi }}_{\mathit{r\beta }}-{\tilde{i}}_{\mathit{s\beta }}{\widehat{\psi }}_{\mathit{r\alpha }}+{\gamma }_1{k}_f{\widehat{s}}_{\omega }\right),$$E54

where the robust term ${\widehat{s}}_{\omega }$ is given from

$${\widehat{s}}_{\omega }=\sqrt{\left({\widehat{\psi }}_{\mathit{r\alpha }}^2+{\widehat{\psi }}_{\mathit{r\beta }}^2\right)\left({\tilde{i}}_{\mathit{s\alpha }}^2+{\tilde{i}}_{\mathit{s\beta }}^2\right)+{\left({\widehat{\psi }}_{\mathit{r\alpha }}{\tilde{i}}_{\mathit{s\alpha }}+{\widehat{\psi }}_{\mathit{r\beta }}{\tilde{i}}_{\mathit{s\beta }}\right)}^2},$$E55

and γ₁ > 0 is the additional gain, $k_f=\mathit{sign}\left({\widehat{\omega }}_r\right)$.

Considering the non-adaptive scheme for rotor speed estimation presented Section 2.2. For the pair of vectors $\left({\widehat{\mathrm{x}}}_k,{\tilde{\mathrm{x}}}_k\right)\equiv \left({\widehat{\mathrm{\psi }}}_r,{\tilde{\mathrm{i}}}_s\right)$, the rotor speed value can be estimated from

$${\widehat{\omega }}_r=\frac{{\tilde{i}}_{\mathit{s\alpha }}{\widehat{\psi }}_{\mathit{r\beta }}-{\tilde{i}}_{\mathit{s\beta }}{\widehat{\psi }}_{\mathit{r\alpha }}+k_f{\widehat{s}}_{\omega }}{2\left({\widehat{\psi }}_{\mathit{r\alpha }}^2+{\widehat{\psi }}_{\mathit{r\beta }}^2\right)},$$E56

where

$${\widehat{s}}_{\omega }=\sqrt{{\left({\tilde{i}}_{\mathit{s\alpha }}{\widehat{\psi }}_{\mathit{r\beta }}-{\tilde{i}}_{\mathit{s\beta }}{\widehat{\psi }}_{\mathit{r\alpha }}\right)}^2+4{\gamma }_1{\left({\widehat{\psi }}_{\mathit{r\alpha }}{\tilde{i}}_{\mathit{s\alpha }}+{\widehat{\psi }}_{\mathit{r\beta }}{\tilde{i}}_{\mathit{s\beta }}\right)}^2}.$$E57

The proposed AFO speed observer was tested on the 5.5 kW induction machine, which was clutched to DC motor. The sensorless control system structure was based on feedback control with the multi-scalar variables shown in [5, 6]. The control system contains four PI controllers of the rotor speed, electromagnetic torque T_e, square of rotor flux ${\psi }_r^2={\psi }_{\mathit{r\alpha }}^2+{\psi }_{\mathit{r\beta }}^2$, and the variables $x_{22}={\widehat{\psi }}_{\mathit{r\alpha }}{\widehat{i}}_{\mathit{s\alpha }}+{\widehat{\psi }}_{\mathit{r\beta }}{\widehat{i}}_{\mathit{s\beta }}$.

The control system was implemented in the interface with a DSP Sharc ADSP21363 floating-point signal processor with Altera Cyclone 2 FPGA. The interrupt time was 6.6 kHz, and the transistor switching frequency was 3.3 kHz. The rotor speed and position were measured by the incremental encoder (11-bits)—only to the accuracy verification of observer structure. The stator current was measured by the current transducers LA 25-NP—in the phases “a” and “b” and transformed to the (αβ) reference frame by using the Park transformation. The nominal parameters of the IM are presented in Table 1.

Parameter	Value	Unit
Nominal power	5.5	kW
Nominal speed	1430	rpm
Nominal voltage (Y)	400	V
Nominal current (Y)	11	A
Nominal frequency	50	Hz
Stator resistance RsN	2.92/0.035	Ω/p.u.
Stator resistance RrN	3.36/0.032	Ω/p.u.
Magnetizing inductance LmN	0.422/1.95	H/p.u.
Stator and rotor inductance Ls, Lr	0.439/2.04	H/p.u.
Ub = Un	0.82	p.u.
$I_b=I_n\sqrt{3}$	0.89	p.u.

Table 1.

Parameters of the IM and references unit.

In Figures 1–3, the following variables are presented:

Figure 1.

The IM is starting up to 1.0 p.u., non-loaded and the rotor speed value is estimated from a) non-adaptive law (56), b) adaptively (54) – Experimental results.

Figure 2.

The IM is reversing from 1.0 to −1.0 p.u., non-loaded and the rotor speed value is estimated from a) non-adaptive law (56), b) adaptively (54) – Experimental results.

Figure 3.

Motoring and regenerating mode of IM for the rotor speed estimated a) from non-adaptive law (56), b) adaptively (54) – Experimental results.

${\widehat{\omega }}_r$ - estimated rotor speed, ${\omega }_{\mathit{rM}}$ - measured rotor speed, ${\tilde{\omega }}_r$ - rotor speed error,${\widehat{s}}_{\omega }$ - additional variables, ${\widehat{\psi }}_r^2={\widehat{\psi }}_{\mathit{r\alpha }}^2+{\widehat{\psi }}_{\mathit{r\beta }}^2$, ${\widehat{T}}_e={\widehat{\psi }}_{\mathit{r\alpha }}{\widehat{i}}_{\mathit{s\beta }}-{\widehat{\psi }}_{\mathit{r\beta }}{\widehat{i}}_{\mathit{s\alpha }}$.

In Figure 1, the IM is starting up from 0.1 to 1.0 p.u. The waveforms of the estimated value of rotor speed, measured rotor speed, square of rotor flux components, electromagnetic torque, and reference rotor speed are presented. The reference value of electromagnetic torque is limited to 0.75 p.u. The reference value of the square of the rotor flux vector components is set to 0.9 p.u. The estimated rotor speed error during the dynamic states is about 0.015 p.u., and for the steady state is smaller than 0.01 p.u. In Figure 1a, the rotor speed is estimated non-adaptively. In Figure 1b, the rotor speed value is estimated from adaptive law.

In Figure 2, the rotor speed reversed from a nominal speed 1.0 p.u. to −1.0 p.u. The IM during this test is loaded at about T_L = 0.08 p.u. The square of rotor flux was set to 0.9 p.u. The value ${\widehat{s}}_{\omega }$is determined from (57). The value of the rotor speed error for the case presented in Figure 2a is smaller than 0.02 in the dynamic states. For the case presented in Figure 2b, the value of rotor speed error is almost the same and smaller than 0.02 p.u.

