A Novel Terminal Sliding Mode Control with Robust Prescribed-Time Stability

Abstract

The present paper investigates a new tool for analyzing stability/convergence properties and robustness against matched perturbations of a class of nonlinear systems. We start with a scalar system, where it is shown that the state can be regulated or stabilized to a prescribed time using time-varying functions. The proof is based on Lyapunov theory. We developed a robust terminal-integral sliding mode controller that guarantees convergence of the system states to a desired equilibrium within a user-defined time, irrespective of initial conditions and under bounded disturbances. The method was extended to a class of second-order nonlinear systems, achieving both fixed-time (prescribed-time) convergence and robustness. Theoretical properties were established via Lyapunov-based analysis, and numerical simulations confirmed the effectiveness of the proposed methods in terms of robustness and convergence.

1. Introduction

1.1. Motivation and Background

Nonlinear systems are ubiquitous in engineering applications such as robotics, aerospace, and automotive systems. Their complex dynamics, combined with uncertainties and external disturbances, pose major challenges for controller design. Sliding mode control (SMC) is widely recognized for its robustness and insensitivity to such uncertainties [1,2], but conventional SMC generally ensures only asymptotic convergence, which may be inadequate for time-critical tasks. Terminal sliding mode control (TSMC) was introduced to achieve finite-time convergence [3,4]. However, classical TSMC designs often suffer from singularity issues in the control law. Integral-type approaches have been proposed to mitigate this problem, yet they do not provide explicit control over the convergence time. Fixed-time stability has recently emerged as a framework ensuring convergence within a bounded time independent of initial conditions [5], while prescribed-time control further enhances this property by guaranteeing convergence within a user-defined time bound [6], regardless of initial states [7,8,9]. Despite these advances, achieving prescribed-time convergence in a robust and computationally efficient manner remains an open challenge.

1.2. Related Works

Recent research has extensively investigated integral terminal sliding mode control and prescribed-time stability for nonlinear systems. In [10], a prescribed-time Lyapunov theorem was proposed, ensuring convergence to the equilibrium point within a fixed, user-defined time, independent of initial conditions. The importance of settling-time guarantees has been emphasized in system dynamics [11]. While finite-time controllers provide convergence in finite but initial condition-dependent time [12], fixed-time controllers guarantee convergence within a bounded time independent of initial conditions [5]; both approaches lack the explicit specification of the convergence horizon. Prescribed-time (PT) control overcomes this limitation by mapping the infinite-time domain into a finite interval, thereby ensuring convergence at an exactly predetermined instant [6,9,13]. Building on this concept, numerous PT controllers have been developed for linear and nonlinear systems. Time-varying approaches proposed by Song et al. [6] demonstrated how explicit convergence bounds can be achieved through time-scale transformations. Extensions of this method have been shown to retain robustness under non-vanishing disturbances [7,14], and generalized time-transformation techniques have broadened its applicability to diverse classes of nonlinear systems [15,16]. Prescribed-time methods have also been applied to systems with matched uncertainties [17], uncertain input gains [18], and autonomous or hybrid autonomous time-varying settings [19,20,21]. Despite these advances, several challenges remain. First, many prescribed-time controllers assume globally bounded disturbances, limiting applicability to systems with state-dependent or unmodeled dynamics. Second, autonomous predefined-time controllers often exhibit chattering near equilibrium, which undermines robustness. Most critically, prescribed-time and fixed-time controllers frequently require highly nonlinear or discontinuous control laws, resulting in control signals with unreasonably large amplitudes. This singularity problem, arising especially in terminal sliding mode control due to the use of fractional powers of states, leads to actuator saturation, high energy consumption, and limited feasibility in practice [22]. While efforts have been made to accelerate convergence by using more complex control structures [23], this is often achieved at the expense of robustness and smoothness. Recent works have also explored prescribed-time extensions beyond control design. For example, prescribed-time differentiators with finite-varying gains were developed to improve robustness against noise and uncertainties [23], while observers with prescribed settling-time guarantees were proposed to enhance state estimation in uncertain environments [24]. Prescribed-time exact tracking for nonlinear systems has further demonstrated the potential of PT methods in high-performance applications [25]. A recent survey [9] provides a comprehensive overview of these developments. However, singularity issues in terminal sliding mode designs (finite-/fixed-time) and restrictive assumptions on uncertainties remain major obstacles.

1.3. Contributions

Motivated by the challenges of controlling nonlinear systems with singularities and undefined convergence times, this paper introduces a robust prescribed-time control framework that ensures both stability and robustness. The main contributions are as follows:

• A novel prescribed-time Lyapunov stability theorem is established, providing a rigorous foundation that guarantees convergence within a fixed, user-defined time, independent of initial conditions.
• An integral terminal sliding mode controller (ITSMC) with bounded, time-varying gains is proposed, which eliminates the traditional reaching phase of sliding mode control and avoids the singularity problem.
• A new terminal sliding mode control scheme is developed for a class of second-order nonlinear systems, extending the ITSMC framework to broader system classes.
• The proposed controllers guarantee that system trajectories reach both the sliding manifold and the desired equilibrium point within the prescribed time, ensuring robustness and high tracking accuracy under matched perturbations and uncertainties.

The theoretical results are derived using Lyapunov-based analysis, and their effectiveness is validated through numerical simulations, including an application to PMSM systems, demonstrating prescribed-time convergence, robustness, and smooth control performance.

1.4. Paper Organization

This paper is organized as follows. The problem statement and preliminaries are presented in Section 2. Section 3 introduces the robust prescribed-time stability results. Section 4 includes (i) robust prescribed-time stability, (ii) integral terminal sliding mode control (ITSMC), and (iii) non-singular terminal sliding mode control for uncertain nonlinear second-order systems. Simulation results are reported in Section 5, followed by concluding remarks in Section 6.

