ECE 636: Systems identification · 2011-12-05 · ECE 636: Systems identification Lectures 23‐24...

ECE 636 Systems yidentificationLectures 23‐24

Identification of closed‐loop systemsNonlinear systemsNonlinear systems

Applications

bull Model order selectionbull Start from simple models and try successively more complex modelsbull Compare model prediction to observations (we shouldnrsquot get zero error) ˆ( ) ( ) ( )my t G q u tΝ= θ

bull Residual testingbull PEM residuals should be white for

bull Whiteness testing Residual autocorrelation function0

ˆN =θ θ 1ˆ ( ) ( ) ( )

N

t tτ

ϕ τ ε ε τminus

= +sum

ˆ ˆˆ( | ) ( ) ( | )N Nt y t y tε = minusθ θ

bull Whiteness testing Residual autocorrelation function

Cross‐correlation function between inputresiduals

Statistical hypothesis testing

1( ) ( ) ( )

tt t

Nεεϕ τ ε ε τ=

= +sum

1

1ˆ ( ) ( ) ( )N

ut

t u tN

τ

εϕ τ ε τminus

=

= minussum2ˆ (0)ϕ λrarrStatistical hypothesis testing (0)

ˆ ( ) 0 0εε

εε

ϕ λϕ τ τ

rarrrarr ne

2 22

1

ˆ ( )ˆ (0)

m

mN

εετεε

ϕ τ χϕ =

rarrsum ˆ ( ) (01)ˆ (0)

N Nεε

εε

ϕ τϕ

rarr

Open loop systems any τ Closed‐loop systems only for τgt0

( ) ( ) ( ) 0u E t u tεϕ τ ε τ= + =

12

ˆ ( )ux ετ

ϕ τ= (0 1)N x Nrarr[ ]12ˆ ˆ(0) ( )uu

xτεεϕ ϕ τ

(01)N x Nτ rarr

bull Cross‐validationbull Sufficient data pointsbull System (almost) time‐invariant

M d l ibull Model comparison

( )

St ti ti l it i

1 2 2 1

2 2

( ) ( ) ( )

MSE MSE p pFMSE N pminus minus

=minus 2 1 2p p N pF minus minus 2 1

2p pχ minus

2ˆ pbull Statistical criteria 2ˆ( ) ( )1 N NpAIC p VN

= +θ

1 ˆ( ) ( )1 N Np NFPE p Vp N

+=

minusθ

bull Recursive identificationU d t th ti t t h ti i t i t f

lnˆ( ) ( )1 N Np NMDL p VN

= +θ

ˆ( 1)θˆ( )θbull Update the estimates at each time point ie get frombull Recursive least‐squares

( 1)t minusθ( )tθ

ˆ ˆ( ) ( 1) ( ) ( )t t t tε= minus +θ θ K( 1) ( ) ( ) ( 1)( ) ( 1)1 ( ) ( 1) ( )

T

T

t t t tt tt t t

minus minus= minus minus

+ minusP φ φ PP P

φ P φˆ( ) ( ) ( ) ( 1)Tt y t t tε = minus minusφ θ

1 ( ) ( 1) ( )t t t+φ P φ( 1) ( )( ) ( ) ( )

1 ( ) ( 1) ( )T

t tt t tt t tminus

= =+ minus

P φK P φφ P φ

bull Time‐varying systemsbull Modified cost function 2

1( ) ( )

tt s

ts

V sλ εminus

=

=sumθ

λ forgetting factor (between 0 and 1) Smaller λ faster adaptation but more sensitivity to noise

ˆ ˆ( ) ( 1) ( ) ( )t t t tε= minus +θ θ K( 1) ( ) ( ) ( 1)1( ) ( 1)

( ) ( 1) ( )

T

T

t t t tt tt t tλ λ

⎡ ⎤minus minus= minus minus⎢ ⎥+ minus⎣ ⎦

P φ φ PP Pφ P φ

ˆ( ) ( ) ( ) ( 1)Tt y t t tε = minus minusφ θ( ) ( ) ( )⎣ ⎦φ φ

( 1) ( )( ) ( ) ( )( ) ( 1) ( )T

t tt t tt t tλminus

= =+ minus

P φK P φφ P φ

ˆ (0) 0=θ(0) 0(0) ρ

==

θP Ι

Closed‐loop systemsbull Many true systems (industrial physiological) operate with feedback

bull Often the open‐loop system may be unstable so experiments are performed in closed‐loop fi ti ( i i d t i l t )configurations (eg in industrial systems)

bull We have already seen (nonparametric identification ndash HW 2) that the presence of feedback may affect our estimates considerablybull Eg the estimation of G from measurements of uy is significantly affected by the fact that the noise d i t i l l t dand input signals are correlated

(t) f l t i t i t i th l tHs(q-1)

e(t)1 1 2 2

1 1

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )s sy t G q u t H q e t E e t

u t F q y t L q v t

λminus minus

minus minus

= + =

= minus +

v(t) reference valuesetpoint or noise entering the regulatorbull Goal Identification of Gs(q-1) Hs(q-1) bull Cases

bull v(t) observable non‐observableF( 1) k k

Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)

bull F(q-1) knownunknownbull Assumptions

bull Gs(0)=0 (no instantaneous effects of u on y)bull The subsystems between ve and y are stable ie LHs and[1+GF] 1 t bl

F(q-1)

-

[1+GF]‐1 are stablebull v(t) is stationary and persistently exciting of sufficient orderbull v(t) and e(s) are independent for any ts

Closed‐loop systemsbull The general relations that describe this system are

1( ) ( ( ) ( ))s sy t G Lv t H e t= +( ) ( ( ) ( ))1

( ) 1 ( ) ( )1 1

s ss

ss

s s

yFG

FG Fu t Lv t H e tFG FG

+

⎡ ⎤= minus minus⎢ ⎥+ +⎣ ⎦

Hs(q-1)

e(t)

Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)

F(q-1)

-

Closed‐loop systemsbull Often control systems applications aim to minimize the difference between a reference signal v(t) and y(t) which implies small u(t) values ndash not ideal for identification purposesbull Spectral analysis

W h b f (HW2) th t th t i ti t f G ( 1) i th f d ibull We have seen before (HW2) that the nonparametric estimate of Gs(q‐1) in the frequency domain (assuming L=1) ie

i ld bi d ti t I th l th bi i

ˆ ( ) ( ) ( ) ( ) ( )ˆ ( ) ˆ ( ) ( )( )yu s vv zz

vv zzuu

G FGω ω ω ω ωω

ω ωωΦ Φ minusΦ

= =Φ +ΦΦ

yields biased estimates In the general case the bias is

where

1 ( ) ( ) ( )ˆ ( ) ( )( ) ( ) ( )

s zzs

vv zz

G FG GFω ω ωω ωω ω ω

+ Φminus = minus

Φ +ΦHs(q-1)

e(t)

For e(t)=0 no bias in the estimateFor v(t)=0 we have

1 1( ) ( ) ( ) ( )sz t F q H q e tminus minus=

Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)

1ˆ ( )( )

GF

ωω

minusF(q-1)

-

Closed‐loop systemsbull Modified spectral analysis

bull Assuming that the external input signal ν(t) is measurable ( ) ( ) ( ) ( ) ( )vy s vu s veG Hω ω ω ω ωΦ = Φ + Φ =

Therefore we can obtain the following (unbiased) estimate

Ti d i t l h f id tifi ti f l d l t

( ) ( )s vuG ω ω= Φ

ˆ ( )ˆ ( ) ˆ ( )vy

vu

Gω

ωω

Φ=Φ

bull Time domain ndash two general approaches for identification of closed‐loop systems arebull Direct identification We treat the system as an open‐loop systembull Indirect identification We assume that the external input reference signal v(t) is measurable and that the feedback law F is known

Hs(q-1)

e(t)

Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

F(q-1)

Closed‐loop systemsbull Direct identification We use the measurements of u(t) and y(t) and the general model structure we have used before

