# State Space Representation

jegues

## Homework Statement

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.

## 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!

jegues
Bump, can someone please clarify this for me?

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)?

jegues
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)?

Here's the question. (see attached)

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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.

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?