# Dynamic Stability of an Electric Monowheel System Using LQG-Based Adaptive Control
## 6 Reference Model Assisted Adaptive Control
### Lyapunov Stability Analysis  
Let our [PERSON] candidate function be  
\[V=\frac{1}{2}\sum_{i=1,2,3}\alpha_{i}e_{i}^{2}+\frac{1}{2}\sum_{i=1,2}\beta_{i} \hat{p}_{i}^{2}. \tag{30}\]  
By differentiating \(V\) with respect to time, we get  
\[\hat{V}=\sum_{i=1,2,3}\alpha_{i}e_{i}\dot{e}_{i}+\sum_{i=1,2}\beta_{i}\hat{p}_ {i}\dot{\hat{p}}_{i}. \tag{31}\]  
Substituting the error dynamics, adaptive update laws, and adaptive control laws (\(\dot{e}_{i}\), \(\dot{\hat{p}}_{i}\), u) in order to cancel out all \(\hat{p}_{i}\), we finally arrive at  
\[\hat{V}=-\frac{B}{M}\alpha_{2}e_{2}^{2}+e_{1}e_{3}(\alpha_{3}g\hat{p}_{2}- \alpha_{1}). \tag{32}\]  
For the function to be Lyapunov stable, \(\hat{V}\) must be negative definite. The term \(-\alpha_{2}\frac{B}{M}e_{2}^{2}\) is a negative definite term. If \(e_{1}e_{3}=0\), then, \(\hat{V}\) is negative definite. When \(e_{1}e_{3}\
eq 0\), we must find a range of values of \(\alpha_{1}\) and \(\alpha_{3}\) for which \(e_{1}e_{3}(\alpha_{3}g\hat{p}_{2}-\alpha_{1})\) is negative at all times.  
If \(e_{1}e_{3}>0\), then  
\[(\alpha_{3}g\hat{p}_{2}-\alpha_{1})<0,\ \ \frac{\alpha_{1}}{\alpha_{3}}>\hat{p}_{ 2}g. \tag{33}\]  
If \(e_{1}e_{3}<0\), then  
\[(\alpha_{3}g\hat{p}_{2}-\alpha_{1})>0,\ \ \frac{\alpha_{1}}{\alpha_{3}}<\hat{p}_{ 2}g. \tag{34}\]  
Thus, the proposed adaptive control structure with the constraints derived in Equations (33) and (34) guarantees \(\hat{V}<0\), which implies that \(V\in L_{\infty}\) which from Equation (30) implies that \(e_{i}=\dot{\hat{x}}_{i}-x_{i}\hat{p}_{i}=\dot{p}_{i}-p_{i}\in L_{\infty}\) and so \(x_{i},\dot{p}_{i}\in L_{\infty}\). The control signal u(t) as given by Equation (29) is also bounded; that is, \(u\in L_{\infty}\). Next, from Equations (26) and (28), we find that \(\dot{e}_{i}\), \(\dot{\hat{p}}_{i}\in L_{\infty}\) because all the signals which are part of those equations are bounded. Additionally, since \(\hat{V}(t)\) is lower-bounded by the negative of the squared error signals, the error signals also satisfy the following property: \(e_{i}\in L_{2}\). Note that such a property is not applicable to the parameter error signals \(\hat{p}_{i}\) in general. In conclusion, since \(e_{i}\in L_{\infty}\cap L_{2}\) and \(\dot{e}_{i}\in L_{\infty}\), the corollary of the Barbalet lemma implies that \(e_{i}(\infty)=0,i=1,2,3\).  
The reference model stabilizer is built using LQR control techniques with a structure  
\[\hat{u}(t)=K_{2}(K_{1}r(t)-\hat{z}), \tag{35}\]  
where \(K_{1}\) and \(K_{2}\) are 3 x 1 and 1 x 3 state control and reference control gain matrices, respectively, as mentioned in Table 4, meaning that the reference signal stabilizes at the desired velocity value inputted by the rider.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Nominal Value** & **Bounds** \\ \hline \(p_{1}\) & 80 kg & 50–120 kg \\ \(p_{2}\) & 1.67 m\({}^{-1}\) & 1.25–2.5 m\({}^{-1}\) \\ \(K_{1}\) & [0.0221 63.223 0.0159] & - \\ \(K_{2}\) & [0; 1; 0] & - \\ \hline \hline \end{tabular}
\end{table}
Table 4: Nominal values of parameters and feedback gains.