Cascade Nonlinear Observer-Based Speed-Sensorless Adaptive Twisting Sliding Mode Control of Linear Induction Motor

Abstract

This paper presents a novel adaptive twisting sliding mode control strategy combined with a speed-sensorless cascade nonlinear observer for the high-performance control of linear induction motors (LIMs). The primary objective is to achieve accurate speed and rotor flux tracking without relying on mechanical sensors, thereby enhancing system reliability and reducing hardware complexity. For this purpose, a cascade nonlinear observer is designed and applied to the class of nonlinear affine systems representing LIM dynamics. Based on the interconnected form of the LIM mathematical model, the observer simultaneously reconstructs both the motor speed and rotor fluxes in real time. The stability of the proposed cascade observer is analyzed using Lyapunov theory, ensuring the convergence of the estimation errors under bounded disturbances. Complementing the observer, two adaptive gain twisting sliding mode controllers are developed: one for speed tracking and another for flux regulation. These controllers are robust against external disturbances and parameter uncertainties, even when the bounds of such disturbances are unknown. This feature significantly enhances the practical applicability of the control system in real-world industrial environments. To validate the performance and robustness of the proposed control scheme, a hardware-in-the-loop (HIL) experiment was conducted. Comparative studies with existing state-of-the-art sensorless control methods demonstrate that the proposed cascade nonlinear observer-based approach achieves faster convergence, higher estimation accuracy, and better disturbance rejection capabilities, while requiring less computational effort.

1. Introduction

The existing linear induction motor (LIM) speed observation algorithm research is mainly based on the LIM state space model, and state reconstruction is the core to realize motor speed observation [1]. However, insufficient consideration of losses and other practical factors has resulted in current methods being inadequate in terms of observation accuracy, robustness, and convergence speed, making it difficult to comprehensively and accurately reflect the system state of the linear induction motor under different operating conditions [2,3,4]. Therefore, it is particularly important to carry out research on high-precision and robust observation technology for LIMs without speed sensors [5].

To address the issue of linear induction motor speed observation, establishing an accurate mathematical model is the primary task in order to achieve effective speed estimation [6]. The most fundamental difference between linear induction motors and rotary induction motors lies in the presence of end effects, which makes motor design, performance calculation, and speed observation more difficult [7]. When the primary and secondary components of the linear induction motor move relative to each other, the primary input end and the secondary output end will induce eddy currents in the secondary conductive layer with the same magnitude and opposite direction as the primary end winding current. The eddy current magnetic field makes the air gap magnetic field weaken at the entry end and strengthen at the exit end [8].

Most of the linear induction motor models are directly adapted from rotating induction motor models, without adequately considering the effects of end effects, which often results in large estimation errors, especially during high-speed operation. Most existing linear induction motor models are based on the linear induction motor model proposed by Ducan considering end effects [9,10]. Reference [11] verified the observability of a linear induction motor and proposed a sliding mode observer scheme; however, this algorithm exhibits slight chattering. Reference [12] proposed a second-order sliding mode observer scheme, but the parameter selection of this scheme is complicated and difficult to apply directly. References [13,14] proposed a Kalman filter scheme for the speed estimation of LIM drives. However, the computational burden of the Kalman filter is relatively high, which may limit its use in commercial applications.

Sliding mode control (SMC) has been widely studied and applied in electromechanical systems due to its strong robustness, insensitivity to parameter variations and external disturbances, and fast dynamic response [15,16]. The design of a sliding mode controller consists of two steps: the first step is to design an appropriate sliding mode surface; the second step is to design an appropriate control rate. The higher-order sliding mode control (HOSMC) method is applied to systems of the relative degree r and drives the output of a system and its r-1 derivatives to zero in a finite time in spite of the bounded disturbances [17,18,19]. The main disadvantage of HOSMC is the chattering phenomenon. If the boundaries of disturbances exist and are not known, the corresponding control gains may be overestimated, which causes increased chattering [20]. To address this issue, a novel adaptive-gain twisting controller was proposed in [21] for bounded perturbed systems using the Lyapunov method with unknown boundaries.

This paper proposes an adaptive twisting controller (ATC)-based cascade nonlinear observer (CNO) scheme for a linear induction motor (LIM), aiming to achieve accurate speed and rotor flux tracking without requiring direct measurements of speed or flux. The cascading observer is renowned for its ability to handle complex systems by decomposing them into simpler subsystems. Similarly, this paper divides the EHA system into three parts and designs observers independently for each part [22]. However, unlike the traditional cascaded observers, our method combines the adaptive gain calculated based on the absolute value of the observation error, thereby improving the convergence speed of the near-equilibrium region and the far-equilibrium region. The control scheme utilizes only the measured stator currents and stator voltages. The observer design for the sensorless LIM is based on the interconnection between observers that satisfy certain required properties [23,24]. The proposed cascade nonlinear observer is made through synthesis of the observer for each subsystem. The main contributions of this paper are presented as follows:

(1)

An observer scheme connected with an estimator is designed in order to reconstruct the LIM speed of a sensorless linear induction motor, whereas the estimator is used to estimate the rotor fluxes.

(2)

Using Lyapunov-like arguments, the exponential convergence of the estimation errors in the designed cascade nonlinear observer is proved.

(3)

Based on estimated variables with the proposed cascade nonlinear observer, two ATCs are designed in order to track desired LIM speed and rotor flux in a finite time, in the presence of the bounded disturbances with unknown boundaries.

The rest of this paper is organized as follows. In Section 2, the state-space equations of the LIM in the inductor part flux reference frame ($\alpha$, $\beta$) are presented. In Section 3, a methodology for designing a cascade nonlinear observer for the linear induction motor is presented and the exponential convergence of the estimation errors in the proposed observer is analyzed and proved. In Section 4, based on the proposed cascade nonlinear observer, the estimated states serve as injection variables, and two adaptive-gain twisting controllers are applied into the LIM system to achieve LIM speed and rotor flux tracking. In Section 5, the performance of the proposed CNO-ATC scheme is evaluated through hardware-in-the-loop experiments. Finally, some conclusions are presented in Section 6.

2. LIM’s State-Space Equation

Taking the stator currents $i_{s\alpha },i_{s\beta }$ and rotor fluxes ${\psi }_{r\alpha },{\psi }_{r\beta }$ as the state variables in the stationary reference frame [11,25], the state-space vector equation of the LIM system is established, with the following form:

$${\dot{i}}_{s\alpha }=-c_1{i}_{s\alpha }+c_{2}w{\psi }_{r\alpha }+c_3{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\beta }+{\displaystyle \frac{u_{s\alpha }}{c_4}}$$

(1)

$${\dot{i}}_{s\beta }=-c_1{i}_{s\beta }+c_{2}w{\psi }_{r\beta }-c_3{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\alpha }+{\displaystyle \frac{u_{s\beta }}{c_4}}$$

(2)

$${\dot{\psi }}_{r\alpha }=c_5{i}_{s\alpha }-c_6(2w-1){\psi }_{r\alpha }-{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\beta }$$

(3)

$${\dot{\psi }}_{r\beta }=c_5{i}_{s\beta }-c_6(2w-1){\psi }_{r\beta }+{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\alpha }$$

(4)

$$\dot{v}=c_7\left(i_{s\beta }{\psi }_{r\alpha }-i_{s\alpha }{\psi }_{r\beta }\right)-{\displaystyle \frac{D}{M}}v-{\displaystyle \frac{T_L}{M}}$$

(5)

where v is the LIM speed; $w={\displaystyle \frac{1}{1-f\left(Q\right)}}={\displaystyle \frac{\left(k/v\right)}{e^{-\left(k/v\right)}-1+\left(k/v\right)}}$ is a speed-dependent parameter; $u_{s\alpha }$ and $u_{s\beta }$ are the stator voltages; $i_{s\alpha }$ and $i_{s\beta }$ are the stator currents; ${\psi }_{r\alpha }$ and ${\psi }_{r\beta }$ are the rotor fluxes; D is viscous friction; M is the motor mass; and $T_L$ is the load torque. The related parameters are given as follows:

$$c_1={\displaystyle \frac{1}{L_{\sigma s}}}\left(R_s+R_r\right),c_2={\displaystyle \frac{R_r}{L_m{L}_{\sigma s}}},c_3={\displaystyle \frac{1}{L_{\sigma s}}}$$

$$c_4=L_{\sigma s},c_5=R_r,c_6={\displaystyle \frac{R_r}{L_m}},c_7={\displaystyle \frac{3n_p\pi }{2Mh}}$$

$$f\left(Q\right)={\displaystyle \frac{1-e^{-Q}}{Q}},Q={\displaystyle \frac{k}{v}},k={\displaystyle \frac{{\tau }_m{R}_r}{L_m}}$$