In Figure 3, the motoring and regenerating modes of the IM are presented. In the AFO speed observer in which the rotor speed is estimated from the classical law of adaptation, for γ₁ = 0 in (54) (for this case, the stabilizing function is omitted), the observer structure is unstable in the regenerating machine mode, what was signaled in [6]. For the adaptive case (Figure 3b), if γ₁ ≠ 0 and the value ${\widehat{s}}_{\omega }$ is estimated from (55). The observer structure is stable during the load torque value change from 0.7 to −0.7 p.u. The rotor speed error is smaller than 0.015 in the dynamic states. The value of estimated electromagnetic torque for the regenerating case is about −0.65 p.u. It means that the electromagnetic torque value is estimated with a small value of the error of about 0.05 p.u. in the stationary state, but the observer structure is stable.

For the non-adaptive case presented in Figure 3a, the estimated rotor speed value has more oscillations than in Figure 3a. It is because the rotor speed is not filtered as in the case of adaptive estimation law. However, the estimated value of electromagnetic torque is −0.7 p.u. (the same as the load torque). Hence, the electromagnetic torque is estimated more accurately than in the case of the adaptive law of rotor speed estimation.

3.1 Extended speed observer of the squirrel-cage induction machine

In [19], the speed observer was proposed, which is based on the extended model of the IM. In the model of the observer structure, an auxiliary variable marked in [5] as “Z” was introduced and defined as follows:

$${\widehat{Z}}_{\alpha }={\widehat{\omega }}_r{\widehat{\psi }}_{\mathit{r\alpha }},$$E58

$${\widehat{Z}}_{\beta }={\widehat{\omega }}_r{\widehat{\psi }}_{\mathit{r\beta }}.$$E59

Based on the introduced auxiliary variables, the observer model can be determined

$$\frac{d{\widehat{i}}_{\mathit{s\alpha }}}{\mathit{d\tau }}=a_1{\widehat{i}}_{\mathit{s\alpha }}+a_2{\widehat{\psi }}_{\mathit{r\alpha }}+a_3{\widehat{Z}}_{\beta }+a_4{u}_{\mathit{s\alpha }}+k_1\left({\widehat{i}}_{\mathit{s\alpha }}-i_{\mathit{s\alpha }}\right),$$E60

$$\frac{d{\widehat{i}}_{\mathit{s\beta }}}{\mathit{d\tau }}=a_1{\widehat{i}}_{\mathit{s\beta }}+a_2{\widehat{\psi }}_{\mathit{r\beta }}-a_3{\widehat{Z}}_{\alpha }+a_4{u}_{\mathit{s\beta }}+k_1\left({\widehat{i}}_{\mathit{s\beta }}-i_{\mathit{s\beta }}\right),$$E61

$$\frac{d{\widehat{\psi }}_{\mathit{r\alpha }}}{\mathit{d\tau }}=a_5{\widehat{\psi }}_{\mathit{r\alpha }}-{\widehat{Z}}_{\beta }+a_6{\widehat{i}}_{\mathit{s\alpha }}+k_2\left({\widehat{Z}}_{\beta }-Z_{\beta }\right),$$E62

$$\frac{d{\widehat{\psi }}_{\mathit{r\beta }}}{\mathit{d\tau }}=a_5{\widehat{\psi }}_{\mathit{r\beta }}+{\widehat{Z}}_{\alpha }+a_6{\widehat{i}}_{\mathit{s\beta }}-k_2\left({\widehat{Z}}_{\alpha }-Z_{\alpha }\right),$$E63

$$\frac{d{\widehat{Z}}_{\alpha }}{\mathit{d\tau }}=-{\widehat{\omega }}_r\left({\widehat{Z}}_{\beta }-a_6{i}_{\mathit{s\alpha }}\right)-a_5{\widehat{Z}}_{\alpha }+k_3\left({\widehat{i}}_{\mathit{s\alpha }}-i_{\mathit{s\alpha }}\right),$$E64

$$\frac{d{\widehat{Z}}_{\beta }}{\mathit{d\tau }}={\widehat{\omega }}_r\left({\widehat{Z}}_{\alpha }+a_6{i}_{\mathit{s\beta }}\right)-a_5{\widehat{Z}}_{\beta }+k_3\left({\widehat{i}}_{\mathit{s\beta }}-i_{\mathit{s\beta }}\right),$$E65

where the derivative of estimated rotor speed can be approximated $\frac{d{\widehat{\omega }}_r}{\mathit{d\tau }}\approx \frac{\Delta {\widehat{\omega }}_r}{\mathit{\Delta T}}\approx 0$ in the small interval time $\mathit{\Delta T}$, and coefficients a₁…a₆ are defined in (45).

The rotor speed can be determined non-adaptively from the dependence [19]:

$${\widehat{\omega }}_r=\frac{{\widehat{Z}}_{\alpha }{\widehat{\psi }}_{\mathit{r\alpha }}+{\widehat{Z}}_{\beta }{\widehat{\psi }}_{\mathit{r\beta }}}{{\widehat{\psi }}_{\mathit{r\alpha }}^2+{\widehat{\psi }}_{\mathit{r\beta }}^2}.$$E66

The experimental results in this section are limited only to the regenerating mode of the IM, in which the observer structure can be unstable. The reference rotor speed is set to 0.1 p.u.

In the first case presented in Figure 4a, (in which the rotor speed is estimated from (66)) after 0.5 s machine is loaded T_L = −0.6 p.u. For the motoring mode (${\widehat{\omega }}_r$ > 0, ${\widehat{T}}_e\ge 0$), the speed observer (60)–(65) is stable. After 1.5 s, when the load torque value is decreased up to −0.2 p.u., the observer system estimates the state variables incorrectly, the error of estimated rotor speed increases up to 0.05 p.u., and after 1.8 s, the electromagnetic torque value achieves its limitation (0.75 p.u.). After 1.9 s, the rotor speed error is higher than 0.05 p.u., and the IM is braking. The observer structure achieves unstable points of operation in which all the estimates do not converge to their real values.

Figure 4.

Regenerating mode of IM for the rotor speed estimated: a) the rotor speed value is estimated from (66), b) the rotor speed is estimated by using the proposed robust law (additional stabilization function) – Experimental results.

