Research ArticleApproximate Design of Optimal DisturbanceRejection for Discrete-Time Systems with Multiple DelayedInputs Application to a Jacket-Type Offshore Structure

Shi-Yuan Han1 Dong Wang1 Yue-Hui Chen1 Gong-You Tang2 and Xi-Xin Yang3

1Shandong Provincial Key Laboratory of Network Based Intelligent Computing University of Jinan336 Nanxinzhuang Road Jinan 250022 China2College of Information Science and Engineering Ocean University of China 238 Songling Road Qingdao 266100 China3College of Software Technical Qingdao University Qingdao 266100 China

Correspondence should be addressed to Dong Wang ise wangdujneducn

Received 28 May 2014 Accepted 26 July 2014

Academic Editor Hak-Keung Lam

Copyright copy 2015 Shi-Yuan Han et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The study is concerned with problem of optimal disturbance rejection for a class of discrete-time systems with multiple delayedinputs In order to avoid the two-point boundary value (TPBV) problem with items of time-delay and time-advance causedby multiple delayed inputs the discrete-time system with multiple delayed inputs is transformed into a delay-free system byintroducing a variable transformation and the original performance index is reformulated as a corresponding form without theexplicit appearance of time-delay itemsThen the approximate optimal disturbance rejection controller (AODRC) is derived fromRiccati equation and Stein equation based on the reduced system and reformulated performance index which is combined withfeedback item of system state feedforward item of disturbances and items of delayed inputs Also the existence and uniquenessof AODRC are proved and the stability of the closed-loop system is analysed Finally numerical examples of disturbance rejectionfor jacket-type offshore structure and pure mathematical model are illustrated to validate the feasibility and effectiveness of theproposed approach

1 Introduction

In practice time-delays are inherent in various engineeringsystems As an efficient control algorithm for the improve-ment of the efficiency reliability and scalability of thedynamic systems time-delays must be taken into considera-tion in process of modeling and controller design from engi-neering and control scientists It is well known that the actua-tors and transmission of the communication information arewith the electrical and electromagnetic characteristicsThere-fore delayed inputs are more general pervasive on dynamicmodelling of engineering systems Especially with the riseof networked control systems (NCSs) and communicationsystems [1 2] multiple delayed inputs are mainly caused bythe online data acquisition filtering calculating controllerand transmitting the control force signals from a computer tothe actuator in engineering systems Recently manymethodshave been proposed to solve the problem for the time-delay

discrete-time systems For example sliding mode control(SMC) is a widely used technique to handle matched dis-turbances by designing sliding surface and reaching motioncontroller for the time-delay process [3ndash5] Smith predictorhas the potential capability to achieve an improvement ofset-point response and disturbance response [6 7] observerbased output feedback control is very powerful for systemswith delayed input by using the predictor feedback whilethe state vectors are measurable or estimated online [8 9]119867infin

control has been intensively applied among the stabilityanalysis stabilization control and robustness control fordiscrete-time systems with multiple delayed inputs [10 11]

Disturbance rejection control has been viewed as atypical issue since the origins of control theory and manyresearchers have been payingmore attention to this field bothacademia and engineersMany physical systems are subjectedto persistent disturbances such as offshore jacket platformssubjected to the waves [12 13] helicopters subjected to the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 474528 10 pageshttpdxdoiorg1011552015474528

2 Mathematical Problems in Engineering

winds [14] and vehicle suspension systems subjected to theroad roughness [15 16] The external disturbances are oftena source of poor performance of underlying control systemsThe active disturbance rejection control has been viewed asan effective solution to solve the above problem Howeverfinding an explicit form of optimal control law for time-delaysystems under disturbances remains difficult [17] Thereforesomemodification or improvement has been done in the con-ventional control schemes For example active disturbancerejection control has been shown to be an effective tool indealing with real world problems of dynamic uncertaintiesdisturbances and nonlinearities [18ndash20] a robust 119867

infinguar-

anteed cost control law is designed for uncertain Markovianjump systems with distributed delays and input delays in[21] by using an inertial actuator for suppression of multipleunknown andor time-varying vibrations an active vibrationcontrol system was proposed in [22] a disturbance estimatordesignmethodwas proposed to design the control scheme foropen-loop and unstable processes with time delay [23] Anoverview of antidisturbance control for engineering systemswith multiple disturbances can be found in [24]

For discrete-time systems with multiple delayed inputunder persistent disturbances the traditional optimal con-trol problem with respect to quadratic performance indexwill induce (TPBV) problem with time-delay and time-advance items Although there have been many methodsto solve the control problem for discrete-time systems withdelayed input it is difficult to find an optimal solutionfor this problem On the one hand augmented methodcould convert a delay system into free-delay ones Howeverwith the multiple delays or large delays the dimension andstability of the delay-free system cannot be ensured even thecomputing work would increase exponentially On the otherhand virtually most of the studies on optimal control fordiscrete-time systemwithmultiple input delays consider onlyapproximate optimal control by using an iterative solutionFor example iterative learning algorithm was proposed toensure the control performance for nonlinear discrete-timesystems with multiple input delays in [25] [26] proposedan optimal controller by using only measured inputoutputdata from a class of linear discrete-time systemswithmultipledelays by using adaptive dynamic programming a successiveapproximation approach (SAA) was proposed in [27] todesign a suboptimal control for discrete linear systems withtime delay in which an iterative procedure is designed tosolve the TPBV problemwith time-delay items and nonlinearitems after that based on SAA and augmented method [28]designed an approximate tracking controller for discrete-time systems with multiple state and input delays

The motivation of this work is to design an AODRC fordiscrete-time systems with multiply delayed inputs underpersistent disturbances with respect to a typical quadricperformance index In order to deal with multiple delayedinputs a variable transformation is proposed so that theoriginal system is transformed into delay-free ones andthe quadratic performance index of the original systemis simplified without the explicit appearance of time-delayitemsThen the AODRC is designed based on the maximumprinciple by solving Riccati and Stein equations which is

composed of a feedback item of system state a feedforwarditem of disturbance state and some controlmemory terms Interm of design of controller compared to the previous work[25 27 28] the proposed AODRC is analytical solution sothat the computation cost and time are reduced In terms ofcontrol performance by analysing the simulation results foran offshore platform and pure mathematical example withdifferent cases in delayed inputs the proposed AODRC cannot only reduce the influence from multiple delayed inputsfor underlying systems but also trade off among rejectioneffect opposed to the multiple delayed inputs persistentdisturbances and energy consumption

The rest of the paper is organized as follows The distur-bance rejection problem for discrete-time systems with mul-tiple delayed inputs is formulated in Section 2 In Section 3the original problem is reformulated as an equivalent one fordelay-free transformed system with respect to a transformedperformance index without the explicit appearance of timedelay Then the proposed AODRC is presented in Section 4in which its existence and uniqueness are proved and thestability of system with multiple delayed inputs is analysedIn Section 5 the effectiveness and implement of the proposedAODRC are evaluated by applying it to numerical examplesof pure mathematical example and a jacket-type offshoreplatform for different cases with delayed inputs Finally ourfindings are concluded in Section 6

2 Problem Statement

Consider the discrete-time systems with multiple delayedinputs under persistent disturbances described by

119909 (119896 + 1) = 119860119909 (119896) +

119873

sum

119894=1

119861119894119906 (119896 minus ℎ

119894) + 119863V (119896)

119910 (119896) = 119862119909 (119896) 119896 = 0 1 2

119909 (0) = 1199090

119906 (119896) = 0 119896 lt 0

(1)

where 119909 isin 119877119899 denotes the state vector 119906 isin 119877

119898 is the controlinput V isin 119877

119901 denotes the persistent disturbances signalℎ119894gt 0 (119894 = 1 2 119873) are positive multiple delayed inputs

respectively 119910 isin 119877119903 is the output vector 119873 is the account of

multiple delayed inputs 119860 119861119894(119894 = 1 2 119873) 119862 and 119863 are

constant matrices of appropriate dimensions

Assumption 1 The pair (119860 1198611) is completely controllable and

the pair (119860 119862) is completely observable where 119862 is definedby 119876 = 119862

119879

119862

Thedynamic characteristics of the persistent disturbancesvector V(119896) are described by the following exosystem

119908 (119896 + 1) = 119866119908 (119896)

V (119896) = 119864119908 (119896)

(2)

where 119908 isin 119877119902 (119901 le 119902 le 119903) is the disturbance state whose

initial condition is unknown 119866 and 119864 are constant matrices

Mathematical Problems in Engineering 3

of appropriate dimensions with full rank It is assumed thatthe pair (119864 119866) is completely observable Also the eigenvaluesof the 119866 satisfy

1003816100381610038161003816120582119894 (119866)1003816100381610038161003816 le 1 119894 = 1 2 119902 (3)

Then exosystem (2) of the persistent disturbances is stablebut may not be asymptotically stable It should be noted thatthe expression of exosystem (2) could describe the variousforms of disturbances with known or unknown dynamiccharacteristics such as the sinusoidal disturbances periodicdisturbances step disturbances and random signals

The performance index could be depending on thedynamic characteristics of persistent disturbances Whenthe persistent disturbances are asymptotically stable thetraditional quadratic performance index could be chosen as

119869 =1

2

infin

sum

119896=0

[119910119879

(119896) 119876119910 (119896) + 119906119879

(119896) 119877119906 (119896)] (4)

On the other hand the persistent disturbances are stable butmay not be asymptotically stable the effect from persistentdisturbance vector V(119896) the steady state of the state vector119909(119896) and the control input 119906(119896) will not converge to zerosynchronouslyTherefore the following infinite-time averageperformance index is chosen

119869 = lim119873rarrinfin

1

119873

119873

sum

119896=0

[119910119879

(119896) 119876119910 (119896) + 119906119879

(119896) 119877119906 (119896)] (5)

where 119876 isin 119877119899times119899 is positive semidefinite and 119877 isin 119877

119898times119898 ispositive definite

The aim of the optimal disturbance rejection is to findan optimal control law 119906

lowast

(119896) for the discrete-time systemsdescribed by (1) and (2) with respect to the performanceindex (4) or (5) that make the performance index (4) or (5)obtain the minimum value

3 Reformulation of DisturbanceRejection Problem

For discrete-time delay systems with quadratic performanceindex the optimal control problem is difficult to be solvedanalytically caused by the TPBV problem with both time-delay and time-advance terms In this section a vector trans-formation is introduced to deal with the multiple delayedinputs and the performance index is reformulated as theequivalent ones without the explicit appearance of time delay

31 Delay-Free Transformation of Control System By propos-ing the following variable transformation

119909 (119896) = 119909 (119896) +

119873

sum

119894=1

119896minus1

sum

119895=119896minusℎ119894

119860119896minusℎ119894minus119895minus1119861

119894119906 (119895) (6)

system (1) with multiple delayed input can be transformedinto the following form

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119863V (119896)

119910 (119896) = 119862(119909 (119896) minus

119873

sum

119894=1

119896minus1

sum

119895=119896minusℎ119894

119860119896minusℎ119894minus119895minus1119906 (119895))

(7)

where 119909(119896) isin 119877119899 is the state vector of the delay-free system

(7) and 119861 = sum119873

119894=1119860minusℎ119894119861119894 Note that the transformed system

(7) is in the form of a delay-free system and the pair (119860 119861)is completely controllable if and only if (119860 119861

1) is completely

controllable [29]

32 Simplification of Performance Index Through the vectortransformation (6) introduced the original system withmultiple delayed inputs (1) could be transformed into system(7) without the explicit appearance of time delay In order toevaluate the effectiveness of the control law more exactly it isnecessary to transform the performance index (4) or (5) intothe equivalent form for the transformed system (7)

By using (6) and (7) the quadratic performance index (4)could be transformed into the following form

119869 =

infin

sum

119896=0

119909119879

(119896) 119862119879

119876119862119909 (119896) + 119906119879

(119896) 119877119906 (119896)

minus 2119880119879

1119862119879

119876119862119909 (119896) + 119880119879

1119862119879

1198761198621198801

(8)

where 1198801= sum119873

119894=1sum119896minus1

119895=119896minusℎ119894

119860119896minusℎ119894minus119895minus1119906(119895)

Based on (2) (6) (7) and (9) one getsinfin

sum

119896=0

119909119879

(119896) 119862119879

1198761198621198801

=

infin

sum

119896=0

119909119879

(119896) 1198621119906 (119896) + 119908

119879

(119896) 1198622119906 (119896)

+

119873

sum

119905=1

ℎ119905

sum

119894=1

ℎ119905

sum

119895=1

((119862119860minus119895

119861119894119906(119896 + 119894 minus 1))

119879

times 119876119862119860119894minus119895minus1

119861119894) 119906 (119896)

infin

sum

119896=0

119880119879

1119862119879

1198761198621198801

=

infin

sum

119896=0

119906119879

(119896)

119873

sum

119894=1

ℎ119894

sum

119895=1

((119862119860minus119895

119861119894)119879

119876119862119860minus119895

119861119894) 119906 (119896)

+ 2

119873

sum

119905=1

ℎ119905

sum

119894=1

ℎ119905

sum

119895=1

((119862119860minus119895

119861119894119906 (119896 + 119894 minus 1))

119879

times 119876119862119860119894minus119895minus1

119861119894) 119906 (119896)

(9)

4 Mathematical Problems in Engineering

where

1198621=

119873

sum

119894=1

((

ℎ119894

sum

119895=1

(119862119860119895

)119879

119876119860119862119860119895minus1minusℎ119894119861

119894))

1198622=

119873

sum

119894=1

ℎ119894

sum

119895=1

ℎ119894minus119895

sum

119905=0

((119862119860119905

119863119864119866119895minus1

)119879

119876119862119860119905+119895minusℎ119894minus1119861

119894)

(10)

Then the quadratic performance index (4) could be reformedas the following form

119869 =1

2

infin

sum

119896=0

119909119879

(119896) 119862119879

119876119862119909 (119896) + 119906119879

(119896) 119906 (119896)

minus2119909119879

(119896) 1198621119906 (119896) minus 2119908

119879

(119896) 1198622119906 (119896)

(11)

where = 119877 + sum119873

119894=1sumℎ119894

119895=1((119862119860minus119895

119861119894)119879

119876119862119860minus119895

119861119894)

Actually the above transformations are suitable for theinfinite-time average quadratic performance index (5) Thereformed performance index could be described as

119869 = lim119873rarrinfin

1

119873

119873

sum

119896=0

119909119879

(119896) 119862119879

119876119862119909 (119896) + 119906119879

(119896) 119906 (119896)

minus2119909119879

(119896) 1198621119906 (119896) minus 2119908

119879

(119896) 1198622119906 (119896)

(12)

Then the original optimal disturbance rejection is refor-mulated as an equivalent optimal regulation problem thatdesigns an approximate control law 119906

lowast

(119896) to make thequadratic performance index (11) or (12) obtain theminimumvalue subject to the discrete-time system (2) and (7) such thatthe negative effects from persistent disturbances andmultipledelayed inputs are eliminated

4 Approximate Design of OptimalDisturbance Rejection Controller

In this section the proposed AODRC is presented in detailand its existence and uniqueness are proved In order to statethe proposed control law clearly the following matrices aredefined

1198601= 119860 + 119861

minus1

119862119879

1 119860

2= 119863119864 + 119861

minus1

119862119879

2

1198761= 119862119879

119876119862 minus 1198621minus1

119862119879

1 119876

2= 1198621minus1

119862119879

2

(13)

Theorem 2 Consider the optimal disturbance rejection con-trol problem for the discrete-time systems given by (1) and(2) with respect to the quadratic performance indexes (4) or

(5) The approximate optimal disturbance rejection controlleruniquely exists and can be formulated as

119906lowast

(119896) = minus1

[119862119879

1minus 119861119879

119860minus119879

1(1198751minus 1198761)]

times [

[

119909 (119896) +

119873

sum

119894=1

119896minus1

sum

119895=119896minusℎ119894

119860119896minusℎ119894minus119895minus1119861119906 (119895)]

]

+ (119862119879

2minus 119861119879

119860minus119879

1(1198752+ 1198762))119908 (119896)

(14)

where 1198751is the unique positive definite solution of the Riccati

matrix equation

1198751= 1198761+ 119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198601

(15)

and 1198752is the unique solution of the Stein matrix equation

1198752+ 1198762= 119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198602

+ 119860119879

1[119868 minus 119860

119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

119861minus1

119861119879

] 1198752119866

(16)

Proof Define the Hamiltonian function for the optimaldisturbance rejection problem as

119867(∙) =1

2119909119879

(119896) 119862119879

119876119862119909 (119896) + 119906119879

(119896) 119906 (119896)

minus 2119909119879

(119896) 1198621119906 (119896) minus 2119908

119879

(119896) 1198622119906 (119896)

+ 120582119879

(119896 + 1) [119860119909 (119896) + 119861119906 (119896) + 119863V (119896)]

(17)

Applying the maximum principle condition 120597119867120597119906 = 0 tothis specific Hamiltonian function (17) yields

119906 (119896) = minus1

[119861119879

120582 (119896 + 1) minus 119862119879

1119909 (119896) minus 119862

119879

2119908 (119896)] (18)

where 120582(119896) is the solution of the following TPBV problem

119909 (119896 + 1) = 1198601119909 (119896) + 119860

2119908 (119896) minus 119861

minus1

119861119879

120582 (119896 + 1)

1198761119909 (119896) minus 119876

2119908 (119896) + 119860

119879

1120582 (119896 + 1) = 120582 (119896)

119909 (0) = 119909 (0)

120582 (infin) = 0

(19)

To solve TPBV problem (19) let

120582 (119896) = 1198751119909 (119896) + 119875

2119908 (119896) (20)

Rearranging (19) one gets

120582 (119896 + 1) = 119860minus119879

1[(1198751minus 1198761) 119909 (119896) + (119875

2+ 1198762) 119908 (119896)]

119909 (119896 + 1) = (119868 + 119861minus1

119861119879

1198751)minus1

times 1198601119909 (119896) + (119860

2minus 119861minus1

119861119879

1198752119866)119908 (119896)

(21)

Mathematical Problems in Engineering 5

Substituting the first formula of (21) into (18) the approxi-mate optimal control law (14) is obtained Note that

120582 (119896) = 1198761119909 (119896) minus 119876

2119908 (119896) + 119860

119879

1120582 (119896 + 1)

= [1198761+ 119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198601] 119909 (119896)

+ 119860119879

1[119868 minus 119860

119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

119861minus1

119861119879

] 1198752119866

+119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198602minus 1198762119908 (119896)

(22)

By comparing the parameters of (20) and (22) the Riccatiequation (15) and Stein equation (16) can be obtained

In the following the existence and uniqueness of theproposed controller will be proved Because 119875

1and 119875

2are

unknown in the parameters of proposed control law (14)the existence and uniqueness of (14) are equivalent to thatof 1198751and 119875

2 For Riccati matrix equation (15) because of

the completely controllable and observable pairs (119860 1198611) and

(119860 119862) the matrix 1198751is existent and the unique solution of

Riccati matrix equation (15) It should be noted that the firstformula of (19) is equivalent to (7) based on the completelycontrollable pair (119860 119861) one gets

100381610038161003816100381610038161003816120582119894[119868 + 119861

minus1

119861119879

1198751]119879

1198601100381610038161003816100381610038161003816lt 1 119894 = 1 2 119899 (23)

Note (3) one gets100381610038161003816100381610038161003816120582119894[119868 + 119861

minus1

119861119879

1198751]119879

1198601100381610038161003816100381610038161003816

10038161003816100381610038161003816120582119895(119866)

10038161003816100381610038161003816lt 1

119894 = 1 2 119899 119895 = 1 2 119902

(24)

Then 1198752is existent and the unique solution of Stein matrix

equation (16) Therefore the proposed approximate con-troller is existent and unique

According to [29] (6) and (14) one gets

119909 (119896) le 119909 (119896) +

119873

sum

119894=1

[ℎ119894max1le119904leℎ119894

10038171003817100381710038171003817119860119896minus119895minus1

10038171003817100381710038171003817

times10038171003817100381710038171003817119861minus1

[119861119879

119860minus119879

1(1198751minus 1198761) minus 119862119879

1]10038171003817100381710038171003817

times 119909 (119896 minus 119904) max1le119904leℎ119894

]

+

119873

sum

119894=1

[ℎ119894max1le119904leℎ119894

10038171003817100381710038171003817119860119896minus119895minus1

10038171003817100381710038171003817119908 (119896 minus 119904)

times10038171003817100381710038171003817119861minus1

[119861119879

119860minus119879

1(1198752+ 1198762) minus 119862119879

2]10038171003817100381710038171003817]

(25)

Because 119909(119896) and 119908(119896) are bounded 119909(119896) is boundedThen the proposed control law (14) is a stabilizing control lawfor discrete-time system (1) with multiple delayed inputsTheproof is completed

Table 1 Parameters of installed AMD device

Name of parameter Variable Value UnitMass 119898

211855 kg

Natural frequency 1205962

233 radsStructural damping ratio 120585

2932 Percent

k2

k1

um2

m1

c2

c1

x2

x1(k)

Figure 1 The sketch of offshore platform with AMD device

5 Simulation

In order to demonstrate the effectiveness and feasibility ofthe proposedAODRC numerous simulations are undertakenin this section In case 1 the vibration control problemfor offshore steel jacket platforms with single delayed inputis simulated by using the proposed AODRC In case 2the disturbance rejection problem for pure mathematicalexample with two delayed inputs is considered

51 Vibration Control for Offshore Jacket Platforms with SingleDelayed Input The jacket type platforms play an importrole in the oil exploration and drilling operations Howeverthe jacket type platforms subject to external disturbancespersistently such as wave force wind and earthquake Inorder to ensure the safety and production efficiency thevibration caused by external disturbances could be rejectedeffectively especially for wave force In this subsection thevibration control problem for offshore jacket platforms withsingle input delay is solved by the proposed control law

Consider the jacket platform located in Bohai Bay thesketch of offshore platform with an AMD device is shown inFigure 1 The structural parameters of offshore platform andinstalled AMD device on the deck of the jacket type platformare listed in Tables 1 and 2 [30] respectively

Based on the JONSWAP spectrum the wave can bedescribed as

119878120578(120596) = [

51198672

119904

161205960

](1205960

120596)

5

exp [minus125(1205960

120596)

minus4

] 120574120573

(26)

6 Mathematical Problems in Engineering

Table 2 Parameters of offshore platform

Name of parameter Variable Value UnitEquivalent characteristic diameter of legs 119863 17 mFirst modal mass 119898

12371100 kg

Natural frequency 1205961

220 radsStructural damping ratio 120585

14 Percent

Shape function of first mode 120593(119904) 1 minus cos ( 120587119904

2119871) 0 le 119904 le 119871

where 119867119904

= 4m is the significant wave height 120596 isthe wave frequency 120574 = 33 is the peakedness param-eter 120596

0= 087 rads is the peak frequency 120573 =

exp[minus(120596 minus 1205960)2

(21205902

1205962

0)] in which 120590 is the shape parameter

120590 = 007 (120596 le 1205960) and 120590 = 009 (120596 gt 120596

0) Then based

on [30] the wave force can be formulated as an exosystemdescribed by (2) with the water depth 119889 = 132m the dragcoefficient119862

119889= 12 and the inertial coefficient119862

119898= 20The

external disturbances from irregular wave force are displayedin Figure 2

By setting the sampling period 119879 = 01 s 119876 =

diag (100 0 100 0) and 119877 = 1 the discrete-time modelof the offshore structure could be established with the sameexpression of (1) based on the previous investigations on thesubject which is described as

119860 =

[[[

[

09790 00002 00989 00000

00205 09789 00047 00952

minus04182 00041 09701 00010

03954 minus04152 00996 08993

]]]

]

119861 =

[[[

[

minus00006

00621

minus00120

12205

]]]

]

119863 =

[[[

[

00006

00000

00126

00006

]]]

]

119862 = [1 0 0 0

0 0 1 0]

(27)