3. Cascade Nonlinear Observer for LIM System

3.1. LIM Model Observability

In this subsection, we will analyze the observability of the states in the proposed LIM mode Equations (1)–(4). The stator voltages $u_{s\alpha }$ and $u_{s\beta }$, as well as the stators $i_{s\alpha }$ and $i_{s\beta }$, are assumed to be measurable. The objective is to determine whether the modified rotor flux components ${\psi }_{r\alpha }$ and ${\psi }_{r\beta }$, the motor speed v, and the speed-dependent parameter w can be estimated solely based on the measured stator quantities. Generally speaking, no information about the LIM load torque is available [26]. To simplify the analysis, it can be assumed that the motor speed varies slowly over time [27].

$$\dot{v}=0$$

(6)

The observability theorem will now be applied to the LIM system described by Equations (1)–(4) and (6), which can be expressed in component form as follows.

$${\dot{i}}_{s\alpha }=-c_1{i}_{s\alpha }+c_{2}w{\psi }_{r\alpha }+c_3{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\beta }+{\displaystyle \frac{u_{s\alpha }}{c_4}}$$

(7)

$${\dot{i}}_{s\beta }=-c_1{i}_{s\beta }+c_{2}w{\psi }_{r\beta }-c_3{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\alpha }+{\displaystyle \frac{u_{s\beta }}{c_4}}$$

(8)

$${\dot{\psi }}_{r\alpha }=c_5{i}_{s\alpha }-c_6(2w-1){\psi }_{r\alpha }-{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\beta }$$

(9)

$${\dot{\psi }}_{r\beta }=c_5{i}_{s\beta }-c_6(2w-1)x_4+{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\alpha }$$

(10)

$$\dot{v}=0$$

(11)

The equations can be rewritten as follows:

$$\left\{\begin{array}{c}\dot{\mathit{x}}=\mathit{f}(\mathit{x},\mathit{u})\\ \mathit{y}=\mathit{h}\left(\mathit{x}\right)\end{array}\right.$$

(12)

where $\mathit{x},\mathit{f}(\mathit{x},\mathit{u}),$ and $\mathit{h}\left(\mathit{x}\right)$ are

$$\mathit{x}=\left[\begin{array}{c}{i}_{s\alpha }\\ i_{s\beta }\\ {\psi }_{r\alpha }\\ {\psi }_{r\beta }\\ v\end{array}\right],\mathit{u}=\left[\begin{array}{c}{u}_{s\alpha }\\ u_{s\beta }\end{array}\right],\mathit{h}\left(\mathit{x}\right)=\left[\begin{array}{c}{i}_{s\alpha }\\ i_{s\beta }\end{array}\right]$$

$$\mathit{f}(\mathit{x},\mathit{u})=\left[\begin{array}{c}{f}_1\\ f_2\\ f_3\\ f_4\\ f_5\end{array}\right]=\left[\begin{array}{c}-c_1{i}_{s\alpha }+c_{2}w{\psi }_{r\alpha }+c_3{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\beta }+{\displaystyle \frac{u_{s\alpha }}{c_4}}\\ -c_1{i}_{s\beta }+c_{2}w{\psi }_{r\beta }-c_3{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\alpha }+{\displaystyle \frac{u_{s\beta }}{c_4}}\\ c_5{i}_{s\alpha }-c_6(2w-1){\psi }_{r\alpha }-{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\beta }\\ c_5{i}_{s\beta }-c_6(2w-1)x_4+{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\alpha }\\ 0\end{array}\right]$$

In this case, the state space has a dimension of $n=5$. Consequently, the result of the Lie derivatives form a vector containing two components.

$${\mathit{L}}_{\mathit{f}}^k\mathit{h}=\left[\begin{array}{c}{\mathit{L}}_{\mathit{f}}^k{h}_1\\ {\mathit{L}}_{\mathit{f}}^k{h}_2\end{array}\right]$$

(13)

$${\mathit{L}}_{\mathit{f}}^0\mathit{h}=\mathit{h}=\left[\begin{array}{c}{i}_{s\alpha }\\ i_{s\beta }\end{array}\right]$$

(14)

$$\begin{array}{c}{\mathit{L}}_{\mathit{f}}\mathit{h}=\left[\begin{array}{c}{\mathit{L}}_{\mathit{f}}^k{h}_1\\ {\mathit{L}}_{\mathit{f}}^k{h}_2\end{array}\right]\hfill \\ =\left[\begin{array}{c}-c_1{i}_{s\alpha }+c_{2}w{\psi }_{r\alpha }+c_3{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\beta }+{\displaystyle \frac{u_{s\alpha }}{c_4}}\\ -c_1{i}_{s\beta }+c_{2}w{\psi }_{r\beta }-c_3{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\alpha }+{\displaystyle \frac{u_{s\beta }}{c_4}}\end{array}\right]\hfill \end{array}$$

(15)

Since the system order is $n=5$, it is required that the Lie derivatives are computed up to the fourth order, $k=4$. This leads to the construction of a criterion matrix of the size $10\times 5$, which is obtained as the Jacobian of the Lie derivatives:

$$\mathit{O}=\left[\begin{array}{ccccc}{\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^0{h}_1}{\partial i_{s\alpha }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^0{h}_1}{\partial i_{s\beta }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^0{h}_1}{\partial {\psi }_{s\alpha }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^0{h}_1}{\partial {\psi }_{s\beta }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^0{h}_1}{\partial v}}\\ {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^0{h}_2}{\partial i_{s\alpha }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^0{h}_2}{\partial i_{s\beta }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^0{h}_2}{\partial {\psi }_{s\alpha }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^0{h}_2}{\partial {\psi }_{s\beta }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^0{h}_2}{\partial v}}\\ {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}{h}_1}{\partial i_{s\alpha }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}{h}_1}{\partial i_{s\beta }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}{h}_1}{\partial {\psi }_{s\alpha }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}{h}_1}{\partial {\psi }_{s\beta }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}{h}_1}{\partial v}}\\ {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}{h}_2}{\partial i_{s\alpha }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}{h}_2}{\partial i_{s\beta }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}{h}_2}{\partial {\psi }_{s\alpha }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}{h}_2}{\partial {\psi }_{s\beta }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}{h}_2}{\partial v}}\\ \begin{array}{c}.\hfill \\ .\hfill \end{array}& \begin{array}{c}.\hfill \\ .\hfill \end{array}& \begin{array}{c}.\hfill \\ .\hfill \end{array}& \begin{array}{c}.\hfill \\ .\hfill \end{array}& \begin{array}{c}.\hfill \\ .\hfill \end{array}\\ {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^4{h}_1}{\partial i_{s\alpha }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^4{h}_1}{\partial i_{s\beta }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^4{h}_1}{\partial {\psi }_{s\alpha }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^4{h}_1}{\partial {\psi }_{s\beta }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^4{h}_1}{\partial v}}\\ {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^4{h}_2}{\partial i_{s\alpha }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^4{h}_2}{\partial i_{s\beta }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^4{h}_2}{\partial {\psi }_{s\alpha }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^4{h}_2}{\partial {\psi }_{s\beta }}}& {\displaystyle \frac{\partial {\mathit{L}}_{\mathit{f}}^4{h}_2}{\partial v}}\end{array}\right]$$

(16)

Based on the previously mentioned observability theorem, the observability matrix O must have full rank in order to ensure that the system is weakly locally observable: [27,28]

$$rank\left\{\mathit{O}\right\}=5$$

(17)

Based on the observability theory analysis of nonlinear systems and in combination with the conditions in linear induction motor systems, we have

$${\displaystyle \frac{d{\psi }_r}{dt}}\ne 0\Rightarrow rank\left\{\mathit{O}\right\}=5$$

(18)

where ${\psi }_r={\psi }_{r\alpha }+j{\psi }_{r\beta }$. A more detailed proof process can be obtained from Reference [27].

3.2. Design of Cascade Nonlinear Observer

In this subsection, the design of a sensorless cascade nonlinear observer for the linear induction motor is introduced. For a given nonlinear system, it is well known that there is no systematic approach to observer design. The linear induction motor model can be seen as an interconnection between two subsystems, where each of them can satisfy related required properties for designing an apposite observer [29].

