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MAE143a Signals & Systems Lecture Notes for Week 1/10 •

PU R PO SE O F CO U R SE & C O URSE OR G A N I ZATI O N

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EX AMPLES O F DYN A MI C SYST EMS AN D T H EI R SI GN A LS ( C H. 1)

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CO N T I N U O US- T I ME SI G N ALS AN D D I SC R ET E - TI ME SA MPLI N G (C H . 1 A N D C H . 15 PP. 542 - 545, 547 -549)

COU R SE WEB SI T E: H T TP :/ /M EC HATRO NI CS. UC SD. EDU /M AE 143 A

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Welcome to our MAE143a Course! Prof. Raymond A. de Callafon is with the Department of Mechanical and Aerospace Engineering (MAE) at the Univ. of California, San Diego (UCSD). q Affiliated within the Dynamic Systems & Control (DS&C) group q Involved in teaching and research that covers many aspects in signal processing, estimation, experiment-based modeling and adaptive control. q Application of research to control of mechanical servo systems, vibration modeling of aerospace systems, adaptive noise cancellation, and active control of power flow in (renewable) energy systems Week 1/10

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Aim of MAE143a Course Understand dynamic aspects of systems and the signals these dynamic systems may process and create. Recognize similarities of dynamic behavior between engineering (mechanical, electrical, aerospace) systems.

Provide introduction to signal analysis and the analysis and design of (linear) dynamic systems

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Aim of MAE143a Course – example biker Why? q Next to static analysis q Important to understand dynamic behavior

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Aim of MAE143a Course – example biker Observer and analyze the signals the system creates q For example: acceleration of biker’s head… Static What is max/min? What is average? Dynamic What is slew rate? Frequency contents? What is it’s position?

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Aim of MAE143a Course – example engineer Why? q Your work as an engineer: email, meetings, spreadsheets, design, coding and lots, lots of (big) data…

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Aim of MAE143a Course – concepts & tools Concepts q Differential equations & difference equations q Continuous-time & discrete-time signals q Input/output systems with “states” q Impulse response & Transfer Function Tools q Fourier transform (frequency analysis) q Laplace and Z-transform (solving differential/difference equations) q Frequency response, Bode plots & Filtering (dynamic analysis)

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MAE143A, 2018 - R.A. DE CALLAFON

Aim of MAE143a Course – illustration/examples We will illustrate concepts/tools on examples throughout the course

Inertial aerospace system

Mechanical (vibration) system

Electric circuit dynamics

General Input/Output Systems

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Course Organization Lectures, Section & Office Hours q Lectures on Monday/Wednesday/Friday 2pm-2:50pm (Peter 108) Used to present concepts, tools & some examples. q Section Hour by TAs on Mondays 8pm-8:50pm (Center 101) Used to go over examples/problems. q Office Hours by TAs on Monday/Wednesday/Friday and posted on the course website http://mechatronics.ucsd.edu/mae143a

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Course Staff Professor q R.A. de Callafon: [email protected] TAs q Raja Annapooranan: [email protected] Behrooz Amini: [email protected] Yangsheng Hu: [email protected] Reader

q Bharath Ramling: [email protected] NOTE: instead of emailing questions on course material, please pose questions during lectures, section & office hours!

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Course Textbook Classic (and good) Text Books q Signals and Systems, Oppenheim, Willsky, and Young, Prentice-Hall, 1983, ISBN:0-13-809731-3 q Fundamentals of Signals and Systems, 1st Edition, M. J. Roberts, McGraw-Hill, 2008. ISBN 978-0073404547. q Signals and Systems: Analysis Using Transform Methods and MATLAB, 2nd edition, M. J. Roberts, McGraw Hill, 2012. Our recommended text book: q Fundamentals of Signals & Systems, B. Boulet, Charles River Media, 2006, ISBN: 1-58450-381-5. https://mlichouri.files.wordpress.com/2013/10/fundamentals-of-signals-and-systems.pdf