According to the previous investigations on the subject therange of delayed input is in the interval (0 1) s Thereforeto demonstrate the effectiveness of the proposed AODRCthe comparison results between the open-loop system thefeedback control and the AODRC with single delayed inputℎ = 6 are compared in which feedback control law is justconsidered the feedback item in AODRC Then the curvesof displacement of offshore structure velocity of offshorestructure and control law are presented in Figures 3 4 and5 respectively

In order to show the comparison results among the sim-ulated case clearly the values of the quadratic performanceindex with different delayed input are compared in Table 3Note that the percentage number given in the parenthesesindicates the reduced amount of the closed-loop response

0 10 20 30 40 50 60 70 80 90 100

0

2

4

6

Wav

e for

ce(k)

(106

N)

minus2

minus4

minus6

kT (s)

Figure 2 The external disturbances of wave force

0 10 20 30 40 50 60 70 80 90 100

0

01

02

03

04

Disp

lace

men

t of o

ffsho

re st

ruct

ure (

m)

minus01

minus02

minus03

minus04

Open-loopFeedback controlAODRC

kT (s)

Figure 3 Curves of the displacement of offshore structure

relative to the open-loop case For instance the value ofperformance index is reduced by 1840when delayed inputℎ = 2 less than that of open-loop one using feedback controlwhile the value of performance index is reduced by 6723less than that of open-loop one by using proposed AODRC

Mathematical Problems in Engineering 7

Table 3 Comparison of values of performance index with different delayed input

ℎ = 2 ℎ = 4 ℎ = 6 ℎ = 8

Open-loop 50953 50953 50953 50953Feedback Control 41578 (1840) 35232 (3085) 30731 (3969) 31232 (3870)AODRC 16695 (6723) 16875 (6688) 17076 (6649) 17366 (6592)

0 10 20 30 40 50 60 70 80 90 100

02

0

04

06

Velo

city

of o

ffsho

re st

ruct

ure (

ms

)

Open-loopFeedback controlAODRC

minus02

minus04

minus06

minus08

kT (s)

Figure 4 Curves of the velocity of offshore structure

0 10 20 30 40 50 60 70 80 90 100

0

02

04

06

08

minus02

minus04

minus06

minus08

Con

trol l

awu(k)

(104

N)

kT (s)

Feedback controlAODRC

Figure 5 Curves of the control law

In order to present the control results more intuitive themaximum displacement the maximum acceleration of off-shore structure and the maximum control law with differentinput delay are shown in Table 4 From Table 4 it can beseen clearly that the maximum displacement the maximumacceleration of offshore structure and the values ofmaximumcontrol law with different input delay are close The resultsshow that the proposed control law could effectively eliminate

the influences fromdifferent input delay andwave forces withhigh value

It can be seen from Figures 3ndash5 and Tables 3-4 thatthe disturbances of wave force for offshore structure isreduced significantly by using the proposed AODRC Withthe increase of delayed input the proposed AODRC is stillable to effectively control the displacement and velocity ofoffshore structure Whatrsquos more the quadratic performanceindex is reduced significantly by using the proposed AODRCcompared by the feedback control and open-loop onesTherefore we can conclude that the proposed AODRC isefficient to reject the disturbances from wave force foroffshore structure as well as being with lower consumption

52 Disturbance Rejection for Pure Mathematical Examplewith Two Delayed Inputs In order to verify the elasticity ofthe proposed AODRC the discrete-time system with delayedinputs ℎ

1and ℎ2under sinusoidal disturbances is considered

The matrices of (1) are chosen as

119860 = [1 04

minus02 09]

119863 = [0

1]

1198611= [

02

05]

1198612= [

01

015]

119862 = [1 0]

119909 (0) = [2 0]119879

(28)

The external disturbance V(119905) is chosen as sinusoidal distur-bance the matrices of (2) are given by

119866 = [04radic5 04

minus05 04radic5]

119864 = [02 0]

V (0) = [1 0]119879

(29)

The curves of sinusoidal disturbances with (29) are shownin Figure 6 Also the quadratic performance index is chosenas (5) with 119876 = 1 119877 = 2

Then the responses of system states 1199091and 119909

2for differ-

ent cases (open-loop feedback control only and proposedAODRC) with delayed inputs ℎ

1= 1 and ℎ

2= 4 are

8 Mathematical Problems in Engineering

Table 4 The maximum displacement the maximum accelerationof offshore structure and the maximum control law

ℎ = 2 ℎ = 4 ℎ = 6 ℎ = 8

Maximumdisplacement (m) 0126 0130 0146 0150

Maximumacceleration (m2s) 0171 0175 0178 0181

Maximum controlinput (104 N) 0520 0523 0582 0588

0 50 100 150 200

0

005

01

015

02

Sinu

soid

al d

istur

banc

e(k)

minus005

minus01

minus015

minus02

kT (s)

Figure 6 Curves of the sinusoidal disturbance V(119896)

simulated and shown in Figures 7 and 8 The curves of theproposed AODRC for different cases (the proposed AODRCand feedback control only) are shown in Figure 9 The valuesof quadratic performance index (5) are shown in Table 3 withtwo delayed inputs (ℎ

1= 1 ℎ

2= 2 ℎ

1= 1 ℎ

2= 4 ℎ

1= 2

ℎ2= 4) Figures 7 and 8 and Table 5 show that the proposed

AODRC is able to effectively ensure the system states in lowervalues and reject the disturbances from external sinusoidalinput

From Figures 1ndash9 and Tables 1ndash5 the simulation resultsshow that the proposed AODRC for the offshore platformand pure mathematical example are effective to attenuate thevibration amplitudes of the system state and the quadraticperformance index of the controlled system under AODRC issmaller than those under the open-loop and feedback controlonly Moreover the proposed AODRC is more scalable fordifferent cases with multiple delayed inputs

6 Conclusion

This paper has presented an interesting approach to solvethe disturbance rejection problem for discrete-time systemswith multiple delayed inputs under persistent external dis-turbances A discrete-time system with multiple input delaysis transformed into a non-delayed system in form and thequadratic performance index has been transformed into arelevant format without the explicit appearance of time delayThen the AODRC is deployed to eliminate the negativeeffects from multiple delayed inputs and persistent external

0 50 100 150 200

0

1

2

3

minus1

minus2

minus3

Curv

es o

f sys

tem

stat

ex1(k

)

Open-loopFeedback controlAODRC

kT (s)

Figure 7 Curves of system state 1199091(119896)

0 50 100 150 200

0

05

1

15

2

25

minus05

minus1

minus15

minus2

Curv

es o

f sys

tem

stat

ex2(k

)

Open-loopFeedback controlAODRC

kT (s)

Figure 8 Curves of system state 1199092(119896)

Table 5 Comparison of values of performance index with twodelayed inputs

ℎ1= 1 ℎ

2= 2 ℎ

1= 1 ℎ

2= 4 ℎ

1= 2 ℎ

2= 4

Open-loop 03422 03422 03422FeedbackControl 00836 (7557) 01055 (6917) 01035 (6975)

AODRC 00372 (8913) 00506 (8521) 00901 (6649)

disturbances On the other hand the disturbance rejectionproblem for jacket-type offshore structure is illustrated tovalidate the feasibility and effectiveness of the proposedapproach

Mathematical Problems in Engineering 9

0 50 100 150 200

01

02

03

04

05

0

Feedback controlAODRC

minus01

minus02

minus03

minus04

minus05

Con

trol l

awu(k)

kT (s)

Figure 9 Curves of the control law

Conflict of Interests

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

Acknowledgments

This work was supported by the Program for Scien-tific Research Innovation Team in Colleges and Universi-ties of Shandong Province Science Foundation of Chinaunder Grants 61201428 61203105 and 61302128 the NaturalScience Foundation of Shandong Province under GrantsZR2011FZ003 ZR2012FQ016 and ZR2011FL022 and theDoctoral Foundation of Shandong Province under GrantBS2014DX015

References

[1] E Altman and T Basar ldquoOptimal rate control for high speedtelecommunication networksrdquo in Proceedings of the 34th IEEEConference on Decision and Control vol 2 pp 1389ndash1394December 1995

[2] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[3] X Yu C Wu F Liu and L Wu ldquoSliding mode control ofdiscrete-time switched systems with time-delayrdquo Journal of theFranklin Institute vol 350 no 1 pp 19ndash33 2013

[4] S Janardhanan and B Bandyopadhyay ldquoOutput feedbackdiscrete-time sliding mode control for time delay systemsrdquo IEEProceedingsmdashControl Theory and Applications vol 153 no 4pp 387ndash396 2006

[5] M Cedomir P-D Branislava and V Boban ldquoDiscrete-timevelocity servo system design using sliding mode controlapproach with disturbance compensationrdquo IEEE Transactionson Industrial Informatics vol 9 no 2 pp 920ndash927 2013

[6] P Albertos and P Garcıa ldquoPredictor-observer-based control ofsystems with multiple inputoutput delaysrdquo Journal of ProcessControl vol 22 no 7 pp 1350ndash1357 2012

[7] E Normey-Rico and E F Camacho ldquoDead-time compensatorsa surveyrdquo Control Engineering Practice vol 16 no 4 pp 407ndash428 2008

[8] B Zhou Z-Y Li and Z Lin ldquoObserver based output feedbackcontrol of linear systems with input and output delaysrdquo Auto-matica vol 49 no 7 pp 2039ndash2052 2013

[9] B Zhou ldquoObserver-based output feedback control of discrete-time linear systems with input and output delaysrdquo InternationalJournal of Control 2014 in press

[10] H Zhang G Duan and L Xie ldquoLinear quadratic regulationfor linear time-varying systems with multiple input delays PartI discrete time caserdquo in Proceedings of the 5th InternationalConference on Control and Automation Budapest Hungary2005

[11] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[12] B-L Zhang L Ma and Q-L Han ldquoSliding mode Hinfin

controlfor offshore steel jacket platforms subject to nonlinear self-excited wave force and external disturbanceurbancerdquoNonlinearAnalysis Real World Applications vol 14 no 1 pp 163ndash1782013

[13] B Zhang Q Han X Zhang and X Yu ldquoIntegral sliding modecontrol for offshore steel jacket platformsrdquo Journal of Sound andVibration vol 331 no 14 pp 3271ndash3285 2012

[14] S R Hall and N MWereley ldquoPerformance of higher harmoniccontrol algorithms for helicopter vibration reductionrdquo Journalof Guidance Control and Dynamics vol 16 no 4 pp 793ndash7971993

[15] S Han G Tang Y Chen and X Yang ldquoOptimal vibrationcontrol for vehicle active suspension discrete-time systems withactuator time delayrdquo Asian Journal of Control vol 15 no 6 pp1579ndash1588 2013

[16] H Gao J Lam andCWang ldquoMulti-objective control of vehicleactive suspension systems via load-dependent controllersrdquoJournal of Sound and Vibration vol 290 no 3ndash5 pp 654ndash6752006

[17] L Mirkin ldquoOn the approximation of distributed-delay controllawsrdquo Systems ampControl Letters vol 51 no 5 pp 331ndash342 2004

[18] Y Huang W C Xue and C Z Zhao ldquoActive disturbance rejec-tion control methodology and theoretical analysisrdquo Journal ofSystems Science and Mathematical Sciences vol 31 no 9 pp1111ndash1129 2011

[19] Y Shi J Huang and B Yu ldquoRobust tracking control ofnetworked control systems application to a networked DCmotorrdquo IEEE Transactions on Industrial Electronics vol 60 no12 pp 5864ndash5874 2013

[20] X Guo and M Bodson ldquoAnalysis and implementation of anadaptive algorithm for the rejection of multiple sinusoidal dis-turbancesrdquo IEEE Transactions on Control Systems Technologyvol 17 no 1 pp 40ndash50 2009

[21] H Zhao Q Chen and S Xu ldquo119867infin

guaranteed cost controlfor uncertain Markovian jump systems with mode-dependentdistributed delays and input delaysrdquo Journal of the FranklinInstitute vol 346 no 10 pp 945ndash957 2009

[22] I D Landau M Alma J J Martinez and G Buche ldquoAdaptivesuppression ofmultiple time-varying unknownvibrations usingan inertial actuatorrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 6 pp 1327ndash1338 2011

[23] M Shamsuzzoha and M Lee ldquoEnhanced disturbance rejectionfor open-loop unstable process with time delayrdquo ISA Transac-tions vol 48 no 2 pp 237ndash244 2009

10 Mathematical Problems in Engineering

[24] L Guo and S Cao ldquoAnti-disturbance control theory for systemswith multiple disturbances a surveyrdquo ISA Transactions vol 53no 4 pp 846ndash849 2014

[25] X Li T W S Chow and J K L Ho ldquoIterative learning controlfor a class of nonlinear discrete-time systems with multipleinput delaysrdquo International Journal of Systems Science vol 39no 4 pp 361ndash369 2008

[26] J Zhang H Zhang Y Luo and T Feng ldquoModel-free optimalcontrol design for a class of linear discrete-time systems withmultiple delays using adaptive dynamic programmingrdquo Neuro-computing vol 135 pp 163ndash170 2014

[27] G-Y Tang and H-H Wang ldquoSuboptimal control for discretelinear systems with time-delay a successive approximationapproachrdquo Acta Automatica Sinica vol 31 no 3 pp 419ndash4262005

[28] H H Wang and G Y Tang ldquoObserver-based optimal outputtracking for discrete-time Systems with multiple state andinput delaysrdquo International Journal of Control Automation andSystems vol 7 no 1 pp 57ndash66 2009

[29] H T Banks M Q Jacobs and M R Latina ldquoThe synthesisof optimal controls for linear time-optimal problems withretarded controlsrdquo Journal of OptimizationTheory and Applica-tions vol 8 pp 319ndash366 1971

[30] H Ma G-Y Tang and Y-D Zhao ldquoFeedforward and feedbackoptimal control for offshore structures subjected to irregularwave forcesrdquo Ocean Engineering vol 33 no 8-9 pp 1105ndash11172006

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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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

2 Mathematical Problems in Engineering

winds [14] and vehicle suspension systems subjected to theroad roughness [15 16] The external disturbances are oftena source of poor performance of underlying control systemsThe active disturbance rejection control has been viewed asan effective solution to solve the above problem Howeverfinding an explicit form of optimal control law for time-delaysystems under disturbances remains difficult [17] Thereforesomemodification or improvement has been done in the con-ventional control schemes For example active disturbancerejection control has been shown to be an effective tool indealing with real world problems of dynamic uncertaintiesdisturbances and nonlinearities [18ndash20] a robust 119867

infinguar-

anteed cost control law is designed for uncertain Markovianjump systems with distributed delays and input delays in[21] by using an inertial actuator for suppression of multipleunknown andor time-varying vibrations an active vibrationcontrol system was proposed in [22] a disturbance estimatordesignmethodwas proposed to design the control scheme foropen-loop and unstable processes with time delay [23] Anoverview of antidisturbance control for engineering systemswith multiple disturbances can be found in [24]

For discrete-time systems with multiple delayed inputunder persistent disturbances the traditional optimal con-trol problem with respect to quadratic performance indexwill induce (TPBV) problem with time-delay and time-advance items Although there have been many methodsto solve the control problem for discrete-time systems withdelayed input it is difficult to find an optimal solutionfor this problem On the one hand augmented methodcould convert a delay system into free-delay ones Howeverwith the multiple delays or large delays the dimension andstability of the delay-free system cannot be ensured even thecomputing work would increase exponentially On the otherhand virtually most of the studies on optimal control fordiscrete-time systemwithmultiple input delays consider onlyapproximate optimal control by using an iterative solutionFor example iterative learning algorithm was proposed toensure the control performance for nonlinear discrete-timesystems with multiple input delays in [25] [26] proposedan optimal controller by using only measured inputoutputdata from a class of linear discrete-time systemswithmultipledelays by using adaptive dynamic programming a successiveapproximation approach (SAA) was proposed in [27] todesign a suboptimal control for discrete linear systems withtime delay in which an iterative procedure is designed tosolve the TPBV problemwith time-delay items and nonlinearitems after that based on SAA and augmented method [28]designed an approximate tracking controller for discrete-time systems with multiple state and input delays

The motivation of this work is to design an AODRC fordiscrete-time systems with multiply delayed inputs underpersistent disturbances with respect to a typical quadricperformance index In order to deal with multiple delayedinputs a variable transformation is proposed so that theoriginal system is transformed into delay-free ones andthe quadratic performance index of the original systemis simplified without the explicit appearance of time-delayitemsThen the AODRC is designed based on the maximumprinciple by solving Riccati and Stein equations which is

composed of a feedback item of system state a feedforwarditem of disturbance state and some controlmemory terms Interm of design of controller compared to the previous work[25 27 28] the proposed AODRC is analytical solution sothat the computation cost and time are reduced In terms ofcontrol performance by analysing the simulation results foran offshore platform and pure mathematical example withdifferent cases in delayed inputs the proposed AODRC cannot only reduce the influence from multiple delayed inputsfor underlying systems but also trade off among rejectioneffect opposed to the multiple delayed inputs persistentdisturbances and energy consumption

The rest of the paper is organized as follows The distur-bance rejection problem for discrete-time systems with mul-tiple delayed inputs is formulated in Section 2 In Section 3the original problem is reformulated as an equivalent one fordelay-free transformed system with respect to a transformedperformance index without the explicit appearance of timedelay Then the proposed AODRC is presented in Section 4in which its existence and uniqueness are proved and thestability of system with multiple delayed inputs is analysedIn Section 5 the effectiveness and implement of the proposedAODRC are evaluated by applying it to numerical examplesof pure mathematical example and a jacket-type offshoreplatform for different cases with delayed inputs Finally ourfindings are concluded in Section 6

2 Problem Statement

Consider the discrete-time systems with multiple delayedinputs under persistent disturbances described by

119909 (119896 + 1) = 119860119909 (119896) +

119873

sum

119894=1

119861119894119906 (119896 minus ℎ

119894) + 119863V (119896)

119910 (119896) = 119862119909 (119896) 119896 = 0 1 2

119909 (0) = 1199090

119906 (119896) = 0 119896 lt 0

(1)

where 119909 isin 119877119899 denotes the state vector 119906 isin 119877

119898 is the controlinput V isin 119877

119901 denotes the persistent disturbances signalℎ119894gt 0 (119894 = 1 2 119873) are positive multiple delayed inputs

respectively 119910 isin 119877119903 is the output vector 119873 is the account of

multiple delayed inputs 119860 119861119894(119894 = 1 2 119873) 119862 and 119863 are

constant matrices of appropriate dimensions

Assumption 1 The pair (119860 1198611) is completely controllable and

the pair (119860 119862) is completely observable where 119862 is definedby 119876 = 119862

119879

119862

Thedynamic characteristics of the persistent disturbancesvector V(119896) are described by the following exosystem

119908 (119896 + 1) = 119866119908 (119896)

V (119896) = 119864119908 (119896)

(2)

where 119908 isin 119877119902 (119901 le 119902 le 119903) is the disturbance state whose

initial condition is unknown 119866 and 119864 are constant matrices

Mathematical Problems in Engineering 3

of appropriate dimensions with full rank It is assumed thatthe pair (119864 119866) is completely observable Also the eigenvaluesof the 119866 satisfy

1003816100381610038161003816120582119894 (119866)1003816100381610038161003816 le 1 119894 = 1 2 119902 (3)

Then exosystem (2) of the persistent disturbances is stablebut may not be asymptotically stable It should be noted thatthe expression of exosystem (2) could describe the variousforms of disturbances with known or unknown dynamiccharacteristics such as the sinusoidal disturbances periodicdisturbances step disturbances and random signals

The performance index could be depending on thedynamic characteristics of persistent disturbances Whenthe persistent disturbances are asymptotically stable thetraditional quadratic performance index could be chosen as

119869 =1

2

infin

sum

119896=0

[119910119879

(119896) 119876119910 (119896) + 119906119879

(119896) 119877119906 (119896)] (4)

On the other hand the persistent disturbances are stable butmay not be asymptotically stable the effect from persistentdisturbance vector V(119896) the steady state of the state vector119909(119896) and the control input 119906(119896) will not converge to zerosynchronouslyTherefore the following infinite-time averageperformance index is chosen

119869 = lim119873rarrinfin

1

119873

119873

sum

119896=0

[119910119879

(119896) 119876119910 (119896) + 119906119879

(119896) 119877119906 (119896)] (5)

where 119876 isin 119877119899times119899 is positive semidefinite and 119877 isin 119877

119898times119898 ispositive definite

The aim of the optimal disturbance rejection is to findan optimal control law 119906

lowast

(119896) for the discrete-time systemsdescribed by (1) and (2) with respect to the performanceindex (4) or (5) that make the performance index (4) or (5)obtain the minimum value

3 Reformulation of DisturbanceRejection Problem

For discrete-time delay systems with quadratic performanceindex the optimal control problem is difficult to be solvedanalytically caused by the TPBV problem with both time-delay and time-advance terms In this section a vector trans-formation is introduced to deal with the multiple delayedinputs and the performance index is reformulated as theequivalent ones without the explicit appearance of time delay

31 Delay-Free Transformation of Control System By propos-ing the following variable transformation

119909 (119896) = 119909 (119896) +

119873

sum

119894=1

119896minus1

sum

119895=119896minusℎ119894

119860119896minusℎ119894minus119895minus1119861

119894119906 (119895) (6)

system (1) with multiple delayed input can be transformedinto the following form

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119863V (119896)

119910 (119896) = 119862(119909 (119896) minus

119873

sum

119894=1

119896minus1

sum

119895=119896minusℎ119894

119860119896minusℎ119894minus119895minus1119906 (119895))

(7)

where 119909(119896) isin 119877119899 is the state vector of the delay-free system

(7) and 119861 = sum119873

119894=1119860minusℎ119894119861119894 Note that the transformed system

(7) is in the form of a delay-free system and the pair (119860 119861)is completely controllable if and only if (119860 119861

1) is completely

controllable [29]

32 Simplification of Performance Index Through the vectortransformation (6) introduced the original system withmultiple delayed inputs (1) could be transformed into system(7) without the explicit appearance of time delay In order toevaluate the effectiveness of the control law more exactly it isnecessary to transform the performance index (4) or (5) intothe equivalent form for the transformed system (7)

By using (6) and (7) the quadratic performance index (4)could be transformed into the following form

119869 =

infin

sum

119896=0

119909119879

(119896) 119862119879

119876119862119909 (119896) + 119906119879

(119896) 119877119906 (119896)

minus 2119880119879

1119862119879

119876119862119909 (119896) + 119880119879

1119862119879

1198761198621198801

(8)

where 1198801= sum119873

119894=1sum119896minus1

119895=119896minusℎ119894

119860119896minusℎ119894minus119895minus1119906(119895)

Based on (2) (6) (7) and (9) one getsinfin

sum

119896=0

119909119879

(119896) 119862119879

1198761198621198801

=

infin

sum

119896=0

119909119879

(119896) 1198621119906 (119896) + 119908

119879

(119896) 1198622119906 (119896)

+

119873

sum

119905=1

ℎ119905

sum

119894=1

ℎ119905

sum

119895=1

((119862119860minus119895

119861119894119906(119896 + 119894 minus 1))

119879

times 119876119862119860119894minus119895minus1

119861119894) 119906 (119896)

infin

sum

119896=0

119880119879

1119862119879

1198761198621198801

=

infin

sum

119896=0

119906119879

(119896)

119873

sum

119894=1

ℎ119894

sum

119895=1

((119862119860minus119895

119861119894)119879

119876119862119860minus119895

119861119894) 119906 (119896)

+ 2

119873

sum

119905=1

ℎ119905

sum

119894=1

ℎ119905

sum

119895=1

((119862119860minus119895

119861119894119906 (119896 + 119894 minus 1))

119879

times 119876119862119860119894minus119895minus1

119861119894) 119906 (119896)

(9)

4 Mathematical Problems in Engineering

where

1198621=

119873

sum

119894=1

((

ℎ119894

sum

119895=1

(119862119860119895

)119879

119876119860119862119860119895minus1minusℎ119894119861

119894))

1198622=

119873

sum

119894=1

ℎ119894

sum

119895=1

ℎ119894minus119895

sum

119905=0

((119862119860119905

119863119864119866119895minus1

)119879

119876119862119860119905+119895minusℎ119894minus1119861

119894)

