Enhanced Trajectory Tracking via Disturbance-Observer-Based Modified Sliding Mode Control

Abstract

Trajectory tracking is a crucial aspect of controlling nonlinear systems and is an important area of research. Researchers have proposed several strategies to perform this task in the presence of perturbations, which are the sum of a system’s uncertainty, modeling errors, and external disturbances. Nonlinear systems, such as robot manipulators, have complex dynamics, and deriving their exact mathematical models is a tedious task. Therefore, the objective of this research is to design a model-free form of control for such systems. To achieve this goal, a sliding mode control (SMC) with a proportional-integral-derivative (PID) sliding surface was designed and integrated with a saturation-function-based extended-state observer (ESO). In an extended-state observer (ESO), the primary concept is to define the system’s perturbation. The ESO estimates the system’s states and perturbation, including the known and unknown dynamics, uncertainties, and external disturbances, which are considered as perturbations. The estimated perturbation is used in a closed loop to cancel the actual perturbation. This perturbation-rejection technique improved the controller’s performance, resulting in reduced position error, reduced sensitivity to low-frequency elements of perturbation, and a small magnitude of switching gain. The designed control algorithm requires minimal information about the system, specifically position feedback, and, therefore, there is no need to identify the system parameters. A mathematical analysis of the designed algorithm was performed in detail, and the algorithm was compared with the existing ESO-based SMC algorithm. Simulations were conducted using MATLAB/SimMechanics on two different systems, and the comparison results validated the performance of the designed algorithm in comparison to previous research.

1. Introduction

Accurately tracking trajectories is a challenging task for control engineers. Designing controllers for real-life problems is demanding due to the nonlinear nature of the system [1,2,3,4,5,6,7,8,9]. Real systems are affected by perturbations, which are the sum of system uncertainties, modeling errors, and external disturbances. The controller’s task is to achieve the desired state in the presence of perturbations. Multi-degree of freedom (DOF) robot manipulators are nonlinear systems, and accurately modeling them mathematically is not an easy task. Researchers have proposed various methods to formulate the mathematical model of the system, including mathematical derivation and system-identification techniques [10,11,12,13]. Typically, identification techniques estimate the linear parameters of the system, whereas the nonlinear parameters, uncertainties, and external disturbances are considered as perturbations.

Nonlinear control techniques, such as sliding mode control (SMC), are robust against perturbations. The SMC technique has been designed and implemented for desired-position tracking in various practical applications, including underwater vehicles, motor drives, and robot-position control [14,15,16,17,18,19,20,21,22,23,24,25,26]. In SMC, the main concept is to design the desired sliding surface and then force the system state to move towards the designed sliding surface. The switching control moves the system state to the sliding surface [27]. There are two phases in SMC: the reaching phase and the sliding phase. The sliding-surface dynamics are affected by perturbations during the reaching phase; therefore, the system’s stability is not guaranteed. To maintain stability while avoiding chattering, the magnitude of the switching gain must be greater than the upper bound of the perturbation. However, determining the upper bound of the perturbation can be difficult, and large switching gains can introduce unacceptable levels of chattering in real-world control problems. Several solutions have been proposed in the literature to mitigate these issues associated with SMC. High-order sliding mode control (HOSMC) was introduced to improve system stability, while integral sliding mode control (ISMC) enhanced the dynamics of the actual sliding variable by incorporating an auxiliary sliding surface [28,29]. Nonetheless, ISMC still exhibits chattering due to the switching function. Researchers have proposed the use of a smooth switching function to reduce chattering, but this may increase the system’s position error [29].

Researchers have proposed a disturbance observer (DO)-based SMC for accurate position tracking [30,31,32,33]. The DO estimates system states and perturbations/disturbances. The control input’s design requires information about the system states, and the estimated disturbance information is used along with the SMC control input to cancel the effects of actual perturbation. The compensation of perturbation in DO-based SMC affects the system dynamics through the estimation error of the perturbation. However, this error is much smaller than the actual perturbation [30], which ensures the system’s stability due to disturbance rejection. The DO-based SMC is also effective in reducing chattering without compromising the system’s performance in terms of position error. Moura et al. [30] proposed the sliding perturbation observer (SPO)-based SMC for accurate trajectory tracking. The SPO is a nonlinear observer. It estimates the system state and perturbation. The estimated perturbation was used in the control feedback to cancel the actual perturbation. The perturbation compensation reduced the magnitude of the switching gains (reduced chattering). The design of the above-mentioned disturbance observer requires the nominal model of the system, which is itself a difficult task. Zhao et al. [31] proposed a disturbance observer (DO)-based sliding mode control (SMC) technique for stabilizing a system that is subjected to mismatched disturbances and time delays. Due to disturbance rejection, the performance of the system was improved. Guo et al. [32] designed a DO-based non-singular terminal sliding mode control for an n-degree-of-freedom robot manipulator. Disturbance rejection reduced the system chattering. Liu et al. [34] proposed an intermediate observer (IO)-based control scheme for the electromagnetic docking of spacecraft in elliptical orbit. The proposed IO estimated the relative motion and the total disturbance.

