## 3 Controller Design
### Stability Analysis  
**Theorem 1**. : _Considering the control-oriented RLV model (1) with input constraints, if the initial conditions satisfies \(-\rho_{i0}<e_{1i}(0)<\rho_{i0}\), the nearly optimal control policy is chosen as (30) with the weight update laws (32) and (33), then the following results can be obtained:_  
* _the sliding variable_ \(\mathbf{s}\)_, the weight estimation errors_ \(\hat{\mathbf{W}}_{a}\) _and_ \(\hat{\mathbf{W}}_{c}\) _are uniformly ultimately bounded (_IUUB_);_
* _the attitude tracking errors_ \(e_{1i}\) _uniformly obey the fixed-time performance envelops in (_8_)._  
**Proof of Theorem 1. ** Consider the Lyapunov function candidate as follows:  
\[V=V(\mathbf{s})+\frac{1}{2}\hat{\mathbf{W}}_{c}^{\top}A_{1}^{-1}\hat{\mathbf{W}}_{c}+\frac {p_{c}}{2p_{a}}\hat{\mathbf{W}}_{a}^{\top}A_{2}^{-1}\hat{\mathbf{W}}_{a} \tag{34}\]  
Figure 1: The block diagram of the proposed control scheme. where \(\tilde{\mathbf{W}}_{c}=\mathbf{W}^{*}-\hat{\mathbf{W}}_{c}\) and \(\tilde{\mathbf{W}}_{a}=\mathbf{W}^{*}-\hat{\mathbf{W}}_{a}\). Taking the first time derivative of \(V\) yields  
\[\dot{V} =\dot{V}(\mathbf{s})+\dot{V}_{c}+\dot{V}_{a}\] (35) with \[\dot{V}_{c} =\tilde{\mathbf{W}}_{c}^{\top}\mathbf{A}_{1}^{-1}\hat{\mathbf{W}}_{c},\,\dot{V }_{a}=\tfrac{p_{e}}{p_{e}}\hat{\mathbf{W}}_{a}^{\top}\mathbf{A}_{2}^{-1}\hat{\mathbf{W}}_{a}\], and \[\dot{V}(\mathbf{s}) =\
abla V_{s}^{\top}\big{(}\mathbf{\Gamma}+\mathbf{D}_{1}+\mathbf{B}_{1}\hat{ \mathbf{M}}_{a}^{*}\big{)}. \tag{36}\]  
By invoking the HJB equation (29), \(\
abla V_{s}^{\top}\Xi\) can be rewritten as:  
\[\
abla V_{s}^{\top}\Xi=-V_{s}-\lambda^{2}\bar{\mathbf{\sigma}}\text{ln}\Big{[}\bm {1}_{c}-\text{tanh}^{2}\Big{(}\hat{\mathbf{M}}_{o}^{*}\Big{)}\Big{]}+\
abla V_{s} ^{\top}\Big{\{}\mathbb{C}\Big{[}\lambda_{1}\text{sig}^{1/2}(s_{i})\Big{]}-\bm {M}_{d}\Big{\}}-\varepsilon_{e} \tag{37}\]  
where \(\varepsilon_{e}=B_{e}+\
abla V_{s}^{\top}\mathbf{D}_{1}\). Substituting (37) into (36) yields  
\[\dot{V}(\mathbf{s})=-V_{s}-\lambda^{2}\bar{\mathbf{\sigma}}\text{ln}\Big{[}\mathbf{1}_{c} -\text{tanh}^{2}\Big{(}\hat{\mathbf{M}}_{o}^{*}\Big{)}\Big{]}-\lambda\
abla V_{s} ^{*\top}\mathbf{B}_{1}\tanh\Big{(}\hat{\mathbf{M}}_{a}^{*}\Big{)}-B_{e}. \tag{38}\]  
Furthermore, according to (17), it can be deduced that  
\[\lambda^{2}\bar{\mathbf{\sigma}}\text{ln}\Big{[}\mathbf{1}_{c}-\text{tanh}^{2}\Big{(} \hat{\mathbf{M}}_{o}^{*}\Big{)}\Big{]}=J(\mathbf{M}_{o}^{*})+\
abla V_{s}^{*\top}\mathbf{ B}_{1}\underline{\mathbf{M}}_{o}^{*} \tag{39}\]  
with \(\underline{\mathbf{M}}_{o}^{*}=-\lambda\tanh\Big{(}\hat{\mathbf{M}}_{o}^{*}\Big{)}\). Equation (38) can be rewritten as:  
\[\dot{V}(\mathbf{s}) =-V_{s}-J(\underline{\mathbf{M}}_{o}^{*})-\
abla V_{s}^{*\top}\mathbf{B} _{1}\underline{\mathbf{M}}_{o}^{*}-B_{e}-\lambda\Big{(}\mathbf{W}^{*\top}\
abla\mathbf{ \sigma}_{s}+\
abla\mathbf{\varepsilon}_{s}^{\top}\Big{)}\mathbf{B}_{1}\tanh\Big{(}\hat {\mathbf{M}}_{a}^{*}\Big{)} \tag{40}\] \[=-V_{s}-J(\underline{\mathbf{M}}_{o}^{*})+\varepsilon_{v}-\lambda \Big{(}\hat{\mathbf{W}}_{a}^{\top}+\tilde{\mathbf{W}}_{a}^{\top}\Big{)}\
abla\mathbf{ \sigma}_{s}\mathbf{B}_{1}\tanh\Big{(}\hat{\mathbf{M}}_{a}^{*}\Big{)}\]  
where \(\varepsilon_{v}=-B_{e}+\
abla V_{s}^{*\top}\mathbf{B}_{1}\underline{\mathbf{M}}_{o}^{ *}-\
abla\mathbf{\varepsilon}_{s}^{\top}\mathbf{B}_{1}\hat{\mathbf{M}}_{a}^{*}\). Noting that \(\tanh(\cdot)\) is an odd function, it can be concluded that \(\lambda\hat{\mathbf{W}}_{a}^{\top}\
abla\sigma_{s}\mathbf{B}_{1}\tanh\Big{(}\hat\mathbf{{M}}_{a}^{*}\Big{)}>0\). Moreover, it is indicated that \(J(\underline{\mathbf{M}}_{o}^{*})>0\) from the definition in (17). Based on the above discussions, (40) can be simplified as:  
Figure 1:0  
Subsequently, incorporating (32) with \(\dot{V}_{c}\) yields  
Figure 1:1  
Recalling (29) and the definition of \(\mathbf{\theta}\), (42) can be rearranged as:  
Figure 1:2 Figure 1:3 Figure 1:4 Figure 1:5 Figure 1:6 Figure 1:7With the aid of Lemma 1 and Lemma 3, (43) can be further simplified as:  
\[