Interpolating numerical function of many variables

Hi everyone,

I'm attempting to develop a lookup function/table based on a set of data.

This data describes the measured airflow through a fan as a function of the fan's PWM command and the position of various ducts in the system (airflow = f(PWM,Vent 1 position, Vent 2 position, Vent 3 position, Vent 4 position). This data only exists for configurations where each duct is either open or closed at 100%.

There are 4 ducts, (V, D, F, and S) and I have the airflow values at each combination of ducts open/closed and each PWM command. Assume that when the direction says something like 'V+F+D', all 3 ducts are open at 100%, and if a duct isn't specified (like S in this example) it is fully closed at 0%.

I would like the ability to specify something like (V open at 40%, S open at 20%, D fully closed, F open at 100%, PWM command = 50) and have the output be an interpolated airflow at this set of door conditions. I'm getting stuck trying to format the data in a way that makes the problem solvable. Any advice and tips would be appreciated!

Please find some sample data attached.

Here's what I have so far, but I haven't figured out how to interpolate at vent states I don't yet have data for.

pwm_values = [0, 24, 34, 44, 64, 85]; % [Val out of 85]

duct_openings = {'Off', 'V', 'F', 'D', 'S', 'V+F', 'V+S', 'V+D', 'F+S', 'F+D', 'S+D', 'V+F+S', 'V+F+D', 'V+D+S', 'F+D+S', 'V+F+D+S'};

airflow_matrix = [0 0 0 0 0 0;

0 80.05 96.175 112.6 146.65 171.55;

0 85.75 104.2 122.875 157.975 176.65;

0 92.575 111.1 130.975 165.175 183.625;

0 53.8 64.525 72.85 87.325 97.6;

0 100.975 126.775 154.075 213.55 240.4;

0 80.875 100.975 122.125 160.075 182.125;

0 95.575 121.75 146.875 193.6 226;

0 81.55 101.275 119.725 159.4 179.65;

0 102.55 132.25 160.525 208.825 232.75;

0 93.55 116.725 142.15 1680.4 194.95;

0 106.825 137.95 171.925 226.675 252.175;

0 114.4 147.925 184.525 243.55 271.075;

0 102.55 132.25 160.525 210.775 236.8;

0 97.45 128.425 156.325 210.775 236.8;

0 116.5 150.775 188.125 248.275 276.4;

];

% Create meshgrid of known vals

[X,Y] = meshgrid(pwm_values, 1:size(airflow_matrix,1));

% Transpose to format for griddedInterpolant

X = X';Y = Y';airflow_matrix = airflow_matrix';

% Create lookup entity

F = griddedInterpolant(X,Y,airflow_matrix,'linear');

% Define the PWM value and duct opening combination to test

pwm = 84;

duct_combination = 'V+D+S';

interp_airflow = F([pwm, find(strcmp(duct_openings,duct_combination))])

[Torsten]

How do you want to interpolate between duct opening combinations ?

There is no such thing as

1/2 * 'Off' + 1/2 * 'V+F'

I guess.

[erictoleratesMATLAB]

Hey Torsten, thanks for your response!

I suppose what I'm looking for is the ability to specify any arbitrary duct opening combination and percentage.

For example: What if the opening percentages for the ducts are V=50%,D=50%,F=100%,S=75% and PWM = 34? (The data I provided shows what happens when each respective duct is either at 0 or 100%)

We have data for V,D,F,S=100% and various other combinations, so shouldn't it be possible to somehow interpolate the requested behavior?

Apologies if I'm still unclear, I can try to clarify my question further if helpful

[Torsten]

For example: What if the opening percentages for the ducts are V=50%,D=50%,F=100%,S=75% and PWM = 34?

And you are optimistic that you can capture such complicated interactions that might occur by percentage openings using a simple interpolation ?

Ok. But then you will have to define V, D, F, S and PWM as independent variables.

What would be an adequate model to get a correlation airflow = f(V,D,F,S,PVM) ?

You have measurements that take interactions between V, D, F and S into account. Thus the model should contain terms

V, D, F, S, V*F, V*S, V*D, F*S, F*D, S*D, V*F*S,V*F*D,V*D*S.

