research article controlling the stochastic sensitivity in...

Research ArticleControlling the Stochastic Sensitivity in NonlinearDiscrete-Time Systems with Incomplete Information

Lev Ryashko and Irina Bashkirtseva

Ural Federal University Lenina 51 Ekaterinburg 620000 Russia

Correspondence should be addressed to Irina Bashkirtseva irinabashkirtsevaurfuru

Received 19 April 2015 Revised 13 September 2015 Accepted 16 September 2015

Academic Editor Zhan Zhou

Copyright copy 2015 L Ryashko and I Bashkirtseva This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

For stochastic nonlinear discrete-time system with incomplete information a problem of the stabilization of equilibrium isconsidered Our approach uses a regulator which synthesizes the required stochastic sensitivity Mathematically this problem isreduced to the solution of some quadratic matrix equations A description of attainability sets and algorithms for regulators designis given The general results are applied to the suppression of unwanted large-amplitude oscillations around the equilibria of thestochastically forced Verhulst model with noisy observations

1 Introduction

Controlling of the complex systems in nature and societyis a challenging and fundamental problem of the modernmathematical theory of nonlinear dynamics and engineeringDiscrete dynamic models because of widely used computer-oriented technologies attract attention of many researchers[1 2] Even in simple discrete models due to nonlinearitya variety of dynamic regimes both regular and chaotic isobserved [3ndash5] An interplay of nonlinearity and stochasticitycan generate new unexpected phenomena [6ndash10]

A lot of nonlinear systems operate in zones of stochastictransitions from order to chaos After the pioneering work[11] a problem of controlling chaos is extensively studied[12ndash14] Most of the reported results are based on the directnumerical simulation A detailed theoretical description ofstochastic attractor is given by the stationary probabilis-tic density function For discrete systems this function isgoverned by Frobenius-Perron equation [15] Unfortunatelyan analytical solution of this equation is possible only invery special cases so a development of the asymptoticapproximations is a highly relevant area of research

For the constructive analysis of the stochastic attractorsof nonlinear discrete-time dynamical systems a stochasticsensitivity functions technique was elaborated [16] This

technique was applied to the analysis on noise-inducedintermittency [17] and neuron excitability [18] On the baseof this technique a new approach for the solution of controlproblems in stochastic discrete-time systems was suggestedin [19] In these studies it was supposed that the completeinformation about current state of the controlled system isknown However in many practical situations the systemdata are far from complete For example only some coor-dinates of the system state are observable and moreoverthese observations contain stochastic errors So the control ofstochastic systems with incomplete information is an urgentresearch domain [20ndash22]

In present paper we further develop a theory for thesynthesis of the stochastic sensitivity for the equilibria ina randomly forced control discrete system with incom-plete information Mathematically presence of noise in theobservations leads to a new algebraic analysis of quadraticmatrix equations In Section 2 we introduce the stochasticsensitivity matrix as a basic probabilistic characteristics forthe randomly forced equilibria A problem of the synthesisof this matrix is considered A important notion of theattainability is discussed here A problem of the stochasticsensitivity matrix synthesis is reduced to the analysis of thecorresponding quadratic matrix equation Results of thistheoretical analysis in the general multidimensional case are

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 658048 5 pageshttpdxdoiorg1011552015658048

2 Discrete Dynamics in Nature and Society

presented in a Theorem This Theorem gives a description ofattainability sets and algorithms for regulators design

One-dimensional case is discussed in details in Section 3In Section 4 we apply the results to the suppression ofunwanted large-amplitude oscillations around the equilib-ria of the stochastically forced Verhulst model with noisyobservations We show that our regulator can be used for thesuppression of chaos

2 Synthesis of Stochastic Sensitivity

Consider a nonlinear controlled discrete-time stochasticsystem

119909119905+1

= 119891 (119909119905 119906119905) + 120576120590 (119909

119905) 120585119905 (1)

where 119909 119891 isin 119877119899 119906 isin 119877

119897 120590 isin 119877119899times119898 and 119906 is a control input

Here 120585119905

isin 119877119898 is an uncorrelated random sequence with

parameters E120585119905

= 0 and E120585119905120585⊤

119905= 119868 and 119868 is the identity

119898 times 119898-matrix and 120576 is a scalar parameter of noise intensityIt is supposed that the corresponding deterministic

uncontrolled system (1) (with 120576 = 0 and 119906 = 0 therein) has anequilibrium 119909 119909 = 119891(119909 0) Stability of 119909 is not assumed

In present paper we consider a case of incompleteinformation when the measurement vector 119910

