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W O R K S HO P O N N O NL I N E A R S Y S T E M I D E N T I F I C A T I O N B E N C H M A R K S
AP R IL 2 5 -2 7 , 2 0 1 6 , BR U S SE LS , BE LG IU M
Organizers:
Jean-Philippe Noël, University of Liège
Maarten Schoukens, Vrije Universiteit Brussel
Scientific Committee:
Gaëtan Kerschen, University of Liège
Carl Edward Rasmussen, University of Cambridge
Thomas Schön, Uppsala Universitet
Johan Schoukens, Vrije Universiteit Brussel
Keith Worden, The University of Sheffield
Organized with the support of:
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P R O G R A M
MONDAY 25/04/2016
08.30 – 09.00 Registration & Coffee 09.00 – 09.30 Welcome &
Introduction J.-P. Noël & M. Schoukens
09.30 – 10.30 Keynote 1 K. Worden – Is System Identification Just Machine Learning?
10.30 – 11.00 Coffee 11.00 – 12.30 Session 1:
Wiener-Hammerstein K. Tiels – PNLSS 1.0 - A polynomial nonlinear state-space Matlab toolbox A. Svensson, F. Lindsten, T.B. Schön – Particle methods for the Wiener-Hammerstein system E. Zhang, M. Schoukens, J. Schoukens – Structural modeling of Wiener-Hammerstein system in the presence of the process noise
12.30 – 13.45 Lunch 13.45 – 14.45 Keynote 2 C.E. Rasmussen – Variational Inference in Gaussian
Processes for Non-Linear Time Series
14.45 – 15.45 Session 2: Cascaded Tanks
G. Holmes, T. Rogers, E.J. Cross, N. Dervilis, G. Manson, R.J. Barthorpe, K. Worden – Cascaded Tanks Benchmark: Parametric and Nonparametric Identification G. Giordano, J. Sjöberg – Cascade Tanks Benchmark
15.45 – 16.15 Coffee 16.15 – 17.15 Session 3:
Bouc-Wen J.-P. Noël, A.F. Esfahani, G. Kerschen, J. Schoukens – A nonlinear state-space solution to a hysteretic benchmark in system identification A.F. Esfahani, P. Dreesen, K. Tiels, J.-P. Noël, J. Schoukens – Using a polynomial decoupling algorithm for state-space identification of a Bouc-Wen system
17.15 – 17.45 Discussion Session 1 Chairs: T. Schön & J. Schoukens
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TUESDAY 26/04/2016
08.30 – 9.30 Keynote 3 J. Schoukens – Data Driven Discrete Time Modeling of Continuous Time Nonlinear Systems: Problems, Challenges, Success Stories
09.30 – 10.30 Session 4: Bouc-Wen
R. Gaasbeek, R. Mohan – Control-focused identification of hysteric systems: Selecting model structures? Think about the final use of the model! A. Bajrić – System identification of a linearized hysteretic system using covariance driven stochastic subspace identification
10.30 – 11.00 Coffee 11.00 – 12.30 Session 5:
Cascaded Tanks R. Relan, K. Tiels, A. Marconato – Identifying an Unstructured Flexible Nonlinear Model for the Cascaded Water-tanks Benchmark: Capabilities and Short-comings
P. Mattson, D. Zachariah, P. Stoica – Identification of a PWARX model for the cascade water tanks
G. Birpoutsoukis, P.Z. Csurcsia – Nonparametric Volterra series estimate of the cascaded tank
12.30 – 13.45 Lunch 13.45 – 14.45 Keynote 4 T. Schön – Solving Nonlinear Inference Problems using
Sequential Monte Carlo
14.45 – 15.15 Session 6
M. Rébillat, K. Ege, N. Mechbal, J. Antoni – Repeated exponential sine sweeps for the autonomous estimation of nonlinearities and bootstrap assessment of uncertainties
15.15 – 15.45 Coffee 15.45 – 16.45 Session 7:
Wiener-Hammerstein M. Schoukens – Identification of Wiener-Hammerstein systems with process noise using an Errors-in-Variables framework K. Worden, G. Manson, R.J. Barthorpe, E.J. Cross, N. Dervilis, G. Holmes, T. Rogers – Wiener-Hammerstein Benchmark with process noise: Parametric and Nonparametric Identification
16.45 – 17.15 Discussion Session 2 Chairs: C.E. Rasmussen & K. Worden
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WEDNESDAY 27/04/2016
08.30 – 9.30 Keynote 5 G. Kerschen – Identification of Nonlinear Mechanical Systems: State of the Art and Recent Trends
09.30 – 10.30 Session 8: Bouc-Wen
G. Manson, R.J. Barthorpe, E.J. Cross, N. Dervilis, G. Holmes, T. Rogers, K. Worden – Bouc-Wen Benchmark: Parametric and Nonparametric Identification E. Louarroudi, S. Vanlanduit, R. Pintelon – Identification of non-linear restoring forces through linear time-periodic approximations
10.30 – 11.00 Coffee 11.00 – 12.00 Session 9:
Cascaded Tanks M. Schoukens, F.G. Scheiwe – Modeling Nonlinear Systems Using a Volterra Feedback Model
A. Svensson, F. Lindsten, T.B. Schön – First principles and black box modeling of the cascaded water tanks
12.00 – 12.30 Discussion Session 3 & Closing
Chairs: J.-P. Noël & M. Schoukens
12.30 – 13.45 Lunch
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Benchmark Setups
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Hysteretic benchmark
with a dynamic nonlinearity
J.P. Noel1, M. Schoukens2
1 Space Structures and Systems LaboratoryAerospace and Mechanical Engineering Department
University of Liege, Liege, Belgium
2 ELEC DepartmentVrije Universiteit Brussel, Brussels, Belgium
1 Introduction
Hysteresis is a phenomenology commonly encountered in very diverse engineering andscience disciplines, ranging from solid mechanics, electromagnetism and aerodynamics [1,2, 3] to biology, ecology and psychology [4, 5, 6]. The defining property of a hystereticsystem is the persistence of an input-output loop as the input frequency approacheszero [7]. Hysteretic systems are inherently nonlinear, as the quasi-static existence of aloop requires an input-output phase shift different from 0 and 180 degrees, which are theonly two options offered by linear theory. The root cause of hysteresis is multistability [8].A hysteretic system possesses multiple stable equilibria, attracting the output dependingon the input history. In this sense, it is appropriate to refer hysteresis as system nonlinearmemory.
This document describes the synthesis of noisy data exhibiting hysteresis behaviour car-ried out by combining the Bouc-Wen differential equations (Section 2) and the Newmarkintegration rules (Section 3). User guidelines to an accurate simulation are provided inSection 4. The test signals and the figures of merit that are used in this benchmark arepresented in Section 5. Anticipated nonlinear system identification challenges associatedwith the present benchmark are listed in Section 6.
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2 The Bouc-Wen model of hysteresis
The Bouc-Wen model [9, 10] has been intensively exploited during the last decades torepresent hysteretic effects in mechanical engineering, especially in the case of randomvibrations. Extensive literature reviews about phenomenological and applied aspects re-lated to Bouc-Wen modelling can be found in Refs. [11, 12].
The vibrations of a single-degree-of-freedom Bouc-Wen system, i.e. a Bouc-Wen oscillatorwith a single mass, is governed by Newton’s law of dynamics written in the form [10]
mL y(t) + r(y, y) + z(y, y) = u(t), (1)
where mL is the mass constant, y the displacement, u the external force, and where anover-dot indicates a derivative with respect to the time variable t. The total restoringforce in the system is composed of a static nonlinear term r(y, y), which only dependson the instantaneous values of the displacement y(t) and velocity y(t), and of a dynamic,i.e. history-dependent, nonlinear term z(y, y), which encodes the hysteretic memory ofthe system. In the present study, the static restoring force contribution is assumed to belinear, that is
r(y, y) = kL y + cL y, (2)
where kL and cL are the linear stiffness and viscous damping coefficients, respectively.The hysteretic force z(y, y) obeys the first-order differential equation
z(y, y) = α y − β(γ |y| |z|ν−1 z + δ y |z|ν
), (3)
where the five Bouc-Wen parameters α, β, γ, δ and ν are used to tune the shape andthe smoothness of the system hysteresis loop. Table 1 lists the values of the physicalparameters selected in this study. The linear modal parameters deduced from mL, cL andkL are given in Table 2. Fig. 1 (a) illustrates the existence of a non-degenerate loop in thesystem input-output plane for quasi-static forcing conditions. In comparison, by settingthe β parameter to 0, a linear behaviour is retrieved in Fig. 1 (b). The excitation u(t) inthese two figures is a sine wave with a frequency of 1 Hz and an amplitude of 120 N . Theresponse exhibits no initial condition transients as it is depicted over 10 cycles in steadystate.
Parameter mL cL kL α β γ δ νValue (in SI unit) 2 10 5 104 5 104 1 103 0.8 -1.1 1
Table 1: Physical parameters of the Bouc-Wen system.
Parameter Natural frequency ω0 (Hz) Damping ratio ζ (%)Value 35.59 1.12
Table 2: Linear modal parameters of the Bouc-Wen system.
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−150 −100 −50 0 50 100 150−1
−0.5
0
0.5
1
Input (N)
Outp
ut (m
m)
(a)
−150 −100 −50 0 50 100 150−1.5
−1
−0.5
0
0.5
1
1.5
Input (N)
Outp
ut (m
m)
(b)
Figure 1: Hysteresis loop in the system input-output plane for quasi-static forcing con-ditions. (a) Non-degenerate loop obtained for the parameters in Table 1; (b) linear be-haviour retrieved when setting the β parameter to 0.
3 Time integration
The Bouc-Wen dynamics in Eqs. (1) and (3) can be effectively integrated in time using aNewmark method. Newmark integration relies on one-step-ahead approximations of thevelocity and displacement fields obtained by applying Taylor expansion and numericalquadrature techniques [13]. Denoting by h the integration time step, these approximationrelations write
y(t+ h) = y(t) + (1− a) h y(t) + a h y(t+ h)y(t+ h) = y(t) + h y(t) +
(12− b)h2 y(t) + b h2 y(t+ h).
(4)
Parameters a and b are typically set to 0.5 and 0.25, respectively. Eqs. (4) are hereinenriched with an integration formula for the variable z(t), which takes the form
z(t+ h) = z(t) + (1− c) h z(t) + c h z(t+ h), (5)
where c, similarly to a, is set to 0.5. Based on Eqs. (4) and (5), a Newmark schemeproceeds in two steps. First, predictions of y(t+ h), y(t+ h) and z(t+ h) are calculatedassuming y(t + h) = 0 and z(t + h) = 0. Second, the initial predictors are corrected viaNewton-Raphson iterations so as to satisfy the dynamic equilibria in Eqs. (1) and (3).
4 User guidelines to an accurate simulation
The Newmark integration of the Bouc-Wen dynamics in Eqs. (1) and (3) is implementedin the Matlab encrypted p-file BoucWen NewmarkIntegration.p. This function features5 inputs, namely:
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• the integration time step h;
• the external force time history u(t);
• the initial value of the displacement y(t = 0);
• the initial value of the velocity y(t = 0);
• the initial value of the hysteretic force z(t = 0).
The single output of the function is the displacement time history y(t).
Based on the authors’ experience with the Newmark integration of the Bouc-Wen systemof Section 2, the following guidelines are formulated:
• it is suggested to consider a working sampling frequency of 750 Hz in order toproperly observe the harmonic components generated by the nonlinearity;
• it is strongly advised to upsample the input force u(t) by a factor 20 during timeintegration to guarantee the accuracy of the resulting displacement time series. Thiscomes down to setting the integration sampling frequency, i.e. 1/h, to 15000 Hz;
• after integration, the output sequence y(t) may be low-pass filtered and downsam-pled using the Matlab command decimate. Note that this command belongs to theMatlab Signal Processing toolbox;
• low-pass filtering may be achieved using a 30-th order FIR filter (argument ‘fir’ ofthe decimate command), paying attention to the inherent edge effects of the filter;
• the decimate command may be called several times breaking the downsamplingargument, e.g. 20, into its prime factors, e.g. 2 - 2 - 5, to enhance numericalprecision;
• initial conditions on y(t), y(t) and z(t) are usually set to 0.