(10)

Then the quadratic performance index (4) could be reformedas the following form

119869 =1

2

infin

sum

119896=0

119909119879

(119896) 119862119879

119876119862119909 (119896) + 119906119879

(119896) 119906 (119896)

minus2119909119879

(119896) 1198621119906 (119896) minus 2119908

119879

(119896) 1198622119906 (119896)

(11)

where = 119877 + sum119873

119894=1sumℎ119894

119895=1((119862119860minus119895

119861119894)119879

119876119862119860minus119895

119861119894)

Actually the above transformations are suitable for theinfinite-time average quadratic performance index (5) Thereformed performance index could be described as

119869 = lim119873rarrinfin

1

119873

119873

sum

119896=0

119909119879

(119896) 119862119879

119876119862119909 (119896) + 119906119879

(119896) 119906 (119896)

minus2119909119879

(119896) 1198621119906 (119896) minus 2119908

119879

(119896) 1198622119906 (119896)

(12)

Then the original optimal disturbance rejection is refor-mulated as an equivalent optimal regulation problem thatdesigns an approximate control law 119906

lowast

(119896) to make thequadratic performance index (11) or (12) obtain theminimumvalue subject to the discrete-time system (2) and (7) such thatthe negative effects from persistent disturbances andmultipledelayed inputs are eliminated

4 Approximate Design of OptimalDisturbance Rejection Controller

In this section the proposed AODRC is presented in detailand its existence and uniqueness are proved In order to statethe proposed control law clearly the following matrices aredefined

1198601= 119860 + 119861

minus1

119862119879

1 119860

2= 119863119864 + 119861

minus1

119862119879

2

1198761= 119862119879

119876119862 minus 1198621minus1

119862119879

1 119876

2= 1198621minus1

119862119879

2

(13)

Theorem 2 Consider the optimal disturbance rejection con-trol problem for the discrete-time systems given by (1) and(2) with respect to the quadratic performance indexes (4) or

(5) The approximate optimal disturbance rejection controlleruniquely exists and can be formulated as

119906lowast

(119896) = minus1

[119862119879

1minus 119861119879

119860minus119879

1(1198751minus 1198761)]

times [

[

119909 (119896) +

119873

sum

119894=1

119896minus1

sum

119895=119896minusℎ119894

119860119896minusℎ119894minus119895minus1119861119906 (119895)]

]

+ (119862119879

2minus 119861119879

119860minus119879

1(1198752+ 1198762))119908 (119896)

(14)

where 1198751is the unique positive definite solution of the Riccati

matrix equation

1198751= 1198761+ 119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198601

(15)

and 1198752is the unique solution of the Stein matrix equation

1198752+ 1198762= 119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198602

+ 119860119879

1[119868 minus 119860

119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

119861minus1

119861119879

] 1198752119866

(16)

Proof Define the Hamiltonian function for the optimaldisturbance rejection problem as

119867(∙) =1

2119909119879

(119896) 119862119879

119876119862119909 (119896) + 119906119879

(119896) 119906 (119896)

minus 2119909119879

(119896) 1198621119906 (119896) minus 2119908

119879

(119896) 1198622119906 (119896)

+ 120582119879

(119896 + 1) [119860119909 (119896) + 119861119906 (119896) + 119863V (119896)]

(17)

Applying the maximum principle condition 120597119867120597119906 = 0 tothis specific Hamiltonian function (17) yields

119906 (119896) = minus1

[119861119879

120582 (119896 + 1) minus 119862119879

1119909 (119896) minus 119862

119879

2119908 (119896)] (18)

where 120582(119896) is the solution of the following TPBV problem

119909 (119896 + 1) = 1198601119909 (119896) + 119860

2119908 (119896) minus 119861

minus1

119861119879

120582 (119896 + 1)

1198761119909 (119896) minus 119876

2119908 (119896) + 119860

119879

1120582 (119896 + 1) = 120582 (119896)

119909 (0) = 119909 (0)

120582 (infin) = 0

(19)

To solve TPBV problem (19) let

120582 (119896) = 1198751119909 (119896) + 119875

2119908 (119896) (20)

Rearranging (19) one gets

120582 (119896 + 1) = 119860minus119879

1[(1198751minus 1198761) 119909 (119896) + (119875

2+ 1198762) 119908 (119896)]

119909 (119896 + 1) = (119868 + 119861minus1

119861119879

1198751)minus1

times 1198601119909 (119896) + (119860

2minus 119861minus1

119861119879

1198752119866)119908 (119896)

(21)

Mathematical Problems in Engineering 5

Substituting the first formula of (21) into (18) the approxi-mate optimal control law (14) is obtained Note that

120582 (119896) = 1198761119909 (119896) minus 119876

2119908 (119896) + 119860

119879

1120582 (119896 + 1)

= [1198761+ 119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198601] 119909 (119896)

+ 119860119879

1[119868 minus 119860

119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

119861minus1

119861119879

] 1198752119866

+119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198602minus 1198762119908 (119896)

(22)

By comparing the parameters of (20) and (22) the Riccatiequation (15) and Stein equation (16) can be obtained

In the following the existence and uniqueness of theproposed controller will be proved Because 119875

1and 119875

2are

unknown in the parameters of proposed control law (14)the existence and uniqueness of (14) are equivalent to thatof 1198751and 119875

2 For Riccati matrix equation (15) because of

the completely controllable and observable pairs (119860 1198611) and

(119860 119862) the matrix 1198751is existent and the unique solution of

Riccati matrix equation (15) It should be noted that the firstformula of (19) is equivalent to (7) based on the completelycontrollable pair (119860 119861) one gets

100381610038161003816100381610038161003816120582119894[119868 + 119861

minus1

119861119879

1198751]119879

1198601100381610038161003816100381610038161003816lt 1 119894 = 1 2 119899 (23)

Note (3) one gets100381610038161003816100381610038161003816120582119894[119868 + 119861

minus1

119861119879

1198751]119879

1198601100381610038161003816100381610038161003816

10038161003816100381610038161003816120582119895(119866)

10038161003816100381610038161003816lt 1

119894 = 1 2 119899 119895 = 1 2 119902

(24)

Then 1198752is existent and the unique solution of Stein matrix

equation (16) Therefore the proposed approximate con-troller is existent and unique

According to [29] (6) and (14) one gets

119909 (119896) le 119909 (119896) +

119873

sum

119894=1

[ℎ119894max1le119904leℎ119894

10038171003817100381710038171003817119860119896minus119895minus1

10038171003817100381710038171003817

times10038171003817100381710038171003817119861minus1

[119861119879

119860minus119879

1(1198751minus 1198761) minus 119862119879

1]10038171003817100381710038171003817

times 119909 (119896 minus 119904) max1le119904leℎ119894

]

+

119873

sum

119894=1

[ℎ119894max1le119904leℎ119894

10038171003817100381710038171003817119860119896minus119895minus1

10038171003817100381710038171003817119908 (119896 minus 119904)

times10038171003817100381710038171003817119861minus1

[119861119879

119860minus119879

1(1198752+ 1198762) minus 119862119879

2]10038171003817100381710038171003817]

(25)

Because 119909(119896) and 119908(119896) are bounded 119909(119896) is boundedThen the proposed control law (14) is a stabilizing control lawfor discrete-time system (1) with multiple delayed inputsTheproof is completed

Table 1 Parameters of installed AMD device

Name of parameter Variable Value UnitMass 119898

211855 kg

Natural frequency 1205962

233 radsStructural damping ratio 120585

2932 Percent

k2

k1

um2

m1

c2

c1

x2

x1(k)

Figure 1 The sketch of offshore platform with AMD device

5 Simulation

In order to demonstrate the effectiveness and feasibility ofthe proposedAODRC numerous simulations are undertakenin this section In case 1 the vibration control problemfor offshore steel jacket platforms with single delayed inputis simulated by using the proposed AODRC In case 2the disturbance rejection problem for pure mathematicalexample with two delayed inputs is considered

51 Vibration Control for Offshore Jacket Platforms with SingleDelayed Input The jacket type platforms play an importrole in the oil exploration and drilling operations Howeverthe jacket type platforms subject to external disturbancespersistently such as wave force wind and earthquake Inorder to ensure the safety and production efficiency thevibration caused by external disturbances could be rejectedeffectively especially for wave force In this subsection thevibration control problem for offshore jacket platforms withsingle input delay is solved by the proposed control law

Consider the jacket platform located in Bohai Bay thesketch of offshore platform with an AMD device is shown inFigure 1 The structural parameters of offshore platform andinstalled AMD device on the deck of the jacket type platformare listed in Tables 1 and 2 [30] respectively

Based on the JONSWAP spectrum the wave can bedescribed as

119878120578(120596) = [

51198672

119904

161205960

](1205960

120596)

5

exp [minus125(1205960

120596)

minus4

] 120574120573

(26)

6 Mathematical Problems in Engineering

Table 2 Parameters of offshore platform

Name of parameter Variable Value UnitEquivalent characteristic diameter of legs 119863 17 mFirst modal mass 119898

12371100 kg

Natural frequency 1205961

220 radsStructural damping ratio 120585

14 Percent

Shape function of first mode 120593(119904) 1 minus cos ( 120587119904

2119871) 0 le 119904 le 119871

where 119867119904

= 4m is the significant wave height 120596 isthe wave frequency 120574 = 33 is the peakedness param-eter 120596

0= 087 rads is the peak frequency 120573 =

exp[minus(120596 minus 1205960)2

(21205902

1205962

0)] in which 120590 is the shape parameter

120590 = 007 (120596 le 1205960) and 120590 = 009 (120596 gt 120596

0) Then based

on [30] the wave force can be formulated as an exosystemdescribed by (2) with the water depth 119889 = 132m the dragcoefficient119862

119889= 12 and the inertial coefficient119862

119898= 20The

external disturbances from irregular wave force are displayedin Figure 2

By setting the sampling period 119879 = 01 s 119876 =

diag (100 0 100 0) and 119877 = 1 the discrete-time modelof the offshore structure could be established with the sameexpression of (1) based on the previous investigations on thesubject which is described as

119860 =

[[[

[

09790 00002 00989 00000

00205 09789 00047 00952

minus04182 00041 09701 00010

03954 minus04152 00996 08993

]]]

]

119861 =

[[[

[

minus00006

00621

minus00120

12205

]]]

]

119863 =

[[[

[

00006

00000

00126

00006

]]]

]

119862 = [1 0 0 0

0 0 1 0]

(27)

According to the previous investigations on the subject therange of delayed input is in the interval (0 1) s Thereforeto demonstrate the effectiveness of the proposed AODRCthe comparison results between the open-loop system thefeedback control and the AODRC with single delayed inputℎ = 6 are compared in which feedback control law is justconsidered the feedback item in AODRC Then the curvesof displacement of offshore structure velocity of offshorestructure and control law are presented in Figures 3 4 and5 respectively

In order to show the comparison results among the sim-ulated case clearly the values of the quadratic performanceindex with different delayed input are compared in Table 3Note that the percentage number given in the parenthesesindicates the reduced amount of the closed-loop response

0 10 20 30 40 50 60 70 80 90 100

0

2

4

6

Wav

e for

ce(k)

(106

N)

minus2

minus4

minus6

kT (s)

Figure 2 The external disturbances of wave force

0 10 20 30 40 50 60 70 80 90 100

0

01

02

03

04

Disp

lace

men

t of o

ffsho

re st

ruct

ure (

m)

minus01

minus02

minus03

minus04

Open-loopFeedback controlAODRC

kT (s)

Figure 3 Curves of the displacement of offshore structure

relative to the open-loop case For instance the value ofperformance index is reduced by 1840when delayed inputℎ = 2 less than that of open-loop one using feedback controlwhile the value of performance index is reduced by 6723less than that of open-loop one by using proposed AODRC

Mathematical Problems in Engineering 7

Table 3 Comparison of values of performance index with different delayed input

ℎ = 2 ℎ = 4 ℎ = 6 ℎ = 8

Open-loop 50953 50953 50953 50953Feedback Control 41578 (1840) 35232 (3085) 30731 (3969) 31232 (3870)AODRC 16695 (6723) 16875 (6688) 17076 (6649) 17366 (6592)

0 10 20 30 40 50 60 70 80 90 100

02

0

04

06

Velo

city

of o

ffsho

re st

ruct

ure (

ms

)

Open-loopFeedback controlAODRC

minus02

minus04

minus06

minus08

kT (s)

Figure 4 Curves of the velocity of offshore structure

0 10 20 30 40 50 60 70 80 90 100

0

02

04

06

08

minus02

minus04

minus06

minus08

Con

trol l

awu(k)

(104

N)

kT (s)

Feedback controlAODRC

Figure 5 Curves of the control law

In order to present the control results more intuitive themaximum displacement the maximum acceleration of off-shore structure and the maximum control law with differentinput delay are shown in Table 4 From Table 4 it can beseen clearly that the maximum displacement the maximumacceleration of offshore structure and the values ofmaximumcontrol law with different input delay are close The resultsshow that the proposed control law could effectively eliminate

the influences fromdifferent input delay andwave forces withhigh value

It can be seen from Figures 3ndash5 and Tables 3-4 thatthe disturbances of wave force for offshore structure isreduced significantly by using the proposed AODRC Withthe increase of delayed input the proposed AODRC is stillable to effectively control the displacement and velocity ofoffshore structure Whatrsquos more the quadratic performanceindex is reduced significantly by using the proposed AODRCcompared by the feedback control and open-loop onesTherefore we can conclude that the proposed AODRC isefficient to reject the disturbances from wave force foroffshore structure as well as being with lower consumption

52 Disturbance Rejection for Pure Mathematical Examplewith Two Delayed Inputs In order to verify the elasticity ofthe proposed AODRC the discrete-time system with delayedinputs ℎ

1and ℎ2under sinusoidal disturbances is considered

The matrices of (1) are chosen as

119860 = [1 04

minus02 09]

119863 = [0

1]

1198611= [

02

05]

1198612= [

01

015]

119862 = [1 0]

119909 (0) = [2 0]119879

(28)

The external disturbance V(119905) is chosen as sinusoidal distur-bance the matrices of (2) are given by

119866 = [04radic5 04

minus05 04radic5]

119864 = [02 0]

V (0) = [1 0]119879

(29)

The curves of sinusoidal disturbances with (29) are shownin Figure 6 Also the quadratic performance index is chosenas (5) with 119876 = 1 119877 = 2

Then the responses of system states 1199091and 119909

2for differ-

ent cases (open-loop feedback control only and proposedAODRC) with delayed inputs ℎ

1= 1 and ℎ

2= 4 are

8 Mathematical Problems in Engineering

Table 4 The maximum displacement the maximum accelerationof offshore structure and the maximum control law

ℎ = 2 ℎ = 4 ℎ = 6 ℎ = 8

Maximumdisplacement (m) 0126 0130 0146 0150

Maximumacceleration (m2s) 0171 0175 0178 0181

Maximum controlinput (104 N) 0520 0523 0582 0588

0 50 100 150 200

0

005

01

015

02

Sinu

soid

al d

istur

banc

e(k)

minus005

minus01

minus015

minus02

kT (s)

Figure 6 Curves of the sinusoidal disturbance V(119896)

simulated and shown in Figures 7 and 8 The curves of theproposed AODRC for different cases (the proposed AODRCand feedback control only) are shown in Figure 9 The valuesof quadratic performance index (5) are shown in Table 3 withtwo delayed inputs (ℎ

1= 1 ℎ

2= 2 ℎ

1= 1 ℎ

2= 4 ℎ

1= 2

ℎ2= 4) Figures 7 and 8 and Table 5 show that the proposed

AODRC is able to effectively ensure the system states in lowervalues and reject the disturbances from external sinusoidalinput

From Figures 1ndash9 and Tables 1ndash5 the simulation resultsshow that the proposed AODRC for the offshore platformand pure mathematical example are effective to attenuate thevibration amplitudes of the system state and the quadraticperformance index of the controlled system under AODRC issmaller than those under the open-loop and feedback controlonly Moreover the proposed AODRC is more scalable fordifferent cases with multiple delayed inputs

6 Conclusion

This paper has presented an interesting approach to solvethe disturbance rejection problem for discrete-time systemswith multiple delayed inputs under persistent external dis-turbances A discrete-time system with multiple input delaysis transformed into a non-delayed system in form and thequadratic performance index has been transformed into arelevant format without the explicit appearance of time delayThen the AODRC is deployed to eliminate the negativeeffects from multiple delayed inputs and persistent external

0 50 100 150 200

0

1

2

3

minus1

minus2

minus3

Curv

es o

f sys

tem

stat

ex1(k

)

Open-loopFeedback controlAODRC

kT (s)

Figure 7 Curves of system state 1199091(119896)

0 50 100 150 200

0

05

1

15

2

25

minus05

minus1

minus15

minus2

Curv

es o

f sys

tem

stat

ex2(k

)

Open-loopFeedback controlAODRC

kT (s)

Figure 8 Curves of system state 1199092(119896)

Table 5 Comparison of values of performance index with twodelayed inputs

ℎ1= 1 ℎ

2= 2 ℎ

1= 1 ℎ

2= 4 ℎ

1= 2 ℎ

2= 4

Open-loop 03422 03422 03422FeedbackControl 00836 (7557) 01055 (6917) 01035 (6975)

AODRC 00372 (8913) 00506 (8521) 00901 (6649)

disturbances On the other hand the disturbance rejectionproblem for jacket-type offshore structure is illustrated tovalidate the feasibility and effectiveness of the proposedapproach

Mathematical Problems in Engineering 9

0 50 100 150 200

01

02

03

04

05

0

Feedback controlAODRC

minus01

minus02

minus03

minus04

minus05

Con

trol l

awu(k)

kT (s)

Figure 9 Curves of the control law

Conflict of Interests

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

Acknowledgments

This work was supported by the Program for Scien-tific Research Innovation Team in Colleges and Universi-ties of Shandong Province Science Foundation of Chinaunder Grants 61201428 61203105 and 61302128 the NaturalScience Foundation of Shandong Province under GrantsZR2011FZ003 ZR2012FQ016 and ZR2011FL022 and theDoctoral Foundation of Shandong Province under GrantBS2014DX015

References

[1] E Altman and T Basar ldquoOptimal rate control for high speedtelecommunication networksrdquo in Proceedings of the 34th IEEEConference on Decision and Control vol 2 pp 1389ndash1394December 1995

[2] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[3] X Yu C Wu F Liu and L Wu ldquoSliding mode control ofdiscrete-time switched systems with time-delayrdquo Journal of theFranklin Institute vol 350 no 1 pp 19ndash33 2013

[4] S Janardhanan and B Bandyopadhyay ldquoOutput feedbackdiscrete-time sliding mode control for time delay systemsrdquo IEEProceedingsmdashControl Theory and Applications vol 153 no 4pp 387ndash396 2006

[5] M Cedomir P-D Branislava and V Boban ldquoDiscrete-timevelocity servo system design using sliding mode controlapproach with disturbance compensationrdquo IEEE Transactionson Industrial Informatics vol 9 no 2 pp 920ndash927 2013

[6] P Albertos and P Garcıa ldquoPredictor-observer-based control ofsystems with multiple inputoutput delaysrdquo Journal of ProcessControl vol 22 no 7 pp 1350ndash1357 2012

[7] E Normey-Rico and E F Camacho ldquoDead-time compensatorsa surveyrdquo Control Engineering Practice vol 16 no 4 pp 407ndash428 2008

[8] B Zhou Z-Y Li and Z Lin ldquoObserver based output feedbackcontrol of linear systems with input and output delaysrdquo Auto-matica vol 49 no 7 pp 2039ndash2052 2013

[9] B Zhou ldquoObserver-based output feedback control of discrete-time linear systems with input and output delaysrdquo InternationalJournal of Control 2014 in press

[10] H Zhang G Duan and L Xie ldquoLinear quadratic regulationfor linear time-varying systems with multiple input delays PartI discrete time caserdquo in Proceedings of the 5th InternationalConference on Control and Automation Budapest Hungary2005

[11] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[12] B-L Zhang L Ma and Q-L Han ldquoSliding mode Hinfin

controlfor offshore steel jacket platforms subject to nonlinear self-excited wave force and external disturbanceurbancerdquoNonlinearAnalysis Real World Applications vol 14 no 1 pp 163ndash1782013

[13] B Zhang Q Han X Zhang and X Yu ldquoIntegral sliding modecontrol for offshore steel jacket platformsrdquo Journal of Sound andVibration vol 331 no 14 pp 3271ndash3285 2012

[14] S R Hall and N MWereley ldquoPerformance of higher harmoniccontrol algorithms for helicopter vibration reductionrdquo Journalof Guidance Control and Dynamics vol 16 no 4 pp 793ndash7971993

[15] S Han G Tang Y Chen and X Yang ldquoOptimal vibrationcontrol for vehicle active suspension discrete-time systems withactuator time delayrdquo Asian Journal of Control vol 15 no 6 pp1579ndash1588 2013

[16] H Gao J Lam andCWang ldquoMulti-objective control of vehicleactive suspension systems via load-dependent controllersrdquoJournal of Sound and Vibration vol 290 no 3ndash5 pp 654ndash6752006

[17] L Mirkin ldquoOn the approximation of distributed-delay controllawsrdquo Systems ampControl Letters vol 51 no 5 pp 331ndash342 2004

[18] Y Huang W C Xue and C Z Zhao ldquoActive disturbance rejec-tion control methodology and theoretical analysisrdquo Journal ofSystems Science and Mathematical Sciences vol 31 no 9 pp1111ndash1129 2011

[19] Y Shi J Huang and B Yu ldquoRobust tracking control ofnetworked control systems application to a networked DCmotorrdquo IEEE Transactions on Industrial Electronics vol 60 no12 pp 5864ndash5874 2013

[20] X Guo and M Bodson ldquoAnalysis and implementation of anadaptive algorithm for the rejection of multiple sinusoidal dis-turbancesrdquo IEEE Transactions on Control Systems Technologyvol 17 no 1 pp 40ndash50 2009

[21] H Zhao Q Chen and S Xu ldquo119867infin

guaranteed cost controlfor uncertain Markovian jump systems with mode-dependentdistributed delays and input delaysrdquo Journal of the FranklinInstitute vol 346 no 10 pp 945ndash957 2009

[22] I D Landau M Alma J J Martinez and G Buche ldquoAdaptivesuppression ofmultiple time-varying unknownvibrations usingan inertial actuatorrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 6 pp 1327ndash1338 2011

[23] M Shamsuzzoha and M Lee ldquoEnhanced disturbance rejectionfor open-loop unstable process with time delayrdquo ISA Transac-tions vol 48 no 2 pp 237ndash244 2009

10 Mathematical Problems in Engineering

[24] L Guo and S Cao ldquoAnti-disturbance control theory for systemswith multiple disturbances a surveyrdquo ISA Transactions vol 53no 4 pp 846ndash849 2014

[25] X Li T W S Chow and J K L Ho ldquoIterative learning controlfor a class of nonlinear discrete-time systems with multipleinput delaysrdquo International Journal of Systems Science vol 39no 4 pp 361ndash369 2008

[26] J Zhang H Zhang Y Luo and T Feng ldquoModel-free optimalcontrol design for a class of linear discrete-time systems withmultiple delays using adaptive dynamic programmingrdquo Neuro-computing vol 135 pp 163ndash170 2014

[27] G-Y Tang and H-H Wang ldquoSuboptimal control for discretelinear systems with time-delay a successive approximationapproachrdquo Acta Automatica Sinica vol 31 no 3 pp 419ndash4262005

[28] H H Wang and G Y Tang ldquoObserver-based optimal outputtracking for discrete-time Systems with multiple state andinput delaysrdquo International Journal of Control Automation andSystems vol 7 no 1 pp 57ndash66 2009

[29] H T Banks M Q Jacobs and M R Latina ldquoThe synthesisof optimal controls for linear time-optimal problems withretarded controlsrdquo Journal of OptimizationTheory and Applica-tions vol 8 pp 319ndash366 1971

