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The Problem 2(47) Nonlinear System Identification: A Palette from Tunn: measurement; Tjock: simulation 5 y 1 0 5 0 1 2 3 4 5 6 7 8 9 10 0.5 Off-white to Pit-black y 2 0 0.5 0.5 0 1 2 3 4 5 6 7 8 9 10 y 3 0

SLIDE 1

Nonlinear System Identification: A Palette from Off-white to Pit-black

Lennart Ljung Automatic Control, ISY, Linköpings Universitet

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET

The Problem

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Missile Dynamics:

1 2 3 4 5 6 7 8 9 10 −5 5 Tunn: measurement; Tjock: simulation y1 1 2 3 4 5 6 7 8 9 10 −0.5 0.5 y2 1 2 3 4 5 6 7 8 9 10 −0.5 0.5 y3 1 2 3 4 5 6 7 8 9 10 −100 100 y4 1 2 3 4 5 6 7 8 9 10 −200 200 y5 [s]

Pulp Buffer Vessel:

100 200 300 400 500 600 5 10 15 20 25 OUTPUT #1 100 200 300 400 500 600 10 15 20 INPUT #1

Forest Crane:

100 200 300 400 500 600

2 4 OUTPUT #1 100 200 300 400 500 600

0.5 1 1.5 INPUT #1

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET

This Presentation ...

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... aims at a display of the essence of the problem of non-linear identification a color-coded overview of typical parametric approaches

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET

A Common Frame

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The world of nonlinear models is very diverse. A common framework: Discrete time observations of inputs and outputs:

Zt = [u(1), u(2), ..., u(t), y(1), y(2), ..., y(t)]

A model is a parameterized predictor of the next output y(t) made at time t − 1:

ˆ y(t|t − 1, θ) = ˆ y(t|θ) = h(Zt−1, θ)

The parameters can be estimated using the prediction error method:

ˆ θ = arg min

θ ∑ t

y(t) − h(Zt−1, θ)2

(could be Maximum Likelihood)

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET

SLIDE 2

What’s Special with Nonlinear Models?

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ˆ y(t|θ) = h(Zt−1, θ) is a nonlinear function of Z. What makes the

nonlinear problem much more difficult and rich than the linear problem? Two major reasons: The richness of the model surface Propagation of noise signals to the output not immediate

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET

The Model Surface

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Let us take Zt−1 = [u(t − 1), u(t − 2)] and a scalar output y(t). A model is then a surface in the space spanned by

[y(t), u(t − 1), u(t − 2)] and the estimation task is to estimate this

surface.

1 2 3 4 5 1 2 3 4 5 6 −1 −0.5 0.5 1 1.5 2 2.5 3 u(t−1)

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

u(t−2) y

Linear:

ˆ y(t) = a1u(t− 1) + a2u(t− 2)

1 2 3 4 5 1 2 3 4 5 6 −4 −3 −2 −1 1 2 3 4 x 10

u(t−1)

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * * * *

u(t−2) y

Nonlinear:

ˆ y(t) = h(u(t − 1), u(t − 2))

The observations Zt are points in this space.

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET

Propagation of Noise Signals

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In linear systems that are cascaded we can always propagate the noise signals to the output: y = Gu + He, where H picks up the coloring obtained by propagating the noise through a linear system. For nonlinear systems, this in generally not possible. Example: A linear system + noise, z = Gu + w is followed by a static nonlinearity f(z). At the output we have

y(t) =f(Gu + w) = f(Gu) + ˜ w ˜ w =f(Gu + w) − f(Gu)

Here, ˜

w is not really a “noise”: It is clearly contaminated with the

input u which will create bias-problems when minimizing the

utput error. Indicates that the calculation of the true predictor

could be challenging.

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET

The Palette of Nonlinear Models

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White: Known model Off-white: Careful Physical Modeling w or w/o noise models Smoke-grey: Semi-physical modeling (Could be used more!) Steel-grey: Local Linear Models Slate-grey: Block-oriented Models. Black: Flexible structures – universal approximators Pit-black: Non-Parametric Smoothing

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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SLIDE 3

Off-white Models: Physical Modeling

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Perform physical modeling (e.g. in MODELICA) and denote unknown physical parameters by θ. Collect the model equations as

˙ x(t)= f(x(t), u(t), θ) y(t)= h(x(t), u(t), θ)

(or in DAE, Differential Algebraic Equations, form.) For each parameter θ this defines a simulated output ˆ

y(t|θ) which is the

parameterized function from sampled data

ˆ y(t|θ) = h(Zt−1, θ) (Zt−1 = ut−1)

in somewhat implicit form. To be a correct predictor this really assumes white measurement noise. Then the estimate is the θ that minimizes the output error fit ∑t y(t) − ˆ

y(t|θ)2

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Example: Missile

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10 inputs, 5 outputs, 16 unknown parameters.