A third-order polynomial of the form

p0*V + p1*D + p2*F + p3*S + p02*V*F + p03*V*S + p01*V*D + p23*F*S + p21*F*D + p31*S*D + p023*V*F*S + p021*V*F*D + p013*V*D*S

comes to mind with coefficients p0,p1,...,p013 to be determined from your measurement data.

But how to integrate PVM here ?

Answer 1

[erictoleratesMATLAB]

Hi everyone!

Thank you for your thoughtful answers and responses to my question. I figured out a good solution to this particular problem that address all the key points I originally wanted.

I started by formulating the problem as a matrix problem, attempting to fit a function (CMH) of 5 variables (Door 1, Door 2, Door 3, Door 4, PWM) to the lowest order polynomial that I could reasonably expect to implement (The end goal is to implement this fucntion in C for another personal project).

I treated door 1 (V) as , door 2 (F) as , D as , S as , PWM as and CMH as . I then looped over each CMH value in my matrix, and for that value of CMH () I record the values of each input . When a door position is open, we record it as 1. When the door is closed, we record it at 0. Doing this records every known door state + PWM and the corresponding CMH. The end result is a matrix that looks something like this (the picture is truncated of course, the matrix is much larger):

I then used fitlm to test a bunch of fits, and pick the one that matches the data best with relatively easy implementation. Here are the results:

Here's the code so anyone can try this themselves if they have a similar problem.

% Goal: To obtain airflow = f(PWM,V,D,F,S)

% Where:

% - PWM = PWM Command

% - V = Vent Position (between 0 and 1)

% - F = Foot Position (between 0 and 1)

% - D = Defrost Position (between 0 and 1)

% - S = Seat Position (between 0 and 1)

% - CMH = Measured Airflow through blower [CMH]

clear;close all;clc

%% Hard Code values to be fit

airflow_matrix = [0 0 0 0 0 0

0 93.4 114.9 136.8 182.2 215.4

0 101 125.6 150.5 197.3 222.2

0 110.1 134.8 161.3 206.9 231.5

0 58.4 72.7 83.8 103.1 116.8

0 121.3 155.7 192.1 271.4 307.2

0 94.5 121.3 149.5 200.1 229.5

0 114.1 149 182.5 244.8 288

0 95.4 121.7 146.3 199.2 226.2

0 123.4 163 200.7 265.1 297

0 111.4 142.3 176.2 227.2 246.6

0 129.1 170.6 215.9 288.9 322.9

0 139.2 183.9 232.7 311.4 348.1

0 123.4 163 200.7 267.7 302.4

0 116.6 157.9 195.1 267.7 302.4

0 142 187.7 237.5 317.7 355.2

];

PWM_array = [0, 24, 34, 44, 64, 85];

duct_openings = {'Off', 'V', 'F', 'D', 'S', 'V+F', 'V+S', 'V+D', 'F+S', 'F+D', 'S+D', 'V+F+S', 'V+F+D', 'V+D+S', 'F+D+S', 'V+F+D+S'};

%% Now we try to save them in a productive format

x1 = []; % V

x2 = []; % F

x3 = []; % D

x4 = []; % S

x5 = []; % PWM

CMH = [];% CMH

for i = 2:numel(duct_openings)

% for j = 2:6

for j = 1:numel(PWM_array)

x1 = [x1 contains(duct_openings{i},'V')];

x2 = [x2 contains(duct_openings{i},'F')];

x3 = [x3 contains(duct_openings{i},'D')];

x4 = [x4 contains(duct_openings{i},'S')];

x5 = [x5 PWM_array(j)];

CMH = [CMH airflow_matrix(i,j)];

end

end

% Configure the independent variables into a useful representation

indep_vars = [x1' x2' x3' x4' x5'];

% full_matrix = [x1' x2' x3' x4' x5' CMH'];

%% Now let's try to fit a model to this data

mdl = fitlm(indep_vars,CMH,'quadratic');

%% Let's test this fit in a way actually usable in C. It'll have to be hardcoded,

% so reducing the precision of the coefficients is top priority.

