Research ArticleOptimal Vibration Control of a Class of Nonlinear StochasticSystems with Markovian Jump

R H Huan R C Hu D Pu and W Q Zhu

Department of Mechanics State Key Laboratory of Fluid Power Transmission and Control Zhejiang UniversityHangzhou 310027 China

Correspondence should be addressed to W Q Zhu wqzhuzjueducn

Received 15 May 2015 Accepted 26 August 2015

Academic Editor Mickael Lallart

Copyright copy 2016 R H Huan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The semi-infinite time optimal control for a class of stochastically excited Markovian jump nonlinear system is investigated Usingstochastic averaging each form of the system is reduced to a one-dimensional partially averaged Ito equation of total energy Afinite set of coupled dynamical programming equations is then set up based on the stochastic dynamical programming principleand Markovian jump rules from which the optimal control force is obtained The stationary response of the optimally controlledsystem is predicted by solving the Fokker-Planck-Kolmogorov (FPK) equation associated with the fully averaged Ito equation Twoexamples are worked out in detail to illustrate the application and effectiveness of the proposed control strategy

1 Introduction

Markovian jump system (MJS) is a class of hybrid systemswhich has different operation modes or forms governed bya Markov process This class of systems can be used torepresent complex real systemswhichmay experience abruptvariation in their structures and parameters It is well knownthat random form switching can give rise to unexpectedphenomena of the whole MJSs even if the dynamics ineach form is simple and well understood Development ofmethodology for analysis of such systems is thus muchdeserving

MJSs have been studiedwith growing interest and activityin recent years since they are first introduced by Kats andKrasovskii [1] For dynamics analysis several importantcriteria for stochastic stability of MJSs have been established[2 3] The stationary response of the stochastically excitednonlinear MJSs is studied by Huan et al [4] For the problemof optimal control Krasosvkii and Lidskii studied the LQRcontrol of Markovian jump linear systems firstly [5] As aconsequence considerable attention has been paid to theoptimal control problem of MJSs [6 7] Sworder solved thejump linear Gaussian problem for finite time horizon [8]The infinite time horizon problem of linear MJSs was studied

by Wonham [9] and Fragoso and Hemerly [10] The optimalcontrol solution for linearMJSs subject to Gaussian input andmeasurement noise is given by Ji and Chizeck [11] Ghoshet al investigated the ergodic control problem of switchingdiffusion [12] However most of the published results areapplicable to linearMJSs Far less is known about the optimalcontrol of nonlinear MJSs particularly for stochasticallyexcited nonlinear MJSs

Recently a nonlinear optimal control strategy of stochas-tic excited nonlinear systems has been proposed by Zhubased on the stochastic averaging method [13] and stochasticdynamical programming principle It is proved that thisstrategy is quite effective for response alleviation of nonlinearstochastic systems The goal of this paper is to extendthe strategy for nonlinear stochastic system to the one fornonlinear stochastic MJSs

An infinite time optimal control problem for singledegree-of-freedom (SDOF) stochastically excited nonlinearMarkovian jump system is investigated in this paper Theorganization of this paper is as follows In Section 2 theequation of SDOF nonlinear Markovian jump system isexamined Stochastic averaging method is applied to thissystem in Section 3 The partially averaged equation of totalenergy for each form of the original system is obtained A

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 9641075 9 pageshttpdxdoiorg10115520169641075

2 Shock and Vibration

finite set of coupled dynamical programming equations isthen set up in Section 4 fromwhich the optimal control forceis determined The Fokker-Planck-Kolmogorov (FPK) equa-tion associated with the fully averaged Ito equation is alsoderived and solved In Section 5 two examples are workedout in detail to illustrate the application and effectiveness ofthe proposed control strategy

2 Formulation of Control Problem

Consider a SDOF controlled stochastically driven nonlinearsystem with Markovian jump governed by

+ 119892 (119909) = 120576119891 (119909 119904 (119905)) + 120576119906 (119909 119904 (119905))

+ 120576

12ℎ

119896(119909 119904 (119905))119882

119896(119905)

119909 (119905

0) = 119909

0

119904 (119905

0) = 119904

0

(1)

where 119892(119909) is nonlinear stiffness 120576 is a small parameter120576119891(119909 119904(119905)) denotes light damping with Markovian jump120576119906(119909 119904(119905)) is the feedback control force 12057612ℎ

119896(119909 119904(119905)) (119896 =

1 2 119898) represent the Markovian jump coefficients ofweakly external and (or) parametric stochastic excitations119882

119896(119905) are independent Gaussian white noises with zero

means and intensities 2119863119896

The parameter 119904(119905) is a continuous-timeMarkovian jumpprocess with finite discrete state-space 119878 = 1 2 119897 andtransition matrix Λ = 120582

119894119895 (119894 119895 isin 119878) where 120582

119894119895are real

numbers such that for 119894 = 119895 120582119894119895gt 0 and for all 119894 isin 119878

120582

119894119894= minussum

119895 =119894120582

119894119895 The transition probability is given by [14 15]

119875 119904 (119905 + Δ119905) = 119895 | 119904 (119905) = 119894

=

120582

119894119895Δ119905 + 119900 (Δ119905) 119894 = 119895

1 + 120582

119894119894Δ119905 + 119900 (Δ119905) 119894 = 119895

(2)

where 119900(Δ119905) is such that limΔ119905rarr0

(119900(Δ119905)Δ119905) = 0Our primary goal here is to design an optimal controller

which will result in the minimum of a specified performanceFor infinite-horizon control problem the performance indexis specified by

119869 (119906)

= lim119905119891rarrinfin

1

119905

119891

119864[int

119905119891

0

119871 (119909 (120591) (120591) 119906 (120591) 119904 (120591)) 119889120591]

(3)

where 119864[sdot] denotes an expectation operation 119905119891is the termi-

nal time of control 119871(119909(120591) (120591) 119906(120591) 119904(120591)) is cost functionIf random excitations 119882

119896(119905) and Markovian jump process

119904(119905) are ergodic then the response of controlled systemapproaches to stationary and ergodic as 119905

119891rarr infin In this

case the performance index (3) is rewritten as

119869 (119906) = lim119905119891rarrinfin

1

119905

119891

int

119905119891

0

119871 (119909 (120591) (120591) 119906 (120591) 119904 (120591)) 119889120591 (4)

Equations (1) and (4) constitute the mathematical for-mulation of the infinite-horizon optimal control problem fora SDOF nonlinear system undergoing Markovian jump Inthis paper we consider the perfect observation case whichmeans that the system state (119909(119905) (119905)) and values of 119904(119905) areall available at each time 119905

3 Partially Averaged Equation

Let 119902 = 119909 and 119901 = (1) can be converted to the following Itostochastic differential equations [11 16]

119889119902 = 119901119889119905

119889119901 = [minus119892 (119902) + 120576119891 (119902 119901 119904 (119905)) + 120576119906 (119902 119901 119904 (119905))]

+ 120576

12120590

119896(119902 119904 (119905)) 119889119861

119896(119905)

(5)

where 119861119896(119905) are independent standard Wiener processes

119902 and 119901 are the generalized displacement and generalizedmomentum respectively 120590

119896(119902 119904(119905)) = radic2119863

119896ℎ

119896(119902 119904(119905)) The

Hamiltonian function 119867 (total energy) of the Hamiltoniansystem associated with system (5) is

119867(119902 119901) =

119901

2

2

+ int

119902

0

119892 (119906) 119889119906(6)

Suppose that 119904(119905) is a slow jump process and independent ofsystemrsquos state That means that for original system (1) or (5)the system jumps among the 119897 forms or modes slowly andindependently We consider one of the forms by letting theMarkovian jumpprocess be arbitrarily fixed at 119904(119905) = 119894 (119894 isin 119878)When 119904(119905) = 119894 denote 119891(119894)(119902 119901) = 119891(119902 119901 119904(119905)) 119906(119894)(119902 119901) =119906(119902 119901 119904(119905)) and 120590(119894)

119896(119902) = 120590

119896(119902 119904(119905)) Since119867 is a function of

119902 and 119901 the Ito equation for 119867 can be derived from (5) byusing Ito differential rule as follows [17]

119889119867 = 120576 [119891

(119894)(119902 119901)

120597119867

120597119901

+

1

2

120590

(119894)

119896(119902) 120590

(119894)

119896(119902)

120597

2119867

120597119901

2

+ 120576119906

(119894)(119902 119901)

120597119867

120597119901

] + 120576

12120590

(119894)

119896(119902)

120597119867

120597119901

119889119861

119896(119905)

(7)

In the case of light damping and weak excitations theHamiltonian 119867 in system (7) is a slowly varying processwhile the generalized displacement 119902 in (5) is a rapidlyvarying process Since the slowly varying process is essentialfor describing the long-term behavior of the system thestochastic averagingmethod is used to average out the rapidlyvarying process and to yield the following partially averagedIto equation for119867 [13 18 19]

119889119867 = 120576 [119898

(119894)(119867) + 119898

(119894)

119906(119867)] 119889119905 + 120576

12120590

(119894)(119867) 119889119861 (119905) (8)

Shock and Vibration 3

where

119898

(119894)(119867) =

1

119879 (119867)

sdot int

Ω

[119891

(119894)(119902 119901)

120597119867

120597119901

+

1

2

120590

(119894)

119896(119902) 120590

(119894)

119896(119902)

120597

2119867

120597119901

2]

sdot (

120597119867

120597119901

)

minus1

119889119902

119898

(119894)

119906(119867) = ⟨119906

(119894)(119902 119901)

120597119867

120597119901

⟩

120590

(119894)(119902 119901)

2

=

1

119879 (119867)

int

Ω

[120590

(119894)

119896(119902) 120590

(119894)

119896(119902) (

120597119867

120597119901

)

2

]

sdot (

120597119867

120597119901

)

minus1

119889119902

⟨sdot⟩ =

1

119879 (119867)

int

Ω

[sdot] (

120597119867

120597119901

)

minus1

119889119902

(9)

The region of integration is Ω = 119902 | 119867(119902 0) le 119867 and theparameter

119879 (119867) = int

Ω

(

120597119867

120597119901

)

minus1

119889119902(10)

Note that the second term 119898(119894)119906(119867) on the right-hand of (8)

is unknown since 119906(119894) are unknown function of 119902 119901 and 119904 atthis stage

Equation (8) is only valid when 119904(119905) = 119894 As Markovianjumps are allowed so that 119904(119905) takes values from 119878 = 1 2 119897 (8) can be extended so that

119889119867 = 120576 [119898 (119867 119904) + 119898

119906(119867 119904)] 119889119905

+ 120576

12120590 (119867 119904) 119889119861 (119905)

(11)

where119898(119867 119904)119898119906(119867 119904) and 120590(119867 119904) change as 119904(119905) jumps so

that119898(119867 119904(119905) = 119894) = 119898(119894)(119867)119898119906(119867 119904(119905) = 119894) = 119898

(119894)

119906(119867) and

120590(119867 119904(119905) = 119894) = 120590

(119894)(119867)

To be consistent with the partially averaged equation (11)performance index (4) is also partially averaged that is

119869 (119906) = lim119905119891rarrinfin

1

119905

119891

int

119905119891

0

119871 (119867 (120591) ⟨119906 (120591)⟩ 119904 (120591)) 119889120591 (12)

The original optimal control problem (1) and (4) has beenconverted to the partially averaged optimal control problem(11) and (12) The partially averaged optimal control problemis much simpler than the original one since (11) is only one-dimensional It has been proved [20] that the optimal controlfor the partially averaged control problem is quasi-optimalcontrol for the original one

4 Optimal Control Law

For the proposed control problem we introduce the valuefunction

119881 (119867 119904) = min119906119869 (119906) (13)

According to the stochastic dynamical programmingprinciple andMarkovian jump rules the following dynamicalprogramming equation can be established [16]

min119906

1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) + 119898

119906(119867 119904)]

120597119881 (119867 119904)

120597119867

+ 119871 (119867 ⟨119906⟩ 119904)

+

119897

sum

119903=1

120582

119903119904119881 (119867 119903) = 0 119904 isin 119878

(14)

where 120582119903119904are the transition rates described in (2) Equation

(14) is a finite set of second-order partial differential equationsindexed by 119904 These equations are coupled through the zero-order terms sum119897

119903=1120582

119903119904119881(119867 119903)

The necessary condition for minimizing the left-handside of (14) is

120597

120597119906

119898

119906(119867 119904)

120597119881

120597119867

+ 119871 (119867 ⟨119906 (119902 119901 119904)⟩ 119904) = 0 (15)

Let the cost function 119871 be of the form

119871 (119867 ⟨119906⟩ 119904) = 119891 (119867 119904) + 119877 ⟨119906

2⟩ (16)

where 119891(119867 119904) ge 0 and 119877 is a positive constant By substitut-ing (16) into (15) and exchanging the order of the derivativewith respect to 119906 and averaging in (15) the expression of theoptimal control force is determined

119906

lowast(119902 119901 119904) = minus

1

2119877

120597119881 (119867 119904)

120597119867

120597119867

120597119901

119904 isin 119878 (17)

Note that 119881(119867 119904) is generally a nonlinear functional of 119867Hence 119906lowast(119902 119901 119904) is a nonlinear function of 119902 and 119901 thatis the optimal feedback control is nonlinear To ensure thatthe control force 119906lowast(119902 119901 119904) is indeed optimal the followingsufficient condition should be satisfied

120597

2

120597119906

2119871 (119867 ⟨119906⟩ 119904) ge 0

(18)

Substituting the expression for 119906lowast(119902 119901 119904) in (17) into (14)yields the final dynamical programming equation

1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) + 119898

lowast

119906(119867 119904)]

120597119881 (119867 119904)

120597119867

+ 119871 (119867 ⟨119906

lowast⟩ 119904) +

119897

sum

119903=1

120582

119903119904119881 (119867 119903) = 0

(19)

4 Shock and Vibration

where

119898

lowast

119906(119867 119904) = ⟨119906

lowast(119902 119901 119904)

120597119867

120597119901

⟩

= minus

1

2119877

⟨

120597119881 (119867 119904)

120597119867

(

120597119867

120597119901

)

2

⟩

(20)

The value function 119881(119867 119904) can be obtained by solving(19) The optimal control force 119906lowast(119902 119901 119904) is then determinedby substituting 119881(119867 119904) into expression (17) By inserting theoptimal control force 119906lowast(119902 119901 119904) into (11) and averaging theterm ⟨119906lowast(119902 119901 119904)(120597119867120597119901)⟩ the following completely averagedIto equation is obtained

119889119867 = 120576 [119898 (119867 119904) + 119898

lowast

119906(119867 119904)] 119889119905

+ 120576

12120590 (119867 119904) 119889119861 (119905)

(21)

The FPK equation associated with the fully averaged Itoequation (21) is [4 21 22]

120597

120597119905

119901 (119867 119904 119905)

= minus

120597

120597119867

[119898 (119867 119904) + 119898

lowast

119906(119867 119904)] 119901 (119867 119904 119905)

+

1

2

120597

2

120597119867

2[120590 (119867 119904)

2119901 (119867 119904 119905)]

minus

119897

sum

119903=1

119903 =119904

[120582

119904119903119901 (119867 119904 119905) minus 120582

119903119904119901 (119867 119903 119905)] 119904 isin 119878

(22)

where 119901(119867 119904 119905) is the probability density function of totalenergy119867 for optimally controlled system (1) while the systemoperating in 119904th form (119901(119867 119904 119905) for uncontrolled system canbe obtained by letting 119906lowast(119902 119901 119904) = 0) The initial condition is

119901 (119867 119904 0) = 119901 (119867

0 119904) (23)

and boundary conditions are

119901 (0 119904 119905) = finite

119901 (119867 119904 119905)

1003816

1003816

1003816

1003816119867rarrinfin997888rarr 0

120597119901 (119867 119904 119905)

120597119867

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119867rarrinfin

997888rarr 0

(24)

The FPK equation (22) is a finite set of second-order par-tially differential equations which are coupled through thezero-order terms Equation (22) does not admit easy solutionanalytically or numerically However in practical applicationwe are more interested in the stationary solution of FPKequation (22) In this case FPK equation (22) is reduced byletting 120597119901120597119905 = 0 Then the stationary probability density119901(119867 119904) is obtained readily from solving (22) numericallyThestationary probability density 119901(119867) can be obtained from119901(119867 119904) as follows

119901 (119867) = 119888

119897

sum

119904=1

119901 (119867 119904) (25)

where 119888 is a normalization constant The stationary probabil-ity density119901(119902) of the generalized displacement and themeanvalue 119864[119867] of the total energy are then obtained as

119901 (119902) = int

infin

minusinfin

119901 (119902 119901) 119889119901

119864 [119867] = int

infin

0

119867119901 (119867) 119889119867

(26)

where

119901 (119902 119901) =

119901 (119867)

119879 (119867)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119867=119867(119902119901)

(27)

is the stationary joint probability density of the generalizeddisplacement and generalized momentum

5 Numerical Example

To demonstrate the application and to assess the effective-ness of the proposed control strategy numerical results areobtained for two examples

51 Example 1 Consider a controlled van der Pol-Rayleighoscillator subjected to both external and parametric randomexcitations which is capable of independentMarkovian jumpand governed by

+ 119909 = 120572 (119904 (119905)) minus [120573 (119904 (119905)) 119909

2+ 120574 (119904 (119905))

2]

+ 119906 (119909 119904 (119905)) + ℎ

1(119904 (119905))119882

1(119905)

+ ℎ

2(119904 (119905)) 119909119882

2(119905)

(28)

where 120572(119904(119905)) 120573(119904(119905)) and 120574(119904(119905)) are Markovian jumpcoefficients of linear and quasi-linear damping respectivelyℎ

1(119904(119905)) and ℎ

2(119904(119905))119909 denote Markovian jump coefficients

of external and parametric random excitations respectively119906(119909 119904(119905)) is the feedback control force 119882

119896(119905) (119896 = 1 2)

are independent Gaussian white noises with zero mean andintensities 2119863

119896 119904(119905) is a finite-state continuous-time Markov

process pointing to the systemrsquos form with the transitionprobability defined in (2)

We consider the 3-form case here that means 119897 = 3 and119878 = 1 2 3 Prescribe the transition rate 120582

119894119895(119894 119895 = 1 2 3)

between the form 119894 and the form 119895 by a transition matrix Λ =[120582

119894119895]Let 119902 = 119909 and 119901 = The Hamiltonian associated with

system (28) is

119867 =

1

2

119901

2+

1

2

119902

2 (29)

As a Hamiltonian system (28) can be expressed as

[

119901

] = [119901 minus 119902

+ 120572 (119904 (119905)) minus [120573 (119904 (119905)) 119902

2+ 120574 (119904 (119905)) 119901

2] 119901

+ 119906 (119902 119901 119904 (119905)) + ℎ

1(119904 (119905))119882

1(119905)

+ ℎ

2(119904 (119905)) 119902119882

2(119905)]

(30)

Shock and Vibration 5

Upon stochastic averaging (11) for system (28) isobtained for whichΩ = 119902 | 119867(119902 0) le 119867 and

119898(119867 119904 (119905) = 119894) = minus [

120573

(119894)

2

+

3

2

120574

(119894)]119867

2+ ℎ

(119894)

1

2

119863

1

+ (120572

(119894)+ ℎ

(119894)

2

2

119863

2)119867

119898

119906(119867 119904 (119905) = 119894) = ⟨119906

(119894)(119902 119901) 119901⟩

120590

2(119867 119904 (119905) = 119894) = 2 ℎ

(119894)

1

2

119863

1119867 + ℎ

(119894)

2

2

119863

2119867

2

119894 = 1 2 3

(31)

For the proposed control strategy the partially averagedcost function 119871 is of the form of (16) with

119891 (119867 119904) = 119888

0(119904) + 119888

1(119904)119867 + 119888

2(119904)119867

2+ 119888

3(119904)119867

3 (32)

Following (17) the expression of the optimal control forceis

119906

lowast(119902 119901 119904) = minus

1

2119877

120597119881 (119867 119904)

120597119867

119901(33)

Substituting 119906lowast(119902 119901 119904) in (33) into the dynamical pro-gramming equation (14) completing the averaging yields thefollowing final coupled dynamical programming equations

1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) minus

1

4119877

120597119881 (119867 119904)

120597119867

⟨(

120597119867

120597119901

)

2

⟩]

120597119881 (119867 119904)

119889119867

+ 119891 (119867 119904) +

3

sum

119903=1

120582

119903119904119881 (119867 119903) = 0 119904 = 1 2 3

(34)

To solve (34) the value function 119881(119867 119904) is supposed tobe of the following form

119881 (119867 119904) = 119887

0(119904) + 119887

1(119904)119867 + 119887

2(119904)119867

2 (35)

Since the optimal control force 119906lowast is function of 120597119881120597119867only the coefficients 119887

1(119904) and 119887

2(119904) are determined by insert-

ing (32) and (35) into (34)

119887

1(119904 = 119894) = 119887

(119894)

1

=

2 (119888

2+ sum

3

119896=1120582

119896119894119887

(119896)

2+ 2119887

(119894)

2120572

(119894)+ 2119863

2ℎ

(119894)

2

2

119887

(119894)

2)119877

2119887

(119894)

2+ 120573

(119894)119877 + 3120574

(119894)119877

119887

2(119904 = 119894) = 119887

(119894)

2

= minus

1

2

120573

(119894)minus

3

2

120574

(119894)119877

+

1

2

radic

(120573

(119894))

