# Multi-Objective Motion Control Optimization for the Bridge Crane System
## 3 Control Design
### Stability Analysis  
**Theorem.**_Independently from the stochastic initial payload swing, the proposed algorithm given by Equation (29) can control the crane to the desired position and suppress the payload swing, i.e.,:_  
\[\underset{l\rightarrow\infty}{lim}\Big{[}~{}x_{T}~{}~{}~{}\dot{x}_{T}~{}~{}~{ }\ddot{x}_{T}~{}~{}~{}\theta~{}~{}~{}\dot{\theta}~{}~{}~{}\ddot{\theta}~{}~{} \Big{]}=\Big{[}~{}~{}x_{d}~{}~{}~{}0~{}~{}~{}0~{}~{}~{}0~{}~{}~{}0~{}~{}~{} \Big{]} \tag{33}\]  
**Proof.** In order to meet the demand of the theorem, a non-negative function is selected as:  
\[V(t)=\frac{1}{2}\dot{L}\dot{\theta}^{2}+\text{g}(1-\cos\theta)\geq 0 \tag{34}\]  
Differentiating Equation (34) and combining Equation (2) with Equation (29), we can get:  
\[\begin{array}{ll}\dot{V}(t)&=\dot{\theta}(L\ddot{\theta}+g\sin\theta)\\ &=-\dot{\theta}\cos\theta(K_{I}\int_{0}^{t}e_{T}dt+K_{D}\dot{e}_{T}-k_{1}x_{T} -k_{2}\dot{x}_{T}-k_{3}\theta-k_{4}\dot{\theta})\end{array} \tag{35}\]  
According to the principle of the arithmetic mean-geometric mean (AM-GM) inequality, Equation (35) can be rewritten as:  
\[\begin{array}{ll}\dot{V}(t)&\leq\frac{1}{2}\dot{\theta}^{2}\cos^{2}\theta+ \frac{1}{2}\Big{(}K_{I}\int_{0}^{t}e_{T}dt+K_{D}\dot{e}_{T}-k_{1}x_{T}-k_{2} \dot{x}_{T}-k_{3}\theta-k_{4}\dot{\theta}\Big{)}^{2}\\ &\leq\frac{1}{2}\dot{\theta}^{2}\cos^{2}\theta+\frac{5}{2}\Big{(}K_{I}^{2} \Big{(}\int_{0}^{t}e_{T}dt\Big{)}^{2}+(K_{D}+k_{2})^{2}\dot{x}_{T}^{2}+k_{1}^{ 2}x_{T}^{2}+k_{3}{}^{2}\theta^{2}+k_{4}{}^{2}\dot{\theta}^{2}\Big{)}\end{array} \tag{36}\]  
Figure 4: Flowchart of multi-objective genetic algorithm.  
By integrating Equation (34), we can further get:  
\[\begin{array}{ll}V(t)&\leq V(0)+\frac{1}{2}\int_{0}^{t}\dot{\theta}^{2}\cos^{2 }\theta dt+\frac{5}{2}K_{1}^{2}\int_{0}^{t}\left(\int_{0}^{t}e_{T}dt\right)^{2} dt\\ &+\frac{5}{2}(K_{D}+k_{2})^{2}\int_{0}^{t}\dot{x}_{T}^{2}dt+\frac{5}{2}k_{1}^{2} \int_{0}^{t}x_{T}^{2}dt+\frac{5}{2}k_{3}^{2}\int_{0}^{t}\dot{\theta}^{2}dt+ \frac{5}{2}k_{4}{}^{2}\int_{0}^{t}\dot{\theta}^{2}dt\end{array} \tag{37}\]  
In Equation (37), we can find that all of the integration items are bounded. Based on Equations (4) and (34), it can prove that:  
\[\frac{1}{2}\int_{0}^{t}\dot{\theta}^{2}\cos^{2}\theta dt\in L_{\infty} \tag{38}\]  
Therefore, Equation (37) can be utilized to show that \(V(t)\in L_{\infty}\), Based on this fact, the following conclusion can be drawn from Equation (2). It can be proven that:  
\[\ddot{\theta}(t),\ddot{x}_{T}(t)\in L_{\infty} \tag{39}\]  
Then, Barbalat lemma [38] can then be directly utilized to show that:  
\[\underset{t\rightarrow\infty}{lim}\ddot{\theta}=0,\underset{t\rightarrow\infty }{lim}\ddot{x}_{T}=0 \tag{40}\]  
And Equation (37) can be rewritten as:  
\[\begin{array}{ll}-\frac{5}{2}\bigg{[}K_{1}^{2}\int_{0}^{t}\left(\int_{0}^{t }e_{T}dt\right)^{2}dt+(K_{D}+k_{2})^{2}\int_{0}^{t}\dot{x}_{T}^{2}dt+k_{1}^{2 }\int_{0}^{t}x_{T}^{2}dt+k_{3}{}^{2}\int_{0}^{t}\theta^{2}dt+k_{4}{}^{2}\int_ {0}^{t}\dot{\theta}^{2}dt\bigg{]}\\ \leq V(0)-V(t)+\frac{1}{2}\int_{0}^{t}\dot{\theta}^{2}\cos^{2}\theta dt\end{array} \tag{41}\]  
Based on \(V(t)\in L_{\infty}\) and Equation (38), Equation (41) can then be employed to conclude that:  
\[\theta(t),\dot{\theta}(t),x_{T}(t),\dot{x}_{T}(t)\in L_{\infty} \tag{42}\]  
According to Equations (37) and (42), it is easy to show that:  
\[\underset{t\rightarrow\infty}{lim}\dot{\theta}=0,\underset{t\rightarrow\infty }{lim}\dot{x}_{T}=0 \tag{43}\]  
Based on Equations (2), (40), and (43), we can go through a similar analysis to show that:  
\[\underset{t\rightarrow\infty}{lim}\sin\theta=0 \tag{44}\]  
Thus, Assumption 3 and Equation (4) can be employed to conclude that:  
\[\underset{t\rightarrow\infty}{lim}\theta=0 \tag{45}\]  
From Equations (29), (41) and (45), it is clear that:  
\[\underset{t\rightarrow\infty}{lim}x_{T}=x_{d} \tag{46}\]  
Now, the designed controller is proved to be asymptotically stable. Next, numerical simulation and experimental validation are performed in order to validate the proposed algorithm.