1.5. Notation

The sign function is the set-valued function defined on $\mathbb{R}$ as

$$\begin{array}{c}\hfill sgn\left(s\right)=\left\{\begin{array}{cc}{\displaystyle \frac{s}{\left|s\right|}}\hfill & \mathrm{if}s\ne 0,\hfill \\ [-1,1]\hfill & \mathrm{if}s=0.\hfill \end{array}\right.\end{array}$$

(1)

2. Preliminaries

In this paper, a PTC is proposed for a scalar system and then applied to robust integral sliding mode control. In this section, we will introduce some definitions related to stability.

First, consider the following nonlinear system:

$$\dot{x}=f(x,t),\mathrm{where}f(0,t)=0,\forall t\ge 0$$

(2)

In control system analysis, three traditional stability concepts—Lyapunov stability, global asymptotic stability, and global exponential stability—are commonly employed. Beyond these asymptotic notions, global finite-time stability and global fixed-time stability are also important and are defined as follows:

Definition 1. 

(Prescribed-Time Stability). A system (2) is said to be prescribed-time stable if there exists a constant $T_c>0$, chosen a priori by the designer and independent of the initial condition $x\left(0\right)$, such that

$$\underset{t\to T_c^{-}}{lim}x\left(t\right)=0andx\left(t\right)\equiv 0,\forall t\ge T_c.$$

(3)

In contrast to fixed-time stability, where the settling time is bounded by a constant independent of initial conditions but determined by the system parameters, prescribed-time stability allows this constant $T_c$ to be arbitrarily specified by the user in advance.

Definition 2. 

[5,12,26,27] Let $x_0$ be the state vector of system (2) at$t=0$ . The zero equilibrium of system (2) is said to be

1. 

Globally finite-time stableif it is globally asymptotically stable and there exists a settling time function $T\left(x_0\right):{\mathbb{R}}^n\to (0,\infty )$ such that for all$t\ge T\left(x_0\right)$, $x\left(t\right)=0$;

2. 

Globally fixed-time stable if it is globally finite-time stable and there exists a constant $\overline{T}>0$ such that $T\left(x_0\right)\le \overline{T}$ for all initial conditions $x_0$, indicating that the convergence time is finite and has a known upper bound.

For systems described by

$$\dot{x}=f(x,t,v),\mathrm{where}f(0,t,v)=0,\forall t\ge 0,\forall v>0$$

(4)

where $v\in (0,\infty )$ is a user-defined parameter, the notion of global prescribed-time stability (GPTS) is defined as follows:

For systems described by

$$\dot{x}=f(x,t,v),\mathrm{where}f(0,t,v)=0,\forall t\ge 0,\forall v>0$$

(5)

where $v\in (0,\infty )$ is a user-defined parameter, the notion of global prescribed-time stability (GPTS) extends Definition 1 by requiring both global convergence and a user-specified settling time.

Definition 3. 

([6,28,29]). Let $x_0$ be the state vector of system (5) at$t=0$. The zero equilibrium of system (5) is said to be globally prescribed-time stable if

1. 

it is globally finite-time stable and

2. 

for every prescribed constant $v>0$, chosen independently of $x_0$, the settling time $T\left(x_0\right)$ satisfies $T\left(x_0\right)\le v$.

This means that the system not only converges in finite time but also allows the convergence time to be arbitrarily specified a priori, as in prescribed-time stability (Definition 1).

In this paper, we study robust prescribed-time stability, where the system is input-to-state-stable with convergence, as presented in [29].

3. Robust Prescribed-Time Stability

In this section, we introduce a time-varying piecewise function $\zeta \left(t\right)$ to achieve GPTS, where $\mathsf{\epsilon }$ and T are decoupled. The function is defined as follows [7]:

$$\zeta \left(t\right)=\left\{\begin{array}{cc}exp\left(\mathsf{\epsilon }(T-t)\right)-1,\hfill & \mathrm{if}t\in [t_0,T)\hfill \\ exp\left(\mathsf{\epsilon }(t-T)\right),\hfill & \mathrm{if}t\in [T,+\infty )\hfill \end{array}\right.$$

(6)

where T is user time and $\mathsf{\epsilon }\in (0,ln(2)/T]$ is a tuning parameter.

Building upon the (6), we contemplate the subsequent perturbed system.

$$\dot{x}\left(t\right)=-((k_1+{\displaystyle \frac{|\dot{\zeta }\left(t\right)|}{\zeta \left(t\right)}})\left|x\left(t\right)\right|-{\displaystyle \frac{k_2}{\zeta \left(t\right)}}+k_3)sgn\left(x\left(t\right)\right)+d\left(t\right).$$

(7)

with $x\left(t\right)\in \mathbb{R}$, $d\left(t\right)\in \mathbb{R}$ such as $\left|d\left(t\right)\right|<\overline{d}$, $k_1$, $k_2$, $k_3$, and $k_4$ are positive scalars such that $k_1>\mathsf{\epsilon }$, $k_2>k_4$, $k_4=\overline{d}-k_3>0$.

Remark 1. 

Compared to existing prescribed-time controllers [6,28], this article presents a more thorough stability analysis of the transient-state process and offers a detailed quantitative basis for parameter selection. Rather than using fixed empirical values, we adjust ε based on a predetermined convergence time. Moreover, the scalar function $t>T$ is replaced by an exponential form to ensure that the system trajectory converges asymptotically after the scheduled settling time. In this regard, we elucidate the relationship between PTS, asymptotic stability, and uniformly ultimately bounded stability for the case where $t>T$.

Assumption 1. 

The perturbation $d_i\left(t\right)$ is bounded with $\left|d\right(t\left)\right|\le \delta$.

Theorem 1. 