The rotor speed value is estimated from (66), which is suitable only for the motoring mode of the machine. This is the same case as for the AFO speed observer structure. The speed estimation law is based on the algebraic eq. (66), which does not guarantee the stability of the observer structure during the regenerating mode of the machine. Some poles of the observer move to an unstable zone (are zero or positive). The reason for this is the form of dependence (66) in which the additional stabilization function does not exist. The stabilization function, proposed in Section 2.2, which is based on Lagrange’s identity, cannot be directly used, because the vectors: ${\widehat{\mathrm{\psi }}}_r$ and $\widehat{\mathrm{Z}}$have the same position and different amplitude only. This is the result of the definition (58)–(59). In this case, it is better to use from (58)–(59), and after few simple transformations, one can be obtain

$${\widehat{Z}}_{\alpha }{\widehat{\psi }}_{\mathit{r\beta }}-{\widehat{Z}}_{\beta }{\widehat{\psi }}_{\mathit{r\alpha }}={\widehat{\omega }}_r{\widehat{\psi }}_{\mathit{r\alpha }}{\widehat{\psi }}_{\mathit{r\beta }}-{\widehat{\omega }}_r{\widehat{\psi }}_{\mathit{r\alpha }}{\widehat{\psi }}_{\mathit{r\beta }}=0.$$E67

This is satisfied for the ideal case in which all estimation errors are equal to zero. In the other case, taking the left side of (67) as

$${\widehat{s}}_{\omega }={\widehat{Z}}_{\alpha }{\widehat{\psi }}_{\mathit{r\beta }}-{\widehat{Z}}_{\beta }{\widehat{\psi }}_{\mathit{r\alpha }},$$E68

and using (66) the update form of non-adaptive speed estimation with the stabilization function (68) can be determined

$${\widehat{\omega }}_r=\frac{{\widehat{Z}}_{\alpha }{\widehat{\psi }}_{\mathit{r\alpha }}+{\widehat{Z}}_{\beta }{\widehat{\psi }}_{\mathit{r\beta }}+k_f{\gamma }_1{\widehat{s}}_{\omega }}{{\widehat{\psi }}_{\mathit{r\alpha }}^2+{\widehat{\psi }}_{\mathit{r\beta }}^2},$$E69

where $k_f=\mathit{sign}\left({\widehat{\omega }}_r\right)$, ${\gamma }_1\ge 0$.

The experimental results of the proposed non-adaptive speed estimation with the stabilization function (68) are presented in Figure 4b. The reference of rotor speed is set to 0.1 p.u. After 1.5 s, the load torque slowly changes from (the motoring mode) 0.7 to −0.7 p.u. (the regenerating mode). The speed observer correctly estimates the electromagnetic torque value with the rotor speed error smaller than 0.015 p.u. during the dynamic states. The square of rotor flux vector components is stabilized on the almost constant value equal to 0.9 p.u. The proposed stabilized function (68) improves the speed observer properties making the speed observer structure more robust in the regenerating mode, which was confirmed in Figure 4b.

4. Speed and position observer of interior permanent magnet machine

This section is concerned with the observer system of interior permanent magnet machines IPMSM and their problems during a sensorless application under disturbances.

4.1 Mathematical models of the IPMSM machines

The mathematical model of IPMSM is often determined in the rotating reference frame (d-q), which is connected to the position of the rotor. The model in (d-q) was presented in [12, 13, 14]. However, sometimes the mathematical model is better considered in the stationary (α-β) reference frame connected to the stator. The model of IPMSM in (α-β) has the following form [12]:

$$\frac{{\mathit{di}}_{\mathit{s\alpha }}}{\mathit{d\tau }}=\frac{{\omega }_r}{L_d}{\lambda }_{\beta }+\left(-R_s{i}_{\mathit{s\alpha }}+u_{\mathit{s\alpha }}\right)L_1+\left(-R_s{i}_{\mathit{s\beta }}+u_{\mathit{s\beta }}\right)L_3,$$E70

$$\frac{{\mathit{di}}_{\mathit{s\beta }}}{\mathit{d\tau }}=-\frac{{\omega }_r}{L_d}{\lambda }_{\alpha }+\left(-R_s{i}_{\mathit{s\alpha }}+u_{\mathit{s\alpha }}\right)L_3+\left(-R_s{i}_{\mathit{s\beta }}+u_{\mathit{s\beta }}\right)L_4,$$E71

$$\frac{d{\omega }_r}{\mathit{d\tau }}=\frac{1}{J}\left({\psi }_{\mathit{f\alpha }}{i}_{\mathit{s\beta }}-{\psi }_{\mathit{f\beta }}{i}_{\mathit{s\alpha }}+\left(L_d-L_q\right)i_{\mathit{s\alpha }}{i}_{\mathit{s\beta }}-T_L\right),$$E72

$$\frac{d{\theta }_r}{\mathit{d\tau }}={\omega }_r,$$E73

where

$${\lambda }_{\alpha }=L_d{L}_q^{-1}{\psi }_{\mathit{f\alpha }}-\left(1-L_d{L}_q^{-1}\right)\left(L_0{i}_{\alpha 2}+L_2{i}_{\mathit{s\alpha }}\right),$$E74

$${\lambda }_{\beta }=L_d{L}_q^{-1}{\psi }_{\mathit{f\beta }}+\left(1-L_d{L}_q^{-1}\right)\left(L_0{i}_{\beta 2}-L_2{i}_{\mathit{s\beta }}\right),$$E75

$$L_0=0.5\left(L_d+L_q\right),,L_1=L_d^{-1}{cos}^2{\theta }_r+L_q^{-1}{sin}^2{\theta }_r$$E76

$$L_2=0.5\left(L_d-L_q\right),,L_3=0.5\left(\frac{1}{L_d}-\frac{1}{L_q}\right)sin\left(2{\theta }_r\right)$$E77

$$L_4=L_d^{-1}{sin}^2{\theta }_r+L_q^{-1}{cos}^2{\theta }_r,$$E78

$$i_{\alpha 2}=i_{\mathit{s\alpha }}cos2{\theta }_r+i_{\mathit{s\beta }}sin2{\theta }_r$$E79

$$i_{\beta 2}=-i_{\mathit{s\alpha }}sin2{\theta }_r+i_{\mathit{s\beta }}cos2{\theta }_r,$$E80

$${\psi }_{\mathit{f\alpha }}={\psi }_{f}cos{\theta }_r,,{\psi }_{\mathit{f\beta }}={\psi }_{f}sin{\theta }_r$$E81

where R_s is the stator resistance, L_d, L_q are the winding inductances, i_sα,_β are the stator currents, u_sα,_β are the stator voltages, ψ_f is the permanent magnet flux linkage, ω_r is the rotor speed, θ_r is the rotor position, J is the rotor inertia, T_L is the load torque.

The design of the IPMSM machine has a significant impact on the properties of the whole drive system. In IPMSM without the skews in the slots, there occur the rotor slot’s harmonics, [20]. The slot’s harmonics cause the non-sinusoidal distribution of the electromagnetic force (EMF) generated in the machine, [21]. These have a negative influence on the quality of the control system of the machine, and in particular, on the speed and position observer. In the literature [12, 13, 14], these negative effects are named by the disturbances in the IPMSM, which have bounded values. These disturbances have a significant impact on the machine rotor speed smaller than 15% of the nominal value and the idling mode of the IPMSM. For the low-speed range, the stator voltage has a small value, and similarly is the stator currents; because of this, the back-EMF value is significant. The example waveform of back-EMF voltage in 3.5 kW IPMSM machine for 10% of nominal rotor speed, registered by using the digital oscilloscope is presented in Figure 5.

