## 2 Development of a Servo Valve Prediction Model
### Rationale for Expectancy Projections
#### 2.2.1 Theoretical Foundations of Exponential Smoothing Forecasting and Fusion Prediction
This is where the recursive magic happens, and \(f_{t-1}\) itself is defined as part of \(d_{t-2}\) and \(f_{t-2}\). In the experimental data sample of this article, the observed value \(f_{t}\) represents the predicted value of our servo valve performance parameter--pressure gain or leakage at time \(t\). \(d_{t}\) represents the experimental data value of the performance parameter at time \(t\). There is a trade-off between learning and memory and between reactivity and stability. If \(a\) is high, the model will place more emphasis on recent demand observations (i.e., the model will learn quickly) and will respond to changes in demand levels. However, it can also be sensitive to outliers and noise. On the other hand, if \(a\) is low, the model will not notice rapid changes in levels, but will also not overreact to noise and outliers. Once the historical period has been left, forecasts for future periods must be populated, and the final projections (those based on recent demand observations) are extrapolated into the future. If \(f\{t^{*}\}\) is defined as the last forecast that can be made on the basis of demand history, just \(f_{t\geq t^{*}}=f_{t^{*}}\), this simple exponential smoothing model is more flexible than the moving average model because it is more intelligent in its weighting of historical demand observations. For the exponential smoothing algorithm, its exponential smoothing statistic lies mainly in the \(\alpha\)-value. Moreover, for the value of \(\alpha\), we can turn to the planning solution of the Excel software. Excel is used to record the servo valve pressure gain and leakage performance parameter data obtained from the next chapter of the experiment and combined in the same data table, and then the predicted value of the data for exponential smoothing calculation, the predicted value by \(f_{t}=\alpha^{*}\,d_{t-1}+(1-\alpha)^{*}f_{t-1}\), is obtained, where \(d\) indicates the actual data value and \(f\) indicates the predicted value, and then \(d_{t}-f_{t}\) is used to get the error value. Then, the error is squared and the average of the whole set of working conditions data is sought to obtain the mean squared error MSE; a smaller MSE means that the value of \(\alpha\) is more appropriate at this time, and thus there is a higher accuracy of the exponential smoothing model. Finally, the data column of the Excel software is used to simulate the analysis option in the planning solution, the target solution goal is set to the minimum value of the MSE, the variable cell is set to the value of \(\alpha\), the requirement is set to make \(0<\alpha<1\), and then the solution is clicked on. After several calculations of pressure gain and leakage, we found that at \(\alpha=0.812\), the relative MSE of each data point was \(0.001070273\), so we took the weight parameter \(\alpha\) of \(0.812\) as the parameter requirement in the exponential smoothing model. Fusion forecasting refers to the technique of combining the results of multiple forecasting models to generate more accurate and stable forecasts. In practice, we often use several different forecasting models to make predictions, such as models based on different algorithms, models with other parameters, and so on. Each of these models has its own strengths and weaknesses, so combining their results can compensate for the shortcomings of individual models and improve prediction accuracy and robustness.