Control Systems Lectures - Time and Frequency Domain

Summary

This lecture introduces the concepts of time domain and frequency domain, emphasizing their physical meaning and applications. It begins with familiar time-related equations, like distance as a function of velocity and time. The lecture then explores harmonic oscillators, showing how sinusoidal motion can be described in the time domain and its corresponding frequency domain representation. Key topics include Fourier series and transform, which convert time domain signals into frequency domain representations, and Laplace transforms, used for analyzing systems with exponential growth or decay, helping solve differential equations in physics and control system design.

Takeaways

• 😀 Time domain equations are intuitive because they reflect familiar concepts like time, velocity, and acceleration.
• 📈 Distance is a function of time and velocity, easily visualized through equations like D = V × T.
• 🌀 A harmonic oscillator follows a sinusoidal motion that can be represented mathematically, and its force is negatively proportional to the distance from the neutral point.
• ⚖️ Newton's second law is used to describe the motion of a mass attached to a spring, leading to a sinusoidal solution in the time domain.
• 🎶 Fourier series allows for the transformation of a repeating time-domain signal into the frequency domain, expressed as an infinite summation of sinusoids.
• 🎸 Combining sinusoids of different frequencies creates complex time-domain signals that can be plotted in the frequency domain.
• 🔄 The Fourier transform extends the Fourier series to non-repeating signals, representing any signal as a summation of sinusoids with frequency, amplitude, and phase.
• 📉 Adding damping to an oscillator introduces an exponential decay term, which adjusts the time-domain response.
• 🧮 The Laplace transform extends the Fourier transform by accounting for exponential growth and decay, transforming signals into the S-domain for easier differential equation solving.
• 🔧 The S-domain is useful for control system design, offering insights into stability margins and simplifying complex time-domain calculations.

Q & A

What is the difference between time domain and frequency domain?

The time domain represents how a signal or physical quantity changes over time, while the frequency domain represents the signal's composition in terms of its frequencies. Time domain equations are often intuitive as they relate to physical concepts like velocity and distance, while the frequency domain helps in analyzing the periodic or harmonic components of a signal.

What is a harmonic oscillator, and how is it described in the time domain?

A harmonic oscillator is a system where the restoring force is proportional to the displacement, such as a spring with a mass attached. In the time domain, its motion can be described by a sinusoidal function representing how the displacement varies over time.

How does Newton's second law apply to the harmonic oscillator?

In the harmonic oscillator, Newton's second law states that the force acting on the mass is equal to its mass times acceleration (F = ma). The restoring force is proportional to the displacement (F = -kx), leading to a differential equation that describes the sinusoidal motion of the system.

What is the purpose of the Fourier transform in signal analysis?

The Fourier transform is used to represent a time-domain signal in the frequency domain. It expresses the signal as a sum of sinusoids at different frequencies, amplitudes, and phases, providing insight into the signal's frequency content.

What is the difference between a Fourier series and a Fourier transform?

A Fourier series represents a periodic time-domain signal as a sum of sinusoids at discrete harmonic frequencies. In contrast, the Fourier transform generalizes this concept to non-periodic signals, representing them as a continuous spectrum of frequencies.

How does damping affect the motion of a harmonic oscillator, and how is it represented mathematically?

Damping causes the amplitude of the harmonic oscillator's motion to decrease over time due to energy loss, usually to friction or resistance. Mathematically, this is represented by an exponential decay term in the time domain, modifying the simple sinusoidal motion.

Why is phase important in the frequency domain representation of a signal?

Phase determines the position of the sinusoidal components relative to each other in time. In control systems, phase is crucial because it can affect system stability and performance, even when the amplitude and frequency content are known.

What is the significance of Joseph Fourier's work in signal processing?

Joseph Fourier showed that any periodic time-domain signal can be represented as a sum of sinusoids at different frequencies. This principle forms the basis of the Fourier series and Fourier transform, which are essential tools for analyzing signals in the frequency domain.

What is the purpose of the Laplace transform, and how does it differ from the Fourier transform?

The Laplace transform is used to analyze systems that exhibit exponential growth or decay in addition to sinusoidal behavior. It extends the Fourier transform by including a real component (σ), which captures exponential behavior, making it useful for solving differential equations in control systems and other physical applications.

How is stability margin quantified in control system design using the S-plane?

