Higher-Order PID-Nested Nonsingular Terminal Sliding Mode Control for Induction Motor Speed Servo Systems

Abstract

This paper presents an approach to the velocity control loop of induction motor drives utilizing the Higher-Order PID-Nested Nonsingular Terminal Sliding Mode (PID-NTSM) method. Here, the PID-NTSM sliding manifold is formulated by the incorporation of both derivative and integral errors of states into the conventional nonsingular terminal sliding mode surface (NTSM). In this manner, the control signals take the higher-order sliding mode control law, obtained by multiple integrals. In this way, such signals are continuous, and the sliding manifold is obtained in finite time; the system’s states asymptotically converge chattering-free to zero at a much faster response time and higher tracking precision while maintaining inherited robustness characteristics. The effectiveness of the proposed method is comprehensively validated both numerically and experimentally.

1. Introduction

The induction motor (IM) plays a crucial role in electric drive systems due to its outstanding advantages, including high durability, low cost, and efficient operation [1]. With a simple structure and low maintenance, IM accounts for the largest proportion of industrial motors, especially in applications requiring high power and continuous operation [1,2,3,4]. Induction motor control is controlled based on two main control methods, Field Oriented Control (FOC) and Direct Torque Control (DTC) [5,6,7,8]. The FOC method uses a cascade control method to control the speed and current of the induction motor independently, with the outermost loop controlling the motor speed, and the inner loops controlling the current and rotor flux. This approach allows precise torque control and fast response in applications requiring high efficiency. The asynchronous motor is chosen as the object of investigation because it is a typical nonlinear system with many practical challenges in control, and is suitable for verifying the effectiveness of the proposed method.

The open literature reports numerous methods for velocity control of induction motor FOC drives, i.e., PI control, adaptive control, and robust control strategies [9,10,11,12,13], including the utilization of sliding mode control approaches, i.e., conventional SMC, terminal SMC, and nonsingular terminal SMC [14,15,16]. The majority of these reported SMC-based induction motor control methods are very much at the simulation level. That is because conventional linear SMC methods suffer from their intrinsic “chattering” phenomenon that limits their application in practical motor drives since these undesired high-frequency oscillating dynamics would result in severe damage to the drive’s associated power electronics circuitry as well as connected mechanical system, i.e., transmission, gearbox, etc. [16,17].

Often, for “chattering” reduction/elimination, low-pass filters or boundary layer approximation are integrated into conventional SMC controllers to smooth the respective control signals [18,19] despite losing SMC’s inherent robustness properties as well as control precision and accuracy losses. In recent years, Terminal Sliding Mode (TSM) approaches to the issues mentioned above have attracted increasing attention from the motion control research community (i.e., [20,21,22]). The work reported in [20] demonstrates that TSM can achieve finite-time convergence and enhanced control precision compared to classical SMC. Building upon this foundation, NTSM [23,24] was introduced to avoid TSM’s singularity. These methods, despite improving accuracy and achieving finite-time convergence, still exhibit the “chattering” phenomenon.

Among popular techniques to overcome this is to rely upon higher-order sliding mode control (HOSMC) methods, i.e., [25,26,27]. The main idea of HOSMC is to place discontinuous switching elements at higher-order derivatives of the control inputs. In this manner, the obtained control inputs are smooth and continuous [21,28,29]. Along with the development of higher-order TSM methods, hierarchical (nested) TSM manifolds are introduced [30]. PI-nested TSM (PI-TSM), which incorporates the proportional–integral errors of states into the TSM sliding manifold, has been introduced, i.e., [31]. This approach, nevertheless, would not be able to formulate a higher-order TSM since the discontinuous switching function, like the TSM and NTSM methods, is still embedded in the first-order derivatives.

Further investigation would propose the introduction of PID (proportional–integral–derivative) errors into the TSM/NTSM manifold, which formulates a PID-nested TSM/NTSM approach (PID-NTSM). Within this structure, incorporating both derivative and integral errors of states into the nonsingular terminal sliding manifolds raises the order of the sliding surface. Here, the NTSM sliding surface is reached in finite time, and the error converges asymptotically to zero, governed by the aforementioned nested PID dynamics.

Within the cascaded control structure of an electrical motor drive, placing PID-nested TSM controllers at the velocity control loop naturally raises the order of the NTSM sliding manifold (being second-order instead of first-order as in classical TSM/NTSM approaches). In such a manner, the discontinuous switching elements are placed at the control inputs’ second derivatives. As a result, the “chattering” phenomenon would be significantly reduced or even eliminated, enabling the application of such PID-NTSM methods in practical applications in induction motor drives. This is the original motivation and contribution of the reported work.

