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Identification of block-oriented nonlinear systems starting from linear approximations: A survey

Maarten Schoukens, Koen Tiels

arXiv:1607.01217v3eess.SY

TL;DR

Identifying block-oriented nonlinear systems involves model-selection challenges that linear approximations can help address in a familiar LTI framework. This survey synthesizes models and algorithms built around best linear approximations, showing that they support modeling a wide range of structures and guide nonlinear identification.

  • Problem

    Block-oriented nonlinear identification still involves model-selection challenges, motivating approaches that address part of this problem within a familiar LTI framework.

  • Method

    The paper surveys block-oriented nonlinear models identifiable through best linear approximations and reviews the identification algorithms developed for them.

  • Results

    Best linear approximation approaches model single-branch, parallel-branch, and nonlinear-feedback systems while providing system information and guiding modeling decisions.

  • Takeaways & Limitations

    Linear approximations can provide useful structure and dynamic-order information and serve as starting points for nonlinear identification across diverse block-oriented models.

  • Takeaways & Limitations

    Very noisy data or strongly nonlinear systems can produce poor best linear approximation estimates and increase the risk of local optimization minima.

Abstract

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Block-oriented nonlinear models are popular in nonlinear system identification because of their advantages of being simple to understand and easy to use. Many different identification approaches were developed over the years to estimate the parameters of a wide range of block-oriented nonlinear models. One class of these approaches uses linear approximations to initialize the identification algorithm. The best linear approximation framework and the $ε$-approximation framework, or equivalent frameworks, allow the user to extract important information about the system, guide the user in selecting good candidate model structures and orders, and prove to be a good starting point for nonlinear system identification algorithms. This paper gives an overview of the different block-oriented nonlinear models that can be identified using linear approximations, and of the identification algorithms that have been developed in the past. A non-exhaustive overview of the most important other block-oriented nonlinear system identification approaches is also provided throughout this paper.

1 Introduction … Noise Framework

The paper surveys input-output identification of block-oriented nonlinear models initialized by linear approximation frameworks. It covers their structures, representations, parameterization issues, and a simplified output-noise framework with known limitations.

  • 1 Introduction: The survey examines identification algorithms for block-oriented nonlinear models using the Best Linear Approximation and ϵ-approximation frameworks.These frameworks initialize all discussed identification algorithms.
  • 1 Introduction: Linear-approximation approaches address part of model selection in an LTI framework and have obtained good results on Wiener-Hammerstein and Silverbox benchmarks.Linear models can also help detect the underlying block-oriented structure.
  • Model Structures: Block-oriented models combine LTI and static nonlinear blocks through series, parallel, and feedback connections, producing single-branch, parallel-branch, and feedback structures.The paper studies structures including Wiener, Hammerstein, Wiener-Hammerstein, Hammerstein-Wiener, parallel, and feedback models.
  • Model Representation: LTI blocks may use frequency-response, impulse-response, state-space, transfer-function, or basis-expansion representations, while static nonlinearities may use nonparametric or parametric forms.The presented methods typically use rational transfer functions and basis-function expansions linear in the parameters.
  • Model Representation: Gain exchanges, delay exchanges, and equivalence transformations create different parameterizations with identical input-output behavior and can make the cost-function Jacobian rank deficient.These indistinguishability issues affect both parameterizations and model structures.
  • Model Representation: Two simple feedback representations have the same input-output behavior when the alternative nonlinearity and LTI block are the inverses of the original blocks.This equivalence assumes both G(q) and g(.) are invertible.
  • Noise Framework: The noise framework places a Gaussian additive colored zero-mean noise source with finite variance σ2 at the system output, independent of the known input.The measured output is y(t), while a subscript 0 denotes the exact unknown value.
  • Noise Framework: The simplified output-noise assumption can bias estimates when additional noise sources occur elsewhere, such as process noise passing through a nonlinear subsystem.More realistic frameworks can introduce multiple noise sources or place noise at different system locations.

