## 5 Experimental Results  
The proposed IP controller was tested on a laser diode _ADL-65052 TL_ under the theoretically established operating conditions and using the trapezoidal integration method. This theoretically guarantees the convergence of the system to \(y_{sp}\) with bounded oscillations (see Figure 5b). Experimental tests were conducted at stable temperatures of \(15,20,25,30\) and \(35\,^{\circ}\)C.  
The experimental emission curves (light-current) of the laser were experimentally obtained (see Figure 10) to estimate \(\eta\) and \(I_{th}\). The theoretical system output at the operating temperature was obtained by replacing these terms in Equations (17) and (18). Therefore, a comparison can be made between the experimental output of the system generated by the controller and the theoretical output. Initially, the response of the system was analyzed with a \(\eta K_{p}\approx 0.64\) and an operating temperature of \(20.28\,^{\circ}\)C with a standard deviation of \(0.02\,^{\circ}\)C as shown in Figure 11. There we observed that the theoretical solution did not coincide with the experimental response of the PI controller. This is because the theoretical model of Equation (17) is only valid if the theoretical control signal (Equation (4)) is greater than the threshold current at the operating temperature which is approximately \(I_{th}=11.5\) mA. Figure 12 shows that, when the theoretical control signal is lower than the threshold current, it generates negative theoretical values in the optical power of the laser (see Figure 11). Figure 11: Comparison of the experimental and theoretical output of the system for a total gain of \(\bar{K}_{p}\approx 0.64\) and an integral time of \(T_{i}=1\) for an operating temperature of \(20\,^{\circ}\)C.  
Figure 10: Experimental laser diode emission curve for a _ADL-65052 TL_ at different operating temperatures. On the other hand, the experimental controller increases the control signal linearly until it exceeds the threshold current. Then, the behavior of the control signal is similar to the theoretical solution, converging the error signal to zero, and the system output to \(y_{sp}\). The comparison between the theoretical and experimental response of the system requires forcing the controller to operate in the region of stimulated emission of the laser with a control signal greater than the threshold current. This is achieved by modifying the integral in the controller by adding a constant term \(u_{0}\) on the integral so that a new control variable, \(u(t)\), is defined as:  
\[u(t)=K_{p}e(t)+\frac{K_{p}}{T_{i}}\bigg{(}u_{0}+\int_{0}^{t}e(\tau)\,d\tau\bigg{)} \tag{27}\]  
The controller of Equation (27) is replaced in the system described by Equation (15). This new differential equation is solved in the way shown previously. Thus, the general solutions of Equations (8) and (9) are applicable to this new control system. The restriction for the particular solution becomes:  
\[y_{sp}=\big{(}y_{sp}-y_{0}\big{)}\big{(}1+\eta K_{p}\big{)}+\frac{\eta K_{p}} {T_{i}}u_{0}-\eta I_{th} \tag{28}\]  
Thus, the particular solution for the optical power of the laser is:  
\[y(t)=y_{sp}-\frac{T_{i}\eta I_{th}+T_{i}y_{sp}-\eta K_{p}u_{0}}{\big{(}1+\eta K _{p}\big{)}T_{i}}e^{-\frac{\eta K_{p}}{(1+\eta K_{p})T_{i}}t} \tag{29}\]  
Equation (15) indicates that, when the control variable equals the threshold current, the system's output becomes zero. Therefore, Equation (29) must be zero at \(t=0\). Thus, the value of \(u_{0}\) for this condition is:  
\[u_{0}=T_{i}\bigg{(}\frac{I_{th}}{K_{p}}-y_{sp}\bigg{)} \tag{30}\]  
The term \(u_{0}\), according to the Equation (30), is added to the experimental controller, thereby, making the system response similar to the theoretical described in Equation (29) (see Figure 13). Additionally, the experimental and theoretical control signal responses (see Figure 14) are similar when the oscillations produced by the initial conditions of the system disappear.