This paper is organized as follows. Materials and methods are presented in Section 2, including the PID-NTSM control method for nonlinear systems, and application to the velocity control loop of induction motor FOC drives, which is the main contribution of this study. Simulation and experimental results are illustrated in Section 3, demonstrating the efficiency and merits of the proposed controller. Finally, Section 4 concludes the paper.

2. Materials and Methods

2.1. Higher-Order PID-NTSM Control for a Nonlinear System

Consider the first-order system as Equation (1).

$$\dot{x}=f(x,t)+g(x,t)u+d(x,t)$$

(1)

Assumption 1.

The second derivative of d(t) in Equation (1) is bounded by

$$\left|\ddot{d}(x,t)\right|\le D_{\max }$$

(2)

where D_max > 0 is a constant. This assumption is quite suitable in practical applications. This study proposes the integration of the PID sliding surface into the NTSM sliding surface for a comprehensive evaluation of system error. The proposed sliding surface is as (3-4), which is the main contribution of this study.

$$l=s+\delta {\dot{s}}^{p/q}$$

(3)

$$s={\zeta }_1{x}_1(t)+{\zeta }_2{\displaystyle \underset{0}{\overset{t}{\int }}{x}_1(t)dt+}{\zeta }_3{\displaystyle \frac{d{x}_1}{dt}}$$

(4)

where p and q are odd positive integers satisfying the condition 1 < p/q < 2, and ζ₁, ζ₂, ζ₃ are adjustment coefficients. ζ₁ > 0;${\omega }_n=\sqrt{{\zeta }_2/{\zeta }_3}$> 0; and $\xi =0.5{\zeta }_1/\sqrt{{\zeta }_2/{\zeta }_3}$ where ω_n is the natural oscillation frequency, and ξ is the damping coefficient.

Theorem 1.

The system dynamics in Equation (1) asymptotically approach zero in finite time when the sliding surface l is chosen according to Equation (3) with s defined as (4) and the control law is designed according to Equations (5)–(7).

$$u=u_{eq}+u_n$$

(5)

$$u_{eq}={\displaystyle \frac{1}{g(x){\zeta }_3}}{\displaystyle \underset{0}{\overset{t}{\int }}\left(-{\zeta }_1{\dot{x}}_1-{\zeta }_2{x}_1-{\zeta }_3\dot{f}(x)\right)} dt$$

(6)

$$u_n={\displaystyle \frac{1}{g(x){\zeta }_3}}{\displaystyle \underset{0}{\overset{t}{\int }}{\displaystyle \underset{0}{\overset{t_1}{\int }}\left[-\left(K+{\zeta }_3{D}_{\max }\right)sign(l)-\mu l-\frac{q}{p}{\gamma }^{-1}{\dot{s}}^{\left(2-\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.\right)}\right]}}d{t}_{1}dt$$

(7)

Proof. 

Choose the Lyapunov function V as follows (8).

$$V={\displaystyle \frac{1}{2}}{l}^2$$

(8)

The first and second derivatives of the sliding surface s are calculated as (9) and (10).

$$\begin{array}{ll}\dot{s}& ={\zeta }_1{\dot{x}}_1+{\zeta }_2{x}_1+{\zeta }_3{\ddot{x}}_1\\ & ={\zeta }_1{\dot{x}}_1+{\zeta }_2{x}_1+{\zeta }_3\left(\dot{f}(x)+g(x)\dot{u}+\dot{d}(t)\right)\end{array}$$

(9)

$$\ddot{s}={\zeta }_{3}g(x){\ddot{u}}_n+{\zeta }_3\ddot{d}(t)$$

(10)

Combining the control law designed as (6) and (7) and the calculations of s, the derivative of the Lyapunov function is rewritten as (11).

$$\begin{array}{l}\dot{V}=l\dot{l}\\ =l\left[\gamma {\displaystyle \frac{p}{q}}{\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}\left(\ddot{s}+{\gamma }^{-1}{\displaystyle \frac{q}{p}}{\dot{s}}^{\left(2-{\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}\right)}\right)\right]\\ =l\left[\gamma {\displaystyle \frac{p}{q}}{\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}\left(-Ksign(l)-\mu l+{\zeta }_3{D}_{\max }\right)\right]\\ =\gamma {\displaystyle \frac{p}{q}}{\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}\left(-K‖l‖-\mu l^2+{\zeta }_3{D}_{\max }l\right)\\ =-\gamma {\displaystyle \frac{p}{q}}{\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}\left((K+{\zeta }_3{D}_{\max })‖l‖+\mu l^2\right)\end{array}$$

(11)

Based on Assumption 1, the sliding gain K is chosen larger than ζ₃D_max. In the case that ${\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}>0$, it can be considered as Equation (12).

$$\dot{V}=-\gamma {\displaystyle \frac{p}{q}}{\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}\left(K‖l‖+\mu l^2\right)<0,\forall l\ne 0$$

(12)

□

Similarly, consider the second-order system as Equation (13).

$$\left\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=f(x)+g(x)u+d(t)\end{array}\right.$$

(13)

Theorem 2.

