Sliding mode control with prescribed performance for a class of mechanical systems subject to uncertainties and disturbances

Abstract

This paper proposes a predefined-time tracking control methodology for a class of nonlinear mechanical servo systems operating in the presence of the external disturbances and system uncertainties. The process begins with the creation of a nonlinear disturbance observer aimed at accurately estimating the lumped disturbance. This observer is designed to ensure that the error in estimation converges towards zero in a time specified by the user. Following this, a new predefined-time sliding-mode surface is presented, which guarantees that the closed-loop system adheres to predefined-time performance while avoiding potential singularities. Leveraging this manifold, a continuous control law is formulated to impose a predefined-time constraint upon the sliding dynamics, guiding the system states into a designated vicinity of the sliding surface within a specified finite duration. This approach eliminates the need for discontinuous switching components, effectively reducing the chattering phenomenon. A rigorous theoretical framework confirms that the proposed control strategy achieves global predefined-time stability. Lastly, extensive numerical simulations validate the effectiveness and enhanced performance of the proposed strategy.

Introduction

The performance of high-precision mechanical servo systems, essential in areas such as precision manufacturing and optical engineering, often faces challenges due to a combination of external disturbances and uncertainties in internal parameters.^1,2 These detrimental factors can significantly impair positioning accuracy, highlighting the urgent need for the development of high-fidelity, robust control systems as a key research focus. In response to this challenge, a variety of sophisticated nonlinear control methods has emerged, which include adaptive compensation techniques,^3,4 robust control frameworks,^5,6 and intelligent algorithms.^7,8 These methods are designed to alleviate the impact of combined uncertainties and friction. A notable accomplishment across these approaches is their capability to ensure that the tracking error is uniformly ultimately bounded (UUB), indicating that it converges within a small vicinity of the origin. Nevertheless, a considerable analytical gap remains. The presence of an unmodeled composite disturbance term, inherent to the system dynamics, often causes these strategies to inadequately deliver a precise, quantitative model for assessing the magnitude of steady-state errors or transient response features, including convergence speed.

Disturbance observer-based control (DOBC) is a powerful strategy for enhancing system robustness. By estimating and compensating for lumped disturbances, DOBC effectively mitigates the adverse effects of model uncertainties and external perturbations. Research in DOBC has progressed along several key directions. One major thrust involves integrating disturbance observers into advanced control frameworks. For instance, ⁹ embeds a disturbance observer within a command-filtered backstepping architecture to simultaneously address inertial uncertainties, external disturbances, and input saturation in attitude control systems. This integrated approach ensures UUB stability while alleviating the “explosion of complexity” inherent in traditional backstepping. Another direction focuses on designing observers for broad classes of disturbances, such as the nonlinear disturbance observer in,^10,11 which demonstrates efficacy against both matched and unmatched disturbances. Concurrently, significant theoretical efforts have aimed to improve observer performance while relaxing restrictive assumptions on the disturbance dynamics. The work in ¹² exemplifies this by achieving exponential convergence of the estimation error without requiring the disturbance to be smooth or have a decaying derivative—a common prerequisite for Extended State Observers (ESOs).^13–15 However, this theoretical advancement has a crucial limitation: for persistently time-varying disturbances with a non-vanishing derivative, the observer only guarantees the estimation error converges to a bounded residual set, rather than achieving predefined-time convergence.

The inherent robustness of sliding-mode control (SMC) against uncertainties has motivated significant research efforts in nonlinear control. A plethora of advanced SMC strategies have been designed^16–19 to address systems subject to both internal uncertainties and external disturbances. The development of these methods has centered on enhancing system convergence properties. For instance, while conventional linear SMC (LSMC)^20–21 is effective for tracking tasks, its asymptotic convergence restricts its applicability in scenarios demanding rapid response. To achieve finite-time convergence, terminal SMC (TSMC)^22,23 were introduced. However, TSMC exhibits a sluggish convergence rate when the system state is far from the equilibrium. This deficiency motivated the development of fast TSMC (FTSMC),^24,25 which significantly enhance the convergence speed. Nevertheless, the convergence time of FTSMC remains dependent on the system's initial conditions. Fixed-time SMC^26–28 resolved this issue by ensuring the convergence time has an upper bound determined solely by controller parameters, irrespective of the initial state. The process of setting this time, however, can be complex due to the involvement of multiple parameters. Building upon this, predefined-time SMC^29–31 methods were proposed to simplify the design, allowing the convergence time to be directly specified via a single parameter. Furthermore, to tackle potential singularity issues arising from negative fractional powers, methods like Nonsingular ³² and Tunable ³³ predefined-time SMC have been developed, enabling the pre-setting of convergence time, and offering flexibility in practical implementation.

Despite the above progress, several limitations remain for practical high-precision servo applications. Most existing disturbance observers and ESOs^9–15 guarantee only asymptotic or exponential convergence of the estimation error. Advanced SMC schemes^16–28 enhance convergence speed from asymptotic to finite-time, fixed-time, or predefined-time regimes. However, the achievable settling time and tracking accuracy are typically coupled with unknown lumped disturbances and modeling uncertainties. As a consequence, the convergence region of the tracking error cannot be quantified explicitly for a given settling time. Moreover, most sliding-mode controllers employ discontinuous switching terms, which induce chattering and may excite unmodeled dynamics. Existing predefined-time SMC methods^29–33 still do not account for precise disturbance reconstruction within a user-defined time and therefore cannot simultaneously guarantee disturbance compensation, prescribed transient behavior and steady-state accuracy. Motivated by the above discussion, a predefined-time tracking control strategy is proposed for a class of nonlinear system. The technical contributions of this paper are briefly summarized as follows:

A predefined-time stability lemma is first established and then used to synthesize a nonlinear disturbance observer (PTNDO). Unlike conventional disturbance observers and ESOs,^9–15 the PTNDO guarantees that the lumped disturbance estimation error converges to zero within a user-specified predefined time. This feature allows the controller to compensate for rapidly varying disturbances.

A nonlinear sliding surface is constructed by embedding the error compensation term into the manifold. This design enforces prescribed tracking precision with predefined-time convergence: the system states are driven into a neighborhood of the manifold within a user-settable time, and the size of the ultimate tracking error set is analytically characterized as an explicit function of the design parameters. In contrast to existing finite-time and fixed-time SMC schemes,^22–28 the convergence region is no longer implicit or dependent on unknown disturbances.

Leveraging the disturbance estimate provided by the PTNDO, a continuous sliding-mode control law is derived. The proposed control law does not rely on discontinuous sign functions, thereby effectively suppressing chattering while maintaining robustness against model uncertainties and external perturbations. The resulting closed-loop system is rigorously proven to be predefined-time stable.

