Prescribed Performance Control for Robotic System with Communication Delays and Disturbances

Abstract

This paper presents a Prescribed Performance Control (PPC) approach for robotic systems experiencing communication delay and disturbances. Under input and feedback delays, a state feedback controller is designed to maintain the output tracking error within prescribed performance specifications. Additionally, a super-twisting algorithm-based sliding-mode observer is proposed to estimate and compensate for external disturbance in the robotic system. Based on the Lyapunov method, appropriate controller parameters and observer gains are selected to ensure the accuracy of output tracking and disturbance estimation. Finally, the effectiveness of the proposed approach is validated through simulations on a nonlinear robotic system. The proposed method remains effective in the simultaneous presence of state measurement delay, control input delay, and disturbance.

1. Introduction

Robotic manipulators have emerged as indispensable tools across a wide spectrum of modern industrial applications. In practical implementations, time delays frequently arise due to the inherent communication latency between control systems and robotic actuators. Consequently, investigating predictive control strategies for robotic manipulators under time delay conditions presents substantial practical significance and research value [1].

Network-induced time delays can significantly degrade the performance of robotic systems, potentially leading to instability. Research in this domain has predominantly focused on mitigating delays in control inputs [2,3,4,5,6,7,8,9,10,11]. A subset of studies is dedicated to state measurement delays [12,13,14], while a smaller body of literature concurrently addresses both types [15,16,17]. However, the predominant focus of existing research has been on ensuring system stability, often with less emphasis on performance metrics.

Evaluating system performance in the context of concurrent control input and state measurement delays continues to be a focal point of contemporary research. Although Model Predictive Control (MPC) incorporates performance metrics [18], its practical application is often hindered by a strong reliance on a precise system model. To overcome the challenge of model uncertainty, methodologies like Prescribed Performance Control (PPC) have been developed for nonlinear systems [19,20]. Nevertheless, a practical limitation of these approaches is their requirement for high-order derivatives of the reference trajectory. Furthermore, their scope is typically restricted to addressing control input delays, leaving state measurement delays unaccounted for.

The existing literature on Prescribed Performance Control (PPC) for robotic systems has not adequately addressed the impact of external disturbances, which poses a significant threat to system stability. To circumvent the lack of end-effector force sensors in many robotic platforms, the method proposed by Ficuciello et al. [21] utilizes force estimation; however, its effectiveness is limited by its reliance on an open-loop estimator that is highly susceptible to noise. In contrast, the disturbance observer represents a well-established alternative for estimating external forces in robotic systems [22,23,24]. For instance, Liu et al. [25] constructed a globally and exponentially stable joint torque observer using only position measurements. Similarly, other researchers have utilized neural networks to develop external force estimators [26,27]. The method by [28] employs reinforcement learning algorithms to address the challenge of unpredictable disturbances in changing environments.

This paper designs a state feedback controller in the presence of a communication delay and disturbance, which is used to keep the output tracking error within the specified performance range. A super-twisting algorithm-based sliding-mode observer is proposed to estimate and compensate for external disturbance in the robotic system. The parameters are selected and adjusted based on the Lyapunov theory. Finally, simulation verification was carried out on a two-degree-of-freedom robotic system.

This paper’s main contributions and innovations include the following:

• This paper introduces a novel PPC method to maintain the tracking performance of a robot system under state measurement delays, control input delays, and external disturbances.
• An integrated super-twisting disturbance observer was designed, which provides accurate and real-time estimates of external disturbances, thereby significantly enhancing the system’s overall robustness.

The rest of this article is organized as follows: Section 2 presents the dynamic model of the robotic system. Then, it details the design of the proposed controller and disturbance observer. After that, a stability analysis is conducted on the closed-loop system. In Section 3, simulation studies are conducted to validate the theoretical findings and demonstrate the effectiveness of the proposed method. Finally, Section 4 concludes the paper with a summary of the key contributions.

2. Materials and Methods

2.1. Dynamic Model of Robotic System

The dynamics of an n-DOF robotic manipulator, subject to external disturbance, can be expressed as follows:

$$M\left(q\right)\ddot{q}+C(q,\dot{q})\dot{q}+G\left(q\right)=F+d$$

(1)

where $q\in {\Re }^n$ and $\dot{q}\in {\Re }^n$, respectively, denote the joint angles and angular velocities of the robotic manipulator, $M\left(q\right)\in {\Re }^{n\times n}$ represents a symmetric positive definite inertia matrix, $C\left(q,\dot{q}\right)\in {\Re }^{n\times n}$ is the matrix of the centripetal and Coriolis torques, $G\left(q\right)\in {\Re }^{n\times n}$ represents the vector of gravitational torques, F denotes input torques, and d is an externally applied disturbance.

Denote $x_1=q,x_2=\dot{q}$. The state space model of system (1) can be described by

$${\dot{x}}_1\left(t\right)=x_2\left(t\right),$$

(2a)

$${\dot{x}}_2\left(t\right)=f\left(x,t\right)+M^{-1}\left(x_1\right)(F+d)$$

(2b)

where $x\left(t\right)={\left[x_1^T\left(t\right)x_2^T\left(t\right)\right]}^T$ represents the system state, $x_1\left(t\right)={[x_{11}{x}_{21}...x_{n1}]}^T$, $x_2\left(t\right)={[x_{12}{x}_{22}...x_{n2}]}^T$, and $f(x,t)=-M^{-1}\left(x_1\right)(C(x_1,x_2)x_2+G\left(x_1\right))$.

Prescribed Performance Control constrains the system tracking error to remain within predefined, time-varying boundaries. These boundaries are delineated by a performance function, which explicitly specifies key performance metrics such as the maximum allowable steady-state error and the minimum convergence rate. The proposed controller is developed for the Prescribed Performance Control problem in the presence of concurrent time delays and external disturbances. This study considers the system described in (2), which is subject to a desired trajectory $x_d\left(t\right)$, a state measurement delay ${\tau }_s\left(t\right)$, and a control input delay ${\tau }_u\left(t\right)$. A state feedback controller is designed to achieve the following objectives:

• The system output $x_1\left(t\right)$ is required to track the desired trajectory $x_d\left(t\right)$ in accordance with prescribed performance specifications, namely, the maximum steady-state error and the minimum convergence rate.
• A disturbance observer is designed to compensate for the external disturbances inherent in practical robotic applications.