The minimal working example file BoucWen ExampleIntegration.m implements all theseguidelines. In this example, a multisine excitation [14] is applied to the Bouc-Wen systemconsidering all excited frequencies in the 5 – 150 Hz band and a frequency resolution f0 =fs/N ∼= 0.09Hz, given a sampling frequency fs = 750Hz and a number of time samplesN = 8192. The root-mean-squared amplitude of the input is 50 N and 5 output periodsare simulated. The sampling rate during integration is set to 15000 Hz. The synthesiseddisplacement time history is low-pass filtered and downsampled back to 750 Hz.
In more details:
• the working and integration sampling frequencies are defined in section Time integration
parameters on line 10;
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• the excitation signal is designed in section Excitation signal design on line 17.Note that the Newmark simulation algorithm supports, in principle, all types ofinput time series;
• initial conditions are set on lines 43, 44 and 45;
• time integration is run on line 48;
• low-pass filtering and downsampling are carried out in section Low-pass filtering
and downsampling on line 51;
• the edge effects of the low-pass filter are addressed by adding an extra period duringtime integration (see lines 28 and 29) and removing it afterwards (see lines 63, 64and 65).
Note that Gaussian noise band-limited in 0 – 375 Hz is automatically added to thesynthesised measurement of y(t) considering a root-mean-squared amplitude of 8 10−3
mm. This provides a realistic signal-to-noise ratio of about 40 dB at 50 N excitationlevel. The input time series u(t) is assumed to be noiseless.
Fig. 2 (a) displays the calculated system output. The exponential decay of the systemtransient response is plotted in Fig. 2 (b) by subtracting the last synthesised period fromthe entire time record. This graph indicates that transients due to initial conditions onlyaffect the first period of measurement, and that the applied periodic input results in aperiodic output. It also demonstrates the high accuracy of the Newmark integration, asthe transient response reaches the Matlab precision of -313 dB in steady state. Remarkthat, in this particular case, no noise was added to the output to focus on integrationaccuracy.
5 Model test and figure of merit
Two fixed test datasets are provided through the benchmark meeting website: a randomphase multisine and a sine-sweep signal. The test datasets are noiseless and a samplingfrequency of 750 Hz is considered. The random phase multisine dataset contains onesteady-state period of 8192 samples. The excited band encompasses all frequencies in 5 –150 Hz, and the RMS input value is 50 N . The sine-sweep dataset is not in steady state,the simulation started with initial conditions equal to zero. In this case, the amplitude ofthe input is 40 N , and the frequency band from 20 to 50 Hz is covered at a sweep rateof 10 Hz/min. These test sets function as a target for the obtained model, the modelshould perform as good as possible on these test datasets. The goal of the benchmark isto estimate a good model on the estimation data. The test data should not be used forany purpose during the estimation.
We expect all participants of the benchmark to report the following figure of merit for all
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0 1 2 3 4 5−3
−2
−1
0
1
2
3
Period
Ou
tpu
t (m
m)
(a)
0 1 2 3 4 5−400
−300
−200
−100
0
PeriodO
utp
ut (d
B)
(b)
Figure 2: System output calculated in response to a multisine input band-limited in 5– 150 Hz. (a) Output over 5 periods, with one specific period highlighted in grey; (b)output in logarithmic scaling (in black) and decay of the transient response (in blue).
test datasets to allow for a fair comparison between different methods:
eRMSt =
√√√√1/Nt
Nt∑t=1
(ymod(t)− yt(t))2, (6)
where ymod is the modeled output, yt is the output provided in the test dataset, Nt is thetotal number of points in yt.
Also mention whether the modeled output ymod is obtained using simulation (only thetest input ut is used to obtain the modeled output ymod(t) = F (ut(1), . . . , ut(t))) orprediction (both the test input ut and the past test output yt are used to obtain themodeled output ymod(t) = F (ut(1), . . . , ut(t), yt(1), . . . , yt(t − 1))). Provide both figuresof merit (simulation and prediction) if the identified model allows for it.
6 Nonlinear system identification challenges
We anticipate the Bouc-Wen benchmark to be associated with 4 major nonlinear systemidentification challenges:
• it possesses a nonlinearity featuring memory, i.e. a dynamic nonlinearity;
• the nonlinearity is governed by an internal variable z(t), which is not measurable;
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• the nonlinear functional form in Eq. (3) is nonlinear in the parameter ν;
• the nonlinear functional form in Eq. (3) does not admit a finite Taylor series expan-sion because of the presence of absolute values.
References
[1] D.J. Morrison, Y. Jia, and J.C. Moosbrugger. Cyclic plasticity of nickel at low plasticstrain amplitude: hysteresis loop shape analysis. Materials Science and Engineering,A314:24–30, 2001.
[2] G. Bertotti. Hysteresis in Magnetism: For Physicists, Materials Scientists, andEngineers. Academic Press, San Diego, CA, 1998.
[3] T.J. Mueller. The influence of laminar separation and transition on low Reynoldsnumber airfoil hysteresis. AIAA Journal of Aircraft, 22(9):763–770, 1985.
[4] D. Angeli, J.E. Ferrell, and E.D. Sontag. Detection of multistability, bifurcations,and hysteresis in a large class of biological positive-feedback systems. Proceedings ofthe National Academy of Sciences of the United States of America, 101(7):1822–1827,2004.
[5] B.E. Beisner, D.T. Haydon, and K. Cuddington. Alternative stable states in ecology.Frontiers in Ecology and the Environment, 1(7):376–382, 2003.
[6] V.S. Ramachandran and S.M. Anstis. Perceptual organization in multistable appar-ent motion. Perception, 14:135–143, 1985.
[7] D.S. Bernstein. Ivory ghost. IEEE Control Systems Magazine, 27:16–17, 2007.
[8] O. Jinhyoung, B. Drincic, and D.S. Bernstein. Nonlinear feedback models of hystere-sis. IEEE Control Systems Magazine, 29(1):100–119, 2009.
[9] R. Bouc. Forced vibrations of a mechanical system with hysteresis. In Proceedingsof the 4th Conference on Nonlinear Oscillations, Prague, Czechoslovakia, 1967.
[10] Y. Wen. Method for random vibration of hysteretic systems. ASCE Journal of theEngineering Mechanics Division, 102(2):249–263, 1976.
[11] M. Ismail, F. Ikhouane, and J. Rodellar. The hysteresis Bouc-Wen model, a survey.Archives of Computational Methods in Engineering, 16:161–188, 2009.
[12] F. Ikhouane and J. Rodellar. Systems with Hysteresis: Analysis, Identification andControl using the Bouc-Wen Model. John Wiley & Sons, Chichester, United King-dom, 2007.
[13] M. Geradin and D.J. Rixen. Mechanical Vibrations: Theory and Application toStructural Dynamics. John Wiley & Sons, Chichester, United Kingdom, 2015.
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[14] R. Pintelon and J. Schoukens. System Identification: A Frequency Domain Approach.IEEE Press, Piscataway, NJ, 2001.
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Wiener-Hammerstein benchmark
with process noise
M. Schoukens1, J.P. Noel2
1 ELEC DepartmentVrije Universiteit Brussel, Brussels, Belgium
2 Space Structures and Systems LaboratoryAerospace and Mechanical Engineering Department
University of Liege, Liege, Belgium
1 Introduction
Process noise is already well studied and modeled in the linear time-invariant (LTI) frame-work. Nonparametric and parametric noise models (Box-Jenkins, ARX, ARMAX) providegood solutions to the LTI process noise problem [1, 2].
Most of the nonlinear modeling approaches only consider additive (colored) noise at theoutput (see, for instance, the methods listed in [3, 4]), or are restricted to an ARX orARMAX like noise model (NARX and NARMAX in [5]). Some recent methods considera more complex noise framework using expectation maximization, particle filter methods,or errors-in-variables approaches [6, 7, 8].
This benchmark presents a Wiener-Hammerstein electronic circuit where the process noiseis the dominant noise distortion.
The next sections describe the Wiener-Hammerstein system (Section 2) and describe thedata restrictions (Section 3). The test data and the figures of merit that are used in thisbenchmark are presented in Section 4. Finally, some of the expected challenges duringthe identification process are listed in Section 5.
2 Wiener-Hammerstein system with process noise
The Wiener-Hammerstein structure is a well known block-oriented system. It contains astatic nonlinearity that is sandwiched in between two LTI blocks (Figure 1). The presenceof the two LTI blocks results in a problem that is harder to identify. The system is quitesimilar to the Wiener-Hammerstein system that is studied in an earlier benchmark [9, 10],the main difference is the presence of the process noise.
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Figure 1: A Wiener-Hammerstein system with process noise. The LTI blocks at theinput and the output are depicted by R(s) and S(s) respectively. f(x) denotes the staticnonlinearity. The process noise ex(t) enters the system before the static nonlinearity.Two smaller (neglectable) noise sources eu(t) and ey(t) are present in the measurementchannels. um(t) and ym(t) are the measured input and output signals.
The first filter R(s) can be described well with a third order lowpass filter. The second LTIsubsystem S(s) is designed as an inverse Chebyshev filter with a stopband attenuationof 40 dB and a cutoff frequency of 5 kHz. The second LTI subsystem has a transmissionzero within the excited frequency range. This makes the inversion of S(s) difficult.
The static nonlinearity f(x) is realized with a diode-resistor network, this results in asaturation nonlinearity.
The additive process noise ex(t) is a filtered white Gaussian noise sequence. The filterednoise is generated starting from a discrete-time 3rd order lowpass Butterworth filter fol-lowed by a zero-order hold reconstruction and an analog low-pass reconstruction filterwith a cut-off frequency of 20 kHz. The noise sources eu(t) and ey(t) account for the mea-surement noise, they can be considered to be white Gaussian noise sources. The dominantnoise source is ex(t), the measurement noise is very small.
3 Data and user guidelines
3.1 Estimation data
The participants are offered the unique opportunity to design the estimation input sig-nals themselves. The measurements are performed at the VUB ELEC department by anexperienced user of the measurement setup during 3 different measurement campaigns.The exact dates of these measurement campaigns will be announced to all subscribedparticipants via e-mail and on the website. All measured input-output data will be of-fered to the participants to obtain a good model of the system. The possibility for theparticipants to perform a short measurement campaign at the ELEC department, VUBcan be discussed.
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• WH Measurement campaign 1: October/November
• WH Measurement campaign 2: January/February
• WH Measurement campaign 3: March
An initial dataset is available on the benchmark meeting website to perform some firstanalysis and tests on the system. All the data measured by the participants will also beavailable to all participants through the benchmark meeting website.
A reference signal will be measured during each new measurement campaign to check thereproducibility of the measurements compared to the previous measurements performedon the system.
3.2 Measurement setup
The inputs and the process noise are generated by an arbitrary zero-order hold waveformgenerator (AWG), the Agilent/HP E1445A, sampling at 78125 Hz. The generated zero-order hold signals are passed through a reconstruction filter (Tektronix Wavetek 432) witha cut-off frequency of 20 kHz. The in- and output signals of the system are measured bythe alias protected acquisition channels (Agilent/HP E1430A) sampling at 78125 Hz. TheAWG and acquisition cards are synchronized with the AWG clock, and hence the acqui-sition is phase coherent to the AWG. Leakage errors are hereby easily avoided. Finally,buffers are added between the acquisition cards and the in- and output of the system toavoid that the measurement equipment would distort the measurements.
3.3 User guidelines
The following restrictions apply for the input signals:
• The input signals should be stored in a .mat file,
• The name of the input signal variable is ’input’,
• The variable ’input’ has the dimension N ×M , where N is the number of points inthe signals and M is the number of signals that needs to be measured,
• The maximum length of the signal is Nmax = 65536,
• The maximum number of signals in one file is Mmax = 100,
• The amplitude of the signals should be between -4 and 4,
• Note that the sampling frequency is fixed: fs = 78125 Hz.
The measurement file contains a structure ’dataMeas’. This structure has 4 fields:
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• r: reference signal, signal loaded into the generator,
• u: measured input signal,
• y: measured output signal,
• fs: the sample frequency.
4 Model test and figure of merit
Two fixed test sets are provided through the benchmark meeting website: a randomphase multisine and a sine-sweep signal. Both signals are measured as periodic signals,the datasets contain one steady-state period of the signal. Both measured input signalshave an rms value of 0.71 Vrms, and they excite the frequencies from DC to 15 kHZ, DCnot included. The sine-sweep signal covers the frequency band from DC to 15 kHz at asweep rate of 4.29 MHz/min.
These test sets function as a target for the obtained model, the model should perform asgood as possible on these test datasets. The goal of the benchmark is to estimate a goodmodel on the estimation data. The test data should not be used for any purpose duringthe estimation. The test sets are measured in the absence of process noise. The noiselesstest sets can be used to evaluate the bias on the estimate since wrong noise assumptionscan lead to a biased estimate of the system under test [11].
