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Stained Glass Window, 1906 Young Family House, Lancaster, England

The Brexitiers’ View of England’s Green & Pleasant Land: Nostalgia for the Lost Empire of 1906?

State-Dependent Parameter Nonlinear Models and a Hydrological Identification Benchmark

Peter Young Lancaster University, U.K. p.young@lancaster.ac.uk ***

With thanks to Alex Janot and Mathieu Brunot of ONERA for their help in checking the presentation and demo m-file

Workshop on Nonlinear System Identification Benchmarks, Liege 2018

1

My Background and Philosophical Approach

• One of my greatest pleasures is that I live in a multi-disciplinary world:

academia; control and systems engineering; statistics; time-series analysis; forecasting; environmental science; and, most recently, electro-mechanical engineering, particularly the modelling and control of robotic systems.

• Predominantly, I am not a mathematician; rather I use mathematics to evolve

computational methods that relate to the defined application objectives, most

• f which are concerned with the modelling, forecasting and control (including

management) of systems in the natural, engineering and socio-economic worlds.

• My philosophy for doing this is enshrined in an approach I call Data-Based

Mechanistic (DBM) modelling. This is an inductive method of modelling directly from time series data that tries to avoid prejudicial hypotheses and, while it exploits ‘black-box’ modelling techniques, also requires the resulting model to have a clear and scientifically acceptable mechanistic interpretation.

2

Data-Based Mechanistic (DBM) Modelling

• DBM modelling is a ‘method theory’, developed over many years. Its name

emphasizes my contention that, while the model should be inferred from the analysis of data in an objective manner, it should also have an obvious mechanistic interpretation and be as simple as possible, consistent with the modelling objectives.

• In other words, I believe that a model should not just explain the time series

data well, it should also provide a clear, transparent and scientifically acceptable mechanistic description of the system under investigation; a description that further enhances confidence in its ability to approximate reality in a meaningful manner and helps to achieve the application objectives.

• Note:

‘benchmark’ data sets, such as those we are discussing here, that emphasise only the modelling of the data, without necessarily considering of the objectives, are not consistent with the DBM philosophy.

3

• In a very real sense, therefore, although DBM modelling exploits some of the

methodology used in ‘black-box’ modelling, it is partly a reaction against the notion and, I believe, over-use of pure black-box models particularly when they are being used in research that is attempting to investigate the real nature of dynamic systems in the natural and man-made world.

• In the case of nonlinear systems, this means that the nonlinearities in the model

should be clearly identified in some form, such as a graphical portrayal, that makes physical sense to practitioners in the scientific or engineering discipline in which the model is being used.

• One nonlinear model identification procedure that facilitates this is State-

Dependent Parameter (SDP) estimation, an approach evolved many years ago (Young, 1968; Mendel, 1969; Young, 1981) and Priestley (1980), who first used the name. The SDP methods used in the later example were developed in the 1990s (see e.g. Young, 2000, 2001b) and these have been applied successfully to a large number of practical systems, in diverse areas of study, since then.

4

State-Dependent Parameter (SDP) Models

An SDP example that will be well known to you is the model of the ‘Silverbox’. My analysis of these data (Young, 2016) suggests that an acceptable system model is the following second order, nonlinear differential equation:

d2x(t) dt2 + a1 dx(t) dt + k1x(t) + k2x2(t) + k3x3(t) = b0u(t))

• r, in SDP form : d2x(t)

dt2 + a1 dx(t) dt + a2{x(t)}.x(t) = b0u(t))

where

a2{x(t)} = k1 + k2x(t) + k3x2(t)

The associated SDP Transfer Function (SDPTF) model is represented as follows:

x(t) = b0 s2 + a1s + a2{x(t)}u(t)

This explains 99.999% of the variance in silverbox output (R2

T = 0.99999), which

should be good enough for almost all applications.

5

General SDP Transfer Function Model

The simplest SISO discrete-time SDPTF model takes the following form: x(k) = B(z−1, v(k)) A(z−1, v(k))u(k − δ); y(k) = x(k) + ξ(k) A(z−1, v(k)) = 1 + a1{v(1, k}z−1 + a2{v(2, k}z−2 + ... + anv(n, k B(z−1, v(k)) = b0{v(n + 1, k} + b1{v(n + 2, k}z−1 + ... + bm{v(n + m + 1, k}z−m and vt is a vector of measured variables (states) on which the parameters may be dependent. Or, in SDARX estimation equation terms: y(k) = z(k)Tp(k) + e(k); where p(k) = [a1{v1,t} . . . bm{vn+m+1,k}]T , zT

t = [−y(k − 1) . . . − y(k − n) u(k − δ) . . . u(k − δ − m)]

The continuous-time SDP model is defined in a similar manner.

