# How to handle cross products in feedback loop?

I'm trying to simplify the following block diagram from

to the following block diagram

However, h(t), H(s) in time domain, has a cross product in it (gyroscopic torque). How can I represent the cross product in h(t) as H(s) so I can simplify the blocks? The goal is to have 1 transfer function to represent the system. Is this possible through linearization or any other techniques? Should H(s), gyroscopic torque, be injected instead of a feedback block? What are common methods to work around cross products?

 More info: The output of G(s) or Y(s) is rate, w. H(s) is ideally calculating gyroscopic torque, w x Iw, where x is cross product and I is inertia.

• I assume that with $h(t)$ you mean the output of the block containing $H(s)$? Is the output of that block a linear function of $Y(s)$? – fibonatic Jan 13 at 23:53
• In time domain, h(t) has a cross product. I'm not sure how to transform it to H(s) in order to simplify the above block diagram in to 1 transfer function for the system. The output of G(s) or Y(s) is rate, w. H(s) is ideally computing the gyroscopic torque and eventually subtracting that from X(s), torque. To compute gyroscopic torque in the time domain, its w x Iw where x is cross product and I is inertia. – fewqo Jan 14 at 0:05
• That is nonlinear and can't be expressed as a transfer function. – fibonatic Jan 14 at 1:06
• The question is if it's possible to condense this to 1 transfer function. Is this not possible through linearization or any other techniques? Should H(s), gyroscopic torque, be injected instead of a feedback block? What are common methods to work around cross products? I don't believe that this is impossible since there are plenty of systems that use cross products. – fewqo Jan 14 at 1:35
• What do you mean by "Cross product". That's a vector term, but all you seem to have are 1 dimensional time domain signals. What vector are involved here? – Hilmar Jan 14 at 13:10