119905is known only

119910119905= 119892 (119909

119905) + 120576120593 (119909

119905) 120578119905 (2)

where 119910 119892 isin 119877119897 120593 isin 119877

119897times119896 Here 120578119905isin 119877119896 is an uncorrelated

random sequence with parameters E120578119905= 0 and E120578

119905120578⊤

119905= 119868

and 119868 is the identity 119896 times 119896-matrixIn this circumstance we consider the following regulator

119906119905= 119870 [119910

119905minus 119892 (119909)] (3)

The dynamics of the closed-loop stochastic system (1) withthe regulator (3) using noisy observations (2) is governed bythe following system

119909119905+1

= 119891 (119909119905 119870 [119892 (119909

119905) + 120576120593 (119909

119905) 120578119905minus 119892 (119909)])

+ 120576120590 (119909119905) 120585119905

(4)

For the asymptotics 119911119905= lim120576rarr0

((119909120576

119905minus119909)120576) of the deviations

of solutions 119909120576

119905of system (4) from the equilibrium 119909 the

following stochastic system can be written

119911119905+1

= (119865 + 119861119870119862) 119911119905+ 119861119870120593120578

119905+ 120590120585119905 (5)

where

119865 =

120597119891

120597119909

(119909 0)

119861 =

120597119891

120597119906

(119909 0)

119862 =

120597119892

120597119909

(119909)

120593 = 120593 (119909)

120590 = 120590 (119909)

(6)

Due to the uncorrelatedness of random terms 120578119905and 120585119905 the

second moments matrix 119872119905

= E(119911119905119911⊤

119905) is governed by the

equation

119872119905+1

= (119865 + 119861119870119862)119872119905(119865 + 119861119870119862)

⊤+ 119861119870Φ119870

⊤119861⊤+ 119878 (7)

where Φ = 120593120593⊤ 119878 = 120590120590

⊤ A set of matrices 119870 that providean exponential stability to the equilibrium 119909 of the closeddeterministic system (4) (with 120576 = 0 therein) has thefollowing form

K = 119870 | 120588 (119865 + 119861119870119862) lt 1 (8)

where 120588(119860) is a spectral radius of the matrix 119860 We supposethat the set K is not empty

For any119870 isin K (7) has a unique stable stationary solution119882 satisfying the equation

119882 = (119865 + 119861119870119862)119882 (119865 + 119861119870119862)⊤+ 119861119870Φ119870

⊤119861⊤+ 119878 (9)

Thismatrix119882 is called the stochastic sensitivity matrix of theequilibrium 119909 for system (4)The stochastic sensitivitymatrix119882 approximates a limit behavior of the second momentsE(119909120576119905minus 119909)(119909

120576

119905minus 119909)⊤ for deviations of solutions 119909120576

119905from 119909

lim119905rarrinfin

E (119909120576

119905minus 119909) (119909

120576

119905minus 119909)⊤

asymp 1205762119882 (10)

So the matrix 119882 characterizes a dispersion of the stationarydistributed random states of system (4) around the equilib-rium 119909

For any 119870 isin K the regulator (3) forms a correspondingstochastic equilibrium of system (4) with the stochasticsensitivity matrix 119882

119870which is a solution of (9)

Consider further the following inverse problem

Problem of Stochastic Sensitivity Synthesis Let M be a set ofsymmetric and positive-definite 119899 times 119899-matrices Let 119882 isin Mbe some assigned matrix The problem is to find a feedbackmatrix119870 isin K of regulator (3) such that the equality119882

119870= 119882

holds Here 119882119870is a solution of (9)

In some cases this problem can be unsolvableThereforewe consider an important notion of the attainability

Definition 1 An element 119882 isin M is said to be attainable forsystem (4) if the equality 119882

119870= 119882 holds for some 119870 isin K

Definition 2 The set of all attainable elements

W = 119882 isin M | exist119870 isin K 119882119870

= 119882 (11)

is called the attainability set for system (4)

As it follows from (9) the attainability analysis is reducedto the study of solvability of the quadratic matrix equation

119861119870 (119862119882119862⊤+ Φ)119870

⊤119861⊤+ 119861119870119862119882119865

⊤+ 119865119882119862

⊤119870⊤119861⊤

+ 119865119882119865⊤+ 119878 minus 119882 = 0

(12)

Rewrite (12) with respect to a new unknown matrix

119871 = 119861119870 (13)

Discrete Dynamics in Nature and Society 3

in the following form

119871 (119862119882119862⊤+ Φ) 119871

⊤+ 119871119862119882119865

⊤+ 119865119882119862

⊤119871⊤+ 119865119882119865

⊤

+ 119878 minus 119882 = 0

(14)