6

IDENTIFICATION and ESTIMATION

• 1. Initial linear and Time-Variable Parameter (TVP) model identification, to

establish whether the model is TVP or SDP, using the CAPTAIN Toolbox discrete-time rivbjid, dtfmopt, dtfm routines; and, if appropriate, the continuous-time rivcbjid routine (see later example and Young, 2011, 2015).

• 2. Initial Non-Parametric Identification of the nonlinear model structure (i.e.

in the sense of the location and nature of the statistically significant SDP nonlinearities) using the CAPTAIN sdp routine. This is then followed by:

• 3. Parameterization of the SDP nonlinearities using whatever method is most

appropriate, if possible one that is transparent and has a physical interpretation.

• 4. Final Parameter Estimation: statistically optimal estimation applied to this

parameterized model (e.g. using NLS or PEM optimization, often applied to a Simulink version of the model, as in the later example).

7

Non-Parametric SDP Identification Algorithm

• 1. Estimate the starting parameters p0

i,k, i = 1, 2, ..., n + m + 1 in the above

SDARX model using LS (i.e. constant LS ARX parameter estimates);

• 2. Backfitting Algorithm: Iterate, i − 1, 2, ..., n + m + 1; t = 1, 2, ..., tc

(i) form the modified ’dependent’ variable yi

k = yk − j=i zj,k.ˆ

pt

j,k|N;

(ii) sort both yi

k and zi,k according to the ascending order of the state si,k

associated with zi,k; (iii) obtain an ML optimized Fixed Interval Smoothing (FIS) estimate ˆ pt

i,k|N of

pi,k in the modified single state dependent variable relationship yi

k = pi,k.zi,k

• 3. Continue 2 until iteration t = tc, when the individual SDPs (which are each

time series of length N) have not changed significantly according to some chosen criterion. Note: this algorithm produces a limited SDP ARX model.

8

Parametric SDP TF Model Optimization

This can be carried out in two stages, although the first is not essential:

• 1. Initial nonlinear model estimation using a multiple input, quasi-linear model

estimated by CAPTAIN rivcbj or rivbj routines1, where the additional inputs are the parameterized nonlinear functions (possibly a series of ‘basis functions’).

• 2. Final model optimization using NLS or PEM, initialised with the parameters

from stage 1. I normally optimize the parameters in a Simulink model, which then serves for subsequent forecasting and control/management studies. The following example (see also the supplied Matlab command-line Demo.), illustrates the identification strategy outlined here and in the previous slides.

1The CAPTAIN Toolbox is available free via http://captaintoolbox.co.uk/Captain_

Toolbox.html/Captain_Toolbox.html

9

A Hydrological Benchmark Example: DBM Modelling of Rainfall-Flow

• A important environmental topic is the modelling and forecasting of flow or

level changes in a river system on the basis of upstream rainfall measurements.

• It is well known that the relationship between rainfall and flow is nonlinear,

although the nature of this nonlinearity is still a topic of research.

• The benchmark example considered here involves daily rainfall and flow

measurements made on the ‘ephemeral’ River Canning in Western Australia (i.e. a river that stops flowing in Summer: see also Young, 2008).

• I will outline the sequential stages of DBM modelling based entirely on these

data, with no prior hypotheses about the nature of the model form and structure, other than that it can be described by a linear or nonlinear transfer function in discrete or continuous-time form.