Based on that, the model of the linear induction motor in (1)–(5) can be rewritten as the following form:

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{\dot{i}}_{s\alpha }\\ \dot{v}\end{array}\right]& =\left[\begin{array}{cc}0& c_3{\displaystyle \frac{n_p\pi }{h}}{\psi }_{r\beta }\\ 0& 0\end{array}\right]\left[\begin{array}{c}{i}_{s\alpha }\\ v\end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{c}-c_1{i}_{s\alpha }+c_2{\displaystyle \frac{\left(k/v\right)}{e^{-\left(k/v\right)}-1+\left(k/v\right)}}{\psi }_{r\alpha }+{\displaystyle \frac{u_{s\alpha }}{c_4}}\\ c_7\left(i_{s\beta }{\psi }_{r\alpha }-i_{s\alpha }{\psi }_{r\beta }\right)-{\displaystyle \frac{D}{M}}v-{\displaystyle \frac{T_L}{M}}\end{array}\right]\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{\dot{i}}_{s\beta }\\ {\dot{\psi }}_{r\alpha }\\ {\dot{\psi }}_{r\beta }\end{array}\right]& =\left[\begin{array}{ccc}-c_1& 0& 0\\ 0& c_6& 0\\ c_5& 0& c_6\end{array}\right]\left[\begin{array}{c}{i}_{s\beta }\\ {\psi }_{r\alpha }\\ {\psi }_{r\beta }\end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{c}{\displaystyle \frac{c_2\left(k/v\right)}{e^{-\left(k/v\right)}-1+\left(k/v\right)}}{\psi }_{r\beta }-c_3{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\alpha }+{\displaystyle \frac{u_{s\beta }}{c_4}}\\ c_5{i}_{s\alpha }-{\displaystyle \frac{2c_6\left(k/v\right)}{e^{-\left(k/v\right)}-1+\left(k/v\right)}}{\psi }_{r\alpha }-{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\beta }\\ -{\displaystyle \frac{2c_6\left(k/v\right)}{e^{-\left(k/v\right)}-1+\left(k/v\right)}}{\psi }_{r\beta }+{\displaystyle \frac{n_p\pi }{h}}v{\psi }_{r\alpha }\end{array}\right]\hfill \end{array}$$

(20)

We assume the following:

(1)

$c_1,c_2,c_3,c_4,c_5,c_6,c_7,$ and k are known and remain constant.

(2)

The load torque $T_L$ is constant.

The above Equations (19) and (20) can be rewritten in a compact interconnected form as follows:

$${\Sigma }_1\left\{\begin{array}{c}{\dot{X}}_1=A_1\left(u,y,X_2\right)X_1+g_1\left(u,y,X_2,X_1\right)\\ y_1=C_1{X}_1\end{array}\right.$$

(21)

$${\Sigma }_2\left\{\begin{array}{c}{\dot{X}}_2=A_2{X}_2+g_2\left(u,y,X_1,X_2\right)\\ y_2=C_2{X}_2\end{array}\right.$$

(22)

where $A_1\left(u,y,X_2\right)$,$A_2$,$g_1\left(u,y,X_2,X_1\right)$,$g_2\left(u,y,X_1,X_2\right)$ are

$$A_1\left(u,y,X_2\right)=\left[\begin{array}{cc}0& c_3{\displaystyle \frac{n_p\pi }{h}}{x}_4\\ 0& 0\end{array}\right],A_2=\left[\begin{array}{ccc}-c_1& 0& 0\\ 0& c_6& 0\\ c_5& 0& c_6\end{array}\right]$$

$$g_1\left(u,y,X_2,X_1\right)=\left[\begin{array}{c}-c_1{x}_1+{\displaystyle \frac{c_2\left(k/x_5\right)}{e^{-\left(k/x_5\right)}-1+\left(k/x_5\right)}}{x}_3+{\displaystyle \frac{u_{s\alpha }}{c_4}}\\ c_7\left(x_2{x}_3-x_1{x}_4\right)-{\displaystyle \frac{D}{M}}{x}_5-{\displaystyle \frac{1}{M}}{T}_L\end{array}\right]$$

$$g_2\left(u,y,X_1,X_2\right)=\left[\begin{array}{c}{\displaystyle \frac{c_2\left(k/x_5\right)}{e^{-\left(k/x_5\right)}-1+\left(k/x_5\right)}}{x}_4-c_3{\displaystyle \frac{n_p\pi }{h}}{x}_5{x}_3+{\displaystyle \frac{u_{s\beta }}{c_4}}\\ c_5{x}_1-{\displaystyle \frac{2c_6\left(k/x_5\right)}{e^{-\left(k/x_5\right)}-1+\left(k/x_5\right)}}{x}_3-{\displaystyle \frac{n_p\pi }{h}}{x}_5{x}_4\\ -{\displaystyle \frac{2c_6\left(k/x_5\right)}{e^{-\left(k/x_5\right)}-1+\left(k/x_5\right)}}{x}_4+{\displaystyle \frac{n_p\pi }{h}}{x}_5{x}_3\end{array}\right]$$

where $X_1={\left[\begin{array}{cc}{x}_1& x_5\end{array}\right]}^{\mathrm{T}}$ is the state of the first subsystem with the components $x_1=i_{s\alpha }$ and $x_5=v$; $X_2={\left[\begin{array}{ccc}{x}_2& x_3& x_4\end{array}\right]}^{\mathrm{T}}$ is the state of the second subsystem with the components $x_2=i_{s\beta }$, $x_3={\psi }_{r\alpha }$, and $x_4={\psi }_{r\beta }$; $u={\left[\begin{array}{cc}{u}_{s\alpha }& u_{s\alpha }\end{array}\right]}^{\mathrm{T}}$ is the control input and $y={\left[\begin{array}{cc}{y}_1& y_2\end{array}\right]}^{\mathrm{T}}$ is the measured output variables with $y_1=i_{s\alpha }$ and $y_2=i_{s\beta }$; and $C_1=\left[\begin{array}{cc}1& 0\end{array}\right]$ and $C_2=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]$.

Remark 1. 

The LIM physical operation domain D is defined by the set of values as follows:

$$\begin{array}{cc}\hfill \mathcal{D}=& \left\{X\in R^5\right.\left|\left|i_{s\alpha }\right|\le I_{\alpha }^{max},\left|i_{s\beta }\right|\le I_{\beta }^{max},\right.\hfill \\ \hfill & \left.\left|{\psi }_{r\alpha }\right|\le {\Psi }_{\alpha }^{max},\left|{\psi }_{r\beta }\right|\le {\Psi }_{\beta }^{max},\left|v\right|\le V^{max}\right\}\hfill \end{array}$$

(23)

where $X={\left[\begin{array}{ccccc}{i}_{s\alpha }& i_{s\beta }& {\psi }_{r\alpha }& {\psi }_{r\beta }& v\end{array}\right]}^T$ and $I_{\alpha }^{max}$, $I_{\beta }^{max}$, ${\Psi }_{\alpha }^{max}$, ${\Psi }_{\beta }^{max}$, and $V^{max}$ are the actual maximum values for stator currents, rotor fluxes, and LIM motor speed, respectively. These maximum values are determined by the linear induction motor specification table.

The main objective of this paper is to design an observer for Subsystem (21), which is based on a class of affine systems defined in the literature, and an estimator for Subsystem (22), by using only the measured stator currents and voltages of the LIM. The designed observer needs reconstruct the motor speed of the LIM, whereas the estimator is applied to estimate the rotor fluxes [29].

Defining $\theta :={\left[u,y,X_2\right]}^T$, Subsystem (21) can be rewritten in the following form:

$$\left\{\begin{array}{c}{\dot{X}}_1=A_1\left(\theta \right)X_1+g_1\left(\theta ,X_1\right)\\ y_1=C_1{X}_1\end{array}\right.$$

(24)

To design an approximate observer for Subsystem (6), the following assumptions are introduced.

Assumption 1. 

The following is clear:

1. θ is bounded and assumed to be regularly persistent to guarantee the observability property of the subsystem.

2. $A_1\left(X_2\right)$ is globally Lipschitz with respect to $X_2$ and uniformly so with respect to $\left(u,y\right)$.

3. $X_1\in {\mathcal{D}}_1$ of ${\Re }^{n_1}$ and $X_2\in {\mathcal{D}}_2$ of ${\Re }^{n_2}$, where $n_1$ and $n_2$ are the dimensions of Subsystems (21) and (22), respectively.

4. $g_1\left(u,y,X_2,X_1\right)$ is globally Lipschitz with respect to $X_1$ and uniformly so with respect to $\left(u,y,X_2\right)$.

