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I am trying to feedback linearize my non-linear system in Matlab. To that end I have to represent it in the following form:

So after I have modeled my system in the form of nonlinear differential equations, , I use the diff function to take the derrivative with respect to u to aquire g, and . This works, of course only if the system can be represented in the mentioned form. However, the diff function rewrites some of my expressions, which then after subtraction lead to non-vanishing factors multiplying u. For example, my input is still present in multiplied with a factor 5.7024e-18. I suspect, there is some quantization error introduced due to diff re-writing the derivative . Can this be somehow avoided?

g_x=[diff(dxdt,[Q_g]) diff(dxdt,[Q_b])];

for i=1:size(g_x,1)

i

f_x(i,:)=simplify(dxdt(i,:)-g_x(i,1)*Q_g-g_x(i,2)*Q_b,'steps',10,'IgnoreAnalyticConstraints',true);

end

Here Q_g and Q_b are my inputs.

An example of the re-writing of expressions:

dxdt = - (1297036692682702848*Q_b*(C_carbv - 47/20000))/15694143981460705 + ..................

g_x=1905022642377719808/9808839988412940625 - (1297036692682702848*C_carbv)/15694143981460705

f_x=Q_b*((1297036692682702848*C_carbv)/15694143981460705 - 1905022642377719808/9808839988412940625) - (1297036692682702848*Q_b*(C_carbv - 47/20000))/15694143981460705 + ..............................

simplify(Q_b*((1297036692682702848*C_carbv)/15694143981460705 - 1905022642377719808/9808839988412940625) - ...

(1297036692682702848*Q_b*(C_carbv - 47/20000))/15694143981460705)

ans =

(47*Q_b)/8242150268041428750

The .................. stands for terms that do not depend on Q_b.

Paul
on 16 Jun 2021

Why not just use collect() and coeffs() on the elements of dxdt? If the system is affine in u, then those should give the desired result directly

syms x u1 u2

dxdt = x^2 + x + x*u1 + x^3*u1 + x^2*u2 + u2;

dxdt = collect(dxdt,[u1 u2])

[c,t] = coeffs(dxdt,[u1 u2])

fofx = c(find(t==1))

gofx = [c(find(t==u1)) c(find(t==u2))]

If the system is not affine in u, then extra terms will show up in t, so you can check numel(t) before doing any other processing to decide how to proceed.

dxdt = x^2 + x + x*u1 + x^3*u1 + x^2*u2 + u2 + u1*u2;

dxdt = collect(dxdt,[u1 u2])

[c,t] = coeffs(dxdt,[u1 u2])

John D'Errico
on 16 Jun 2021

This is a floating point problem, but it comes when MATLAB takes your constants and turns them into symbolic numbers. Do you see all of those large integers? For example...

X = sym(1.34536363425474)

But that ratio is not exactly the same value as the original number. It is typically a close approximation.

vpa(X,40)

So you get trash in there, down in digit 17 and below.

My question is, why is this a problem?

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