The rest of the paper is organized as follows. Section “Preliminaries and problem statement” provides the statements of problem and preliminaries. The control design procedure is shown in Section Design of predefined-time control law. In Section Simulation study, the addressed method is applied to ball-plate system and robotic manipulator to test their effectiveness via simulation. Finally, some conclusions are drawn in Section Conclusions.

Preliminaries and problem statement

System description

Consider the following the uncertain mechanical system with actuator faults:

$$\mathit{M}\ddot{\mathit{x}}+\mathit{F}(\mathit{x},\dot{\mathit{x}},\ddot{\mathit{x}})+\mathit{D}=\mathit{u}$$

(1)

where $\mathit{x}\in {\mathbb{R}}^n$ is the system state vector; $\mathit{M}\in {\mathbb{R}}^{n\times n}$ is unknown inertia matrix or unknown mass matrix, and it is composed of known nominal values and unknown perturbations, written as $\mathit{M}={\mathit{M}}_0+\mathrm{\Delta }\mathit{M}$ ; $\mathit{F}\left(\mathit{x},\dot{\mathit{x}},\ddot{\mathit{x}}\right)\in {\mathbb{R}}^n$ represents denotes the nonlinear dynamics of the controlled system; $\mathit{D}\in {\mathbb{R}}^n$ represents the unknown external disturbance; $\mathit{u}\in {\mathbb{R}}^n$ is system input vector. Based on the controlled system (1), the following Assumptions that can be adopted in the stability analysis of the closed-loop system.

Assumption 1. Bounded measurements of the system's position, velocity, and acceleration are assumed to be available from a fault-free sensor suite.

Assumption 2. The each element of $\mathit{D}$ and $\mathit{F}$ are bound, and the each element of the command signal ${\mathit{x}}_{\mathit{d}}\in {\mathbb{R}}^n$ and its first and second derivatives are bound.

Remark 1. The system model (1) serves as a general representation for a broad class of nonlinear systems, including linear motor positioner, ² robotic manipulators, ⁵ and micro-electro-mechanical system. ³⁴ This inherent generality ensures that the controller designed herein can be seamlessly implemented across these different platforms.

Let ${\mathit{x}}_1=\mathit{x}$ , ${\dot{\mathit{x}}}_1={\mathit{x}}_2$ and $\mathit{\omega }={\mathit{M}}_o^{-1}\left(-\mathrm{\Delta }\mathit{M}\ddot{\mathit{x}}-\mathit{F}-\mathit{D}\right)$ , the previous model (1) can be expressed as

$$\{\begin{array}{c}{\dot{\mathit{x}}}_1={\mathit{x}}_2\\ {\dot{\mathit{x}}}_2=\omega +{\mathit{M}}_0^{-1}u\end{array}$$

(2)

We first present two lemmas that are essential for the subsequent controller synthesis.

Lemma 1 . ⁵ Let $\delta \in {\mathbb{R}}^{+}$ and $y\in (-\delta ,\delta )$ . Then,

$$\log {\displaystyle \frac{{\delta }^2}{{\delta }^2-y^2}\le {\displaystyle \frac{y^2}{{\delta }^2-y^2}}}$$

(3)

Lemma 2 . Consider the nonlinear function $\psi (t):[0,\mathrm{\infty })\to [0,1]$ given by

$$\psi (t)=\{\begin{array}{c}1-{\displaystyle \frac{\varsigma {(T_F-t)}^3}{e^{t-T_F}},if0\le t<T_F}\\ 1,ift\ge T_F\end{array}$$

(4)

with parameter $T_F\in {\mathbb{R}}^{+}$ is a settling time and the parameter $\varsigma$ is given by $\varsigma =\frac{e^{-T_F}}{T_F^3}$ . This function satisfies:

1. $\psi (0)=0$ and $\psi (t)>0$ for all $t>0$ ;

2. $\psi (t)=1$ for all $t\ge T_F$ ;

3. $\dot{\psi }(t)$ is continuous and bounded on $[0,\mathrm{\infty })$ .

Proof : See Appendix A.

From (4), the derivative of $\psi$ is

$$\dot{\psi }(t)=\{\begin{array}{c}\theta \cdot e^{T_F-t},if0\le t<T_F\\ 0,if{T}_F\le t\end{array}$$

(5)

where $\theta =\varsigma (T_F-t{)}^2\cdot (3+T_F-t)$ .

Design of predefined-time control law

To facilitate the analysis of the closed-loop system, the following definition is introduced.

Definition 1. ³⁰ For the following system:

$$\dot{z}=f(z),z_0\triangleq z(0)$$

(6)

in which $z\in {\mathbb{R}}^n$ and $f(z)$ is a nonlinear function, the origin of system (6) is said to be prescribed time stable if for prespecified positive constants $T_R$ (the predefine time) and $\epsilon *$ (prescribed bound), the solution $\Vert z\Vert <\epsilon *$ for $\mathrm{\forall }t\ge T_R$ .

Lemma 3 . Consider system (6) and a continuous Lyapunov function $V(z)$ , if

$$\dot{V}\le -{\displaystyle \frac{\kappa \pi }{rT}\left(V^{1-\frac{r}{2}}+V^{1+\frac{r}{2}}+2V\right)+\xi }$$

(7)

where $\kappa >2$ , $T>0$ , $0<r<1$ are design parameters and $\xi$ is a bounded function (or constant), then system (6) is practically predefined-time stable, and state z satisfies

$$\begin{array}{rl}& z\in {\mathrm{\Omega }}_z\triangleq \left\{z|V\le max\left\{{\beta }_Z^{\frac{2}{(2-r)}},{\beta }_Z^{\frac{2}{(2+r)}},{\displaystyle \frac{(\kappa -1){\beta }_Z}{\kappa -2}}\right\}\right\}\\ & fort\ge T.\end{array}$$

(8)

where ${\beta }_Z=\frac{\xi rT}{((\kappa -1)\pi )}$ .

