Interval Uncertainty Optimization Method for Electromagnetic Orbital Launcher
Abstract
:1. Introduction
2. Description of the Optimization Object
- The increasing phase lasts from 0 to 0.5 ms, during which the current rises from 0 to 0.32 MA;
- The constant phase extends from 0.5 ms to 2.0 ms, with the current maintaining a steady value of 0.32 MA;
- The decreasing phase occurs from 2.0 ms to 2.5 ms, where the current gradually decays from 0.32 MA to 0.1 MA.
3. Uncertainty Analysis Based on Interval Theory
3.1. Description of the Interval Optimization Model
3.2. Interval Order Relationship Conversion Model
- (1)
- Interval order relationship :expresses a preference for the upper and lower boundaries of the interval.
- (2)
- Interval order relationship :expresses a preference for the midpoint and radius of the interval.
- (3)
- Interval order relationship :expresses a preference for the lower bound and midpoint of the interval.
- (4)
- Interval order relationship :expresses a preference for the lower bound of the interval.
- (5)
- Interval order relationship :expresses a preference for the upper bound of the interval.
3.3. Transformation of Uncertainty Optimization Problems
4. Process of Interval Uncertainty Optimization Method
4.1. Interval Uncertainty Optimization System
4.2. Algorithm Implementation Process
- (1)
- Sampling in the mixed space (design space and uncertainty domain) to obtain multiple sets of samples consisting of design variables X and uncertainty variables U.
- (2)
- For any sample , the uncertain objective function value is calculated by the finite element simulation model, which constitutes the sample set of the inner layer proxy model.
- (3)
- The design variables and uncertainty variables are used as inputs, and the uncertain objective function values are used as outputs to build an inner layer high-precision proxy model.
- (4)
- For each sample X, two inner-level optimizations of andare performed to find the lower bound and upper bound of the objective function, respectively. The optimization solution process is shown in Figure 5.
- (5)
- An outer proxy model is established, with the design vector X as input and the lower bound and the upper bound of the uncertain objective function at X as output, to realize the nonlinear mapping between the input and output.
- (6)
- The trained outer proxy model is combined with the optimization algorithm to solve the transformed Equation (18) problem, as shown in Figure 6. For the design vector individuals generated by the PSO optimization algorithm, the interval of the objective function is obtained directly by the outer proxy model, and then the optimal solution or optimal solution set is selected by calculating the fitness value based on the multi-objective evaluation function.
5. Multi-Objective Optimization Algorithm Example
6. Conclusions
- 1.
- The uncertainty of variables is described using interval numbers, and the conversion of uncertainty optimization problems into deterministic optimization problems is achieved based on interval order relations.
- 2.
- The DBN-DNN algorithm is used to build a high-precision proxy model to predict the performance of electromagnetic orbital launchers. It is also combined with the particle swarm optimization algorithm to solve the transformed deterministic optimization problem.
- 3.
- The results of the multi-objective optimization algorithm for the electromagnetic orbital launcher show that the optimized solution set obtained by the method contains interval rather than deterministic values, it is able to handle interval optimization problems with uncertain parameters, and the performance is improved compared with that before optimization.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Variable | /mm | /mm | /mm | /mm | /mm | A/m) |
---|---|---|---|---|---|---|
Initial | 7 | 10 | 6 | 7.5 | 2.93 | 1.46 |
Structure Parameter/mm | Range of Values | Step Length |
---|---|---|
Armature tail width | 7.0∼10.0 | 0.5 |
Armature length | 9.0∼12.0 | 1.0 |
Armature tail length | 5.0∼8.0 | 1.0 |
Armature tail height | 6.0∼8.0 | 0.5 |
Objective Function | MAPE/(%) | MSE/ | |
---|---|---|---|
lower bound | 1.04 | 2.11 | 0.99 |
upper bound | 1.12 | 2.28 | 0.99 |
lower bound | 1.36 | 3.21 | 0.99 |
upper bound | 1.27 | 3.04 | 0.99 |
Optimal Compromise Solution | Design Vector X/mm | ||||
---|---|---|---|---|---|
1 | (9.44581, 9.75918, 5.98861, 7.92596) | −1.54101 | 0.66790 | [3.08201, 3.19037] | [1.30423, 1.33580] |
2 | (9.55383, 9.60918, 5.86247, 7.90717) | −1.55151 | 0.66781 | [3.10302, 3.19253] | [1.30599, 1.33561] |
3 | (9.38883, 11.09869, 7.61396, 7.49150) | −1.60515 | 0.66531 | [3.21029, 3.31429] | [1.31387, 1.33062] |
4 | (9.53728, 11.09491, 7.49053, 7.12074) | −1.65271 | 0.67824 | [3.30542, 3.42124] | [1.33519, 1.35648] |
5 | (9.89567, 0.86087, 7.54290, 6.47364) | −1.71869 | 0.70619 | [3.43738, 3.49947] | [1.39587, 1.41237] |
Multi-Objective Evaluation Function | /mm | /mm | /mm | /mm | Predicted Value | Actual Value | Error/% |
---|---|---|---|---|---|---|---|
9.89567 | 0.86087 | 7.54290 | 6.47364 | 1.71869 | −1.69429 | 1.44 | |
9.89567 | 0.86087 | 7.54290 | 6.47364 | 0.70618 | 0.71744 | 1.57 |
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Jin, L.; Liu, L.; Song, J.; Yan, Y.; Zhang, X. Interval Uncertainty Optimization Method for Electromagnetic Orbital Launcher. Appl. Sci. 2023, 13, 8806. https://doi.org/10.3390/app13158806
Jin L, Liu L, Song J, Yan Y, Zhang X. Interval Uncertainty Optimization Method for Electromagnetic Orbital Launcher. Applied Sciences. 2023; 13(15):8806. https://doi.org/10.3390/app13158806
Chicago/Turabian StyleJin, Liang, Lu Liu, Juheng Song, Yingang Yan, and Xinchen Zhang. 2023. "Interval Uncertainty Optimization Method for Electromagnetic Orbital Launcher" Applied Sciences 13, no. 15: 8806. https://doi.org/10.3390/app13158806
APA StyleJin, L., Liu, L., Song, J., Yan, Y., & Zhang, X. (2023). Interval Uncertainty Optimization Method for Electromagnetic Orbital Launcher. Applied Sciences, 13(15), 8806. https://doi.org/10.3390/app13158806