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

1. Sep 6, 2011

### serbring

Hi all,

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*ddxm+kl*(xm-xl)+ku*(xm-xu)+cl*(dxm-dxl)+cu*(dxmdxu)

where ddxm is the mass acceleration
dxm is the mass speed
xm is the mass position
xl is the lower spring end position
dxl is the lower spring end velocity
xu is the lower spring end position
dxu 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:
y1=xm
y2=dxm

and I derived the following equation:

dy2=-y1*(kl/m+ku/m)-y2*(cl/m+cu/m)+kl/m*xl+ku/m*xu+cl/m*xl+cu/m*xu.

And this should be the A matrix:

A=[0 1 ;
-kl/m-ku/m -cl/m-cu/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

Last edited: Sep 6, 2011
2. Sep 10, 2011

### serbring

none can help me?