Black Box Identification of Systems

An exercise often given to students in introductory engineering circuit laboratory:

The instructor builds a simple circuit, and hiding it within a black box allows only the input and output terminals to be exposed. The students must determine the structure of the circuit and component values by exciting the circuit with test signals and observing the response.

This procedure is often referred to as "Black Box Identification". The method applies not only to electric circuits, but to any linear or near-linear system including mechanical, chemical, biological, economic, etc. ... In any case, to solve the contents of the black box, one must assume structure and degree of complexity, then test whether measured output data fits the modeled response. If the general structure is known beforehand, the task of identifying parameters is greatly simplified. In practical engineering applications the science of black box identification has greatly advanced with specialized instruments which use signal processing and estimation techniques. These instruments are known as multi-port real time dynamic signal analyzers.

The Dynamic Signal Analyzer evolved from earlier instrumentation, specifically spectrum analyzers and network analyzers. Hewlett Packard [1] makes the following distinction:

Applications of Dynamic Signal Analyzers

Today, spectrum analyzers are most often used for RF and microwave applications, and for the analysis and design of feedback control systems, dynamic signal analyzers are primarily used.

Excitation

Dynamic signal analyzers typically include one or more types of excitation signal sources within the instrument. These sources are automatically adjusted to excite frequencies within the selected measurement range. Excitation sources are provided on an external port which can be connected to control system inputs.

Although swept sine excitation is often available in these instruments, if the system under measurement is near linear, random excitation is a better choice, as it provides the fastest measurement, providing equal energy across the measurement spectrum concurrent with time.

Measurement Process

Measurement processing can be categorized into 4 major steps:

  1.  
  2. Recording Signals are over-sampled at a high rate and processed with an adjustable bandwidth digital decimating filter to prevent aliasing.
  3. Analysis Recorded signals are processed through FIR windowing filters and an FFT algorithm to produce instantaneous spectra of the signals.
  4. Averaging Autospectra and cross spectra are calculated and time averaged.
  5. Post Processing First, the frequency response and coherance function are calculated. From frequency response, autocorrelation, impulse response, cross correlation, and autocorrelation functions are calculated. Results are displayed as Bode, Nyquist, Nichols or impulse response.

Some analyzers provide curve fitting and direct solution of the pole-zero form of the system transfer function. The coherance function is typically used as a weighting function in the estimate. Math functions are often provided allowing manipulation on sets of processed data. One example is the calculation of the open loop transfer function from the closed loop transfer function[2].

Bruel & Kjaer's[3] 2032 and 2034 analyzers provide two built-in methods for calculation of the complex transfer function. The transfer function, H1( f ), is defined as the ratio of the cross spectra from input to output over the input autospectra, and the transfer function, H2( f ), as the ratio of the output autospectra over the cross spectra from output to input. H1( f ) and H2( f ) have the following properties:

Measurement Confidence Level

The Coherance function is a measure of how closely the input and output signals are linearly correlated at each frequency. Nearly all dynamic signal analyzers provide coherance function calculation as a means for determining a confidence level for transfer function measurements over the frequency range. A Coherance of 1 represents perfect correlation. Coherance less than 1 at any given frequency can be an indication of any one or more of the following:

Product Information

Most dynamic signal analyzers exist as dedicated stand-alone instruments, however some manufacturers provide PC based solutions.

References:

[1] The Fundamentals of Signal Analysis Application Note 243 Hewlett Packard Corp.
[2] Feedback Control System Measurements Application Note 240-1 Hewlett Packard Corp.
[3] Dual Channel FFT Analysis (Part 1) Technical Review, A Bruel & Kjaer Publication No. 1, 1984
[4] Dual Channel FFT Analysis (Part 2) Technical Review, A Bruel & Kjaer Publication No. 2, 1984

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