How to derive the space state form of this system?

[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*ddx_m+k_l*(x_m-x_l)+k_u*(x_m-x_u)+c_l*(dx_m-dx_l)+c_u*(dx_mdx_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:
y₁=x_m
y₂=dx_m

and I derived the following equation:

dy₂=-y₁*(k_l/m+k_u/m)-y₂*(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

 

[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*ddx_m+k_l*(x_m-x_l)+k_u*(x_m-x_u)+c_l*(dx_m-dx_l)+c_u*(dx_mdx_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:
y₁=x_m
y₂=dx_m

and I derived the following equation:

dy₂=-y₁*(k_l/m+k_u/m)-y₂*(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

none can help me?

 

[NatSusan]

Hi there,

It seems like you have made good progress in deriving the equations for your system. To define the input matrix, you will need to consider what the inputs are and how they affect the system. In your case, you have two inputs: the position and velocity of the lower spring end. These inputs will affect the acceleration and velocity of the mass.

One way to define the input matrix is to consider the inputs as disturbances to the system. In other words, how does a change in the position and velocity of the lower spring end affect the acceleration and velocity of the mass? Once you have determined this relationship, you can use it to define the input matrix.

Another approach is to consider the inputs as control inputs. In this case, you will need to design a controller that can manipulate the position and velocity of the lower spring end to achieve a desired response from the mass. The input matrix will then depend on the control strategy you use.

I hope this helps. If you have any further questions, please don't hesitate to ask. Good luck with your system!