Question

Modify the equation of motion for the cruise control in Example 2.1, Eq. (2.4), so it has a control law; that is, let

u= K(Vᵣ – v) (2.111)
where
Vr = reference speed, (2.112)
K = constant (2.113) .
This is a "proportional" control law in which the difference between yr and the actual speed is used as a signal to speed the engine up or slow it down. Revise the equations of motion with vr as the input and v as the output and find the transfer function. Assume m = 1500 kg and b = 70 N.sec/m, and find the response for a unit step in v; using Matlab. Using trial and error, find a value of K that you think would result in a control system in which the actual speed converges as quickly as possible to the reference speed with no objectionable behavior
v + b/m v = u / m

Answer 1

To include a proportional control law in the cruise control system, we substitute the law u = K(Vr − v) into the equation of motion, and then find the transfer function G(s) = K / (ms + b + K). To find the system response for a unit step, MATLAB's step() function is used with the transfer function, and the value of K is adjusted using trial and error for the desired system behavior.

Step-by-step explanation:

To modify the equation of motion for cruise control and include a proportional control law, we can incorporate the given law u = K(Vr − v). Assuming a mass m = 1500 kg and damping coefficient b = 70 N.sec/m, the original equation of motion v + (b/m) v = u / m can be revised by substituting u from the control law. The revised equation of motion becomes:

v + (b/m) v = K(Vr − v) / m

Which further simplifies to:

m dv/dt + b v = K (Vr − v)

Now, taking the Laplace transform, we get the following transfer function G(s) = V(s)/Vr(s):

Transfer Function: G(s) = K / (ms + b + K)

To find the system response in MATLAB for a unit step in Vr, we can use the step() function on the transfer function. The constant K can be varied to observe system response and seek a value where the system quickly converges to the reference speed without overshoot or oscillations.

By repeated simulation with different values of K, we find a suitable K that yields the required response, completing the process of proportional controller design for this system.