Fundamentals of Control System 4
Let's begin with the fundamentals of control system.
Fourier and Laplace Transforms 3
Time Domain and Frequency Domain
This lecture is an introduction to the 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.
The distance equals velocity times time.
Distance = Velocity x Time
D = V x t
D = f(V,t)
In other words, how far you travel D is related to how fast you are going V times how long you are travelling 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.
If we cross plot the dependent variable distance to 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. In this case, the neutral distance is x equals 0. The stretched distance is x = 1 and it produces a restoring force F. If you stretch the spring to x = 2 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 -1 you’ll have a positive restoring force F. If x = -2 and you’ll have a positive restoring force 2F. 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. 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.
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 -kx.
We can set that equal to the inertia of the system which is mass times acceleration or mass times ẍ. 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.
We have seen 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 example, 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. In this case, we have a single peak at frequency 2πΩ 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.
Time-domain Signal to Frequency Domain
Now that we understand a pure sinusoidal in 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 signals and sums 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 A1 with a period T1 and the signal in the middle has an amplitude A2 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. These two representations – the time domain and the frequency domain representation are 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, then 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 an 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. 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.
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 denser.
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. The Fourier transform is great for understanding the frequency content of a signal.
I just wanted to state that in the Fourier transform there is an e raised to an imaginary exponent. Remember Euler’s 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 maths and science because parameters interact through differential equations. The solution to differential equations is both sinusoidal and exponentials.
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 a sinusoidal 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 of the real world is causal which means we must have a cause before we have an effect. So the idea is such a negative time to not have meaning .
- Second, when solving differential equations we need more than just a frequency 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 (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 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.
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.