Research ArticleOptimal Preview Control for a Class of Linear ContinuousStochastic Control Systems in the Infinite Horizon

Jiang Wu1 Fucheng Liao1 and Jiamei Deng2

1School of Mathematics and Physics University of Science and Technology Beijing Beijing 100083 China2Leeds Sustainability Institute Leeds Beckett University Leeds LS2 9EN UK

Correspondence should be addressed to Fucheng Liao fcliaoustbeducn

Received 1 July 2016 Accepted 26 September 2016

Academic Editor Andrzej Swierniak

Copyright copy 2016 Jiang Wu et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper discusses the optimal preview control problem for a class of linear continuous stochastic control systems in the infinitehorizon based on the augmented error system method Firstly an assistant system is designed and the state equation is translatedto the assistant systemThen an integrator is introduced to construct a stochastic augmented error system As a result the trackingproblem is converted to a regulation problem Secondly the optimal regulator is solved based on dynamic programming principlefor the stochastic system and the optimal preview controller of the original system is obtained Compared with the finite horizonwe simplify the performance index We also study the stability of the stochastic augmented error system and design the observerfor the original stochastic system Finally the simulation example shows the effectiveness of the conclusion in this paper

1 Introduction

Future reference signals or disturbance signals are knownin certain circumstances All the known future informationcan be utilized by preview control theory to improve theperformance of the dynamic system The original idea ofpreview control theory is that in order to minimize the errorbetween the reference signals and the controlled terms notonly the past and present information but also the futureinformation should be concentrated on [1ndash5] An augmentederror system is constructed while designing the optimalpreview controller for the discrete systemAnd also a group ofidentical equations of future reference signals is added to theaugmented error system [6ndash9] Since the continuous systemis of infinite dimensions the method in dealing with thediscrete condition is no longer useful In [10] the augmentederror system was constructed by differentiating the stateequation on both sides combining with the error equationAccording to the maximum principle the optimal previewcontroller was obtained by solving a differential equationon reference signals backward in time This method wasextended to systems with previewable disturbance signalsin [11] and to singular continuous systems in [12] The

application of preview control method to continuous systemswith time delay is studied in [13]

According to automatic control system theory the con-trolled systems can be regarded as falling into two categoriesdeterministic systems and stochastic systems Stochastic sys-tems are a collection of dynamic systems that contain internalstochastic parameters external stochastic disturbances orobservation noises [14] A typical stochastic system is thestochastic differential equation It is a class of differentialequations driven by one or more stochastic processes There-fore the solution of a stochastic differential equation is also astochastic process Up to the present a phenomenon such asstockmarket volatility or thermalmotion in a physical systemis usually described by a stochastic differential equationTypically a white noise stochastic variable represented by thedifferential form of Brownian motion or Weiner process iscontained in a stochastic differential equation

In this paper preview control theory is applied to a classof linear continuous stochastic control systems in the infinitehorizon In the finite horizon [15] an assistant system isdesigned and the state equation is translated to the assistantsystem Then an integrator is introduced to construct astochastic augmented error system As a result the tracking

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 7679165 9 pageshttpdxdoiorg10115520167679165

2 Mathematical Problems in Engineering

problem is converted to a regulation problem The optimalregulator is obtained based on the dynamic programmingprinciple for stochastic systems which means the optimalpreview controller of the original stochastic system is gainedFor such a system when the integrator approaches zeroat infinity the error also approaches zero Therefore thisproperty can be utilized to simplify the performance indexin the infinite horizon Due to the fact that the relative termsof reference signals are included in the stochastic augmentederror system the conclusion in the infinite horizon cannotbe directly employed when solving the optimal regulationproblem in this paper Firstly the corresponding optimalregulation problem is solved in the finite horizon withthe new performance index and then the time was set toapproach infinity The stability of the stochastic augmentederror system is studied and the sufficient and necessarycriteria which guarantee that there exists a unique semipos-itive definite solution to the corresponding Riccati equationhave been met Also an observer for the original stochasticsystem is designed The introduction of the integrator caneliminate the static error and the simulation example showsthe effectiveness of the conclusion in this paper [16]

Notation The notations are standard (Ω 119865 119875) is a completeprobability space Adaptive procedure 119865119905 is 120590 algebra gener-ated by 119861119904 119904 le 119905 and 119861119904 denotes Brownian motion of 1198981015840dimensions 119860 isin 119877119898times119899 denotes 119898 times 119899 matrix 119875 gt 0 (119875 ge 0)denotes a positive definite (semidefinite) matrix 119868 denotesthe unitmatrixThe symbollowast denotes the symmetric terms ina symmetric matrix 1198641199050 1199090 denotes the expectation of process(119905 119909) with initial time 1199050 and state 1199090 tr(sdot) denotes the traceof a matrix

2 Problem Statement

Consider the following stochastic control system119889119909 (119905) = [119872119909 (119905) + 119873119906 (119905)] 119889119905 + 120596 (119905) 119889119861119905119910 (119905) = 119862119909 (119905)

119909 (1199050) = 1199090 119905 isin [1199050infin) (1)

where 119909(119905) is the state vector of 119898 dimensions 119910(119905) is theoutput vector of 119901 dimensions 119906(119905) is the 119865119905 adaptive input of119899 dimensions and 119861119905 is the Brownian motion of 119865119905 adaptiveof 1198981015840 dimensions119872 isin 119877119898times119898119873 isin 119877119898times119899 and 119862 isin 119877119901times119898 areconstant matrices 120596(119905) isin 119877119898times1198981015840

First of all the following assumptions are introduced

Assumption 1 Suppose the matrices pair (119872119873) is stabiliz-able

Assumption 2 Suppose thematrices pair (119862119872) is detectableand the matrix 119862 is of full row rank

By denoting the reference signals of 119901 dimensions with119903(119905) another assumption about 119903(119905) is neededAssumption 3 Suppose the reference signal 119903(119905) is a piecewisecontinuously differentiable function defined on (119905 ge 1199050)

Moreover 119903(120588) (119905 le 120588 le 119905 + 119897119903) is available at any moment119905 and when 120588 gt 119905 + 119897119903 119903(120588) = 119903(119905 + 119897119903) that is 119903(120588) = 0 119897119903 isthe preview length of the reference signal

Remark 4 Since there are few effects on the system when thereference signals exceed the previewable range it is alwaysassumed that the signals are usually constant [11]

The tracking error signal 119890(119905) is defined as the differencebetween the output vector 119910(119905) and the reference signal 119903(119905)that is

119890 (119905) = 119910 (119905) minus 119903 (119905) (2)

In this paper an optimal controller will be designed tomake the output 119910(119905) in (1) tracking the reference signal 119903(119905)as accurately as possible without static error

Then an assistant system is defined The followingassumption is needed

Assumption 5 Suppose

rank [119872 119873119862 0 ] = 119898 + 119901 (full row rank) (3)

With Assumption 5 the following lemma holds

Lemma 6 (see [15]) There exist matrices pairs (Γ 120574) whichsolve the following

119872Γ +119873120574 = 0119862Γ = 119868 (4)

if and only if Assumption 5 holds where Γ isin 119877119898times119901 and 120574 isin119877119899times119901Definition 7 (see [17]) If matrix 119860 isin 119877119898times119899 and the rank of119860 is equal to 119899 then 119860 has a left inverse 119861 isin 119877119899times119898 such that119861119860 = 119868Remark 8 If 119901 = 119899 there exists unique solution of matricespair (Γ 120574) while if 119901 lt 119899 according to Definition 7 anyleft inverse of matrix [119872 119873119862 0 ] can be applied to calculate thesolution of (4) and the solutions are infinite In this paper onlythe square full-rank case is used

Let the state vector and input vector be denoted by 119909lowast(119905)and 119906lowast(119905) respectively Let 119909lowast(119905) = Γ119903(119905 + 119897119903) and 119906lowast(119905) =120574119903(119905 + 119897119903) According to Lemma 6 at the moment 119904 for anytime 119905 (119904 le 119905 le 119905120591) 119909lowast(119905)meets

119889119909lowast (119905) = Γ 119903 (119905 + 119897119903) 119889119905119862119909lowast (119905) = 119862Γ119903 (119905 + 119897119903) (5)

Since [119872Γ + 119873120574]119903(119905 + 119897119903) = 0 and 119862Γ119903(119905 + 119897119903) = 119903(119905 + 119897119903)according to (4) substituting these two equations into (5)yields

119889119909lowast (119905) = [119872Γ + 119873120574] 119903 (119905 + 119897119903) + Γ 119903 (119905 + 119897119903) 119889119905119903 (119905 + 119897119903) = 119862119909lowast (119905) (6)

Mathematical Problems in Engineering 3

that is

119889119909lowast (119905) = [119872119909lowast (119905) + 119873119906lowast (119905)] + Γ 119903 (119905 + 119897119903) 119889119905119903 (119905 + 119897119903) = 119862119909lowast (119905) (7)

Define (119905) = 119909 (119905) minus 119909lowast (119905) (119905) = 119906 (119905) minus 119906lowast (119905) (119905) = 119910 (119905) minus 119862119909lowast (119905) (119905) = 119903 (119905) minus 119903 (119905 + 119897119903)

(8)

Subtracting (7) by (1) from both sides the following could beobtained119889 (119905) = [119872 (119905) + 119873 (119905) + (minusΓ) 119903 (119905 + 119897119903)] 119889119905

+ 120596 (119905) 119889119861119905 (119905) = 119862 (119905)

(1199050) = 1199090 minus 119909lowast (1199050) 119905 isin [1199050infin) (9)

Now an integrator is introduced Integrating the trackingerror signal 119890(119905) in the interval of 1199050 to 119905 yields

119902 (119905) = int1199051199050

119890 (119904) 119889119904 = int1199051199050

[119910 (119904) minus 119903 (119904)] 119889119904 (10)

Obviously 119902(1199050) = 0 Let 119902lowast(119905) = int1199051199050[119862119909lowast(119904) minus 119903(119904 + 119897119903)]119889119904 andthen 119902lowast(119905) = 0 holds Define (119905) = 119902(119905)minus119902lowast(119905) and the differ-ential form of (119905) becomes

119889 (119905) = 119890 (119905) 119889119905 = [119862 (119905) minus (119905)] 119889119905 (11)

Remark 9 According to Assumption 3 the term (119905) in (11)is also previewable while (120588) (119905 le 120588 le 119905 + 119897119903) In particularwhile 120588 ge 119905 + 119897119903 (120588) = 0

In order to make the output 119910(119905) tracking the referencesignal 119903(119905) as accurately as possible the performance index 119869is designed as

119869 (1199050 1199090 (sdot))= 1198641199050 1199090 12 int

infin

1199050

[119879 (119905) 119876 (119905) + 119879 (119905) 119877 (119905)] 119889119905 (12)

where 119876 = 119876119879 gt 0 and 119877 = 119877119879 gt 0 are matrices of properdimensions

Remark 10 Compared with the finite horizon lim119905rarrinfin119890(119905) =0 can also be obtained in the infinite horizon whilelim119905rarrinfin119902(119905) = 0 Therefore the term 119890119879(119905)119876119890119890(119905) in theperformance index of [15] can be replaced by 119879(119905)119876(119905)in (12) Furthermore since there exists unique semipositivedefinite solution of the corresponding Riccati equation when119876 gt 0 no other terms need to be included in the perform-ance index which simplifies the calculation

To solve the optimal control problem of (9) with the per-formance index (12) the augmented error systemmethod canbe employed

3 Construction of the Stochastic AugmentedError System

Combining (9) and (11) the following holds

119889119911 (119905) = [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905+ 120596 (119905) 119889119861119905

(13)

where

119911 (119905) = [ (119905) (119905)]

= [0 1198620 119872]

= [ 0119873]

= [minus1198680 ]

Γ = [ 0minusΓ]

(14)

Referring to the output equation of (9) and the referencesignal the output of (13) can be defined as

(119905) = 119911 (119905) (15)

where = [119868 0] Joining (13) and (15) together the followinghold

119889119911 (119905) = [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905+ 120596 (119905) 119889119861119905

(119905) = 119911 (119905) 119911 (1199050) = [ (1199050)0 (1199050)] = 1199110

(16)

Equation (16) is the needed stochastic augmented errorsystem

4 Design of the Optimal Controller forthe Stochastic System

Denoting the performance index (12) with the related vari-ables in (16) yields

(1199050 1199110 (sdot)) = 11986411990501199110 12sdot intinfin1199050

[119911119879 (119905) (119879119876) 119911 (119905) + 119879 (119905) 119877 (119905)] 119889119905 (17)

Due to the existence of the term (119905) + Γ 119903(119905 + 119897119903) in(16) the conclusion of the optimal regulation problem in

4 Mathematical Problems in Engineering

dealingwith the infinite horizon cannot be directly employedTherefore similar to the condition in deterministic systems[11] the optimal regulator of the stochastic augmented errorsystem (16) can be obtained through the following three steps

Firstly (17) is revised to

= lim119905120591rarrinfin

11986411990501199110 12sdot int1199051205911199050

[119911119879 (119905) 119879119876119911 (119905) + 119879 (119905) 119877 (119905)] 119889119905 (18)

Secondly the finite-time horizon optimal regulationproblem with the performance index

119869 = 1198641199050 1199110 12sdot int1199051205911199050

[119911119879 (119905) 119879119876119911 (119905) + 119879 (119905) 119877 (119905)] 119889119905(19)

is solved where 119905120591 is the terminal timeCompared with the performance index in [15] the two

terms 119890119879(119905)119876119890119890(119905) and 119890119879(119905120591)119876119890(119905120591)119890(119905120591) are removed in (19)Therefore according to the dynamic programming principlefor stochastic systems [16 18] the following lemma about thefinite horizon optimal control problemwith the terminal time119905120591 can be obtained by letting 119876119890 = 119876119890(119905120591) = 0 in Theorem 1 in[15]

Lemma 11 (see [15]) If Assumptions 3 and 5 hold the optimalcontroller of (16) with the performance index (19) is

(119905) = minus119877minus11198791 119875 (119905) 119911 (119905) + 120579 (119905) (20)

where 119875(119905) ge 0 isin 119877(119898+119901)times(119898+119901) satisfies the Riccati differentialequation

minus (119905) = 119879119875 (119905) + 119875 (119905) minus 119875 (119905) 119877minus1119879119875119879 (119905)+ 119879119876

(21)

With the terminal condition 119875(119905120591) = 0 and 120579(119905) the differentialequation is obtained

(119905) + (119879 minus 119875 (119905) 119877minus1119879) 120579 (119905)+ (119875 (119905) (119905) + 119875 (119905) Γ 119903 (119905 + 119897119903)) = 0

(22)

with the terminal condition 120579(119905120591) = 0Thirdly Theorem 12 is received by letting 119905120591 rarrinfin

Theorem 12 If Assumptions 1 to 5 hold the optimal controller(119905) of (16) with the performance index (17) can be expressedas

(119905) = minus119877minus1119879119875119911 (119905)minus 119877minus1119879int119905+119897119903

119905exp [(1199050 minus 119905) ( minus 119877minus1119879119875)119879]

sdot (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904(23)

where 119875 isin 119877times is the unique semipositive definite solution ofRiccati differential equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (24)

Proof In terms of the performance index (17) if ( ) isstabilizable and (11987612

) is detectable according to the

known conclusion there exists a unique semipositive definitesolution 119875 of Riccati algebra equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (25)

And when 119905 rarr infin

lim119905rarrinfin

119875 (119905) = 119875 (26)

As a result the solving of 120579(119905) in (22) is also simplifiedDenote119872119888 = minus119877minus1119879119875 Equation (22) and its term-

inal condition can be expressed as

(119905) + 119872119879119888 120579 (119905) + (119875 (119905) + 119875Γ 119903 (119905 + 119897119903)) = 0120579 (119905120591) = 0

(27)

Based on linear system theory [19] the fundamental solutionmatrix of the homogeneous system corresponding to (27) is

Φ (119905 1199050) = exp [minusint1199051199050

119872119879119888 119889119905] = exp [(1199050 minus 119905)119872119879119888 ] (28)

with the property

Φminus1 (1199050 119905) = Φ (119905 1199050) Φ (119905 1199050) = Φ (119905 119905119904) Φ (119905119904 1199050)

1199050 le 119904 le 119905(29)

Therefore the solution of (27) is

120579 (119905) = Φ (119905 1199050) 120579 (1199050)minus int1199051199050

Φ (119905 119904) (119875 (119905) + 119875Γ 119903 (119905 + 119897119903)) 119889119904 (30)

and the backward form is

120579 (119905120591) = Φ (119905120591 119905) 120579 (119905)minus int119905120591119905Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(31)

With the terminal condition 120579(119905120591) = 0 the following will beobtained

120579 (119905)= Φminus1 (119905120591 119905) int119905120591

119905Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904 (32)

Mathematical Problems in Engineering 5

According to the property of the reference signals in Assump-tion 3 120579(119905) can be expressed as

120579 (119905) = Φminus1 (119905120591 119905)sdot int119905120591and(119905+119897119903)119905

Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904= Φ (119905 119905120591)sdot int119905120591and(119905+119897119903)119905

Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904= int119905120591and(119905+119897119903)119905

Φ (119905 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(33)

Substituting (33) into (20) yields

(119905) = minus119877minus1119879119875119911 (119905) minus 119877minus11198791 int119905120591and(119905+119897119903)119905

