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I am reading the book "Model-Based Fault Diagnosis Techniques" by Steven X. Ding. Here, they describe a system with the equation $$ ẋ = (A + ΔA_F)x +(B + ΔB_F)u+E_f f$$ Then, $$ ΔA_F = A_i θ{_A}{_i} $$ is given

Then, they proceed with finding the partial derivative of $ẋ$ with respect to $θ{_A}{_i}$ for which they get the answer as $ A \frac {∂x}{∂θ{_A}{_i}} +A_ix $.

I tried doing it myself, and this is how I do it.

$$ \frac {∂ẋ}{∂θ{_A}{_i}} = A \frac {∂x}{∂θ{_A}{_i}} + A_i x \frac {∂θ{_A}{_i}}{∂θ{_A}{_i}} + A_i θ{_A}{_i} \frac {∂x}{∂θ{_A}{_i}} $$

Which can be simplified as

$$ \frac {∂ẋ}{∂θ{_A}{_i}} = A \frac {∂x}{∂θ{_A}{_i}} + A_i x + A_i θ{_A}{_i} \frac {∂x}{∂θ{_A}{_i}} $$

Essentially, the last term is surplus and I am not sure how I can get the answer as in the book.

Just to add some context, will the product rule not apply for the last term ? As $x$ and $θ{_A}{_i}$ are both functions of $θ{_A}{_i}$.

Dipsy2000
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1 Answers1

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I agree with you, and I think this is simply an inconsequential mistake from the book, since in the proof, this result is only an intermediate step, and it is later evaluated at $\theta_{A_i} = 0$.

gdcvdqpl
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