2

119877

2+ 6120573

(119894)119877

2120574

(119894)+ 9 (120574

(119894))

2

119877

2+ 4119888

3119877

119894 = 1 2 3

(36)

Uncontrolled

Optimally controlled

1 2 3 4 500

1

2

3

4

5

6

7

H

p(H

)

Figure 1 Stationary probability density 119901(119867) of total energy ofuncontrolled and optimally controlled system (28) The lines areobtained fromnumerical solution of (37) while the dots are obtainedfrom direct simulation of original system (28)

By substituting (35) into (17) with the coefficients deter-mined by (36) the optimal control force is determinedThenthe following completely averaged FPK equation is obtained

120597

120597119905

119901 (119867 119904 119905)

= minus

120597

120597119867

[119898 (119867 119904) + 119898

lowast

119906(119867 119904)] 119901 (119867 119904 119905)

+

1

2

120597

2

120597119867

2[120590 (119867 119904)

2119901 (119867 119904 119905)]

minus

3

sum

119903=1

119903 =119904

[120582

119904119903119901 (119867 119904 119905) minus 120582

119903119904119901 (119867 119903 119905)]

119904 = 1 2 3

(37)

where

119898

lowast

119906(119867 119904 = 119894) = minus

1

2119877

(119887

(119894)

1+ 2119887

(119894)

2119867)119867 (38)

Solving (37) numerically with 120597119901120597119905 = 0 the stationaryprobability density 119901(119867 119904) of optimal controlled system canbe obtained The stationary probability densities 119901(119867) of thetotal energy 119901(119902) of the generalized displacement and themean value 119864[119867] of the total energy are then obtained using(25) and (26)

Some numerical results are obtained as shown in Figures1ndash4 for system parameters 119863

1= 003 119863

2= 003 119877 = 10

6 Shock and Vibration

0 2 40

02

04

06

08

1

12

Uncontrolled

Optimally controlled

q

minus4 minus2

p(q)

Figure 2 Stationary probability density 119901(119902) of displacement ofuncontrolled and optimally controlled system (28) The lines areobtained fromnumerical solution of (37) while the dots are obtainedfrom direct simulation of original system (28)

1 15 2 25 30

05

1

15

2

Uncontrolled

Optimally controlled

E[H

]

1205821112058222

Figure 3Mean value119864[119867] of total energy of uncontrolled and opti-mally controlled system (28) as functions of the ratio 120582

11120582

22(120582

22=

120582

33) The lines are obtained from numerical solution of (37) while

the dots are obtained from direct simulation of original system (28)

2000 2100 2200 2300 2400 2500 2600

0

2

4

6

Uncontrolled

Optimally controlled

q

t

minus4

minus6

minus2

Figure 4 The sample time history of displacement of uncontrolledand optimally controlled system (28)

119904

1= 119904

2= 119904

3= 10 Λ = [

minus12058211 120582112 120582112

120582222 minus12058222 120582222

120582332 120582332 minus12058233

] 12058211= 120582

22= 120582

33=

30 and

120572

(1)= 001

120573

(1)= 001

120574

(1)= 001

ℎ

(1)

1= 20

ℎ

(1)

2= 20

120572

(2)= 001

120573

(2)= 002

120574

(2)= 002

ℎ

(2)

1= 20

ℎ

(2)

2= 10

120572

(3)= 001

120573

(3)= 004

120574

(3)= 004

ℎ

(3)

1= 10

ℎ

(3)

2= 10

(39)

The stationary probability density 119901(119867) of the totalenergy for uncontrolled and optimally controlled system(28) is shown in Figure 1 Obviously as 119867 increases 119901(119867)for optimally controlled system (28) decreases much fasterthan that for uncontrolled system The stationary probabilitydensity 119901(119902) of the displacement is plotted in Figure 2 withthe same parameters as in Figure 1 For optimally controlledsystem there is a much higher probability that 119902 will belocated near their equilibrium Hence 119901(119902) has much largermode and smaller dispersion around 119902 = 0 for optimallycontrolled system than those for uncontrolled system Thisimplies that the proposed control strategy has high effective-ness for attenuating the response of system (28)

Figure 3 shows the mean value 119864[119867] of the total energyof uncontrolled and optimally controlled system as functionsof ratio 120582

11120582

22(120582

22= 120582

33) Obviously higher value of ratio

120582

11120582

22implies lower probability for the system operating in

form 119904(119905) = 1 and higher probability for the system oper-ating in forms 119904(119905) = 2 and 119904(119905) = 3 When the system oper-ates in form 119904(119905) = 1 the system has the smallest dampingcoefficients and the largest amplitudes of stochastic excita-tions than those when it operates in forms 119904(119905) = 2 and119904(119905) = 3 As a consequence 119864[119867] for uncontrolled systemdecreases monotonously while the ratio 120582

11120582

22increases as

shown in Figure 3 However 119864[119867] for optimally controlledsystem always keeps a very low value despite changing of

Shock and Vibration 7

the ratio 12058211120582

22 which indicates that the proposed control

strategy is very effective and quite robust with respect tothe ratio 120582

11120582

22 In Figures 1ndash3 lines are obtained by

numerical solution of (37) while dots are obtained by directsimulation of system (28)Observe that the dotsmatch closelywith the corresponding lines demonstrating the validity andaccuracy of the proposed method The displacements of theuncontrolled and optimally controlled system (28) are shownin Figure 4 from which the effect of the optimal control onthe displacement can be visualized intuitively

52 Example 2 Consider a controlled stochastically excitedDuffing oscillator withMarkovian jumpwhich is governed by

+ 120596

2119909 + 120572119909

3= 120573 (119904 (119905)) + 119906 (119909 119904 (119905))

+ ℎ (119904 (119905)) 120585 (119905)

(40)

where 120573(119904(119905)) is Markovian jump coefficient of linear damp-ing ℎ(119904(119905)) is Markovian jump coefficient of external randomexcitation 119906(119909 119904(119905)) is the feedback control force 120585(119905) isGaussian white noise with zero mean and intensity 2119863 Herewe consider the 2-form case 119897 = 2 and 119878 = 1 2

Upon the stochastic averaging method the original jumpsystem (40) can be approximately substituted by the partiallyaveraged system (11) with

119898(119867 119904 = 119894) = ℎ

(119894)

2

119863 minus 120573

(119894)119866 (119867)

119898

119906(119867 119904 = 119894) =

4

119879 (119867)

int

119860(119867)

0

119906

(119894)(119902 119901) 119901 119889119902

120590 (119867 119904 = 119894)

2= 2 ℎ

(119894)

2

119863119866 (119867)

119866 (119867) =

4

119879 (119867)

int

119860(119867)

0

(2119867 minus 120596

2119902

2minus

120572119902

4

2

)

12

119889119902

119879 (119867) = 4int

119860(119867)

0

(2119867 minus 120596

2119902

2minus

120572119902

4

2

)

minus12

119889119902

119860 (119867) = [

(120596 + 4120572119867)

12minus 120596

2

120572

]

12

119894 = 1 2

(41)

Following (17) the expression of the optimal control forceis

119906

lowast(119902 119901 119904) = minus

1

2119877

120597119881 (119867 119904)

120597119867

119901(42)

Then the following final dynamical programming equa-tion is obtained1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) minus

1

4119877

120597119881 (119867 119904)

120597119867

119866 (119867)]

120597119881 (119867 119904)

119889119867

+ 119891 (119867 119904) +

2

sum

119903=1

120582

119903119904119881 (119867 119903) = 0 119904 = 1 2

(43)

120597119881120597119867 can be obtained from solving (43) numericallyThen the optimal control force is determined by substituting120597119881120597119867 into (17) Substituting the optimal control force (17)into the partially averaged equation (11) and completingthe averaging yield the following completely averaged FPKequation

120597

120597119905

119901 (119867 119904 119905)

= minus

120597

120597119867

[119898 (119867 119904) minus

1

2119877

120597119881 (119867 119904)

120597119867

119866 (119867)]

sdot 119901 (119867 119904 119905) +

1

2

120597

2

120597119867

2[120590

2(119867 119904) 119901 (119867 119904 119905)]

minus

2

sum

119903=1

119903 =119904

[120582

119904119903119901 (119867 119904 119905) minus 120582

119903119904119901 (119867 119903 119905)] 119904 = 1 2

(44)

Solving (44) numerically with 120597119901120597119905 = 0 the statistics119901(119867) 119901(119902) and 119864[119867] of the stationary response are thenobtained from (25) and (26)

The numerical results shown in Figures 5ndash8 are for systemwith parameters 120596 = 10 120572 = 05 119863 = 002 Λ = [ minus12058211 12058211

12058222 minus120582122]

120582

11= 120582

22= 20 and

120573

(1)= 001

ℎ

(1)= 20

120573

(2)= 002

ℎ

(2)= 10

(45)

The stationary probability densities 119901(119867) of total energyand 119901(119902) of the displacement of uncontrolled and optimallycontrolled system (40) are evaluated and plotted in Figures5 and 6 respectively The mean value 119864[119867] of total energyof uncontrolled and optimally controlled system (40) asfunctions of ratio 120582

11120582

22is shown in Figure 7 It is seen that

the optimal control force alleviates the response of system(40) significantly This effect can be seen more apparentlyfrom Figure 8 Again the analytical results obtained fromsolving (44) match closely with those from digital simulationof original system (40)

6 Conclusions

In this paper we proposed a nonlinear stochastic optimalcontrol strategy for SDOF stochastically excited Markovianjump systems where the jump parameter takes on values in afinite discrete set In the slow jump case the original systemwas first reduced to one governed by a finite set of one-dimensional partially averaged Ito equation for total energyupon stochastic averaging Using the stochastic dynami-cal programming principle and Markovian jump rules afinite set of coupled dynamical programming equations wasderived from which the optimal control force was obtainedThe obtained optimal control force has been proved to be the

8 Shock and Vibration

0 1 2 3 4 50

2

4

6

8

10

Uncontrolled

Optimally controlled

H

p(H

)

Figure 5 Stationary probability density 119901(119867) of total energy ofuncontrolled and optimally controlled system (40) The lines areobtained fromnumerical solution of (44)while the dots are obtainedfrom direct simulation of original system (40)

0 1 2 30

02

04

06

08

1

12

14

16

Uncontrolled

Optimally controlled

q

minus3 minus2 minus1

p(q)

Figure 6 Stationary probability density 119901(119902) of displacement ofuncontrolled and optimally controlled system (40) The lines areobtained fromnumerical solution of (44)while the dots are obtainedfrom direct simulation of original system (40)

quasi-optimal control for the original system The strategywas applied to two nonlinear oscillators that were capableof independent Markovian jumps Numerical results showedthat the strategy is fairly robust and effective in reductionof stationary response of the controlled system Thus it ispotentially promising for practical control applications afterfurther research

The proposed method has the potential to be extended tothe optimal control forMulti-DOFMarkovian jump systemsHowever to obtain the concrete conclusionmuchwork has tobe done Mathematical tools should be developed to solve thedynamical programming equation (14) for Multi-DOF case

1 15 2 25 30

1

2

3

4

5

Uncontrolled

Optimally controlled

E[H

]

1205821112058222

Figure 7 Mean value 119864[119867] of total energy of uncontrolled andoptimally controlled system (40) as functions of the ratio 120582

11120582

22

The lines are obtained fromnumerical solution of (44) while the dotsare obtained from direct simulation of original system (40)

2000 2050 2100 2150 2200

0

5

Uncontrolled

Optimally controlled

q

t

minus5

Figure 8The sample time history of displacements of uncontrolledand optimally controlled system (40)

Disclaimer

Opinions findings and conclusions expressed in this paperare those of the authors and do not necessarily reflect theviews of the sponsors

Conflict of Interests

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

Acknowledgments

This work was supported by the Natural Science Foundationof China (through Grants no 11272279 11372271 1143201211321202 and 11272201) and 973 Program (no 2011CB711105)

Shock and Vibration 9

References

[1] L Kats and N N Krasovskii ldquoOn the stability of systemswith random parametersrdquo Journal of Applied Mathematics andMechanics vol 24 no 5 pp 1225ndash1246 1960

[2] M Mariton ldquoAlmost sure and moments stability of jump linearsystemsrdquo Systems amp Control Letters vol 11 no 5 pp 393ndash3971988

[3] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[4] R H Huan W-Q Zhu F Ma and Z-G Ying ldquoStationaryresponse of a class of nonlinear stochastic systems undergoingMarkovian jumpsrdquo Journal of Applied Mechanics vol 82 no 5Article ID 051008 6 pages 2015

[5] N N Krasosvkii and E A Lidskii ldquoAnalytical design of con-trollers in systems with random attributes Irdquo Automation andRemote Control vol 22 pp 1021ndash1025 1961

[6] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967

[7] O L V Costa M D Fragoso and M G Todorov Continuous-Time Markov Jump Linear Systems Springer New York NYUSA 2013

[8] D D Sworder ldquoFeedback control of a class of linear systemswith jump parametersrdquo IEEE Transactions on Automatic Con-trol vol 14 no 1 pp 9ndash14 1969

[9] W M Wonham ldquoRandom differential equations in controltheoryrdquoProbabilisticMethods in AppliedMathematics vol 2 pp131ndash213 1971

[10] M D Fragoso and E M Hemerly ldquoOptimal control for a classof noisy linear systemswithMarkovian jumping parameters andquadratic costrdquo International Journal of Systems Science vol 22no 12 pp 2553ndash2561 1991

[11] Y D Ji and H J Chizeck ldquoJump linear quadratic Gaussian con-trol in continuous timerdquo IEEE Transactions on AutomaticControl vol 37 no 12 pp 1884ndash1892 1992

[12] M K Ghosh A Arapostathis and S I Marcus ldquoErgodiccontrol of switching diffusionsrdquo SIAM Journal on Control andOptimization vol 35 no 6 pp 1952ndash1988 1997

[13] W Q Zhu ldquoNonlinear stochastic dynamics and control inHamiltonian formulationrdquo Applied Mechanics Reviews vol 59no 4 pp 230ndash248 2006

[14] E K Boukas and H Yang ldquoStability of stochastic systems withjumpsrdquoMathematical Problems in Engineering vol 3 no 2 pp173ndash185 1996

[15] X B Feng K A Loparo Y D Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[16] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions Springer New York NY USA 1993

[17] K Ito ldquoOn stochastic differential equationsrdquo Memoirs of theAmerican Mathematical Society vol 4 pp 289ndash302 1951

[18] W Q Zhu and Z G Ying ldquoOptimal nonlinear feedbackcontrol of quasi-Hamiltonian systemsrdquo Science in China SeriesA Mathematics vol 42 no 11 pp 1213ndash1219 1999

[19] W Q Zhu Z G Ying and T T Soong ldquoAn optimal nonlinearfeedback control strategy for randomly excited structural sys-temsrdquo Nonlinear Dynamics vol 24 no 1 pp 31ndash51 2001

[20] H J Kushner andD S Clark Stochastic ApproximationMethodsfor Constrained andUnconstrained Systems Springer NewYorkNY USA 1978

[21] YW Fang Y LWu andHQWangOptimal ControlTheory ofStochastic Jump System National Defense Inustry Press BeijingChina 2012

[22] S T WuTheTheory of Stochastic Jump System and Its Applica-tion Science Press Beijing China 2007

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2 Shock and Vibration

finite set of coupled dynamical programming equations isthen set up in Section 4 fromwhich the optimal control forceis determined The Fokker-Planck-Kolmogorov (FPK) equa-tion associated with the fully averaged Ito equation is alsoderived and solved In Section 5 two examples are workedout in detail to illustrate the application and effectiveness ofthe proposed control strategy

2 Formulation of Control Problem

Consider a SDOF controlled stochastically driven nonlinearsystem with Markovian jump governed by

+ 119892 (119909) = 120576119891 (119909 119904 (119905)) + 120576119906 (119909 119904 (119905))

+ 120576

12ℎ

119896(119909 119904 (119905))119882

119896(119905)

119909 (119905

0) = 119909

0

119904 (119905

0) = 119904

0

(1)

where 119892(119909) is nonlinear stiffness 120576 is a small parameter120576119891(119909 119904(119905)) denotes light damping with Markovian jump120576119906(119909 119904(119905)) is the feedback control force 12057612ℎ

119896(119909 119904(119905)) (119896 =

1 2 119898) represent the Markovian jump coefficients ofweakly external and (or) parametric stochastic excitations119882

119896(119905) are independent Gaussian white noises with zero

means and intensities 2119863119896

The parameter 119904(119905) is a continuous-timeMarkovian jumpprocess with finite discrete state-space 119878 = 1 2 119897 andtransition matrix Λ = 120582

119894119895 (119894 119895 isin 119878) where 120582

119894119895are real

numbers such that for 119894 = 119895 120582119894119895gt 0 and for all 119894 isin 119878

120582

119894119894= minussum

119895 =119894120582

119894119895 The transition probability is given by [14 15]

119875 119904 (119905 + Δ119905) = 119895 | 119904 (119905) = 119894

=

120582

119894119895Δ119905 + 119900 (Δ119905) 119894 = 119895

1 + 120582

119894119894Δ119905 + 119900 (Δ119905) 119894 = 119895

(2)

where 119900(Δ119905) is such that limΔ119905rarr0

(119900(Δ119905)Δ119905) = 0Our primary goal here is to design an optimal controller

which will result in the minimum of a specified performanceFor infinite-horizon control problem the performance indexis specified by

119869 (119906)

= lim119905119891rarrinfin

1

119905

119891

119864[int

119905119891

0

119871 (119909 (120591) (120591) 119906 (120591) 119904 (120591)) 119889120591]

(3)

where 119864[sdot] denotes an expectation operation 119905119891is the termi-

nal time of control 119871(119909(120591) (120591) 119906(120591) 119904(120591)) is cost functionIf random excitations 119882

119896(119905) and Markovian jump process

119904(119905) are ergodic then the response of controlled systemapproaches to stationary and ergodic as 119905

119891rarr infin In this

case the performance index (3) is rewritten as

119869 (119906) = lim119905119891rarrinfin

1

119905

119891

int

119905119891

0

119871 (119909 (120591) (120591) 119906 (120591) 119904 (120591)) 119889120591 (4)

Equations (1) and (4) constitute the mathematical for-mulation of the infinite-horizon optimal control problem fora SDOF nonlinear system undergoing Markovian jump Inthis paper we consider the perfect observation case whichmeans that the system state (119909(119905) (119905)) and values of 119904(119905) areall available at each time 119905

3 Partially Averaged Equation

Let 119902 = 119909 and 119901 = (1) can be converted to the following Itostochastic differential equations [11 16]

119889119902 = 119901119889119905

119889119901 = [minus119892 (119902) + 120576119891 (119902 119901 119904 (119905)) + 120576119906 (119902 119901 119904 (119905))]

+ 120576

12120590

119896(119902 119904 (119905)) 119889119861

119896(119905)

(5)

where 119861119896(119905) are independent standard Wiener processes

119902 and 119901 are the generalized displacement and generalizedmomentum respectively 120590

119896(119902 119904(119905)) = radic2119863

119896ℎ

119896(119902 119904(119905)) The

Hamiltonian function 119867 (total energy) of the Hamiltoniansystem associated with system (5) is

119867(119902 119901) =

119901

2

2

+ int

119902

0

119892 (119906) 119889119906(6)

Suppose that 119904(119905) is a slow jump process and independent ofsystemrsquos state That means that for original system (1) or (5)the system jumps among the 119897 forms or modes slowly andindependently We consider one of the forms by letting theMarkovian jumpprocess be arbitrarily fixed at 119904(119905) = 119894 (119894 isin 119878)When 119904(119905) = 119894 denote 119891(119894)(119902 119901) = 119891(119902 119901 119904(119905)) 119906(119894)(119902 119901) =119906(119902 119901 119904(119905)) and 120590(119894)

119896(119902) = 120590

119896(119902 119904(119905)) Since119867 is a function of

119902 and 119901 the Ito equation for 119867 can be derived from (5) byusing Ito differential rule as follows [17]

119889119867 = 120576 [119891

(119894)(119902 119901)

120597119867

120597119901

+

1

2

120590

(119894)

119896(119902) 120590

(119894)

119896(119902)

120597

2119867

120597119901

2

+ 120576119906

(119894)(119902 119901)

120597119867

120597119901

] + 120576

12120590

(119894)

119896(119902)

120597119867

120597119901

119889119861

119896(119905)

(7)

In the case of light damping and weak excitations theHamiltonian 119867 in system (7) is a slowly varying processwhile the generalized displacement 119902 in (5) is a rapidlyvarying process Since the slowly varying process is essentialfor describing the long-term behavior of the system thestochastic averagingmethod is used to average out the rapidlyvarying process and to yield the following partially averagedIto equation for119867 [13 18 19]

119889119867 = 120576 [119898

(119894)(119867) + 119898

(119894)

119906(119867)] 119889119905 + 120576

12120590

(119894)(119867) 119889119861 (119905) (8)

Shock and Vibration 3

where

119898

(119894)(119867) =

1

119879 (119867)

sdot int

Ω

[119891

(119894)(119902 119901)

120597119867

120597119901

+

1

2

120590

(119894)

119896(119902) 120590

(119894)

119896(119902)

120597

2119867

120597119901

2]

sdot (

120597119867

120597119901

)

minus1

119889119902

119898

(119894)

119906(119867) = ⟨119906

(119894)(119902 119901)

120597119867

120597119901

⟩

120590

(119894)(119902 119901)

2

=

1

119879 (119867)

int

Ω

[120590

(119894)