We expect all participants of the benchmark to report the following figure of merit for alltest datasets to allow for a fair comparison between different methods:
eRMSt =
√√√√1/Nt
Nt∑t=1
(ymod(t) − yt(t))2, (1)
where ymod is the modeled output, yt is the output provided in the test dataset, Nt is thetotal number of points in yt.
Also mention whether the modeled output ymod is obtained using simulation (only thetest input ut is used to obtain the modeled output ymod(t) = F (ut(1), . . . , ut(t))) orprediction (both the test input ut and the past test output yt are used to obtain themodeled output ymod(t) = F (ut(1), . . . , ut(t), yt(1), . . . , yt(t − 1))). Provide both figuresof merit (simulation and prediction) if the identified model allows for it.
5 Nonlinear system identification challenges
We anticipate the Wiener-Hammerstein benchmark to be associated with 3 major non-linear system identification challenges:
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• the process noise that is present in the system,
• the static nonlinearity which is not directly accessible from neither the measuredinput or output,
• the output dynamics are difficult to invert due to the presence of a transmissionzero.
References
[1] R. Pintelon and J. Schoukens. System Identification: A Frequency Domain Approach.Wiley-IEEE Press, Hoboken, New Jersey, 2nd edition, 2012.
[2] L. Ljung. System Identification: Theory for the User (second edition). Prentice Hall,Upper Saddle River, New Jersey, 1999.
[3] F. Giri and E.W. Bai, editors. Block-oriented Nonlinear System Identification, vol-ume 404 of Lecture Notes in Control and Information Sciences. Springer, BerlinHeidelberg, 2010.
[4] J. Paduart, L. Lauwers, J. Swevers, K. Smolders, J. Schoukens, and R. Pintelon.Identification of nonlinear systems using polynomial nonlinear state space models.Automatica, 46(4):647–656, 2010.
[5] S.A. Billings. Nonlinear System Identification: NARMAX Methods in the Time,Frequency, and Spatio-Temporal Domains. John Wiley & Sons, Ltd., West Sussex,United Kingdom, 2013.
[6] T.B. Schon, A. Wills, and B. Ninness. System identification of nonlinear state-spacemodels. Automatica, 47(1):39–49, 2011.
[7] F. Lindsten, T.B. Schon, and M.I. Jordan. Bayesian semiparametric Wiener systemidentification. Automatica, 49(7):2053 – 2063, 2013.
[8] B. Wahlberg, J. Welsh, and L. Ljung. Identification of Wiener systems with processnoise is a nonlinear errors-in-variables problem. In 53rd IEEE Conference on Decisionand Control (CDC), pages 3328–3333, Dec. 2014.
[9] J. Schoukens, J.A.K. Suykens, and L. Ljung. Wiener-Hammerstein benchmark. In15th IFAC Symposium on System Identification (SYSID), Saint-Malo, France, July2009.
[10] G. Vandersteen. Identification of linear and nonlinear systems in an errors-in-variables least squares and total least squares framework. PhD thesis, Vrije Uni-versiteit Brussel, 1997.
[11] A. Hagenblad, L. Ljung, and A. Wills. Maximum likelihood identification of Wienermodels. Automatica, 44(11):2697–2705, 2008.
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Cascaded tanks benchmark
combining soft and hard nonlinearities
M. Schoukens1, P. Mattsson2, T. Wigren2, J.P. Noel3
1 ELEC DepartmentVrije Universiteit Brussel, Brussels, Belgium
2 Division of Systems and ControlDepartment of Information TechnologyUppsala University, Uppsala, Sweden
3 Space Structures and Systems LaboratoryAerospace and Mechanical Engineering Department
University of Liege, Liege, Belgium
1 Introduction
Many systems exhibit a quasi linear or weakly nonlinear behavior during normal operation,and a hard saturation effect for high peaks of the input signal. The proposed benchmarkis an example of this type of nonlinear system. On top of this, only a short data recordis available for the parameter estimation step.
The next sections describe the cascaded tanks system (Section 2) and introduce the esti-mation and test data (Section 3). The figures of merit that are used in this benchmark arepresented in Section 4. Finally, some of the expected challenges during the identificationprocess are listed in Section 5.
2 Cascaded tanks system
The cascaded tanks system is a fluid level control system consisting of two tanks with freeoutlets fed by a pump. The input signal controls a water pump that pumps the waterfrom a reservoir into the upper water tank. The water of the upper water tank flowsthrough a small opening into the lower water tank, and finally through a small openingfrom the lower water tank back into the reservoir. This process is shown in Figure 1.
The relation between (1) the water flowing from the upper tank to the lower tank and (2)the water flowing from the lower tank into the reservoir are weakly nonlinear functions.
20
2
Figure 1: The cascaded tanks system: the water is pumped from a reservoir in the uppertank, flows to the lower tank and finally flows back into the reservoir. The input is thepump voltage, the output is the water level of the lower tank.
However, when the amplitude of the input signal is too large, an overflow can happen inthe upper tank, and with a delay also in the lower tank. When the upper tank overflows,part of the water goes into the lower tank, the rest flows directly into the reservoir. Thiseffect is partly stochastic, hence it acts as an input-dependent process noise source. Theoverflow saturation nonlinear behavior of the lower tank is clearly visible in the timedomain representation of the output signals (see Figure 2). A video of such an overflowsituation can be found on the benchmark website.
Without considering the overflow effect, the following input-output model can be con-structed based on Bernoulli’s principle and conservation of mass:
x1(t) = −k1√
x1(t) + k4u(t) + w1(t), (1)
x2(t) = k2√
x1(t) − k3√x2(t) + w2(t), (2)
y(t) = x2(t) + e(t), (3)
where u(t) is the input signal, x1(t) and x2(t) are the states of the system, w1(t), w2(t)and e(t) are additive noise sources, and k1, k2, k3 and k4 are constants depending on thesystem properties.
3 Estimation and test data
The input signals are multisine signals which are 1024 points long, and excite the frequencyrange from 0 to 0.0144 Hz, both for the estimation and test case. The lowest frequencieshave a higher amplitude then the higher frequencies (see Figure 2). The sample periodTs is equal to 4 s. The input signals are zeroth-order hold input signals.
The process is controlled from a Matlab interface to the A/D and D/A converters attachedto the water level sensor and the pump actuator. The water level is measured using
21
3
time (s)0 1000 2000 3000 4000
mag
nitu
de (
V)
0
2
4
6
8input signals
(a)
time (s)0 1000 2000 3000 4000
mag
nitu
de (
V)
2
4
6
8
10output signals
(b)
frequency (Hz)0 0.02 0.04 0.06 0.08 0.1 0.12
mag
nitu
de (
dB)
-400
-300
-200
-100
0
100input signals
(c)
frequency (Hz)0 0.02 0.04 0.06 0.08 0.1 0.12
mag
nitu
de (
dB)
-100
-50
0
50output signals
(d)
Figure 2: Input (a,c) and output (b,d) signals of the estimation (blue) and test (red) datarecords in the time (a,b) and frequency (c,d) domain.
capacitive water level sensors, the measured output signals have a signal-to-noise ratiothat is close to 40 dB. The water level sensors are considered to be part of the system,they are not calibrated and can introduce an extra source of nonlinear behavior.
Note that the system was not in steady state during the measurements. The system stateshave an unknown initial value at the start of the measurements. This unknown state isthe same for both the estimation and the test data record.
4 Figure of merit
The goal of the benchmark is to estimate a good model on the estimation data. The testdata should not be used for any purpose during the estimation.
We expect all participants of the benchmark to report the following figure of merit for alltest datasets to allow for a fair comparison between different methods:
eRMSt =
√√√√1/Nv
Nt∑t=1
(ymod(t) − yt(t))2, (4)
where ymod is the modeled output, yt is the output provided in the test data set, Nt is thetotal number of points in yt.
22
4
Also mention whether the modeled output ymod is obtained using simulation (only thetest input ut is used to obtain the modeled output ymod(t) = F (ut(1), . . . , ut(t))) orprediction (both the test input ut and the past test output yt are used to obtain themodeled output ymod(t) = F (ut(1), . . . , ut(t), yt(1), . . . , yt(t − 1))). Provide both figuresof merit (simulation and prediction) if the identified model allows for it.
5 Nonlinear system identification challenges
We anticipate the cascaded tanks benchmark to be associated with 4 major nonlinearsystem identification challenges:
• the hard saturation nonlinearity combined with the weakly nonlinear behavior ofthe system in normal operation,
• the overflow from the upper to the lower tank, this effect also introduces input-dependent process noise,
• the relatively short estimation data record,
• the unknown initial values of the states.
23
Workshop Contributions
24
PNLSS 1.0 — A polynomial nonlinear state-space Matlab toolbox
Koen TielsVrije Universiteit Brussel
1 Introduction
A polynomial nonlinear state-space (PNLSS) model [1] cancapture many nonlinear behaviors and has been successfullyused in a large range of applications. This abstract presentsPNLSS 1.0 [2], a Matlab toolbox to identify PNLSS modelsfrom measured data. The usage of the toolbox is illustratedhere on the Wiener-Hammerstein benchmark [3], showingalso some of the limitations of the PNLSS approach. ThePNLSS approach is also illustrated on the Bouc-Wen hys-teretic system in [4] and on the cascaded tanks system in [5].
2 PNLSS modeling
A PNLSS model extends a linear discrete-time state-spacemodel with polynomial terms in the state update and the out-put equation:
x(t +1) = Ax(t)+Bu(t)+Eζ (x(t),u(t)) (1)y(t) =Cx(t)+Du(t)+Fη(x(t),u(t)) (2)
where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the inputvector, y(t)∈Rp is the output vector, and ζ (x(t),u(t))∈Rnζ
and η(x(t),u(t))∈Rnη contain nonlinear monomials in x(t)and u(t) up to a user-chosen degree.
The PNLSS model is identified from measured input/outputdata in three steps [1]: 1) Estimate the best linear approx-imation (BLA) [6] nonparametrically. 2) Estimate a para-metric linear model on the BLA using the frequency domainsubspace method in [7, 8] and a Levenberg-Marquardt non-linear optimization of the obtained linear model. 3) Estimatethe PNLSS model starting from the linear model using aLevenberg-Marquardt optimization. The difference betweenthe measured and the modeled output spectrum is minimizedin (weighted) least-squares sense.
3 Wiener-Hammerstein benchmark results
The BLA, a 6th-order linear subspace model, and thePNLSS model are estimated using 2 steady-state periods of9 realizations of a multisine signal with 4096 samples andan rms level of 0.8. The PNLSS model contains all possiblemonomials in the state update and output equation up to de-gree 3. The 10th realization of the multisine is used duringestimation of the PNLSS model to check the stability of themodel and during validation to select the best PNLSS model.Table 1 reports the rms simulation error of the linear and thePNLSS model on the multisine and swept sine test data. The
Table 1: Rms simulation error of the linear and PNLSS model onthe multisine and swept sine test data.
multisine swept sinelinear 0.03817 0.02319PNLSS 0.03817 0.02258
PNLSS model does not perform better than the linear modelon the multisine data and only slightly better on the sweptsine data. This can be attributed to the heavy process noisethat is ignored in the PNLSS modeling approach, in whichthe output error is minimized.
Acknowledgment
Many members of the nonlinear system identification teamof the ELEC Department have contributed to this software.
References[1] J. Paduart, L. Lauwers, J. Swevers, K. Smolders,J. Schoukens, and R. Pintelon. Identification of nonlinearsystems using Polynomial Nonlinear State Space models.Automatica, vol. 46, pp. 647–656, 2010.
[2] available at homepages.vub.ac.be/˜ktiels/pnlss.html.
[3] M. Schoukens and J.-P. Noel. Wiener-Hammersteinbenchmark with process noise. Technical report, home-pages.vub.ac.be/˜mschouke/benchmark2016.html, 2015.
[4] A.F. Esfahani, P. Dreesen, K. Tiels, J.-P. Noel, andJ. Schoukens. Using a polynomial decoupling algorithm forstate-space identification of a Bouc-Wen system. Workshopon Nonlinear System Identification Benchmarks, Apr. 2016.
[5] R. Relan and K. Tiels. Identifying an unstructuredflexible nonlinear model for the cascaded water-tanksbenchmark: capabilities and short-comings. Workshop onNonlinear System Identification Benchmarks, Apr. 2016.