The extended-state observer (ESO) estimates system states and perturbation. In the ESO, the idea is to define the system uncertainty, modeling error, and known and unknown dynamics as perturbations. This defined perturbation is then assumed to be an extended state of the system. The observer (ESO)’s task is to estimate the defined perturbation and the system states. This estimated-perturbation information is useful for perturbation rejection. The design of the ESO requires a minimal amount of information about the system, and it does not require the nominal model of the system. It is easy to implement. The researchers designed and implemented an ESO-based SMC for real-life applications [35,36,37]. The idea is to estimate the perturbation and use the estimated information in feedback (along the control input) to cancel the actual perturbation. Ren et al. [35] designed an ESO-based SMC for friction compensation in a three-wheeled omnidirectional mobile robot. The frictional modeling was a tedious task, but this information was necessary for the desired control performance. They defined the friction model as part of the perturbation. By utilizing the known dynamics of the mobile robot, they proposed a reduced-order ESO to estimate system states and perturbation. This estimated perturbation helped to achieve the desired performance. Rsetam et al. [37] proposed a cascade-ESO-based SMC for the accurate control of underactuated flexible-joint robots. The ESO was utilized to estimate the unmeasurable states and lumped disturbance. The chattering was reduced due to disturbance rejection. Although that work used DOs (ESO and others) to estimate the system uncertainty and external disturbance, its design required the nominal model of the system. The estimation techniques are necessary to estimate the nominal system’s model. Implementing such techniques requires experiments, which in turn increases the computational cost. Additionally, it should be noted that system parameters vary with time and the position of the system (e.g., gravity changes with the position).

Saad et al. [38] proposed a model-free control algorithm (ESO-based SMC) for the accurate trajectory tracking of a multi-DOF virtual robot manipulator. The algorithm was implemented on a virtual simulator developed in MATLAB (Simulink), and the desired trajectory was assigned through an external joystick. Although the controller’s performance was satisfactory, there was a need to further reduce the error. The objective of the current research is to further reduce the trajectory-tracking error and design a model-free control for the nonlinear system. To address this issue, a new algorithm is discussed below.

In this research, an SMC with a proportional integral derivative (PID) sliding surface was designed and integrated with a saturation function-based ESO. The ESO estimates system states and perturbation, and the estimated state information was used in the design of the control algorithm. The estimated perturbation was combined with the PID-SMC input and utilized as a feedback term to cancel the effect of the actual perturbation. The utilization of a PID-type sliding surface enhanced the system’s performance, leading to the following benefits:

• Reduced tracking error compared to the existing method;
• Effectiveness in mitigating low-frequency perturbations when compared to the existing method;
• The design requires minimal information about the system, making it a model-free control approach.

A mathematical analysis of the proposed control algorithm was performed in detail and compared to the previous method [38]. The proposed algorithm, the ESO-based PIDSMC, and the previous algorithm (the ESO-based SMC) were implemented on MATLAB/Simulink. Two different simulations were performed. Firstly, simulations were performed on a second-order system to validate the mathematical analysis of the algorithm in detail. Different cases were considered in the simulations based on the reference trajectory, and the performance and limitations of the algorithm were discussed in detail. In the second simulation, the algorithms were implemented on a two-DOF virtual robot manipulator using SimScape/MATLAB. The trajectory-tracking error of the proposed algorithm was smaller than that of the ESO-based SMC, and simulation results verified the proposed algorithm’s performance.

This manuscript is organized as follows. Section 2 presents the control design and a detailed analysis of the proposed control logic. Section 3 features the simulation and discussions. Finally, Section 4 concludes the paper.

2. Control Design

In this section, the control theory is elaborated in detail. The purpose of the proposed method is to achieve the desired state even in the presence of perturbations. The concept of the proposed method can be observed in Figure 1. The goal of the designed algorithm is to effectively reach the desired state despite the presence of perturbations. The extended-state observer (ESO) is used to estimate both the system state and the perturbation. The estimated-perturbation information is then utilized in conjunction with the PIDSMC (proportional-integral-derivative sliding mode control) input to achieve the desired state in a satisfactory manner. In Section 2.1, a saturation-function-based ESO is designed for general second-order systems. In Section 2.2, the ESO-based PIDSMC is developed and discussed in detail. Section 2.2 describes the performance characteristics of the proposed algorithm.

2.1. Extended-State Observer

The ESO estimates system states and perturbation. The estimated perturbation is sum of system modeled and unmodeled dynamics, nonlinearity, and external disturbance. It is easy to design ESO for the nonlinear system as it requires minimal information about the system. The goal of this research is to design model-free control for the system. Consider the following second-order system:

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

(1)

where x presents the system states [$x_1{x}_2$], $g\left(x,\dot{x},t\right)$ represents the system dynamics, $∆g\left(x,\dot{x},t\right)$ is the uncertainty in system dynamics, d is the external disturbance, and u is the control input. The control objective is to achieve the desired position in satisfactory time.

Assumption 1.