If $k_1>\mathsf{\epsilon }$, $k_2>k_4$, and $k_4=\overline{d}-k_3>0$, there exists a positive-definite Lyapunov function $V(x,t):{\mathbb{R}}^n\times {\mathbb{R}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$ of the system (7) satisfying

$$\begin{array}{c}\hfill \dot{V}\le -\left(k_1+{\displaystyle \frac{|\dot{\zeta }\left(t\right)|}{\zeta \left(t\right)}}\right)V\left(t\right)+{\displaystyle \frac{k_2}{\zeta \left(t\right)}}+k_4\end{array}$$

(8)

Then, the system (7) is robust GPTS. Thus, for $t\ge T$, the system trajectory will reach the convergence in the domain $\mathsf{\Omega }=\left\{x\left(t\right)\right|V\left(t\right)\le k_2/\mathsf{\epsilon }\}$.

Proof. 

To establish the prescribed-time stability of system (7), it is imperative to first demonstrate the inequality (8). After that we follow the same logic proposed in [6,7], proving the global prescribed-time stability of the origin within system (7). Building upon the analytical framework presented in [7], it can be methodically substantiated that the origin of the system is GPTS, indeed, globally finite-time stable. We choose the Lyapunov function candidate $V\left(t\right)=\left|x\right(t\left)\right|$ (for more details, please see [5]), whose derivative is

$$\begin{array}{cc}\hfill \dot{V}& =\dot{x}\left(t\right)sgn\left(x\left(t\right)\right)\hfill \\ \hfill & =-(k_1+{\displaystyle \frac{|\dot{\zeta }\left(t\right)|}{\zeta \left(t\right)}})\left|x\left(t\right)\right|+{\displaystyle \frac{k_2}{\zeta \left(t\right)}}-k_3+d\left(t\right)sgn\left(x\left(t\right)\right)\hfill \\ \hfill & \le -(k_1+{\displaystyle \frac{|\dot{\zeta }\left(t\right)|}{\zeta \left(t\right)}})\left|x\left(t\right)\right|+{\displaystyle \frac{k_2}{\zeta \left(t\right)}}-k_3+\overline{d}\hfill \end{array}$$

(9)

Using $V\left(t\right)=\left|x\right(t\left)\right|$ and Assumption 1, one has

$$\begin{array}{cc}\hfill \dot{V}& \le -(k_1+{\displaystyle \frac{|\dot{\zeta }\left(t\right)|}{\zeta \left(t\right)}})V\left(t\right)+{\displaystyle \frac{k_2}{\zeta \left(t\right)}}+k_4\hfill \end{array}$$

(10)

The form of the Lyapunov function with its derivative is proven. In the next step, we demonstrate that the Lyapunov function is bounded with $k_2/\mathsf{\epsilon }$.

For $t\in [0,T)$, the time-derivative of $V/\zeta \left(t\right)$ is derived by integrating (10)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{V}{\zeta \left(t\right)}}\right)& \le {\displaystyle \frac{\dot{V}}{\zeta \left(t\right)}}+{\displaystyle \frac{|\dot{\zeta }\left(t\right)|}{\zeta {\left(t\right)}^2}}V\hfill \\ \hfill & \le -k_1{\displaystyle \frac{V}{\zeta \left(t\right)}}+{\displaystyle \frac{k_2}{\zeta {\left(t\right)}^2}}+{\displaystyle \frac{k_4}{\zeta \left(t\right)}}.\hfill \end{array}$$

(11)

The uniformly ultimately bounded stability can be guaranteed by applying the Lyapunov theorem. Moreover, with the comparison theorem, integrating both sides of (11) yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{V\left(t\right)}{\zeta \left(t\right)}}& \le exp(-k_{1}t){\displaystyle \frac{V\left(0\right)}{\zeta \left(0\right)}}+exp(-k_1(t-T))\times \hfill \\ \hfill & \left[(k_4-k_2)(T-t)+{\displaystyle \frac{k_2}{\mathsf{\epsilon }}}{\displaystyle \frac{\zeta \left(t\right)+1}{\zeta \left(t\right)}}-{\displaystyle \frac{k_4-k_2}{\mathsf{\epsilon }}}ln\zeta \left(t\right)\right].\hfill \end{array}$$

(12)

We first consider the system evolution over the interval $t\in [t_0,T)$. We define the auxiliary function $\zeta \left(t\right)$ as

$$\zeta \left(t\right)=exp(-\mathsf{\epsilon }(T-t))-1,\mathrm{with}\mathsf{\epsilon }\in \left(0,{\displaystyle \frac{ln\left(2\right)}{T}}\right].$$

Then, we have

$$ln\left(\zeta \right(t\left)\right)=-\mathsf{\epsilon }(T-t)\le 0,$$

which implies that $\zeta \left(t\right)$ is strictly decreasing and satisfies $\zeta \left(t\right)\in (0,1]$ for $t\in [t_0,T)$.

Under the Lyapunov condition specified for $t\in [t_0,T)$:

$$\dot{V}\left(t\right)\le {\displaystyle \frac{k_{2}ln\left(\zeta \left(t\right)\right)}{\zeta \left(t\right)}}<0,$$

it follows that the Lyapunov function $V\left(t\right)$ is monotonically decreasing over $[t_0,T)$. Moreover, as $t\to T$, we have

$$\zeta \left(t\right)\to 0,\mathrm{and}\zeta \left(t\right)ln\zeta \left(t\right)\to 0,$$

which implies

$$V\left(T\right)=\underset{t\to T}{lim}V\left(t\right)\le {\displaystyle \frac{k_2}{\mathsf{\epsilon }}}.$$

Therefore, the system state enters the compact region

$$\mathsf{\Omega }=\left\{x\in {\mathbb{R}}^n|V\left(x\right)\le {\displaystyle \frac{k_2}{\mathsf{\epsilon }}}\right\}$$

at the prescribed time $t=T$. Notably, the size of the region $\mathsf{\Omega }$ depends only on the user-defined parameters $(k_2,\mathsf{\epsilon })$, not on the system dynamics. Thus, the convergence precision can be explicitly specified according to task requirements.