Figure 5.

The phase “A” back-EMF voltage.

In Figure 5, there are visible 18 slot’s harmonics in the waveform. The IPMSM nominal parameters are shown in Table 2.

Parameter	Value	Unit
Nominal power	3.5	kW
Nominal speed	1500	rpm
Nominal voltage (Y)	285	V
Nominal current (Y)	7.5	A
Stator resistance Rs	0.023	p.u.
Inductance LdN	0.28	p.u.
Inductance LqN	0.82	p.u.
Rotor flux linkage	0.89	p.u.

Table 2.

IPMSM nominal parameters and reference units.

In the next section, the speed and position observer is proposed for the IPMSM in which the disturbances have occurred and the back-EMF voltage has an almost trapezoidal distribution (Figure 5).

4.2 Adaptive speed and position observer of IPMSM

As was mentioned in 4.1, the IPMSM has disturbances in the form of trapezoidal back-EMF voltage with slot’s harmonics. The classical structures of the observer in (d-q) are not stable if the rotor speed is smaller than 15–20% of the nominal speed. One of the solutions to overcome this problem is to implement a dedicated algorithm in which additional high or low-frequency signals are injected into the stator voltage or current [13, 14]. However, the sensorless control system is then more complicated than classical FOC, and the observer structure contains a few low-passes or bandwidth filters [22]. The procedure of selecting the settings of the observer and the PI controllers in the control system is difficult. Therefore in this section, a new form of the speed and position observer is proposed, which is based on the mathematical model in the stationary reference frame presented in Section 4.1. Considering the procedure of design of the observer stabilization function from Section 2, the AFO speed observer of IPMSM can be determined

$$\frac{d{\widehat{i}}_{\mathit{s\alpha }}}{\mathit{d\tau }}=\frac{{\widehat{\omega }}_r}{L_d}{\widehat{\lambda }}_{\beta }+\left(-R_s{\widehat{i}}_{\mathit{s\alpha }}+u_{\mathit{s\alpha }}\right)L_1+\left(-R_s{\widehat{i}}_{\mathit{s\beta }}+u_{\mathit{s\beta }}\right)L_3+v_{\alpha },$$E82

$$\frac{d{\widehat{i}}_{\mathit{s\beta }}}{\mathit{d\tau }}=-\frac{{\widehat{\omega }}_r}{L_d}{\widehat{\lambda }}_{\alpha }+\left(-R_s{\widehat{i}}_{\mathit{s\alpha }}+u_{\mathit{s\alpha }}\right)L_3+\left(-R_s{\widehat{i}}_{\mathit{s\beta }}+u_{\mathit{s\beta }}\right)L_4+v_{\beta },$$E83

$$\frac{d{\widehat{\theta }}_r}{\mathit{d\tau }}={\widehat{\omega }}_r+v_{\theta },$$E84

where “^” denotes estimated values; v_α,β, and v_θ are stabilizing functions introduced to (82)–(83).

The values of the rotor flux vector components can be obtained by using (74)–(75) as follows:

$${\widehat{\lambda }}_{\alpha }=L_d{L}_q^{-1}{\widehat{\psi }}_{\mathit{f\alpha }}-\left(1-L_d{L}_q^{-1}\right)\left(L_0{\widehat{i}}_{\alpha 2}+L_2{\widehat{i}}_{\mathit{s\alpha }}\right),$$E85

$${\widehat{\lambda }}_{\beta }=L_d{L}_q^{-1}{\widehat{\psi }}_{\mathit{f\beta }}+\left(1-L_d{L}_q^{-1}\right)\left(L_0{\widehat{i}}_{\beta 2}-L_2{\widehat{i}}_{\mathit{s\beta }}\right),$$E86

where

$$L_0=0.5\left(L_d+L_q\right),,L_1=L_d^{-1}{cos}^2{\widehat{\theta }}_r+L_q^{-1}{sin}^2{\widehat{\theta }}_r$$E87

$$L_2=0.5\left(L_d-L_q\right),,L_3=0.5\left(\frac{1}{L_d}-\frac{1}{L_q}\right)sin\left(2{\widehat{\theta }}_r\right)$$E88

$$L_4=L_d^{-1}{sin}^2{\widehat{\theta }}_r+L_q^{-1}{cos}^2{\widehat{\theta }}_r,$$E89

$${\widehat{i}}_{\alpha 2}={\widehat{i}}_{\mathit{s\alpha }}cos2{\widehat{\theta }}_r+{\widehat{i}}_{\mathit{s\beta }}sin2{\widehat{\theta }}_r$$E90

$${\widehat{i}}_{\beta 2}=-{\widehat{i}}_{\mathit{s\alpha }}sin2{\widehat{\theta }}_r+{\widehat{i}}_{\mathit{s\beta }}cos2{\widehat{\theta }}_r,$$E91

$${\widehat{\psi }}_{\mathit{f\alpha }}={\psi }_{f}cos{\widehat{\theta }}_r,.{\widehat{\psi }}_{\mathit{f\beta }}={\psi }_{f}sin{\widehat{\theta }}_r$$E92

To stabilize the observer structure (82)–(84), the appropriate form of the stabilization functions v_α,β and v_θ should be determined to satisfy the Lyapunov theorem. The Lyapunov candidate function has the form

$$V=0.5\left(\left({\tilde{i}}_{\mathit{s\alpha }}^2+{\tilde{i}}_{\mathit{s\beta }}^2\right)+{\tilde{\theta }}_r^2+{\gamma }^{-1}{\tilde{\omega }}_r^2\right),$$E93

where

$${\tilde{i}}_{\mathit{s\alpha },\beta }={\widehat{i}}_{\mathit{s\alpha },\beta }-i_{\mathit{s\alpha },\beta },,.{\tilde{\omega }}_r={\widehat{\omega }}_r-{\omega }_r{\tilde{\theta }}_r={\widehat{\theta }}_r-{\theta }_r$$E94

The derivative of the Lyapunov function can be determined by using the estimation errors (94) and the proposed observer structure as

$$\dot{V}=-\left(c_{\alpha }{\tilde{i}}_{\mathit{s\alpha }}^2+c_{\alpha }{\tilde{i}}_{\mathit{s\beta }}^2\right)+{\tilde{\omega }}_r\left(-\frac{1}{L_d}{\widehat{\lambda }}_{\beta }{\tilde{i}}_{\mathit{s\alpha }}+\frac{1}{L_d}{\widehat{\lambda }}_{\alpha }{\tilde{i}}_{\mathit{s\beta }}+\frac{1}{\gamma }{\dot{\tilde{\omega }}}_r\right)+{\tilde{\theta }}_r\left({\tilde{\omega }}_r+v_{\theta }\right)\le 0.$$E95