In (1), it is assumed that system dynamics, its uncertainty, and external disturbance are unknown. Therefore, the unknown terms are defined as the extended state of the system:

$$x_3=g\left(x,\dot{x},t\right)+∆g\left(x,\dot{x},t\right)+d,$$

(2)

where${\dot{x}}_3=G$is assumed to be an unknown but bounded function. The system (1) can be rewritten as:

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

(3)

$${\dot{x}}_2=u+x_3,$$

(4)

$${\dot{x}}_3=G.$$

(5)

Now the system has three states: $x_1$, $x_2$, and $x_3$. The observer’s task is to estimate the system states. The mathematical structure of nonlinear ESO is expressed as:

$${\dot{\widehat{x}}}_1={\widehat{x}}_2+l_1\mathrm{s}\mathrm{a}\mathrm{t}\left({\tilde{x}}_1\right),$$

(6)

$${\dot{\widehat{x}}}_2={\widehat{x}}_3+l_2\mathrm{s}\mathrm{a}\mathrm{t}\left({\tilde{x}}_1\right)+u,$$

(7)

$${\dot{\widehat{x}}}_3=l_3\mathrm{s}\mathrm{a}\mathrm{t}\left({\tilde{x}}_1\right),$$

(8)

where ${\widehat{x}}_1$, ${\widehat{x}}_2$, and ${\widehat{x}}_3$ are the estimated state of the system; $l_1$, $l_2$, and $l_3$ are the observer gains selected through pole-placement method [38]; and ${\tilde{x}}_1$ is the estimation error of the state (${\tilde{x}}_1=x_1-{\widehat{x}}_1$). Sat is the saturation function, defined as:

$$\mathrm{s}\mathrm{a}\mathrm{t}\left({\tilde{x}}_1\right)=\left\{\begin{array}{c}\raisebox{1ex}{${\tilde{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\left|{\tilde{x}}_1\right|$}\right.,\left|{\tilde{x}}_1\right|\ge {\epsilon }_o\\ \raisebox{1ex}{${\tilde{x}}_1$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_o$}\right.,\left|{\tilde{x}}_1\right|<{\epsilon }_o\end{array},\right.$$

(9)

where ${\epsilon }_o$ is the boundary-layer thickness. By using Equations (3)–(8), the estimation error of the states can be calculated as:

$${\dot{\tilde{x}}}_1={\tilde{x}}_2-l_1\mathrm{s}\mathrm{a}\mathrm{t}\left({\tilde{x}}_1\right),$$

(10)

$${\dot{\tilde{x}}}_2={\tilde{x}}_3-l_2\mathrm{s}\mathrm{a}\mathrm{t}\left({\tilde{x}}_1\right),$$

(11)

$${\dot{\tilde{x}}}_3=G-l_3\mathrm{s}\mathrm{a}\mathrm{t}\left({\tilde{x}}_1\right),$$

(12)

Assumption 2.

In order to calculate the observer gain, it is assumed that the estimation error (${\tilde{x}}_1$) is within the boundary layers ($\left|{\tilde{x}}_1\right|<{\epsilon }_o$). Therefore, the saturation function can be written as (according to Equation (9)):

$$\mathrm{s}\mathrm{a}\mathrm{t}\left({\tilde{x}}_1\right)=\frac{{\tilde{x}}_1}{{\epsilon }_o},$$

(13)

based on Assumption 2, Equations (10)–(12) can be updated as follows:

$${\dot{\tilde{x}}}_1={\tilde{x}}_2-l_1\left(\frac{{\tilde{x}}_1}{{\epsilon }_o}\right),$$

(14)

$${\dot{\tilde{x}}}_2={\tilde{x}}_3-l_2\left(\frac{{\tilde{x}}_1}{{\epsilon }_o}\right),$$

(15)

$${\dot{\tilde{x}}}_3=G-l_3\left(\frac{{\tilde{x}}_1}{{\epsilon }_o}\right).$$

(16)

The state-space dynamics of the above Equations, (14)–(16), can be presented as:

$$\dot{\tilde{x}}=A\tilde{x}+EG,$$

(17)

where

$$A=\left[\begin{array}{ccc}\raisebox{1ex}{$-l_1$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_o$}\right.& 1& 0\\ \raisebox{1ex}{$-l_2$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_o$}\right.& 0& 1\\ \raisebox{1ex}{$-l_3$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_o$}\right.& 0& 0\end{array}\right], E=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right].$$

(18)

The characteristic Equation of matrix A can be derived as:

$$\left|\lambda I-A\right|=\left|\begin{array}{ccc}\lambda +\raisebox{1ex}{$l_1$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_o$}\right.& -1& 0\\ \raisebox{1ex}{$l_2$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_o$}\right.& \lambda & -1\\ \raisebox{1ex}{$l_3$}\!\left/ \!\raisebox{-1ex}{$\delta {\epsilon }_o$}\right.& 0& \lambda \end{array}\right|={\lambda }^3+\left(\raisebox{1ex}{$l_1$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_o$}\right.\right){\lambda }^2+\left(\raisebox{1ex}{$l_2$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_o$}\right.\right)\lambda +\raisebox{1ex}{$l_3$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_o$}\right..$$

(19)

The values of the above gains are calculated using pole-placement method:

$${(s+\lambda )}^3=s^3+3\lambda s^2+3{\lambda }^{2}s+{\lambda }^3.$$

(20)

By comparing the coefficient of (19), and (20), the following relations are derived as:

$$l_1=3\lambda {\epsilon }_o, l_2=3{\lambda }^2{\epsilon }_o, l_3={\lambda }^3{\epsilon }_o.$$

(21)

