3 PI Control of a First Order Plant
You are a control engineer starting a new job at a company that makes ski lifts for clients in
Tahoe. You are tasked with designing a controller that regulates the chair speed through a torque
applied on the main wheel that turns the cable. You represent this plant as a first order system
below, where $v$ is the speed of the chair, $u$ is the input torque, and $m$, $b$, and $R$ are model
parameters:
$m\dot{v} + bv = Ru$
You decide to design a proportional-integral (PI) controller of the form
$u = K_pe + K_i \int e$
and choose the negative feedback architecture below, where $C(s)$ is the controller transfer function
and $P(s)$ is the plant transfer function:
(a) What is the order of the plant model ($u$ to $v$)? What is the order of the controller ($r$ to $u$)?
(b) Use the block diagram and the two equations above to find the closed-loop transfer function
from $r$ to $v$ (i.e., $v = H(s)r$). This should be in terms of $K_p$, $K_i$, $R$, $m$, and $b$. What is the
order of the closed-loop model ($r$ to $v$)?
(c) You are told the closed-loop system must have a minimum 10-90% rise time of 5 seconds
and a maximum percent overshoot of 5%. Using the parameter values $m = 1000$, $b = 100$,
and $R = 6$, find the necessary values of $K_p$ and $K_i$.
(d) Use Simulink to plot the closed-loop response to a step reference with steady-state value
$r_{ss} = 10$ (with all initial conditions set to 0. Use the plant differential equation as your
starting point, as in previous homeworks, and use the control gains you found in part
3. Remember, the control input $u(t)$ can be directly expressed as the following, where
$e(t) = r(t) - y(t)$:
$u(t) = K_pe(t) + K_i \int_0^t e(\tau)d\tau$
Confirm that your controller is stable and meets the rise time and overshoot design goals.
(e) Consider the real-world effect of increasing/decreasing the rise time and percent overshoot
on your chairlift's riders. Write a few sentences considering what your riders may prefer,
and how this informs your control design $K_p$ and $K_i$.