119896(119902) 120590

(119894)

119896(119902) (

120597119867

120597119901

)

2

]

sdot (

120597119867

120597119901

)

minus1

119889119902

⟨sdot⟩ =

1

119879 (119867)

int

Ω

[sdot] (

120597119867

120597119901

)

minus1

119889119902

(9)

The region of integration is Ω = 119902 | 119867(119902 0) le 119867 and theparameter

119879 (119867) = int

Ω

(

120597119867

120597119901

)

minus1

119889119902(10)

Note that the second term 119898(119894)119906(119867) on the right-hand of (8)

is unknown since 119906(119894) are unknown function of 119902 119901 and 119904 atthis stage

Equation (8) is only valid when 119904(119905) = 119894 As Markovianjumps are allowed so that 119904(119905) takes values from 119878 = 1 2 119897 (8) can be extended so that

119889119867 = 120576 [119898 (119867 119904) + 119898

119906(119867 119904)] 119889119905

+ 120576

12120590 (119867 119904) 119889119861 (119905)

(11)

where119898(119867 119904)119898119906(119867 119904) and 120590(119867 119904) change as 119904(119905) jumps so

that119898(119867 119904(119905) = 119894) = 119898(119894)(119867)119898119906(119867 119904(119905) = 119894) = 119898

(119894)

119906(119867) and

120590(119867 119904(119905) = 119894) = 120590

(119894)(119867)

To be consistent with the partially averaged equation (11)performance index (4) is also partially averaged that is

119869 (119906) = lim119905119891rarrinfin

1

119905

119891

int

119905119891

0

119871 (119867 (120591) ⟨119906 (120591)⟩ 119904 (120591)) 119889120591 (12)

The original optimal control problem (1) and (4) has beenconverted to the partially averaged optimal control problem(11) and (12) The partially averaged optimal control problemis much simpler than the original one since (11) is only one-dimensional It has been proved [20] that the optimal controlfor the partially averaged control problem is quasi-optimalcontrol for the original one

4 Optimal Control Law

For the proposed control problem we introduce the valuefunction

119881 (119867 119904) = min119906119869 (119906) (13)

According to the stochastic dynamical programmingprinciple andMarkovian jump rules the following dynamicalprogramming equation can be established [16]

min119906

1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) + 119898

119906(119867 119904)]

120597119881 (119867 119904)

120597119867

+ 119871 (119867 ⟨119906⟩ 119904)

+

119897

sum

119903=1

120582

119903119904119881 (119867 119903) = 0 119904 isin 119878

(14)

where 120582119903119904are the transition rates described in (2) Equation

(14) is a finite set of second-order partial differential equationsindexed by 119904 These equations are coupled through the zero-order terms sum119897

119903=1120582

119903119904119881(119867 119903)

The necessary condition for minimizing the left-handside of (14) is

120597

120597119906

119898

119906(119867 119904)

120597119881

120597119867

+ 119871 (119867 ⟨119906 (119902 119901 119904)⟩ 119904) = 0 (15)

Let the cost function 119871 be of the form

119871 (119867 ⟨119906⟩ 119904) = 119891 (119867 119904) + 119877 ⟨119906

2⟩ (16)

where 119891(119867 119904) ge 0 and 119877 is a positive constant By substitut-ing (16) into (15) and exchanging the order of the derivativewith respect to 119906 and averaging in (15) the expression of theoptimal control force is determined

119906

lowast(119902 119901 119904) = minus

1

2119877

120597119881 (119867 119904)

120597119867

120597119867

120597119901

119904 isin 119878 (17)

Note that 119881(119867 119904) is generally a nonlinear functional of 119867Hence 119906lowast(119902 119901 119904) is a nonlinear function of 119902 and 119901 thatis the optimal feedback control is nonlinear To ensure thatthe control force 119906lowast(119902 119901 119904) is indeed optimal the followingsufficient condition should be satisfied

120597

2

120597119906

2119871 (119867 ⟨119906⟩ 119904) ge 0

(18)

Substituting the expression for 119906lowast(119902 119901 119904) in (17) into (14)yields the final dynamical programming equation

1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) + 119898

lowast

119906(119867 119904)]

120597119881 (119867 119904)

120597119867

+ 119871 (119867 ⟨119906

lowast⟩ 119904) +

119897

sum

119903=1

120582

119903119904119881 (119867 119903) = 0

(19)

4 Shock and Vibration

where

119898

lowast

119906(119867 119904) = ⟨119906

lowast(119902 119901 119904)

120597119867

120597119901

⟩

= minus

1

2119877

⟨

120597119881 (119867 119904)

120597119867

(

120597119867

120597119901

)

2

⟩

(20)

The value function 119881(119867 119904) can be obtained by solving(19) The optimal control force 119906lowast(119902 119901 119904) is then determinedby substituting 119881(119867 119904) into expression (17) By inserting theoptimal control force 119906lowast(119902 119901 119904) into (11) and averaging theterm ⟨119906lowast(119902 119901 119904)(120597119867120597119901)⟩ the following completely averagedIto equation is obtained

119889119867 = 120576 [119898 (119867 119904) + 119898

lowast

119906(119867 119904)] 119889119905

+ 120576

12120590 (119867 119904) 119889119861 (119905)

(21)

The FPK equation associated with the fully averaged Itoequation (21) is [4 21 22]

120597

120597119905

119901 (119867 119904 119905)

= minus

120597

120597119867

[119898 (119867 119904) + 119898

lowast

119906(119867 119904)] 119901 (119867 119904 119905)

+

1

2

120597

2

120597119867

2[120590 (119867 119904)

2119901 (119867 119904 119905)]

minus

119897

sum

119903=1

119903 =119904

[120582

119904119903119901 (119867 119904 119905) minus 120582

119903119904119901 (119867 119903 119905)] 119904 isin 119878

(22)

where 119901(119867 119904 119905) is the probability density function of totalenergy119867 for optimally controlled system (1) while the systemoperating in 119904th form (119901(119867 119904 119905) for uncontrolled system canbe obtained by letting 119906lowast(119902 119901 119904) = 0) The initial condition is

119901 (119867 119904 0) = 119901 (119867

0 119904) (23)

and boundary conditions are

119901 (0 119904 119905) = finite

119901 (119867 119904 119905)

1003816

1003816

1003816

1003816119867rarrinfin997888rarr 0

120597119901 (119867 119904 119905)

120597119867

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119867rarrinfin

997888rarr 0

(24)

The FPK equation (22) is a finite set of second-order par-tially differential equations which are coupled through thezero-order terms Equation (22) does not admit easy solutionanalytically or numerically However in practical applicationwe are more interested in the stationary solution of FPKequation (22) In this case FPK equation (22) is reduced byletting 120597119901120597119905 = 0 Then the stationary probability density119901(119867 119904) is obtained readily from solving (22) numericallyThestationary probability density 119901(119867) can be obtained from119901(119867 119904) as follows

119901 (119867) = 119888

119897

sum

119904=1

119901 (119867 119904) (25)

where 119888 is a normalization constant The stationary probabil-ity density119901(119902) of the generalized displacement and themeanvalue 119864[119867] of the total energy are then obtained as

119901 (119902) = int

infin

minusinfin

119901 (119902 119901) 119889119901

119864 [119867] = int

infin

0

119867119901 (119867) 119889119867

(26)

where

119901 (119902 119901) =

119901 (119867)

119879 (119867)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119867=119867(119902119901)

(27)

is the stationary joint probability density of the generalizeddisplacement and generalized momentum

5 Numerical Example

To demonstrate the application and to assess the effective-ness of the proposed control strategy numerical results areobtained for two examples

51 Example 1 Consider a controlled van der Pol-Rayleighoscillator subjected to both external and parametric randomexcitations which is capable of independentMarkovian jumpand governed by

+ 119909 = 120572 (119904 (119905)) minus [120573 (119904 (119905)) 119909

2+ 120574 (119904 (119905))

2]

+ 119906 (119909 119904 (119905)) + ℎ

1(119904 (119905))119882

1(119905)

+ ℎ

2(119904 (119905)) 119909119882

2(119905)

(28)

where 120572(119904(119905)) 120573(119904(119905)) and 120574(119904(119905)) are Markovian jumpcoefficients of linear and quasi-linear damping respectivelyℎ

1(119904(119905)) and ℎ

2(119904(119905))119909 denote Markovian jump coefficients

of external and parametric random excitations respectively119906(119909 119904(119905)) is the feedback control force 119882

119896(119905) (119896 = 1 2)

are independent Gaussian white noises with zero mean andintensities 2119863

119896 119904(119905) is a finite-state continuous-time Markov

process pointing to the systemrsquos form with the transitionprobability defined in (2)

We consider the 3-form case here that means 119897 = 3 and119878 = 1 2 3 Prescribe the transition rate 120582

119894119895(119894 119895 = 1 2 3)

between the form 119894 and the form 119895 by a transition matrix Λ =[120582

119894119895]Let 119902 = 119909 and 119901 = The Hamiltonian associated with

system (28) is

119867 =

1

2

119901

2+

1

2

119902

2 (29)

As a Hamiltonian system (28) can be expressed as

[

119901

] = [119901 minus 119902

+ 120572 (119904 (119905)) minus [120573 (119904 (119905)) 119902

2+ 120574 (119904 (119905)) 119901

2] 119901

+ 119906 (119902 119901 119904 (119905)) + ℎ

1(119904 (119905))119882

1(119905)

+ ℎ

2(119904 (119905)) 119902119882

2(119905)]

(30)

Shock and Vibration 5

Upon stochastic averaging (11) for system (28) isobtained for whichΩ = 119902 | 119867(119902 0) le 119867 and

119898(119867 119904 (119905) = 119894) = minus [

120573

(119894)

2

+

3

2

120574

(119894)]119867

2+ ℎ

(119894)

1

2

119863

1

+ (120572

(119894)+ ℎ

(119894)

2

2

119863

2)119867

119898

119906(119867 119904 (119905) = 119894) = ⟨119906

(119894)(119902 119901) 119901⟩

120590

2(119867 119904 (119905) = 119894) = 2 ℎ

(119894)

1

2

119863

1119867 + ℎ

(119894)

2

2

119863

2119867

2

119894 = 1 2 3

(31)

For the proposed control strategy the partially averagedcost function 119871 is of the form of (16) with

119891 (119867 119904) = 119888

0(119904) + 119888

1(119904)119867 + 119888

2(119904)119867

2+ 119888

3(119904)119867

3 (32)

Following (17) the expression of the optimal control forceis

119906

lowast(119902 119901 119904) = minus

1

2119877

120597119881 (119867 119904)

120597119867

119901(33)

Substituting 119906lowast(119902 119901 119904) in (33) into the dynamical pro-gramming equation (14) completing the averaging yields thefollowing final coupled dynamical programming equations

1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) minus

1

4119877

120597119881 (119867 119904)

120597119867

⟨(

120597119867

120597119901

)

2

⟩]

120597119881 (119867 119904)

119889119867

+ 119891 (119867 119904) +

3

sum

119903=1

120582

119903119904119881 (119867 119903) = 0 119904 = 1 2 3

(34)

To solve (34) the value function 119881(119867 119904) is supposed tobe of the following form

119881 (119867 119904) = 119887

0(119904) + 119887

1(119904)119867 + 119887

2(119904)119867

2 (35)

Since the optimal control force 119906lowast is function of 120597119881120597119867only the coefficients 119887

1(119904) and 119887

2(119904) are determined by insert-

ing (32) and (35) into (34)

119887

1(119904 = 119894) = 119887

(119894)

1

=

2 (119888

2+ sum

3

119896=1120582

119896119894119887

(119896)

2+ 2119887

(119894)

2120572

(119894)+ 2119863

2ℎ

(119894)

2

2

119887

(119894)

2)119877

2119887

(119894)

2+ 120573

(119894)119877 + 3120574

(119894)119877

119887

2(119904 = 119894) = 119887

(119894)

2

= minus

1

2

120573

(119894)minus

3

2

120574

(119894)119877

+

1

2

radic

(120573

(119894))

2

119877

2+ 6120573

(119894)119877

2120574

(119894)+ 9 (120574

(119894))

2

119877

2+ 4119888

3119877

119894 = 1 2 3

(36)

Uncontrolled

Optimally controlled

1 2 3 4 500

1

2

3

4

5

6

7

H

p(H

)

Figure 1 Stationary probability density 119901(119867) of total energy ofuncontrolled and optimally controlled system (28) The lines areobtained fromnumerical solution of (37) while the dots are obtainedfrom direct simulation of original system (28)

By substituting (35) into (17) with the coefficients deter-mined by (36) the optimal control force is determinedThenthe following completely averaged FPK equation is obtained

120597

120597119905

119901 (119867 119904 119905)

= minus

120597

120597119867

[119898 (119867 119904) + 119898

lowast

119906(119867 119904)] 119901 (119867 119904 119905)

+

1

2

120597

2

120597119867

2[120590 (119867 119904)

2119901 (119867 119904 119905)]

minus

3

sum

119903=1

119903 =119904

[120582

119904119903119901 (119867 119904 119905) minus 120582

119903119904119901 (119867 119903 119905)]

119904 = 1 2 3

(37)

where

119898

lowast

119906(119867 119904 = 119894) = minus

1

2119877

(119887

(119894)

1+ 2119887

(119894)

2119867)119867 (38)

Solving (37) numerically with 120597119901120597119905 = 0 the stationaryprobability density 119901(119867 119904) of optimal controlled system canbe obtained The stationary probability densities 119901(119867) of thetotal energy 119901(119902) of the generalized displacement and themean value 119864[119867] of the total energy are then obtained using(25) and (26)

Some numerical results are obtained as shown in Figures1ndash4 for system parameters 119863

1= 003 119863

2= 003 119877 = 10

6 Shock and Vibration

0 2 40

02

04

06

08

1

12

Uncontrolled

Optimally controlled

q

minus4 minus2

p(q)

Figure 2 Stationary probability density 119901(119902) of displacement ofuncontrolled and optimally controlled system (28) The lines areobtained fromnumerical solution of (37) while the dots are obtainedfrom direct simulation of original system (28)

1 15 2 25 30

05

1

15

2

Uncontrolled

Optimally controlled

E[H

]

1205821112058222

Figure 3Mean value119864[119867] of total energy of uncontrolled and opti-mally controlled system (28) as functions of the ratio 120582

11120582

22(120582

22=

120582

33) The lines are obtained from numerical solution of (37) while

the dots are obtained from direct simulation of original system (28)

2000 2100 2200 2300 2400 2500 2600

0

2

4

6

Uncontrolled

Optimally controlled

q

t

minus4

minus6

minus2

Figure 4 The sample time history of displacement of uncontrolledand optimally controlled system (28)

119904

1= 119904

2= 119904

3= 10 Λ = [

minus12058211 120582112 120582112

120582222 minus12058222 120582222

120582332 120582332 minus12058233

] 12058211= 120582

22= 120582

33=

30 and

120572

(1)= 001

120573

(1)= 001

120574

(1)= 001

ℎ

(1)

1= 20

ℎ

(1)

2= 20

120572

(2)= 001

120573

(2)= 002

120574

(2)= 002

ℎ

(2)

1= 20

ℎ

(2)

2= 10

120572

(3)= 001

120573

(3)= 004

120574

(3)= 004

ℎ

(3)

1= 10

ℎ

(3)

2= 10

(39)

The stationary probability density 119901(119867) of the totalenergy for uncontrolled and optimally controlled system(28) is shown in Figure 1 Obviously as 119867 increases 119901(119867)for optimally controlled system (28) decreases much fasterthan that for uncontrolled system The stationary probabilitydensity 119901(119902) of the displacement is plotted in Figure 2 withthe same parameters as in Figure 1 For optimally controlledsystem there is a much higher probability that 119902 will belocated near their equilibrium Hence 119901(119902) has much largermode and smaller dispersion around 119902 = 0 for optimallycontrolled system than those for uncontrolled system Thisimplies that the proposed control strategy has high effective-ness for attenuating the response of system (28)

Figure 3 shows the mean value 119864[119867] of the total energyof uncontrolled and optimally controlled system as functionsof ratio 120582

11120582

22(120582

22= 120582

33) Obviously higher value of ratio

120582

11120582

22implies lower probability for the system operating in

form 119904(119905) = 1 and higher probability for the system oper-ating in forms 119904(119905) = 2 and 119904(119905) = 3 When the system oper-ates in form 119904(119905) = 1 the system has the smallest dampingcoefficients and the largest amplitudes of stochastic excita-tions than those when it operates in forms 119904(119905) = 2 and119904(119905) = 3 As a consequence 119864[119867] for uncontrolled systemdecreases monotonously while the ratio 120582

11120582

22increases as

shown in Figure 3 However 119864[119867] for optimally controlledsystem always keeps a very low value despite changing of

Shock and Vibration 7

the ratio 12058211120582

22 which indicates that the proposed control

strategy is very effective and quite robust with respect tothe ratio 120582

11120582

22 In Figures 1ndash3 lines are obtained by

numerical solution of (37) while dots are obtained by directsimulation of system (28)Observe that the dotsmatch closelywith the corresponding lines demonstrating the validity andaccuracy of the proposed method The displacements of theuncontrolled and optimally controlled system (28) are shownin Figure 4 from which the effect of the optimal control onthe displacement can be visualized intuitively

52 Example 2 Consider a controlled stochastically excitedDuffing oscillator withMarkovian jumpwhich is governed by

+ 120596

2119909 + 120572119909

3= 120573 (119904 (119905)) + 119906 (119909 119904 (119905))

+ ℎ (119904 (119905)) 120585 (119905)

(40)

where 120573(119904(119905)) is Markovian jump coefficient of linear damp-ing ℎ(119904(119905)) is Markovian jump coefficient of external randomexcitation 119906(119909 119904(119905)) is the feedback control force 120585(119905) isGaussian white noise with zero mean and intensity 2119863 Herewe consider the 2-form case 119897 = 2 and 119878 = 1 2

Upon the stochastic averaging method the original jumpsystem (40) can be approximately substituted by the partiallyaveraged system (11) with

119898(119867 119904 = 119894) = ℎ

(119894)

2

119863 minus 120573

(119894)119866 (119867)

119898

119906(119867 119904 = 119894) =

4

119879 (119867)

int

119860(119867)

0

119906

(119894)(119902 119901) 119901 119889119902

120590 (119867 119904 = 119894)

2= 2 ℎ

(119894)

2

119863119866 (119867)

119866 (119867) =

4

119879 (119867)

int

119860(119867)

0

(2119867 minus 120596

2119902

2minus

120572119902

4

2

)

12

119889119902

119879 (119867) = 4int

119860(119867)

0

(2119867 minus 120596

2119902

2minus

120572119902

4

2

)

minus12

119889119902

119860 (119867) = [

(120596 + 4120572119867)

12minus 120596

2

120572

]

12

119894 = 1 2

(41)

Following (17) the expression of the optimal control forceis

119906

lowast(119902 119901 119904) = minus

1

2119877

120597119881 (119867 119904)

120597119867

119901(42)

Then the following final dynamical programming equa-tion is obtained1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) minus

1

4119877

120597119881 (119867 119904)

120597119867

119866 (119867)]

120597119881 (119867 119904)

119889119867

+ 119891 (119867 119904) +

2

sum

119903=1

120582

119903119904119881 (119867 119903) = 0 119904 = 1 2

(43)

120597119881120597119867 can be obtained from solving (43) numericallyThen the optimal control force is determined by substituting120597119881120597119867 into (17) Substituting the optimal control force (17)into the partially averaged equation (11) and completingthe averaging yield the following completely averaged FPKequation

120597

120597119905

119901 (119867 119904 119905)

= minus

120597

120597119867

[119898 (119867 119904) minus

1

2119877

120597119881 (119867 119904)

120597119867

119866 (119867)]

sdot 119901 (119867 119904 119905) +

1

2

120597

2

120597119867

2[120590

2(119867 119904) 119901 (119867 119904 119905)]

minus

2

sum

119903=1

119903 =119904

[120582

119904119903119901 (119867 119904 119905) minus 120582

119903119904119901 (119867 119903 119905)] 119904 = 1 2

(44)

Solving (44) numerically with 120597119901120597119905 = 0 the statistics119901(119867) 119901(119902) and 119864[119867] of the stationary response are thenobtained from (25) and (26)

The numerical results shown in Figures 5ndash8 are for systemwith parameters 120596 = 10 120572 = 05 119863 = 002 Λ = [ minus12058211 12058211

12058222 minus120582122]

120582

11= 120582

22= 20 and

120573

(1)= 001

ℎ

(1)= 20

120573

(2)= 002

ℎ

(2)= 10

(45)

The stationary probability densities 119901(119867) of total energyand 119901(119902) of the displacement of uncontrolled and optimallycontrolled system (40) are evaluated and plotted in Figures5 and 6 respectively The mean value 119864[119867] of total energyof uncontrolled and optimally controlled system (40) asfunctions of ratio 120582

11120582

22is shown in Figure 7 It is seen that

the optimal control force alleviates the response of system(40) significantly This effect can be seen more apparentlyfrom Figure 8 Again the analytical results obtained fromsolving (44) match closely with those from digital simulationof original system (40)

6 Conclusions

In this paper we proposed a nonlinear stochastic optimalcontrol strategy for SDOF stochastically excited Markovianjump systems where the jump parameter takes on values in afinite discrete set In the slow jump case the original systemwas first reduced to one governed by a finite set of one-dimensional partially averaged Ito equation for total energyupon stochastic averaging Using the stochastic dynami-cal programming principle and Markovian jump rules afinite set of coupled dynamical programming equations wasderived from which the optimal control force was obtainedThe obtained optimal control force has been proved to be the