The system dynamics in Equation (13) asymptotically approach zero in finite time when the sliding surface l is chosen according to Equation (3) with s defined as (4) and the control law is designed according to Equations (14)–(16).

$$u=u_{eq}+u_n$$

(14)

$$u_{eq}={\displaystyle \frac{1}{g(x){\zeta }_3}}\left(-{\zeta }_1{x}_2-{\zeta }_2{x}_1-{\zeta }_{3}f(x)\right)$$

(15)

$$u_n={\displaystyle \frac{1}{g(x){\zeta }_3}}{\displaystyle \underset{0}{\overset{t}{\int }}\left(-\left(K+{\zeta }_3{D}_{\max }\right)sign(l)-\mu l-\frac{q}{p}{\gamma }^{-1}{\dot{s}}^{\left(2-\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.\right)}\right)}dt$$

(16)

Similarly to the proof for Theorem 1, the Lyapunov function is chosen as (), considering the time derivative of the Lyapunov function V, and ensuring that $\dot{V}$ < 0, the time derivative of V can be rewritten as Equation (17).

$$\begin{array}{l}\dot{V}=l\dot{l}\\ =l\left[\gamma {\displaystyle \frac{p}{q}}{\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}\left(\ddot{s}+{\gamma }^{-1}{\displaystyle \frac{q}{p}}{\dot{s}}^{\left(2-{\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}\right)}\right)\right]\\ =l\left[\gamma {\displaystyle \frac{p}{q}}{\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}\left(-Ksign(l)-\mu l+{\zeta }_3{D}_{\max }\right)\right]\\ =\gamma {\displaystyle \frac{p}{q}}{\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}\left(-K‖l‖-\mu l^2+{\zeta }_3{D}_{\max }l\right)\\ =-\gamma {\displaystyle \frac{p}{q}}{\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}\left((K+{\zeta }_3{D}_{\max })‖l‖+\mu l^2\right)\end{array}$$

(17)

In the case that ${\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}>0$, $\dot{V}$ can be considered as Equation (18).

$$\dot{V}=-\gamma {\displaystyle \frac{p}{q}}{\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}\left(K‖l‖+\mu l^2\right)<0,\forall l\ne 0$$

(18)

This proves that the sliding manifold l = 0 first, then s = 0 in finite time, and the system dynamics in the equation asymptotically approach zero in finite time. On the other hand, if the sliding surface contains only PI, it does not raise the order of the NSTM sliding surface. The main idea of PI-nested TSM is to incorporate integral errors of states into the terminal sliding manifolds. Nevertheless, like NTSM and TSM methods, the discontinuous switching function is embedded in the first-order derivatives. Assuming that the PI error-nested NTSM is modified from Equations (3) and (4), the manifold becomes (19) and (20).

$$l=s+\delta {\dot{s}}^{p/q}$$

(19)

$$s={\zeta }_{1}x(t)+{\zeta }_2{\displaystyle \underset{0}{\overset{t}{\int }}x(t)dt}$$

(20)

For the first-order system introduced as Formula (1), the control law is designed as (21)–(23).

$$u=u_{eq}+u_n$$

(21)

$$u_{eq}={\displaystyle \frac{1}{g(x){\zeta }_1}}\left({\zeta }_{3}f(x)-{\zeta }_{2}x\right)$$

(22)

$$u_n={\displaystyle \frac{1}{g(x){\zeta }_1}}\left(-Ksign(l)-\mu l-{\displaystyle \frac{q}{p}}{\gamma }^{-1}{\dot{s}}^{\left(2-{\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}\right)}\right)$$

(23)

From Equations (21)–(23), the u_n is not continuous, and the problem of chattering is not solved.

