Determining the transfer function of cruise control given characteristic equations

Question

During my time revising for my control engineering module, i've come across this question

The automatic cruise speed control for a vehicle is described by the following equations

$\bigl( 1\bigl)$ $m \frac{dv_0(t)}{dt} + Fv_0(t)=u(t)$

$\bigl( 2\bigl)$ $u(t)=K[v_0(t)-v_r(t)]$

where m is the mass of the car, 1000$kg$, F is the frictional constant 50 $Nm^{-1}s$ and $u(t)$ is the throttle actuator force and K is the proportional gain control for the actuator. Also $V_0(t)$ is the actual speed while $V_r(t)$ is the desired speed.

There are two parts to this question, and im a bit confused on both of them.

part 1 says to draw a block diagram of the speed control system. In the control engineering module we generally draw the block diagrams as related to s, so i thought i'd get the laplace transform of the function (to get the transfer function) and draw it from the general model of a control system

This coincided with part 2, which wants me to show that the transfer function in terms of K is given by

$\frac{V_0(s)}{V_r(s)} = \frac{K}{1000s+50+K}$

I went about working it out and eventually ended up with this

$\frac{V_0(s)}{V_r(s)} = \frac{-K}{1000s+50-K}$

Can anyone help me figure out where i went wrong?

Also, i attempted making a block diagram in the t domain for this system, however since there are two equations with equal outputs (not summed) i was completely stumped on how i would display that as a model that is accurate, without making assumptions like equating K with the system gain of the other equation

Answer 1

If you are using $(2)$ as your definition for $u(t)$ then your expression is correct. However in control it is often the convention to use negative feedback. In this case equation two would become

$$ u(t) = K [v_r(t) - v_0(t)] \tag{2} $$

which is equivalent to negating your value of $K$. This would yield the same as the given result.

Answer 2

If it was me, and bearing in mind what the other answer said about negative feedback, I would do something like this:

Change the gain blocks from 1 to the correct parameters. vr is obviously your input/reference speed profile. I don't think there's any need to compute the transfer function to do a block diagram.

Answer 3

If you take the Laplace transform of the system you obtain:

$$msV_0(s)+FV_0(s)=U(s)$$ $$U(s)=K\left[V_0(s)-V_\text{r}(s) \right]$$

Plug the second equation into the first equation and solve for $V_0(s)/V_\text{r}(s)$:

$$\dfrac{V_0(s)}{V_\text{r}(s)}=\dfrac{-K}{ms+(F-K)},$$

So your answer seems to be correct.

Now, the block diagram (in time domain) is given by in the following picture.