Φ (119905 119904)sdot (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(34)

where 119905120591 and (119905 + 119897119903) = min119905120591 119905 + 119897119903The conclusion is that when 119905120591 rarrinfin the following theo-

rem can be obtained

Theorem 13 If Assumptions 1 to 5 hold the optimal controller(119905) of (16) with the performance index (17) can be expressedas

(119905)= minus119877minus1119879119875119911 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(35)

where 119875 isin 119877times is the unique semipositive definite solution ofRiccati differential equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (36)

Then let the matrix 119875 be separated into

119875 = [1198751 1198752] (37)

where 1198751 and 1198752 are matrices of times119901 and times119898 According to(8) (35) 119911(119905) = [ (119905)

(119905)] and int119905+119897119903

119905Φ(119905 119904)119875Γ 119903(119904 + 119897119903)119889119904 = 0 the

following will be obtained

119906 (119905) = minus119877minus11198791198751119902 (119905) minus 119877minus11198791198752119909 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) 119875 (119904) 119889119904 + 119906lowast (119905)

+ 119877minus11198791198752119902lowast (119905) + 119877minus11198791198752119909lowast (119905) (38)

where119875 ge 0 isin 119877times satisfies Riccati differential equation (24)

Due to the fact that 119909lowast(119905) = Γ119903lowast(119905+ 119897119903) 119906lowast(119905) = 120574119903lowast(119905+ 119897119903)and 119902lowast(119905) = 0 the following corollary is obtainedCorollary 14 If Assumptions 1 to 5 hold the optimal previewcontroller 119906(119905) of (1) with the performance index (12) can beexpressed as

119906 (119905) = minus119877minus11198791198751119902 (119905) minus 119877minus11198791198752119909 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) 119875 (119904) 119889119904

+ [120574 + 119877minus11198791198752Γ] 119903 (119905 + 119897119903) (39)

From (39) it can be seen that there are four parts con-tained in the preview controller of (1) the first part is the inte-gral of tracking error term minus119877minus11198791198751119902(119905) stemmed from theinducing of the integrator The second part is the state feed-back minus119877minus11198791198752119909(119905) The third one minus119877minus1119879 int119905+119897119903

119905Φ(119905

119904)119875(119904)119889119904 is the reference preview compensation The lastone [120574 + 119877minus11198791198752Γ]119903(119905 + 119897119903) is the previewable complementof the assistant system

5 Stability of Closed-Loop System

With 120596(119905)119889119861119905 being an external disturbance the stability anddetectability of system (16) are studied only when 120596(119905) = 0Here (16) becomes the deterministic system

119889119911 (119905) = [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905 (40)

As a result the sufficient and necessary criteria which guar-antee that there exists a unique semipositive definite solutionto the Riccati equation (24) can be gained by the known con-clusion as follows

Lemma 15 (see [11]) Suppose the following conditions aresatisfied

(1) The matrices 119876 and 119877 are both positive definite

(2) The matrix

Π = [[119872 119873119862 0]]

(41)

is of full row rank

(3) The matrices pair (119872119873) is stabilizable(4) The matrices pair (119862119872) is detectableThen there exists unique semipositive definite solution in

the Riccati equation (24) and there exists the asymptoticallystable coefficient matrix119872119888 in the closed-loop system (27)

Therefore the following theorem can be received

6 Mathematical Problems in Engineering

Theorem 16 If 119876 and 119877 are both positive definite matricesand Assumptions 1 to 5 hold the full regulation is achieved inthe closed-loop system

119889119911 (119905)= [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905+ 120596 (119905) 119889119861119905

(119905)= minus119877minus1119879119875119911 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(119905) = 119911 (119905)119911 (1199050) = [ (1199050)0 (1199050)] = 1199110

(42)

namely

lim119905rarrinfin

(119905) = 0 (43)

Furthermore the following will be obtained

lim119905rarrinfin

119890 (119905) = 0 (44)

6 State Observer

If the state vector 119909(119905) in (1) cannot be measured directly theoptimal preview controller (39) cannot be realized In orderto solve this problem the state observer

119889 (119905) = [119872 (119905) + 119873119906 (119905) + 119871 (119910 (119905) minus 119862 (119905))] 119889119905+ 120596 (119905) 119889119861119905 119909 (1199050) = 1199090 119905 isin [1199050infin) (45)

could be designedSubtracting (45) from (1) on both sides yields

119889119909 (119905) = [(119872 minus 119871119862) 119909 (119905)] 119889119905 (46)

where 119909(119905) = 119909(119905) minus (119905)So if (119872 minus 119871119862) is stable the following will hold

lim119905rarrinfin

119909 (119905) = 0 (47)

which means that the state vector (119905) in the observerequation (45) approximates the state vector 119909(119905) in (1) when119905 rarr infin Based on linear system theory [19] it is known thatif (119862 119860) is detectable (119872 minus 119871119862) is stable Then the followingwill be obtained

Theorem 17 If the conditions in Theorem 16 hold the closed-loop system can be received

119889119909 (119905) = [119872119909 (119905) + 119873119906 (119905)] 119889119905 + 120596 (119905) 119889119861119905119889 (119905) = [119872 (119905) + 119873119906 (119905) + 119871 (119910 (119905) minus 119862 (119905))] 119889119905

+ 120596 (119905) 119889119861119905119910 (119905) = 119862119909 (119905) 119906 (119905) = minus119877minus11198791198751 int119905

1199050

(119862 (119904) minus 119903 (119904)) 119889119904minus 119877minus11198791198752 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) 119875 (119904) 119889119904

+ [120574 + 119877minus11198791198752Γ] 119903 (119905 + 119897119903)

(48)

where (119872 minus 119871119862) is stable and (48) achieve the completeregulation

7 Numerical Simulation

Example 1 Consider the stochastic control system

119889[1199091 (119905)1199092 (119905)] = [[0 1minus1 minus1][

1199091 (119905)1199092 (119905)] + [

01] 119906 (119905)] 119889119905

+ 01 119889119861119905119910 (119905) = [1 0] [1199091 (119905)1199092 (119905)]

(49)

where the coefficient matrices are

119872 = [ 0 1minus1 minus1]

119873 = [01] 119862 = [1 0]

120596 (119905) = [0101]

(50)

respectively Therefore the coefficient matrices in (16) are

= [[[

0 1 00 0 10 minus1 minus1

]]]

= [[[

001]]]

Mathematical Problems in Engineering 7

Reference signal r(t)lr = 0 s output y(t)

minus02

0

02

04

06

08

1

12

y amp

r

5 10 15 20 25 30 35 400t (s)

Figure 1 Closed-loop output response with nonpreview and refer-ence signal 119903(119905)

= [[[

minus1 00 minus10 0

]]]

(51)

and Brownian motion 119861119905 satisfies119861119905 sim 119873 (0 001) (52)

Let the initial state of (1) be

119909 (0) = [00] (53)

where 1199091 and 1199092 are mutually independent The previewlengths of the reference signals are 119897119894119903 = 0 05 075 s (119894 =1 2 3) respectively The weight matrices of the performanceindex are

119876 = 1119877 = 1 (54)

and the solution of Riccati equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (55)

is

119875 = [[[

21533 18184 100018184 19157 11531000 1153 0818

]]] (56)

which is a positive definite matrix The coefficient matrix119872119888in (25) is

119872119888 = minus 119877minus1119879119875 = [[[

0 1 00 0 1minus1 minus2153 minus1818

]]] (57)

Reference signal r(t)lr = 05 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

0

02

04

06

08

1

12

y amp

r

Figure 2 Closed-loop output response with 119897119903 = 05 s preview andreference signal

Reference signal r(t)lr = 075 s output y(t)

minus02

0

02

04

06

08

1

12

y amp

r

5 10 15 20 25 30 35 400t (s)

Figure 3 Closed-loop output response with 119897119903 = 075 s preview andreference signal 119903(119905)

When the reference signal is

119903 (119905) =

0 0 le 119905 le 502 (119905 minus 5) 5 lt 119905 le 101 119905 gt 10

(58)

Figures 1ndash3 can be obtainedFigure 1 shows the tracking controller with nonpreview

Figure 2 shows the tracking controller with 119897119903 = 05 s previewFigure 3 shows the tracking controller with 119897119903 = 075 s pre-view Comparing the above three figures it can be seen that

8 Mathematical Problems in Engineering

Reference signal r(t)lr = 0 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

minus015

minus01

minus005

0

005

01

015

02

y amp

r

Figure 4 Closed-loop output response with nonpreview andreference signal 119903(119905)

5 10 15 20 25 30 35 400t (s)

minus015

minus01

minus005

0

005

01

015

02

y amp

r

Reference signal r(t)lr = 05 s output y(t)

Figure 5 Closed-loop output response with 119897119903 = 05 s preview andreference signal

based on the preview controller the output signals can trackthe reference signals much faster and with less tracking error

When the reference signal is

119903 (119905) =

0 0 lt 119905 le 501 sin 05 (119905 minus 5) 5 lt 119905 le 5 + 51205870 119905 gt 5 + 5120587

(59)

Figures 4ndash6 can be obtainedFigure 4 shows the tracking controller with nonpreview

Figure 5 shows the tracking controller with 119897119903 = 05 s preview

Reference signal r(t)lr = 075 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

minus015

minus01

minus005

0

005

01

015

y amp

r

Figure 6 Closed-loop output response with 119897119903 = 075 s preview andreference signal 119903(119905)

Figure 6 shows the tracking controller with 119897119903 = 075 s pre-view Comparing the above three figures it can be seen thatbased on the preview controller the output signals can trackthe reference signals much faster and with less tracking error

8 Conclusion

This paper has studied the optimal preview control problemfor a class of continuous stochastic system in the infinitehorizon By introducing the integrator and the assistantsystem the stochastic augmented error system is constructedCompared with the finite horizon the performance indexis simplified The stability of the stochastic augmented errorsystem is also studied and the observer for the originalstochastic system is designed Finally the simulation exampleshows the effectiveness of the conclusion in this paper

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 61174209)

References

[1] M Tomizuka ldquoOptimal continuous finite preview problemrdquoIEEE Transactions on Automatic Control vol 20 no 3 pp 362ndash365 1975

[2] Y Xu F Liao L Cui and J Wu ldquoPreview control for a class oflinear continuous time-varying systemsrdquo International Journalof Wavelets Multiresolution and Information Processing vol 11no 2 Article ID 1350018 1350018 14 pages 2013

[3] F LiaoM Cao Z Hu and P An ldquoDesign of an optimal previewcontroller for linear discrete time causal descriptor systemsrdquo

Mathematical Problems in Engineering 9

International Journal of Control vol 85 no 10 pp 1616ndash16242012

[4] F Liao K Takaba T Katayama and J Katsuura ldquoDesign of anoptimal preview servomechanism for discrete-time systems ina multirate settingrdquoDynamics of Continuous Discrete amp Impul-sive Systems Series B Applications amp Algorithms vol 10 no 5pp 727ndash744 2003

[5] T Katayama T Ohki T Inoue and T Kato ldquoDesign of anoptimal controller for a discrete-time system subject to pre-viewable demandrdquo International Journal of Control vol 41 no3 pp 677ndash699 1985

[6] F Liao Y Guo and Y Y Tang ldquoDesign of an optimal previewcontroller for linear time-varying discrete systems in amultiratesettingrdquo International Journal of Wavelets Multiresolution andInformation Processing vol 13 no 6 Article ID 15500501550050 19 pages 2015

[7] T Tsuchiya andT EgamiDigital Preview andPredictive ControlTranslated by Fucheng Liao Beijing Science and TechnologyPress Beijing China 1994

[8] M Cao and F Liao ldquoDesign of an optimal preview controller forlinear discrete-time descriptor systems with state delayrdquo Inter-national Journal of Systems Science vol 46 no 5 pp 932ndash9432015

[9] Z Y Zhen Z S Wang and D B Wang ldquoInformation fusionestimation based preview control for a discrete linear systemrdquoActa Automatica Sinica vol 36 no 2 pp 347ndash352 2010

[10] T Katayama and T Hirono ldquoDesign of an optimal servomech-anism with preview action and its dual problemrdquo InternationalJournal of Control vol 45 no 2 pp 407ndash420 1987

[11] F Liao Y Y Tang H Liu and Y Wang ldquoDesign of an optimalpreview controller for continuous-time systemsrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 9 no 4 pp 655ndash673 2011

[12] F Liao Z Ren M Tomizuka and J Wu ldquoPreview controlfor impulse-free continuous-time descriptor systemsrdquo Interna-tional Journal of Control vol 88 no 6 pp 1142ndash1149 2015

[13] F Liao andHXu ldquoApplication of the preview controlmethod tothe optimal tracking control problem for continuous-time sys-tems with time-delayrdquo Mathematical Problems in Engineeringvol 2015 Article ID 423580 8 pages 2015

[14] A Astrom Introduction to Stochastic ControlTheory Translatedby Yuhuan Pan Science Press Beijing China 1983

[15] JWu F Liao andM Tomizuka ldquoOptimal preview control for alinear continuous-time stochastic control system in finite-timehorizonrdquo International Journal of Systems Science vol 48 no 1pp 129ndash137 2017

[16] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 6th edition2005

[17] T H Cormen C E Leiserson R Rivest and C Stein Introduc-tion to Algorithms MIT Press and McGraw-Hill Boston MassUSA 2nd edition 2001

[18] J Yong and X Y Zhou Stochastic Control Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[19] D Zheng Linear System Theory Tsinghua University PressBeijing China 2002

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Mathematical Problems in Engineering

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Function Spaces

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

problem is converted to a regulation problem The optimalregulator is obtained based on the dynamic programmingprinciple for stochastic systems which means the optimalpreview controller of the original stochastic system is gainedFor such a system when the integrator approaches zeroat infinity the error also approaches zero Therefore thisproperty can be utilized to simplify the performance indexin the infinite horizon Due to the fact that the relative termsof reference signals are included in the stochastic augmentederror system the conclusion in the infinite horizon cannotbe directly employed when solving the optimal regulationproblem in this paper Firstly the corresponding optimalregulation problem is solved in the finite horizon withthe new performance index and then the time was set toapproach infinity The stability of the stochastic augmentederror system is studied and the sufficient and necessarycriteria which guarantee that there exists a unique semipos-itive definite solution to the corresponding Riccati equationhave been met Also an observer for the original stochasticsystem is designed The introduction of the integrator caneliminate the static error and the simulation example showsthe effectiveness of the conclusion in this paper [16]

Notation The notations are standard (Ω 119865 119875) is a completeprobability space Adaptive procedure 119865119905 is 120590 algebra gener-ated by 119861119904 119904 le 119905 and 119861119904 denotes Brownian motion of 1198981015840dimensions 119860 isin 119877119898times119899 denotes 119898 times 119899 matrix 119875 gt 0 (119875 ge 0)denotes a positive definite (semidefinite) matrix 119868 denotesthe unitmatrixThe symbollowast denotes the symmetric terms ina symmetric matrix 1198641199050 1199090 denotes the expectation of process(119905 119909) with initial time 1199050 and state 1199090 tr(sdot) denotes the traceof a matrix

2 Problem Statement

Consider the following stochastic control system119889119909 (119905) = [119872119909 (119905) + 119873119906 (119905)] 119889119905 + 120596 (119905) 119889119861119905119910 (119905) = 119862119909 (119905)

119909 (1199050) = 1199090 119905 isin [1199050infin) (1)

where 119909(119905) is the state vector of 119898 dimensions 119910(119905) is theoutput vector of 119901 dimensions 119906(119905) is the 119865119905 adaptive input of119899 dimensions and 119861119905 is the Brownian motion of 119865119905 adaptiveof 1198981015840 dimensions119872 isin 119877119898times119898119873 isin 119877119898times119899 and 119862 isin 119877119901times119898 areconstant matrices 120596(119905) isin 119877119898times1198981015840

First of all the following assumptions are introduced

Assumption 1 Suppose the matrices pair (119872119873) is stabiliz-able

Assumption 2 Suppose thematrices pair (119862119872) is detectableand the matrix 119862 is of full row rank

By denoting the reference signals of 119901 dimensions with119903(119905) another assumption about 119903(119905) is neededAssumption 3 Suppose the reference signal 119903(119905) is a piecewisecontinuously differentiable function defined on (119905 ge 1199050)

Moreover 119903(120588) (119905 le 120588 le 119905 + 119897119903) is available at any moment119905 and when 120588 gt 119905 + 119897119903 119903(120588) = 119903(119905 + 119897119903) that is 119903(120588) = 0 119897119903 isthe preview length of the reference signal

Remark 4 Since there are few effects on the system when thereference signals exceed the previewable range it is alwaysassumed that the signals are usually constant [11]

The tracking error signal 119890(119905) is defined as the differencebetween the output vector 119910(119905) and the reference signal 119903(119905)that is

119890 (119905) = 119910 (119905) minus 119903 (119905) (2)

In this paper an optimal controller will be designed tomake the output 119910(119905) in (1) tracking the reference signal 119903(119905)as accurately as possible without static error

Then an assistant system is defined The followingassumption is needed

Assumption 5 Suppose

rank [119872 119873119862 0 ] = 119898 + 119901 (full row rank) (3)

With Assumption 5 the following lemma holds

Lemma 6 (see [15]) There exist matrices pairs (Γ 120574) whichsolve the following