3 Bussgang’s Theorem and Separable Processes · 4 Linear Approximations of Nonlinear Systems

Bussgang’s theorem links nonlinear outputs to linear correlation and spectral relationships for Gaussian and related excitation classes. The BLA and ϵ-approximation frameworks use these properties to represent nonlinear systems through LTI approximations under specified excitation and system conditions.

  • 3.1 Bussgang’s Theorem: For zero-mean stationary Gaussian input, Bussgang’s theorem states that the input–output cross-correlation is proportional to the input autocorrelation.In the frequency domain, the corresponding relation uses the crosspower spectrum of y(t) and the autopower spectrum of u(t).
  • 3.2 Riemann Equivalence Class of Asymptotically Normally Distributed Excitation Signals: The Riemann equivalence class extends beyond Gaussian noise to periodic random-phase multisines, for which Bussgang’s theorem still applies.Random-phase multisines provide user control over the amplitude spectrum and become asymptotically Gaussian as N →∞.
  • 3.3 Separable Processes: Bussgang’s theorem has also been extended through separable processes to Gaussian, sine-wave, phase-modulated, elliptically symmetric, and some random-phase multisine signals.The cited extensions include random-phase multisines with flat amplitude spectra.
  • 4 Linear Approximations of Nonlinear Systems: Linear approximations can explain significant behavior in weakly nonlinear systems while providing insight into system behavior and motivating linear time-invariant modeling.The framework approximates a nonlinear system under well-chosen, system-specific boundary conditions.
  • 4.1 System Class: The BLA and ϵ-approximation consider PISPO systems, whose periodic steady-state outputs have the same period as their periodic inputs.The system class includes saturation, clipping, and dead-zone nonlinearities but excludes chaos and bifurcations generating sub-harmonics.
  • 4.2 Best Linear Approximation: The BLA represents a nonlinear system by an LTI model together with additive noise and unmodeled nonlinear distortion, optimized in mean square sense for a fixed input class.Its result depends on the input class, which determines both signal probability density and power spectrum; for zero-mean Gaussian input and an even static nonlinearity, the BLA is zero.
  • 4.3 ϵ-Approximation: The ϵ-approximation uses asymptotically normally distributed, limited-variance excitations converging toward zero, enabling cascade linearization when sufficiently small signals can be measured.Such measurements are often infeasible in noise, but may be possible during high-quality mechanical vibration tests or electronic-circuit measurements.

5 Cost Function

The model parameters are estimated by minimizing a least-squares cost function based on the modeled output. Although this cost function yields a maximum likelihood estimate under white Gaussian output noise, it is generally non-convex and requires careful initialization for reliable optimization.

  • Cost-function formulation: Least-squares minimization estimates the model parameters using the modeled output ˆy(t, θ).The modeled output depends on the parameter vector θ.
  • Maximum-likelihood interpretation: Under white Gaussian additive output noise satisfying Assumption 1, the cost function produces a maximum likelihood estimate.Additive colored Gaussian noise instead requires a weighted cost function, based on either a nonparametric or monic parametric noise model.
  • Maximum-likelihood interpretation: For additive colored Gaussian noise, a weighted cost function is required to obtain a sample maximum likelihood estimate.The estimated sample noise variance can serve as a weighting function in the frequency domain.
  • Optimization: Because the cost function is usually non-convex in θ, Levenberg-Marquardt optimization converges to a local minimum, making good initial parameter values important.The algorithm is used for numerically stable minimization and to support convergence toward the global minimum.

6 Initialization Methods: an Overview · 7 Single Branch Models

The paper surveys initialization methods beyond linear-approximation approaches, including overparametrization, alternating least squares, inverse estimation, and other structure-exploiting techniques. For single-branch models, the BLA reveals scaled linear dynamics and supports LTI-based pole-zero model selection.