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Course Material and Course Website q Recommended Text Book (Fundamentals of Signals & Systems, B. Boulet, 2006) q Lectures notes for each week o One set of lecture notes/week, covering the 3 lectures of that week. o References to “Fundamentals of Signals & Systems, B. Boulet, Charles River Media, 2006”. o Posted on/before start of Week (Monday’s)

q Homework and Homework Solutions

q Midterm and Midterm Solutions Additional course material is posted on course website: http://mechatronics.ucsd.edu/mae143a Week 1/10

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Course Lecture Notes/Topics by Week q Week 1 (start of Ch. 1 & 15) q Purpose of course & Course organization q Dynamic systems and their signals (Ch. 1) q Continuous-time signals and discrete-time sampling (Ch. 1 and Ch. 15, pp. 542)

q Week 2 (Ch. 1) q Fundamental signals in continuous- and discrete-time q Elementary (continuous- & discrete-time) signal operations q Properties of deterministic and stochastic signals

q Week 3 (Ch. 1 & 2) q Linear Time Invariant (LTI) systems (Ch. 2) q Convolution Sum and Integral (Ch. 2) q Basic properties and connections of LTI systems (Ch. 1, pp. 35 - 39)

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Course Lecture Notes/Topics by Week q Week 4 (Ch. 3 & 10) q LTI systems: differential & difference equations (Ch. 3) q LTI systems: state space equations (Ch. 10, pp. 352) q Definition of the impulse response and notion of stability (Ch. 3)

q Week 5 (Ch. 4, 5 & 13) q Fourier Series for Periodic Signals (Ch. 4) q Fourier transform of continuous-time signals (Ch. 5) q Discrete Fourier transform of discrete-time signals (Ch. 13)

q Week 6 (Ch 4, 5 & 13) q Spectrum of a signal, Parseval's Theorem and Sampling Theorem (Ch. 4) q LTI systems - sinusoidal response (Ch. 4, page 155) q Application of Filtering and Frequency Analysis of signals (Ch. 5, Ch. 13)

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Course Lecture Notes/Topics by Week q Week 7 (Ch. 6) q La Place transform for continuous-time signals q La Place transform of common signals and Final Value Theorem q La Place transform for solving differential equations

q Week 8 (Ch. 7) q Application of La Place Transform to Linear Time Invariant systems q Continuous-time transfer function, state space and stability q Frequency Response, Bode Diagram and Filtering

q Week 9 (Ch. 13) q Z-transform: La Place for discrete-time signals and systems q Discrete-time transfer function, state space, stability q Discrete-time frequency response, Bode diagram and Filtering

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Course Lecture Notes/Topics by Week q Week 10 (Ch. 14) q Concept of Filtering and Control q Applications in sound and noise cancellation q Applications in servo and vibration control

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Grading & Course Evaluation Grading based on: q 4 homework sets, every two weeks (20%) o Posted at end of week 2, 4, 6 & 8 (Friday’s). o Due and solutions posted at end of week 3, 5, 7 & 9 (Friday’s). o O.K. to work together but everyone must hand in their own unique solutions (no photo copies or print copies allowed)

q 2 open-book/open-notes Pop Quizzes (5%) q 1 open-book/open-notes Midterm Exam (25%)

q 1 open-book/open-notes Final Exam (50%) Homework + (pop quiz) solutions is posted on course website: http://mechatronics.ucsd.edu/mae143a Week 1/10

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Questions?

Any Questions on Course Aim, Course Organization or Grading?

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Dynamic systems and their signals We start our course with Chapter 1, Fundamentals of Signals & Systems, B. Boulet, Charles River Media, 2006. “Elementary Continuous-Time and Discrete-Time Signals and Systems”

Review of the concepts of: q Systems q Continuous-time

q Discrete-time q Elementary operations & properties of (dynamic) signals

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Dynamic systems and their signals Concept of a Dynamic System/Model Definition of dynamic system depends on discipline: q Sound engineering: varying degrees of volume (loudness) in music

q Physics: the interrelationships among the elements of space, time and force/energy q Bioengineering: the biomechanical aspects of the human body in motion.

q Dance: dance element which relates to how a movement is done.