Stability margin in control systems is quantified using the S-plane, where the system's poles and zeros are mapped. The distance of the poles from the imaginary axis indicates the stability of the system. Systems with poles closer to the left half of the S-plane are more stable.

Outlines

00:00

⏳ Introduction to Time and Frequency Domains

This paragraph introduces the concepts of time domain and frequency domain. It begins by explaining the physical meaning of time domain equations, such as how distance is a function of velocity and time. A real-world example is given, describing a journey to meet someone by walking a specific distance within a set time. The paragraph then shifts to a harmonic oscillator example, explaining the linear relationship between force and distance in a spring system. The motion of a mass attached to a spring is described using Newton's second law, which leads to the derivation of a sinusoidal motion equation in the time domain. The transition from time domain to frequency domain is then mentioned, focusing on how a pure sinusoid is represented by a single frequency in the frequency domain.

05:01

📊 Fourier Series and Frequency Representation

This section discusses the mathematical approach to representing signals in the frequency domain, introducing the Fourier series, which can decompose any periodic signal into a sum of sinusoids. Joseph Fourier's 1807 equation is mentioned, explaining that any repeating signal can be represented by a series of sinusoids at increasing frequencies. The paragraph explains how the frequency domain representation of a signal changes as the period increases, leading to a continuous frequency spectrum in non-repeating signals. This forms the basis of the Fourier transform, which transforms time-domain signals into frequency-domain representations. The Fourier transform can be applied to any signal, capturing its frequency, amplitude, and phase.

10:02

🔄 Fourier Transform and Real-World Applications

The final paragraph touches upon the broader utility of the Fourier transform and how it leads to the Laplace transform for more complex physical systems. While the Fourier transform helps analyze the frequency content of a signal, it falls short in scenarios involving differential equations with exponential terms, such as systems with damping. The introduction of the Laplace transform, which incorporates both the frequency and exponential components, allows for more comprehensive analysis. This paragraph briefly introduces the S-domain, explaining how the Laplace transform aids in solving differential equations and control system design by simplifying complex time-domain convolutions into algebraic expressions in the S-plane. The paragraph concludes by stating that future lessons will delve deeper into the Laplace transform.

Keywords

💡Time Domain

The time domain refers to the representation of a signal or function in relation to time. In the video, the time domain is used to show how variables like distance and velocity change over time, such as the example where walking a certain distance takes a specific amount of time. This representation is crucial for understanding motion and changes as they occur in the physical world.

💡Frequency Domain

The frequency domain represents how much of a signal lies within each given frequency band, rather than over time. The video introduces this concept by explaining how a sinusoidal function can be transformed from the time domain into the frequency domain, making it easier to analyze systems based on their frequency components rather than their behavior over time.

💡Harmonic Oscillator

A harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement. In the video, the spring and mass system is an example of a simple harmonic oscillator, where the restoring force is proportional to the stretched or compressed distance from a neutral point, illustrating sinusoidal motion in the time domain.

💡Restoring Force

Restoring force is the force that brings a system back to its equilibrium position after being displaced. The video uses the example of a spring where the restoring force increases as the spring is stretched or compressed. This force is crucial in systems like harmonic oscillators, where it causes the back-and-forth motion that can be described mathematically in both time and frequency domains.

💡Fourier Transform

The Fourier Transform is a mathematical technique used to transform a time-domain signal into its frequency-domain representation. The video explains how this transform allows us to break down a complex time-domain signal, such as a repeating wave, into a sum of sinusoidal components with different frequencies. This transformation is essential for analyzing non-repeating signals as well.

💡Sine Wave

A sine wave is a smooth, periodic oscillation that represents a pure frequency. In the video, the motion of the mass-spring system produces a sinusoidal pattern, and this type of wave is key to understanding the time-domain representation of many physical systems. In the frequency domain, sine waves correspond to specific frequencies and amplitudes.

💡Laplace Transform

The Laplace Transform is an extension of the Fourier Transform that incorporates both frequency and exponential growth or decay. The video introduces this concept as a way to deal with systems where damping or energy loss is present, such as a damped harmonic oscillator. The Laplace Transform is particularly useful for solving differential equations in control systems.

💡Damping

Damping refers to the reduction in the amplitude of an oscillation due to energy loss, often in the form of friction or resistance. In the video, damping is introduced when a damping term is added to the harmonic oscillator, resulting in a decaying oscillation over time. This concept is important when considering real-world systems that don't exhibit perfect, undamped motion.