The structure of the closed-loop system is shown in Figure 1.

Our work is based on the following assumptions:

Assumption 1.

The communication delays ${\tau }_s$ and ${\tau }_u$ are assumed to be bounded and continuously differentiable. Furthermore, ${\tau }_s$ is assumed to be known, whereas ${\tau }_u$ is unknown but upper-bounded by a positive constant ${\overline{\tau }}_u$. Their derivatives are bounded, satisfying ${\dot{\tau }}_s,{\dot{\tau }}_u\in L_{\infty }$ and ${\dot{\tau }}_s<1,{\dot{\tau }}_u<1$. Moreover, there exist positive constants ${\overline{\tau }}_u$ and $\overline{\tau }$ such that for all $t>-\overline{\tau }$, the inequality ${\tau }_s\left(t\right)+{\overline{\tau }}_u\le \overline{\tau }$ is satisfied.

Remark 1.

By comparing the timestamp of the generated data and the timestamp of the data reaching the controller, the state measurement delay ${\tau }_s$ can be estimated relatively accurately. The control input delay ${\tau }_u$ is generally impossible to measure. Therefore, this is why it is assumed that ${\tau }_s$ is known, while ${\tau }_u$ is unknown. The derivative of the time delay being less than 1 is a condition for the stability of the system. In this case, the system is still capable of “catching up” with the delay. If the derivative of the delay is greater than 1, it indicates that the growth rate of the delay is faster than the rate of time passage. This causes the control system to violate the law of causality to some extent.

Assumption 2.

The components of the desired trajectory, $x_{di}\left(t\right)$ for $i=1,\dots ,n$, are known and belong to the interval $[-\overline{\tau },+\infty )$. Furthermore, these trajectory components are bounded, i.e., $x_{d,i}\left(t\right)\in L_{\infty }$.

Assumption 3.

The disturbance d and its time derivative $\dot{d}$ are bounded.

2.2. Controller Design

2.2.1. Design of the Controller

The design of the proposed controller builds upon the framework presented in [14]. Let $T\left(\eta \right):(-1,1)\to R$ be a strictly increasing function that satisfies the limit conditions ${lim}_{\eta \to 1^{-}}T\left(\eta \right)=+\infty$ and ${lim}_{\eta \to -1^{+}}T\left(\eta \right)=-\infty$. For notational simplicity, the following definitions are introduced for the indices $i=1,\dots ,n$ and $j=1,2$.

The function $T\left(\eta \right)$ is chosen as follows:

$$T\left(\eta \right)=ln{\displaystyle \frac{1+\eta }{1-\eta }}$$

(3)

Next, denote

$${\theta }_s\left(t\right)=t-{\tau }_s$$

(4)

The controller design commences by considering a disturbance-free dynamic model of an n-DOF robotic manipulator system. Accordingly, the joint control torque $u_i$ is defined as

$$u_i\left(x,t\right)=-sgn\left({\left(M^{-1}\right)}^{*}\right){\displaystyle \frac{2k_{i2}T\left({\sigma }_i\left(t\right)\right)}{{\lambda }_{i2}\left({\theta }_s\left(t\right)\right)\left(1-{\sigma }_i^2\left(t\right)\right)}}$$

(5)

where

$${\sigma }_i\left(t\right)={\displaystyle \frac{x_{i2}\left({\theta }_s\left(t\right)\right)+sgn\left({\left(M^{-1}\right)}^{*}\right){\int }_{{\theta }_s\left(t\right)-{\overline{\tau }}_u}^t{u}_i\left(s\right)ds-{\mu }_i\left(t\right)}{{\lambda }_{i2}\left({\theta }_s\left(t\right)\right)}}$$

(6)

and ${\left(M^{-1}\right)}^{*}={\displaystyle \frac{1}{2}}\left[M^{-1}+{\left(M^{-1}\right)}^T\right]$ is sign-definite with known sign donated by $sgn\left({\left(M^{-1}\right)}^{*}\right)$, where $sgn\left({\left(M^{-1}\right)}^{*}\right)=1$ if $M^{-1}$ is positive definite, and $sgn\left({\left(M^{-1}\right)}^{*}\right)=-1$ if $M^{-1}$ is negative definite.

Remark 2.

For a robot model with n degrees of freedom, M is symmetric and positive definite. So $sgn\left({\left(M^{-1}\right)}^{*}\right)=1$. The proposed method can also be applied to other nonlinear multi-input multi-output systems to handle the issues of time delay and disturbance. At this point, it is necessary to determine the sign of $sgn\left({\left(M^{-1}\right)}^{*}\right)$. This is a strong controllability condition which is imposed on system (2) [29].

Next, the intermediate variable is introduced below, where $k_{ij}$ denotes the control gain.

$${\mu }_i\left(t\right)=-k_{i1}T\left({\displaystyle \frac{x_{i1}\left({\theta }_s\right)-x_{di}\left({\theta }_s\right)}{{\lambda }_{i1}\left({\theta }_s\right)}}\right)$$

(7)

The prescribed performance function is formulated as

$${\lambda }_{ij}\left(t\right)=\left({\lambda }_{ij}^0-{\lambda }_{ij}^{\infty }\right)e^{-{\phi }_{ij}t}+{\lambda }_{ij}^{\infty }$$

(8)

where ${\lambda }_{ij}^0$ is a constant and ${\lambda }_{ij}^{\infty }$ and ${\phi }_{ij}$, respectively, specify the maximum allowable steady-state error and the minimum convergence rate, which collectively govern the transient and steady-state performance of the tracking error. For this study, the parameters are selected as follows:

$${\lambda }_{ij}^{\infty }>0,{\phi }_{ij}>0,{\lambda }_{ij}^0>{\displaystyle \frac{\left|x_{ij}\left({\theta }_s^0\right)-x_{di}\left({\theta }_s^0\right)+\left(e^{{\phi }_{ij}\overline{\tau }}-1\right){\lambda }_{ij}^{\infty }\right|}{e^{{\phi }_{ij}\tilde{\tau }}}}$$

(9)

The performance function ${\lambda }_{i1}\left(t\right)$ is judiciously selected to enforce the constraint $|x_{i1}\left(t\right)-x_{di}\left(t\right)|<{\lambda }_{i1}\left(t\right)$ on the output tracking error. This constraint ensures that the system’s tracking performance adheres to predefined specifications, namely, the maximum steady-state error and the minimum convergence rate.

