CH-5120: Modern Control Theory Assignment 3 solution

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Original Work ?

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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., �(�) = !’!”.\$ &
“#\$%

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.