# State Space Representation

1. Sep 26, 2013

### jegues

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

A first-order dynamic system is represented by the differential equation,

$$5\frac{dx(t)}{dt} + x(t) = u(t).$$

Find the corresponding transfer function and state space reprsentation.

2. Relevant equations

3. The attempt at a solution

Putting the equation in the Laplace domain yields,

$$5sX(s) + X(s) = U(s)$$

$$\Rightarrow G(s) = \frac{X(s)}{U(s)} = \frac{1}{1+5s}$$

For the state space equations,

$$\frac{dx(t)}{dt} = -0.2x(t) + 0.2u(t)$$

$$\frac{dx(t)}{dt} = -0.2x(t) + 0.5u(t), \quad y(t) = 0.4x(t)$$

How did they 0.5u(t) and how did they know that y(t) = 0.4x(t)?

Thanks again!

2. Sep 29, 2013

### jegues

Bump, can someone please clarify this for me?

3. Sep 29, 2013

### Staff: Mentor

Hi jeques. I don't know if I can help to clarify your problem. The definition of the first-order system made no mention of y(t), so it's a mystery to me where it came from at the end! Is there something missing from the problem statement that might tie-in with y(t)?

4. Sep 29, 2013

### jegues

Here's the question. (see attached)

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5. Sep 30, 2013

### Staff: Mentor

Hmm. Nope, that doesn't help me The transfer function bit is clear enough, but I don't "get" the introduction of the y(t) stuff. I'll see if I can find someone who recognizes the problem type.

6. Sep 30, 2013

### Staff: Mentor

Upon reflection and discussion with another Homework Helper, it occurred to me that the problem would make more sense to me if the variable used in the dynamic system differential equation was y rather than x.

Is it possible that we should take the given system D.E. to represent the form of the equation describing the system rather than an equation of the state variables?