  • 6 Initialization Methods: an Overview: Block-oriented nonlinear systems can be initialized using multiple estimation methods, with a broader literature overview covering approaches beyond linear approximations.The paper notes that two or more estimation methods are often combined to identify the system under test.
  • 6.1 Overparametrization: Overparametrization first increases the parameter count to make identification linear, then reduces the estimated set to recover its underlying low-rank structure.Singular value decomposition is given as an example of the reduction step.
  • 6.1 Overparametrization: Overparametrization can increase parameter-estimate variance because its first step requires estimating many parameters, while nuclear-norm and kernel methods reduce variance and user interaction.These methods improve classical approaches by requiring less user interaction and reducing variance.
  • 6.2 Alternating Least Squares Methods: Alternating least squares solves multilinear identification problems and was among the first successful methods for Hammerstein systems, whose parameters form a bilinear problem.Separable least squares has also been applied to Hammerstein and Wiener models.
  • 6.3 Inverse Estimation: Inverse estimation can simplify difficult identification problems by estimating an invertible part of the system, providing initialization for Wiener and Hammerstein-Wiener models.For Wiener systems, identifying the inverse static nonlinearity makes the problem linear in the parameters, but output noise causes bias.
  • 6.4 Other Approaches: Other approaches include subspace, frequency-domain, designed-input, Gaussian-input, and convex-optimization methods for exploiting block-oriented structure and nonlinear behavior.Designed inputs include steps, sine waves, sinesweeps, and phase-coupled multisines; Gaussian inputs support use of Bussgang’s Theorem.
  • 7 Single Branch Models: For single-branch Hammerstein, Wiener, and Wiener-Hammerstein systems, the BLA equals a scaled version of the system’s linear dynamics and supports pole-zero selection in the LTI framework.This provides valuable information for model selection before nonlinear identification.

Hammerstein Structure … Wiener identification using the BLA

The paper surveys Hammerstein, Wiener, and Wiener-Hammerstein structures and shows how BLA-based methods initialize consistent identification by separating linear dynamics from static nonlinearities. For Wiener-Hammerstein systems, the BLA reveals scaled dynamics, while the two-step Hammerstein and Wiener procedures estimate model components sequentially.

  • Hammerstein Structure: A Hammerstein structure places a static nonlinearity f(u) before an LTI block S(q), modeling systems with input-side nonlinear behavior.Examples include nonlinear actuators followed by linear processes, as well as some chemical and physiological processes.
  • Hammerstein Structure: Hammerstein identification algorithms include early methods, later improvements, and extensions covering kernel-based, mixed parametric-nonparametric, feedback, errors-in-variables, MIMO, and dynamic nonlinearities.The survey presents a non-exhaustive classification of these approaches by their properties.
  • Wiener Structure: A Wiener structure places an LTI block G(q) before a static nonlinearity g(x), modeling systems whose nonlinear behavior occurs at the output.Examples include sensor nonlinearities, overflow valves, and some physiological systems; identification methods span nonparametric and parametric approaches.
  • Wiener-Hammerstein Structure: A Wiener-Hammerstein model sandwiches a static nonlinearity f(x) between LTI blocks G(q) and S(q), making identification harder because the dynamics must be separated.Reported approaches include nonparametric estimation, Volterra and tensor decompositions, iterative, recursive EIV, and evolutionary methods.
  • Best Linear Approximation: For Wiener-Hammerstein systems excited by inputs in SU, the BLA is a scaled version of the LTI dynamics present in G(q)S(q).Theorem 2 gives the asymptotic BLA as the number of data tends to infinity.
  • Hammerstein identification using the BLA: BLA-based Hammerstein identification decouples estimation of the LTI block from estimation of the static nonlinear block.The BLA first provides a consistent estimate of the LTI block up to an unknown gain exchange α.
  • Hammerstein identification using the BLA: Replacing S(q) with its estimate makes the static-nonlinearity problem linear in parameters γ_i, enabling linear least squares estimation.This yields a consistent but not efficient estimator; nonlinear optimization can provide an efficient estimate.
  • Wiener identification using the BLA: BLA-based Wiener identification estimates G(q) first, then identifies the static nonlinearity from the estimated intermediate signal x̂(t) = Ĝ(q)[u(t)] and the output.Polynomial, neural-network, or other preferred structures can describe the nonlinearity, yielding a consistent but not efficient estimator.