Proof : Define the auxiliary set

$${\overline{\mathrm{\Omega }}}_z=\left\{z|V>max\left\{{\beta }_Z^{\frac{2}{(2-r)}},{\beta }_Z^{\frac{2}{(2+r)}},{\displaystyle \frac{(\kappa -1){\beta }_Z}{\kappa -2}}\right\}\right\}$$

(9)

From (7) and $\kappa >2$ , the derivative $\dot{V}$ can be equivalently written as

$$\{\begin{array}{c}\dot{V}\le -{\displaystyle \frac{\pi }{rT}\left(V^{1-\frac{r}{2}}+V^{1+\frac{r}{2}}+2V\right)-{\displaystyle \frac{(\kappa -1)\pi }{rT}{V}^{1-\frac{r}{2}}+\xi ,}}\\ or\dot{V}\le -{\displaystyle \frac{\pi }{rT}\left(V^{1-\frac{r}{2}}+V^{1+\frac{r}{2}}+2V\right)-{\displaystyle \frac{(\kappa -1)\pi }{rT}{V}^{1+\frac{r}{2}}+\xi ,}}\\ or\dot{V}\le -{\displaystyle \frac{\pi }{rT}\left(V^{1-\frac{r}{2}}+V^{1+\frac{r}{2}}+2V\right)-{\displaystyle \frac{(\kappa -2)\pi }{rT}V+\xi .}}\end{array}$$

(10)

For any $z\in {\overline{\mathrm{\Omega }}}_z$ , definition (9) ensures $\frac{(\kappa -1)\pi }{rT}{V}^{1-\frac{r}{2}}>\xi$ , $\frac{(\kappa -1)\pi }{rT}{V}^{1+\frac{r}{2}}>\xi$ and $\frac{(\kappa -2)\pi }{rT}V>\xi$ , and each line of (10) reduces to the common bound

$$\dot{V}\le -{\displaystyle \frac{\pi }{rT}\left(V^{1-\frac{r}{2}}+V^{1+\frac{r}{2}}+2V\right)}$$

(11)

Let $T*\triangleq inf\{t*|z(t)\in {\mathrm{\Omega }}_{z}forallt\ge t*\}$ denote the first entrance time into ${\mathrm{\Omega }}_z$ and let the initial condition be $V(0)={\nu }_0>0$ . For all $t\in [0,T*)$ we have $z\in {\overline{\mathrm{\Omega }}}_z$ , thus (11) holds and yields the separable inequality

$$-{\displaystyle \frac{\pi }{rT}dt\ge {\displaystyle \frac{dV}{V^{1-\frac{r}{2}}+V^{1+\frac{r}{2}}+2V}}}$$

(12)

Integrating (12) from 0 to $T*$ gives

$$-{\displaystyle \frac{\pi }{rT}\underset{0}{\overset{T*}{\int }}dt\ge \underset{V(0)}{\overset{V(T*)}{\int }}{\displaystyle \frac{dV}{V^{1-\frac{r}{2}}+V^{1+\frac{r}{2}}+2V}.}}$$

(13)

Hence,

$${\displaystyle \frac{\pi }{rT}\underset{0}{\overset{T*}{\int }}dt\le \underset{V(T*)}{\overset{V(0)}{\int }}{\displaystyle \frac{dV}{V^{1-\frac{r}{2}}+V^{1+\frac{r}{2}}+2V}=\underset{V(T*)}{\overset{V(0)}{\int }}{\displaystyle \frac{2d{V}^{r/2}}{r{(1+V^{r/2})}^2}\le \underset{V(T*)}{\overset{V(0)}{\int }}{\displaystyle \frac{2d{V}^{r/2}}{r(1+V)}.}}}}$$

(14)

Based on (14), one has

$${\displaystyle \frac{\pi }{rT}T*\le {\displaystyle \frac{2}{r}\cdot {\displaystyle \frac{\pi }{2}={\displaystyle \frac{\pi }{r}.}}}}$$

(15)

Equation (15) implies that $T*\le T$ , such that the trajectory z enters and remains within the set ${\mathrm{\Omega }}_z$ for all $t\ge T$ . The proof is thus complete.

Disturbance observer design

To compensate the uncertain lumped disturbance $\mathit{\omega }$ in (2), conventional control methods often employ high-gain feedback for suppression. However, this approach typically leads to an overly conservative design and may induce high-frequency chattering. In this work, we propose a predefined-time disturbance reconstruction strategy. Specifically, an online estimator is designed to accurately approximate $\mathit{\omega }$ , and the control law is augmented with the disturbance estimate to actively compensate for its adverse effects on the system.

To facilitate disturbance estimation, we first introduce the following auxiliary states:

$$\{\begin{array}{c}\hat{\mathit{\omega }}=\mu +\dot{\mathit{s}},\\ s={\mathit{x}}_2-\eta ,\end{array}$$

(16)

where $\hat{\mathit{\omega }}$ is employed to estimate $\mathit{\omega }$ , with $\tilde{\mathit{\omega }}=\mathit{\omega }-\hat{\mathit{\omega }}$ is defined as the disturbance estimation error, while $\mathit{\mu }$ and $\mathit{\eta }$ are updated according to the following equations.

$$\{\begin{array}{c}\dot{\mathit{\eta }}={\mathit{M}}_0^{-1}u+s,\\ \dot{\mathit{\mu }}=\dot{\mathit{s}}+{\displaystyle \frac{{\kappa }_O\pi }{T_O{r}_o}\left(2^{\frac{r_o}{2}-1}\cdot \mathit{s}\mathit{i}{\mathit{g}}^{1-r_o}(\tilde{\mathit{\mu }})+2^{-1-\frac{r_o}{2}}\cdot \mathit{s}\mathit{i}{\mathit{g}}^{1+r_o}(\tilde{\mathit{\mu }})+\tilde{\mathit{\mu }}\right).}\end{array}$$

(17)

in which, $\mathit{s}\mathit{i}\mathit{g}*(\tilde{\mathit{\mu }})\triangleq [|{\tilde{\mu }}_1|*sign({\tilde{\mu }}_1)$ , $|{\tilde{\mu }}_2|*sign({\tilde{\mu }}_2),\dots ,|{\tilde{\mu }}_3|*$ $sign({\tilde{\mu }}_3){]}^T$ , $\tilde{\mathit{\mu }}=\mathit{s}-\mathit{\mu }=[{\tilde{\mu }}_1,{\tilde{\mu }}_2,\dots ,{\tilde{\mu }}_n{]}^T\in {\mathbb{R}}^n$ , $0<r_o<1$ and

${\kappa }_O>2$ are the design parameters. The validity of the disturbance observer structure in (16)–(17), as well as the desired conclusions, can be justified by the proof given below.

Theorem 1 . For the uncertain mechanical system (2), consider the disturbance observer defined by (16)-(17). The resulting disturbance estimation error $\tilde{\mathit{\omega }}$ is guaranteed to converge to zero within a predefined-time $T_O$ .