1 1 2 2ˆ ˆˆ( ) ( ) ( ) ( ) ( ) ( )y t G q u t H q e t E e t λminus minus= + =θ θ

bull The goal is to fine the parameter vector that minimizes the prediction error eg

[ ]22

1 1

1 1 ˆ( ) ( ) ( ) ( | )N N

Nt t

V t y t y tN N

ε= =

= = minussum sumθ θ θ 1 1 1ˆˆ( ) ( )[ ( ) ( ) ( )]t H q y t G q u tε minus minus minus= minusθ

bull The question is whether for this parametrization the closed loop system is identifiable ie whether there exists θ such that

ˆ ˆs sG G H H= =

bull This depends on the parametrization and the true system form (eg feedback law)

bull Example Let the system

Model

2 2( ) ( 1) ( 1) ( ) ( )( ) ( )y t ay t bu t e t E e tu t fy t

λ+ minus = minus + == minus

ˆˆ( ) ( 1) ( 1) ( )y t ay t bu t tε+ minus = minus +

( ) ( ) ( 1) ( )y t a fb y t e t+ + minus =

If we use eg least squares we can estimate only the linear combination a+fb

Closed‐loop systemsbull Assume we have two different regulators f1 and f2 which are active for fractions γ1 and γ2 of the total time ie

bull As before the system is2 2( ) ( 1) ( 1) ( ) ( )

( ) ( )y t ay t bu t e t E e t

fλ+ minus = minus + =

and the model( ) ( )iu t f y t= minus

ˆˆ( ) ( 1) ( 1) ( )y t ay t bu t tε+ minus = minus +( ) ( ) ( 1) ( )

ˆˆ( ) ( ) ( ) ( 1)i iy t a bf y t e t

t y t a bf y tε

+ + minus =

= + + minus

therefore

bull Cost functionN

( ) ( ) ( ) ( 1)i i it y t a bf y tε + +2

2 22

ˆˆ( ) ( ) 11 ( )

i ii

i

a bf a bfE ta bf

ε λ⎡ ⎤+ minus minus

= +⎢ ⎥minus +⎢ ⎥⎣ ⎦

2

1

2 22 2 21 1 2 2

1 22 2

1ˆˆlim ( ) ( )

ˆ ˆˆ ˆ( ) ( )1 ( ) 1 ( )

N

N Nt

V a b tN

a bf a bf a bf a bfa bf a bf

ε

λ γ λ γ λ

rarrinfin=

=

+ minus minus + minus minus= + +

minus + minus +

sum θ

bull We can see that ‐minimum

1 21 ( ) 1 ( )a bf a bf+ +

2ˆˆ( ) ( )N NV a b V a b λge =

Closed‐loop systemsbull Global minimum Solution of

bull unique solution for f1nef2

bull Indirect identificationbull Knowledge of v(t) and F(q‐1) L(q-1) is requiredbull Using one of the examined methods (LS PEM) we identify the total system between v ybull Using knowledge of L F we can get estimates of Gs HsUsing knowledge of L F we can get estimates of Gs Hs

Hs(q-1)

e(t)1( ) ( ( ) ( ))

1

( ) 1 ( ) ( )

s ss

s

y t G Lv t H e tFG

FG Fu t Lv t H e t

= ++

⎡ ⎤⎢ ⎥

bullComparing this to the general LTI structure

we can get estimates of

Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

( ) 1 ( ) ( )1 1

ss

s s

u t Lv t H e tFG FG

= minus minus⎢ ⎥+ +⎣ ⎦

1 1ˆ ˆ( ) ( ) ( ) ( ) ( )y t G q v t H q e tminus minus= +θ θ1we can get estimates of

and use knowledge of LF to obtain GsHs

F(q-1)

11

11

ss

ss

G G LFG

H HFG

=+

=+

Nonlinear systems

S(τ) y(t)x(t) S(Ax1+Bx2)neAS(x1)+BS(x2)

bull Most real‐life systems are nonlinear linearity is an approximation that may or may not yield satisfactory resultsbull Nonlinear models identification methods

N t i (V lt Wi i )bull Nonparametric (VolterraWiener series)bull Parametric (nonlinear ARX ARMAX models nonlinear differential equations)bull Block diagrams Neural network models

yu z

A b f th l i t ti t th t d l f IO d t

Gyu z

21 2

( ) ( ) ( )( ) ( ) ( )z t g t u ty t c z t c z t

=

= +

bull As before the goal is to estimate the system model from IO databull More complicated problem ndash number of free parameters much higherbull No simple relations exist in the frequency domain as for linear systems (the output of a nonlinear system to a sinusoidal signal is not a sinusoidal signal with the same frequency but hibit h i t l )exhibits harmonic components also)

bull Generally more difficult to obtain theoretical results for convergence consistency etc compared to linear systems

Nonlinear systemsM MemoryQ Ordersum sum sum

=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(

Volterra‐Wiener seriesbull Generalization of convolution sum bull Goal estimate Volterra kernels kn from IO measurements

system memory

k 1(m

)k 2

(m1m

2)k

system memory

k

131313m

m1m2 m1 m2


=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(

bull Estimationbull Least‐squares we can write the above relation as a linear regression problems ‐ very large number of free parameters (order ΜQ) bull Cross‐correlation method Estimate high‐order cross‐correlation functions between input output eg

As we saw for linear systems also long data records are typically needed to obtain accurate estimates for these cross‐correlation functions

1 2 1 2( ) ( ) ( ) ( )xxy E y t x t x tϕ τ τ τ τ= + +

Nonlinear systems⎫⎧

sum sum sum=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(

bull Efficient Volterra kernel estimationF ti i R d ti f f tbull Function expansions Reduction of free parameters

bull One of the most widely used basis set is the Laguerre basis( ) 2 1 2

0( ) (1 ) ( 1) (1 ) 0 1

jj m j m m

jm

jb a

m mτ τ

τ α α α αminus minus

=

⎛ ⎞⎛ ⎞= minus minus minus lt lt⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠sum

03

04

05α =01α =02α =04

1 1 1

1

1 10 0

( ) ( ) ( )q

q

L L

q q j j j j qj j

k m m c b m b m= =

= sum sum

V + b

0

01

02y=Vc+ε vj=xbj

cest=[VTV]-1VTy

bull Orthonormal basis in [0infin) ndash well‐

-03

-02

-01Orthonormal basis in [0 infin) wellconditioned estimation of causal systems

bull Exponential decay suitable for finite memory systems

0 5 10 15 20 25 30 35 40 45 50-05

-04

Time lags

y ybull α critical for model accuracy

parsimony

Nonlinear systems

bull Polynomial activation ffunctionsbull Free parameters ~ QMbull Equivalent to Volterra model

M

sum=

minus=j

jii jnxwnv0

)()(miiiiiii zazaazf 110 )( +++=

H

sum=

=H

iijijijmiimm m

wwwacjjjk1

21 )(11

Nonlinear systems

bull Combination of neural networks with functionexpansions (Laguerre‐Volterra networks)

x(n)

expansions (Laguerre Volterra networks)

sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1

bull Drastic reduction of free parameters

vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L

Drastic reduction of free parameters (Q+L+1)∙K+2 bull K number of hidden units f1 fK

hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy

bull LVN trainingndash Iterative gradient descent (backpropagation)ndash Even the Laguerre parameter α can be trained

bull Recursive relations for Laguerre filter outputsbull Training

⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ

( 1) ( ) ( ) ( ) ( 1) r r r r rJd dβ β β β γ μ β β α+ minus⎛ ⎞part= + = minus + =⎜ ⎟

bull Equivalence to Volterra modelK L L

r

d dβ ββ β β β γ μ β β αβ

= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(

Nonlinear systemsbull Each input convolved

with different filterx1(n) xI(n)

Nonlinear systems

with different filter bankndash Distinct α parameters

diff i d i)((1)