Denote 119866(119882) = (119862119882119862⊤

+ Φ)12 Suppose that the matrix

119866(119882) is positive-definite (119866(119882) ≻ 0) A substitution 119873 =

119871119866(119882) transforms (14) into the following equation

(119873 + 1198651) (119873 + 119865

1)⊤

= 119877 (119882) (15)

where1198651= 119865119882119862

⊤119866minus1

(119882)

119877 (119882) = 119865119882119862⊤(119862119882119862

⊤+ Φ)

minus1

119862119882119865⊤minus 119865119882119865

⊤minus 119878

+ 119882

(16)

A necessary condition of (15) solvability is in the nonnegativedefiniteness of the matrix 119877(119882)

119877 (119882) = 119865119882119862⊤(119862119882119862

⊤+ Φ)

minus1

119862119882119865⊤minus 119865119882119865

⊤minus 119878

+ 119882 ⪰ 0

(17)

Let condition (17) be fulfilled Then quadratic equation (15)is equivalent to the linear equation

119873 + 1198651= 11987712

(119882) 119869 (18)

where 119869 is an arbitrary orthogonal 119899times119899-matrix It follows from(18) that the feedback matrix 119870 of the regulator (3) whichsynthesizes the stochastic sensitivitymatrix119882 satisfies to thelinear matrix equation

119861119870 = (11987712

(119882) 119869 minus 119865119882119862⊤119866minus1

(119882))119866minus1

(119882) (19)

In the following theorem we summarize our theoreticalresults

Theorem 3 Let noises in system (1) and observations (2) benonsingular (119878 ≻ 0 Φ ≻ 0)

(a) If the matrix 119861 is quadratic and nonsingular (rank119861 =

119899 = 119897) then

W = 119882 isin M | 119877 (119882) ⪰ 0 (20)

and for any matrix 119882 isin W (19) has a solution

119870 = 119861minus1

(11987712

(119882) 119869 minus 119865119882119862⊤119866minus1

(119882))119866minus1

(119882)

isin K

(21)

(b) If rank(119861) lt 119899 then

W = 119882 isin M |

119877 (119882) ⪰ 0 119875 (11987712

(119882) 119869 minus 119865119882119862⊤119866minus1

(119882)) = 0

(22)


119870 = 119861+(11987712

(119882) 119869 minus 119865119882119862⊤119866minus1

(119882))119866minus1

(119882) isin K (23)

Here 119869 is an arbitrary orthogonal 119899 times 119899-matrix 119875 = 119868 minus 119861119861+

is a projective matrix and a ldquo+rdquo sign means a pseudoinversion[23]

3 Controlling of One-DimensionalStochastic System

Consider one-dimensional discrete stochastic controlled sys-tem

119909119905+1

= 119891 (119909119905) + 119906 + 120576120590120585

119905(24)

with noisy observations

119910119905= 119909119905+ 120576120593120578

119905 (25)

Here 119909 119910 119906 are scalar variables of state output and controlinput 120585

119905 120578119905are uncorrelated random scalar sequences with

parameters E120585119905= 0 E1205852

119905= 1 E120578

119905= 0 and E1205782

119905= 1 and 120576 120590

120593 are scalar parameters of noise intensitiesIt is supposed that the corresponding deterministic

uncontrolled system (24) with 120576 = 0 and 119906 = 0 therein hasan equilibrium 119909 119909 = 119891(119909) In what follows we use theregulator

119906119905= 119896 (119910

119905minus 119909) (26)

For the synthesis of the assigned scalar stochastic sensitivity119908 of the equilibrium 119909 we apply theoretical results presentedabove

At first describe the attainability set for the consideredexampleThe function 119877(119882) from (17) has here the followingrepresentation

119877 (119908) =

11988621199082

119908 + 1205932minus 1198862119908 minus 120590

2+ 119908

=

1199082minus ((1198862minus 1) 120593

2+ 1205902)119908 minus 120590

21205932

119908 + 1205932

119886 = 1198911015840(119909)

(27)

The attainability condition 119877(119908) ge 0 (see Theorem 3 inSection 2) is equivalent to quadratic inequality

1199082minus ((1198862minus 1) 120593

2+ 1205902)119908 minus 120590

21205932ge 0 (28)

Thus all attainable values of 119908 have to satisfy the inequality

119908 ge 119908lowast=

1

2

[(1198862minus 1) 120593

2+ 1205902

+ radic((1198862minus 1) 120593

2+ 1205902)2

+ 412059021205932]