5. $g_2\left(u,y,X_2,X_1\right)$ is globally Lipschitz with respect to $X_2$ and uniformly so with respect to $\left(u,y,X_1\right)$.

From Assumption 1, we introduce an observer for the form of Subsystem ${\Sigma }_1$ as follows:

$${\Sigma }_{O1}\left\{\begin{array}{c}\begin{array}{c}{\dot{Z}}_1=A_1\left(\theta \right)Z_1+g_1\left(\theta ,Z_1\right)+N\left(\theta \right)C_1\left(X_1-Z_1\right)\hfill \end{array}\\ {\widehat{y}}_1=C_1{Z}_1\end{array}\right.$$

(25)

where $Z_1={\left[\begin{array}{cc}{z}_1& z_5\end{array}\right]}^{\mathrm{T}}$ with $z_1={\widehat{i}}_{s\alpha }$, $z_5=\widehat{v}$, and the gains of the designed observer are given by

$$N\left(\theta \right)C_1={\mathsf{\Gamma }}^{-1}\left(\theta \right){\Delta }_{\kappa }^{-1}K{C}_1$$

(26)

where $\mathsf{\Gamma }=diag\left[\begin{array}{cc}1& \zeta \left(\theta \right)\end{array}\right]$ and ${\Delta }_{\kappa }=diag\left[\begin{array}{cc}{\displaystyle \frac{1}{\kappa }}& {\displaystyle \frac{1}{{\kappa }^2}}\end{array}\right]$ with $\zeta \left(\theta \right)=c_3{\displaystyle \frac{n_p\pi }{h}}{\widehat{x}}_4$, $\kappa >0$, and the matrix $K={\left[\begin{array}{cc}{K}_1& K_2\end{array}\right]}^T$ is such that the matrix $\left(\overline{A}-K{C}_1\right)$ is stable where $\overline{A}=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$.

From Assumption 1, Subsystem (24)’s observability property is satisfied; thus $x_4={\psi }_{r\beta }$ is not equal to zero except for a very short period of time (the linear induction motor requires to be fluxed with electromechanical energy conversion). In order to avoid a large gain of ${\mathsf{\Gamma }}^{-1}$, a discontinuity offset at zero for $x_4={\psi }_{r\beta }$ is introduced to LIM system.

Remark 2. 

The rotor fluxes $x_3={\psi }_{r\alpha }$ and $x_4={\psi }_{r\beta }$ are not measured variables. Then, we estimate them by the estimator of Subsystem (22) and the actual variables of rotor fluxes are replaced by their estimated values.

The observer for the subsystem ${\Sigma }_2$ is given by the following form:

$${\Sigma }_{O2}\left\{\begin{array}{c}{\dot{Z}}_2=A_2{Z}_2+g_2\left(u,y,Z_1,Z_2\right)\\ {\widehat{y}}_2=C_2{Z}_2\end{array}\right.$$

(27)

where $Z_2={\left[\begin{array}{ccc}{z}_2& z_3& z_4\end{array}\right]}^{\mathrm{T}}$ with $z_2={\widehat{i}}_{s\beta }$, $z_3={\widehat{\psi }}_{r\alpha }$, $z_4={\widehat{\psi }}_{r\beta }$.

The main idea of this article is to construct an estimator for the whole system $\Sigma$, given by Equations (21) and (22), from the separate observer design for each subsystem ${\Sigma }_i$, using the ${\Sigma }_{Oi}$ subsystem, which is an exponential observer for ${\Sigma }_i$, for $i=1,2$. After that, the interconnected LIM observer system ${\Sigma }_O$ is constituted by two cascade observers, ${\Sigma }_{Oi}$. Figure 1 presents the block diagram structure for the cascade nonlinear observer, where the cascaded structure of the observer can be seen.

3.3. Analysis of Cascade Nonlinear Observer Stability

In this subsection, we present the stability analysis of the designed cascade nonlinear observer. We define the estimate errors $e_1$ and $e_2$ as $e_1=X_1-Z_1$, $e_2=X_2-Z_2$. The first derivatives of $e_1$ and $e_2$ are given by

$$\begin{array}{cc}\hfill {\dot{e}}_1=& \left[A_1\left(u,y,Z_2\right)-N\left(u,y,Z_2\right)C_1\right]e_1\hfill \\ \hfill & +g_1\left(u,y,X_2,X_1\right)-g_1\left(u,y,Z_2,Z_1\right)\hfill \\ \hfill & +\left[A_1\left(u,y,X_2\right)-A_1\left(u,y,Z_2\right)\right]X_1\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill {\dot{e}}_2=& A_2{e}_2+g_2\left(u,y,X_1,X_2\right)-g_2\left(u,y,Z_1,Z_2\right)\hfill \end{array}$$

(29)

We make the definitions ${\mathrm{E}}_1=\mathsf{\Gamma }\left(u,y,Z_2\right){\Delta }_{\kappa }{e}_1$ and ${\mathrm{E}}_2={\displaystyle \frac{1}{{\kappa }^{n_1}}}{e}_2$. The first derivatives of ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$ are

$$\begin{array}{cc}\hfill {\dot{\mathrm{E}}}_1=& \left[\kappa \left(\overline{A}-K{C}_1\right)\right]{\mathrm{E}}_1+{\mathrm{T}}_1+{\mathrm{T}}_2+{\mathrm{T}}_3\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {\dot{\mathrm{E}}}_2=& A_2{\mathrm{E}}_2+{\displaystyle \frac{1}{{\kappa }^{n_1}}}\left[g_2\left(u,y,X_1,X_2\right)-g_2\left(u,y,Z_1,Z_2\right)\right]\hfill \end{array}$$

(31)

where ${\mathrm{T}}_1,{\mathrm{T}}_2,$ and ${\mathrm{T}}_3$ are

$$\begin{array}{cc}\hfill {\mathrm{T}}_1=& \mathsf{\Gamma }\left(u,y,Z_2\right){\Delta }_{\kappa }\left[g_1\left(u,y,X_2,X_1\right)-g_1\left(u,y,Z_2,Z_1\right)\right]\hfill \\ \hfill {\mathrm{T}}_2=& \mathsf{\Gamma }\left(u,y,Z_2\right){\Delta }_{\kappa }\left\{\left[A_1\left(u,y,Z_2\right)-A_1\left(u,y,Z_2\right)C_1\right]X_1\right\}\hfill \\ \hfill {\mathrm{T}}_3=& \dot{\mathsf{\Gamma }}\left(u,y,Z_2\right){\mathsf{\Gamma }}^{-1}\left(u,y,Z_2\right){\mathrm{E}}_1\hfill \end{array}$$

Assumption 2. 

Assume that

$$∥\dot{\mathsf{\Gamma }}\left(u,y,Z_2\right){\mathsf{\Gamma }}^{-1}\left(u,y,Z_2\right){\mathrm{E}}_1∥\le \rho$$

(32)

where ${\mathsf{\Gamma }}^{-1}$ is defined in Equation (26) and $\dot{\mathsf{\Gamma }}\left(u,y,Z_2\right)=diag\left[\begin{array}{cc}0& \lambda \end{array}\right]$, and λ is

$$\begin{array}{cc}\hfill \lambda =& c_3{\displaystyle \frac{n_p\pi }{h}}\left\{c_5{x}_2-c_6\left({\displaystyle \frac{2\left(k/{\widehat{x}}_5\right)}{e^{-\left(k/{\widehat{x}}_5\right)}-1+\left(k/{\widehat{x}}_5\right)}}-1\right){\widehat{x}}_4\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{n_p\pi }{h}}{\widehat{x}}_5{\widehat{x}}_3\right\}\hfill \end{array}$$

(33)

In this paper, the notation denotes the Euclidean norm (L2 norm) for vectors, defined as $∥x∥=\sqrt{{\sum }_{i=1}^n{x}_i^2}$ for the vector $x={\left[\begin{array}{cccc}{x}_1& x_2& \cdots & x_n\end{array}\right]}^T$. For matrices, $∥·∥$ is defined as ${∥\mathrm{A}∥}_F=\sqrt{{\sum }_{i=1}^m{\sum }_{j=1}^n{\left|a_{ij}\right|}^2}$ for the matrix $\mathrm{A}\in R^{m\times n}$. This assumption is verified because of the bounds of the considered state variables (Assumption 2) and the persistent. The norm notation $∥·∥$ used throughout this paper denotes the Euclidean norm for vectors and the Frobenius norm for matrices, unless otherwise specified.

Lemma 1. 

