## Description

## Problem 1: Model Conversion (3 + 3 points)

Consider the first-order system from previous assignment: �(�) = !

“#$%

which can be rewritten in the convenient form:

�(�) = 5�

� + � , � = 1/�

This can be converted into observable canonical form:

��

�� = (−�)� + (5�)�, � = (1)�

The terms in red-color font are the terms �&, �&, �& in the state-space model.

Recall that the value of � was based on the last digit of your roll number. If the last digit of

your roll number is a, then � = 0.5(1 + �) and � = 1/�.

Thus, if your roll number is CH20D000, the a = 0, � = 0.5 and � = 2.

Thus, if your roll number is CH20D999, the a = 9, � = 5 and � = 0.2.

Question 1: Discretize the above model (highlighted in yellow) and obtain a discrete-time state space

model. Use sampling time of Δ� = 0.5. Please do hand-calculations and avoid using MATLAB.

Question 2: As in Assignment-2, repeat the above for an input delay of 1.5, i.e., �(�) = !’!”.$ &

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## Problem 2: Model Linearization (4 + 4 points)

Consider a reactor as shown in the adjoining figure. The

CVs are liquid level and concentration whereas inlet flowrate, Fin

is the MV. The model of the form �̇ = �(�, �) is given by:

�ℎ

�� = 1

� >�() − �√ℎB, � = 0.2; � = 0.5

��

�� = �()

�ℎ (�() − �) − ��*, � = 1.5.

As we have been doing in earlier assignments, we will choose the

inlet concentration based on the last digit of your roll number “a” as: �() = 5 + 0.2�. Thus, if your

roll number is CH20D999, then Cin = 5+0.2*9 = 6.8.

Fin

Question 3

The system is to be linearized around the steady state. The first step is to compute the steady

state values of h and C. Consider the operating point at Fin = 0.5 and compute the steady state.

Question 4

Linearize the model around the steady state values, (�##, �##) computed in the above problem.

You can linearize using Taylor’s series expansion:

�(�, �) = �(�##, �##) +

��

��

G

(,&&,.&&)

(� − �##) +

��

��

G

(,&&,.&&)

(� − �##)

Defining deviation variables, �0 = � − �##, �0 = � − �##, �0 = � − �##, obtain the linear

state space model of the form:

��0

�� = �&�0 + �&�0

, �0 = ��0

## Problem 3: MIMO System (6 points)

Consider the MIMO system from Assignment-2:

�(�) =

⎣

⎢

⎢

⎢

⎡ 2

40�* + 16� + 1

0.5

20�* + 7� + 1

1.2

10�* + 5� + 1

1

36�* + 12� + 1⎦

⎥

⎥

⎥

⎤

The following steps were performed in MATLAB (you will do something similar next week):

• We used the tf function to obtain the above four transfer functions: Gp11, Gp12, Gp21, Gp22.

• We then used the ss function to obtain the four state-space models: Gc11, Gc12, Gc21, Gc22.

• We used c2d with Δ� = 2 to obtain discrete-time state space models: G11, G12, G21, G22.

�%% = 2

40�* + 16� + 1

⇒ �%% = T

0.42 −0.27

0.17 0.96 V, �%% = T 0.68

0.097V, �%% = [0 0.8]

�%* = 0.5

20�* + 7� + 1

⇒ �%* = T

0.43 −0.28

0.35 0.92 V, �%* = T

0.35

0.1 V, �%* = [0 0.4]

�*% = 1.2

10�* + 5� + 1

⇒ �*% = T

0.27 −0.47

0.30 0.86 V, �*% = T

0.59

0.18V, �*% = [0 0.96]

�** = 1

36�* + 12� + 1

⇒ �** = T

0.48 −0.32

0.18 0.96 V, �** = T

0.72

0.1 V, �** = [0 0.44]

Question 5

Combine the above four SISO state-space models to obtain a single MIMO model of the form

�(� + 1) = ��(�) + ��(�), �(�) = ��(�)

Hint: Since you have 2 CVs and 2 MVs, � ∈ ℝ*, � ∈ ℝ*. When you combine the four models, you

will get eight states in the model, i.e., � ∈ ℝ1.