engg 1015 tutorial system functionculei/engg1015_12/engg1015_tutorial_5.pdf · engg 1015 tutorial...
TRANSCRIPT
-
1
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
2
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
3
Find the Pole(s)
� Let . . Determine the pole(s) of H3 and the pole(s)
of .
4
-
Finding Equations and Poles
� For k = 0.9
5
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”.
6
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.
7
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.
8
-
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?
9
Feedback (3)
� Let E represent the output of the adder. Then
10
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:
11
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
12
-
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 ?
13
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.
14
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.)
15
Personal Savings (3)
16
-
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
17 18
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
19
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
20
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− = =
-
21
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
22
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)
23
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