8 Shock and Vibration

0 1 2 3 4 50

2

4

6

8

10

Uncontrolled

Optimally controlled

H

p(H

)

Figure 5 Stationary probability density 119901(119867) of total energy ofuncontrolled and optimally controlled system (40) The lines areobtained fromnumerical solution of (44)while the dots are obtainedfrom direct simulation of original system (40)

0 1 2 30

02

04

06

08

1

12

14

16

Uncontrolled

Optimally controlled

q

minus3 minus2 minus1

p(q)

Figure 6 Stationary probability density 119901(119902) of displacement ofuncontrolled and optimally controlled system (40) The lines areobtained fromnumerical solution of (44)while the dots are obtainedfrom direct simulation of original system (40)

quasi-optimal control for the original system The strategywas applied to two nonlinear oscillators that were capableof independent Markovian jumps Numerical results showedthat the strategy is fairly robust and effective in reductionof stationary response of the controlled system Thus it ispotentially promising for practical control applications afterfurther research

The proposed method has the potential to be extended tothe optimal control forMulti-DOFMarkovian jump systemsHowever to obtain the concrete conclusionmuchwork has tobe done Mathematical tools should be developed to solve thedynamical programming equation (14) for Multi-DOF case

1 15 2 25 30

1

2

3

4

5

Uncontrolled

Optimally controlled

E[H

]

1205821112058222

Figure 7 Mean value 119864[119867] of total energy of uncontrolled andoptimally controlled system (40) as functions of the ratio 120582

11120582

22

The lines are obtained fromnumerical solution of (44) while the dotsare obtained from direct simulation of original system (40)

2000 2050 2100 2150 2200

0

5

Uncontrolled

Optimally controlled

q

t

minus5

Figure 8The sample time history of displacements of uncontrolledand optimally controlled system (40)

Disclaimer

Opinions findings and conclusions expressed in this paperare those of the authors and do not necessarily reflect theviews of the sponsors

Conflict of Interests

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

Acknowledgments

This work was supported by the Natural Science Foundationof China (through Grants no 11272279 11372271 1143201211321202 and 11272201) and 973 Program (no 2011CB711105)

Shock and Vibration 9

References

[1] L Kats and N N Krasovskii ldquoOn the stability of systemswith random parametersrdquo Journal of Applied Mathematics andMechanics vol 24 no 5 pp 1225ndash1246 1960

[2] M Mariton ldquoAlmost sure and moments stability of jump linearsystemsrdquo Systems amp Control Letters vol 11 no 5 pp 393ndash3971988

[3] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[4] R H Huan W-Q Zhu F Ma and Z-G Ying ldquoStationaryresponse of a class of nonlinear stochastic systems undergoingMarkovian jumpsrdquo Journal of Applied Mechanics vol 82 no 5Article ID 051008 6 pages 2015

[5] N N Krasosvkii and E A Lidskii ldquoAnalytical design of con-trollers in systems with random attributes Irdquo Automation andRemote Control vol 22 pp 1021ndash1025 1961

[6] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967

[7] O L V Costa M D Fragoso and M G Todorov Continuous-Time Markov Jump Linear Systems Springer New York NYUSA 2013

[8] D D Sworder ldquoFeedback control of a class of linear systemswith jump parametersrdquo IEEE Transactions on Automatic Con-trol vol 14 no 1 pp 9ndash14 1969

[9] W M Wonham ldquoRandom differential equations in controltheoryrdquoProbabilisticMethods in AppliedMathematics vol 2 pp131ndash213 1971

[10] M D Fragoso and E M Hemerly ldquoOptimal control for a classof noisy linear systemswithMarkovian jumping parameters andquadratic costrdquo International Journal of Systems Science vol 22no 12 pp 2553ndash2561 1991

[11] Y D Ji and H J Chizeck ldquoJump linear quadratic Gaussian con-trol in continuous timerdquo IEEE Transactions on AutomaticControl vol 37 no 12 pp 1884ndash1892 1992

[12] M K Ghosh A Arapostathis and S I Marcus ldquoErgodiccontrol of switching diffusionsrdquo SIAM Journal on Control andOptimization vol 35 no 6 pp 1952ndash1988 1997

[13] W Q Zhu ldquoNonlinear stochastic dynamics and control inHamiltonian formulationrdquo Applied Mechanics Reviews vol 59no 4 pp 230ndash248 2006

[14] E K Boukas and H Yang ldquoStability of stochastic systems withjumpsrdquoMathematical Problems in Engineering vol 3 no 2 pp173ndash185 1996

[15] X B Feng K A Loparo Y D Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[16] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions Springer New York NY USA 1993

[17] K Ito ldquoOn stochastic differential equationsrdquo Memoirs of theAmerican Mathematical Society vol 4 pp 289ndash302 1951

[18] W Q Zhu and Z G Ying ldquoOptimal nonlinear feedbackcontrol of quasi-Hamiltonian systemsrdquo Science in China SeriesA Mathematics vol 42 no 11 pp 1213ndash1219 1999

[19] W Q Zhu Z G Ying and T T Soong ldquoAn optimal nonlinearfeedback control strategy for randomly excited structural sys-temsrdquo Nonlinear Dynamics vol 24 no 1 pp 31ndash51 2001

[20] H J Kushner andD S Clark Stochastic ApproximationMethodsfor Constrained andUnconstrained Systems Springer NewYorkNY USA 1978

[21] YW Fang Y LWu andHQWangOptimal ControlTheory ofStochastic Jump System National Defense Inustry Press BeijingChina 2012

[22] S T WuTheTheory of Stochastic Jump System and Its Applica-tion Science Press Beijing China 2007

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International Journal of

Shock and Vibration 3

where

119898

(119894)(119867) =

1

119879 (119867)

sdot int

Ω

[119891

(119894)(119902 119901)

120597119867

120597119901

+

1

2

120590

(119894)

119896(119902) 120590

(119894)

119896(119902)

120597

2119867

120597119901

2]

sdot (

120597119867

120597119901

)

minus1

119889119902

119898

(119894)

119906(119867) = ⟨119906

(119894)(119902 119901)

120597119867

120597119901

⟩

120590

(119894)(119902 119901)

2

=

1

119879 (119867)

int

Ω

[120590

(119894)

119896(119902) 120590

(119894)

119896(119902) (

120597119867

120597119901

)

2

]

sdot (

120597119867

120597119901

)

minus1

119889119902

⟨sdot⟩ =

1

119879 (119867)

int

Ω

[sdot] (

120597119867

120597119901

)

minus1

119889119902

(9)

The region of integration is Ω = 119902 | 119867(119902 0) le 119867 and theparameter

119879 (119867) = int

Ω

(

120597119867

120597119901

)

minus1

119889119902(10)

Note that the second term 119898(119894)119906(119867) on the right-hand of (8)

is unknown since 119906(119894) are unknown function of 119902 119901 and 119904 atthis stage

Equation (8) is only valid when 119904(119905) = 119894 As Markovianjumps are allowed so that 119904(119905) takes values from 119878 = 1 2 119897 (8) can be extended so that

119889119867 = 120576 [119898 (119867 119904) + 119898

119906(119867 119904)] 119889119905

+ 120576

12120590 (119867 119904) 119889119861 (119905)

(11)

where119898(119867 119904)119898119906(119867 119904) and 120590(119867 119904) change as 119904(119905) jumps so

that119898(119867 119904(119905) = 119894) = 119898(119894)(119867)119898119906(119867 119904(119905) = 119894) = 119898

(119894)

119906(119867) and

120590(119867 119904(119905) = 119894) = 120590

(119894)(119867)

To be consistent with the partially averaged equation (11)performance index (4) is also partially averaged that is

119869 (119906) = lim119905119891rarrinfin

1

119905

119891

int

119905119891

0

119871 (119867 (120591) ⟨119906 (120591)⟩ 119904 (120591)) 119889120591 (12)

The original optimal control problem (1) and (4) has beenconverted to the partially averaged optimal control problem(11) and (12) The partially averaged optimal control problemis much simpler than the original one since (11) is only one-dimensional It has been proved [20] that the optimal controlfor the partially averaged control problem is quasi-optimalcontrol for the original one

4 Optimal Control Law

For the proposed control problem we introduce the valuefunction

119881 (119867 119904) = min119906119869 (119906) (13)

According to the stochastic dynamical programmingprinciple andMarkovian jump rules the following dynamicalprogramming equation can be established [16]

min119906

1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) + 119898

119906(119867 119904)]

120597119881 (119867 119904)

120597119867

+ 119871 (119867 ⟨119906⟩ 119904)

+

119897

sum

119903=1

120582

119903119904119881 (119867 119903) = 0 119904 isin 119878

(14)

where 120582119903119904are the transition rates described in (2) Equation

(14) is a finite set of second-order partial differential equationsindexed by 119904 These equations are coupled through the zero-order terms sum119897

119903=1120582

119903119904119881(119867 119903)

The necessary condition for minimizing the left-handside of (14) is

120597

120597119906

119898

119906(119867 119904)

120597119881

120597119867

+ 119871 (119867 ⟨119906 (119902 119901 119904)⟩ 119904) = 0 (15)

Let the cost function 119871 be of the form

119871 (119867 ⟨119906⟩ 119904) = 119891 (119867 119904) + 119877 ⟨119906

2⟩ (16)

where 119891(119867 119904) ge 0 and 119877 is a positive constant By substitut-ing (16) into (15) and exchanging the order of the derivativewith respect to 119906 and averaging in (15) the expression of theoptimal control force is determined

119906

lowast(119902 119901 119904) = minus

1

2119877

120597119881 (119867 119904)

120597119867

120597119867

120597119901

119904 isin 119878 (17)

Note that 119881(119867 119904) is generally a nonlinear functional of 119867Hence 119906lowast(119902 119901 119904) is a nonlinear function of 119902 and 119901 thatis the optimal feedback control is nonlinear To ensure thatthe control force 119906lowast(119902 119901 119904) is indeed optimal the followingsufficient condition should be satisfied

120597

2

120597119906

2119871 (119867 ⟨119906⟩ 119904) ge 0

(18)

Substituting the expression for 119906lowast(119902 119901 119904) in (17) into (14)yields the final dynamical programming equation

1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) + 119898

lowast

119906(119867 119904)]

120597119881 (119867 119904)

120597119867

+ 119871 (119867 ⟨119906

lowast⟩ 119904) +

119897

sum

119903=1

120582

119903119904119881 (119867 119903) = 0

(19)

4 Shock and Vibration

where

119898

lowast

119906(119867 119904) = ⟨119906

lowast(119902 119901 119904)

120597119867

120597119901

⟩

= minus

1

2119877

⟨

120597119881 (119867 119904)

120597119867

(

120597119867

120597119901

)

2

⟩

(20)

The value function 119881(119867 119904) can be obtained by solving(19) The optimal control force 119906lowast(119902 119901 119904) is then determinedby substituting 119881(119867 119904) into expression (17) By inserting theoptimal control force 119906lowast(119902 119901 119904) into (11) and averaging theterm ⟨119906lowast(119902 119901 119904)(120597119867120597119901)⟩ the following completely averagedIto equation is obtained

119889119867 = 120576 [119898 (119867 119904) + 119898

lowast

119906(119867 119904)] 119889119905

+ 120576

12120590 (119867 119904) 119889119861 (119905)

(21)

The FPK equation associated with the fully averaged Itoequation (21) is [4 21 22]

120597

120597119905

119901 (119867 119904 119905)

= minus

120597

120597119867

[119898 (119867 119904) + 119898

lowast

119906(119867 119904)] 119901 (119867 119904 119905)

+

1

2

120597

2

120597119867

2[120590 (119867 119904)

2119901 (119867 119904 119905)]

minus

119897

sum

119903=1

119903 =119904

[120582

119904119903119901 (119867 119904 119905) minus 120582

119903119904119901 (119867 119903 119905)] 119904 isin 119878

(22)

where 119901(119867 119904 119905) is the probability density function of totalenergy119867 for optimally controlled system (1) while the systemoperating in 119904th form (119901(119867 119904 119905) for uncontrolled system canbe obtained by letting 119906lowast(119902 119901 119904) = 0) The initial condition is

119901 (119867 119904 0) = 119901 (119867

0 119904) (23)

and boundary conditions are

119901 (0 119904 119905) = finite

119901 (119867 119904 119905)

1003816

1003816

1003816

1003816119867rarrinfin997888rarr 0

120597119901 (119867 119904 119905)

120597119867

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119867rarrinfin

997888rarr 0

(24)

The FPK equation (22) is a finite set of second-order par-tially differential equations which are coupled through thezero-order terms Equation (22) does not admit easy solutionanalytically or numerically However in practical applicationwe are more interested in the stationary solution of FPKequation (22) In this case FPK equation (22) is reduced byletting 120597119901120597119905 = 0 Then the stationary probability density119901(119867 119904) is obtained readily from solving (22) numericallyThestationary probability density 119901(119867) can be obtained from119901(119867 119904) as follows

119901 (119867) = 119888

119897

sum

119904=1

119901 (119867 119904) (25)

where 119888 is a normalization constant The stationary probabil-ity density119901(119902) of the generalized displacement and themeanvalue 119864[119867] of the total energy are then obtained as

119901 (119902) = int

infin

minusinfin

119901 (119902 119901) 119889119901

119864 [119867] = int

infin

0

119867119901 (119867) 119889119867

(26)

where

119901 (119902 119901) =

119901 (119867)

119879 (119867)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119867=119867(119902119901)

(27)

is the stationary joint probability density of the generalizeddisplacement and generalized momentum

5 Numerical Example

To demonstrate the application and to assess the effective-ness of the proposed control strategy numerical results areobtained for two examples

51 Example 1 Consider a controlled van der Pol-Rayleighoscillator subjected to both external and parametric randomexcitations which is capable of independentMarkovian jumpand governed by

+ 119909 = 120572 (119904 (119905)) minus [120573 (119904 (119905)) 119909

2+ 120574 (119904 (119905))

2]

+ 119906 (119909 119904 (119905)) + ℎ

1(119904 (119905))119882

1(119905)

+ ℎ

2(119904 (119905)) 119909119882

2(119905)

(28)

where 120572(119904(119905)) 120573(119904(119905)) and 120574(119904(119905)) are Markovian jumpcoefficients of linear and quasi-linear damping respectivelyℎ

1(119904(119905)) and ℎ

2(119904(119905))119909 denote Markovian jump coefficients

of external and parametric random excitations respectively119906(119909 119904(119905)) is the feedback control force 119882

119896(119905) (119896 = 1 2)

are independent Gaussian white noises with zero mean andintensities 2119863

119896 119904(119905) is a finite-state continuous-time Markov

process pointing to the systemrsquos form with the transitionprobability defined in (2)

We consider the 3-form case here that means 119897 = 3 and119878 = 1 2 3 Prescribe the transition rate 120582

119894119895(119894 119895 = 1 2 3)

between the form 119894 and the form 119895 by a transition matrix Λ =[120582

119894119895]Let 119902 = 119909 and 119901 = The Hamiltonian associated with

system (28) is

119867 =

1

2

119901

2+

1

2

119902

2 (29)

As a Hamiltonian system (28) can be expressed as

[

119901

] = [119901 minus 119902

+ 120572 (119904 (119905)) minus [120573 (119904 (119905)) 119902

2+ 120574 (119904 (119905)) 119901

2] 119901

+ 119906 (119902 119901 119904 (119905)) + ℎ

1(119904 (119905))119882

1(119905)

+ ℎ

2(119904 (119905)) 119902119882

2(119905)]

(30)

Shock and Vibration 5

Upon stochastic averaging (11) for system (28) isobtained for whichΩ = 119902 | 119867(119902 0) le 119867 and

119898(119867 119904 (119905) = 119894) = minus [

120573

(119894)

2

+

3

2

120574

(119894)]119867

2+ ℎ

(119894)

1

2

119863

1

+ (120572

(119894)+ ℎ

(119894)

2

2

119863

2)119867

119898

119906(119867 119904 (119905) = 119894) = ⟨119906

(119894)(119902 119901) 119901⟩

120590

2(119867 119904 (119905) = 119894) = 2 ℎ

(119894)

1

2

119863

1119867 + ℎ

(119894)

2

2

119863

2119867

2

119894 = 1 2 3

(31)

For the proposed control strategy the partially averagedcost function 119871 is of the form of (16) with

119891 (119867 119904) = 119888

0(119904) + 119888

1(119904)119867 + 119888

2(119904)119867

2+ 119888

3(119904)119867

3 (32)

Following (17) the expression of the optimal control forceis

119906

lowast(119902 119901 119904) = minus

1

2119877

120597119881 (119867 119904)

120597119867

119901(33)

Substituting 119906lowast(119902 119901 119904) in (33) into the dynamical pro-gramming equation (14) completing the averaging yields thefollowing final coupled dynamical programming equations

1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) minus

1

4119877

120597119881 (119867 119904)

120597119867

⟨(

120597119867

120597119901

)

2

⟩]

120597119881 (119867 119904)

119889119867

+ 119891 (119867 119904) +

3

sum

119903=1

120582

119903119904119881 (119867 119903) = 0 119904 = 1 2 3

(34)

To solve (34) the value function 119881(119867 119904) is supposed tobe of the following form

119881 (119867 119904) = 119887

0(119904) + 119887

1(119904)119867 + 119887

2(119904)119867

2 (35)

Since the optimal control force 119906lowast is function of 120597119881120597119867only the coefficients 119887

1(119904) and 119887

2(119904) are determined by insert-

ing (32) and (35) into (34)

119887

1(119904 = 119894) = 119887

(119894)

1

=

2 (119888

2+ sum

3

119896=1120582

119896119894119887

(119896)

2+ 2119887

(119894)

2120572

(119894)+ 2119863

2ℎ

(119894)

2

2

119887

(119894)

2)119877

2119887

(119894)

2+ 120573

(119894)119877 + 3120574

(119894)119877

119887

2(119904 = 119894) = 119887

(119894)

2

= minus

1

2

120573

(119894)minus

3

2

120574

(119894)119877

+

1

2

radic

(120573

(119894))

2

119877

2+ 6120573

(119894)119877

2120574

(119894)+ 9 (120574

(119894))

2

119877

2+ 4119888

3119877

119894 = 1 2 3

(36)

Uncontrolled

Optimally controlled

1 2 3 4 500

1

2

3

4

5

6

7

H

p(H

)

Figure 1 Stationary probability density 119901(119867) of total energy ofuncontrolled and optimally controlled system (28) The lines areobtained fromnumerical solution of (37) while the dots are obtainedfrom direct simulation of original system (28)

By substituting (35) into (17) with the coefficients deter-mined by (36) the optimal control force is determinedThenthe following completely averaged FPK equation is obtained

120597

120597119905

119901 (119867 119904 119905)

= minus

120597

120597119867

[119898 (119867 119904) + 119898

lowast

119906(119867 119904)] 119901 (119867 119904 119905)

+

1

2

120597

2

120597119867

2[120590 (119867 119904)

2119901 (119867 119904 119905)]

minus

3

sum

119903=1

119903 =119904

[120582

119904119903119901 (119867 119904 119905) minus 120582

119903119904119901 (119867 119903 119905)]

119904 = 1 2 3

(37)

where

119898

lowast

119906(119867 119904 = 119894) = minus

1

2119877

(119887

(119894)

1+ 2119887

(119894)

2119867)119867 (38)

Solving (37) numerically with 120597119901120597119905 = 0 the stationaryprobability density 119901(119867 119904) of optimal controlled system canbe obtained The stationary probability densities 119901(119867) of thetotal energy 119901(119902) of the generalized displacement and themean value 119864[119867] of the total energy are then obtained using(25) and (26)

Some numerical results are obtained as shown in Figures1ndash4 for system parameters 119863

1= 003 119863

2= 003 119877 = 10

6 Shock and Vibration

0 2 40

02

04

06

08

1

12

Uncontrolled

Optimally controlled

q

minus4 minus2

p(q)

Figure 2 Stationary probability density 119901(119902) of displacement ofuncontrolled and optimally controlled system (28) The lines areobtained fromnumerical solution of (37) while the dots are obtainedfrom direct simulation of original system (28)

1 15 2 25 30

05

1

15

2

Uncontrolled

Optimally controlled

E[H

]

1205821112058222

Figure 3Mean value119864[119867] of total energy of uncontrolled and opti-mally controlled system (28) as functions of the ratio 120582

11120582

22(120582

22=

120582

33) The lines are obtained from numerical solution of (37) while

the dots are obtained from direct simulation of original system (28)

2000 2100 2200 2300 2400 2500 2600

0

2

4

6

Uncontrolled

Optimally controlled

q

t

minus4

minus6

minus2

Figure 4 The sample time history of displacement of uncontrolledand optimally controlled system (28)

119904

1= 119904

2= 119904

3= 10 Λ = [

minus12058211 120582112 120582112

120582222 minus12058222 120582222

120582332 120582332 minus12058233

] 12058211= 120582

22= 120582

33=

30 and

120572

(1)= 001

120573

(1)= 001

120574

(1)= 001

ℎ

(1)

1= 20

ℎ

(1)

2= 20

120572

(2)= 001

120573

(2)= 002

120574

(2)= 002

ℎ

(2)

1= 20

ℎ

(2)

2= 10

120572

(3)= 001

120573

(3)= 004

120574

(3)= 004

ℎ

(3)