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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The Equations

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function [dx, y] = missile(t, x, p, u); MISSILE A non-linear missile system. Output equation. y = [x(1); ... x(2); ... x(3); ...

p(18)*u(4)*(p(1)*x(5)+p(2)*u(3))/p(22); ...
p(18)*u(4)*(p(3)*x(4)+p(4)*u(2))/p(22) ... ];

State equations. dx = [1/p(19)(p(17)p(18)(p(5)x(5)+0.5p(6)p(17)x(1)/u(5)+ ... % Angular velocity around x-axis. p(7)u(1))u(4)-(p(21)-p(20))x(2)x(3))+ ... p(23)(u(6)-x(1)); ... 1/p(20)(p(17)p(18)(p(8)x(4)+0.5p(9)p(17)*x(2)/u(5)+ ... ... p(1)-p(25) unknown parameters u, y : measured inputs and outputs

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Initial Fit between Model and Data

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1 2 3 4 5 6 7 8 9 10 −20 20 Tunn: measurement; Tjock: simulation y1 1 2 3 4 5 6 7 8 9 10 −1 1 y2 1 2 3 4 5 6 7 8 9 10 −5 5 y3 1 2 3 4 5 6 7 8 9 10 −100 100 y4 1 2 3 4 5 6 7 8 9 10 −500 500 y5 [s] Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET

SLIDE 4

Adjusted Fit between Model and Data

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1 2 3 4 5 6 7 8 9 10 −5 5 Tunn: measurement; Tjock: simulation y1 1 2 3 4 5 6 7 8 9 10 −0.5 0.5 y2 1 2 3 4 5 6 7 8 9 10 −0.5 0.5 y3 1 2 3 4 5 6 7 8 9 10 −100 100 y4 1 2 3 4 5 6 7 8 9 10 −200 200 y5 [s] Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET

Off-white Models with Noise Models

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The (output error, off-white) approach is conceptually simple, but could be very demanding in practice. A main shortcoming is the use of the output error criterion, which really assumes white measurement noise. Noise signals in nonlinear models cannot really be propagated to the output. If the size of the noise is non-trivial, more careful noise modeling should be done:

˙ x(t)= f(x(t), u(t), w(t), θ) y(t)= h(x(t), u(t), θ) + e(t)

where w and e are white noises.To find correctly predicted outputs

ˆ y(t|Zt−1, θ) = E(y(t)|Zt−1, θ) is then the well-known “intractable”

problem of nonlinear filtering. Often one has to resort to some simplistic observer.

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Probabilistic Learning

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Recently however, with the increasing computing power, new computing intensive simulation based methods have been developed for nonlinear filtering problem, and hence for applying the Maximum Likelihood method to non-linear state space models. Particle filtering, Markov Chain Monte Carlo, MCMC, Sequential Monte Carlo .... Loosely, and briefly, these are based on simulation of the noisy state-space model, and evaluating the state probabilities, focusing on paths that give the measured output sequence. It is a central current research area, Probabilistic Learning , to make these calculations as efficient as possible. See e.g. Thomas B. Schön, Andreas Svensson, Lawrence Murray, and Fredrik Lindsten: Probabilistic learning of nonlinear dynamical systems using sequential Monte Carlo, ArXiv

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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The Palette of Nonlinear Models

16(47)

White: Known model Off-white: Careful Physical Modeling w or w/o noise models Smoke-grey: Semi-physical modeling (Could be used more!) Steel-grey: Local Linear Models Slate-grey: Block-oriented Models. Black: Flexible structures – universal approximators Pit-black: Non-Parametric Smoothing

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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SLIDE 5

Smoke-grey: Semi-physical Models

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Apply non-linear transformations to the measured data, so that the transformed data stand a better chance to describe the system in a linear relationship. “Rules: Only high-school physics and max 10 minutes” Toy Example: Immersion heater: Input: voltage to the heater. Output: temperature of the fluid. . . ..

. . .. Square the voltage! Sense morale: No excuse for not thinking

ver the basic physical facts!