💡Newton's Second Law

Newton's Second Law states that the force acting on an object is equal to the mass of the object times its acceleration (F = ma). The video references this law in the context of a free body diagram for the mass-spring system, where the restoring force from the spring is set equal to the mass times the acceleration of the system. This forms the basis for the differential equation governing the system’s motion.

💡Sinusoidal Function

A sinusoidal function describes a smooth, repetitive oscillation, such as a sine or cosine wave. The video discusses how the motion of a harmonic oscillator can be modeled using a sinusoidal function, which is essential for understanding both the time domain and frequency domain representations of the system's behavior.

Highlights

Introduction to the time domain and frequency domain.

Understanding physical meaning of time domain equations using algebraic variables like time, velocity, and acceleration.

Example of distance as a function of time, illustrating travel and arrival over time.

Explanation of harmonic oscillator and its relationship with springs and restoring force.

Mathematical description of motion in time domain using Newton’s second law.

Introduction to frequency domain representation of sinusoidal functions.

Description of amplitude and frequency representation in the frequency domain.

Discussion on Fourier series and its role in transforming time domain signals into frequency domain.

Example of superposition of two sinusoidal signals and its representation in both time and frequency domains.

Introduction to Joseph Fourier’s work on representing periodic signals through infinite summation of sinusoids.

Explanation of harmonic frequencies and their role in forming complex waveforms like sawtooth waves.

Introduction to Fourier transform, its role in representing both repeating and non-repeating signals.

Introduction to exponential terms and damping effects in systems with energy loss.

Transition from Fourier transform to Laplace transform for solving differential equations and including exponential decay.

Introduction to the S-domain and its importance for analyzing system stability and control design.

