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ENGG 1015 Tutorial

� Systems, Control and Computer Arithmetic

� 11 Dec

� Learning Objectives

� Analyse systems and control systems

� Interpret computer arithmetic

� News

� HW3 deadline (Dec 3)

� Project Competition (Dec 5)

� Ack.: MIT OCW 6.01, 6.003

System Function

� Write an expression for the system function for this

whole system, in terms of n1, d1, n2, d2, n3, d3

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Difference Equations

� Consider the system represented by the following

difference equation

� � = � � +�

�5� � − 1 + 5� � − 2

where x[n] and y[n] represent the nth samples of the input

and output signals, respectively.

� Pole(s) of this system: 3 and -0.5

� Does the unit-sample response of the system converge

or diverge as n→∞? Diverge

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Find the Pole(s)

� Let . . Determine the pole(s) of H3 and the pole(s)

of .

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Finding Equations and Poles

� For k = 0.9

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Conversion between Block Diagrams

(1)� The system that is represented by the following

difference equation y[n] = x[n] + y[n − 1] + 2y[n − 2]

can also be represented by the left block diagram. It is

possible to choose coefficients for the right block

diagram so that the systems represented by the left and

right block diagrams are “equivalent”.

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Conversion between Block Diagrams

(2)� For the left diagram,

�

=

�

�������

� For the right diagram, ,�

= �

�

�����+ �

�

�����

� The two systems are equivalent

if �

�������= �

�

�����+ �

�

�����

� Equating denominators

and numerators,

�� = 2,�� = −1,

A = 2/3 and B = 1/3.

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Feedback (1)

� Let H represent a system with input X and output.

Assume that the system function for H can be written as

a ratio of polynomials in R with constant, real-valued,

coefficients. In this problem, we investigate when the

system H is equivalent to the following feedback system

where F is also a ratio of polynomials in R with constant,

real-valued coefficients.

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Feedback (2)

� Example 1: Systems 1 and 2 are equivalent when

� Example 2: Systems 1 and 2 are equivalent when

� Which expressions for F guarantees equivalence of

Systems 1 and 2?

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Feedback (3)

� Let E represent the output of the adder. Then

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E

What’s Cooking (1)

� Sous vide ("under vacuum") cooking involves cooking

food at a very precise, fixed temperature T (low enough

to keep it moist, but high enough to kill any pathogens).

� In this problem, we model the behavior of the heater and

water bath used for such cooking. Let I be the current

going into the heater, and c be the proportionality

constant such that Ic is the rate of heat input.

� The system is thus described by the following diagram:

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What’s Cooking (2)

� Difference equation of the system:

� � = �� − �� � � − 1 + ����� � − 2 + �� �

� The system function: � =�

�=

��!�" �#!�"

� $ = �� + ��$% ⇒ $ =�

��!�"

� � = $ − ���% ⇒ � ='

�#!�"=

�

�#!�"

�

��!�"

� Let k1 = 0.5, k2 = 3, and c = 1. Determine the poles of H.

� Poles at 0.5 and -3

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What’s Cooking (3)

� Let the system start at rest (all signals are zero).

Suppose I[0]= 100 and I[n]= 0 for n>0.

� What is the plot when k1 = 0.5 and k2 = 0?

� What is the plot when k1 = 1 and k2 = 0.5 ?

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Personal Savings (1)

� You and your friend Waverly have accounts in rival

banks. Each month, your bank deposits your interest

from last month into your account, leaving your new

balance equal to α times your old balance. Waverly’s

bank is similar but the constant is γ instead of α.

� Each month, you make an additional deposit (into your

account) of x[n] dollars plus β times the balance in Waverly’s account from last month. Each month,

Waverly withdraws (from her account) δ times the balance in your account from last month.

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Personal Savings (2)

� We wish to describe the balances in these bank

accounts as a linear system. Let y[n] and w[n] represent

last month’s balances in your account and in Waverly’s

account, respectively. Let x[n] represent the input to the

system, and let w[n] represent the output.

� Determine a system function to describe the relation

between the signals X and W. (The system function

should not depend on Y.)

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Personal Savings (3)

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Personal Savings (4)

� Determine if Waverly’s balance oscillates and diverges

� α = 0.1, β = 0.5, γ = 0.1, δ = 0.5

Oscillates over time; Magnitude converges

� α = 1.1, β = 1.1, γ = 1.1, δ = 1.5

Oscillates over time; Magnitude diverges

� α = 0.5, β = 0.1, γ = 1, δ = 0.1

Not oscillates over time; Magnitude converges

� α = 1.5, β = 0.1, γ = 1, δ = 0

Not oscillates over time; Magnitude diverges

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Computer Arithmetic (1)

� Convert the following decimal values to

binary:a) 205 b) 2133

� Perform the following operations in the 2’s complement system. Use eight bits (including the sign bit) for each number.

a) add +9 to +6 b) add +14 to -17 c) add +19 to -24

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Computer Arithmetic (2)

� Convert the following decimal values to binary:

a) 205 b) 2133

20510 = 1 x 27 + 1 x 26 + 1 x 23

+ 1 x 22 + 1 x 20

= 110011012

213310 = 1 x 211 + 1 x 26

+1 x 24 + 1 x 22 + 1 x 20

= 1000010101012

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Computer Arithmetic (3)

� Perform the following operations in the 2’s

complement system. Use eight bits (including the sign

bit) for each number.

a) add +9 to +6 b) add +14 to -17 c) add +19 to -24

00001001 9

00000110 6

00001111 15

= +

= +

= +

00001110 14

11101111 17

11111101 3

= +

= −

= −

00010011 19

11101000 24

11111011 5

= +

= −

= −

24 00011000= (1's complement) (2's complement)11100111 1024 111 000− = =

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Overflow

� Overflow: Add two positive numbers to get a negative

number or two negative numbers to get a positive

number

For 2’s complement,

(+1)+(+6)

= +7 � OK

(+1)+(+7)

= -8 � Overflow

(-1)+(-8)

= +7 � Overflow

(-6)+(+7)

= -1 � OK

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Addition using 2’s Complement (1)

� Perform the following computations.

� Indicate on your answer if an overflow has

occurred.

� 01000000 + 01000001 (64 + 65)

� 00000111 − 11111001 (7 - -7)

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Addition using 2’s Complement (2)

� 01000000 (64)

+ 01000001 (65)

----------------

10000001 (-127) � Overflow

� 00000111 (7)

+ 00000111 (7)

----------------

00001110 (14) � No Overflow

� 00000111 - 11111001= 00000111 + (-11111001) = 00000111+00000111