Remark 3.

As can be seen from (5)–(7), when designing the controller, the design of each joint does not require information from other joints. Therefore, there is no issue of coupling.

To simplify the subsequent proof, the coordinate transformation is defined as follows:

$$z_{i1}={\displaystyle \frac{x_{i1}\left({\theta }_s\right)-x_{di}\left({\theta }_s\right)}{{\lambda }_{i1}\left({\theta }_s\right)}}$$

(10a)

$$z_{i2}={\displaystyle \frac{x_{i2}\left({\theta }_s\right)+sgn\left({\left(M^{-1}\right)}^{*}\right){\int }_{{\theta }_s-{\overline{\tau }}_u}^t{u}_i\left(s\right)ds}{{\lambda }_{i2}\left({\theta }_s\right)}}$$

(10b)

The transformed error is defined as

$$e_{ij}=T\left(z_{ij}\right)$$

(11)

Define

$$r_{i1}={\displaystyle \frac{\mathrm{d}T\left(z_{ij}\right)}{\mathrm{d}{z}_{ij}}}{\lambda }_{i1}^{-1}\left({\theta }_s\right)={\displaystyle \frac{2{\lambda }_{i1}^{-1}\left({\theta }_s\right)}{1-z_{i1}^2}}$$

Finally, the intermediate variable (5) and control input (7) become

$${\mu }_j=-k_j{e}_j,j=1,2$$

(12)

$$u=-sgn\left({\left(M^{-1}\right)}^{*}\right)k_2{r}_2{e}_2$$

(13)

where ${\mu }_j={\left[{\mu }_{1j}\dots {\mu }_{nj}\right]}^T$, $u={\left[u_1\dots u_n\right]}^T$, $k_j=diag\left(k_{1j},\dots ,k_{nj}\right)$, $r_j=diag\left(r_{1j},\dots ,r_{nj}\right)$, and $e_j={\left[e_{1j}\dots e_{nj}\right]}^T$.

2.2.2. Design of the Disturbance Observer

According to (1) and (2)

$${\dot{x}}_2=-M^{-1}\left(x_1\right)\left[C(x_1,x_2)x_2+G\left(x_1\right)-F-d\right]$$

(14)

where $x_j={\left[x_{1j}\dots x_{nj}\right]}^T$, ${\dot{x}}_j={\left[{\dot{x}}_{1j}\dots {\dot{x}}_{nj}\right]}^T$.

The super-twisting algorithm-based disturbance observer is defined as follows [30]:

$${\dot{\widehat{x}}}_2=-M^{-1}\left(x_1\right)\left[C(x_1,x_2)x_2+G\left(x_1\right)-F\right]+{\alpha }_1\left|x_2-{\widehat{x}}_2\right|sgn(x_2-{\widehat{x}}_2)+{\alpha }_2(x_2-{\widehat{x}}_2)+\beta \widehat{\varphi }$$

(15)

$$\dot{\widehat{\varphi }}={\alpha }_{3}sgn(x_2-{\widehat{x}}_2)+{\alpha }_4(x_2-{\widehat{x}}_2)-{\alpha }_5\widehat{\varphi }$$

(16)

where ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5$ and $\beta$ are positive definite constants. $\beta \widehat{\varphi }$ is the estimate of $\beta \varphi =M^{-1}\left(x_1\right)d$. Therefore, the estimate of the disturbance is

$$\widehat{d}=\beta M\left(x_1\right)\widehat{\varphi }$$

(17)

The estimation error of disturbance observer is defined as $e_{d1}=x_2-{\widehat{x}}_2$ and $e_{d2}=\varphi -\widehat{\varphi }$. According to (14) to (17)

$${\dot{e}}_{d1}=-{\alpha }_1\left|x_2-{\widehat{x}}_2\right|sgn(x_2-{\widehat{x}}_2)-{\alpha }_2(x_2-{\widehat{x}}_2)+\beta {\dot{e}}_{d2}$$

(18a)

$${\dot{e}}_{d2}=-\beta {\alpha }_{3}sgn(x_2-{\widehat{x}}_2)-\beta {\alpha }_4(x_2-{\widehat{x}}_2)-\beta {\alpha }_5{e}_{d2}+\dot{\varphi }+\beta {\alpha }_5\varphi$$

(18b)

With the application of the disturbance observer, the system input is given as follows:

$$F=u(t-{\tau }_u)-\widehat{d}$$

(19)

where u has been designed in (13).

Remark 4.

The proposed control scheme exhibits low complexity, requiring neither prior knowledge of the system’s nonlinearities nor approximation/adaptive techniques to acquire such knowledge. Additionally, derivative information of the desired reference trajectory is not required. Consequently, control signal generation is completely free from complex computations, whether analytical or numerical. Regarding energy efficiency, the designed controller only employs some basic operations, such as proportion and integration. This requires a relatively low computing power and does not involve significant energy consumption.

Theorem 1.

Consider the n-DOF robotic system (2) subject to state measurement delay ${\tau }_s\left(t\right)$, control input delay ${\tau }_u\left(t\right)$, and external bounded disturbance $d\left(t\right)$. Under Assumptions 1–3, with the controller defined by (19) and the disturbance observer by (15)–(16), the system output tracking error $x_{i1}\left(t\right)-x_{di}\left(t\right)$ is strictly confined within the bounds defined by the prescribed performance function ${\lambda }_{i1}\left(t\right)$, such that

$$\left|x_{i1}\left(t\right)-x_{di}\left(t\right)\right|<{\lambda }_{i1}\left(t\right)$$

(20)

Remark 5.

