Fixed-time bounded H infinity tracking control of a single-joint manipulator system with input saturation

ABSTRACT Based on Lyapunov finite-time stability theory and backstepping strategy, we put forward a novel fixed-time bounded H infinity tracking control scheme for a single-joint manipulator system with input saturation. The main control objective is to maintain that the system output variable tracks the desired signal at fixed time. The advantages of this paper are the settling time of the tracking error converging to the origin is independent of the initial conditions, and its convergence speed is more faster. Meanwhile, bounded H infinity control is adopted to suppress the influence of external disturbances on the controlled system. At the same time, the problem of input saturation control is considered, which effectively reduces the input energy consumption. Theoretical analysis shows that the tracking error of the closed-loop system converges to a small neighbourhood of the origin within a fixed time. In the end, a simulation example is presented to demonstrate the effectiveness of the proposed scheme.


Introduction
With the rapid development of modern industrial manufacturing, agricultural production and space exploration, the applications of robotic arms have become more extensive. As the difficulty of the control tasks increases, the requirement of highprecision is also required accordingly. Especially in some specific and complex task environments, it is particularly important to realize accurate control of the manipulators, such as assembly (Wardeh & Frimpong, 2020), welding (X. Liu et al., 2020), sorting and so on. To accomplish some complex and demanding tasks, the manipulators need to accurately track the desired trajectories especially. It should be noted that there are many research methods for robotic arms under certain circumstances. The most widely used approaches are to combine the backstepping control strategy with adaptive control. In X. H. Li et al. (2021), the authors designed an adaptive neural finite-time controller. An adaptive H infinity fault-tolerant controller was proposed by Lei and Chen (2020). Jiao et al. (2020) and H. Q.  combined manipulator systems and adaptive fuzzy logic systems to design the controllers. The other tracking control research was based on the robust control method (Bu et al., 2020;Gholipour & Fateh, 2020;Guo & Meng, 2021; J. J. Liu & Guo, 2022).
With the rapid development of modern industry, the control requirements of manipulator systems are higher and higher. However, most of the manipulator systems have the characteristics of an increasing number of components and high nonlinear order. These characteristics lead to more and more ways for CONTACT Ming Chen cm8061@sina.com School of Electronic and Information Engineering, University of Science and Technology Liaoning, Anshan 114051, People's Republic of China disturbances to enter the manipulator systems. It is worth mentioning that these disturbances will reduce the control performance and even directly destroy the stability of the manipulator systems. It is essential for the stable operation of the manipulator systems to suppress external disturbances. In articles (Xiao et al., 2020;Zhou et al., 2020), the disturbance observers are designed to eliminate the influence of disturbances on the manipulator systems. Unfortunately, it is difficult to observe the disturbances in practical systems. And it is not easy to construct the disturbance observers and compensate the systems with disturbances. Bounded H infinity control reduces the influence of external disturbances on the systems by adjusting disturbance suppression coefficients. In recent years, there have been a lot of research results on bounded H infinity control.
In actual industrial productions, the inputs of the manipulators should not be infinite. Too large control signals will destroy the actuators. The control signals should be kept within the pre-designed range. Therefore, it is necessary to consider the input saturation of the manipulator systems. And it has been a research hotspot. After a long time of exploration by scholars at home and abroad, the research of input saturation constraint control has made great progress by Park et al. (2018), Dastres et al. (2020), and Q. .
There have been many articles about the combination of the manipulator systems and finite-time control (Bao et al., 2020;X. Li et al., 2018;Yu et al., 2020). However, although the traditional finite-time control can ensure the convergence of the systems in a limited time, its convergence time is usually affected by the initial states of the systems. When the initial states of the systems are far from the equilibrium points of the systems, the convergence time of the systems will become very long. In the finite-time stability theory, Polyakov et al. (2015) first proposed the problem of fixed-time convergence control and gave the definition of fixed-time stability. Fixed-time control is an effective tool to improve transient performance, which ensures the rapid convergence of the systems. And there is an upper bound on the convergence time of fixed-time control. It is worth mentioning that its convergence time is independent of the initial states of the system. In recent years, fixedtime control has received extensive attention and many research schemes have been proposed. Sun and Zhang (2020) showed the problem of fuzzy logic adaptive fixed-time control for nonlinear switched systems. A fixed-time scheme is designed by combining adaptive neural network with Euler-Lagrange systems (ELSs) in G. Zhu and Du (2020). Yang and Niu (2020) studied the problem of the fixed-time tracking control for a class of uncertain strict-feedback non-linear system based on event-triggering mechanism. Note that in practical manipulator applications, many systems require fixed operating time, so it is very important to control the manipulator systems to achieve control objectives within the fixed time. Unfortunately, few of the articles on robotic arms above take this into account.
Compared with many existing achievements, the major contributions of the paper can be briefly stated as follows: (i) Unlike most of the existing results, bounded H infinity control is combined with fixed-time control for the first time in this paper. The significant virtue of the combination of these two algorithms lies in the improvement of the system's rapidity and anti-interference. Particularly, the influence of external disturbances on the system is reduced effectively. (ii) Compared with the traditional finite-time control, fixedtime control has more significant advantages. With the introduction of the fixed-time control, the response speed of the system has been improved by adjusting the design parameters. Especially, its convergence time has nothing to do with the initial states of the system but only depends on its design parameters. (iii) This design scheme combines fixed-time and H infinity to design a tracking controller for a single-joint manipulator system with the unknown disturbances, and the problem of input saturation control is also investigated, which effectively reduces the input energy consumption. In addition, it ensures that the output variable tracks the desired output signal in fixed time, and all the signals in the closed-loop system are bounded.
The rest of this paper is arranged as follows: systems description and some mathematical preparation are introduced in Section 2. In the following section, a novel fixed-time input saturation tracking controller is proposed by using the backstepping strategy. In Section 4, the effectiveness of the proposed method is testified through a simulation example. At last, conclusions are drawn in Section5.

