Data Science and Computing with Python for Pilots and Flight Test Engineers
Laplace Transform
Introduction
In this lesson we introduce the Laplace transform, which is a very useful tool for solving differential equations, such as the equations of motion of a dynamic system, and performing system analysis, such as studying the system’s stability characteristics.
The Laplace transform is a linear transform that will allow us to turn a differential equation into an algebraic equation, which then becomes much easier to solve. After obtaining a solution in the Laplace domain, we will then need to apply the inverse Laplace transform to obtain the solution in the original time domain (the inverse transform and useful associated algebraic techniques are covered in the next lesson). The stability characteristics can be studied directly in the Laplace domain.
We will use real variable \(t\) to denote the variable in the time domain, and complex variable \(s\) to be the variable in the Laplace domain. The Laplace domain is therefore often also referred to as the \(s\)-plane.
Definition
The Laplace transform is defined as
\begin{equation} F(s) \equiv \mathcal{L}[f(t)] := \int_{-\infty}^\infty f(t)e^{-st}\,dt \end{equation}
where \(s=\sigma + i\omega\) is a complex number with real part denoted by \(\sigma\) and imaginary part \(\omega\).
By convention, we denote functions in the time domain with lowercase letters. These functions take the variable \(t\) as their argument. The Laplace transform of such a function is denoted by the corresponding uppercase letter; this Laplace transform takes the complex variable \(s\) as its argument.
This lowercase/uppercase notation convention is not always followed. Sometimes the same letter may be used (either lowercase or uppercase) for both the function and its Laplace transform, and the distinction is made only by explicitly writing its argument (\(t\) or \(s\)). For instance, for the unit impulse we may write \(\delta(t)\) for the function in the time domain and \(\delta(s)\) for its Laplace transform in the Laplace domain.
Utility of the Laplace Transform
As we will see, if we set \(s=i\omega\) (i.e. \(\sigma=0\), the above definition of the Laplace transform becomes identical to the definition of the Fourier transform. The Laplace transform is therefore a generalization of the Fourier transform. The Fourier domain is essentially the imaginary axis of the \(s\)-plane. This generalization is convenient, because with the Laplace transform, we will be able to:
- turn the differential equations describing the motion of a system into algebraic equations (thus getting rid of the derivatives), which makes them much easier to solve (we have outlined this process in an earlier lesson briefly already),
- perform stability analysis of these equations in the Laplace domain (s-plane),
- characterize exponentially decaying or growing sinusoidal solutions in the time domain by the location of the poles of their Laplace transform.
In order to obtain the frequency response of a system, we will still use the Fourier transform though, simply by setting \(s=i\omega\) in the Laplace transform of a function.
Mathematical Properties
Some useful mathematical properties of the Laplace transform of any function are the following (either remember them or remember to be able to look them up here). Let \(x(t)\) and \(y(t)\) be two functions in the time domain, and \(X(s)\) and \(Y(s)\) denote their Laplace transforms, respectively. And let \(\lambda\) be a real scalar (i.e. a real number). Then the Laplace transform has the following properties.
| Property | Time Domain | Laplace Transform |
|---|---|---|
| Linearity 1 | \(x(t)+y(t)\) | \(X(s)+Y(s)\) |
| Linearity 2 | \(\lambda x(t)\) | \(\lambda X(s)\) |
| First Derivative | \(\dot x(t)\) | \(sX(s)-x(0)\) |
| Second Derivative | \(\ddot x(t)\) | \(s^2X(s)-sx(0)-\dot x(0)\) |
| Integral | \(\int_0^T x(t) \, dt\) | \(\frac{X(s)}{s}\) |
| Time Delay | \(x(t+T)\) | \(X(s)e^{-sT}\) |
| Unit Impulse | \(\delta(t)\) | 1 |
| Unit Step | \(\theta(t)\) | \frac{1}{s} |
In the table above, \(\delta(t)\) denotes the Dirac delta function (or \(\delta\) distribution), which is a distribution (generalized function) with the interesting property of being nonzero only at \(t=0\) yet still having an integral equal to 1, \(\int_{-\infty}^{\infty} \delta(t)\,dt = 1\). \(\theta(t)\) denotes the Heaviside step function (or unit step function), which is zero for \(t<0\) and 1 for \(t\ge 1\). Note that the Heaviside step function is sometimes also denoted by \(H(t)\) (which will not use to avoid confusion with the system’s transfer function, which we will denote by \(H(s)\)) or by \(u(t)\) (which we may use occasionally, if \(theta\) is needed for pitch angle (one of the Euler angles describing aircraft attitude)).
Initial Value Theorem (IVT) and Final Value Theorem (FVT)
The initial value theorem (IVT) and the final value theorem (FVT) let us compute the values of function \(x(t)\) at initial time \(t=0\) and at final time infinitely in the future \(t\rightarrow\infty\), respectively, from the Laplace transform \(X(s)\) directly, without having to compute the inverse Laplace transform from \(X(s)\) to \(x(t)\). This makes these theorems very convenient, if we are only interested in the value of the solution at the very beginning and at the very end.