The derivative of the Lyapunov function (95) is negative if the stabilizing functions are chosen

$$v_{\alpha }=-c_{\alpha }{R}_s{L}_1{\tilde{i}}_{\mathit{s\alpha }}+c_{\lambda }{L}_d^{-1}{\widehat{\omega }}_r{\widehat{\lambda }}_{\beta }{\tilde{i}}_{\mathit{s\alpha }},$$E96

$$v_{\beta }=-c_{\alpha }{R}_s{L}_4{\tilde{i}}_{\mathit{s\beta }}-c_{\lambda }{L}_d^{-1}{\widehat{\omega }}_r{\widehat{\lambda }}_{\alpha }{\tilde{i}}_{\mathit{s\beta }},$$E97

$$v_{\theta }=-c_{\theta }{\tilde{\theta }}_r,$$E98

where (c_α, c_θ) > 0 and $c_{\lambda }\le \frac{R_s{L}_1{\widehat{\lambda }}_{\beta }-R_s{L}_4{\widehat{\lambda }}_{\alpha }}{L_d^{-1}\left|{\widehat{\omega }}_r\right|\left({\widehat{\lambda }}_{\alpha }^2+{\widehat{\lambda }}_{\beta }^2\right)}$.

The rotor speed value can be estimated by using the classical adaptation law

$${\dot{\widehat{\omega }}}_r=\gamma L_d^{-1}\left({\widehat{\lambda }}_{\beta }{\tilde{i}}_{\mathit{s\alpha }}-{\widehat{\lambda }}_{\alpha }{\tilde{i}}_{\mathit{s\beta }}\right),$$E99

where γ > 0.

However, under Assumption 3 from Section 2.1 in order to improve the quality of reconstruction of the rotor speed value, it is better to introduce the additional stabilization function to the open-integrator (99)

$${\dot{\widehat{\omega }}}_r=\gamma L_d^{-1}\left({\widehat{\lambda }}_{\beta }{\tilde{i}}_{\mathit{s\alpha }}-{\widehat{\lambda }}_{\alpha }{\tilde{i}}_{\mathit{s\beta }}+k_f{\widehat{s}}_{\omega }\right),$$E100

where the value of the stabilizing function can be obtained by using the approach presented in Section 2.1:

$${\widehat{s}}_{\omega }=\sqrt{\left({\widehat{\lambda }}_{\alpha }^2+{\widehat{\lambda }}_{\beta }^2\right)\left({\tilde{i}}_{\mathit{s\alpha }}^2+{\tilde{i}}_{\mathit{s\beta }}^2\right)+{\left({\widehat{\lambda }}_{\alpha }{\tilde{i}}_{\mathit{s\alpha }}+{\widehat{\lambda }}_{\beta }{\tilde{i}}_{\mathit{s\beta }}\right)}^2}.$$E101

For $\left({\widehat{\lambda }}_{\alpha }^2+{\widehat{\lambda }}_{\beta }^2\right)\left({\tilde{i}}_{\mathit{s\alpha }}^2+{\tilde{i}}_{\mathit{s\beta }}^2\right)\ll {\left({\widehat{\lambda }}_{\alpha }{\tilde{i}}_{\mathit{s\alpha }}+{\widehat{\lambda }}_{\beta }{\tilde{i}}_{\mathit{s\beta }}\right)}^2$, the value of (101) can be determined from the simplified form

$${\widehat{s}}_{\omega }={\widehat{\lambda }}_{\alpha }{\tilde{i}}_{\mathit{s\alpha }}+{\widehat{\lambda }}_{\beta }{\tilde{i}}_{\mathit{s\beta }}.$$E102

The rotor position can be obtained directly from (84), and the stabilizing function v_θ from (98).

In (98) there is the rotor position error, which is defined ${\tilde{\theta }}_r={\widehat{\theta }}_r-{\theta }_r$, where ${\theta }_r$ means the real (measured) value of rotor speed. However, in the speed observer structure, the rotor speed is not measured but only estimated. Therefore, it is proposed to replace the deviation ${\tilde{\theta }}_r$ by ${\tilde{\theta }}_{\lambda }$ and (98) is rewritten as

$$v_{\theta }=-c_{\theta }{\tilde{\theta }}_{\lambda },$$E103

where ${\tilde{\theta }}_{\lambda }$ can be defined as the angle between the rotor flux vector components λ_{α, β} and their estimated values ${\widehat{\lambda }}_{\alpha }{,}_{\beta }$. Values of deviation ${\tilde{\theta }}_{\lambda }$ can be determined as

$${\tilde{\theta }}_{\lambda }={tan}^{-1}\left(\phi \right),$$E104

where $\phi =\left({\lambda }_{\alpha }{\widehat{\lambda }}_{\beta }-{\lambda }_{\beta }{\widehat{\lambda }}_{\alpha }\right){\left({\lambda }_{\alpha }{\widehat{\lambda }}_{\alpha }+{\lambda }_{\beta }{\widehat{\lambda }}_{\beta }\right)}^{-1}$. The rotor flux vector components, λ_{α, β}, can be determined from (74)–(75) in which it is assumed ${\theta }_r\approx {\widehat{\theta }}_r$ and the measured values of i_sα,_β are used; also,$\left({\lambda }_{\alpha }{\widehat{\lambda }}_{\alpha }+{\lambda }_{\beta }{\widehat{\lambda }}_{\beta }\right)\ne 0$.

Value of ${\tilde{\theta }}_{\lambda }$ should be projected using

$${\tilde{\theta }}_{\lambda }=\left\{\begin{array}{l}{\tilde{\theta }}_{\lambda }-\pi /2,\mathit{\text{if}}\phi >0\\ {\tilde{\theta }}_{\lambda }+\pi /2\mathit{\text{if}}\phi <0\end{array}\right\},$$E105

It gives the values ${\tilde{\theta }}_{\lambda }$ in a steady state close to zero, and it can be assumed that ${\tilde{\theta }}_{\lambda }\approx {\tilde{\theta }}_r$. The proposed stabilizing function improves the estimated value of the rotor position, particularly in the dynamic states of the IPMSM. The stabilizing function is necessary in the case of IPMSM with the described above disturbances.

Remark 4. The value of rotor flux vector components must be estimated from (74)–(75), however, by using the estimated rotor speed position ${\widehat{\theta }}_r$ in (76)–(81).