2.2. Mathematical Model of Induction Motor Speed Servo Systems

Mathematical modeling of an induction motor in different coordinate systems is necessary to design a speed motor controller for precise control. In this study, the dynamics of stator current, rotor flux, and rotor speed in the dq coordinate system are described as in the study of Yong Feng et al. [17], and the mathematical model of the induction motor is written as Equation (24).

$$\left\{\begin{array}{l}{\dot{i}}_{sd}=\xi {\displaystyle \frac{1}{T_r}}{\psi }_r-\lambda i_{sd}+{\omega }_s{i}_{sq}+K{u}_{sd}\\ {\dot{i}}_{sq}=-\xi \omega {\psi }_r-{\omega }_s{i}_{sd}-\lambda i_{sq}+K{u}_{sq}\\ {\dot{\psi }}_r=-{\displaystyle \frac{{\widehat{L}}_m}{T_r^2\left({\omega }_s-\omega \right)}}{i}_{sq}+{\displaystyle \frac{{\widehat{L}}_m}{T_r}}{i}_{sd}\\ {\dot{\omega }}_m={\displaystyle \frac{n_p{\widehat{L}}_m}{\widehat{J}{\widehat{L}}_r}}{i}_{sq}{\psi }_r-{\displaystyle \frac{1}{\widehat{J}}}\left(T_L+d(t)\right)\end{array}\right.$$

(24)

where T_L is the load torque; $T_r={\displaystyle \frac{{\widehat{L}}_r}{{\widehat{R}}_r}}$ is the rotor time constant; $\sigma =1-{\displaystyle \frac{{\widehat{L}}_m^2}{{\widehat{L}}_s{\widehat{L}}_r}}$ is the leakage coefficient; $K={\displaystyle \frac{1}{\sigma {\widehat{L}}_s}}$; $\xi =K{\displaystyle \frac{{\widehat{L}}_m}{{\widehat{L}}_r}}$; and $\lambda =K\left({\widehat{R}}_s+{\widehat{R}}_r{\displaystyle \frac{{\widehat{L}}_m^2}{{\widehat{L}}_r^2}}\right)$. R_s and R_r are the resistances of the stator and rotor, respectively. L_m, L_r, and L_s are the mutual inductance between stator and rotor, stator inductance, and rotor inductance, respectively. ω_m is the mechanical angular speed of the rotor; ω_s is the synchronous angular speed; ω is the slip angular frequency; ψ_r is the rotor flux; u_sd and u_sq are the stator voltages; i_sd and i_sq are the stator currents; and d(t) is the sum of model uncertainties and disturbances.

The influence of heat generation during operation or mechanical changes in the system during the operating process makes the motor parameters no longer accurate [32]. The parameters are rewritten as Equation (25) to completely model the IM electric motor model.

$$\left\{\begin{array}{l}{R}_s={\widehat{R}}_s+\Delta R_s; L_s={\widehat{L}}_s+\Delta L_s;L_m={\widehat{L}}_m+\Delta L_m\\ R_r={\widehat{R}}_r+\Delta R_r; L_r={\widehat{L}}_r+\Delta L_r;J=\widehat{J}+\Delta J\end{array}\right.,$$

(25)

where ${\widehat{R}}_s,{\widehat{R}}_r,{\widehat{L}}_s,{\widehat{L}}_r,{\widehat{L}}_m, \widehat{J}$ are the initially estimated values of electrical resistance, inductance, and moment of inertia; R_s, R_r, L_s, L_r, L_m, J are the actual values; and ΔR_s, ΔR_r, ΔL_s, ΔL_r, ΔL_m, ΔJ denote their corresponding deviation values, which arise from modeling inaccuracies, parameter variations, or measurement uncertainties. The motor speed error is defined as the deviation between the reference speed and the actual speed of the motor, and it is defined in Equation (26).

$$e_{\omega }(t)={\omega }_{mref}-{\omega }_m$$

(26)

The derivative of velocity error is defined as follows (27):

$${\dot{e}}_{\omega }={\dot{\omega }}_{ref}-{\alpha }_1{i}_{sq}+{\alpha }_2\rho (t)$$

(27)

where ${\alpha }_1={\displaystyle \frac{n_p{\widehat{L}}_m{\psi }_r}{\widehat{J}{\widehat{L}}_r}}$; ${\alpha }_2={\displaystyle \frac{1}{\widehat{J}}}$; and $\rho (t)=T_L+d(t)$ is the sum of the external forces acting on the system that cannot be determined. This parameter contains both the T_L, disturbances, and model uncertainties as ΔR_s, ΔR_r, ΔL_s, ΔL_r, ΔL_m, and ΔJ. From Formula (4), the acceleration of the velocity error is calculated in Formula (28).

$${\ddot{e}}_{\omega }={\ddot{\omega }}_{ref}-{\alpha }_1{\dot{i}}_{sq}+{\alpha }_2\dot{\rho }(t),$$

(28)

2.3. Higher-Order PID-Nested Nonsingular Terminal Sliding Mode Control for Speed Controller

In this section, a PID-NTSM is proposed to effectively solve the stability and chattering problems for the speed of the IM drives, which is the main contribution of this study. The speed of induction motors is an outermost loop, which is affected by strong external factors such as load, disturbances, or uncertainty. In contrast, the current control loop adjusts the stator current i_d and i_q, directly affecting the rotor flux and torque, which means it is a linear first-order system, especially with the inductive component of the stator coil, making the current unable to change instantaneously. So, the linear controllers are used to reduce the complexity of the controller and its applicability to practical models. The overall diagram of the proposed FOC controller applied in this case is shown in Figure 1, which illustrates the transformation between coordinate systems [25].