119872Γ +119873120574 = 0119862Γ = 119868 (4)

if and only if Assumption 5 holds where Γ isin 119877119898times119901 and 120574 isin119877119899times119901Definition 7 (see [17]) If matrix 119860 isin 119877119898times119899 and the rank of119860 is equal to 119899 then 119860 has a left inverse 119861 isin 119877119899times119898 such that119861119860 = 119868Remark 8 If 119901 = 119899 there exists unique solution of matricespair (Γ 120574) while if 119901 lt 119899 according to Definition 7 anyleft inverse of matrix [119872 119873119862 0 ] can be applied to calculate thesolution of (4) and the solutions are infinite In this paper onlythe square full-rank case is used

Let the state vector and input vector be denoted by 119909lowast(119905)and 119906lowast(119905) respectively Let 119909lowast(119905) = Γ119903(119905 + 119897119903) and 119906lowast(119905) =120574119903(119905 + 119897119903) According to Lemma 6 at the moment 119904 for anytime 119905 (119904 le 119905 le 119905120591) 119909lowast(119905)meets

119889119909lowast (119905) = Γ 119903 (119905 + 119897119903) 119889119905119862119909lowast (119905) = 119862Γ119903 (119905 + 119897119903) (5)

Since [119872Γ + 119873120574]119903(119905 + 119897119903) = 0 and 119862Γ119903(119905 + 119897119903) = 119903(119905 + 119897119903)according to (4) substituting these two equations into (5)yields

119889119909lowast (119905) = [119872Γ + 119873120574] 119903 (119905 + 119897119903) + Γ 119903 (119905 + 119897119903) 119889119905119903 (119905 + 119897119903) = 119862119909lowast (119905) (6)

Mathematical Problems in Engineering 3

that is

119889119909lowast (119905) = [119872119909lowast (119905) + 119873119906lowast (119905)] + Γ 119903 (119905 + 119897119903) 119889119905119903 (119905 + 119897119903) = 119862119909lowast (119905) (7)

Define (119905) = 119909 (119905) minus 119909lowast (119905) (119905) = 119906 (119905) minus 119906lowast (119905) (119905) = 119910 (119905) minus 119862119909lowast (119905) (119905) = 119903 (119905) minus 119903 (119905 + 119897119903)

(8)

Subtracting (7) by (1) from both sides the following could beobtained119889 (119905) = [119872 (119905) + 119873 (119905) + (minusΓ) 119903 (119905 + 119897119903)] 119889119905

+ 120596 (119905) 119889119861119905 (119905) = 119862 (119905)

(1199050) = 1199090 minus 119909lowast (1199050) 119905 isin [1199050infin) (9)

Now an integrator is introduced Integrating the trackingerror signal 119890(119905) in the interval of 1199050 to 119905 yields

119902 (119905) = int1199051199050

119890 (119904) 119889119904 = int1199051199050

[119910 (119904) minus 119903 (119904)] 119889119904 (10)

Obviously 119902(1199050) = 0 Let 119902lowast(119905) = int1199051199050[119862119909lowast(119904) minus 119903(119904 + 119897119903)]119889119904 andthen 119902lowast(119905) = 0 holds Define (119905) = 119902(119905)minus119902lowast(119905) and the differ-ential form of (119905) becomes

119889 (119905) = 119890 (119905) 119889119905 = [119862 (119905) minus (119905)] 119889119905 (11)

Remark 9 According to Assumption 3 the term (119905) in (11)is also previewable while (120588) (119905 le 120588 le 119905 + 119897119903) In particularwhile 120588 ge 119905 + 119897119903 (120588) = 0

In order to make the output 119910(119905) tracking the referencesignal 119903(119905) as accurately as possible the performance index 119869is designed as

119869 (1199050 1199090 (sdot))= 1198641199050 1199090 12 int

infin

1199050

[119879 (119905) 119876 (119905) + 119879 (119905) 119877 (119905)] 119889119905 (12)

where 119876 = 119876119879 gt 0 and 119877 = 119877119879 gt 0 are matrices of properdimensions

Remark 10 Compared with the finite horizon lim119905rarrinfin119890(119905) =0 can also be obtained in the infinite horizon whilelim119905rarrinfin119902(119905) = 0 Therefore the term 119890119879(119905)119876119890119890(119905) in theperformance index of [15] can be replaced by 119879(119905)119876(119905)in (12) Furthermore since there exists unique semipositivedefinite solution of the corresponding Riccati equation when119876 gt 0 no other terms need to be included in the perform-ance index which simplifies the calculation

To solve the optimal control problem of (9) with the per-formance index (12) the augmented error systemmethod canbe employed

3 Construction of the Stochastic AugmentedError System

Combining (9) and (11) the following holds

119889119911 (119905) = [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905+ 120596 (119905) 119889119861119905

(13)

where

119911 (119905) = [ (119905) (119905)]

= [0 1198620 119872]

= [ 0119873]

= [minus1198680 ]

Γ = [ 0minusΓ]

(14)

Referring to the output equation of (9) and the referencesignal the output of (13) can be defined as

(119905) = 119911 (119905) (15)

where = [119868 0] Joining (13) and (15) together the followinghold

119889119911 (119905) = [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905+ 120596 (119905) 119889119861119905

(119905) = 119911 (119905) 119911 (1199050) = [ (1199050)0 (1199050)] = 1199110

(16)

Equation (16) is the needed stochastic augmented errorsystem

4 Design of the Optimal Controller forthe Stochastic System

Denoting the performance index (12) with the related vari-ables in (16) yields

(1199050 1199110 (sdot)) = 11986411990501199110 12sdot intinfin1199050

[119911119879 (119905) (119879119876) 119911 (119905) + 119879 (119905) 119877 (119905)] 119889119905 (17)

Due to the existence of the term (119905) + Γ 119903(119905 + 119897119903) in(16) the conclusion of the optimal regulation problem in

4 Mathematical Problems in Engineering

dealingwith the infinite horizon cannot be directly employedTherefore similar to the condition in deterministic systems[11] the optimal regulator of the stochastic augmented errorsystem (16) can be obtained through the following three steps

Firstly (17) is revised to

= lim119905120591rarrinfin

11986411990501199110 12sdot int1199051205911199050

[119911119879 (119905) 119879119876119911 (119905) + 119879 (119905) 119877 (119905)] 119889119905 (18)

Secondly the finite-time horizon optimal regulationproblem with the performance index

119869 = 1198641199050 1199110 12sdot int1199051205911199050

[119911119879 (119905) 119879119876119911 (119905) + 119879 (119905) 119877 (119905)] 119889119905(19)

is solved where 119905120591 is the terminal timeCompared with the performance index in [15] the two

terms 119890119879(119905)119876119890119890(119905) and 119890119879(119905120591)119876119890(119905120591)119890(119905120591) are removed in (19)Therefore according to the dynamic programming principlefor stochastic systems [16 18] the following lemma about thefinite horizon optimal control problemwith the terminal time119905120591 can be obtained by letting 119876119890 = 119876119890(119905120591) = 0 in Theorem 1 in[15]

Lemma 11 (see [15]) If Assumptions 3 and 5 hold the optimalcontroller of (16) with the performance index (19) is

(119905) = minus119877minus11198791 119875 (119905) 119911 (119905) + 120579 (119905) (20)

where 119875(119905) ge 0 isin 119877(119898+119901)times(119898+119901) satisfies the Riccati differentialequation

minus (119905) = 119879119875 (119905) + 119875 (119905) minus 119875 (119905) 119877minus1119879119875119879 (119905)+ 119879119876

(21)

With the terminal condition 119875(119905120591) = 0 and 120579(119905) the differentialequation is obtained

(119905) + (119879 minus 119875 (119905) 119877minus1119879) 120579 (119905)+ (119875 (119905) (119905) + 119875 (119905) Γ 119903 (119905 + 119897119903)) = 0

(22)

with the terminal condition 120579(119905120591) = 0Thirdly Theorem 12 is received by letting 119905120591 rarrinfin

Theorem 12 If Assumptions 1 to 5 hold the optimal controller(119905) of (16) with the performance index (17) can be expressedas

(119905) = minus119877minus1119879119875119911 (119905)minus 119877minus1119879int119905+119897119903

119905exp [(1199050 minus 119905) ( minus 119877minus1119879119875)119879]

sdot (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904(23)

where 119875 isin 119877times is the unique semipositive definite solution ofRiccati differential equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (24)

Proof In terms of the performance index (17) if ( ) isstabilizable and (11987612

) is detectable according to the

known conclusion there exists a unique semipositive definitesolution 119875 of Riccati algebra equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (25)

And when 119905 rarr infin

lim119905rarrinfin

119875 (119905) = 119875 (26)

As a result the solving of 120579(119905) in (22) is also simplifiedDenote119872119888 = minus119877minus1119879119875 Equation (22) and its term-

inal condition can be expressed as

(119905) + 119872119879119888 120579 (119905) + (119875 (119905) + 119875Γ 119903 (119905 + 119897119903)) = 0120579 (119905120591) = 0

(27)

Based on linear system theory [19] the fundamental solutionmatrix of the homogeneous system corresponding to (27) is

Φ (119905 1199050) = exp [minusint1199051199050

119872119879119888 119889119905] = exp [(1199050 minus 119905)119872119879119888 ] (28)

with the property

Φminus1 (1199050 119905) = Φ (119905 1199050) Φ (119905 1199050) = Φ (119905 119905119904) Φ (119905119904 1199050)

1199050 le 119904 le 119905(29)

Therefore the solution of (27) is

120579 (119905) = Φ (119905 1199050) 120579 (1199050)minus int1199051199050

Φ (119905 119904) (119875 (119905) + 119875Γ 119903 (119905 + 119897119903)) 119889119904 (30)

and the backward form is

120579 (119905120591) = Φ (119905120591 119905) 120579 (119905)minus int119905120591119905Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(31)

With the terminal condition 120579(119905120591) = 0 the following will beobtained

120579 (119905)= Φminus1 (119905120591 119905) int119905120591

119905Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904 (32)

Mathematical Problems in Engineering 5

According to the property of the reference signals in Assump-tion 3 120579(119905) can be expressed as

120579 (119905) = Φminus1 (119905120591 119905)sdot int119905120591and(119905+119897119903)119905

Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904= Φ (119905 119905120591)sdot int119905120591and(119905+119897119903)119905

Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904= int119905120591and(119905+119897119903)119905

Φ (119905 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(33)

Substituting (33) into (20) yields

(119905) = minus119877minus1119879119875119911 (119905) minus 119877minus11198791 int119905120591and(119905+119897119903)119905

Φ (119905 119904)sdot (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(34)

where 119905120591 and (119905 + 119897119903) = min119905120591 119905 + 119897119903The conclusion is that when 119905120591 rarrinfin the following theo-

rem can be obtained

Theorem 13 If Assumptions 1 to 5 hold the optimal controller(119905) of (16) with the performance index (17) can be expressedas

(119905)= minus119877minus1119879119875119911 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(35)

where 119875 isin 119877times is the unique semipositive definite solution ofRiccati differential equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (36)

Then let the matrix 119875 be separated into

119875 = [1198751 1198752] (37)

where 1198751 and 1198752 are matrices of times119901 and times119898 According to(8) (35) 119911(119905) = [ (119905)

(119905)] and int119905+119897119903

119905Φ(119905 119904)119875Γ 119903(119904 + 119897119903)119889119904 = 0 the

following will be obtained

119906 (119905) = minus119877minus11198791198751119902 (119905) minus 119877minus11198791198752119909 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) 119875 (119904) 119889119904 + 119906lowast (119905)

+ 119877minus11198791198752119902lowast (119905) + 119877minus11198791198752119909lowast (119905) (38)

where119875 ge 0 isin 119877times satisfies Riccati differential equation (24)

Due to the fact that 119909lowast(119905) = Γ119903lowast(119905+ 119897119903) 119906lowast(119905) = 120574119903lowast(119905+ 119897119903)and 119902lowast(119905) = 0 the following corollary is obtainedCorollary 14 If Assumptions 1 to 5 hold the optimal previewcontroller 119906(119905) of (1) with the performance index (12) can beexpressed as

119906 (119905) = minus119877minus11198791198751119902 (119905) minus 119877minus11198791198752119909 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) 119875 (119904) 119889119904

+ [120574 + 119877minus11198791198752Γ] 119903 (119905 + 119897119903) (39)

From (39) it can be seen that there are four parts con-tained in the preview controller of (1) the first part is the inte-gral of tracking error term minus119877minus11198791198751119902(119905) stemmed from theinducing of the integrator The second part is the state feed-back minus119877minus11198791198752119909(119905) The third one minus119877minus1119879 int119905+119897119903

119905Φ(119905

119904)119875(119904)119889119904 is the reference preview compensation The lastone [120574 + 119877minus11198791198752Γ]119903(119905 + 119897119903) is the previewable complementof the assistant system

5 Stability of Closed-Loop System

With 120596(119905)119889119861119905 being an external disturbance the stability anddetectability of system (16) are studied only when 120596(119905) = 0Here (16) becomes the deterministic system

119889119911 (119905) = [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905 (40)

As a result the sufficient and necessary criteria which guar-antee that there exists a unique semipositive definite solutionto the Riccati equation (24) can be gained by the known con-clusion as follows

Lemma 15 (see [11]) Suppose the following conditions aresatisfied

(1) The matrices 119876 and 119877 are both positive definite

(2) The matrix

Π = [[119872 119873119862 0]]

(41)

is of full row rank

(3) The matrices pair (119872119873) is stabilizable(4) The matrices pair (119862119872) is detectableThen there exists unique semipositive definite solution in

the Riccati equation (24) and there exists the asymptoticallystable coefficient matrix119872119888 in the closed-loop system (27)

Therefore the following theorem can be received

6 Mathematical Problems in Engineering

Theorem 16 If 119876 and 119877 are both positive definite matricesand Assumptions 1 to 5 hold the full regulation is achieved inthe closed-loop system

119889119911 (119905)= [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905+ 120596 (119905) 119889119861119905

(119905)= minus119877minus1119879119875119911 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(119905) = 119911 (119905)119911 (1199050) = [ (1199050)0 (1199050)] = 1199110

(42)

namely

lim119905rarrinfin

(119905) = 0 (43)

Furthermore the following will be obtained

lim119905rarrinfin

119890 (119905) = 0 (44)

6 State Observer

If the state vector 119909(119905) in (1) cannot be measured directly theoptimal preview controller (39) cannot be realized In orderto solve this problem the state observer

119889 (119905) = [119872 (119905) + 119873119906 (119905) + 119871 (119910 (119905) minus 119862 (119905))] 119889119905+ 120596 (119905) 119889119861119905 119909 (1199050) = 1199090 119905 isin [1199050infin) (45)

could be designedSubtracting (45) from (1) on both sides yields

119889119909 (119905) = [(119872 minus 119871119862) 119909 (119905)] 119889119905 (46)

where 119909(119905) = 119909(119905) minus (119905)So if (119872 minus 119871119862) is stable the following will hold

lim119905rarrinfin

119909 (119905) = 0 (47)

which means that the state vector (119905) in the observerequation (45) approximates the state vector 119909(119905) in (1) when119905 rarr infin Based on linear system theory [19] it is known thatif (119862 119860) is detectable (119872 minus 119871119862) is stable Then the followingwill be obtained

Theorem 17 If the conditions in Theorem 16 hold the closed-loop system can be received

119889119909 (119905) = [119872119909 (119905) + 119873119906 (119905)] 119889119905 + 120596 (119905) 119889119861119905119889 (119905) = [119872 (119905) + 119873119906 (119905) + 119871 (119910 (119905) minus 119862 (119905))] 119889119905

+ 120596 (119905) 119889119861119905119910 (119905) = 119862119909 (119905) 119906 (119905) = minus119877minus11198791198751 int119905

1199050

(119862 (119904) minus 119903 (119904)) 119889119904minus 119877minus11198791198752 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) 119875 (119904) 119889119904

+ [120574 + 119877minus11198791198752Γ] 119903 (119905 + 119897119903)

(48)

where (119872 minus 119871119862) is stable and (48) achieve the completeregulation

7 Numerical Simulation

Example 1 Consider the stochastic control system

119889[1199091 (119905)1199092 (119905)] = [[0 1minus1 minus1][

1199091 (119905)1199092 (119905)] + [

01] 119906 (119905)] 119889119905

+ 01 119889119861119905119910 (119905) = [1 0] [1199091 (119905)1199092 (119905)]

(49)

where the coefficient matrices are

119872 = [ 0 1minus1 minus1]

119873 = [01] 119862 = [1 0]

120596 (119905) = [0101]

(50)

respectively Therefore the coefficient matrices in (16) are

= [[[

0 1 00 0 10 minus1 minus1

]]]

= [[[

001]]]

Mathematical Problems in Engineering 7

Reference signal r(t)lr = 0 s output y(t)

minus02

0

02

04

06

08

1

12

y amp

r

5 10 15 20 25 30 35 400t (s)

Figure 1 Closed-loop output response with nonpreview and refer-ence signal 119903(119905)

= [[[

minus1 00 minus10 0

]]]

(51)

and Brownian motion 119861119905 satisfies119861119905 sim 119873 (0 001) (52)

Let the initial state of (1) be

119909 (0) = [00] (53)

where 1199091 and 1199092 are mutually independent The previewlengths of the reference signals are 119897119894119903 = 0 05 075 s (119894 =1 2 3) respectively The weight matrices of the performanceindex are

