From linearity to the Laplace transform

English translation

Control Engineering I · Week 3

This week's objective · Assess LTI assumptions and transform time-domain differential equations.

Run: After signing in to MathWorks, open .mlx in your workspace. Click inside the code → Live Editor → Run Section to run one section at a time. Run runs everything. After changing a value, rerun the relevant section. Edit the text in ⟦enter here⟧ response fields; do not execute them. You can use Ctrl+Enter on Windows/Linux and ⌘+Enter on Mac.

1. Linearity combines two conditions

For linearity, adding inputs must add their outputs, and scaling an input by a must scale the output by a. $y=3x$ is linear, but $y=3x+1$ is not, because input 0 gives output 1.

Linearity combines two conditions

$y=x^2$ gives 4 for input 2, but the sum of the outputs for two separate inputs of 1 is 2. One counterexample can establish nonlinearity, but a few numerical examples cannot prove linearity for all inputs.

2. Distinguish a time shift from a system change

Time invariance means that delaying the same input delays the output by the same amount. $y=3x(t)$ is time-invariant, while $y=\cos(t)x(t)$ is time-varying because its multiplier changes with the input time.

Distinguish a time shift from a change in the system

An input varying over time differs from a time-varying system. Differentiation is a linear, time-invariant operation under the usual signal assumptions.

3. Approximate near the operating point

$\sin x\approx x$ approximates x measured in radians near 0. Near the operating point $\mathbf x_0$ , write f(x)≈f( $\mathbf x_0$ )+f′( $\mathbf x_0$ )(x− $\mathbf x_0$ ). Plotting the error helps assess the valid range.

Approximation near the operating point

Let δx=x− $\mathbf x_0$ and δy=f(x)−f( $\mathbf x_0$ ). The preceding equation becomes δy≈f′( $\mathbf x_0$ )δx. For example, near $\mathbf x_0$ =2 for f(x)=x², the slope is 4, so an input increase of 0.1 predicts an output increase of about 0.4. The actual change is 2.1²−2²=0.41. Such an approximation describes small changes near the reference point; it does not retain the same accuracy far away.

4. Convert derivatives into algebraic expressions

In the Laplace transform, derivatives are expressed using s and initial values: L{x′}=sX−x(0). With all initial conditions equal to 0, it becomes easier to obtain the input–output ratio from a differential equation.

Convert derivatives into algebraic expressions

Check the results using laplace and ilaplace in Symbolic Math Toolbox. Specify t and s explicitly, and verify the inverse transform by substituting it into the original equation.

Check your understanding

What values should go into the code, and what does each row and column of the resulting array represent? If calculating physical quantities, have you also checked the units?
Can you identify which parts to rerun when you change one condition?
Do the results match your expectations? If they differ, can you explain why using the output values or plots?

Official MATLAB documentation · Control System Toolbox

Practice materials

Control_Engineering_1_Week03_LtiLaplace.mlxPreview mlx script · Download mlx script