4.3 Non-adaptive speed estimation of the IPMSM

The rotor speed value can be estimated by using the non-adaptive estimation scheme. Consider the non-adaptive scheme for the rotor speed estimation presented in Section 2.2 for the pair of vectors, $\left({\widehat{\mathrm{x}}}_k,{\tilde{\mathrm{x}}}_k\right)\equiv \left({\widehat{\lambda }}_r,{\tilde{\mathrm{i}}}_s\right)$ the rotor speed value can be estimated from

$${\widehat{\omega }}_r=\frac{{\tilde{i}}_{\mathit{s\alpha }}{\widehat{\lambda }}_{\beta }-{\tilde{i}}_{\mathit{s\beta }}{\widehat{\lambda }}_{\alpha }+k_f{\widehat{s}}_{\omega }}{2\left({\widehat{\lambda }}_{\alpha }^2+{\widehat{\lambda }}_{\beta }^2\right)},$$E106

where

$${\widehat{s}}_{\omega }=\sqrt{{\left({\tilde{i}}_{\mathit{s\alpha }}{\widehat{\lambda }}_{\beta }-{\tilde{i}}_{\mathit{s\beta }}{\widehat{\lambda }}_{\alpha }\right)}^2+4{\gamma }_1{\left({\widehat{\lambda }}_{\alpha }{\tilde{i}}_{\mathit{s\alpha }}+{\widehat{\lambda }}_{\beta }{\tilde{i}}_{\mathit{s\beta }}\right)}^2},$$E107

and $k_f=\mathit{sign}\left({\widehat{\omega }}_r\right)$, ${\gamma }_1\ge 0$.

4.4 Simulation and experimental results of the speed and position observer of IPMSM

In this section, the chosen waveforms from the simulation and the experiment setup are shown. The nominal parameters of the IPMSM are shown in Table 2. The experimental validations were carried out on 3.5 kW IPMSM. The stator of IPMSM has 18 slots, which are visible in the waveform of EMF from Figure 5. The machine is controlled by using the classical FOC control presented in [12, 13, 22]. There are three PI controllers for the rotor speed, i_sq, and i_sd stator vector components. Additionally, the MTPA algorithm [12] was applied.

In Figure 6, the waveform of the simulation results is shown. The estimated rotor speed ${\widehat{\omega }}_r$, stator current vector components ${\widehat{i}}_{\mathit{sd}}{,}_q$ estimated rotor speed error ${\tilde{\omega }}_r$, and the estimated rotor position ${\widehat{\theta }}_r$ are presented. In Figure 6a, the machine is starting up to 1.0 p.u. and after 600 ms loaded to about 0.6 p.u. The error of the rotor position is smaller than 0.05 p.u. during the dynamic states, the rotor speed error is smaller than 0.01 p.u. In Figure 6b, the machine is reversing to −1.0 p.u. The position error is smaller than 0.1 p.u. during the dynamic states, and the error of rotor speed is smaller than 0.05 p.u. In Figure 6b, the measured value of rotor position θ_rM is shown.

Figure 6.

a) Machine is starting up to 1.0 p.u and b) reversing to −1.0 p.u. – Simulation results.

In Figure 7a, after 100 ms the machine is loaded T_L = 1.0 p.u. and after 600 ms the regenerating mode is applied and T_L = −1.0 p.u. The rotor reference speed is equal to 0.1 p.u. The estimated electromagnetic torque ${\widehat{T}}_e$, rotor position error ${\tilde{\theta }}_r$, ${\tilde{\theta }}_{\lambda }$ defined in (104), s_ω and rotor speed error, and the estimated ${\widehat{i}}_{\mathit{sd}}$ stator current component are presented. It is worth noticing that the waveforms of ${\tilde{\theta }}_r$ as well as ${\widehat{s}}_{\omega }$ and ${\tilde{\omega }}_r$ are converged on each other.

Figure 7.

a) the load torque T_L is changed from 1.0 to – 1.0 p.u., b) parameters of the machine are changed in the sensorless control system (parameters uncertainties test) – Simulation.

The experimental waveforms are presented in Figure 8. The machine’s reversal from 1.0 to −1.0 p.u. is shown in Figure 8a. The estimated rotor speed ${\widehat{\omega }}_r$, stator current vector components ${\widehat{i}}_{\mathit{sd}}{,}_q$ estimated rotor speed error ${\tilde{\omega }}_r$, and the estimated rotor position ${\widehat{\theta }}_r$ and the stator current module i_m are presented. In Figure 8b, the same waveforms but for the measured value from the encoder of the rotor speed and rotor position are presented (for comparison). During the machine reverse, the i_sd value is about 0.25 p.u. It results from the MTPA algorithm [12].

Figure 8.

Machine is reversing from 1.0 to −1.0 p.u. for a) the sensorless control system with the proposed observer, b) the control system with measured rotor speed and position values – Experimental results.

In Figure 9, the regenerating mode of IPMSM is shown. The rotor speed was set to 0.025 p.u., and the machine was loaded at about −0.5 p.u. In Figure 9a, the stabilizing function is k_f = 0.5 in (106), and the value of the error of estimated rotor position is increased to 0.075 p.u., which is visible in Figure 9a. The value of the stator current component ${\widehat{i}}_{\mathit{sq}}$was incorrectly estimated (due to rotor position error). In Figure 9b, the same case is shown, however for $k_f=2.5$ p.u. The rotor position error is almost minimized to zero, and the ${\widehat{i}}_{\mathit{sq}}$ value is about −0. 5 p.u. (the same as the referenced).

Figure 9.

Regenerating mode of IPMSM machine for the low reference speed 0.025 p.u., T_L = −0.5 p.u. for the cases a) in (106)k_f = 0.5, b) in (106)k_f = 2.5 p.u. – Experimental results.

In this section, the presented simulation and experimental results confirmed that the introduced stabilization function into the speed adaptation scheme leads to the improvement of the properties of the observer system and robustness of the occurred disturbances. In this case, these are the non-sinusoidal EMF and slot’s harmonics.

5. Speed and position observer of doubly fed induction generator

In this section, the speed and position observer of the doubly fed induction generator DFIG is considered. The rotor is connected to a voltage source converter, and the stator is directly connected to three phases AC-grid. The field-oriented control FOC is used to control the active and reactive stator powers presented in [23]. The rotor speed will be estimated by using a non-adaptive estimation scheme only.