2.2. ESO-Based PIDSMC

The ESO estimates system states and perturbation. The estimated-state information is used in the control design. The perturbation is combined with PIDSMC input and utilized as a feedback term to compensate the actual perturbation (Figure 2). The estimated PID sliding surface ($\widehat{s}$) is designed as:

$$\widehat{s}=\dot{\widehat{e}}+c_1·\widehat{e}+c_2·\int \widehat{e},$$

(22)

where $\widehat{e}$ is the estimated error ($\widehat{e}={\widehat{x}}_1-x_d$), and $c_1$ and $c_2$ are constant. The actual PID sliding surface (s) is defined as:

$$s=\dot{e}+c_1·e+c_2·\int e,$$

(23)

where e is the error ($e=x-x_d$), and $c_1$ and $c_2$ are constant (similar to Equation (22)). Equation (22) can be written as:

$$\widehat{s}={\dot{\widehat{x}}}_1-{\dot{x}}_d+c_1\left({\widehat{x}}_1-x_d\right)+c_2·\int \widehat{e},$$

(24)

by using Equation (6) in (24):

$$\widehat{s}={\widehat{x}}_2+l_1\mathrm{s}\mathrm{a}\mathrm{t}\left({\tilde{x}}_1\right)-{\dot{x}}_d+c_1\left({\widehat{x}}_1-x_d\right)+c_2\int \widehat{e},$$

(25)

based on Assumption 2, Equation (25) can be updated as:

$$\widehat{s}={\widehat{x}}_2+l_1\left(\raisebox{1ex}{${\tilde{x}}_1$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_o$}\right.\right)-{\dot{x}}_d+c_1\left({\widehat{x}}_1-x_d\right)+c_2·\int \widehat{e},$$

(26)

to satisfy the stability condition ($\widehat{s}·\dot{\widehat{s}}\le 0$.), the control input is calculated as follows:

$$\dot{\widehat{s}}=-K^{\prime }·\mathrm{s}\mathrm{a}\mathrm{t}\left(\widehat{s}\right),$$

(27)

where $K^{\prime }$ is the gain and sat is the saturation function, defined as:

$$\mathrm{s}\mathrm{a}\mathrm{t}\left(\widehat{s}\right)=\left\{\begin{array}{c}\raisebox{1ex}{$\widehat{s}$}\!\left/ \!\raisebox{-1ex}{$\left|\widehat{s}\right|$}\right.,\left|\widehat{s}\right|\ge {ϵ}_c\\ \raisebox{1ex}{$\widehat{s}$}\!\left/ \!\raisebox{-1ex}{${ϵ}_c$}\right.,\left|\widehat{s}\right|<{ϵ}_c\end{array}\right.,$$

(28)

where ${ϵ}_c$ is the boundary-layer thickness. The derivative of Equation (26) can be written as

$$\dot{\widehat{s}}={\dot{\widehat{x}}}_2+\left(\frac{l_1}{{ϵ}_0}\right){\dot{\tilde{x}}}_1-{\ddot{x}}_d+c_1\left({\dot{\widehat{x}}}_1-{\dot{x}}_d\right)+c_2·\widehat{e},$$

(29)

by using Equations (6) and (7) in (29):

$$\dot{\widehat{s}}={\widehat{x}}_3+\left(\frac{l_2}{{ϵ}_0}\right){\tilde{x}}_1+u+\left(\frac{l_1}{{ϵ}_0}\right){\dot{\tilde{x}}}_1-{\ddot{x}}_d+c_1\left({\widehat{x}}_2+\left(\frac{l_1}{{ϵ}_0}\right){\tilde{x}}_1-{\dot{x}}_d\right)+c_2·\widehat{e},$$

(30)

Equation (30) can be written as:

$$\dot{\widehat{s}}={\widehat{x}}_3+\left(\frac{l_2}{{ϵ}_o}+\frac{c_1·l_1}{{ϵ}_o}\right){\tilde{x}}_1+u+\left(\frac{l_1}{{ϵ}_o}\right){\tilde{x}}_2-{\ddot{x}}_d+c_1\left({\widehat{x}}_2-{\dot{x}}_d\right)+c_2·\widehat{e},$$

(31)

Assumption 3.

The estimated dynamics error is assumed to be:

$${\tilde{x}}_2=0,$$

(32)

based on Assumption 3, Equation (31) can be rewrite as:

$$\dot{\widehat{s}}={\widehat{x}}_3+\left(\frac{l_2}{{ϵ}_o}+\frac{c_1·l_1}{{ϵ}_o}\right){\tilde{x}}_1+u-{\ddot{x}}_d+c_1\left({\widehat{x}}_2-{\dot{x}}_d\right)+c_2·\widehat{e},$$

(33)

to calculate the control input (u), Equation (33) is substituted into Equation (27):

$${\widehat{x}}_3+\left(\frac{l_2}{{ϵ}_o}+\frac{c_1·l_1}{{ϵ}_o}\right){\tilde{x}}_1+u-{\ddot{x}}_d+c_1\left({\widehat{x}}_2-{\dot{x}}_d\right)+c_2·\widehat{e}=-K^{\prime }·\mathrm{s}\mathrm{a}\mathrm{t}(\widehat{s}),$$

(34)

the control input (u) is calculated as:

$$u=-K^{\prime }·\mathrm{s}\mathrm{a}\mathrm{t}\left(\widehat{s}\right)-\left(\frac{l_2}{{ϵ}_o}+\frac{c_1·l_1}{{ϵ}_o}\right){\tilde{x}}_1+{\ddot{x}}_d-c_1\left({\widehat{x}}_2-{\dot{x}}_d\right)-c_2·\widehat{e}-{\widehat{x}}_3,$$