1= 10

ℎ

(3)

2= 10

(39)

The stationary probability density 119901(119867) of the totalenergy for uncontrolled and optimally controlled system(28) is shown in Figure 1 Obviously as 119867 increases 119901(119867)for optimally controlled system (28) decreases much fasterthan that for uncontrolled system The stationary probabilitydensity 119901(119902) of the displacement is plotted in Figure 2 withthe same parameters as in Figure 1 For optimally controlledsystem there is a much higher probability that 119902 will belocated near their equilibrium Hence 119901(119902) has much largermode and smaller dispersion around 119902 = 0 for optimallycontrolled system than those for uncontrolled system Thisimplies that the proposed control strategy has high effective-ness for attenuating the response of system (28)

Figure 3 shows the mean value 119864[119867] of the total energyof uncontrolled and optimally controlled system as functionsof ratio 120582

11120582

22(120582

22= 120582

33) Obviously higher value of ratio

120582

11120582

22implies lower probability for the system operating in

form 119904(119905) = 1 and higher probability for the system oper-ating in forms 119904(119905) = 2 and 119904(119905) = 3 When the system oper-ates in form 119904(119905) = 1 the system has the smallest dampingcoefficients and the largest amplitudes of stochastic excita-tions than those when it operates in forms 119904(119905) = 2 and119904(119905) = 3 As a consequence 119864[119867] for uncontrolled systemdecreases monotonously while the ratio 120582

11120582

22increases as

shown in Figure 3 However 119864[119867] for optimally controlledsystem always keeps a very low value despite changing of

Shock and Vibration 7

the ratio 12058211120582

22 which indicates that the proposed control

strategy is very effective and quite robust with respect tothe ratio 120582

11120582

22 In Figures 1ndash3 lines are obtained by

numerical solution of (37) while dots are obtained by directsimulation of system (28)Observe that the dotsmatch closelywith the corresponding lines demonstrating the validity andaccuracy of the proposed method The displacements of theuncontrolled and optimally controlled system (28) are shownin Figure 4 from which the effect of the optimal control onthe displacement can be visualized intuitively

52 Example 2 Consider a controlled stochastically excitedDuffing oscillator withMarkovian jumpwhich is governed by

+ 120596

2119909 + 120572119909

3= 120573 (119904 (119905)) + 119906 (119909 119904 (119905))

+ ℎ (119904 (119905)) 120585 (119905)

(40)

where 120573(119904(119905)) is Markovian jump coefficient of linear damp-ing ℎ(119904(119905)) is Markovian jump coefficient of external randomexcitation 119906(119909 119904(119905)) is the feedback control force 120585(119905) isGaussian white noise with zero mean and intensity 2119863 Herewe consider the 2-form case 119897 = 2 and 119878 = 1 2

Upon the stochastic averaging method the original jumpsystem (40) can be approximately substituted by the partiallyaveraged system (11) with

119898(119867 119904 = 119894) = ℎ

(119894)

2

119863 minus 120573

(119894)119866 (119867)

119898

119906(119867 119904 = 119894) =

4

119879 (119867)

int

119860(119867)

0

119906

(119894)(119902 119901) 119901 119889119902

120590 (119867 119904 = 119894)

2= 2 ℎ

(119894)

2

119863119866 (119867)

119866 (119867) =

4

119879 (119867)

int

119860(119867)

0

(2119867 minus 120596

2119902

2minus

120572119902

4

2

)

12

119889119902

119879 (119867) = 4int

119860(119867)

0

(2119867 minus 120596

2119902

2minus

120572119902

4

2

)

minus12

119889119902

119860 (119867) = [

(120596 + 4120572119867)

12minus 120596

2

120572

]

12

119894 = 1 2

(41)

Following (17) the expression of the optimal control forceis

119906

lowast(119902 119901 119904) = minus

1

2119877

120597119881 (119867 119904)

120597119867

119901(42)

Then the following final dynamical programming equa-tion is obtained1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) minus

1

4119877

120597119881 (119867 119904)

120597119867

119866 (119867)]

120597119881 (119867 119904)

119889119867

+ 119891 (119867 119904) +

2

sum

119903=1

120582

119903119904119881 (119867 119903) = 0 119904 = 1 2

(43)

120597119881120597119867 can be obtained from solving (43) numericallyThen the optimal control force is determined by substituting120597119881120597119867 into (17) Substituting the optimal control force (17)into the partially averaged equation (11) and completingthe averaging yield the following completely averaged FPKequation

120597

120597119905

119901 (119867 119904 119905)

= minus

120597

120597119867

[119898 (119867 119904) minus

1

2119877

120597119881 (119867 119904)

120597119867

119866 (119867)]

sdot 119901 (119867 119904 119905) +

1

2

120597

2

120597119867

2[120590

2(119867 119904) 119901 (119867 119904 119905)]

minus

2

sum

119903=1

119903 =119904

[120582

119904119903119901 (119867 119904 119905) minus 120582

119903119904119901 (119867 119903 119905)] 119904 = 1 2

(44)

Solving (44) numerically with 120597119901120597119905 = 0 the statistics119901(119867) 119901(119902) and 119864[119867] of the stationary response are thenobtained from (25) and (26)

The numerical results shown in Figures 5ndash8 are for systemwith parameters 120596 = 10 120572 = 05 119863 = 002 Λ = [ minus12058211 12058211

12058222 minus120582122]

120582

11= 120582

22= 20 and

120573

(1)= 001

ℎ

(1)= 20

120573

(2)= 002

ℎ

(2)= 10

(45)

The stationary probability densities 119901(119867) of total energyand 119901(119902) of the displacement of uncontrolled and optimallycontrolled system (40) are evaluated and plotted in Figures5 and 6 respectively The mean value 119864[119867] of total energyof uncontrolled and optimally controlled system (40) asfunctions of ratio 120582

11120582

22is shown in Figure 7 It is seen that

the optimal control force alleviates the response of system(40) significantly This effect can be seen more apparentlyfrom Figure 8 Again the analytical results obtained fromsolving (44) match closely with those from digital simulationof original system (40)

6 Conclusions

In this paper we proposed a nonlinear stochastic optimalcontrol strategy for SDOF stochastically excited Markovianjump systems where the jump parameter takes on values in afinite discrete set In the slow jump case the original systemwas first reduced to one governed by a finite set of one-dimensional partially averaged Ito equation for total energyupon stochastic averaging Using the stochastic dynami-cal programming principle and Markovian jump rules afinite set of coupled dynamical programming equations wasderived from which the optimal control force was obtainedThe obtained optimal control force has been proved to be the

8 Shock and Vibration

0 1 2 3 4 50

2

4

6

8

10

Uncontrolled

Optimally controlled

H

p(H

)

Figure 5 Stationary probability density 119901(119867) of total energy ofuncontrolled and optimally controlled system (40) The lines areobtained fromnumerical solution of (44)while the dots are obtainedfrom direct simulation of original system (40)

0 1 2 30

02

04

06

08

1

12

14

16

Uncontrolled

Optimally controlled

q

minus3 minus2 minus1

p(q)

Figure 6 Stationary probability density 119901(119902) of displacement ofuncontrolled and optimally controlled system (40) The lines areobtained fromnumerical solution of (44)while the dots are obtainedfrom direct simulation of original system (40)

quasi-optimal control for the original system The strategywas applied to two nonlinear oscillators that were capableof independent Markovian jumps Numerical results showedthat the strategy is fairly robust and effective in reductionof stationary response of the controlled system Thus it ispotentially promising for practical control applications afterfurther research

The proposed method has the potential to be extended tothe optimal control forMulti-DOFMarkovian jump systemsHowever to obtain the concrete conclusionmuchwork has tobe done Mathematical tools should be developed to solve thedynamical programming equation (14) for Multi-DOF case

1 15 2 25 30

1

2

3

4

5

Uncontrolled

Optimally controlled

E[H

]

1205821112058222

Figure 7 Mean value 119864[119867] of total energy of uncontrolled andoptimally controlled system (40) as functions of the ratio 120582

11120582

22

The lines are obtained fromnumerical solution of (44) while the dotsare obtained from direct simulation of original system (40)

2000 2050 2100 2150 2200

0

5

Uncontrolled

Optimally controlled

q

t

minus5

Figure 8The sample time history of displacements of uncontrolledand optimally controlled system (40)

Disclaimer

Opinions findings and conclusions expressed in this paperare those of the authors and do not necessarily reflect theviews of the sponsors

Conflict of Interests

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

Acknowledgments

This work was supported by the Natural Science Foundationof China (through Grants no 11272279 11372271 1143201211321202 and 11272201) and 973 Program (no 2011CB711105)

Shock and Vibration 9

References

[1] L Kats and N N Krasovskii ldquoOn the stability of systemswith random parametersrdquo Journal of Applied Mathematics andMechanics vol 24 no 5 pp 1225ndash1246 1960

[2] M Mariton ldquoAlmost sure and moments stability of jump linearsystemsrdquo Systems amp Control Letters vol 11 no 5 pp 393ndash3971988

[3] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[4] R H Huan W-Q Zhu F Ma and Z-G Ying ldquoStationaryresponse of a class of nonlinear stochastic systems undergoingMarkovian jumpsrdquo Journal of Applied Mechanics vol 82 no 5Article ID 051008 6 pages 2015

[5] N N Krasosvkii and E A Lidskii ldquoAnalytical design of con-trollers in systems with random attributes Irdquo Automation andRemote Control vol 22 pp 1021ndash1025 1961

[6] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967

[7] O L V Costa M D Fragoso and M G Todorov Continuous-Time Markov Jump Linear Systems Springer New York NYUSA 2013

[8] D D Sworder ldquoFeedback control of a class of linear systemswith jump parametersrdquo IEEE Transactions on Automatic Con-trol vol 14 no 1 pp 9ndash14 1969

[9] W M Wonham ldquoRandom differential equations in controltheoryrdquoProbabilisticMethods in AppliedMathematics vol 2 pp131ndash213 1971

[10] M D Fragoso and E M Hemerly ldquoOptimal control for a classof noisy linear systemswithMarkovian jumping parameters andquadratic costrdquo International Journal of Systems Science vol 22no 12 pp 2553ndash2561 1991

[11] Y D Ji and H J Chizeck ldquoJump linear quadratic Gaussian con-trol in continuous timerdquo IEEE Transactions on AutomaticControl vol 37 no 12 pp 1884ndash1892 1992

[12] M K Ghosh A Arapostathis and S I Marcus ldquoErgodiccontrol of switching diffusionsrdquo SIAM Journal on Control andOptimization vol 35 no 6 pp 1952ndash1988 1997

[13] W Q Zhu ldquoNonlinear stochastic dynamics and control inHamiltonian formulationrdquo Applied Mechanics Reviews vol 59no 4 pp 230ndash248 2006

[14] E K Boukas and H Yang ldquoStability of stochastic systems withjumpsrdquoMathematical Problems in Engineering vol 3 no 2 pp173ndash185 1996

[15] X B Feng K A Loparo Y D Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[16] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions Springer New York NY USA 1993

[17] K Ito ldquoOn stochastic differential equationsrdquo Memoirs of theAmerican Mathematical Society vol 4 pp 289ndash302 1951

[18] W Q Zhu and Z G Ying ldquoOptimal nonlinear feedbackcontrol of quasi-Hamiltonian systemsrdquo Science in China SeriesA Mathematics vol 42 no 11 pp 1213ndash1219 1999

[19] W Q Zhu Z G Ying and T T Soong ldquoAn optimal nonlinearfeedback control strategy for randomly excited structural sys-temsrdquo Nonlinear Dynamics vol 24 no 1 pp 31ndash51 2001

[20] H J Kushner andD S Clark Stochastic ApproximationMethodsfor Constrained andUnconstrained Systems Springer NewYorkNY USA 1978

[21] YW Fang Y LWu andHQWangOptimal ControlTheory ofStochastic Jump System National Defense Inustry Press BeijingChina 2012

[22] S T WuTheTheory of Stochastic Jump System and Its Applica-tion Science Press Beijing China 2007

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International Journal of

4 Shock and Vibration

where

119898

lowast

119906(119867 119904) = ⟨119906

lowast(119902 119901 119904)

120597119867

120597119901

⟩

= minus

1

2119877

⟨

120597119881 (119867 119904)

120597119867

(

120597119867

120597119901

)

2

⟩

(20)

The value function 119881(119867 119904) can be obtained by solving(19) The optimal control force 119906lowast(119902 119901 119904) is then determinedby substituting 119881(119867 119904) into expression (17) By inserting theoptimal control force 119906lowast(119902 119901 119904) into (11) and averaging theterm ⟨119906lowast(119902 119901 119904)(120597119867120597119901)⟩ the following completely averagedIto equation is obtained

119889119867 = 120576 [119898 (119867 119904) + 119898

lowast

119906(119867 119904)] 119889119905

+ 120576

12120590 (119867 119904) 119889119861 (119905)

(21)

The FPK equation associated with the fully averaged Itoequation (21) is [4 21 22]

120597

120597119905

119901 (119867 119904 119905)

= minus

120597

120597119867

[119898 (119867 119904) + 119898

lowast

119906(119867 119904)] 119901 (119867 119904 119905)

+

1

2

120597

2

120597119867

2[120590 (119867 119904)

2119901 (119867 119904 119905)]

minus

119897

sum

119903=1

119903 =119904

[120582

119904119903119901 (119867 119904 119905) minus 120582

119903119904119901 (119867 119903 119905)] 119904 isin 119878

(22)

where 119901(119867 119904 119905) is the probability density function of totalenergy119867 for optimally controlled system (1) while the systemoperating in 119904th form (119901(119867 119904 119905) for uncontrolled system canbe obtained by letting 119906lowast(119902 119901 119904) = 0) The initial condition is

119901 (119867 119904 0) = 119901 (119867

0 119904) (23)

and boundary conditions are

119901 (0 119904 119905) = finite

119901 (119867 119904 119905)

1003816

1003816

1003816

1003816119867rarrinfin997888rarr 0

120597119901 (119867 119904 119905)

120597119867

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119867rarrinfin

997888rarr 0

(24)

The FPK equation (22) is a finite set of second-order par-tially differential equations which are coupled through thezero-order terms Equation (22) does not admit easy solutionanalytically or numerically However in practical applicationwe are more interested in the stationary solution of FPKequation (22) In this case FPK equation (22) is reduced byletting 120597119901120597119905 = 0 Then the stationary probability density119901(119867 119904) is obtained readily from solving (22) numericallyThestationary probability density 119901(119867) can be obtained from119901(119867 119904) as follows

119901 (119867) = 119888

119897

sum

119904=1

119901 (119867 119904) (25)

where 119888 is a normalization constant The stationary probabil-ity density119901(119902) of the generalized displacement and themeanvalue 119864[119867] of the total energy are then obtained as

119901 (119902) = int

infin

minusinfin

119901 (119902 119901) 119889119901

119864 [119867] = int

infin

0

119867119901 (119867) 119889119867

(26)

where

119901 (119902 119901) =

119901 (119867)

119879 (119867)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119867=119867(119902119901)

(27)

is the stationary joint probability density of the generalizeddisplacement and generalized momentum

5 Numerical Example

To demonstrate the application and to assess the effective-ness of the proposed control strategy numerical results areobtained for two examples

51 Example 1 Consider a controlled van der Pol-Rayleighoscillator subjected to both external and parametric randomexcitations which is capable of independentMarkovian jumpand governed by

+ 119909 = 120572 (119904 (119905)) minus [120573 (119904 (119905)) 119909

2+ 120574 (119904 (119905))

2]

+ 119906 (119909 119904 (119905)) + ℎ

1(119904 (119905))119882

1(119905)

+ ℎ

2(119904 (119905)) 119909119882

2(119905)

(28)

where 120572(119904(119905)) 120573(119904(119905)) and 120574(119904(119905)) are Markovian jumpcoefficients of linear and quasi-linear damping respectivelyℎ

1(119904(119905)) and ℎ

2(119904(119905))119909 denote Markovian jump coefficients

of external and parametric random excitations respectively119906(119909 119904(119905)) is the feedback control force 119882

119896(119905) (119896 = 1 2)

are independent Gaussian white noises with zero mean andintensities 2119863

119896 119904(119905) is a finite-state continuous-time Markov

process pointing to the systemrsquos form with the transitionprobability defined in (2)

We consider the 3-form case here that means 119897 = 3 and119878 = 1 2 3 Prescribe the transition rate 120582

119894119895(119894 119895 = 1 2 3)

between the form 119894 and the form 119895 by a transition matrix Λ =[120582

119894119895]Let 119902 = 119909 and 119901 = The Hamiltonian associated with

system (28) is

119867 =

1

2

119901

2+

1

2

119902

2 (29)

As a Hamiltonian system (28) can be expressed as

[

119901

] = [119901 minus 119902

+ 120572 (119904 (119905)) minus [120573 (119904 (119905)) 119902

2+ 120574 (119904 (119905)) 119901

2] 119901

+ 119906 (119902 119901 119904 (119905)) + ℎ

1(119904 (119905))119882

1(119905)

+ ℎ

2(119904 (119905)) 119902119882

2(119905)]

(30)

Shock and Vibration 5

Upon stochastic averaging (11) for system (28) isobtained for whichΩ = 119902 | 119867(119902 0) le 119867 and

119898(119867 119904 (119905) = 119894) = minus [

120573

(119894)

2

+

3

2

120574

(119894)]119867

2+ ℎ

(119894)

1

2

119863

1

+ (120572

(119894)+ ℎ

(119894)

2

2

119863

2)119867

119898

119906(119867 119904 (119905) = 119894) = ⟨119906

(119894)(119902 119901) 119901⟩

120590

2(119867 119904 (119905) = 119894) = 2 ℎ

(119894)

1

2

119863

1119867 + ℎ

(119894)

2

2

119863

2119867

2

119894 = 1 2 3

(31)

For the proposed control strategy the partially averagedcost function 119871 is of the form of (16) with

119891 (119867 119904) = 119888

0(119904) + 119888

1(119904)119867 + 119888

2(119904)119867

2+ 119888

3(119904)119867

3 (32)

Following (17) the expression of the optimal control forceis

119906

lowast(119902 119901 119904) = minus

1

2119877

120597119881 (119867 119904)

120597119867

119901(33)

Substituting 119906lowast(119902 119901 119904) in (33) into the dynamical pro-gramming equation (14) completing the averaging yields thefollowing final coupled dynamical programming equations

1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) minus

1

4119877

120597119881 (119867 119904)

120597119867

⟨(

120597119867

120597119901

)

2

⟩]

120597119881 (119867 119904)

119889119867

+ 119891 (119867 119904) +

3

sum

119903=1

120582

119903119904119881 (119867 119903) = 0 119904 = 1 2 3

(34)

To solve (34) the value function 119881(119867 119904) is supposed tobe of the following form

119881 (119867 119904) = 119887

0(119904) + 119887

1(119904)119867 + 119887

2(119904)119867

2 (35)

Since the optimal control force 119906lowast is function of 120597119881120597119867only the coefficients 119887

1(119904) and 119887

2(119904) are determined by insert-

ing (32) and (35) into (34)

119887

1(119904 = 119894) = 119887

(119894)

1

=

2 (119888

2+ sum

3

119896=1120582

119896119894119887

(119896)

2+ 2119887

(119894)

2120572

(119894)+ 2119863

2ℎ

(119894)

2

2

119887

(119894)

2)119877

2119887

(119894)

2+ 120573

(119894)119877 + 3120574

(119894)119877

119887

2(119904 = 119894) = 119887

(119894)

2

= minus

1

2

120573

(119894)minus

3

2

120574

(119894)119877

+

1

2

radic

(120573

(119894))

2

119877

2+ 6120573

(119894)119877

2120574

(119894)+ 9 (120574

(119894))

2

119877

2+ 4119888

3119877

119894 = 1 2 3

(36)

Uncontrolled

Optimally controlled

1 2 3 4 500

1

2

3

4

5

6

7

H

p(H

)

Figure 1 Stationary probability density 119901(119867) of total energy ofuncontrolled and optimally controlled system (28) The lines areobtained fromnumerical solution of (37) while the dots are obtainedfrom direct simulation of original system (28)

By substituting (35) into (17) with the coefficients deter-mined by (36) the optimal control force is determinedThenthe following completely averaged FPK equation is obtained

120597

120597119905

119901 (119867 119904 119905)

= minus

120597

120597119867

[119898 (119867 119904) + 119898

lowast

119906(119867 119904)] 119901 (119867 119904 119905)

+

1

2

120597

2

120597119867

2[120590 (119867 119904)

2119901 (119867 119904 119905)]

minus

3

sum

119903=1

119903 =119904

[120582

119904119903119901 (119867 119904 119905) minus 120582

119903119904119901 (119867 119903 119905)]

119904 = 1 2 3

(37)

where

119898

lowast

119906(119867 119904 = 119894) = minus

1

2119877

(119887

(119894)

1+ 2119887

(119894)

2119867)119867 (38)

Solving (37) numerically with 120597119901120597119905 = 0 the stationaryprobability density 119901(119867 119904) of optimal controlled system canbe obtained The stationary probability densities 119901(119867) of thetotal energy 119901(119902) of the generalized displacement and themean value 119864[119867] of the total energy are then obtained using(25) and (26)

Some numerical results are obtained as shown in Figures1ndash4 for system parameters 119863

1= 003 119863

2= 003 119877 = 10

6 Shock and Vibration

0 2 40

02

04

06

08

1

12

Uncontrolled

Optimally controlled

q

minus4 minus2

p(q)

