Learn more about System Identification Toolbox

Estimating Delays in the Multiple-Input System

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Why Estimate Delays? Estimating Delays Using the ARX Model Structure Estimating Delays Using Alternative Methods

Why Estimate Delays?

To identify parametric black-box models, you must specify the input/output delay as part of the model order.

If you do not know the input/output delays for your system from the experiment, you can use the System Identification Toolbox software to estimate the delay.

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Estimating Delays Using the ARX Model Structure

In the case of single-input systems, you can read the delay on the impulse-response plot. However, in the case of multiple-input systems, such as the one in this tutorial, you might be unable to tell which input caused the initial change in the output and you can use the delayest command instead.

The delayest command estimates the time delay in a dynamic system by estimating a low-order, discrete-time ARX model with a range of delays, and then choosing the delay that corresponding to the best fit.

The ARX model structure is one of the simplest black-box parametric structures. In discrete-time, the ARX structure is a difference equation with the following form:

y(t) represents the output at time t, u(t) represents the input at time t, n_a is the number of poles, n_b is the number of b parameters (equal to the number of zeros plus 1), n_k is the number of samples before the input affects output of the system (called the delay or dead time of the model), and e(t) is the white-noise disturbance.

delayest assumes that n_a=n_b=2 and that the noise e is white or insignificant, and estimates n_k.

To estimate the delay in this system, type the following command in the MATLAB Command Window:

delayest(ze)

System Identification Toolbox software responds with the following:

ans =

     5    10

This result includes two numbers because there are two inputs: the estimated delay for the first input is 5 data samples, and the estimated delay for the second input is 10 data samples. Because the sampling interval for the experiments is 0.5 min, this corresponds to a 2.5-min delay before the first input affects the output, and a 5.0-min delay before the second input affects the output.

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Estimating Delays Using Alternative Methods

There are two alternative methods for estimating the time delay in the system:

• Plot the time plot of the input and output data and read the time difference between the first change in the input and the first change in the output. This method is practical only for single-input/single-output system; in the case of multiple-input systems, you might be unable to tell which input caused the initial change in the output.

• Plot the impulse response of the data with a 1-standard-deviation confidence region. You can estimate the time delay using the time when the impulse response is first outside the confidence region.

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Estimating Step- and Frequency-Response Models

Estimating Model Orders Using an ARX Model Structure

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