## 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?