10

The Canning River, WA: Location

11

The Canning River When Flowing

Nutrient fractions and loads in upper Canning River

Total phosphorus (TP) concentrations remained in total nitrogen (TN)

• ccurred over the reporting
• period. This was verified by

statistical analysis, which detected no trends over the short- or long-term (2007–11 and 2002–11 respectively). Both 2006 and 2010 were extremely dry years and this is shown by the absence of high values for these two years. Until 2001 the upper Canning River was passing

• nly the short-term TN
• target. Since then it has

been passing both the Approximately one-third of the nitrogen (N) in the upper dissolved inorganic N (DIN, consisting of ammonium – NH and N oxides – NOx). This form of N is readily available for plant and algal uptake and is most likely sourced from fertilisers used in agricultural and urban and animal waste. The

• rganic N which consists of

both dissolved (DON) and particulate (PON) fractions. DON largely comprises

• rganic compounds leached

from peaty subsoils and degrading plant and animal matter and is available for uptake by plants, algae and

• bacteria. PON is composed
• f plant and animal debris

and needs to be further broken down to become available to plants and algae. average TN load (2007–11) sits centrally, contributing the ninth-largest load of the 14 catchments with fmow data. The load per unit area is low, the second lowest of all the catchments. Approximately two-thirds of the phosphorus (P) in the the form of particulate P, which includes sediment- bound forms of P and degrading plant and animal

• matter. This form of P is not

readily available for plant and algal uptake, but some may become available as particles decompose and bound phosphate is released. The remainder of the P is present as soluble reactive phosphorus (SRP) which is derived from fertilisers used in the catchment, animal waste and septic tank leachate. SRP is readily available for plant and algal uptake. has the fjfth-smallest average TP load (2007–11) of the 14 catchments with fmow data. The load per unit area is the second-smallest (same as Susannah Brook) overall and the smallest of the catchments that discharge into the Canning Estuary. short- and the long-term targets. steady during the reporting period, except for the peak starting in late 2010. Statistical analysis found no long-term trend (2002–11) and an emerging increasing short-term trend (2001–11)

• f 0.001 mg/L/yr.

has been passing both the short- and long-term TP began in 1987.

The Canning River fmowing through paddocks with only a thin band of riparian vegetation present. Photo: Water Science Branch.

12

The Daily Data Set

1985.4 1985.6 1985.8 1986 1986.2 1986.4 1986.6 1986.8 1987 Flow (cumecs) 2 4 Flow, Rainfall and Temperature: Canning River, W.A., 1985.2-1987.1 1985.4 1985.6 1985.8 1986 1986.2 1986.4 1986.6 1986.8 1987 Rainfall (mm) 20 40 60 80 Date 1985.4 1985.6 1985.8 1986 1986.2 1986.4 1986.6 1986.8 1987 Temperature (deg.C) 20 40 13

The Best Linear Models: All Unacceptable

Date 1985.4 1985.6 1985.8 1986 1986.2 1986.4 1986.6 1986.8 1987 Flow (cumecs)

• 0.5

0.5 1 1.5 2 2.5 3 3.5 4 Initial Linear Modelling

DT [2 3 2] Model CT [2 3 2] model DT [1 1 1] model Measured

14

Off-Line Time Variable Parameter Estimation

• Off-line estimation of time variable parameters is more powerful than the

normal on-line ‘filtering’ (RLS, RPEM RRIV) methods because it can exploit Fixed Interval Smoothing to yield zero lag, lower variance TVP estimates.

• The dtfmopt and dtfm routines in CAPTAIN use this approach for discrete-time
• models. In the present example the simplest first order [1 1 1], discrete-time

TVP model is considered: y(k) = b1(k) 1 + a1(k)u(k − 1) + ξ(k)

• The first results (see next slide) are obtained allowing both parameters to vary.

However, this suggests that only b1(k) varies significantly, so the subsequent TVP estimation is based on this, with a1 assumed to be constant.

15

DTFM: Time Variable Parameter Estimation

Date 1985.5 1986 1986.5 1987 TVP Esimate a1(t)

• 0.87
• 0.865
• 0.86
• 0.855
• 0.85
• 0.845
• 0.84
• 0.835
• 0.83

TVP Estimate a 1(t)

Standard error bounds TVP Estimate Constant parameter estimate

Date 1985.5 1986 1986.5 1987 TVP Esimate b1(t) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 TVP Estimate b 1(t)

Standard error bounds TVP Estimate (free estimation) TVP Estimate b1(t) (a1 constant)

16

TVP Model with b1(k) Varying; a1 Constant

Date 1985.2 1985.4 1985.6 1985.8 1986 1986.2 1986.4 1986.6 1986.8 1987 1987.2 Flow (cumecs) 0.5 1 1.5 2 2.5 3 3.5 4 Time variable Parameter Modelling

Model Measured

17

TVP b1(t) Highly Correlated with Flow

100 200 300 400 500 600 700 2 4 6 8 X(Red) Y(Blue) Cross Correlation Between TVP Estimate and Flow

• 24
• 12

12 24 0.2 0.4 0.6 0.8 CCF between X(t) Y(t-k) CCF between X(t-k) Y(t) 18

State-Dependent Parameter Nonlinear Model: Non-parametric (NP) Identification

• Based on a comparison [1 1 1] and [1 1 0] SDP models and later results, the

non-parametric SDP estimation is based on the [1 1 0] SDP model with an SDP input parameter b0(y(t)) and constant a1 parameter.

