分析噪声数据

发布于 2024-08-15 20:22:51 字数 2977 浏览 6 评论 0原文

我最近发射了一枚带有气压高度计的火箭，精确度约为 10 英尺（通过飞行过程中获取的数据计算）。记录的数据以每个样本 0.05 秒的时间增量进行，高度与时间的关系图看起来与缩小整个飞行过程时的情况非常相似。

问题是，当我尝试根据数据计算其他值（例如速度或加速度）时，测量的准确性使计算值几乎毫无价值。我可以使用哪些技术来平滑数据，以便计算（或近似）合理的速度和加速度值？重要的是，重大事件要及时保持原状，最值得注意的是第一次进入的 0 和飞行期间的最高点 (2707)。

随后是高度数据，并以地面以上英尺为单位进行测量。第一次为 0.00，每个样本都在前一个样本之后 0.05 秒。飞行开始时的尖峰是由于升空过程中发生的技术问题造成的，去除尖峰是最佳选择。

我最初尝试使用线性插值，对附近的数据点进行平均，但需要多次迭代才能平滑数据以进行积分，并且曲线的平坦化消除了重要的远地点和地面事件。

非常感谢所有帮助。请注意，这不是完整的数据集，我正在寻找有关更好的数据分析方法的建议，而不是让某人回复转换后的数据集。最好在未来的火箭上使用一种算法，该算法可以在不知道完整飞行数据的情况下预测当前的高度/速度/加速度，但这不是必需的。

原文

I recently launched a rocket with a barometric altimeter that is accurate to roughly 10 ft (calculated via data acquired during flight). The recorded data is in time increments of 0.05 sec per sample and a graph of altitude vs. time looks pretty much like it should when zoomed out over the entire flight.

The problem is when I try to calculate other values such as velocity or acceleration from the data, the accuracy of the measurements makes the calculated values pretty much worthless. What techniques can I use to smooth out the data so that I can calculate (or approximate) reasonable values for the velocity and acceleration? It is important that major events remain in place in time, most notably the 0 for for the first entry and the highest point during flight (2707).

The altitude data follows and is measured in ft above ground level. The first time would be 0.00 and each sample is 0.05 seconds after the previous sample. The spike at the beginning of the flight is due to a technical problem that occurred during liftoff and removing the spike is optimal.

I originally tried using linear interpolation, averaging nearby data points, but it took many iterations to smooth the data enough for integration and the flattening of the curve removed the important apogee and ground level events.

All help is greatly appreciated. Please note this is not the complete data set and I am looking for suggestions on better ways to analyze the data, not for someone to reply with a transformed data set. It would be nice to use an algorithm on board future rockets which can predict current altitude/velocity/acceleration without knowing the full flight data, though that is not required.

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各自安好 2024-08-22 20:22:51

这是我的解决方案，使用卡尔曼滤波器。如果您想或多或少地平滑，您将需要调整参数（甚至+/-数量级）。

#!/usr/bin/env octave

% Kalman filter to smooth measures of altitude and estimate
% speed and acceleration. The continuous time model is more or less as follows:
% derivative of altitude := speed
% derivative of speed := acceleration
% acceleration is a Wiener process

%------------------------------------------------------------
% Discretization of the continuous-time linear system
% 
%   d  |x|   | 0 1 0 | |x|
%  --- |v| = | 0 0 1 | |v|   + "noise"
%   dt |a|   | 0 0 0 | |a|
%
%   y = [1 0 0] |x|     + "measurement noise"
%               |v|
%               |a|
%
st = 0.05;    % Sampling time
A = [1  st st^2/2;
     0  1  st    ;
     0  0  1];
C = [1 0 0];

%------------------------------------------------------------
% Fine-tune these parameters! (in particular qa and R)
% The acceleration follows a "random walk". The greater is the variance qa,
% the more "reactive" the system is expected to be, i.e.
% the more the acceleration is expected to vary
% The greater is R, the more noisy is your measurement instrument
% (less "accuracy" of the barometric altimeter);
% if you increase R, you will smooth the estimate more
qx = 1.0;                      % Variance of model noise for position
qv = 1.0;                      % Variance of model noise for speed
qa = 50.0;                     % Variance of model noise for acceleration
Q  = diag([qx, qv, qa]);
R  = 100.0;                    % Variance of measurement noise
                               % (10^2, if 10ft is the standard deviation)

load data.txt  % Put your measures in this file

est_position     = zeros(length(data), 1);
est_speed        = zeros(length(data), 1);
est_acceleration = zeros(length(data), 1);

