# A Robust \(H_{\infty}\) Application for Motor-Link Control Systems of Industrial Manipulators
## 5 A New Implementation Approach
The new weighting functions are then determined as:  
\[W^{*}_{S}(s)=s\frac{1}{(s+\lambda)}\frac{1}{S^{*}(s)} \tag{20}\]  
\[W^{*}_{T}(s)=\frac{1}{T^{*}(s)} \tag{21}\]  
As already pointed out, the most important advantage of this new formulation of the weighting functions is the partial maintenance of the plant dynamics when closing the loop. In the standard procedure, the zeros of the nominal plant are introduced among the poles of the controller, thus causing their cancellation when multiplying the transfer functions of the controller and the plant. The \(H_{\infty}\) approach builds the controller starting from a _generalized plant_, which is substantially a feedback system relating \(W^{*}_{S}(s),W^{*}_{T}(s)\) and the original plant \(P(s)\). With the new formulation of the two weighting functions, the expression of \(W^{*}_{T}(s)\) includes the zeros of the plant among its poles. This choice leads to the presence of the zeros of the plant also among the zeros of the controller, resulting in their cancellation in the controller expression. When closing the loop, by multiplying \(P(s)\) and \(G_{c}(s)\) the dynamics associated to the zeros of the plant will be re-introduced. This way, the action of the controller will automatically counteract the effects of the vibration modes of the link and attenuate them as any disturbance acting on the system. In other words, since it is not possible to directly measure the link vibrations, because only sensors on the motor side are available, their influence on the controlled plant is kept alive to allow the controller to attenuate it, as a disturbance. The control optimization problem is solved again by means of the LMI Matlab toolbox, using the new weighting functions (20) and (21). The application to the considered system, given in (12), and a subsequent simplification of the resulting controller, as suggested in Section 3 (i.e., eliminating the very high frequency poles and substituting a low-frequency pole in about 0.009 with a pole at zero), allows us to obtain the following controller expression (to be applied instead of (13)):  
\[G^{*}_{cv}(s)=\frac{0.037419(s+0.269)(s^{2}+5.785s+6005)(s^{2}+6232s+9.898\times 1 0^{6})}{s(s+800)(s^{2}+100.7s+8094)} \tag{22}\]  
By following similar steps for the position control loop, the position controller for system (12) is achieved as:  
\[G_{cp}(s)=\frac{1096(s+458.6)(s+166.2)(s^{2}+57.27s+903.5)(s^{2}+37.41s+7437)} {(s^{2}+100.7s+8095)(s^{2}+33.19s+6375)(s^{2}+1820s+1.342\times 10^{6})} \tag{23}\]  
The application of controllers (22) and (23) provides much better results on both the motor and the link side, as it can be seen in Figures 13 and 14, where the velocity and position time responses are reported, respectively. In particular, by comparing Figures 11 and 13, showing the velocity step responses resulting by the application of the two control solutions, i.e., with the standard controller (13) and the modified one (22), respectively, it is possible to appreciate how, in the second case, the unacceptable oscillations of the link previously obtained have completely disappeared. Similar good results are achieved for the position, too, as shown in Figure 14.  
In order to prove the robustness of the control system to model uncertainties, the same variation in the link inertia considered in Section 3, corresponding to the increase of 30% of its original value, has been applied. Figures 15 and 16 show that, in spite of the important load variation introduced, satisfying velocity and position time responses are achieved on both the motor and the link side. The reader is invited to compare, in particular, the velocity time response reported in Figure 15 with the one previously obtained in \[T(s)=\frac{\omega_{n}^{2}}{s^{2}+2\zeta\omega_{n}s+\omega_{n}^{2}} \tag{16}\]0  
After having achieved satisfying velocity and position time step responses for both the motor and the link, a backward/forward reference trajectory has been applied to the axis, corresponding to the motion of the link between two desired positions.