(35)

the control input mentioned above is the sum of the input from PIDSMC ($\overline{u}$) and the estimated perturbation from ESO (${\widehat{x}}_3$):

$$u=\overline{u}-{\widehat{x}}_3,$$

(36)

where

$$\overline{u}=-K^{\prime }sat\left(\widehat{s}\right)-\left(\frac{l_2}{{ϵ}_o}+\frac{c_1·l_1}{{ϵ}_o}\right){\tilde{x}}_1+{\ddot{x}}_d-c_1\left({\widehat{x}}_2-{\dot{x}}_d\right)-c_2·\widehat{e}.$$

(37)

The actual control input is u, which is the sum of the PIDSMC input $\overline{u}$ and the estimated perturbation (${\widehat{x}}_3$). Integrating the estimated perturbation with the control input is helpful in achieving the desired control response. The estimated perturbation consists of system dynamics, uncertainty in dynamics, nonlinearity, and external disturbance. The accuracy of the perturbation estimation depends on the observer gains. The higher the magnitude of gains, the better the estimation accuracy. In the next subsection, a detailed mathematical analysis of the designed control logic is performed. The analysis includes system stability (Section 2.2.1), the effect of perturbation on the sliding variable (Section 2.2.2), and the relationship between the error and perturbation (Section 2.2.3).

2.2.1. System Stability

Stability is an important feature of control algorithms. The control stability is analyzed using the Lyapunov stability theorem:

$$\widehat{s}·\dot{\widehat{s}}\le 0,$$

(38)

by using Equation (29) in above relation:

$$\widehat{s}·\dot{\widehat{s}}\le \left|\widehat{s}\right|({\dot{\widehat{x}}}_2+\left(\frac{l_1}{{ϵ}_0}\right){\dot{\tilde{x}}}_1-{\ddot{x}}_d+c_1\left({\dot{\widehat{x}}}_1-{\dot{x}}_d\right)+c_2·\widehat{e})\le 0,$$

(39)

by using Equation (7) in (39):

$$\widehat{s}·\dot{\widehat{s}}\le \left|\widehat{s}\right|({\widehat{x}}_3+\left(\frac{l_2}{{ϵ}_0}\right){\tilde{x}}_1+u+\left(\frac{l_1}{{ϵ}_0}\right){\tilde{x}}_2-{\ddot{x}}_d+c_1\left({\dot{\widehat{x}}}_1-{\dot{x}}_d\right)+c_2·\widehat{e})\le 0,$$

(40)

$$\widehat{s}·\dot{\widehat{s}}\le \left|\widehat{s}\right|({\widehat{x}}_3+\left(\frac{l_2}{{ϵ}_o}+\frac{c_1·l_1}{{ϵ}_o}\right){\tilde{x}}_1+u+\left(\frac{l_1}{{ϵ}_o}\right){\tilde{x}}_2-{\ddot{x}}_d+c_1\left({\widehat{x}}_2-{\dot{x}}_d\right)+c_2·\widehat{e})\le 0,$$

(41)

after using the control input Equation (35) in (41):

$$\begin{array}{c}\widehat{s}·\dot{\widehat{s}}\le \left|\widehat{s}\right|({\widehat{x}}_3+\left(\frac{l_2}{{ϵ}_o}+\frac{c_1·l_1}{{ϵ}_o}\right){\tilde{x}}_1-K^{\prime }·\mathrm{s}\mathrm{a}\mathrm{t}\left(\widehat{s}\right)-\left(\frac{l_2}{{ϵ}_o}+\frac{c_1·l_1}{{ϵ}_o}\right){\tilde{x}}_1+{\ddot{x}}_d-c_1\left({\widehat{x}}_2-{\dot{x}}_d\right)-\\ c_2·\widehat{e}-{\widehat{x}}_3+\left(\frac{l_1}{{ϵ}_o}\right){\tilde{x}}_2-{\ddot{x}}_d+c_1\left({\widehat{x}}_2-{\dot{x}}_d\right)+c_2·\widehat{e})\le 0,\end{array}$$

(42)

after the cancelation, Equation (42) can be written as:

$$\widehat{s}·\dot{\widehat{s}}\le \left|\widehat{s}\right|(-K^{\prime }·\mathrm{s}\mathrm{a}\mathrm{t}\left(\widehat{s}\right)+\left(\frac{l_1}{{ϵ}_o}\right){\tilde{x}}_2)\le 0,$$

(43)

in order to enforce the stability, the magnitude of gain $K^{\prime }$ should satisfy the following condition (44):

$$K^{\prime }>\frac{l_1}{{ϵ}_0}·\left|{\tilde{x}}_2\right|.$$

(44)

2.2.2. Perturbation Effects on Sliding Variable

The actual sliding-variable dynamics ($\dot{s}$) are affected by the estimation error of perturbation (${\tilde{x}}_3$) (Appendix C):

$$\dot{s}=-K^{\prime }\mathrm{s}\mathrm{a}\mathrm{t}\left(s\right)+{\tilde{x}}_3,$$

(45)

Assumption 4.