In this approach, the PID-NTSM manifold is formulated by the incorporation of both derivative and integral errors of states into the conventional nonsingular terminal sliding mode surface. The proposed sliding surface is introduced according to Equation (29), with s defined in Equation (30).

$$l_{\omega }=s_{\omega }+\delta {{\dot{s}}_{\omega }}^{p/q}$$

(29)

$$s_{\omega }={\zeta }_1{e}_{\omega }(t)+{\zeta }_2{\displaystyle \underset{0}{\overset{t}{\int }}{e}_{\omega }(t)dt}+{\zeta }_3{\displaystyle \frac{d{e}_{\omega }}{dt}}$$

(30)

where p and q are odd positive integers satisfying the condition 1 < p/q < 2, and ζ₁, ζ₂, ζ₃ are adjustment coefficients. ζ₁ > 0; ${\omega }_n=\sqrt{{\zeta }_2/{\zeta }_3}$> 0; and $\xi =0.5{\zeta }_1/\sqrt{{\zeta }_2/{\zeta }_3}$ with ω_n as the natural oscillation frequency, and ξ as the damping coefficient.

Theorem 3.

The sliding surface s approaches zero in finite time, and the motor’s speed error dynamics in Equations (26)–(28) approach zero asymptotically if the sliding surface is chosen using Equations (29) and (30) and the control law is designed according to Equations (31)–(33).

$$i_{sq}=i_{sqeq}+i_{sqn}$$

(31)

$$i_{sqeq}={\displaystyle \frac{1}{{\xi }_3{\alpha }_1}}{\displaystyle \underset{0}{\overset{t}{\int }}\left({\xi }_1{\dot{e}}_{\omega }+{\xi }_2{e}_{\omega }+{\xi }_3{\ddot{\omega }}_{mref}\right)}dt$$

(32)

$$i_{sqn}={\displaystyle \frac{1}{{\xi }_3{\alpha }_1}}{\displaystyle \underset{0}{\overset{t}{\int }}{\displaystyle \underset{0}{\overset{t_1}{\int }}\left[Ksign(l)+{\mu }_{1}l+\frac{q}{p}{\gamma }^{-1}{\dot{s}}^{\left(2-\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.\right)}\right]}}d{t}_{1}dt$$

(33)

Proof. 

Consider the following Lyapunov function (34).

$$V={\displaystyle \frac{1}{2}}{l}^2$$

(34)

Based on the sliding surface defined in Equations (29) and (30), the derivatives of l and s concerning time t are calculated as Equations (35) and (36).

$$\begin{array}{l}\dot{l}=\dot{s}+\gamma {\displaystyle \frac{p}{q}}{\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}\ddot{s}\\ =\gamma {\displaystyle \frac{p}{q}}{\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}\left(\ddot{s}+{\gamma }^{-1}{\displaystyle \frac{q}{p}}{\dot{s}}^{\left(2-{\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}\right)}\right)\end{array}$$

(35)

$$\begin{array}{l}\dot{s}={\xi }_1{\dot{e}}_{\omega }+{\xi }_2{e}_{\omega }+{\xi }_3{\ddot{e}}_{\omega }\\ ={\xi }_1{\dot{e}}_{\omega }+{\xi }_2{e}_{\omega }+{\xi }_3({\ddot{\omega }}_{ref}-{\alpha }_1{\dot{i}}_{sq}+{\alpha }_2\dot{\rho })\end{array}$$

(36)

From the control law (31)–(33), the derivative of s can be rewritten as Equation (37), and the second derivative of s is calculated using Equation (38).

$$\dot{s}={\xi }_3{\alpha }_1{\dot{i}}_{sqn}+{\xi }_3{\alpha }_2\dot{\rho }$$

(37)

$$\ddot{s}={\xi }_3{\alpha }_1{\ddot{i}}_{sqn}+{\xi }_3{\alpha }_2\ddot{\rho }$$

(38)

Consider the time derivative of the Lyapunov function V and ensure that $\dot{V}$< 0. From Equations (30)–(33), the time derivative of V can be rewritten as Equation (39).

$$\begin{array}{ll}\dot{V}& =l\left[\gamma {\displaystyle \frac{p}{q}}{\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}\left(\ddot{s}+{\gamma }^{-1}{\displaystyle \frac{q}{p}}{\dot{s}}^{\left(2-{\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}\right)}\right)\right]\\ & =\gamma {\displaystyle \frac{p}{q}}{\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}\left[-K‖l‖-{\mu }_1{l}^2+{\xi }_3{\alpha }_2\ddot{\rho }l\right]\end{array}$$

(39)

Assumption 2.