(29)

Note that the value 119908lowastis a minimal value of the stochastic

sensitivity that we can provide by this regulatorOur regulator will synthesize any assign stochastic sensi-

tivity 119908 ge 119908lowastif we will take (see Theorem 3 in Section 2) a

feedback coefficient as follows

119896 =

radic1199082minus ((1198862minus 1) 120593

2+ 1205902) 119908 minus 120590

21205932minus 119886119908

119908 + 1205932

(30)

Note that the optimal regulator synthesizing the minimalvalue of the stochastic sensitivity 119908

lowasthas the feedback coeffi-

cient

119896lowast= minus

119886119908lowast

119908lowast+ 1205932 (31)


3 3525120583

02

04

06

08

x

(a)

35

D

10minus2

10minus3

10minus4

325120583

(b)

0

minus1

minus2

minus3

32 34 363120583

Λ

(c)

Figure 1 Stochastic Verhulst model for 120576 = 001 120590 = 1 and 120593 = 01without control (red) and with optimal control (blue) (a) random states(b) dispersion (c) largest Lyapunov exponent

4 Example Controlling StochasticVerhulst System

Consider stochastically forced well-known Verhulst systemwith control and noisy observations

119909119905+1

= 120583119909119905(1 minus 119909

119905) + 119906 + 120576120590120585

119905

119910119905= 119909119905+ 120576120593120578

119905

(32)

The corresponding deterministic uncontrolled system (32)has a nontrivial equilibrium 119909 = 1 minus 1120583 This equilibriumis stable for 1 lt 120583 lt 3 and unstable for 3 lt 120583 lt 4

At first consider the influence of random disturbances forsystem (32) without control (119906 = 0) Under the stochastic dis-turbances for random states of this system some probabilisticdistribution is formed [24] In Figure 1(a) random states ofsystem (32) with 119906 = 0 for 120576 = 001 120590 = 1 calculated bydirect numerical simulation are plotted by red color

As the parameter 120583 crosses the bifurcation value 120583 = 3

from the left to right a dispersion of these random statessharply increases Let 119863(120583) be a mean square deviation ofrandom states from the equilibrium 119909(120583) This function isplotted in Figure 1(b) by red color

Consider now abilities of control 119906119905= 119896(119910

119905minus 119909) The aim

of the control is to stabilize the equilibrium 119909(120583) and providea small dispersion of random states in a wide range of theparameter 120583 For the solution of this problem we will use theoptimal regulator that minimizes a stochastic sensitivity ofthe equilibrium 119909(120583)

For the Verhulst system 119886(120583) = 2 minus 120583 and the minimalvalue 119908

lowastof the stochastic sensitivity (see (29)) that we can

provide by the regulator has the following explicit parametricrepresentation

119908lowast(120583) =

1

2

[(1205832minus 4120583 + 3) 120593

2+ 1205902

+ radic((1205832minus 4120583 + 3) 120593

2+ 1205902)2

+ 412059021205932]

(33)

For the optimal regulator that synthesizes this minimal value119908lowast(120583) the feedback coefficient

119896lowast=

(120583 minus 2)119908lowast(120583)

119908lowast(120583) + 120593

2 (34)

Results of the control by this regulator are shown in Figure 1by blue color As one can see in Figure 1(a) random states of


the controlled system are well localized near the equilibrium119909(120583) regardless of whether this equilibrium is stable ornot This optimal control provides the smaller and uniformdispersion (see Figure 1(b))

Along with the spatial description of the stochastic Ver-hulst system consider dynamics of the mutual arrangementof random trajectories in stochastic flows For quantitativedescription of this dynamics the largest Lyapunov exponentΛ is traditionally used [25] In Figure 1(c) plots of Λ(120583) areshown for uncontrolled (red) and controlled (blue) Verhulstsystem A change of Λ(120583) sign from minus to plus justifies atransition of the uncontrolled Verhulst system from order tochaos

As one can see the optimal regulator essentially decreasesvalues of Λ(120583) and provides their negativeness for the wholezone of the parameter 120583 So our regulator suppresses chaosand provides a structural stabilization of the system

5 Conclusion

In this paper a problem of the stabilization of randomstates for the nonlinear discrete-time stochastic system withincomplete information was studied To solve this problemthe stochastic sensitivity technique was developed Mathe-matically the considered control problem was reduced tothe analysis of the corresponding quadratic matrix equationFor this equation we have analyzed its solvability and sug-gested a constructive method of the solution A descriptionof attainability sets and algorithms for regulators designwere accumulated in Theorem 3 A constructiveness of thesegeneral theoretical results was demonstrated for the stochas-tically forced Verhulst model It is shown that our methodsuppresses unwanted large-amplitude oscillations around theequilibria and transforms the system dynamics from chaoticto regular It is worth noting that the elaborated techniqueis readily applicable to the control of higher dimensionaldiscrete-time stochastic models