[6] R. Pintelon and J. Schoukens. System identification:A frequency domain approach. Wiley-IEEE Press, 2nd edi-tion, 2012.
[7] T. McKelvey, H. Akcay, and L. Ljung. Subspace-based multivariable system identification from frequency re-sponse data. IEEE Trans. Autom. Control, vol. 41 pp. 960–979, 1996.
[8] R. Pintelon. Frequency-domain subspace systemidentification using nonparametric noise models. Automat-ica, vol. 38, pp. 1295–1311, 2002.
25
Particle methods for the Wiener-Hammerstein system
Andreas SvenssonUppsala University
Fredrik LindstenUppsala University
Thomas B. SchonUppsala University
1 Introduction
We investigate how to apply particle filter-based methods forthe Wiener-Hammerstein identification problem. As we willsee, the presence of only one significant noise source (in-stead of both process and measurement noise) makes the ap-plication of particle filter methods complicated. This mightat a first sight perhaps seem counterintuitive (less noise =more information), but we will explain why.
We propose two different workarounds for this problemby somehow assuming presence of noise, and we obtainpromising results for both methods.
2 Why this problem is hard – the role of the latent statevariables in particle filters
The particle filter represents the unobserved states xt in astate space model by a number of particles {xi
t}Ni=1. For a
perfectly known model θ , it basically proceeds in the fol-lowing fashion:
1. Weight update. Evaluate ‘how likely’ each particleis and attach a weight wi to particle xi
t correspondingto that: wi
t ∝ pθ (yt | xit ,ut).
2. Resampling. Remove the particles with low weights,and duplicate the particles with high weights.
3. Time update. Sample N particles for next time stepby simulating from the model for pθ (xi
t+1 | xit ,ut).
(cf. step (1) and (3) with the Kalman filter)
For the purpose of system identification, this procedure canbe entangled with a parameter estimation method [5, 3].With noise present, the weights in the weight update effec-tively becomes a measure of proximity between the actualmeasurement yt and the measurement particle xi
t would havegenerated, had it been the true state. The problem with thepresent Wiener-Hammerstein problem is that pθ (yt | xi
t ,ut)essentially is a Dirac mass, because of the (relatively) lackof measurement noise, rendering only binary weights.
In order to obtain particles with non-zero weights in the caseof almost zero measurement noise, we thus would have tosimulate particles until at least one particle ‘hits’ the actualmeasurement yt exactly. This would not be practically feasi-ble, and illustrates the challenges in applying particle filter-based methods to this problem.
3 Proposed solutions
To handle the discussed challenges, we consider two differ-ent schemes of introducing noise in the model (not the data).The results will be presented in detail at the workshop.
3.1 Particle GibbsWe use the particle Gibbs with ancestor sampling [4]to identify a nonlinear state-space model with Wiener-Hammerstein structure. In the model, we assume zero-meannoise is present for the time update of each state space com-ponent, and can thereby resort to [6, Example IV.B] for do-ing closed-form updates of the unknown parameters.
3.2 Particle Metropolis-HastingsWe use Metropolis-Hastings (MH) to learn the parameters.MH amounts to randomly propose a parameter θ ′, and ac-cept the update with a probability related to p(θ ′ | y1:T ). To(approximately) evaluate this density, a particle filter can beused. This way, a valid PMCMC algorithm is obtained [1].However, to be able to run the particle filter, measurementnoise has once again to be assumed. To converge to a modelessentially without measurement noise, we experiment withiteratively decreasing the assumed measurement noise. Wealso discuss the connections of this solution to the frame-work of approximate Bayesian computations (ABC, [2]).
References
[1] C. Andrieu, A. Doucet, and R. Holenstein. Particle Markov chainMonte Carlo methods. Journal of the Royal Statistical Society: Se-ries B (Statistical Methodology), 72(3):269–342, 2010.
[2] M. A. Beaumont, W. Zhang, and D. J. Balding. Approximate Bayesiancomputation in population genetics. Genetics, 162(4):2025–2035,2002.
[3] N. Kantas, A. Doucet, S. S. Singh, J. M. Maciejowski, and N. Chopin.On particle methods for parameter estimation in state-space models.Statistical Science, 30(3):328–351, 2015.
[4] F. Lindsten, M. I. Jordan, and T. B. Schon. Particle Gibbs with ancestorsampling. The Journal of Machine Learning Research, 15(1):2145–2184, 2014.
[5] T. B. Schon, F. Lindsten, J. Dahlin, J. Wagberg, C. A. Naesseth,A. Svensson, and L. Dai. Sequential Monte Carlo methods for systemidentification. In Proceedings of the 17th IFAC Symposium on SystemIdentification (SYSID), pages 775–786, Beijing, China, Oct. 2015.
[6] A. Svensson, T. B. Schon, A. Solin, and S. Sarkka. Nonlinear statespace model identification using a regularized basis function expan-sion. In Proceedings of the 6th IEEE international workshop on com-putational advances in multi-sensor adaptive processing (CAMSAP),pages 493–496, Cancun, Mexico, Dec. 2015.
26
Structural modeling of Wiener-Hammestein sytem in the presence ofthe process noise
Erliang ZhangSchool of Mechanical Engineering, Zhengzhou University, China
Maarten Schoukens, Johan SchoukensDepartment ELEC, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
[email protected], [email protected]
1 Introduction
The identification of Wiener-Hammerstein (WH) systemshas received intensive attention without the presence of theprocess noise, see the Special Section in Control Engineer-ing Practice [1] and the algorithms reported therein. TheWH model identification with the process noise precedingthe static nonlinearity is recently considered with a costfunction of totally different formulation [2]. So it is primaryto determine the location where the process noise enters intothe WH model prior to implementing any algorithm for con-sistent identification. The goal of this paper is to propose asimple protocol to differentiate Case I and Case II (see Fig.1), providing a good block-oriented model structure for theWH model identification.
R(q)
ex(t)
f(t) S(q)u(t) x(t) r(t) y(t)
ey(t)
case IIcase I
ym(t)
Figure 1: Wiener-Hammerstein model with the process noise.
2 Basic idea & principle
The kernel observation is that the component of the outputerrors, caused by the process noise preceding the static non-linearity, depends on the excitation signal of the WH system.So the basic idea is to use a specially designed nonstationaryinput signal [4], which is periodic and nonstationary withinone period, to reveal whether the process noise precedes thestatic nonlinearity assuming that the measurement noise isstationary.
The periodicity of the input excitation makes the output dis-turbance (measurement noise, process noise) free of the sys-tem nonlinear distortions, and its nonstationarity enables todifferentiate the output error component contributed by theprocess noise from the measurement noise when the processnoise precedes the static nonlinearity. Simply, the 1st and
2nd order statistics of the output error is used for the detec-tion of the process noise.
3 Illustration & conclusion
The proposed protocol is illustrated on the Wiener-Hammerstein benchmark [3]. Fig. 2 suggests that the pro-cess noise precedes the static nonlinearity. In summary, a
0 0.005 0.01 0.015 0.02 0.025
0.08
0.09
0.1
σe(t)
Time (s)
Figure 2: Estimated variance of the output error, σ2e (t), in one
period.
simple approach is proposed for the structural modeling ofthe WH system with the process noise. Moreover, it canprovide insight on the static nonlinearity which the processnoise precedes, and can be straightforwardly applied to otherblock-oriented models.
References[1] H. Hjalmarsson, C. R. Rojas, D. E. Rivera, Systemidentification: A Wiener-Hammerstein benchmark, ControlEngineering Practice 20 (2012) 1095 – 1096.
[2] A. Wills, T. B. Schon, L. Ljung, B.Ninness, Iden-tification of Hammerstein-Wiener models, Automatica 49(2013) 70–81.
[3] M. Schoukens, J. P. Noel, Wiener-Hammersteinbenchmark with the process noise, in: Workshop on Non-linear System Identification Benchmarks, Belgium, 2016.
[4] E. Zhang, J. Antoni, R. Pintelon, J. Schoukens, Fastdetection of system nonlinearity using nonstationary signals,Mechanical Systems and Signal Processing 24 (2010) 2065– 2075.
27
Cascaded Tanks Benchmark: Parametric and NonparametricIdentification
G. Holmes, T. Rogers, E.J. Cross, N. Dervilis, G. Manson, R.J. Barthorpe, K. WordenDynamics Research Group, Department of Mechanical Engineering, University of Sheffield,
Mappin Street, Sheffield S1 3JD, [email protected]
Abstract
Cascaded tanks or liquid level systems have a long historyas benchmark systems for system identification. The bench-mark considered in this paper is a two-tank system whichpresents a number of challenges for identification:
• The dynamics of the system is encoded in the two tanklevels; however, only one state is available in the data.
• Overflow of the upper tank means that the systemswitches between two types of (nonlinear) behaviour.
• Only a short data record is provided.
The approach taken in the current paper is to apply an evolu-tionary approach, similar to that in [1]. In fact, the problemshares with hysteretic systems, similar difficulties to the firsttwo above. The main issue caused by the short data recordis uncertainty in the parameter estimates. However, it isshown that model uncertainty is also an issue, arising fromthe omission of significant physics from the model equa-tions. For comparison, a nonparametric approach, based onGaussian process NARX models is presented. Some dis-cussion on how the GP-NARX models may be extended toinclude switching models is presented.
References[1] Worden (K.) & Manson (G.) 2012 Mechanical Sys-tems and Signal Processing 29 pp.201-212. On the identi-fication of hysteretic systems, Part I: fitness landscapes andevolutionary identification.
28
Cascade Tanks Benchmark
Giuseppe GiordanoChalmers University of Technology
Jonas SjobergChalmers University of Technology
Table 1: Black-Box Model Structures used for identification. Thelast column reports the total numbers of parameters.
Model na nb nc nk NL NparOE 2 2 - 2 - 4
ARX 2 2 - 2 - 4ARMAX 2 2 2 2 - 6
Nonlinear OE 2 2 - 2 3 Sigmoids 19Nonlinear ARX 2 2 - 2 3 Sigmoids 19
1 Black- and White- Box Approaches
In this contribution, we considered the Cascade Tanks (C-T) system. For this system, the main issue is the combina-tion of weakly nonlinear behaviour with hard saturations.Our contribution for the identification of the C-T systemis the study of several linear/nonlinear black-box/white-boxmodel structures. Simulation and prediction errors havebeen compared and the results show that, while the bestmodel for simulation performance is a nonlinear state-spacemodel based on the physical modeling of the system (8 pa-rameters), the best prediction error is achieved by a linearARMAX model (6 parameters). In Table 1, the propertiesof the tested black-box models are shown. The integers na,nb, nc, and nk indicate the order of the system, the degree ofthe numerator, the degree of the noise model, and the delayin the system. For both NOE and NARX, the nonlinearityconsists of a feed-forward net composed by 3 sigmoid func-tions. Several white-box models have been estimated, basedon the physical model given in Eq. (1)-(2), where x1, x2 arethe discrete-time states, u is the input, e is the model resid-ual and ki, i = 1, · · · ,7, x1MAX are the parameters. xover
1 is theoverflow of the first tank. The output is y = x2. Initial con-ditions x1(0) and x2(0) are also estimated. Specifications ofeach tested model are reported in Table 2. Finally, in Fig-ures 1 and 2, the root mean squares of the simulation andprediction errors on validation data are plotted.
x+1 =
{x1− k1
√x1 + k4u+ k5e, if x1 ≤ x1MAX
x1MAX + k6e, otherwise(1)
x+2 =
{x2 + k2
√x1− k3
√x2 + k8xover
1 + k7e, if x2 ≤ x2MAX
x2MAX, otherwise(2)
Table 2: Nonlinear State-Space Models used for identification,see Equations (1)-(2); e means parameter is estimated;k1, k2, k3, k4 are always estimated.
Model k5 k6 k7 k8 x1MAX x2MAX Nparm1 0 0 0 0 ∞ ∞ 6m2 0 0 0 0 ∞ 10 6m3 0 0 0 e e 10 8m4 0 e 0 e e 10 9m5 e (= k5) (= k5) e e 10 9m6 e e (= k5) e e 10 10
Figure 1: RMSE on Simulation: the best simulation model onvalidation data is m3, i.e. nonlinear state-space modelbased on physical modeling with description of theoverflow effect, see Table 2.
Figure 2: RMSE on Prediction: the best prediction model on val-idation data is a linear ARMAX with 5 parameters.
