Linear Time Invariant Systems

By Bolun Dai | Jan 1st 2021

To understand control theory, we need a medium. We can see how different motor torques affect the walking motion of a biped robot, but the correlation would be convoluted for a beginner. The default choice of such a medium for control theory noobs is a linear time-invariant (LTI) system. The next question would be: what makes LTI systems special? Let’s start with some of their properties. An LTI system has three main properties: homogeneity, additivity (commonly known as superposition), and, as the name suggests, time invariance. One may ask: I see the time invariance, where is the linear? The “linear” part is constructed from homogeneity and additivity. In the following sections, we will look into these three properties.

First, let’s take a look at homogeneity. What this means is that for an LTI system, if the output of the system is \(y(t)\) when given input \(x(t)\), then the output will be \(ay(t)\) when the input is \(ax(t)\). For example, if I weigh \(70kg\), the scale will show that I weigh \(70kg\); if I weigh \(2\times70 = 140kg\), the scale will show that I weigh \(2\times70 = 140kg\).

Next, let’s talk about additivity. What this means is that for an LTI system, if the system outputs \(y_1(t)\) when given input \(x_1(t)\) and outputs \(y_2(t)\) when given input \(x_2(t)\), then when given input \(x_1(t) + x_2(t)\), it will output \(y_1(t) + y_2(t)\). Also using the scale example, if two people of weight \(70kg\) each stand on the scale at the same time, the scale will show \(70 + 70 = 140kg\).

Finally, time invariance means that if the output of the system is \(y(t)\) when given input \(x(t)\), then the output will be \(y(t-a)\) when given input \(x(t-a)\). Using the scale example, if I maintained the same weight of \(70kg\) throughout the day, the scale will show \(70kg\) no matter if it is in the morning or 11pm.

For a system that is subject to a unit impulse \(\delta(t)\), we can have its output as \(y(t)\). If the system is subject to three impulses \(\delta(t)\), \(2\delta(t-1)\), and \(3\delta(t-2)\), using the properties of an LTI system, we can say the output at time \(t\) will be

\[\mathrm{output}(t) = y(t) + 2y(t-1) + 3y(t-2).\]

For a continuous input signal \(x(t)\), we can see it as applying the impulse \(x(a)\delta(t-a)\) at time \(t = a\). Then, also using the properties of an LTI system, we have

\[\mathrm{output}(t) = \int_{-\infty}^{\infty}x(a)y(t-a)da\]

with \(x(a) = 0\) for \(a \leq 0\) and \(y(t-a) = 0\) for \(a \geq 0\). This accounts for the control signal being \(0\) for all negative time, and the output being \(0\) for all negative time. This operation is called a convolution and is usually denoted as \(x*y\).