• The next plot compares the SDP estimates with those obtained by ‘free’

estimation with both parameters considered as functions of y(t).

• This confirms that b0(y(t)) is the most significant SDP parameter with −ˆ

a1 estimated as constant at 0.894. This is a ‘Hammerstein’ Nonlinear Model, i.e. with only an input nonlinearity.

• NB: this SDP estimation is mainly for nonlinear structure identification
• purposes. The NP estimates can be used directly (e.g. as a look-up table)

but are mainly a step to the final parametric stage of SDP model estimation.

19

Initial Non-Parametric SDP Estimates

Flow (acting as surrogate for catchment storage) 1 2 3 SDP Estimate of -a1 Parameter 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Comparison of Free (black) and Constrained (red) SDP estimates

Confidence bounds SDP a1 estimate (RW) SDP a1 estimate (location) Constant a1 estimate SDP a1 estimate (IRW)

Flow (acting as surrogate for catchment storage) 1 2 3 SDP Estimate of b0 Parameter

• 0.01

0.01 0.02 0.03 0.04 0.05 0.06

Confidence bounds SDP b0 estimate (free estimation) SDP b0estimate with a1 constant

20

State-Dependent Parameter Nonlinear Model: Final Parametric Estimation

• There are various ways in which the SDP b0{y(t)}, shown in the right hand

panel of the previous plot, could be parameterized (see e.g. Beven et al., 2011) but I always feel that the simpler and more physically plausible it is, the better.

• This is because complex ‘black box’ parameterizations are often more difficult

and dangerous to extrapolate into regions not explored during SDP estimation (or any other nonlinear modelling, for that matter) based on limited data.

• In this case simple power (i.e.

b0 = y(t)γ) and exponential (i.e. b0 = 1 − exp{−γy(t)}) laws, defined by only a single parameter, yield good results and are acceptable on hydrological grounds. However, it is clear that other simple, or a little more complex, parameterizations are possible.

21

• At this stage, it should be recalled that a simple [1 1 0] model was used to

explore non-parametric SDP possibilities. But now, in the final stage of SDP estimation, it makes sense to evaluate whether other, higher order, models may provide a superior explanation of the flow changes.

• This is made simpler by the Hammerstein nature of the identified SDP model

because the ‘effective’ rainfall input can be defined as ue(t) = b0{y(t)}u(t) and then this can be used as the input for linear estimation using either DT or CT models. In this case, the former is identified as a [2 3 0 5 0] linear model.

• I have estimated the final SDP model with ue(t) = {y(t)γ}u(t) incorporated

into a lsqnonlin optimization routine, where the rivbj routine is used for DT linear model estimation based this ue(t) (later, I consider alternative CT model estimation of the same kind using the rivcbj routine).

• The plot below compares the ˆ

b0{y(t)}, estimated in this way, with the non- parametric estimate and the estimate obtained with an exponential SDP law.

22

Non-Parametric and Parametric SDP Estimates

Flow (acting as surrogate for catchment storage) 0.5 1 1.5 2 2.5 3 3.5 State Dependent Parameter estimate

• 0.01

0.01 0.02 0.03 0.04 0.05 0.06 0.07 SDP Estimation of Effective Rainfall Coefficient

Confidence interval Nonparametric (NP) estimate NP estimate: exact location Exponential law Power law

23

DBM Nonlinear DT Model Response: 1985-1987

Date 1985.4 1985.6 1985.8 1986 1986.2 1986.4 1986.6 1986.8 1987 Flow (cumecs) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 DBM Model Estimation: 1985.2-1987.1: 95.6% of Flow Explained

Model Measured Note: the residuals of the AR(5) noise model are serially uncorrelated and not significantly correlated with the input signal or the temper- ature series, as required.