Let $\ddot{\rho }$ be bounded by a positive coefficient D_max in Equation (40).

$$‖\ddot{\rho }‖\le D_{\max }$$

(40)

By designing and selecting an appropriate control strategy, the system ensures that disturbances and uncertainties are bounded within a range, allowing the system to converge to the proposed sliding manifold. This ensures that the sliding surface l approaches zero, maintaining the stability and performance of the system. The coefficient K is chosen to eliminate the disturbances as in Formula (41).

$$K>{\xi }_3{\alpha }_2{D}_{\max }$$

(41)

With the coefficient K designed to be as strong as (41), the derivative of the Lyapunov function V is rewritten as (42).

$$\dot{V}=-\gamma {\displaystyle \frac{p}{q}}{\dot{s}}^{\left({\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1\right)}\left(K‖l‖+{\mu }_1{l}^2\right)\le 0,\forall l\ne 0$$

(42)

□

Remark 1

.Suppose t_l is the time when l reaches the value 0 starting from time t₀, where l(t) = 0 for all t_l > t₀.

With${\dot{s}}^{{\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1}>0, \exists \delta >0, and {\dot{s}}^{{\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}-1}\ge \delta \forall t\ge t_0$, the derivative of V is rewritten as (43) and (44).

$$\dot{V}\le -{\displaystyle \frac{p}{q}}\gamma \delta K‖l‖$$

(43)

$$\begin{array}{l}{\displaystyle \underset{t_0}{\overset{t_l}{\int }}\frac{\dot{V}}{\sqrt{2V}}}dt\le {\displaystyle \underset{t_0}{\overset{t_l}{\int }}-\frac{p}{q}\gamma \delta K}dt\\ -\sqrt{2V(t_0)}\le -{\displaystyle \frac{p}{q}}\gamma \delta K\left(t_l-t_0\right)\\ t_l\le t_0+{\displaystyle \frac{q}{p\gamma \delta K}}‖l\left(t_0\right)‖\end{array}$$

(44)

Remark 2.

Suppose t_s is the time to $\mathit{s}\mathbf{\to } \mathbf{0}$ and $\dot{\mathit{s}}\mathbf{\to }\mathbf{0}$ to make the error e approach zero, meaning a very small value (asymptote to zero). At this point, the trajectory of the system has entered a state near the equilibrium point, with t_s calculated in (45). The system can be modeled as a quadratic linear system as follows (46).

$$t_s=t_0+{\displaystyle \frac{p}{p-q}}{\gamma }^{{\displaystyle \raisebox{1ex}{$q$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}}{\left|s\left(t_0\right)\right|}^{\left(p-q\right)/p}$$

(45)

$${\ddot{e}}_{\omega }(t)+2\xi {\omega }_n{\dot{e}}_{\omega }(t)+{\omega }_n^2{e}_{\omega }(t)=0,\forall t\ge t_s,$$

(46)

where ω_n is the natural oscillation frequency of the system, and ξ is the damping coefficient commonly used in the range (0.7, 1]. In the case where the damping coefficient is chosen to be ζ = 1, the solution has the form (47).

$$e_{\omega }(t)=\left[e_{\omega }(t_s)+\left({\dot{e}}_{\omega }(t_s)+{\omega }_n{e}_{\omega }(t_s)\right)\right]\exp \left[-{\omega }_n(t-t_s)\right]$$

(47)

which asymptotically converges to zero (i.e., a very small value ε, close to zero) within $t_c-t_s=-{\displaystyle \frac{\ln \epsilon }{{\omega }_n}}$.

The total convergence time t_c is rewritten as Equation (48).

$$t_c=t_0+{\displaystyle \frac{p}{p-q}}{\gamma }^{{\displaystyle \raisebox{1ex}{$q$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}}{\left|s\left(t_0\right)\right|}^{\left(p-q\right)/p}-{\displaystyle \frac{\ln \epsilon }{{\omega }_n}}$$

(48)

The current and magnetic flux feedback is faster in the inner loop, where simple linear controllers are often used to meet the system’s response time faster than the outer loop [33]. In this approach, the current i_sd is used to control the rotor flux. From Equation (24), it can be seen that the relationship between i_sd and ψ_r is approximately linear and the response is fast, so the simple structure of the PI controller is suitable and ensures the tracking of the desired flux. Nonlinear controllers result in an increase in the level and complexity of calculations. The PI controller for d-axis current control is designed as Formula (49).

$$u_{sd}=K_p^{id}{e}_{isd}(t)+K_d^{id}{\displaystyle \underset{0}{\overset{t}{\int }}{e}_{isd}(t)}dt$$

(49)

where $K_p^{id},K_i^{id}>0$ are the control gains of the current in the d-axis.

