How to Convert Mechanical System Equations to State Space Form?

[zoom1]

JL*QL'' + BL*QL' + k(QL - Qm) = 0
Jm*Qm'' + Bm*Qm' - k(QL - Qm) = u

This is the equation set I have for a motor with a load.
QL'' means second derivative and QL' means first derivative.

I need to be able to obtain the state space representation of this model where X = [QL;QL';Qm;Qm'] (This is of course a column array)

I tried my best but couldn't obtain it.
Started off with QL' = [-JL*QL'' - k(QL - Qm)]/BL

But when I try to represent this, I don't have the QL'' term in my state variable array X, so couldn't proceed further.

How am I supposed to approach this problem ?

 

[donpacino]

as you are most likely aware, state space representation is usually of the form

X'=A*X+B
Y=C*X+D

your first goal is to define your X array.
then you need to define your A and B matrix.

I would define each element of your X array.
SO you already stated
QL = X1
QL' =X2
Qm=X3
Qm'=X4

so right off the bat you know X1'=X2 and X3'=X4, that gives you half of your A matrix

So you need to find your A matrix rows for X2' and X4'.

So use equation one to solve for X2', then use equation 2 to solve for X4'. as you can see there is only one input which is in equation 2, so there will only be one nonzero element in the B matrix.

 

[zoom1]

as you are most likely aware, state space representation is usually of the form

X'=A*X+B
Y=C*X+D

your first goal is to define your X array.
then you need to define your A and B matrix.

I would define each element of your X array.
SO you already stated
QL = X1
QL' =X2
Qm=X3
Qm'=X4

so right off the bat you know X1'=X2 and X3'=X4, that gives you half of your A matrix

So you need to find your A matrix rows for X2' and X4'.

So use equation one to solve for X2', then use equation 2 to solve for X4'. as you can see there is only one input which is in equation 2, so there will only be one nonzero element in the B matrix.

Perfect !

Thanks a lot