Figure 2 Stationary probability density 119901(119902) of displacement ofuncontrolled and optimally controlled system (28) The lines areobtained fromnumerical solution of (37) while the dots are obtainedfrom direct simulation of original system (28)

1 15 2 25 30

05

1

15

2

Uncontrolled

Optimally controlled

E[H

]

1205821112058222

Figure 3Mean value119864[119867] of total energy of uncontrolled and opti-mally controlled system (28) as functions of the ratio 120582

11120582

22(120582

22=

120582

33) The lines are obtained from numerical solution of (37) while

the dots are obtained from direct simulation of original system (28)

2000 2100 2200 2300 2400 2500 2600

0

2

4

6

Uncontrolled

Optimally controlled

q

t

minus4

minus6

minus2

Figure 4 The sample time history of displacement of uncontrolledand optimally controlled system (28)

119904

1= 119904

2= 119904

3= 10 Λ = [

minus12058211 120582112 120582112

120582222 minus12058222 120582222

120582332 120582332 minus12058233

] 12058211= 120582

22= 120582

33=

30 and

120572

(1)= 001

120573

(1)= 001

120574

(1)= 001

ℎ

(1)

1= 20

ℎ

(1)

2= 20

120572

(2)= 001

120573

(2)= 002

120574

(2)= 002

ℎ

(2)

1= 20

ℎ

(2)

2= 10

120572

(3)= 001

120573

(3)= 004

120574

(3)= 004

ℎ

(3)

1= 10

ℎ

(3)

2= 10

(39)

The stationary probability density 119901(119867) of the totalenergy for uncontrolled and optimally controlled system(28) is shown in Figure 1 Obviously as 119867 increases 119901(119867)for optimally controlled system (28) decreases much fasterthan that for uncontrolled system The stationary probabilitydensity 119901(119902) of the displacement is plotted in Figure 2 withthe same parameters as in Figure 1 For optimally controlledsystem there is a much higher probability that 119902 will belocated near their equilibrium Hence 119901(119902) has much largermode and smaller dispersion around 119902 = 0 for optimallycontrolled system than those for uncontrolled system Thisimplies that the proposed control strategy has high effective-ness for attenuating the response of system (28)

Figure 3 shows the mean value 119864[119867] of the total energyof uncontrolled and optimally controlled system as functionsof ratio 120582

11120582

22(120582

22= 120582

33) Obviously higher value of ratio

120582

11120582

22implies lower probability for the system operating in

form 119904(119905) = 1 and higher probability for the system oper-ating in forms 119904(119905) = 2 and 119904(119905) = 3 When the system oper-ates in form 119904(119905) = 1 the system has the smallest dampingcoefficients and the largest amplitudes of stochastic excita-tions than those when it operates in forms 119904(119905) = 2 and119904(119905) = 3 As a consequence 119864[119867] for uncontrolled systemdecreases monotonously while the ratio 120582

11120582

22increases as

shown in Figure 3 However 119864[119867] for optimally controlledsystem always keeps a very low value despite changing of

Shock and Vibration 7

the ratio 12058211120582

22 which indicates that the proposed control

strategy is very effective and quite robust with respect tothe ratio 120582

11120582

22 In Figures 1ndash3 lines are obtained by

numerical solution of (37) while dots are obtained by directsimulation of system (28)Observe that the dotsmatch closelywith the corresponding lines demonstrating the validity andaccuracy of the proposed method The displacements of theuncontrolled and optimally controlled system (28) are shownin Figure 4 from which the effect of the optimal control onthe displacement can be visualized intuitively

52 Example 2 Consider a controlled stochastically excitedDuffing oscillator withMarkovian jumpwhich is governed by

+ 120596

2119909 + 120572119909

3= 120573 (119904 (119905)) + 119906 (119909 119904 (119905))

+ ℎ (119904 (119905)) 120585 (119905)

(40)

where 120573(119904(119905)) is Markovian jump coefficient of linear damp-ing ℎ(119904(119905)) is Markovian jump coefficient of external randomexcitation 119906(119909 119904(119905)) is the feedback control force 120585(119905) isGaussian white noise with zero mean and intensity 2119863 Herewe consider the 2-form case 119897 = 2 and 119878 = 1 2

Upon the stochastic averaging method the original jumpsystem (40) can be approximately substituted by the partiallyaveraged system (11) with

119898(119867 119904 = 119894) = ℎ

(119894)

2

119863 minus 120573

(119894)119866 (119867)

119898

119906(119867 119904 = 119894) =

4

119879 (119867)

int

119860(119867)

0

119906

(119894)(119902 119901) 119901 119889119902

120590 (119867 119904 = 119894)

2= 2 ℎ

(119894)

2

119863119866 (119867)

119866 (119867) =

4

119879 (119867)

int

119860(119867)

0

(2119867 minus 120596

2119902

2minus

120572119902

4

2

)

12

119889119902

119879 (119867) = 4int

119860(119867)

0

(2119867 minus 120596

2119902

2minus

120572119902

4

2

)

minus12

119889119902

119860 (119867) = [

(120596 + 4120572119867)

12minus 120596

2

120572

]

12

119894 = 1 2

(41)

Following (17) the expression of the optimal control forceis

119906

lowast(119902 119901 119904) = minus

1

2119877

120597119881 (119867 119904)

120597119867

119901(42)

Then the following final dynamical programming equa-tion is obtained1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) minus

1

4119877

120597119881 (119867 119904)

120597119867

119866 (119867)]

120597119881 (119867 119904)

119889119867

+ 119891 (119867 119904) +

2

sum

119903=1

120582

119903119904119881 (119867 119903) = 0 119904 = 1 2

(43)

120597119881120597119867 can be obtained from solving (43) numericallyThen the optimal control force is determined by substituting120597119881120597119867 into (17) Substituting the optimal control force (17)into the partially averaged equation (11) and completingthe averaging yield the following completely averaged FPKequation

120597

120597119905

119901 (119867 119904 119905)

= minus

120597

120597119867

[119898 (119867 119904) minus

1

2119877

120597119881 (119867 119904)

120597119867

119866 (119867)]

sdot 119901 (119867 119904 119905) +

1

2

120597

2

120597119867

2[120590

2(119867 119904) 119901 (119867 119904 119905)]

minus

2

sum

119903=1

119903 =119904

[120582

119904119903119901 (119867 119904 119905) minus 120582

119903119904119901 (119867 119903 119905)] 119904 = 1 2

(44)

Solving (44) numerically with 120597119901120597119905 = 0 the statistics119901(119867) 119901(119902) and 119864[119867] of the stationary response are thenobtained from (25) and (26)

The numerical results shown in Figures 5ndash8 are for systemwith parameters 120596 = 10 120572 = 05 119863 = 002 Λ = [ minus12058211 12058211

12058222 minus120582122]

120582

11= 120582

22= 20 and

120573

(1)= 001

ℎ

(1)= 20

120573

(2)= 002

ℎ

(2)= 10

(45)

The stationary probability densities 119901(119867) of total energyand 119901(119902) of the displacement of uncontrolled and optimallycontrolled system (40) are evaluated and plotted in Figures5 and 6 respectively The mean value 119864[119867] of total energyof uncontrolled and optimally controlled system (40) asfunctions of ratio 120582

11120582

22is shown in Figure 7 It is seen that

the optimal control force alleviates the response of system(40) significantly This effect can be seen more apparentlyfrom Figure 8 Again the analytical results obtained fromsolving (44) match closely with those from digital simulationof original system (40)

6 Conclusions

In this paper we proposed a nonlinear stochastic optimalcontrol strategy for SDOF stochastically excited Markovianjump systems where the jump parameter takes on values in afinite discrete set In the slow jump case the original systemwas first reduced to one governed by a finite set of one-dimensional partially averaged Ito equation for total energyupon stochastic averaging Using the stochastic dynami-cal programming principle and Markovian jump rules afinite set of coupled dynamical programming equations wasderived from which the optimal control force was obtainedThe obtained optimal control force has been proved to be the

8 Shock and Vibration

0 1 2 3 4 50

2

4

6

8

10

Uncontrolled

Optimally controlled

H

p(H

)

Figure 5 Stationary probability density 119901(119867) of total energy ofuncontrolled and optimally controlled system (40) The lines areobtained fromnumerical solution of (44)while the dots are obtainedfrom direct simulation of original system (40)

0 1 2 30

02

04

06

08

1

12

14

16

Uncontrolled

Optimally controlled

q

minus3 minus2 minus1

p(q)

Figure 6 Stationary probability density 119901(119902) of displacement ofuncontrolled and optimally controlled system (40) The lines areobtained fromnumerical solution of (44)while the dots are obtainedfrom direct simulation of original system (40)

quasi-optimal control for the original system The strategywas applied to two nonlinear oscillators that were capableof independent Markovian jumps Numerical results showedthat the strategy is fairly robust and effective in reductionof stationary response of the controlled system Thus it ispotentially promising for practical control applications afterfurther research

The proposed method has the potential to be extended tothe optimal control forMulti-DOFMarkovian jump systemsHowever to obtain the concrete conclusionmuchwork has tobe done Mathematical tools should be developed to solve thedynamical programming equation (14) for Multi-DOF case

1 15 2 25 30

1

2

3

4

5

Uncontrolled

Optimally controlled

E[H

]

1205821112058222

Figure 7 Mean value 119864[119867] of total energy of uncontrolled andoptimally controlled system (40) as functions of the ratio 120582

11120582

22

The lines are obtained fromnumerical solution of (44) while the dotsare obtained from direct simulation of original system (40)

2000 2050 2100 2150 2200

0

5

Uncontrolled

Optimally controlled

q

t

minus5

Figure 8The sample time history of displacements of uncontrolledand optimally controlled system (40)

Disclaimer

Opinions findings and conclusions expressed in this paperare those of the authors and do not necessarily reflect theviews of the sponsors

Conflict of Interests

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

Acknowledgments

This work was supported by the Natural Science Foundationof China (through Grants no 11272279 11372271 1143201211321202 and 11272201) and 973 Program (no 2011CB711105)

Shock and Vibration 9

References

[1] L Kats and N N Krasovskii ldquoOn the stability of systemswith random parametersrdquo Journal of Applied Mathematics andMechanics vol 24 no 5 pp 1225ndash1246 1960

[2] M Mariton ldquoAlmost sure and moments stability of jump linearsystemsrdquo Systems amp Control Letters vol 11 no 5 pp 393ndash3971988

[3] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[4] R H Huan W-Q Zhu F Ma and Z-G Ying ldquoStationaryresponse of a class of nonlinear stochastic systems undergoingMarkovian jumpsrdquo Journal of Applied Mechanics vol 82 no 5Article ID 051008 6 pages 2015

[5] N N Krasosvkii and E A Lidskii ldquoAnalytical design of con-trollers in systems with random attributes Irdquo Automation andRemote Control vol 22 pp 1021ndash1025 1961

[6] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967

[7] O L V Costa M D Fragoso and M G Todorov Continuous-Time Markov Jump Linear Systems Springer New York NYUSA 2013

[8] D D Sworder ldquoFeedback control of a class of linear systemswith jump parametersrdquo IEEE Transactions on Automatic Con-trol vol 14 no 1 pp 9ndash14 1969

[9] W M Wonham ldquoRandom differential equations in controltheoryrdquoProbabilisticMethods in AppliedMathematics vol 2 pp131ndash213 1971

[10] M D Fragoso and E M Hemerly ldquoOptimal control for a classof noisy linear systemswithMarkovian jumping parameters andquadratic costrdquo International Journal of Systems Science vol 22no 12 pp 2553ndash2561 1991

[11] Y D Ji and H J Chizeck ldquoJump linear quadratic Gaussian con-trol in continuous timerdquo IEEE Transactions on AutomaticControl vol 37 no 12 pp 1884ndash1892 1992

[12] M K Ghosh A Arapostathis and S I Marcus ldquoErgodiccontrol of switching diffusionsrdquo SIAM Journal on Control andOptimization vol 35 no 6 pp 1952ndash1988 1997

[13] W Q Zhu ldquoNonlinear stochastic dynamics and control inHamiltonian formulationrdquo Applied Mechanics Reviews vol 59no 4 pp 230ndash248 2006

[14] E K Boukas and H Yang ldquoStability of stochastic systems withjumpsrdquoMathematical Problems in Engineering vol 3 no 2 pp173ndash185 1996

[15] X B Feng K A Loparo Y D Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[16] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions Springer New York NY USA 1993

[17] K Ito ldquoOn stochastic differential equationsrdquo Memoirs of theAmerican Mathematical Society vol 4 pp 289ndash302 1951

[18] W Q Zhu and Z G Ying ldquoOptimal nonlinear feedbackcontrol of quasi-Hamiltonian systemsrdquo Science in China SeriesA Mathematics vol 42 no 11 pp 1213ndash1219 1999

[19] W Q Zhu Z G Ying and T T Soong ldquoAn optimal nonlinearfeedback control strategy for randomly excited structural sys-temsrdquo Nonlinear Dynamics vol 24 no 1 pp 31ndash51 2001

[20] H J Kushner andD S Clark Stochastic ApproximationMethodsfor Constrained andUnconstrained Systems Springer NewYorkNY USA 1978

[21] YW Fang Y LWu andHQWangOptimal ControlTheory ofStochastic Jump System National Defense Inustry Press BeijingChina 2012

[22] S T WuTheTheory of Stochastic Jump System and Its Applica-tion Science Press Beijing China 2007

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Shock and Vibration 5

Upon stochastic averaging (11) for system (28) isobtained for whichΩ = 119902 | 119867(119902 0) le 119867 and

119898(119867 119904 (119905) = 119894) = minus [

120573

(119894)

2

+

3

2

120574

(119894)]119867

2+ ℎ

(119894)

1

2

119863

1

+ (120572

(119894)+ ℎ

(119894)

2

2

119863

2)119867

119898

119906(119867 119904 (119905) = 119894) = ⟨119906

(119894)(119902 119901) 119901⟩

120590

2(119867 119904 (119905) = 119894) = 2 ℎ

(119894)

1

2

119863

1119867 + ℎ

(119894)

2

2

119863

2119867

2

119894 = 1 2 3

(31)

For the proposed control strategy the partially averagedcost function 119871 is of the form of (16) with

119891 (119867 119904) = 119888

0(119904) + 119888

1(119904)119867 + 119888

2(119904)119867

2+ 119888

3(119904)119867

3 (32)

Following (17) the expression of the optimal control forceis

119906

lowast(119902 119901 119904) = minus

1

2119877

120597119881 (119867 119904)

120597119867

119901(33)

Substituting 119906lowast(119902 119901 119904) in (33) into the dynamical pro-gramming equation (14) completing the averaging yields thefollowing final coupled dynamical programming equations

1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) minus

1

4119877

120597119881 (119867 119904)

120597119867

⟨(

120597119867

120597119901

)

2

⟩]

120597119881 (119867 119904)

119889119867

+ 119891 (119867 119904) +

3

sum

119903=1

120582

119903119904119881 (119867 119903) = 0 119904 = 1 2 3

(34)

To solve (34) the value function 119881(119867 119904) is supposed tobe of the following form

119881 (119867 119904) = 119887

0(119904) + 119887

1(119904)119867 + 119887

2(119904)119867

2 (35)

Since the optimal control force 119906lowast is function of 120597119881120597119867only the coefficients 119887

1(119904) and 119887

2(119904) are determined by insert-

ing (32) and (35) into (34)

119887

1(119904 = 119894) = 119887

(119894)

1

=

2 (119888

2+ sum

3

119896=1120582

119896119894119887

(119896)

2+ 2119887

(119894)

2120572

(119894)+ 2119863

2ℎ

(119894)

2

2

119887

(119894)

2)119877

2119887

(119894)

2+ 120573

(119894)119877 + 3120574

(119894)119877

119887

2(119904 = 119894) = 119887

(119894)

2

= minus

1

2

120573

(119894)minus

3

2

120574

(119894)119877

+

1

2

radic

(120573

(119894))

2

119877

2+ 6120573

(119894)119877

2120574

(119894)+ 9 (120574

(119894))

2

119877

2+ 4119888

3119877

119894 = 1 2 3

(36)

Uncontrolled

Optimally controlled

1 2 3 4 500

1

2

3

4

5

6

7

H

p(H

)

Figure 1 Stationary probability density 119901(119867) of total energy ofuncontrolled and optimally controlled system (28) The lines areobtained fromnumerical solution of (37) while the dots are obtainedfrom direct simulation of original system (28)

By substituting (35) into (17) with the coefficients deter-mined by (36) the optimal control force is determinedThenthe following completely averaged FPK equation is obtained

120597

120597119905

119901 (119867 119904 119905)

= minus

120597

120597119867

[119898 (119867 119904) + 119898

lowast

119906(119867 119904)] 119901 (119867 119904 119905)

+

1

2

120597

2

120597119867

2[120590 (119867 119904)

2119901 (119867 119904 119905)]

minus

3

sum

119903=1

119903 =119904

[120582

119904119903119901 (119867 119904 119905) minus 120582

119903119904119901 (119867 119903 119905)]

119904 = 1 2 3

(37)

where

119898

lowast

119906(119867 119904 = 119894) = minus

1

2119877

(119887

(119894)

1+ 2119887

(119894)

2119867)119867 (38)

Solving (37) numerically with 120597119901120597119905 = 0 the stationaryprobability density 119901(119867 119904) of optimal controlled system canbe obtained The stationary probability densities 119901(119867) of thetotal energy 119901(119902) of the generalized displacement and themean value 119864[119867] of the total energy are then obtained using(25) and (26)

Some numerical results are obtained as shown in Figures1ndash4 for system parameters 119863

1= 003 119863

2= 003 119877 = 10

6 Shock and Vibration

0 2 40

02

04

06

08

1

12

Uncontrolled

Optimally controlled

q

minus4 minus2

p(q)

Figure 2 Stationary probability density 119901(119902) of displacement ofuncontrolled and optimally controlled system (28) The lines areobtained fromnumerical solution of (37) while the dots are obtainedfrom direct simulation of original system (28)

1 15 2 25 30

05

1

15

2

Uncontrolled

Optimally controlled

E[H

]

1205821112058222

Figure 3Mean value119864[119867] of total energy of uncontrolled and opti-mally controlled system (28) as functions of the ratio 120582

11120582

22(120582

22=

120582

33) The lines are obtained from numerical solution of (37) while

the dots are obtained from direct simulation of original system (28)

2000 2100 2200 2300 2400 2500 2600

0

2

4

6

Uncontrolled

Optimally controlled

q

t

minus4

minus6

minus2

Figure 4 The sample time history of displacement of uncontrolledand optimally controlled system (28)

119904

1= 119904

2= 119904

3= 10 Λ = [

minus12058211 120582112 120582112

120582222 minus12058222 120582222

120582332 120582332 minus12058233

] 12058211= 120582

22= 120582

33=

30 and

120572

(1)= 001

120573

(1)= 001

120574

(1)= 001

ℎ

(1)

1= 20

ℎ

(1)

2= 20

120572

(2)= 001

120573

(2)= 002

120574

(2)= 002

ℎ

(2)

1= 20

ℎ

(2)

2= 10

120572

(3)= 001

120573

(3)= 004

120574

(3)= 004

ℎ

(3)

1= 10

ℎ

(3)

2= 10

(39)

The stationary probability density 119901(119867) of the totalenergy for uncontrolled and optimally controlled system(28) is shown in Figure 1 Obviously as 119867 increases 119901(119867)for optimally controlled system (28) decreases much fasterthan that for uncontrolled system The stationary probabilitydensity 119901(119902) of the displacement is plotted in Figure 2 withthe same parameters as in Figure 1 For optimally controlledsystem there is a much higher probability that 119902 will belocated near their equilibrium Hence 119901(119902) has much largermode and smaller dispersion around 119902 = 0 for optimallycontrolled system than those for uncontrolled system Thisimplies that the proposed control strategy has high effective-ness for attenuating the response of system (28)

Figure 3 shows the mean value 119864[119867] of the total energyof uncontrolled and optimally controlled system as functionsof ratio 120582

11120582

22(120582

22= 120582

33) Obviously higher value of ratio

120582

11120582

22implies lower probability for the system operating in

form 119904(119905) = 1 and higher probability for the system oper-ating in forms 119904(119905) = 2 and 119904(119905) = 3 When the system oper-ates in form 119904(119905) = 1 the system has the smallest dampingcoefficients and the largest amplitudes of stochastic excita-tions than those when it operates in forms 119904(119905) = 2 and119904(119905) = 3 As a consequence 119864[119867] for uncontrolled systemdecreases monotonously while the ratio 120582

11120582

22increases as

shown in Figure 3 However 119864[119867] for optimally controlledsystem always keeps a very low value despite changing of

Shock and Vibration 7

the ratio 12058211120582

22 which indicates that the proposed control

strategy is very effective and quite robust with respect tothe ratio 120582

11120582

22 In Figures 1ndash3 lines are obtained by

numerical solution of (37) while dots are obtained by directsimulation of system (28)Observe that the dotsmatch closelywith the corresponding lines demonstrating the validity andaccuracy of the proposed method The displacements of theuncontrolled and optimally controlled system (28) are shownin Figure 4 from which the effect of the optimal control onthe displacement can be visualized intuitively

52 Example 2 Consider a controlled stochastically excitedDuffing oscillator withMarkovian jumpwhich is governed by

+ 120596

2119909 + 120572119909

3= 120573 (119904 (119905)) + 119906 (119909 119904 (119905))

