Neural Network Predictive Control for Autonomous Underwater Vehicle … · 2019. 7. 30. · Neural...
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Research Article Neural Network Predictive Control for Autonomous Underwater Vehicle with Input Delay Jiemei Zhao School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan 430023, China Correspondence should be addressed to Jiemei Zhao; [email protected] Received 6 January 2018; Accepted 24 April 2018; Published 29 May 2018 Academic Editor: Jongrae Kim Copyright © 2018 Jiemei Zhao. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A path tracking controller is designed for an autonomous underwater vehicle (AUV) with input delay based on neural network (NN) predictive control algorithm. To compensate for the time-delay in control system and realize the purpose of path tracking, a predictive control algorithm is proposed. An NN is used to estimate the nonlinear uncertainty of AUV induced by hydrodynamic coefficients and the coupling of the surge, sway, and yaw angular velocity. By Lyapunov theorem, stability analysis is also given. Simulation results show the effectiveness of the proposed control strategy. 1. Introduction With the rapid development of demands for the resources, countries around the world have attached importance to the exploration and application of the marine resources. Autonomous underwater vehicle (AUV) is a mobile carrier which is small in size and convenient in controllability as the special equipment for resource exploration, environmental monitoring, and ocean investigation. It has the ability for long-time navigating and great-weight carrying and satisfies the different demands of the fields of military science and economics (see [1–3] and references therein). In recent years, control problems of AUV such as setpoint stabilization, trajectory tracking control, and path tracking control have been actively considered by many researchers. Based on nonlinear control theory, several control methods have been proposed, such as sliding mode control [4–6], adaptive control [7–11], and predictive control [12–15]. How- ever, a common problem of the above literatures is that the time-delays are not taken into account. In practical systems, time-delays are unavoidable in information acquisition and transmission. Time-delay phenomenon is oſten a source of instability and poor performance [16–18]. From this point of view, considerable amount of attention has been paid to the problem of stabilization and control of time-delay systems. Predictive control is a good method with the ability to handle constraints and time-delays [19–23]. Now, it has become one of the most popular control methodologies no matter in theory or the reality (see [24–27]). e NN predictive control for nonlinear dynamic systems with input delay was studied in [24], but the considered predictive model is required for linear ones and this condition is removed in this paper. e predictor-based control algorithm for an uncertain input delay Euler-Lagrange system was studied in [26], but the controller is an iteration form. To overcome the problem of input delay in Euler-Lagrange dynamical systems directly, a predictor with uncertain system dynamics was proposed in [27]. Recently, predictive control has been applied in many kinds of practical systems [28–31]. Up to now, only a few papers have considered this problem because of its complexity. Paper [32] addressed the control problem with input delay and synthesized a robust controller for underwater vehicles which requires only knowledge of mass matrix. e region tracking problem for AUV with input delay based on predictive control was studied in [33], but it assumes that all the states are known in advance. erefore, it is a very challenging and significant work to investigate the path tracking control of AUV with input delay. In this paper, a novel controller is investigated for path tracking control of AUV with input delay. Because of the hydrodynamic coefficients and the surge, sway, and yaw angular velocity coupling, an NN is used to identify the Hindawi Journal of Control Science and Engineering Volume 2018, Article ID 2316957, 8 pages https://doi.org/10.1155/2018/2316957
Transcript of Neural Network Predictive Control for Autonomous Underwater Vehicle … · 2019. 7. 30. · Neural...
Research ArticleNeural Network Predictive Control for Autonomous UnderwaterVehicle with Input Delay
Jiemei Zhao
School of Mathematics and Computer Science Wuhan Polytechnic University Wuhan 430023 China
Correspondence should be addressed to Jiemei Zhao jiemeizhao163com
Received 6 January 2018 Accepted 24 April 2018 Published 29 May 2018
Academic Editor Jongrae Kim
Copyright copy 2018 Jiemei Zhao This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A path tracking controller is designed for an autonomous underwater vehicle (AUV) with input delay based on neural network(NN) predictive control algorithm To compensate for the time-delay in control system and realize the purpose of path tracking apredictive control algorithm is proposed An NN is used to estimate the nonlinear uncertainty of AUV induced by hydrodynamiccoefficients and the coupling of the surge sway and yaw angular velocity By Lyapunov theorem stability analysis is also givenSimulation results show the effectiveness of the proposed control strategy
1 Introduction
With the rapid development of demands for the resourcescountries around the world have attached importance tothe exploration and application of the marine resourcesAutonomous underwater vehicle (AUV) is a mobile carrierwhich is small in size and convenient in controllability as thespecial equipment for resource exploration environmentalmonitoring and ocean investigation It has the ability forlong-time navigating and great-weight carrying and satisfiesthe different demands of the fields of military science andeconomics (see [1ndash3] and references therein)
