# Cooperative Control of Interconnected Air Suspension Based on Model Predictive Control
## 3 Cooperative Control Design Based on MPC
### Rule of Best Suspension Force Distribution for IMA and OMA
If the pressure balance interconnection on two sides of the interconnection pipeline is closed when the air spring on the low-pressure side reaches the target air pressure, it follows that the low-pressure air spring must be deflated by the OMA to achieve the target air pressure. Moreover, the air spring on the high-pressure side may be charged and discharged by OMA. The energy consumption of adjusting the air spring pressure at the high-pressure and low-pressure sides to the target pressure are set as \(J_{\rm Hd}\) and \(J_{\rm Ld}\), respectively. This is due to the time required for IMA. At this point, the problem becomes comparing the size of \(J_{\rm H0d}\) and \(J_{\rm Hd}+J_{\rm Ld}\). In the above discussion, the initial state and the end state of the two interconnected air springs are fixed. In other words, in case 2, IMA composed of two air springs must discharge a certain amount of gas to ensure a terminal state. It has nothing to do with the process, therefore \(J_{\rm H0d}=J_{\rm Hd}+J_{\rm Ld}\). However, since the continued IMA consumes a certain amount of energy, as soon as the air spring on the low-pressure side reaches the target pressure in case 2, the IMA shall be immediately disconnected, as in case 3. In Figure 4, we see the selection of actions under different states for the IMA and OMA. \begin{table}
\begin{tabular}{c c} \hline \hline
**Number** & **Case** \\ \hline
1 & The target pressure cannot be achieved. \\
2 & The low-pressure air spring reaches the target pressure first, while the high-pressure air spring must continue discharging to reach the target pressure. \\
3 & The air spring on the high-pressure side reaches the target air pressure first, while the air spring on the low-pressure side needs to inflate further to reach the target air pressure. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Three categories of pressure in the air suspension. Figure 4: Action of solenoid valves in interconnected mode activation by judging different pressure condition in two-side of interconnected air suspension. In which, \(\Delta P\) (L, H) represents the pressure difference between the high- and low-pressure sides; L represents low-pressure side; H represents high-pressure; \(\uparrow\) represents increasing pressure; \(\downarrow\) represents decreasing pressure; \(\surd\) represents reaching targeted pressure. The discussion above clarifies the differing responsibilities of IMA and OMA, both of which are capable of modifying air spring pressure. Figure 4 also serves as a constraint for the calculation of energy costs in IMA, if interconnection can be turned on at this time. Afterward, the energy consumption cost can be computed normally; if the interconnection should be closed at this time, the energy consumption cost is set to positive infinity. Accordingly, if suspension control force \(\mathbf{u}\) is discretized into \(k\) parts, the energy consumption costs of the OMA and the IMA control are expressed as follows:  
\[j_{i}^{k}=G_{i}\left(E_{opt}^{k}\right) \tag{21}\]  
It is evident that the control unit that consumes the least energy is selected to generate this portion of suspension force, namely,  
\[i^{*k}=\min_{i}j_{i}^{k} \tag{22}\]  
The control unit corresponding to the best suspension of each part can be obtained from Equation (23) when \(k\) gradually increases from \(1\) to \(n\). Accordingly, the force generated for each control system is  
\[F_{si}=\sum_{k=1}^{n}F_{si}^{k} \tag{23}\]  
in which  
\[F_{si}^{k}=\left\{ \begin{array}{ll}F_{\text{opt}}^{k}&i=i^{*}\\ 0&i\
eq i^{*}\end{array} \right. \tag{24}\]  
The specific calculation process for an optimal distribution of suspension force increment with optimal energy consumption is shown in Algorithm 1. In other words, the nonlinear optimization with constraints is simplified to \(N\) times energy consumption cost calculation, comparison, and summation calculation through discretization, which not only reduces the calculation effort, but also ensures the lowest energy consumption costs under certain conditions.