Proof : According to the definition of $\tilde{\mathit{\mu }}$ and (16)-(17), we obtain

$$\begin{array}{r}\dot{\tilde{\mu }}=\dot{s}-\dot{\mu }=-\frac{\pi }{T_o{r}_o}(2^{\frac{r_o}{2-1}}\cdot si{g}^{1-r_o}(\tilde{\mu })+2^{\frac{-1-r_o}{2}}\cdot si{g}^{1+r_o}(\tilde{\mu })+\tilde{\mu })\end{array}$$

(18)

Construct Lyapunov function:

$$V_O=0.5{\tilde{\mathit{\mu }}}^T\tilde{\mathit{\mu }}$$

(19)

Taking its time derivative along (18) yields

$$\begin{array}{rl}{\dot{V}}_o& ={\tilde{\mu }}^T\dot{\tilde{\mu }}\\ ={\tilde{\mu }}^T(-\frac{{\kappa }_o\pi }{T_o{r}_o}(2^{\frac{r_o}{2-1}}{\mathrm{sig}}^{1-r_o}(\tilde{\mu })+2^{\frac{-1-r_o}{2}}{\mathrm{sig}}^{1+r_o}(\tilde{\mu })+\tilde{\mu }))\\ & \le -\frac{\pi }{T_o{r}_o}(V_o^{\frac{1-r_o}{2}}+V_o^{\frac{1+r_o}{2}}+2V_o).\end{array}$$

(20)

The differential inequality (20) has the same structure as (7) in Lemma 3. Hence, by repeating the arguments used in (11)–(15), it follows that $\tilde{\mathit{\mu }}$ is driven to the origin within a prescribed time $T_O$ . Moreover, using the relations in (16) and the plant dynamics, one obtains

$$\tilde{\mathit{\omega }}=\mathit{\omega }-\hat{\mathit{\omega }}=\mathit{\omega }-\mathit{\mu }-{\dot{\mathit{x}}}_2+\dot{\mathit{\eta }}=\tilde{\mathit{\mu }}$$

(21)

Thus, it follows that $\tilde{\mathit{\omega }}$ converges to the origin by the time $\tilde{\mathit{\mu }}$ , which completes the proof of Theorem 1.

Control law design

For the purpose of controller design, we first define the tracking error as

$${\mathit{e}}_1={\mathit{x}}_1-{\mathit{x}}_d$$

(22)

where ${\mathit{x}}_d\in {\mathbb{R}}^n$ is the desired command signal vector, subsequently, a nonlinear sliding-mode surface $\mathit{\sigma }=[{\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n{]}^T\in {\mathbb{R}}^n$ is constructed as

$$\mathit{\sigma }={\dot{\mathit{e}}}_1+{\displaystyle \frac{{\mathit{e}}_1}{2}+{\displaystyle \frac{{\kappa }_c\pi }{r_c{T}_C}({\alpha }_1\mathbf{\Xi }+{\alpha }_2\mathit{S}\mathit{i}{\mathit{g}}^{1+r_c}({\mathit{e}}_1)+{\mathit{e}}_1)}}$$

(23)

where $\mathit{S}\mathit{i}{\mathit{g}}^{1+r_c}({\mathit{e}}_1)=[|e_{11}{|}^{1+r_c}sign(e_{11}),|e_{12}{|}^{1+r_c}sign(e_{12})$ $,\dots ,|e_{1n}{|}^{1+r_c}sign(e_{1n}){]}^T$ ; ${\kappa }_C>2$ , $0<r_C<1$ , $T_C>0$ are design parameters, ${\alpha }_1=2^{\frac{r}{2}-2}$ , ${\alpha }_2=2^{-2-\frac{r}{2}}$ , moreover $\mathbf{\Xi }=[{\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2,\dots ,{\mathrm{\Xi }}_n{]}^T$ , the each element of $\mathbf{\Xi }$ is defined as follows ³¹ :

$${\mathrm{\Xi }}_i=\{\begin{array}{c}{|e_{1i}|}^{1-r_c}sign(e_{1i})for|e_{1i}|>\delta \\ c_1\cdot e_{1i}+c_2{e}_{1i}^{2}sign(e_{1i})for|e_{1i}|\le \delta \end{array}$$

(24)

in which $\delta >0$ , $c_1=(1+r_c){\delta }^{-r_c}$ and $c_2=-r_c{\delta }^{-1-r_c}$ . From (23), ${\sigma }_i$ can be written as follows:

$$\begin{array}{rl}& {\sigma }_i={\dot{e}}_{1i}+{\displaystyle \frac{e_{1i}}{2}+{\displaystyle \frac{{\kappa }_C\pi }{r_C{T}_C}({\alpha }_1{\varpi }_i+{\alpha }_2{|e_{1i}|}^{1+r_c}sign(e_{1i})+e_{1i})}}\end{array}$$

(25)

Remark 2 . The selected parameters $c_i$ , $(i=1,2)$ ensure that the function ${\varpi }_i$ is continuously differentiable ( $C^1$ ). This is confirmed by examining the limits at $e_{1i}=\pm \delta$ :

$$\{\begin{array}{c}\underset{e_{1i}\to {\delta }^{+}}{lim}{\varpi }_i=\underset{e_{1i}\to {\delta }^{-}}{lim}{\varpi }_i={\delta }^{1-r},\\ \underset{e_{1i}\to -{\delta }^{+}}{lim}{\varpi }_i=\underset{e_{1i}\to -{\delta }^{-}}{lim}{\varpi }_i=-{\delta }^{1-r},\\ \underset{e_{1i}\to {\delta }^{+}}{lim}{\dot{\varpi }}_i=\underset{e_{1i}\to {\delta }^{-}}{lim}{\dot{\varpi }}_i=(1-r){\delta }^{-r}{\dot{e}}_{1i},\\ \underset{e_{1i}\to -{\delta }^{+}}{lim}{\dot{\varpi }}_i=\underset{e_{1i}\to -{\delta }^{-}}{lim}{\dot{\varpi }}_i=(1-r){\delta }^{-r}{\dot{e}}_{1i}.\end{array}$$

(26)

The $C^1$ continuity of ${\varpi }_i$ ensures that the sliding surface ${\sigma }_i$ and its derivative ${\dot{\sigma }}_i$ are also continuous. Consequently, any potential singularity in the control computation is avoided.

Lemma 4 . For a sliding variable ${\sigma }_i$ satisfying $|{\sigma }_i|\le \xi$ , where $\xi >0$ is a unknown constant, the tracking error $e_{1i}$ converge to a residual set ${\mathrm{\Omega }}_E$ in a predefined-time $T_C$ , the set ${\mathrm{\Omega }}_E$ is given by:

$${\mathrm{\Omega }}_E=\{e_{1i}\in \mathbb{R}|V_i=0.5e_i^2\le V_{max}\}$$

(27)

where $V_{max}=max\{{\beta }_E^{\frac{2}{2-r}},{\beta }_E^{\frac{2}{2+r}},\frac{(\kappa -1){\beta }_E}{\kappa -2},0.5{\delta }^2\}$ with ${\beta }_E=0.5\frac{{\xi }^2{r}_C{T}_C}{(({\kappa }_C-1)\pi )}$ .