Conflict of Interests

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

Acknowledgment

The work was supported by Act 211 Government of theRussian Federation Contract 02A03210006

References

[1] J T Sandefur Discrete Dynamical Modeling Oxford UniversityPress Oxford UK 1993

[2] S Elaydi ldquoIs theworld evolving discretelyrdquoAdvances in AppliedMathematics vol 31 no 1 pp 1ndash9 2003

[3] R L Devaney An Introduction to Chaotic Dynamical SystemsWestview Press 2003

[4] H G Schuster Deterministic Chaos Physik Weinheim Ger-many 1984

[5] B Ibarz JMCasado andMA F Sanjuan ldquoMap-basedmodelsin neuronal dynamicsrdquo Physics Reports vol 501 no 1-2 pp 1ndash74 2011

[6] W Horsthemke and R Lefever Noise-Induced TransitionsSpringer Series in Synergetics Springer Berlin Germany 1984

[7] K Matsumoto and I Tsuda ldquoNoise-induced orderrdquo Journal ofStatistical Physics vol 31 no 1 pp 87ndash106 1983

[8] F Gassmann ldquoNoise-induced chaos-order transitionsrdquo PhysicalReview E vol 55 article 2215 1997

[9] J B Gao S K Hwang and J M Liu ldquoWhen can noise inducechaosrdquo Physical Review Letters vol 82 no 6 pp 1132ndash11351999

[10] N Hritonenko A Rodkina and Y Yatsenko ldquoStability analysisof stochastic Ricker population modelrdquo Discrete Dynamics inNature and Society vol 2006 Article ID 64590 13 pages 2006

[11] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquo PhysicalReview Letters vol 64 no 11 pp 1196ndash1199 1990

[12] A L Fradkov and A Y Pogromsky Introduction to Control ofOscillations and Chaos vol 35World Scientific River Edge NJUSA 1998

[13] G Chen and X Yu Chaos Control Theory and ApplicationsSpringer New York NY USA 2003

[14] M Sanjuan and C Grebogi Recent Progress in ControllingChaos World Scientific Singapore 2010

[15] A Lasota and M C Mackey Chaos Fractals and NoiseStochastic Aspects of Dynamics Springer-Verlag New York NYUSA 1994

[16] I Bashkirtseva L Ryashko and I Tsvetkov ldquoSensitivity analysisof stochastic equilibria and cycles for the discrete dynamic sys-temsrdquoDynamics of Continuous Discrete and Impulsive SystemsSeries A Mathematical Analysis vol 17 no 4 pp 501ndash515 2010

[17] I Bashkirtseva and L Ryashko ldquoStochastic sensitivity analysisof noise-induced intermittency and transition to chaos in onemdashdimensional discrete-time systemsrdquo Physica A vol 392 no 2pp 295ndash306 2013

[18] I Bashkirtseva ldquoStochastic phenomena in one-dimensionalRulkov model of neuronal dynamicsrdquo Discrete Dynamics inNature and Society vol 2015 Article ID 495417 7 pages 2015

[19] I Bashkirtseva and L Ryashko ldquoControl of equilibria fornonlinear stochastic discrete-time systemsrdquo IEEE Transactionson Automatic Control vol 56 no 9 pp 2162ndash2166 2011

[20] K J Astrom Introduction to the Stochastic Control TheoryAcademic Press New York NY USA 1970

[21] B Shen ZWang andH ShuNonlinear Stochastic Systems withIncomplete Information Filtering and Control Springer 2013

[22] H Dong Z Wang and H Gao Filtering Control and FaultDetection with Randomly Occurring Incomplete InformationJohn Wiley amp Sons Ltd New York NY USA 2013

[23] A Albert Regression and the Moore-Penrose PseudoinverseAcademic Press New York NY USA 1972

[24] J P Crutchfield J D Farmer and B A Huberman ldquoFluctua-tions and simple chaotic dynamicsrdquo Physics Reports A ReviewSection of Physics Letters vol 92 no 2 pp 45ndash82 1982

[25] O Martin ldquoLyapunov exponents of stochastic dynamical sys-temsrdquo Journal of Statistical Physics vol 41 no 1-2 pp 249ndash2611985