It is assumed that sliding variable is within boundary layer ($s<\left|{ϵ}_c\right|$); therefore, saturation function in Equation (45) can be defined as:

$$\mathrm{s}\mathrm{a}\mathrm{t}\left(s\right)=\frac{s}{{ϵ}_c},$$

(46)

based on Assumption 4, Equation (45) can be written as:

$$\dot{s}=-K^{\prime }\left(\frac{s}{{ϵ}_c}\right)+{\tilde{x}}_3,$$

(47)

Apply Laplace transformation to Equation (47):

$$sp=-K^{\prime }\left(\frac{s}{{ϵ}_c}\right)+{\tilde{x}}_3,$$

(48)

$$s\left(p+\frac{K^{\prime }}{{ϵ}_c}\right)={\tilde{x}}_3,$$

(49)

the relation between the estimated error of perturbation (${\tilde{x}}_3$) and the actual perturbation ($x_3$) is defined as (Appendix B):

$${\tilde{x}}_3=\frac{{ϵ}_0{p}^3+p^2{l}_1+p{l}_2}{{ϵ}_0{p}^3+p^2{l}_1+p{l}_2+l_3}·x_3,$$

(50)

after using Equation (50) in (49):

$$s\left(p+\frac{K^{\prime }}{{ϵ}_c}\right)=\frac{{ϵ}_0{p}^3+p^2{l}_1+p{l}_2}{{ϵ}_0{p}^3+p^2{l}_1+p{l}_2+l_3}·x_3,$$

(51)

$$\frac{s}{x_3}=\frac{{ϵ}_0{p}^3+p^2{l}_1+p{l}_2}{({ϵ}_0{p}^3+p^2{l}_1+p{l}_2+l_3)\left(p+\frac{K^{\prime }}{{ϵ}_c}\right)},$$

(52)

$$\frac{s}{x_3}=\frac{{ϵ}_0{p}^3+l_1{p}^2+l_{2}p}{{ϵ}_0{p}^4+A_1{p}^3+A_2{p}^2+A_{3}p+A_4},$$

(53)

where $A_1=l_1+\raisebox{1ex}{$K^{\prime }·{ϵ}_0$}\!\left/ \!\raisebox{-1ex}{${ϵ}_c$}\right.$, $A_2=l_2+\raisebox{1ex}{$K^{\prime }·l_1$}\!\left/ \!\raisebox{-1ex}{${ϵ}_c$}\right.$, $A_3=l_3+\raisebox{1ex}{$K^{\prime }·l_2$}\!\left/ \!\raisebox{-1ex}{${ϵ}_c$}\right.$, and $A_4=\raisebox{1ex}{$K^{\prime }·l_3$}\!\left/ \!\raisebox{-1ex}{${ϵ}_c$}\right.$.

It can be understood from the Bode plot of the relation (53) that the designed algorithm effectively attenuates the low- and high-frequency elements of the perturbation (Figure 3) ($a=30, K=20, c_1=5, c_2=2.5, {ϵ}_c=1, {ϵ}_o=1)$.

2.2.3. Perturbation Effects on Output Error

It is important to understand the effects of perturbation ($x_3$) on the output error (e). The relation between these two variables is derived and explained in this section (Figure 4). By taking the Laplace transform of Equation (23), the relation between the output error (e) and sliding variable (s) is derived as follows:

$$s=pe+c_{1}e+c_2\raisebox{1ex}{$e$}\!\left/ \!\raisebox{-1ex}{$p$}\right.,$$

(54)

$$s=e(p+c_1+\raisebox{1ex}{$c_2$}\!\left/ \!\raisebox{-1ex}{$p$}\right.),$$

(55)

$$s=e\left(\frac{p^2+c_{1}p+c_2}{p}\right),$$

(56)

$$\frac{e}{s}=\frac{p}{p^2+c_{1}p+c_2},$$

(57)

by using Equations (57) and (53), the relation between error (e) and perturbation ($x_3$) is calculated as ($\raisebox{1ex}{$e$}\!\left/ \!\raisebox{-1ex}{$x_3$}\right.$):

$$\frac{e}{x_3}=\frac{s}{x_3}·\frac{e}{s}=\frac{e}{x_3},$$

(58)

$$\frac{e}{x_3}=\frac{{ϵ}_0{p}^3+l_1{p}^2+l_{2}p}{{ϵ}_0{p}^4+A_1{p}^3+A_2{p}^2+A_{3}p+A_4}·\frac{p}{p^2+c_{1}p+c_2},$$

(59)

$$\frac{e}{x_3}=\frac{{ϵ}_0{p}^4+l_1{p}^3+l_2{p}^2}{{ϵ}_0{p}^6+B_1{p}^5+B_2{p}^4+B_3{p}^3+B_4{p}^2+B_{5}p+B_6},$$

(60)

where $B_1=A_1+\delta c_1$, $B_2=A_2+A_1{c}_1+\delta c_2$, $B_3=A_3+A_2{c}_1+A_1{c}_2$, $B_4=A_4+A_3{c}_1+A_2{c}_2$, $B_5=A_5{c}_1+A_3{c}_2$, and $B_6=A_4{c}_2$.

For further explanation, the relation between the error and perturbation for the ESO based SMC is also presented and plotted in this section. This is helpful for explaining the superiority of the designed algorithm (ESO based PIDSMC). The relation of ESO-based SMC is derived as:

$$\frac{e}{x_3}=\frac{{ϵ}_0{p}^3+l_1{p}^2+l_{2}p}{{ϵ}_0{p}^5+D_1{p}^4+D_2{p}^3+D_3{p}^2+D_{4}p+D_5},$$

(61)

where $D_1=A_1+\delta c_1$, $D_2=A_2+A_1{c}_1$, $D_3=A_3+A_2{c}_1$, $D_4=A_4+A_3{c}_1$, and $D_5=A_4{c}_1$.