29
A nonlinear state-space solutionto a hysteretic benchmark in system identification
J.P. Noel, G. KerschenSpace Structures and Systems Laboratory
University of Liege, Liege, Belgiumjp.noel, [email protected]
A.F. Esfahani, J. SchoukensELEC Department
Vrije Universiteit Brussel, Brussels, Belgiumalireza.fakhrizadeh.esfahani, [email protected]
1 Introduction and model structure
The present contribution addresses the identification of thehysteretic benchmark introduced in Ref. [1]. To this end, adiscrete-time nonlinear state-space model is built consider-ing multivariate polynomial terms in the state equation,i.e.
{
x(t +1) = Ax(t)+Bu(t)+Ee(x(t))y(t) = Cx(t)+Du(t),
(1)
where x ∈ Rn is the state vector,u ∈ R
q the input vec-tor, y ∈ R
l the output vector,n the model order, and whereA ∈R
n×n, B ∈Rn×q, C ∈R
l×n andD ∈Rl×q are the linear
state, input, output and feedthrough matrices, respectively.The vectore∈ R
ne includes all monomial combinations ofthe state variables up to degreed. The associated coeffi-cients are arranged in matrixE ∈ R
n×ne .
2 Identification methodology
A two-step methodology was proposed in Ref. [2] to identifythe parameters of the model structure in Eqs. (1). First, ini-tial estimates of the linear system matrices(A,B,C,D) arecalculated by measuring and fitting the best linear approxi-mation (BLA) of the system under test. Second, assumingzero initial values for the nonlinear coefficients inE, a fullnonlinear model is obtained using optimisation.
3 Identification results and discussion
The frequency-domain behaviour of the validation mod-elling error is studied in Fig. 1, where the output spectrumin grey is compared with linear and nonlinear fitting errorlevels in orange and blue, respectively. Details regardingthe excitation signals and noise assumptions underlying theconstruction of this figure are to be found in Ref. [3]. Us-ing monomials of degree 3, 5 and 7 is found to reduce thelinear modelling error by a factor of 30dB, whilst requiring217 model parameters. It should be remarked that an exactpolynomial description of the nonlinearities in the system
demands, in principle, an infinite series of terms, prevent-ing the nonlinear error in the figure from reaching the noiselevel depicted in black.
5 25 50 75 100 125 150−160
−140
−120
−100
−80
Frequency (Hz)
Am
plit
ud
e (
dB
)
Figure 1: Validation modelling error, featuring the output spec-trum (in grey), the linear (in orange) and nonlinear (inblue) fitting error levels, and the noise level (in black)considering monomials of degree 3, 5 and 7.
Acknowledgement
This work was supported by the ERC Advanced grant SNL-SID, under contract 320378.
References
[1] J.P. Noel and M. Schoukens, Hysteretic benchmarkwith a dynamic nonlinearity, unpublished document, 2016.
[2] J. Paduart, L. Lauwers, J. Swevers, K. Smolders, J.Schoukens, R. Pintelon, Identification of nonlinear systemsusing Polynomial Nonlinear State Space models, 2010, Au-tomatica 46, 647-656.
[3] J.P. Noel, A.F. Esfahani, G. Kerschen, J. Schoukens,A nonlinear state-space approach to hysteresis identifica-tion, 2016, Mechanical Systems and Signal Processing, inreview.
30
Using a polynomial decoupling algorithm for state-spaceidentification of a Bouc-Wen system
Alireza Fakhrizadeh Esfahani, Philippe Dreesen, Koen Tiels, Jean-Philippe Noel?, Johan [email protected] [email protected]
Vrije Universiteit Brussel, Dept. ELEC ?University of Liege
1 IntroductionThe polynomial nonlinear state space (PNLSS) approach[1] is a powerful tool for modeling nonlinear systems. APNLSS model consists of a discrete-time linear state spacemodel, extended with polynomials in the state and the outputequation: x(t +1) = Ax(t)+Bu(t)+Eζ (t) (1)
y(t) = Cx(t)+Du(t)+Fη(t) (2)
where ζ (x(t),u(t)) and η(x(t),u(t)) are both vectors withmonomials in the states x(t) and the inputs u(t). The ma-trices E and F contain the polynomial coefficients. ThePNLSS model is very flexible as it can capture many dif-ferent types of nonlinear behavior, such as nonlinear feed-back and hysteresis. This flexibility generally comes at thecost of a large number of parameters. Increasing the orderof the polynomials for example leads to a combinatorial in-crease of the number of parameters due to the multivariatenature of the polynomials Eζ (x(t),u(t)) and Fη(x(t),u(t)).In this study, the PNLSS approach is used to model a Bouc-Wen hysteretic system [2]. The multivariate polynomialsEζ (x(t),u(t)) and Fη(x(t),u(t)) are decoupled using themethod in [3]. Like this, the nonlinearity in the PNLSSmodel is described in terms of univariate polynomials forwhich increasing their order is not so parameter expensive.
2 MethodologyIn a first step, we estimate the best linear approximation(BLA) [4] of the system. A linear state-space model esti-mated on the BLA serves as an initial guess for the PNLSSmodel in (1) and (2), which is optimized using a Levenberg-Marquardt approach. In a second step, the multivariate poly-nomials Eζ (t) and Fη(t) are decoupled using the decompo-sition method in [3]:
Eζ (x(t),u(t))≈Wxg(
V T[
x(t)u(t)
])(3)
Fη(x(t),u(t))≈Wyg(
V T[
x(t)u(t)
])(4)
where the matrix V transforms the states and inputs in newvariables ξ = V T
[x(t) u(t)
]T . The function g is a col-lection of univariate polynomials gi(ξi) for i = 1,2, · · · ,r :g(ξ ) =
[g1(ξ1) g2(ξ2) · · · gr(ξr)
]T that act as basisfunctions for the decoupled state-space model. The matricesWx and Wy contain the corresponding basis function coeffi-cients.
3 ResultsThe Bouc-Wen model is excited with a random-phase mul-tisine of 8192 samples, once with a standard deviation (std)
folder/Gcoeff4LowerAmp.png
Figure 1: The validation output spectrum (in blue), the error of linearmodel (cyan) and the error of PNLSS (green) for 2nd and 3rd
degree monomials of states and inputs in state updates (F =0), and the error for the decoupled model with 11th degreepolynomials and 3 branches (in red).
of 6.8130 N and once with a std of 4.6419 N. Twenty real-izations and 5 steady-state periods are used to estimate theBLA, a 3rd order linear model, and the full and decoupledPNLSS model. The results on the validation data (using amultisine with a std of 4.6419 N) are plotted in Figure 1. Therms error on the test data are 1.8703×10−5 (multisine) and1.2024×10−5 (swept sine) for the full PNLSS model and3.7406×10−4 (multisine) and 3.8806×10−4 (swept sine)for the decoupled model. The decoupled model has an rmserror higher than that of the linear model on the test data.
4 ConclusionA PNLSS model can capture the behavior of a Bouc-Wensystem. On the lower amplitude data, a decoupled PNLSSmodel reaches a similar accuracy, but has less than two thirdof the number of parameters (67 instead of 106). The orderof the polynomials in the decoupled model can also be in-creased without blowing up the number of parameters, as itis the case for the full PNLSS model. On the higher ampli-tude benchmark data, the decoupling with 3 branches fails.
AcknowledgementThis work has been supported by the ERC Advanced grantSNLSID, under contract 320378.
References[1] J. Paduart, L. Lauwers, J. Swevers, K. Smolders, J. Schoukens, andR. Pintelon, “Identification of nonlinear systems using Polynomial Nonlin-ear State Space models,” Automatica 46(4) 2010: 647-656.[2] J.P. Noel, and M. Schoukens, “Hysteretic benchmark with a dynamicnonlinearity,” Technical report,http://homepages.vub.ac.be/∼mschouke/benchmark2016.html, 2015[3] Ph. Dreesen, M. Ishteva, and J. Schoukens, “Decoupling multivariatepolynomials using first-order information and tensor decoupling,” SIAMJournal on Matrix Analysis and Applications 36, pp. 864-879, 2015.[4] R. Pintelon, and J. Schoukens, “System Identification: a Frequency Do-main Approach,” John Wiley & Sons, 2012.
31
Control-focused identification of hysteric systemsSelecting model structures? Think about the final use of the model!
Rolf GaasbeekEindhoven University of Technology
Rishi MohanEindhoven University of Technology
System identification techniques utilize data to create a mathemat-ical model of a dynamical system. Typically, during the identifi-cation process specific model structures or basis-functions are se-lected. It is important to realize that identified models are often af-terwards used to design or analyse a controller. Therefore, it is wiseto use information about the controller -structure and -objectives toselect an appropriate model structure.In this work, identification of hysteric systems is discussed. It isproposed to stay limited to the class of invertible models, as in-verted hysteric models are often used in control procedures forsystems that suffer from hysteresis. Lastly, an identification-basedanalysis is proposed to compare performance of invertible models.
1 Proposed ApproachTo illustrate how the inverse of a mathematical model can beused in control design, an example control-scheme is shownin Figure 1.
The proposed control approach relies on the availability ofan accurate model γ of the system γ , of which an (approxi-mate) inverse γ∗−1 exist. Thus, to allow for control-focusedidentification, only invertible model structures should beconsidered. For simplicity, this work limits itself to theDuhem model [1]; a nonlinear dynamic hysteresis modelthat is dependent on only 4 parameters. In [2] an approx-imate inverse of the Duhem model has been derived.
Although less suited for the Bouc-Wen model, the model-functions are adopted from [1], and thus are optimized forShape Memory Alloy actuation. More research should bedone on suitable invertible Bouc-Wen models.
2 System IdentificationThe lsqcurvefit function in Matlab is used to (sub)optimalfind the parameters of the Duhem model. The validation setfor the Shape Memory Alloy actuator is shown in Figure2; the hysteresis behavior is clearly visible. The identifiedmathematical model is able to predict the output of the actu-ator with a maximum RMS error of 0.05 mm. Unfortunately,the model structure chosen in Section 1 proved to be not richenough to capture the behavior of the Bouc-Wen model.
3 Identification-based AnalysisThe control approach as derived in Section 1 is dependenton the quality of both the model as well as the inverse. Inpractice, both are not perfect. Thus, the mapping between uand y (denoted G), is not equal to identity.
The robust method [3] is used to find both a linear approx-imation (black ◦) and non-linear distortions (grey 4) of G.If the linear approximation is not equal to 0dB, the inverseis degraded; the feedback controller can be designed ac-cordingly. However, non-linear distortions are significantlymore difficult to take into account with the controller-design.Therefore, from a control perspective, the proposed inversevalidation is preffered over classical inverse analysis ap-proaches, such as looking at the RMS value on the error.
References[1] S. M. Dutta and F. H. Ghorbel, ”Differential hysteresis modeling ofa shape memory alloy wire actuator,” Mechatronics, IEEE/ASME Transac-tions on, vol. 10, no. 2, pp. 189-197, 2005.
[2] S. M. Dutta, F. H. Ghorbel, and J. B. Dabney, ”Modeling and con-trol of a shape memory alloy actuator,” Intelligent Control, 2005.
[3] R. Pintelon and J. Schoukens, ”System Identification: A FrequencyDomain Approach,” IEEE Press, Piscataway, NJ, 2001.
32
Workshop on Nonlinear System Identification Benchmarks, Brussels, Belgium, April 25-27, 2016
System identification of a linearized hysteretic system using covariancedriven stochastic subspace identification
Anela Bajric
Department of Mechanical EngineeringTechnical University of Denmark
Kongens Lyngby, Denmark
A single mass Bouc-Wen oscillator with linear static restoring force contribution is approximated by an equivalentlinear system. The aim of the linearized model is to emulate the correct force-displacement response of the Bouc-Wenmodel with characteristic hysteretic behaviour. The linearized model has been evaluated by the root-mean-square errorbetween simulated response and the response history from two sets of experimental test data.
Hysteretic behaviour is encountered in engineering structures exposed to severe cyclic environmental loads, as well asin vibration mitigation systems such as magneto-rheological dampers. The mathematical representation of a hystereticforce can be obtained using the non-linear first-order differential equation
z(y, y) = α y − β(γ |y| z |z|ν−1 + δ y |z|ν
)(1)
known as the Bouc-Wen model [1, 2]. The shape and smoothness of the hysteresis loop is controlled by model parameters:α, β, γ, δ and ν.
The non-linear hysteretic force governed by the Bouc-Wen model in (1) is approximated by a linear model of the form
z(y, y) = λ y(t) + κ y(t) (2)
The individual coefficients in (2) are determined by assuming sinusoidal motion constant over one period with amplitudeA, applying harmonic averaging and integration over the full vibration period with angular frequency ω,
λ =9π2 ε
32A2 µ+ 9π2, κ =
12Aµπε
ω(32A2µ2 + 9π2)where ε = α− β δ , µ = β γ (3)
These approximations are exact for vanishing non-linearity in z, associated with the power coefficient ν approachingunity.