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Stochastic AnalysisInternational Journal of


presented in a Theorem This Theorem gives a description ofattainability sets and algorithms for regulators design

One-dimensional case is discussed in details in Section 3In Section 4 we apply the results to the suppression ofunwanted large-amplitude oscillations around the equilib-ria of the stochastically forced Verhulst model with noisyobservations We show that our regulator can be used for thesuppression of chaos

2 Synthesis of Stochastic Sensitivity

Consider a nonlinear controlled discrete-time stochasticsystem

119909119905+1

= 119891 (119909119905 119906119905) + 120576120590 (119909

119905) 120585119905 (1)

where 119909 119891 isin 119877119899 119906 isin 119877

119897 120590 isin 119877119899times119898 and 119906 is a control input

Here 120585119905

isin 119877119898 is an uncorrelated random sequence with

parameters E120585119905

= 0 and E120585119905120585⊤

119905= 119868 and 119868 is the identity

119898 times 119898-matrix and 120576 is a scalar parameter of noise intensityIt is supposed that the corresponding deterministic

uncontrolled system (1) (with 120576 = 0 and 119906 = 0 therein) has anequilibrium 119909 119909 = 119891(119909 0) Stability of 119909 is not assumed

In present paper we consider a case of incompleteinformation when the measurement vector 119910

119905is known only

119910119905= 119892 (119909

119905) + 120576120593 (119909

119905) 120578119905 (2)

where 119910 119892 isin 119877119897 120593 isin 119877

119897times119896 Here 120578119905isin 119877119896 is an uncorrelated

random sequence with parameters E120578119905= 0 and E120578

119905120578⊤

119905= 119868

and 119868 is the identity 119896 times 119896-matrixIn this circumstance we consider the following regulator

119906119905= 119870 [119910

119905minus 119892 (119909)] (3)

The dynamics of the closed-loop stochastic system (1) withthe regulator (3) using noisy observations (2) is governed bythe following system

119909119905+1

= 119891 (119909119905 119870 [119892 (119909

119905) + 120576120593 (119909

119905) 120578119905minus 119892 (119909)])

+ 120576120590 (119909119905) 120585119905

(4)

For the asymptotics 119911119905= lim120576rarr0

((119909120576

119905minus119909)120576) of the deviations

of solutions 119909120576

119905of system (4) from the equilibrium 119909 the

following stochastic system can be written

119911119905+1

= (119865 + 119861119870119862) 119911119905+ 119861119870120593120578

119905+ 120590120585119905 (5)

where

119865 =

120597119891

120597119909

(119909 0)

119861 =

120597119891

120597119906

(119909 0)

119862 =

120597119892

120597119909

(119909)

120593 = 120593 (119909)

120590 = 120590 (119909)

(6)

Due to the uncorrelatedness of random terms 120578119905and 120585119905 the

second moments matrix 119872119905

= E(119911119905119911⊤

119905) is governed by the

equation

119872119905+1

= (119865 + 119861119870119862)119872119905(119865 + 119861119870119862)

⊤+ 119861119870Φ119870

⊤119861⊤+ 119878 (7)

where Φ = 120593120593⊤ 119878 = 120590120590

⊤ A set of matrices 119870 that providean exponential stability to the equilibrium 119909 of the closeddeterministic system (4) (with 120576 = 0 therein) has thefollowing form

K = 119870 | 120588 (119865 + 119861119870119862) lt 1 (8)

where 120588(119860) is a spectral radius of the matrix 119860 We supposethat the set K is not empty

For any119870 isin K (7) has a unique stable stationary solution119882 satisfying the equation

119882 = (119865 + 119861119870119862)119882 (119865 + 119861119870119862)⊤+ 119861119870Φ119870

⊤119861⊤+ 119878 (9)

Thismatrix119882 is called the stochastic sensitivity matrix of theequilibrium 119909 for system (4)The stochastic sensitivitymatrix119882 approximates a limit behavior of the second momentsE(119909120576119905minus 119909)(119909

120576

119905minus 119909)⊤ for deviations of solutions 119909120576

119905from 119909

lim119905rarrinfin

E (119909120576

119905minus 119909) (119909

120576

119905minus 119909)⊤

asymp 1205762119882 (10)

So the matrix 119882 characterizes a dispersion of the stationarydistributed random states of system (4) around the equilib-rium 119909

For any 119870 isin K the regulator (3) forms a correspondingstochastic equilibrium of system (4) with the stochasticsensitivity matrix 119882

119870which is a solution of (9)

