Stimulus and Acquisition Considerations in the System Identification Process

Background

The identification of a system involves a number of choices with regard to the system output signals to measure and the input signals to manipulate. Choices in how to manipulate system inputs, types of signal conditioning, signal ranges, and sampling behavior play a large role in the validity of the obtained model. Different modeling techniques can be used on the same experimental data set, but if the data does not capture the behavior of interest then another test will be necessary. Since performing an identification experiment is often time consuming and possibly costly it makes sense to give some thought to the design of the experiment and/or experiments prior to performing them. We will address the various considerations and attempt to provide some guidance on the tradeoffs involved with each choice.

System Stimulus-Response Signal Selection

Of utmost importance to the identification process is knowledge of the process to be stimulated. This provides the basis for determining what signals are considered outputs for determining sensor placement and which signals are inputs that may be used to excite the system. Simple tests may be necessary to determine influences and coupling, time delays, and time constants to aid in the modeling effort.

Also we should consider signals that while not directly capable of being manipulated still affect the system and should be included as inputs to the system model. An example of this would be the effect of wind gusts on the pitch dynamics of an airplane. The airplane responds in pitch to its elevator angle as a direct input. A wind gust provides additional pitching moments that can influence the dynamics, but is not directly controllable. A model of the airplane dynamics might include wind gusts as an input variable.

Stimulus Choice

The choice of stimulus signals plays an important role in the observed system behavior and the “goodness” of the model. These signals determine the operating points of the system and the modes that undergo excitation. While the choice of signals is often limited by the system under test there are a variety of characteristics that an input signal should exhibit to produce an experiment that provides the desired information for developing a model.

These criteria can be summarized as follows:

· To obtain the most information possible, experiments should be carried out under conditions similar to operating conditions and in the same operating range that the model is going to be used. This reduces the introduction of bias into the system model resulting in a better model. This is extremely important for non-linear systems.

· The inputs to the system under test should excite the system. This excitation is dependent on the spectrum of the input signal and not the actual waveform. Hence the frequency range of the input should be chosen so most of its energy is within the frequency ranges important to the system.

· Input value levels should exercise the entire desired range of variation.

· The input signal should deliver as much input power (as determined by the mean square signal level) into the system as possible. This waveform property is defined in terms of the crest factor,. The crest factor is defined as follows



The smaller the crest factor the better the signal excitation resulting in larger total energy delivery and enhanced signal-to-noise ratio. The theoretical lower bound for crest factor is 1.

Common Stimulus Signal Types

Filtered Gaussian White Noise

Filtered Gaussian white noise is a simple signal that can generate virtually any signal spectra in conjunction with the proper linear filtering. The theoretical crest factor for a Gaussian is infinite, but practical concerns required clipping the Gaussian amplitude to the input signal limits resulting in a corresponding reduction in crest factor while minimally affecting the generated spectrum.


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Random Binary Signal

A random binary signal is a random process that assumes two possible values. A simple method of generating the signal is to take Gaussian white noise, filter it for the desired spectra and then generate the appropriate levels by taking the sign of the filtered signal. You can now adjust the signal to any desired binary level. The resulting signal has a minimum crest factor of 1. Some differences in the resulting spectra are expected so off-line analysis of the signal should be performed.

Binary signals are very suitable for identifying linear systems. If a system is non-linear, an interval of input corresponding to the desired operating point should be used. It is often necessary to work with more than two input levels in these cases. Multiple binary signals of different levels can be combined to form the stimulus signal.

Pseudo-Random Binary Sequence (PRBS - Also known as Maximal Length Sequence (MLS))

A Pseudo-Random Binary Sequence is a periodic, deterministic signal with white-noise-like properties. They are often generated using an n bit shift register with feedback through an exclusive-OR logic. While appearing random in actually the sequence repeats every 2n-1 values. When using a whole period of signals the PRBS has special mathematical advantages that make it attractive as a stimulus signal. In particular, variations in response signals between two periods of the stimulus can be attributable to noise due to the periodic nature of the signal. Also, like white random binary noise it has an optimal crest factor.


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Chirp (Swept Sine)

The chirp is a sinusoid with a frequency that varies continuously over a certain range of values  for a specific period of time. The resulting signal has a crest factor ofand is easily modified to excite specific signal spectra. An illustration of a chirp is shown in the following figure.


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Sampling Rate Selection

The selection of a sampling rate is coupled to a system’s time constants. Sampling at rates substantially greater than system bandwidth leads to data redundancy, numerical issues and modeling of high frequency artifacts likely due to noise. Sampling at rates slower than system dynamics leads to difficulties determining the “correct” system model as well as the problems introduced by aliasing. This aliasing should be addressed by an anti-aliasing filter as discussed in the next section. A common rule of thumb is to sample signals at 10 times the bandwidth of the system (or bandwidth of interest) for the model. If uncertainty exists in the system bandwidth and a very fast data acquisition environment is available, it is useful to sample as fast as possible and then used digital pre-filtering and decimation to reduce the sampling rate to the desired value.

Anti-Aliasing Filters

According to the Nyquist sampling theorem the sampling rate should be at least twice the maximum frequency component of the signal of interest. In other words, the maximum frequency of the input signal should be less than or equal to half of the sampling rate. But how do you ensure that this is definitely the case in practice? Even if you are sure that the signal being measured has an upper limit on its frequency, pickup from stray signals (such as power line frequency or from local radio stations) could contain frequencies higher than the Nyquist frequency. These frequencies may then alias into the appropriate frequency range and thus give erroneous results.

To be sure that the frequency content of the input signal is limited, a low pass filter (a filter that passes low frequencies but attenuates the high frequencies) is added before the sampler and the ADC. This filter is an anti-alias filter because by attenuating the higher frequencies (greater than the Nyquist frequency), it prevents the aliasing components from being sampled. Because at this stage (before the sampler and the ADC) you are still in the analog world, the anti-aliasing filter is an analog filter.

A similar concept can be used in the very fast acquisition where the acquired data is post processed using a digital filter to remove frequency content above the system bandwidth and then decimated to the desired sampling rate.

Conclusion

System identification methods can greatly reduce the amount of effort needed to develop a model for use in a control system design effort. Choosing the proper stimulus signal, signal conditioning and sampling rate greatly aid the overall accuracy of the resulting identification.

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The defination of crest factor used in this well written note is not clear. Perhaps it is a more precise representation of: Cf = (peak amplitude)/(rms value)
- J. Karl Meinhardt, Miradia Inc.. kmeinhardt@miradia.com - Jan 5, 2007