Once Assumptions 1 and 2 hold, then the system in (25)–(27) acts as an asymptotic observer for the system in (24)–(22). Furthermore, the convergence ratio of the estimate error $e={\left[\begin{array}{cc}{e}_1& e_2\end{array}\right]}^{\mathrm{T}}$ can be made as fast as the error dynamics governed by the subsystem estimator (27).

Proof. 

Consider the candidate Lyapunov function V as follows:

$$V=V_1+V_2$$

(34)

where $V_1={\mathrm{E}}_1^{T}P{\mathrm{E}}_1$ and $V_2={\mathrm{E}}_2^T{\mathrm{E}}_2$, and $P=P^T>0$; ${\left({\overline{A}}_1-K{C}_1\right)}^{T}P+P\left({\overline{A}}_1-K{C}_1\right)=-Q$; and $Q=Q^T>0$.

We calculate the first time derivative of V and we obtain

$$\begin{array}{cc}\hfill \dot{V}=& \kappa {\mathrm{E}}_1^T{\left(\overline{A}-K{C}_1\right)}^{T}P{\mathrm{E}}_1+\kappa {\mathrm{E}}_1^{T}P\left(\overline{A}-K{C}_1\right){\mathrm{E}}_1\hfill \\ \hfill & +2{\mathrm{E}}_1^{T}P\left({\mathrm{T}}_1+{\mathrm{T}}_2+{\mathrm{T}}_3\right)+{\mathrm{E}}_2^T\left(A_2+A_2^T\right){\mathrm{E}}_2\hfill \\ \hfill & +{\displaystyle \frac{2}{{\kappa }^{n_1}}}{\mathrm{E}}_2^T\left[g_2\left(u,y,X_1,X_2\right)-g_2\left(u,y,Z_1,Z_2\right)\right]\hfill \\ \hfill =& -\kappa {\mathrm{E}}_1^{T}Q{\mathrm{E}}_1\hfill \\ \hfill & +2{\mathrm{E}}_1^{T}P\left({\mathrm{T}}_1+{\mathrm{T}}_2+{\mathrm{T}}_3\right)+{\mathrm{E}}_2^T\left(A_2+A_2^T\right){\mathrm{E}}_2\hfill \\ \hfill & +{\displaystyle \frac{2}{{\kappa }^{n_1}}}{\mathrm{E}}_2^T\left[g_2\left(u,y,X_1,X_2\right)-g_2\left(u,y,Z_1,Z_2\right)\right]\hfill \end{array}$$

(35)

With Assumptions 1 and 2 we get

$$\begin{array}{cc}\hfill \dot{V}\le & -\kappa {\mathrm{E}}_1^{T}Q{\mathrm{E}}_1\hfill \\ \hfill & +2∥{\mathrm{E}}_1∥P\left({\displaystyle \frac{k_1}{{\kappa }^{n_1}}}∥e_1∥+{\displaystyle \frac{k_2{k}_x}{{\kappa }^{n_1}}}∥e_2∥+\rho ∥{\mathrm{E}}_1∥\right)\hfill \\ \hfill & +{\mathrm{E}}_2^T\left(A_2+A_2^T\right){\mathrm{E}}_2+{\displaystyle \frac{2}{{\kappa }^{n_1}}}∥{\mathrm{E}}_2∥\left(l_2∥e_2∥\right)\hfill \\ \hfill \le & -\kappa {\mathrm{E}}_1^{T}Q{\mathrm{E}}_1\hfill \\ \hfill & +2∥{\mathrm{E}}_1∥P\left(k_1{l}_1∥{\mathrm{E}}_1∥+k_2{k}_x∥{\mathrm{E}}_2∥+\rho ∥{\mathrm{E}}_1∥\right)\hfill \\ \hfill & +{\mathrm{E}}_2^T\left(A_2+A_2^T\right){\mathrm{E}}_2+{\displaystyle \frac{2}{{\kappa }^{n_1}}}∥{\mathrm{E}}_2∥\left({\kappa }^{n_1}{l}_2∥{\mathrm{E}}_2∥\right)\hfill \\ \hfill =& -\kappa {\mathrm{E}}_1^{T}Q{\mathrm{E}}_1\hfill \\ \hfill & +2∥{\mathrm{E}}_1∥P\left[\left(k_1{l}_1+\rho \right)∥{\mathrm{E}}_1∥+k_2{k}_x∥{\mathrm{E}}_2∥\right]\hfill \\ \hfill & +{\mathrm{E}}_2^T\left(A_2+A_2^T\right){\mathrm{E}}_2+2∥{\mathrm{E}}_2∥\left(l_2∥{\mathrm{E}}_2∥\right)\hfill \end{array}$$

(36)

Then

$$\begin{array}{cc}\hfill \dot{V}\le & -\left[\kappa {\mu }_1-2\left(k_1{l}_1+\rho \right)\right]∥P∥{∥{\mathrm{E}}_1∥}^2\hfill \\ \hfill & +\left(k_2{k}_x∥P∥\right)∥{\mathrm{E}}_1∥∥{\mathrm{E}}_2∥-\left({\mu }_2-2l_2\right){∥{\mathrm{E}}_2∥}^2\hfill \end{array}$$

(37)

where $\mu ,{\mu }_1,l_1,l_2,k_1,k_2,$ and $k_x$ are positive constants that are chosen to satisfy the Lipschitz conditions, the inequalities in (37), and the following equation:

$${\mu }_2=min\left[\begin{array}{cc}2c_1& 2c_6\left({\displaystyle \frac{2\left(k/x_5\right)}{e^{-\left(k/x_5\right)}-1+\left(k/x_5\right)}}-1\right)\end{array}\right]>0$$

We rewrite the terms of the Lyapunov functions $V_1$ and $V_2$ as follows:

$$\begin{array}{cc}\hfill \dot{V}\le & -\left[\kappa {\mu }_1-2\left(k_1{l}_1+\rho \right)\right]∥P∥V_1\hfill \\ \hfill & +\left(k_2{k}_x∥P∥\right)k_p\sqrt{V_1}\sqrt{V_2}-\left({\mu }_2-2l_2\right)V_2\hfill \end{array}$$

(38)

The above inequalities can be expressed with the following compact form:

$$\dot{V}\le -r_1{V}_1+r_2\sqrt{V_1}\sqrt{V_2}-r_3{V}_2$$

(39)

with

$$\begin{array}{cc}\hfill r_1=& \left[\kappa {\mu }_1-2\left(k_1{l}_1+\rho \right)\right]∥P∥>0\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill r_2=& \left(k_2{k}_x∥P∥\right)k_p>0\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill r_3=& \left({\mu }_2-2l_2\right)>0\hfill \end{array}$$

(42)

Finally, with $\alpha >0$, we obtain

$$\begin{array}{cc}\hfill \dot{V}\le & -\alpha V_2-r_1{V}_1+r_2\sqrt{V_1}\sqrt{V_2}-\left(r_3-\alpha \right)V_2\hfill \\ \hfill \le & -\alpha V_2-\left(r_1-{\displaystyle \frac{r_2^2}{4{\left(r_3-\alpha \right)}^2}}\right)V_1\hfill \\ \hfill & -{\left[{\displaystyle \frac{r_2}{2\left(r_3-\alpha \right)}}\sqrt{V_1}-\sqrt{\left(r_3-\alpha \right)}\sqrt{V_2}\right]}^2\hfill \end{array}$$

(43)

where $\alpha$ is such that $\left(r_3-\alpha \right)>0$.

Taking $\beta =\left(r_1-{\displaystyle \frac{r_2^2}{4{\left(r_3-\alpha \right)}^2}}\right)>0$, and letting $\epsilon =min\left(\alpha ,\beta \right)$, we then have

$$\begin{array}{cc}\hfill \dot{V}\le & -\epsilon \left(V_1+V_2\right)\hfill \\ \hfill \le & -\epsilon V\hfill \end{array}$$

(44)

That is the end of the proof. □

4. Adaptive Twisting Controller Design

In the LIM system, the mathematical relationship between $i_{s\alpha },i_{s\beta },{\psi }_{r\alpha },{\psi }_{r\beta },u_{s\alpha },$ and $u_{s\beta }$ and $i_{ds},i_{qs},{\psi }_{dr},{\psi }_{qr},u_{ds},$ and $u_{qs}$ can be expressed as

$$\left[\begin{array}{c}{i}_{ds}\\ i_{qs}\end{array}\right]=\left[\begin{array}{cc}cos\rho & sin\rho \\ -sin\rho & cos\rho \end{array}\right]\left[\begin{array}{c}{i}_{s\alpha }\\ i_{s\beta }\end{array}\right]$$

(45)

$$\left[\begin{array}{c}{\psi }_{dr}\\ {\psi }_{qr}\end{array}\right]=\left[\begin{array}{cc}cos\rho & sin\rho \\ -sin\rho & cos\rho \end{array}\right]\left[\begin{array}{c}{\psi }_{r\alpha }\\ {\psi }_{r\beta }\end{array}\right]$$

(46)

$$\left[\begin{array}{c}{u}_{ds}\\ u_{qs}\end{array}\right]=\left[\begin{array}{cc}cos\rho & sin\rho \\ -sin\rho & cos\rho \end{array}\right]\left[\begin{array}{c}{u}_{s\alpha }\\ u_{s\beta }\end{array}\right]$$

(47)

where $\rho$ is the induced-part flux angle, and ${\omega }_{mr}=\dot{\rho }={\displaystyle \frac{n_p\pi }{h}}v+c_5{\displaystyle \frac{i_{qs}}{{\psi }_{dr}}}$ is the induced-part flux vector rotational speed.

