Forced Response Prof. Seungchul Lee Industrial AI Lab. Outline - - PowerPoint PPT Presentation

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Forced Response Prof. Seungchul Lee Industrial AI Lab. Outline - - PowerPoint PPT Presentation

Jan 27, 2024 117 likes β€’1.12k views

Forced Response Prof. Seungchul Lee Industrial AI Lab. Outline LTI Systems Time Response to Constant Input Time Response to Singularity Function Inputs Response to General Inputs (in Time) Response to Sinusoidal Input (in

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Forced Response

  • Prof. Seungchul Lee

Industrial AI Lab.

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Outline

  • LTI Systems
  • Time Response to Constant Input
  • Time Response to Singularity Function Inputs
  • Response to General Inputs (in Time)
  • Response to Sinusoidal Input (in Frequency)
  • Response to Periodic Input (in Frequency)
  • Response to General Input (in Frequency)
  • Fourier Transform

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Linear Time-Invariant (LTI) Systems

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Systems

  • 𝐼 is a transformation (a rule or formula) that maps an input signal 𝑦(𝑒) into a time output signal 𝑧(𝑒)
  • System examples

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Linear Systems

  • A system 𝐼 is linear if it satisfies the following two properties:
  • Scaling
  • Additivity

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Time-Invariant Systems

  • A system 𝐼 processing infinite-length signals is time-invariant (shift-invariant) if a time shift of the

input signal creates a corresponding time shift in the output signal

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Linear Time-Invariant (LTI) Systems

  • We will only consider Linear Time-Invariant (LTI) systems
  • Examples

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Time Response to Constant Input

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Natural Response

  • So far, natural response of zero input with non-zero initial conditions are examined

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Response to Non-Zero Constant Input

  • Assume all the systems are stable
  • Inhomogeneous ODE
  • Same dynamics, but it reaches different steady state
  • Good enough to sketch

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Response to Non-Zero Constant Input

  • Dynamic system response = transient + steady state
  • Transient response is present in the short period of time immediately after the system is turned on

– It will die out if the system is stable

  • The system response in the long run is determined by its steady state component only
  • In steady state, all the transient responses go to zero

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Example

  • Example

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Response to Non-Zero Constant Input

  • Think about mass-spring-damper system in horizontal setting

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Response to Non-Zero Constant Input

  • Mass-spring-damper system in vertical setting
  • 𝑧 0 = 0 no initial displacement
  • ሢ

𝑧 0 = 0 initially at rest

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Response to Non-Zero Constant Input

  • Shift the origin of 𝑧 axis to the static equilibrium point, then act like a natural response with
  • 𝑧 0 = βˆ’

𝑛𝑕 𝑙 and ሢ

𝑧 0 = 0 as initial conditions

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Time Response to Singularity Function Inputs

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Time Response to General Inputs

  • We studied output response 𝑧(𝑒) when input 𝑦(𝑒) is constant
  • Ultimate Goal: output response of 𝑧(𝑒) to general input 𝑦(𝑒)
  • Consider singularity function inputs first

– Step function – Impulse function (Delta Dirac function)

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Step Function

  • Step function

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Step Response

  • Start with a step response example
  • Or
  • The solution is given:

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Step Response

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Impulse

  • Impulse: difficult to image
  • The unit-impulse signal acts as a pulse with unit area but zero width
  • The unit-impulse function is represented by an arrow with the number 1, which represents its area
  • It has two seemingly contradictory properties :

– It is nonzero only at 𝑒 = 0 and – Its definite integral (βˆ’βˆž, ∞) is 1

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Properties of Dirac Delta Function

  • The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere

except at the origin, where it is infinite, and which is also constrained to satisfy the identity

  • Sifting property

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Impulse Response

  • Impulse response: difficult to image
  • Question: how to realize initial velocity of 𝑀0 β‰  0
  • Momentum and impulse in physics I

– Consider an "impulse" which is a sudden increase in momentum 0 β†’ 𝑛𝑀 of an object applied at time 0 – To model this, – where force 𝑔(𝑒) is strongly peaked at time 0 – Actually the details of the shape of the peak are not important, what is important is the area under the curve – This is the motivation that mathematician and physicist invented the delta Dirac function

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Impulse Response

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Impulse Response to LTI system

  • Later, we will discuss why the impulse response is so important to understand an LTI system

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Impulse Response to LTI system

  • Example: now think about the impulse response
  • The solution is given: (why?)
  • Impulse input can be equivalently changed to zero input with non-zero initial condition

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Step Response Again

  • Relationship between impulse response and unit-step response
  • Impulse response is the derivative of the step response

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Response to General Inputs (in Time)

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Response to a General Input (in Time)

  • Finally, think about response to a "general input" in time
  • The solution is given
  • If this is true, we can compute output response to any general input if an impulse response is given

– Impulse response = LTI system

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Convolution: Definition

  • 𝑧(𝑒) is the integral of the product of two functions after one is reversed and shifted by 𝑒

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Easier Way to Understand Continuous Time Signal

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Easier Way to Understand Continuous Time Signal

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Structure of Superposition

  • If a system is linear and time-invariant (LTI) then its output is the integral of weighted and shifted unit-

impulse responses.

