## 3 Control Scheme Using Quasi-Velocities
\bigg{]}N_{3}^{-1}\tau_{r0}\] \[+R_{1}^{T}(k_{1}z_{1}+k_{2}z_{2})\bigg{)}, \tag{29}\]  
_with \(k_{1}>0\), \(k_{2}>0\), and then origin of the dynamical system of the system is globally exponentially stable._  
Proof: Taking into account (23), the LFC (20) is radially unbounded and positive definite function. Applying the controller (29) into (24), the closed-loop time derivative as well as the LFC lead to  
\[\dot{V}=-A(z_{1},z_{2}). \tag{30}\]  
Thus, from [PERSON]'s direct method (a simple convergence lemma [35] (p. 91)), one obtains  
\[\dot{V}(t) \leq-\alpha V(t), \tag{31}\] \[V(t) \leq V(0)\,e^{-\alpha\,t}, \tag{32}\]  
where \(\alpha\) is a real positive number. Hence, it can be concluded that the origin of the error system is globally exponentially stable. Zero dynamics question.Because \(\zeta_{3}=r\), the analysis of the zero dynamics is thus analogous as in [22] but not direct. So, let us consider this issue. The angular kinetic energy of the hovercraft is given in the form  
\[V_{r}=\frac{1}{2}r. \tag{33}\]  
Using (8), (11), the controller (29) and calculating the time derivative of \(\dot{V}_{r}\), one obtains  
\[\dot{V}_{r} =rN_{3}^{-1}(-mx_{g}ur+my_{g}vr-d_{ur}u-d_{vr}v-d_{r0}\text{sign }\,r-d_{r}r\] \[-ab_{T}T\sin\psi+Y_{13}b_{T}T\cos\psi+Y_{23}b_{T}T\sin\psi). \tag{34}\]Taking into account \(z_{2}\) (18), \(\delta\), and denoting \(e_{2}=[ \begin{array}{cc}0&1\end{array} ]^{T}\), it can be written as follows:  
\[v=e_{2}^{T}R_{1}^{T}(z_{2}+\dot{p}_{d})-\delta_{x}r. \tag{35}\]  
In addition, the inverse matrix to matrix \(B\) is of the form  
\[B^{-1}=b_{T}^{-1}\left[ \begin{array}{cc}\tilde{p}_{11}&\tilde{p}_{12}\\ \tilde{p}_{21}&\tilde{p}_{22}\end{array} \right], \tag{36}\]  
where \(\tilde{\beta}_{11}=\beta_{22}/P\), \(\tilde{\beta}_{12}=-\beta_{12}/P\), \(\tilde{\beta}_{21}=-\beta_{21}/P\), \(\tilde{\beta}_{22}=\beta_{11}/P\) (\(P=\beta_{11}\beta_{22}-\beta_{12}\beta_{21}\)). The sway velocity \(v\) (35) is applied in the time derivative \(\dot{V}_{r}\). Thus, (34) can be rewritten as follows:  
\[\dot{V}_{r}=-N_{3}^{-1}r\,d_{r0}\text{sign}\,r+\left(\begin{array} []{cc}-N_{3}^{-1}d_{r}r^{2}+N_{3}^{-1}r(-mx_{g}ur^{2}+my_{g}vr^{2}-d_{w}u-d_{ w}v)\\ \\ -N_{3}^{-1}r[1\;1]\left[ \begin{array}{cc}\text{Y}_{13}&0\\ 0&(\text{Y}_{23}-a)\end{array} \right]\left[ \begin{array}{cc}\beta_{11}& \beta_{12}\\ \beta_{21}&\beta_{22}\end{array} \right]\left(m^{-1}\left[ \begin{array}{cc} mx_{g}r^{2}-d_{w0}\text{sign}(u)-d_{w}u-d_{w}r\\ \\ -my_{g}r^{2}-d_{w0}\text{sign}(v)-d_{w}v-d_{w}r\end{array} \right]\] \[+R_{1}^{T}(k_{1}z_{1}+k_{2}z_{2}-\dot{p}_{d})-\left[ \begin{array} []{cc}\delta_{x}\\ 0\end{array} \right]^{2}+\left[ \begin{array}{cc}\text{Y}_{13}\\ \text{Y}_{23}+\delta_{x}\end{array} \right]N_{3}^{-1}(-mx_{g}ur+my_{g}v\pi\] \[-d_{w}u-d_{w}v-d_{w0}\text{sign}(r)-d_{w}r))\Bigg{)}. \tag{37}\]  
Denoting now  
\[\Xi=-N_{3}^{-1}d_{r}r^{2}+N_{3}^{-1}r(-mx_{g}ur^{2}+my_{g}vr^{2} -d_{w}u-d_{w}v)\] \[-N_{3}^{-1}r[1\;1]\left[ \begin{array}{cc}\text{Y}_{13}&0\\ 0&(\text{Y}_{23}-a)\end{array} \right]\left[ \begin{array}{cc}\beta_{11}& \beta_{12}\\ \tilde{p}_{21}&\tilde{p}_{22}\end{array} \right]\left(m^{-1}\left[ \begin{array} []{cc}mx_{g}r^{2}-d_{w0}\text{sign}(u)-d_{w}u-d_{w}r\\ \\ -my_{g}r^{2}-d_{w0}\text{sign}(v)-d_{w}v-d_{w}r\end{array} \right]\] \[+R_{1}^{T}(k_{1}z_{1}+k_{2}z_{2}-\dot{p}_{d})-\left[ \begin{array} []{cc}\delta_{x}\\ 0\end{array} \right]^{2}+\left[ \begin{array}{cc}\text{Y}_{13}\\ \text{Y}_{23}+\delta_{x}\end{array} \right]N_{3}^{-1}(-mx_{g}ur+my_{g}v\pi\] \[-d_{w}u-d_{w}v-d_{w0}\text{sign}(r)-d_{w}r)\Big{)}, \tag{38}\]  
one obtains  
\[\dot{V}_{r}=-N_{3}^{-1}r\,d_{r0}\text{sign}\,r+\Xi. \tag{39}\]  
The following theorem can now be proposed. Theorem 2: _Consider the closed loop consisting of the system described by (12)-(15) together with the trajectory tracking controller (29).