## 6 Comparison with the Classical PD and PID Controllers
### Comparison with the Classical PID Controller  
Consider the classical PID controller given by the expression below:  
\[u=\mathcal{K}_{p}\dot{\phi}+\mathcal{K}_{o}\dot{\phi}+\mathcal{K}_{i}\int_{0}^{t} \phi\,d\bar{\zeta} \tag{103}\]  
The position gain, velocity gain, and integral position gain are represented, respectively, by the expressions \(\mathcal{K}_{p}\), \(\mathcal{K}_{v}\), and \(\mathcal{K}_{i}\) in expression (103). Using Expression (103) the PID controller \(u\) becomes as follows:  
\[\left\{ \begin{array}{l}u=\mathcal{K}_{p}\dot{\phi}+\mathcal{K}_{v}\dot{\phi} +\mathcal{K}_{i}\chi\\ \dot{\chi}=\phi\end{array} \right. \tag{104}\]  
Consequently, by implementing the classical PID controller (104), the regulated dynamics (9) of robotic system are modified to the following:  
\[\left\{ \begin{array}{l}\
u_{\star}\ddot{\phi}=-f_{v}^{\star}\dot{\phi}+ \mathcal{K}_{p}\dot{\phi}+\mathcal{K}_{v}\dot{\phi}+\mathcal{K}_{i}\chi+a_{ \star}\cos(\phi+\theta_{d})-f_{s}^{\star}\text{sgn}(\dot{\phi})+\Delta_{t}\\ \dot{\chi}=\phi\end{array} \right. \tag{105}\]  
At the equilibrium state of the system (105), where \(\ddot{\phi}=0\), \(\dot{\phi}=0\), and \(\dot{\chi}=0\), and for which \(\Delta_{t}=0\), it follows from System (105) that \(\phi=0\); therefore, the PID controller (104) will be utilized to regulate the position of the knee exoskeleton at the required point \(\theta_{d}\). Moreover, when the target point \(\theta_{d}\) of stability for the knee exoskeleton has been reached, the dynamic model (105) leads to the following relation:  
\[\mathcal{K}_{i}\chi+a_{\star}\cos(\theta_{d})=0 \tag{106}\]  
Hence, using relation (106), the augmented variable \(\chi\) at the desired position \(\theta_{d}\) becomes as follows:  
\[\chi^{\infty}=\chi_{d}=-\mathcal{K}_{i}^{-1}a_{\star}\cos(\theta_{d}) \tag{107}\]  
It follows that at the desired point \(\theta_{d}\), and since \(\phi=0\) and \(\dot{\phi}=0\), and relying on Expressions (104), (107), and (17), the PID controller \(u\) becomes equal to the following:  
\[u^{\infty}=u_{d}=-a_{\star}\cos(\theta_{d}) \tag{108}\]  
As a result, the proposed LMI-based affine state-feedback controller (25) proposed in the present work and the classical PID controller (103) lead to the same final value \(u^{\infty}\) at the desired point \(\theta_{d}\). Nevertheless, the impact of the PID controller on the trajectory stability to this desired point is still not realized. It will achieved in the sequel using the LMI approach. Let us reconsider first the controlled dynamical system (105) and the PID controller (104). Let us also consider the term \(\chi^{\infty}\) (or \(\chi_{d}\)) and its expression (107). By making a new variable change \(\phi_{X}=\chi-\chi_{d}\), and as \(\mathcal{K}_{i}\chi_{d}=u_{d}\), the dynamical system (105) is recast like so:  
\[\left\{ \begin{array}{l}v_{\star}\dot{\phi}=-f_{x}^{\star}\dot{\phi}+\mathcal{ K}_{p}\dot{\phi}+\mathcal{K}_{v}\dot{\phi}+\mathcal{K}_{i}\dot{\phi}_{X}-a_{ \star}\cos(\theta_{d})+a_{\star}\cos(\phi+\theta_{d})-f_{s}^{\star}\text{sgn}( \dot{\phi})+\Delta_{t}\\ \dot{\chi}_{\chi}=\dot{\phi}\end{array} \right. \tag{109}\]  
Moreover, the PID controller (104) becomes as follows:  
\[u=u_{d}+\mathcal{K}_{p}\dot{\phi}+\mathcal{K}_{v}\dot{\phi}+\mathcal{K}_{i} \dot{\phi}_{X} \tag{110}\]Let us consider the new state vector \(\mathcal{X}=\left[ \begin{array}{c}\phi\\ \dot{\phi}\\ \phi_{\chi}\end{array} \right]\). Then, expression (110) of the PID controller \(u\) is rewritten in the form of (25), where the term \(\Delta u\) is the state feedback controller defined by expression (26), and where, in the present case, the gain \(\mathcal{K}\in\mathbb{R}^{3}\), and is given as follows:  
\[\mathcal{K}=\left[ \begin{array}{cc}\mathcal{K}_{p}&\mathcal{K}_{v}&\mathcal{ K}_{i}\end{array} \right] \tag{111}\]  
In order to approach the development achieved in the previous parts and then to obtain the same form of the initial nonlinear dynamical model (21), and since \(u=u_{d}+\Delta u\), the system (109) can be rewritten like so:  
\[\left\{