# State-space modeling of a control system

1. Sep 8, 2015

### yaro99

1. The problem statement, all variables and given/known data

Here is the problem:

And this is the system in question:

2. Relevant equations
G(s) = θ_0(s)/θ_i(s) = output/input (open-loop transfer function)

3. The attempt at a solution

I've been staring at this problem and looking through my textbook for over an hour but honestly I'm not sure how to begin here. I think I'm supposed to create a vector [x1, x2, x3] and multiply it by a matrix in order to get the forward path. I'm still confused on how to determine this matrix, as well as what to do about x2, since the problem defines it as angular velocity ω_m, and I don't see that anywhere on the diagram (I'm assuming it's the derivative of θ_m?).

From looking at my notes, I believe I'm supposed to have something in the form of x'=Ax + Bu (A and B are matrices, x and u are vectors). The only problems of this kind that I've seen in class were circuit problems, but that doesn't help me here.

I'm not too worried about the MATLAB part, if I can solve the first part I believe I can figure it out.

2. Sep 10, 2015

### donpacino

For 2.2 A,

you want to have 3 states
1.applied voltage E(a)
2.velocity ω_m
3. position θ_m

just name E(a)=X1
ω_m=X2
θ_m=X3

so X=[E(a) w_m θ_m]'

X'=AX+BU
A will have to be a 3x3 matrix,

you assumed correct... knowing that what is the second row of A?
also B should be fairly easy (hint: there is only one state that will be directly affected by the input