0 nv

hellip hellip (1)1Lb

(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL

hellip hellip (I)ILb

(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j

different input dynamics

ndash Nonlinear interactions between inputs modeled

N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK

bull Number of free parameters

⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs

Some examples ndash biological systems identificationp g y

bull Noninvasive accurate measurement of a wide variety of biological signals and programmable micropumps for pharmacological substance infusion

bull Rich spontaneous variability in physiological signals

bull We can obtain information about the function of biological physiological systemsndash Homeostatic mechanisms eg pressure regulation blood flow in the brain

ndash Functional connectivity in the brain (EEGMEGfMRI measurements)

bull Control of physiological signals based on mathematical models (model‐predictive control))

ndash Glucose control in diabetics with insulin micropumps

bull Obtain biomarkers for the early diagnosis of pathophysiological condition better understanding of the mechanisms that cause these conditions

Cerebral blood flow regulation

Autoregulation homeostatic regulation of own blood flow Brain extremely effective autoregulationBrain extremely effective autoregulation

2 of body mass15 of total cardiac output 20 O2 consumption

Assessment of dynamic autoregulationAssessment of dynamic autoregulation

Step response to inducedABP CO2 changesABP CO2 changes

Thigh cuff deflationHypercapnia hypocapnia

Spontaneous variability MABP

CBFV

Spontaneous variabilityNormal conditions all naturally occurring frequenciesCBFV variability correlated toΜΑBP

MABP

y CO2 variabilityLinear methods

High pass ABP‐CBFV characteristic[Zhang et al Amer J Physiol 1998]Low coherence lt 007 Hz Nonlinearities influence of CO2

Nonlinearmethods

22

Nonlinearmethods Larger fraction of CBFV variability explained [Mitsis et al Ann Biomed Eng 2002]

Nonlinear model of cerebral h d ihemodynamics

ΑBP CO2

Nonlinear two‐input model of cerebral hemodynamics ΑBP


(1)jb

(1)0b hellip hellip (2)

ILb(2)jb

(2)0b

CO2of cerebral hemodynamicsInputs ABP PETCO2variations

Dynamic pressureautoregulation

DynamicCO2 reactivity

a at o s

Output CBFV variations

Simultaneous assessment of fKhellipf1dynamic pressure

autoregulation CO2reactivity

+

CBFV

reactivity

Includes MABP‐CO2interactions

23

Cerebral hemodynamics under resting conditions

Experimental data14 subjects

conditions

Resting conditions 45 minsTraining data 6 min (360 points)Validation data 1 minValidation data 1 min

Systemorder

NMSE []MABP CO2 ΜΑBP amp CO2

2424

orderLinear (Q=1) 328plusmn132 716plusmn121 248plusmn111

Nonlinear (Q=3) 200plusmn92 515plusmn104 145plusmn69

Cerebral hemodynamics under resting conditions Model predictionsconditions ndash Model predictions

Dynamic range VLF 0005‐004 Hz LF 004‐015 Hz HF 015‐030 HzNonlinearities prominent in VLFCO2 accounts mainly for VLF CBFV variations (lt004Hz) MABP dominates in HF

Cerebral hemodynamics under resting conditions ndashLinear componentsLinear components

MABP HP characteristicSlow MABP changes regulated more g geffectively

PETCO2 LP characteristicSlow CO2 changes have more effect on CBFon CBF

26

Cerebral hemodynamics under resting conditions ndashNonlinear components

Second order kernels

Nonlinear components

Second‐order kernelsMost power in VLF LF

Relative contribution of NL terms more significantsignificant

for PETCO2

MABP PETCO2

Nonlinear to linear terms power ratio

031plusmn013 118plusmn045

27

Cerebral hemodynamics under resting conditions ‐nonstationarity

k1MABP k1 CO2

nonstationarity

1CO2

Tracking 6 min sliding data segments

28

Tracking 6‐min sliding data segments

From systemic to regional hemodynamics ‐functional MRIfunctional MRI

Neural activationRegional changes inblood flow oxygenationOxy Deoxygenated hemoglobin Different magnetic properties ‐ BOLD signalPrecise nature of neurovascular

2929

neurovascular coupling not known

Respiratory network imaging Voxel‐wise l ianalysis

RestingRestingRestingResting

30EndEnd--tidal tidal forcingforcing

Modeling of regional CO2 reactivityModeling of regional CO2 reactivity

bull Definition of anatomical and functional regions ofand functional regions of interest

bull CO2 reactivity Averaged O i i i hi OBOLD time‐series within ROI

AV thalamus

31

Regional dynamic CO2 reactivityRegional dynamic CO2 reactivity

bull Significant regional variabilitybull Differential responses to small large CO changes (undershoot

32

bull Differential responses to small large CO2 changes (undershoot during resting only)

Functional brain connectivityFunctional brain connectivity

bull Brain function relies on multiple complex interconnections between different regions that may may not be task specific

bull We can assess functional connectivity from EEGMEGfMRI measurements by quantifying h d h ll d (l l ) h d l

33

their dynamic interactions with well‐suited (linear or nonlinear) methods eg correlation coherence directed coherence mutual information etc

bull This problem is also particularly relevant in neurological disorders such as epilepsy (seizure prediction)


bull Brain function relies on multiple complex interconnections between different regionsbull Brain function relies on multiple complex interconnections between different regions that may may not be task specific

bull We can assess functional connectivity from EEGMEGfMRI measurements by quantifying their dynamic interactions with well‐suited (linear or nonlinear) methods eg correlation coherence directed coherence mutual information etc

34

coherence directed coherence mutual information etcbull Combination with graph‐theoretic measures bull This problem is also particularly relevant in neurological disorders such as epilepsy

(seizure prediction)

Metabolic system ndash Diabetes and glucose b limetabolism

bull Diabetes mellitusndash Type 1 β cells do not produce insulinyp pndash Type 2 Insulin not utilized properlyndash Many long‐term implications

bull Interaction between key variables y(glucose insulin glucagon)ndash Currently ODE models specialized

experimental protocols (IVGTT)experimental protocols (IVGTT)ndash Continuous glucose sensors

insulin micropumpsbull Extraction of richer informationbull Extraction of richer informationbull Detection of subtle changes (Insulin secretion pattern sensitivity) improved early diagnosis

bull Model based glucose control DelaypreventionModel based glucose control Delayprevention of long‐term implications

35

Glucose metabolism models in diabetic patientsGlucose metabolism models in diabetic patients

Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1

100Sensor Glucose [mg dl]

20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50

Insulin uptake [microUnitsml]

-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36

Glucose controlGlucose control

37

bull Model order selectionbull Start from simple models and try successively more complex modelsbull Compare model prediction to observations (we shouldnrsquot get zero error) ˆ( ) ( ) ( )my t G q u tΝ= θ

bull Residual testingbull PEM residuals should be white for

bull Whiteness testing Residual autocorrelation function0

ˆN =θ θ 1ˆ ( ) ( ) ( )

N

t tτ

ϕ τ ε ε τminus

= +sum

ˆ ˆˆ( | ) ( ) ( | )N Nt y t y tε = minusθ θ

bull Whiteness testing Residual autocorrelation function

Cross‐correlation function between inputresiduals

Statistical hypothesis testing

1( ) ( ) ( )

tt t

Nεεϕ τ ε ε τ=

= +sum

1

1ˆ ( ) ( ) ( )N

ut

t u tN

τ

εϕ τ ε τminus

=

= minussum2ˆ (0)ϕ λrarrStatistical hypothesis testing (0)

ˆ ( ) 0 0εε

εε

ϕ λϕ τ τ

rarrrarr ne

2 22

1

ˆ ( )ˆ (0)

m

mN

εετεε

ϕ τ χϕ =

rarrsum ˆ ( ) (01)ˆ (0)

N Nεε

εε

ϕ τϕ

rarr

Open loop systems any τ Closed‐loop systems only for τgt0

( ) ( ) ( ) 0u E t u tεϕ τ ε τ= + =

12

ˆ ( )ux ετ

ϕ τ= (0 1)N x Nrarr[ ]12ˆ ˆ(0) ( )uu

xτεεϕ ϕ τ

(01)N x Nτ rarr



( )