Wiener-Hammerstein identification using the BLA · Best Linear Approximation · Identification using the BLA

The BLA supports Wiener-Hammerstein identification by turning separation of the two LTI blocks into pole-zero allocation, while for Hammerstein-Wiener systems it provides an initialization despite weaker direct interpretability. The approaches enumerate or optimize candidate allocations and estimate nonlinearities, with nonlinear refinement reducing estimate variance.

  • Wiener-Hammerstein identification using the BLA: Wiener-Hammerstein identification reformulates separation of the front and back dynamics as assigning poles and zeros to G(q) or S(q).The BLA is first estimated and parametrized; under no pole-zero cancellations, its poles and zeros correspond to those of the two LTI blocks.
  • Wiener-Hammerstein identification using the BLA: Binary parameters β encode whether each pole or zero belongs to G(q) or S(q), with complex-conjugate pairs assigned together.Pairwise assignment preserves real-coefficient transfer functions, but parametrization can combine two real poles or zeros into a complex-conjugate pair.
  • Wiener-Hammerstein identification using the BLA: Each candidate allocation is evaluated by estimating a basis-function static nonlinearity through linear least squares, after which the lowest-error model is selected.The finite candidate set is scanned by fitting a static nonlinearity for every realization.
  • Wiener-Hammerstein identification using the BLA: The brute-force allocation scan becomes computationally expensive because the number of realizations increases rapidly with model order and equals the required number of least-squares regressions.Discrete optimization, fractional approaches, and higher-order correlation methods such as QBLA can reduce or avoid exhaustive allocation evaluation.
  • Wiener-Hammerstein identification using the BLA: Although the BLA-based Wiener-Hammerstein approach yields consistent parameter estimates, subsequent nonlinear optimization of all parameters can reduce their variance.The nonlinear optimization follows the linear initialization step.
  • Best Linear Approximation: For Hammerstein-Wiener systems, the BLA cannot generally be simplified to a direct function of the system dynamics because the intermediate signal is non-Gaussian.The ϵ-approximation instead gives a scaled version of the LTI dynamics G(q), while the BLA is often close to a scaled G(q) in practical cases.
  • Identification using the BLA: Hammerstein-Wiener identification uses the BLA or ϵ-approximation to initialize an estimate of G(q), then identifies the front nonlinearity and inverse back nonlinearity.Representing the nonlinearities with basis functions makes the problem linear in the parameters and solvable by total least squares.

8 Parallel Branch Model Structures · Best Linear Approximation

Parallel branch models extend single-branch structures to represent multiple signal paths and increase modeling capacity. The BLA helps identify branch counts and LTI dynamics, with poles preserved while zeros and gains may vary with input conditions.

  • 8 Parallel Branch Model Structures: Parallel Hammerstein, Wiener, and Wiener-Hammerstein structures extend single-branch models through parallel signal paths and increased modeling capacity.Their use can be motivated by multiple signal paths inside the system under test.
  • 8 Parallel Branch Model Structures: Combining BLAs measured at different system setpoints supports selecting the number of parallel branches and the numbers of poles and zeros.
  • 8 Parallel Branch Model Structures: The BLA can initialize orthonormal basis functions for system dynamics using user-selected pole locations, naturally starting from BLA poles.
  • 8.1 Parallel Hammerstein and Parallel Wiener Structures: Parallel Hammerstein and parallel Wiener models connect more than one Hammerstein or Wiener system in parallel and are generalized forms of those structures.Parallel Hammerstein models are also called Uryson, memory polynomial, or generalized Hammerstein models.
  • 8.1 Parallel Hammerstein and Parallel Wiener Structures: A common parallel Hammerstein identification method feeds nonlinear basis functions into FIR filters and solves the resulting LTI multiple-input-single-output problem.
  • 8.1 Parallel Hammerstein and Parallel Wiener Structures: Earlier parallel Wiener methods may require very large datasets for Volterra-kernel estimation or be restricted to FIR linear subsystems.These limitations can result in an unwanted high n…
  • Best Linear Approximation: For parallel Hammerstein and Wiener systems, the BLA is a simple function of the LTI dynamics, and its poles are also poles of the system’s LTI blocks.The BLA of a parallel Hammerstein system is obtained by summing the BLAs of its separate branches.
  • Best Linear Approximation: BLA zeros may change with input variance or offset, while parallel Wiener gains also depend on input coloring; the denominator remains unchanged as gains vary.

