Midterm Examination
Control Systems
MIDTERM EXAMINATION PAPER
CONTROL SYSTEMS
2021 – 2022 Academic Year
Complete all tasks.
Electronic calculators may be used, provided that they cannot store text.
Midterm Examination
Control Systems
Task 1
Consider a linear dynamic system whose transfer function is given as follows:
a. Specify the number of poles and zeros and their locations in the 𝑠 – plane.
b. The input signal applied to this system is described as the following function of time:
Specify the values of 𝑎 > 0and 𝑏 > 0 such that the steady – state output is equal to 2.5.
c. Suppose this system is placed in feedback with a proportional – only controller with gain
𝑘𝑃 . Sketch the root locus of the closed – loop poles for 0 ≤ 𝑘𝑃 < ∞.
Task 2
Consider the closed – loop control system shown in Figure 1.
Figure 1. Closed – Loop Control System
a. Show that the steady – state error resulting from a step change in set – point, denoted
as 𝑟 in Figure 1, is equal to zero for any finite and positive value of 𝑘𝑝 .
b. Determine the range of values of 𝑘𝑝 such that the resulting closed – loop system
exhibits stable non – oscillatory response.
Midterm Examination
Control Systems
Task 3
Consider the closed – loop system shown in Figure 2.
a. Sketch the root locus of the closed – loop system poles for 𝑘𝑃 > 0. Comment on
whether the closed – loop system is stable for any 𝑘𝑃 > 0.
b. Derive the Sensitivity Transfer Function for the closed – loop system.
Figure 2. Closed – loop System.
Task 4
A plant has transfer function
Suppose that the plant is placed in negative feedback with a PI controller whose transfer
function is
The integral time 𝑇𝑖 is chosen so that the zero of 𝐶 (𝑠) cancels the low frequency pole of 𝐺 (𝑠).
What value of 𝑘𝑃 gives closed – loop damping ratio 𝜁 = 0.7?
Midterm Examination
Control Systems
Task 5
As shown in Figure 3.1, a system that has the transfer function
is placed in negative feedback with the PI controller
The system 𝐺 (𝑠)has a limited input range with the following saturation characteristics:
Figure 3.1 Closed – Loop Control System
a. Figure 3.2 shows the closed – loop system response 𝑦(𝑡) to two step changes in the
reference signal 𝑟(𝑡). Sketch the controller output signal 𝒖𝟏 (𝒕) and the plant input
signal 𝒖𝟐 (𝒕).
Midterm Examination
Control Systems
Figure 3.2
b. As shown in Figure 3.3, a dead zone function is added in a negative feedback loop. The
dead zone function has the following characteristics:
Figure 3.3
Midterm Examination
Control Systems
The reference signal 𝑟(𝑡) is shown in Figure 3.4. Briefly state the benefits of the modified
diagram with the dead zone function and then sketch the system response 𝒚(𝒕).
Figure 3.4
Task 6
Find the z – transform of the following signals (see below a and b). In your solution show the
“long – hand” and “closed” forms of the z – transform.
a. Unit impulse function occurring at 𝑡 = 𝑘𝑇.
b. Ramp signal.
Task 7
A sampled – data system has the following transfer function
𝐺 (𝑠 ) =
1
𝑠+2
The sampling time is one second and the system is subject to a unit impulse input signal. In this
task, for the sake of simplicity, a zero – order hold can be omitted.
Determine the discrete – time response using:
a. Infinite power series method.
b. Difference equation method.
END OF MIDTERM EXAMINATION PAPER