System Identification Toolbox

Terms to Characterize the Model Properties

The properties of an input-output relationship like the ARX model follow from the numerical values of the coefficients, and the number of delays used. This is however a fairly implicit way of talking about the model properties. Instead a number of different terms are used in practice:

Impulse Response

The impulse response of a dynamical model is the output signal that results when the input is an impulse, i.e., u(t) is zero for all values of t except t=0, where u(0)=1. It can be computed as in the equation following (ARX), by letting t be equal to 0, 1, 2, ... and taking y(-T)=y(-2T)=0 and u(0)=1.

Step Response

The step response is the output signal that results from a step input, i.e., u(t) is zero for negative values of t and equal to one for positive values of t. The impulse and step responses together are called the model's transient response.

Frequency Response

The frequency response of a linear dynamic model describes how the model reacts to sinusoidal inputs. If we let the input u(t) be a sinusoid of a certain frequency, then the output y(t) will also be a sinusoid of this frequency. The amplitude and the phase (relative to the input) will however be different. This frequency response is most often depicted by two plots; one that shows the amplitude change as a function of the sinusoid's frequency and one that shows the phase shift as function of frequency. This is known as a Bode plot.

Zeros and Poles

The zeros and the poles are equivalent ways of describing the coefficients of a linear difference equation like the ARX model. The poles relate to the "output-side" and the zeros relate to the "input-side" of this equation. The number of poles (zeros) is equal to the number of sampling intervals between the most and least delayed output (input). In the ARX example in the beginning of this section, there are consequently two poles and one zero.

How to Interpret the Noise Source

The Basic Steps of System Identification