St ti ti l it i

1 2 2 1

2 2

( ) ( ) ( )



2p pχ minus


= +θ


+=

minusθ



= +θ


( 1)t minusθ( )tθ


T

T

t t t tt tt t t


+ minusP φ φ PP P


1 ( ) ( 1) ( )t t t+φ P φ( 1) ( )( ) ( ) ( )

1 ( ) ( 1) ( )T

t tt t tt t tminus

= =+ minus

P φK P φφ P φ


1( ) ( )

tt s

ts

V sλ εminus

=

=sumθ



( ) ( 1) ( )

T

T



P φ φ PP Pφ P φ


( 1) ( )( ) ( ) ( )( ) ( 1) ( )T


= =+ minus

P φK P φφ P φ

ˆ (0) 0=θ(0) 0(0) ρ

==

θP Ι





e(t)1 1 2 2

1 1

( ) ( ) ( ) ( ) ( ) ( )


u t F q y t L q v t

λminus minus

minus minus

= + =

= minus +



Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)



F(q-1)

-



1( ) ( ( ) ( ))s sy t G Lv t H e t= +( ) ( ( ) ( ))1

( ) 1 ( ) ( )1 1

s ss

ss

s s

yFG


+


Hs(q-1)

e(t)

Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)

F(q-1)

-




ˆ ( ) ( ) ( ) ( ) ( )ˆ ( ) ˆ ( ) ( )( )yu s vv zz

vv zzuu



= =Φ +ΦΦ


where

1 ( ) ( ) ( )ˆ ( ) ( )( ) ( ) ( )

s zzs

vv zz


+ Φminus = minus

Φ +ΦHs(q-1)

e(t)



Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)

1ˆ ( )( )

GF

ωω

minusF(q-1)

-





( ) ( )s vuG ω ω= Φ

ˆ ( )ˆ ( ) ˆ ( )vy

vu

Gω

ωω

Φ=Φ


Hs(q-1)

e(t)

Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

F(q-1)




[ ]22

1 1

1 1 ˆ( ) ( ) ( ) ( | )N N

Nt t

V t y t y tN N

ε= =



ˆ ˆs sG G H H= =



Model




( ) ( ) ( 1) ( )y t a fb y t e t+ + minus =








ˆˆ( ) ( ) ( ) ( 1)i iy t a bf y t e t

t y t a bf y tε

+ + minus =

= + + minus

therefore

bull Cost functionN

( ) ( ) ( ) ( 1)i i it y t a bf y tε + +2

2 22

ˆˆ( ) ( ) 11 ( )

i ii

i

a bf a bfE ta bf


= +⎢ ⎥minus +⎢ ⎥⎣ ⎦

2

1

2 22 2 21 1 2 2

1 22 2

1ˆˆlim ( ) ( )

ˆ ˆˆ ˆ( ) ( )1 ( ) 1 ( )

N

N Nt

V a b tN


ε

λ γ λ γ λ

rarrinfin=

=


minus + minus +

sum θ


1 21 ( ) 1 ( )a bf a bf+ +





Hs(q-1)

e(t)1( ) ( ( ) ( ))

1

( ) 1 ( ) ( )

s ss

s

y t G Lv t H e tFG

FG Fu t Lv t H e t

= ++

⎡ ⎤⎢ ⎥



Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

( ) 1 ( ) ( )1 1

ss

s s

u t Lv t H e tFG FG




F(q-1)

11

11

ss

ss

G G LFG

H HFG

=+

=+

Nonlinear systems




yu z


Gyu z

21 2

( ) ( ) ( )( ) ( ) ( )z t g t u ty t c z t c z t

=

= +




=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(


system memory

k 1(m

)k 2

(m1m

2)k

system memory

k

131313m

m1m2 m1 m2


=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(





sum sum sum=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(



0( ) (1 ) ( 1) (1 ) 0 1

jj m j m m

jm

jb a

m mτ τ


=


⎝ ⎠⎝ ⎠sum

03

04

05α =01α =02α =04

1 1 1

1

1 10 0

( ) ( ) ( )q

q

L L

q q j j j j qj j

k m m c b m b m= =

= sum sum

V + b

0

01

02y=Vc+ε vj=xbj

cest=[VTV]-1VTy


-03

-02



0 5 10 15 20 25 30 35 40 45 50-05

-04

Time lags


parsimony

Nonlinear systems


M

sum=

minus=j

jii jnxwnv0


H

sum=

=H

iijijijmiimm m

wwwacjjjk1

21 )(11

Nonlinear systems


x(n)


sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1


vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L


hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37



( )

St ti ti l it i

1 2 2 1

2 2

( ) ( ) ( )



2p pχ minus


= +θ


+=

minusθ



= +θ


( 1)t minusθ( )tθ


T

T

t t t tt tt t t


+ minusP φ φ PP P


1 ( ) ( 1) ( )t t t+φ P φ( 1) ( )( ) ( ) ( )

1 ( ) ( 1) ( )T

t tt t tt t tminus

= =+ minus

P φK P φφ P φ


1( ) ( )

tt s

ts

V sλ εminus

=

=sumθ



( ) ( 1) ( )

T

T



P φ φ PP Pφ P φ


( 1) ( )( ) ( ) ( )( ) ( 1) ( )T


= =+ minus

P φK P φφ P φ

ˆ (0) 0=θ(0) 0(0) ρ

==

θP Ι





e(t)1 1 2 2

1 1

( ) ( ) ( ) ( ) ( ) ( )


u t F q y t L q v t

λminus minus

minus minus

= + =

= minus +



Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)



F(q-1)

-



1( ) ( ( ) ( ))s sy t G Lv t H e t= +( ) ( ( ) ( ))1

( ) 1 ( ) ( )1 1

s ss

ss

s s

yFG


+


Hs(q-1)

e(t)

Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)

F(q-1)

-




ˆ ( ) ( ) ( ) ( ) ( )ˆ ( ) ˆ ( ) ( )( )yu s vv zz

vv zzuu



= =Φ +ΦΦ


where

1 ( ) ( ) ( )ˆ ( ) ( )( ) ( ) ( )

s zzs

vv zz


+ Φminus = minus

Φ +ΦHs(q-1)

e(t)



Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)

1ˆ ( )( )

GF

ωω

minusF(q-1)

-





( ) ( )s vuG ω ω= Φ

ˆ ( )ˆ ( ) ˆ ( )vy

vu

Gω

ωω

Φ=Φ


Hs(q-1)

e(t)

Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

F(q-1)




[ ]22

1 1

1 1 ˆ( ) ( ) ( ) ( | )N N

Nt t

V t y t y tN N

ε= =



ˆ ˆs sG G H H= =



Model




( ) ( ) ( 1) ( )y t a fb y t e t+ + minus =








ˆˆ( ) ( ) ( ) ( 1)i iy t a bf y t e t

t y t a bf y tε

+ + minus =

= + + minus

therefore

bull Cost functionN

( ) ( ) ( ) ( 1)i i it y t a bf y tε + +2

2 22

ˆˆ( ) ( ) 11 ( )

i ii

i

a bf a bfE ta bf


= +⎢ ⎥minus +⎢ ⎥⎣ ⎦

2

1

2 22 2 21 1 2 2

1 22 2

1ˆˆlim ( ) ( )

ˆ ˆˆ ˆ( ) ( )1 ( ) 1 ( )

N

N Nt

V a b tN


ε

λ γ λ γ λ

rarrinfin=

=


minus + minus +

sum θ


1 21 ( ) 1 ( )a bf a bf+ +





Hs(q-1)

e(t)1( ) ( ( ) ( ))

1

( ) 1 ( ) ( )

s ss

s

y t G Lv t H e tFG

FG Fu t Lv t H e t

= ++

⎡ ⎤⎢ ⎥



Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

( ) 1 ( ) ( )1 1

ss

s s

u t Lv t H e tFG FG




F(q-1)

