## 4 Control Application of Mechatronic Devices: Adaptive Controller
### Adaptive Controller  
To test the adaptability of the technique, the I-PD control scheme is implemented. This control system is a modification of the PID control scheme [20]. The I-PD control scheme is shown in Figure 6, where \(K_{p}\), \(K_{i}\), and \(K_{d}\) are the proportional, integral, and derivative gains of the regulator. R(s) is the reference signal, and Y(s) is the output signal of the system.  
In this type of PID controller, the integral and derivative actions operate on the output signal of the system, rather than on the error signal as it occurs with classical PID controllers. The basic idea of I-PD control is to avoid large control signals which causes saturation phenomenon. By bringing the proportional and derivative control actions to the feedback path, values for \(K_{p}\) and \(K_{d}\) can be chosen with a greater range than those chosen when using a PID control scheme. This facilitates the convergence of the adaptive controller.  
On the other hand, one disadvantage of this type of controller is their slower rise time compared to a classic PID controller when using the same gains values. In addition, these controllers become unstable more easily, so this should be taken into account when tuning them.  
Figure 6: Scheme of the I-PD controller.  
The equivalent transfer function of the hole system can be written as follows:  
\[G_{sys}(s)=\frac{\frac{A\cdot K_{i}}{1+A\cdot K_{d}}}{s^{2}+\frac{B+A\cdot K_{p}} {1+A\cdot K_{d}}s+\frac{A\cdot K_{i}}{1+A\cdot K_{d}}}. \tag{2}\]  
It should be pointed out that the same transfer function is obtained for both modified and classical PID controllers, but without the zeros of the classical PID. Equation (2) can be compared with a second-order system, following the expression presented in [20]:  
\[G_{2^{0}}(s)=\frac{\omega_{n}{}^{2}}{s^{2}+2\cdot\omega_{n}\cdot\zeta\cdot s+ \omega_{n}{}^{2}}, \tag{3}\]  
where \(\omega_{n}\) is the natural frequency of the system, and \(\zeta\) the damping ratio. Notice that parameters are only used to calculate the coefficients; they do not have physical meaning.  
By rearranging Equations (2) and (3) and adjusting the derivative gain based on stability criteria \(K_{d}\), the I-PD gains can be calculated as follows:  
\[K_{i}=\frac{\omega_{n}{}^{2}\cdot(1+A\cdot K_{d})}{A}, \tag{4}\]  
\[K_{p}=\frac{2\cdot\omega_{n}\cdot\zeta\cdot(1+A\cdot K_{d})-B}{A}. \tag{5}\]  
These adaptation laws are used to calculate the regulator gains according to the changing parameters of the plant.  
Since three equations are obtained with two unknown parameters, the value for \(K_{d}\) was established empirically depending on the plant using Ziger Nichols Techniques [21], taking into account the stability rules to obtain a control action with a fast response and reducing the noise. However, \(K_{d}\) will be bounded with the stability condition of the system.  
Thus, from the values obtained from the plant via online identification, it is possible to calculate the regulator gains according to the natural frequency \(\omega_{n}\) and the damping ratio \(\zeta\), a second-order system with the desired specifications.