System Identification Toolbox

How to Interpret the Noise Source

In many cases of system identification, the effects of the noise on the output are insignificant compared to those of the input. With good signal-to-noise ratios (SNR), it is less important to have an accurate disturbance model. Nevertheless it is important to understand the role of the disturbances and the noise source e(t), whether it appears in the ARX model or in the general descriptions given above.

There are three aspects of the disturbances that should be stressed:

• Understanding white noise
• Interpreting the noise source
• Using the noise source when working with the model

These aspects are discussed one by one.

How can we understand white noise? From a formal point of view, the noise source e will normally be regarded as white noise. This means that it is entirely unpredictable. In other words, it is impossible to guess the value of e(t) no matter how accurately we have measured past data up to time t-1.

How can we interpret the noise source? The actual disturbance contribution to the output, H e, has real significance. It contains all the influences on the measured y, known and unknown, that are not contained in the input u. It explains and captures the fact that even if an experiment is repeated with the same input, the output signal will typically be somewhat different. However, the noise source e need not have a physical significance. In the airplane example mentioned earlier, the disturbance effects are wind gusts and turbulence. Describing these as arising from a white noise source via a transfer function H, is just a convenient way of capturing their character.

How can we deal with the noise source when using the model? If the model is used just for simulation, i.e., the responses to various inputs are to be studied, then the disturbance model plays no immediate role. Since the noise source e(t) for new data will be unknown, it is taken as zero in the simulations, so as to study the effect of the input alone (a noise-free simulation). Making another simulation with e being arbitrary white noise will reveal how reliable the result of the simulation is, but it will not give a more accurate simulation result for the actual system's response. It is a different thing when the model is used for prediction: Predicting future outputs from inputs and previously measured outputs, means that also future disturbance contributions have to be predicted. A known, or estimated, correlation structure (which really is the disturbance model) for the disturbances, will allow predictions of future disturbances, based on the previously measured values.

The need and use of the noise model can be summarized as follows:

• It is, in most cases, required to obtain a better estimate for the dynamics, G.
• It indicates how reliable noise-free simulations are.
• It is required for reliable predictions and stochastic control design.

Variants of Model Descriptions

Terms to Characterize the Model Properties