## 4 Stability Analysis  
In this section, the convergence of the error signals and the stability of the resulting control system are analyzed using the Lyapunov stability theory. **Theorem 1**. : _Consider the proposed RESOs (8) and (9) for the single-link FJM described by (2) with matched and mismatched disturbances. If Assumption 2 is satisfied, then the error norm of the RESOs is bounded by_  
\[\|e_{o}\|\leq\frac{\max\big{(}\bar{d}_{i}\big{)}}{\min(|\beta_{i}|)}(i=1,2) \tag{39}\]  
Proof. : Consider the Lyapunov function candidate \(V_{o}\)  
\[V_{o}=\frac{1}{2}\epsilon_{o}^{\top}\mathbf{P}e_{o} \tag{40}\]  
where \(\mathbf{P}\) is a positive definite matrix. Since \(\mathbf{A}_{o}\) is a Hurwitz matrix, there exists a positive definite matrix \(\mathbf{Q}\) such that  
\[\mathbf{A}_{o}^{\top}\mathbf{P}+\mathbf{P}\mathbf{A}_{o}=-\mathbf{Q} \tag{41}\]  
Taking the time derivative of \(V_{o}\) along (14) and (41) yields  
\[ \begin{split}\dot{V}_{o}&=\frac{1}{2}\epsilon_{o}^ {\top}\mathbf{P}e_{o}+\frac{1}{2}\epsilon_{o}^{\top}\mathbf{P}\dot{e}_{o}\\ &=\frac{1}{2}\epsilon_{o}^{\top}\Big{(}A_{o}^{\top}\mathbf{P}+\mathbf{P} \mathbf{A}_{o}\Big{)}\mathbf{e}_{o}+\mathbf{e}_{o}^{\top}\mathbf{P}\dot{d}\\ &=-\frac{1}{2}\epsilon_{o}^{\top}\mathbf{Q}\mathbf{e}_{o}+\mathbf{e}_{o}^{ \top}\mathbf{P}\dot{d}\\ &\leq-\frac{1}{2}\|e_{o}\|^{2}\mathbf{Q}+\|e_{o}\|\|\mathbf{P}\|\max\big{(} \bar{d}_{i}\big{)}\\ &\leq-\|\mathbf{e}_{o}\|\big{[}\|e_{o}\|\|\mathbf{P}\|\min(|\beta_{i}|)- \|\mathbf{P}\|\max\big{(}\bar{d}_{i}\big{)}\big{]}\end{split}  \tag{42}\]  
where \(\min(|\beta_{i}|)(i=1,2)\) represents the smallest eigenvalue of \(\mathbf{A}_{o}\). Therefore, within finite time, the norm of the estimation errors is bounded by  
\[\|e_{o}\|\leq\frac{\max\big{(}\bar{d}_{i}\big{)}}{\min(|\beta_{i}|)}(i=1,2) \tag{43}\]  
This completes the proof of Theorem 1. **Theorem 2**. : _Consider the single-link FJM described by (2) with matched and mismatched disturbances. Suppose that Assumptions 1 and 2 are satisfied. The RESOs (8) and (9), the error compensation system (16), and the SCFB controller (38) guarantee that all the signals in the control system are uniformly ultimately bounded, and the tracking errors converge to a small neighborhood around zero._  
Figure 3: Block diagram of the composite control system for the single–link FJM control system. Proof. : Based on the design procedure in Section 3.2, the final Lyapunov function candidate for the feedback control is  
\[V_{f}=\frac{1}{2}v_{1}^{2}+\frac{I}{2}v_{2}^{2}+\frac{1}{2}v_{3}^{2}+\frac{I}{2}v _{4}^{2} \tag{44}\]  
According to (37), the time derivative of \(V_{f}\) can be described as  
\[\dot{V}_{f}=-k_{1}v_{1}^{2}-k_{2}v_{2}^{2}-k_{3}v_{3}^{2}-k_{4}v_{4}^{2}-lv_{2} \tilde{d}_{1}-Jv_{4}\tilde{d}_{2} \tag{45}\]  
Using the [PERSON]'s inequality, we have  
\[ \begin{split}\left|{{Iv_{2}\tilde{d}_{1}}}\right|& \leq\frac{I}{2}v_{2}^{2}+\frac{I}{2}\tilde{d}_{1}^{2}\\ \left|{{Jv_{4}\tilde{d}_{2}}}\right|&\leq\frac{I}{2 }v_{4}^{2}+\frac{I}{2}\tilde{d}_{2}^{2}\end{split}  \tag{46}\]  
Substituting (46) into (45) yields  
\[\dot{V}_{f}\leq-k_{1}v_{1}^{2}-\left(k_{2}-\frac{I}{2}\right)v_{2}^{2}-k_{3}v _{3}^{2}-\left(k_{4}-\frac{I}{2}\right)v_{4}^{2}-\frac{I}{2}\tilde{d}_{1}^{2} -\frac{I}{2}\tilde{d}_{2}^{2} \tag{47}\]  
Rewriting inequality (47) in a compact form, we have  
\[\dot{V}_{f}\leq-\alpha_{0}V_{f}+D \tag{48}\]  
where  
\[ \begin{split}\alpha_{0}&=\min\left(2k_{1},2k_{2}- I,2k_{3},2k_{4}-J\right)\\ D&=-\frac{I}{2}\tilde{d}_{1}^{2}-\frac{I}{2}\tilde {d}_{2}^{2}\end{split}  \tag{49}\]  
Selecting the design parameters \(k_{1}>0,k_{2}>I/2,k_{3}>0,k_{4}>J/2\) to ensure \(\alpha_{0}>0\). Then, according to Lemma 2, the solution of (48) is  
\[V_{f}(t)\leq\frac{D}{\alpha_{0}}+\left[V_{f}(0)-\frac{D}{\alpha_{0}}\right]e ^{-\alpha_{0}t} \tag{50}\]  
which means that \(V_{f}(t)\) converges exponentially to the upper bound of \(\frac{D}{\alpha_{0}}\), i.e., as \(t\rightarrow\infty\), \(V_{f}(t)\leq\frac{D}{\alpha_{0}}\). As a result, the compensated errors \(v_{i}(i=1,2,3,4)\) are bounded.