+ ℎ (119904 (119905)) 120585 (119905)

(40)

where 120573(119904(119905)) is Markovian jump coefficient of linear damp-ing ℎ(119904(119905)) is Markovian jump coefficient of external randomexcitation 119906(119909 119904(119905)) is the feedback control force 120585(119905) isGaussian white noise with zero mean and intensity 2119863 Herewe consider the 2-form case 119897 = 2 and 119878 = 1 2

Upon the stochastic averaging method the original jumpsystem (40) can be approximately substituted by the partiallyaveraged system (11) with

119898(119867 119904 = 119894) = ℎ

(119894)

2

119863 minus 120573

(119894)119866 (119867)

119898

119906(119867 119904 = 119894) =

4

119879 (119867)

int

119860(119867)

0

119906

(119894)(119902 119901) 119901 119889119902

120590 (119867 119904 = 119894)

2= 2 ℎ

(119894)

2

119863119866 (119867)

119866 (119867) =

4

119879 (119867)

int

119860(119867)

0

(2119867 minus 120596

2119902

2minus

120572119902

4

2

)

12

119889119902

119879 (119867) = 4int

119860(119867)

0

(2119867 minus 120596

2119902

2minus

120572119902

4

2

)

minus12

119889119902

119860 (119867) = [

(120596 + 4120572119867)

12minus 120596

2

120572

]

12

119894 = 1 2

(41)

Following (17) the expression of the optimal control forceis

119906

lowast(119902 119901 119904) = minus

1

2119877

120597119881 (119867 119904)

120597119867

119901(42)

Then the following final dynamical programming equa-tion is obtained1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) minus

1

4119877

120597119881 (119867 119904)

120597119867

119866 (119867)]

120597119881 (119867 119904)

119889119867

+ 119891 (119867 119904) +

2

sum

119903=1

120582

119903119904119881 (119867 119903) = 0 119904 = 1 2

(43)

120597119881120597119867 can be obtained from solving (43) numericallyThen the optimal control force is determined by substituting120597119881120597119867 into (17) Substituting the optimal control force (17)into the partially averaged equation (11) and completingthe averaging yield the following completely averaged FPKequation

120597

120597119905

119901 (119867 119904 119905)

= minus

120597

120597119867

[119898 (119867 119904) minus

1

2119877

120597119881 (119867 119904)

120597119867

119866 (119867)]

sdot 119901 (119867 119904 119905) +

1

2

120597

2

120597119867

2[120590

2(119867 119904) 119901 (119867 119904 119905)]

minus

2

sum

119903=1

119903 =119904

[120582

119904119903119901 (119867 119904 119905) minus 120582

119903119904119901 (119867 119903 119905)] 119904 = 1 2

(44)

Solving (44) numerically with 120597119901120597119905 = 0 the statistics119901(119867) 119901(119902) and 119864[119867] of the stationary response are thenobtained from (25) and (26)

The numerical results shown in Figures 5ndash8 are for systemwith parameters 120596 = 10 120572 = 05 119863 = 002 Λ = [ minus12058211 12058211

12058222 minus120582122]

120582

11= 120582

22= 20 and

120573

(1)= 001

ℎ

(1)= 20

120573

(2)= 002

ℎ

(2)= 10

(45)

The stationary probability densities 119901(119867) of total energyand 119901(119902) of the displacement of uncontrolled and optimallycontrolled system (40) are evaluated and plotted in Figures5 and 6 respectively The mean value 119864[119867] of total energyof uncontrolled and optimally controlled system (40) asfunctions of ratio 120582

11120582

22is shown in Figure 7 It is seen that

the optimal control force alleviates the response of system(40) significantly This effect can be seen more apparentlyfrom Figure 8 Again the analytical results obtained fromsolving (44) match closely with those from digital simulationof original system (40)

6 Conclusions

In this paper we proposed a nonlinear stochastic optimalcontrol strategy for SDOF stochastically excited Markovianjump systems where the jump parameter takes on values in afinite discrete set In the slow jump case the original systemwas first reduced to one governed by a finite set of one-dimensional partially averaged Ito equation for total energyupon stochastic averaging Using the stochastic dynami-cal programming principle and Markovian jump rules afinite set of coupled dynamical programming equations wasderived from which the optimal control force was obtainedThe obtained optimal control force has been proved to be the

8 Shock and Vibration

0 1 2 3 4 50

2

4

6

8

10

Uncontrolled

Optimally controlled

H

p(H

)

Figure 5 Stationary probability density 119901(119867) of total energy ofuncontrolled and optimally controlled system (40) The lines areobtained fromnumerical solution of (44)while the dots are obtainedfrom direct simulation of original system (40)

0 1 2 30

02

04

06

08

1

12

14

16

Uncontrolled

Optimally controlled

q

minus3 minus2 minus1

p(q)

Figure 6 Stationary probability density 119901(119902) of displacement ofuncontrolled and optimally controlled system (40) The lines areobtained fromnumerical solution of (44)while the dots are obtainedfrom direct simulation of original system (40)

quasi-optimal control for the original system The strategywas applied to two nonlinear oscillators that were capableof independent Markovian jumps Numerical results showedthat the strategy is fairly robust and effective in reductionof stationary response of the controlled system Thus it ispotentially promising for practical control applications afterfurther research

The proposed method has the potential to be extended tothe optimal control forMulti-DOFMarkovian jump systemsHowever to obtain the concrete conclusionmuchwork has tobe done Mathematical tools should be developed to solve thedynamical programming equation (14) for Multi-DOF case

1 15 2 25 30

1

2

3

4

5

Uncontrolled

Optimally controlled

E[H

]

1205821112058222

Figure 7 Mean value 119864[119867] of total energy of uncontrolled andoptimally controlled system (40) as functions of the ratio 120582

11120582

22

The lines are obtained fromnumerical solution of (44) while the dotsare obtained from direct simulation of original system (40)

2000 2050 2100 2150 2200

0

5

Uncontrolled

Optimally controlled

q

t

minus5

Figure 8The sample time history of displacements of uncontrolledand optimally controlled system (40)

Disclaimer

Opinions findings and conclusions expressed in this paperare those of the authors and do not necessarily reflect theviews of the sponsors

Conflict of Interests

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

Acknowledgments

This work was supported by the Natural Science Foundationof China (through Grants no 11272279 11372271 1143201211321202 and 11272201) and 973 Program (no 2011CB711105)

Shock and Vibration 9

References

[1] L Kats and N N Krasovskii ldquoOn the stability of systemswith random parametersrdquo Journal of Applied Mathematics andMechanics vol 24 no 5 pp 1225ndash1246 1960

[2] M Mariton ldquoAlmost sure and moments stability of jump linearsystemsrdquo Systems amp Control Letters vol 11 no 5 pp 393ndash3971988

[3] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[4] R H Huan W-Q Zhu F Ma and Z-G Ying ldquoStationaryresponse of a class of nonlinear stochastic systems undergoingMarkovian jumpsrdquo Journal of Applied Mechanics vol 82 no 5Article ID 051008 6 pages 2015

[5] N N Krasosvkii and E A Lidskii ldquoAnalytical design of con-trollers in systems with random attributes Irdquo Automation andRemote Control vol 22 pp 1021ndash1025 1961

[6] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967

[7] O L V Costa M D Fragoso and M G Todorov Continuous-Time Markov Jump Linear Systems Springer New York NYUSA 2013

[8] D D Sworder ldquoFeedback control of a class of linear systemswith jump parametersrdquo IEEE Transactions on Automatic Con-trol vol 14 no 1 pp 9ndash14 1969

[9] W M Wonham ldquoRandom differential equations in controltheoryrdquoProbabilisticMethods in AppliedMathematics vol 2 pp131ndash213 1971

[10] M D Fragoso and E M Hemerly ldquoOptimal control for a classof noisy linear systemswithMarkovian jumping parameters andquadratic costrdquo International Journal of Systems Science vol 22no 12 pp 2553ndash2561 1991

[11] Y D Ji and H J Chizeck ldquoJump linear quadratic Gaussian con-trol in continuous timerdquo IEEE Transactions on AutomaticControl vol 37 no 12 pp 1884ndash1892 1992

[12] M K Ghosh A Arapostathis and S I Marcus ldquoErgodiccontrol of switching diffusionsrdquo SIAM Journal on Control andOptimization vol 35 no 6 pp 1952ndash1988 1997

[13] W Q Zhu ldquoNonlinear stochastic dynamics and control inHamiltonian formulationrdquo Applied Mechanics Reviews vol 59no 4 pp 230ndash248 2006

[14] E K Boukas and H Yang ldquoStability of stochastic systems withjumpsrdquoMathematical Problems in Engineering vol 3 no 2 pp173ndash185 1996

[15] X B Feng K A Loparo Y D Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[16] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions Springer New York NY USA 1993

[17] K Ito ldquoOn stochastic differential equationsrdquo Memoirs of theAmerican Mathematical Society vol 4 pp 289ndash302 1951

[18] W Q Zhu and Z G Ying ldquoOptimal nonlinear feedbackcontrol of quasi-Hamiltonian systemsrdquo Science in China SeriesA Mathematics vol 42 no 11 pp 1213ndash1219 1999

[19] W Q Zhu Z G Ying and T T Soong ldquoAn optimal nonlinearfeedback control strategy for randomly excited structural sys-temsrdquo Nonlinear Dynamics vol 24 no 1 pp 31ndash51 2001

[20] H J Kushner andD S Clark Stochastic ApproximationMethodsfor Constrained andUnconstrained Systems Springer NewYorkNY USA 1978

[21] YW Fang Y LWu andHQWangOptimal ControlTheory ofStochastic Jump System National Defense Inustry Press BeijingChina 2012

[22] S T WuTheTheory of Stochastic Jump System and Its Applica-tion Science Press Beijing China 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

0 2 40

02

04

06

08

1

12

Uncontrolled

Optimally controlled

q

minus4 minus2

p(q)

Figure 2 Stationary probability density 119901(119902) of displacement ofuncontrolled and optimally controlled system (28) The lines areobtained fromnumerical solution of (37) while the dots are obtainedfrom direct simulation of original system (28)

1 15 2 25 30

05

1

15

2

Uncontrolled

Optimally controlled

E[H

]

1205821112058222

Figure 3Mean value119864[119867] of total energy of uncontrolled and opti-mally controlled system (28) as functions of the ratio 120582

11120582

22(120582

22=

120582

33) The lines are obtained from numerical solution of (37) while

the dots are obtained from direct simulation of original system (28)

2000 2100 2200 2300 2400 2500 2600

0

2

4

6

Uncontrolled

Optimally controlled

q

t

minus4

minus6

minus2

Figure 4 The sample time history of displacement of uncontrolledand optimally controlled system (28)

119904

1= 119904

2= 119904

3= 10 Λ = [

minus12058211 120582112 120582112

120582222 minus12058222 120582222

120582332 120582332 minus12058233

] 12058211= 120582

22= 120582

33=

30 and

120572

(1)= 001

120573

(1)= 001

120574

(1)= 001

ℎ

(1)

1= 20

ℎ

(1)

2= 20

120572

(2)= 001

120573

(2)= 002

120574

(2)= 002

ℎ

(2)

1= 20

ℎ

(2)

2= 10

120572

(3)= 001

120573

(3)= 004

120574

(3)= 004

ℎ

(3)

1= 10

ℎ

(3)

2= 10

(39)

The stationary probability density 119901(119867) of the totalenergy for uncontrolled and optimally controlled system(28) is shown in Figure 1 Obviously as 119867 increases 119901(119867)for optimally controlled system (28) decreases much fasterthan that for uncontrolled system The stationary probabilitydensity 119901(119902) of the displacement is plotted in Figure 2 withthe same parameters as in Figure 1 For optimally controlledsystem there is a much higher probability that 119902 will belocated near their equilibrium Hence 119901(119902) has much largermode and smaller dispersion around 119902 = 0 for optimallycontrolled system than those for uncontrolled system Thisimplies that the proposed control strategy has high effective-ness for attenuating the response of system (28)

Figure 3 shows the mean value 119864[119867] of the total energyof uncontrolled and optimally controlled system as functionsof ratio 120582

11120582

22(120582

22= 120582

33) Obviously higher value of ratio

120582

11120582

22implies lower probability for the system operating in

form 119904(119905) = 1 and higher probability for the system oper-ating in forms 119904(119905) = 2 and 119904(119905) = 3 When the system oper-ates in form 119904(119905) = 1 the system has the smallest dampingcoefficients and the largest amplitudes of stochastic excita-tions than those when it operates in forms 119904(119905) = 2 and119904(119905) = 3 As a consequence 119864[119867] for uncontrolled systemdecreases monotonously while the ratio 120582

11120582

22increases as

shown in Figure 3 However 119864[119867] for optimally controlledsystem always keeps a very low value despite changing of

Shock and Vibration 7

the ratio 12058211120582

22 which indicates that the proposed control

strategy is very effective and quite robust with respect tothe ratio 120582

11120582

22 In Figures 1ndash3 lines are obtained by

numerical solution of (37) while dots are obtained by directsimulation of system (28)Observe that the dotsmatch closelywith the corresponding lines demonstrating the validity andaccuracy of the proposed method The displacements of theuncontrolled and optimally controlled system (28) are shownin Figure 4 from which the effect of the optimal control onthe displacement can be visualized intuitively

52 Example 2 Consider a controlled stochastically excitedDuffing oscillator withMarkovian jumpwhich is governed by

+ 120596

2119909 + 120572119909

3= 120573 (119904 (119905)) + 119906 (119909 119904 (119905))

+ ℎ (119904 (119905)) 120585 (119905)

(40)

where 120573(119904(119905)) is Markovian jump coefficient of linear damp-ing ℎ(119904(119905)) is Markovian jump coefficient of external randomexcitation 119906(119909 119904(119905)) is the feedback control force 120585(119905) isGaussian white noise with zero mean and intensity 2119863 Herewe consider the 2-form case 119897 = 2 and 119878 = 1 2

Upon the stochastic averaging method the original jumpsystem (40) can be approximately substituted by the partiallyaveraged system (11) with

119898(119867 119904 = 119894) = ℎ

(119894)

2

119863 minus 120573

(119894)119866 (119867)

119898

119906(119867 119904 = 119894) =

4

119879 (119867)

int

119860(119867)

0

119906

(119894)(119902 119901) 119901 119889119902

120590 (119867 119904 = 119894)

2= 2 ℎ

(119894)

2

119863119866 (119867)

119866 (119867) =

4

119879 (119867)

int

119860(119867)

0

(2119867 minus 120596

2119902

2minus

120572119902

4

2

)

12

119889119902

119879 (119867) = 4int

119860(119867)

0

(2119867 minus 120596

2119902

2minus

120572119902

4

2

)

minus12

119889119902

119860 (119867) = [

(120596 + 4120572119867)

12minus 120596

2

120572

]

12

119894 = 1 2

(41)

Following (17) the expression of the optimal control forceis

119906

lowast(119902 119901 119904) = minus

1

2119877

120597119881 (119867 119904)

120597119867

119901(42)

Then the following final dynamical programming equa-tion is obtained1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) minus

1

4119877

120597119881 (119867 119904)

120597119867

119866 (119867)]

120597119881 (119867 119904)

119889119867

+ 119891 (119867 119904) +

2

sum

119903=1

120582

119903119904119881 (119867 119903) = 0 119904 = 1 2

(43)

120597119881120597119867 can be obtained from solving (43) numericallyThen the optimal control force is determined by substituting120597119881120597119867 into (17) Substituting the optimal control force (17)into the partially averaged equation (11) and completingthe averaging yield the following completely averaged FPKequation

120597

120597119905

119901 (119867 119904 119905)

= minus

120597

120597119867

[119898 (119867 119904) minus

1

2119877

120597119881 (119867 119904)

120597119867

119866 (119867)]

sdot 119901 (119867 119904 119905) +

1

2

120597

2

120597119867

2[120590

2(119867 119904) 119901 (119867 119904 119905)]

minus

2

sum

119903=1

119903 =119904

[120582

119904119903119901 (119867 119904 119905) minus 120582

119903119904119901 (119867 119903 119905)] 119904 = 1 2

(44)

Solving (44) numerically with 120597119901120597119905 = 0 the statistics119901(119867) 119901(119902) and 119864[119867] of the stationary response are thenobtained from (25) and (26)

The numerical results shown in Figures 5ndash8 are for systemwith parameters 120596 = 10 120572 = 05 119863 = 002 Λ = [ minus12058211 12058211

12058222 minus120582122]

120582

11= 120582

22= 20 and

120573

(1)= 001

ℎ

(1)= 20

120573

(2)= 002

ℎ

(2)= 10

(45)

The stationary probability densities 119901(119867) of total energyand 119901(119902) of the displacement of uncontrolled and optimallycontrolled system (40) are evaluated and plotted in Figures5 and 6 respectively The mean value 119864[119867] of total energyof uncontrolled and optimally controlled system (40) asfunctions of ratio 120582

11120582

22is shown in Figure 7 It is seen that

the optimal control force alleviates the response of system(40) significantly This effect can be seen more apparentlyfrom Figure 8 Again the analytical results obtained fromsolving (44) match closely with those from digital simulationof original system (40)

6 Conclusions

In this paper we proposed a nonlinear stochastic optimalcontrol strategy for SDOF stochastically excited Markovianjump systems where the jump parameter takes on values in afinite discrete set In the slow jump case the original systemwas first reduced to one governed by a finite set of one-dimensional partially averaged Ito equation for total energyupon stochastic averaging Using the stochastic dynami-cal programming principle and Markovian jump rules afinite set of coupled dynamical programming equations wasderived from which the optimal control force was obtainedThe obtained optimal control force has been proved to be the

8 Shock and Vibration

0 1 2 3 4 50

2

4

6

8

10

Uncontrolled

Optimally controlled

H

p(H

)

Figure 5 Stationary probability density 119901(119867) of total energy ofuncontrolled and optimally controlled system (40) The lines areobtained fromnumerical solution of (44)while the dots are obtainedfrom direct simulation of original system (40)

0 1 2 30

02

04

06

08

1

12

14

16

Uncontrolled

Optimally controlled

q

minus3 minus2 minus1

p(q)

Figure 6 Stationary probability density 119901(119902) of displacement ofuncontrolled and optimally controlled system (40) The lines areobtained fromnumerical solution of (44)while the dots are obtainedfrom direct simulation of original system (40)

quasi-optimal control for the original system The strategywas applied to two nonlinear oscillators that were capableof independent Markovian jumps Numerical results showedthat the strategy is fairly robust and effective in reductionof stationary response of the controlled system Thus it ispotentially promising for practical control applications afterfurther research

The proposed method has the potential to be extended tothe optimal control forMulti-DOFMarkovian jump systemsHowever to obtain the concrete conclusionmuchwork has tobe done Mathematical tools should be developed to solve thedynamical programming equation (14) for Multi-DOF case

1 15 2 25 30

1

2

3

4

5

Uncontrolled

Optimally controlled

E[H

]

1205821112058222

Figure 7 Mean value 119864[119867] of total energy of uncontrolled andoptimally controlled system (40) as functions of the ratio 120582

11120582

22

The lines are obtained fromnumerical solution of (44) while the dotsare obtained from direct simulation of original system (40)

2000 2050 2100 2150 2200

0

5

Uncontrolled

Optimally controlled

q

t

minus5

Figure 8The sample time history of displacements of uncontrolledand optimally controlled system (40)

Disclaimer

Opinions findings and conclusions expressed in this paperare those of the authors and do not necessarily reflect theviews of the sponsors

Conflict of Interests

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

Acknowledgments

This work was supported by the Natural Science Foundationof China (through Grants no 11272279 11372271 1143201211321202 and 11272201) and 973 Program (no 2011CB711105)

Shock and Vibration 9

References

[1] L Kats and N N Krasovskii ldquoOn the stability of systemswith random parametersrdquo Journal of Applied Mathematics andMechanics vol 24 no 5 pp 1225ndash1246 1960

[2] M Mariton ldquoAlmost sure and moments stability of jump linearsystemsrdquo Systems amp Control Letters vol 11 no 5 pp 393ndash3971988

[3] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[4] R H Huan W-Q Zhu F Ma and Z-G Ying ldquoStationaryresponse of a class of nonlinear stochastic systems undergoingMarkovian jumpsrdquo Journal of Applied Mechanics vol 82 no 5Article ID 051008 6 pages 2015

[5] N N Krasosvkii and E A Lidskii ldquoAnalytical design of con-trollers in systems with random attributes Irdquo Automation andRemote Control vol 22 pp 1021ndash1025 1961

[6] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967

[7] O L V Costa M D Fragoso and M G Todorov Continuous-Time Markov Jump Linear Systems Springer New York NYUSA 2013

[8] D D Sworder ldquoFeedback control of a class of linear systemswith jump parametersrdquo IEEE Transactions on Automatic Con-trol vol 14 no 1 pp 9ndash14 1969

[9] W M Wonham ldquoRandom differential equations in controltheoryrdquoProbabilisticMethods in AppliedMathematics vol 2 pp131ndash213 1971

[10] M D Fragoso and E M Hemerly ldquoOptimal control for a classof noisy linear systemswithMarkovian jumping parameters andquadratic costrdquo International Journal of Systems Science vol 22no 12 pp 2553ndash2561 1991

[11] Y D Ji and H J Chizeck ldquoJump linear quadratic Gaussian con-trol in continuous timerdquo IEEE Transactions on AutomaticControl vol 37 no 12 pp 1884ndash1892 1992

