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R7003E 2022-01-13

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Automatic Control
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Allowed aids:
Mathematics book Beta
Not allowed aids:
The textbooks
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On duty teacher (complete telephone number):
Khalid Atta
+46 (0) 760 579 7 41
On duty teacher (complete telephone number):
On duty teacher (complete telephone number):
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Number of questions and total score:
3 - 12 points
4 - 16 points
5 - 20 points
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Problem 1
(2 + 2 + 2 = 6 points)
The linearized differential equations governing the fluid-flow dynamics for the two cascaded tanks in Fig. 1 are
∆ḣ1 + σ∆h1 = ∆u
∆ḣ2 + σ∆h2 = σ∆h1
where ∆h1 = deviation of depth in tank 1 from the nominal level, ∆h2 = deviation of
depth in tank 2 from the nominal level, ∆u = deviation in fluid in flow rate to tank 1
Figur 1: Coupled tanks for Problem Problem 1.
a) Level Controller for Two Cascaded Tanks: Using state feedback of the form
∆u = −K1 ∆h1 − K2 ∆h2 ,
choose values of K1 and K2 that will place the closed-loop eigenvalues at
s = −2σ(1 ± j).
b) Level Estimator for Two Cascaded Tanks: Suppose that only the deviation in the
level of tank 2 is measured (that is, y = ∆h2 ). Using this measurement, design an
estimator that will give continuous, smooth estimates of the deviation in levels of
tank 1 and tank 2 , with estimator error poles at −8σ(1 ± j).
c) Estimator/Controller for Two Cascaded Tanks: Sketch a block diagram (showing individual integrators) of the closed-loop system obtained by combining the estimator
of part (b) with the controller of part (a).
Problem 2
(2 + 2 + 2 = 6 points)
consider the following proper system:
G(s) =
s2 + 4s + 2
s2 + 2s + 1
a) Write the continuous time state space and discretize it with sample time Ts = 1
b) Design a deadbeat controller,
c) Extend the controller to track a reference input r(t).
Note that the transfer function is not strictly proper. In order to use the standard canonical
forms, the transfer function should e proper. Hint: Long division can be used!
Problem 3
(2 + 2 + 2 = 6 points)
Explain the meaning of the following:
a) Uncontrollable system.
b) Non minimum phase system
c) Asymptotic stability
Problem 4
(2 + 2 + 2 = 6 points)
The block diagram of a feedback system is shown in Fig. 2. The system state is x =
and the dimensions of the matrices are as follows:
A = n × n, L = n × 1
B = n × 1, x = 2n × 1
C = 1 × n, r = 1 × 1
K = 1 × n, y = 1 × 1
Figur 2: Block diagram for Problem Problem 4
a) Write state equations for the system.
b) Let x = Tz, where
I 0
Show that the system is not controllable. What does this mean?
c) Find the transfer function of the system from r to y.