Now consider the evolution for $t\ge T$. Substituting (6) into condition (10) yields

$$\dot{V}\left(t\right)\le -(k_1+\mathsf{\epsilon })V\left(t\right)+2k_2.$$

(13)

This is a linear differential inequality. Applying the integrating factor method and integrating over $[T,t]$, we obtain

$$\begin{array}{cc}\hfill V\left(t\right)& \le e^{-(k_1+\mathsf{\epsilon })(t-T)}V\left(T\right)\hfill \\ \hfill & +{\int }_T^t{e}^{-(k_1+\mathsf{\epsilon })(t-\tau )}·2k_{2}d\tau \hfill \\ \hfill & =e^{-(k_1+\mathsf{\epsilon })(t-T)}V\left(T\right)+{\displaystyle \frac{2k_2}{k_1+\mathsf{\epsilon }}}\left(1-e^{-(k_1+\mathsf{\epsilon })(t-T)}\right).\hfill \end{array}$$

(14)

Using the upper bound $V\left(T\right)\le {\displaystyle \frac{k_2}{\mathsf{\epsilon }}}$, we get

$$\begin{array}{cc}\hfill V\left(t\right)& \le {\displaystyle \frac{k_2}{\mathsf{\epsilon }}}{e}^{-(k_1+\mathsf{\epsilon })(t-T)}\hfill \\ \hfill & +{\displaystyle \frac{2k_2}{k_1+\mathsf{\epsilon }}}\left(1-e^{-(k_1+\mathsf{\epsilon })(t-T)}\right)\hfill \\ \hfill & <{\displaystyle \frac{k_2}{\mathsf{\epsilon }}},\forall t\ge T.\hfill \end{array}$$

(15)

Hence, the system trajectory remains within the region $\mathsf{\Omega }$ for all future times $t>T$. This completes the proof. □

Remark 2. 

The piecewise-defined function in (6) possesses several fundamental properties that are crucial for achieving prescribed-time stability over the interval $t\in [t_0,T)$.

1. 

The function $\zeta \left(t\right)$ is strictly decreasing and remains strictly positive within the entire interval.

2. 

It converges to zero if and only if t tends to T.

3. 

The limit ${lim}_{\zeta \to 0}\zeta ln\zeta$ is equal to zero.

As a result, any smooth function $\zeta \left(t\right)\in C^{\infty }$ that satisfies these conditions can be adopted as a substitute in (6).

For practical implementation, the time derivative $\dot{\zeta }\left(t\right)$ as $t\to T$ plays a significant role in control design. The parameters ε, $k_1$, and $k_2$ are central to realizing prescribed-time (PT) stability. Notably, increasing ε or $k_1$ (or, equivalently, reducing $k_2$) can enhance the convergence precision. However, this improvement typically comes at the cost of increased control effort, potentially leading to high control gains. Therefore, a balance must be struck in selecting these parameters, taking into account real-world constraints such as communication bandwidth limitations and actuator saturation levels.

Example 1. 

Consider system (7), with $d\left(t\right)=sin\left(10t\right)$, $T=1$, $k_1=3$, $\mathsf{\epsilon }=ln\left(2\right)/T$, $k_2=0.02$, and $k_3=0.99$.

The simulation results are presented in Figure 1, Figure 2, Figure 3 and Figure 4, which collectively demonstrate the effectiveness of the proposed control scheme as established in Theorem 1. Figure 1 shows the state trajectories converging in fixed time, while Figure 2 displays the corresponding control efforts. The zoomed-in views in Figure 3 and Figure 4 provide detailed insight into the convergence behavior near the prescribed settling time.

4. Application to SMC

4.1. Integral TSMC

Let us consider the system with perturbations.

$$\left\{\begin{array}{c}{\dot{x}}_{i1}\left(t\right)=f_1(x,t)+g_1(x,t)u_1\left(t\right)+d_1\left(t\right)\hfill \\ {\dot{x}}_{i2}\left(t\right)=f_2(x,t)+g_2(x,t)u_2\left(t\right)+d_2\left(t\right)\hfill \\ ⋮\hfill \\ {\dot{x}}_n\left(t\right)=f_n(x,t)+g_n(x,t)u_n\left(t\right)+d_n\left(t\right)\hfill \end{array}\right.$$

(16)

$$x\left(0\right)=x_0,y_i=x_i\left(t\right),$$

where $x\in {\mathbb{R}}^n$ denotes the state vector, $u\in {\mathbb{R}}^n$ the input, and $y\in {\mathbb{R}}^n$ the output. The functions $f_i$ and $g_i$ are known nonlinear mappings, with $g_i\ne 0$ for all $x\in {\mathbb{R}}^n$. The term $d_i\in {\mathbb{R}}^n$ represents a perturbation, which is assumed to be bounded but with an unknown upper bound, satisfying $|d_i\left(t\right)|\le {\overline{d}}_i$ for each $i\in I$, where $I=\{1,\dots ,n\}$. The mathematical model represented by Equation (16) holds significance across various crucial engineering applications, including but not limited to multi-agent systems [30], synchronization of complex dynamical networks [31], recurrent neural networks [32], and beyond. This study aimed to develop a fixed-time integral sliding mode control strategy wherein all system states converge to zero within a predetermined settling time, while ensuring that this convergence is independent of initial conditions and bounded by control parameters.

