# Dynamic Stability of an Electric Monowheel System Using LQG-Based Adaptive Control
## 6 Reference Model Assisted Adaptive Control
### Simulation Results and Discussion  
Next, we carry out a simulation for the proposed adaptive control environment in the monowheel system. Parameter \(p_{1}\) is varied from 80 kg to 110 kg at 50 s of the simulation; i.e., the load mass is increased by 30 kg. Figure 18 shows a comparison between the graph of the reference model and the corresponding nominal model outputs. It is to be noted that the reference and nominal models were set at different initial conditions. The initial conditions of reference model are considered to be \([z_{10},z_{20},z_{30}]=[0,0,0]\) and \([\hat{z}_{10},\hat{z}_{20},\hat{z}_{30}]=[-1.5,4,2]\). The values of the nominal parameters are as mentioned in Table 4. Due to the unequal initial conditions, the nominal system state responses take  
Figure 17: Model assisted adaptive control structure for Monowheel System. Figure 18: The reference and the nominal states. around 30 s to completely stabilize into the values of the reference system state response. Due to the introduction of disturbance in parameter \(p_{1}\) at 50 s, we observe a spike in the nominal state responses, which takes around 20 s to stabilize completely at \(t=70\) s. The corresponding state errors are shown in Figure 19.  
Once the reference system is stabilized, the next task is to develop the adaptation laws. This mechanism consists of the tracking-error dynamics \(\dot{e}_{i}\), parameter estimation dynamics \(\dot{p}_{i}\), and adaptive control law \(u(t)\) for \(i=1,2,3\), as established in Equations (24)-(30), respectively. This mechanism is designed such that all the tracking errors converge to zero asymptotically. In addition, the parameter estimates \(\dot{p}_{i}\), \(i=1,2\) converge to the steady-state values within the bounds mentioned in Table 4 and satisfy the constraints obtained in Equations (33) and (34). The Lyapunov stability analysis is utilized to obtain the parameter constraints and to choose the tuning parameters (i.e., \(\alpha_{i},\beta_{i}>0,i=1,2,3\)). The chosen values of such tuning parameters are presented in Table 5.  
Based on these adaptation gain values, Equations (32)-(34) were implemented on MATLAB Simulink, and the corresponding responses were obtained. Figure 20 demonstrates that for the chosen values of adaptation gain values, the derivative of the Lyapunov candidate function is always negative definite, as it satisfies the condition in Equation (33). Using these values, the tracking-error dynamics are stabilized around zero as shown in Figure 19. The proposed controller can adapt to the change, which is introduced at 50 s and ensures the  
Figure 19: State errors. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(\alpha_{1}\) & 0.23 \\ \(\alpha_{2}\) & 2.8 \\ \(\alpha_{3}\) & \(10^{-8}\) \\ \(\beta_{1}\) & 0.1 \\ \(\beta_{2}\) & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Adaptation gains. convergence of the error between the two responses to zero within a short time of 20 s. The error, which was already zero before the change in parameters, re-converges to zero after short transience, thereby demonstrating the controller's robustness, as shown in Figure 19; that is, the tracking error is dynamically stable around 0, and it is at zero except for a brief period of transience immediately following the change in a parameter. It is equally important to obtain the bounded steady-state response of the estimated parameters \((\hat{p}_{i}(t),i=1,2,3)\) of the reference model. Hence, using the heuristic approach and the knowledge of the monowheel system, the parametric update laws in Equation (28) and the control law in Equation (29) have been tuned. The response of estimated parameters is shown in Figure 21, and the corresponding parameter errors are as shown in Figure 22. The oscillations and damping are observed in the response of \(\hat{p}_{i}(t),i=1,2\) due to the presence of tracking errors in the dynamics of the parameter update laws in Equation (28). Figure 21: Parameter estimates.