UQ Robotics Teachingrobotics.itee.uq.edu.au/~elec3004/2015/lectures/L13...SECaTs: Some Lessons in...

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Estimation + Information Theory

© 2014 School of Information Technology and Electrical Engineering at The University of Queensland

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http://elec3004.org

ELEC 3004: Systems 1 June 2015 2

Thank you!

http://itee.uq.edu.au/~metr4202/

http://itee.uq.edu.au/~metr4202/

http://creativecommons.org/licenses/by-nc-sa/3.0/au/deed.en_US

http://elec3004.com/

http://elec3004.com/

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Schedule

1 June 2015 ELEC 3004: Systems 3

Week Date Lecture Title

1 2-Mar Introduction

3-Mar Systems Overview

2 9-Mar Signals as Vectors & Systems as Maps

10-Mar [Signals]

3 16-Mar Sampling & Data Acquisition & Antialiasing Filters

17-Mar [Sampling]

4 23-Mar System Analysis & Convolution

24-Mar [Convolution & FT]

5 30-Mar Discrete Systems & Z-Transforms

31-Mar [Z-Transforms]

6 13-Apr Frequency Response & Filter Analysis

14-Apr [Filters]

7 20-Apr Digital Filters

21-Apr [Digital Filters]

8 27-Apr Discrete Systems Analysis

28-Apr [Feedback]

9 4-May Introduction to (Digital) Control

5-May [Digitial Control]

10 11-May Digital Control Design

12-May [Introduction to State-Space]

11 18-May State-Space - Analysis

19-May [Stability]

12 25-May Digital Control Systems: Shaping the Dynamic Response

26-May [Applications in Industry]

13 1-Jun System Identification & [Summary and Course Review]

2-Jun Estimation + Information Theory

Lab 4


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Final Exam Review

• Monday, June 22, 2015

• From: 2-4p

• In Hawken: 50-T105

• Tuesday June 9, 2015

• From 2-4p

• In TBA

ELEC 3004: Systems 1 June 2015 5

Estimation:

Yet another way

to beat the noise


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Along multiple dimensions


State Space • We collect our set of uncertain variables into a vector …

x = [x1, x2,…, xN]T

• The set of values that x might take on is termed the state space

• There is a single true value for x,

but it is unknown


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State Space Dynamics


Measured versus True • Measurement errors are inevitable

• So, add Noise to State... – State Dynamics becomes:

• Can represent this as a “Normal” Distribution


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Recovering The Truth • Numerous methods

• Termed “Estimation” because we are trying to estimate the

truth from the signal

• A strategy discovered by Gauss

• Least Squares in Matrix Representation


Recovering the Truth: Terminology


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General Problem…


Duals and Dual Terminology


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Estimation Process in Pictures


Kalman Filter Process


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KF Process in Equations


KF Considerations


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Ex: Kinematic KF: Tracking • Consider a System with Constant Acceleration


In Summary

• KF: – The true state (x) is separate from the measured (z)

– Lets you combine prior controls knowledge with

measurements to filter signals and find the truth

– It regulates the covariance (P) • As P is the scatter between z and x

• So, if P 0, then z x (measurements truth)

• EKF: – Takes a Taylor series approximation to get a local “F” (and

“G” and “H”)


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“The fundamental problem of communication is that of reproducing at one point,

either exactly or approximately, a message selected at another point.”

On the transmission of information over a noisy channel: • An information source that produces a message

• A transmitter that operates on the message to create a signal which can be sent through a

channel

• A channel, which is the medium over which the signal, carrying the information that

composes the message, is sent

• A receiver, which transforms the signal back into the message intended for delivery

• A destination, which can be a person or a machine, for whom or which the message is

intended

Shannon Information Theory


Three “problems” in Communication:

• The technical problem: how

accurately can the message be

transmitted?

• The semantic problem: how precisely

is the meaning “conveyed”?

• The effectiveness problem: how

effectively does the received meaning

affect behaviour?