The mathematical model described by Equation (7) plays an important role in a wide range of key engineering fields, such as multi-agent systems [30], synchronization in complex dynamical networks [31], and recurrent neural networks [32], among others.

Problem 1. 

The objective of this study was to develop a fixed-time integral sliding mode control strategy that ensures that all system states converge to zero within a predetermined settling time. This convergence is independent of initial conditions and bounded by control parameters, providing robust and reliable control for the system described by Equation (7).

This paper introduces the following novel integral terminal sliding surface:

$$\begin{array}{cc}\hfill s_i\left(t\right)& =e_i\left(t\right)+{\int }_0^t((k_{1i}+{\displaystyle \frac{|\dot{{\zeta }_1}\left(t\right)\left(\tau \right)|}{{\zeta }_1\left(\tau \right)}})\left|e_i\left(\tau \right)\right|-{\displaystyle \frac{k_{2i}}{{\zeta }_1\left(\tau \right)}}+k_{3i})\times \hfill \\ \hfill & sgn\left(e_i\left(\tau \right)\right)d\tau \hfill \end{array}$$

(17)

where

$$e_i\left(t\right)=x_i\left(t\right)-x_{ir}\left(t\right)$$

(18)

$x_{ir}$ denotes the reference value.

$${\zeta }_1\left(t\right)=\left\{\begin{array}{cc}exp\left({\mathsf{\epsilon }}_1(T_1-t)\right)-1,\hfill & \mathrm{if}t\in [t_0,T_1)\hfill \\ exp\left({\mathsf{\epsilon }}_1(t-T_1)\right),\hfill & \mathrm{if}t\in [T_1,+\infty )\hfill \end{array}\right.$$

(19)

where $T_1$ is user time and ${\mathsf{\epsilon }}_1\in (0,ln\left(2\right)/T_1]$ is a tuning parameter.

We propose the following control law:

$$\begin{array}{cc}\hfill u_i\left(e\right)& =-g_i{\left(x\right)}^{-1}\left(f_i\left(x\right)+\left(k_{1i}+{\displaystyle \frac{|{\dot{\zeta }}_i\left(t\right)|}{{\zeta }_1\left(t\right)}}\right)\left|e_i\left(t\right)\right|-{\displaystyle \frac{k_{2i}}{{\zeta }_1\left(t\right)}}\right.\hfill \\ \hfill & +k_{3i}sign\left(e_i\left(t\right)\right)-{\dot{x}}_{ir}\left(t\right)+\left(h_{1i}+{\displaystyle \frac{|{\dot{\zeta }}_2\left(t\right)|}{{\zeta }_2\left(t\right)}}\right)\left|s_i\left(t\right)\right|\hfill \\ \hfill & \left.-{\displaystyle \frac{h_{2i}}{{\zeta }_2\left(t\right)}}+h_{3i}sign\left(s_i\left(t\right)\right)\right)\hfill \end{array}$$

(20)

where

$${\zeta }_2\left(t\right)=\left\{\begin{array}{cc}exp\left({\mathsf{\epsilon }}_2(T_2-t)\right)-1,\hfill & \mathrm{if}t\in [t_0,T_2)\hfill \\ exp\left({\mathsf{\epsilon }}_2(t-T_2)\right),\hfill & \mathrm{if}t\in [T_2,+\infty )\hfill \end{array}\right.$$

(21)

where $T_2$ is user time and ${\mathsf{\epsilon }}_2\in (0,ln\left(2\right)/T_2]$ is a tuning parameter.

We propose the following theorem.

Theorem 2. 

If $h_{i1}>{\mathsf{\epsilon }}_2$, $h_{i2}>h_{i4}$, and $h_{i4}={\overline{d}}_i-h_{i3}>0$, then the global closed-loop systems (7), (18), (17), and (20) are PP stable and bounded by settling time.

$$T_{max}=T_1+T_2,$$

(22)

Proof. 

Consider the sliding manifold

$$\mathcal{S}=\left\{e_i\left(t\right)\in \mathbb{R}:s_i\left(e\right)=0\right\}.$$

We have

$$\begin{array}{cc}\hfill {\dot{s}}_i\left(e\right)& =(k_{1i}+{\displaystyle \frac{|{\dot{\zeta }}_i\left(t\right)|}{{\zeta }_1\left(t\right)}})\left|e_i\left(t\right)\right|-{\displaystyle \frac{k_{2i}}{{\zeta }_1\left(t\right)}}+k_{3i}sign\left(e_i\left(t\right)\right)-{\dot{x}}_{ir}\left(t\right)\hfill \\ \hfill & +f_i\left(x\right)+g_i\left(x\right)u_i\left(e\right)+d_i\left(t\right),\hfill \end{array}$$

(23)

Substituting the input yields

$${\dot{s}}_i\left(e\right)=-\left((h_{1i}+{\displaystyle \frac{|{\dot{\zeta }}_i\left(t\right)|}{{\zeta }_1\left(t\right)}})\left|s_i\left(t\right)\right|-{\displaystyle \frac{h_{2i}}{{\zeta }_1\left(t\right)}}+h_{3i}\right)sign\left(s_i\left(t\right)\right)+d_i\left(t\right),$$

Applying Theorem 1, the sliding variable (17) is GPTS with a bounded settling time $T_2$ given in advance.