%------------------------------------------------------------
% Kalman filter
xhat = [0;0;0];     % Initial estimate
P    = zeros(3,3);  % Initial error variance
for i=1:length(data),
   y = data(i);
   xpred = A*xhat;                                    % Prediction
   Ppred = A*P*A' + Q;                                % Prediction error variance
   Lambdainv = 1/(C*Ppred*C' + R);
   xhat  = xpred + Ppred*C'*Lambdainv*(y - C*xpred);  % Update estimation
   P = Ppred - Ppred*C'*Lambdainv*C*Ppred;            % Update estimation error variance
   est_position(i)     = xhat(1);
   est_speed(i)        = xhat(2);
   est_acceleration(i) = xhat(3);
end

%------------------------------------------------------------
% Plot
figure(1);
hold on;
plot(data, 'k');               % Black: real data
plot(est_position, 'b');       % Blue:  estimated position
plot(est_speed, 'g');          % Green: estimated speed
plot(est_acceleration, 'r');   % Red:   estimated acceleration
pause

Here is my solution, using a Kalman filter. You will need to tune the parameters (even +- orders of magnitude) if you want to smooth more or less.

#!/usr/bin/env octave

% Kalman filter to smooth measures of altitude and estimate
% speed and acceleration. The continuous time model is more or less as follows:
% derivative of altitude := speed
% derivative of speed := acceleration
% acceleration is a Wiener process

%------------------------------------------------------------
% Discretization of the continuous-time linear system
% 
%   d  |x|   | 0 1 0 | |x|
%  --- |v| = | 0 0 1 | |v|   + "noise"
%   dt |a|   | 0 0 0 | |a|
%
%   y = [1 0 0] |x|     + "measurement noise"
%               |v|
%               |a|
%
st = 0.05;    % Sampling time
A = [1  st st^2/2;
     0  1  st    ;
     0  0  1];
C = [1 0 0];

%------------------------------------------------------------
% Fine-tune these parameters! (in particular qa and R)
% The acceleration follows a "random walk". The greater is the variance qa,
% the more "reactive" the system is expected to be, i.e.
% the more the acceleration is expected to vary
% The greater is R, the more noisy is your measurement instrument
% (less "accuracy" of the barometric altimeter);
% if you increase R, you will smooth the estimate more
qx = 1.0;                      % Variance of model noise for position
qv = 1.0;                      % Variance of model noise for speed
qa = 50.0;                     % Variance of model noise for acceleration
Q  = diag([qx, qv, qa]);
R  = 100.0;                    % Variance of measurement noise
                               % (10^2, if 10ft is the standard deviation)

load data.txt  % Put your measures in this file

est_position     = zeros(length(data), 1);
est_speed        = zeros(length(data), 1);
est_acceleration = zeros(length(data), 1);

%------------------------------------------------------------
% Kalman filter
xhat = [0;0;0];     % Initial estimate
P    = zeros(3,3);  % Initial error variance
for i=1:length(data),
   y = data(i);
   xpred = A*xhat;                                    % Prediction
   Ppred = A*P*A' + Q;                                % Prediction error variance
   Lambdainv = 1/(C*Ppred*C' + R);
   xhat  = xpred + Ppred*C'*Lambdainv*(y - C*xpred);  % Update estimation
   P = Ppred - Ppred*C'*Lambdainv*C*Ppred;            % Update estimation error variance
   est_position(i)     = xhat(1);
   est_speed(i)        = xhat(2);
   est_acceleration(i) = xhat(3);
end

%------------------------------------------------------------
% Plot
figure(1);
hold on;
plot(data, 'k');               % Black: real data
plot(est_position, 'b');       % Blue:  estimated position
plot(est_speed, 'g');          % Green: estimated speed
plot(est_acceleration, 'r');   % Red:   estimated acceleration
pause

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