119876 = 1119877 = 1 (54)

and the solution of Riccati equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (55)

is

119875 = [[[

21533 18184 100018184 19157 11531000 1153 0818

]]] (56)

which is a positive definite matrix The coefficient matrix119872119888in (25) is

119872119888 = minus 119877minus1119879119875 = [[[

0 1 00 0 1minus1 minus2153 minus1818

]]] (57)

Reference signal r(t)lr = 05 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

0

02

04

06

08

1

12

y amp

r

Figure 2 Closed-loop output response with 119897119903 = 05 s preview andreference signal

Reference signal r(t)lr = 075 s output y(t)

minus02

0

02

04

06

08

1

12

y amp

r

5 10 15 20 25 30 35 400t (s)

Figure 3 Closed-loop output response with 119897119903 = 075 s preview andreference signal 119903(119905)

When the reference signal is

119903 (119905) =

0 0 le 119905 le 502 (119905 minus 5) 5 lt 119905 le 101 119905 gt 10

(58)

Figures 1ndash3 can be obtainedFigure 1 shows the tracking controller with nonpreview

Figure 2 shows the tracking controller with 119897119903 = 05 s previewFigure 3 shows the tracking controller with 119897119903 = 075 s pre-view Comparing the above three figures it can be seen that

8 Mathematical Problems in Engineering

Reference signal r(t)lr = 0 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

minus015

minus01

minus005

0

005

01

015

02

y amp

r

Figure 4 Closed-loop output response with nonpreview andreference signal 119903(119905)

5 10 15 20 25 30 35 400t (s)

minus015

minus01

minus005

0

005

01

015

02

y amp

r

Reference signal r(t)lr = 05 s output y(t)

Figure 5 Closed-loop output response with 119897119903 = 05 s preview andreference signal

based on the preview controller the output signals can trackthe reference signals much faster and with less tracking error

When the reference signal is

119903 (119905) =

0 0 lt 119905 le 501 sin 05 (119905 minus 5) 5 lt 119905 le 5 + 51205870 119905 gt 5 + 5120587

(59)

Figures 4ndash6 can be obtainedFigure 4 shows the tracking controller with nonpreview

Figure 5 shows the tracking controller with 119897119903 = 05 s preview

Reference signal r(t)lr = 075 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

minus015

minus01

minus005

0

005

01

015

y amp

r

Figure 6 Closed-loop output response with 119897119903 = 075 s preview andreference signal 119903(119905)

Figure 6 shows the tracking controller with 119897119903 = 075 s pre-view Comparing the above three figures it can be seen thatbased on the preview controller the output signals can trackthe reference signals much faster and with less tracking error

8 Conclusion

This paper has studied the optimal preview control problemfor a class of continuous stochastic system in the infinitehorizon By introducing the integrator and the assistantsystem the stochastic augmented error system is constructedCompared with the finite horizon the performance indexis simplified The stability of the stochastic augmented errorsystem is also studied and the observer for the originalstochastic system is designed Finally the simulation exampleshows the effectiveness of the conclusion in this paper

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 61174209)

References

[1] M Tomizuka ldquoOptimal continuous finite preview problemrdquoIEEE Transactions on Automatic Control vol 20 no 3 pp 362ndash365 1975

[2] Y Xu F Liao L Cui and J Wu ldquoPreview control for a class oflinear continuous time-varying systemsrdquo International Journalof Wavelets Multiresolution and Information Processing vol 11no 2 Article ID 1350018 1350018 14 pages 2013

[3] F LiaoM Cao Z Hu and P An ldquoDesign of an optimal previewcontroller for linear discrete time causal descriptor systemsrdquo

Mathematical Problems in Engineering 9

International Journal of Control vol 85 no 10 pp 1616ndash16242012

[4] F Liao K Takaba T Katayama and J Katsuura ldquoDesign of anoptimal preview servomechanism for discrete-time systems ina multirate settingrdquoDynamics of Continuous Discrete amp Impul-sive Systems Series B Applications amp Algorithms vol 10 no 5pp 727ndash744 2003

[5] T Katayama T Ohki T Inoue and T Kato ldquoDesign of anoptimal controller for a discrete-time system subject to pre-viewable demandrdquo International Journal of Control vol 41 no3 pp 677ndash699 1985

[6] F Liao Y Guo and Y Y Tang ldquoDesign of an optimal previewcontroller for linear time-varying discrete systems in amultiratesettingrdquo International Journal of Wavelets Multiresolution andInformation Processing vol 13 no 6 Article ID 15500501550050 19 pages 2015

[7] T Tsuchiya andT EgamiDigital Preview andPredictive ControlTranslated by Fucheng Liao Beijing Science and TechnologyPress Beijing China 1994

[8] M Cao and F Liao ldquoDesign of an optimal preview controller forlinear discrete-time descriptor systems with state delayrdquo Inter-national Journal of Systems Science vol 46 no 5 pp 932ndash9432015

[9] Z Y Zhen Z S Wang and D B Wang ldquoInformation fusionestimation based preview control for a discrete linear systemrdquoActa Automatica Sinica vol 36 no 2 pp 347ndash352 2010

[10] T Katayama and T Hirono ldquoDesign of an optimal servomech-anism with preview action and its dual problemrdquo InternationalJournal of Control vol 45 no 2 pp 407ndash420 1987

[11] F Liao Y Y Tang H Liu and Y Wang ldquoDesign of an optimalpreview controller for continuous-time systemsrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 9 no 4 pp 655ndash673 2011

[12] F Liao Z Ren M Tomizuka and J Wu ldquoPreview controlfor impulse-free continuous-time descriptor systemsrdquo Interna-tional Journal of Control vol 88 no 6 pp 1142ndash1149 2015

[13] F Liao andHXu ldquoApplication of the preview controlmethod tothe optimal tracking control problem for continuous-time sys-tems with time-delayrdquo Mathematical Problems in Engineeringvol 2015 Article ID 423580 8 pages 2015

[14] A Astrom Introduction to Stochastic ControlTheory Translatedby Yuhuan Pan Science Press Beijing China 1983

[15] JWu F Liao andM Tomizuka ldquoOptimal preview control for alinear continuous-time stochastic control system in finite-timehorizonrdquo International Journal of Systems Science vol 48 no 1pp 129ndash137 2017

[16] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 6th edition2005

[17] T H Cormen C E Leiserson R Rivest and C Stein Introduc-tion to Algorithms MIT Press and McGraw-Hill Boston MassUSA 2nd edition 2001

[18] J Yong and X Y Zhou Stochastic Control Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[19] D Zheng Linear System Theory Tsinghua University PressBeijing China 2002

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

that is

119889119909lowast (119905) = [119872119909lowast (119905) + 119873119906lowast (119905)] + Γ 119903 (119905 + 119897119903) 119889119905119903 (119905 + 119897119903) = 119862119909lowast (119905) (7)

Define (119905) = 119909 (119905) minus 119909lowast (119905) (119905) = 119906 (119905) minus 119906lowast (119905) (119905) = 119910 (119905) minus 119862119909lowast (119905) (119905) = 119903 (119905) minus 119903 (119905 + 119897119903)

(8)

Subtracting (7) by (1) from both sides the following could beobtained119889 (119905) = [119872 (119905) + 119873 (119905) + (minusΓ) 119903 (119905 + 119897119903)] 119889119905

+ 120596 (119905) 119889119861119905 (119905) = 119862 (119905)

(1199050) = 1199090 minus 119909lowast (1199050) 119905 isin [1199050infin) (9)

Now an integrator is introduced Integrating the trackingerror signal 119890(119905) in the interval of 1199050 to 119905 yields

119902 (119905) = int1199051199050

119890 (119904) 119889119904 = int1199051199050

[119910 (119904) minus 119903 (119904)] 119889119904 (10)

Obviously 119902(1199050) = 0 Let 119902lowast(119905) = int1199051199050[119862119909lowast(119904) minus 119903(119904 + 119897119903)]119889119904 andthen 119902lowast(119905) = 0 holds Define (119905) = 119902(119905)minus119902lowast(119905) and the differ-ential form of (119905) becomes

119889 (119905) = 119890 (119905) 119889119905 = [119862 (119905) minus (119905)] 119889119905 (11)

Remark 9 According to Assumption 3 the term (119905) in (11)is also previewable while (120588) (119905 le 120588 le 119905 + 119897119903) In particularwhile 120588 ge 119905 + 119897119903 (120588) = 0

In order to make the output 119910(119905) tracking the referencesignal 119903(119905) as accurately as possible the performance index 119869is designed as

119869 (1199050 1199090 (sdot))= 1198641199050 1199090 12 int

infin

1199050

[119879 (119905) 119876 (119905) + 119879 (119905) 119877 (119905)] 119889119905 (12)

where 119876 = 119876119879 gt 0 and 119877 = 119877119879 gt 0 are matrices of properdimensions

Remark 10 Compared with the finite horizon lim119905rarrinfin119890(119905) =0 can also be obtained in the infinite horizon whilelim119905rarrinfin119902(119905) = 0 Therefore the term 119890119879(119905)119876119890119890(119905) in theperformance index of [15] can be replaced by 119879(119905)119876(119905)in (12) Furthermore since there exists unique semipositivedefinite solution of the corresponding Riccati equation when119876 gt 0 no other terms need to be included in the perform-ance index which simplifies the calculation

To solve the optimal control problem of (9) with the per-formance index (12) the augmented error systemmethod canbe employed

3 Construction of the Stochastic AugmentedError System

Combining (9) and (11) the following holds

119889119911 (119905) = [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905+ 120596 (119905) 119889119861119905

(13)

where

119911 (119905) = [ (119905) (119905)]

= [0 1198620 119872]

= [ 0119873]

= [minus1198680 ]

Γ = [ 0minusΓ]

(14)

Referring to the output equation of (9) and the referencesignal the output of (13) can be defined as

(119905) = 119911 (119905) (15)

where = [119868 0] Joining (13) and (15) together the followinghold

119889119911 (119905) = [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905+ 120596 (119905) 119889119861119905

(119905) = 119911 (119905) 119911 (1199050) = [ (1199050)0 (1199050)] = 1199110

(16)

Equation (16) is the needed stochastic augmented errorsystem

4 Design of the Optimal Controller forthe Stochastic System

Denoting the performance index (12) with the related vari-ables in (16) yields

(1199050 1199110 (sdot)) = 11986411990501199110 12sdot intinfin1199050

[119911119879 (119905) (119879119876) 119911 (119905) + 119879 (119905) 119877 (119905)] 119889119905 (17)

Due to the existence of the term (119905) + Γ 119903(119905 + 119897119903) in(16) the conclusion of the optimal regulation problem in

4 Mathematical Problems in Engineering

dealingwith the infinite horizon cannot be directly employedTherefore similar to the condition in deterministic systems[11] the optimal regulator of the stochastic augmented errorsystem (16) can be obtained through the following three steps

Firstly (17) is revised to

= lim119905120591rarrinfin

11986411990501199110 12sdot int1199051205911199050

[119911119879 (119905) 119879119876119911 (119905) + 119879 (119905) 119877 (119905)] 119889119905 (18)

Secondly the finite-time horizon optimal regulationproblem with the performance index

119869 = 1198641199050 1199110 12sdot int1199051205911199050

[119911119879 (119905) 119879119876119911 (119905) + 119879 (119905) 119877 (119905)] 119889119905(19)

is solved where 119905120591 is the terminal timeCompared with the performance index in [15] the two

terms 119890119879(119905)119876119890119890(119905) and 119890119879(119905120591)119876119890(119905120591)119890(119905120591) are removed in (19)Therefore according to the dynamic programming principlefor stochastic systems [16 18] the following lemma about thefinite horizon optimal control problemwith the terminal time119905120591 can be obtained by letting 119876119890 = 119876119890(119905120591) = 0 in Theorem 1 in[15]

Lemma 11 (see [15]) If Assumptions 3 and 5 hold the optimalcontroller of (16) with the performance index (19) is

(119905) = minus119877minus11198791 119875 (119905) 119911 (119905) + 120579 (119905) (20)

where 119875(119905) ge 0 isin 119877(119898+119901)times(119898+119901) satisfies the Riccati differentialequation

minus (119905) = 119879119875 (119905) + 119875 (119905) minus 119875 (119905) 119877minus1119879119875119879 (119905)+ 119879119876

(21)

With the terminal condition 119875(119905120591) = 0 and 120579(119905) the differentialequation is obtained

(119905) + (119879 minus 119875 (119905) 119877minus1119879) 120579 (119905)+ (119875 (119905) (119905) + 119875 (119905) Γ 119903 (119905 + 119897119903)) = 0

(22)

with the terminal condition 120579(119905120591) = 0Thirdly Theorem 12 is received by letting 119905120591 rarrinfin

Theorem 12 If Assumptions 1 to 5 hold the optimal controller(119905) of (16) with the performance index (17) can be expressedas

(119905) = minus119877minus1119879119875119911 (119905)minus 119877minus1119879int119905+119897119903

119905exp [(1199050 minus 119905) ( minus 119877minus1119879119875)119879]

sdot (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904(23)

where 119875 isin 119877times is the unique semipositive definite solution ofRiccati differential equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (24)

Proof In terms of the performance index (17) if ( ) isstabilizable and (11987612

) is detectable according to the

known conclusion there exists a unique semipositive definitesolution 119875 of Riccati algebra equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (25)

And when 119905 rarr infin

lim119905rarrinfin

119875 (119905) = 119875 (26)

As a result the solving of 120579(119905) in (22) is also simplifiedDenote119872119888 = minus119877minus1119879119875 Equation (22) and its term-

inal condition can be expressed as

(119905) + 119872119879119888 120579 (119905) + (119875 (119905) + 119875Γ 119903 (119905 + 119897119903)) = 0120579 (119905120591) = 0

(27)

Based on linear system theory [19] the fundamental solutionmatrix of the homogeneous system corresponding to (27) is

Φ (119905 1199050) = exp [minusint1199051199050

119872119879119888 119889119905] = exp [(1199050 minus 119905)119872119879119888 ] (28)

with the property

Φminus1 (1199050 119905) = Φ (119905 1199050) Φ (119905 1199050) = Φ (119905 119905119904) Φ (119905119904 1199050)

1199050 le 119904 le 119905(29)

Therefore the solution of (27) is

120579 (119905) = Φ (119905 1199050) 120579 (1199050)minus int1199051199050

Φ (119905 119904) (119875 (119905) + 119875Γ 119903 (119905 + 119897119903)) 119889119904 (30)

and the backward form is

120579 (119905120591) = Φ (119905120591 119905) 120579 (119905)minus int119905120591119905Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(31)

With the terminal condition 120579(119905120591) = 0 the following will beobtained

120579 (119905)= Φminus1 (119905120591 119905) int119905120591

119905Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904 (32)

Mathematical Problems in Engineering 5

According to the property of the reference signals in Assump-tion 3 120579(119905) can be expressed as

120579 (119905) = Φminus1 (119905120591 119905)sdot int119905120591and(119905+119897119903)119905

Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904= Φ (119905 119905120591)sdot int119905120591and(119905+119897119903)119905

Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904= int119905120591and(119905+119897119903)119905

Φ (119905 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(33)

Substituting (33) into (20) yields

(119905) = minus119877minus1119879119875119911 (119905) minus 119877minus11198791 int119905120591and(119905+119897119903)119905

Φ (119905 119904)sdot (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(34)

where 119905120591 and (119905 + 119897119903) = min119905120591 119905 + 119897119903The conclusion is that when 119905120591 rarrinfin the following theo-

rem can be obtained

Theorem 13 If Assumptions 1 to 5 hold the optimal controller(119905) of (16) with the performance index (17) can be expressedas

(119905)= minus119877minus1119879119875119911 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(35)

where 119875 isin 119877times is the unique semipositive definite solution ofRiccati differential equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (36)

Then let the matrix 119875 be separated into

119875 = [1198751 1198752] (37)

where 1198751 and 1198752 are matrices of times119901 and times119898 According to(8) (35) 119911(119905) = [ (119905)

(119905)] and int119905+119897119903

119905Φ(119905 119904)119875Γ 119903(119904 + 119897119903)119889119904 = 0 the

following will be obtained

119906 (119905) = minus119877minus11198791198751119902 (119905) minus 119877minus11198791198752119909 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) 119875 (119904) 119889119904 + 119906lowast (119905)

+ 119877minus11198791198752119902lowast (119905) + 119877minus11198791198752119909lowast (119905) (38)

where119875 ge 0 isin 119877times satisfies Riccati differential equation (24)

Due to the fact that 119909lowast(119905) = Γ119903lowast(119905+ 119897119903) 119906lowast(119905) = 120574119903lowast(119905+ 119897119903)and 119902lowast(119905) = 0 the following corollary is obtainedCorollary 14 If Assumptions 1 to 5 hold the optimal previewcontroller 119906(119905) of (1) with the performance index (12) can beexpressed as

119906 (119905) = minus119877minus11198791198751119902 (119905) minus 119877minus11198791198752119909 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) 119875 (119904) 119889119904

+ [120574 + 119877minus11198791198752Γ] 119903 (119905 + 119897119903) (39)

From (39) it can be seen that there are four parts con-tained in the preview controller of (1) the first part is the inte-gral of tracking error term minus119877minus11198791198751119902(119905) stemmed from theinducing of the integrator The second part is the state feed-back minus119877minus11198791198752119909(119905) The third one minus119877minus1119879 int119905+119897119903