When ${\tau }_s={\tau }_u=0$, the system is under the delay-free condition, so ${\int }_{{\theta }_s\left(t\right)-{\overline{\tau }}_u}^{t}u\left(s\right)ds=0$, and the equation ${\sigma }_n\left(t\right)$ reduces to [31]:

$${\sigma }_i\left(t\right)={\displaystyle \frac{x_{i2}\left({\theta }_s\left(t\right)\right)-{\mu }_i\left(t\right)}{{\lambda }_{i2}\left({\theta }_s\left(t\right)\right)}}$$

(21)

Remark 6.

The intermediate performance function is selected independently of any specific indices. This function must be continuously differentiable, strictly positive, and bounded on the interval $\left[0,+\infty \right)$, while also satisfying Condition (9). Selecting a conservative performance function alongside a smaller control gain $k_{ij}$ enhances the overall controller performance. Furthermore, selecting sufficiently large values for the disturbance observer gains, namely ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5$ and β, guarantees that the disturbance estimation errors, $e_{d1}$ and $e_{d2}$, converge to zero within a finite time. The selection of relevant parameters is elaborately explained in references [30,32]. It can be concluded that the parameters only need to be large enough. The conclusion drawn from Lyapunov theory is rather conservative. In fact, a slightly smaller value can also be used.

Proof of Theorem 1. 

(a)

Finite time stability of the disturbance observer.

Consider a Lyapunov function candidate as follows [33]:

$$V=\sum _{i=1}^n{V}_i$$

(22)

$$V_i={\displaystyle \frac{1}{2}}{\xi }_i^{T}Q{\xi }_i$$

(23)

where ${\xi }_i^T=\left[\begin{array}{ccc}|e_{d1,i}{|}^{\frac{1}{2}}sgn\left(e_{d1,i}\right)& e_{d1,i}& e_{d2,i}\end{array}\right]$,

$$Q=\left[\begin{array}{ccc}4\beta {\alpha }_3+{\alpha }_1^2& {\alpha }_1{\alpha }_2& -{\alpha }_1\\ {\alpha }_1{\alpha }_2& 2\beta {\alpha }_4+{\alpha }_2^2& -{\alpha }_2\\ -{\alpha }_1& -{\alpha }_2& 2\end{array}\right]$$

Taking the time derivative of $V_i$ yields

$$\dot{V_i}=-{\displaystyle \frac{1}{{\left|e_{d1,i}\right|}^{\frac{1}{2}}}}{\xi }_i^T{Q}_1{\xi }_i-{\xi }_i^T{Q}_2{\xi }_i-{\alpha }_1(\dot{\varphi }+\beta {\alpha }_5\varphi ){\left|e_{d1,i}\right|}^{{\displaystyle \frac{1}{2}}}sgn\left(e_{d1,i}\right)-{\alpha }_2\left(\dot{\varphi }+\beta {\alpha }_5\varphi \right)\left|e_{d1}\right|$$

(24)

where

$$Q_1={\displaystyle \frac{{\alpha }_1}{2}}\left[\begin{array}{ccc}2\beta {\alpha }_3+{\alpha }_1^2& 0& -{\alpha }_1\\ 0& 2\beta {\alpha }_4+5{\alpha }_2^2& -3{\alpha }_2\\ -{\alpha }_1& -3{\alpha }_2& 1\end{array}\right]$$

$$Q_2={\alpha }_2\left[\begin{array}{ccc}\beta {\alpha }_3+2{\alpha }_1^2& 0& -{\displaystyle \frac{1}{2}}\beta {\alpha }_1{\alpha }_5\\ 0& \beta {\alpha }_4+{\alpha }_2^2& -{\alpha }_2\\ -{\displaystyle \frac{1}{2}}\beta {\alpha }_1{\alpha }_5& -{\alpha }_2& 1\end{array}\right]$$

By applying Condition (24), there is

$$-{\alpha }_1(\dot{\varphi }+\beta {\alpha }_5\varphi ){\left|e_{d1,i}\right|}^{{\displaystyle \frac{1}{2}}}sgn\left(e_{d,1i}\right)-{\alpha }_2\left(\dot{\varphi }+\beta {\alpha }_5\varphi \right)\left|e_{d1}\right|<{\alpha }_1\delta {\left|e_{d1,i}\right|}^{{\displaystyle \frac{1}{2}}}+{\alpha }_2\delta \left|e_{d1}\right|$$

(25)

Thus, the upper bound of $\dot{V}$ is determined as

$$\dot{V}<-\sum _{i=1}^n\left({\displaystyle \frac{1}{{\left|e_{d1,i}\right|}^{\frac{1}{2}}}}{\xi }_i^T{Q}_3{\xi }_i+{\xi }_i^T{Q}_4{\xi }_i\right)$$

(26)

where

$$Q_3={\displaystyle \frac{{\alpha }_1}{2}}\left[\begin{array}{ccc}2\beta {\alpha }_3+{\alpha }_1^2-{\alpha }_1\delta & 0& -{\alpha }_1\\ 0& 2\beta {\alpha }_4+5{\alpha }_2^2& -3{\alpha }_2\\ -{\alpha }_1& -3{\alpha }_2& 1\end{array}\right]$$

$$Q_4={\alpha }_2\left[\begin{array}{ccc}\beta {\alpha }_3+2{\alpha }_1^2-{\alpha }_2\delta & 0& -{\displaystyle \frac{1}{2}}\beta {\alpha }_1{\alpha }_5\\ 0& \beta {\alpha }_4+{\alpha }_2^2& -{\alpha }_2\\ -{\displaystyle \frac{1}{2}}\beta {\alpha }_1{\alpha }_5& -{\alpha }_2& 1\end{array}\right]$$

Should the gains selected be sufficiently large to ensure that the matrices Q, $Q_3$, and $Q_4$ are positive definite, then there exist positive constants $c_1$ and $c_2$ such that

$$\dot{V}<-c_1{V}^{{\displaystyle \frac{1}{2}}}-c_{2}V$$

(27)

Based on the conparison lemma, it follows that $e_{d1}$ and $e_{d2}$ converge to $e_{d1}=0$ and $e_{d2}=0$ in finite time which is less than $T={\displaystyle \frac{2V^{\frac{1}{2}}}{c_1}}$.

