# state space representation of system with a disturbance

my question is how you can convert the bloc diagram which have a disturbance input to a state space representation and how you can calculate the controllability matrix then

if you can give me any textbook that explains that i googled it a lot but i guess i dont have the right vocabulary because i couldn't find anything

the system of issue : Mx is the disturbance

It is basic block diagram algebra.

First, you write out the algebraic equations and solve for the unknowns $$\x1\$$ and $$\x2\$$.

$$\text{x1}=\frac{0.05 (\text{Ua}-0.1 \text{x2})}{0.01 s+1}$$ $$\text{x2}=\frac{-\text{Mx}+\text{x1}-2 Y}{0.5 s+1}$$

Next you write the output equation as $$\Y= \frac{1}{s} x2\$$ and solve for $$\Y\$$ in terms of the two inputs $$\Ua\$$ and $$\Mx\$$. This will give you the transfer function model from the two inputs to the output $$\Y\$$.

Then obtain a state-space realization.

Since $$\Mx\$$ is a disturbance, you want to remove the column of the input matrix that corresponds to that before computing the controllability matrix.

• thank you so much Commented Oct 16, 2019 at 18:47