[30] H Ma G-Y Tang and Y-D Zhao ldquoFeedforward and feedbackoptimal control for offshore structures subjected to irregularwave forcesrdquo Ocean Engineering vol 33 no 8-9 pp 1105ndash11172006

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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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 3

of appropriate dimensions with full rank It is assumed thatthe pair (119864 119866) is completely observable Also the eigenvaluesof the 119866 satisfy

1003816100381610038161003816120582119894 (119866)1003816100381610038161003816 le 1 119894 = 1 2 119902 (3)

Then exosystem (2) of the persistent disturbances is stablebut may not be asymptotically stable It should be noted thatthe expression of exosystem (2) could describe the variousforms of disturbances with known or unknown dynamiccharacteristics such as the sinusoidal disturbances periodicdisturbances step disturbances and random signals

The performance index could be depending on thedynamic characteristics of persistent disturbances Whenthe persistent disturbances are asymptotically stable thetraditional quadratic performance index could be chosen as

119869 =1

2

infin

sum

119896=0

[119910119879

(119896) 119876119910 (119896) + 119906119879

(119896) 119877119906 (119896)] (4)

On the other hand the persistent disturbances are stable butmay not be asymptotically stable the effect from persistentdisturbance vector V(119896) the steady state of the state vector119909(119896) and the control input 119906(119896) will not converge to zerosynchronouslyTherefore the following infinite-time averageperformance index is chosen

119869 = lim119873rarrinfin

1

119873

119873

sum

119896=0

[119910119879

(119896) 119876119910 (119896) + 119906119879

(119896) 119877119906 (119896)] (5)

where 119876 isin 119877119899times119899 is positive semidefinite and 119877 isin 119877

119898times119898 ispositive definite

The aim of the optimal disturbance rejection is to findan optimal control law 119906

lowast

(119896) for the discrete-time systemsdescribed by (1) and (2) with respect to the performanceindex (4) or (5) that make the performance index (4) or (5)obtain the minimum value

3 Reformulation of DisturbanceRejection Problem

For discrete-time delay systems with quadratic performanceindex the optimal control problem is difficult to be solvedanalytically caused by the TPBV problem with both time-delay and time-advance terms In this section a vector trans-formation is introduced to deal with the multiple delayedinputs and the performance index is reformulated as theequivalent ones without the explicit appearance of time delay

31 Delay-Free Transformation of Control System By propos-ing the following variable transformation

119909 (119896) = 119909 (119896) +

119873

sum

119894=1

119896minus1

sum

119895=119896minusℎ119894

119860119896minusℎ119894minus119895minus1119861

119894119906 (119895) (6)

system (1) with multiple delayed input can be transformedinto the following form

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119863V (119896)

119910 (119896) = 119862(119909 (119896) minus

119873

sum

119894=1

119896minus1

sum

119895=119896minusℎ119894

119860119896minusℎ119894minus119895minus1119906 (119895))

(7)

where 119909(119896) isin 119877119899 is the state vector of the delay-free system

(7) and 119861 = sum119873

119894=1119860minusℎ119894119861119894 Note that the transformed system

(7) is in the form of a delay-free system and the pair (119860 119861)is completely controllable if and only if (119860 119861

1) is completely

controllable [29]

32 Simplification of Performance Index Through the vectortransformation (6) introduced the original system withmultiple delayed inputs (1) could be transformed into system(7) without the explicit appearance of time delay In order toevaluate the effectiveness of the control law more exactly it isnecessary to transform the performance index (4) or (5) intothe equivalent form for the transformed system (7)

By using (6) and (7) the quadratic performance index (4)could be transformed into the following form

119869 =

infin

sum

119896=0

119909119879

(119896) 119862119879

119876119862119909 (119896) + 119906119879

(119896) 119877119906 (119896)

minus 2119880119879

1119862119879

119876119862119909 (119896) + 119880119879

1119862119879

1198761198621198801

(8)

where 1198801= sum119873

119894=1sum119896minus1

119895=119896minusℎ119894

119860119896minusℎ119894minus119895minus1119906(119895)

Based on (2) (6) (7) and (9) one getsinfin

sum

119896=0

119909119879

(119896) 119862119879

1198761198621198801

=

infin

sum

119896=0

119909119879

(119896) 1198621119906 (119896) + 119908

119879

(119896) 1198622119906 (119896)

+

119873

sum

119905=1

ℎ119905

sum

119894=1

ℎ119905

sum

119895=1

((119862119860minus119895

119861119894119906(119896 + 119894 minus 1))

119879

times 119876119862119860119894minus119895minus1

119861119894) 119906 (119896)

infin

sum

119896=0

119880119879

1119862119879

1198761198621198801

=

infin

sum

119896=0

119906119879

(119896)

119873

sum

119894=1

ℎ119894

sum

119895=1

((119862119860minus119895

119861119894)119879

119876119862119860minus119895

119861119894) 119906 (119896)

+ 2

119873

sum

119905=1

ℎ119905

sum

119894=1

ℎ119905

sum

119895=1

((119862119860minus119895

119861119894119906 (119896 + 119894 minus 1))

119879

times 119876119862119860119894minus119895minus1

119861119894) 119906 (119896)

(9)

4 Mathematical Problems in Engineering

where

1198621=

119873

sum

119894=1

((

ℎ119894

sum

119895=1

(119862119860119895

)119879

119876119860119862119860119895minus1minusℎ119894119861

119894))

1198622=

119873

sum

119894=1

ℎ119894

sum

119895=1

ℎ119894minus119895

sum

119905=0

((119862119860119905

119863119864119866119895minus1

)119879

119876119862119860119905+119895minusℎ119894minus1119861

119894)

(10)

Then the quadratic performance index (4) could be reformedas the following form

119869 =1

2

infin

sum

119896=0

119909119879

(119896) 119862119879

119876119862119909 (119896) + 119906119879

(119896) 119906 (119896)

minus2119909119879

(119896) 1198621119906 (119896) minus 2119908

119879

(119896) 1198622119906 (119896)

(11)

where = 119877 + sum119873

119894=1sumℎ119894

119895=1((119862119860minus119895

119861119894)119879

119876119862119860minus119895

119861119894)

Actually the above transformations are suitable for theinfinite-time average quadratic performance index (5) Thereformed performance index could be described as

119869 = lim119873rarrinfin

1

119873

119873

sum

119896=0

119909119879

(119896) 119862119879

119876119862119909 (119896) + 119906119879

(119896) 119906 (119896)

minus2119909119879

(119896) 1198621119906 (119896) minus 2119908

119879

(119896) 1198622119906 (119896)

(12)

Then the original optimal disturbance rejection is refor-mulated as an equivalent optimal regulation problem thatdesigns an approximate control law 119906

lowast

(119896) to make thequadratic performance index (11) or (12) obtain theminimumvalue subject to the discrete-time system (2) and (7) such thatthe negative effects from persistent disturbances andmultipledelayed inputs are eliminated

4 Approximate Design of OptimalDisturbance Rejection Controller

In this section the proposed AODRC is presented in detailand its existence and uniqueness are proved In order to statethe proposed control law clearly the following matrices aredefined

1198601= 119860 + 119861

minus1

119862119879

1 119860

2= 119863119864 + 119861

minus1

119862119879

2

1198761= 119862119879

119876119862 minus 1198621minus1

119862119879

1 119876

2= 1198621minus1

119862119879

2

(13)

Theorem 2 Consider the optimal disturbance rejection con-trol problem for the discrete-time systems given by (1) and(2) with respect to the quadratic performance indexes (4) or

(5) The approximate optimal disturbance rejection controlleruniquely exists and can be formulated as

119906lowast

(119896) = minus1

[119862119879

1minus 119861119879

119860minus119879

1(1198751minus 1198761)]

times [

[

119909 (119896) +

119873

sum

119894=1

119896minus1

sum

119895=119896minusℎ119894

119860119896minusℎ119894minus119895minus1119861119906 (119895)]

]

+ (119862119879

2minus 119861119879

119860minus119879

1(1198752+ 1198762))119908 (119896)

(14)

where 1198751is the unique positive definite solution of the Riccati

matrix equation

1198751= 1198761+ 119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198601

(15)

and 1198752is the unique solution of the Stein matrix equation

1198752+ 1198762= 119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198602

+ 119860119879

1[119868 minus 119860

119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

119861minus1

119861119879

] 1198752119866

(16)

Proof Define the Hamiltonian function for the optimaldisturbance rejection problem as

119867(∙) =1

2119909119879

(119896) 119862119879

119876119862119909 (119896) + 119906119879

(119896) 119906 (119896)

minus 2119909119879

(119896) 1198621119906 (119896) minus 2119908

119879

(119896) 1198622119906 (119896)

+ 120582119879

(119896 + 1) [119860119909 (119896) + 119861119906 (119896) + 119863V (119896)]

(17)

Applying the maximum principle condition 120597119867120597119906 = 0 tothis specific Hamiltonian function (17) yields

119906 (119896) = minus1

[119861119879

120582 (119896 + 1) minus 119862119879

1119909 (119896) minus 119862

119879

2119908 (119896)] (18)

where 120582(119896) is the solution of the following TPBV problem

119909 (119896 + 1) = 1198601119909 (119896) + 119860

2119908 (119896) minus 119861

minus1

119861119879

120582 (119896 + 1)

1198761119909 (119896) minus 119876

2119908 (119896) + 119860

119879

1120582 (119896 + 1) = 120582 (119896)

119909 (0) = 119909 (0)

120582 (infin) = 0

(19)

To solve TPBV problem (19) let

120582 (119896) = 1198751119909 (119896) + 119875

2119908 (119896) (20)

Rearranging (19) one gets

120582 (119896 + 1) = 119860minus119879

1[(1198751minus 1198761) 119909 (119896) + (119875

2+ 1198762) 119908 (119896)]

119909 (119896 + 1) = (119868 + 119861minus1

119861119879

1198751)minus1

times 1198601119909 (119896) + (119860

2minus 119861minus1

119861119879

1198752119866)119908 (119896)

(21)

Mathematical Problems in Engineering 5

Substituting the first formula of (21) into (18) the approxi-mate optimal control law (14) is obtained Note that

120582 (119896) = 1198761119909 (119896) minus 119876

2119908 (119896) + 119860

119879

1120582 (119896 + 1)

= [1198761+ 119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198601] 119909 (119896)

+ 119860119879

1[119868 minus 119860

119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

119861minus1

119861119879

] 1198752119866

+119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198602minus 1198762119908 (119896)

(22)

By comparing the parameters of (20) and (22) the Riccatiequation (15) and Stein equation (16) can be obtained

In the following the existence and uniqueness of theproposed controller will be proved Because 119875

1and 119875

2are

unknown in the parameters of proposed control law (14)the existence and uniqueness of (14) are equivalent to thatof 1198751and 119875

2 For Riccati matrix equation (15) because of

the completely controllable and observable pairs (119860 1198611) and

(119860 119862) the matrix 1198751is existent and the unique solution of

Riccati matrix equation (15) It should be noted that the firstformula of (19) is equivalent to (7) based on the completelycontrollable pair (119860 119861) one gets

100381610038161003816100381610038161003816120582119894[119868 + 119861

minus1

119861119879

1198751]119879

1198601100381610038161003816100381610038161003816lt 1 119894 = 1 2 119899 (23)

Note (3) one gets100381610038161003816100381610038161003816120582119894[119868 + 119861

minus1

119861119879

1198751]119879

1198601100381610038161003816100381610038161003816

10038161003816100381610038161003816120582119895(119866)

10038161003816100381610038161003816lt 1

119894 = 1 2 119899 119895 = 1 2 119902

(24)

Then 1198752is existent and the unique solution of Stein matrix

equation (16) Therefore the proposed approximate con-troller is existent and unique

According to [29] (6) and (14) one gets

119909 (119896) le 119909 (119896) +

119873

sum

119894=1

[ℎ119894max1le119904leℎ119894

10038171003817100381710038171003817119860119896minus119895minus1

10038171003817100381710038171003817

times10038171003817100381710038171003817119861minus1

[119861119879

119860minus119879

1(1198751minus 1198761) minus 119862119879

1]10038171003817100381710038171003817

times 119909 (119896 minus 119904) max1le119904leℎ119894

]

+

119873

sum

119894=1

[ℎ119894max1le119904leℎ119894

10038171003817100381710038171003817119860119896minus119895minus1

10038171003817100381710038171003817119908 (119896 minus 119904)

times10038171003817100381710038171003817119861minus1

[119861119879

119860minus119879

1(1198752+ 1198762) minus 119862119879

2]10038171003817100381710038171003817]

(25)

Because 119909(119896) and 119908(119896) are bounded 119909(119896) is boundedThen the proposed control law (14) is a stabilizing control lawfor discrete-time system (1) with multiple delayed inputsTheproof is completed

Table 1 Parameters of installed AMD device

Name of parameter Variable Value UnitMass 119898

211855 kg

Natural frequency 1205962

233 radsStructural damping ratio 120585

2932 Percent

k2

k1

um2

m1

c2

c1

x2

x1(k)

Figure 1 The sketch of offshore platform with AMD device

5 Simulation

In order to demonstrate the effectiveness and feasibility ofthe proposedAODRC numerous simulations are undertakenin this section In case 1 the vibration control problemfor offshore steel jacket platforms with single delayed inputis simulated by using the proposed AODRC In case 2the disturbance rejection problem for pure mathematicalexample with two delayed inputs is considered

51 Vibration Control for Offshore Jacket Platforms with SingleDelayed Input The jacket type platforms play an importrole in the oil exploration and drilling operations Howeverthe jacket type platforms subject to external disturbancespersistently such as wave force wind and earthquake Inorder to ensure the safety and production efficiency thevibration caused by external disturbances could be rejectedeffectively especially for wave force In this subsection thevibration control problem for offshore jacket platforms withsingle input delay is solved by the proposed control law

Consider the jacket platform located in Bohai Bay thesketch of offshore platform with an AMD device is shown inFigure 1 The structural parameters of offshore platform andinstalled AMD device on the deck of the jacket type platformare listed in Tables 1 and 2 [30] respectively

Based on the JONSWAP spectrum the wave can bedescribed as

119878120578(120596) = [

51198672

119904

161205960

](1205960

120596)

5

exp [minus125(1205960

120596)

minus4

] 120574120573

(26)

6 Mathematical Problems in Engineering

Table 2 Parameters of offshore platform

Name of parameter Variable Value UnitEquivalent characteristic diameter of legs 119863 17 mFirst modal mass 119898

12371100 kg

Natural frequency 1205961

220 radsStructural damping ratio 120585

14 Percent

Shape function of first mode 120593(119904) 1 minus cos ( 120587119904

2119871) 0 le 119904 le 119871

where 119867119904

= 4m is the significant wave height 120596 isthe wave frequency 120574 = 33 is the peakedness param-eter 120596

0= 087 rads is the peak frequency 120573 =

exp[minus(120596 minus 1205960)2

(21205902

1205962

0)] in which 120590 is the shape parameter

120590 = 007 (120596 le 1205960) and 120590 = 009 (120596 gt 120596

0) Then based

on [30] the wave force can be formulated as an exosystemdescribed by (2) with the water depth 119889 = 132m the dragcoefficient119862

119889= 12 and the inertial coefficient119862

119898= 20The

external disturbances from irregular wave force are displayedin Figure 2

By setting the sampling period 119879 = 01 s 119876 =

diag (100 0 100 0) and 119877 = 1 the discrete-time modelof the offshore structure could be established with the sameexpression of (1) based on the previous investigations on thesubject which is described as

119860 =

[[[

[

09790 00002 00989 00000

00205 09789 00047 00952

minus04182 00041 09701 00010

03954 minus04152 00996 08993

]]]

]

119861 =

[[[

[

minus00006

00621

minus00120

12205

]]]

]

119863 =

[[[

[

00006

00000

00126

00006

]]]

]

119862 = [1 0 0 0

0 0 1 0]

(27)

According to the previous investigations on the subject therange of delayed input is in the interval (0 1) s Thereforeto demonstrate the effectiveness of the proposed AODRCthe comparison results between the open-loop system thefeedback control and the AODRC with single delayed inputℎ = 6 are compared in which feedback control law is justconsidered the feedback item in AODRC Then the curvesof displacement of offshore structure velocity of offshorestructure and control law are presented in Figures 3 4 and5 respectively

In order to show the comparison results among the sim-ulated case clearly the values of the quadratic performanceindex with different delayed input are compared in Table 3Note that the percentage number given in the parenthesesindicates the reduced amount of the closed-loop response

0 10 20 30 40 50 60 70 80 90 100

0

2

4

6

Wav

e for

ce(k)

(106

N)

minus2

minus4

minus6

kT (s)

Figure 2 The external disturbances of wave force

0 10 20 30 40 50 60 70 80 90 100

0

01

02

03

04

Disp

lace

men

t of o

ffsho

re st

ruct

ure (

m)

minus01

minus02

minus03

minus04

Open-loopFeedback controlAODRC

kT (s)

Figure 3 Curves of the displacement of offshore structure

relative to the open-loop case For instance the value ofperformance index is reduced by 1840when delayed inputℎ = 2 less than that of open-loop one using feedback controlwhile the value of performance index is reduced by 6723less than that of open-loop one by using proposed AODRC

Mathematical Problems in Engineering 7

Table 3 Comparison of values of performance index with different delayed input

ℎ = 2 ℎ = 4 ℎ = 6 ℎ = 8

Open-loop 50953 50953 50953 50953Feedback Control 41578 (1840) 35232 (3085) 30731 (3969) 31232 (3870)AODRC 16695 (6723) 16875 (6688) 17076 (6649) 17366 (6592)

0 10 20 30 40 50 60 70 80 90 100

02

0

04

06

Velo

city

of o

ffsho

re st

ruct

ure (

ms

)

Open-loopFeedback controlAODRC

minus02

minus04

minus06

minus08

kT (s)

Figure 4 Curves of the velocity of offshore structure

0 10 20 30 40 50 60 70 80 90 100

0

02

04

06

08

minus02

minus04

minus06

minus08

Con

trol l

awu(k)

(104

N)

kT (s)

Feedback controlAODRC

Figure 5 Curves of the control law

In order to present the control results more intuitive themaximum displacement the maximum acceleration of off-shore structure and the maximum control law with differentinput delay are shown in Table 4 From Table 4 it can beseen clearly that the maximum displacement the maximumacceleration of offshore structure and the values ofmaximumcontrol law with different input delay are close The resultsshow that the proposed control law could effectively eliminate

the influences fromdifferent input delay andwave forces withhigh value

It can be seen from Figures 3ndash5 and Tables 3-4 thatthe disturbances of wave force for offshore structure isreduced significantly by using the proposed AODRC Withthe increase of delayed input the proposed AODRC is stillable to effectively control the displacement and velocity ofoffshore structure Whatrsquos more the quadratic performanceindex is reduced significantly by using the proposed AODRCcompared by the feedback control and open-loop onesTherefore we can conclude that the proposed AODRC isefficient to reject the disturbances from wave force foroffshore structure as well as being with lower consumption

52 Disturbance Rejection for Pure Mathematical Examplewith Two Delayed Inputs In order to verify the elasticity ofthe proposed AODRC the discrete-time system with delayedinputs ℎ

1and ℎ2under sinusoidal disturbances is considered

The matrices of (1) are chosen as

119860 = [1 04

minus02 09]

119863 = [0

1]

1198611= [

02

05]

1198612= [

01

015]

119862 = [1 0]

119909 (0) = [2 0]119879

(28)

The external disturbance V(119905) is chosen as sinusoidal distur-bance the matrices of (2) are given by

119866 = [04radic5 04

minus05 04radic5]

119864 = [02 0]

V (0) = [1 0]119879

(29)

The curves of sinusoidal disturbances with (29) are shownin Figure 6 Also the quadratic performance index is chosenas (5) with 119876 = 1 119877 = 2

Then the responses of system states 1199091and 119909

2for differ-

ent cases (open-loop feedback control only and proposedAODRC) with delayed inputs ℎ

1= 1 and ℎ

2= 4 are

8 Mathematical Problems in Engineering

Table 4 The maximum displacement the maximum accelerationof offshore structure and the maximum control law

ℎ = 2 ℎ = 4 ℎ = 6 ℎ = 8

Maximumdisplacement (m) 0126 0130 0146 0150

Maximumacceleration (m2s) 0171 0175 0178 0181

Maximum controlinput (104 N) 0520 0523 0582 0588

0 50 100 150 200

0

005

01

015

02

Sinu

soid

al d

istur

banc

e(k)

minus005

minus01

minus015

minus02

kT (s)

Figure 6 Curves of the sinusoidal disturbance V(119896)

simulated and shown in Figures 7 and 8 The curves of theproposed AODRC for different cases (the proposed AODRCand feedback control only) are shown in Figure 9 The valuesof quadratic performance index (5) are shown in Table 3 withtwo delayed inputs (ℎ

1= 1 ℎ

2= 2 ℎ

1= 1 ℎ

2= 4 ℎ

1= 2

ℎ2= 4) Figures 7 and 8 and Table 5 show that the proposed

AODRC is able to effectively ensure the system states in lowervalues and reject the disturbances from external sinusoidalinput

From Figures 1ndash9 and Tables 1ndash5 the simulation resultsshow that the proposed AODRC for the offshore platformand pure mathematical example are effective to attenuate thevibration amplitudes of the system state and the quadraticperformance index of the controlled system under AODRC issmaller than those under the open-loop and feedback controlonly Moreover the proposed AODRC is more scalable fordifferent cases with multiple delayed inputs

6 Conclusion

This paper has presented an interesting approach to solvethe disturbance rejection problem for discrete-time systemswith multiple delayed inputs under persistent external dis-turbances A discrete-time system with multiple input delaysis transformed into a non-delayed system in form and thequadratic performance index has been transformed into arelevant format without the explicit appearance of time delayThen the AODRC is deployed to eliminate the negativeeffects from multiple delayed inputs and persistent external

0 50 100 150 200

0

1

2

3

minus1

minus2

minus3

Curv

es o

f sys

tem

stat

ex1(k

)

Open-loopFeedback controlAODRC

kT (s)

Figure 7 Curves of system state 1199091(119896)

0 50 100 150 200

0

05

1

15

2

25

minus05

minus1

minus15

minus2

Curv

es o

f sys

tem

stat

ex2(k

)

Open-loopFeedback controlAODRC

kT (s)

Figure 8 Curves of system state 1199092(119896)

Table 5 Comparison of values of performance index with twodelayed inputs

ℎ1= 1 ℎ

2= 2 ℎ

1= 1 ℎ

2= 4 ℎ

1= 2 ℎ

2= 4

Open-loop 03422 03422 03422FeedbackControl 00836 (7557) 01055 (6917) 01035 (6975)

AODRC 00372 (8913) 00506 (8521) 00901 (6649)

disturbances On the other hand the disturbance rejectionproblem for jacket-type offshore structure is illustrated tovalidate the feasibility and effectiveness of the proposedapproach

Mathematical Problems in Engineering 9

0 50 100 150 200

01

02

03

04

05

0

Feedback controlAODRC

minus01

minus02

minus03

minus04

minus05

Con

trol l

awu(k)

kT (s)

Figure 9 Curves of the control law

Conflict of Interests

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

Acknowledgments

This work was supported by the Program for Scien-tific Research Innovation Team in Colleges and Universi-ties of Shandong Province Science Foundation of Chinaunder Grants 61201428 61203105 and 61302128 the NaturalScience Foundation of Shandong Province under GrantsZR2011FZ003 ZR2012FQ016 and ZR2011FL022 and theDoctoral Foundation of Shandong Province under GrantBS2014DX015

References

[1] E Altman and T Basar ldquoOptimal rate control for high speedtelecommunication networksrdquo in Proceedings of the 34th IEEEConference on Decision and Control vol 2 pp 1389ndash1394December 1995

[2] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[3] X Yu C Wu F Liu and L Wu ldquoSliding mode control ofdiscrete-time switched systems with time-delayrdquo Journal of theFranklin Institute vol 350 no 1 pp 19ndash33 2013