11

11

ss

ss

G G LFG

H HFG

=+

=+

Nonlinear systems




yu z


Gyu z

21 2

( ) ( ) ( )( ) ( ) ( )z t g t u ty t c z t c z t

=

= +




=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(


system memory

k 1(m

)k 2

(m1m

2)k

system memory

k

131313m

m1m2 m1 m2


=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(





sum sum sum=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(



0( ) (1 ) ( 1) (1 ) 0 1

jj m j m m

jm

jb a

m mτ τ


=


⎝ ⎠⎝ ⎠sum

03

04

05α =01α =02α =04

1 1 1

1

1 10 0

( ) ( ) ( )q

q

L L

q q j j j j qj j

k m m c b m b m= =

= sum sum

V + b

0

01

02y=Vc+ε vj=xbj

cest=[VTV]-1VTy


-03

-02



0 5 10 15 20 25 30 35 40 45 50-05

-04

Time lags


parsimony

Nonlinear systems


M

sum=

minus=j

jii jnxwnv0


H

sum=

=H

iijijijmiimm m

wwwacjjjk1

21 )(11

Nonlinear systems


x(n)


sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1


vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L


hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37


1( ) ( )

tt s

ts

V sλ εminus

=

=sumθ



( ) ( 1) ( )

T

T



P φ φ PP Pφ P φ


( 1) ( )( ) ( ) ( )( ) ( 1) ( )T


= =+ minus

P φK P φφ P φ

ˆ (0) 0=θ(0) 0(0) ρ

==

θP Ι





e(t)1 1 2 2

1 1

( ) ( ) ( ) ( ) ( ) ( )


u t F q y t L q v t

λminus minus

minus minus

= + =

= minus +



Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)



F(q-1)

-



1( ) ( ( ) ( ))s sy t G Lv t H e t= +( ) ( ( ) ( ))1

( ) 1 ( ) ( )1 1

s ss

ss

s s

yFG


+


Hs(q-1)

e(t)

Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)

F(q-1)

-




ˆ ( ) ( ) ( ) ( ) ( )ˆ ( ) ˆ ( ) ( )( )yu s vv zz

vv zzuu



= =Φ +ΦΦ


where

1 ( ) ( ) ( )ˆ ( ) ( )( ) ( ) ( )

s zzs

vv zz


+ Φminus = minus

Φ +ΦHs(q-1)

e(t)



Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)

1ˆ ( )( )

GF

ωω

minusF(q-1)

-





( ) ( )s vuG ω ω= Φ

ˆ ( )ˆ ( ) ˆ ( )vy

vu

Gω

ωω

Φ=Φ


Hs(q-1)

e(t)

Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

F(q-1)




[ ]22

1 1

1 1 ˆ( ) ( ) ( ) ( | )N N

Nt t

V t y t y tN N

ε= =



ˆ ˆs sG G H H= =



Model




( ) ( ) ( 1) ( )y t a fb y t e t+ + minus =








ˆˆ( ) ( ) ( ) ( 1)i iy t a bf y t e t

t y t a bf y tε

+ + minus =

= + + minus

therefore

bull Cost functionN

( ) ( ) ( ) ( 1)i i it y t a bf y tε + +2

2 22

ˆˆ( ) ( ) 11 ( )

i ii

i

a bf a bfE ta bf


= +⎢ ⎥minus +⎢ ⎥⎣ ⎦

2

1

2 22 2 21 1 2 2

1 22 2

1ˆˆlim ( ) ( )

ˆ ˆˆ ˆ( ) ( )1 ( ) 1 ( )

N

N Nt

V a b tN


ε

λ γ λ γ λ

rarrinfin=

=


minus + minus +

sum θ


1 21 ( ) 1 ( )a bf a bf+ +





Hs(q-1)

e(t)1( ) ( ( ) ( ))

1

( ) 1 ( ) ( )

s ss

s

y t G Lv t H e tFG

FG Fu t Lv t H e t

= ++

⎡ ⎤⎢ ⎥



Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

( ) 1 ( ) ( )1 1

ss

s s

u t Lv t H e tFG FG




F(q-1)

11

11

ss

ss

G G LFG

H HFG

=+

=+

Nonlinear systems




yu z


Gyu z

21 2

( ) ( ) ( )( ) ( ) ( )z t g t u ty t c z t c z t

=

= +




=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(


system memory

k 1(m

)k 2

(m1m

2)k

system memory

k

131313m

m1m2 m1 m2


=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(





sum sum sum=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(



0( ) (1 ) ( 1) (1 ) 0 1

jj m j m m

jm

jb a

m mτ τ


=


⎝ ⎠⎝ ⎠sum

03

04

05α =01α =02α =04

1 1 1

1

1 10 0

( ) ( ) ( )q

q

L L

q q j j j j qj j

k m m c b m b m= =

= sum sum

V + b

0

01

02y=Vc+ε vj=xbj

cest=[VTV]-1VTy


-03

-02



0 5 10 15 20 25 30 35 40 45 50-05

-04

Time lags


parsimony

Nonlinear systems


M

sum=

minus=j

jii jnxwnv0


H

sum=

=H

iijijijmiimm m

wwwacjjjk1

21 )(11

Nonlinear systems


x(n)


sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1


vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L


hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37





e(t)1 1 2 2

1 1

( ) ( ) ( ) ( ) ( ) ( )


u t F q y t L q v t

λminus minus

minus minus

= + =

= minus +



Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)



F(q-1)

-



1( ) ( ( ) ( ))s sy t G Lv t H e t= +( ) ( ( ) ( ))1

( ) 1 ( ) ( )1 1

s ss

ss

s s

yFG


+


Hs(q-1)

e(t)

Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)

F(q-1)

-




ˆ ( ) ( ) ( ) ( ) ( )ˆ ( ) ˆ ( ) ( )( )yu s vv zz

vv zzuu



= =Φ +ΦΦ


where

1 ( ) ( ) ( )ˆ ( ) ( )( ) ( ) ( )

s zzs

vv zz


+ Φminus = minus

Φ +ΦHs(q-1)

e(t)



Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)

1ˆ ( )( )

GF

ωω

minusF(q-1)

-





( ) ( )s vuG ω ω= Φ

ˆ ( )ˆ ( ) ˆ ( )vy

vu

Gω

ωω

Φ=Φ


Hs(q-1)

e(t)

Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

F(q-1)




[ ]22

1 1

1 1 ˆ( ) ( ) ( ) ( | )N N

Nt t

V t y t y tN N

ε= =



ˆ ˆs sG G H H= =



Model




( ) ( ) ( 1) ( )y t a fb y t e t+ + minus =








ˆˆ( ) ( ) ( ) ( 1)i iy t a bf y t e t

t y t a bf y tε

+ + minus =

= + + minus

therefore

bull Cost functionN

( ) ( ) ( ) ( 1)i i it y t a bf y tε + +2

2 22

ˆˆ( ) ( ) 11 ( )

i ii

i

a bf a bfE ta bf


= +⎢ ⎥minus +⎢ ⎥⎣ ⎦

2

1

2 22 2 21 1 2 2

1 22 2

1ˆˆlim ( ) ( )

ˆ ˆˆ ˆ( ) ( )1 ( ) 1 ( )

N

N Nt

V a b tN


ε

λ γ λ γ λ

rarrinfin=

=


minus + minus +

sum θ


1 21 ( ) 1 ( )a bf a bf+ +





Hs(q-1)

e(t)1( ) ( ( ) ( ))

1

( ) 1 ( ) ( )

s ss

s

y t G Lv t H e tFG

FG Fu t Lv t H e t

= ++

⎡ ⎤⎢ ⎥



Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

( ) 1 ( ) ( )1 1

ss

s s

u t Lv t H e tFG FG




F(q-1)

