Forced Response

Outline

• LTI Systems

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• Time Response to Constant Input
• Time Response to Singularity Function Inputs
• Response to General Inputs (in Time)

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• Response to Sinusoidal Input (in Frequency)
• Response to Periodic Input (in Frequency)
• Response to General Input (in Frequency)

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• Fourier Transform

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Linear Time-Invariant (LTI) Systems

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Systems

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Linear Systems

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Time-Invariant Systems

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Linear Time-Invariant (LTI) Systems

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

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• 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

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• Assume all the systems are stable
• Inhomogeneous ODE

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• 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

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• 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

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• The system response in the long run is determined by its steady state component only

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• 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

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Response to Non-Zero Constant Input

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Time Response to Singularity Function Inputs

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Time Response to General Inputs

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Step Function

• Step function

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Step Response

• Start with a step response example

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• Or

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• The solution is given:

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Step Response

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Impulse

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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,

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and which is also constrained to satisfy the identity

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• Sifting property

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Impulse Response

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Impulse Response

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Impulse Response to LTI system

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• 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

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• The solution is given: (why?)

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• 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

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• 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

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• The solution is given

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• 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

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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: MATLAB (1/2)

• Example

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• The solution is given:

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Response to Arbitrary Input: MATLAB (2/2)

• Example

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• The solution is given:

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Response to Sinusoidal Input (in Frequency)

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Response to a Sinusoidal Input

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Fourier Transform

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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)

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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

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• Underlying properties

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Harmonic Representations

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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)

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• Periodic input: Fourier series → sum of complex exponentials

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• Complex exponentials: eigenfunctions of LTI system

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• Output: same eigenfunctions, but amplitudes and phase are adjusted by the LTI system

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• The output of an LTI system is a “filtered” version of the input

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Output is a “Filtered” Version of Input

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Output is a “Filtered” Version of Input

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Output is a “Filtered” Version of Input

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Output is a “Filtered” 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

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• The output response of LTI

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Response to a Square Wave Input: MATLAB

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Response to a Square Wave Input: MATLAB

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Response to a Square Wave Input: MATLAB

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Response to General Input (in Frequency)

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

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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

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Fourier Transform

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Fourier Transform

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Fourier Transform

• Definition: Fourier transform

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Response to General Input

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Magic of Impulse Response

• Fourier transform of Dirac delta function

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• Dirac delta function contains all the frequency components with 1
• Convolution in time
• Filtering in frequency

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Magic of Impulse Response

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LTI

LTI

Frequency Response (Frequency Sweep)

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LTI

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

• Input frequency near resonance frequency
• Resonance frequency is generally different from natural frequency, but they often are close enough

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Resonance frequency

Summary

• To understand LTI system

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• Impulse response

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• Frequency sweep

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