[4] S Janardhanan and B Bandyopadhyay ldquoOutput feedbackdiscrete-time sliding mode control for time delay systemsrdquo IEEProceedingsmdashControl Theory and Applications vol 153 no 4pp 387ndash396 2006

[5] M Cedomir P-D Branislava and V Boban ldquoDiscrete-timevelocity servo system design using sliding mode controlapproach with disturbance compensationrdquo IEEE Transactionson Industrial Informatics vol 9 no 2 pp 920ndash927 2013

[6] P Albertos and P Garcıa ldquoPredictor-observer-based control ofsystems with multiple inputoutput delaysrdquo Journal of ProcessControl vol 22 no 7 pp 1350ndash1357 2012

[7] E Normey-Rico and E F Camacho ldquoDead-time compensatorsa surveyrdquo Control Engineering Practice vol 16 no 4 pp 407ndash428 2008

[8] B Zhou Z-Y Li and Z Lin ldquoObserver based output feedbackcontrol of linear systems with input and output delaysrdquo Auto-matica vol 49 no 7 pp 2039ndash2052 2013

[9] B Zhou ldquoObserver-based output feedback control of discrete-time linear systems with input and output delaysrdquo InternationalJournal of Control 2014 in press

[10] H Zhang G Duan and L Xie ldquoLinear quadratic regulationfor linear time-varying systems with multiple input delays PartI discrete time caserdquo in Proceedings of the 5th InternationalConference on Control and Automation Budapest Hungary2005

[11] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[12] B-L Zhang L Ma and Q-L Han ldquoSliding mode Hinfin

controlfor offshore steel jacket platforms subject to nonlinear self-excited wave force and external disturbanceurbancerdquoNonlinearAnalysis Real World Applications vol 14 no 1 pp 163ndash1782013

[13] B Zhang Q Han X Zhang and X Yu ldquoIntegral sliding modecontrol for offshore steel jacket platformsrdquo Journal of Sound andVibration vol 331 no 14 pp 3271ndash3285 2012

[14] S R Hall and N MWereley ldquoPerformance of higher harmoniccontrol algorithms for helicopter vibration reductionrdquo Journalof Guidance Control and Dynamics vol 16 no 4 pp 793ndash7971993

[15] S Han G Tang Y Chen and X Yang ldquoOptimal vibrationcontrol for vehicle active suspension discrete-time systems withactuator time delayrdquo Asian Journal of Control vol 15 no 6 pp1579ndash1588 2013

[16] H Gao J Lam andCWang ldquoMulti-objective control of vehicleactive suspension systems via load-dependent controllersrdquoJournal of Sound and Vibration vol 290 no 3ndash5 pp 654ndash6752006

[17] L Mirkin ldquoOn the approximation of distributed-delay controllawsrdquo Systems ampControl Letters vol 51 no 5 pp 331ndash342 2004

[18] Y Huang W C Xue and C Z Zhao ldquoActive disturbance rejec-tion control methodology and theoretical analysisrdquo Journal ofSystems Science and Mathematical Sciences vol 31 no 9 pp1111ndash1129 2011

[19] Y Shi J Huang and B Yu ldquoRobust tracking control ofnetworked control systems application to a networked DCmotorrdquo IEEE Transactions on Industrial Electronics vol 60 no12 pp 5864ndash5874 2013

[20] X Guo and M Bodson ldquoAnalysis and implementation of anadaptive algorithm for the rejection of multiple sinusoidal dis-turbancesrdquo IEEE Transactions on Control Systems Technologyvol 17 no 1 pp 40ndash50 2009

[21] H Zhao Q Chen and S Xu ldquo119867infin

guaranteed cost controlfor uncertain Markovian jump systems with mode-dependentdistributed delays and input delaysrdquo Journal of the FranklinInstitute vol 346 no 10 pp 945ndash957 2009

[22] I D Landau M Alma J J Martinez and G Buche ldquoAdaptivesuppression ofmultiple time-varying unknownvibrations usingan inertial actuatorrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 6 pp 1327ndash1338 2011

[23] M Shamsuzzoha and M Lee ldquoEnhanced disturbance rejectionfor open-loop unstable process with time delayrdquo ISA Transac-tions vol 48 no 2 pp 237ndash244 2009

10 Mathematical Problems in Engineering

[24] L Guo and S Cao ldquoAnti-disturbance control theory for systemswith multiple disturbances a surveyrdquo ISA Transactions vol 53no 4 pp 846ndash849 2014

[25] X Li T W S Chow and J K L Ho ldquoIterative learning controlfor a class of nonlinear discrete-time systems with multipleinput delaysrdquo International Journal of Systems Science vol 39no 4 pp 361ndash369 2008

[26] J Zhang H Zhang Y Luo and T Feng ldquoModel-free optimalcontrol design for a class of linear discrete-time systems withmultiple delays using adaptive dynamic programmingrdquo Neuro-computing vol 135 pp 163ndash170 2014

[27] G-Y Tang and H-H Wang ldquoSuboptimal control for discretelinear systems with time-delay a successive approximationapproachrdquo Acta Automatica Sinica vol 31 no 3 pp 419ndash4262005

[28] H H Wang and G Y Tang ldquoObserver-based optimal outputtracking for discrete-time Systems with multiple state andinput delaysrdquo International Journal of Control Automation andSystems vol 7 no 1 pp 57ndash66 2009

[29] H T Banks M Q Jacobs and M R Latina ldquoThe synthesisof optimal controls for linear time-optimal problems withretarded controlsrdquo Journal of OptimizationTheory and Applica-tions vol 8 pp 319ndash366 1971

[30] H Ma G-Y Tang and Y-D Zhao ldquoFeedforward and feedbackoptimal control for offshore structures subjected to irregularwave forcesrdquo Ocean Engineering vol 33 no 8-9 pp 1105ndash11172006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

where

1198621=

119873

sum

119894=1

((

ℎ119894

sum

119895=1

(119862119860119895

)119879

119876119860119862119860119895minus1minusℎ119894119861

119894))

1198622=

119873

sum

119894=1

ℎ119894

sum

119895=1

ℎ119894minus119895

sum

119905=0

((119862119860119905

119863119864119866119895minus1

)119879

119876119862119860119905+119895minusℎ119894minus1119861

119894)

(10)

Then the quadratic performance index (4) could be reformedas the following form

119869 =1

2

infin

sum

119896=0

119909119879

(119896) 119862119879

119876119862119909 (119896) + 119906119879

(119896) 119906 (119896)

minus2119909119879

(119896) 1198621119906 (119896) minus 2119908

119879

(119896) 1198622119906 (119896)

(11)

where = 119877 + sum119873

119894=1sumℎ119894

119895=1((119862119860minus119895

119861119894)119879

119876119862119860minus119895

119861119894)

Actually the above transformations are suitable for theinfinite-time average quadratic performance index (5) Thereformed performance index could be described as

119869 = lim119873rarrinfin

1

119873

119873

sum

119896=0

119909119879

(119896) 119862119879

119876119862119909 (119896) + 119906119879

(119896) 119906 (119896)

minus2119909119879

(119896) 1198621119906 (119896) minus 2119908

119879

(119896) 1198622119906 (119896)

(12)

Then the original optimal disturbance rejection is refor-mulated as an equivalent optimal regulation problem thatdesigns an approximate control law 119906

lowast

(119896) to make thequadratic performance index (11) or (12) obtain theminimumvalue subject to the discrete-time system (2) and (7) such thatthe negative effects from persistent disturbances andmultipledelayed inputs are eliminated

4 Approximate Design of OptimalDisturbance Rejection Controller

In this section the proposed AODRC is presented in detailand its existence and uniqueness are proved In order to statethe proposed control law clearly the following matrices aredefined

1198601= 119860 + 119861

minus1

119862119879

1 119860

2= 119863119864 + 119861

minus1

119862119879

2

1198761= 119862119879

119876119862 minus 1198621minus1

119862119879

1 119876

2= 1198621minus1

119862119879

2

(13)

Theorem 2 Consider the optimal disturbance rejection con-trol problem for the discrete-time systems given by (1) and(2) with respect to the quadratic performance indexes (4) or

(5) The approximate optimal disturbance rejection controlleruniquely exists and can be formulated as

119906lowast

(119896) = minus1

[119862119879

1minus 119861119879

119860minus119879

1(1198751minus 1198761)]

times [

[

119909 (119896) +

119873

sum

119894=1

119896minus1

sum

119895=119896minusℎ119894

119860119896minusℎ119894minus119895minus1119861119906 (119895)]

]

+ (119862119879

2minus 119861119879

119860minus119879

1(1198752+ 1198762))119908 (119896)

(14)

where 1198751is the unique positive definite solution of the Riccati

matrix equation

1198751= 1198761+ 119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198601

(15)

and 1198752is the unique solution of the Stein matrix equation

1198752+ 1198762= 119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198602

+ 119860119879

1[119868 minus 119860

119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

119861minus1

119861119879

] 1198752119866

(16)

Proof Define the Hamiltonian function for the optimaldisturbance rejection problem as

119867(∙) =1

2119909119879

(119896) 119862119879

119876119862119909 (119896) + 119906119879

(119896) 119906 (119896)

minus 2119909119879

(119896) 1198621119906 (119896) minus 2119908

119879

(119896) 1198622119906 (119896)

+ 120582119879

(119896 + 1) [119860119909 (119896) + 119861119906 (119896) + 119863V (119896)]

(17)

Applying the maximum principle condition 120597119867120597119906 = 0 tothis specific Hamiltonian function (17) yields

119906 (119896) = minus1

[119861119879

120582 (119896 + 1) minus 119862119879

1119909 (119896) minus 119862

119879

2119908 (119896)] (18)

where 120582(119896) is the solution of the following TPBV problem

119909 (119896 + 1) = 1198601119909 (119896) + 119860

2119908 (119896) minus 119861

minus1

119861119879

120582 (119896 + 1)

1198761119909 (119896) minus 119876

2119908 (119896) + 119860

119879

1120582 (119896 + 1) = 120582 (119896)

119909 (0) = 119909 (0)

120582 (infin) = 0

(19)

To solve TPBV problem (19) let

120582 (119896) = 1198751119909 (119896) + 119875

2119908 (119896) (20)

Rearranging (19) one gets

120582 (119896 + 1) = 119860minus119879

1[(1198751minus 1198761) 119909 (119896) + (119875

2+ 1198762) 119908 (119896)]

119909 (119896 + 1) = (119868 + 119861minus1

119861119879

1198751)minus1

times 1198601119909 (119896) + (119860

2minus 119861minus1

119861119879

1198752119866)119908 (119896)

(21)

Mathematical Problems in Engineering 5

Substituting the first formula of (21) into (18) the approxi-mate optimal control law (14) is obtained Note that

120582 (119896) = 1198761119909 (119896) minus 119876

2119908 (119896) + 119860

119879

1120582 (119896 + 1)

= [1198761+ 119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198601] 119909 (119896)

+ 119860119879

1[119868 minus 119860

119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

119861minus1

119861119879

] 1198752119866

+119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198602minus 1198762119908 (119896)

(22)

By comparing the parameters of (20) and (22) the Riccatiequation (15) and Stein equation (16) can be obtained

In the following the existence and uniqueness of theproposed controller will be proved Because 119875

1and 119875

2are

unknown in the parameters of proposed control law (14)the existence and uniqueness of (14) are equivalent to thatof 1198751and 119875

2 For Riccati matrix equation (15) because of

the completely controllable and observable pairs (119860 1198611) and

(119860 119862) the matrix 1198751is existent and the unique solution of

Riccati matrix equation (15) It should be noted that the firstformula of (19) is equivalent to (7) based on the completelycontrollable pair (119860 119861) one gets

100381610038161003816100381610038161003816120582119894[119868 + 119861

minus1

119861119879

1198751]119879

1198601100381610038161003816100381610038161003816lt 1 119894 = 1 2 119899 (23)

Note (3) one gets100381610038161003816100381610038161003816120582119894[119868 + 119861

minus1

119861119879

1198751]119879

1198601100381610038161003816100381610038161003816

10038161003816100381610038161003816120582119895(119866)

10038161003816100381610038161003816lt 1

119894 = 1 2 119899 119895 = 1 2 119902

(24)

Then 1198752is existent and the unique solution of Stein matrix

equation (16) Therefore the proposed approximate con-troller is existent and unique

According to [29] (6) and (14) one gets

119909 (119896) le 119909 (119896) +

119873

sum

119894=1

[ℎ119894max1le119904leℎ119894

10038171003817100381710038171003817119860119896minus119895minus1

10038171003817100381710038171003817

times10038171003817100381710038171003817119861minus1

[119861119879

119860minus119879

1(1198751minus 1198761) minus 119862119879

1]10038171003817100381710038171003817

times 119909 (119896 minus 119904) max1le119904leℎ119894

]

+

119873

sum

119894=1

[ℎ119894max1le119904leℎ119894

10038171003817100381710038171003817119860119896minus119895minus1

10038171003817100381710038171003817119908 (119896 minus 119904)

times10038171003817100381710038171003817119861minus1

[119861119879

119860minus119879

1(1198752+ 1198762) minus 119862119879

2]10038171003817100381710038171003817]

(25)

Because 119909(119896) and 119908(119896) are bounded 119909(119896) is boundedThen the proposed control law (14) is a stabilizing control lawfor discrete-time system (1) with multiple delayed inputsTheproof is completed

Table 1 Parameters of installed AMD device

Name of parameter Variable Value UnitMass 119898

211855 kg

Natural frequency 1205962

233 radsStructural damping ratio 120585

2932 Percent

k2

k1

um2

m1

c2

c1

x2

x1(k)

Figure 1 The sketch of offshore platform with AMD device

5 Simulation

In order to demonstrate the effectiveness and feasibility ofthe proposedAODRC numerous simulations are undertakenin this section In case 1 the vibration control problemfor offshore steel jacket platforms with single delayed inputis simulated by using the proposed AODRC In case 2the disturbance rejection problem for pure mathematicalexample with two delayed inputs is considered

51 Vibration Control for Offshore Jacket Platforms with SingleDelayed Input The jacket type platforms play an importrole in the oil exploration and drilling operations Howeverthe jacket type platforms subject to external disturbancespersistently such as wave force wind and earthquake Inorder to ensure the safety and production efficiency thevibration caused by external disturbances could be rejectedeffectively especially for wave force In this subsection thevibration control problem for offshore jacket platforms withsingle input delay is solved by the proposed control law

Consider the jacket platform located in Bohai Bay thesketch of offshore platform with an AMD device is shown inFigure 1 The structural parameters of offshore platform andinstalled AMD device on the deck of the jacket type platformare listed in Tables 1 and 2 [30] respectively

Based on the JONSWAP spectrum the wave can bedescribed as

119878120578(120596) = [

51198672

119904

161205960

](1205960

120596)

5

exp [minus125(1205960

120596)

minus4

] 120574120573

(26)

6 Mathematical Problems in Engineering

Table 2 Parameters of offshore platform

Name of parameter Variable Value UnitEquivalent characteristic diameter of legs 119863 17 mFirst modal mass 119898

12371100 kg

Natural frequency 1205961

220 radsStructural damping ratio 120585

14 Percent

Shape function of first mode 120593(119904) 1 minus cos ( 120587119904

2119871) 0 le 119904 le 119871

where 119867119904

= 4m is the significant wave height 120596 isthe wave frequency 120574 = 33 is the peakedness param-eter 120596

0= 087 rads is the peak frequency 120573 =

exp[minus(120596 minus 1205960)2

(21205902

1205962

0)] in which 120590 is the shape parameter

120590 = 007 (120596 le 1205960) and 120590 = 009 (120596 gt 120596

0) Then based

on [30] the wave force can be formulated as an exosystemdescribed by (2) with the water depth 119889 = 132m the dragcoefficient119862

119889= 12 and the inertial coefficient119862

119898= 20The

external disturbances from irregular wave force are displayedin Figure 2

By setting the sampling period 119879 = 01 s 119876 =

diag (100 0 100 0) and 119877 = 1 the discrete-time modelof the offshore structure could be established with the sameexpression of (1) based on the previous investigations on thesubject which is described as

119860 =

[[[

[

09790 00002 00989 00000

00205 09789 00047 00952

minus04182 00041 09701 00010

03954 minus04152 00996 08993

]]]

]

119861 =

[[[

[

minus00006

00621

minus00120

12205

]]]

]

119863 =

[[[

[

00006

00000

00126

00006

]]]

]

119862 = [1 0 0 0

0 0 1 0]

(27)

According to the previous investigations on the subject therange of delayed input is in the interval (0 1) s Thereforeto demonstrate the effectiveness of the proposed AODRCthe comparison results between the open-loop system thefeedback control and the AODRC with single delayed inputℎ = 6 are compared in which feedback control law is justconsidered the feedback item in AODRC Then the curvesof displacement of offshore structure velocity of offshorestructure and control law are presented in Figures 3 4 and5 respectively

In order to show the comparison results among the sim-ulated case clearly the values of the quadratic performanceindex with different delayed input are compared in Table 3Note that the percentage number given in the parenthesesindicates the reduced amount of the closed-loop response

0 10 20 30 40 50 60 70 80 90 100

0

2

4

6

Wav

e for

ce(k)

(106

N)

minus2

minus4

minus6

kT (s)

Figure 2 The external disturbances of wave force

0 10 20 30 40 50 60 70 80 90 100

0

01

02

03

04

Disp

lace

men

t of o

ffsho

re st

ruct

ure (

m)

minus01

minus02

minus03

minus04

Open-loopFeedback controlAODRC

kT (s)

Figure 3 Curves of the displacement of offshore structure

relative to the open-loop case For instance the value ofperformance index is reduced by 1840when delayed inputℎ = 2 less than that of open-loop one using feedback controlwhile the value of performance index is reduced by 6723less than that of open-loop one by using proposed AODRC

Mathematical Problems in Engineering 7

Table 3 Comparison of values of performance index with different delayed input

ℎ = 2 ℎ = 4 ℎ = 6 ℎ = 8

Open-loop 50953 50953 50953 50953Feedback Control 41578 (1840) 35232 (3085) 30731 (3969) 31232 (3870)AODRC 16695 (6723) 16875 (6688) 17076 (6649) 17366 (6592)

0 10 20 30 40 50 60 70 80 90 100

02

0

04

06

Velo

city

of o

ffsho

re st

ruct

ure (

ms

)

Open-loopFeedback controlAODRC

minus02

minus04

minus06

minus08

kT (s)

Figure 4 Curves of the velocity of offshore structure

0 10 20 30 40 50 60 70 80 90 100

0

02

04

06

08

minus02

minus04

minus06

minus08

Con

trol l

awu(k)

(104

N)

kT (s)

Feedback controlAODRC

Figure 5 Curves of the control law

In order to present the control results more intuitive themaximum displacement the maximum acceleration of off-shore structure and the maximum control law with differentinput delay are shown in Table 4 From Table 4 it can beseen clearly that the maximum displacement the maximumacceleration of offshore structure and the values ofmaximumcontrol law with different input delay are close The resultsshow that the proposed control law could effectively eliminate

the influences fromdifferent input delay andwave forces withhigh value

It can be seen from Figures 3ndash5 and Tables 3-4 thatthe disturbances of wave force for offshore structure isreduced significantly by using the proposed AODRC Withthe increase of delayed input the proposed AODRC is stillable to effectively control the displacement and velocity ofoffshore structure Whatrsquos more the quadratic performanceindex is reduced significantly by using the proposed AODRCcompared by the feedback control and open-loop onesTherefore we can conclude that the proposed AODRC isefficient to reject the disturbances from wave force foroffshore structure as well as being with lower consumption

52 Disturbance Rejection for Pure Mathematical Examplewith Two Delayed Inputs In order to verify the elasticity ofthe proposed AODRC the discrete-time system with delayedinputs ℎ

1and ℎ2under sinusoidal disturbances is considered

The matrices of (1) are chosen as

119860 = [1 04

minus02 09]

119863 = [0

1]

1198611= [

02

05]

1198612= [

01

015]

119862 = [1 0]

119909 (0) = [2 0]119879

(28)

The external disturbance V(119905) is chosen as sinusoidal distur-bance the matrices of (2) are given by

119866 = [04radic5 04

minus05 04radic5]

119864 = [02 0]

V (0) = [1 0]119879

(29)

The curves of sinusoidal disturbances with (29) are shownin Figure 6 Also the quadratic performance index is chosenas (5) with 119876 = 1 119877 = 2

Then the responses of system states 1199091and 119909

2for differ-

ent cases (open-loop feedback control only and proposedAODRC) with delayed inputs ℎ

1= 1 and ℎ

2= 4 are

8 Mathematical Problems in Engineering

Table 4 The maximum displacement the maximum accelerationof offshore structure and the maximum control law

ℎ = 2 ℎ = 4 ℎ = 6 ℎ = 8

Maximumdisplacement (m) 0126 0130 0146 0150

Maximumacceleration (m2s) 0171 0175 0178 0181

Maximum controlinput (104 N) 0520 0523 0582 0588

0 50 100 150 200

0

005

01

015

02

Sinu

soid

al d

istur

banc

e(k)

minus005

minus01

minus015

minus02

kT (s)

Figure 6 Curves of the sinusoidal disturbance V(119896)

simulated and shown in Figures 7 and 8 The curves of theproposed AODRC for different cases (the proposed AODRCand feedback control only) are shown in Figure 9 The valuesof quadratic performance index (5) are shown in Table 3 withtwo delayed inputs (ℎ

1= 1 ℎ

2= 2 ℎ

1= 1 ℎ

2= 4 ℎ

1= 2

ℎ2= 4) Figures 7 and 8 and Table 5 show that the proposed

AODRC is able to effectively ensure the system states in lowervalues and reject the disturbances from external sinusoidalinput

From Figures 1ndash9 and Tables 1ndash5 the simulation resultsshow that the proposed AODRC for the offshore platformand pure mathematical example are effective to attenuate thevibration amplitudes of the system state and the quadraticperformance index of the controlled system under AODRC issmaller than those under the open-loop and feedback controlonly Moreover the proposed AODRC is more scalable fordifferent cases with multiple delayed inputs

6 Conclusion

This paper has presented an interesting approach to solvethe disturbance rejection problem for discrete-time systemswith multiple delayed inputs under persistent external dis-turbances A discrete-time system with multiple input delaysis transformed into a non-delayed system in form and thequadratic performance index has been transformed into arelevant format without the explicit appearance of time delayThen the AODRC is deployed to eliminate the negativeeffects from multiple delayed inputs and persistent external

0 50 100 150 200

0

1

2

3

minus1

minus2

minus3

Curv

es o

f sys

tem

stat

ex1(k

)

Open-loopFeedback controlAODRC

kT (s)

Figure 7 Curves of system state 1199091(119896)

0 50 100 150 200

0

05

1

15

2

25

minus05

minus1

minus15

minus2

Curv

es o

f sys

tem

stat

ex2(k

)

Open-loopFeedback controlAODRC

kT (s)

Figure 8 Curves of system state 1199092(119896)

Table 5 Comparison of values of performance index with twodelayed inputs

ℎ1= 1 ℎ

2= 2 ℎ

1= 1 ℎ

2= 4 ℎ

1= 2 ℎ

2= 4

Open-loop 03422 03422 03422FeedbackControl 00836 (7557) 01055 (6917) 01035 (6975)

AODRC 00372 (8913) 00506 (8521) 00901 (6649)

disturbances On the other hand the disturbance rejectionproblem for jacket-type offshore structure is illustrated tovalidate the feasibility and effectiveness of the proposedapproach

Mathematical Problems in Engineering 9

0 50 100 150 200

01

02

03

04

05

0

Feedback controlAODRC

minus01

minus02

minus03

minus04

minus05

Con

trol l

awu(k)

kT (s)

Figure 9 Curves of the control law

Conflict of Interests

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

Acknowledgments

This work was supported by the Program for Scien-tific Research Innovation Team in Colleges and Universi-ties of Shandong Province Science Foundation of Chinaunder Grants 61201428 61203105 and 61302128 the NaturalScience Foundation of Shandong Province under GrantsZR2011FZ003 ZR2012FQ016 and ZR2011FL022 and theDoctoral Foundation of Shandong Province under GrantBS2014DX015