Shannon and Weaver: Models of Communication

Source:

http://en.wikipedia.org/wiki/File:Transactional_comm_model.jpg


http://en.wikipedia.org/wiki/Signal_(electrical_engineering)

http://en.wikipedia.org/wiki/Semantic



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Today’s Lecture is Brought To You By the Number 5


The course is has a huge mandate:

• It is really 3 ∙ ½ courses in one ! – Linear Systems

– Signal Processing

– Controls & Digital Controls

• ∴ It is b r o a d !!

• There is a logic to it – They share the same

mathematical nature (poles & zeros)

– The math is common

to more than just circuits!

SECATs: Let’s look back at the topic list from Lecture 1


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Lots of Stuff To Cover… • Systems

• Signal Abstractions

• Signals as Vectors / Systems as Maps

• Linear Systems and Their Properties

• LTI Systems

• Autonomous Linear Dynamical Systems

• Convolution

• FIR & IIR Systems

• Frequency domain

• Fourier Transform (CT)

• Fourier Transform (DT)

• Even and Odd Signals

• Likelihood

• Causality

• Impulse Response

• Root Locus

• Bode Functions

• Left-hand Plane

• Frequency Response

• Discrete Time

• Continuous Time

• Laplace Transformation

• Feedback and Control

• Additional Applications

• Linear Functions

• Linear Algebra Review

• Least Squares

• Least Squares Problems

• Least Squares Applications

• Matrix Decomposition and Linear

Algebra

• Regularized Least Squares

• Least-squares

• Least-squares applications

• Orthonormal sets of vectors

• Eigenvectors and diagonalization

• Linear dynamical systems with inputs

and outputs

• Symmetric matrices, quadratic forms,

matrix norm, and SVD

• Controllability and state transfer

• Observability and state estimation

• And that, of course,

Linear Systems are Cool!

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Yes, this is Hard! Why? • Breath

– Books, books, everywhere, yet we’re all on Wikipedia!!

– Authors tend to be “too generalizable”

• Assumptions: – Numerous conditions that need to be remembered

• Tacit Details: The need for examples (but these are few and always seem the same)

• Time consuming


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• I shall only use my own slides

• Less is more! – Smaller assignments

– More time for Examples

• Better organization – Better tutorials

– More examples!!

– I get that. But, we’ve come a long way

To make this happen I need your support!

SECaTs: Some Lessons in the Works for Next Year


• We’re at the End. It’s (the) final!

• Thank you folks!

Next Time in Linear Systems ….


Week Date Lecture Title

1 2-Mar Introduction

3-Mar Systems Overview

2 9-Mar Signals as Vectors & Systems as Maps

10-Mar [Signals]

3 16-Mar Sampling & Data Acquisition & Antialiasing Filters

17-Mar [Sampling]

4 23-Mar System Analysis & Convolution

24-Mar [Convolution & FT]

5 30-Mar Discrete Systems & Z-Transforms

31-Mar [Z-Transforms]

6 13-Apr Frequency Response & Filter Analysis

14-Apr [Filters]

7 20-Apr Digital Filters

21-Apr [Digital Filters]

8 27-Apr Discrete Systems Analysis

28-Apr [Feedback]

9 4-May Introduction to (Digital) Control

5-May [Digitial Control]

10 11-May Digital Control Design

12-May [Introduction to State-Space]

11 18-May State-Space - Analysis

19-May [Stability]

12 25-May Digital Control Systems: Shaping the Dynamic Response

26-May [Applications in Industry]

13 1-Jun System Identification & [Summary and Course Review]

2-Jun Estimation + Information Theory

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Now Finally Some Philosophy (I am a Dr of it!!!)

Systems: Signals, Controls…A Fundamental Yearn!

© National Geographic. Mount Everest at night (the lights along the apex are the headlamps of other mountaineers)

We keep moving forward, opening new doors, and doing new things because we're curious and curiosity keeps leading us down new paths. -Walt Disney


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