Another example: . . .

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Buffer Vessel Dynamics

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100 200 300 400 500 600 5 10 15 20 25 OUTPUT #1 100 200 300 400 500 600 10 15 20 INPUT #1 100 200 300 400 500 600 20 40 60 80 100 200 300 400 500 600 50 100 150 200

κ-number of outflow,

flow

κ-number of inflow,

volume

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Linear Model Based on Raw Data

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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10

−8 −6 −4 −2 2 4 6 y1 Measured Output and Simulated Model Output Measured Output mraw Fit: 21.11%

Dashed line: κ-number after the vessel, actual measurements. Solid line: Simulated κ-number using the input only and a process model estimated using the first 200 data points. G(s) =

0.818 1+676se−480s

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Now it’s time to

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Think: .... No mixing (“Plug flow”): The vessel is then just a pure time delay for the pulp flow: Delay time: Vessel Volume/Pulp Flow (dimension time.) Perfect mixing in tank: A text-book first order system with gain=1 and time constant = Volume/Flow So if Volume and Flow are changing, we have a non-linear system! The natural time variable is really Volume/Flow, (which we have measured). Let us re-sample the observed data according to this natural time variable.

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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SLIDE 6

Re-sample Data

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z = [y,u]; pf = flow./level; t = 1:length(z) newt = interp1([cumsum(pf),t],[pf(1):sum(pf)]’); newz = interp1([t,z], newt);

10 20 30 40 50 60 70 80 5 10 15 20 25

κ−number of Inflow

10 20 30 40 50 60 70 80 8 10 12 14 16 18 20 22

κ−number of Outflow

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Semi-physical Model

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G(s) = 0.8116 1 + 110.28se−369.58s

5000 10000 15000 −8 −6 −4 −2 2 4 6 y1 Measured Output and Simulated Model Output Measured Output mves Fit: 60.39%

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4 −8 −6 −4 −2 2 4 6 y1 Measured Output and Simulated Model Output Measured Output mraw Fit: 21.11%

Recall Linear model.

The semi-physical model gives a sufficiently good de- scription of the buffer, to al- low proper time-marking of the pulp before and after.

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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The Palette of Nonlinear Models

23(47)

White: Known model Off-white: Careful Physical Modeling w or w/o noise models Smoke-grey: Semi-physical modeling (Could be used more!) Steel-grey: Local Linear Models Slate-grey: Block-oriented Models. Black: Flexible structures – universal approximators Pit-black: Non-Parametric Smoothing

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Steel-Grey: Composite Local Models

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Non-linear systems are often handled by linearization around a working point. The idea behind Composite Local (Local Linear) Models is to deal with the nonlinearities by selecting or averaging over relevant linearized models. Example: Tank with inflow u and free outflow y and level h: (Bernoulli’s) equations:

˙ h = − √ h + u; y = √ h

Linearize around level h∗ with corresponding flows u∗ = y∗ =

√ h∗: ˙ h = − 1 2 √ h∗ (h − h∗) + (u − u∗); y = y∗ + 1 2 √ h∗ (h − h∗)

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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SLIDE 7

Tank Example, ctd

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Sampled data model around level h∗ (Sampling time Ts):

y(t) = γ(h∗) + α(h∗)y(t − Ts) + β(h∗)u(t − Ts) = θT(h∗)ϕ(t)

An ARX-model with level-dependent parameters. Now compute linearized model for d different levels, h1, h2, . . . , hd. Total model: select or average over these local models

ˆ y(t) =

d

∑

k=1

wk(h(t), hk)θT(hk)ϕ(t)

Choices of weights wk : . . ..

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Data and Linear Model

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Measured data: Linear Model (d = 1)

10 20 30 40 50 60 70 80 0.2 0.4 0.6 0.8 1 1.2 1.4 Inflöde u 10 20 30 40 50 60 70 80 0.5 1 1.5 Nivå h 10 20 30 40 50 60 70 80 0.2 0.4 0.6 0.8 1 1.2 1.4

Thick line: Model. Thin: Measured.

