ME 360
Modeling, Analysis, and Control of Dynamic Systems
Winter 2026
Homework Assignment #10
Due Tuesday, April 21st, 2026 at 11:59 PM EST on Canvas
Problem 1 (10 Points): Consider the control system represented in the block diagram below, where the plant is
the mass-spring-damper system shown below (with input 𝑧 and output 𝑥), where 𝑘 = 1 N/m, 𝑏 = 1 Ns/m, and 𝑚 =
1 kg. The signal 𝑅(𝑠) represents a setpoint (reference), whereas the signal 𝑁(𝑠) represents measurement noise.
Furthermore, suppose that the controller’s transfer function is given by 𝐶(𝑠) =
𝐾(𝑠+𝑎)
.
𝑠
(a) Assuming that 𝑎 ≠ 0, what type of controller is this (e.g., proportional, PI, PID, PD, etc.)?
(b) Suppose that 𝑎 = 2. What is the range of 𝐾 for which the closed-loop system is stable? If 𝑟 is a unit step
input, what will be the size of the steady-state error (𝑒𝑠𝑠 )? Assume that 𝑛 = 0.
(c) Suppose that 𝑎 = 0. What type of controller is this now? What is the range of 𝐾 for which the closed-loop
system is stable? If 𝑟(𝑡) is a unit step input, what will be the size of the steady-state error (𝑒𝑠𝑠 )?
(d) Conceptually, this problem illustrates a very common tradeoff associated with utilizing an integrator in a
controller. In one sentence, describe conceptually what this tradeoff entails (the sentence can be
something like “Using an integral term in a controller can achieve [fill in the blank] at the expense of [fill
in the blank].”
Problem 2 (15 Points): For this problem, we will work with the same block diagram and physical system as in
Problem 1. However, we will also assume that 𝐶(𝑠) =
𝐾(𝑎1 𝑠2 +𝑎2 𝑠+1)
.
𝑠
𝑋(𝑠)
(a) Compute a symbolic expression for the transfer function from 𝑟 to 𝑥 (i.e., compute 𝑅(𝑠)).
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(b) Assuming that 𝑛 = 0, and assuming that 𝑎1 = 0, 𝑎2 = 4, and 𝐾 = 2, plot the response of 𝑥(𝑡) to a unit
step in 𝑟(𝑡). What type of controller is this (e.g., proportional, PI, PID, PD, etc.)?
(c) Assuming that 𝑛 = 0, and assuming that 𝑎1 = 4, 𝑎2 = 4, and 𝐾 = 2, plot the response of 𝑥(𝑡) to a unit
step in 𝑟(𝑡). What type of controller is this (e.g., proportional, PI, PID, PD, etc.)?
𝑍(𝑠)
(d) Compute a symbolic expression for the transfer function from 𝑛 to 𝑧 (i.e., compute 𝑁(𝑠)).
(e) Using MATLAB, generate Bode plots corresponding to the transfer function in (d), for the parameters
from part (b) in the problem and the parameters in part (c).
(f) In many control systems, noise commonly manifests itself as a high-frequency input. Given this fact, the
results from (b) and (c), and the Bode plots from part (e), describe a tradeoff that arises from the use of
derivative control (this tradeoff is in fact very common and not just specific to this particular system).
Your answer can be a short sentence in the form of “Derivative control helps to achieve improved [fill in
the blank] at the expense of [fill in the blank].”
Problem 3 (10 Points): Consider the inner-loop, outer-loop control system shown below, where 𝐼 = 20 kg m2 and
𝑐 = 10 Nms/rad. Select values for 𝐾𝑃 and 𝐾2 that achieve the following requirements:
●
●
●
Zero steady-state error in response to a step input in the setpoint, 𝜃𝑟 (𝑡);
Steady-state error of no more than 0.1 radians in response to a unit step in the disturbance, 𝑇𝑑 (𝑡);
Closed-loop time constants of 0.1 seconds or faster.
Problem 4 (15 Points): Consider the mass-spring-damper system below, where 𝑘1 = 1000 𝑁/𝑚, 𝑘2 =
2000 𝑁/𝑚, 𝑏 = 4000 𝑁𝑠/𝑚, and 𝑚 = 1000 𝑘𝑔.
𝑌(𝑠)
(a) Compute the transfer function from 𝑍(𝑠) to 𝑌(𝑠), i.e., compute 𝑍(𝑠).
(b) Suppose a controller has been designed, where 𝑧(𝑡) is the control signal (as supplied by a linear actuator)
and the control law has the following form:
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𝑍(𝑠) =
𝐾𝑝
𝑠+1
(𝑌𝑑𝑒𝑠 (𝑠) − 𝑌(𝑠)).
Furthermore, suppose that the response of 𝑦(𝑡) to a unit step in 𝑦𝑑𝑒𝑠 (𝑡) is given by the plot below. Based
on that plot, what is the value of 𝐾𝑝 ?
(c) Now that you have the value of 𝐾𝑝 , compare the step response above to the one you obtain under a simpler
proportional control law, 𝑍(𝑠) = 𝐾𝑝 (𝑌𝑑𝑒𝑠 (𝑠) − 𝑌(𝑠)) (i.e., plot the response of 𝑦(𝑡) to a unit step in
𝑦𝑑𝑒𝑠 (𝑡) using MATLAB, under the same value of 𝐾𝑝 as in (b)).
(d) You should have found that the plot from (c) exhibited less overshoot, faster settling time, and the same
amount of steady-state error (you need integral control to get rid of that) – so, in other words, everything
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looks better, and the controller is simpler! What purpose, then, could the 𝑠+1 in the original controller
1
transfer function possibly serve? Hints: Think about the Bode plot of 𝑠+1. What sort of signal gets
attenuated based on that Bode plot?
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