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Impulse Response to LTI System

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Time-invariant Linear (scaling)

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Response to Arbitrary Input π’š(𝒖) (1/2)

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Response to Arbitrary Input π’š(𝒖) (2/2)

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Response to Arbitrary Input: MATLAB (1/2)

  • Example
  • The solution is given:

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Response to Arbitrary Input: MATLAB (2/2)

  • Example
  • The solution is given:

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Response to Sinusoidal Input (in Frequency)

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Response to a Sinusoidal Input

  • When the input 𝑦 𝑒 = π‘“π‘˜πœ•π‘’ to an LTI system

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Fourier Transform

  • Definition: Fourier transform
  • 𝐼 π‘˜πœ• π‘“π‘˜πœ•π‘’ rotates the same angular velocity πœ•

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Response to a Sinusoidal Input: MATLAB

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Response to a Sinusoidal Input: MATLAB

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Response to Periodic Input (in Frequency)

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Response to a Periodic Input (in Frequency Domain)

  • Periodic signal: Definition
  • Fourier series represent periodic signals in terms of sinusoids (or complex exponential of π‘“π‘˜πœ•π‘’)
  • Fourier series represent periodic signals by their harmonic components

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Response to a Periodic Input (in Frequency)

  • Fourier series represent periodic signals by their harmonic components

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Response to a Periodic Input (in Frequency)

  • What signals can be represented by sums of harmonic components?

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Harmonic Representations

  • It is possible to represent all periodic signals with harmonics
  • Question: how to separate harmonic components given a periodic signal
  • Underlying properties

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Harmonic Representations

  • Assume that 𝑦(𝑒) is periodic in π‘ˆ and is composed of a weighted sum of harmonics of πœ•0 =

2𝜌 π‘ˆ

  • Then

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Fourier Series

  • Fourier series: determine harmonic components of a periodic signal

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Example: Triangle Waveform

  • One can visualize convergence of the Fourier Series by incrementally adding terms.

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Example: Triangle Waveform

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Example: Triangle Waveform

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Example: Triangle Waveform

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Example: Triangle Waveform

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Example: Triangle Waveform

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Example: Triangle Waveform

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Example: Triangle Waveform

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Example: Triangle Waveform

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Example: Triangle Waveform: MATLAB

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Example: Square Waveform

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Example: Square Waveform

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Example: Square Waveform

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Example: Square Waveform

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Example: Square Waveform

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Example: Square Waveform

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Example: Square Waveform

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Example: Square Waveform

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Example: Square Waveform

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Example: Square Waveform: MATLAB

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Response to a Periodic Input (Filtering)

  • Periodic input: Fourier series β†’ sum of complex exponentials
  • Complex exponentials: eigenfunctions of LTI system
  • Output: same eigenfunctions, but amplitudes and phase are adjusted by the LTI system
  • The output of an LTI system is a β€œfiltered” version of the input

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Output is a β€œFiltered” Version of Input

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Output is a β€œFiltered” Version of Input

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Output is a β€œFiltered” Version of Input

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Output is a β€œFiltered” Version of Input

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Response to a Square Wave Input: MATLAB

  • Decompose a square wave to a linear combination of sinusoidal signals
  • The output response of LTI

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Response to a Square Wave Input: MATLAB

  • Decompose a square wave to a linear combination of sinusoidal signals
  • The output response of LTI
  • Given input π‘“π‘˜πœ•π‘’
  • 𝑧 = π΅π‘“π‘˜(πœ•π‘’+𝜚)

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Response to a Square Wave Input: MATLAB

  • Linearity: input Οƒ 𝑏𝑙𝑦𝑙(𝑒) produces Οƒ 𝑏𝑙𝑧𝑙(𝑒)

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Response to a Square Wave Input: MATLAB

  • Linearity: input Οƒ 𝑏𝑙𝑦𝑙(𝑒) produces Οƒ 𝑏𝑙𝑧𝑙(𝑒)

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Response to General Input (in Frequency)

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Response to a General Input (Aperiodic Signal) in Frequency Domain

  • An aperiodic signal can be thought of as periodic with infinite period
  • Let 𝑦(𝑒) represent an aperiodic signal
  • Periodic extension
  • Then

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Example: Periodic Square Wave

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Example: Periodic Square Wave

  • Doubling period doubles # of harmonics in given frequency interval

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Example: Periodic Square Wave

  • As π‘ˆ β†’ ∞, discrete harmonic amplitudes β†’ a continuum π‘Œ(π‘˜πœ•)
  • As alternative way of interpreting is as samples of an envelope function, specifically
  • That is, with πœ• thought of as a continuous variable, the set of Fourier series coefficients approaches

the envelop function as π‘ˆ β†’ ∞

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Fourier Transform

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Fourier Transform

  • As π‘ˆ β†’ ∞, synthesis sum β†’ integral
  • Aperiodic signal has all the frequency

components instead of discrete harmonic components

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Fourier Transform

  • Definition: Fourier transform

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Response to General Input

  • Response to LTI system with impulse response β„Ž(𝑒)

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Magic of Impulse Response

  • Fourier transform of Dirac delta function
  • Dirac delta function contains all the frequency components with 1

– Convolution in time – Filtering in frequency

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Magic of Impulse Response

  • Impulse basically excites a system

with all the frequency of π‘“π‘˜πœ•π‘’

  • Impulse response contains the

information on how much magnitude and phase are filtered via the LTI system at all the frequency

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LTI LTI

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Frequency Response (Frequency Sweep)

  • Frequency sweeping is another way to collect LTI system characteristics

– (same as the impulse response)

  • Given input π‘“π‘˜πœ•π‘’
  • 𝑧 = π΅π‘“π‘˜(πœ•π‘’+𝜚)

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LTI

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The First Order ODE: MATLAB

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Example: The Second Order ODE

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The Second Order ODE: MATLAB

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The Second Order ODE: MATLAB

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Experiment: The Second Order ODE

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Experiment: The Second Order ODE

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Resonance

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  • Input frequency near resonance frequency
  • Resonance frequency is generally different from natural frequency, but

they often are close enough

Resonance frequency

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Summary

  • To understand LTI system
  • Impulse response
  • Frequency sweep

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