The input-output test data has been provided in [3], wherein the individual data sets are described. The randomphase multi-sine data set contains the steady-state response, while the sine-sweep data set is excited such that theresponse is not in steady state. The natural frequency and damping ratio of the test systems have been estimated fromthe displacement time history, using a covariance driven stochastic subspace identification algorithm (COV-SSI) [4],implemented in MATLAB. The number of block-rows in the Toeplizt matrix and the model order have been selectedbased on a minimization of the squared residual between the estimated natural frequency and damping ratio, and thecorresponding equivalent parameters obtained from the decay of the unbiased covariance estimate.
The response has been simulated using a fixed time-step fourth-order Runge-Kutta time integration scheme, with thesample interval of 1/20 of the period and with zero initial conditions. The root-mean-square errors erms, relative tothe model and the test output time series are 7.9 · 10−5 m and 1.1 · 10−4 m for the multi-sine and sweep-sine data sets,respectively. The error measure erms has been scaled by the number of points in the time series, in this case 8192 and153000 for the multi- and sweep-sine data sets, respectively.
The present approach is suitable for simulations of steady state response of a single-degree-of-freedom system, wherethe main advantage lies in its simplicity. For simulations of unsteady response the assumption of sinusoidal motionis obviously violated. Furthermore, the assumptions of the estimated covariance function will also be violated and inparticular the estimates of the damping ratio by the COV-SSI will be erroneous.
References
[1] Bouc R. Forced vibrations of a mechanical system with hysteresis. In Proceedings of the 4th Conference on NonlinearOscillations, Prague, Czechoslovakia, 1967.
[2] Wen Y. Method for random vibration of hysteretic systems. ASCE Journal of the Engineering Mechanics Division,102(2):249 263, 1976.
[3] Noel J.P. and Schoukens M. Hysteretic benchmark with a dynamic nonlinearity, Brussels, Belgium, 2016.
[4] Peeters B., and De Roeck G. Reference-based stochastic subspace identification for output-only modal analy-sis.Mechanical systems and signal processing, 13(6): 855-878, 1999.
33
Identifying an Unstructured Flexible Nonlinear Model for theCascaded Water-tanks Benchmark: Capabilities and Short-comings
Rishi RelanDepartment ELEC, VUB
Koen TielsDepartment ELEC, VUB
Anna MarconatoDepartment ELEC, VUB
1 Introduction: Cascaded Water Tanks BenchmarkIn this case study, we will illustrate the capabilities, flexi-bility and short-comings of an unstructured nonlinear mod-elling approach applied to the cascased tanks system bench-mark problem. This benchmark combines soft and hardnonlinearities to be identified based on relatively short datarecords. The major identification challenges associated withthis benchmark problem are: 1. the hard saturation nonlin-earity combined with the weakly nonlinear behavior of thesystem in normal operation, 2. the overflow from the upperto the lower tank, this effect also introduces input dependentprocess noise, 3. the relatively short estimation data record,4. the unknown initial values of the states.
2 Nonlinear modellingIn order to identify the discrete-time nonlinear model for thecascaded water tank benchmark, the polynomial nonlinearstate-space (PNLSS) model structure [1] is selected:
x(t +1) = Ax(t)+Bu(t)+Eζ (t) (1)y(t) =Cx(t)+Du(t)+Fη(t)+ e(t) (2)
The coefficients of the linear terms in x(t)∈Rna and u(t) aregiven by the matrices A∈Rna×na and B∈Rna×nu in the stateequation, C ∈Rny×na and D∈Rny×nu in the output equation.The vectors ζ (t) ∈ Rnζ and η(t) ∈ Rnη contain nonlinearmonomials in x(t) and u(t) of degree two up to a chosendegree P . The coefficients associated with these nonlinearterms are given by the matrices E ∈Rna×nζ and F ∈Rny×nη .
2.1 Identification ProcedureFor the identification of the PNLSS model the estimationdata was divided further into estimation (70%) and valida-tion data (30%) sets. A 3rd−order discrete-time parametriclinear model is fitted on nonparametric Best Linear Approxi-mation (BLA) obtained using the methods described in. Thenonparametric BLA was calculated utilising the first 60 fre-quency lines (20 frequency lines for the detrended case) ofthe input and output spectrum respectively. Later on, thislinear discrete-time model was converted into a state-spaceform and further optimized to identify in least-square sensea PNLSS model. In addition, an improved initialization ap-proach [2] to identify a NLSS model was explored, wherethe problem was divided into two separate problems namely:nonlinear estimation of states and static regression of thenonlinear functions. This initialized model was further non-linearly optimized to identify the full NLSS2 model.
3 ResultsTable 1 shows the performance of linear, the PNLSS andthe NLSS2 model on different datasets (with and withoutmeans). Figure 1 shows a comparison between the outputof the PNLSS and NLSS2 model in time domain on thetest/validation dataset.
Figure 1: Output error comparison on test dataset
Table 1: Linear vs. PNLSS Model on different datasetsPNLSS LINEAR PNLSS LINEAR PNLSS-I NLSS2
(M) (M) (NM) (NM) (NM) (NM)Est 0.033861 8.588 0.1133 0.54708 0.032393 0.0973Val 0.62178 7.1259 0.75063 0.75743 0.44984 0.4622Test 0.63691 8.6814 0.69737 0.75331 0.44984 0.4622
4 Conclusion
It can be concluded that the flexible PNLSS model performsbetter than the estimated linear model. It is able to capturethe dominant part of the water-tanks system nonlinear dy-namics. It performs well on the estimation dataset but itsperformance degrades on the validation & test datasets re-spectively. The main causes of this performance degrada-tion can be attributed to the curse of dimensionality, shortdata records and the underlying assumption on the modelstructure which ignores the influence of process noise.
References[1] J. Paduart, L. Lauwers, J. Swevers, K. Smolders,J. Schoukens, and R. Pintelon, “Identification of nonlinearsystems using polynomial nonlinear state space models,”Automatica, vol. 46, no. 4, pp. 647 – 656, 2010.
[2] A. Marconato, J. Sjoberg, J. A. Suykens, andJ. Schoukens, “Improved initialization for nonlinear state-space modeling,” IEEE Transactions on Instrumentationand Measurement, vol. 63, no. 4, pp. 972–980, 2014.
34
Identification of a PWARX model for the cascade water tanks
Per MattssonUppsala University
Dave ZachariahUppsala University
Petre StoicaUppsala University
1 Introduction
We consider the identification of a piecewise affine (PWA)model for the cascade water tanks. An affine ARX modelcan be written as
y(t) = φ>(t)ϑ + e(t).
where ϑ is the parameter vector, e(t) is modelled as whitenoise with variance σ2, and the regressor vector is given by
φ(t) = [−y(t−1) · · · − y(t−na)
u(t−1) · · · u(t−nb) 1]>.(1)
In order to define the PWA models considered here, letf (φ(t)) : Rd→Rn f , with d = na +nb +1, be a known func-tion and partition Rn f into nr regions R1, . . .Rnr . The PWAmodel is then defined as
y(t) = φ>(t)ϑi + e(t), if f (φ(t)) ∈Ri. (2)
A common choice here is to let f (φ(t)) = φ(t), which givesus the popular PWARX model, but as we will see this choicemight not be the most suitable for the cascade water tank.The identification problem is then to find both the regionsRi and the parameter vector ϑi in each region.
2 Proposed solution
Note that if the regions in (2) are known, then the parame-ter vectors ϑi can be found using e.g. linear least squares.However, the regions are of course unknown. The solutiontaken here is to make a fine partition of Rn f , thus effectivelyoverparameterizing the problem.
We then reparametrize the regions by picking one referenceregion R? with a corresponding parameter vector ϑ?. Theparameters of the remaining regions are then formed by aset of differences {δ j}nr−1
j=1 from ϑ?. An example of this isgiven in Fig. 1. In this way it makes intuitive sense that thatδ =
[δ>1 · · · δ>nr−1
]>should be sparse.
Assigning a normally distributed prior with zero mean andcovariance Pδ to δ , it can be shown that simultaneouslymaximizing the posterior distribution p(δ |Y,ϑ?) and themarginalized likelihood p(Y |ϑ?) is equivalent to minimiz-ing
σ−2‖Y −Fϑ?−Gδ‖2
2 +‖δ‖2P−1
δ
+ ln |C|
for F and G that can easily be computed from φ(t), t =1, . . . ,N, and C = GPδ G> + σ2IN . In order to get a con-vex problem, we linearize the concave function ln |C| aroundPδ = 0 and some arbitrary variance σ2 = c, to get ln |C| ≈1c tr{C}+K. It can be shown that, if Pδ is assumed to be di-agonal and we concentrate out σ and Pδ , then the resultingoptimization problem can be written as
minϑ?, δ
‖Y −Fϑ?−Gδ‖2 +‖w�δ‖1 (3)
w =1√N
[‖g1‖2 . . . ‖g(nr−1)d‖2
]>.
where gi is the ith column of G. This is a convex problem,with no tuning parameters, that can be solved recursively.
ϑ⋆ + δ4ϑ⋆
ϑ⋆ + δ1
ϑ⋆ + δ2
ϑ⋆ + δ3
ϑ⋆ + δ1 + δ5 ϑ⋆ + δ1 + δ6
ϑ⋆ + δ2 + δ7 ϑ⋆ + δ2 + δ8
Figure 1: Stylized example of the linearization regions and pa-rameterization used, where nr = 9 (3×3 grid).
3 The water tank data
In order to apply the proposed method to the benchmark datathe region selection function f (·) in (2) must be chosen. Asmentioned, a standard choice is to let f (φ(t)) = φ(t). Thiswould probably work well if we only had overflow in thelower tank. However, the upper tank also overflows, lead-ing to an increased inflow into the lower tank. Our intu-itive idea is that this might be possible to detect by studyingy(t)− y(t− k) for some k > 0. Hence, for this data set welet f (φ(t)) =
[y(t−1) y(t−1)− y(t− k)
]. What is left is
then to choose the parameters na, nb, k and a fine 2D-grid,and then the proposed method will estimate the model bysolving (3). Initial results have been promising, and the pro-posed method outperforms e.g. wavelet networks using thesystem identification toolbox in Matlab.
35
Nonparametric Volterra series estimate of the cascaded tank
Georgios BirpoutsoukisVrije Universiteit Brussel, Department of
Fundamental Electricity and InstrumentationB-1050 Elsene, Pleinlaan 2, [email protected]
Peter Zoltan CsurcsiaVrije Universiteit Brussel, Department of
Fundamental Electricity and InstrumentationB-1050 Elsene, Pleinlaan 2, Belgium
1 Introduction
In this work an efficient nonparametric time domain non-linear system identification method applied to the cascadedtanks measurement benchmark is presented.
2 The nonparametric identification method
2.1 The model structureIn this abstract, we present a method to estimate efficientlyfinite Volterra kernels without the need of long records,based on the regularization methods that have been devel-oped for impulse response estimates of LTI systems [2].Assume that the dynamics of an underlying nonlinear sys-tem can be described by the following finite discrete timeVolterra series of degree M[1]:
y[n] = h0+
M
∑m=1
nm−1
∑τ1=0· · ·
nm−1
∑τm=0
hm[τ1, . . ,τm]m
∏j=0
u[n− τ j]+ e[n] (1)
where u[n] denotes the input, y[n] represents the measuredoutput signal, e[n] is zero mean i.i.d. white noise with finitevariance, hm[τ1, . . ,τm] is the Volterra kernel of order m, andτi, i = 1, ...,m denotes the lag variables.
2.2 The cost functionEquation (1) can be rewritten into a vectorial form asY = Kθ +E, where θ contains the Volterra coefficients andK is the observation matrix (the regressor), Y contains themeasured output, and E contains the measurement noise.The kernel coefficients are obtained by minimizing the fol-lowing (regularized least square) cost function [3]:
θreg = ||Y −θK||22 +θT Dθ (2)
where the block diagonal matrix D contains (M+1) subma-trices penalizing the coefficitients of the Volterra kernels. Inthis work, prior information about the smoothness and theexponential decay of the higher dimensional kernels is usedduring the identification step by proper construction of ma-trix D [3].