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Dynamic systems and their signals Concept of a Dynamic System/Model in our Course q Mechanical/Aerospace Engineering: the study of the action of forces on bodies and the changes in motion they produce

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Dynamic systems and their signals Concept of a Dynamic System/Model in our Course q Electrical/Sound Engineering: the study of the action of voltage on circuits and the changes in current/power it produces

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Dynamic systems and their signals We will use Models to describe the Dynamic System q Model: description/approximation of reality or object q Physical Models: q Scaled representation

q Description of observed physical phenomena

q Solid/computer model

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Dynamic systems and their signals Mathematical Model: combination of ◦ A set of assumptions (axioms) ◦ A set of equations and algorithms that describes how a physical system or “reality” would behave ◦ A set of parameters for numerical values of physical properties

Dynamic models typically a differential equation. Example: ◦ Any object with a mass m is subjected to a gravitational force F F = m× g

where

g = 9.81 m / s 2

◦ The time dependent position x(t) of an object with a constant mass m subjected to a time dependent force F(t) is described by 2nd Newton’s law d2 m 2 x( t) = F ( t) dt Week 1/10

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Dynamic systems and their signals Example: free moving cart

å F (t ) = Ma (t )

Consider a constant input force: F (t ) = c ◦ 2nd Newton’s law F (t) = c = Ma (t ) yields: a (t ) = c M we have: v(t ) = c × t M c 2 1 ×t ◦ With v (t ) = c × t we have: x (t ) = M 2M ◦ With a(t ) = c

M

Conclusion: constant input (force) leads to quadratic output (position) Week 1/10

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Dynamic systems and their signals Example: cart with spring Consider constant (zero) input force.

d2 å F (t ) = Ma (t ) Þ - Kx(t ) = M dt 2 x (t ) Solution to this differential equation is given by: x (t ) =

M K t cos K M

Why is this a solution? Conclusion: constant input (force) leads to sinusoidal output (position) Week 1/10

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Dynamic systems and their signals Example: Voltage over RC circuit Consider closing switch S: equivalent to step-wise change on input voltage ! (").

Voltage over capacitor # given by $

*

" = 1/# ∫ & " '" or & " = # *+

This makes voltage

,

over resistor

-#

*

*+ $

,

" +

$ (")

" = -& " = -#

$ (")

=

*

(") *+ $

so:

! (")

Again a (1st order) differential equation…

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MAE143A, 2018 - R.A. DE CALLAFON

Dynamic systems and their signals Example: Voltage over RC circuit Consider closing switch S: equivalent to step-wise change on input voltage ! (").

Solution to -#

*

*+ $

" + $

$ (")

=

!

"

when

" = (1 − 2

3

4 56

!

" = :

)

Why is this a solution? Conclusion: constant input (voltage) leads to exponential output (voltage) Week 1/10

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Dynamic systems and their signals Signals do not necessarily have to be a continuous function of time t Examples: q Stocks q Money in your bank account q Signals measured by a computer (analog to digital)

q Discrete-time signals may use index [n] or [k] instead of time (t) Week 1/10

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Dynamic systems and their signals Example: Money in your savings account Let 7[8] be money at each month 8 = 1,2, … , 12 Let ;[8] be monthly deposit with ;[8] > 0 or monthly withdrawal with ;[8] < 0. Let @ > 0 be your interest rate., e.g. 5% interest means @ = 0.05

Dynamics of your savings account described by difference equation: 7 8 = ; 8 + 1 + @ 7[8 − 1] What is now solution if ; 0 = ;C and ; 8 = 0, 8 > 0 or when ; 8 = ; (constant)? Week 1/10

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Dynamic systems and their signals Example: Money in your savings account The dynamics is described by a difference equation: 7 8 = ; 8 + 1 + @ 7[8 − 1] Solution when ; 0 = ;C and ; 8 = 0, 8 > 0: q q q q q

8 = 0: 7 0 = ; 0 = ;C 8 = 1: 7 1 = ; 1 + 1 + @ 7[0] = 1 + @ ;C 8 = 2: 7 2 = ; 2 + 1 + @ 7[1] = (1 + @)D ;C 8 = 3: 7 3 = ; 3 + 1 + @ 7[2] = (1 + @)F ;C …

