Improved adaptive Kalman filtering algorithm

Abstract

The invention provides an improved variable forgetting factor recursive least square and Kalman filtering coupled real-time correction method, and the defects of a traditional Kalman filtering algorithm are effectively overcome. By taking the real-time correction of the flood forecast of the maple dam reservoir as an example, the algorithm is applied, and compared with other methods, the calculation result shows that the algorithm has a better simulation effect.

Description

Technical Field

[0001] While the Kalman filter method possesses a complete and rigorous theoretical framework and has achieved excellent results in fields such as automatic control, one drawback is that it is based on the conditional expectations of all past observations and assigns uniform weights to these observations, meaning that new and old data are given the same confidence level. As time progresses and more data is collected, the amount of information obtained from new data relatively decreases, and the algorithm gradually loses its corrective ability, resulting in the so-called "data saturation" phenomenon. At this point, the parameter estimates may still deviate significantly from the true values ​​before they can be updated, or, for time-varying structures, the parameter estimates may fail to track changes in time-varying parameters. To overcome the "data saturation" phenomenon, a variable forgetting factor is introduced into the filtering process. By reducing the confidence level of older data, the algorithm is modified to form an adaptive filter. Background Technology

[0002] Improved recursive algorithm for least squares parameter estimation of variable forgetting factor The variable forgetting factor least squares recursive algorithm proposed by TRFortescue is introduced into real-time correction, and the algorithm is improved by adding confidence to the data at different time points using a discount factor. The improved variable forgetting factor least squares parameter estimation recursive algorithm is as follows: Correction: (1) Parameter estimation: (2) Gain factor: (3) Covariance matrix: (4) Variable forgetting factor: (5) In the above expression: for Real-time correction (adjustment) value at any given moment. and They are respectively time, The model calculation value at time 10:00. for The measured value at that moment; , and They are respectively time, The model prediction error at time; ,in and For time-varying coefficients. As can be seen from equation (3), when the gain factor formula... When changed to 1, the above-mentioned variable forgetting factor least squares improved recursive algorithm is transformed into the recursive algorithm proposed by TRFortescue. When the value is constant, the formula is transformed into a least-squares recursive algorithm for the constant forgetting factor. When the value is set to 1.0, it transforms into a sequential recursive least squares algorithm. Therefore, the improved variable forgetting factor recursive least squares algorithm is a higher-level synthesis of the various recursive least squares algorithms mentioned above, and has a more general form. The improved variable forgetting factor least squares method has a strong ability to track real-time systems. It can adaptively adjust its forgetting factor according to changes in measured values ​​to achieve the best tracking parameter effect and improve prediction accuracy. Summary of the Invention

[0003] Adaptive Feedback Kalman Filter Real-Time Correction Algorithm When applying the Kalman filter method for real-time model correction calculations, the forecast system model must first be expressed as the system's state equations and observation equations. This patent is based on an established second-order residual regression forecast model, and further derives a Kalman filter real-time correction calculation method. The structure of the second-order residual regression correction method with constant terms removed is as follows: (6) The second-order regression residual prediction with the constant term removed can be expressed as: (7) In the above expression: , They are respectively Real-time correction values ​​and model calculation values ​​at any given time; and for time, The model prediction error at time; and is a constant coefficient. The second-order regression residual prediction model in equation (7) can be rewritten in the form of the Kalman filter state equation and observation equation, resulting in: Equations of state: (8) It can be further simplified to: (9) The state variables, noise assignment matrix, and state transition matrix are respectively: , , As can be seen from equation (9), the state transition matrix The value is constant and does not change over time. It is based on the conditional expectation of all past observations and assigns uniform weights to these observations, that is, giving the same confidence level to new and old data. As time goes by, more and more data are collected, and the amount of information obtained from the new data will decrease relatively. The algorithm will gradually lose its ability to correct and the so-called "data saturation" phenomenon will occur. At this time, the parameter estimate may still be far from the true value and cannot be updated. Or, for time-varying structures, it will cause the parameter estimate to fail to track the changes of time-varying parameters. To address this problem, a forgetting factor is introduced into the filtering process to modify the algorithm by reducing the confidence level of old data. As mentioned above, the variable forgetting factor least squares method has a strong ability to track hydrological systems in real time. It can adaptively adjust its forgetting factor according to the changes of actual floods in each field to achieve the best tracking effect of parameters and improve the accuracy of flood forecasting. Therefore, consider using the time-varying parameters in equation (2). For the Kalman filter state transition matrix in equation (9) After modification, the Kalman filter state equation can be written as: (10) in, , and These are time-varying coefficients; This is the state transition matrix; This is the system noise vector. Thus, it is obtained through... Real-time tracking enables real-time correction of the Kalman filter algorithm, effectively overcoming the "data saturation" phenomenon of traditional Kalman filtering methods. Specific application results are shown in Table 1, and the algorithm flowchart is shown in the attached diagram of the manual. Figure 1 The flood forecasting model for Fengshuba Reservoir was analyzed and calculated using the above method, and compared with the improved variable forgetting factor recursive least squares algorithm and the traditional Kalman filter algorithm. The results are shown in Table 1 and the accompanying drawings in the manual. Figure 2 — Figure 4 As shown in Table 1. Summary Table of Flood Forecasting Correction Model and Algorithm Results for Fengshuba Reservoir. Model method categories Coefficient of certainty (%) Flood peak pass rate (%) Peak-to-peak time difference pass rate (%) Forecast model (before correction) 71.79 55.56 55.56 Improved variable forgetting factor recursive least squares 90.85 100 77.78 Kalman filter real-time correction algorithm 90.59 77.78 77.78 Adaptive Feedback Kalman Filter Real-Time Correction Algorithm 91.11 100 77.78 Figure 1 The flowchart shows the algorithm. After the initial parameter values ​​are selected, the model is calculated. The state transition matrix of the Kalman filter is corrected in real time using the variable forgetting factor least squares method. The algorithm is adaptively adjusted according to the changes in the measured values ​​at different times to achieve the best tracking effect of the parameters and improve the prediction accuracy. Figure 2 The results are calculated using the traditional Kalman filter method, with the vertical axis representing the flow rate (m³). 3The x-axis represents the time period (h), and the three data sequences are the measured value, the model calculated value, and the corrected value (using the Kalman filter method). Figure 3 The results of the improved variable forgetting factor recursive least squares method are shown on the ordinate, representing the flow rate (m). 3 The x-axis represents the time period (h), and the three data sequences are the measured value, the model calculated value, and the corrected value (using the improved variable forgetting factor recursive least squares method). Figure 4 The results of the improved adaptive Kalman filter method are shown on the ordinate, representing the flow rate (m³). 3 The x-axis represents the time period (h), and the three data sequences are the measured value, the model calculated value, and the corrected value (using an improved adaptive Kalman filter method).

Claims

1. An improved recursive least squares algorithm for variable forgetting factor was developed, resulting in an improved recursive algorithm for parameter estimation of variable forgetting factor least squares.

2. By introducing a variable forgetting factor into the filtering process, the algorithm is modified by reducing the reliability of old data, thus forming an adaptive filter and overcoming the "data saturation" phenomenon.

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