24

First Validation: 1977-1978

Date 1977.4 1977.5 1977.6 1977.7 1977.8 1977.9 1978 1978.1 Flow (cumecs) 0.5 1 1.5 2 2.5 3 3.5 Predictive Validation: 1977-1978.5: 95.1% of Flow Explained

Model Measured

25

Second Validation: 1979-1980

Date 1979.4 1979.5 1979.6 1979.7 1979.8 1979.9 1980 Flow (cumecs) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Cross-Validation: 1979-1980.5: 92.6% of Flow Explained

Model Measured

26

The Modelling Objective: Explaining the Catchment Characteristics

• The optimized SDP model has no time-delay and the rainfall is very difficult

to forecast into the future, so it is not particularly useful as a forecasting tool (although the introduction of a ‘false’ time delay would allow this).

• Its main use is as an indicator of the catchment dynamic behaviour and

associated hydrological interpretation. A Continuous-Time Model is more appropriate for this: it is (i) unique (not dependent on the sampling interval) and (ii) immediately interpretable in physically meaningful terms.

• As a final exercise, therefore, the model is estimated in a continuous time

stochastic form, using the same lsqnonlin optimization routine but with a CT

• model. The DBM model obtained in this manner is shown on the next slide.

27

Continuous-Time State-Dependent Parameter Nonlinear Stochastic Model

x(t) = −0.06459s2 + 0.2473s + 0.0214 s2 + 0.439s + 0.02084 ue(t) ue(t) = {ys(t)0.8}u(t) ξ(k) = 1 1 − 0.8674z−1 + 0.3238z−2 − 0.1670z−3e(k); e(k) = N(0, σ2{y(k)}), where σ2{y(k)} is an SD function of y(k) y(k) = x(k) + ξ(k) R2

T = 0.958

Note: ys(t) denotes here y(t) as a ‘surrogate’ for catchment wetness (CW) and is not a feedback of y(t). Consequently, the model cannot be used for on-line simulation unless ys(t) is replaced by some estimate of CW based, for example,

• n rainfall and/or temperature: see ‘HI-DBM modelling’, Young (2003, 2013).

28

Credibility Demonstrated by Model Decomposition

Effective Rainfall SDP Nonlinearity

+

• 6.3%

Quick Surface Processes

First Order TF with Residence Time

Tq = 2.6 days 0.252 s + 0.385 Tq = 2.6 days 0.252 s + 0.385

63.8% Measured Noisy Flow Output y(k)

y(k)

Measured Rainfall Input Effective Rainfall Heteroscedastic WhiteNoise e(k)

e(k)

+

Noise-Free Output

ue(k)

Instantaneous Effect

Negative Gain =-0.065 suggesting Losses in system

Sub-Surface Processes

First Order TF with Residence Time

Ts = 18.47 days 0.0236 s + 0.0541 Ts = 18.47 days 0.0236 s + 0.0541

42.5%

u(k) xq(t) xs(t) xI (t)

Sampler

x(k) x(t) 1 1-0.867z−1 + 0.324z−2-0.167z−3

AR(3) Noise Model

ξ(k) 29

Monte Carlo Uncertainty Analysis

• Given the above stochastic model, Monte Carlo uncertainty analysis can be

used for estimating the uncertainty in parameters and the model response.

• In particular, this allows us to evaluate the uncertainty in the derived, physically

meaningful parameters (the residence times and partition percentages) and the uncertainty bounds on the model response. This is in the form of the histograms obtained from the stochastic realizations of the derived parameters; and the 95 percentile uncertainty bounds on the response.

• One

problem with the latter computation in this example is the heteroscedasticity in the estimated noise sequence ξ(k), whose variations are clearly proportional to the variations in the flow. This is handled by estimating a linear relationship between FIS estimated variations in the standard deviations

• f the final residuals e(k) and the flow y(k); effectively defining a state-

dependent white noise input with variance σ2{y(k)}.