Proof . From (25), the error dynamics can be written as

$${\dot{e}}_{1i}=-{\displaystyle \frac{{\kappa }_C\pi }{r_C{T}_C}({\alpha }_1{\varpi }_i+{\alpha }_2{|e_{1i}|}^{1+r_c}sign(e_{1i})+e_{1i})-{\displaystyle \frac{e_{1i}}{2}+{\sigma }_i}}$$

(28)

Consider the Lyapunov function candidate

$$V_i={\displaystyle \frac{1}{2}{e}_i^2}$$

(29)

Taking the time derivative of $V_i$ and substituting (28) yields

$${\dot{V}}_i=e_i{\dot{e}}_i=-{\displaystyle \frac{{\kappa }_C\pi e_i}{r_c{T}_C}({\alpha }_1{\varpi }_i+{\alpha }_2{|e_{1i}|}^{1+r_c}sign(e_{1i})+e_{1i})-{\displaystyle \frac{e_{1i}^2}{2}+e_i{\sigma }_i}}$$

(30)

For $|e_i|>\delta$ , using Young's inequality implies yields $e_i{\sigma }_i\le 0.5e_i^2+0.5{\sigma }_i^2=0.5e_i^2+0.5{\xi }^2$ , then Equation (30) becomes

$${\dot{V}}_i\le -{\displaystyle \frac{{\kappa }_C\pi }{r_c{T}_C}({\alpha }_1{|e_{1i}|}^{2-r_c}+{\alpha }_2{|e_{1i}|}^{2+r_c}+e_{1i}^2)+0.5{\xi }^2\le -{\displaystyle \frac{{\kappa }_C\pi }{r_c{T}_C}\left(V_i^{1-\frac{r}{2}}+V_i^{1+\frac{r}{2}}+2V_i\right)+0.5{\xi }^2.}}$$

(31)

According to Lemma 3, we can obtain that $e_{1i}$ is confined to a compact set ${\mathrm{\Omega }}_E$ within a given time $T_C$ . Note that this case $|e_i|\le \delta$ (i.e. $V_i\le 0.5{\delta }^2$ ) is included in ${\mathrm{\Omega }}_E$ . This completes the proof.

It follows from Theorem 1 that as long as the sliding variable ${\sigma }_i$ is bounded, the tracking error $e_{1i}$ is guaranteed to enter a small residual set in a prescribed time. Furthermore, a precise convergence region for $e_{1i}$ can be determined once the convergence region for ${\sigma }_i$ is known. In the following analysis, we will establish the known boundary on the sliding-mode variable ${\sigma }_i$ within a predefined time.

Firstly, we define a new variable as

$$\mathit{S}=[S_1,S_2,\dots ,S_n{]}^T=\psi \mathit{\sigma }$$

(32)

Then, it follows from (4) and (25) that the dynamics of the variable $\mathit{S}$ is

$$\begin{array}{rl}\}dotS& =\dot{\psi }\sigma +\psi \dot{\sigma }=\dot{\psi }\sigma +\psi (\omega +M_0^{-1}u-{\ddot{x}}_d+0.5e_1)\\ & +\frac{{\dot{e}}_1}{2}+\frac{{\kappa }_C\pi }{r_c{T}_C}({\alpha }_1\dot{\mathrm{\Xi }}+{\alpha }_2(1+r_c)|e{|}^{r_c}\dot{e}+{\dot{e}}_1)=\mathrm{\Phi }+\psi (\omega +M_0^{-1}u).\end{array}$$

(33)

where $\mathbf{\Phi }=\dot{\psi }\mathit{\sigma }+\psi \left(0.5{\mathit{e}}_1-{\ddot{\mathit{x}}}_d\right)+0.5{\dot{\mathit{e}}}_1+\frac{{\kappa }_c\pi }{r_c{T}_C}({\alpha }_1\dot{\mathbf{\Xi }}$ $+{\alpha }_2(1+r_c)\cdot {|\mathit{e}|}^{r_c}\cdot \dot{\mathit{e}}+{\dot{\mathit{e}}}_1)$ . To obtain the following results, a compact set is defined as

$${\mathrm{\Omega }}_S=\{\mathit{S}|{\mathit{S}}^T\mathit{S}<{\lambda }^2\}$$

(34)

The following barrier Lyapunov function $V_S$ is considered in ${\mathrm{\Omega }}_S$ :

$$V_S={\displaystyle \frac{1}{2}\log {\displaystyle \frac{{\lambda }^2}{{\lambda }^2-{\mathit{S}}^T\mathit{S}}}}$$

(35)

Differentiating $V_S$ along (35) yields

$${\dot{V}}_S={\displaystyle \frac{{\mathit{S}}^T}{{\lambda }^2-{\mathit{S}}^T\mathit{S}}(\mathbf{\Phi }+\psi (\mathit{\omega }+{\mathit{M}}_0^{-1}\mathit{u}))}$$

(36)

Next, the control input $\mathit{u}$ with parameter updating law is designed as follows:

$$\mathit{u}=-{\mathit{M}}_0\left(\hat{\mathit{\omega }}+{\psi }^{-1}(\mathbf{\Phi }+0.5{\kappa }_C\mathit{S})+{\displaystyle \frac{\psi \mathit{S}}{4({\lambda }^2-{\mathit{S}}^T\mathit{S})}}\right)$$

(37)

Substituting (37) into (36), it can be shown that

$${\dot{V}}_S={\displaystyle \frac{{\mathit{S}}^T\psi \tilde{\mathit{\omega }}}{{\lambda }^2-{\mathit{S}}^T\mathit{S}}-{\displaystyle \frac{1}{2}{\displaystyle \frac{{\kappa }_C{\mathit{S}}^T\mathit{S}}{{\lambda }^2-{\mathit{S}}^T\mathit{S}}-{\displaystyle \frac{{\psi }^2{\mathit{S}}^T\mathit{S}}{4{({\lambda }^2-{\mathit{S}}^T\mathit{S})}^2}}}}}$$

(38)

Based on Young's inequality, ⁵ we have

$${\displaystyle \frac{{\mathit{S}}^T\psi \tilde{\mathit{\omega }}}{{\lambda }^2-{\mathit{S}}^T\mathit{S}}\le {\displaystyle \frac{{\psi }^2{\mathit{S}}^T\mathit{S}}{4{({\lambda }^2-{\mathit{S}}^T\mathit{S})}^2}+||\tilde{\mathit{\omega }}|{|}_2^2,}}$$

(39)

Since $||\tilde{\mathit{\omega }}||\in L_{\mathrm{\infty }}$ , it implies that there exists a positive constant ${\mathrm{\Delta }}_S$ satisfying $||\tilde{\mathit{\omega }}|{|}_2^2\le {\mathrm{\Delta }}_S$ , by using inequality (39), Equation (38) is fulfilled:

$${\dot{V}}_S\le -{\displaystyle \frac{{\kappa }_C{\mathit{S}}^T\mathit{S}}{2({\lambda }^2-{\mathit{S}}^T\mathit{S})}+{\mathrm{\Delta }}_S\le -{\displaystyle \frac{{\kappa }_C}{2}\cdot \log {\displaystyle \frac{{\lambda }^2}{{\lambda }^2-{\mathit{S}}^T\mathit{S}}+{\mathrm{\Delta }}_S=-{\kappa }_C\cdot V_S+{\mathrm{\Delta }}_S.}}}$$

(40)

Integrating the differential inequality (40) yields the solution for the Lyapunov function:

$$V_S\le {\displaystyle \frac{{\mathrm{\Delta }}_S}{{\kappa }_C}+\left(V_S(0)-{\displaystyle \frac{{\mathrm{\Delta }}_S}{{\kappa }_C}}\right)e^{-{\kappa }_{C}t}}$$

(41)

Recalling the definition of the barrier Lyapunov function (35), Equation (41) implies that $\frac{1}{2}\log \frac{{\lambda }^2}{{\lambda }^2-{\mathit{S}}^T\mathit{S}}\le \frac{{\mathrm{\Delta }}_S}{{\kappa }_C}+V_S(0)$ , and solving this inequality for ${\mathit{S}}^T\mathit{S}$ by exponentiating both sides and rearranging terms, we obtain

$${\mathit{S}}^T\mathit{S}\le {\lambda }^2\left(1-e^{-2\left(V_S(0)+\frac{{\mathrm{\Delta }}_S}{{\kappa }_C}\right)}\right)<{\lambda }^2$$

(42)

According to the above analysis results, the constraint ${\mathit{S}}^T\mathit{S}<{\lambda }^2$ is satisfied for $\mathrm{\forall }t>0$ . From (42), we know that ${\mathrm{\Delta }}_S$ contains an upper bound of the disturbance–estimation error. This bound forces each component $|S_i|$ to stay close to $\lambda \sqrt{\left(1-e^{-2\left(V_S(0)+\frac{{\mathrm{\Delta }}_S}{{\kappa }_C}\right)}\right)}$ . Using the definition of $\mathit{S}$ , for $i=1,2,\dots ,n,$ we obtain $|{\sigma }_i|<\frac{\lambda \sqrt{\left(1-e^{-2\left(V_S(0)+\frac{{\mathrm{\Delta }}_S}{{\kappa }_C}\right)}\right)}}{\psi (t)}$ for $\mathrm{\forall }t>0$ . In particular, since $\psi (t)=1$ for $\mathrm{\forall }t>T_F$ , this reduces to $|{\sigma }_i|<\lambda \sqrt{\left(1-e^{-2\left(V_S(0)+\frac{{\mathrm{\Delta }}_S}{{\kappa }_C}\right)}\right)}$ for $\mathrm{\forall }t>T_F$ . For the theoretical analysis, we further introduce a conservative bound $|S_i|<\lambda$ , this directly implies $|{\sigma }_i|<\lambda$ for $\mathrm{\forall }t>T_F$ . According to Lemma 4, the upper bound of ${\sigma }_i$ determines the size of the error–convergence region. Hence, the tracking error converges to the residual set ${\mathrm{\Omega }}_E$ within predefined-time $T_C>T_F$ , i.e.

$$\begin{array}{r}{e}_{1i}\in {\mathrm{\Omega }}_E\triangleq \left\{e_{1i}|V_i\le max\left\{{\beta }_E^{\frac{2}{(2-r_c)}},{\beta }_E^{\frac{2}{(2+r_c)}},{\displaystyle \frac{({\kappa }_C-1){\beta }_E}{({\kappa }_C-2)},{\displaystyle \frac{{\delta }^2}{2}}}\right\}\right\}\end{array}$$

(43)

where ${\beta }_E=0.5\frac{{\lambda }^2{r}_c{T}_C}{(({\kappa }_C-1)\pi )}$ .

Remark 3 . The design parameters for the predefined-time disturbance observer and the proposed controller are selected from the theoretical analysis: (1) $T_O$ and $T_C$ are chosen from the desired settling time, with $T_O<T_C$ , so that the disturbance estimate converges before the tracking error reaches its predefined-time bound. ${\kappa }_C>2$ , $\lambda$ , $\delta$ , and $0<r_C<1$ govern the ultimate tracking accuracy: from (42) and (43), choosing relatively large gains ${\kappa }_C$ , and small coefficients $\lambda$ , $\delta$ , and $r_C$ yields a higher tracking accuracy in the convergence phase, while simultaneously amplifying the control-input magnitude.

From the above control law design, the structure diagram of the control system is shown in Figure 1, and we can state the following theorem.

Figure 1.

The control system block diagram.

Theorem 2 . For the system model (2) confronted with uncertain parameters and external disturbances, and provided that Assumptions 1–2 are met, the designed closed-loop system, incorporating the controller (37) and observer (16)-(17), achieves the following objectives:

1. The sliding variables ${\sigma }_i$ , $(i=1,2,\dots ,n)$ are kept within a bounded operational range, ultimately settling inside the prescribed region $|{\sigma }_i|<\lambda$ after time $\mathrm{\forall }t>T_F$ .

2. The tracking error $e_{1i}$ is suppressed to the predefined set ${\mathrm{\Omega }}_E$ within a predefined settling time $T_C$ .

Remark 4 . The auxiliary variables are defined on the BLF domain ${\mathrm{\Omega }}_S=\{\mathit{S}|{\mathit{S}}^T\mathit{S}<{\lambda }^2\}$ where $\lambda >0$ is a design parameter. From (41), the solution of this inequality yields an upper bound, i.e. $V_s(t)\le {\overline{V}}_s$ for $\mathrm{\forall }t>0$ . Since $V_S$ is a barrier Lyapunov function that is strictly increasing as $\Vert \mathit{S}\Vert \to \lambda$ , the level set $\left\{S|V_S\le {\stackrel{-}{V}}_S\right\}$ is contained in ${\mathrm{\Omega }}_S$ . Hence ${\mathrm{\Omega }}_S$ is forward invariant and ${\mathit{S}}^T\mathit{S}<{\lambda }^2$ for $\mathrm{\forall }t\ge 0$ . This guarantees that no finite-time escape or singularity occurs and all closed-loop signals remain bounded. Moreover, based on $\psi (0)=0$ and $\mathit{S}=\psi \mathit{\sigma }$ , and therefore $\lambda$ can be chosen arbitrarily large, so that the domain ${\mathrm{\Omega }}_S$ can always be selected to contain the given initial tracking error.

Remark 5 . According to (33), the auxiliary variable dynamics can be rewritten by using plant (2) and the disturbance observer error (21), namely, $\dot{\mathit{S}}=\mathbf{\Phi }+\psi (\mathit{\omega }+{\mathit{M}}_0^{-1}\mathit{u})$ given in (33). Using the Lyapunov function (35) and applying standard inequalities, we derive the inequality (42). The proposed fixed-time auxiliary sliding-mode observer analyzed in Theorem 1 ensures that the estimation error is uniformly bounded. A larger disturbance estimation error bound leads to a larger residual sliding-mode variable (See the derivation given in the discussion between (42) and (43)), but the closed-loop signals remain uniformly bounded and the predefined-time convergence to a residual set is preserved.

Remark 6 . The predefined-time DO–SMC law in (37) is synthesized by neglecting explicit amplitude constraints on the control input. The Lyapunov inequalities that lead to the predefined-time convergence and boundedness properties are valid provided that the physical actuators can realize the control signal generated by (37). In the presence of severe actuator saturation, the effective input deviates from the designed law, and the theoretical predefined-time guarantees may no longer hold globally. In such a case, the tracking error remains bounded in practice, but the actual convergence time and residual error will depend on the saturation level. A rigorous treatment of hard input constraints would require integrating the present framework with anti-windup or input-constrained prescribed-performance techniques, which is an important topic for future work.

Simulation study

Example A:

The efficacy of the proposed controller is demonstrated on a linear motor positioning system governed by the following dynamic model ¹ :

$$\{\begin{array}{c}{\dot{x}}_1=x_2,\\ {\dot{x}}_2={\displaystyle \frac{{\omega }_c}{\zeta }-{\displaystyle \frac{\phi }{\zeta }-\chi .}}\end{array}$$

(44)

Here, the state variables $x_1$ and $x_2$ represent the position and linear velocity of the positioning stage, respectively. The term $\zeta$ denotes the mass of the stage, and ${\omega }_c$ is the control input. The term $\chi$ is a lumped disturbance that encapsulates unmodeled dynamics and external forces. The friction is modeled by $\phi ={\iota }_v{x}_2+{\iota }_{c}sign(x_2)$ where ${\iota }_v$ and ${\iota }_c$ are the respective friction coefficients.

The model parameters are based on a system with mass $\zeta =3.5kg$ . The friction is characterized by parameters ${\iota }_v={\overline{\iota }}_v+\mathrm{\Delta }{\iota }_v$ and ${\iota }_c={\overline{\iota }}_c+\mathrm{\Delta }{\iota }_c$ , which are decomposed into their nominal parts ( $({\overline{\iota }}_v=9\mathrm{N}\mathrm{s}/\mathrm{m}\text{and}{\overline{\iota }}_c=12\mathrm{N})$ and ${\overline{\iota }}_c=12N$ ) and uncertain parts ( $\mathrm{\Delta }{\iota }_v$ and $\mathrm{\Delta }{\iota }_c$ ). The specific profiles for the time-varying uncertainties $\mathrm{\Delta }{\iota }_v$ and $\mathrm{\Delta }{\iota }_c$ are introduced later. For the closed-loop system, the design parameters are chosen to be ${\kappa }_C={\kappa }_O=4$ , $r_o=r_c=0.5$ , $\delta =0.1$ , $T_S=2$ , $T_C=1$ , and $\lambda =0.5$ . The system is tasked with tracking the reference $x_d=0.5\sin (\frac{\pi t}{4})\times \sin (\frac{\pi t}{8}+\frac{\pi }{5})$ and the system is initialized with the state $x(0)=[0.12,0.1{]}^T$ . An investigation into the controller's robustness is conducted through three case studies. These cases are structured to systematically evaluate the system's ability to maintain desired performance metrics when confronted with significant parametric uncertainty and persistent external disturbances.

Case 1: $\mathrm{\Delta }{\iota }_v=0.5\cdot \underset{v}{\overset{-}{\iota }}$ , $\mathrm{\Delta }{\iota }_c=0.2\cdot \underset{c}{\overset{-}{\iota }}$ , $\chi =0.25\cdot \sin (\pi t)+0.5\cdot rand$ (rand represents a random interference signal, whose value is confined to the interval (0, 1]).

Case 2: $\mathrm{\Delta }{\iota }_v=\underset{v}{\overset{-}{\iota }}$ , $\mathrm{\Delta }{\iota }_c=0.45\cdot \underset{c}{\overset{-}{\iota }}$ , $\chi =0.4\cdot \sin (\pi t)+$ $0.2\cdot \cos (t)+rand$ .

Case 3: $\mathrm{\Delta }{\iota }_v=(1+rand)\cdot \underset{v}{\overset{-}{\iota }}$ , $\mathrm{\Delta }{\iota }_c=(0.8+rand)\cdot \underset{c}{\overset{-}{\iota }}$ , $\chi =0.8\cdot \sin (\pi t)+0.5\cdot \cos (5t)+3\cdot rand$ .

Figures 2–4 depict the system's position tracking response using the proposed controller under three conditions, verifying its capacity for high-speed reference following. This rapid convergence is a direct consequence of the controller's predefined-time stability properties, which also confer significant robustness against composite disturbances. The position tracking error, shown in Figure 5, diminishes rapidly. For the three cases, the error settles into a steady-state residual set bounded by 0.02 for all t > 2 s.

Figure 2.

Trajectory tracking under Case 1.

Figure 3.

Trajectory tracking under Case 2.

Figure 4.

Trajectory tracking under Case 3.

Figure 5.

Tracking errors under different cases.

The proposed method's capacity for disturbance rejection is validated in Figures 6 and 7. Figure 6 displays the performance of the disturbance estimation algorithm, confirming its ability to track the temporal evolution of a composite disturbance signal corrupted by stochastic noise. This accurate, online reconstruction enables the control law to generate a timely compensatory action. The resulting control effort is shown in Figure 7. The synergy between the high-fidelity estimation and the compensatory control action is critical, endowing the closed-loop system with enhanced resilience against a broad class of unknown disturbances.

Figure 6.

The estimation of the lumped disturbance.

Figure 7.

Control input under different cases.

Example B: Robot Manipulator System

The proposed control strategy is implemented on a two-degree of freedom robot manipulator system whose dynamic model ³⁵ is given by:

$$\mathit{M}(\mathit{z})\ddot{\mathit{z}}+\mathit{C}(\mathit{z},\dot{\mathit{z}})\dot{\mathit{z}}+\mathit{G}(\mathit{z})=\mathit{\tau }+\mathit{d}$$

(45)

In (45), the system dynamics are characterized by the variables $\mathit{z}$ , velocity $\dot{\mathit{z}}$ and acceleration $\ddot{\mathit{z}}$ ; the system matrices $\mathit{M}(\mathit{z})={\mathit{M}}_0+\mathrm{\Delta }\mathit{M}$ , $\mathit{C}(\mathit{z},\dot{\mathit{z}})={\mathit{C}}_0+\mathrm{\Delta }\mathit{C}$ , and $\mathit{G}(\mathit{z})={\mathit{G}}_0+\mathrm{\Delta }\mathit{G}$ are defined in, ³⁵ $\mathit{\tau }$ and $\mathit{d}$ are the control inputs and the unknown disturbance vector, respectively. Let ${\mathit{x}}_1=[x_{11},x_{12}]=\mathit{z}$ and ${\mathit{x}}_2=[x_{21},x_{22}]=\dot{\mathit{z}}$ , then (45) becomes

$$\{\begin{array}{c}{\dot{\mathit{x}}}_1={\mathit{x}}_2\\ {\dot{\mathit{x}}}_2=\omega +{\mathit{M}}_0^{-1}u\end{array}$$

(46)

where ${\mathit{M}}_0$ represents the nominal symmetric inertia matrix, $\mathit{\omega }=-{\mathit{M}}_0^{-1}({\mathit{C}}_0{\mathit{x}}_2+{\mathit{G}}_0)+{\mathit{M}}_0^{-1}(\mathit{d}-\mathrm{\Delta }\mathit{M}$ ${\dot{\mathit{x}}}_2-\mathrm{\Delta }\mathit{C}{\mathit{x}}_2-\mathrm{\Delta }\mathit{G})$ is the lumped disturbance. The system is initialized from the state $[x_{11},x_{12},x_{21},x_{22}]=[1,1.5,0,0]$ and the desired command signal ${\mathit{x}}_d=[1-\frac{7e^{-t}}{5}+\frac{7e^{-4t}}{20},$ $1+e^{-t}-\frac{e^{-4t}}{4}{]}^T$ . The system parameters of the controlled plant are adopted from, ³⁶ while the control parameters are chosen to be ${\kappa }_C={\kappa }_O=40$ , $r_o=r_c=0.5$ , $\delta =0.01$ , $T_F=5$ , $T_C=8$ , and $\lambda =0.1$ . To validate the effectiveness of the proposed algorithm, two cases are considered: Case 1 involves a combination of the lumped disturbances and fixed-level parametric uncertainties matrix; Case 2 examines the control performance of the closed-loop system under random disturbances and randomly varying parametric uncertainties, i.e.

• Case 1: $\mathit{d}=[2\sin (4t)+0.5\sin (200\pi t)+rand,2\cos$ $(4t)+0.5\sin (200\pi t)+rand{]}^T$ (rand represents a random interference signal, whose value is confined to the interval (0, 1]), $\mathrm{\Delta }\mathit{M}=0.2{\mathit{M}}_0$ , $\mathrm{\Delta }\mathit{C}=0.3{\mathit{C}}_0$ , and $\mathrm{\Delta }\mathit{G}=0.3{\mathit{G}}_0$ ;

• Case 2: $\mathit{d}=[3\cdot rand,3\cdot rand{]}^T$ , $\mathrm{\Delta }\mathit{M}=rand\cdot {\mathit{M}}_0$ , $\mathrm{\Delta }\mathit{C}=rand\cdot {\mathit{C}}_0$ , and $\mathrm{\Delta }\mathit{G}=rand\cdot {\mathit{G}}_0$ .

The robustness of the designed closed-loop system against parameter uncertainties and external disturbances is validated through the simulation results presented in Figures 8–17. Specifically, Figures 8–11 illustrate the robot manipulator's superior trajectory tracking performance, where it precisely follows the desired command signals despite these adverse conditions. Based on the rigorous theoretical derivation, the ultimate bound of the tracking error can be explicitly calculated. The analysis confirms that after a predefined-time $T_C=8$ s, the tracking error is confined to $|e_{1i}|\le 0.045$ under different cases. As depicted in Figures 12–13, the control system exhibits a prompt transient response at the initial stage. Furthermore, the steady-state tracking error is consistently maintained below $4\times {10}^{-3}$ under both tested cases. The simulation results provide a clear validation of the theoretical analysis, demonstrating a strong correspondence between the predicted performance and the observed results. The control system demonstrates remarkable robustness, maintaining its predefined tracking accuracy even when the plant is simultaneously subjected to two distinct types of disturbances and stochastic parameter variations as showed in Figures 14 and 15.

Figure 8.

Trajectory tracking $x_{11}and{x}_{1d}$ under Case 1.

Figure 9.

Trajectory tracking $x_{12}and{x}_{2d}$ under Case 1.

Figure 10.

Trajectory tracking $x_{11}and{x}_{1d}$ under Case 2.

Figure 11.

Trajectory tracking $x_{12}and{x}_{2d}$ under Case 2.

Figure 12.

Tracking error $e_{11}$ under different cases.

Figure 13.

Tracking error $e_{12}$ under different cases.

Figure 14.

The estimation of the lumped disturbance under Case 1.

Figure 15.

The estimation of the lumped disturbance under Case 2.

Figure 16.

Control input under Case 1.

Figure 17.

Control input under Case 2.

Conclusions

This paper presents a predefined-time SMC strategy for a class of mechanical systems. To counteract the influence of unknown dynamics and external disturbances, a predefined-time disturbance observer is synthesized, which facilitates the rapid and precise estimation of lumped system uncertainties. The core of the proposed strategy is a piecewise nonlinear sliding manifold, which is augmented with an error compensation term to guarantee convergence to a prescribed steady-state accuracy. Subsequently, a predefined-time SMC law is formulated to steer the system states into a predefined vicinity of this manifold within a user-specified time. A rigorous Lyapunov-based analysis confirms the predefined-time convergence of the tracking error. The numerical case studies on a linear motor positioner and a two-DOF robotic manipulator further illustrate these quantitative properties. Under simultaneous parametric perturbations and composite external disturbances, the proposed DO–SMC scheme drives all tracking errors $|e_i|$ below 0.0450 after the prescribed time $T_C=8s$ and maintains the steady-state tracking accuracy at the level of $3-4\times {10}^{-3}$ in both cases, while the reconstructed disturbance closely follows its true value. These results confirm that the predefined-time design achieves high-precision tracking with predictable robustness and an explicitly characterizable performance bound. Future work will focus on the following aspects: (i) The proposed predefined-time DO–SMC has been assessed only in simulation on a linear motor and a two-DOF manipulator; future work will implement it on real platforms to examine robustness to sensor noise, actuator saturation, ³⁷ and unmodeled dynamics^38,39; (ii) State and input constraints are not explicitly handled; incorporating saturation or prescribed-performance mechanisms into the predefined-time framework while preserving the convergence guarantees in (43) is another direction.