2.3 ResultsDue to the limited number of available data samples, thehighest considered Volterra order is 3. The following ta-ble shows the results for different scenarios varying from a
simple FIR model to a 3rd degree Volterra series (M = 3).In each case, the transients were removed by a specialmethod detailed in [4]. As expected, the higher the degreeof Volterra series is, the better the fit.
Table 1: Result of identification for various scenarios
type validationerror
simple FIR model 0.84631st degree Volterra 0.59922nd degree Volterra 0.55513rd degree Volterra 0.5410
3 Conclusion
Modeling the cascaded tank system is challenging because:1. The number of samples is limited,2. The underlying system exhibits a hard saturation ef-
fect for high peaks of the input signal, and3. The measurement contains transients.
In order to properly describe the saturation effect, an in-creased order of Volterra series should be used. Due to thelimited number of samples, the highest order is kept to 3.
4 Acknowledgments
This work was supported by FWO-Vlaanderen, theMethusalem project, IAP VII) Program, and by the ERC ad-vanced grant SNLSID, under contract 320378
References[1] M. Schetzen, The Volterra and Wiener theories ofnonlinear systems., Vol. 1, New York: Wiley, 1980
[2] G. Pillonetto, F. Dinuzzo, T. Chen, G. De Nicolao,and L. Ljung, Kernel methods in system identification, ma-chine learning and function estimation: A survey, Automat-ica, vol. 50, no. 3, pp. 657-682, 2014
[3] G. Birpoutsoukis and J. Schoukens, Regularized non-parametric Volterra kernel estimation, 2015 IEEE Interna-tional Instrumentation and
[4] P. Z. Csurcsia, ”Nonparametric identification of lineartime-varying systems”, PhD thesis, ISBN 978-94-6197-326-9, Uitgeverij University Press, 2015
36
Repeated exponential sine sweeps for the autonomous estimation of nonlinearities and bootstrapassessment of uncertainties
M. Rebillat1, K. Ege2, N. Mechbal1, J. Antoni21: DYSCO, PIMM, ENSAM, Paris, France, 2: LVA, INSA Lyon, Lyon, France
Systems and structures are generally assumed to behave lin-early and in a noise-free environment. This is in practice notperfectly the case. First, nonlinear phenomena can appearand second, the presence of noise is unavoidable for all ex-perimental measurements. Nonlinearities can be consideredas a deterministic process in the sense that in the absenceof noise the output signal depends only on the input signal.Noise is purely stochastic: in the absence of an input sig-nal, the output signal is not null and cannot be predicted atany arbitrary instant. It turns out that these two issues arecoupled: all the noise that is not correctly removed from themeasurements could be misinterpreted as nonlinearities, andif nonlinearities are not accurately estimated, they will endup within the noise signal and information about the systemunder study will be lost. The underlying idea consists herein extracting the maximum of available linear and nonlineardeterministic information from measurements without mis-interpreting noise.
Figure 1: A cascade of Hammerstein models
The first problem addressed here is related to the estimationof nonlinear models [1]. We choose here to rely on black-box models having a given block-oriented structure. A classthat is particularly interesting is the class of parallel Ham-merstein models (see Fig. 1) as it is shown to possess a gooddegree of generality [1]. Moreover, thanks to exponentialsine sweeps, nonparametric versions of such models can bevery easily and rapidly estimated [2]. The second problemaddressed here is related to the estimation of uncertainties inthe context of nonlinear system estimation. The use of mul-tiple exponential sine sweeps allows for a more robust andefficient estimation of nonlinear models of vibrating struc-tures as estimation uncertainties can be simultaneously as-
Figure 2: Illustration of the frequency gains of kernels estimatedby synchronously averaging different periods of the ex-citation signal together with the mean square uncer-tainty obtained by bootstrap [3].
sessed using repetitions of the input signal as the input of abootstrap procedure (see Fig. 2) [3].
Our contribution to this workshop would thus be to assessthe efficiency of this methodology [3]. In our opinion thismethod is interesting as it is in practice really easy to im-plement and do not rely on complex and computationallydemanding signal processing or optimization steps. How-ever this method is strongly linked with parallel Hammer-stein models which cannot represent all the nonlinear sys-tems and some limitations are expected here.
References[1] R. K. Pearson. Discrete-Time Dynamic Models. Ox-ford University Press, 1999.
[2] M. Rebillat, R. Hennequin, E. Corteel, and B. F.G.Katz. Identification of cascade of hammerstein models forthe description of nonlinearities in vibrating devices. Jour-nal of sound and vibration, 330:1018–1038, 2011.
[3] M. Rebillat, K. Ege, M. Gallo, and J. Antoni. Re-peated exponential sine sweeps for the autonomous estima-tion of nonlinearities and bootstrap assessment of uncertain-ties. Proceedings of the Institution of Mechanical Engineers,Part C: Journal of Mechanical Engineering Science, 2015.
37
Identification of Wiener-Hammerstein systems with process noiseusing an Errors-in-Variables framework
Maarten SchoukensVrije Universiteit Brussel
1 Introduction
This abstract evaluates the possibility to treat the identifica-tion of a Wiener-Hammerstein system with process noise asan Errors-in-Variables (EIV) problem, as has been suggestedin [4] for the identification of Wiener systems. The proposedapproach is applied to the Wiener-Hammerstein benchmarkwith process noise [2].
2 The Errors-in-Variables Formulation
An extended EIV cost function is minimized to obtain themodel parameters:
VN(θ) =1N
N
∑t=1
((u(t)− u(t))2
σ2eu
+(y(t)− y(t))2
σ2ey
)+λ (σeu −σeu)
2 , (1)
where u(t) and y(t) are the measured input and output re-spectively, σ2
eu and σ2ey are the variances of the noise sources
that are present at the input and the output of the system,while σeu is the standard deviation of (u(t)− u(t)), λ is auser defined factor.
From the estimation point of view, the noise source at the in-put comprises also the process noise (filtered by the inverseof R(s)) that enters the system. The discrete-time model out-put y(t) is given by:
y(t) = S(q)[r(t)], (2)
r(t) = f (x(t)), (3)
y(t) = R(q)[u(t)], (4)
where q−1 is the backwards shift operator, and R, S andf are the estimates of the two linear time-invariant blocksand the static nonlinearity that are present in the Wiener-Hammerstein structure (see Figure 1).
The optimization of the EIV cost function is initialized usingthe Best Linear Approximation [1] and an output error (OE)based identification method based described in [3].
3 Benchmark Results
The proposed approach is applied on the Wiener-Hammerstein benchmark with process noise [2]. The ini-tialization is performed using 10 different realizations of a
Figure 1: A Wiener-Hammerstein system with process noise: astatic nonlinearity f (x) sandwiched in between two LTIblocks R(s) and S(s).
random phase multisine. The actual EIV cost function opti-mization is performed for a small number of multisine real-izations at a time for computational reasons. The obtainedresults indicate that the EIV optimization results in a smallimprovement on the noiseless multisine validation dataset,as is shown in Table 1. The results vary from one realizationto another, sometimes even resulting in a worse validationerror.
Table 1: Results of the Wiener-Hammerstein EIV model com-pared with a Wiener-Hammerstein OE model and anLTI model on the Wiener-Hammerstein benchmark. TheEIV results obtained with different multisine realizationsare compared on the noiseless multisine test signal. TheLTI and Wiener-Hammerstein OE results are obtainedfor the 10 multisine realizations combined.
LTI WH OE WH EIVRealization 1 0.055875 0.031387 0.022458Realization 2 0.055875 0.031387 0.023080Realization 3 0.055875 0.031387 0.040724Realization 1-3 0.055875 0.031113 0.025004
References[1] R. Pintelon and J. Schoukens. System Identification: A Fre-quency Domain Approach. Wiley-IEEE Press, Hoboken, New Jer-sey, 2nd edition, 2012.[2] M. Schoukens and J.-P. Noel. Wiener-Hammersteinbenchmark with process noise. Technical report, home-pages.vub.ac.be/mschouke/benchmark2016.html, 2015.[3] J. Sjoberg and J. Schoukens. Initializing Wiener-Hammerstein models based on partitioning of the best linear ap-proximation. Automatica, 48(2):353–359, 2012.[4] B. Wahlberg, J. Welsh, and L. Ljung. Identificationof Wiener systems with process noise is a nonlinear errors-in-variables problem. In 53rd IEEE Conference on Decision and Con-trol (CDC), pages 3328–3333, Dec. 2014.
38
Wiener-Hammerstein Benchmark with process noise: Parametricand Nonparametric Identification
K. Worden, G. Manson, R.J. Barthorpe, E.J. Cross, N. Dervilis, G. Holmes, T. RogersDynamics Research Group, Department of Mechanical Engineering, University of Sheffield,
Mappin Street, Sheffield S1 3JD, [email protected]
Abstract
Block-structured systems have long been regarded as a ver-satile model structure for system identification in the Elec-trical Engineering and Control communities [1]. Attentionhas generally focused on certain ‘canonical’ forms: Wiener,Hammerstein and Wiener-Hammerstein; each of which isa combination of linear system blocks, positioned with re-spect to a static nonlinearity. Such systems have not receiveda great deal of attention from the Mechanical Engineeringcommunity as they do not reflect a realistic physical repre-sentation of the vast majority of mechanical systems. How-ever, the systems can be regarded as a useful nonparametricmodel structure in the Mechanical context; having certainatttractive features, like simplicity of their Volterra repre-sentation.
The current paper considers a benchmark data set for aWiener-Hammerstein system (sequence: linear block/ staticnonlinearity/linear block) where considerable coloured pro-cess noise has been introduced directly before the nonlinearelement. This represents a difficult problem in the contextof parametric identification, as the coloured noise is likelyto induce bias in the parameter estimates. In fact, it is recog-nised that standard least-squares parameter estimation willlead to bias and alternative approaches are advocated [2, 3].
The approach proposed in this paper is an evolutionaryscheme, typified by [4], which frames the identificationproblem as an optimisation problem. The advantages of theapproach are that it allows a direct attack on nonlinear least-squares problems and that it can accommodate a range ofobjective functions i.e. it is relatively straightforward to gobeyond ordinary least-squares.
As a contrast to the evolutionary approach, a nonparamet-ric approach based on Gaussian process NARX models isapplied. It is shown that the high process noise in the bench-mark data set presents difficulties for the approach.
References
[1] Giri (F.) & Bai (E.) (Eds.) 2010 Block-oriented Non-linear System Identification. Springer.
[2] Hagenblad (A.), Ljung (L.) & Wills (A.) 2008 Auto-
matica 44 pp.2697-2705. Maximum likelihood identifica-tion of Wiener models.
[3] Wahlberg (B.), Welsh (J.) & Ljung (L.) 2014 53rd
IEEE Conference on Decision and Control (CDC) pp.3328-3333. Identification of Wiener systems with process noise isa nonlinear errors-in-variables problem.
[4] Worden (K.) & Manson (G.) 2012 Mechanical Sys-tems and Signal Processing 29 pp.201-212. On the identi-fication of hysteretic systems, Part I: fitness landscapes andevolutionary identification.
39
Bouc-Wen Benchmark: Parametric and NonparametricIdentification
G. Manson, R.J. Barthorpe, E.J. Cross, N. Dervilis, G. Holmes, T. Rogers, K. WordenDynamics Research Group, Department of Mechanical Engineering, University of Sheffield,
Mappin Street, Sheffield S1 3JD, [email protected]
Abstract
The Bouc-Wen model has long been recognised as a versa-tile ‘grey-box’ model of hysteresis. In its simplest form itcombines the standard physical model of a Single-Degree-of-Freedom oscillator, with an unphysical supplementarystate equation which allows the representation of a largerange of hysteresis loop possibilities. If the physical compo-nent of the model is taken as a standard SDOF linear system,the Bouc-Wen equations of motion take the form,
my+ cy+ ky+ z(y, y) = x(t) (1)
The hysteretic component is defined via an additional equa-tion of motion,
z = Ay−α|y|zn −β y|zn| (2)
for n odd, or,
z = Ay−α|y|zn−1|z|−β yzn (3)
for n even.
The parameters α , β and n govern the shape and smoothnessof the hysteresis loop. As a system identification problem,this set of equations presents a number of difficulties, fore-most are:
• The variables available from measurement will gen-erally be the input x and some form of response: dis-placement, velocity or acceleration. In this paper theresponse variable will be assumed to be displacementy, although the identification problem can just as eas-ily be formulated in terms of velocity or acceleration.Even if all the response variables mentioned are avail-able, the state z is not measurable and therefore it isnot possible to use equations (2) or (3) directly in aleast-squares formulation.
