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Assignment ON Optimal Control Theory: Submitted By: Rabiya Azmat M.TECH (CIS) - 13

This document is an assignment submission on optimal control theory. It contains a question regarding a chemical mixing process involving two tanks. The student is asked to write the differential equations of the system using the state variables h1(t), h2(t), v1(t), and v2(t), which represent the head and volume of dye in each tank. The student provides the differential equations as: dh1/dt = w1(t)/a1 + m(t)/a1 – k.[h1(t) – h2(t)]/a1 dh2/dt = w2(t)/a2 – k.[h1(t) – h2(t)]

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54 views3 pages

Assignment ON Optimal Control Theory: Submitted By: Rabiya Azmat M.TECH (CIS) - 13

This document is an assignment submission on optimal control theory. It contains a question regarding a chemical mixing process involving two tanks. The student is asked to write the differential equations of the system using the state variables h1(t), h2(t), v1(t), and v2(t), which represent the head and volume of dye in each tank. The student provides the differential equations as: dh1/dt = w1(t)/a1 + m(t)/a1 – k.[h1(t) – h2(t)]/a1 dh2/dt = w2(t)/a2 – k.[h1(t) – h2(t)]

Uploaded by

alshifa
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© © All Rights Reserved
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ASSIGNMENT

ON
OPTIMAL CONTROL THEORY

SUBMITTED BY:

RABIYA AZMAT

M.TECH(CIS) -13
Question No. 1.6 :

A chemical mixing process is shown in fig. below. Water enters the tanks at rates
of w1(t) and w2(t) ft3/min and m(t) ft3/min of dye enters tank 1. v1(t) and v2(t) ft3
of dye are present in tanks 1 and 2 at time t. The tanks have cross-sectional areas
a1 and a2. Assume that flow rate between the two tanks q(t), is proportional to the
difference in head with proportionality constant k ft3/ft-min, and that the mixtures
in the tanks are homogeneous. Write the differential equations of the system ,
using h1(t),h2(t), v1(t) and v2(t) as state variables.

Solution:

Taking h1(t),h2(t), v1(t) and v2(t) as state variables,

dh1/dt = w1(t)/a1 + m(t)/a1 k.[h1(t) h2(t)]/a1

dh2/dt = w2(t)/a2 k.[h1(t) h2(t)]/a2

dv1(t)/dt = { m(t) k.v1(t). [h1(t) h2(t)]/a1.h1(t) for h1(t) >=h2(t)

m(t) + k.v2(t). [h2(t) h1(t)]/a2.h2(t) for h2(t) >=h1(t) }

dv2(t)/dt = { k.v1(t). [h1(t) h2(t)]/a1.h1(t) for h1(t) >=h2(t)

- k.v2(t). [h2(t) h1(t)]/a2.h2(t) for h2(t) >=h1(t) }

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