Let us now prove that when $e_i\in \mathcal{S}$, we have

$${\dot{e}}_i\left(t\right)=-((k_{1i}+{\displaystyle \frac{|\dot{\zeta }{\left(t\right)}_i\left(t\right)|}{{\zeta }_1\left(t\right)}})\left|e_i\left(t\right)\right|-{\displaystyle \frac{k_{2i}}{{\zeta }_1\left(t\right)}}+k_{3i})sgn\left(e_i\left(t\right)\right).$$

(24)

Applying Theorem 1, the tracking error is PPT stable and bounded by the settling time $T_1$, which concludes the proof. □

4.2. Non-Singular Terminal Sliding Mode Control

Consider the following uncertain nonlinear second-order system described by

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

(25)

where $x={[x_1,x_2]}^T\in {\mathbb{R}}^2$ is the system state, $f\left(x\right)$ and $g\left(x\right)$ are continuous, non-linear functions of x, $g\left(x\right)\ne 0:\forall x\in {\mathbb{R}}^2$, $u\in \mathbb{R}$ is the control input, and $d\left(t\right)$ is the lumped disturbance including unmodeled dynamics, model uncertainties, and external disturbances.

To address the previous issue, we introduce a novel sliding variable approach. Consider the sliding variable defined by

$$s\left(x\right)=x_2+\left(\left(k_{1x}+{\displaystyle \frac{|{\dot{\zeta }}_x\left(t\right)|}{{\zeta }_x\left(t\right)}}\right)\left|x_1\right|-{\displaystyle \frac{k_{2x}}{{\zeta }_x\left(t\right)}}+k_{3x}\right)x_1,$$

(26)

where $k_{1x},k_{2x},k_{3x}$ are positive constants and ${\zeta }_x\left(t\right)$ is a positive, time-varying function with a bounded derivative. Here, the state vector is $x={(x_1,x_2)}^{\top }\in {\mathbb{R}}^2$.

To simplify notation, define the time-dependent coefficient

$$A(t,x_1):=\left(k_{1x}+{\displaystyle \frac{|{\dot{\zeta }}_x\left(t\right)|}{{\zeta }_x\left(t\right)}}\right)\left|x_1\right|-{\displaystyle \frac{k_{2x}}{{\zeta }_x\left(t\right)}}+k_{3x}.$$

(27)

Thus, the sliding variable can be expressed compactly as

$$s\left(x\right)=x_2+A(t,x_1)x_1.$$

Next, we compute the time derivative of $s\left(x\right)$. Applying the product rule, we have

$$\dot{s}\left(x\right)={\dot{x}}_2+{\displaystyle \frac{d}{dt}}\left(A(t,x_1)x_1\right)={\dot{x}}_2+\dot{A}(t,x_1)x_1+A(t,x_1){\dot{x}}_1.$$

(28)

Using the system’s state relations

$${\dot{x}}_1=x_2,{\dot{x}}_2={\ddot{x}}_1,$$

Equation (28) becomes

$$\dot{s}\left(x\right)={\ddot{x}}_1+\dot{A}(t,x_1)x_1+A(t,x_1)x_2.$$

(29)

Now, consider the system’s second-order dynamics given by

$${\ddot{x}}_1=f\left(x\right)+g\left(x\right)u+d,$$

(30)

where $f\left(x\right)$ is the known system drift, $g\left(x\right)$ is the input gain, u is the control input, and d denotes bounded disturbances or uncertainties.

Substituting (30) into (29), we obtain

$$\dot{s}\left(x\right)=f\left(x\right)+g\left(x\right)u+d+\dot{A}(t,x_1)x_1+A(t,x_1)x_2.$$

(31)

To complete the expression for $\dot{s}\left(x\right)$, we compute the time derivative of the coefficient $A(t,x_1)$. From (27), since $k_{1x},k_{2x},k_{3x}$ are constants, we have

$$\dot{A}(t,x_1)={\displaystyle \frac{d}{dt}}\left(\left(k_{1x}+{\displaystyle \frac{|{\dot{\zeta }}_x\left(t\right)|}{{\zeta }_x\left(t\right)}}\right)\left|x_1\right|-{\displaystyle \frac{k_{2x}}{{\zeta }_x\left(t\right)}}+k_{3x}\right).$$

(32)

Expanding (32), we get

$$\dot{A}=\left(k_{1x}+{\displaystyle \frac{|{\dot{\zeta }}_x|}{{\zeta }_x}}\right){\displaystyle \frac{d}{dt}}|x_1|+|x_1|{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{|{\dot{\zeta }}_x|}{{\zeta }_x}}\right)+k_{2x}{\displaystyle \frac{d}{dt}}\left(-{\displaystyle \frac{1}{{\zeta }_x}}\right).$$

(33)

Recall the derivatives of each term involved:

$${\displaystyle \frac{d}{dt}}\left|x_1\right|=sgn\left(x_1\right){\dot{x}}_1=sgn\left(x_1\right)x_2,$$

$${\displaystyle \frac{d}{dt}}\left(-{\displaystyle \frac{1}{{\zeta }_x}}\right)={\displaystyle \frac{{\dot{\zeta }}_x}{{\zeta }_x^2}},$$

and denote

$$H\left(t\right):={\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{|{\dot{\zeta }}_x\left(t\right)|}{{\zeta }_x\left(t\right)}}\right),$$

which depends on the known or bounded function ${\zeta }_x\left(t\right)$.

Substituting these results into (33) gives

$$\dot{A}=\left(k_{1x}+{\displaystyle \frac{|{\dot{\zeta }}_x|}{{\zeta }_x}}\right)sgn\left(x_1\right)x_2+\left|x_1\right|H\left(t\right)+k_{2x}{\displaystyle \frac{{\dot{\zeta }}_x}{{\zeta }_x^2}}.$$

(34)

Finally, substituting (34) back into (31), we obtain the complete expression for the time derivative of the sliding variable:

$$\begin{array}{cc}\hfill \dot{s}& =f\left(x\right)+g\left(x\right)u+d\hfill \\ \hfill & +\left[\left(k_{1x}+{\displaystyle \frac{|{\dot{\zeta }}_x|}{{\zeta }_x}}\right)sgn\left(x_1\right)x_2+\left|x_1\right|H\left(t\right)+k_{2x}{\displaystyle \frac{{\dot{\zeta }}_x}{{\zeta }_x^2}}\right]x_1+A(t,x_1)x_2,\hfill \end{array}$$

(35)

where $A(t,x_1)$ is given in (27).