30

MCS Analysis: Residence Time Distributions

Slow Residence Time (hours) 15 20 25 Frequency 50 100 150 200 250 300 Mean Estimate=18.5 days Quick Residence Time (hours) 2 2.5 3 Frequency 50 100 150 200 250 300 Mean Estimate=2.6 days 31

MCS Analysis: Partition % Distributions

Groundwater Partition (%) 30 35 40 45 50 55 Frequency 50 100 150 200 250 Mean Estimate=42.5% Surface Partition (%) 55 60 65 70 75 Frequency 50 100 150 200 250 Mean Estimate=63.8% 32

MCS Analysis: Uncertainty in Model Response

Date 1985.4 1985.6 1985.8 1986 1986.2 1986.4 1986.6 1986.8 1987 Flow (cumecs) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

95%confidence bounds Model output Measured Flow

33

Zoom into Section of Previous Plot

Date 1985.64 1985.66 1985.68 1985.7 1985.72 1985.74 1985.76 1985.78 1985.8 Flow (cumecs) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

95%confidence bounds Model output Measured Flow

34

‘Transparent SDP’ or ‘Black Box’ Models

• An important aspect of DBM modelling is the definition of the modelling
• bjectives.

I feel the concentration on the identification of overly-complex black-box nonlinear models that can explain the data too well, without evaluation of how they function in terms of these modelling objectives, is questionable and needs to be justified in practical terms.

• Based on experience in multiple different disciplines, I believe that many

(most?) scientists and engineers mistrust totally ‘black-box’ models because they are unable to visualise and interpret the nature of the nonlinearities.

• They also feel that the models can be over-parameterised, as typified by

the published research using data from experiments on a hydraulic actuator controlling a robot arm (see e.g. Sj¨

• berg et al., 1995; Hu et al., 2001), where

most models appear to be totally black box, with too many parameters.

35

• The SDP Hammerstein model of these robot data (see Young, 2001a, 2006b)

has only 15 parameters: a 10 parameter radial basis function model for the nonlinearity and 5 in the [3 2 1] linear TF. The model has reasonable explanatory power, cf other models, with R2

T = 0.96 (error RMS=0.318).

• Finally, there is evidence that the system is not only nonlinear but also non-

stationary so that an adaptive SDP model, in which just the parameters of the linear component in the Hammerstein model parameters are allowed to vary, might well be required. An adaptive extension is quite straightforward because of the model’s simplicity. This is discussed in Young (2006b) where the dtfmopt/dtfm routines are applied in on-line ‘filtering’ mode, producing an adaptive model that shows a significant improvement in validation terms.

• Even without this, the validation error for the constant parameter SDP model is

marginally less than Hu et al’s 102 parameter black box ‘Quasi-ARMAX’ model and, in contrast to the black-box model, it provides a graphical visualisation that makes rather obvious physical sense, as we see on the next slide.

36

Input Variable

• 1
• 0.5

0.5 1 SDP Transformed Input Variable

• 4
• 3
• 2
• 1

1 2 Estimates of SDP Input Nonlinearity

Confidence bounds Non-parametric estimate Parametric estimate

Time (number of Samples) 100 200 300 400 500 Output Variables

• 4
• 2

2 4 Measured and SDP Model Outputs

Measured output Model output

Time (number of Samples) 100 200 300 400 500 Input

• 1.5
• 1
• 0.5

0.5 1 1.5 Measured Input Variable

37

A ‘Simple’ but Flawed Alternative Approach to SDP Model Estimation

• As mentioned previously, an alternative method of SDP TF model estimation

is to use standard rivbj or rivcbj estimation with additional, nonlinearly defined ‘basis function’ inputs. While inherently flawed (e.g. biased basis function parameter estimates; can be over-parameterized; etc.), it can be helpful.

• In this rainfall-flow example, an exponential law can be represented as a series

expansion in a finite number of terms. This produces reasonable CT model results with five terms in the expansion, yielding a model with better, but possibly misleading, explanatory power of R2

T = 0.966 (see the Matlab Demo).

• However, this 5-input TF model, with the additional TFs having first and second
• rder numerator polynomials, and the common denominator polynomial, has

12 parameters, twice as many as the 6 parameters of the simple exponential model; and extrapolation of the resulting input nonlinearity is questionable.

38

Concluding Comments

• The DBM approach to system identification is to identify the simplest,

mechanistically transparent model that satisfies the defined objectives (normally

connected with scientific understanding, forecasting or control/management system design).

• In the multi-disciplines where I work, the scientists, practicing engineers and

economists are not normally impressed by ‘black-box’ models, no matter how well they perform, unless these can be interpreted in a manner that is credible to them. In other words, a nonlinear SDP model needs to have nonlinearities that, wherever possible, have a clear and physically meaningful form.

• I have found that SDP nonlinear models are widely applicable and can

normally satisfy such scientific and practical engineering requirements, including nonlinear SDP control (see e.g. Young, 2001b, 2006b,a; Janot et al., 2017).

