# Robust Multi-Performances Control for Four-Link Manipulator Arm
## 1 Introduction
On the other hand, the planar underactuated manipulator is not constrained by gravity, so any position of the manipulator arm is the equilibrium point and the linear Jacobian approximated model at any equilibrium point is uncontrollable [7]. The control approaches for the vertical underactuated manipulator cannot be applied for the planar underactuated manipulator. Studies on the controlling planar underactuated manipulator are extensive and some important methods, such as the nilpotent approximated method [26], the converting-chained form method [27] and the order reduction method [28], have been proposed to perform position control for the planar underactuated manipulator. However, the aforementioned approaches of controlling the planar underactuated manipulator are only valid for the nominal plant model. In real systems, nonlinear acting factors need to be considered [29] and the aforementioned approaches cannot be applied for the practical planar underactuated manipulator. So far, it is obvious to see that the robust tracking controller design for manipulator arms is still a challenging subject due to the strict global stability requirement and disturbance reduction involving nonlinear system dynamics. Stimulated by these points, we apply the feedback linearized approach to construct the robust tracking controller of a manipulator arm with the almost disturbance decoupling, adjustable convergence rate and \"globally\" exponentially stable performances, taking the place of the \"locally\" Jacobian linearized method. The feedback linearized approach has contributed to many significant studies [30; 31; 32; 33; 34] in industrial applications, such as the dual parallel-PMSM system [30], grid-tied packed e-cell inverter [31], PHEV charging station [32], artificial pancreas [33] and weak AC grid integration [34]. References [35; 36] have exploited the fact that the stricter tracking error condition of almost disturbance decoupling performance including the absolute error, integral error and input-to-state-stable error is involved to reduce the disturbance effect. However, the almost disturbance decoupling performance of a nonlinear system can only be achieved without the disturbance being multiplied by nonlinearity and in the presence of non-Lipschitz nonlinearity. Therefore, the almost disturbance decoupling performance cannot be achieved for the following nonlinear system:  
\[\dot{x}_{1}(t)=\tan^{-1}(x_{2})+w_{noise}(t) \tag{1}\]  
\[\dot{x}_{2}(t)=u_{fb},\ y_{o1}=x_{1}\equiv\sigma_{o1} \tag{2}\]  
where \(\sigma_{o1}\) and \(u_{fb}\) are the output and control input, respectively. On the contrary, almost disturbance decoupling performance is more likely to be achieved by the proposed method. The main novelty of this study is in its design of a robust tracking controller for a nonlinear complex FLMA. The major contributions of this study are summarized as follows:  
(i) This study is the \"first\" to present the convergence rate formula of a nonlinear FLMA. (ii) The FLMA is the first to be designed by applying the feedback linearized approach with almost disturbance decoupling performance, which takes the place of the traditional posture-energy approach and avoids some torque chattering change behaviour in the swing-up space. Moreover, the proposed approach takes the place of the LQR approach without the necessary trial and error operations. (iii) A robust tracking controller is presented to obtain global exponential stability without solving the Hamilton-Jacobi equation that must to be solved for the famous H-infinity approach. (iv) This study proposes a new approach to improve the shortcomings of traditional fuzzy function approximators without requiring a lot of design experience and knowledge. (v) The implications of this proposed method are \"globally\" valid, whereas the Jacobian linearized approach is \"locally\" valid.