Description
1. A second-order process G(s) = 10
s
2 + 7s + 10
is in negative feedback with what is known as a PI
controller Gc(s) = Kc +
KI
s
.
(a) Determine the characteristic equation of the closed-loop (CL) system Gcl(s) = Y (s)/R(s).
(b) Identify the admissible regions of Kc and KI that guarantee CL stability.
(c) Is tracking of set-point guaranteed for any admissible values of Kc and KI?
(d) Demonstrate your findings in SIMULINK by simulating the CL system for a unit step change
in set-point, for two different settings of Kc and KI , one from the admissible and another
from the non-admissible region. Report the chosen values and step responses in each case.
2. A process is given by the transfer function G(s) = 10(s − 4)
s
2 + 7s + 10
e
−3s
.
For this process,
(a) Compute the impulse and step response of the system. Sketch these responses by hand.
(b) Determine the large-time response of the process to the input u(t) = 2 sin(5t) + 3 cos(0.1t)
(c) Construct the Bode plot by hand. Show the working details neatly.
(d) Determine the LTI system that has the same magnitude at all ω but has the lowest phase.
(e) Verify your answers to all parts using MATLAB.
3. The dynamic behavior of the liquid level in a leg of a manometer tube, responding to a change in
pressure, is given by
d
2h
0
dt2
+
6µ
R2ρ
dh0
dt +
3
2
g
L
h
0 =
3
4ρL
p
0
(t)
where h(t) is the level of fluid measured with respect to the initial steady-state value, p(t) is the
pressure change, and R, L, g, ρ, and µ are constants.
(a) Rearrange this equation into standard gain-time constant form and find expressions for K, τ, ζ
in terms of the physical constants.
(b) For what values of the physical constants does the manometer response oscillate?
(c) How would you change the length L of the manometer leg so as to make the response more
oscillatory, or less? Repeat the analysis for an increase in µ (viscosity).
4. The transfer function that relates the change in blood pressure y to change in u the infusion rate
of a drug (sodium nitroprusside) is given by
Gp(s) = Ke−D1s
(1 + αe−D2s
)
τs + 1
The two time delays result from the blood recirculation that occurs in the body, and α is the
recirculation coefficient. The following parameter values are available:
K = −1.2
mm Hg
ml/h ,
α = 0.4, D1 = 30 s, D2 = 45 s, and τ = 40 s
Use Simulink to construct the block diagram and simulate the blood pressure response to a unit
step change (u = 1) in sodium nitroprusside infusion rate. Is it similar to other responses discusses
in the class?