Transcripts

this lecture is an introduction to time

domain and frequency domain it's easy to

see the physical meaning of time domain

equations because most likely you have

been using algebraic variables to

represent concepts like time velocity

and acceleration for many years distance

equals velocity times time or in other

words how far you travel D is related to

how fast you are going V times how long

you are traveling for T this can also be

said D is a function of V and T say for

example when you want to meet someone

for dinner in three hours at their house

which is a distance D from you you start

walking in a straight line and after one

hour you've made it a third of the way

another hour passes and another third of

the distance passes so that after three

hours you arrive at the house right on

time although you've walked in a

straight line and therefore in a single

dimension in your perspective you've

also passed through time which can be

thought of as a second dimension in this

case and if we cross plot the dependent

variable distance in the independent

variable time we can get a relationship

between how your distance has changed

with time or another way of saying that

is that distance is a function of time

let's try this again and this time let's

use a simple harmonic oscillator say you

were able to design a spring that had

these unique characteristics every time

you doubled the stretched distance of

the spring the restoring force also

doubled so in this case the neutral

distance is x equals zero the stretched

distance is x equals one and it produces

a restoring force F if you stretch the

spring to X equal two the restoring

force also doubles similarly when you

compress the spring by the same distance

that you stretched it the force will be

equal and opposite so again if you

compress the spring to x equals minus

one you'll have a positive restoring

force F x equals minus two and you'll

have a positive restoring force to

in other words you've designed a linear

spring in which the force is negatively

proportional to the distance from the

neutral point if we attach a mass M to

the end of the spring and then we set

the mass in motion by hitting it with a

hammer

that is we impart an impulse into the

system which corresponds to

instantaneous velocity we can observe

the resulting motion over time and if

you're familiar with the motion of the

jack in the box after it pops out of the

box you'll recognize a familiar bobbing

motion that might resemble a sinusoid

now mathematically we can describe the

motion of this system in the time domain

using Newton's second law a free body

diagram of the mass shows that the only

force acting on it is the restoring

force from the spring minus K times X

and we can set that equal to the inertia

of the system which is mass times

acceleration or mass times x double-dot

rearranging this equation produces a

differential equation that describes the

motion of this system it can be shown

that the general solution of this

differential equation is truly a

sinusoid in the form the amplitude times

the sine of the natural frequency times

time plus the phase this is the

description of the resulting motion in

the time domain but how would you

describe this in the frequency domain

since this is a pure sinusoidal the

resulting frequency domain

representation is straightforward we can

plot the amplitude and frequencies

across the spectrum of the sinusoids

that make up the signal so in this case

we have a single peak at frequency 2 pi

Omega with a height corresponding to

amplitude a the rest of the spectrum

would have zero amplitude often phase is

discarded with this representation and

only amplitude and frequency are looked

at however as it will become obvious in

later lessons phase is crucial in

designing a control system now that we

understand a pure sinusoidal the

frequency domain the next question is

how to represent a more complex time

domain signal or function in the

frequency domain

if we take two sinusoidal sum them

together we can create a signal that is

a superposition of the two frequencies

and the resulting waveform would have

two frequency inputs with corresponding

amplitudes in this example the signal on

the Left has amplitude ay-one with a

period t1 and the signal in the middle

has an amplitude ay-two with period t2

if we plot this against frequency or one

over the period you would see two

distinct frequencies with two separate

amplitudes as you'd expect now these two

representations the time and the

frequency representation are both

equivalent mathematician Joseph Fourier

in 1807 published an equation that

stated that if you have a signal that

repeats over the period T or the

frequency of the repeating pattern is 1

over T that that time domain signal can

be represented by an infinite summation

of sinusoids at ever increasing

frequencies the equation is a bit too

long to write down here and I don't want

to go too far into the math in this

lecture so I will just say that a

Fourier series will transform it from

the time domain on the left to the

frequency domain on the right even

though this is a summation of infinite

number of frequencies not every

frequency as possible the key here is

that each sinusoidal of the lowest

frequency usually called the first

harmonic and then by multiplying the

first harmonic frequency by an integer

in you can get the second third and

fourth harmonic frequencies and so on

all the way up to infinity

for example this sawtooth wave input

does not look anything like the smooth

contours of the sine wave

yet by adding an infinite series of

these harmonics together you can produce

a sawtooth wave in the time domain the

equivalent frequency domain

representation would look something like

this on the right so now if you let the

period t of the repeating signal

increase the first harmonic frequency

would get smaller and smaller and

therefore the discrete frequencies in

the time domain

that described the signal would get more

dense now if you take the limit as the

period approaches infinity essentially

making it a non repeating function you

can see that the first harmonic

frequency would approach zero and then

every frequency is possible this turns

the discrete Fourier summation into a

continuous Fourier integral this is

called the Fourier transform the Fourier

transform is capable of representing any

signal repeating or not into an infinite

summation of sinusoids that includes

frequencies amplitudes and phase and the

Fourier transform is great for

understanding the frequency content of a

signal now my screen capture program

jumped here so I just wanted to state

that in the Fourier transform there is

an e raised to an imaginary exponent now

remember oilers formula that states that

when you raise an exponential to an

imaginary number you get cosine T plus J

sine T and so that's how sinusoidal a

with the Fourier transform now the

Fourier transform is a very general

approach to understanding a linear

system through its frequency response

however it's a bit limiting for many

applications in math and science because

parameters interact through differential

equations and the solution to

differential equations are both

sinusoidal x' if we take the simple

harmonic oscillator from above and add a

damping term to it with coefficient B it

can be shown that the general solution

also includes an exponential term namely

the energy loss in the system to the

damper therefore the time domain

response might look something like this

plot on the right where there's an

exponential term which is due to the

damping and assign your soil term which

is due to the spring constant now we can

start tweaking the Fourier transform to

aid us in solving differential equations

for physical world problems first off

the real world is causal which means we

must have a cause before we have an

effect so the idea is such as negative

time to not have meaning and second when

solving differential equations we need

more than just a free

Quincy content of the function we also

need that exponential content and to get

that we can take another step past the

Fourier transform to the Laplace

transform the Laplace transform takes

into account the exponential growth and

decay of a signal by including a real

component Sigma in the equation which is

the orange part of the equation when you

pre multiply the Fourier transform by e

to this negative Sigma T you can combine

the exponents to produce a complex

exponent Sigma plus J Omega or the real

part plus the imaginary part this is

traditionally called s and the resulting

transformation is said to transform from

the time domain into the S domain or s

plane now both the frequency domain and

the S domain are just as physically real

as the time domain once you get familiar

with them having different ways of

looking at the same physical system is

very valuable the S plane allows us to

quantify concepts such as stability

margin in a control design and it also

reduces complex convolution integrals

that need to take place in the time

domain to just simple algebraic steps in

the S plane now I know this was just a

very brief very fast introduction to the

frequency domain a future lecture will

be devoted to understanding the Laplace

transform in detail and how we use it to

design control systems