(b)

Overall stability of the closed-loop system.

Considering a case where Equation (19) is applied to system (2), the robotic arm system dynamics can be reformulated as

$${\dot{x}}_1=x_2$$

(28a)

$${\dot{x}}_2=f\left({\overline{x}}_2,{\theta }_s\right)+M^{-1}(u\left(t-{\tau }_u\left(t\right)\right)-\widehat{d}+d)$$

(28b)

According to (10), the time derivatives of $z_{i,1},z_{i,2}$ are given by

$$\begin{array}{cc}\hfill {\dot{z}}_1& ={\lambda }_1^{-1}\left[\left(1-{\dot{\tau }}_s\right)\left({\lambda }_2{z}_2-{\dot{\mu }}_0\right)-\left(1-{\dot{\tau }}_s\right)({\dot{\lambda }}_1{\dot{z}}_1+k_1{e}_1\right)]\hfill \end{array}$$

(29a)

$$\begin{array}{cc}\hfill {\dot{z}}_2& ={\lambda }_2^{-1}\left[\left(1-{\dot{\tau }}_s\right)\left(f\left({\overline{x}}_2,{\theta }_s\right)+M^{-1}(u_2-\widehat{d}+d)-{\dot{\lambda }}_2{z}_2\right)\right.\hfill \\ \hfill & \left.-{\dot{\mu }}_1-k_2{r}_2{e}_2-sgn\left({\left(M^{-1}\right)}^{*}\right)\left(1-{\dot{\tau }}_s\right)u_{\overline{\tau }}\right]\hfill \end{array}$$

(29b)

where $z_j={\left[z_{1j}\dots z_{nj}\right]}^T$, ${\lambda }_j=diag({\lambda }_{1j}\left({\theta }_s\right),\dots ,{\lambda }_{nj}\left({\theta }_s\right))$, ${\dot{\lambda }}_j=diag({\dot{\lambda }}_{1j}\left({\theta }_s\right),\dots ,{\dot{\lambda }}_{nj}\left({\theta }_s\right))$, ${\dot{\mu }}_j={[{\dot{\mu }}_{1j}\dots {\dot{\mu }}_{nj}]}^T$, $u_{\tau }={[u_1({\theta }_s-{\tau }_u\left(t\right))\dots u_n({\theta }_s-{\tau }_u\left(t\right))]}^T$, $u_{\overline{\tau }}={[u_1({\theta }_s-{\overline{\tau }}_u\left(t\right))\dots u_n({\theta }_s-{\overline{\tau }}_u\left(t\right))]}^T$.

Further, define

$$\overline{z}={\left[z_1^T{z}_2^T\right]}^T$$

Finally, the open set is defined as follows:

$${\Omega }_z={\left(-1,1\right)}^{2n}\subset R^{2n}$$

Step 1: A Lyapunov function is chosen as follows

$$V_1={\displaystyle \frac{1}{2}}{e}_1^T{e}_1$$

According to ${\dot{z}}_1$, ${\dot{z}}_2$, the derivative of ${\dot{V}}_1$ is given by

$$\begin{array}{cc}\hfill {\dot{V}}_1& =e_1^{\mathrm{T}}{r}_1\left[(1-{\dot{\tau }}_s)({\lambda }_2{z}_2-{\dot{\lambda }}_1{z}_1-{\dot{\mu }}_0-k_1{e}_1)\right]\hfill \end{array}$$

Let

$$a_{11}\triangleq \mathrm{diag}\left(a_{11}^1,\dots ,a_{11}^n\right)=(1-{\dot{\tau }}_s)E_2$$

(30a)

$$a_{21}\triangleq {\left[a_{21}^1\dots a_{21}^n\right]}^T=(1-{\dot{\tau }}_s)\left({\lambda }_2{z}_2-(1-{\dot{\tau }}_s){\dot{\lambda }}_1{z}_1-{\dot{\mu }}_0\right)$$

(30b)

where $E_n$ represents the n-th order identity matrix.

Applying Equation (30), ${\dot{V}}_1$ becomes

$${\dot{V}}_1=-e_1^T{r}_1{a}_{11}{k}_1{e}_1+e_1^T{r}_1{a}_{21}$$

From Assumption 1, it follows that $1-\dot{\tau }>0$, which consequently renders $a_{11}$ positive definite. Furthermore, the boundedness of ${\dot{\lambda }}_1$, ${\lambda }_2$, ${\dot{\mu }}_0$, $z_1$, $z_2$, and $1-\dot{\tau }$ ensures that $a_{21}$ is also bounded. Moreover, since ${\lambda }_1$ is positive definite by construction and $\overline{z}\in {\Omega }_z$, it follows that $r_1$ is also positive definite. This boundedness of $a_{21}$ guarantees the existence of positive constants ${\overline{a}}_{21}^i$ for $i=1,\dots ,m$, such that $\left|a_{21}^i\right|\le {\overline{a}}_{21}^i$.

