25-apr holiday 21-apr fft 28-apr feedback 18-apr...
TRANSCRIPT
1
Signals as Vectors Systems as Maps
& Data Acquisition
© 2014 School of Information Technology and Electrical Engineering at The University of Queensland
TexPoint fonts used in EMF.
Read the TexPoint manual before you delete this box.: AAAAA
http://elec3004.com
Lecture Schedule: Week Date Lecture Title
29-Feb Introduction
3-Mar Systems Overview
7-Mar Systems as Maps & Signals as Vectors
10-Mar Data Acquisition & Sampling14-Mar Sampling Theory
17-Mar Antialiasing Filters
21-Mar Discrete System Analysis
24-Mar Convolution Review
28-Mar
31-Mar
11-Apr Digital Filters
14-Apr Digital Filters
18-Apr Digital Windows
21-Apr FFT
25-Apr Holiday
28-Apr Feedback
3-May Introduction to Feedback Control
5-May Servoregulation/PID
9-May Introduction to (Digital) Control
12-May Digitial Control
16-May Digital Control Design
19-May Stability
23-May Digital Control Systems: Shaping the Dynamic Response & Estimation
26-May Applications in Industry
30-May System Identification & Information Theory
31-May Summary and Course Review
Holiday
1
13
7
8
9
10
11
12
6
2
3
4
10 March 2016 ELEC 3004: Systems 2
2
Systems as Maps
10 March 2016 ELEC 3004: Systems 3
• A basis of a vector space is a sequence of vectors that has two
properties at once:
1. The vectors are linearly independent
2. The vectors span the space
The vectors v1, …, vn are a basis for ℝn exactly when they are
the columns of a n×n invertible matrix. Thus ℝn has infinitely
many bases.
Basis Spaces of a Signal
10 March 2016 ELEC 3004: Systems 4
3
Then a System is a MATRIX
10 March 2016 ELEC 3004: Systems 6
• Linear & Time-invariant (of course - tautology!)
• Impulse response: h(t)=F(δ(t))
• Why? – Since it is linear the output response (y) to any input (x) is:
• The output of any continuous-time LTI system is the convolution of
input u(t) with the impulse response F(δ(t)) of the system.
Linear Time Invariant
LTI
h(t)=F(δ(t)) u(t) y(t)=u(t)*h(t)
10 March 2016 ELEC 3004: Systems 7
4
≡ LTI systems for which the input & output are linear ODEs
• Total response = Zero-input response + Zero-state response
Linear Dynamic [Differential] System
Initial conditions External Input
10 March 2016 ELEC 3004: Systems 8
• Linear system described by differential equation
Linear Systems and ODE’s
• Which using Laplace Transforms can be written as
0 1 0 1
n m
n mn m
dy d y dx d xa y a a b x b b
dt dt dt dt
)()()()(
)()()()()()( 1010
sXsBsYsA
sXsbssXbsXbsYsassYasYa m
m
n
n
where A(s) and B(s) are polynomials in s
10 March 2016 ELEC 3004: Systems 9
5
• δ(t): Impulsive excitation
• h(t): characteristic mode terms
Unit Impulse Response
LTI
F(δ(t)) δ(t) h(t)=F(δ(t))
Ex:
10 March 2016 ELEC 3004: Systems 10
Consider the following system:
• How to model and predict (and control the output)?
This can help simplify matters… An Example
Source: EE263 (s.1-13)
10 March 2016 ELEC 3004: Systems 11
6
Consider the following system:
• How to model and predict (and control the output)?
This can help simplify matters… An Example
Source: EE263 (s.1-13)
10 March 2016 ELEC 3004: Systems 12
• Consider the following system:
• x(t) ∈ ℝ8, y(t) ∈ ℝ1 8-state, single-output system
• Autonomous: No input yet! ( u(t) = 0 )
This can help simplify matters… An Example
Source: EE263 (s.1-13)
10 March 2016 ELEC 3004: Systems 13
7
• Consider the following system:
This can help simplify matters… An Example
Source: EE263 (s.1-13)
10 March 2016 ELEC 3004: Systems 14
This can help simplify matters… An Example
Source: EE263 (s.1-13)
10 March 2016 ELEC 3004: Systems 15
8
Expand the system to have a control input…
• B∈ ℝ8×2, C ∈ ℝ2×8 (note: the 2nd dimension of C)
• Problem: Find u such that ydes(t)=(1,-2)
• A simple (and rational) approach: – solve the above equation!
– Assume: static conditions (u, x, y constant)
Solve for u:
Example: Let’s consider the control…
10 March 2016 ELEC 3004: Systems 16
Example: Apply u=ustatic and presto!
• Note: It takes 1500 seconds for the y(t) to converge …
but that’s natural … can we do better? Source: EE263 (s.1-13)
10 March 2016 ELEC 3004: Systems 17
9
• How about:
Example: Yes we can!
Source: EE263 (s.1-13)
10 March 2016 ELEC 3004: Systems 18
• How about:
• Converges in 50 seconds (3.3% of the time )
Example: How? How about a more clever input?
Source: EE263 (s.1-13)
10 March 2016 ELEC 3004: Systems 19
10
• Converges in 20 seconds (1.3% of the time )
Example: Can we beat it? Larger inputs & LDS
Source: EE263 (s.1-13)
10 March 2016 ELEC 3004: Systems 20
• Matlab: deconvwnr
Ex: Deblurring
10 March 2016 ELEC 3004: Systems 23
11
What about …
10 March 2016 ELEC 3004: Systems 24
• For small current inputs, neuron membrane potential output
response is surprisingly linear.
