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I need to derive the space state form of this simple system:

http://imageshack.us/photo/my-images/856/system.png/

The two springs end are moving.

I derived the equation fo motion:

m*ddx_{m}+k_{l}*(x_{m}-x_{l})+k_{u}*(x_{m}-x_{u})+c_{l}*(dx_{m}-dx_{l})+c_{u}*(dx_{m}dx_{u})

where ddx_{m}is the mass acceleration

dx_{m}is the mass speed

xm is the mass position

x_{l}is the lower spring end position

dx_{l}is the lower spring end velocity

x_{u}is the lower spring end position

dx_{u}is the lower spring end velocity

My system has two inputs and one output and my problem is to understand how to manage them.

so I changed the variables in this way:

y_{1}=x_{m}

y_{2}=dx_{m}

and I derived the following equation:

dy_{2}=-y_{1}*(k_{l}/m+k_{u}/m)-y_{2}*(c_{l}/m+c_{u}/m)+k_{l}/m*x_{l}+k_{u}/m*x_{u}+c_{l}/m*x_{l}+c_{u}/m*x_{u}.

And this should be the A matrix:

A=[0 1 ;

-k_{l}/m-k_{u}/m -c_{l}/m-c_{u}/m]

How should I define the input matrix since I have speed and velocity in the input and they are related each other? Hopefully to have properly explained my doubt, if not don't hesitate to ask me please

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# How to derive the space state form of this system?

**Physics Forums | Science Articles, Homework Help, Discussion**