## 2 Model-Based Robust Control Design
### Mathematical Model Using Bond Graph
\begin{table}
\begin{tabular}{c c c} \hline
**Variable** & **Component** & **Description** & **Value** \\ \hline _Se: e\_mhj_ & Electrical & Source of Effort, i.e., Voltage Source & 12 V \\ _R: Ra\_mhj_ & Electrical & Armature Resistance (Ra) of Linear Actuator & 1 \(\Omega\) \\ _I: la\_mhj_ & Electrical & Armature Inductance (Ia) of Linear Actuator & 1 mH \\ _GY: Gyration\_Ratio\_mhj_ & Mechanical & Gyration Ratio & \(5\times 10^{-2}\) \\ _R: Dash\_mhj_ & Mechanical & Damping of the Armature Shaft (sft) & \(5.7286\times 10^{-7}\) Ns/m \\ _I: J\_nsf\_mhj_ & Mechanical & Moment of Inertia of Armature Shaft (sft) & \(2\times 10^{-4}\) kgm\({}^{2}\) \\ _TF: al\_mhj_ & Mechanical & Angular to Linear (al) & \(2\times 10^{-2}\) \\ _R: Fvsfnt\_mhj_ & Mechanical & Viscous Friction between Shaft (sft) and Nut (nt) & \(1\times 10^{-4}\) Ns/m \\ _C: Ksnt\_mhj_ & Mechanical & Spring Constant of Nut (nt) & \(1\times 10^{-5}\) \\ _R: Fu\_fiffiffoftgd_ & Mechanical & Viscous Friction between joints (Femur (f), Fibula and Tibia & \(1\times 10^{-4}\) Ns/m \\ _I: Mass\_ffffloft_ & Mechanical & Mass of remaining actuators, links (l), and human leg & \(10\) \\ _C: Ks\_fiffoftgd_ & Mechanical & Spring Constant of remaining actuators, links and human leg & \(1\times 10^{-4}\) \\ _Se: g\_Mass\_ffffloft_ & Mechanical & Gravitational Force (g) associated with remaining structure & \(98\) \\ \hline \end{tabular}
\end{table} 
Table 1: Variables of the composite model and their details. Figure 5: Lead-compensated \(H_{\infty}\) reference gain controller. * _The control law design takes the form;_ \(u=-k_{\rm C}\hat{x}(t)\)_. The reference gain is adjusted to achieve acceptable steady-state response/outcome for the step and impulse inputs but for the sinusoidal signal a phase delay exists in the output._  
A lead compensator is proposed to be integrated with the \(H_{\infty}\) controller to overcome the phase delay. The lead compensator is a passive network used to overcome the disadvantages of ideal differentiation and still retain the ability to improve the transient response. The standard form of the lead compensator is given in Equation (5):  
\[G_{\rm C}(s)=\frac{1}{\beta}\Bigg{(}\frac{s+\frac{1}{T}}{s+\frac{1}{\beta T}} \Bigg{)} \tag{5}\]  
where \(\beta<1\). The control design assumptions for the \(H_{\infty}\) controller gain were checked and controller \(K_{\rm C}(s)\) is obtained by following the algorithmic steps. The feedback configuration of the controlled plant was simulated in MATLAB(r). The test bound for \(\gamma\) is set as given in Equation (6):  
\[0.0001<\gamma\leq 100 \tag{6}\]  
After fifteen iterations, a satisfactory solution was obtained, and the value of gamma achieved is 0.0062. The value of \(\beta=2.75\times 10^{-3}\) for the maximum phase shift \(\varphi_{max}\) through the designed compensator value 83.9962\({}^{\circ}\). The lead compensator break frequencies \(\frac{1}{T}=0.025\) and \(\frac{1}{\beta T}=9.09\). Hence, the compensator is described as Equation (7):  
\[G_{\rm C}(s)=\frac{1}{2.75\times 10^{-3}}\Bigg{(}\frac{s+0.025}{s+9.09} \Bigg{)} \tag{7}\]  
A gain is added at the reference signal '\(K_{R}\)' to adjust the tracking error for the sinusoidal signal. The control law takes the form:  
\[u=-k_{\rm C}\hat{x}(t)+k_{R}r \tag{8}\]  
\[u_{p}=u+d_{i} \tag{9}\]  
The compensator \(G_{\rm C}(s)\) is cascaded with the plant \(G_{P}(s)\). The \(H_{\infty}\) controller ensures stability, and the combination of compensator and reference gain ensures transient and tracking performance.