By using the indirect field-oriented control (IRFOC) strategy [30] in the LIM system (${\psi }_{qr}=0,{\psi }_{dr}={\psi }_r$), the state-space Equations (1)–(5) of the LIM can be expressed in the inductor-part flux reference frame ($d,q$ axis) as follows:

$${\dot{i}}_{ds}=-c_1{i}_{ds}+{\displaystyle \frac{n_p\pi }{h}}v{i}_{qs}+c_5{\displaystyle \frac{i_{qs}^2}{{\psi }_r}}+c_{2}w{\psi }_r+{\displaystyle \frac{u_{ds}}{c_4}}$$

(48)

$${\dot{i}}_{qs}=-c_1{i}_{qs}-{\displaystyle \frac{n_p\pi }{h}}v{i}_{ds}-c_5{\displaystyle \frac{i_{ds}{i}_{qs}}{{\psi }_r}}-c_3{\displaystyle \frac{n_p\pi }{h}}v{\psi }_r+{\displaystyle \frac{u_{qs}}{c_4}}$$

(49)

$${\dot{\psi }}_r=c_5{i}_{ds}-c_6(2w-1){\psi }_r$$

(50)

$$\dot{\rho }={\displaystyle \frac{n_p\pi }{h}}v+c_5{\displaystyle \frac{i_{qs}}{{\psi }_r}}$$

(51)

$$\dot{v}=c_7{\psi }_r{i}_{qs}-{\displaystyle \frac{D}{M}}v-{\displaystyle \frac{1}{M}}{T}_L$$

(52)

The control objective is to force the LIM speed v and rotor flux ${\psi }_r$ to track the desired LIM speed ${\psi }_r^{*}$ and rotor flux $v^{*}$. By using IRFOC theory, twisting sliding mode controllers with an adaptation algorithm [21] are successfully applied to the LIM system. Here, we define two tracking errors, $s_{\psi 1}$ and $s_{v1}$, as

$$\begin{array}{cc}\hfill s_{\psi 1}& ={\psi }_r-{\psi }_r^{*}\hfill \\ \hfill & ={\psi }_r-{\widehat{\psi }}_r+{\widehat{\psi }}_r-{\psi }_r^{*}\hfill \\ \hfill & ={\widehat{s}}_{\psi 1}+e_{\psi }\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill s_{v1}& =v-v^{*}\hfill \\ \hfill & =v-\widehat{v}+\widehat{v}-v^{*}\hfill \\ \hfill & ={\widehat{s}}_{v1}+e_v\hfill \end{array}$$

(54)

where ${\psi }_r^{*}$, $v^{*}$ are the desired rotor flux and desired LIM speed, respectively; ${\widehat{s}}_{\psi 1}={\widehat{\psi }}_r-{\psi }_r^{*}$ and ${\widehat{s}}_{v1}=\widehat{v}-v^{*}$; $e_v=v-\widehat{v}$ is the estimated speed error; and $e_{\psi }={\psi }_r-{\widehat{\psi }}_r$ is the estimated flux error, where ${\psi }_r$ and ${\widehat{\psi }}_r$ can be calculated:

$${\psi }_r=\sqrt{{\psi }_{r\alpha }^2+{\psi }_{r\beta }^2}$$

(55)

$${\widehat{\psi }}_r=\sqrt{{\widehat{\psi }}_{r\alpha }^2+{\widehat{\psi }}_{r\beta }^2}$$

(56)

From the stability analysis of the cascade nonlinear observer designed above, $\underset{t\to \infty }{lim}{e}_{{\psi }_{r\alpha }}\left(t\right)={\psi }_{r\alpha }\left(t\right)-{\widehat{\psi }}_{r\alpha }\left(t\right)=0$; $\underset{t\to \infty }{lim}{e}_{{\psi }_{r\beta }}\left(t\right)={\psi }_{r\beta }\left(t\right)-{\widehat{\psi }}_{r\beta }\left(t\right)=0$; and $\underset{t\to \infty }{lim}{e}_v\left(t\right)=v\left(t\right)-\widehat{v}\left(t\right)=0$. Obviously, we obtain $\underset{t\to \infty }{lim}{e}_{\psi }\left(t\right)={\psi }_r\left(t\right)-{\widehat{\psi }}_r\left(t\right)=0$ and $\underset{t\to \infty }{lim}{e}_v\left(t\right)=e_5\left(t\right)=0$.

The first derivatives of $s_{\psi 1}$ and $s_{v1}$ are

$$\begin{array}{cc}\hfill {\dot{s}}_{\psi 1}=& {\dot{\psi }}_r-{\dot{\psi }}_r^{*}\hfill \\ \hfill =& c_5{i}_{ds}-c_6(2w-1)s_{\psi 1}-c_6(2w-1){\psi }_r^{*}-{\dot{\psi }}_r^{*}\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill {\dot{s}}_{v1}=& \dot{v}-{\dot{v}}^{*}\hfill \\ \hfill =& c_7{\psi }_r{i}_{qs}-{\displaystyle \frac{D}{M}}{s}_{v1}-{\displaystyle \frac{D}{M}}{v}^{*}-{\displaystyle \frac{1}{M}}{T}_L-{\dot{v}}^{*}\hfill \end{array}$$

(58)

The second derivatives of $s_{\psi 1}$ and $s_{v1}$ are

$$\begin{array}{cc}\hfill {\ddot{s}}_{\psi 1}=& c_5{\dot{i}}_{ds}-c_6(2w-1){\dot{s}}_{\psi 1}-c_6(2w-1){\dot{\psi }}_r^{*}-{\ddot{\psi }}_r^{*}\hfill \\ \hfill =& -c_6(2w-1){\dot{s}}_{\psi 1}+c_2{c}_{5}w\left(s_{\psi 1}+{\psi }_r^{*}\right)\hfill \\ \hfill & +c_5^2{\displaystyle \frac{i_{qs}^2}{\left(s_{\psi 1}+{\psi }_r^{*}\right)}}-c_1{c}_5{i}_{ds}+c_5{\displaystyle \frac{n_p\pi }{h}}v{i}_{qs}\hfill \\ \hfill & -c_6(2w-1){\dot{\psi }}_r^{*}-{\ddot{\psi }}_r^{*}+{\displaystyle \frac{c_5}{c_4}}{u}_{ds}\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill {\ddot{s}}_{v1}=& c_7{\dot{\psi }}_r{i}_{qs}+c_7{\psi }_r{\dot{i}}_{qs}-{\displaystyle \frac{D}{M}}{\dot{s}}_{v1}-{\displaystyle \frac{D}{M}}{\dot{v}}^{*}-{\displaystyle \frac{1}{M}}{\dot{T}}_L-{\ddot{v}}^{*}\hfill \\ \hfill =& -{\displaystyle \frac{D}{M}}{\dot{s}}_{v1}-\left[c_7{\displaystyle \frac{n_p\pi }{h}}{i}_{ds}{\psi }_r+c_3{c}_7{\displaystyle \frac{n_p\pi }{h}}{\psi }_r^2\right]s_{v1}\hfill \\ \hfill & -\left[c_7{\displaystyle \frac{n_p\pi }{h}}{i}_{ds}{\psi }_r+c_3{c}_7{\displaystyle \frac{n_p\pi }{h}}{\psi }_r^2\right]v^{*}\hfill \\ \hfill & -c_6{c}_7(2w-1)i_{qs}{\psi }_r-c_1{c}_7{i}_{qs}{\psi }_r\hfill \\ \hfill & -{\displaystyle \frac{D}{M}}{\dot{v}}^{*}-{\displaystyle \frac{1}{M}}{\dot{T}}_L-{\ddot{v}}^{*}+{\displaystyle \frac{c_7{\psi }_r}{c_4}}{u}_{qs}\hfill \end{array}$$