References

[1] E Altman and T Basar ldquoOptimal rate control for high speedtelecommunication networksrdquo in Proceedings of the 34th IEEEConference on Decision and Control vol 2 pp 1389ndash1394December 1995

[2] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[3] X Yu C Wu F Liu and L Wu ldquoSliding mode control ofdiscrete-time switched systems with time-delayrdquo Journal of theFranklin Institute vol 350 no 1 pp 19ndash33 2013

[4] S Janardhanan and B Bandyopadhyay ldquoOutput feedbackdiscrete-time sliding mode control for time delay systemsrdquo IEEProceedingsmdashControl Theory and Applications vol 153 no 4pp 387ndash396 2006

[5] M Cedomir P-D Branislava and V Boban ldquoDiscrete-timevelocity servo system design using sliding mode controlapproach with disturbance compensationrdquo IEEE Transactionson Industrial Informatics vol 9 no 2 pp 920ndash927 2013

[6] P Albertos and P Garcıa ldquoPredictor-observer-based control ofsystems with multiple inputoutput delaysrdquo Journal of ProcessControl vol 22 no 7 pp 1350ndash1357 2012

[7] E Normey-Rico and E F Camacho ldquoDead-time compensatorsa surveyrdquo Control Engineering Practice vol 16 no 4 pp 407ndash428 2008

[8] B Zhou Z-Y Li and Z Lin ldquoObserver based output feedbackcontrol of linear systems with input and output delaysrdquo Auto-matica vol 49 no 7 pp 2039ndash2052 2013

[9] B Zhou ldquoObserver-based output feedback control of discrete-time linear systems with input and output delaysrdquo InternationalJournal of Control 2014 in press

[10] H Zhang G Duan and L Xie ldquoLinear quadratic regulationfor linear time-varying systems with multiple input delays PartI discrete time caserdquo in Proceedings of the 5th InternationalConference on Control and Automation Budapest Hungary2005

[11] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[12] B-L Zhang L Ma and Q-L Han ldquoSliding mode Hinfin

controlfor offshore steel jacket platforms subject to nonlinear self-excited wave force and external disturbanceurbancerdquoNonlinearAnalysis Real World Applications vol 14 no 1 pp 163ndash1782013

[13] B Zhang Q Han X Zhang and X Yu ldquoIntegral sliding modecontrol for offshore steel jacket platformsrdquo Journal of Sound andVibration vol 331 no 14 pp 3271ndash3285 2012

[14] S R Hall and N MWereley ldquoPerformance of higher harmoniccontrol algorithms for helicopter vibration reductionrdquo Journalof Guidance Control and Dynamics vol 16 no 4 pp 793ndash7971993

[15] S Han G Tang Y Chen and X Yang ldquoOptimal vibrationcontrol for vehicle active suspension discrete-time systems withactuator time delayrdquo Asian Journal of Control vol 15 no 6 pp1579ndash1588 2013

[16] H Gao J Lam andCWang ldquoMulti-objective control of vehicleactive suspension systems via load-dependent controllersrdquoJournal of Sound and Vibration vol 290 no 3ndash5 pp 654ndash6752006

[17] L Mirkin ldquoOn the approximation of distributed-delay controllawsrdquo Systems ampControl Letters vol 51 no 5 pp 331ndash342 2004

[18] Y Huang W C Xue and C Z Zhao ldquoActive disturbance rejec-tion control methodology and theoretical analysisrdquo Journal ofSystems Science and Mathematical Sciences vol 31 no 9 pp1111ndash1129 2011

[19] Y Shi J Huang and B Yu ldquoRobust tracking control ofnetworked control systems application to a networked DCmotorrdquo IEEE Transactions on Industrial Electronics vol 60 no12 pp 5864ndash5874 2013

[20] X Guo and M Bodson ldquoAnalysis and implementation of anadaptive algorithm for the rejection of multiple sinusoidal dis-turbancesrdquo IEEE Transactions on Control Systems Technologyvol 17 no 1 pp 40ndash50 2009

[21] H Zhao Q Chen and S Xu ldquo119867infin

guaranteed cost controlfor uncertain Markovian jump systems with mode-dependentdistributed delays and input delaysrdquo Journal of the FranklinInstitute vol 346 no 10 pp 945ndash957 2009

[22] I D Landau M Alma J J Martinez and G Buche ldquoAdaptivesuppression ofmultiple time-varying unknownvibrations usingan inertial actuatorrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 6 pp 1327ndash1338 2011

[23] M Shamsuzzoha and M Lee ldquoEnhanced disturbance rejectionfor open-loop unstable process with time delayrdquo ISA Transac-tions vol 48 no 2 pp 237ndash244 2009

10 Mathematical Problems in Engineering

[24] L Guo and S Cao ldquoAnti-disturbance control theory for systemswith multiple disturbances a surveyrdquo ISA Transactions vol 53no 4 pp 846ndash849 2014

[25] X Li T W S Chow and J K L Ho ldquoIterative learning controlfor a class of nonlinear discrete-time systems with multipleinput delaysrdquo International Journal of Systems Science vol 39no 4 pp 361ndash369 2008

[26] J Zhang H Zhang Y Luo and T Feng ldquoModel-free optimalcontrol design for a class of linear discrete-time systems withmultiple delays using adaptive dynamic programmingrdquo Neuro-computing vol 135 pp 163ndash170 2014

[27] G-Y Tang and H-H Wang ldquoSuboptimal control for discretelinear systems with time-delay a successive approximationapproachrdquo Acta Automatica Sinica vol 31 no 3 pp 419ndash4262005

[28] H H Wang and G Y Tang ldquoObserver-based optimal outputtracking for discrete-time Systems with multiple state andinput delaysrdquo International Journal of Control Automation andSystems vol 7 no 1 pp 57ndash66 2009

[29] H T Banks M Q Jacobs and M R Latina ldquoThe synthesisof optimal controls for linear time-optimal problems withretarded controlsrdquo Journal of OptimizationTheory and Applica-tions vol 8 pp 319ndash366 1971

[30] H Ma G-Y Tang and Y-D Zhao ldquoFeedforward and feedbackoptimal control for offshore structures subjected to irregularwave forcesrdquo Ocean Engineering vol 33 no 8-9 pp 1105ndash11172006

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 5

Substituting the first formula of (21) into (18) the approxi-mate optimal control law (14) is obtained Note that

120582 (119896) = 1198761119909 (119896) minus 119876

2119908 (119896) + 119860

119879

1120582 (119896 + 1)

= [1198761+ 119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198601] 119909 (119896)

+ 119860119879

1[119868 minus 119860

119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

119861minus1

119861119879

] 1198752119866

+119860119879

11198751(119868 + 119861

minus1

119861119879

1198751)minus1

1198602minus 1198762119908 (119896)

(22)

By comparing the parameters of (20) and (22) the Riccatiequation (15) and Stein equation (16) can be obtained

In the following the existence and uniqueness of theproposed controller will be proved Because 119875

1and 119875

2are

unknown in the parameters of proposed control law (14)the existence and uniqueness of (14) are equivalent to thatof 1198751and 119875

2 For Riccati matrix equation (15) because of

the completely controllable and observable pairs (119860 1198611) and

(119860 119862) the matrix 1198751is existent and the unique solution of

Riccati matrix equation (15) It should be noted that the firstformula of (19) is equivalent to (7) based on the completelycontrollable pair (119860 119861) one gets

100381610038161003816100381610038161003816120582119894[119868 + 119861

minus1

119861119879

1198751]119879

1198601100381610038161003816100381610038161003816lt 1 119894 = 1 2 119899 (23)

Note (3) one gets100381610038161003816100381610038161003816120582119894[119868 + 119861

minus1

119861119879

1198751]119879

1198601100381610038161003816100381610038161003816

10038161003816100381610038161003816120582119895(119866)

10038161003816100381610038161003816lt 1

119894 = 1 2 119899 119895 = 1 2 119902

(24)

Then 1198752is existent and the unique solution of Stein matrix

equation (16) Therefore the proposed approximate con-troller is existent and unique

According to [29] (6) and (14) one gets

119909 (119896) le 119909 (119896) +

119873

sum

119894=1

[ℎ119894max1le119904leℎ119894

10038171003817100381710038171003817119860119896minus119895minus1

10038171003817100381710038171003817

times10038171003817100381710038171003817119861minus1

[119861119879

119860minus119879

1(1198751minus 1198761) minus 119862119879

1]10038171003817100381710038171003817

times 119909 (119896 minus 119904) max1le119904leℎ119894

]

+

119873

sum

119894=1

[ℎ119894max1le119904leℎ119894

10038171003817100381710038171003817119860119896minus119895minus1

10038171003817100381710038171003817119908 (119896 minus 119904)

times10038171003817100381710038171003817119861minus1

[119861119879

119860minus119879

1(1198752+ 1198762) minus 119862119879

2]10038171003817100381710038171003817]

(25)

Because 119909(119896) and 119908(119896) are bounded 119909(119896) is boundedThen the proposed control law (14) is a stabilizing control lawfor discrete-time system (1) with multiple delayed inputsTheproof is completed

Table 1 Parameters of installed AMD device

Name of parameter Variable Value UnitMass 119898

211855 kg

Natural frequency 1205962

233 radsStructural damping ratio 120585

2932 Percent

k2

k1

um2

m1

c2

c1

x2

x1(k)

Figure 1 The sketch of offshore platform with AMD device

5 Simulation

In order to demonstrate the effectiveness and feasibility ofthe proposedAODRC numerous simulations are undertakenin this section In case 1 the vibration control problemfor offshore steel jacket platforms with single delayed inputis simulated by using the proposed AODRC In case 2the disturbance rejection problem for pure mathematicalexample with two delayed inputs is considered

51 Vibration Control for Offshore Jacket Platforms with SingleDelayed Input The jacket type platforms play an importrole in the oil exploration and drilling operations Howeverthe jacket type platforms subject to external disturbancespersistently such as wave force wind and earthquake Inorder to ensure the safety and production efficiency thevibration caused by external disturbances could be rejectedeffectively especially for wave force In this subsection thevibration control problem for offshore jacket platforms withsingle input delay is solved by the proposed control law

Consider the jacket platform located in Bohai Bay thesketch of offshore platform with an AMD device is shown inFigure 1 The structural parameters of offshore platform andinstalled AMD device on the deck of the jacket type platformare listed in Tables 1 and 2 [30] respectively

Based on the JONSWAP spectrum the wave can bedescribed as

119878120578(120596) = [

51198672

119904

161205960

](1205960

120596)

5

exp [minus125(1205960

120596)

minus4

] 120574120573

(26)

6 Mathematical Problems in Engineering

Table 2 Parameters of offshore platform

Name of parameter Variable Value UnitEquivalent characteristic diameter of legs 119863 17 mFirst modal mass 119898

12371100 kg

Natural frequency 1205961

220 radsStructural damping ratio 120585

14 Percent

Shape function of first mode 120593(119904) 1 minus cos ( 120587119904

2119871) 0 le 119904 le 119871

where 119867119904

= 4m is the significant wave height 120596 isthe wave frequency 120574 = 33 is the peakedness param-eter 120596

0= 087 rads is the peak frequency 120573 =

exp[minus(120596 minus 1205960)2

(21205902

1205962

0)] in which 120590 is the shape parameter

120590 = 007 (120596 le 1205960) and 120590 = 009 (120596 gt 120596

0) Then based

on [30] the wave force can be formulated as an exosystemdescribed by (2) with the water depth 119889 = 132m the dragcoefficient119862

119889= 12 and the inertial coefficient119862

119898= 20The

external disturbances from irregular wave force are displayedin Figure 2

By setting the sampling period 119879 = 01 s 119876 =

diag (100 0 100 0) and 119877 = 1 the discrete-time modelof the offshore structure could be established with the sameexpression of (1) based on the previous investigations on thesubject which is described as

119860 =

[[[

[

09790 00002 00989 00000

00205 09789 00047 00952

minus04182 00041 09701 00010

03954 minus04152 00996 08993

]]]

]

119861 =

[[[

[

minus00006

00621

minus00120

12205

]]]

]

119863 =

[[[

[

00006

00000

00126

00006

]]]

]

119862 = [1 0 0 0

0 0 1 0]

(27)

According to the previous investigations on the subject therange of delayed input is in the interval (0 1) s Thereforeto demonstrate the effectiveness of the proposed AODRCthe comparison results between the open-loop system thefeedback control and the AODRC with single delayed inputℎ = 6 are compared in which feedback control law is justconsidered the feedback item in AODRC Then the curvesof displacement of offshore structure velocity of offshorestructure and control law are presented in Figures 3 4 and5 respectively

In order to show the comparison results among the sim-ulated case clearly the values of the quadratic performanceindex with different delayed input are compared in Table 3Note that the percentage number given in the parenthesesindicates the reduced amount of the closed-loop response

0 10 20 30 40 50 60 70 80 90 100

0

2

4

6

Wav

e for

ce(k)

(106

N)

minus2

minus4

minus6

kT (s)

Figure 2 The external disturbances of wave force

0 10 20 30 40 50 60 70 80 90 100

0

01

02

03

04

Disp

lace

men

t of o

ffsho

re st

ruct

ure (

m)

minus01

minus02

minus03

minus04

Open-loopFeedback controlAODRC

kT (s)

Figure 3 Curves of the displacement of offshore structure

relative to the open-loop case For instance the value ofperformance index is reduced by 1840when delayed inputℎ = 2 less than that of open-loop one using feedback controlwhile the value of performance index is reduced by 6723less than that of open-loop one by using proposed AODRC

Mathematical Problems in Engineering 7

Table 3 Comparison of values of performance index with different delayed input

ℎ = 2 ℎ = 4 ℎ = 6 ℎ = 8

Open-loop 50953 50953 50953 50953Feedback Control 41578 (1840) 35232 (3085) 30731 (3969) 31232 (3870)AODRC 16695 (6723) 16875 (6688) 17076 (6649) 17366 (6592)

0 10 20 30 40 50 60 70 80 90 100

02

0

04

06

Velo

city

of o

ffsho

re st

ruct

ure (

ms

)

Open-loopFeedback controlAODRC

minus02

minus04

minus06

minus08

kT (s)

Figure 4 Curves of the velocity of offshore structure

0 10 20 30 40 50 60 70 80 90 100

0

02

04

06

08

minus02

minus04

minus06

minus08

Con

trol l

awu(k)

(104

N)

kT (s)

Feedback controlAODRC

Figure 5 Curves of the control law

In order to present the control results more intuitive themaximum displacement the maximum acceleration of off-shore structure and the maximum control law with differentinput delay are shown in Table 4 From Table 4 it can beseen clearly that the maximum displacement the maximumacceleration of offshore structure and the values ofmaximumcontrol law with different input delay are close The resultsshow that the proposed control law could effectively eliminate

the influences fromdifferent input delay andwave forces withhigh value

It can be seen from Figures 3ndash5 and Tables 3-4 thatthe disturbances of wave force for offshore structure isreduced significantly by using the proposed AODRC Withthe increase of delayed input the proposed AODRC is stillable to effectively control the displacement and velocity ofoffshore structure Whatrsquos more the quadratic performanceindex is reduced significantly by using the proposed AODRCcompared by the feedback control and open-loop onesTherefore we can conclude that the proposed AODRC isefficient to reject the disturbances from wave force foroffshore structure as well as being with lower consumption

52 Disturbance Rejection for Pure Mathematical Examplewith Two Delayed Inputs In order to verify the elasticity ofthe proposed AODRC the discrete-time system with delayedinputs ℎ

1and ℎ2under sinusoidal disturbances is considered

The matrices of (1) are chosen as

119860 = [1 04

minus02 09]

119863 = [0

1]

1198611= [

02

05]

1198612= [

01

015]

119862 = [1 0]

119909 (0) = [2 0]119879

(28)

The external disturbance V(119905) is chosen as sinusoidal distur-bance the matrices of (2) are given by

119866 = [04radic5 04

minus05 04radic5]

119864 = [02 0]

V (0) = [1 0]119879

(29)

The curves of sinusoidal disturbances with (29) are shownin Figure 6 Also the quadratic performance index is chosenas (5) with 119876 = 1 119877 = 2

Then the responses of system states 1199091and 119909

2for differ-

ent cases (open-loop feedback control only and proposedAODRC) with delayed inputs ℎ

1= 1 and ℎ

2= 4 are

8 Mathematical Problems in Engineering

Table 4 The maximum displacement the maximum accelerationof offshore structure and the maximum control law

ℎ = 2 ℎ = 4 ℎ = 6 ℎ = 8

Maximumdisplacement (m) 0126 0130 0146 0150

Maximumacceleration (m2s) 0171 0175 0178 0181

Maximum controlinput (104 N) 0520 0523 0582 0588

0 50 100 150 200

0

005

01

015

02

Sinu

soid

al d

istur

banc

e(k)

minus005

minus01

minus015

minus02

kT (s)

Figure 6 Curves of the sinusoidal disturbance V(119896)

simulated and shown in Figures 7 and 8 The curves of theproposed AODRC for different cases (the proposed AODRCand feedback control only) are shown in Figure 9 The valuesof quadratic performance index (5) are shown in Table 3 withtwo delayed inputs (ℎ

1= 1 ℎ

2= 2 ℎ

1= 1 ℎ

2= 4 ℎ

1= 2

ℎ2= 4) Figures 7 and 8 and Table 5 show that the proposed

AODRC is able to effectively ensure the system states in lowervalues and reject the disturbances from external sinusoidalinput

From Figures 1ndash9 and Tables 1ndash5 the simulation resultsshow that the proposed AODRC for the offshore platformand pure mathematical example are effective to attenuate thevibration amplitudes of the system state and the quadraticperformance index of the controlled system under AODRC issmaller than those under the open-loop and feedback controlonly Moreover the proposed AODRC is more scalable fordifferent cases with multiple delayed inputs

6 Conclusion

This paper has presented an interesting approach to solvethe disturbance rejection problem for discrete-time systemswith multiple delayed inputs under persistent external dis-turbances A discrete-time system with multiple input delaysis transformed into a non-delayed system in form and thequadratic performance index has been transformed into arelevant format without the explicit appearance of time delayThen the AODRC is deployed to eliminate the negativeeffects from multiple delayed inputs and persistent external

0 50 100 150 200

0

1

2

3

minus1

minus2

minus3

Curv

es o

f sys

tem

stat

ex1(k

)

Open-loopFeedback controlAODRC

kT (s)

Figure 7 Curves of system state 1199091(119896)

0 50 100 150 200

0

05

1

15

2

25

minus05

minus1

minus15

minus2

Curv

es o

f sys

tem

stat

ex2(k

)

Open-loopFeedback controlAODRC

kT (s)

Figure 8 Curves of system state 1199092(119896)

Table 5 Comparison of values of performance index with twodelayed inputs

ℎ1= 1 ℎ

2= 2 ℎ

1= 1 ℎ

2= 4 ℎ

1= 2 ℎ

2= 4

Open-loop 03422 03422 03422FeedbackControl 00836 (7557) 01055 (6917) 01035 (6975)

AODRC 00372 (8913) 00506 (8521) 00901 (6649)

disturbances On the other hand the disturbance rejectionproblem for jacket-type offshore structure is illustrated tovalidate the feasibility and effectiveness of the proposedapproach

Mathematical Problems in Engineering 9

0 50 100 150 200

01

02

03

04

05

0

Feedback controlAODRC

minus01

minus02

minus03

minus04

minus05

Con

trol l

awu(k)

kT (s)

Figure 9 Curves of the control law

Conflict of Interests

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

Acknowledgments

This work was supported by the Program for Scien-tific Research Innovation Team in Colleges and Universi-ties of Shandong Province Science Foundation of Chinaunder Grants 61201428 61203105 and 61302128 the NaturalScience Foundation of Shandong Province under GrantsZR2011FZ003 ZR2012FQ016 and ZR2011FL022 and theDoctoral Foundation of Shandong Province under GrantBS2014DX015

References

[1] E Altman and T Basar ldquoOptimal rate control for high speedtelecommunication networksrdquo in Proceedings of the 34th IEEEConference on Decision and Control vol 2 pp 1389ndash1394December 1995

[2] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[3] X Yu C Wu F Liu and L Wu ldquoSliding mode control ofdiscrete-time switched systems with time-delayrdquo Journal of theFranklin Institute vol 350 no 1 pp 19ndash33 2013

[4] S Janardhanan and B Bandyopadhyay ldquoOutput feedbackdiscrete-time sliding mode control for time delay systemsrdquo IEEProceedingsmdashControl Theory and Applications vol 153 no 4pp 387ndash396 2006

[5] M Cedomir P-D Branislava and V Boban ldquoDiscrete-timevelocity servo system design using sliding mode controlapproach with disturbance compensationrdquo IEEE Transactionson Industrial Informatics vol 9 no 2 pp 920ndash927 2013

[6] P Albertos and P Garcıa ldquoPredictor-observer-based control ofsystems with multiple inputoutput delaysrdquo Journal of ProcessControl vol 22 no 7 pp 1350ndash1357 2012

[7] E Normey-Rico and E F Camacho ldquoDead-time compensatorsa surveyrdquo Control Engineering Practice vol 16 no 4 pp 407ndash428 2008

[8] B Zhou Z-Y Li and Z Lin ldquoObserver based output feedbackcontrol of linear systems with input and output delaysrdquo Auto-matica vol 49 no 7 pp 2039ndash2052 2013

[9] B Zhou ldquoObserver-based output feedback control of discrete-time linear systems with input and output delaysrdquo InternationalJournal of Control 2014 in press

[10] H Zhang G Duan and L Xie ldquoLinear quadratic regulationfor linear time-varying systems with multiple input delays PartI discrete time caserdquo in Proceedings of the 5th InternationalConference on Control and Automation Budapest Hungary2005

[11] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[12] B-L Zhang L Ma and Q-L Han ldquoSliding mode Hinfin

controlfor offshore steel jacket platforms subject to nonlinear self-excited wave force and external disturbanceurbancerdquoNonlinearAnalysis Real World Applications vol 14 no 1 pp 163ndash1782013

[13] B Zhang Q Han X Zhang and X Yu ldquoIntegral sliding modecontrol for offshore steel jacket platformsrdquo Journal of Sound andVibration vol 331 no 14 pp 3271ndash3285 2012

[14] S R Hall and N MWereley ldquoPerformance of higher harmoniccontrol algorithms for helicopter vibration reductionrdquo Journalof Guidance Control and Dynamics vol 16 no 4 pp 793ndash7971993

[15] S Han G Tang Y Chen and X Yang ldquoOptimal vibrationcontrol for vehicle active suspension discrete-time systems withactuator time delayrdquo Asian Journal of Control vol 15 no 6 pp1579ndash1588 2013

[16] H Gao J Lam andCWang ldquoMulti-objective control of vehicleactive suspension systems via load-dependent controllersrdquoJournal of Sound and Vibration vol 290 no 3ndash5 pp 654ndash6752006

[17] L Mirkin ldquoOn the approximation of distributed-delay controllawsrdquo Systems ampControl Letters vol 51 no 5 pp 331ndash342 2004

[18] Y Huang W C Xue and C Z Zhao ldquoActive disturbance rejec-tion control methodology and theoretical analysisrdquo Journal ofSystems Science and Mathematical Sciences vol 31 no 9 pp1111ndash1129 2011

[19] Y Shi J Huang and B Yu ldquoRobust tracking control ofnetworked control systems application to a networked DCmotorrdquo IEEE Transactions on Industrial Electronics vol 60 no12 pp 5864ndash5874 2013

[20] X Guo and M Bodson ldquoAnalysis and implementation of anadaptive algorithm for the rejection of multiple sinusoidal dis-turbancesrdquo IEEE Transactions on Control Systems Technologyvol 17 no 1 pp 40ndash50 2009

[21] H Zhao Q Chen and S Xu ldquo119867infin

guaranteed cost controlfor uncertain Markovian jump systems with mode-dependentdistributed delays and input delaysrdquo Journal of the FranklinInstitute vol 346 no 10 pp 945ndash957 2009