The initial value theorem states:
\begin{equation} \lim_{t\rightarrow0} x(t) = \lim_{s\rightarrow\infty} X(s) \end{equation}
The final value theorem is:
\begin{equation}\lim_{t\rightarrow\infty} x(t) = \lim_{s\rightarrow0} X(s) \end{equation}
List of Laplace Transforms for Some Basic Functions
When we are faced with having to do a Laplace transform of a function, we will try to bring it into the form of some basic function types, for which the Laplace transform and its inverse have been tabulated. This is much more convenient than having to compute the integral in the definition of the Laplace transform analytically by hand. We will learn several techniques in the next lesson, how to bring functions into a sum of basic function types that are tabulated, but let us give the Laplace transforms of some basics function types here first. Note that this list is not comprehensive by any means, the reader is encouraged to look at more extensive Laplace transform tables in other sources. Yet the list here will suffice to solve most or all of the problems in this course.
| Function Name | Function in Time Domain | Laplace Transform |
|---|---|---|
| Unit Impulse | \(\delta(t)\) | 1 |
| Unit Step | \(\theta(t)\) | \(\frac{1}{s}\) |
| Unit Ramp | \(t\) | \(\frac{1}{s^2}\) |
| Polynomial | \(t^n\) | \(\frac{n!}{s^{n+1}}\) |
| Exponential | \(e^{-\sigma t}\) | \(\frac{1}{s+\sigma}\) |
| Polynomial Exponential | \(t^{n-1}e^{-\sigma t}\) | \(\frac{1}{(s+\sigma)^n}\) |
| Sine Wave | \(\sin (\omega t)\) | \(\frac{\omega}{s^2+\omega^2}\) |
| Cosine Wave | \(\cos (\omega t)\) | \(\frac{s}{s^2+\omega^2}\) |
| Damped Sine Wave | \(e^{-\sigma t}\sin (\omega t)\) | \(\frac{\omega}{(s+\sigma)^2+\omega^2}\) |
| Damped Cosine Wave | \(e^{-\sigma t}\cos (\omega t)\) | \(\frac{s+\sigma}{(s+\sigma)^2+\omega^2}\) |
The parameters (\sigma\) and \(\omega\) appearing in the table above are not to be confused with the notation for the real and imaginary part of \(s\), as in \(s=\sigma+i\omega\). The value of \(s\) is independent of these parameters \(\sigma\) and \(\omega\) appearing in the table above. In particular, notice the minus sign in front of the \(\sigma\) in the exponentials.
Relationship of Laplace Transform to Fourier Transform
With the above definition, the familiar reader will recognize immediately that the Laplace transform is a generalization of the Fourier transform. The Fourier transform transforms a function from a time or spatial domain to the frequency domain (this can be a temporal or spatial frequency). Even though all functions (with suitable mathematical properties such as integrability) can be expressed as a Fourier transform, and the Laplace transform is not needed, this generalization allows the Laplace transform to express even solutions which are exponentially growing or decaying sinusoidal waves in the Laplace domain in a way where their properties are immediately apparent: by the location of their poles in the Laplace domain. (The Fourier transform does not do this, because it “sees” only the imaginary axis of the \(s\) plane. Since in engineering applications dynamical systems are often exponentially decaying or growing, this makes the Laplace transform particularly convenient. In fact, it will enable us to use it for stability analysis.
As mentioned already, in order to obtain the Fourier transform from the Laplace transform, we just need to set \(s=i\omega\), where \(\omega\) is a real number denoting the angular frequency. With other words, the Fourier transform is obtained as a special case of the Laplace transform by setting \(\sigma=0\) in \(s=\sigma+i\omega\).
To illustrate the above, imagine we have a sine wave with just one frequency, for instance \(f(t)=\sin(3t)\), then its Fourier transform will have the shape of a Dirac delta function sitting on angular frequency \(\omega=3\), because this is the only frequency present.
But imagine that we have a sinusoidal wave with one frequency, yet with an exponentially decaying amplitude, such as is often the case for dynamic systems. Say the function looks like \(f(t)=e^{-2t}\sin(3t)\). We can still express this function in Fourier space, but because it is not a pure sine wave anymore and is exponentially decaying, this function has to be expressed in Fourier space as an infinite sum of many frequencies of non-zero amplitude. It is hard to see the property of the function (e.g. its damping, i.e. decrease in amplitude) from this.
In the Laplace domain we see the same thing on the imaginary axis (since the Fourier domain is just the imaginary axis of the s-plane), but in addition we have the whole rest of the complex \(s\) plane to observe. And here we see that the Laplace transform of the above function \(f(t)\) has a pole at the point \(s=\sigma+i\omega\), where \(\sigma=-2\) and \(\omega=3\), and another one at \(s=\sigma-i\omega\). This complex conjugate pair of poles tells us immediately, that it is a sinusoidal wave with angular frequency \(\omega=3\) and total damping \(\sigma=-2\). This is very convenient, also for the purpose of stability analysis, which can be performed directly and easily visibly in the \(s\)-plane. We will see many examples of this in our course.
Nonetheless, the frequency response of a system is a very important insight. It will be depicted in the so called Bode plots. And when we want to see the frequency response, we do not want the full Laplace transform, but just the Fourier transform. And the rule to remember is that we need to replace \(s\) everywhere with \(i\omega\) to achieve that. In particular, we will see than to compute the Bode plots, i.e. the gain (output amplitude to input amplitude ratio) and phase shift for every angular frequency \(\omega\), we will have substitute \(s=i\omega\) everywhere.