(60)

We define two new variables, ${\widehat{s}}_{\psi 2}$ and ${\widehat{s}}_{v2}$, as ${\widehat{s}}_{\psi 2}={\dot{\widehat{s}}}_{\psi 1}={\dot{\widehat{\psi }}}_r-{\dot{\psi }}_r^{*}$ and ${\widehat{s}}_{v2}={\dot{\widehat{s}}}_{v1}=\dot{\widehat{v}}-{\dot{v}}^{*}$; we have

$$\begin{array}{cc}\hfill {\dot{\widehat{s}}}_{\psi 1}=& {\widehat{s}}_{\psi 2}\hfill \\ \hfill {\dot{\widehat{s}}}_{\psi 2}=& {\delta }_{\psi }\left(t,{\widehat{s}}_{\psi 1},{\widehat{s}}_{\psi 2}\right)+{\displaystyle \frac{c_5}{c_4}}{u}_{ds}\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill {\dot{\widehat{s}}}_{v1}=& {\widehat{s}}_{v2}\hfill \\ \hfill {\dot{\widehat{s}}}_{v2}=& {\delta }_v\left(t,{\widehat{s}}_{v1},{\widehat{s}}_{v2}\right)+{\displaystyle \frac{c_7{\psi }_r}{c_4}}{u}_{qs}\hfill \end{array}$$

(62)

where the disturbances ${\delta }_{\psi }\left(t,{\widehat{s}}_{\psi 1},{\widehat{s}}_{\psi 2}\right)$ and ${\delta }_v\left(t,{\widehat{s}}_{v1},{\widehat{s}}_{v2}\right)$ are

$$\begin{array}{cc}\hfill {\delta }_{\psi }\left(t,{\widehat{s}}_{\psi 1},{\widehat{s}}_{\psi 2}\right)=& c_2{c}_{5}w{\widehat{s}}_{\psi 1}-c_6(2w-1){\widehat{s}}_{\psi 2}\hfill \\ \hfill & +c_2{c}_{5}w\left(e_{\psi }+{\psi }_r^{*}\right)+c_5^2{\displaystyle \frac{i_{qs}^2}{\left(s_{\psi 1}+{\psi }_r^{*}\right)}}\hfill \\ \hfill & -c_6(2w-1)\left({\dot{e}}_{\psi }+{\dot{\psi }}_r^{*}\right)-c_1{c}_5{i}_{ds}\hfill \\ \hfill & +c_5{\displaystyle \frac{n_p\pi }{h}}v{i}_{qs}-{\ddot{\psi }}_r^{*}\hfill \end{array}$$

(63)

$$\begin{array}{cc}\hfill {\delta }_v\left(t,{\widehat{s}}_{v1},{\widehat{s}}_{v2}\right)=& -\left[c_7{\displaystyle \frac{n_p\pi }{h}}{i}_{ds}{\psi }_r+c_3{c}_7{\displaystyle \frac{n_p\pi }{h}}{\psi }_r^2\right]{\widehat{s}}_{v1}\hfill \\ \hfill & -{\displaystyle \frac{D}{M}}{\widehat{s}}_{v2}-{\displaystyle \frac{D}{M}}\left({\dot{e}}_v+{\dot{v}}^{*}\right)-c_1{c}_7{i}_{qs}{\psi }_r\hfill \\ \hfill & -\left[c_7{\displaystyle \frac{n_p\pi }{h}}{i}_{ds}{\psi }_r+c_3{c}_7{\displaystyle \frac{n_p\pi }{h}}{\psi }_r^2\right]\left(e_v+v^{*}\right)\hfill \\ \hfill & -c_6{c}_7(2w-1)i_{qs}{\psi }_r-{\displaystyle \frac{1}{M}}{\dot{T}}_L-{\ddot{v}}^{*}\hfill \end{array}$$

(64)

where ${\delta }_{\psi }\left(t,{\widehat{s}}_{\psi 1},{\widehat{s}}_{\psi 2}\right)$ and ${\delta }_v\left(t,{\widehat{s}}_{v1},{\widehat{s}}_{v2}\right)$ are two bounded disturbances with the unknown boundaries $d_1\left(t\right)\in \left[0,D_1\right]$, $D_1>0$ and $d_2\left(t\right)\in \left[0,D_2\right]$, $D_2>0$, which satisfy

$$\left|{\delta }_{\psi }\left(t,{\widehat{s}}_{\psi 1},{\widehat{s}}_{\psi 2}\right)\right|\le d_1\left(t\right)\le D_1$$

(65)

$$\left|{\delta }_v\left(t,{\widehat{s}}_{v1},{\widehat{s}}_{v2}\right)\right|\le d_2\left(t\right)\le D_2$$

(66)

The ATCs $u_{ds}$ and $u_{qs}$ are designed as follows:

$$u_{ds}=-{\alpha }_d\left[sign\left({\widehat{s}}_{\psi 1}\right)+0.5sign\left({\widehat{s}}_{\psi 2}\right)\right]$$

(67)

$$u_{qs}=-{\alpha }_q\left[sign\left({\widehat{s}}_{v1}\right)+0.5sign\left({\widehat{s}}_{v2}\right)\right]$$

(68)

where the adaptive gains ${\alpha }_d$ and ${\alpha }_q$ are obtained through [21]

$${\dot{\alpha }}_d=\left\{\begin{array}{c}\begin{array}{ccc}{\chi }_{1d},& if& {\alpha }_d\ge {\alpha }_{min,d}\end{array}\\ \begin{array}{ccc}{\chi }_{2d},& if& {\alpha }_d<{\alpha }_{min,d}\end{array}\end{array}\right.$$

(69)

$${\dot{\alpha }}_q=\left\{\begin{array}{c}\begin{array}{ccc}{\chi }_{1q},& if& {\alpha }_q\ge {\alpha }_{min,q}\end{array}\\ \begin{array}{ccc}{\chi }_{2q},& if& {\alpha }_q<{\alpha }_{min,q}\end{array}\end{array}\right.$$

(70)

with

$$\begin{array}{cc}\hfill {\chi }_{1d}=& {\displaystyle \frac{\frac{{\omega }_{1d}}{\sqrt{2{\gamma }_{1d}}}}{\frac{1}{{\gamma }_{1d}}-\frac{2{\alpha }_d{\widehat{s}}_{\psi 1}^2+\left|{\widehat{s}}_{\psi 1}\right|s_{\psi 2}^2}{{\left({\alpha }_d^{*}-{\alpha }_d\right)}^3}}}sign\left(V_0\left({\widehat{s}}_{\psi 1},{\widehat{s}}_{\psi 2}\right)-{\mu }_d\right)\hfill \end{array}$$

(71)

$$\begin{array}{cc}\hfill {\chi }_{1q}=& {\displaystyle \frac{\frac{{\omega }_{1q}}{\sqrt{2{\gamma }_{1q}}}}{\frac{1}{{\gamma }_{1q}}-\frac{2{\alpha }_q{\widehat{s}}_{v1}^2+\left|{\widehat{s}}_{v1}\right|e_{v2}^2}{{\left({\alpha }_q^{*}-{\alpha }_q\right)}^3}}}sign\left(V_0\left({\widehat{s}}_{v1},{\widehat{s}}_{v2}\right)-{\mu }_q\right)\hfill \end{array}$$

(72)

where

$$\begin{array}{cc}\hfill V_0\left({\widehat{s}}_{\psi 1},{\widehat{s}}_{\psi 2}\right)=& {\alpha }_d^2{\widehat{s}}_{\psi 1}^2+{\gamma }_d{\left|{\widehat{s}}_{\psi 1}\right|}^{3/2}sign\left({\widehat{s}}_{\psi 1}\right){\widehat{s}}_{\psi 2}\hfill \\ \hfill & +{\alpha }_d\left|{\widehat{s}}_{\psi 1}\right|{\widehat{s}}_{\psi 2}^2+{\displaystyle \frac{1}{4}}{\widehat{s}}_{\psi 2}^4\hfill \end{array}$$

(73)

$$\begin{array}{cc}\hfill V_0\left({\widehat{s}}_{v1},{\widehat{s}}_{v2}\right)=& {\alpha }_q^2{\widehat{s}}_{v1}^2+{\gamma }_q{\left|{\widehat{s}}_{v1}\right|}^{3/2}sign\left({\widehat{s}}_{v1}\right){\widehat{s}}_{v2}\hfill \\ \hfill & +{\alpha }_q\left|{\widehat{s}}_{v1}\right|{\widehat{s}}_{v2}^2+{\displaystyle \frac{1}{4}}{\widehat{s}}_{v2}^4\hfill \end{array}$$

(74)

By choosing appropriate parameters for the adaptation algorithm, the ATCs can achieve finite-time convergence to the vicinity of zero, despite the presence of two bounded disturbances, ${\delta }_{\psi }\left(t,{\widehat{s}}_{\psi 1},{\widehat{s}}_{\psi 2}\right)$ and ${\delta }_v\left(t,{\widehat{s}}_{v1},{\widehat{s}}_{v2}\right)$, with unknown boundaries [21].