[12] M K Ghosh A Arapostathis and S I Marcus ldquoErgodiccontrol of switching diffusionsrdquo SIAM Journal on Control andOptimization vol 35 no 6 pp 1952ndash1988 1997

[13] W Q Zhu ldquoNonlinear stochastic dynamics and control inHamiltonian formulationrdquo Applied Mechanics Reviews vol 59no 4 pp 230ndash248 2006

[14] E K Boukas and H Yang ldquoStability of stochastic systems withjumpsrdquoMathematical Problems in Engineering vol 3 no 2 pp173ndash185 1996

[15] X B Feng K A Loparo Y D Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[16] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions Springer New York NY USA 1993

[17] K Ito ldquoOn stochastic differential equationsrdquo Memoirs of theAmerican Mathematical Society vol 4 pp 289ndash302 1951

[18] W Q Zhu and Z G Ying ldquoOptimal nonlinear feedbackcontrol of quasi-Hamiltonian systemsrdquo Science in China SeriesA Mathematics vol 42 no 11 pp 1213ndash1219 1999

[19] W Q Zhu Z G Ying and T T Soong ldquoAn optimal nonlinearfeedback control strategy for randomly excited structural sys-temsrdquo Nonlinear Dynamics vol 24 no 1 pp 31ndash51 2001

[20] H J Kushner andD S Clark Stochastic ApproximationMethodsfor Constrained andUnconstrained Systems Springer NewYorkNY USA 1978

[21] YW Fang Y LWu andHQWangOptimal ControlTheory ofStochastic Jump System National Defense Inustry Press BeijingChina 2012

[22] S T WuTheTheory of Stochastic Jump System and Its Applica-tion Science Press Beijing China 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

the ratio 12058211120582

22 which indicates that the proposed control

strategy is very effective and quite robust with respect tothe ratio 120582

11120582

22 In Figures 1ndash3 lines are obtained by

numerical solution of (37) while dots are obtained by directsimulation of system (28)Observe that the dotsmatch closelywith the corresponding lines demonstrating the validity andaccuracy of the proposed method The displacements of theuncontrolled and optimally controlled system (28) are shownin Figure 4 from which the effect of the optimal control onthe displacement can be visualized intuitively

52 Example 2 Consider a controlled stochastically excitedDuffing oscillator withMarkovian jumpwhich is governed by

+ 120596

2119909 + 120572119909

3= 120573 (119904 (119905)) + 119906 (119909 119904 (119905))

+ ℎ (119904 (119905)) 120585 (119905)

(40)

where 120573(119904(119905)) is Markovian jump coefficient of linear damp-ing ℎ(119904(119905)) is Markovian jump coefficient of external randomexcitation 119906(119909 119904(119905)) is the feedback control force 120585(119905) isGaussian white noise with zero mean and intensity 2119863 Herewe consider the 2-form case 119897 = 2 and 119878 = 1 2

Upon the stochastic averaging method the original jumpsystem (40) can be approximately substituted by the partiallyaveraged system (11) with

119898(119867 119904 = 119894) = ℎ

(119894)

2

119863 minus 120573

(119894)119866 (119867)

119898

119906(119867 119904 = 119894) =

4

119879 (119867)

int

119860(119867)

0

119906

(119894)(119902 119901) 119901 119889119902

120590 (119867 119904 = 119894)

2= 2 ℎ

(119894)

2

119863119866 (119867)

119866 (119867) =

4

119879 (119867)

int

119860(119867)

0

(2119867 minus 120596

2119902

2minus

120572119902

4

2

)

12

119889119902

119879 (119867) = 4int

119860(119867)

0

(2119867 minus 120596

2119902

2minus

120572119902

4

2

)

minus12

119889119902

119860 (119867) = [

(120596 + 4120572119867)

12minus 120596

2

120572

]

12

119894 = 1 2

(41)

Following (17) the expression of the optimal control forceis

119906

lowast(119902 119901 119904) = minus

1

2119877

120597119881 (119867 119904)

120597119867

119901(42)

Then the following final dynamical programming equa-tion is obtained1

2

120590

2(119867)

120597

2119881 (119867 119904)

120597119867

2

+ [119898 (119867 119904) minus

1

4119877

120597119881 (119867 119904)

120597119867

119866 (119867)]

120597119881 (119867 119904)

119889119867

+ 119891 (119867 119904) +

2

sum

119903=1

120582

119903119904119881 (119867 119903) = 0 119904 = 1 2

(43)

120597119881120597119867 can be obtained from solving (43) numericallyThen the optimal control force is determined by substituting120597119881120597119867 into (17) Substituting the optimal control force (17)into the partially averaged equation (11) and completingthe averaging yield the following completely averaged FPKequation

120597

120597119905

119901 (119867 119904 119905)

= minus

120597

120597119867

[119898 (119867 119904) minus

1

2119877

120597119881 (119867 119904)

120597119867

119866 (119867)]

sdot 119901 (119867 119904 119905) +

1

2

120597

2

120597119867

2[120590

2(119867 119904) 119901 (119867 119904 119905)]

minus

2

sum

119903=1

119903 =119904

[120582

119904119903119901 (119867 119904 119905) minus 120582

119903119904119901 (119867 119903 119905)] 119904 = 1 2

(44)

Solving (44) numerically with 120597119901120597119905 = 0 the statistics119901(119867) 119901(119902) and 119864[119867] of the stationary response are thenobtained from (25) and (26)

The numerical results shown in Figures 5ndash8 are for systemwith parameters 120596 = 10 120572 = 05 119863 = 002 Λ = [ minus12058211 12058211

12058222 minus120582122]

120582

11= 120582

22= 20 and

120573

(1)= 001

ℎ

(1)= 20

120573

(2)= 002

ℎ

(2)= 10

(45)

The stationary probability densities 119901(119867) of total energyand 119901(119902) of the displacement of uncontrolled and optimallycontrolled system (40) are evaluated and plotted in Figures5 and 6 respectively The mean value 119864[119867] of total energyof uncontrolled and optimally controlled system (40) asfunctions of ratio 120582

11120582

22is shown in Figure 7 It is seen that

the optimal control force alleviates the response of system(40) significantly This effect can be seen more apparentlyfrom Figure 8 Again the analytical results obtained fromsolving (44) match closely with those from digital simulationof original system (40)

6 Conclusions

In this paper we proposed a nonlinear stochastic optimalcontrol strategy for SDOF stochastically excited Markovianjump systems where the jump parameter takes on values in afinite discrete set In the slow jump case the original systemwas first reduced to one governed by a finite set of one-dimensional partially averaged Ito equation for total energyupon stochastic averaging Using the stochastic dynami-cal programming principle and Markovian jump rules afinite set of coupled dynamical programming equations wasderived from which the optimal control force was obtainedThe obtained optimal control force has been proved to be the

8 Shock and Vibration

0 1 2 3 4 50

2

4

6

8

10

Uncontrolled

Optimally controlled

H

p(H

)

Figure 5 Stationary probability density 119901(119867) of total energy ofuncontrolled and optimally controlled system (40) The lines areobtained fromnumerical solution of (44)while the dots are obtainedfrom direct simulation of original system (40)

0 1 2 30

02

04

06

08

1

12

14

16

Uncontrolled

Optimally controlled

q

minus3 minus2 minus1

p(q)

Figure 6 Stationary probability density 119901(119902) of displacement ofuncontrolled and optimally controlled system (40) The lines areobtained fromnumerical solution of (44)while the dots are obtainedfrom direct simulation of original system (40)

quasi-optimal control for the original system The strategywas applied to two nonlinear oscillators that were capableof independent Markovian jumps Numerical results showedthat the strategy is fairly robust and effective in reductionof stationary response of the controlled system Thus it ispotentially promising for practical control applications afterfurther research

The proposed method has the potential to be extended tothe optimal control forMulti-DOFMarkovian jump systemsHowever to obtain the concrete conclusionmuchwork has tobe done Mathematical tools should be developed to solve thedynamical programming equation (14) for Multi-DOF case

1 15 2 25 30

1

2

3

4

5

Uncontrolled

Optimally controlled

E[H

]

1205821112058222

Figure 7 Mean value 119864[119867] of total energy of uncontrolled andoptimally controlled system (40) as functions of the ratio 120582

11120582

22

The lines are obtained fromnumerical solution of (44) while the dotsare obtained from direct simulation of original system (40)

2000 2050 2100 2150 2200

0

5

Uncontrolled

Optimally controlled

q

t

minus5

Figure 8The sample time history of displacements of uncontrolledand optimally controlled system (40)

Disclaimer

Opinions findings and conclusions expressed in this paperare those of the authors and do not necessarily reflect theviews of the sponsors

Conflict of Interests

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

Acknowledgments

This work was supported by the Natural Science Foundationof China (through Grants no 11272279 11372271 1143201211321202 and 11272201) and 973 Program (no 2011CB711105)

Shock and Vibration 9

References

[1] L Kats and N N Krasovskii ldquoOn the stability of systemswith random parametersrdquo Journal of Applied Mathematics andMechanics vol 24 no 5 pp 1225ndash1246 1960

[2] M Mariton ldquoAlmost sure and moments stability of jump linearsystemsrdquo Systems amp Control Letters vol 11 no 5 pp 393ndash3971988

[3] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[4] R H Huan W-Q Zhu F Ma and Z-G Ying ldquoStationaryresponse of a class of nonlinear stochastic systems undergoingMarkovian jumpsrdquo Journal of Applied Mechanics vol 82 no 5Article ID 051008 6 pages 2015

[5] N N Krasosvkii and E A Lidskii ldquoAnalytical design of con-trollers in systems with random attributes Irdquo Automation andRemote Control vol 22 pp 1021ndash1025 1961

[6] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967

[7] O L V Costa M D Fragoso and M G Todorov Continuous-Time Markov Jump Linear Systems Springer New York NYUSA 2013

[8] D D Sworder ldquoFeedback control of a class of linear systemswith jump parametersrdquo IEEE Transactions on Automatic Con-trol vol 14 no 1 pp 9ndash14 1969

[9] W M Wonham ldquoRandom differential equations in controltheoryrdquoProbabilisticMethods in AppliedMathematics vol 2 pp131ndash213 1971

[10] M D Fragoso and E M Hemerly ldquoOptimal control for a classof noisy linear systemswithMarkovian jumping parameters andquadratic costrdquo International Journal of Systems Science vol 22no 12 pp 2553ndash2561 1991

[11] Y D Ji and H J Chizeck ldquoJump linear quadratic Gaussian con-trol in continuous timerdquo IEEE Transactions on AutomaticControl vol 37 no 12 pp 1884ndash1892 1992

[12] M K Ghosh A Arapostathis and S I Marcus ldquoErgodiccontrol of switching diffusionsrdquo SIAM Journal on Control andOptimization vol 35 no 6 pp 1952ndash1988 1997

[13] W Q Zhu ldquoNonlinear stochastic dynamics and control inHamiltonian formulationrdquo Applied Mechanics Reviews vol 59no 4 pp 230ndash248 2006

[14] E K Boukas and H Yang ldquoStability of stochastic systems withjumpsrdquoMathematical Problems in Engineering vol 3 no 2 pp173ndash185 1996

[15] X B Feng K A Loparo Y D Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[16] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions Springer New York NY USA 1993

[17] K Ito ldquoOn stochastic differential equationsrdquo Memoirs of theAmerican Mathematical Society vol 4 pp 289ndash302 1951

[18] W Q Zhu and Z G Ying ldquoOptimal nonlinear feedbackcontrol of quasi-Hamiltonian systemsrdquo Science in China SeriesA Mathematics vol 42 no 11 pp 1213ndash1219 1999

[19] W Q Zhu Z G Ying and T T Soong ldquoAn optimal nonlinearfeedback control strategy for randomly excited structural sys-temsrdquo Nonlinear Dynamics vol 24 no 1 pp 31ndash51 2001

[20] H J Kushner andD S Clark Stochastic ApproximationMethodsfor Constrained andUnconstrained Systems Springer NewYorkNY USA 1978

[21] YW Fang Y LWu andHQWangOptimal ControlTheory ofStochastic Jump System National Defense Inustry Press BeijingChina 2012

[22] S T WuTheTheory of Stochastic Jump System and Its Applica-tion Science Press Beijing China 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

0 1 2 3 4 50

2

4

6

8

10

Uncontrolled

Optimally controlled

H

p(H

)

Figure 5 Stationary probability density 119901(119867) of total energy ofuncontrolled and optimally controlled system (40) The lines areobtained fromnumerical solution of (44)while the dots are obtainedfrom direct simulation of original system (40)

0 1 2 30

02

04

06

08

1

12

14

16

Uncontrolled

Optimally controlled

q

minus3 minus2 minus1

p(q)

Figure 6 Stationary probability density 119901(119902) of displacement ofuncontrolled and optimally controlled system (40) The lines areobtained fromnumerical solution of (44)while the dots are obtainedfrom direct simulation of original system (40)

quasi-optimal control for the original system The strategywas applied to two nonlinear oscillators that were capableof independent Markovian jumps Numerical results showedthat the strategy is fairly robust and effective in reductionof stationary response of the controlled system Thus it ispotentially promising for practical control applications afterfurther research

The proposed method has the potential to be extended tothe optimal control forMulti-DOFMarkovian jump systemsHowever to obtain the concrete conclusionmuchwork has tobe done Mathematical tools should be developed to solve thedynamical programming equation (14) for Multi-DOF case

1 15 2 25 30

1

2

3

4

5

Uncontrolled

Optimally controlled

E[H

]

1205821112058222

Figure 7 Mean value 119864[119867] of total energy of uncontrolled andoptimally controlled system (40) as functions of the ratio 120582

11120582

22

The lines are obtained fromnumerical solution of (44) while the dotsare obtained from direct simulation of original system (40)

2000 2050 2100 2150 2200

0

5

Uncontrolled

Optimally controlled

q

t

minus5

Figure 8The sample time history of displacements of uncontrolledand optimally controlled system (40)

Disclaimer

Opinions findings and conclusions expressed in this paperare those of the authors and do not necessarily reflect theviews of the sponsors

Conflict of Interests

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

Acknowledgments

This work was supported by the Natural Science Foundationof China (through Grants no 11272279 11372271 1143201211321202 and 11272201) and 973 Program (no 2011CB711105)

Shock and Vibration 9

References

[1] L Kats and N N Krasovskii ldquoOn the stability of systemswith random parametersrdquo Journal of Applied Mathematics andMechanics vol 24 no 5 pp 1225ndash1246 1960

[2] M Mariton ldquoAlmost sure and moments stability of jump linearsystemsrdquo Systems amp Control Letters vol 11 no 5 pp 393ndash3971988

[3] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[4] R H Huan W-Q Zhu F Ma and Z-G Ying ldquoStationaryresponse of a class of nonlinear stochastic systems undergoingMarkovian jumpsrdquo Journal of Applied Mechanics vol 82 no 5Article ID 051008 6 pages 2015

[5] N N Krasosvkii and E A Lidskii ldquoAnalytical design of con-trollers in systems with random attributes Irdquo Automation andRemote Control vol 22 pp 1021ndash1025 1961

[6] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967

[7] O L V Costa M D Fragoso and M G Todorov Continuous-Time Markov Jump Linear Systems Springer New York NYUSA 2013

[8] D D Sworder ldquoFeedback control of a class of linear systemswith jump parametersrdquo IEEE Transactions on Automatic Con-trol vol 14 no 1 pp 9ndash14 1969

[9] W M Wonham ldquoRandom differential equations in controltheoryrdquoProbabilisticMethods in AppliedMathematics vol 2 pp131ndash213 1971

[10] M D Fragoso and E M Hemerly ldquoOptimal control for a classof noisy linear systemswithMarkovian jumping parameters andquadratic costrdquo International Journal of Systems Science vol 22no 12 pp 2553ndash2561 1991

[11] Y D Ji and H J Chizeck ldquoJump linear quadratic Gaussian con-trol in continuous timerdquo IEEE Transactions on AutomaticControl vol 37 no 12 pp 1884ndash1892 1992

[12] M K Ghosh A Arapostathis and S I Marcus ldquoErgodiccontrol of switching diffusionsrdquo SIAM Journal on Control andOptimization vol 35 no 6 pp 1952ndash1988 1997

[13] W Q Zhu ldquoNonlinear stochastic dynamics and control inHamiltonian formulationrdquo Applied Mechanics Reviews vol 59no 4 pp 230ndash248 2006

[14] E K Boukas and H Yang ldquoStability of stochastic systems withjumpsrdquoMathematical Problems in Engineering vol 3 no 2 pp173ndash185 1996

[15] X B Feng K A Loparo Y D Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[16] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions Springer New York NY USA 1993

[17] K Ito ldquoOn stochastic differential equationsrdquo Memoirs of theAmerican Mathematical Society vol 4 pp 289ndash302 1951

[18] W Q Zhu and Z G Ying ldquoOptimal nonlinear feedbackcontrol of quasi-Hamiltonian systemsrdquo Science in China SeriesA Mathematics vol 42 no 11 pp 1213ndash1219 1999

[19] W Q Zhu Z G Ying and T T Soong ldquoAn optimal nonlinearfeedback control strategy for randomly excited structural sys-temsrdquo Nonlinear Dynamics vol 24 no 1 pp 31ndash51 2001

[20] H J Kushner andD S Clark Stochastic ApproximationMethodsfor Constrained andUnconstrained Systems Springer NewYorkNY USA 1978

[21] YW Fang Y LWu andHQWangOptimal ControlTheory ofStochastic Jump System National Defense Inustry Press BeijingChina 2012

[22] S T WuTheTheory of Stochastic Jump System and Its Applica-tion Science Press Beijing China 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

References

[1] L Kats and N N Krasovskii ldquoOn the stability of systemswith random parametersrdquo Journal of Applied Mathematics andMechanics vol 24 no 5 pp 1225ndash1246 1960

[2] M Mariton ldquoAlmost sure and moments stability of jump linearsystemsrdquo Systems amp Control Letters vol 11 no 5 pp 393ndash3971988

[3] O L V Costa and M D Fragoso ldquoStability results for discrete-time linear systems with Markovian jumping parametersrdquoJournal of Mathematical Analysis and Applications vol 179 no1 pp 154ndash178 1993

[4] R H Huan W-Q Zhu F Ma and Z-G Ying ldquoStationaryresponse of a class of nonlinear stochastic systems undergoingMarkovian jumpsrdquo Journal of Applied Mechanics vol 82 no 5Article ID 051008 6 pages 2015

[5] N N Krasosvkii and E A Lidskii ldquoAnalytical design of con-trollers in systems with random attributes Irdquo Automation andRemote Control vol 22 pp 1021ndash1025 1961

[6] H J Kushner Stochastic Stability and Control Academic PressNew York NY USA 1967

[7] O L V Costa M D Fragoso and M G Todorov Continuous-Time Markov Jump Linear Systems Springer New York NYUSA 2013

[8] D D Sworder ldquoFeedback control of a class of linear systemswith jump parametersrdquo IEEE Transactions on Automatic Con-trol vol 14 no 1 pp 9ndash14 1969

[9] W M Wonham ldquoRandom differential equations in controltheoryrdquoProbabilisticMethods in AppliedMathematics vol 2 pp131ndash213 1971

[10] M D Fragoso and E M Hemerly ldquoOptimal control for a classof noisy linear systemswithMarkovian jumping parameters andquadratic costrdquo International Journal of Systems Science vol 22no 12 pp 2553ndash2561 1991

[11] Y D Ji and H J Chizeck ldquoJump linear quadratic Gaussian con-trol in continuous timerdquo IEEE Transactions on AutomaticControl vol 37 no 12 pp 1884ndash1892 1992

[12] M K Ghosh A Arapostathis and S I Marcus ldquoErgodiccontrol of switching diffusionsrdquo SIAM Journal on Control andOptimization vol 35 no 6 pp 1952ndash1988 1997

[13] W Q Zhu ldquoNonlinear stochastic dynamics and control inHamiltonian formulationrdquo Applied Mechanics Reviews vol 59no 4 pp 230ndash248 2006

[14] E K Boukas and H Yang ldquoStability of stochastic systems withjumpsrdquoMathematical Problems in Engineering vol 3 no 2 pp173ndash185 1996

[15] X B Feng K A Loparo Y D Ji and H J Chizeck ldquoStochasticstability properties of jump linear systemsrdquo IEEE Transactionson Automatic Control vol 37 no 1 pp 38ndash53 1992

[16] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions Springer New York NY USA 1993

[17] K Ito ldquoOn stochastic differential equationsrdquo Memoirs of theAmerican Mathematical Society vol 4 pp 289ndash302 1951

[18] W Q Zhu and Z G Ying ldquoOptimal nonlinear feedbackcontrol of quasi-Hamiltonian systemsrdquo Science in China SeriesA Mathematics vol 42 no 11 pp 1213ndash1219 1999

[19] W Q Zhu Z G Ying and T T Soong ldquoAn optimal nonlinearfeedback control strategy for randomly excited structural sys-temsrdquo Nonlinear Dynamics vol 24 no 1 pp 31ndash51 2001

[20] H J Kushner andD S Clark Stochastic ApproximationMethodsfor Constrained andUnconstrained Systems Springer NewYorkNY USA 1978

[21] YW Fang Y LWu andHQWangOptimal ControlTheory ofStochastic Jump System National Defense Inustry Press BeijingChina 2012

[22] S T WuTheTheory of Stochastic Jump System and Its Applica-tion Science Press Beijing China 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of