## Description

Exercise

1. Consider a household storage geyser that provides hot fluid stream to the user by heating the

incoming cold water. Do / answer each of the following:

(a) Classify the process as a continuous / batch / fed-batch process.

(b) Identify the controlled variables, manipulated variables, and disturbance variables.

(c) Propose a feedback control method and sketch the schematic diagram.

(d) Suggest a feed-forward control method and sketch the schematic diagram.

2. An input-output dynamical system is governed by

d

2y

dt2

+ a1

dy

dt + a0y(t) = b0u(t), a1 = 8, a0 = 15, b0 = 3

(a) Re-write the ODE in terms of deviations from steady-state, u˜(t) = u(t) − u(0), y˜(t) =

y(t) − y(0), where u(0) and y(0) are the steady-state input and output, respectively.

(b) AahaOohu wishes to increase the steady-state value of output by 2 units. To realize this

objective, he decides to change the input by a fixed value. Determine the amount of

change in input that is required.

(c) Due to uncertainties in a0 and b0, AahaOohu wishes to deploy a feedback strategy by

changing the input proportional to the error u˜(t) = Kce(t), where e(t) = 2 − y˜(t). Will

the control objective be achieved for any finite value of Kc > 0?

(d) In a different scheme of feedback control, AahaOohu changes the rate of input as u˜˙(t) =

Kce˙(t) + KIe(t). Will the control objective be achieved for any value of Kc and KI?

3. Consider the following model of 2-stage absorption model:

dw

dt = −

L + V a

M

w +

V a

M

z

dz

dt =

L

M

w −

L + V a

M

z +

V

M

zf

where w and z are liquid concentrations on stage 1 and 2, respectively. L and V are the liquid

and vapour molar flow rates, zf si the concentration of the vapour stream entering the column.

The steady-state input values are L = 80 gmol inert liquid/min and V = 100 gmol inert

vapour/min. The parameter values are M = 20 gmol inert liquid, a = 0.5 and zf = 0.1 gmol

solute / gmol inert vapour.

Do / answer each of the following:

(a) Find the steady-state values of w and z.

(b) Obtain a linearized state-space model around the normal steady-state operation assuming

that L and V are the inputs.

(c) Find the eigenvalues of the system. What are the expected “slowest” and “fastest” initial

condition directions of the system?

(d) Set up the non-linear system in MATLAB. Solve for steady-state and obtain a linearized

model using the linear analysis tools in MATLAB/SIMULINK.

(e) Plot and compare the step responses of the non-linear system with that of the linearized

model for two different magnitudes of steps (i) 5% and (ii) 15% change in the flow rate.

4. Answer the following

(a) Find the Laplace Transform of the signal x(t) =

t − 2 0 ≤ t < 3

1 3 ≤ t < 4

− cos(3π(t − 4)) 4 ≤ t < 5

e

−2(t−5) cos(5π(t − 5)) t ≥ 5

.

(b) Find the inverse Laplace transform of X(s) = (s − 2)

s(τ

2s

2 + 2ζτs + 1), where τ > 0. Consider three different cases: (i) ζ > 1, (ii) ζ = 1 and (iii) 0 ≤ ζ < 1