How to derive the space state form of this system?

AI Thread Summary
The discussion revolves around deriving the state-space representation of a mechanical system involving two moving springs. The equation of motion has been established, leading to the formulation of the A matrix, which describes the system's dynamics. The user seeks guidance on defining the input matrix due to the complexity of having both speed and velocity as inputs, which are interrelated. Clarification on how to appropriately manage these inputs in the state-space model is requested. The conversation highlights the challenges of modeling systems with multiple inputs and their relationships.
serbring
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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:
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serbring said:
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

none can help me?
 

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