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

N/A.

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,

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

The answer they provide is,

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

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

Thanks again!

 

You haven't given any hint about what y(t) is, so how can we possibly tell you where it came from?

Is u(t) the unit step function?

What happened to the initial conditions when you took the Laplace transform?

 

In state-space convention y(t) is often taken to be the output in response to a given input u(t). The form of u(t) is arbitrary.

The state space form would typically be ...

\(\text{\dot{x}(t)=Ax(t)+Bu(t)}\)
\(\text{y(t)=Cx(t)+Du(t)}\)

The state space variable is x(t)

I find a useful cross check of the solution is to draw an equivalent block diagram with one or more pure integrator blocks, gain/attenuator blocks and summing junctions.

 

You haven't given any hint about what y(t) is, so how can we possibly tell you where it came from?

I wasn't explicity given y(t) in the question, they provide a different y(t) in each of the multiple choice answers provided for the question.

Is u(t) the unit step function?

Yes.

What happened to the initial conditions when you took the Laplace transform?

When you are solving for the transfer function, \(G(s) = \frac{X(s)}{U(s)}\) by definition you neglect the initial conditions.

EDIT: Attached is the question as it is read from the book.

 

Would there be any loss of generality if one simply wrote ...?

\(\text{ \dot{x}=-0.2x+u(t) \\ y=0.2x}\)

Why does u(t) need to be the unit step? Doesn't the state space method assume u(t) to be any arbitrary input which has an equivalent 's' domain or Laplace function U(s).

 

Would there be any loss of generality if one simply wrote ...?

\(\text{ \dot{x}=-0.2x+u(t) \\ y=0.2x}\)

Why does u(t) need to be the unit step? Doesn't the state space method assume u(t) to be any arbitrary input which has an equivalent 's' domain or Laplace function U(s).

I don't understand how when they are isolating for, \(\dot{x}\) they don't end up with \(0.2u(t)\) on the RHS of the equation.

 

I don't understand how when they are isolating for, \(\dot{x}\) they don't end up with \(0.2u(t)\) on the RHS of the equation.

A reasonable query on your part. I suspect they (whoever "they" are) are either being annoyingly perverse or trying to test your perceptiveness in the matter. If you think about it there are numerous solutions one can dream up. See my attachment.

Edit: The block H(s) is intended to produce a time domain response for a given 's' domain transfer function - which it does in the application I use to create the block diagram models. The implementation producing output V2 would correspond to the SS model given as the "correct" answer in the original question.