Hello!

I am trying to find a space state model for the following system. The input of the system its the applied voltage V(t) and the output of the system is the velocity of of the armature x(t). The force in the inductor its proportional to the velocity of the armature eo=-kt*(dx)/(dt)
F=kt*i


I considered that the system is linear, its proprierties are invariant in time and its parameters are concentrated.


I came up with the following differential equation in order to represent the system. Note that ^^ represent dots, that is, ^^=.. (2 dots)
^=. (1 dot)

m*y^^(t)=f(t)-C(t)*y^(t)-k*y(t)

y^^(t)=((f(t)-c(t)*y^(t)-k*y(t))/(m)


So the matrix that represent the space state model of the system is


[y^^(t);y^(t)]=[0 1 ; (-k)/(m) -(b)/(m)] [y^(t);y(t)] +[0; (1)/(m)] *f(t)


Is this correct? What would be my output equation in this case?


Thanks

 

Hi,

Wow, that's a bit hard to read with the notation being used

Did you happen to look up how to reduce a differential equation to a system of ODE's ?
That would be the way to go
It involves starting with a number of derivatives, then reducing.
You may have done it right, but it's hard to tell with that notation.
Be aware that you can upload gif's and jpgs here, so if you type out your result into a paint program you can then save the file and upload to this thread. You can use one dot, two dots, or any number of dots and they will look like dots. That way we can all read it much easier.

The output equation relates the state variables and the inputs to the output. So for example if you had a capacitor voltage X as state variable and the output was divided by 2 through the use of a voltage divider, then the output statement would be very simple:
y=X/2

If some of the input reaches the output directly, then the other term Du comes into play. It relates the input signal to the output when there is a direct path. So if the input was A and the gain was 3, then the whole thing would look like:
y=X/2+3*A

BTW, small point, it is usually called "State Space", not space state

 

Hi again,

Here is an outline of a step by step procedure...


The original equation:
a*dx^2/d^2t+b*dx/dt+c*x=d

with simplified notation where x' is 1st derivative, x" second:
a*x''+b*x'+c*x=d

step 1 (isolate term with highest derivative)
a*x''=d-b*x'-c*x

step 2 (divide by coefficient)
x''=d/a-b/a*x'-c/a*x

step 3 (define successive state variables)
x1=x
x2=x'
x3=x''

step 4 (equate derivatives to state variables)
x1'=x2
x2'=x3

step 5 (rewrite in terms of derivatives)
x1'=x2
x2'=d/a-(b/a)*x'-(c/a)*x

step 6 (replace right side derivatives with equivalent state variables)
x1'=x2
x2'=d/a-(b/a)*x2-(c/a)*x1

In the upper case/lower case notation this would look like this:
x1=X2
x2=d/a-(b/a)*X2-(c/a)*X1

where X1 and X2 are the state variables.