Parallel Wiener and parallel Hammerstein identification using the BLA - SVD Approach

The BLA–SVD approach identifies parallel Wiener and Hammerstein systems by estimating BLAs under varied operating conditions, jointly parametrizing their dynamics, and decomposing them through singular value decomposition. The decomposition estimates branch dynamics and branch count, after which static nonlinearities are obtained by linear least squares.

  • BLA estimation: The approach estimates the system’s BLA under multiple operating conditions created by varying input power-spectrum magnitude, offset, or coloring.These conditions provide distinct BLA measurements for subsequent parametrization.
  • BLA parametrization: BLAs from all operating conditions are parametrized with separate LTI models sharing a common denominator, yielding consistent estimates of the overall dynamics.The common denominator is justified because the poles of the measured BLAs are identical.
  • SVD decomposition: The stacked numerator-coefficient matrix is decomposed using SVD, whose right singular vectors estimate the numerator coefficients of the individual parallel branches.The singular-vector matrices provide orthonormal bases for the relevant row and column spaces.
  • Branch-count estimation: The number of parallel branches is inferred from the estimated rank of the decomposed matrix using its singular values, after whitening the matrix beforehand.Whitening is required before rank estimation.
  • Nonlinearity estimation: After estimating the LTI blocks, the static nonlinearities are identified by linear least squares, while joint nonlinear optimization can reduce estimator variance.The BLA–SVD procedure provides consistent system-parameter estimates before optional simultaneous nonlinear optimization.

Best Linear Approximation · Identification using the BLA · Best Linear Approximation

The survey presents the BLA as a practical bridge from linear approximations to block-oriented nonlinear identification. It uses BLA dynamics to guide basis construction and describes tradeoffs among orthogonality, conditioning, experiments, and parameter growth.

  • Best Linear Approximation: For a Wiener-Schetzen model, the BLA is a linear combination of the dynamics in its different branches, so its poles are also poles of the system’s LTI dynamics.This follows because the Wiener-Schetzen model is a parallel Wiener model with a MISO polynomial nonlinearity.
  • Identification using the BLA: Orthogonal basis functions are stable, discrete-time, proper, rational transfer functions, and their shared-input outputs are orthogonal when the input is white.For non-white inputs, general orthogonality claims do not hold, although the considered bases are robust to input coloring.
  • Identification using the BLA: Laguerre, Kautz, and Takenaka-Malmquist functions are popular orthogonal basis choices constructed from stable poles ξ_i with |ξ_i| < 1.All poles at the origin yield FIR models; identical real poles yield Laguerre bases; alternating two poles yields Kautz bases, while periodically repeated finite pole sets yield GOBFs.
  • Identification using the BLA: Choosing basis poles equal to the underlying linear dynamics is optimal for accurate representation with few basis functions, making BLA poles excellent candidates.Pole-estimation mismatches can be compensated by periodically repeating the estimated pole set.
  • Identification using the BLA: After constructing the bases, estimating polynomial coefficients becomes a linear regression problem that is optimally conditioned for white Gaussian inputs with Hermite polynomials.The orthogonality is exact only for infinitely long records; finite records provide approximate orthogonality.
  • Identification using the BLA: Compared with the BLA/SVD approach, this method needs only one experiment but introduces many parameters because polynomial coefficients grow combinatorially with basis count and nonlinear degree.Dimension-reduction techniques can reduce the number of basis functions and parameters.
  • Best Linear Approximation: Parallel Wiener-Hammerstein structures increase flexibility beyond parallel Hammerstein and Wiener models, which emphasize dominant output or input dynamics, respectively.Earlier methods studied subclasses of the parallel Wiener-Hammerstein structure.
  • Best Linear Approximation: For parallel Wiener-Hammerstein systems, the BLA is obtained by summing the BLAs of the individual parallel branches.Thus, the BLA remains a simple function of the LTI dynamics present in those branches, asymptotically as data length grows.