The Bode plot of relations (61) and (60) can be observed in Figure 5. In the low-frequency (LF) region, the proposed algorithm (blue dotted line) can effectively work against perturbation, whereas in the mid-frequency (MF) and high-frequency (HF) region, the performances of both algorithms are same. The designer can change the frequency areas, depending upon the values of the controller parameters.

3. Simulation and Discussion

This section presents the simulations and a discussion. Two different simulation results are presented in this research, since two different DO-based control logics were implemented: the proposed approach and the previous approach [38]. In the first simulation, the control logics were implemented on a second-order transfer function. The simulations were performed in MATLAB, and three different cases were studied. In the second simulation, the control algorithms were implemented on two-DOF virtual robot manipulators, and the simulations were performed using SimScape multibody (SimMechanics).

3.1. Simulations on Second-Order System

The simulations were performed in MATLAB/Simulink. Different cases were studied to verify the performance of the designed logic (ESO-based PIDSMC) compared with the previous logic (ESO based SMC) [38]. The simulations were performed on a second-order system.

$$\ddot{x}=u+f\left(x,\dot{x},t\right)+∆f\left(x,\dot{x},t\right)+d,$$

(62)

where x is the system state, u is the control input, $f\left(x,\dot{x},t\right)=5\dot{x}\mathrm{c}\mathrm{o}\mathrm{s}\left(x\right)$, $∆f\left(x,\dot{x},t\right)$ is uncertainty in the system dynamics, and d is the external disturbance. The system’s known dynamics, its uncertainty, and its external disturbance were considered as perturbations:

$$x_3=f\left(x,\dot{x},t\right)+∆f\left(x,\dot{x},t\right)+d,$$

(63)

$$\ddot{x}=u+x_3.$$

(64)

The controller’s task is to reach the desired state in the presence of perturbation. The objective of the research was to minimize the effect of perturbation on the output error. A block diagram of the simulation is shown in Figure 6. The controller parameters are shown in

Table 1. Controllers’ parameters for simulations.

ESO Based SMC	ESO Based PIDSMC
a = 15	a = 15
K = 50	$K^{\prime }$ = 50
${ϵ}_o=1$	${ϵ}_o=1$
${ϵ}_c=1$	${ϵ}_c=1$
$c_1=5$	$c_1=5$
	$c_2=2.5$

. The control switching gain (K and $K^{\prime }$) was selected on the basis of Equation (44). The other controller parameters were selected based on a trial-and-error method.

Three different cases were considered. The performances of the proposed and previous controllers are discussed in the next subsection.

3.1.1. Case 1 (Faster Error Convergence)

In first case, there was no disturbance in the system ($d=0$). The constant (1 rad/s) was set as the desired trajectory. The output error of the system can be observed in Figure 7. The proposed control’s logic converged faster than that of the ESO-based SMC. The faster convergence was due to the addition of integral term in sliding surface. The convergence depends upon the value of the integral gain ($c_2$). But it should be selected optimally. Larger values of integral gain can reduce the increase time, but this creates overshoot in the system, which is not desirable. In this research, the value of the integral gain ($c_2$) was selected by hit and trail. The control input can be observed in Figure 8. Figure 9 depicts the estimation accuracy of the ESO. The absolute average error of the ESO-based PIDSMC was less than that of the ESO-based SMC (

Table 2. Absolute average error in degree (case 1).

ESO-Based SMC	ESO-Based PIDSMC
2.404196	1.547471

).

3.1.2. Case 2 (Effectiveness against Low-Frequency Perturbation)

In the second case, the simulation was performed with disturbance (d). The sinewave was introduced as disturbance (Figure 10). The frequency of the sinewave was 0.9 rad/s (low frequency). As described in Section 2.2.3, our proposed algorithm is effective against low-frequency perturbation. The proposed method has the ability to minimize the effect of low-frequency perturbation (Figure 11). The control input can be observed in Figure 12. The input disturbance had a low frequency. Therefore, the ESO estimated it accurately (Figure 13). The absolute average error of the ESO-based PIDSMC was smaller than that of the ESO-based SMC (

Table 3. Absolute average error in degree (case 2).

ESO-Based SMC	ESO-Based PIDSMC
0.279947	0.182187

).

3.1.3. Case 3 (More Effective against Variable Input)

In the third case, a variable reference (ramp and sine wave (1 rad/s)) trajectory (Figure 14) was given to the system in the presence of disturbance (Figure 15). It was observed that the performance of the designed controller was better than that of the ESO-based SMC (Figure 16). The control input can be observed in Figure 17. The input trajectory had a low frequency. Therefore, the output error (Figure 16) of the proposed algorithm is small (

Table 4. Absolute average error in degree (case 3).

ESO-Based SMC	ESO-Based PIDSMC
0.212769	0.139154

). The perturbation-estimation accuracy of the ESO can be observed in Figure 18.