[22] I D Landau M Alma J J Martinez and G Buche ldquoAdaptivesuppression ofmultiple time-varying unknownvibrations usingan inertial actuatorrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 6 pp 1327ndash1338 2011

[23] M Shamsuzzoha and M Lee ldquoEnhanced disturbance rejectionfor open-loop unstable process with time delayrdquo ISA Transac-tions vol 48 no 2 pp 237ndash244 2009

10 Mathematical Problems in Engineering

[24] L Guo and S Cao ldquoAnti-disturbance control theory for systemswith multiple disturbances a surveyrdquo ISA Transactions vol 53no 4 pp 846ndash849 2014

[25] X Li T W S Chow and J K L Ho ldquoIterative learning controlfor a class of nonlinear discrete-time systems with multipleinput delaysrdquo International Journal of Systems Science vol 39no 4 pp 361ndash369 2008

[26] J Zhang H Zhang Y Luo and T Feng ldquoModel-free optimalcontrol design for a class of linear discrete-time systems withmultiple delays using adaptive dynamic programmingrdquo Neuro-computing vol 135 pp 163ndash170 2014

[27] G-Y Tang and H-H Wang ldquoSuboptimal control for discretelinear systems with time-delay a successive approximationapproachrdquo Acta Automatica Sinica vol 31 no 3 pp 419ndash4262005

[28] H H Wang and G Y Tang ldquoObserver-based optimal outputtracking for discrete-time Systems with multiple state andinput delaysrdquo International Journal of Control Automation andSystems vol 7 no 1 pp 57ndash66 2009

[29] H T Banks M Q Jacobs and M R Latina ldquoThe synthesisof optimal controls for linear time-optimal problems withretarded controlsrdquo Journal of OptimizationTheory and Applica-tions vol 8 pp 319ndash366 1971

[30] H Ma G-Y Tang and Y-D Zhao ldquoFeedforward and feedbackoptimal control for offshore structures subjected to irregularwave forcesrdquo Ocean Engineering vol 33 no 8-9 pp 1105ndash11172006

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

Table 2 Parameters of offshore platform

Name of parameter Variable Value UnitEquivalent characteristic diameter of legs 119863 17 mFirst modal mass 119898

12371100 kg

Natural frequency 1205961

220 radsStructural damping ratio 120585

14 Percent

Shape function of first mode 120593(119904) 1 minus cos ( 120587119904

2119871) 0 le 119904 le 119871

where 119867119904

= 4m is the significant wave height 120596 isthe wave frequency 120574 = 33 is the peakedness param-eter 120596

0= 087 rads is the peak frequency 120573 =

exp[minus(120596 minus 1205960)2

(21205902

1205962

0)] in which 120590 is the shape parameter

120590 = 007 (120596 le 1205960) and 120590 = 009 (120596 gt 120596

0) Then based

on [30] the wave force can be formulated as an exosystemdescribed by (2) with the water depth 119889 = 132m the dragcoefficient119862

119889= 12 and the inertial coefficient119862

119898= 20The

external disturbances from irregular wave force are displayedin Figure 2

By setting the sampling period 119879 = 01 s 119876 =

diag (100 0 100 0) and 119877 = 1 the discrete-time modelof the offshore structure could be established with the sameexpression of (1) based on the previous investigations on thesubject which is described as

119860 =

[[[

[

09790 00002 00989 00000

00205 09789 00047 00952

minus04182 00041 09701 00010

03954 minus04152 00996 08993

]]]

]

119861 =

[[[

[

minus00006

00621

minus00120

12205

]]]

]

119863 =

[[[

[

00006

00000

00126

00006

]]]

]

119862 = [1 0 0 0

0 0 1 0]

(27)

According to the previous investigations on the subject therange of delayed input is in the interval (0 1) s Thereforeto demonstrate the effectiveness of the proposed AODRCthe comparison results between the open-loop system thefeedback control and the AODRC with single delayed inputℎ = 6 are compared in which feedback control law is justconsidered the feedback item in AODRC Then the curvesof displacement of offshore structure velocity of offshorestructure and control law are presented in Figures 3 4 and5 respectively

In order to show the comparison results among the sim-ulated case clearly the values of the quadratic performanceindex with different delayed input are compared in Table 3Note that the percentage number given in the parenthesesindicates the reduced amount of the closed-loop response

0 10 20 30 40 50 60 70 80 90 100

0

2

4

6

Wav

e for

ce(k)

(106

N)

minus2

minus4

minus6

kT (s)

Figure 2 The external disturbances of wave force

0 10 20 30 40 50 60 70 80 90 100

0

01

02

03

04

Disp

lace

men

t of o

ffsho

re st

ruct

ure (

m)

minus01

minus02

minus03

minus04

Open-loopFeedback controlAODRC

kT (s)

Figure 3 Curves of the displacement of offshore structure

relative to the open-loop case For instance the value ofperformance index is reduced by 1840when delayed inputℎ = 2 less than that of open-loop one using feedback controlwhile the value of performance index is reduced by 6723less than that of open-loop one by using proposed AODRC

Mathematical Problems in Engineering 7

Table 3 Comparison of values of performance index with different delayed input

ℎ = 2 ℎ = 4 ℎ = 6 ℎ = 8

Open-loop 50953 50953 50953 50953Feedback Control 41578 (1840) 35232 (3085) 30731 (3969) 31232 (3870)AODRC 16695 (6723) 16875 (6688) 17076 (6649) 17366 (6592)

0 10 20 30 40 50 60 70 80 90 100

02

0

04

06

Velo

city

of o

ffsho

re st

ruct

ure (

ms

)

Open-loopFeedback controlAODRC

minus02

minus04

minus06

minus08

kT (s)

Figure 4 Curves of the velocity of offshore structure

0 10 20 30 40 50 60 70 80 90 100

0

02

04

06

08

minus02

minus04

minus06

minus08

Con

trol l

awu(k)

(104

N)

kT (s)

Feedback controlAODRC

Figure 5 Curves of the control law

In order to present the control results more intuitive themaximum displacement the maximum acceleration of off-shore structure and the maximum control law with differentinput delay are shown in Table 4 From Table 4 it can beseen clearly that the maximum displacement the maximumacceleration of offshore structure and the values ofmaximumcontrol law with different input delay are close The resultsshow that the proposed control law could effectively eliminate

the influences fromdifferent input delay andwave forces withhigh value

It can be seen from Figures 3ndash5 and Tables 3-4 thatthe disturbances of wave force for offshore structure isreduced significantly by using the proposed AODRC Withthe increase of delayed input the proposed AODRC is stillable to effectively control the displacement and velocity ofoffshore structure Whatrsquos more the quadratic performanceindex is reduced significantly by using the proposed AODRCcompared by the feedback control and open-loop onesTherefore we can conclude that the proposed AODRC isefficient to reject the disturbances from wave force foroffshore structure as well as being with lower consumption

52 Disturbance Rejection for Pure Mathematical Examplewith Two Delayed Inputs In order to verify the elasticity ofthe proposed AODRC the discrete-time system with delayedinputs ℎ

1and ℎ2under sinusoidal disturbances is considered

The matrices of (1) are chosen as

119860 = [1 04

minus02 09]

119863 = [0

1]

1198611= [

02

05]

1198612= [

01

015]

119862 = [1 0]

119909 (0) = [2 0]119879

(28)

The external disturbance V(119905) is chosen as sinusoidal distur-bance the matrices of (2) are given by

119866 = [04radic5 04

minus05 04radic5]

119864 = [02 0]

V (0) = [1 0]119879

(29)

The curves of sinusoidal disturbances with (29) are shownin Figure 6 Also the quadratic performance index is chosenas (5) with 119876 = 1 119877 = 2

Then the responses of system states 1199091and 119909

2for differ-

ent cases (open-loop feedback control only and proposedAODRC) with delayed inputs ℎ

1= 1 and ℎ

2= 4 are

8 Mathematical Problems in Engineering

Table 4 The maximum displacement the maximum accelerationof offshore structure and the maximum control law

ℎ = 2 ℎ = 4 ℎ = 6 ℎ = 8

Maximumdisplacement (m) 0126 0130 0146 0150

Maximumacceleration (m2s) 0171 0175 0178 0181

Maximum controlinput (104 N) 0520 0523 0582 0588

0 50 100 150 200

0

005

01

015

02

Sinu

soid

al d

istur

banc

e(k)

minus005

minus01

minus015

minus02

kT (s)

Figure 6 Curves of the sinusoidal disturbance V(119896)

simulated and shown in Figures 7 and 8 The curves of theproposed AODRC for different cases (the proposed AODRCand feedback control only) are shown in Figure 9 The valuesof quadratic performance index (5) are shown in Table 3 withtwo delayed inputs (ℎ

1= 1 ℎ

2= 2 ℎ

1= 1 ℎ

2= 4 ℎ

1= 2

ℎ2= 4) Figures 7 and 8 and Table 5 show that the proposed

AODRC is able to effectively ensure the system states in lowervalues and reject the disturbances from external sinusoidalinput

From Figures 1ndash9 and Tables 1ndash5 the simulation resultsshow that the proposed AODRC for the offshore platformand pure mathematical example are effective to attenuate thevibration amplitudes of the system state and the quadraticperformance index of the controlled system under AODRC issmaller than those under the open-loop and feedback controlonly Moreover the proposed AODRC is more scalable fordifferent cases with multiple delayed inputs

6 Conclusion

This paper has presented an interesting approach to solvethe disturbance rejection problem for discrete-time systemswith multiple delayed inputs under persistent external dis-turbances A discrete-time system with multiple input delaysis transformed into a non-delayed system in form and thequadratic performance index has been transformed into arelevant format without the explicit appearance of time delayThen the AODRC is deployed to eliminate the negativeeffects from multiple delayed inputs and persistent external

0 50 100 150 200

0

1

2

3

minus1

minus2

minus3

Curv

es o

f sys

tem

stat

ex1(k

)

Open-loopFeedback controlAODRC

kT (s)

Figure 7 Curves of system state 1199091(119896)

0 50 100 150 200

0

05

1

15

2

25

minus05

minus1

minus15

minus2

Curv

es o

f sys

tem

stat

ex2(k

)

Open-loopFeedback controlAODRC

kT (s)

Figure 8 Curves of system state 1199092(119896)

Table 5 Comparison of values of performance index with twodelayed inputs

ℎ1= 1 ℎ

2= 2 ℎ

1= 1 ℎ

2= 4 ℎ

1= 2 ℎ

2= 4

Open-loop 03422 03422 03422FeedbackControl 00836 (7557) 01055 (6917) 01035 (6975)

AODRC 00372 (8913) 00506 (8521) 00901 (6649)

disturbances On the other hand the disturbance rejectionproblem for jacket-type offshore structure is illustrated tovalidate the feasibility and effectiveness of the proposedapproach

Mathematical Problems in Engineering 9

0 50 100 150 200

01

02

03

04

05

0

Feedback controlAODRC

minus01

minus02

minus03

minus04

minus05

Con

trol l

awu(k)

kT (s)

Figure 9 Curves of the control law

Conflict of Interests

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

Acknowledgments

This work was supported by the Program for Scien-tific Research Innovation Team in Colleges and Universi-ties of Shandong Province Science Foundation of Chinaunder Grants 61201428 61203105 and 61302128 the NaturalScience Foundation of Shandong Province under GrantsZR2011FZ003 ZR2012FQ016 and ZR2011FL022 and theDoctoral Foundation of Shandong Province under GrantBS2014DX015

References

[1] E Altman and T Basar ldquoOptimal rate control for high speedtelecommunication networksrdquo in Proceedings of the 34th IEEEConference on Decision and Control vol 2 pp 1389ndash1394December 1995

[2] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[3] X Yu C Wu F Liu and L Wu ldquoSliding mode control ofdiscrete-time switched systems with time-delayrdquo Journal of theFranklin Institute vol 350 no 1 pp 19ndash33 2013

[4] S Janardhanan and B Bandyopadhyay ldquoOutput feedbackdiscrete-time sliding mode control for time delay systemsrdquo IEEProceedingsmdashControl Theory and Applications vol 153 no 4pp 387ndash396 2006

[5] M Cedomir P-D Branislava and V Boban ldquoDiscrete-timevelocity servo system design using sliding mode controlapproach with disturbance compensationrdquo IEEE Transactionson Industrial Informatics vol 9 no 2 pp 920ndash927 2013

[6] P Albertos and P Garcıa ldquoPredictor-observer-based control ofsystems with multiple inputoutput delaysrdquo Journal of ProcessControl vol 22 no 7 pp 1350ndash1357 2012

[7] E Normey-Rico and E F Camacho ldquoDead-time compensatorsa surveyrdquo Control Engineering Practice vol 16 no 4 pp 407ndash428 2008

[8] B Zhou Z-Y Li and Z Lin ldquoObserver based output feedbackcontrol of linear systems with input and output delaysrdquo Auto-matica vol 49 no 7 pp 2039ndash2052 2013

[9] B Zhou ldquoObserver-based output feedback control of discrete-time linear systems with input and output delaysrdquo InternationalJournal of Control 2014 in press

[10] H Zhang G Duan and L Xie ldquoLinear quadratic regulationfor linear time-varying systems with multiple input delays PartI discrete time caserdquo in Proceedings of the 5th InternationalConference on Control and Automation Budapest Hungary2005

[11] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[12] B-L Zhang L Ma and Q-L Han ldquoSliding mode Hinfin

controlfor offshore steel jacket platforms subject to nonlinear self-excited wave force and external disturbanceurbancerdquoNonlinearAnalysis Real World Applications vol 14 no 1 pp 163ndash1782013

[13] B Zhang Q Han X Zhang and X Yu ldquoIntegral sliding modecontrol for offshore steel jacket platformsrdquo Journal of Sound andVibration vol 331 no 14 pp 3271ndash3285 2012

[14] S R Hall and N MWereley ldquoPerformance of higher harmoniccontrol algorithms for helicopter vibration reductionrdquo Journalof Guidance Control and Dynamics vol 16 no 4 pp 793ndash7971993

[15] S Han G Tang Y Chen and X Yang ldquoOptimal vibrationcontrol for vehicle active suspension discrete-time systems withactuator time delayrdquo Asian Journal of Control vol 15 no 6 pp1579ndash1588 2013

[16] H Gao J Lam andCWang ldquoMulti-objective control of vehicleactive suspension systems via load-dependent controllersrdquoJournal of Sound and Vibration vol 290 no 3ndash5 pp 654ndash6752006

[17] L Mirkin ldquoOn the approximation of distributed-delay controllawsrdquo Systems ampControl Letters vol 51 no 5 pp 331ndash342 2004

[18] Y Huang W C Xue and C Z Zhao ldquoActive disturbance rejec-tion control methodology and theoretical analysisrdquo Journal ofSystems Science and Mathematical Sciences vol 31 no 9 pp1111ndash1129 2011

[19] Y Shi J Huang and B Yu ldquoRobust tracking control ofnetworked control systems application to a networked DCmotorrdquo IEEE Transactions on Industrial Electronics vol 60 no12 pp 5864ndash5874 2013

[20] X Guo and M Bodson ldquoAnalysis and implementation of anadaptive algorithm for the rejection of multiple sinusoidal dis-turbancesrdquo IEEE Transactions on Control Systems Technologyvol 17 no 1 pp 40ndash50 2009

[21] H Zhao Q Chen and S Xu ldquo119867infin

guaranteed cost controlfor uncertain Markovian jump systems with mode-dependentdistributed delays and input delaysrdquo Journal of the FranklinInstitute vol 346 no 10 pp 945ndash957 2009

[22] I D Landau M Alma J J Martinez and G Buche ldquoAdaptivesuppression ofmultiple time-varying unknownvibrations usingan inertial actuatorrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 6 pp 1327ndash1338 2011

[23] M Shamsuzzoha and M Lee ldquoEnhanced disturbance rejectionfor open-loop unstable process with time delayrdquo ISA Transac-tions vol 48 no 2 pp 237ndash244 2009

10 Mathematical Problems in Engineering

[24] L Guo and S Cao ldquoAnti-disturbance control theory for systemswith multiple disturbances a surveyrdquo ISA Transactions vol 53no 4 pp 846ndash849 2014

[25] X Li T W S Chow and J K L Ho ldquoIterative learning controlfor a class of nonlinear discrete-time systems with multipleinput delaysrdquo International Journal of Systems Science vol 39no 4 pp 361ndash369 2008

[26] J Zhang H Zhang Y Luo and T Feng ldquoModel-free optimalcontrol design for a class of linear discrete-time systems withmultiple delays using adaptive dynamic programmingrdquo Neuro-computing vol 135 pp 163ndash170 2014

[27] G-Y Tang and H-H Wang ldquoSuboptimal control for discretelinear systems with time-delay a successive approximationapproachrdquo Acta Automatica Sinica vol 31 no 3 pp 419ndash4262005

[28] H H Wang and G Y Tang ldquoObserver-based optimal outputtracking for discrete-time Systems with multiple state andinput delaysrdquo International Journal of Control Automation andSystems vol 7 no 1 pp 57ndash66 2009

[29] H T Banks M Q Jacobs and M R Latina ldquoThe synthesisof optimal controls for linear time-optimal problems withretarded controlsrdquo Journal of OptimizationTheory and Applica-tions vol 8 pp 319ndash366 1971

[30] H Ma G-Y Tang and Y-D Zhao ldquoFeedforward and feedbackoptimal control for offshore structures subjected to irregularwave forcesrdquo Ocean Engineering vol 33 no 8-9 pp 1105ndash11172006

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

Table 3 Comparison of values of performance index with different delayed input

ℎ = 2 ℎ = 4 ℎ = 6 ℎ = 8

Open-loop 50953 50953 50953 50953Feedback Control 41578 (1840) 35232 (3085) 30731 (3969) 31232 (3870)AODRC 16695 (6723) 16875 (6688) 17076 (6649) 17366 (6592)

0 10 20 30 40 50 60 70 80 90 100

02

0

04

06

Velo

city

of o

ffsho

re st

ruct

ure (

ms

)

Open-loopFeedback controlAODRC

minus02

minus04

minus06

minus08

kT (s)

Figure 4 Curves of the velocity of offshore structure

0 10 20 30 40 50 60 70 80 90 100

0

02

04

06

08

minus02

minus04

minus06

minus08

Con

trol l

awu(k)

(104

N)

kT (s)

Feedback controlAODRC

Figure 5 Curves of the control law

In order to present the control results more intuitive themaximum displacement the maximum acceleration of off-shore structure and the maximum control law with differentinput delay are shown in Table 4 From Table 4 it can beseen clearly that the maximum displacement the maximumacceleration of offshore structure and the values ofmaximumcontrol law with different input delay are close The resultsshow that the proposed control law could effectively eliminate

the influences fromdifferent input delay andwave forces withhigh value

It can be seen from Figures 3ndash5 and Tables 3-4 thatthe disturbances of wave force for offshore structure isreduced significantly by using the proposed AODRC Withthe increase of delayed input the proposed AODRC is stillable to effectively control the displacement and velocity ofoffshore structure Whatrsquos more the quadratic performanceindex is reduced significantly by using the proposed AODRCcompared by the feedback control and open-loop onesTherefore we can conclude that the proposed AODRC isefficient to reject the disturbances from wave force foroffshore structure as well as being with lower consumption

52 Disturbance Rejection for Pure Mathematical Examplewith Two Delayed Inputs In order to verify the elasticity ofthe proposed AODRC the discrete-time system with delayedinputs ℎ

1and ℎ2under sinusoidal disturbances is considered

The matrices of (1) are chosen as

119860 = [1 04

minus02 09]

119863 = [0

1]

1198611= [

02

05]

1198612= [

01

015]

119862 = [1 0]

119909 (0) = [2 0]119879

(28)

The external disturbance V(119905) is chosen as sinusoidal distur-bance the matrices of (2) are given by

119866 = [04radic5 04

minus05 04radic5]

119864 = [02 0]

V (0) = [1 0]119879

(29)

The curves of sinusoidal disturbances with (29) are shownin Figure 6 Also the quadratic performance index is chosenas (5) with 119876 = 1 119877 = 2

Then the responses of system states 1199091and 119909

2for differ-

ent cases (open-loop feedback control only and proposedAODRC) with delayed inputs ℎ

1= 1 and ℎ

2= 4 are

8 Mathematical Problems in Engineering

Table 4 The maximum displacement the maximum accelerationof offshore structure and the maximum control law

ℎ = 2 ℎ = 4 ℎ = 6 ℎ = 8

Maximumdisplacement (m) 0126 0130 0146 0150

Maximumacceleration (m2s) 0171 0175 0178 0181

Maximum controlinput (104 N) 0520 0523 0582 0588

0 50 100 150 200

0

005

01

015

02

Sinu

soid

al d

istur

banc

e(k)

minus005

minus01

minus015

minus02

kT (s)

Figure 6 Curves of the sinusoidal disturbance V(119896)

simulated and shown in Figures 7 and 8 The curves of theproposed AODRC for different cases (the proposed AODRCand feedback control only) are shown in Figure 9 The valuesof quadratic performance index (5) are shown in Table 3 withtwo delayed inputs (ℎ

1= 1 ℎ

2= 2 ℎ

1= 1 ℎ

2= 4 ℎ

1= 2

ℎ2= 4) Figures 7 and 8 and Table 5 show that the proposed

AODRC is able to effectively ensure the system states in lowervalues and reject the disturbances from external sinusoidalinput

From Figures 1ndash9 and Tables 1ndash5 the simulation resultsshow that the proposed AODRC for the offshore platformand pure mathematical example are effective to attenuate thevibration amplitudes of the system state and the quadraticperformance index of the controlled system under AODRC issmaller than those under the open-loop and feedback controlonly Moreover the proposed AODRC is more scalable fordifferent cases with multiple delayed inputs

6 Conclusion

This paper has presented an interesting approach to solvethe disturbance rejection problem for discrete-time systemswith multiple delayed inputs under persistent external dis-turbances A discrete-time system with multiple input delaysis transformed into a non-delayed system in form and thequadratic performance index has been transformed into arelevant format without the explicit appearance of time delayThen the AODRC is deployed to eliminate the negativeeffects from multiple delayed inputs and persistent external

0 50 100 150 200

0

1

2

3

minus1

minus2

minus3

Curv

es o

f sys

tem

stat

ex1(k

)

Open-loopFeedback controlAODRC

kT (s)

Figure 7 Curves of system state 1199091(119896)

0 50 100 150 200

0

05

1

15

2

25

minus05

minus1

minus15

minus2

Curv

es o

f sys

tem

stat

ex2(k

)

Open-loopFeedback controlAODRC

kT (s)

Figure 8 Curves of system state 1199092(119896)

Table 5 Comparison of values of performance index with twodelayed inputs

ℎ1= 1 ℎ

2= 2 ℎ

1= 1 ℎ

2= 4 ℎ

1= 2 ℎ

2= 4

Open-loop 03422 03422 03422FeedbackControl 00836 (7557) 01055 (6917) 01035 (6975)

AODRC 00372 (8913) 00506 (8521) 00901 (6649)

disturbances On the other hand the disturbance rejectionproblem for jacket-type offshore structure is illustrated tovalidate the feasibility and effectiveness of the proposedapproach

Mathematical Problems in Engineering 9

0 50 100 150 200

01

02

03

04

05

0

Feedback controlAODRC

minus01

minus02

minus03

minus04

minus05

Con

trol l

awu(k)

kT (s)

Figure 9 Curves of the control law

Conflict of Interests

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

Acknowledgments

This work was supported by the Program for Scien-tific Research Innovation Team in Colleges and Universi-ties of Shandong Province Science Foundation of Chinaunder Grants 61201428 61203105 and 61302128 the NaturalScience Foundation of Shandong Province under GrantsZR2011FZ003 ZR2012FQ016 and ZR2011FL022 and theDoctoral Foundation of Shandong Province under GrantBS2014DX015

References

[1] E Altman and T Basar ldquoOptimal rate control for high speedtelecommunication networksrdquo in Proceedings of the 34th IEEEConference on Decision and Control vol 2 pp 1389ndash1394December 1995