Identification using the BLA - SVD Approach … Identification using the BLA

The paper presents BLA-based identification procedures for parallel Wiener-Hammerstein and feedback block-oriented models. These methods exploit BLA or ε-approximations for initialization and model selection, while facing computational and approximation limitations in feedback settings.

  • Identification using the BLA - SVD Approach: BLA decomposition first splits the dynamics across parallel branches, after which a brute-force scan allocates each branch’s poles and zeros to the front or back and estimates its static nonlinearity.The parallel Wiener-Hammerstein algorithm combines the procedures for parallel Hammerstein and parallel Wiener identification.
  • Identification using the BLA - SVD Approach: A subsequent nonlinear optimization of all parameters can reduce estimate variance, although the SVD approach has high computational load and introduces a MIMO static nonlinearity into an intrinsically decoupled model.The MIMO nonlinearity results from equivalence transforms and the SVD-based dynamics split.
  • 9 Feedback Model Structures: Feedback structures are motivated by nonlinear feedback phenomena such as resonance frequencies that depend on excitation variance, and they can increase modeling capacity.Nonlinear feedback is also often encountered in biological systems.
  • 9 Feedback Model Structures: BLA or ε-approximation frameworks support selecting the number of poles and zeros in feedback-system dynamics, under the assumption that the considered systems are BIBO stable for the selected input class.Feedback models commonly include at least one sample delay to avoid nonlinear algebraic loops; under band-limited assumptions, the resulting approximation error can be made arbitrarily low with sufficiently high sampling frequency.
  • 9.1 Simple Feedback Structure: In a simple feedback structure, an LTI block and a static nonlinear block are connected in feedback, with either block allowed in the feedforward path because the two arrangements are input-output equivalent.BLA-based identification algorithms have been proposed for this structure.
  • Best Linear Approximation: For simple feedback systems, the BLA generally lacks a simple dynamics-only expression because the nonlinear block’s input is non-Gaussian, whereas a sufficiently long linear memory can make the approximation relevant.The approximation treats the nonlinearity as a gain α, and its quality depends on how non-Gaussian the nonlinear input is.
  • Best Linear Approximation: Unlike the BLA, the ε-approximation gives a simple exact analytical expression in terms of system dynamics, with α replaced by γ; both gains may depend on input variance, power spectrum, and offset.In feedback, only the poles of the ε-approximation shift when gain γ changes.
  • Identification using the BLA: For simple feedback identification, the BLA initializes the linear subsystem up to equivalence transforms, the static nonlinearity is estimated using basis functions, and joint nonlinear optimization produces a consistent estimate.The initial nonlinear estimate uses the noisy output as a regressor and works well for high signal-to-noise ratios; output noise is assumed outside the feedback loop.

Best Linear Approximation · Identification using the BLA

For Wiener-Hammerstein feedback structures, the BLA is often approximated by a system-dynamics expression under practical assumptions, whereas the ε-approximation provides an exact analytical expression. BLA-based identification estimates dynamics from multiple setpoints, separates linear blocks, and jointly refines all parameters through nonlinear optimization.