This expression serves as the basis for the design of the control input u to enforce the sliding condition $s\left(x\right)=0$ and guarantee robust convergence in the presence of disturbance d.

In order to find the equivalent control law, we impose the sliding condition $\dot{s}=0$ while assuming that the disturbance d is negligible (i.e., $d\approx 0$). From Equation (35), setting $\dot{s}=0$ yields

$$0=f\left(x\right)+g\left(x\right)u_{eq}+\dot{A}(t,x_1)x_1+A(t,x_1)x_2,$$

(36)

where $u_{eq}$ denotes the equivalent control input. Solving for $u_{eq}$, we obtain

$$u_{eq}=-g{\left(x\right)}^{-1}\left[f\left(x\right)+\dot{A}(t,x_1)x_1+A(t,x_1)x_2\right].$$

(37)

This control law represents the continuous control action that maintains the system state on the sliding manifold defined by $s\left(x\right)=0$, ensuring that the sliding condition is met in the absence of disturbances.

To ensure stability of the sliding variable s, we propose the following switching control law:

$$u_s=-g{\left(x\right)}^{-1}\left(A_s(t,s)sign\left(s\right)\right),$$

(38)

where $A_s(t,s)>0$ is a design gain function that can depend on time and the sliding variable s, such as

$$A_s(t,s):=\left(k_{1s}+{\displaystyle \frac{|{\dot{\zeta }}_s\left(t\right)|}{{\zeta }_s\left(t\right)}}\right)\left|s\right|-{\displaystyle \frac{k_{2s}}{{\zeta }_s\left(t\right)}}+k_{3s}.$$

$${\zeta }_s\left(t\right)=\left\{\begin{array}{cc}exp\left({\mathsf{\epsilon }}_s(T_s-t)\right)-1,\hfill & \mathrm{if}t\in [t_0,T_s)\hfill \\ exp\left({\mathsf{\epsilon }}_s(t-T_s)\right),\hfill & \mathrm{if}t\in [T_s,+\infty )\hfill \end{array}\right.$$

(39)

This discontinuous control term is responsible for driving the system state trajectories towards the sliding manifold $s\left(x\right)=0$ and maintaining sliding motion in the presence of disturbances and model uncertainties.

Combining the equivalent control law $u_{eq}$ given by (37) and the switching term $u_s$, the total control input can be expressed as

$$u=u_{eq}+u_s.$$

(40)

This composite control law guarantees both the existence of the sliding mode and robustness against perturbations.

The following theorem summarizes the stability properties of the closed-loop system under the proposed sliding mode control law.

Theorem 3. 

Consider the system (25) with the sliding variable $s\left(x\right)$ defined in (26). Suppose that the control input u is designed as in (40), where $u_{eq}$ is given by (37) and the switching gain $A_s(t,s)$ in (38) is chosen to satisfy

$$A_s(t,s)>\left|d\left(t\right)\right|,$$

with $d\left(t\right)$ being the bounded disturbance.

Then, the closed-loop system is predefined-time stable with a given settling time $T_c$.

Proof. 

Starting from the dynamics of the sliding variable s, we have

$$\dot{s}=f\left(x\right)+g\left(x\right)u+d\left(t\right)+\dot{A}(t,x_1)x_1+A(t,x_1)x_2.$$

Choosing the control input as

$$u=u_{eq}+u_s,$$

where the equivalent control $u_{eq}$ cancels the nominal terms

$$f\left(x\right)+g\left(x\right)u_{eq}+\dot{A}(t,x_1)x_1+A(t,x_1)x_2=0,$$

and the switching control is defined by

$$u_s=-g{\left(x\right)}^{-1}{A}_s(t,s)sign\left(s\right),$$

substituting into the expression for $\dot{s}$ yields

$$\dot{s}=g\left(x\right)u_s+d\left(t\right)=-A_s(t,s)sign\left(s\right)+d\left(t\right).$$

Applying Theorem 1 with $k_{1s}>{\mathsf{\epsilon }}_s$, $k_{2s}>k_{4s}$, and $k_{4s}=\overline{d}-k_{3s}>0$, s converges to zero in $T_s$. On the sliding manifold, we have

$${\dot{x}}_1=-\left(\left(k_{1x}+{\displaystyle \frac{|{\dot{\zeta }}_x\left(t\right)|}{{\zeta }_x\left(t\right)}}\right)\left|x_1\right|-{\displaystyle \frac{k_{2x}}{{\zeta }_x\left(t\right)}}+k_{3x}\right)x_1,$$

Applying Theorem 1, the state $x_1$ converges to zero within the predefined time $T_x$, which is characterized by the function ${\zeta }_x$. Consequently, the closed-loop system is predefined-time stable, with a settling time bounded by

$$T_c\le T_s+T_x,$$

which completes the proof. □

Remark 3. 

(Chattering Reduction). One of the known drawbacks of classical sliding mode control is the presence of chattering in the control input due to the discontinuous nature of the sign function. This chattering effect can potentially excite unmodeled dynamics or lead to mechanical wear in practical implementations. To mitigate this issue, one effective approach is to replace the discontinuous sign function with a continuous approximation such as the hyperbolic tangent function, i.e.,

$$\mathit{sign}\left(s\right)⟶tanh\left({\displaystyle \frac{s}{ϵ}}\right),$$

where $ϵ>0$ is a small positive constant that controls the smoothness of the approximation. This modification allows for smoother control action while preserving the finite-time convergence and robustness properties within a boundary layer. The effectiveness of such approximations has been widely validated in the literature and will be considered in future experimental implementations.