• Though this has limits …
neurons “spike” are (quite) nonlinear (truly)
What about …
Source: (s.3-49)
10 March 2016 ELEC 3004: Systems 25
12
Linear Dynamical Systems Review
10 March 2016 ELEC 3004: Systems 26
Linear Differential Systems
10 March 2016 ELEC 3004: Systems 27
13
• In practice: m ≤ n
∵ if m > n:
then the system is an
(m - n)th -order differentiator of high-frequency signals!
• Derivatives magnify noise!
Linear Differential System Order
y(t)=P(D)/Q(D) f(t)
P(D): M
Q(D): N
(yes, N is deNominator)
10 March 2016 ELEC 3004: Systems 28
Second Order Systems
10 March 2016 ELEC 3004: Systems 29
14
Second Order Systems
10 March 2016 ELEC 3004: Systems 30
Three Types:
• I: Underdamped:
Second Order Response
10 March 2016 ELEC 3004: Systems 31
15
Three Types:
• II: Critically Damped:
Second Order Response
10 March 2016 ELEC 3004: Systems 32
Three Types:
• III: Over Damped:
Second Order Response
10 March 2016 ELEC 3004: Systems 33
16
Unit-Step Response
Second Order Response
Normalize
10 March 2016 ELEC 3004: Systems 34
Second Order Response Envelope Curves
10 March 2016 ELEC 3004: Systems 35
17
• Delay time, td: The time required for the response to reach half the final value
• Rise time, tr: The time required for the response to rise from 10% to 90%
• Peak time, tp:The time required for the response to reach the first peak of the overshoot
• Maximum (percent) overshoot, Mp:
• Settling time, ts: The time to be within 2-5% of the final value
Second Order Response Unit Step Response Terms
10 March 2016 ELEC 3004: Systems 36
Example: Quarter-Car Model
10 March 2016 ELEC 3004: Systems 37
18
Example: Quarter-Car Model (2)
10 March 2016 ELEC 3004: Systems 38
Data Acquisition
10 March 2016 ELEC 3004: Systems 39
19
Digital Signal • Representation of a signal against a discrete set
• The set is fixed in by computing hardware
• Can be scaled or normalized … but is limited
• Time is also discretized
10 March 2016 ELEC 3004: Systems 40
Analog vs Digital
• Analog Signal: An analog or analogue signal is any
variable signal continuous in both time and
amplitude
• Digital Signal: A digital signal is a signal that is both
discrete and quantized
E.g. Music stored in a
CD: 44,100 Samples
per second and 16 bits
to represent amplitude
10 March 2016 ELEC 3004: Systems 41
20
Representation of Signal
• Time Discretization • Digitization
0 5 10 150
100
200
300
400
500
600
time (s)
Ex
pec
ted
sig
nal (m
V)
Coarse time discretization
True signal
Discrete time sampled points
0 5 10 150
100
200
300
400
500
600
time (s)
Ex
pec
ted
sig
nal
(mV
)
Coarse signal digitization
True signal
Digitization
10 March 2016 ELEC 3004: Systems 42
Signal: A carrier of (desired) information [1]
• Need NOT be electrical: • Thermometer
• Clock hands
• Automobile speedometer
• Need NOT always being given
– “Abnormal” sounds/operations
– Ex: “pitch” or “engine hum” during machining as an
indicator for feeds and speeds
10 March 2016 ELEC 3004: Systems 43
21
Signal: A carrier of (desired) information [2]
• Electrical signals
– Voltage
– Current
• Digital signals
– Convert analog electrical signals to an appropriate
digital electrical message
– Processing by a microcontroller or microprocessor
10 March 2016 ELEC 3004: Systems 44
Transduction (sensor to an electrical signal) • Sensor reacts to environment (physics)
• Turn this into an electrical signal: – V: voltage source
– I: current source
• Measure this signal – Resistance
– Capacitance
– Inductance
10 March 2016 ELEC 3004: Systems 45
22
Ex: Current-to-voltage conversion
• simple:
Precision Resistor
• better:
Use an “op amp”
10 March 2016 ELEC 3004: Systems 46
BUT there is Noise …
Note: this picture illustrates the concepts but it is not quantitatively precise
Source: Prof. M. Siegel, CMU 10 March 2016 ELEC 3004: Systems 47
23
• Cross-coupled measurements
• Cross-talk (at a restaurant or even a lecture)
• A bright sunny day obstructing picture subject
• Strong radio station near weak one
• observation-to-observation variation – Measurement fluctuates (ex: student)
– Instrument fluctuates (ex: quiz !)
• Unanticipated effects / variation (Temperature)
• One man’s noise might be another man’s signal
Noise: “Unwanted” Signals Carrying Errant Information
10 March 2016 ELEC 3004: Systems 48
Noise: Fundamental Natural Sources • Voltage (EMF) – Capacitive & Inductive Pickup
• Johnson Noise – thermal / Brownian
• 1/f (Hooge Noise)
• Shot noise (interval-to-interval statistical count)
10 March 2016 ELEC 3004: Systems 49
24
SNR : Signal to Noise Ratio
10 March 2016 ELEC 3004: Systems 50
-6 -4 -2 0 2 4 6
-1
-0.5
0
0.5
1
t
sin(10 t) + 0.1 sin(100 t)
-6 -4 -2 0 2 4 6
-1
-0.5
0
0.5
1
t
sin(10 t) + 0.1 sin(100 t)
Derivatives magnify noise!
•
•
•
•
-6 -4 -2 0 2 4 6
-20
-15
-10
-5
0
5
10
15
20
t
10 cos(10 t) + 10 cos(100 t)-6 -4 -2 0 2 4 6
-10
-5
0
5
10
t
10 cos(10 t)
10 March 2016 ELEC 3004: Systems 51
25
• Register on Platypus
• Try the practise assignment
• We will talk about Data Acquisition / Sampling
Next Time…
10 March 2016 ELEC 3004: Systems 52