119905Φ(119905

119904)119875(119904)119889119904 is the reference preview compensation The lastone [120574 + 119877minus11198791198752Γ]119903(119905 + 119897119903) is the previewable complementof the assistant system

5 Stability of Closed-Loop System

With 120596(119905)119889119861119905 being an external disturbance the stability anddetectability of system (16) are studied only when 120596(119905) = 0Here (16) becomes the deterministic system

119889119911 (119905) = [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905 (40)

As a result the sufficient and necessary criteria which guar-antee that there exists a unique semipositive definite solutionto the Riccati equation (24) can be gained by the known con-clusion as follows

Lemma 15 (see [11]) Suppose the following conditions aresatisfied

(1) The matrices 119876 and 119877 are both positive definite

(2) The matrix

Π = [[119872 119873119862 0]]

(41)

is of full row rank

(3) The matrices pair (119872119873) is stabilizable(4) The matrices pair (119862119872) is detectableThen there exists unique semipositive definite solution in

the Riccati equation (24) and there exists the asymptoticallystable coefficient matrix119872119888 in the closed-loop system (27)

Therefore the following theorem can be received

6 Mathematical Problems in Engineering

Theorem 16 If 119876 and 119877 are both positive definite matricesand Assumptions 1 to 5 hold the full regulation is achieved inthe closed-loop system

119889119911 (119905)= [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905+ 120596 (119905) 119889119861119905

(119905)= minus119877minus1119879119875119911 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(119905) = 119911 (119905)119911 (1199050) = [ (1199050)0 (1199050)] = 1199110

(42)

namely

lim119905rarrinfin

(119905) = 0 (43)

Furthermore the following will be obtained

lim119905rarrinfin

119890 (119905) = 0 (44)

6 State Observer

If the state vector 119909(119905) in (1) cannot be measured directly theoptimal preview controller (39) cannot be realized In orderto solve this problem the state observer

119889 (119905) = [119872 (119905) + 119873119906 (119905) + 119871 (119910 (119905) minus 119862 (119905))] 119889119905+ 120596 (119905) 119889119861119905 119909 (1199050) = 1199090 119905 isin [1199050infin) (45)

could be designedSubtracting (45) from (1) on both sides yields

119889119909 (119905) = [(119872 minus 119871119862) 119909 (119905)] 119889119905 (46)

where 119909(119905) = 119909(119905) minus (119905)So if (119872 minus 119871119862) is stable the following will hold

lim119905rarrinfin

119909 (119905) = 0 (47)

which means that the state vector (119905) in the observerequation (45) approximates the state vector 119909(119905) in (1) when119905 rarr infin Based on linear system theory [19] it is known thatif (119862 119860) is detectable (119872 minus 119871119862) is stable Then the followingwill be obtained

Theorem 17 If the conditions in Theorem 16 hold the closed-loop system can be received

119889119909 (119905) = [119872119909 (119905) + 119873119906 (119905)] 119889119905 + 120596 (119905) 119889119861119905119889 (119905) = [119872 (119905) + 119873119906 (119905) + 119871 (119910 (119905) minus 119862 (119905))] 119889119905

+ 120596 (119905) 119889119861119905119910 (119905) = 119862119909 (119905) 119906 (119905) = minus119877minus11198791198751 int119905

1199050

(119862 (119904) minus 119903 (119904)) 119889119904minus 119877minus11198791198752 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) 119875 (119904) 119889119904

+ [120574 + 119877minus11198791198752Γ] 119903 (119905 + 119897119903)

(48)

where (119872 minus 119871119862) is stable and (48) achieve the completeregulation

7 Numerical Simulation

Example 1 Consider the stochastic control system

119889[1199091 (119905)1199092 (119905)] = [[0 1minus1 minus1][

1199091 (119905)1199092 (119905)] + [

01] 119906 (119905)] 119889119905

+ 01 119889119861119905119910 (119905) = [1 0] [1199091 (119905)1199092 (119905)]

(49)

where the coefficient matrices are

119872 = [ 0 1minus1 minus1]

119873 = [01] 119862 = [1 0]

120596 (119905) = [0101]

(50)

respectively Therefore the coefficient matrices in (16) are

= [[[

0 1 00 0 10 minus1 minus1

]]]

= [[[

001]]]

Mathematical Problems in Engineering 7

Reference signal r(t)lr = 0 s output y(t)

minus02

0

02

04

06

08

1

12

y amp

r

5 10 15 20 25 30 35 400t (s)

Figure 1 Closed-loop output response with nonpreview and refer-ence signal 119903(119905)

= [[[

minus1 00 minus10 0

]]]

(51)

and Brownian motion 119861119905 satisfies119861119905 sim 119873 (0 001) (52)

Let the initial state of (1) be

119909 (0) = [00] (53)

where 1199091 and 1199092 are mutually independent The previewlengths of the reference signals are 119897119894119903 = 0 05 075 s (119894 =1 2 3) respectively The weight matrices of the performanceindex are

119876 = 1119877 = 1 (54)

and the solution of Riccati equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (55)

is

119875 = [[[

21533 18184 100018184 19157 11531000 1153 0818

]]] (56)

which is a positive definite matrix The coefficient matrix119872119888in (25) is

119872119888 = minus 119877minus1119879119875 = [[[

0 1 00 0 1minus1 minus2153 minus1818

]]] (57)

Reference signal r(t)lr = 05 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

0

02

04

06

08

1

12

y amp

r

Figure 2 Closed-loop output response with 119897119903 = 05 s preview andreference signal

Reference signal r(t)lr = 075 s output y(t)

minus02

0

02

04

06

08

1

12

y amp

r

5 10 15 20 25 30 35 400t (s)

Figure 3 Closed-loop output response with 119897119903 = 075 s preview andreference signal 119903(119905)

When the reference signal is

119903 (119905) =

0 0 le 119905 le 502 (119905 minus 5) 5 lt 119905 le 101 119905 gt 10

(58)

Figures 1ndash3 can be obtainedFigure 1 shows the tracking controller with nonpreview

Figure 2 shows the tracking controller with 119897119903 = 05 s previewFigure 3 shows the tracking controller with 119897119903 = 075 s pre-view Comparing the above three figures it can be seen that

8 Mathematical Problems in Engineering

Reference signal r(t)lr = 0 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

minus015

minus01

minus005

0

005

01

015

02

y amp

r

Figure 4 Closed-loop output response with nonpreview andreference signal 119903(119905)

5 10 15 20 25 30 35 400t (s)

minus015

minus01

minus005

0

005

01

015

02

y amp

r

Reference signal r(t)lr = 05 s output y(t)

Figure 5 Closed-loop output response with 119897119903 = 05 s preview andreference signal

based on the preview controller the output signals can trackthe reference signals much faster and with less tracking error

When the reference signal is

119903 (119905) =

0 0 lt 119905 le 501 sin 05 (119905 minus 5) 5 lt 119905 le 5 + 51205870 119905 gt 5 + 5120587

(59)

Figures 4ndash6 can be obtainedFigure 4 shows the tracking controller with nonpreview

Figure 5 shows the tracking controller with 119897119903 = 05 s preview

Reference signal r(t)lr = 075 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

minus015

minus01

minus005

0

005

01

015

y amp

r

Figure 6 Closed-loop output response with 119897119903 = 075 s preview andreference signal 119903(119905)

Figure 6 shows the tracking controller with 119897119903 = 075 s pre-view Comparing the above three figures it can be seen thatbased on the preview controller the output signals can trackthe reference signals much faster and with less tracking error

8 Conclusion

This paper has studied the optimal preview control problemfor a class of continuous stochastic system in the infinitehorizon By introducing the integrator and the assistantsystem the stochastic augmented error system is constructedCompared with the finite horizon the performance indexis simplified The stability of the stochastic augmented errorsystem is also studied and the observer for the originalstochastic system is designed Finally the simulation exampleshows the effectiveness of the conclusion in this paper

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 61174209)

References

[1] M Tomizuka ldquoOptimal continuous finite preview problemrdquoIEEE Transactions on Automatic Control vol 20 no 3 pp 362ndash365 1975

[2] Y Xu F Liao L Cui and J Wu ldquoPreview control for a class oflinear continuous time-varying systemsrdquo International Journalof Wavelets Multiresolution and Information Processing vol 11no 2 Article ID 1350018 1350018 14 pages 2013

[3] F LiaoM Cao Z Hu and P An ldquoDesign of an optimal previewcontroller for linear discrete time causal descriptor systemsrdquo

Mathematical Problems in Engineering 9

International Journal of Control vol 85 no 10 pp 1616ndash16242012

[4] F Liao K Takaba T Katayama and J Katsuura ldquoDesign of anoptimal preview servomechanism for discrete-time systems ina multirate settingrdquoDynamics of Continuous Discrete amp Impul-sive Systems Series B Applications amp Algorithms vol 10 no 5pp 727ndash744 2003

[5] T Katayama T Ohki T Inoue and T Kato ldquoDesign of anoptimal controller for a discrete-time system subject to pre-viewable demandrdquo International Journal of Control vol 41 no3 pp 677ndash699 1985

[6] F Liao Y Guo and Y Y Tang ldquoDesign of an optimal previewcontroller for linear time-varying discrete systems in amultiratesettingrdquo International Journal of Wavelets Multiresolution andInformation Processing vol 13 no 6 Article ID 15500501550050 19 pages 2015

[7] T Tsuchiya andT EgamiDigital Preview andPredictive ControlTranslated by Fucheng Liao Beijing Science and TechnologyPress Beijing China 1994

[8] M Cao and F Liao ldquoDesign of an optimal preview controller forlinear discrete-time descriptor systems with state delayrdquo Inter-national Journal of Systems Science vol 46 no 5 pp 932ndash9432015

[9] Z Y Zhen Z S Wang and D B Wang ldquoInformation fusionestimation based preview control for a discrete linear systemrdquoActa Automatica Sinica vol 36 no 2 pp 347ndash352 2010

[10] T Katayama and T Hirono ldquoDesign of an optimal servomech-anism with preview action and its dual problemrdquo InternationalJournal of Control vol 45 no 2 pp 407ndash420 1987

[11] F Liao Y Y Tang H Liu and Y Wang ldquoDesign of an optimalpreview controller for continuous-time systemsrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 9 no 4 pp 655ndash673 2011

[12] F Liao Z Ren M Tomizuka and J Wu ldquoPreview controlfor impulse-free continuous-time descriptor systemsrdquo Interna-tional Journal of Control vol 88 no 6 pp 1142ndash1149 2015

[13] F Liao andHXu ldquoApplication of the preview controlmethod tothe optimal tracking control problem for continuous-time sys-tems with time-delayrdquo Mathematical Problems in Engineeringvol 2015 Article ID 423580 8 pages 2015

[14] A Astrom Introduction to Stochastic ControlTheory Translatedby Yuhuan Pan Science Press Beijing China 1983

[15] JWu F Liao andM Tomizuka ldquoOptimal preview control for alinear continuous-time stochastic control system in finite-timehorizonrdquo International Journal of Systems Science vol 48 no 1pp 129ndash137 2017

[16] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 6th edition2005

[17] T H Cormen C E Leiserson R Rivest and C Stein Introduc-tion to Algorithms MIT Press and McGraw-Hill Boston MassUSA 2nd edition 2001

[18] J Yong and X Y Zhou Stochastic Control Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[19] D Zheng Linear System Theory Tsinghua University PressBeijing China 2002

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

dealingwith the infinite horizon cannot be directly employedTherefore similar to the condition in deterministic systems[11] the optimal regulator of the stochastic augmented errorsystem (16) can be obtained through the following three steps

Firstly (17) is revised to

= lim119905120591rarrinfin

11986411990501199110 12sdot int1199051205911199050

[119911119879 (119905) 119879119876119911 (119905) + 119879 (119905) 119877 (119905)] 119889119905 (18)

Secondly the finite-time horizon optimal regulationproblem with the performance index

119869 = 1198641199050 1199110 12sdot int1199051205911199050

[119911119879 (119905) 119879119876119911 (119905) + 119879 (119905) 119877 (119905)] 119889119905(19)

is solved where 119905120591 is the terminal timeCompared with the performance index in [15] the two

terms 119890119879(119905)119876119890119890(119905) and 119890119879(119905120591)119876119890(119905120591)119890(119905120591) are removed in (19)Therefore according to the dynamic programming principlefor stochastic systems [16 18] the following lemma about thefinite horizon optimal control problemwith the terminal time119905120591 can be obtained by letting 119876119890 = 119876119890(119905120591) = 0 in Theorem 1 in[15]

Lemma 11 (see [15]) If Assumptions 3 and 5 hold the optimalcontroller of (16) with the performance index (19) is

(119905) = minus119877minus11198791 119875 (119905) 119911 (119905) + 120579 (119905) (20)

where 119875(119905) ge 0 isin 119877(119898+119901)times(119898+119901) satisfies the Riccati differentialequation

minus (119905) = 119879119875 (119905) + 119875 (119905) minus 119875 (119905) 119877minus1119879119875119879 (119905)+ 119879119876

(21)

With the terminal condition 119875(119905120591) = 0 and 120579(119905) the differentialequation is obtained

(119905) + (119879 minus 119875 (119905) 119877minus1119879) 120579 (119905)+ (119875 (119905) (119905) + 119875 (119905) Γ 119903 (119905 + 119897119903)) = 0

(22)

with the terminal condition 120579(119905120591) = 0Thirdly Theorem 12 is received by letting 119905120591 rarrinfin

Theorem 12 If Assumptions 1 to 5 hold the optimal controller(119905) of (16) with the performance index (17) can be expressedas

(119905) = minus119877minus1119879119875119911 (119905)minus 119877minus1119879int119905+119897119903

119905exp [(1199050 minus 119905) ( minus 119877minus1119879119875)119879]

sdot (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904(23)

where 119875 isin 119877times is the unique semipositive definite solution ofRiccati differential equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (24)

Proof In terms of the performance index (17) if ( ) isstabilizable and (11987612

) is detectable according to the

known conclusion there exists a unique semipositive definitesolution 119875 of Riccati algebra equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (25)

And when 119905 rarr infin

lim119905rarrinfin

119875 (119905) = 119875 (26)

As a result the solving of 120579(119905) in (22) is also simplifiedDenote119872119888 = minus119877minus1119879119875 Equation (22) and its term-

inal condition can be expressed as

(119905) + 119872119879119888 120579 (119905) + (119875 (119905) + 119875Γ 119903 (119905 + 119897119903)) = 0120579 (119905120591) = 0

(27)

Based on linear system theory [19] the fundamental solutionmatrix of the homogeneous system corresponding to (27) is

Φ (119905 1199050) = exp [minusint1199051199050

119872119879119888 119889119905] = exp [(1199050 minus 119905)119872119879119888 ] (28)

with the property

Φminus1 (1199050 119905) = Φ (119905 1199050) Φ (119905 1199050) = Φ (119905 119905119904) Φ (119905119904 1199050)

1199050 le 119904 le 119905(29)

Therefore the solution of (27) is

120579 (119905) = Φ (119905 1199050) 120579 (1199050)minus int1199051199050

Φ (119905 119904) (119875 (119905) + 119875Γ 119903 (119905 + 119897119903)) 119889119904 (30)

and the backward form is

120579 (119905120591) = Φ (119905120591 119905) 120579 (119905)minus int119905120591119905Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(31)

With the terminal condition 120579(119905120591) = 0 the following will beobtained

120579 (119905)= Φminus1 (119905120591 119905) int119905120591

119905Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904 (32)

Mathematical Problems in Engineering 5

According to the property of the reference signals in Assump-tion 3 120579(119905) can be expressed as

120579 (119905) = Φminus1 (119905120591 119905)sdot int119905120591and(119905+119897119903)119905

Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904= Φ (119905 119905120591)sdot int119905120591and(119905+119897119903)119905

Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904= int119905120591and(119905+119897119903)119905

Φ (119905 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(33)

Substituting (33) into (20) yields

(119905) = minus119877minus1119879119875119911 (119905) minus 119877minus11198791 int119905120591and(119905+119897119903)119905

Φ (119905 119904)sdot (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(34)

where 119905120591 and (119905 + 119897119903) = min119905120591 119905 + 119897119903The conclusion is that when 119905120591 rarrinfin the following theo-

rem can be obtained

Theorem 13 If Assumptions 1 to 5 hold the optimal controller(119905) of (16) with the performance index (17) can be expressedas

(119905)= minus119877minus1119879119875119911 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(35)

where 119875 isin 119877times is the unique semipositive definite solution ofRiccati differential equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (36)

Then let the matrix 119875 be separated into

119875 = [1198751 1198752] (37)

where 1198751 and 1198752 are matrices of times119901 and times119898 According to(8) (35) 119911(119905) = [ (119905)

(119905)] and int119905+119897119903

119905Φ(119905 119904)119875Γ 119903(119904 + 119897119903)119889119904 = 0 the

following will be obtained

119906 (119905) = minus119877minus11198791198751119902 (119905) minus 119877minus11198791198752119909 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) 119875 (119904) 119889119904 + 119906lowast (119905)

+ 119877minus11198791198752119902lowast (119905) + 119877minus11198791198752119909lowast (119905) (38)

where119875 ge 0 isin 119877times satisfies Riccati differential equation (24)