11

11

ss

ss

G G LFG

H HFG

=+

=+

Nonlinear systems




yu z


Gyu z

21 2

( ) ( ) ( )( ) ( ) ( )z t g t u ty t c z t c z t

=

= +




=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(


system memory

k 1(m

)k 2

(m1m

2)k

system memory

k

131313m

m1m2 m1 m2


=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(





sum sum sum=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(



0( ) (1 ) ( 1) (1 ) 0 1

jj m j m m

jm

jb a

m mτ τ


=


⎝ ⎠⎝ ⎠sum

03

04

05α =01α =02α =04

1 1 1

1

1 10 0

( ) ( ) ( )q

q

L L

q q j j j j qj j

k m m c b m b m= =

= sum sum

V + b

0

01

02y=Vc+ε vj=xbj

cest=[VTV]-1VTy


-03

-02



0 5 10 15 20 25 30 35 40 45 50-05

-04

Time lags


parsimony

Nonlinear systems


M

sum=

minus=j

jii jnxwnv0


H

sum=

=H

iijijijmiimm m

wwwacjjjk1

21 )(11

Nonlinear systems


x(n)


sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1


vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L


hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37


1( ) ( ( ) ( ))s sy t G Lv t H e t= +( ) ( ( ) ( ))1

( ) 1 ( ) ( )1 1

s ss

ss

s s

yFG


+


Hs(q-1)

e(t)

Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)

F(q-1)

-




ˆ ( ) ( ) ( ) ( ) ( )ˆ ( ) ˆ ( ) ( )( )yu s vv zz

vv zzuu



= =Φ +ΦΦ


where

1 ( ) ( ) ( )ˆ ( ) ( )( ) ( ) ( )

s zzs

vv zz


+ Φminus = minus

Φ +ΦHs(q-1)

e(t)



Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)

1ˆ ( )( )

GF

ωω

minusF(q-1)

-





( ) ( )s vuG ω ω= Φ

ˆ ( )ˆ ( ) ˆ ( )vy

vu

Gω

ωω

Φ=Φ


Hs(q-1)

e(t)

Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

F(q-1)




[ ]22

1 1

1 1 ˆ( ) ( ) ( ) ( | )N N

Nt t

V t y t y tN N

ε= =



ˆ ˆs sG G H H= =



Model




( ) ( ) ( 1) ( )y t a fb y t e t+ + minus =








ˆˆ( ) ( ) ( ) ( 1)i iy t a bf y t e t

t y t a bf y tε

+ + minus =

= + + minus

therefore

bull Cost functionN

( ) ( ) ( ) ( 1)i i it y t a bf y tε + +2

2 22

ˆˆ( ) ( ) 11 ( )

i ii

i

a bf a bfE ta bf


= +⎢ ⎥minus +⎢ ⎥⎣ ⎦

2

1

2 22 2 21 1 2 2

1 22 2

1ˆˆlim ( ) ( )

ˆ ˆˆ ˆ( ) ( )1 ( ) 1 ( )

N

N Nt

V a b tN


ε

λ γ λ γ λ

rarrinfin=

=


minus + minus +

sum θ


1 21 ( ) 1 ( )a bf a bf+ +





Hs(q-1)

e(t)1( ) ( ( ) ( ))

1

( ) 1 ( ) ( )

s ss

s

y t G Lv t H e tFG

FG Fu t Lv t H e t

= ++

⎡ ⎤⎢ ⎥



Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

( ) 1 ( ) ( )1 1

ss

s s

u t Lv t H e tFG FG




F(q-1)

11

11

ss

ss

G G LFG

H HFG

=+

=+

Nonlinear systems




yu z


Gyu z

21 2

( ) ( ) ( )( ) ( ) ( )z t g t u ty t c z t c z t

=

= +




=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(


system memory

k 1(m

)k 2

(m1m

2)k

system memory

k

131313m

m1m2 m1 m2


=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(





sum sum sum=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(



0( ) (1 ) ( 1) (1 ) 0 1

jj m j m m

jm

jb a

m mτ τ


=


⎝ ⎠⎝ ⎠sum

03

04

05α =01α =02α =04

1 1 1

1

1 10 0

( ) ( ) ( )q

q

L L

q q j j j j qj j

k m m c b m b m= =

= sum sum

V + b

0

01

02y=Vc+ε vj=xbj

cest=[VTV]-1VTy


-03

-02



0 5 10 15 20 25 30 35 40 45 50-05

-04

Time lags


parsimony

Nonlinear systems


M

sum=

minus=j

jii jnxwnv0


H

sum=

=H

iijijijmiimm m

wwwacjjjk1

21 )(11

Nonlinear systems


x(n)


sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1


vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L


hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37




ˆ ( ) ( ) ( ) ( ) ( )ˆ ( ) ˆ ( ) ( )( )yu s vv zz

vv zzuu



= =Φ +ΦΦ


where

1 ( ) ( ) ( )ˆ ( ) ( )( ) ( ) ( )

s zzs

vv zz


+ Φminus = minus

Φ +ΦHs(q-1)

e(t)



Gs(q-1) y(t)u(t)+

v(t) z(t)

+

s(q )

L(q-1)

1ˆ ( )( )

GF

ωω

minusF(q-1)

-





( ) ( )s vuG ω ω= Φ

ˆ ( )ˆ ( ) ˆ ( )vy

vu

Gω

ωω

Φ=Φ


Hs(q-1)

e(t)

Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

F(q-1)




[ ]22

1 1

1 1 ˆ( ) ( ) ( ) ( | )N N

Nt t

V t y t y tN N

ε= =



ˆ ˆs sG G H H= =



Model




( ) ( ) ( 1) ( )y t a fb y t e t+ + minus =








ˆˆ( ) ( ) ( ) ( 1)i iy t a bf y t e t

t y t a bf y tε

+ + minus =

= + + minus

therefore

bull Cost functionN

( ) ( ) ( ) ( 1)i i it y t a bf y tε + +2

2 22

ˆˆ( ) ( ) 11 ( )

i ii

i

a bf a bfE ta bf


= +⎢ ⎥minus +⎢ ⎥⎣ ⎦

2

1

2 22 2 21 1 2 2

1 22 2

1ˆˆlim ( ) ( )

ˆ ˆˆ ˆ( ) ( )1 ( ) 1 ( )

N

N Nt

V a b tN


ε

λ γ λ γ λ

rarrinfin=

=


minus + minus +

sum θ


1 21 ( ) 1 ( )a bf a bf+ +





Hs(q-1)

e(t)1( ) ( ( ) ( ))

1

( ) 1 ( ) ( )

s ss

s

y t G Lv t H e tFG

FG Fu t Lv t H e t

= ++

⎡ ⎤⎢ ⎥



Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

( ) 1 ( ) ( )1 1

ss

s s

u t Lv t H e tFG FG




F(q-1)

11

11

ss

ss

G G LFG

H HFG

=+

=+

Nonlinear systems




yu z


Gyu z

21 2

( ) ( ) ( )( ) ( ) ( )z t g t u ty t c z t c z t

=

= +




=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(


system memory

k 1(m

)k 2

(m1m

2)k

system memory

k

131313m

m1m2 m1 m2


=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(





sum sum sum=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(



0( ) (1 ) ( 1) (1 ) 0 1

jj m j m m

jm

jb a

m mτ τ


=


⎝ ⎠⎝ ⎠sum

03

04

05α =01α =02α =04

1 1 1

1

1 10 0

( ) ( ) ( )q

q

L L

q q j j j j qj j

k m m c b m b m= =

= sum sum

V + b

0

01

02y=Vc+ε vj=xbj

cest=[VTV]-1VTy


-03

-02



0 5 10 15 20 25 30 35 40 45 50-05

-04

Time lags


parsimony

Nonlinear systems


M

sum=

minus=j

jii jnxwnv0


H

sum=

=H

iijijijmiimm m

wwwacjjjk1

21 )(11

Nonlinear systems


x(n)


sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1


vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L


hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37





( ) ( )s vuG ω ω= Φ

ˆ ( )ˆ ( ) ˆ ( )vy

vu

Gω

ωω

Φ=Φ


Hs(q-1)

e(t)

Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

F(q-1)




[ ]22

1 1

1 1 ˆ( ) ( ) ( ) ( | )N N

Nt t

V t y t y tN N

ε= =



ˆ ˆs sG G H H= =



Model




( ) ( ) ( 1) ( )y t a fb y t e t+ + minus =








ˆˆ( ) ( ) ( ) ( 1)i iy t a bf y t e t

t y t a bf y tε

+ + minus =

= + + minus

therefore

bull Cost functionN

( ) ( ) ( ) ( 1)i i it y t a bf y tε + +2

2 22

ˆˆ( ) ( ) 11 ( )

i ii

i

a bf a bfE ta bf


= +⎢ ⎥minus +⎢ ⎥⎣ ⎦

2

1

2 22 2 21 1 2 2

1 22 2

1ˆˆlim ( ) ( )

ˆ ˆˆ ˆ( ) ( )1 ( ) 1 ( )

N

N Nt

V a b tN


ε

λ γ λ γ λ

rarrinfin=

=


minus + minus +

sum θ


1 21 ( ) 1 ( )a bf a bf+ +





Hs(q-1)

e(t)1( ) ( ( ) ( ))

1

( ) 1 ( ) ( )

s ss

s

y t G Lv t H e tFG

FG Fu t Lv t H e t

= ++

⎡ ⎤⎢ ⎥



Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

( ) 1 ( ) ( )1 1

ss

s s

u t Lv t H e tFG FG




F(q-1)

11

11

ss

ss

G G LFG

H HFG

=+

=+

Nonlinear systems




yu z


Gyu z

21 2

( ) ( ) ( )( ) ( ) ( )z t g t u ty t c z t c z t

=

= +




=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(


system memory

k 1(m

)k 2

(m1m

2)k

system memory

k

131313m

m1m2 m1 m2


=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(





sum sum sum=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(



0( ) (1 ) ( 1) (1 ) 0 1

jj m j m m

jm

jb a

m mτ τ


=


⎝ ⎠⎝ ⎠sum

03

04

05α =01α =02α =04

1 1 1

1

1 10 0

( ) ( ) ( )q

q

L L

q q j j j j qj j

k m m c b m b m= =

= sum sum

V + b

0

01

02y=Vc+ε vj=xbj

cest=[VTV]-1VTy


-03

-02



0 5 10 15 20 25 30 35 40 45 50-05

-04

Time lags


parsimony

Nonlinear systems


M

sum=

minus=j

jii jnxwnv0


H

sum=

=H

iijijijmiimm m

wwwacjjjk1

21 )(11

Nonlinear systems


x(n)


sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1


vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L


hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37




[ ]22

1 1

1 1 ˆ( ) ( ) ( ) ( | )N N

Nt t

V t y t y tN N

ε= =



ˆ ˆs sG G H H= =



Model




( ) ( ) ( 1) ( )y t a fb y t e t+ + minus =








ˆˆ( ) ( ) ( ) ( 1)i iy t a bf y t e t

t y t a bf y tε

+ + minus =

= + + minus

therefore

bull Cost functionN

( ) ( ) ( ) ( 1)i i it y t a bf y tε + +2

2 22

ˆˆ( ) ( ) 11 ( )

i ii

i

a bf a bfE ta bf


= +⎢ ⎥minus +⎢ ⎥⎣ ⎦

2

1

2 22 2 21 1 2 2

1 22 2

1ˆˆlim ( ) ( )

ˆ ˆˆ ˆ( ) ( )1 ( ) 1 ( )

N

N Nt

V a b tN


ε

λ γ λ γ λ

rarrinfin=

=


minus + minus +

sum θ


1 21 ( ) 1 ( )a bf a bf+ +





Hs(q-1)

e(t)1( ) ( ( ) ( ))

1

( ) 1 ( ) ( )

s ss

s

y t G Lv t H e tFG

FG Fu t Lv t H e t

= ++

⎡ ⎤⎢ ⎥



Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

( ) 1 ( ) ( )1 1

ss

s s

u t Lv t H e tFG FG




F(q-1)

11

11

ss

ss

G G LFG

H HFG

=+

=+

Nonlinear systems




yu z


Gyu z

21 2

( ) ( ) ( )( ) ( ) ( )z t g t u ty t c z t c z t

=

= +




=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(


system memory

k 1(m

)k 2

(m1m

2)k

system memory

k

131313m

m1m2 m1 m2


=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(





sum sum sum=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(



0( ) (1 ) ( 1) (1 ) 0 1

jj m j m m

jm

jb a

m mτ τ


=


⎝ ⎠⎝ ⎠sum

03

04

05α =01α =02α =04

1 1 1

1

1 10 0

( ) ( ) ( )q

q

L L

q q j j j j qj j

k m m c b m b m= =

= sum sum

V + b

0

01

02y=Vc+ε vj=xbj

cest=[VTV]-1VTy


-03

-02



0 5 10 15 20 25 30 35 40 45 50-05

-04

Time lags


parsimony

Nonlinear systems


M

sum=

minus=j

jii jnxwnv0


H

sum=

=H

iijijijmiimm m

wwwacjjjk1

21 )(11

Nonlinear systems


x(n)


sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1


vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L


hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37







ˆˆ( ) ( ) ( ) ( 1)i iy t a bf y t e t

t y t a bf y tε

+ + minus =

= + + minus

therefore

bull Cost functionN

( ) ( ) ( ) ( 1)i i it y t a bf y tε + +2

2 22

ˆˆ( ) ( ) 11 ( )

i ii

i

a bf a bfE ta bf


= +⎢ ⎥minus +⎢ ⎥⎣ ⎦

2

1

2 22 2 21 1 2 2

1 22 2

1ˆˆlim ( ) ( )

ˆ ˆˆ ˆ( ) ( )1 ( ) 1 ( )

N

N Nt

V a b tN


ε

λ γ λ γ λ

rarrinfin=

=


minus + minus +

sum θ


1 21 ( ) 1 ( )a bf a bf+ +





Hs(q-1)

e(t)1( ) ( ( ) ( ))

1

( ) 1 ( ) ( )

s ss

s

y t G Lv t H e tFG

FG Fu t Lv t H e t

= ++

⎡ ⎤⎢ ⎥



Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

( ) 1 ( ) ( )1 1

ss

s s

u t Lv t H e tFG FG




F(q-1)

11

11

ss

ss

G G LFG

H HFG

=+

=+

Nonlinear systems




yu z


Gyu z

21 2

( ) ( ) ( )( ) ( ) ( )z t g t u ty t c z t c z t

=

= +




=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(


system memory

k 1(m

)k 2

(m1m

2)k

system memory

k

131313m

m1m2 m1 m2


=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(





sum sum sum=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(



0( ) (1 ) ( 1) (1 ) 0 1

jj m j m m

jm

jb a

m mτ τ


=


⎝ ⎠⎝ ⎠sum

03

04

05α =01α =02α =04

1 1 1

1

1 10 0

( ) ( ) ( )q

q

L L

q q j j j j qj j

k m m c b m b m= =

= sum sum

V + b

0

01

02y=Vc+ε vj=xbj

cest=[VTV]-1VTy


-03

-02



0 5 10 15 20 25 30 35 40 45 50-05

-04

Time lags


parsimony

Nonlinear systems


M

sum=

minus=j

jii jnxwnv0


H

sum=

=H

iijijijmiimm m

wwwacjjjk1

21 )(11

Nonlinear systems


x(n)


sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1


vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L


hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37




Hs(q-1)

e(t)1( ) ( ( ) ( ))

1

( ) 1 ( ) ( )

s ss

s

y t G Lv t H e tFG

FG Fu t Lv t H e t

= ++

⎡ ⎤⎢ ⎥



Gs(q-1) y(t)u(t)+

v(t) z(t)

-+

L(q-1)

( ) 1 ( ) ( )1 1

ss

s s

u t Lv t H e tFG FG




F(q-1)

11

11

ss

ss

G G LFG

H HFG

=+

=+

Nonlinear systems




yu z


Gyu z

21 2

( ) ( ) ( )( ) ( ) ( )z t g t u ty t c z t c z t

=

= +




=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(


system memory

k 1(m

)k 2

(m1m

2)k

system memory

k

131313m

m1m2 m1 m2


=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(





sum sum sum=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(



0( ) (1 ) ( 1) (1 ) 0 1

jj m j m m

jm

jb a

m mτ τ


=


⎝ ⎠⎝ ⎠sum

03

04

05α =01α =02α =04

1 1 1

1

1 10 0

( ) ( ) ( )q

q

L L

q q j j j j qj j

k m m c b m b m= =

= sum sum

V + b

0

01

02y=Vc+ε vj=xbj

cest=[VTV]-1VTy


-03

-02



0 5 10 15 20 25 30 35 40 45 50-05

-04

Time lags


parsimony

Nonlinear systems


M

sum=

minus=j

jii jnxwnv0


H

sum=

=H

iijijijmiimm m

wwwacjjjk1

21 )(11

Nonlinear systems


x(n)


sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1


vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L


hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37

Nonlinear systems




yu z


Gyu z

21 2

( ) ( ) ( )( ) ( ) ( )z t g t u ty t c z t c z t

=

= +




=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(


system memory

k 1(m

)k 2

(m1m

2)k

system memory

k

131313m

m1m2 m1 m2


=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(





sum sum sum=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(



0( ) (1 ) ( 1) (1 ) 0 1

jj m j m m

jm

jb a

m mτ τ


=


⎝ ⎠⎝ ⎠sum

03

04

05α =01α =02α =04

1 1 1

1

1 10 0

( ) ( ) ( )q

q

L L

q q j j j j qj j

k m m c b m b m= =

= sum sum

V + b

0

01

02y=Vc+ε vj=xbj

cest=[VTV]-1VTy


-03

-02



0 5 10 15 20 25 30 35 40 45 50-05

-04

Time lags


parsimony

Nonlinear systems


M

sum=

minus=j

jii jnxwnv0


H

sum=

=H

iijijijmiimm m

wwwacjjjk1

21 )(11

Nonlinear systems


x(n)


sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1


vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L


hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37


=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(


system memory

k 1(m

)k 2

(m1m

2)k

system memory

k

131313m

m1m2 m1 m2


=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(





sum sum sum=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(



0( ) (1 ) ( 1) (1 ) 0 1

jj m j m m

jm

jb a

m mτ τ


=


⎝ ⎠⎝ ⎠sum

03

04

05α =01α =02α =04

1 1 1

1

1 10 0

( ) ( ) ( )q

q

L L

q q j j j j qj j

k m m c b m b m= =

= sum sum

V + b

0

01

02y=Vc+ε vj=xbj

cest=[VTV]-1VTy


-03

-02



0 5 10 15 20 25 30 35 40 45 50-05

-04

Time lags


parsimony

Nonlinear systems


M

sum=

minus=j

jii jnxwnv0


H

sum=

=H

iijijijmiimm m

wwwacjjjk1

21 )(11

Nonlinear systems


x(n)


sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1


vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L


hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37


=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(





sum sum sum=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(



0( ) (1 ) ( 1) (1 ) 0 1

jj m j m m

jm

jb a

m mτ τ


=


⎝ ⎠⎝ ⎠sum

03

04

05α =01α =02α =04

1 1 1

1

1 10 0

( ) ( ) ( )q

q

L L

q q j j j j qj j

k m m c b m b m= =

= sum sum

V + b

0

01

02y=Vc+ε vj=xbj

cest=[VTV]-1VTy


-03

-02



0 5 10 15 20 25 30 35 40 45 50-05

-04

Time lags


parsimony

Nonlinear systems


M

sum=

minus=j

jii jnxwnv0


H

sum=

=H

iijijijmiimm m

wwwacjjjk1

21 )(11

Nonlinear systems


x(n)


sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1


vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L


hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37


sum sum sum=

minus

=

minus

= ⎭⎬⎫

⎩⎨⎧

minusminus=Q

n

M

m

M

mnnn

n

mnxmnxmmkny0

1

0

1

011

1

)()()()(



0( ) (1 ) ( 1) (1 ) 0 1

jj m j m m

jm

jb a

m mτ τ


=


⎝ ⎠⎝ ⎠sum

03

04

05α =01α =02α =04

1 1 1

1

1 10 0

( ) ( ) ( )q

q

L L

q q j j j j qj j

k m m c b m b m= =

= sum sum

V + b

0

01

02y=Vc+ε vj=xbj

cest=[VTV]-1VTy


-03

-02



0 5 10 15 20 25 30 35 40 45 50-05

-04

Time lags


parsimony

Nonlinear systems


M

sum=

minus=j

jii jnxwnv0


H

sum=

=H

iijijijmiimm m

wwwacjjjk1

21 )(11

Nonlinear systems


x(n)


sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1


vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L


hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37

Nonlinear systems


M

sum=

minus=j

jii jnxwnv0


H

sum=

=H

iijijijmiimm m

wwwacjjjk1

21 )(11

Nonlinear systems


x(n)


sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1


vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L


hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37

Nonlinear systems


x(n)


sum=

=Q

m

mkkmkk nucnuf

1 ))(())((

m 1


vL(n)b0 bj bL

v0(n) vj(n)

hellip hellip

wwK0

w1jw1L


hellip

w10wKL

K0 wKj

+

y(n)

y0

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37

Nonlinear systemsy



⎟⎞

⎜⎛ partJ )1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rjkw

rjkw

rjk

rjk

rjk

rjk dw

wJwdwww μγ

)1(

)(

)(

)(

)1(

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

minus=+= rkmc

kc

rkm

rkm

rkm

rkm dc

cJcdccc μγ

⎠⎝ part rkmc

)1(0

0

)(0

)(0

)(0

)1(0

minus+ +⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

minus=+= ry

ry

rrrr dyyJydyyy μγ



r


= + = minus + =⎜ ⎟part⎝ ⎠

sum sum sum= = =

=K

k

L

j

L

jnjjjkjkknnn

n

nnmbmbwwcmmk

1 0 011

1

11)()()(



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37



Nonlinear systems


diff i d i)((1)

0 nv


(1)jb(1)

0b

)((I)0 nv )((I) nv j )((I)

I nvL


(I)jb(I)

0b)((1)

1 nvL

hellip)((1) nv j



N b f f

(1)01w

(1) 1

w LK(1)

0wK(1)

w jK

(1)1w j

(1)1 1w L

)( )(j )(I(I)

01w

(I)0wK

(I)w jK

(I)1w j

(I)1 Iw L

(I)w LIK


⎞⎛ I

fKhellipf1

11

++sdot⎟⎠

⎞⎜⎝

⎛++sum

=

NKNQLI

ii

+ y0+

y(n)

I number of inputs
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37
















CBFV


MABP



Nonlinearmethods

22



ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37


ΑBP CO2



(1)jb


ILb(2)jb

(2)0b




a at o s




+

CBFV

reactivity


23



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37



conditions


Systemorder


2424








26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37






26






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37






for PETCO2

MABP PETCO2



27


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37


k1MABP k1 CO2

nonstationarity

1CO2


28




2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37



2929








AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37







AV thalamus

31



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37



32





33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37




33






34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37




34









35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37







35


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37


Mode 1 11NL 1

-10

0

10

0 1

0

01

02Mode 1

v1

z1

v1

z1NL 1

150

-8 -6 -4 -2 0 2 4-30

-20

0 100 200 300 400 500-03

-02

-01

Time [min]Insulin

v1

0 2 4 Glucose

v1


20

40

60

[ ]

-0 1

-005

0Mode 2

v2

z2

v2

z2NL 2

0 50 100 150 200 250 300 350 400 450 5000

50


-6 -4 -2 0-20

0

20

0 100 200 300 400 500-02

-015

01

v2

-6 -4 -2 00Time [min]

v2

0 50 100 150 200 250 300 350 400 450 500Time [min]

Time [min]

36


37


37

ECE 636: Systems identification · 2011-12-05 · ECE 636: Systems identification Lectures 23‐24...

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Transcript of ECE 636: Systems identification · 2011-12-05 · ECE 636: Systems identification Lectures 23‐24...