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Practice Sheet 1

The document contains 7 questions that provide linear system models and ask to derive state space representations and transfer functions. Question 1 provides a second order differential equation model of a DC motor and asks to express it in state variable form and find the transfer function between input voltage and shaft angle. Question 2 linearizes the equation of motion for a simple pendulum and asks for the state space model and transfer function. Question 3 gives a second order transfer function of an aircraft and asks for the state space model.

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50 views4 pages

Practice Sheet 1

The document contains 7 questions that provide linear system models and ask to derive state space representations and transfer functions. Question 1 provides a second order differential equation model of a DC motor and asks to express it in state variable form and find the transfer function between input voltage and shaft angle. Question 2 linearizes the equation of motion for a simple pendulum and asks for the state space model and transfer function. Question 3 gives a second order transfer function of an aircraft and asks for the state space model.

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apeelec10337
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Linear System Theory

Practice Sheet 1

Question #1:
The dynamics of motion for the DC motor shown in the figure above are expressed by the
following second order differential equation.

𝐊𝐭𝐊𝐞 𝐊𝐭
𝐉𝐦 𝛉̈𝐦 + 𝐛 + 𝛉̇𝐦 = 𝐯
𝐑𝐚 𝐑𝐚 𝐚

Assume that

Moment of inertia of the rotor Jm = 0.01 kg.m2


Damping ratio of the mechanical system b = 0.001 N.m.sec
Electromotive force constant K=Ke=Kt = 0.02 Nm/Amp
Electric resistance Ra = 1 ohm
Electric inductance L = 0.5 H
a) Express the above equation in state-variable form.
b) Find the transfer function between the applied voltage 𝐯𝐚 and the shaft
angle 𝛉𝐦

Question #2:
The equation of motion for the simple pendulum is

Tc
Given that 𝜃 is very small the system can be linearized, find state space model and transfer
function..
Question #3: (20)
Consider the second order transfer function of aircraft as

Find the state space model.

Question #4:
You wish to control the elevation of the satellite-tracking
antenna shown in Figure.

The antenna and drive parts have a moment of inertia J and


damping B; these arise to some extent from bearing and
aerodynamic friction but mostly from the back EMF of the
DC drive motor. The equations of motion are
J   B  Tc
Find the state space model.

where Tc is the torque from the drive motor, assume that


J = 680,000 kg.m2, B = 10,000 N.m.sec
Question #5: (10)
Inverted pendulum, shown below, is an important bench mark in
control theory where a mass is stabilized at an upward equilibrium with
a cart movement. The linearized mathematical model of the inverted
pendulum is:
(𝐼 + 𝑚𝑙 )𝜃̈ − 𝑚𝑔𝑙𝜃 = 𝑚𝑙𝑥̈
(𝑀 + 𝑚)𝑥̈ + 𝑏𝑥̇ − 𝑚𝑙𝜃̈ = 𝐹
where, 𝜽 is the tilt of the beam and the desired output, 𝜽̇ is the angular
velocity of the beam, 𝒙 is the linear position of the cart, 𝒙̇ = 𝒗 is the
velocity of the cart and 𝑭 is the force applied to the cart. Find the state-
space 𝑨, 𝑩, 𝑪, 𝑫 matrices for the following state vector.
𝒙 = [𝒙𝟏 𝒙𝟐 𝒙𝟑 ] = [𝜽 𝜽̇ 𝒗]
Assume that:
𝐼 + 𝑚𝑙 = 1; 𝑀 + 𝑚 = 1; 𝑚𝑔𝑙 = 2; 𝑚𝑙 = 0.5; 𝑏 = 1

Question #6:
a) For the two-tank fluid flow system shown in figure below. Find the differential equation
relating the flow into the first tank to the flow out of the second tank.

in

out

where
A1, A2 = area of the tanks. R1, R2 = cross-section of outlet valves.
h1, h2 = m/Aρ = height of water tanks. m1, m2 = mass of water in tanks.
ρ = density of the water. g = force of gravity.

b) Linearize the following equation around the point  = 0.


(l  R )  R 2  g sin   
c) Find the state-space representation for the following model.
 
J11  J 21  b 1  2  k 1   2   Tc
 
J12  J 22  b 2  1  k  2  1   0

Question #7:
Model for the pitch control of an aircraft is given by the state variable equations. Find the transfer
function.

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