Due to the fact that 119909lowast(119905) = Γ119903lowast(119905+ 119897119903) 119906lowast(119905) = 120574119903lowast(119905+ 119897119903)and 119902lowast(119905) = 0 the following corollary is obtainedCorollary 14 If Assumptions 1 to 5 hold the optimal previewcontroller 119906(119905) of (1) with the performance index (12) can beexpressed as

119906 (119905) = minus119877minus11198791198751119902 (119905) minus 119877minus11198791198752119909 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) 119875 (119904) 119889119904

+ [120574 + 119877minus11198791198752Γ] 119903 (119905 + 119897119903) (39)

From (39) it can be seen that there are four parts con-tained in the preview controller of (1) the first part is the inte-gral of tracking error term minus119877minus11198791198751119902(119905) stemmed from theinducing of the integrator The second part is the state feed-back minus119877minus11198791198752119909(119905) The third one minus119877minus1119879 int119905+119897119903

119905Φ(119905

119904)119875(119904)119889119904 is the reference preview compensation The lastone [120574 + 119877minus11198791198752Γ]119903(119905 + 119897119903) is the previewable complementof the assistant system

5 Stability of Closed-Loop System

With 120596(119905)119889119861119905 being an external disturbance the stability anddetectability of system (16) are studied only when 120596(119905) = 0Here (16) becomes the deterministic system

119889119911 (119905) = [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905 (40)

As a result the sufficient and necessary criteria which guar-antee that there exists a unique semipositive definite solutionto the Riccati equation (24) can be gained by the known con-clusion as follows

Lemma 15 (see [11]) Suppose the following conditions aresatisfied

(1) The matrices 119876 and 119877 are both positive definite

(2) The matrix

Π = [[119872 119873119862 0]]

(41)

is of full row rank

(3) The matrices pair (119872119873) is stabilizable(4) The matrices pair (119862119872) is detectableThen there exists unique semipositive definite solution in

the Riccati equation (24) and there exists the asymptoticallystable coefficient matrix119872119888 in the closed-loop system (27)

Therefore the following theorem can be received

6 Mathematical Problems in Engineering

Theorem 16 If 119876 and 119877 are both positive definite matricesand Assumptions 1 to 5 hold the full regulation is achieved inthe closed-loop system

119889119911 (119905)= [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905+ 120596 (119905) 119889119861119905

(119905)= minus119877minus1119879119875119911 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(119905) = 119911 (119905)119911 (1199050) = [ (1199050)0 (1199050)] = 1199110

(42)

namely

lim119905rarrinfin

(119905) = 0 (43)

Furthermore the following will be obtained

lim119905rarrinfin

119890 (119905) = 0 (44)

6 State Observer

If the state vector 119909(119905) in (1) cannot be measured directly theoptimal preview controller (39) cannot be realized In orderto solve this problem the state observer

119889 (119905) = [119872 (119905) + 119873119906 (119905) + 119871 (119910 (119905) minus 119862 (119905))] 119889119905+ 120596 (119905) 119889119861119905 119909 (1199050) = 1199090 119905 isin [1199050infin) (45)

could be designedSubtracting (45) from (1) on both sides yields

119889119909 (119905) = [(119872 minus 119871119862) 119909 (119905)] 119889119905 (46)

where 119909(119905) = 119909(119905) minus (119905)So if (119872 minus 119871119862) is stable the following will hold

lim119905rarrinfin

119909 (119905) = 0 (47)

which means that the state vector (119905) in the observerequation (45) approximates the state vector 119909(119905) in (1) when119905 rarr infin Based on linear system theory [19] it is known thatif (119862 119860) is detectable (119872 minus 119871119862) is stable Then the followingwill be obtained

Theorem 17 If the conditions in Theorem 16 hold the closed-loop system can be received

119889119909 (119905) = [119872119909 (119905) + 119873119906 (119905)] 119889119905 + 120596 (119905) 119889119861119905119889 (119905) = [119872 (119905) + 119873119906 (119905) + 119871 (119910 (119905) minus 119862 (119905))] 119889119905

+ 120596 (119905) 119889119861119905119910 (119905) = 119862119909 (119905) 119906 (119905) = minus119877minus11198791198751 int119905

1199050

(119862 (119904) minus 119903 (119904)) 119889119904minus 119877minus11198791198752 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) 119875 (119904) 119889119904

+ [120574 + 119877minus11198791198752Γ] 119903 (119905 + 119897119903)

(48)

where (119872 minus 119871119862) is stable and (48) achieve the completeregulation

7 Numerical Simulation

Example 1 Consider the stochastic control system

119889[1199091 (119905)1199092 (119905)] = [[0 1minus1 minus1][

1199091 (119905)1199092 (119905)] + [

01] 119906 (119905)] 119889119905

+ 01 119889119861119905119910 (119905) = [1 0] [1199091 (119905)1199092 (119905)]

(49)

where the coefficient matrices are

119872 = [ 0 1minus1 minus1]

119873 = [01] 119862 = [1 0]

120596 (119905) = [0101]

(50)

respectively Therefore the coefficient matrices in (16) are

= [[[

0 1 00 0 10 minus1 minus1

]]]

= [[[

001]]]

Mathematical Problems in Engineering 7

Reference signal r(t)lr = 0 s output y(t)

minus02

0

02

04

06

08

1

12

y amp

r

5 10 15 20 25 30 35 400t (s)

Figure 1 Closed-loop output response with nonpreview and refer-ence signal 119903(119905)

= [[[

minus1 00 minus10 0

]]]

(51)

and Brownian motion 119861119905 satisfies119861119905 sim 119873 (0 001) (52)

Let the initial state of (1) be

119909 (0) = [00] (53)

where 1199091 and 1199092 are mutually independent The previewlengths of the reference signals are 119897119894119903 = 0 05 075 s (119894 =1 2 3) respectively The weight matrices of the performanceindex are

119876 = 1119877 = 1 (54)

and the solution of Riccati equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (55)

is

119875 = [[[

21533 18184 100018184 19157 11531000 1153 0818

]]] (56)

which is a positive definite matrix The coefficient matrix119872119888in (25) is

119872119888 = minus 119877minus1119879119875 = [[[

0 1 00 0 1minus1 minus2153 minus1818

]]] (57)

Reference signal r(t)lr = 05 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

0

02

04

06

08

1

12

y amp

r

Figure 2 Closed-loop output response with 119897119903 = 05 s preview andreference signal

Reference signal r(t)lr = 075 s output y(t)

minus02

0

02

04

06

08

1

12

y amp

r

5 10 15 20 25 30 35 400t (s)

Figure 3 Closed-loop output response with 119897119903 = 075 s preview andreference signal 119903(119905)

When the reference signal is

119903 (119905) =

0 0 le 119905 le 502 (119905 minus 5) 5 lt 119905 le 101 119905 gt 10

(58)

Figures 1ndash3 can be obtainedFigure 1 shows the tracking controller with nonpreview

Figure 2 shows the tracking controller with 119897119903 = 05 s previewFigure 3 shows the tracking controller with 119897119903 = 075 s pre-view Comparing the above three figures it can be seen that

8 Mathematical Problems in Engineering

Reference signal r(t)lr = 0 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

minus015

minus01

minus005

0

005

01

015

02

y amp

r

Figure 4 Closed-loop output response with nonpreview andreference signal 119903(119905)

5 10 15 20 25 30 35 400t (s)

minus015

minus01

minus005

0

005

01

015

02

y amp

r

Reference signal r(t)lr = 05 s output y(t)

Figure 5 Closed-loop output response with 119897119903 = 05 s preview andreference signal

based on the preview controller the output signals can trackthe reference signals much faster and with less tracking error

When the reference signal is

119903 (119905) =

0 0 lt 119905 le 501 sin 05 (119905 minus 5) 5 lt 119905 le 5 + 51205870 119905 gt 5 + 5120587

(59)

Figures 4ndash6 can be obtainedFigure 4 shows the tracking controller with nonpreview

Figure 5 shows the tracking controller with 119897119903 = 05 s preview

Reference signal r(t)lr = 075 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

minus015

minus01

minus005

0

005

01

015

y amp

r

Figure 6 Closed-loop output response with 119897119903 = 075 s preview andreference signal 119903(119905)

Figure 6 shows the tracking controller with 119897119903 = 075 s pre-view Comparing the above three figures it can be seen thatbased on the preview controller the output signals can trackthe reference signals much faster and with less tracking error

8 Conclusion

This paper has studied the optimal preview control problemfor a class of continuous stochastic system in the infinitehorizon By introducing the integrator and the assistantsystem the stochastic augmented error system is constructedCompared with the finite horizon the performance indexis simplified The stability of the stochastic augmented errorsystem is also studied and the observer for the originalstochastic system is designed Finally the simulation exampleshows the effectiveness of the conclusion in this paper

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 61174209)

References

[1] M Tomizuka ldquoOptimal continuous finite preview problemrdquoIEEE Transactions on Automatic Control vol 20 no 3 pp 362ndash365 1975

[2] Y Xu F Liao L Cui and J Wu ldquoPreview control for a class oflinear continuous time-varying systemsrdquo International Journalof Wavelets Multiresolution and Information Processing vol 11no 2 Article ID 1350018 1350018 14 pages 2013

[3] F LiaoM Cao Z Hu and P An ldquoDesign of an optimal previewcontroller for linear discrete time causal descriptor systemsrdquo

Mathematical Problems in Engineering 9

International Journal of Control vol 85 no 10 pp 1616ndash16242012

[4] F Liao K Takaba T Katayama and J Katsuura ldquoDesign of anoptimal preview servomechanism for discrete-time systems ina multirate settingrdquoDynamics of Continuous Discrete amp Impul-sive Systems Series B Applications amp Algorithms vol 10 no 5pp 727ndash744 2003

[5] T Katayama T Ohki T Inoue and T Kato ldquoDesign of anoptimal controller for a discrete-time system subject to pre-viewable demandrdquo International Journal of Control vol 41 no3 pp 677ndash699 1985

[6] F Liao Y Guo and Y Y Tang ldquoDesign of an optimal previewcontroller for linear time-varying discrete systems in amultiratesettingrdquo International Journal of Wavelets Multiresolution andInformation Processing vol 13 no 6 Article ID 15500501550050 19 pages 2015

[7] T Tsuchiya andT EgamiDigital Preview andPredictive ControlTranslated by Fucheng Liao Beijing Science and TechnologyPress Beijing China 1994

[8] M Cao and F Liao ldquoDesign of an optimal preview controller forlinear discrete-time descriptor systems with state delayrdquo Inter-national Journal of Systems Science vol 46 no 5 pp 932ndash9432015

[9] Z Y Zhen Z S Wang and D B Wang ldquoInformation fusionestimation based preview control for a discrete linear systemrdquoActa Automatica Sinica vol 36 no 2 pp 347ndash352 2010

[10] T Katayama and T Hirono ldquoDesign of an optimal servomech-anism with preview action and its dual problemrdquo InternationalJournal of Control vol 45 no 2 pp 407ndash420 1987

[11] F Liao Y Y Tang H Liu and Y Wang ldquoDesign of an optimalpreview controller for continuous-time systemsrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 9 no 4 pp 655ndash673 2011

[12] F Liao Z Ren M Tomizuka and J Wu ldquoPreview controlfor impulse-free continuous-time descriptor systemsrdquo Interna-tional Journal of Control vol 88 no 6 pp 1142ndash1149 2015

[13] F Liao andHXu ldquoApplication of the preview controlmethod tothe optimal tracking control problem for continuous-time sys-tems with time-delayrdquo Mathematical Problems in Engineeringvol 2015 Article ID 423580 8 pages 2015

[14] A Astrom Introduction to Stochastic ControlTheory Translatedby Yuhuan Pan Science Press Beijing China 1983

[15] JWu F Liao andM Tomizuka ldquoOptimal preview control for alinear continuous-time stochastic control system in finite-timehorizonrdquo International Journal of Systems Science vol 48 no 1pp 129ndash137 2017

[16] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 6th edition2005

[17] T H Cormen C E Leiserson R Rivest and C Stein Introduc-tion to Algorithms MIT Press and McGraw-Hill Boston MassUSA 2nd edition 2001

[18] J Yong and X Y Zhou Stochastic Control Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[19] D Zheng Linear System Theory Tsinghua University PressBeijing China 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

According to the property of the reference signals in Assump-tion 3 120579(119905) can be expressed as

120579 (119905) = Φminus1 (119905120591 119905)sdot int119905120591and(119905+119897119903)119905

Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904= Φ (119905 119905120591)sdot int119905120591and(119905+119897119903)119905

Φ (119905120591 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904= int119905120591and(119905+119897119903)119905

Φ (119905 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(33)

Substituting (33) into (20) yields

(119905) = minus119877minus1119879119875119911 (119905) minus 119877minus11198791 int119905120591and(119905+119897119903)119905

Φ (119905 119904)sdot (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(34)

where 119905120591 and (119905 + 119897119903) = min119905120591 119905 + 119897119903The conclusion is that when 119905120591 rarrinfin the following theo-

rem can be obtained

Theorem 13 If Assumptions 1 to 5 hold the optimal controller(119905) of (16) with the performance index (17) can be expressedas

(119905)= minus119877minus1119879119875119911 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(35)

where 119875 isin 119877times is the unique semipositive definite solution ofRiccati differential equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (36)

Then let the matrix 119875 be separated into

119875 = [1198751 1198752] (37)

where 1198751 and 1198752 are matrices of times119901 and times119898 According to(8) (35) 119911(119905) = [ (119905)

(119905)] and int119905+119897119903

119905Φ(119905 119904)119875Γ 119903(119904 + 119897119903)119889119904 = 0 the

following will be obtained

119906 (119905) = minus119877minus11198791198751119902 (119905) minus 119877minus11198791198752119909 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) 119875 (119904) 119889119904 + 119906lowast (119905)

+ 119877minus11198791198752119902lowast (119905) + 119877minus11198791198752119909lowast (119905) (38)

where119875 ge 0 isin 119877times satisfies Riccati differential equation (24)

Due to the fact that 119909lowast(119905) = Γ119903lowast(119905+ 119897119903) 119906lowast(119905) = 120574119903lowast(119905+ 119897119903)and 119902lowast(119905) = 0 the following corollary is obtainedCorollary 14 If Assumptions 1 to 5 hold the optimal previewcontroller 119906(119905) of (1) with the performance index (12) can beexpressed as

119906 (119905) = minus119877minus11198791198751119902 (119905) minus 119877minus11198791198752119909 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) 119875 (119904) 119889119904

+ [120574 + 119877minus11198791198752Γ] 119903 (119905 + 119897119903) (39)

From (39) it can be seen that there are four parts con-tained in the preview controller of (1) the first part is the inte-gral of tracking error term minus119877minus11198791198751119902(119905) stemmed from theinducing of the integrator The second part is the state feed-back minus119877minus11198791198752119909(119905) The third one minus119877minus1119879 int119905+119897119903

119905Φ(119905

119904)119875(119904)119889119904 is the reference preview compensation The lastone [120574 + 119877minus11198791198752Γ]119903(119905 + 119897119903) is the previewable complementof the assistant system

5 Stability of Closed-Loop System

With 120596(119905)119889119861119905 being an external disturbance the stability anddetectability of system (16) are studied only when 120596(119905) = 0Here (16) becomes the deterministic system

119889119911 (119905) = [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905 (40)

As a result the sufficient and necessary criteria which guar-antee that there exists a unique semipositive definite solutionto the Riccati equation (24) can be gained by the known con-clusion as follows

Lemma 15 (see [11]) Suppose the following conditions aresatisfied

(1) The matrices 119876 and 119877 are both positive definite

(2) The matrix

Π = [[119872 119873119862 0]]

(41)

is of full row rank

(3) The matrices pair (119872119873) is stabilizable(4) The matrices pair (119862119872) is detectableThen there exists unique semipositive definite solution in

the Riccati equation (24) and there exists the asymptoticallystable coefficient matrix119872119888 in the closed-loop system (27)

Therefore the following theorem can be received

6 Mathematical Problems in Engineering

Theorem 16 If 119876 and 119877 are both positive definite matricesand Assumptions 1 to 5 hold the full regulation is achieved inthe closed-loop system

119889119911 (119905)= [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905+ 120596 (119905) 119889119861119905

(119905)= minus119877minus1119879119875119911 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(119905) = 119911 (119905)119911 (1199050) = [ (1199050)0 (1199050)] = 1199110

(42)

namely

lim119905rarrinfin

(119905) = 0 (43)

Furthermore the following will be obtained

lim119905rarrinfin

119890 (119905) = 0 (44)

6 State Observer

If the state vector 119909(119905) in (1) cannot be measured directly theoptimal preview controller (39) cannot be realized In orderto solve this problem the state observer

119889 (119905) = [119872 (119905) + 119873119906 (119905) + 119871 (119910 (119905) minus 119862 (119905))] 119889119905+ 120596 (119905) 119889119861119905 119909 (1199050) = 1199090 119905 isin [1199050infin) (45)

could be designedSubtracting (45) from (1) on both sides yields

119889119909 (119905) = [(119872 minus 119871119862) 119909 (119905)] 119889119905 (46)

where 119909(119905) = 119909(119905) minus (119905)So if (119872 minus 119871119862) is stable the following will hold

lim119905rarrinfin

119909 (119905) = 0 (47)

which means that the state vector (119905) in the observerequation (45) approximates the state vector 119909(119905) in (1) when119905 rarr infin Based on linear system theory [19] it is known thatif (119862 119860) is detectable (119872 minus 119871119862) is stable Then the followingwill be obtained