5. Hardware-in-the-Loop Experiment

The proposed cascade nonlinear observer-based adaptive twisting control scheme for the LIM was tested in a hardware-in-the-loop (HIL) experiment. The overall architecture of the proposed CNO-based ATC scheme for the LIM is shown in Figure 2. It mainly consists of two DS1104 Real-Time (RT) board cards: one for the LIM controller/observer application and the other for the emulation of the LIM [31,32]. The HIL experiment test bench was implemented in real time with the sampling frequency of 10 kHz. The nominal parameters of the LIM are given in

Table 1. Parameters of linear induction motor.

Symbol	Quantity	Conversion from Gaussian and CGS EMU to ${\mathit{SI}}^{\mathit{a}}$
$R_s$	inductor resistance	$11\Omega$
$R_r$	induced-part resistance	$32.57\Omega$
$L_s$	inductor inductances	$0.6376$ H
$L_m$	3-phase magnetizing inductance	$0.5175$ H
M	primary mass	20 Kg
D	vicious friction	20 m/s2
$n_p$	pole pairs	3
${\tau }_m$	inductor length	$0.15$ m
h	pole pitch	$0.1$ m

.

In the LIM system, the initial values of the state variables were set as $x_1\left(0\right)=i_{s\alpha }\left(0\right)=0.1$ A; $x_2\left(0\right)=i_{s\beta }\left(0\right)=0.1$ A; $x_3\left(0\right)={\psi }_{r\alpha }\left(0\right)=0.1$ Wb; and $x_4\left(0\right)={\psi }_{r\beta }\left(0\right)=0.1$ Wb. In the proposed cascade nonlinear observer, the designed parameters were chosen as $K_1=0.1$; $K_2=0.01$; and $\kappa =100$.

In the rotor flux adaptive twisting controller, the initial value of ${\alpha }_d$ was set as ${\alpha }_d\left(0\right)=50$, and the designed parameters of the adaptation algorithm were chosen as ${\gamma }_d=2$; ${\alpha }_d^{*}=100$; ${\omega }_{1d}=200$; ${\alpha }_{min,d}=20$; ${\gamma }_{1d}=20$; and ${\mu }_d=0.001$, ${\chi }_{2d}=20$. In the LIM speed adaptive twisting controller, the initial value of ${\alpha }_q$ was set as ${\alpha }_q\left(0\right)=50$, and the designed parameters of the adaptation algorithm were chosen as ${\gamma }_q=2$; ${\alpha }_q^{*}=100$; ${\omega }_{1q}=200$; ${\alpha }_{min,q}=20$; ${\gamma }_{1q}=20$; ${\mu }_q=0.001$; and ${\chi }_{2q}=20$.

5.1. CNO Performance

To better reflect the observer performance of the designed cascade nonlinear observer, the initial values of the estimated currents and fluxes were chosen differently from those of the initial state variables, and their values were set as ${\widehat{x}}_1\left(0\right)={\widehat{i}}_{s\alpha }\left(0\right)=0.5$ A; ${\widehat{x}}_2\left(0\right)={\widehat{i}}_{s\beta }\left(0\right)=0.6$ A; ${\widehat{x}}_3\left(0\right)={\widehat{\psi }}_{r\alpha }\left(0\right)=0.45$ Wb; and ${\widehat{x}}_4\left(0\right)={\widehat{\psi }}_{r\beta }\left(0\right)=0.55$ Wb.

Figure 3a,b show the actual and the estimated rotor fluxes, obtained with the proposed cascade nonlinear observer, under speed square references of $0.5$ m/s. Correspondingly, the LIM speed estimation performance is shown in Figure 3c. In this comparative study, we conducted a systematic experimental analysis of the performance of three types of observation algorithms, namely the local cascade observation algorithm, the sliding mode observer (SMO), and the Luenberger observer. This cascade observation algorithm has higher estimation accuracy. This advantage mainly stems from its innovative multi-level signal coupling mechanism, which effectively suppresses noise interference and improves the accuracy of parameter identification.

5.2. CNO-ATC Performance

In this subsection, we detail how the designed CNO-based STC scheme was implemented on a hardware-in-the-loop test bench. The reference speed signal was set as $0.4$ m/s, and the reference flux modulus signal was set as $1.533$ Wb². The initial estimated states in the proposed cascade observer were set as ${\widehat{i}}_{s\alpha }\left(0\right)=0.5$ A; ${\widehat{i}}_{s\beta }\left(0\right)=0.6$ A; ${\widehat{\psi }}_{r\alpha }\left(0\right)=0.45$ Wb; and ${\widehat{\psi }}_{r\beta }\left(0\right)=0.55$ Wb.

To test the robustness property of the proposed CNO-ATC scheme, the value of the load torque $T_L$ was defined as follows:

$$T_L=\left\{\begin{array}{c}0\mathrm{N},0\le t\le 3\mathrm{s}\\ 100\mathrm{N},3\mathrm{s}<t\le 5\mathrm{s}\\ 40\mathrm{N},5\mathrm{s}<t\le 8\mathrm{s}\end{array}\right.$$

(75)

5.2.1. Nominal System

Figure 4a presents the trajectories of the reference speed, the real speed, and the estimated speed in the proposed observer-based controller scheme. One can see that the estimated speed and tracking speed reached the steady state with a great dynamic response in less than 0.2 s. There existed little speed variation at $t=3$ s and $t=5$ s due to abrupt load variation, while the estimated speed accurately tracked the real speed with high estimation accuracy.

Figure 4b presents the trajectories of the reference modulus flux, the real modulus flux, and the estimated modulus flux in the proposed cascade nonlinear observer-based ATC scheme. One can see that estimated modulus flux and tracking modulus flux reached the steady state with a good dynamic response in less than 0.8 s. Similarly, there existed little modulus flux variation at $t=3$ s and $t=5$ s, although the variation was very small and the estimation accuracy remained high.

5.2.2. Perturbed System

In this subsection, considering the influence of parametric uncertainties, we assume the actual ${\overline{R}}_s$ to be $R_s+\Delta R_s$, where $\Delta R_s=0.3R_s$.

Figure 5a presents the trajectories of the reference speed, the real speed, and the estimated speed in the proposed cascade nonlinear observer-based ATC scheme. One can see that the estimated speed and tracking speed reached the steady state with a great dynamic response in less than 0.2 s. There existed small speed variation at $t=3$ s and $t=5$ s due to abrupt load variation, while the estimated speed accurately tracked the real speed with high estimation accuracy.

Figure 5b presents the trajectories of the reference modulus flux, the real modulus flux, and the estimated modulus flux in the proposed cascade nonlinear observer-based ATC scheme. One can see that estimated modulus flux and tracking modulus flux reached the steady state with a good dynamic response in less than 0.15 s. Similarly, there existed small modulus flux variation at $t=3$ s and $t=5$ s, although the variation was very small and the estimation accuracy was very high.

6. Conclusions

In this paper, two adaptive twisting controllers were designed for the LIM drive without mechanical sensors (speed and load torque sensors). A cascade observer was designed and applied to the linear induction motor, where the rotor fluxes and LIM speed were estimated. The proposed cascade nonlinear observer was utilized by the adaptive twisting controller in a closed loop to achieve LIM speed and flux tracking for the linear induction motor without requiring measurements of LIM speed and rotor fluxes. A detailed analysis of the convergence of the cascade nonlinear observer was presented. The CNO was tested and validated with the ATC in a closed-loop configuration using a hardware-in-the-loop experiment test bench. HIL results showed that the convergence of the designed observer was achieved, and excellent performance was obtained in the presence of variations in the rotor resistance. Moreover, the ATC-based CNO methodology can achieve accurate LIM speed and rotor flux tracking in the absence of mechanical sensors under different operating conditions.