  • Best Linear Approximation: The BLA of a Wiener-Hammerstein feedback structure cannot generally be simplified for inputs belonging to SU, but is often approximated by a rational expression involving the system dynamics.This approximation is considered relevant when the cascade G[2](q)G[3](q)G[1](q) has sufficiently large memory length.
  • Best Linear Approximation: Unlike the BLA, the ε-approximation gives a simple and exact analytical expression by replacing α with γ in the same form.The resulting gain can depend on system characteristics and on the input variance, power spectrum, and DC offset.
  • Identification using the BLA: The BLA-based algorithm first identifies the system’s BLA at R ≥2 different setpoints.The setpoint-specific approximations use gains α[j] for j = 1, . . . , R.
  • Identification using the BLA: A nonparametric initial estimate of the feedback block G[3](ejωTs) is obtained by averaging the inverses of the setpoint-specific BLA estimates.The inverse-BLA relationship uses β = 1 α.
  • Identification using the BLA: At each setpoint, the method estimates a scaled inverse of S(ejωTs) = G[1](ejωTs)G[2](ejωTs), then collects these estimates across frequencies and setpoints in a matrix.The matrix enables extraction of the shared dynamics across operating points.
  • Identification using the BLA: SVD identifies the estimate ˆS−1(ejωkTs) as the first left singular vector, whose inverse yields G[1](ejωTs)G[2](ejωTs) after ignoring the scaling factor.The scaling factor is accounted for when estimating the static nonlinear block.
  • Identification using the BLA: The product dynamics ˆS(ejωkTs) are split into front and back linear blocks using methods from Sections 7.1 and 9.1.This produces estimates for the individual linear blocks of the Wiener-Hammerstein feedback system.
  • Identification using the BLA: Finally, nonlinear optimization jointly refines all closed-loop Wiener-Hammerstein feedback parameters to obtain a consistent estimate.The feedback mechanism is assumed to act on the noise-free signal.

Best Linear Approximation

For general LFR systems, the BLA is not easily expressible from system dynamics because nonlinear feedback makes the nonlinearity input non-Gaussian and prevents direct use of Bussgang’s theorem. The ε-approximation instead provides a simple exact analytical expression, while both approximation gains can depend on input-signal properties.

  • Best Linear Approximation: For general LFR systems, the BLA cannot be simplified to an easy function of system dynamics, even with SU input signals.The nonlinearity input is non-Gaussian because of the nonlinear feedback loop, so Bussgang’s theorem cannot be applied.
  • Best Linear Approximation: The approximation based on linearizing the nonlinearity to gain α is considered fair in many practical cases.This assumption leads to an approximately gain-scaled form for the BLA.
  • Best Linear Approximation: The approximation in (56) is relevant when G[4](q) has a sufficiently large memory length.This condition is stated for the approximation in (56).
  • Best Linear Approximation: The ε-approximation yields a simple and exact analytical expression of the linear approximation in terms of system dynamics.Its expression for Gε(ejωTs) has the same form as the BLA expression, with α replaced by γ.
  • Best Linear Approximation: The gains α and γ depend not only on system characteristics but also on input variance, power spectrum, and offset.The offset is specified as the input signal’s DC value.

Identification using the BLA · 10 Pros and cons of the model structures · 11 Guidelines for the User

The paper describes BLA-based identification procedures, compares block-oriented model structures by approximation power, parameter count, estimation difficulty, and limitations, and gives guidelines for model selection and input design. The BLA also serves as a low-cost analysis tool for detecting nonlinear effects and informing model choices.