[2] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[3] X Yu C Wu F Liu and L Wu ldquoSliding mode control ofdiscrete-time switched systems with time-delayrdquo Journal of theFranklin Institute vol 350 no 1 pp 19ndash33 2013

[4] S Janardhanan and B Bandyopadhyay ldquoOutput feedbackdiscrete-time sliding mode control for time delay systemsrdquo IEEProceedingsmdashControl Theory and Applications vol 153 no 4pp 387ndash396 2006

[5] M Cedomir P-D Branislava and V Boban ldquoDiscrete-timevelocity servo system design using sliding mode controlapproach with disturbance compensationrdquo IEEE Transactionson Industrial Informatics vol 9 no 2 pp 920ndash927 2013

[6] P Albertos and P Garcıa ldquoPredictor-observer-based control ofsystems with multiple inputoutput delaysrdquo Journal of ProcessControl vol 22 no 7 pp 1350ndash1357 2012

[7] E Normey-Rico and E F Camacho ldquoDead-time compensatorsa surveyrdquo Control Engineering Practice vol 16 no 4 pp 407ndash428 2008

[8] B Zhou Z-Y Li and Z Lin ldquoObserver based output feedbackcontrol of linear systems with input and output delaysrdquo Auto-matica vol 49 no 7 pp 2039ndash2052 2013

[9] B Zhou ldquoObserver-based output feedback control of discrete-time linear systems with input and output delaysrdquo InternationalJournal of Control 2014 in press

[10] H Zhang G Duan and L Xie ldquoLinear quadratic regulationfor linear time-varying systems with multiple input delays PartI discrete time caserdquo in Proceedings of the 5th InternationalConference on Control and Automation Budapest Hungary2005

[11] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[12] B-L Zhang L Ma and Q-L Han ldquoSliding mode Hinfin

controlfor offshore steel jacket platforms subject to nonlinear self-excited wave force and external disturbanceurbancerdquoNonlinearAnalysis Real World Applications vol 14 no 1 pp 163ndash1782013

[13] B Zhang Q Han X Zhang and X Yu ldquoIntegral sliding modecontrol for offshore steel jacket platformsrdquo Journal of Sound andVibration vol 331 no 14 pp 3271ndash3285 2012

[14] S R Hall and N MWereley ldquoPerformance of higher harmoniccontrol algorithms for helicopter vibration reductionrdquo Journalof Guidance Control and Dynamics vol 16 no 4 pp 793ndash7971993

[15] S Han G Tang Y Chen and X Yang ldquoOptimal vibrationcontrol for vehicle active suspension discrete-time systems withactuator time delayrdquo Asian Journal of Control vol 15 no 6 pp1579ndash1588 2013

[16] H Gao J Lam andCWang ldquoMulti-objective control of vehicleactive suspension systems via load-dependent controllersrdquoJournal of Sound and Vibration vol 290 no 3ndash5 pp 654ndash6752006

[17] L Mirkin ldquoOn the approximation of distributed-delay controllawsrdquo Systems ampControl Letters vol 51 no 5 pp 331ndash342 2004

[18] Y Huang W C Xue and C Z Zhao ldquoActive disturbance rejec-tion control methodology and theoretical analysisrdquo Journal ofSystems Science and Mathematical Sciences vol 31 no 9 pp1111ndash1129 2011

[19] Y Shi J Huang and B Yu ldquoRobust tracking control ofnetworked control systems application to a networked DCmotorrdquo IEEE Transactions on Industrial Electronics vol 60 no12 pp 5864ndash5874 2013

[20] X Guo and M Bodson ldquoAnalysis and implementation of anadaptive algorithm for the rejection of multiple sinusoidal dis-turbancesrdquo IEEE Transactions on Control Systems Technologyvol 17 no 1 pp 40ndash50 2009

[21] H Zhao Q Chen and S Xu ldquo119867infin

guaranteed cost controlfor uncertain Markovian jump systems with mode-dependentdistributed delays and input delaysrdquo Journal of the FranklinInstitute vol 346 no 10 pp 945ndash957 2009

[22] I D Landau M Alma J J Martinez and G Buche ldquoAdaptivesuppression ofmultiple time-varying unknownvibrations usingan inertial actuatorrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 6 pp 1327ndash1338 2011

[23] M Shamsuzzoha and M Lee ldquoEnhanced disturbance rejectionfor open-loop unstable process with time delayrdquo ISA Transac-tions vol 48 no 2 pp 237ndash244 2009

10 Mathematical Problems in Engineering

[24] L Guo and S Cao ldquoAnti-disturbance control theory for systemswith multiple disturbances a surveyrdquo ISA Transactions vol 53no 4 pp 846ndash849 2014

[25] X Li T W S Chow and J K L Ho ldquoIterative learning controlfor a class of nonlinear discrete-time systems with multipleinput delaysrdquo International Journal of Systems Science vol 39no 4 pp 361ndash369 2008

[26] J Zhang H Zhang Y Luo and T Feng ldquoModel-free optimalcontrol design for a class of linear discrete-time systems withmultiple delays using adaptive dynamic programmingrdquo Neuro-computing vol 135 pp 163ndash170 2014

[27] G-Y Tang and H-H Wang ldquoSuboptimal control for discretelinear systems with time-delay a successive approximationapproachrdquo Acta Automatica Sinica vol 31 no 3 pp 419ndash4262005

[28] H H Wang and G Y Tang ldquoObserver-based optimal outputtracking for discrete-time Systems with multiple state andinput delaysrdquo International Journal of Control Automation andSystems vol 7 no 1 pp 57ndash66 2009

[29] H T Banks M Q Jacobs and M R Latina ldquoThe synthesisof optimal controls for linear time-optimal problems withretarded controlsrdquo Journal of OptimizationTheory and Applica-tions vol 8 pp 319ndash366 1971

[30] H Ma G-Y Tang and Y-D Zhao ldquoFeedforward and feedbackoptimal control for offshore structures subjected to irregularwave forcesrdquo Ocean Engineering vol 33 no 8-9 pp 1105ndash11172006

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

Table 4 The maximum displacement the maximum accelerationof offshore structure and the maximum control law

ℎ = 2 ℎ = 4 ℎ = 6 ℎ = 8

Maximumdisplacement (m) 0126 0130 0146 0150

Maximumacceleration (m2s) 0171 0175 0178 0181

Maximum controlinput (104 N) 0520 0523 0582 0588

0 50 100 150 200

0

005

01

015

02

Sinu

soid

al d

istur

banc

e(k)

minus005

minus01

minus015

minus02

kT (s)

Figure 6 Curves of the sinusoidal disturbance V(119896)

simulated and shown in Figures 7 and 8 The curves of theproposed AODRC for different cases (the proposed AODRCand feedback control only) are shown in Figure 9 The valuesof quadratic performance index (5) are shown in Table 3 withtwo delayed inputs (ℎ

1= 1 ℎ

2= 2 ℎ

1= 1 ℎ

2= 4 ℎ

1= 2

ℎ2= 4) Figures 7 and 8 and Table 5 show that the proposed

AODRC is able to effectively ensure the system states in lowervalues and reject the disturbances from external sinusoidalinput

From Figures 1ndash9 and Tables 1ndash5 the simulation resultsshow that the proposed AODRC for the offshore platformand pure mathematical example are effective to attenuate thevibration amplitudes of the system state and the quadraticperformance index of the controlled system under AODRC issmaller than those under the open-loop and feedback controlonly Moreover the proposed AODRC is more scalable fordifferent cases with multiple delayed inputs

6 Conclusion

This paper has presented an interesting approach to solvethe disturbance rejection problem for discrete-time systemswith multiple delayed inputs under persistent external dis-turbances A discrete-time system with multiple input delaysis transformed into a non-delayed system in form and thequadratic performance index has been transformed into arelevant format without the explicit appearance of time delayThen the AODRC is deployed to eliminate the negativeeffects from multiple delayed inputs and persistent external

0 50 100 150 200

0

1

2

3

minus1

minus2

minus3

Curv

es o

f sys

tem

stat

ex1(k

)

Open-loopFeedback controlAODRC

kT (s)

Figure 7 Curves of system state 1199091(119896)

0 50 100 150 200

0

05

1

15

2

25

minus05

minus1

minus15

minus2

Curv

es o

f sys

tem

stat

ex2(k

)

Open-loopFeedback controlAODRC

kT (s)

Figure 8 Curves of system state 1199092(119896)

Table 5 Comparison of values of performance index with twodelayed inputs

ℎ1= 1 ℎ

2= 2 ℎ

1= 1 ℎ

2= 4 ℎ

1= 2 ℎ

2= 4

Open-loop 03422 03422 03422FeedbackControl 00836 (7557) 01055 (6917) 01035 (6975)

AODRC 00372 (8913) 00506 (8521) 00901 (6649)

disturbances On the other hand the disturbance rejectionproblem for jacket-type offshore structure is illustrated tovalidate the feasibility and effectiveness of the proposedapproach

Mathematical Problems in Engineering 9

0 50 100 150 200

01

02

03

04

05

0

Feedback controlAODRC

minus01

minus02

minus03

minus04

minus05

Con

trol l

awu(k)

kT (s)

Figure 9 Curves of the control law

Conflict of Interests

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

Acknowledgments

This work was supported by the Program for Scien-tific Research Innovation Team in Colleges and Universi-ties of Shandong Province Science Foundation of Chinaunder Grants 61201428 61203105 and 61302128 the NaturalScience Foundation of Shandong Province under GrantsZR2011FZ003 ZR2012FQ016 and ZR2011FL022 and theDoctoral Foundation of Shandong Province under GrantBS2014DX015

References

[1] E Altman and T Basar ldquoOptimal rate control for high speedtelecommunication networksrdquo in Proceedings of the 34th IEEEConference on Decision and Control vol 2 pp 1389ndash1394December 1995

[2] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[3] X Yu C Wu F Liu and L Wu ldquoSliding mode control ofdiscrete-time switched systems with time-delayrdquo Journal of theFranklin Institute vol 350 no 1 pp 19ndash33 2013

[4] S Janardhanan and B Bandyopadhyay ldquoOutput feedbackdiscrete-time sliding mode control for time delay systemsrdquo IEEProceedingsmdashControl Theory and Applications vol 153 no 4pp 387ndash396 2006

[5] M Cedomir P-D Branislava and V Boban ldquoDiscrete-timevelocity servo system design using sliding mode controlapproach with disturbance compensationrdquo IEEE Transactionson Industrial Informatics vol 9 no 2 pp 920ndash927 2013

[6] P Albertos and P Garcıa ldquoPredictor-observer-based control ofsystems with multiple inputoutput delaysrdquo Journal of ProcessControl vol 22 no 7 pp 1350ndash1357 2012

[7] E Normey-Rico and E F Camacho ldquoDead-time compensatorsa surveyrdquo Control Engineering Practice vol 16 no 4 pp 407ndash428 2008

[8] B Zhou Z-Y Li and Z Lin ldquoObserver based output feedbackcontrol of linear systems with input and output delaysrdquo Auto-matica vol 49 no 7 pp 2039ndash2052 2013

[9] B Zhou ldquoObserver-based output feedback control of discrete-time linear systems with input and output delaysrdquo InternationalJournal of Control 2014 in press

[10] H Zhang G Duan and L Xie ldquoLinear quadratic regulationfor linear time-varying systems with multiple input delays PartI discrete time caserdquo in Proceedings of the 5th InternationalConference on Control and Automation Budapest Hungary2005

[11] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[12] B-L Zhang L Ma and Q-L Han ldquoSliding mode Hinfin

controlfor offshore steel jacket platforms subject to nonlinear self-excited wave force and external disturbanceurbancerdquoNonlinearAnalysis Real World Applications vol 14 no 1 pp 163ndash1782013

[13] B Zhang Q Han X Zhang and X Yu ldquoIntegral sliding modecontrol for offshore steel jacket platformsrdquo Journal of Sound andVibration vol 331 no 14 pp 3271ndash3285 2012

[14] S R Hall and N MWereley ldquoPerformance of higher harmoniccontrol algorithms for helicopter vibration reductionrdquo Journalof Guidance Control and Dynamics vol 16 no 4 pp 793ndash7971993

[15] S Han G Tang Y Chen and X Yang ldquoOptimal vibrationcontrol for vehicle active suspension discrete-time systems withactuator time delayrdquo Asian Journal of Control vol 15 no 6 pp1579ndash1588 2013

[16] H Gao J Lam andCWang ldquoMulti-objective control of vehicleactive suspension systems via load-dependent controllersrdquoJournal of Sound and Vibration vol 290 no 3ndash5 pp 654ndash6752006

[17] L Mirkin ldquoOn the approximation of distributed-delay controllawsrdquo Systems ampControl Letters vol 51 no 5 pp 331ndash342 2004

[18] Y Huang W C Xue and C Z Zhao ldquoActive disturbance rejec-tion control methodology and theoretical analysisrdquo Journal ofSystems Science and Mathematical Sciences vol 31 no 9 pp1111ndash1129 2011

[19] Y Shi J Huang and B Yu ldquoRobust tracking control ofnetworked control systems application to a networked DCmotorrdquo IEEE Transactions on Industrial Electronics vol 60 no12 pp 5864ndash5874 2013

[20] X Guo and M Bodson ldquoAnalysis and implementation of anadaptive algorithm for the rejection of multiple sinusoidal dis-turbancesrdquo IEEE Transactions on Control Systems Technologyvol 17 no 1 pp 40ndash50 2009

[21] H Zhao Q Chen and S Xu ldquo119867infin

guaranteed cost controlfor uncertain Markovian jump systems with mode-dependentdistributed delays and input delaysrdquo Journal of the FranklinInstitute vol 346 no 10 pp 945ndash957 2009

[22] I D Landau M Alma J J Martinez and G Buche ldquoAdaptivesuppression ofmultiple time-varying unknownvibrations usingan inertial actuatorrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 6 pp 1327ndash1338 2011

[23] M Shamsuzzoha and M Lee ldquoEnhanced disturbance rejectionfor open-loop unstable process with time delayrdquo ISA Transac-tions vol 48 no 2 pp 237ndash244 2009

10 Mathematical Problems in Engineering

[24] L Guo and S Cao ldquoAnti-disturbance control theory for systemswith multiple disturbances a surveyrdquo ISA Transactions vol 53no 4 pp 846ndash849 2014

[25] X Li T W S Chow and J K L Ho ldquoIterative learning controlfor a class of nonlinear discrete-time systems with multipleinput delaysrdquo International Journal of Systems Science vol 39no 4 pp 361ndash369 2008

[26] J Zhang H Zhang Y Luo and T Feng ldquoModel-free optimalcontrol design for a class of linear discrete-time systems withmultiple delays using adaptive dynamic programmingrdquo Neuro-computing vol 135 pp 163ndash170 2014

[27] G-Y Tang and H-H Wang ldquoSuboptimal control for discretelinear systems with time-delay a successive approximationapproachrdquo Acta Automatica Sinica vol 31 no 3 pp 419ndash4262005

[28] H H Wang and G Y Tang ldquoObserver-based optimal outputtracking for discrete-time Systems with multiple state andinput delaysrdquo International Journal of Control Automation andSystems vol 7 no 1 pp 57ndash66 2009

[29] H T Banks M Q Jacobs and M R Latina ldquoThe synthesisof optimal controls for linear time-optimal problems withretarded controlsrdquo Journal of OptimizationTheory and Applica-tions vol 8 pp 319ndash366 1971

[30] H Ma G-Y Tang and Y-D Zhao ldquoFeedforward and feedbackoptimal control for offshore structures subjected to irregularwave forcesrdquo Ocean Engineering vol 33 no 8-9 pp 1105ndash11172006

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

0 50 100 150 200

01

02

03

04

05

0

Feedback controlAODRC

minus01

minus02

minus03

minus04

minus05

Con

trol l

awu(k)

kT (s)

Figure 9 Curves of the control law

Conflict of Interests

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

Acknowledgments

This work was supported by the Program for Scien-tific Research Innovation Team in Colleges and Universi-ties of Shandong Province Science Foundation of Chinaunder Grants 61201428 61203105 and 61302128 the NaturalScience Foundation of Shandong Province under GrantsZR2011FZ003 ZR2012FQ016 and ZR2011FL022 and theDoctoral Foundation of Shandong Province under GrantBS2014DX015

References

[1] E Altman and T Basar ldquoOptimal rate control for high speedtelecommunication networksrdquo in Proceedings of the 34th IEEEConference on Decision and Control vol 2 pp 1389ndash1394December 1995

[2] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[3] X Yu C Wu F Liu and L Wu ldquoSliding mode control ofdiscrete-time switched systems with time-delayrdquo Journal of theFranklin Institute vol 350 no 1 pp 19ndash33 2013

[4] S Janardhanan and B Bandyopadhyay ldquoOutput feedbackdiscrete-time sliding mode control for time delay systemsrdquo IEEProceedingsmdashControl Theory and Applications vol 153 no 4pp 387ndash396 2006

[5] M Cedomir P-D Branislava and V Boban ldquoDiscrete-timevelocity servo system design using sliding mode controlapproach with disturbance compensationrdquo IEEE Transactionson Industrial Informatics vol 9 no 2 pp 920ndash927 2013

[6] P Albertos and P Garcıa ldquoPredictor-observer-based control ofsystems with multiple inputoutput delaysrdquo Journal of ProcessControl vol 22 no 7 pp 1350ndash1357 2012

[7] E Normey-Rico and E F Camacho ldquoDead-time compensatorsa surveyrdquo Control Engineering Practice vol 16 no 4 pp 407ndash428 2008

[8] B Zhou Z-Y Li and Z Lin ldquoObserver based output feedbackcontrol of linear systems with input and output delaysrdquo Auto-matica vol 49 no 7 pp 2039ndash2052 2013

[9] B Zhou ldquoObserver-based output feedback control of discrete-time linear systems with input and output delaysrdquo InternationalJournal of Control 2014 in press

[10] H Zhang G Duan and L Xie ldquoLinear quadratic regulationfor linear time-varying systems with multiple input delays PartI discrete time caserdquo in Proceedings of the 5th InternationalConference on Control and Automation Budapest Hungary2005

[11] H Zhang L Xie and G Duan ldquo119867infin

control of discrete-time systems with multiple input delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 2 pp 271ndash283 2007

[12] B-L Zhang L Ma and Q-L Han ldquoSliding mode Hinfin

controlfor offshore steel jacket platforms subject to nonlinear self-excited wave force and external disturbanceurbancerdquoNonlinearAnalysis Real World Applications vol 14 no 1 pp 163ndash1782013

[13] B Zhang Q Han X Zhang and X Yu ldquoIntegral sliding modecontrol for offshore steel jacket platformsrdquo Journal of Sound andVibration vol 331 no 14 pp 3271ndash3285 2012

[14] S R Hall and N MWereley ldquoPerformance of higher harmoniccontrol algorithms for helicopter vibration reductionrdquo Journalof Guidance Control and Dynamics vol 16 no 4 pp 793ndash7971993

[15] S Han G Tang Y Chen and X Yang ldquoOptimal vibrationcontrol for vehicle active suspension discrete-time systems withactuator time delayrdquo Asian Journal of Control vol 15 no 6 pp1579ndash1588 2013

[16] H Gao J Lam andCWang ldquoMulti-objective control of vehicleactive suspension systems via load-dependent controllersrdquoJournal of Sound and Vibration vol 290 no 3ndash5 pp 654ndash6752006

[17] L Mirkin ldquoOn the approximation of distributed-delay controllawsrdquo Systems ampControl Letters vol 51 no 5 pp 331ndash342 2004

[18] Y Huang W C Xue and C Z Zhao ldquoActive disturbance rejec-tion control methodology and theoretical analysisrdquo Journal ofSystems Science and Mathematical Sciences vol 31 no 9 pp1111ndash1129 2011

[19] Y Shi J Huang and B Yu ldquoRobust tracking control ofnetworked control systems application to a networked DCmotorrdquo IEEE Transactions on Industrial Electronics vol 60 no12 pp 5864ndash5874 2013

[20] X Guo and M Bodson ldquoAnalysis and implementation of anadaptive algorithm for the rejection of multiple sinusoidal dis-turbancesrdquo IEEE Transactions on Control Systems Technologyvol 17 no 1 pp 40ndash50 2009

[21] H Zhao Q Chen and S Xu ldquo119867infin

guaranteed cost controlfor uncertain Markovian jump systems with mode-dependentdistributed delays and input delaysrdquo Journal of the FranklinInstitute vol 346 no 10 pp 945ndash957 2009

[22] I D Landau M Alma J J Martinez and G Buche ldquoAdaptivesuppression ofmultiple time-varying unknownvibrations usingan inertial actuatorrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 6 pp 1327ndash1338 2011

[23] M Shamsuzzoha and M Lee ldquoEnhanced disturbance rejectionfor open-loop unstable process with time delayrdquo ISA Transac-tions vol 48 no 2 pp 237ndash244 2009

10 Mathematical Problems in Engineering

[24] L Guo and S Cao ldquoAnti-disturbance control theory for systemswith multiple disturbances a surveyrdquo ISA Transactions vol 53no 4 pp 846ndash849 2014

[25] X Li T W S Chow and J K L Ho ldquoIterative learning controlfor a class of nonlinear discrete-time systems with multipleinput delaysrdquo International Journal of Systems Science vol 39no 4 pp 361ndash369 2008

[26] J Zhang H Zhang Y Luo and T Feng ldquoModel-free optimalcontrol design for a class of linear discrete-time systems withmultiple delays using adaptive dynamic programmingrdquo Neuro-computing vol 135 pp 163ndash170 2014

[27] G-Y Tang and H-H Wang ldquoSuboptimal control for discretelinear systems with time-delay a successive approximationapproachrdquo Acta Automatica Sinica vol 31 no 3 pp 419ndash4262005

[28] H H Wang and G Y Tang ldquoObserver-based optimal outputtracking for discrete-time Systems with multiple state andinput delaysrdquo International Journal of Control Automation andSystems vol 7 no 1 pp 57ndash66 2009

[29] H T Banks M Q Jacobs and M R Latina ldquoThe synthesisof optimal controls for linear time-optimal problems withretarded controlsrdquo Journal of OptimizationTheory and Applica-tions vol 8 pp 319ndash366 1971

[30] H Ma G-Y Tang and Y-D Zhao ldquoFeedforward and feedbackoptimal control for offshore structures subjected to irregularwave forcesrdquo Ocean Engineering vol 33 no 8-9 pp 1105ndash11172006

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

10 Mathematical Problems in Engineering

[24] L Guo and S Cao ldquoAnti-disturbance control theory for systemswith multiple disturbances a surveyrdquo ISA Transactions vol 53no 4 pp 846ndash849 2014

[25] X Li T W S Chow and J K L Ho ldquoIterative learning controlfor a class of nonlinear discrete-time systems with multipleinput delaysrdquo International Journal of Systems Science vol 39no 4 pp 361ndash369 2008

[26] J Zhang H Zhang Y Luo and T Feng ldquoModel-free optimalcontrol design for a class of linear discrete-time systems withmultiple delays using adaptive dynamic programmingrdquo Neuro-computing vol 135 pp 163ndash170 2014

[27] G-Y Tang and H-H Wang ldquoSuboptimal control for discretelinear systems with time-delay a successive approximationapproachrdquo Acta Automatica Sinica vol 31 no 3 pp 419ndash4262005

[28] H H Wang and G Y Tang ldquoObserver-based optimal outputtracking for discrete-time Systems with multiple state andinput delaysrdquo International Journal of Control Automation andSystems vol 7 no 1 pp 57ndash66 2009

[29] H T Banks M Q Jacobs and M R Latina ldquoThe synthesisof optimal controls for linear time-optimal problems withretarded controlsrdquo Journal of OptimizationTheory and Applica-tions vol 8 pp 319ndash366 1971

[30] H Ma G-Y Tang and Y-D Zhao ldquoFeedforward and feedbackoptimal control for offshore structures subjected to irregularwave forcesrdquo Ocean Engineering vol 33 no 8-9 pp 1105ndash11172006

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