**4. Multi-Objective Optimization Method Description**

The design parameters are searched by the MOO method, instead of manually adjusting, to improve design efficiency. Various objectives usually conflict with each other, which means improving one objective usually comes at the expense of reducing the character of another. The MOO method can search for the solution set that can not improve any one objectives without reducing other objectives, which usually is called the Pareto front.

Various MOO methods have been proposed, including the multi-objective particle swarm optimization (MOPSO) algorithm Coello et al. [19], the fast non-dominated sorting genetic algorithm (NSGA-2) method Deb et al. [20], DEB [21], andthe Pareto-frontier differential evolution (PDE) approach Abbass et al. [22], among others. In the present study, the (MOPSO) method is employed as it has the advantage of very good global searching capability.

The pseudo-code of the MOPSO method in the present study can be described as:


$$\begin{split} w\_i(t+1) &= wv\_i(t) + c\_1 r\_1(p\_i^{\text{BEST}}(t) - x\_i(t)) \\ &+ c\_2 r\_2(R\_h(t) - x\_i(t)), \end{split} \tag{3}$$

where *pBEST <sup>i</sup>* is the best position in the whole search history; *Rh* is the selected leader from the repository; *w* represent the inertia coefficient of velocity; *c*<sup>1</sup> and *c*<sup>2</sup> are local and social coefficients, respectively; and *r*<sup>1</sup> and *r*<sup>2</sup> are two random values in the range [0, 1]. Then, the position is updated at each iteration based on the velocity.

$$S\_i^{\text{POP}}(t) = S\_i^{\text{POP}}(t) + v\_i(t). \tag{4}$$

7. Determine whether the maximum count of iterations has been reached or not. If the maximum number of iterations has not been reached, go back to the step 2. Otherwise, stop the optimization process and return the Pareto front stored in the repository.

## **5. Optimization Results and Discussion**

A case of EHA system optimization is presented, in order to validate the feasibility of the proposed method. The design requirements of a typical control surface are listed in the Table 1. The maximum voltage is *U*max = 270 VDC and maximum current is set to 50 A. The maximum pressure of EHA is set to 35 MPa. In the optimization process, the optimization method generates the radius of the lever (*R*) and the displacement of the pump (*Dm*). Then the parameters of the EHA (i.e., the area of the piston, the stroke of the cylinder, the rotational inertia of the motor and pump, and the torque constant of the motor) can be calculated. These parameters are transferred to the AMESim model and the simulation is run. Then, the simulation results are obtained and the dynamic objectives can be analyzed automatically by the program.

**Table 1.** Control surface requirements.


Consider the application condition, where the range of the two optimization parameters are set as shown in Table 2. These two parameters decide the major characteristics of the EHA. A longer lever means the force requirement of the EHA is small but the stroke should be longer, in a slender form. On the contrary, a shorter lever means a large force requirement and short stroke. A larger displacement of the pump requires low rotary speeds for a desired flow rate, which will make the motor and pump more heavy, but can improve the efficiency and may make the EHA response quicker. These conflicting objectives make it hard to find a solution to trade off all performances. It usually required of MOO design methods to find the Pareto Front to help the engineer in the design process.

**Table 2.** Range of optimization parameters.


The parameters of MOPSO, in the present study, include the number of particle population *Np* = 100, maximum repository capacity of the Pareto front is 100, *w* = 0.9, *c*<sup>1</sup> = *c*<sup>2</sup> = 1.1, and the maximum iteration count is 10. The Pareto fronts of each of the two objectives are shown in Figure 7. These figures show that the weight and energy consumption trade off against each other. The results also indicate that the rise time and weight also trade off against each other. The rise time and energy consumption are positively correlated. The relationships between dynamic stiffness and the other performances are more complicated, as they are not monotonous functions. These relationships between objectives are useful for designers to balance different performances. The results also indicate that the proposed method can search the Pareto front solutions intelligently, which can save huge effort in adjustment by the engineer.

The relationships between the parameters and the objectives are shown in Figure 8. The weights of the solutions in the Pareto front, with design parameters, are shown in the upper right of Figure 8. The lightest solution is with the lever length about 125 mm, and a displacement of about 0.6 mL/rev. Compared to the energy consumption and rise time, the lightest solution had the biggest energy consumption and slowest response. The most efficient solution is locating at *Dm* = 4 mL/rev and lever length of 103 mm. However, this solution also had a high weight and low dynamic stiffness. The optimization results indicate that using a smaller displacement pump is beneficial for reducing weight and increasing dynamic stiffness. Using a bigger displacement pump will improve the efficiency and make the EHA response quicker, but will make the EHA more heavy. The designer can get all design parameters and performance indices, as in Figure 8, of the solutions in the Pareto front, then they can choose one solution which is most satisfactory for the application. Thus, the proposed method can offer significant support for the engineer.

The time domain performances of four typical solutions in the Pareto front (the lightest solution, the quickest solution, the most efficient solution, and the best stiffness solution) are simulated. The simulation results are illustrated in the Figure 9. All of these four solutions obtain an acceptable control performance, only having a little difference in dynamic response; but the weight and energy consumption had a large disparity. This means the weight and energy consumption can be reduced, if appropriate parameters are given. The Pareto front obtained by the proposed method is very helpful for the engineer to determine the design parameters.

**Figure 7.** Scatter graph of the optimization results, including the Pareto front of each two objectives and the distribution of the each objective.

**Figure 8.** Objective locations in the design space of the Pareto front.

**Figure 9.** Dynamic simulation results of four types of typical design parameters.