5. Simulations

To verify the effectiveness of the proposed integral terminal sliding mode control with robust prescribed-time stability, a PMSM control system model was constructed in MATLAB 2024B/Simulink. The simulation was run with an initial speed of 0 rpm, sampling time $\Delta t=0.001$ s and the solver (MATLAB ODE45), and a total duration of 0.4 s.

Table 1. The PMSM parameters in the simulation.

Parameter	Symbol	Value
Resistance of the stator	$R_s$	0.148 $\mathsf{\Omega }$
Inductance of the d-q-axis	$L_s$	0.192 mH
Flux linkage	${\psi }_f$	0.0082 Wb
Inertia moment	J	12 kg·mm2
Coefficient of damping	B	$5.59\times {10}^{-5}$ N·m·s/rad
Number of pole pairs	$P_n$	4

contains a list of the PMSM parameters that were used. The results of the simulation show how well the suggested controller worked. The ITSMC guaranteed zero steady-state inaccuracy and quick, smooth convergence to the target speed within the allotted time. The durability of the system was demonstrated by its capacity to retain stability and swiftly converge in the face of a sudden load torque. The controller’s ability to adjust and maintain precise tracking without experiencing performance degradation under simultaneous changes in reference speed and external load validated its resilience and efficacy in nonlinear PMSM control.

We considered a permanent magnet synchronous motor (PMSM) described by the model [33]

$$\left\{\begin{array}{cc}{\dot{x}}_1\hfill & ={\lambda }_1(x_2-x_1)\hfill \\ {\dot{x}}_2\hfill & =-x_2-x_3{x}_1+{\lambda }_2{x}_1\hfill \\ {\dot{x}}_3\hfill & =-x_3+x_1{x}_2\hfill \end{array}\right.$$

(41)

where $x_1$, $x_2$, and $x_3$ represent the angle speed, quadrature-axis, and direct-axis currents of the motor, respectively. Chaotic behavior in PMSMs can threaten stability, which was addressed by applying the proposed controller to stabilize chaos in the PMSM model. We considered the PMSM system with the inputs and perturbations with ${\lambda }_1=2.5$ and ${\lambda }_2=25$ as

$$\left\{\begin{array}{cc}{\dot{x}}_1\hfill & =2.5(x_2-x_1)+u_1+d_1\left(t\right)\hfill \\ {\dot{x}}_2\hfill & =-x_2-x_3{x}_1+25x_1+u_2+d_2\left(t\right)\hfill \\ {\dot{x}}_3\hfill & =-x_3+x_1{x}_2+u_3+d_3\left(t\right)\hfill \end{array}\right.$$

(42)

with the perturbations

$$\left\{\begin{array}{cc}{d}_1\left(t\right)\hfill & =sin\left(10t\right)\hfill \\ d_2\left(t\right)\hfill & =cos\left(10t\right)\hfill \\ d_3\left(t\right)\hfill & =cos\left(10t\right)·sin\left(4t\right).\hfill \end{array}\right.$$

(43)

We considered the following control parameters:

$$\begin{array}{c}{\mathsf{\epsilon }}_1={\displaystyle \frac{ln\left(2\right)}{T_{i1}}},{\mathsf{\epsilon }}_2={\displaystyle \frac{ln\left(2\right)}{T_{i2}}},k_{i1}=3,k_{i2}=0.02,k_{i3}=0.99,\\ h_{i1}=3,h_{i2}=0.04,h_{i3}=0.99.\end{array}$$

The following settling time was chosen: $T_{max}=T_1+T_2=2$ s.

The simulation results are plotted in Figure 5, Figure 6 and Figure 7. The state variables of the PMSM converged before the specified time $T_{max}=2$ seconds, as shown in Figure 5. This indicated that the system achieved the desired performance within the predetermined time limit. Additionally, the sliding variables, which were crucial for maintaining system stability and robustness, also converged within the same time frame, as illustrated in Figure 7. This convergence of sliding variables confirmed the effectiveness of the control strategy in achieving the desired performance. The input signals required to drive the system are depicted in Figure 6, showing how they varied over time to meet the performance specifications and ensure that the system operated within the defined constraints.

Figure 8 further illustrates the convergence process on the sliding surface. It shows the sliding variable $s_1\left(t\right)$ approaching zero, which defined the sliding manifold, and the state $x_1\left(t\right)$ converging to the equilibrium point within the prescribed time. This figure strongly supports the theoretical claims about the system’s ability to reach the sliding manifold and the desired equilibrium in the prescribed time interval.

Remark 4. 

While the proposed methods offer significant advantages in terms of robustness and user-defined convergence time, they inherently require larger control efforts, especially during the initial transient phase. This is a well-known trade-off in prescribed-time and fixed-time control schemes: ensuring rapid convergence within a predefined time bound often necessitates high control gains, which, in turn, can lead to large-magnitude control inputs (as observed in Figure 6, where $u_1$ exceeds 20). In practical applications, such large inputs may approach actuator saturation limits or increase energy consumption. Therefore, the prescribed-time stability property can be viewed as being “purchased” at the cost of increased control effort, and, in real-world implementations, this trade-off must be carefully considered when selecting control parameters.

6. Conclusions

In this paper, we proposed three theorems addressing robustness and convergence properties for scalar systems and a class of nonlinear systems. The proofs were established using Lyapunov theory. The core idea was to introduce a time-varying function in the feedback control design and then extend this concept to the design of sliding surface variables. A major drawback of terminal sliding mode control is the singularity problem (arising from the use of state powers), which was avoided in this work by employing the time-varying function. Simulation results were presented for each theorem, and an application to a PMSM system was provided, demonstrating both robustness and fast convergence. Future work will focus on extending the approach to more complex systems.