Theorem 17 If the conditions in Theorem 16 hold the closed-loop system can be received

119889119909 (119905) = [119872119909 (119905) + 119873119906 (119905)] 119889119905 + 120596 (119905) 119889119861119905119889 (119905) = [119872 (119905) + 119873119906 (119905) + 119871 (119910 (119905) minus 119862 (119905))] 119889119905

+ 120596 (119905) 119889119861119905119910 (119905) = 119862119909 (119905) 119906 (119905) = minus119877minus11198791198751 int119905

1199050

(119862 (119904) minus 119903 (119904)) 119889119904minus 119877minus11198791198752 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) 119875 (119904) 119889119904

+ [120574 + 119877minus11198791198752Γ] 119903 (119905 + 119897119903)

(48)

where (119872 minus 119871119862) is stable and (48) achieve the completeregulation

7 Numerical Simulation

Example 1 Consider the stochastic control system

119889[1199091 (119905)1199092 (119905)] = [[0 1minus1 minus1][

1199091 (119905)1199092 (119905)] + [

01] 119906 (119905)] 119889119905

+ 01 119889119861119905119910 (119905) = [1 0] [1199091 (119905)1199092 (119905)]

(49)

where the coefficient matrices are

119872 = [ 0 1minus1 minus1]

119873 = [01] 119862 = [1 0]

120596 (119905) = [0101]

(50)

respectively Therefore the coefficient matrices in (16) are

= [[[

0 1 00 0 10 minus1 minus1

]]]

= [[[

001]]]

Mathematical Problems in Engineering 7

Reference signal r(t)lr = 0 s output y(t)

minus02

0

02

04

06

08

1

12

y amp

r

5 10 15 20 25 30 35 400t (s)

Figure 1 Closed-loop output response with nonpreview and refer-ence signal 119903(119905)

= [[[

minus1 00 minus10 0

]]]

(51)

and Brownian motion 119861119905 satisfies119861119905 sim 119873 (0 001) (52)

Let the initial state of (1) be

119909 (0) = [00] (53)

where 1199091 and 1199092 are mutually independent The previewlengths of the reference signals are 119897119894119903 = 0 05 075 s (119894 =1 2 3) respectively The weight matrices of the performanceindex are

119876 = 1119877 = 1 (54)

and the solution of Riccati equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (55)

is

119875 = [[[

21533 18184 100018184 19157 11531000 1153 0818

]]] (56)

which is a positive definite matrix The coefficient matrix119872119888in (25) is

119872119888 = minus 119877minus1119879119875 = [[[

0 1 00 0 1minus1 minus2153 minus1818

]]] (57)

Reference signal r(t)lr = 05 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

0

02

04

06

08

1

12

y amp

r

Figure 2 Closed-loop output response with 119897119903 = 05 s preview andreference signal

Reference signal r(t)lr = 075 s output y(t)

minus02

0

02

04

06

08

1

12

y amp

r

5 10 15 20 25 30 35 400t (s)

Figure 3 Closed-loop output response with 119897119903 = 075 s preview andreference signal 119903(119905)

When the reference signal is

119903 (119905) =

0 0 le 119905 le 502 (119905 minus 5) 5 lt 119905 le 101 119905 gt 10

(58)

Figures 1ndash3 can be obtainedFigure 1 shows the tracking controller with nonpreview

Figure 2 shows the tracking controller with 119897119903 = 05 s previewFigure 3 shows the tracking controller with 119897119903 = 075 s pre-view Comparing the above three figures it can be seen that

8 Mathematical Problems in Engineering

Reference signal r(t)lr = 0 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

minus015

minus01

minus005

0

005

01

015

02

y amp

r

Figure 4 Closed-loop output response with nonpreview andreference signal 119903(119905)

5 10 15 20 25 30 35 400t (s)

minus015

minus01

minus005

0

005

01

015

02

y amp

r

Reference signal r(t)lr = 05 s output y(t)

Figure 5 Closed-loop output response with 119897119903 = 05 s preview andreference signal

based on the preview controller the output signals can trackthe reference signals much faster and with less tracking error

When the reference signal is

119903 (119905) =

0 0 lt 119905 le 501 sin 05 (119905 minus 5) 5 lt 119905 le 5 + 51205870 119905 gt 5 + 5120587

(59)

Figures 4ndash6 can be obtainedFigure 4 shows the tracking controller with nonpreview

Figure 5 shows the tracking controller with 119897119903 = 05 s preview

Reference signal r(t)lr = 075 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

minus015

minus01

minus005

0

005

01

015

y amp

r

Figure 6 Closed-loop output response with 119897119903 = 075 s preview andreference signal 119903(119905)

Figure 6 shows the tracking controller with 119897119903 = 075 s pre-view Comparing the above three figures it can be seen thatbased on the preview controller the output signals can trackthe reference signals much faster and with less tracking error

8 Conclusion

This paper has studied the optimal preview control problemfor a class of continuous stochastic system in the infinitehorizon By introducing the integrator and the assistantsystem the stochastic augmented error system is constructedCompared with the finite horizon the performance indexis simplified The stability of the stochastic augmented errorsystem is also studied and the observer for the originalstochastic system is designed Finally the simulation exampleshows the effectiveness of the conclusion in this paper

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 61174209)

References

[1] M Tomizuka ldquoOptimal continuous finite preview problemrdquoIEEE Transactions on Automatic Control vol 20 no 3 pp 362ndash365 1975

[2] Y Xu F Liao L Cui and J Wu ldquoPreview control for a class oflinear continuous time-varying systemsrdquo International Journalof Wavelets Multiresolution and Information Processing vol 11no 2 Article ID 1350018 1350018 14 pages 2013

[3] F LiaoM Cao Z Hu and P An ldquoDesign of an optimal previewcontroller for linear discrete time causal descriptor systemsrdquo

Mathematical Problems in Engineering 9

International Journal of Control vol 85 no 10 pp 1616ndash16242012

[4] F Liao K Takaba T Katayama and J Katsuura ldquoDesign of anoptimal preview servomechanism for discrete-time systems ina multirate settingrdquoDynamics of Continuous Discrete amp Impul-sive Systems Series B Applications amp Algorithms vol 10 no 5pp 727ndash744 2003

[5] T Katayama T Ohki T Inoue and T Kato ldquoDesign of anoptimal controller for a discrete-time system subject to pre-viewable demandrdquo International Journal of Control vol 41 no3 pp 677ndash699 1985

[6] F Liao Y Guo and Y Y Tang ldquoDesign of an optimal previewcontroller for linear time-varying discrete systems in amultiratesettingrdquo International Journal of Wavelets Multiresolution andInformation Processing vol 13 no 6 Article ID 15500501550050 19 pages 2015

[7] T Tsuchiya andT EgamiDigital Preview andPredictive ControlTranslated by Fucheng Liao Beijing Science and TechnologyPress Beijing China 1994

[8] M Cao and F Liao ldquoDesign of an optimal preview controller forlinear discrete-time descriptor systems with state delayrdquo Inter-national Journal of Systems Science vol 46 no 5 pp 932ndash9432015

[9] Z Y Zhen Z S Wang and D B Wang ldquoInformation fusionestimation based preview control for a discrete linear systemrdquoActa Automatica Sinica vol 36 no 2 pp 347ndash352 2010

[10] T Katayama and T Hirono ldquoDesign of an optimal servomech-anism with preview action and its dual problemrdquo InternationalJournal of Control vol 45 no 2 pp 407ndash420 1987

[11] F Liao Y Y Tang H Liu and Y Wang ldquoDesign of an optimalpreview controller for continuous-time systemsrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 9 no 4 pp 655ndash673 2011

[12] F Liao Z Ren M Tomizuka and J Wu ldquoPreview controlfor impulse-free continuous-time descriptor systemsrdquo Interna-tional Journal of Control vol 88 no 6 pp 1142ndash1149 2015

[13] F Liao andHXu ldquoApplication of the preview controlmethod tothe optimal tracking control problem for continuous-time sys-tems with time-delayrdquo Mathematical Problems in Engineeringvol 2015 Article ID 423580 8 pages 2015

[14] A Astrom Introduction to Stochastic ControlTheory Translatedby Yuhuan Pan Science Press Beijing China 1983

[15] JWu F Liao andM Tomizuka ldquoOptimal preview control for alinear continuous-time stochastic control system in finite-timehorizonrdquo International Journal of Systems Science vol 48 no 1pp 129ndash137 2017

[16] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 6th edition2005

[17] T H Cormen C E Leiserson R Rivest and C Stein Introduc-tion to Algorithms MIT Press and McGraw-Hill Boston MassUSA 2nd edition 2001

[18] J Yong and X Y Zhou Stochastic Control Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[19] D Zheng Linear System Theory Tsinghua University PressBeijing China 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Theorem 16 If 119876 and 119877 are both positive definite matricesand Assumptions 1 to 5 hold the full regulation is achieved inthe closed-loop system

119889119911 (119905)= [119911 (119905) + (119905) + (119905) + Γ 119903 (119905 + 119897119903)] 119889119905+ 120596 (119905) 119889119861119905

(119905)= minus119877minus1119879119875119911 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) (119875 (119904) + 119875Γ 119903 (119904 + 119897119903)) 119889119904

(119905) = 119911 (119905)119911 (1199050) = [ (1199050)0 (1199050)] = 1199110

(42)

namely

lim119905rarrinfin

(119905) = 0 (43)

Furthermore the following will be obtained

lim119905rarrinfin

119890 (119905) = 0 (44)

6 State Observer

If the state vector 119909(119905) in (1) cannot be measured directly theoptimal preview controller (39) cannot be realized In orderto solve this problem the state observer

119889 (119905) = [119872 (119905) + 119873119906 (119905) + 119871 (119910 (119905) minus 119862 (119905))] 119889119905+ 120596 (119905) 119889119861119905 119909 (1199050) = 1199090 119905 isin [1199050infin) (45)

could be designedSubtracting (45) from (1) on both sides yields

119889119909 (119905) = [(119872 minus 119871119862) 119909 (119905)] 119889119905 (46)

where 119909(119905) = 119909(119905) minus (119905)So if (119872 minus 119871119862) is stable the following will hold

lim119905rarrinfin

119909 (119905) = 0 (47)

which means that the state vector (119905) in the observerequation (45) approximates the state vector 119909(119905) in (1) when119905 rarr infin Based on linear system theory [19] it is known thatif (119862 119860) is detectable (119872 minus 119871119862) is stable Then the followingwill be obtained

Theorem 17 If the conditions in Theorem 16 hold the closed-loop system can be received

119889119909 (119905) = [119872119909 (119905) + 119873119906 (119905)] 119889119905 + 120596 (119905) 119889119861119905119889 (119905) = [119872 (119905) + 119873119906 (119905) + 119871 (119910 (119905) minus 119862 (119905))] 119889119905

+ 120596 (119905) 119889119861119905119910 (119905) = 119862119909 (119905) 119906 (119905) = minus119877minus11198791198751 int119905

1199050

(119862 (119904) minus 119903 (119904)) 119889119904minus 119877minus11198791198752 (119905)minus 119877minus1119879int119905+119897119903

119905Φ (119905 119904) 119875 (119904) 119889119904

+ [120574 + 119877minus11198791198752Γ] 119903 (119905 + 119897119903)

(48)

where (119872 minus 119871119862) is stable and (48) achieve the completeregulation

7 Numerical Simulation

Example 1 Consider the stochastic control system

119889[1199091 (119905)1199092 (119905)] = [[0 1minus1 minus1][

1199091 (119905)1199092 (119905)] + [

01] 119906 (119905)] 119889119905

+ 01 119889119861119905119910 (119905) = [1 0] [1199091 (119905)1199092 (119905)]

(49)

where the coefficient matrices are

119872 = [ 0 1minus1 minus1]

119873 = [01] 119862 = [1 0]

120596 (119905) = [0101]

(50)

respectively Therefore the coefficient matrices in (16) are

= [[[

0 1 00 0 10 minus1 minus1

]]]

= [[[

001]]]

Mathematical Problems in Engineering 7

Reference signal r(t)lr = 0 s output y(t)

minus02

0

02

04

06

08

1

12

y amp

r

5 10 15 20 25 30 35 400t (s)

Figure 1 Closed-loop output response with nonpreview and refer-ence signal 119903(119905)

= [[[

minus1 00 minus10 0

]]]

(51)

and Brownian motion 119861119905 satisfies119861119905 sim 119873 (0 001) (52)

Let the initial state of (1) be

119909 (0) = [00] (53)

where 1199091 and 1199092 are mutually independent The previewlengths of the reference signals are 119897119894119903 = 0 05 075 s (119894 =1 2 3) respectively The weight matrices of the performanceindex are

119876 = 1119877 = 1 (54)

and the solution of Riccati equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (55)

is

119875 = [[[

21533 18184 100018184 19157 11531000 1153 0818

]]] (56)

which is a positive definite matrix The coefficient matrix119872119888in (25) is

119872119888 = minus 119877minus1119879119875 = [[[

0 1 00 0 1minus1 minus2153 minus1818

]]] (57)

Reference signal r(t)lr = 05 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

0

02

04

06

08

1

12

y amp

r

Figure 2 Closed-loop output response with 119897119903 = 05 s preview andreference signal

Reference signal r(t)lr = 075 s output y(t)

minus02

0

02

04

06

08

1

12

y amp

r

5 10 15 20 25 30 35 400t (s)

Figure 3 Closed-loop output response with 119897119903 = 075 s preview andreference signal 119903(119905)

When the reference signal is

119903 (119905) =

0 0 le 119905 le 502 (119905 minus 5) 5 lt 119905 le 101 119905 gt 10

(58)

Figures 1ndash3 can be obtainedFigure 1 shows the tracking controller with nonpreview

Figure 2 shows the tracking controller with 119897119903 = 05 s previewFigure 3 shows the tracking controller with 119897119903 = 075 s pre-view Comparing the above three figures it can be seen that

8 Mathematical Problems in Engineering

Reference signal r(t)lr = 0 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

minus015

minus01

minus005

0

005

01

015

02

y amp

r

Figure 4 Closed-loop output response with nonpreview andreference signal 119903(119905)

5 10 15 20 25 30 35 400t (s)

minus015

minus01

minus005

0

005

01

015

02

y amp

r

Reference signal r(t)lr = 05 s output y(t)

Figure 5 Closed-loop output response with 119897119903 = 05 s preview andreference signal

based on the preview controller the output signals can trackthe reference signals much faster and with less tracking error

When the reference signal is

119903 (119905) =

0 0 lt 119905 le 501 sin 05 (119905 minus 5) 5 lt 119905 le 5 + 51205870 119905 gt 5 + 5120587

(59)

Figures 4ndash6 can be obtainedFigure 4 shows the tracking controller with nonpreview

Figure 5 shows the tracking controller with 119897119903 = 05 s preview

Reference signal r(t)lr = 075 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

minus015

minus01

minus005

0

005

01

015

y amp

r

Figure 6 Closed-loop output response with 119897119903 = 075 s preview andreference signal 119903(119905)

Figure 6 shows the tracking controller with 119897119903 = 075 s pre-view Comparing the above three figures it can be seen thatbased on the preview controller the output signals can trackthe reference signals much faster and with less tracking error

8 Conclusion

This paper has studied the optimal preview control problemfor a class of continuous stochastic system in the infinitehorizon By introducing the integrator and the assistantsystem the stochastic augmented error system is constructedCompared with the finite horizon the performance indexis simplified The stability of the stochastic augmented errorsystem is also studied and the observer for the originalstochastic system is designed Finally the simulation exampleshows the effectiveness of the conclusion in this paper

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 61174209)

References

[1] M Tomizuka ldquoOptimal continuous finite preview problemrdquoIEEE Transactions on Automatic Control vol 20 no 3 pp 362ndash365 1975

[2] Y Xu F Liao L Cui and J Wu ldquoPreview control for a class oflinear continuous time-varying systemsrdquo International Journalof Wavelets Multiresolution and Information Processing vol 11no 2 Article ID 1350018 1350018 14 pages 2013

[3] F LiaoM Cao Z Hu and P An ldquoDesign of an optimal previewcontroller for linear discrete time causal descriptor systemsrdquo

Mathematical Problems in Engineering 9

International Journal of Control vol 85 no 10 pp 1616ndash16242012

[4] F Liao K Takaba T Katayama and J Katsuura ldquoDesign of anoptimal preview servomechanism for discrete-time systems ina multirate settingrdquoDynamics of Continuous Discrete amp Impul-sive Systems Series B Applications amp Algorithms vol 10 no 5pp 727ndash744 2003

[5] T Katayama T Ohki T Inoue and T Kato ldquoDesign of anoptimal controller for a discrete-time system subject to pre-viewable demandrdquo International Journal of Control vol 41 no3 pp 677ndash699 1985

[6] F Liao Y Guo and Y Y Tang ldquoDesign of an optimal previewcontroller for linear time-varying discrete systems in amultiratesettingrdquo International Journal of Wavelets Multiresolution andInformation Processing vol 13 no 6 Article ID 15500501550050 19 pages 2015

[7] T Tsuchiya andT EgamiDigital Preview andPredictive ControlTranslated by Fucheng Liao Beijing Science and TechnologyPress Beijing China 1994

