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.