• The current SDP algorithms need to be improved: (e.g. a Multi-State Dependent

Parameter (MSDP) tool is in development (Sadeghi et al., 2010; Mindham et al., 2018)).

39

References

• K. J. Beven, D. T. Leedal, P. J. Smith, and P. C. Young.

Identification and representation of state dependent non-linearities in flood forecasting using the DBM methodology. In L. Wang, H. Garnier, and A. J. Jakeman, editors, System Identification, Environmetric Modelling and Control, Berlin, 2011. Springer-Verlag (Festschrift celebrating Peter Young’s 70th birthday, 2009). Hu, J., K. Kumamaru, and K. Hirasawa (2001). A quasi-ARMAX approach to modelling nonlinear systems. International Journal of Control 74, 1754–1766. Janot, A., P. C. Young, and M. Gautier (2017). Identification and control

• f electro-mechanical systems using state-dependent parameter estimation.

International Journal of Control 90(4), 643–660. Mendel, J. M. (1969). A priori and a posteriori identification of time varying

• parameters. In Prodeedings, 2nd Hawaii Conference on System Sciences.

40

Mindham, D. A., W. Tych, and N. A. Chappell (2018). Extended state dependent parameter modelling with a data-based mechanistic approach to nonlinear model structure identification. Environmental Modelling & Software 104, 81 – 93. Priestley, M. B. (1980). State-dependent models: A general approach to non- linear time series analysis. Journal of Time Series Analysis 1, 47–71. Sadeghi, J., W. Tych, A. Chotai, and P. C. Young (2010). Multi-state dependent parameter model identification and estimation for control system

• design. Electronics Letters 46(18), 1265–1266.

Sj¨

• berg, J., Q. Zhang, L. Ljung, A. Benveniste, B. Deylon, P. Y. glorennec,
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identification : a unified overview. Automatica 31(12), 1691–1724. Young, P. C. (1968). Adaptive pitch autostabilization of an air-surface missile,. In Proc. Southwestern IEEE Conf.,San Antonio, Texas, New York. IEEE. Young, P. C. (1981). A second generation adaptive autostabilization system for

41

airborne vehicles. Automatica 17, 459–470. Young, P. C. (2000). Stochastic, dynamic modelling and signal processing: time variable and state dependent parameter estimation. In W. J. Fitzgerald,

• A. Walden, R. Smith, and P. C. Young (Eds.), Nonlinear and Nonstationary Signal

Processing, pp. 74–114. Cambridge University Press: Cambridge. Young, P. C. (2001a). Comment on ‘a quasi-ARMAX approach to modelling nonlinear systems’ by J. Hu et al. International Journal of Control 74, 1767–1771. Young, P. C. (2001b). The identification and estimation of nonlinear stochastic

• systems. In A. I. Mees (Ed.), Nonlinear Dynamics and Statistics, pp. 127–166.

Birkhauser: Boston. Young, P. C. (2003). Top-down and data-based mechanistic modelling of rainfall- flow dynamics at the catchment scale. Hydrological Processes 17, 2195–2217. Young, P. C. (2006a). New approaches to volcanic time series analysis. In

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Volcanology, Bath, UK, pp. 143–160. Geological Society of London. Young, P. C. (2006b). Non-parametric model structure identification and parametric efficiency in nonlinear state dependent parameter models. In IEEE International Symposium on Evolving Fuzzy Systems, Ambleside, UK, pp. 349– 354. Young, P. C. (2008). Real-time flow forecasting. In H. Wheater, S. Sarooshian, and K. D. Sharma (Eds.), Hydrological modelling in arid and semi-arid areas, Cambridge, UK, pp. 113–138. Cambridge University Press. Young, P. C. (2011). Recursive Estimation and Time-Series Analysis: An Introduction for the Student and Practitioner. Springer-Verlag, Berlin., 2011. Young, P. C. (2013). Hypothetico-inductive data-based mechanistic modeling of hydrological systems. Water Resources Research 49(2), 915–935.

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Young, P. C. (2015). Refined instrumental variable estimation: Maximum likelihood optimization of a unified Box-Jenkins model. Automatica, 52:35–46, 2015. Young, P. C. (2016). Data-based mechanistic modelling and control of the silverbox nonlinear circuit. Technical Report TN/PCY/3 (2016), Lancaster Environment Centre, Lancaster University, UK.

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