System description
Consider the dynamic model of a single-joint manipulator system with a DC motor described by (see Si et al., 2017) Mq + N sin q + Bq = I + I , (1) among them: q,q,q are the angular position, speed and acceleration of joint, respectively; I andİ are the angle and speed motor for the motor shaft, respectively; I is the interference item of torque; V is the control voltage, which represents the torque of the motor; V is the disturbance of the external environment to voltage; M is the mechanical inertia; B is the coefficient of viscous friction in the joint; N is a normal number, which is the weight and gravity coefficient of the load; L is the armature inductance; R is the armature resistance; K B is the back electromotive force. By introducing the state variables: x 1 = q, x 2 =q, x 3 = I, the dynamic equation (1) can be written as In view of the control input saturation, the signal V is described as Zhao et al. (2020) where Q represents the input variable of the saturation nonlinearity, and u max > 0 and u min < 0 denote the known constants. Now, introduce a smooth piecewise function to approximate the input saturation function and given by and h(Q) satisfying In addition, with the help of the mean value theorem, there exists a constant μ (0 < μ < 1) and the following equation holds.
Remark 2.1: As for g Q μ in (7), we assume that where g m is an known positive constant.

Mathematical preparation
To facilitate the discussion of the main results, it is necessary to provide some assumptions and lemmas.

Remark 2.2:
For the convenience of proof, we set the positive real number in (16) equal to 1. Meanwhile, all the time variables t will be omitted in the following.

Design procedure
In this section, we propose a fixed-time H infinity controller for the system described by (2). To do this, we introduce the following coordinate transformations where y d is a desired output signal, and α i , i = 1, 2, are the virtual control signals.
Step 1. According to the first subsystem state equation defined in (2) and (19), we havė The first Lyapunov function is designed as Based on (20), the time derivative of V 1 iṡ The virtual control law α 1 can be given as where k 11 , k 12 are the positive design parameters, ε 1 > 0. According to (23), the term z 1 α 1 in (22) can be expressed as Substituting (24) and (25) into (22), we can geṫ Step 2. Similar to Step 1, it is easy to obtain thaṫ where
Substituting (34)-(37) and (39) into (33) results iṅ Step 3. The following derivation for the derivative of z 3 can be carried ouṫ where The Lyapunov function is selected for The time derivative of V 3 iṡ Based on the Young's inequality and (6), as for the terms z 3 d 3 , −z 3 ∂α 2 ∂x 2 d 3 and z 3 h(Q) L in (43), it is not difficult to obtain Similarly, by using Lemma 2.1, we obtain the following four inequations.
where σ = √ 2σ . So (74) illustrates that the system (1) meets the performance index of the fixed-time bounded H infinity.

Simulation results
In the section, we will verify the availability of our proposed fixed-time bounded H infinity controller with input saturation by using Theorem 3.1. The following parameter values are the same as in the literature (Si et al., 2017): B = 1 N·ms/rad, N = 10, M = 1 kg/m 2 , L = 0.1 H, R = 0.5, K m = 10 N·m/A.     The initial states are set as x 1 (0) = 0.2, x 2 (0) = 0.3 and x 3 (0) = 0.1. Our target is to ensure that the output signal can track the desired signal at fixed time. The desired signal is given as y d = 0.5 sin t. The external disturbances and the input saturation voltage to are chosen as d 2 = 5 cos(5πt) e −0.5t , d 3 = 5 cos(4πt) e −0.5t , u max = 15 and u min = −15.
It is essential for the performance of the closed-loop system to choose the suitable design parameters. In the process of simulation, it can be discovered that the larger the parameters k 11 , k 12 , k 21 , k 22 , k 31 , k 32 , the better tracking performance. In the meantime, it is necessary to point out that the choice of ε 1 , ε 2 , ε 3 , andσ correctly has also great influence on the performance of the closed-loop system. By the trial, we choose k 11 = k 12 = k 21 = k 22 = k 31 = k 32 = 36,σ = 0.8, ε 1 = ε 2 = ε 3 = 2, g m = 1.
According to Theorem 3.1, the response curves are shown in Figures 1-5. Figure 1 shows the tracking error. From Figure 2, we can find that the system output signal x 1 can follow the desired reference signal very well in fixed time. The response curves of the states x 1 , x 2 and x 3 are illustrated in Figures 3 and 4, respectively, which illustrate distinctly that all the signals are bounded. Figure 5 shows the control law u. From the above simulation results, it clearly shows that our proposed method has a fast convergence speed, good tracking effect and robustness to the external interferences. It is also remarkable that the tracking performance with our proposed method is better than that of other methods. Therefore, our proposed scheme is verified to be effective.

Conclusion
This paper has developed a fixed-time backstepping scheme for a single-joint manipulator system with input saturation. The proposed controller can ensure that the tracking error converges to the origin at a fixed time, and its convergence time is independent of the initial states of the controlled system. At the same time, the problem of the bounded H infinity control and the input saturation control are considered. Eventually, some simulation results have been provided to testify the effectiveness of the devised approach. In addition, the issues of the unmodeled parts and the failures in the systems are not considered in the present study. These problems are interesting and significant and will be investigated under the framework of the existing research results in future work.