Consider further the following inverse problem

Problem of Stochastic Sensitivity Synthesis Let M be a set ofsymmetric and positive-definite 119899 times 119899-matrices Let 119882 isin Mbe some assigned matrix The problem is to find a feedbackmatrix119870 isin K of regulator (3) such that the equality119882

119870= 119882

holds Here 119882119870is a solution of (9)

In some cases this problem can be unsolvableThereforewe consider an important notion of the attainability

Definition 1 An element 119882 isin M is said to be attainable forsystem (4) if the equality 119882

119870= 119882 holds for some 119870 isin K

Definition 2 The set of all attainable elements

W = 119882 isin M | exist119870 isin K 119882119870

= 119882 (11)

is called the attainability set for system (4)

As it follows from (9) the attainability analysis is reducedto the study of solvability of the quadratic matrix equation

119861119870 (119862119882119862⊤+ Φ)119870

⊤119861⊤+ 119861119870119862119882119865

⊤+ 119865119882119862

⊤119870⊤119861⊤

+ 119865119882119865⊤+ 119878 minus 119882 = 0

(12)

Rewrite (12) with respect to a new unknown matrix

119871 = 119861119870 (13)



119871 (119862119882119862⊤+ Φ) 119871

⊤+ 119871119862119882119865

⊤+ 119865119882119862

⊤119871⊤+ 119865119882119865

⊤

+ 119878 minus 119882 = 0

(14)

Denote 119866(119882) = (119862119882119862⊤




(119873 + 1198651) (119873 + 119865

1)⊤

= 119877 (119882) (15)

where1198651= 119865119882119862

⊤119866minus1

(119882)

119877 (119882) = 119865119882119862⊤(119862119882119862

⊤+ Φ)

minus1

119862119882119865⊤minus 119865119882119865

⊤minus 119878

+ 119882

(16)


119877 (119882) = 119865119882119862⊤(119862119882119862

⊤+ Φ)

minus1

119862119882119865⊤minus 119865119882119865

⊤minus 119878

+ 119882 ⪰ 0

(17)


119873 + 1198651= 11987712

(119882) 119869 (18)


119861119870 = (11987712

(119882) 119869 minus 119865119882119862⊤119866minus1

(119882))119866minus1

(119882) (19)




119899 = 119897) then

W = 119882 isin M | 119877 (119882) ⪰ 0 (20)


119870 = 119861minus1

(11987712

(119882) 119869 minus 119865119882119862⊤119866minus1

(119882))119866minus1

(119882)

isin K

(21)

(b) If rank(119861) lt 119899 then

W = 119882 isin M |

119877 (119882) ⪰ 0 119875 (11987712

(119882) 119869 minus 119865119882119862⊤119866minus1

(119882)) = 0

(22)


119870 = 119861+(11987712

(119882) 119869 minus 119865119882119862⊤119866minus1

(119882))119866minus1

(119882) isin K (23)





119909119905+1

= 119891 (119909119905) + 119906 + 120576120590120585

119905(24)


119910119905= 119909119905+ 120576120593120578

119905 (25)



parameters E120585119905= 0 E1205852

119905= 1 E120578

119905= 0 and E1205782

119905= 1 and 120576 120590



119906119905= 119896 (119910

119905minus 119909) (26)



119877 (119908) =

11988621199082

119908 + 1205932minus 1198862119908 minus 120590

2+ 119908

=

1199082minus ((1198862minus 1) 120593

2+ 1205902)119908 minus 120590

21205932

119908 + 1205932

119886 = 1198911015840(119909)

(27)


1199082minus ((1198862minus 1) 120593

2+ 1205902)119908 minus 120590

21205932ge 0 (28)


119908 ge 119908lowast=

1

2

[(1198862minus 1) 120593

2+ 1205902

+ radic((1198862minus 1) 120593

2+ 1205902)2

+ 412059021205932]

(29)





119896 =


2+ 1205902) 119908 minus 120590

21205932minus 119886119908

119908 + 1205932

(30)



cient

119896lowast= minus

119886119908lowast

119908lowast+ 1205932 (31)


3 3525120583

02

04

06

08

x

(a)

35

D

10minus2

10minus3

10minus4

325120583

(b)

0

minus1

minus2

minus3

32 34 363120583

Λ

(c)




119909119905+1

= 120583119909119905(1 minus 119909

119905) + 119906 + 120576120590120585

119905

119910119905= 119909119905+ 120576120593120578

119905

(32)











119908lowast(120583) =

1

2

[(1205832minus 4120583 + 3) 120593

2+ 1205902

+ radic((1205832minus 4120583 + 3) 120593

2+ 1205902)2

+ 412059021205932]