Similarly to i_sd current controller, i_sq current controller is designed as in Equation (50).

$$u_{sq}=K_p^{iq}{e}_{isq}(t)+K_i^{iq}{\displaystyle \underset{0}{\overset{t}{\int }}{e}_{isq}(t)dt}$$

(50)

where $K_p^{iq},K_i^{iq}>0$ are the control gains of the current in the q-axis.

The controller needs to be easy to set up to optimize the processing speed, so the PI controller is suitable for the inner control loop. The PI controller is designed according to Formula (51), with coefficients selected to meet the Ziegler–Nichols criteria. In FOC, torque and flux are separated and controlled independently. The integral component in the PI controller helps to compensate for slow-changing or constant disturbances, so PI achieves good performance in flux control.

$$i_{sd}=K_p^{\psi }{e}_{\psi }(t)+K_i^{\psi }{\displaystyle \underset{0}{\overset{t}{\int }}{e}_{\psi }(t)dt}$$

(51)

where $K_p^{\psi },K_i^{\psi }>0$ are control gains of the rotor flux.

3. Results and Discussions

To evaluate the effectiveness of the control methods, a simulation environment was established using MATLAB Simulink ver. 2023a with a sampling time of 0.5 ms. The parameters of the three-phase asynchronous motor are presented in

Table 1. Parameters of the used induction motor.

Parameter	Symbol	Value	Unit
Power	P	180	W
Number of pole pairs	p	2	-
Stator resistance	Rs	4.5	Ohm
Rotor resistance	Rr	2	Ohm
Mutual flux	Lm	0.0491	H
Stator inductance	Ls	0.0034	H
Rotor inductance	Lr	0.0034	H
Moment of inertia	J	3.5170 × 10−4	Kg.m2
Friction constant	B	4.2490 × 10−4	N.m.s

, with the schematic diagram of the FOC controller in the simulation environment shown in Figure 2. Conventional sliding mode (SLM) and nonsingular terminal sliding mode (NTSM) controllers with classical sliding surfaces are introduced as Formulas (52) and (53), where the NTSM controller adopts the parameters p = 5, q = 3, γ = 0.008, and K = 1000, while the corresponding SLM controller utilizes its respective gain K = 1000.

$$l_{SLM}=c{e}_{\omega }(t)+{\displaystyle \frac{d{e}_{\omega }}{dt}}$$

(52)

$$l_{NTSM}=e_{\omega }(t)+\gamma {{\dot{e}}_{\omega }}^{{\displaystyle \raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}}$$

(53)

The gain values are chosen based on Theorem 1 and Remarks 1 and 2. With ξ = 1 and ω_n = 500, the parameter of the PID-NTSM is calculated as shown in

Table 2. Parameters of PID-NTSM controllers.

Parameter	Value
p	5
q	3
ζ1	1
ζ2	250
ζ3	0.001
γ	0.008
K	1000

. The coefficient γ in the NTSM sliding surface is chosen to adjust the nonlinear convergence rate of the system, which is adjusted through simulation to achieve a balance between the convergence rate and the smoothness of the control signal. The PI controller for the current and rotor flux is designed, and the basic gain parameters are selected using the Ziegler–Nichols standard adjustment method to help the system achieve a fast response and reduce state error.

Figure 3 presents the motor speed responses with SLM, NTSM, and the proposed PID-NTSM controllers. The SLM method shows the slowest convergence and the strongest chattering, approximately ±2 rpm. NTSM improves transient performance and reduces chattering to about ±0.8 rpm but still exhibits small oscillations. In contrast, the proposed PID-NTSM achieves the fastest response and nearly eliminates chattering while maintaining robustness. Load disturbances applied at 5 s and 6 s further confirm the superior disturbance rejection capability of the proposed scheme.

The errors of the controllers are shown in Figure 4, which demonstrates the effectiveness of the proposed controller. The Integral of Absolute Error (IAE) and Integral of Squared Error (ISE) are calculated as in

Table 3. Comparison of ISE and IAE between SLM, NTSM, and the proposed controllers.

	IAE (10−3)	ISE (10−5)
Proposed	0.001	0.003
NTSM	0.438	0.822
SLM	1.933	2.272

to show that the chattering is almost eliminated. The fact that both error indices of the proposed method are considerably lower than those of SLM and NTSM highlights the ability of the proposed controller to minimize steady-state error and eliminate the chattering phenomenon. This shows that the PID-NTSM achieves a highly accurate, stable, and robust response to disturbances and uncertainties.