• The parameter n enters the state equations (2) and(3) in a nonlinear way; this means that a linear least-squares approach is not applicable to the estimation
of the full parameter set, some iterative nonlinear leastsquares approach is needed or an evolutionary schemecan be used as in [3].
In the main part of this paper, an evolutionary identifica-tion approach to the Bouc-Wen benchmark problem is takenbased on JADE, an adaptive variant of Differential Evolution(as opposed to the SADE approach used in [3]).
Although the evolutionary schemes are extremely effectiveat parameter estimation, and overcome the difficulties indi-cated above, they do not naturally provide confidence in-tervals for the estimates. To overcome this limitation, thepaper demonstrates the parametric Bayesian approach toBouc-Wen system identification introduced in [4], based onMarkov-chain Monte Carlo analysis.
Finally, as a contrast, a nonparametric approach based onGaussian process NARX models, is presented.
References[1] Bouc (R.) 1967 Proceedings of 4th Conference onNonlinear Oscillation, Prague, Czechoslovakia. Forced vi-bration of mechanical system with hysteresis.
[2] Wen (Y.) 1976 ASCE Journal of the Engineering Me-chanics Division XX pp.249-263. Method for random vibra-tion of hysteretic systems.
[3] Worden (K.) & Manson (G.) 2012 Mechanical Sys-tems and Signal Processing 29 pp.201-212. On the identi-fication of hysteretic systems, Part I: fitness landscapes andevolutionary identification.
[4] Worden (K.) & Hensman (J.J.) 2012 Mechanical Sys-tems and Signal Processing 32 pp.153-169. Parameter esti-mation and model selection for a class of hysteretic systemsusing Bayesian inference.
40
Identification of non-linear restoring forces throughlinear time-periodic approximations
Ebrahim Louarroudi & Steve VanlanduitOp3Mech group, Universiteit Antwerpen
Salesianenlaan 90, 2660 [email protected]
Rik PintelonDept. ELEC, Vrije Universiteit Brussel
Pleinlaan 2, 1050 [email protected]
1 Introduction
The experimental characterization of restoring forces isof crucial importance in non-linear (NL) mechanics [1].Since most applications in physics are driven by har-monic excitations u(t)= u(t), the NL system’s dynamics{
x = f (x,u)y = g(x,u) (1)
can be linearized around its periodic state orbit x(t+T)=x(t) and input u(t+T) = u(t). Hence, after perturbationof (1) around its periodic orbit (see Fig. 2), we obtain forsmall deviations a linear time-periodic (LTP) model{δx = δ f = ∂ f
∂xδx + ∂ f∂uδu = A(t)δx + B(t)δu
δy = δg = ∂g∂xδx + ∂g
∂uδu = C(t)δx + D(t)δu(2)
with X (t+T) = X (t), X = (A,B,C,D) time-periodic ma-trices and where the partial derivatives in (2) are eval-uated at the periodic orbit x = x and input u = u. TheLTP model in (2) is preferred over the NL one (1) as themodel to be identified remains linear in its input-output.
2 Methodology
Figure 1: Identification methodology of NL systems underharmonic inputs using LTP approximations.
As shown schematically in Fig. 1, the identification ofa NL system boils down to an LTP estimation problemwhen small deviations is considered around the largeharmonic excitation. From two dedicated experiments,the NL characteristics can be obtained by applying theLTP techniques in [2] on the perturbed response δy (2).
3 Results: Duffing oscillator
We will demonstrate the methodology in Fig. 1 on a NLsystem with a static nonlinearity – the Duffing oscillator
mx+ cx+ (kLx+kNLx3)= u (3)
where m = 1, c = 0.02, kL = 1 and kNL = 0.25. The goal isto estimate the restoring force k(x)= (
kLx+kNLx3). The
identification method is tested on noiseless data for fourperturbation multi-sine signals with different distortionratios: 20log {δuRMS/uRMS} = −(10,20,40,60) dB. As in-dicated by the results in in Fig. 2, significant deviationsfrom the true restoring force can be detected when thedistortion ratio is 10 dB. For moderate distortions therestoring force can be well-identified. Note that the truerestoring force in (3) is the integral of dk(x)
dx in Fig. 2.
Figure 2: Left: differentiated restoring force of the Duffingoscillator (3). Right: phase plane.
Extension of this work to dynamic non-linear systemswith hysteresis is still under investigation. Preliminaryresults on the benchmark data of the Bouc-Wen systemwill be provided at the workshop.
References[1] M. W. Sracic and M. S. Allen. Identifying param-eters of multi-degree-of-freedom nonlinear structural dy-namic systems using linear time periodic approxima-tions. Mech. Syst. & Sig. Proc., 26(2), 325-343, 2014.
[2] E. Louarroudi (2014). Frequency domain mea-surement and identification of weakly nonlinear time-periodic systems, PhD, Vrije Universiteit Brussel, Bel-gium, ISBN: 978-4619721-2-5.
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Modeling Nonlinear Systems Using a Volterra Feedback Model
Maarten SchoukensVrije Universiteit Brussel
Fritjof Griesing ScheiweTechnische Universitat Ilmenau
1 Introduction
This abstract increases the modeling flexibility of the simplefeedback structure [2] by introducing Volterra kernels basedblocks instead of static nonlinear blocks. The simple feed-back structure is given by a linear time invariant block witha static nonlinearity feedback loop around it, as is depictedin Figure 1. The developed identification algorithm is ap-plied to both the Bouc-Wen hysteretic system [1] and thecascaded tanks system [4].
2 LTI-Volterra Feedback Identification
The proposed identification algorithm is an extension of themethod presented in [2]. Firstly, the linear time-invariant(LTI) block that is present in the model structure is identi-fied using a linear approximation of the nonlinear system.Secondly, the nonlinear block is identified.
There is no unique representation of the simple feedbacksystem. A linear part γ can be subtracted from the nonlinearfeedback block and added to the linear subsystem resultingin the same input-output behavior of the overall system, butwith a different system representation. This observation al-lows one to use the Best Linear Approximation [3] of theoverall system as a model of the LTI block that is present inthe Volterra Feedback model.
The nonlinear function f (y) is assumed to be represented bya linear combination of nonlinear basis functions. Previouswork [2] assumes f (y) to be a polynomial static nonlinear-ity, in this work we represent f (y) using a truncated Volterraseries with memory depth nV and a maximum degree nd .Once an estimate G(q) of the linear block G(q) is obtained,the identification of the nonlinear function f (y) becomes lin-ear in the parameters:
y(t)− G(q)[u(t)] =n f
∑i=1
ciG[ fi(y(t), . . . ,y(t −nv))], (1)
where n f is the total number of basis functions used to rep-resent f (y), and q−1 is the backwards shift operator. Theparameters ci can hence be estimated using a linear least-squares estimator.
3 Benchmark Results
The validation results on both the Bouc-Wen and the Cas-caded Tanks benchmark are reported in Table 1. It can be
Figure 1: The simple feedback structure: a LTI subsystem con-nected with a nonlinearity in feedback.
observed that the rms simulation error of the Volterra Feed-back model is approximately 50% lower and 30% lower thanthe linear model (after offset removal) for the Bouc-Wen andthe Cascaded Tanks benchmark respectively. Replacing thepolynomial static nonlinearity by the Volterra nonlinearityresults in a significant decrease of the model error.
The Volterra Feedback model for the Bouc-Wen benchmarkhas a LTI block of order 3, with a one-sample delay, theVolterra subsystem has 2 taps (y(t),y(t − 1)) and all odddegree kernels from degree 1 to 7 are included. The LTImodel is a rational transfer function of order (2,3) (numera-tor,denominator). The Volterra Feedback model for the Cas-caded tanks benchmark benchmark has a LTI block of order2, with a one sample delay, the Volterra subsystem has 2 tapsand all kernels up to degree 3. The LTI model is a rationaltransfer function of order (1,2), with a one-sample delay.
Table 1: Results of the Volterra Feedback model compared with aPolynomial Feedback model and an LTI model on boththe Bouc-Wen (multisine (MS) and swept sine (SS)) andCascaded tanks benchmark.
B-W (MS) B-W (SS) Casc. TanksLTI 0.15105 10−3 0.17145 10−3 0.58781Poly. FB 0.11983 10−3 0.13817 10−3 0.48715Volt. FB 0.08409 10−3 0.05601 10−3 0.3972
References[1] J.-P. Noel and M. Schoukens. Hysteretic bench-mark with a dynamic nonlinearity. Technical report, home-pages.vub.ac.be/mschouke/benchmark2016.html, 2015.[2] J. Paduart, G. Horvath, and J. Schoukens. Fast identificationof systems with nonlinear feedback. In 6th IFAC Symposium onNonlinear Control Systems (NOLCOS 2004), Sept. 2004.[3] R. Pintelon and J. Schoukens. System Identification: A Fre-quency Domain Approach. Wiley-IEEE Press, Hoboken, New Jer-sey, 2nd edition, 2012.[4] M. Schoukens, P. Mattson, T. Wigren, and J.-P. Noel. Cascaded tanks benchmark combining softand hard nonlinearities. Technical report, home-pages.vub.ac.be/mschouke/benchmark2016.html, 2015.
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First principles and black box modeling of the cascaded water tanks
Andreas SvenssonUppsala University
Fredrik LindstenUppsala University
Thomas B. SchonUppsala University
1 Introduction
We investigate how to apply particle filter-based methods[4, 2] for the cascaded water tank problem. As noted by [5],the nontrivial physical behavior together with the relativelyshort data record constitutes an interesting and challengingproblem. The ‘hard’ nonlinearities seem to make it surpris-ingly hard to perform better than linear models, and the shortdata record is clearly challenging the model’s ability to sen-sibly generalize the training data.
2 Proposed solutions
We consider two different approaches: a black-box ap-proach based on [6], and model derived using first princi-ples with a set of unknown, but physically interpretable, pa-rameters to estimate. In both cases, particle filter play animportant role in the identification algorithm.
2.1 Black box identification with Gaussian processesWe make use of the general state space model presentedin [6], and adapt it to the water tank problem by encodingrelevant smoothness assumptions using Gaussian processes(‘soft nonlinearities’) and presence of discontinuity points(‘hard nonlinearities’). This may be considered as a black-box model, however with explicit assumptions on the non-linear functions involved. The smoothness assumptions helpin the crucial task of generalize, and not overfit, the trainingdata. The identification procedure itself is performed usingparticle Gibbs with ancestor sampling [3]. A preliminaryversion with promising result is available in [6, Example5.3].
2.2 Physical modeling with unknown parametersWe consider the physically derived model presented by [7],and extend it to cover the case of overflow, introducing afew more unknown parameters. To estimate the parameters,we apply particle Metropolis-Hastings [1], which amountsto explore the parameter space using a Metropolis-Hastingsprocedure, and use a particle filter as an approximate, butunbiased, estimator of the likelihood for the parameters.
References
[1] C. Andrieu, A. Doucet, and R. Holenstein. ParticleMarkov chain Monte Carlo methods. Journal of theRoyal Statistical Society: Series B (Statistical Method-ology), 72(3):269–342, 2010.
[2] N. Kantas, A. Doucet, S. S. Singh, J. M. Maciejowski,and N. Chopin. On particle methods for parameter esti-mation in state-space models. Statistical Science, 30(3):328–351, 2015.
[3] F. Lindsten, M. I. Jordan, and T. B. Schon. ParticleGibbs with ancestor sampling. The Journal of MachineLearning Research, 15(1):2145–2184, 2014.
[4] T. B. Schon, F. Lindsten, J. Dahlin, J. Wagberg, C. A.Naesseth, A. Svensson, and L. Dai. Sequential MonteCarlo methods for system identification. In Proceedingsof the 17th IFAC Symposium on System Identification(SYSID), pages 775–786, Beijing, China, Oct. 2015.
[5] M. Schoukens, P. Mattson, T. Wigren, and J.-P. Noel. Cascaded tanks benchmark combin-ing soft and hard nonlinearities. Available:homepages.vub.ac.be/˜mschouke/benchmark2016.html,2015.
[6] A. Svensson and T. B. Schon. A flexible state spacemodel for learning nonlinear dynamical systems. arXivpreprint arXiv:1603.05486, 2016. Submitted for publi-cation.
[7] T. Wigren and J. Schoukens. Three free data sets for de-velopment and benchmarking in nonlinear system iden-tification. In Proceedings of the 2013 European ControlConference (ECC), pages 2933–2938, Zurich, Switzer-land, July 2013.
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