Therefore, there is

$$\begin{array}{cc}\hfill {\dot{V}}_1& \le \sum _{i=1}^m\left({\overline{a}}_{21}^i{r}_{i1}|e_{i1}|-a_{11}^i{r}_{i1}{k}_{i1}{\left|e_{i1}\right|}^2\right)\hfill \\ \hfill & =\sum _{i=1}^m{r}_{i1}\left|e_{i1}\right|\left({\overline{a}}_{21}^i-a_{11}^i{k}_{i1}\left|e_{i1}\right|\right)\hfill \end{array}$$

Under condition $\left|e_{i1}\right|>{\displaystyle \frac{{\overline{a}}_{21}^i}{a_{11}^i{k}_{i1}}},i=1,\dots ,m$, the above expression is negative. Hence

$$|e_{i1}|\le {\overline{e}}_{i1}\triangleq max\left\{|e_{i1}\left(0\right)|,{\displaystyle \frac{{\overline{a}}_{21}^i}{a_{11}^i{k}_{i1}}}\right\},i=1,\dots ,m.$$

Taking the inverse of function T, yields

$$-1<T^{-1}(-{\overline{e}}_{i1})\le z_{i1}\le T^{-1}\left({\overline{e}}_{i1}\right)<1.$$

(31)

Step 2: A Lyapunov function is chosen as follows

$$V_2={\displaystyle \frac{1}{2}}{e}_2^T{e}_2$$

According to ${\dot{z}}_1$, ${\dot{z}}_2$, the derivative of ${\dot{V}}_2$ is given by

$$\begin{array}{cc}\hfill {\dot{V}}_2=& -e_2^T{k}_2^2{r}_2^2{e}_2+e_2^T{k}_2{r}_2\left[(1-{\dot{\tau }}_s)\left(f({\overline{x}}_2,{\theta }_s)+M^{-1}(u_{\tau }-\widehat{d}+d)-{\dot{\lambda }}_2{z}_2\right)\right.\hfill \\ \hfill & \left.-{\dot{a}}_1-sgn\left({\left(M^{-1}\right)}^{*}\right)(1-{\dot{\tau }}_s)u_{\overline{\tau }}\right].\hfill \end{array}$$

Let

$$\begin{array}{c}\hfill a_{22}\triangleq (1-{\dot{\tau }}_s)\left(f({\overline{x}}_2,{\theta }_s)+M^{-1}(u_{\tau }-\widehat{d}+d)-{\dot{\lambda }}_2{z}_2\right)-{\dot{a}}_1-sgn\left({\left(M^{-1}\right)}^{*}\right)(1-{\dot{\tau }}_s)u_{\overline{\tau }}\end{array}$$

(32)

Applying Equation (32), ${\dot{V}}_2$ becomes

$${\dot{V}}_2=-e_2^T{k}_2^2{r}_2^2{e}_2+e_2^T{k}_2{r}_2{a}_{22}.$$

Based on the continuity of $M^{-1}$, applying the Extreme Value Theorem yields a constant ${\overline{M}}^{-1}>0$ such that $∥M^{-1}∥\le {\overline{M}}^{-1}$. According to Assumption 1, $1-\dot{\tau }$ is bounded. Recalling the proof in Step 1, the boundedness of $f_2,{\dot{\lambda }}_2,z_2,{\dot{\mu }}_1$ are established. The disturbance observer has proven that $d-\widehat{d}$ converges to zero in finite time. Considering all the above conditions, the boundedness of $a_{22}$ can be concluded. There exists a constant ${\overline{a}}_{22}>0$ satisfying $∥a_{22}∥\le {\overline{a}}_{22}$. Hence

$$\begin{array}{cc}\hfill {\dot{V}}_2& \le \sum _{i=1}^m\left(|e_{i2}|k_{i2}{r}_{i2}{\overline{a}}_{22}-{\left|e_{i2}\right|}^2{k}_{i2}^2{r}_{i2}^2\right)\hfill \\ \hfill & \le \sum _{i=1}^m\left|e_{i2}\right|k_{i2}{r}_{i2}\left({\overline{a}}_{22}-{\displaystyle \frac{2k_{i2}}{{\lambda }_{i2}^0}}\left|e_{i2}\right|\right)\hfill \end{array}$$

Under condition $\left|e_{i2}\right|>{\displaystyle \frac{{\overline{a}}_{22}{\lambda }_{i2}^0}{2k_{i2}}},i=1,\dots ,m$, the above expression is negative. Hence

$$|e_{i2}|\le {\overline{e}}_{i2}^2\triangleq max\left\{|e_{i2}\left(0\right)|,{\displaystyle \frac{{\overline{a}}_{22}^i{\lambda }_{i2}^0}{2k_{i2}}}\right\},i=1,\dots ,m.$$

Taking the inverse of function T, yields

$$-1<T^{-1}(-{\overline{e}}_{i2}^2)\le z_{i2}\le T^{-1}\left({\overline{e}}_{i2}^2\right)<1.$$

(33)

Considering (13) and (33), along with the established properties of $\left|{\lambda }_{i2}\left(t\right)\right|$, the boundedness of $u_i$ and ${\int }_{t-{\tau }_s-{\overline{\tau }}_u}^t{u}_i\left(s\right)ds$ can be derived. In summary, based on (31) and (33), the analysis from Step 1 and Step 2, and recalling (10) and (11), it can be concluded that for all $t\ge 0$

$$\left|x_{i1}\left(t\right)-x_{di}\left(t\right)\right|<{\lambda }_{i1}\left(t\right)$$

(34a)

$$\left|x_{i2}\left(t\right)-{\mu }_i(t+{\tau }_s\left(t\right))\right|<{\lambda }_{i2}\left(t\right)$$

(34b)

Consequently, it is established that all closed-loop signals remain bounded, and the output tracks its desired trajectory with the desired performance specifications, which completes the proof of Theorem 1. □

3. Simulation

3.1. Simulation Conditions

To verify the performance of the proposed controller and disturbance observer, we conduct simulation studies on the following dynamics-based 2-DOF robotic manipulator system:

• where

$q={\left[q_1{q}_2\right]}^T\left(\mathrm{rad}\right),\dot{q}={\left[{\dot{q}}_1{\dot{q}}_2\right]}^T(\mathrm{rad}/\mathrm{s}),u={\left[u_1{u}_2\right]}^T\left(\mathrm{N}\right)$

$M\left(q\right)=\left[\begin{array}{cc}{M}_{11}& M_{12}\\ M_{21}& M_{22}\end{array}\right]$, $C(q,\dot{q})=\left[\begin{array}{cc}{C}_{11}& C_{12}\\ C_{21}& C_{22}\end{array}\right]$, $G\left(q\right)=\left[\begin{array}{c}{G}_1\\ G_2\end{array}\right]$