Figure 5 shows the control signals of the SLM, NTSM, and the PID-NTSM controllers. The SLM controller exhibits chattering, which is the major drawback of conventional sliding mode control. The NTSM controller moderately reduces the chattering amplitude compared to the SLM, and chattering still exists in the control signals. The PID-NTSM increases the order of the sliding surface, so the control signal becomes smoother and more stable. The MAE and RMS indices indicate that the proposed controller reduces chattering by 90.82% compared with NTSM and 97.66% compared with SLM in terms of MAE, and by 88.46% and 97.13%, respectively, in terms of RMS, as shown in

Table 4. Assessment of chattering suppression based on MAR and RMS indexes for SLM, NTSM, and proposed controllers.

	MAE	RMS
Proposed	0.0097	0.0169
NTSM	0.1057	0.1464
SLM	0.4144	0.5893

. This substantial reduction quantitatively demonstrates the superior smoothness and effectiveness of the proposed PID-NTSM in eliminating chattering.

To further strengthen the conclusions about the robustness of the PID-NTSM, a test scenario of uncertainties about the motor parameters is performed. The moment of inertia and viscous friction coefficient of the motor are multiplied by five times from the nominal value in the controller design. The system response is shown in Figure 6 for different controllers, including SLM, NTSM, and PID-NTSM. It shows the robustness of SMC in real situations with uncertain parameters. In this case, PID-NTSM still maintains its robustness and responds better than conventional SLM and NTSM, and the chattering phenomenon is almost eliminated in the proposed controller.

The test bench is described in Figure 7, which includes a power supply that provides stable input power to the entire system. The main control circuit is built on the TI C2000 platform, and the output current and voltage of the inverter are monitored through voltage and current sensors. The rotor velocity is measured directly by an encoder mounted directly on the rotor shaft with a resolution of 1000 pulses/revolution, with three channels. A fixed load arrangement system and a mechanical braking mechanism allow simulating different operating conditions—from no load and fixed load to variable and uncertain load—to simulate many industrial scenarios. The fixed load is designed in the form of a uniform disk to calculate the load moment, and the unmeasured noise is put into the system by a brake disk to describe the external forces in natural conditions, such as changes in coefficients due to heat, misalignment during operation, and friction. To protect mechanical devices and motors, experimental results were only evaluated on the PID-NTSM controller.

Figure 8 shows a small delay response and no overshoot, and the chattering phenomenon is eliminated. At approximately 3.5 and 6 seconds, a mechanical disturbance due to the brake impact is introduced into the system to demonstrate the ability to quickly eliminate the deviation and restore the actual speed to stability in a short time. This demonstrates the effectiveness of the proposed control strategy, not only in tracking the speed but also in responding quickly to disturbances, and especially in eliminating chattering. In Figure 9, the error of the speed shows that the response of the proposed controller tracks well with the set values. To strengthen the conclusions, abrupt changes in the set values are performed, and noise from the brake is randomly added to demonstrate the good tracking ability, which helps the disturbances to be compensated quickly due to the robustness of SMC.

Figure 10 shows the experimental speed response of the system when using the proposed controller with full load. The actual speed increases rapidly, and the system reaches steady state in a very short time, which demonstrates that the controller is capable of eliminating chattering and is robust against noise and full load. The current on the q-axis approaches zero after a short time, showing the ability to reduce torque and control speed effectively, as shown in Figure 11a. The current on the d-axis is constant, which maintains a stable value around the set value, proving the ability to keep the rotor flux stable according to the proposed control strategy. In Figure 11c, the waveforms of the three-phase currents i_a, i_b, and i_c have stable amplitude and phase, proving that the stator currents are modulated in the correct period. In practice, only two-phase currents, i_a and i_b, are measured, since the three-phase currents satisfy the balanced condition and ic is computed as i_c = 1 – (i_a + i_b). Figure 11d shows that the stator current trajectory in the αβ system forms a nearly perfect circle, confirming the maintenance of constant current amplitude and uniformity of rotating flux, a key factor in vector control.

4. Conclusions

This paper proposed a PID-NTSM control method for the velocity control loop of an induction motor FOC servo drive. Here, the PID-NTSM sliding manifold is formulated as a composite surface that incorporates both the derivatives and integrals of the states into the nonsingular terminal sliding manifold. It helps to raise the associated NTSM manifold to a higher order. In this manner, the discontinuous switching elements are placed at the second derivatives of the control inputs; thus, significant “chattering” reduction/elimination is obtained. Simulation and experimental results provide comprehensive benchmarks for the proposed PID-NTSM methods against conventional SLM and NTSM approaches, demonstrating their excellent characteristics, including better tracking precision, faster response, and chattering elimination.