In order to design control algorithms for dynamic systems, first, the plant must be identified. Modeling can be performed based on physical components (such as springs, dampers, etc.), or by using a “black box model” entailing mathematical equations for the system model. The first method requires deep knowledge of the physical connections and proper formulation of the differential equations, which is referred to as “white-box modeling.” In this case, after setting up the differential equations, the physical constants must be determined. On the other hand, a second method, called black-box modeling, only considers an abstract system transfer function with a number of parameters, so no detailed modeling of the system’s inner workings occurs. Of course, there are mixed methods (“gray-box modeling”) that depend on the depth of knowledge available about the system, too.
For all methods, one typically relies on measurement data emanating from the system. As a result, the input signal (AKA the excitation signal) needs to be chosen to support selected model identification and best capture the dynamics of the system being examined.
As a simple example, let us consider a bridge. After completion of a long and careful design and following the construction, bridges need to be tested before opening them to the public. Such tests need to be performed; otherwise, defects may appear (as happened with the Volgograd Bridge (Image 1). Load tests with trucks or buses may be used to measure static behavior. The dynamic behavior of such constructs can be evaluated in various ways. One solution is to use a vibroseis truck, which can apply a variable frequency sinusoid force to the bridge. Such behavior can also be stimulated by troops marching in lockstep (which, incidentally, is why troops marching across bridges is typically prohibited). For modal analysis, one common method is to apply a single impulse generated by an impulse hammer or a hydraulic hammer to a structure.
Much less complex measuring equipment is needed to check other dynamic systems, such as robotic arms, motors, thermodynamic or chemical systems. In such systems, the excitation signal could be voltage, force, a heat source, or even the injection of a chemical substance.
This section introduces the properties of the most commonly used excitation signals, including time and frequency domain characteristics; practical aspects will also be taken into consideration. A system may have restrictions on the type of signal, the maximum allowed amplitude, or the highest applied frequency able to excite the system with a significant power output. In addition, the signal measured may be affected by various noise sources, which can negatively influence the measurement results.
The following signals are most commonly applied:
Impulse response (Dirac impulse): Ideal systems are easily characterized by their impulse response. This can be directly measured using an impulse signal as an input. Although it is convenient in theory, in practice, this is not the best solution as it is impossible to generate an ideal impulse, and the amplitude of the system might be restrained at the input.
Unit step: This is a very commonly used system identification technique as it is easy to generate, and the impulse response can be approximated from the output signal being measured. An ideal step function is hard to realize—a similar limitation as before.
Sinusoidal: The generation of sinusoidal signals is straightforward and a technique used quite often to measure transfer function at a specific frequency. The measurement time for multiple sinusoids can be a problem.
Chirp signal: In the case of a chirp signal, frequency increases or decreases according to a function of time. It can be used to characterize the dynamic system in a given frequency band. Measurements, however, can be negatively affected by noise as each frequency is only present for a shorter time.
Pseudorandom binary sequence (PRBS): Such signals are popular for systems whose input system amplitude is either +1 or –1. For this method, frequency domain behavior needs to be optimized.
Multisine signal: This is the most suitable sign for frequency domain identification, as it can excite a specific selected frequency band of the dynamic system with multiple sinusoidal signals of the same amplitude at the same time. Its disadvantage is the selection of the phases for the sinusoids to create a low crest factor (CF) signal, which in turn increases the signal-to-noise ratio (SNR) of the measurement. The multisine signal can be expressed as:
where and are the amplitude, frequency, and phase values of the k sinusoidal component of the signal,is the sampling period, anddenotes the number of sinusoidal components. The CF of the signal can be expressed as the ratio between the maximum amplitude and the root mean square (RMS) of the signal:
Solving the optimization problem involves finding a subset of the values which minimizes the CF of the signal. As a result of using such signals, high SNR measurements can be achieved.
The various signals in the time and frequency domain are depicted in the figure below. It is apparent that each signal has its own advantages and disadvantages, considering the amplitude variations and localization properties in both domains. Choosing the most suitable signal highly depends on the system being examined and the limitations imposed on its input signal.
Optimization of multisine signals
Numerous methods have been proposed to reduce the CF of amultisine signal. A single-shot, closed formula has been presented by Schroeder (1970) where the phase values are determined by an analytical function, resulting in an acceptable CF. Iterative methods are more computationally expensive but can achieve better CF values. Van der Ouderaa, Schoukens, and Renneboog (1988) have described an FFT-based method where a continuous exchange between time- and frequency-domains is applied with nonlinear time-domain clipping and frequency-domain filtering. Guillaume, Schoukens, Pintelon, and Kollár (1991) proposed another method in which Lp norms of the time-domain multisine signal are used as an optimization objective for the Gauss–Newton algorithm.
Various attempts have been made to improve on the results of Schroeder’s and Van der Ouderaa’s algorithms. Yang et al. (2015) presented an improved clipping algorithm in which the clipping threshold increases logarithmically across iterations. Ojarand and Min (2017) utilize a combination of analytical formula and clipping to provide better variance to avoid local minima for sparse frequency distributions. However, their optimization was carried out for a relatively low number of sinusoidal components.
A particle swarm optimization algorithm using an artificial bee colony (ABC) was applied by Janeiro, Hu, and Ramos (2020) to determine a global minimum; however, due to the computational load required and slow convergence speed, this method also has serious limitations in connection with the number of multisine components.
Read the full research paper on our newest suggestion for improving the peak-to-average power ratio (“PAPR”) of multisine signals. The paper published in the Automatica Journal is open-access and can be downloaded here.
Further reading materials:
If you are interested in the full system identification process, including excitation signal design, modeling, and nonlinear effects, please visit Johan Schoukens’ website, which includes numerous sample codes and Livescript files.
Frequency domain identification toolbox: The Frequency Domain System Identification Toolbox (“FDIDENT”) provides specialized tools for identifying linear dynamic single-input/single-output (SISO) systems from the time responses from or measurements of a system’s frequency response. Frequency domain methods support continuous-time modeling, which can be a powerful and highly accurate complement to the more commonly used discrete-time methods. The methods in the toolbox can be applied to problems such as the modeling of electronic-, mechanical-, and acoustical systems.
System identification toolbox: The System Identification Toolbox (SysID) provides MATLAB functions, Simulink blocksets, and an interactive MATLAB app for dynamic system modeling, time-series analysis, and prediction. The toolbox helps you to model linear and nonlinear dynamics using various models such as Hammerstein-Wiener and Nonlinear ARX. Additionally, deep learning is also supported to capture nonlinear system behavior and dynamics.