% Truncated from mdl.Coefficients.Estimate to have only 5 digits. This was

% determined as an acceptable precision for maintaining prediction accuracy.

truncated = [-21.03

11.326

0

32.222

-0.790

3.5869

0.0383

-25.16

8.0716

0.8849

-24.70

-4.286

0.9407

1.2133

0.8871

0.0972

0

19.155

0

0

-0.022

];

% Let's create our fit function using our hard coded values

fit_function = @(x1,x2,x3,x4,x5) truncated(1).*1 + truncated(2).*x1 + truncated(3).*x2 + truncated(4).*x3 + truncated(5).*x4 + truncated(6).*x5 + truncated(7).*x1*x2 + truncated(8).*x1*x3 + truncated(9).*x1*x4 + truncated(10).*x1*x5 + truncated(11).*x2*x3 + truncated(12).*x2*x4 + truncated(13).*x2*x5 + truncated(14).*x3*x4 + truncated(15).*x3*x5 + truncated(16).*x4*x5 + truncated(17).*x1.^2 + truncated(18).*x2.^2 + truncated(19).*x3.^2 + truncated(20).*x4.^2 + truncated(21).*x5.^2;

%% Let's compare the set of predictions to the actual data

x1_test = []; % V

x2_test = []; % F

x3_test = []; % D

x4_test = []; % S

x5_test = []; % PWM

CMH_Predicted = [];% CMH

CMH_Predicted_C = [];

for i = 2:numel(duct_openings)

% for j = 2:6

for j = 1:numel(PWM_array)

x1_test = contains(duct_openings{i},'V');

x2_test = contains(duct_openings{i},'F');

x3_test = contains(duct_openings{i},'D');

x4_test = contains(duct_openings{i},'S');

x5_test = PWM_array(j);

pred = feval(mdl, x1_test,x2_test,x3_test,x4_test, x5_test);

pred = min(365, max(0, pred));

C_pred = fit_function(x1_test,x2_test,x3_test,x4_test, x5_test);

C_pred = min(365, max(0, C_pred));

CMH_Predicted = [CMH_Predicted pred];

CMH_Predicted_C = [CMH_Predicted_C C_pred];

end

end

%% Plot comparison

figure;

err = CMH_Predicted-CMH;

neg = (err<0).*err;

pos = (err>=0).*err;

errorbar(1:numel(CMH),CMH, neg,pos,'LineStyle','none','Color','red');hold on

plot(1:numel(CMH),CMH,'db',1:numel(CMH_Predicted),CMH_Predicted,'dr');title(sprintf("RMSE: %0.2f",rms(err)))

xlabel("Num");ylabel("CMH");grid minor;legend("Prediction Error","Data","Prediction");

fprintf("RMSE: %0.2f\n",rms(err))

% figure;plot(mdl);grid minor;improvePlot

%% Plot comparison using a lower fidelity fit (keeping less data points)

figure;

err2 = CMH_Predicted_C-CMH;

neg = (err2<0).*err2;

pos = (err2>=0).*err2;

errorbar(1:numel(CMH),CMH, neg,pos,'LineStyle','none','Color','red');hold on

plot(1:numel(CMH),CMH,'db',1:numel(CMH_Predicted_C),CMH_Predicted_C,'dr');title(sprintf("RMSE: %0.2f",rms(err2)))

xlabel("HVAC Config Index [-]");ylabel("Blower Airflow Prediction [CMH]");grid minor;legend("Prediction Error","Data","Prediction");

fprintf("RMSE: %0.2f\n",rms(err2))

% figure;plot(mdl);grid minor;improvePlot

%% Plot the 2 fits against eachother

figure;

plot(1:numel(CMH),CMH,'dg',1:numel(CMH_Predicted),CMH_Predicted,'db',1:numel(CMH_Predicted_C),CMH_Predicted_C,'dr');title("Comparing High vs Low Precision Data Fit")

xlabel("Num");ylabel("CMH");grid minor;legend("Real Data","High Precision Fit","Low Precision Fit","Prediction");

Answer 2

[Sulaymon Eshkabilov]

You should consider here some sort of unit compatibe value for U, S, D, V. In other words, 100% (and 80%, 60%, 40%, 20%) open U corresponds to ??% of S opening, ??? % D and ??% of V or vice verse 100% open S corresponds to ?% of U, ?% D, ?%V. So then you shall be able to compute V+F, V+S, V+F+D, etc., make up the states of PWM.