## 4 Multi-Objective Optimization and Experimental Verification  
The multi-objective optimization problem involves determining a set of optimization variables in the applicable area to synchronously minimize a set of contradictory objective functions. In this study, the objective of prolonging the life of the unloader and valve was achieved by reducing the ejection and withdrawal speeds of the unloader. However, from the dynamic equation of the unloader (1), we observed that the power and resistance sources of withdrawal and ejection were opposite, and the objective functions were contradictory. Moreover, owing to the coupling characteristics of the system, the decrease in the withdrawal speed of the unloader and valve plate caused deviations in the flow control and gas power, and there was also a contradiction between the front and back of the setup. Therefore, the problem studied in this paper is a typical multi-objective optimization problem, which is mathematically described as:  
\[\begin{array}{l}\min Y(a)=(Error\_V(a),Error\_W(a),\
u_{mn}^{\sigma}(a), \
u_{mn}^{\sigma j}(a))^{T}\\ s.t.\\ g_{1}(a)=K_{mu}\times(x_{mu\_0}+x_{mu})-(p_{in}\times A_{hp}+G_{mu}-f)<0\\ g_{2}(a)=(p_{bf}\times A_{hp}+G_{mu}-f)-K_{mu}\times(x_{mu\_0}+x_{mu})<0\\ a=\Big{\{}a\Big{|}a=(K_{mu},p_{in},p_{bf})T,K_{mu}\in[46,000,80,000],p_{in}\in [60,80],p_{bf}\in[0,12]\Big{\}}\end{array} \tag{18}\]  
Figure 10: Relationship between withdrawal speed and control deviation under different loads: (a) 80% load; (b) 60% load; (c) 40% load.  
where:  
\(Y(a)\) is expressed as objective functions; \(Error\_V(a)\), \(Error\_W(a)\), \(\
u_{m}^{ce}(a)\) and \(\
u_{m}^{ej}(a)\) are respectively expressed as flow deviation, gas work deviation, unloader withdrawal speed, unloader ejection speed;  
\(g_{1}(a)\) and \(g_{2}(a)\) are inequality constrained functions; \(g_{1}(a)\) is the constraint condition for the unloader to maintain the lower limit; and \(g_{2}(a)\) is the constraint condition for the unloader to maintain the upper limit.  
\(\alpha\) is the optimized parameter vector; \(K_{mn}\) is the stiffness of the reset spring at \(46,000\) N/m \(\leq\) K\({}_{mn}\)\(\leq\) 80,000 N/m; \(p_{in}\) is the intake pressure, \(60\) bar \(\leq\)\(p_{in}\)\(\leq\) 80 bar; and \(p_{bf}\) is the return pressure, \(0\) bar \(\leq\)\(p_{bf}\)\(\leq\) 12 bar.  
For multi-objective optimization, the traditional method transforms the multi-objective problem into a single-objective one by assigning weights to the objectives. However, the second-generation non-dominant sequencing genetic algorithm, NSGA-II, is one of the most effective methods for applying the Pareto optimum concept to multi-objective optimization [44]. The specific procedure of this method can be found in the literature [45]. Based on NSGA-II, this paper proposes a dual optimization method for the optimization of key parameters; it establishes a flow chart of optimization (Figure 11) and combines an ISIGHT-embedded NSGA-II algorithm with AMESim software to optimize the process. First, DOE experiments of the flow regulation system under a full load were performed. Second, an approximate model between the key parameters and objective function was established, which replaced the calculation of the collaborative simulation model. Third, according to the principle of maximum descent ratio and minimum regulation deviation, the optimal design parameters under different loads were obtained through the first and second optimizations. Finally, the optimum design parameters satisfying the full-load optimization were determined according to the simulation experiments, and the actual requirements were determined.  
Figure 11: Process of key parameter optimization.