# A Robust \(H_{\infty}\) Application for Motor-Link Control Systems of Industrial Manipulators
## 1 Introduction
In 2012, a study on the development of an \(H_{\infty}\) controller with LMI region schemes for trajectory tracking and vibration control of a planar single-flexible link was proposed in [5]. In 2013, the implementation of two controllers by using both the Loop-\(H_{\infty}\)-shaping approach and the Mixed \(H_{\infty}\) one has been presented in [6]. Model order reduction of controllers is also discussed, and a comparison between the developed methodology and the canonical PID control shows the advantages of using robust approaches. In 2015, a study about a different implementation of the standard \(H_{\infty}\) theory was developed in [7]. Here, the key idea is to add to the original control law an auxiliary PID input that attenuates the deviation due to modeling uncertainties and unknown disturbances. In 2016, the control of a non-linear robotic manipulator by using the \(H_{\infty}\) technique was discussed in [8]. The \(H_{\infty}\) control is also compared to a Model Predictive controller, where the optimization of the control action is derived by considering past and future behaviors of the system under analysis. In 2017, two non-linear \(H_{\infty}\) control schemes for multi-DOF robotic manipulators were proposed in [9]. The authors formulated the control problem as a min-max differential problem, and they designed the controller through the solution of a Riccati equation. In 2018, an adapting computed-torque controller was proposed in [10], requiring few parameters to be set. This is a very important result, since it allows one to realize and analyze the entire architecture in an easy way. Moreover, the controller does not need any specific initial information, and it automatically adapts to the real system dynamics. The \(H_{\infty}\) technique is used as proof of the robustness of the proposed controller, whose validity is confirmed by experimental results for a 3 DOF and a 7 DOF robot manipulator. The \(H_{\infty}\) algorithm is also used to achieve high-precision tracking performance in [11]. On the basis of theoretical results on robust finite-time stability for uncertain systems, a robust controller is developed by means of the back-stepping approach, achieving high precision, strong robustness, and fast response. Simulations and experimental results confirm the effectiveness of the proposed controller. More recently, a work emphasizing the design of robust control for a 3 DOF robotic manipulator under uncertainties was carried out in [12]. The \(H_{\infty}\) synthesis, under which the controller was developed, is compared with PID, proving how the \(H_{\infty}\) controller achieved better noise rejection, disturbance rejection, performance, and stability robustness characteristics. A new and interesting approach combining identification and \(H_{\infty}\) control design for controlling multiple-link elastic joint robots in the presence of model uncertainties is developed in [13]. The controller has two degrees of freedom, and it is able to withstand uncertainties and variations in model parameters as well as to track reference trajectories. The last aspect is simplified by the presence of a feed-forward action that anticipates the possible future trajectories. In [14], the robust control of robot manipulators is investigated comparing the performances achieved by robust PID controllers, tuned through the Quantitative Feedback Theory (QFT) approach, and the ones provided by the application of \(H_{\infty}\) control, by adopting a standard procedure for the design of the weighting functions. The results achieved by the first control solution seem better, but it must be underlined that only simulation tests were performed, without taking into account the different behavior on the motor and the link side that occurs in practice. Furthermore, very recent results on the application of robust control techniques were published in 2021, but they do not seem mature for an actual industrial application yet.