(33)


119896lowast=

(120583 minus 2)119908lowast(120583)

119908lowast(120583) + 120593

2 (34)






5 Conclusion




Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119871 (119862119882119862⊤+ Φ) 119871

⊤+ 119871119862119882119865

⊤+ 119865119882119862

⊤119871⊤+ 119865119882119865

⊤

+ 119878 minus 119882 = 0

(14)

Denote 119866(119882) = (119862119882119862⊤




(119873 + 1198651) (119873 + 119865

1)⊤

= 119877 (119882) (15)

where1198651= 119865119882119862

⊤119866minus1

(119882)

119877 (119882) = 119865119882119862⊤(119862119882119862

⊤+ Φ)

minus1

119862119882119865⊤minus 119865119882119865

⊤minus 119878

+ 119882

(16)


119877 (119882) = 119865119882119862⊤(119862119882119862

⊤+ Φ)

minus1

119862119882119865⊤minus 119865119882119865

⊤minus 119878

+ 119882 ⪰ 0

(17)


119873 + 1198651= 11987712

(119882) 119869 (18)


119861119870 = (11987712

(119882) 119869 minus 119865119882119862⊤119866minus1

(119882))119866minus1

(119882) (19)




119899 = 119897) then

W = 119882 isin M | 119877 (119882) ⪰ 0 (20)


119870 = 119861minus1

(11987712

(119882) 119869 minus 119865119882119862⊤119866minus1

(119882))119866minus1

(119882)

isin K

(21)

(b) If rank(119861) lt 119899 then

W = 119882 isin M |

119877 (119882) ⪰ 0 119875 (11987712

(119882) 119869 minus 119865119882119862⊤119866minus1

(119882)) = 0

(22)


119870 = 119861+(11987712

(119882) 119869 minus 119865119882119862⊤119866minus1

(119882))119866minus1

(119882) isin K (23)





119909119905+1

= 119891 (119909119905) + 119906 + 120576120590120585

119905(24)


119910119905= 119909119905+ 120576120593120578

119905 (25)



parameters E120585119905= 0 E1205852

119905= 1 E120578

119905= 0 and E1205782

119905= 1 and 120576 120590



119906119905= 119896 (119910

119905minus 119909) (26)



119877 (119908) =

11988621199082

119908 + 1205932minus 1198862119908 minus 120590

2+ 119908

=

1199082minus ((1198862minus 1) 120593

2+ 1205902)119908 minus 120590

21205932

119908 + 1205932

119886 = 1198911015840(119909)

(27)


1199082minus ((1198862minus 1) 120593

2+ 1205902)119908 minus 120590

21205932ge 0 (28)


119908 ge 119908lowast=

1

2

[(1198862minus 1) 120593

2+ 1205902

+ radic((1198862minus 1) 120593

2+ 1205902)2

+ 412059021205932]

(29)





119896 =


2+ 1205902) 119908 minus 120590

21205932minus 119886119908

119908 + 1205932

(30)



cient

119896lowast= minus

119886119908lowast

119908lowast+ 1205932 (31)


3 3525120583

02

04

06

08

x

(a)

35

D

10minus2

10minus3

10minus4

325120583

(b)

0

minus1

minus2

minus3

32 34 363120583

Λ

(c)




119909119905+1

= 120583119909119905(1 minus 119909

119905) + 119906 + 120576120590120585

119905

119910119905= 119909119905+ 120576120593120578

119905

(32)











119908lowast(120583) =

1

2

[(1205832minus 4120583 + 3) 120593

2+ 1205902

+ radic((1205832minus 4120583 + 3) 120593

2+ 1205902)2

+ 412059021205932]

(33)


119896lowast=

(120583 minus 2)119908lowast(120583)

119908lowast(120583) + 120593

2 (34)






5 Conclusion




Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










3 3525120583

02

04

06

08

x

(a)

35

D

10minus2

10minus3

10minus4

325120583

(b)

0

minus1

minus2

minus3

32 34 363120583

Λ

(c)




119909119905+1

= 120583119909119905(1 minus 119909

119905) + 119906 + 120576120590120585

119905

119910119905= 119909119905+ 120576120593120578

119905

(32)











119908lowast(120583) =

1

2

[(1205832minus 4120583 + 3) 120593

2+ 1205902

+ radic((1205832minus 4120583 + 3) 120593

2+ 1205902)2

+ 412059021205932]

(33)


119896lowast=

(120583 minus 2)119908lowast(120583)

119908lowast(120583) + 120593

2 (34)






5 Conclusion




Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













5 Conclusion




Acknowledgment


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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