Conclusion: 7 8 = (1 + @)G ;C and with 1 + @ > 1 we see 7[8] → ∞ Week 1/10

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Dynamic systems and their signals Example: Measurement Averaging Consider a measurement with noise To reduce effect of noise, consider: q Take last 5 measurements q Compute Average This is known as: Moving Average (MA) over 5 samples. NOTE: the MA over 5 samples is described by a difference equation: J

7 8 = (; 8 + ; 8 − 1 + ;[8 − 2]+ ; 8 − 3 + ;[8 − 4]) K

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Dynamic systems and their signals Example: Measurement Averaging Observe the filtering effct of the MA over 5 samples, described by: J

7 8 = (; 8 + ; 8 − 1 + ;[8 − 2]+ ; 8 − 3 + ;[8 − 4]) K

u[k]

u[k] y[k]

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Dynamic systems and their signals Note the distinction in the mathematics when describing a q Continuous-time dynamic system with a differential equation. General (1st order) example: ' ' 7 " + MJ 7 " = NC ; " + NJ ;(") '" '" q Discrete-time dynamic system with a difference equation. General (1st order) example: 7 8 + MJ 7 8 − 1 = NC ; 8 + NJ ;[8 − 1]

Worth noting: q Order is highest derivative or largest # in time shift q Dynamic behavior: determined by order of equation and numerical value of coefficients in equation. Week 1/10

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Dynamic systems and their signals Concept of dynamic systems with the signals they produce can be given an abstract layer of an “input-output” mapping:

continuous-time dynamic system

The input/output mapping can be continuous- or discrete-time:

discrete-time dynamic system

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Dynamic systems and their signals Example: Voltage over RC circuit as Input/Output system Apply arbitrary input voltage output voltage $ " = 7 "

!

" = ; " and measure resulting

Possible model for dynamic input/output relation: -#

* *+

7 " + 7(") = ;(")

Can be considered a “filter” or “circuit” O. What kind of filter? Week 1/10

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Dynamic systems and their signals Concept also applicable to mechanical systems: Acceleration/vibration Input/Output system Apply arbitrary base acceleration top/suspension acceleration " !

! and measure resulting

Possible model for dynamic input/output relation: #

$% " $& %

$

! + ' $& " ! + ("(!) = # (!)

Can also be considered a “filter” or “circuit” ,. What kind of filter? Week 1/10

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Dynamic systems and their signals The abstract layer of an “input-output” mapping will be described by q Differential equations for continuous-time systems. '' " ! + Ȃ + / " ! + /- " ! -01 '! - '! ' ' ! + /- ! = 23 - ! + Ȃ + 2-01 '! '! q Difference equation for discrete-time systems. " ( + /1 " ( − 1 + Ȃ + /- " ( − 6 = 23 ( + 21 ( − 1 + Ȃ + 2- ( − 6

General Input/Output Systems Week 1/10

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Dynamic systems and their signals Given possible model for dynamics (continuous- or discrete-time): $7 " $& 7

! + Ȃ + /-01 $

2-01 $&

! + /-

$ " $&

! + /- " ! = 23

$7 $& 7

! + Ȃ+

!

or " ( + /1 " ( − 1 + Ȃ + /- " ( − 6 = 23 ( + 21 ( − 1 + Ȃ + 2- ( − 6 Analysis & Design of a Dynamical System: q Development of a mechanical/aerospace/electrical system with some desired parameters/coefficients or dynamic behavior. q It will be shown that such design can be done by combining dynamic systems with feedback control (also: MAE143b) Week 1/10

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Dynamic systems and their signals Combining dynamic systems with feedback control: Dynamics of each individual system may be significantly different than the total combined dynamics.

The dynamic system 81 is called the “uncontrolled” system The dynamic system 89 is called the “controller”

The feedback connection is called the “controlled” system. We will come back to this towards the end of the course… Week 1/10

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Dynamic systems and their signals Examples of combining dynamic systems with feedback control:

What is the “uncontrolled” system and what/who is the “controller”?

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Dynamic systems and their signals Before the Design of a Dynamical System: q Development of a mechanical/aerospace/electrical system with some desired parameters/coefficients or dynamic behavior. q Perform such design can be done by combining dynamic systems with feedback control We mu...