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Local Linear Models

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Two levels (models) (d=2) Five levels (models) (d = 5)

10 20 30 40 50 60 70 80 0.2 0.4 0.6 0.8 1 1.2 1.4 10 20 30 40 50 60 70 80 0.2 0.4 0.6 0.8 1 1.2 1.4

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Composite Local Models: General Comments

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Let the measured working point variable (tank level in example) be denoted by ρ(t) (sometimes called regime variable or scheduling variable). If the regime variable is partitioned into d values ρk, and model output according to value ρk is ˆ

y(k)(t) the predicted output will

be

ˆ y(t) =

d

∑

k=1

wk(ρ(t), ρk)ˆ y(k)(t)

If the prediction ˆ

y(k)(t) corresponding to ρk is linear in the

parameters, ˆ

y(k)(t) = ϕT(t)θ(k), and the weights w are fixed, the

whole model will be a linear regression. Important connections to active research areas LPV (Linear Parameter-Varying) Models Hybrid Models (≈ w(·, ·) is estimated too.)

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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SLIDE 8

LPV State Space Models

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x(t + 1) = A(ρ)x(t) + B(ρ)u(t) y(t) = C(ρ)x(t)

is a linear model for each fixed ρ. If ρ ∈ Ω = {ρ1, . . . , ρd} it is a set of local linear models.If ρ = ρ(t) is time varying, we have a Linear Parameter Varying model. A basic difficulty is to find a common state basis from input/output

bservations and to manage the time variable in ρ(t) in relation to

the observations.

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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The Palette of Nonlinear Models

30(47)

White: Known model Off-white: Careful Physical Modeling w or w/o noise models Smoke-grey: Semi-physical modeling (Could be used more!) Steel-grey: Local Linear Models Slate-grey: Block-oriented Models Black: Flexible structures – universal approximators Pit-black: Non-Parametric Smoothing

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Slate Grey: Block-oriented Models

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Building Blocks: Linear Dynamic System

G(s)

Nonlinear static function

f(u)

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Common Models

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Wiener Hammerstein Hammerstein- Wiener

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SLIDE 9

Other Combinations

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Active Research Field: With the linear blocks parameterized as a linear dynamic system and the static blocks parameterized as a function (“curve”), this gives a parameterization of the output as

ˆ y(t|θ) = h(Zt−1, θ)

and the general approach of model fitting can be applied. However, in this contexts many algorithmic variants have been suggested.

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Example: Hydraulic Crane Data

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These are data from a forest harvest machine:

100 200 300 400 500 600

2 4 OUTPUT #1 100 200 300 400 500 600

0.5 1 1.5 INPUT #1

Input: Hydraulic Pressure. Output: Tip Position

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Linear Model

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Black: Measured Output Blue: Model Simulated Output

200 400 600 800 1000 1200 −4 −3 −2 −1 1 2 3 4

Linear model: Fit 41.71 %

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Hammerstein Model of the Hydraulic Crane

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200 400 600 800 1000 1200 −6 −5 −4 −3 −2 −1 1 2 3 4

Hammerstein model: Fit 71.61 %

The Hammer- stein Model gives a good fit. The extra flexibility

ffered

by the input nonlinearity is quite useful, (even though no direct physical explanation is

bvious.)

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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SLIDE 10

Noise Effects in Hammerstein-Wiener?

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There is frequently reason to assume that some noise enters before the output nonlinearity g.

G(q, ϑ) ut wt xt yt Hv(q, η) Hw(q, µ) vt et f(·, α) g(·, β) zg

νt

What happens if we propagate that noise to the output and apply an Output Error criterion to the above 2 input 2 output system?

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Output Error Method for the HW Model

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Bode for linear system Input NLs Output NLs

−2

−1

10 −10 −5 5 10 15 G11 10

−2

−1

10 −10 −5 5 10 15 G12 10

−2

−1

10 −25 −20 −15 −10 −5 5 10 G21 10

−2

−1

10 −25 −20 −15 −10 −5 5 10 G22 −2 2 −1.5 −1 −0.5 0.5 1 1.5 OE Method −2 2 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 OE Method −2 2 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 OE Method −2 2 −1.5 −1 −0.5 0.5 1 1.5 OE Method −1.5 −0.5

Blue curve: Plots for the true system Red curves: Median and standard deviations for estimated systems

ver 80 Monte Carlo runs

Number of observed data: 2000

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Maximum Likelihood (EM) for HW Models

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It is clear that more effort must be paid to the noise structure. We turn to the Maximum Likelihood method for the HW model structure. It is a complication that the ML criterion cannot easily be formed. But if the unmeasured noise zw

t were known it is easy to compute the ML

criterion. So, treat is as “incomplete data” X and apply the EM

algorithm, which iterates between estimating X and estimating the model for this X. Ref: Adrian Wills, Thomas B. Schön, Lennart Ljung , Brett Ninness: Identification of Hammerstein-Wiener models, Automatica 2013