• and

$M_{11}=I_1+I_2+L_1^2\left({\displaystyle \frac{m_1}{4}}+m_2\right)+m_2\left({\displaystyle \frac{L_2^2}{4}}+L_1{L}_{2}cos\left(q_2\right)\right)$

$M_{12}=M_{21}=I_2+m_2\left({\displaystyle \frac{L_2^2}{4}}+{\displaystyle \frac{1}{2}}{L}_1{L}_{2}cos\left(q_2\right)\right)$

$M_{22}=I_2+m_2{\displaystyle \frac{L_2^2}{4}}$

$C_{11}=-{\displaystyle \frac{1}{2}}{m}_2{L}_1{L}_2{\dot{q}}_{2}sin\left(q_2\right)$

$C_{12}=-{\displaystyle \frac{1}{2}}{m}_2{L}_1{L}_2({\dot{q}}_1+{\dot{q}}_2)sin\left(q_2\right)$

$C_{21}={\displaystyle \frac{1}{2}}{m}_2{L}_1{L}_2{\dot{q}}_{1}sin\left(q_2\right)$, $C_{22}=0$

$G_1={\displaystyle \frac{1}{2}}{m}_{1}g{L}_{1}cos\left(q_1\right)+m_{2}g\left(L_{1}cos\left(q_1\right)+{\displaystyle \frac{1}{2}}{L}_{2}cos\left(q_1+q_2\right)\right)$

$G_2={\displaystyle \frac{1}{2}}{m}_{2}g{L}_{2}cos\left(q_1+q_2\right)$

• where $m_i$, $L_i$, $I_i$, $i=1,2$, and g, respectively, represent the mass, the length of each link, the moment of inertia, and the acceleration due to gravity. The system parameters are set as $m_1$ = 0.8 [kg], $m_2$ = 0.5 [kg], $L_1$ = 0.4 [m], $L_2$ = 0.2 [m], $I_1$ = 0.6 [kg· m²], $I_2$ = 0.4 [kg· m²], and g = 9.81 [m/s²].

The control objective is to make the outputs $q_1$ and $q_2$ of the robotic arm system track, respectively, the given desired trajectories $q_{d1}={\displaystyle \frac{\pi }{12}}sin\left(0.2\pi t\right)$ and $q_{d2}={\displaystyle \frac{\pi }{12}}sin\left(0.2\pi t+{\displaystyle \frac{\pi }{2}}\right)$. For tracking performance, the criteria are set as follows: the steady-state error shall not exceed 0.05 (rad), and the convergence speed shall be no less than the minimum value of $e^{-3t}$. For $\dot{q_1}$ and $\dot{q_2}$ tracking $\dot{q_{d1}}$ and $\dot{q_{d2}}$, the performance parameters are similarly defined as 2(rad) and $e^{-2t}$. The above performance can be achieved by setting the following parameters, and expressions we finally obtained for the output performance index function are as follows:

$${\lambda }_{i1}\left(t\right)=\left(2-0.05\right)e^{-3t}+0.05,{\lambda }_{i2}\left(t\right)=\left(4-2\right)e^{-2t}+2$$

The different time delays used in the simulation are shown in

Table 1. Delay functions.

Delay	${\mathit{\tau }}_{\mathit{s}}\left[\mathit{s}\right]$	${\mathit{\tau }}_{\mathit{u}}\left[\mathit{s}\right]$	${\overline{\mathit{\tau }}}_{\mathit{u}}\left[\mathit{s}\right]$
Delay 1	0.008	0.008	0.015
Delay 2	0.004 (1 + sin(0.5$\pi$t))	0.006 (1 + sin(0.5$\pi$t))	0.015
Delay 3	0.004 (1 + sin($\pi$t))	0.006 (1 + sin($\pi$t))	0.015
Delay 4	0.004 (1 + sin(2$\pi$t))	0.006 (1 + sin(2$\pi$t))	0.015

. Unless otherwise specified, the default delay for the simulation is Delay 1.

The disturbances are $d_1={\displaystyle \frac{\pi }{6}}sin\left(0.2\pi t\right)\left(\mathrm{N}\right)$ and $d_2=0.2\left(\mathrm{N}\right)$, concurrently, and delays are given by ${\tau }_s={\tau }_u=0.008\left(\mathrm{s}\right)$. The results demonstrate that our controller guarantees effectiveness under the simultaneous presence of disturbances and delays. Notably, the system output $q_i\left(t\right)$, $i=1,2$ tracks not the performance signal $q_{di}\left(t\right)$, but rather $q_{di}\left(t+{\tau }_s\left(t\right)\right)$. Moreover, for both ${\tau }_s+{\tau }_u\le 0.03\left(\mathrm{s}\right)$ and ${\tau }_u<0.015\left(\mathrm{s}\right)$, the following holds for all $t>0$, with the system initial state given by $q_1\left(t\right)=q_2\left(t\right)=0$, ${\dot{q}}_1\left(t\right)={\dot{q}}_2\left(t\right)=0$, $t\in [-0.03,0)$. The following are also set: $k_{11}$ = 2, $k_{21}$ = 0.25, $k_{12}$ = 25, $k_{22}$ = 10.

3.2. Simulation Results

Finally, the tracking performance of outputs $q_1\left(t\right)$ and $q_2\left(t\right)$ with respect to $q_{d1}\left(t\right)$ and $q_{d2}\left(t\right)$ are illustrated in Figure 2. The tracking performance of outputs ${\dot{q}}_1\left(t\right)$ and ${\dot{q}}_2\left(t\right)$ with respect to ${\dot{q}}_{d1}\left(t\right)$ and ${\dot{q}}_{d2}\left(t\right)$ are illustrated in Figure 3.

The tracking performance bounds ${\lambda }_1,-{\lambda }_1$ and the tracking errors $q_1\left(t\right)-q_{d1}\left(t\right)$, $q_2\left(t\right)-q_{d2}\left(t\right)$ are shown in Figure 4; the tracking performance bounds ${\lambda }_2,-{\lambda }_2$ and the medial tracking errors ${\dot{q}}_1\left(t\right)-{\dot{q}}_{d1}\left(t\right)$, ${\dot{q}}_2\left(t\right)-{\dot{q}}_{d2}\left(t\right)$ are shown in Figure 5.

As depicted in the figure, the controller achieves an excellent tracking performance, ensuring that the tracking errors are strictly confined within the prescribed performance bounds. Subsequently, Figure 6 illustrates the effect of Prescribed Performance Control when the minimum convergence rate and the maximum steady-state error are increased. Case a: ${\lambda }_{i1}\left(t\right)=\left(2-0.05\right)e^{-3t}+0.05$; case b: ${\lambda }_{i1}\left(t\right)=\left(2-0.1\right)e^{-8t}+0.1$. As shown in the figure, the error still satisfies the prescribed performance requirements.

Figure 7 illustrates the input torque errors for the four time delays listed in Table 1, with the no delay case serving as the baseline. To demonstrate the robustness of the proposed controller against noise, some random noises $w_d\sim N\left(0,{\sigma }^2\right)$ are added to system (1). The tracking results of joint angle under different noise levels are presented in Figure 8.

To validate its effectiveness, the performance of the proposed PPC method is benchmarked against that of Bechlious’s PPC method [34]. The comparison is presented in Figure 9. Based on the results, it can be observed that both methods demonstrate a satisfactory tracking performance in the absence of time delays. When both the control input delay and state measurement delay are fixed, Bechlious’s PPC method exhibits a slightly inferior tracking performance compared with our proposed PPC method. In scenarios with time-varying delays affecting both control input and state measurements, Bechlious’s PPC method fails to maintain stability, whereas our proposed predictive control method maintains a superior tracking performance. This demonstrates the robustness of our PPC control method across diverse delay conditions.

To evaluate the disturbance observer, the following gains were selected: ${\alpha }_1=3,{\alpha }_2=1,{\alpha }_3=1,{\alpha }_4=1,{\alpha }_5=0.15$, and $\beta =1.1$. The estimation performance of the observer against the disturbance signals $d_1\left(t\right)={\displaystyle \frac{\pi }{6}}sin\left(0.2\pi t\right)\mathrm{N}$ and $d_2\left(t\right)=0.2\mathrm{N}$ is presented in Figure 10.

To validate its effectiveness, the performance of the proposed disturbance observer is benchmarked against that of Chen’s disturbance observer [35], a method widely employed in nonlinear systems. The comparative analysis is conducted under the following disturbance signals: $d_1\left(t\right)={\displaystyle \frac{\pi }{6}}sin\left(0.2\pi t\right)\mathrm{N}$ and $d_2\left(t\right)=0.2\mathrm{N}$. The results demonstrate that the estimation error of the proposed observer converges rapidly to a small neighborhood of zero. This indicates a significantly superior estimation performance compared with Chen’s disturbance observer. A detailed comparison of the estimation errors for the disturbances $d_1\left(t\right)$ and $d_2\left(t\right)$ is presented in Figure 11 and Figure 12, respectively.

Finally, we calculated the root mean square error of the controller and the disturbance observer. The root mean square error values of the tracking effect of the controller and the root mean square error values of the disturbance observer regarding the disturbance influence are shown in

Table 2. Comparison of root mean square error.

Methods	RMSE of Joint 1	RMSE of Joint 2
Proposed PPC	0.008992	0.071109
Bechliouslis’s PPC	0.052262	0.13875
Proposed DOB	0.030268	0.032637
Chen’s DOB	0.102193	0.038853

.

The initial selection of the controller gains are $k_{11}=2$, $k_{21}=0.25$, $k_{12}=25$, and $k_{22}=10$. The set noise has a variance of ${10}^{-6}$ and a mean of 0. In

Table 3. The sensitivity of the controller gains to noise.

Controller Gains	RMSE1	RMSE2	RMSE3	RMSE4	RMSE5
$k_{11}$	0.008992	0.010542	0.011263	System instability	System instability
$k_{21}$	0.008992	0.010846	0.013394	0.013675	0.013368
$k_{12}$	0.008992	0.013124	0.013097	0.012446	0.010172
$k_{22}$	0.008992	0.012238	0.014691	0.010984	0.013757

, the RMSE for the tracking effect of joint 1 is presented under the conditions of unchanged gain, 10% increase, 25% increase, 50% increase, and 100% increase. Among them, when $k_{11}=3$, the system comes unstable; when $k_{21}=1$, the system becomes unstable; $k_{21}=1$ and $k_{21}=1$ have a relatively small impact on noise; and when $k_{21}$ = 300, the system can still remain stable.

Then we conducted an analysis of the discrete time effect of the system. A discretization sensitivity analysis was conducted by using zero-order retainers. We systematically changed the sampling period of the zero-order hold device and conducted simulations for each sampling period to observe and evaluate its potential impact on the system performance. The results show that when the steady-state error of the performance function is 0.1, the sampling period reaches and exceeds 0.008 s, and the system exhibits continuous and increasingly amplified oscillations, eventually leading to system instability.

4. Discussion

This paper presents a low-complexity Prescribed Performance Control (PPC) scheme for robotic systems, which guarantees that the output tracking error adheres to predefined performance bounds—namely, maximum steady-state error and minimum convergence rate—even in the presence of time delays. The controller is specifically designed to handle a known state measurement delay and an unknown but bounded control input delay. To enhance the robustness against external disturbances inherent in practical applications, a sliding-mode disturbance observer is integrated into the control framework. The simulation results on a two-degree-of-freedom (2-DOF) robotic manipulator validate the efficacy of the proposed composite control strategy, demonstrating its effectiveness under the combined challenges of state measurement and control input delays. The main limitation is that the delay cannot be too large. In the future, we will continue to study this issue and we will make conducting physical verification our top priority. Meanwhile, we will also take into account the application of large language models. For instance, reference [36] employed large language models to handle abnormal data inputs in autonomous driving. This can be applied to the processing of complex disturbances in the environment by our disturbance observer.