5.1 Mathematical models of the DFIG

The mathematical model of DFIG can be determined in rotating or stationary reference frames (x-y). Considering the rotor and stator current vector components, the differential equations have the form [24]:

$$\frac{{\mathit{di}}_{\mathit{sx}}}{\mathit{d\tau }}=-\frac{L_s}{w_{\sigma }}\left(R_s{i}_{\mathit{sx}}-u_{\mathit{sx}}\right)+\frac{L_m}{w_{\sigma }}\left({\omega }_r\left(L_m{i}_{\mathit{sy}}+L_r{i}_{\mathit{ry}}\right)+R_r{i}_{\mathit{rx}}-u_{\mathit{rx}}\right),$$E108

$$\frac{{\mathit{di}}_{\mathit{sy}}}{\mathit{d\tau }}=-\frac{L_r}{w_{\sigma }}\left(R_s{i}_{\mathit{sy}}-u_{\mathit{sy}}\right)-\frac{L_m}{w_{\sigma }}\left({\omega }_r\left(L_m{i}_{\mathit{sx}}+L_R{i}_{\mathit{rx}}\right)-R_r{i}_{\mathit{ry}}+u_{\mathit{ry}}\right),$$E109

$$\frac{{\mathit{di}}_{\mathit{rx}}}{\mathit{d\tau }}=\frac{L_s}{w_{\sigma }}\left(-{\omega }_r\left(L_r{i}_{\mathit{ry}}+L_m{i}_{\mathit{sy}}\right)-R_r{i}_{\mathit{rx}}+u_{\mathit{rx}}\right)+\frac{L_m}{w_{\sigma }}\left(R_s{i}_{\mathit{sx}}-u_{\mathit{sx}}\right),$$E110

$$\frac{{\mathit{di}}_{\mathit{ry}}}{\mathit{d\tau }}=\frac{L_s}{w_{\sigma }}\left({\omega }_r\left(L_r{i}_{\mathit{rx}}+L_m{i}_{\mathit{sx}}\right)-R_r{i}_{\mathit{ry}}+u_{\mathit{ry}}\right)+\frac{L_m}{w_{\sigma }}\left(R_s{i}_{\mathit{sy}}-u_{\mathit{sy}}\right),$$E111

$$\frac{d{\omega }_r}{\mathit{d\tau }}=\frac{L_m}{{\mathit{JL}}_r}\left(i_{\mathit{rx}}{i}_{\mathit{sy}}-i_{\mathit{ry}}{i}_{\mathit{sx}}\right)-\frac{1}{J}\left(T_L+f_r{\omega }_r\right).$$E112

where the (x-y) coordinate system is associated with any angular speed, and it is assumed that (108)–(109) are connected to the stationary stator windings so the angular speed of the (x,y) system is ${\omega }_a=0$, f_r is the friction.

It is assumed that all the DFIG parameters are known and constant. The components u_rx, u_ry are treated as the control vector variables, and u_sx, u_sy, i_sx, i_sy and i_rx, i_ry components are treated as measured and transformed to the adequate (x-y) reference frame.

To design the observer structure, it is proposed to introduce new auxiliary variables, which are defined

$$H_x={\omega }_r\left(L_m{i}_{\mathit{sx}}+L_r{i}_{\mathit{rx}}\right),$$E113

$$H_y={\omega }_r\left(L_m{i}_{\mathit{sy}}+L_r{i}_{\mathit{ry}}\right).$$E114

Considering (113)–(114) and (108)–(111), the observer structure can be determined

$$\frac{d{\widehat{i}}_{\mathit{rx}}}{\mathit{d\tau }}=-\frac{L_s}{w_{\sigma }}\left({\widehat{H}}_y+R_r{i}_{\mathit{rx}}+u_{\mathit{rx}}\right)+\frac{L_m}{w_{\sigma }}\left(R_s{i}_{\mathit{sx}}-u_{\mathit{sx}}\right)+v_{\mathit{rx}},$$E115

$$\frac{d{\widehat{i}}_{\mathit{ry}}}{\mathit{d\tau }}=\frac{L_s}{w_{\sigma }}\left({\widehat{H}}_x-R_r{i}_{\mathit{ry}}+u_{\mathit{ry}}\right)+\frac{L_m}{w_{\sigma }}\left(R_s{i}_{\mathit{sy}}-u_{\mathit{sy}}\right)+v_{\mathit{ry}},$$E116

$$\frac{d{\widehat{H}}_x}{\mathit{d\tau }}={\widehat{\omega }}_r\left(-{\widehat{H}}_y-R_r{i}_{\mathit{rx}}+u_{\mathit{rx}}\right)+v_{\mathit{Hx}},$$E117

$$\frac{d{\widehat{H}}_y}{\mathit{d\tau }}={\widehat{\omega }}_r\left({\widehat{H}}_x-R_r{i}_{\mathit{ry}}+u_{\mathit{ry}}\right)+v_{\mathit{Hy}},$$E118

$$\frac{d{\widehat{\theta }}_r}{\mathit{d\tau }}={\widehat{\omega }}_r+v_{\theta },$$E119

where estimated state variables are marked by “^” and $\frac{d{\widehat{\omega }}_r}{\mathit{d\tau }}\approx \frac{\mathrm{\Delta }{\widehat{\omega }}_r}{\mathrm{\Delta }T}$, the observer contains the stabilization functions v_rx, v_ry and v_Hx, v_Hy, v_θ .

According to the design procedure of the observer presented in Section 2, in order to stabilize the observer structure (115)–(119), appropriate form of the stabilization functions v_rx, v_ry and v_Hx, v_Hy, v_θ should be determined to satisfy the Lyapunov theorem. The Lyapunov function has the form

$$V=\frac{1}{2}\left({\tilde{i}}_{\mathit{rx}}^2+{\tilde{i}}_{\mathit{ry}}^2+{\tilde{H}}_x^2+{\tilde{H}}_y^2+{\tilde{\theta }}_r^2\right)>0,$$E120

where.

$${\tilde{i}}_{\mathit{rx}}={\widehat{i}}_{\mathit{rx}}-i_{\mathit{rx}},{\tilde{i}}_{\mathit{ry}}={\widehat{i}}_{\mathit{ry}}-i_{\mathit{ry}},{\tilde{H}}_x={\widehat{H}}_x-H_x,{\tilde{H}}_y={\widehat{H}}_y-H_y\text{and}{\tilde{\theta }}_r={\widehat{\theta }}_r-{\theta }_r.$$E121

The proposed observer structure will be asymptotically stable if ${\dot{V}}_2<0$ and if the stabilizing functions introduced to the structure are determined as

$$v_{\mathit{rx}}=-c_x{\tilde{i}}_{\mathit{rx}},$$E122

$$v_{\mathit{ry}}=-c_y{\tilde{i}}_{\mathit{ry}},$$E123

$$v_{\mathit{Hx}}=c_{\mathit{Hx}}\left(-\frac{L_s}{w_{\sigma }}{\tilde{i}}_{\mathit{ry}}-{\widehat{\omega }}_r{R}_r{\tilde{i}}_{\mathit{rx}}\right),$$E124

$$v_{\mathit{Hy}}=c_{\mathit{Hy}}\left(\frac{L_s}{w_{\sigma }}{\tilde{i}}_{\mathit{rx}}+{\widehat{\omega }}_r{R}_r{\tilde{i}}_{\mathit{ry}}\right),$$E125

where (c_x, c_y, c_Hx, c_Hy, c_θ) > 0 are the observer tuning gains and

$$v_{\theta }=-c_{\theta }{\tilde{\theta }}_r.$$E126

The speed observer structure will be asymptotically stable if (122)–(126) is satisfied. In the sensorless control, the rotor speed is not measured, therefore the deviation ${\tilde{\theta }}_r$ in (126) should be replaced by ${\tilde{\theta }}_H$. This means that the deviation between the estimated values of H_x and H_y, calculated from (113)–(114) and estimated from the observer structure in (117)–(118) is as follows:

$${\tilde{\theta }}_H={tan}^{-1}\left(\vartheta \right),$$E127

where

$$\vartheta =\frac{H_x{\widehat{H}}_y-H_y{\widehat{H}}_x}{H_x{\widehat{H}}_x+H_y{\widehat{H}}_y}\text{and}\left(H_x{\widehat{H}}_x+H_y{\widehat{H}}_y\right)\ne 0.$$E128