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Results for the ML-EM Method for HW Models

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Bode for linear system Input NLs Output NLs

−2

−1

10 −10 −5 5 10 15 G11 10

−2

−1

10 −10 −5 5 10 15 G12 10

−2

−1

10 −25 −20 −15 −10 −5 5 10 G21 10

−2

−1

10 −25 −20 −15 −10 −5 5 10 G22 −2 2 −1.5 −1 −0.5 0.5 1 1.5 ML−EM Method −2 2 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 ML−EM Method −2 2 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 ML−EM Method −2 2 −1.5 −1 −0.5 0.5 1 1.5 ML−EM Method

Blue curve: Plots for the true system Red curves: Median and standard deviations for estimated systems

ver 80 Monte Carlo runs

Number of observed data: 2000

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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SLIDE 11

The Palette of Nonlinear Models

41(47)

White: Known model Off-white: Careful Physical Modeling w or w/o noise models Smoke-grey: Semi-physical modeling (Could be used more!) Steel-grey: Local Linear Models Slate-grey: Block-oriented Models. Black: Flexible structures – universal approximators Pit-black: Non-Parametric Smoothing

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Black-box Models

42(47)

A general way to generate very flexible mappings from Zt−1 to ˆ

y is x(t + 1) = f(x(t), u(t), y(t), θ) ˆ y(t|θ) = h(x(t), θ)

where f and/or h are flexible functions e.g. in terms of basis function expansions. Working with both f and h may be too general, and a very common special case is the NLARX model:

x(t) = ϕ(t) = [y(t − 1), . . . , y(t − na), u(t − 1), . . . u(t − nb)]T ˆ y(t|θ) = h(ϕ(t), θ) =

d

∑

k=1

αkκk(ϕ(t))

for some basis functions κk

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Basis Functions

43(47)

It is natural to think of Taylor expansions: κk(ϕ) = ϕk. If na = 0 (NLFIR), this becomes the classical Volterra series expansion. But note that if dimϕ = r, then ϕk has rk components! A more common choice is to form all the basis functions κk from one mother function κ and scale and position the argument differently:

κk(ϕ) = κ(βk(ϕ − γk)) ˆ y(t|θ) =

d

∑

k=1

αkκ(βk(ϕ − γk)), θ = {αk, βk, γk}

Intuitive picture: Think of a scalar ϕ and let κ(z) be a unit pulse for

0 ≤ z ≤ 1. Then κ(β(ϕ − γ)) is a pulse of width 1/β starting in ϕ = γ. The sum above is then a piecewise constant function,

capable of approximation "any" function arbitrary well for large enough d.

⇒ ANN, LS-SVM etc (Sjöberg et al, Automatica 1995)

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Focusing on f

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x(t + 1) = f(x(t), u(t), θ) ˆ y(t|θ) = h(x(t), θ)

Basis functions for f, h: Polynomial expansion (Paduart et al, Automatica, 2010) Gaussian Process (GP) model for f (Rasmussen, inverted pendulum experiments; x measured); [Basis expansion in terms

f the eigenfunctions associated with the kernel (covariance

function for the GP)] Sine basis; If process noise affects x, particle filtering must be applied to find the predictor (Svensson and Schön, Automatica 2017)

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET

SLIDE 12

The Palette of Nonlinear Models

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White: Known model Off-white: Careful Physical Modeling w or w/o noise models Smoke-grey: Semi-physical modeling (Could be used more!) Steel-grey: Local Linear Models Slate-grey: Block-oriented Models. Black: Flexible structures – universal approximators Pit-black: Non-Parametric Smoothing

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

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Pit-black Models: Non-Parametric Smoothing Methods

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1 2 3 4 5 1 2 3 4 5 6 −4 −3 −2 −1 1 2 3 4 x 10

u(t−1)

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * * * *

u(t−2) y

Form the model surface h(ϕ(t)) by smoothing over the observation points in the space! Even Blacker! Huge literature – Mostly in the statistical community and now also in machine learning Important to find lower dimensional manifolds (– counterpart of PCA in linear modelling). Concepts like Manifold Learning and Local Linear Embedding become central.

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET

Conclusions

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Confusingly many approaches! A user-oriented roadmap would be excellent!

Lennart Ljung Brussels Workshop, April 25, 2017 Nonlinear System Identification

AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET