research article a robust nonlinear observer for a class...

Research ArticleA Robust Nonlinear Observer for a Class of Neural Mass Models

Xian Liu Dongkai Miao and Qing Gao

Key Lab of Industrial Computer Control Engineering of Hebei Province Institute of Electrical Engineering Yanshan UniversityQinhuangdao 066004 China

Correspondence should be addressed to Xian Liu liuxianysueducn

Received 12 November 2013 Accepted 31 January 2014 Published 20 March 2014

Academic Editors G Cheron and X Fan

Copyright copy 2014 Xian Liu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A new method of designing a robust nonlinear observer is presented for a class of neural mass models by using the Lurrsquoe systemtheory and the projection lemmaThe observer is robust towards input uncertainty andmeasurement noise It is applied to estimatethe unmeasured membrane potential of neural populations from the electroencephalogram (EEG) produced by the neural massmodels An illustrative example shows the effectiveness of the proposed method

1 Introduction

Mathematical modelling provides a powerful tool for study-ing mechanisms involved in the generation of different elec-troencephalogram (EEG) rhythms and neuronal processesof neurological disorders There are two types of approachesto model neural signals One is on the basis of networksbuilt with a large number of elementary cells to describe theactivity of a given system The other is a lumped-parameterapproach in which neural populations are modeled as non-linear oscillators Neural mass models are based on the latterapproach These models comprise macrocolumns or corticalareas and represent themean activity of the whole populationby using one or two state variables It is seldom tractable tomodel EEG signals at the neuronal level due to the complexityof real neural networks The use of neural mass models hasbeen the preferred approach since 1970s Neural mass modelsoriginated from the seminal work of Lopes da Silva et al foralpha rhythm generation [1] and redesigned by Jansen andRit to represent the generation of evoked potentials in thevisual cortex [2] The dynamical analysis [3ndash6] and control[7ndash9] of the neural mass models have been widely studiedover the years Despite the existence of these neural massmodels for simulating distinct rhythms in EEG signals neuralactivity is always measured through observing just a singlevariable such as voltage A combination of noise in neuronsand amplifiers aswell as uncertainties in recording equipmentleads to uncertainty of the measurement The observationof states therefore plays significant roles in neuroscientificstudies for better understanding of the human brain [10]

In general neural mass models can be expressed asnonlinear systems of Lurrsquoe type [11] Observer design fornonlinear systems of Lurrsquoe type [12ndash14] and the neural massmodels [15] has been widely investigated over the years Wehere introduce a newmethod of designing a robust nonlinearobserver for the neural mass modelsThe Lurrsquoe system theoryand new tools in linear matrix inequality (LMI) method[16] are used to obtain the new reformulation We shouldmention that this new reformulation takes input uncertaintyand measurement noise into account The superiority of theproposedmethod is demonstrated in the last section which isdevoted to numerical comparisonsNotation The identity matrix is denoted by 119868 The symmetricblock component of a symmetric matrix is denoted by ⋆ Thevector norm is denoted by | sdot | The 119871

2norm is denoted by

sdot 2 The set of positive real numbers is denoted by R

+

2 Problem Formulation

Let us consider a class of neural mass models that can beformulated as the following mathematical structure

= 119860119909 + 119861119891 (119867119909) + 1198611119906

119911 = 119862119909 + 119863119908

(1)

where 119909 isin R119899 is the state vector 119906 isin R119901 is the input 119911 isin R119902

is the measurement output 119908 isin R119904 is the measurementnoise 119860 isin R119899times119899 119861 isin R119899times119898 119861

1isin R119899times119901 119862 isin R119902times119899

119867 isin R119898times119899 and 119863 isin R119902times119904 are constant matrices and

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 215943 5 pageshttpdxdoiorg1011552014215943

2 The Scientific World Journal

119891(sdot) R119899 rarr R119898 is a memoryless nonlinear vector valuedfunction which is continuously differentiable on R119898 Eachentry of the state-dependent nonlinearity119891(119867119909) is a functionof a linear combination of the states

119891119897= 119891119897(

119899

sum

119895=1

ℎ119897119895119909119895) = 119891

119897(ℎ119897119909) 119897 = 1 2 119898 (2)

where ℎ119897= [ℎ1198971 sdot sdot sdot ℎ

119897119899] It satisfies certain slope-restrictedcondition

0 le 1198911015840

119897(120590) le 120575

119897 forall120590 isin R 119897 = 1 2 119898 (3)

where 120575119897

ge 0 The models in David and Friston [4]Goodfellow et al [6] Jansen and Rit [2] and Wendling et al[3] can all be expressed as the form of (1) Let us construct thefollowing observer for plant (1)

119909 = 119860119909 + 119861119891 [119867119909 + 119870 (119862119909 minus 119911)] + 119871 (119862119909 minus 119911) + 1198611(119906 + 119889)

(4)

where 119909 is the estimation of state 119889 isin R119901 is the disturbanceof input and 119870 isin R119898times119902 119871 isin R119899times119902 are the observer matricesto be designed Defining the observer error as 119890 = 119909 minus 119909 thedynamics of it are governed by

119890 = (119860 + 119871119862) 119890 + 1198611119889 minus 119871119863119908 + 119861120578 (119890 119905) (5)

where 120578(119890 119905) = 119891(119881) minus 119891(119880) 119881 = 119867119909 + 119870(119862119909 minus 119911) and119880 = 119867119909 Note from (3) that each entry of the nonlinearity120578(119890 119905) satisfies

0 le120578119897(119890 119905)

V119897minus 119906119897

=119891119897(V119897) minus 119891119897(119906119897)

V119897minus 119906119897

le 120575119897 forall119890 isin R

forall119905 isin R+ 119897 = 1 2 119898

(6)

The observer design for (1) consists in finding observermatrices 119870 and 119871 such that the observer error 119890 satisfies thefollowing property for all 119905 ge 0

1198902 le 12058110038161003816100381610038161198900

1003816100381610038161003816 + 1205881198891198892 + 120588

1199081199082 (7)

where scalars 120581 gt 0 120588119889ge 0 and 120588

119908ge 0 The disturbance

gains from 119889 and 119908 to 119890 are 120588119889and 120588119908

3 Main Results

Theorem 1 Consider plant (1) and observer (4) Under theslope restrictions (3) if there exist a matrix 119875 = 119875

119879gt 0 a

diagonal matrix 119872 = diag(1198981 119898

119898) ge 0 matrices 119870 and

119871 nonsingular matrices 119866 and 119865 with appropriate dimensionsand scalar constants 120576 gt 0 120583

119908ge 0 and 120583

119889ge 0 such that

[[[[[[[[[[[[

[

119866 + 119866119879

Γ1

119866119861 1198661198611

minus119866119871119863

⋆ Γ2

Γ3

1198651198611

minus119865119871119863

⋆ ⋆ minus119872 0 minus1

2Δ119872119870119863

⋆ ⋆ ⋆ minus120583119889119868 0

⋆ ⋆ ⋆ ⋆ minus120583119908119868

]]]]]]]]]]]]

]

le 0

Γ1= 119866 (119860 + 119871119862) + 119875 minus 119865

119879

Γ2= 119865 (119860 + 119871119862) + (119860 + 119871119862)

119879119865119879+ 120576119868

Γ3= 119865119861 +

1

2(119867 + 119870119862)

119879119872Δ

(8)

then the observer error 119890 satisfies (7) for all 119905 ge 0 where Δ =

diag(1205751 120575

119898) 120581 = radic120582max(119875)120576 120588119889 = radic120583

119889120576 and 120588

119908=

radic120583119908120576

Proof The inequality (8) can be written as

Ω + 1198811[119866

119865]1198812+ 119881119879

2[119866

119865]

119879

119881119879

1le 0 (9)

where

Ω =

[[[[[[[[[[[[[

[

0 119875 0 0 0

⋆ 1205761198681

2(119867 + 119870119862)

119879119872Δ 0 0


2Δ119872119870119863

⋆ ⋆ ⋆ minus120583119889119868 0

⋆ ⋆ ⋆ ⋆ minus120583119908119868

]]]]]]]]]]]]]

]

1198811= [

119868 0 0 0 0

0 119868 0 0 0]

119879

1198812= [minus119868 119860 + 119871119862 119861 119861

1minus119871119863]

(10)

By using the well-known projection lemma in LMI method[16] (9) can be transformed into

[[[[[[[[[

[

Π1

Π2

1198751198611

minus119875119871119863

⋆ minus119872 0 minus1

2Δ119872119870119863

⋆ ⋆ minus120583119889119868 0

⋆ ⋆ ⋆ minus120583119908119868

]]]]]]]]]

]

le 0 (11)

where

Π1= (119860 + 119871119862)

119879119875 + 119875 (119860 + 119871119862) + 120576119868

Π2= 119875119861 +

1

2(119867 + 119870119862)

119879119872Δ

(12)

The Scientific World Journal 3

The derivative of 119881(119890) = 119890119879119875119890 is given by

= 119890119879119875119890 + 119890

119879119875 119890 le 119890

119879119875119890 + 119890

119879119875 119890

minus

119898

sum

119897=1

119898119897120578119897(119890 119905) [120578

119897(119890 119905) minus 120575

119897(V119897minus 119906119897)]

(13)

Applying (11) we have

le minus120576119890119879119890 + 120583119889119889119879119889 + 120583119908119908119879119908 (14)

from which it follows that

1198902le radic

120582max(119875)

120576

100381610038161003816100381611989001003816100381610038161003816 +

radic120583119889

1205761198892 + radic

120583119908

1205761199082

(15)

Hence (7) results from 120581 = radic120582max(119875)120576 120588119889 = radic120583119889120576 and

120588119908= radic120583119908120576

Theorem 1 shows that the observer design for (1) consistsin finding observer matrices 119870 and 119871 to satisfy (8) witha symmetric matrix 119875 gt 0 a diagonal matrix 119872 ge 0nonsingularmatrices119866119865 and scalar constants 120576 gt 0 120583

119908ge 0

and 120583119889ge 0 The feasible solution of (8) can be obtained by

solving the following optimization problem

minmax 120583119908 120583119889

st (8) 119875 gt 0 119872 ge 0 120576 gt 0 120583119908ge 0 120583

119889ge 0

(16)

Efficient numerical tools such as YALMIP in MATLAB areavailable for this task Once the values of 120583

119908and 120583

119889are

computed the disturbance gains 120588119908

and 120588119889can also be

derived When no input uncertainty and measurement noiseare taken into account Theorem 1 is simplified as follows

Theorem 2 Consider plant (1) and observer (4) with 119889 =

0 and 119908 = 0 Under the slope restrictions (3) if thereexist a matrix 119875 = 119875

119879gt 0 a diagonal matrix 119872 =

diag(1198981 119898

119898) ge 0 matrices119870 119871 nonsingular matrices 119866

119865 and scalar constants 120576 gt 0 120583119908ge 0 such that

[[[[

[

119866 + 119866119879

Γ1

119866119861

⋆ Γ2

Γ3

⋆ ⋆ minus119872

]]]]

]

le 0 (17)

where Γ1 Γ2 and Γ

3are defined as Theorem 1 then the origin

of the observer error system (5) is globally exponentially stable

4 Simulations

Let us consider a neural mass model developed by Jansenand Rit [2] This type of single cortical column model withaltered parameters is able to generate realistic patterns suchas alpha rhythms and epileptiform spikes in EEG It can beformulated as the form of (1) with the state vector 119909 =

[11990911199092119909311990941199095119909611990971199098]119879 where 119909

119894(119894 = 1 3 5 7) are

the mean membrane postsynaptic potentials and 119909119895(119895 =

2 4 6 8) are their time derivatives The input 119906 is the afferentinfluence from neighbouring or more distant columns and ismodeled by a Gaussian white noise with mean value 90 andstandard deviation 30The output 119911 is the EEGmeasurementavailable to the observer The system matrices are as follows

119860 = diag (1198601 119860

4)

119860119894= [

[

0 1

minus1205812

119894minus2120581119894

]

]

1205811= 1205812= 119886 120581

3= 119887

1205814= 119886119889

1198611= [0 0 0 120579

119886119886 0 0 0 0]

119879

119862 = [0 0 1 0 minus1 0 0 0] 119863 = 1

119861 =

[[[[

[

0 120579119886119886 0 0 0 0 0 120579

119886119886119889

0 0 0 12057911988611988611986220 0 0 0

0 0 0 0 0 12057911988711988711986240 0

]]]]

]

119879

119867 =

[[[[

[

0 0 1 0 minus1 0 0 0

11986210 0 0 0 0 0 0

11986230 0 0 0 0 0 0

]]]]

]

119891 (119867119909) = [119878 (1199093 minus 1199095) 119878 (119862

11199091) 119878 (119862

31199091)]119879

119878 (V) =21198900

1 + 119890119903(V0minusV)

(18)

The function 119878(sdot) satisfies (3) with 120575119897

= (12)1198900119903 (119897 =

1 2 3) All values of the constants in the model are set on aphysiological interpretation basis which can be found in [2]The standard values of these constants are given anatomicallyas

120579119886= 325mV 120579

119887= 22mV 119886 = 100 sminus1 119887 = 50 sminus1

V0= 6mV 119890

0= 25 sminus1

119903 = 056mVminus1 119886119889= 33 sminus1 119862

1= 135

1198622= 108 119862

3= 3375 119862

4= 3375

(19)

We design the robust nonlinear observer (4) for the neuralmass model The performance of the observer obtained fromTheorem 1 is presented in what follows Input disturbance119889 sim 119873(0 01

2) and measurement noise 119908 sim 119873(0 09

2) are

introduced in the design of robust nonlinear observer For therobust nonlinear observer we solve the optimization problem(16) to obtain119870 and 119871 The computed disturbance gains 120588

119908=

596 and 120588119889= 353 are derived by using the YALMIP toolbox

in MATLAB They are much less than the values given in[15] In the following simulations the initial states of theneural mass model and the observer are chosen as 119909(0) =

[1 05 1 05 1 05 1 05]119879 and 119909(0) = [0 0 0 0 0 0 0 0]

119879respectively Figure 1 presents the time evolutions of the states


0 05 1 15 2minus05

0

05

1

15

18 19 2005

01

015

0 05 1 15 2minus40

minus30

minus20

minus10

0

10

18 19 2minus2

minus1

0

1

2

0 05 1 15 2minus20

minus10

0

10

20

30

18 19 222

23

24

25

26

0 05 1 15 2minus500

0

500

1000

18 19 2minus400minus200

0200400

0 05 1 15 2minus10

0

10

20

30

40

18 19 210

15

20

0 05 1 15 2minus1000

minus500

0

500

1000

1500

18 19 2minus200

0

200

x1

and

its es

timat

ion

x2

and

its es

timat

ion

x3

and

its es

timat

ion

x4

and

its es

timat

ion

x5

and

its es

timat

ion

x6

and

its es

timat

ion

t (s)

t (s)

t (s) t (s)

t (s)

t (s)

0 05 1 15 2minus05

0

05

1

15

18 19 203

032

034

036

0 05 1 15 2minus15

minus10

minus5

0

5

10

18 19 2minus2

0

2

x7

and

its es

timat

ion

x8

and

its es

timat

ion

t (s) t (s)

Figure 1 The time evolutions of the states 1199091ndash1199098and their estimations


1199091ndash1199098(black lines) and their estimations that is the states

of observer (4) proposed in this study (red lines) and in [15](blue line) Insets are given to show the zoom-in on dataFigure 1 shows that the states of observer (4) obtained fromTheorem 1 do converge to a neighbourhoodof the states of theneural mass model It also shows that the observer proposedin this study performs better than that proposed in [15]

5 Conclusions

We have designed a robust nonlinear observer for a classof neural mass models by using the Lurrsquoe system theoryand the projection lemma The resulting observer inhibitsinput uncertainty and measurement noise We apply thisobserver to the neural mass model that generates alpharhythms to estimate the mean membrane potential of neuralpopulations from the EEG measurement We show that theproposed observer performs better than some existing onesThe proposed method can also be applied to other types ofneural models that have the typical structure of Lurrsquoe systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China (61004050 61172095)

References

[1] F H Lopes da Silva A Hoeks H Smits and L H ZetterbergldquoModel of brain rhythmic activityrdquo Biological Cybernetics vol15 no 1 pp 27ndash37 1974

[2] B H Jansen and V G Rit ldquoElectroencephalogram and visualevoked potential generation in a mathematical model of cou-pled cortical columnsrdquo Biological Cybernetics vol 73 no 4 pp357ndash366 1995

[3] F Wendling J J Bellanger F Bartolomei and P ChauvelldquoRelevance of nonlinear lumped-parameter models in theanalysis of depth-EEG epileptic signalsrdquo Biological Cyberneticsvol 83 no 4 pp 367ndash378 2000

[4] O David andK J Friston ldquoA neuralmassmodel forMEGEEGcoupling and neuronal dynamicsrdquo NeuroImage vol 20 no 3pp 1743ndash1755 2003

[5] A Babajani and H Soltanian-Zadeh ldquoIntegrated MEGEEGand fMRI model based on neural massesrdquo IEEE Transactionson Biomedical Engineering vol 53 no 9 pp 1794ndash1801 2006

[6] M Goodfellow K Schindler and G Baier ldquoSelf-organisedtransients in a neural mass model of epileptogenic tissuedynamicsrdquo NeuroImage vol 59 no 3 pp 2644ndash2660 2012

[7] X Liu H J Liu Y G Tang and Q Gao ldquoFuzzy PID controlof epileptiform spikes in a neural mass modelrdquo NonlinearDynamics vol 71 no 1-2 pp 13ndash23 2013

[8] X Liu Q Gao B W Ma J J Du and W J Ren ldquoAnalysis andcontrol of epileptiform spikes in a class of neural mass modelsrdquoJournal of Applied Mathematics vol 2013 Article ID 792507 11pages 2013

[9] X Liu and Q Gao ldquoParameter estimation and control fora neural mass model based on the unscented Kalman filterrdquoPhysical Review E vol 88 no 4 Article ID 042905 2013

[10] S J Schiff Computational Neuroscience Neural Control Engi-neering The Emerging Intersection Between Control Theory andNeuroscience The MIT Press London UK 2011

[11] G A Leonov D V Ponomarenko and V B Smirnova Fre-quency-Domain Methods For Nonlinear Analysis Theory andApplications World Scientific Singapore 1996

[12] M Arcak and P Kokotovic ldquoNonlinear observers a circlecriterion design and robustness analysisrdquo Automatica vol 37no 12 pp 1923ndash1930 2001

[13] X Fan and M Arcak ldquoObserver design for systems withmultivariable monotone nonlinearitiesrdquo Systems and ControlLetters vol 50 no 4 pp 319ndash330 2003

[14] A Zemouche and M Boutayeb ldquoA unified 119867infin

adaptiveobserver synthesis method for a class of systems with bothLipschitz and monotone nonlinearitiesrdquo Systems and ControlLetters vol 58 no 4 pp 282ndash288 2009

[15] M Chong R Postoyan D Nesic L Kuhlmann and AVarsavsky ldquoA robust circle criterion observer with applicationto neural mass modelsrdquo Automatica vol 48 no 11 pp 2986ndash2989 2012

[16] S Boyd L E Ghaoui E Feron and V Balakrishnan Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia USA 1994

Submit your manuscripts athttpwwwhindawicom

Neurology Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Alzheimerrsquos DiseaseHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


BioMed Research International


Research and TreatmentSchizophrenia

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Neural Plasticity


Parkinsonrsquos Disease


Research and TreatmentAutism

Sleep DisordersHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Neuroscience Journal

Epilepsy Research and TreatmentHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Psychiatry Journal


Computational and Mathematical Methods in Medicine

Depression Research and TreatmentHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Brain ScienceInternational Journal of

StrokeResearch and TreatmentHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Neurodegenerative Diseases


Journal of

Cardiovascular Psychiatry and NeurologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


119891(sdot) R119899 rarr R119898 is a memoryless nonlinear vector valuedfunction which is continuously differentiable on R119898 Eachentry of the state-dependent nonlinearity119891(119867119909) is a functionof a linear combination of the states

119891119897= 119891119897(

119899

sum

119895=1

ℎ119897119895119909119895) = 119891

119897(ℎ119897119909) 119897 = 1 2 119898 (2)

where ℎ119897= [ℎ1198971 sdot sdot sdot ℎ

119897119899] It satisfies certain slope-restrictedcondition

0 le 1198911015840

119897(120590) le 120575

119897 forall120590 isin R 119897 = 1 2 119898 (3)

where 120575119897

ge 0 The models in David and Friston [4]Goodfellow et al [6] Jansen and Rit [2] and Wendling et al[3] can all be expressed as the form of (1) Let us construct thefollowing observer for plant (1)

119909 = 119860119909 + 119861119891 [119867119909 + 119870 (119862119909 minus 119911)] + 119871 (119862119909 minus 119911) + 1198611(119906 + 119889)

(4)

where 119909 is the estimation of state 119889 isin R119901 is the disturbanceof input and 119870 isin R119898times119902 119871 isin R119899times119902 are the observer matricesto be designed Defining the observer error as 119890 = 119909 minus 119909 thedynamics of it are governed by

119890 = (119860 + 119871119862) 119890 + 1198611119889 minus 119871119863119908 + 119861120578 (119890 119905) (5)

where 120578(119890 119905) = 119891(119881) minus 119891(119880) 119881 = 119867119909 + 119870(119862119909 minus 119911) and119880 = 119867119909 Note from (3) that each entry of the nonlinearity120578(119890 119905) satisfies

0 le120578119897(119890 119905)

V119897minus 119906119897

=119891119897(V119897) minus 119891119897(119906119897)

V119897minus 119906119897

le 120575119897 forall119890 isin R

forall119905 isin R+ 119897 = 1 2 119898

(6)

The observer design for (1) consists in finding observermatrices 119870 and 119871 such that the observer error 119890 satisfies thefollowing property for all 119905 ge 0

1198902 le 12058110038161003816100381610038161198900

1003816100381610038161003816 + 1205881198891198892 + 120588

1199081199082 (7)

where scalars 120581 gt 0 120588119889ge 0 and 120588

119908ge 0 The disturbance

gains from 119889 and 119908 to 119890 are 120588119889and 120588119908

3 Main Results

Theorem 1 Consider plant (1) and observer (4) Under theslope restrictions (3) if there exist a matrix 119875 = 119875

119879gt 0 a

diagonal matrix 119872 = diag(1198981 119898

119898) ge 0 matrices 119870 and

119871 nonsingular matrices 119866 and 119865 with appropriate dimensionsand scalar constants 120576 gt 0 120583

119908ge 0 and 120583

119889ge 0 such that

[[[[[[[[[[[[

[

119866 + 119866119879

Γ1

119866119861 1198661198611

minus119866119871119863

⋆ Γ2

Γ3

1198651198611

minus119865119871119863


2Δ119872119870119863

⋆ ⋆ ⋆ minus120583119889119868 0

⋆ ⋆ ⋆ ⋆ minus120583119908119868

]]]]]]]]]]]]

]

le 0

Γ1= 119866 (119860 + 119871119862) + 119875 minus 119865

119879

Γ2= 119865 (119860 + 119871119862) + (119860 + 119871119862)

119879119865119879+ 120576119868

Γ3= 119865119861 +

1

2(119867 + 119870119862)

119879119872Δ

(8)

then the observer error 119890 satisfies (7) for all 119905 ge 0 where Δ =

diag(1205751 120575

119898) 120581 = radic120582max(119875)120576 120588119889 = radic120583

119889120576 and 120588

119908=

radic120583119908120576

Proof The inequality (8) can be written as

Ω + 1198811[119866

119865]1198812+ 119881119879

2[119866

119865]

119879

119881119879

1le 0 (9)

where

Ω =

[[[[[[[[[[[[[

[

0 119875 0 0 0

⋆ 1205761198681

2(119867 + 119870119862)

119879119872Δ 0 0


2Δ119872119870119863

⋆ ⋆ ⋆ minus120583119889119868 0

⋆ ⋆ ⋆ ⋆ minus120583119908119868

]]]]]]]]]]]]]

]

1198811= [

119868 0 0 0 0

0 119868 0 0 0]

119879

1198812= [minus119868 119860 + 119871119862 119861 119861

1minus119871119863]

(10)

By using the well-known projection lemma in LMI method[16] (9) can be transformed into

[[[[[[[[[

[

Π1

Π2

1198751198611

minus119875119871119863

⋆ minus119872 0 minus1

2Δ119872119870119863

⋆ ⋆ minus120583119889119868 0

⋆ ⋆ ⋆ minus120583119908119868

]]]]]]]]]

]

le 0 (11)

where

Π1= (119860 + 119871119862)

119879119875 + 119875 (119860 + 119871119862) + 120576119868

Π2= 119875119861 +

1

2(119867 + 119870119862)

119879119872Δ

(12)



= 119890119879119875119890 + 119890

119879119875 119890 le 119890

119879119875119890 + 119890

119879119875 119890

minus

119898

sum

119897=1

119898119897120578119897(119890 119905) [120578

119897(119890 119905) minus 120575

119897(V119897minus 119906119897)]

(13)


le minus120576119890119879119890 + 120583119889119889119879119889 + 120583119908119908119879119908 (14)


1198902le radic

120582max(119875)

120576

100381610038161003816100381611989001003816100381610038161003816 +

radic120583119889

1205761198892 + radic

120583119908

1205761199082

(15)


120588119908= radic120583119908120576


119908ge 0



minmax 120583119908 120583119889

st (8) 119875 gt 0 119872 ge 0 120576 gt 0 120583119908ge 0 120583

119889ge 0

(16)


119908and 120583

119889are







diag(1198981 119898



[[[[

[

119866 + 119866119879

Γ1

119866119861

⋆ Γ2

Γ3

⋆ ⋆ minus119872

]]]]

]

le 0 (17)




4 Simulations


[11990911199092119909311990941199095119909611990971199098]119879 where 119909

119894(119894 = 1 3 5 7) are



119860 = diag (1198601 119860

4)

119860119894= [

[

0 1

minus1205812

119894minus2120581119894

]

]

1205811= 1205812= 119886 120581

3= 119887

1205814= 119886119889

1198611= [0 0 0 120579

119886119886 0 0 0 0]

119879

119862 = [0 0 1 0 minus1 0 0 0] 119863 = 1

119861 =

[[[[

[

0 120579119886119886 0 0 0 0 0 120579

119886119886119889

0 0 0 12057911988611988611986220 0 0 0

0 0 0 0 0 12057911988711988711986240 0

]]]]

]

119879

119867 =

[[[[

[

0 0 1 0 minus1 0 0 0

11986210 0 0 0 0 0 0

11986230 0 0 0 0 0 0

]]]]

]

119891 (119867119909) = [119878 (1199093 minus 1199095) 119878 (119862

11199091) 119878 (119862

31199091)]119879

119878 (V) =21198900

1 + 119890119903(V0minusV)

(18)


= (12)1198900119903 (119897 =


120579119886= 325mV 120579

119887= 22mV 119886 = 100 sminus1 119887 = 50 sminus1

V0= 6mV 119890

0= 25 sminus1

119903 = 056mVminus1 119886119889= 33 sminus1 119862

1= 135

1198622= 108 119862

3= 3375 119862

4= 3375

(19)



2) are


119908=



[1 05 1 05 1 05 1 05]119879 and 119909(0) = [0 0 0 0 0 0 0 0]



0 05 1 15 2minus05

0

05

1

15

18 19 2005

01

015

0 05 1 15 2minus40

minus30

minus20

minus10

0

10

18 19 2minus2

minus1

0

1

2

0 05 1 15 2minus20

minus10

0

10

20

30

18 19 222

23

24

25

26

0 05 1 15 2minus500

0

500

1000


0200400

0 05 1 15 2minus10

0

10

20

30

40

18 19 210

15

20

0 05 1 15 2minus1000

minus500

0

500

1000

1500

18 19 2minus200

0

200

x1

and

its es

timat

ion

x2

and

its es

timat

ion

x3

and

its es

timat

ion

x4

and

its es

timat

ion

x5

and

its es

timat

ion

x6

and

its es

timat

ion

t (s)

t (s)

t (s) t (s)

t (s)

t (s)

0 05 1 15 2minus05

0

05

1

15

18 19 203

032

034

036

0 05 1 15 2minus15

minus10

minus5

0

5

10

18 19 2minus2

0

2

x7

and

its es

timat

ion

x8

and

its es

timat

ion

t (s) t (s)





5 Conclusions




Acknowledgment


References






























Neural Plasticity










Psychiatry Journal









Journal of




= 119890119879119875119890 + 119890

119879119875 119890 le 119890

119879119875119890 + 119890

119879119875 119890

minus

119898

sum

119897=1

119898119897120578119897(119890 119905) [120578

119897(119890 119905) minus 120575

119897(V119897minus 119906119897)]

(13)


le minus120576119890119879119890 + 120583119889119889119879119889 + 120583119908119908119879119908 (14)


1198902le radic

120582max(119875)

120576

100381610038161003816100381611989001003816100381610038161003816 +

radic120583119889

1205761198892 + radic

120583119908

1205761199082

(15)


120588119908= radic120583119908120576


119908ge 0



minmax 120583119908 120583119889

st (8) 119875 gt 0 119872 ge 0 120576 gt 0 120583119908ge 0 120583

119889ge 0

(16)


119908and 120583

119889are







diag(1198981 119898



[[[[

[

119866 + 119866119879

Γ1

119866119861

⋆ Γ2

Γ3

⋆ ⋆ minus119872

]]]]

]

le 0 (17)




4 Simulations


[11990911199092119909311990941199095119909611990971199098]119879 where 119909

119894(119894 = 1 3 5 7) are



119860 = diag (1198601 119860

4)

119860119894= [

[

0 1

minus1205812

119894minus2120581119894

]

]

1205811= 1205812= 119886 120581

3= 119887

1205814= 119886119889

1198611= [0 0 0 120579

119886119886 0 0 0 0]

119879

119862 = [0 0 1 0 minus1 0 0 0] 119863 = 1

119861 =

[[[[

[

0 120579119886119886 0 0 0 0 0 120579

119886119886119889

0 0 0 12057911988611988611986220 0 0 0

0 0 0 0 0 12057911988711988711986240 0

]]]]

]

119879

119867 =

[[[[

[

0 0 1 0 minus1 0 0 0

11986210 0 0 0 0 0 0

11986230 0 0 0 0 0 0

]]]]

]

119891 (119867119909) = [119878 (1199093 minus 1199095) 119878 (119862

11199091) 119878 (119862

31199091)]119879

119878 (V) =21198900

1 + 119890119903(V0minusV)

(18)


= (12)1198900119903 (119897 =


120579119886= 325mV 120579

119887= 22mV 119886 = 100 sminus1 119887 = 50 sminus1

V0= 6mV 119890

0= 25 sminus1

119903 = 056mVminus1 119886119889= 33 sminus1 119862

1= 135

1198622= 108 119862

3= 3375 119862

4= 3375

(19)



2) are


119908=



[1 05 1 05 1 05 1 05]119879 and 119909(0) = [0 0 0 0 0 0 0 0]



0 05 1 15 2minus05

0

05

1

15

18 19 2005

01

015

0 05 1 15 2minus40

minus30

minus20

minus10

0

10

18 19 2minus2

minus1

0

1

2

0 05 1 15 2minus20

minus10

0

10

20

30

18 19 222

23

24

25

26

0 05 1 15 2minus500

0

500

1000


0200400

0 05 1 15 2minus10

0

10

20

30

40

18 19 210

15

20

0 05 1 15 2minus1000

minus500

0

500

1000

1500

18 19 2minus200

0

200

x1

and

its es

timat

ion

x2

and

its es

timat

ion

x3

and

its es

timat

ion

x4

and

its es

timat

ion

x5

and

its es

timat

ion

x6

and

its es

timat

ion

t (s)

t (s)

t (s) t (s)

t (s)

t (s)

0 05 1 15 2minus05

0

05

1

15

18 19 203

032

034

036

0 05 1 15 2minus15

minus10

minus5

0

5

10

18 19 2minus2

0

2

x7

and

its es

timat

ion

x8

and

its es

timat

ion

t (s) t (s)





5 Conclusions




Acknowledgment


References






























Neural Plasticity










Psychiatry Journal









Journal of



0 05 1 15 2minus05

0

05

1

15

18 19 2005

01

015

0 05 1 15 2minus40

minus30

minus20

minus10

0

10

18 19 2minus2

minus1

0

1

2

0 05 1 15 2minus20

minus10

0

10

20

30

18 19 222

23

24

25

26

0 05 1 15 2minus500

0

500

1000


0200400

0 05 1 15 2minus10

0

10

20

30

40

18 19 210

15

20

0 05 1 15 2minus1000

minus500

0

500

1000

1500

18 19 2minus200

0

200

x1

and

its es

timat

ion

x2

and

its es

timat

ion

x3

and

its es

timat

ion

x4

and

its es

timat

ion

x5

and

its es

timat

ion

x6

and

its es

timat

ion

t (s)

t (s)

t (s) t (s)

t (s)

t (s)

0 05 1 15 2minus05

0

05

1

15

18 19 203

032

034

036

0 05 1 15 2minus15

minus10

minus5

0

5

10

18 19 2minus2

0

2

x7

and

its es

timat

ion

x8

and

its es

timat

ion

t (s) t (s)





5 Conclusions




Acknowledgment


References






























Neural Plasticity










Psychiatry Journal









Journal of





5 Conclusions




Acknowledgment


References






























Neural Plasticity










Psychiatry Journal









Journal of














Neural Plasticity










Psychiatry Journal









Journal of


research article a robust nonlinear observer for a class...

Documents