The value ${\widehat{H}}_x,{\widehat{H}}_y$ can be estimated from

$${\widehat{H}}_x={\widehat{\omega }}_r\left(L_m{i}_{\mathit{sx}}+L_r{\widehat{i}}_{\mathit{rx}}\right),$$E129

$${\widehat{H}}_y={\widehat{\omega }}_r\left(L_m{i}_{\mathit{sy}}+L_r{\widehat{i}}_{\mathit{ry}}\right).$$E130

In the observer structure, the rotor speed is not estimated adaptively, therefore the rotor speed error in (120) is not considered. The rotor speed value can be estimated from the non-adaptive scheme. From (129)–(130), after some calculation, the rotor speed value can be determined from

$${\widehat{\omega }}_r=\frac{{\widehat{H}}_x{\widehat{\psi }}_{\mathit{rx}}+{\widehat{H}}_y{\widehat{\psi }}_{\mathit{ry}}-c_f{s}_{\omega }}{{\widehat{\psi }}_{\mathit{rx}}^2+{\widehat{\psi }}_{\mathit{ry}}^2},$$E131

where

$$s_{\omega }={\widehat{H}}_x{\widehat{\psi }}_{\mathit{ry}}-{\widehat{H}}_y{\widehat{\psi }}_{\mathit{rx}},$$E132

$${\widehat{\psi }}_{\mathit{rx}}=L_m{i}_{\mathit{sx}}+L_r{\widehat{i}}_{\mathit{rx}},$$E133

$${\widehat{\psi }}_{\mathit{ry}}=L_m{i}_{\mathit{sy}}+L_r{\widehat{i}}_{\mathit{ry}},$$E134

c_f ≥ 0 and $\left({\widehat{\psi }}_{\mathit{rx}}^2+{\widehat{\psi }}_{\mathit{ry}}^2\right)\ne 0$.

In Figure 10a, the responses of DFIG on the active and reactive power changes are shown. After 0.1 s, the active power s_p value is changed from −0.1 to −0.35, and reactive power is set to −0.6 p.u. and changed at the same time. After 0.4 s, the active and reactive powers are changed s_p to 0.35 and s_q to 0.2 p.u. The rotor speed estimation error is smaller than 0.015 in the dynamic states, the same as the rotor position error.

Figure 10.

a) the changes of the active and reactive power, b) sub and super-synchronous working modes – Simulation results.

In Figure 10b, the active power s_p is set to 0.02 p.u. and reactive power s_q is set to −0.6 p.u. The rotor speed of the DFIG is changed from the sub-synchronous to super-synchronous mode. Close to synchronous rotor speed (1.0 p.u.), the estimated speed error was smaller than 0.01 p.u., and it is increasing when the speed is growing. The rotor position error has almost the same value.

In Figure 11a, the active power is changed from −0.1 to −0.35 p.u. The rotor speed estimation error is smaller than 0.05 p.u. and the rotor position is smaller than 0.1 p.u. during these changes (in the experimental results). In Figure 11b, the reactive power is changed from −0.7 to −0.4 p.u., the rotor speed error is smaller than 0.035 p.u., and the rotor position error value is changed from 0.05 to −0.05 p.u.

Figure 11.

The changes: a) the active power from −0.1 to −0.35 p.u., b) reactive power from −0.7 to −0.4 p.u. – Experimental results.

In Figure 12a, the rotor speed crosses from −1.1 (super-synchronous mode) to −0.7 p.u. (sub-synchronous mode). The estimated value of rotor position is growing and close to synchronous speed (−1.0 p.u.) achieving the value of about 0.12. The reactive power value was set to −0.6 p.u. With LPF, the filtered value of stabilizing function ${\widehat{s}}_{\mathit{\omega f}}$ is presented. The value of this function is about 0.005 for the super-synchronous mode and about 0.001 p.u. for the sub-synchronous mode. The estimated rotor speed error is about 0.05 p.u. during the crossing through the synchronous speed.

Figure 12.

The waveforms of chosen variables in a) the rotor speed are changed from the super-synchronous to sub-synchronous mode, b) the steady state – Experimental results.

6. Conclusions

In this chapter, robust mechanism for different structures of speed observers or rotor position was presented. The solution was tagged “robust mechanism” because of the introduction of stabilizing function in the speed or rotor position estimation schemes. The additional stabilization law prevents the unstable working range of the speed observer structure (positive poles of the observer). In this chapter, the stability analysis based on the Lyapunov function was presented. The introduced additional stabilizing function to the observer structure is based on Lagrange’s identity, which is the main contribution of this chapter. The form of the proposed robust mechanism is based mainly on the vector and scalar product of the two chosen vectors in the observer system. The mutual position of these vectors directly influences the position of the estimated vectors of the observer and also influences the estimated rotor speed value or the rotor position. The mutual position of a vector influences the value of the estimated electromagnetic torque of the machine. The proposed solution has significant meaning during the low speed of the IM and IPMSM (due to the unstable working points of the observer structure), as well as during the synchronous rotor speed of the DFIG system. The proposed robust mechanism for the speed estimation scheme can be applied to each observer structure, which is based on the space vector form of the mathematical model of an observer system.

Nomenclature

“^”

estimated values,

“∼”

error of estimated values,

i_sx,_y

stator current vector components,

i_rx,_y

stator current vector components,

u_rα,_β

rotor voltage vector components,

u_sα,_β

stator voltage vector components,

ω_r

rotor angular speed,

${\theta }_r$

rotor position,

R_r, R_s

rotor and stator resistances,

L_m

mutual-flux inductance,

L_s, L_r

stator and rotor inductances,

T_e

electromagnetic torque,

T_L

load torque,

J

machine torque of inertia,

τ

relative time,

${\widehat{\theta }}_r$

estimated rotor position,

${\widehat{\omega }}_r$

estimated rotor electrical speed,

${\tilde{\omega }}_r$

rotor speed error,

${\tilde{\theta }}_r$

rotor position error,

(x-y)

coordinate system associated with any angular speed,

(d-q)

coordinate system associated with the rotor angular speed,

L_d, L_q

stator winding inductances,

ψ_f

permanent magnet flux linkage,

IM

induction machine,

IPMSM

interior permanent magnet synchronous machine,

DFIG

doubly fed induction generator,

AFO

adaptive full-order observer.