[8] M Cao and F Liao ldquoDesign of an optimal preview controller forlinear discrete-time descriptor systems with state delayrdquo Inter-national Journal of Systems Science vol 46 no 5 pp 932ndash9432015

[9] Z Y Zhen Z S Wang and D B Wang ldquoInformation fusionestimation based preview control for a discrete linear systemrdquoActa Automatica Sinica vol 36 no 2 pp 347ndash352 2010

[10] T Katayama and T Hirono ldquoDesign of an optimal servomech-anism with preview action and its dual problemrdquo InternationalJournal of Control vol 45 no 2 pp 407ndash420 1987

[11] F Liao Y Y Tang H Liu and Y Wang ldquoDesign of an optimalpreview controller for continuous-time systemsrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 9 no 4 pp 655ndash673 2011

[12] F Liao Z Ren M Tomizuka and J Wu ldquoPreview controlfor impulse-free continuous-time descriptor systemsrdquo Interna-tional Journal of Control vol 88 no 6 pp 1142ndash1149 2015

[13] F Liao andHXu ldquoApplication of the preview controlmethod tothe optimal tracking control problem for continuous-time sys-tems with time-delayrdquo Mathematical Problems in Engineeringvol 2015 Article ID 423580 8 pages 2015

[14] A Astrom Introduction to Stochastic ControlTheory Translatedby Yuhuan Pan Science Press Beijing China 1983

[15] JWu F Liao andM Tomizuka ldquoOptimal preview control for alinear continuous-time stochastic control system in finite-timehorizonrdquo International Journal of Systems Science vol 48 no 1pp 129ndash137 2017

[16] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 6th edition2005

[17] T H Cormen C E Leiserson R Rivest and C Stein Introduc-tion to Algorithms MIT Press and McGraw-Hill Boston MassUSA 2nd edition 2001

[18] J Yong and X Y Zhou Stochastic Control Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[19] D Zheng Linear System Theory Tsinghua University PressBeijing China 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Reference signal r(t)lr = 0 s output y(t)

minus02

0

02

04

06

08

1

12

y amp

r

5 10 15 20 25 30 35 400t (s)

Figure 1 Closed-loop output response with nonpreview and refer-ence signal 119903(119905)

= [[[

minus1 00 minus10 0

]]]

(51)

and Brownian motion 119861119905 satisfies119861119905 sim 119873 (0 001) (52)

Let the initial state of (1) be

119909 (0) = [00] (53)

where 1199091 and 1199092 are mutually independent The previewlengths of the reference signals are 119897119894119903 = 0 05 075 s (119894 =1 2 3) respectively The weight matrices of the performanceindex are

119876 = 1119877 = 1 (54)

and the solution of Riccati equation

119879119875 + 119875 minus 119875119877minus1119879119875 + 119879119876 = 0 (55)

is

119875 = [[[

21533 18184 100018184 19157 11531000 1153 0818

]]] (56)

which is a positive definite matrix The coefficient matrix119872119888in (25) is

119872119888 = minus 119877minus1119879119875 = [[[

0 1 00 0 1minus1 minus2153 minus1818

]]] (57)

Reference signal r(t)lr = 05 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

0

02

04

06

08

1

12

y amp

r

Figure 2 Closed-loop output response with 119897119903 = 05 s preview andreference signal

Reference signal r(t)lr = 075 s output y(t)

minus02

0

02

04

06

08

1

12

y amp

r

5 10 15 20 25 30 35 400t (s)

Figure 3 Closed-loop output response with 119897119903 = 075 s preview andreference signal 119903(119905)

When the reference signal is

119903 (119905) =

0 0 le 119905 le 502 (119905 minus 5) 5 lt 119905 le 101 119905 gt 10

(58)

Figures 1ndash3 can be obtainedFigure 1 shows the tracking controller with nonpreview

Figure 2 shows the tracking controller with 119897119903 = 05 s previewFigure 3 shows the tracking controller with 119897119903 = 075 s pre-view Comparing the above three figures it can be seen that

8 Mathematical Problems in Engineering

Reference signal r(t)lr = 0 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

minus015

minus01

minus005

0

005

01

015

02

y amp

r

Figure 4 Closed-loop output response with nonpreview andreference signal 119903(119905)

5 10 15 20 25 30 35 400t (s)

minus015

minus01

minus005

0

005

01

015

02

y amp

r

Reference signal r(t)lr = 05 s output y(t)

Figure 5 Closed-loop output response with 119897119903 = 05 s preview andreference signal

based on the preview controller the output signals can trackthe reference signals much faster and with less tracking error

When the reference signal is

119903 (119905) =

0 0 lt 119905 le 501 sin 05 (119905 minus 5) 5 lt 119905 le 5 + 51205870 119905 gt 5 + 5120587

(59)

Figures 4ndash6 can be obtainedFigure 4 shows the tracking controller with nonpreview

Figure 5 shows the tracking controller with 119897119903 = 05 s preview

Reference signal r(t)lr = 075 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

minus015

minus01

minus005

0

005

01

015

y amp

r

Figure 6 Closed-loop output response with 119897119903 = 075 s preview andreference signal 119903(119905)

Figure 6 shows the tracking controller with 119897119903 = 075 s pre-view Comparing the above three figures it can be seen thatbased on the preview controller the output signals can trackthe reference signals much faster and with less tracking error

8 Conclusion

This paper has studied the optimal preview control problemfor a class of continuous stochastic system in the infinitehorizon By introducing the integrator and the assistantsystem the stochastic augmented error system is constructedCompared with the finite horizon the performance indexis simplified The stability of the stochastic augmented errorsystem is also studied and the observer for the originalstochastic system is designed Finally the simulation exampleshows the effectiveness of the conclusion in this paper

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 61174209)

References

[1] M Tomizuka ldquoOptimal continuous finite preview problemrdquoIEEE Transactions on Automatic Control vol 20 no 3 pp 362ndash365 1975

[2] Y Xu F Liao L Cui and J Wu ldquoPreview control for a class oflinear continuous time-varying systemsrdquo International Journalof Wavelets Multiresolution and Information Processing vol 11no 2 Article ID 1350018 1350018 14 pages 2013

[3] F LiaoM Cao Z Hu and P An ldquoDesign of an optimal previewcontroller for linear discrete time causal descriptor systemsrdquo

Mathematical Problems in Engineering 9

International Journal of Control vol 85 no 10 pp 1616ndash16242012

[4] F Liao K Takaba T Katayama and J Katsuura ldquoDesign of anoptimal preview servomechanism for discrete-time systems ina multirate settingrdquoDynamics of Continuous Discrete amp Impul-sive Systems Series B Applications amp Algorithms vol 10 no 5pp 727ndash744 2003

[5] T Katayama T Ohki T Inoue and T Kato ldquoDesign of anoptimal controller for a discrete-time system subject to pre-viewable demandrdquo International Journal of Control vol 41 no3 pp 677ndash699 1985

[6] F Liao Y Guo and Y Y Tang ldquoDesign of an optimal previewcontroller for linear time-varying discrete systems in amultiratesettingrdquo International Journal of Wavelets Multiresolution andInformation Processing vol 13 no 6 Article ID 15500501550050 19 pages 2015

[7] T Tsuchiya andT EgamiDigital Preview andPredictive ControlTranslated by Fucheng Liao Beijing Science and TechnologyPress Beijing China 1994

[8] M Cao and F Liao ldquoDesign of an optimal preview controller forlinear discrete-time descriptor systems with state delayrdquo Inter-national Journal of Systems Science vol 46 no 5 pp 932ndash9432015

[9] Z Y Zhen Z S Wang and D B Wang ldquoInformation fusionestimation based preview control for a discrete linear systemrdquoActa Automatica Sinica vol 36 no 2 pp 347ndash352 2010

[10] T Katayama and T Hirono ldquoDesign of an optimal servomech-anism with preview action and its dual problemrdquo InternationalJournal of Control vol 45 no 2 pp 407ndash420 1987

[11] F Liao Y Y Tang H Liu and Y Wang ldquoDesign of an optimalpreview controller for continuous-time systemsrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 9 no 4 pp 655ndash673 2011

[12] F Liao Z Ren M Tomizuka and J Wu ldquoPreview controlfor impulse-free continuous-time descriptor systemsrdquo Interna-tional Journal of Control vol 88 no 6 pp 1142ndash1149 2015

[13] F Liao andHXu ldquoApplication of the preview controlmethod tothe optimal tracking control problem for continuous-time sys-tems with time-delayrdquo Mathematical Problems in Engineeringvol 2015 Article ID 423580 8 pages 2015

[14] A Astrom Introduction to Stochastic ControlTheory Translatedby Yuhuan Pan Science Press Beijing China 1983

[15] JWu F Liao andM Tomizuka ldquoOptimal preview control for alinear continuous-time stochastic control system in finite-timehorizonrdquo International Journal of Systems Science vol 48 no 1pp 129ndash137 2017

[16] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 6th edition2005

[17] T H Cormen C E Leiserson R Rivest and C Stein Introduc-tion to Algorithms MIT Press and McGraw-Hill Boston MassUSA 2nd edition 2001

[18] J Yong and X Y Zhou Stochastic Control Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[19] D Zheng Linear System Theory Tsinghua University PressBeijing China 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Reference signal r(t)lr = 0 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

minus015

minus01

minus005

0

005

01

015

02

y amp

r

Figure 4 Closed-loop output response with nonpreview andreference signal 119903(119905)

5 10 15 20 25 30 35 400t (s)

minus015

minus01

minus005

0

005

01

015

02

y amp

r

Reference signal r(t)lr = 05 s output y(t)

Figure 5 Closed-loop output response with 119897119903 = 05 s preview andreference signal

based on the preview controller the output signals can trackthe reference signals much faster and with less tracking error

When the reference signal is

119903 (119905) =

0 0 lt 119905 le 501 sin 05 (119905 minus 5) 5 lt 119905 le 5 + 51205870 119905 gt 5 + 5120587

(59)

Figures 4ndash6 can be obtainedFigure 4 shows the tracking controller with nonpreview

Figure 5 shows the tracking controller with 119897119903 = 05 s preview

Reference signal r(t)lr = 075 s output y(t)

5 10 15 20 25 30 35 400t (s)

minus02

minus015

minus01

minus005

0

005

01

015

y amp

r

Figure 6 Closed-loop output response with 119897119903 = 075 s preview andreference signal 119903(119905)

Figure 6 shows the tracking controller with 119897119903 = 075 s pre-view Comparing the above three figures it can be seen thatbased on the preview controller the output signals can trackthe reference signals much faster and with less tracking error

8 Conclusion

This paper has studied the optimal preview control problemfor a class of continuous stochastic system in the infinitehorizon By introducing the integrator and the assistantsystem the stochastic augmented error system is constructedCompared with the finite horizon the performance indexis simplified The stability of the stochastic augmented errorsystem is also studied and the observer for the originalstochastic system is designed Finally the simulation exampleshows the effectiveness of the conclusion in this paper

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 61174209)

References

[1] M Tomizuka ldquoOptimal continuous finite preview problemrdquoIEEE Transactions on Automatic Control vol 20 no 3 pp 362ndash365 1975

[2] Y Xu F Liao L Cui and J Wu ldquoPreview control for a class oflinear continuous time-varying systemsrdquo International Journalof Wavelets Multiresolution and Information Processing vol 11no 2 Article ID 1350018 1350018 14 pages 2013

[3] F LiaoM Cao Z Hu and P An ldquoDesign of an optimal previewcontroller for linear discrete time causal descriptor systemsrdquo

Mathematical Problems in Engineering 9

International Journal of Control vol 85 no 10 pp 1616ndash16242012

[4] F Liao K Takaba T Katayama and J Katsuura ldquoDesign of anoptimal preview servomechanism for discrete-time systems ina multirate settingrdquoDynamics of Continuous Discrete amp Impul-sive Systems Series B Applications amp Algorithms vol 10 no 5pp 727ndash744 2003

[5] T Katayama T Ohki T Inoue and T Kato ldquoDesign of anoptimal controller for a discrete-time system subject to pre-viewable demandrdquo International Journal of Control vol 41 no3 pp 677ndash699 1985

[6] F Liao Y Guo and Y Y Tang ldquoDesign of an optimal previewcontroller for linear time-varying discrete systems in amultiratesettingrdquo International Journal of Wavelets Multiresolution andInformation Processing vol 13 no 6 Article ID 15500501550050 19 pages 2015

[7] T Tsuchiya andT EgamiDigital Preview andPredictive ControlTranslated by Fucheng Liao Beijing Science and TechnologyPress Beijing China 1994

[8] M Cao and F Liao ldquoDesign of an optimal preview controller forlinear discrete-time descriptor systems with state delayrdquo Inter-national Journal of Systems Science vol 46 no 5 pp 932ndash9432015

[9] Z Y Zhen Z S Wang and D B Wang ldquoInformation fusionestimation based preview control for a discrete linear systemrdquoActa Automatica Sinica vol 36 no 2 pp 347ndash352 2010

[10] T Katayama and T Hirono ldquoDesign of an optimal servomech-anism with preview action and its dual problemrdquo InternationalJournal of Control vol 45 no 2 pp 407ndash420 1987

[11] F Liao Y Y Tang H Liu and Y Wang ldquoDesign of an optimalpreview controller for continuous-time systemsrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 9 no 4 pp 655ndash673 2011

[12] F Liao Z Ren M Tomizuka and J Wu ldquoPreview controlfor impulse-free continuous-time descriptor systemsrdquo Interna-tional Journal of Control vol 88 no 6 pp 1142ndash1149 2015

[13] F Liao andHXu ldquoApplication of the preview controlmethod tothe optimal tracking control problem for continuous-time sys-tems with time-delayrdquo Mathematical Problems in Engineeringvol 2015 Article ID 423580 8 pages 2015

[14] A Astrom Introduction to Stochastic ControlTheory Translatedby Yuhuan Pan Science Press Beijing China 1983

[15] JWu F Liao andM Tomizuka ldquoOptimal preview control for alinear continuous-time stochastic control system in finite-timehorizonrdquo International Journal of Systems Science vol 48 no 1pp 129ndash137 2017

[16] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 6th edition2005

[17] T H Cormen C E Leiserson R Rivest and C Stein Introduc-tion to Algorithms MIT Press and McGraw-Hill Boston MassUSA 2nd edition 2001

[18] J Yong and X Y Zhou Stochastic Control Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[19] D Zheng Linear System Theory Tsinghua University PressBeijing China 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

International Journal of Control vol 85 no 10 pp 1616ndash16242012

[4] F Liao K Takaba T Katayama and J Katsuura ldquoDesign of anoptimal preview servomechanism for discrete-time systems ina multirate settingrdquoDynamics of Continuous Discrete amp Impul-sive Systems Series B Applications amp Algorithms vol 10 no 5pp 727ndash744 2003

[5] T Katayama T Ohki T Inoue and T Kato ldquoDesign of anoptimal controller for a discrete-time system subject to pre-viewable demandrdquo International Journal of Control vol 41 no3 pp 677ndash699 1985

[6] F Liao Y Guo and Y Y Tang ldquoDesign of an optimal previewcontroller for linear time-varying discrete systems in amultiratesettingrdquo International Journal of Wavelets Multiresolution andInformation Processing vol 13 no 6 Article ID 15500501550050 19 pages 2015

[7] T Tsuchiya andT EgamiDigital Preview andPredictive ControlTranslated by Fucheng Liao Beijing Science and TechnologyPress Beijing China 1994

[8] M Cao and F Liao ldquoDesign of an optimal preview controller forlinear discrete-time descriptor systems with state delayrdquo Inter-national Journal of Systems Science vol 46 no 5 pp 932ndash9432015

[9] Z Y Zhen Z S Wang and D B Wang ldquoInformation fusionestimation based preview control for a discrete linear systemrdquoActa Automatica Sinica vol 36 no 2 pp 347ndash352 2010

[10] T Katayama and T Hirono ldquoDesign of an optimal servomech-anism with preview action and its dual problemrdquo InternationalJournal of Control vol 45 no 2 pp 407ndash420 1987

[11] F Liao Y Y Tang H Liu and Y Wang ldquoDesign of an optimalpreview controller for continuous-time systemsrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 9 no 4 pp 655ndash673 2011

[12] F Liao Z Ren M Tomizuka and J Wu ldquoPreview controlfor impulse-free continuous-time descriptor systemsrdquo Interna-tional Journal of Control vol 88 no 6 pp 1142ndash1149 2015

[13] F Liao andHXu ldquoApplication of the preview controlmethod tothe optimal tracking control problem for continuous-time sys-tems with time-delayrdquo Mathematical Problems in Engineeringvol 2015 Article ID 423580 8 pages 2015

[14] A Astrom Introduction to Stochastic ControlTheory Translatedby Yuhuan Pan Science Press Beijing China 1983

[15] JWu F Liao andM Tomizuka ldquoOptimal preview control for alinear continuous-time stochastic control system in finite-timehorizonrdquo International Journal of Systems Science vol 48 no 1pp 129ndash137 2017

[16] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 6th edition2005

[17] T H Cormen C E Leiserson R Rivest and C Stein Introduc-tion to Algorithms MIT Press and McGraw-Hill Boston MassUSA 2nd edition 2001

[18] J Yong and X Y Zhou Stochastic Control Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[19] D Zheng Linear System Theory Tsinghua University PressBeijing China 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of