A linearized model for the water level in a system of two interconnected water tanks is 2 * = [ -1 -²] * + [i] * x u 0-2 a) Find the transfer function for the system with input u when the output, y, is the level of water in the lower tank. [Hint: you should be able to decide which state that represents the level in the lower tank from the dynamic equations.] b) Assume that we wish to control the level of water in the lower tank with a proportional controller U(s) = K (R(s) - Y(s)). What K> 0 will result in oscillations following a step response? c) What will be the period of the oscillations when K = 5? d) We now set K = 2 and assume that initially the system is at rest with x₁(0) = x₂(0) = 0. What is the maximal size of a step (i.e. r(t) = al(t) where 1 is the heaviside function) that we can make without water spilling from the lower tank which has a capacity to hold 1 unit of water? (We assume that the upper tank has unlimited capacity.)
A linearized model for the water level in a system of two interconnected water tanks is 2 * = [ -1 -²] * + [i] * x u 0-2 a) Find the transfer function for the system with input u when the output, y, is the level of water in the lower tank. [Hint: you should be able to decide which state that represents the level in the lower tank from the dynamic equations.] b) Assume that we wish to control the level of water in the lower tank with a proportional controller U(s) = K (R(s) - Y(s)). What K> 0 will result in oscillations following a step response? c) What will be the period of the oscillations when K = 5? d) We now set K = 2 and assume that initially the system is at rest with x₁(0) = x₂(0) = 0. What is the maximal size of a step (i.e. r(t) = al(t) where 1 is the heaviside function) that we can make without water spilling from the lower tank which has a capacity to hold 1 unit of water? (We assume that the upper tank has unlimited capacity.)
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Explain the state space function
![A linearized model for the water level in a system of two interconnected water tanks is
[2/2] x + [i] "
-2
=
[
-1
a) Find the transfer function for the system with input u when the output, y, is the level of water
in the lower tank. [Hint: you should be able to decide which state that represents the level in the
lower tank from the dynamic equations.]
b) Assume that we wish to control the level of water in the lower tank with a proportional controller
U(s) = K(R(s) – Y(s)). What K > 0 will result in oscillations following a step response?
c) What will be the period of the oscillations when K = 5?
=
d) We now set K 2 and assume that initially the system is at rest with x₁(0) = x₂(0) = 0. What is
the maximal size of a step (i.e. r(t) = al(t) where 1 is the heaviside function) that we can make
without water spilling from the lower tank which has a capacity to hold 1 unit of water? (We
assume that the upper tank has unlimited capacity.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F776c0bff-c672-4363-b707-790350ffaf88%2F20cfd3b4-6240-4edd-8629-8d825d7b6307%2Fuxyipa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A linearized model for the water level in a system of two interconnected water tanks is
[2/2] x + [i] "
-2
=
[
-1
a) Find the transfer function for the system with input u when the output, y, is the level of water
in the lower tank. [Hint: you should be able to decide which state that represents the level in the
lower tank from the dynamic equations.]
b) Assume that we wish to control the level of water in the lower tank with a proportional controller
U(s) = K(R(s) – Y(s)). What K > 0 will result in oscillations following a step response?
c) What will be the period of the oscillations when K = 5?
=
d) We now set K 2 and assume that initially the system is at rest with x₁(0) = x₂(0) = 0. What is
the maximal size of a step (i.e. r(t) = al(t) where 1 is the heaviside function) that we can make
without water spilling from the lower tank which has a capacity to hold 1 unit of water? (We
assume that the upper tank has unlimited capacity.)
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