3.1.4. Case 4 (Limitation of Proposed Algorithm)

In this case, the frequency of the disturbance was variable and had a large magnitude (Figure 19). In the first five seconds, the frequency of the disturbance was 9 rad/s (this frequency lies in the MF region). In the last five seconds, the frequency of the disturbance was 24 rad/s (this frequency lies in the HF region). As discussed in Section 2.2.3, in the MF and HF regions, the designed controller’s performance was similar to that of the ESO-based SMC (Figure 20).

The performance of the designed controller depended upon the accuracy of the estimation of the perturbation. The relation between the actual and the estimated perturbation was a low-pass filter. By increasing the observer gain, the estimation accuracy was improved. In case 4, the frequency of the disturbance was high (close to and greater than the breaking frequency of the low-pass filter). The designed algorithm did not effectively estimate the high-frequency perturbation (Figure 21).

The absolute average error of the designed algorithm (in the first three cases) was lower than that of the ESO-based SMC (Figure 22). This suggests that the developed technique is more robust against the low-frequency elements of perturbation.

3.2. Simulations on Two -DOF Virtual Robot Manipulator

The two-DOF virtual robot manipulator (RR robot manipulator) was designed in the MATLAB/SimScape multibody toolbox (Figure 23). The SimScape multibody was used to generate the CAD model of the robot manipulator. Control algorithms were implemented to track the reference trajectory. A polynomial reference trajectory was assigned to both joints of the manipulator (Figure 24). The tracking-error results are presented in this section. The tracking error of joint 1 can be observed in Figure 25. The ESO-based PIDSMC error was better than that of the ESO-based SMC. Figure 26 depicts the control input of joint 1. Similarly, the ESO-based PIDSMC had a better performance for joint 2 (Figure 27). Figure 28 presents the control input of joint 2. The absolute average error of the proposed method was much smaller than that of the previous method. The results indicate that the ESO-based PIDSMC is effective against perturbation. The controller parameters for both joints are presented in

Table 5. Controller parameters.

ESO-Based SMC	ESO-Based PIDSMC
$a_1=20, a_2=18$	$a_1=20, a_2=18$
$K_1=100, K_2=88$	$K_1^{\prime }=100, K_2^{\prime }=88$
${ϵ}_{o1}={ϵ}_{o2}=1$	${ϵ}_{o1}={ϵ}_{o2}=1$
${ϵ}_{c1}={ϵ}_{c2}=1$	${ϵ}_{o1}={ϵ}_{o2}=1$
$c_{11}=10, c_{12}=5$	$c_{11}=10, c_{12}=5$
	$c_{21}=2.5, c_{22}=2.5$

. The observer gain (a) was selected through a trial-and-error method. The values of the switching gains (K and K′) were chosen based on Equation (44). The other control parameters were selected through a trial-and-error method. The parameter values for both algorithms were the same, where m₁ = 2 kg and m₂ = 1.5 kg. The dynamics of the two-DOF robot manipulator are well known in the literature. The dynamics of the first joint of the robot manipulator can be expressed as follows:

$${\tau }_1=M_1{\ddot{q}}_1+C_1{\dot{q}}_1+G_1,$$

(65)

$${\ddot{q}}_1=\frac{1}{M_1}({\tau }_1-C_1{\dot{q}}_1-G_1),$$

(66)

where $M_1$ is the inertia, $C_1$ is Coriolis, $G_1$ is gravity, ${\tau }_1$ is the torque, and $q_1$ is the robot-joint angle of link 1.

$${\ddot{q}}_1=\frac{{\tau }_1}{M_1}+\left(\frac{-C_1{\dot{q}}_1-G_1}{M_1}\right),$$

(67)

the perturbation is defined as:

$$q_{31}=\left(\frac{-C_1{\dot{q}}_1-G_1}{M_1}\right),$$

(68)

$${\ddot{q}}_1=M_1^{-1}{\tau }_1+q_{31},$$

(69)

the value of $M_1^{-1}$ was assumed as unity ($M_1^{-1}=1$) in the control design,

$${\ddot{q}}_1={\tau }_1+q_{31}.$$

(70)

Similarly, the dynamics of the second joint can be written as:

$${\ddot{q}}_2={\tau }_2+q_{32},$$

(71)

The overall comparison between the designed method and the existing method is presented in

Table 6. Concluding summary.

Performance Characteristics	ESO-Based SMC	ESO-Based PIDSMC
Small tracking error	No	Yes
Effective against perturbation	No	Yes
Model-free control	Yes	Yes

. The designed algorithm is a classical control technique. In future research, intelligent control techniques (neural network, reinforcement learning) will be integrated into the designed algorithm. The intelligent technique will be helpful in the selection of the optimal values of the control parameters (K and c). The intelligent control algorithm will be implemented to tackle fractional-order dynamical systems [39,40,41,42].

4. Conclusions

The compensation/rejection of perturbations is crucial for achieving accurate trajectory tracking. In this study, a disturbance-rejection algorithm was developed and thoroughly analyzed. The algorithm combines sliding mode control (SMC) with a proportional integral derivative (PID) sliding surface and extended-state observer (ESO). The ESO was utilized to estimate the system state and the perturbation, with the estimated perturbation used to reject any disturbances. This perturbation rejection led to improved system performance, including small tracking errors and effectiveness against the low-frequency elements of the perturbation. The control design only requires minimal information about the system, making it easy to implement. The simulation results in MATLAB/Simulink demonstrated the superiority of the proposed algorithm.