  • Identification using the BLA: LFR identification estimates BLAs at two setpoints, transforms them into state-space models, scans Riccati-equation solutions, selects the best-fit model, and jointly refines all parameters.The final nonlinear optimization yields a consistent estimate under the stated assumptions, while the feedback mechanism acts on the noise-free signal.
  • 10 Pros and cons of the model structures: Model structures trade descriptive power against parameter count and estimation difficulty: complex structures can model broader behavior, whereas simpler structures are more parameter-parsimonious but less flexible.Independent parameters increase estimation variance, while nonlinear optimization requires good initial estimates because its cost function is non-convex.
  • 10 Pros and cons of the model structures: Only parallel Wiener, Wiener-Schetzen, and parallel Wiener-Hammerstein structures are universal approximators; feedforward structures cannot capture chaotic behavior, sub-harmonics, or hysteresis.Feedback is required for those behaviors, and LFR is the most general one-static-nonlinearity structure, containing the other listed structures as special cases.
  • 10 Pros and cons of the model structures: Single-branch and parallel Hammerstein models use relatively few parameters, while parallel Wiener and Wiener-Schetzen models typically require many because of their MISO static nonlinear blocks.BLA-based methods generally provide consistent initial estimates, but feedback-related methods and Hammerstein-Wiener identification have consistency limitations involving noise-corrupted signals or reconstructed internal signals.
  • 10 Pros and cons of the model structures: Identification scans are mildly expensive for single Wiener-Hammerstein branches and LFR, but become computationally expensive for parallel Wiener-Hammerstein models because every branch requires scanning and MIMO nonlinear estimation.Feedforward stability is easy to check when subblocks are BIBO stable, whereas feedback stability checks are more involved.
  • 10 Pros and cons of the model structures: Very noisy data or strongly nonlinear systems can degrade the BLA and increase the likelihood that nonlinear optimization reaches a local minimum; even nonlinearities make the BLA zero and typically yield poor linear-block estimates.Weighted PCA is reported as an alternative for Wiener-system identification with an even nonlinearity.
  • 11 Guidelines for the User: The BLA framework provides low-cost insight into nonlinearity, the need for a nonlinear model, and observed nonlinear effects, while setpoint changes can reveal feedback loops through pole shifts and parallel paths through zero shifts.It also estimates nonlinear and output-noise disturbances in the frequency domain for quantifying their levels and relative dominance.
  • 11 Guidelines for the User: Input signals should represent realistic amplitude, frequency, and amplitude-distribution conditions; periodic random signals and random-phase multisines support BLA analysis, while more realizations reduce distortion variance.Small multisines can be superimposed on realistic inputs, and the identification algorithms remain fairly robust to deviations from their preferred excitation class.

12 Benchmark and Practical Results · 13 Future Research Directions · 14 Conclusion

BLA-based block-oriented identification methods show strong performance across benchmark and real-life systems, while structural choices remain application-dependent. The survey concludes that BLA frameworks support modeling decisions and initialization, but highlights open challenges involving dynamic nonlinearities, complex structures, noise, limited data, model errors, and usability.

  • 12 Benchmark and Practical Results: BLA-based methods achieved excellent results on Wiener-Hammerstein and Silverbox benchmarks and were successfully applied to coupled electric drives.The surveyed approaches also covered additional benchmark and practical systems.
  • 12 Benchmark and Practical Results: BLA-based approaches successfully modeled diverse practical systems, including insulin-glucose regulation, RF crystal detection, logarithmic amplification, and valve audio amplification.The cited model classes included Wiener, Wiener-Schetzen, LFR, Wiener-Hammerstein feedback, parallel Wiener, Hammerstein, and parallel Hammerstein models.
  • 12 Benchmark and Practical Results: Including structural prior knowledge improves the trade-off between modeling performance and parameter parsimony, although model-structure selection remains application-dependent.For control applications, restricting the model class to invertible models can be appropriate.
  • 13 Future Research Directions: An open research question is incorporating dynamic nonlinearities into the block-oriented modeling toolbox, with initial BLA-based steps reported for Hammerstein and Wiener models.Prior work addressed backlash, backlash-inverse, and hysteresis nonlinearities.
  • 13 Future Research Directions: Further challenges include flexible methods for complex user-generated structures and MIMO extensions that preserve block-oriented structure.Some efforts had already been made toward identifying MIMO block-oriented structures.
  • 13 Future Research Directions: Because most methods assume output-error noise, developing identification methods for more complex noise frameworks remains an ongoing research effort.The output-error assumption is described as a simplified representation of reality.
  • 13 Future Research Directions: Open problems also concern model-error characterization, poorly excited or short-record data, and integrating prior knowledge, with regularization proposed as a possible solution in several cases.Prior knowledge can include structure, static-nonlinearity type, poles, zeros, and stability; BLA results may guide treatment of model errors.
  • 14 Conclusion: The survey covers a wide range of BLA-identifiable structures and concludes that BLA extracts system information, guides modeling decisions, and initializes nonlinear identification algorithms.Covered structures include single-branch, parallel-branch, and nonlinear-feedback systems; future and ongoing research directions are also discussed.
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