Linear System Identification in a Nonlinear Setting: Nonparametric Analysis of the Nonlinear Distortions and Their Impact on the Best Linear Approximation

Abstract

Linear system identification [1]-[4] is a basic step in modern control design approaches. Starting from experimental data, a linear dynamic time-invariant model is identified to describe the relationship between the reference signal and the output of the system. At the same time, the power spectrum of the unmodeled disturbances is identified to generate uncertainty bounds on the estimated model. Linear system identification is also used in other disciplines, for example vibrational analysis of mechanical systems, where it is called modal analysis [5], [6]. Because linear time-invariant models are a basic model structure, linear system identification is frequently used in electrical [7]-[10], electronic, chemical [11], civil [12], and also in biomedical applications [13]. It provides valuable information to the design engineers in all phases of the design process.Starting from the late 1960s, system identification tools have been developed to obtain parametric models to describe the dynamic behavior of systems. A formal framework is set up to study the theoretical properties of the system identification algorithms [1]- [3]. The consistency (does the estimated model converge to the true system as the amount of data grows?) and the efficiency (is the uncertainty of the estimated model as small as possible?) are analyzed in detail. Underlying all these results are the assumptions that the system to be modeled is linear and time invariant.It is clear that these assumptions are often (mostly?) not met in real-life applications. Most systems are only linear to a first approximation. Depending on the excitation level, the output is disturbed by nonlinear distortions so that the linearity assumption no longer holds. This immediately raises doubts about the validity of the results obtained and validated by the linear system identification framework. The term nonlinear distortions indicates that nonlinear systems with a (dominant) linear term are considered. The deviations from the linear behavior are called nonlinear distortions.

Multimodal shunt damping of mechanical structures using multiple digital vibration absorbers

“…To obtain the dynamic elastances E p (cf equation ( 8)) between the voltage V p and the charge q p of the transducer in figure 10, a multisine excitation over the full frequency range was used and realized via currents injected in the patches by the DVAs [31]. A state-space model of the measured dynamic elastances for each patch was then obtained through the PolyMAX modal parameter estimation method [32].…”

A Damaged-Informed Lung Ventilator Model for Ventilator Waveforms

“…In clinical situations, the patient data are variable and often nonstationary because of interventions, patient-ventilator interactions, changes in health, etc., leading to complex parameter estimation issues. Moreover, the model we develop here is not likely to be structurally identifiable (Westwick and Kearney, 2003;Schoukens et al, 2016;Albers et al, 2019c). However, formally computing identifiability properties here is subtle because many parameters in the model functionally affect only part of the breath.…”

Uninterrupted real‐time cerebral stress level monitoring using wearable biosensors: A review

“…The nonlinear domain specifies exact or accurate time‐ and spectral‐domain estimations made from one process. Among all the various indices, the nonlinear domain has the best probability of accuracy as it is unaffected by instability, unlike linear indices 47 …”

“…Among all the various indices, the nonlinear domain has the best probability of accuracy as it is unaffected by instability, unlike linear indices. 47 Though using ECG to measure HRV for stress detection is one of the most accurate and popular methods, it has its certain drawbacks like ECG being expensive and nonportable. Hence, it is not preferred to use ECG for wearable and portable stress-detecting biosensors.…”