In recent years control problems of AUV such as setpointstabilization trajectory tracking control and path trackingcontrol have been actively considered by many researchersBased on nonlinear control theory several control methodshave been proposed such as sliding mode control [4ndash6]adaptive control [7ndash11] and predictive control [12ndash15] How-ever a common problem of the above literatures is that thetime-delays are not taken into account In practical systemstime-delays are unavoidable in information acquisition andtransmission Time-delay phenomenon is often a source ofinstability and poor performance [16ndash18] From this pointof view considerable amount of attention has been paidto the problem of stabilization and control of time-delaysystems Predictive control is a good method with the ability
to handle constraints and time-delays [19ndash23] Now it hasbecome one of the most popular control methodologiesno matter in theory or the reality (see [24ndash27]) The NNpredictive control for nonlinear dynamic systems with inputdelaywas studied in [24] but the considered predictivemodelis required for linear ones and this condition is removedin this paper The predictor-based control algorithm for anuncertain input delay Euler-Lagrange system was studied in[26] but the controller is an iteration form To overcomethe problem of input delay in Euler-Lagrange dynamicalsystems directly a predictor with uncertain system dynamicswas proposed in [27] Recently predictive control has beenapplied in many kinds of practical systems [28ndash31] Up tonow only a few papers have considered this problem becauseof its complexity Paper [32] addressed the control problemwith input delay and synthesized a robust controller forunderwater vehicles which requires only knowledge of massmatrix The region tracking problem for AUV with inputdelay based on predictive control was studied in [33] but itassumes that all the states are known in advance Thereforeit is a very challenging and significant work to investigate thepath tracking control of AUV with input delay
In this paper a novel controller is investigated for pathtracking control of AUV with input delay Because of thehydrodynamic coefficients and the surge sway and yawangular velocity coupling an NN is used to identify the
HindawiJournal of Control Science and EngineeringVolume 2018 Article ID 2316957 8 pageshttpsdoiorg10115520182316957
2 Journal of Control Science and Engineering
nonlinear part of AUV at first Then predictive control algo-rithm is employed to compensate for the delay produced ininput channel The proposed predictive model is a nonlinearmodel Stability of the closed-loop system is guaranteed basedon Lyapunov stability theory Finally a simulation example ispresented to show the effectiveness of the proposed controlstrategy
The remainder of this paper is organized as follows Theproblem of path tracking for AUV is formulated in Section 2Section 3 is devoted to identification of AUV system by NNStability analysis for the boundness of error state and NNweight estimation error are also performed The predictorand the corresponding control are derived in Section 4 Theproblems of dealing with the time-delay and stability analysisare illustrated in Section 5 Section 6 validates the feasibilityand performance of the proposed control law by simulationexperiment Some conclusions are given in Section 7
2 Problem Formulation
In the horizontal plane a 3-DOF AUV with input delay canbe modeled as
120578 = 119869 (120578) ]119872] + 119862 (]) ] + 119863 (]) ] + 119892 (120578) = 120591 (119905 minus 119889)
ℎ = 120578(1)
where 120578 = [119909 119910 120595]119879 denotes the vehicle location andorientation in the earth-fixed frameThe vector ] = [119906 V 119903]119879is the velocities expressed in the body-fixed frame 119872 =119872119877119861+119872119860 is the inertia matrix of rigid body119872119877119861 with addedmass119872119860Thematrix119862(]) is skew symmetrical and it denotedthe Coriolis and centripetal forces Linear and quadraticdamping forces are considered in the total hydrodynamicdamping matrix 119863(]) The vector 119892(120578) is the combinedgravitational and buoyancy forces in the body-fixed frame120591 is the input of the system and the vector of the forcesand moments on AUV induced by the input and fins 119889 isa known constant time-delay ℎ is the output of the systemThe kinematic transformation matrix transformation fromthe body-fixed frame to earth-fixed frame is denoted by 119869(120578)and
119869 (120578) = 119877 (120595) = [[[
cos120595 minus sin120595 0sin120595 cos120595 00 0 1
]]]
(2)
Let1205851 = 1205781205852 = 119869 (120578) ] (3)
Then system (1) changed to1205851 = 12058521205852 = 119891 (1205851 1205852) + 119869 (1205851)119872minus1120591 (119905 minus 119889) 119910 = 1205851
(4)
where nonlinear uncertain function
119891 (1205851 1205852) = 119869 (1205851) 119869minus1 (1205851) 1205852 + 119869 (1205851)sdot 119872minus1 [minus119862 (119869minus1 (1205851) 1205852)minus 119863 (119869minus1 (1205851) 1205852) 119869minus1 (1205851) 1205852 minus 119892 (1205851)]
(5)
The objective of this paper is that the output 119910 of system(4) tracks a desired trajectory 120578119889 with all internal signals andcontrol commands remaining bounded For this purpose wemake the following assumption
Assumption 1 The desired trajectory vector 120577119889 =[120578119889 120578119889 120578119889]119879 is available for measurement and 120578119889 and120578119889 are bounded3 Identification of AUV System
There are two steps to design the output feedback controllerfor AUVwith input delay First an NN is designed to identifysystem (4) Then we will use predictive control algorithmto compensate for the delay that presents in communicationchannel of AUV
Let
119860 = [ 0 11986831198601 1198602]
119861 = [01198683]
119862 = [1198683 0]119879
(6)
where 1198683 denotes the identity matrix matrices1198601 and1198602 areparameters that can be chosen such that matrix 119860 is stableThen there exist symmetric positive definite matrices 119875 and119876 such that Lyapunovmatrix equations119860119879119875+119875119860 = minus119876 hold
Hence system (4) can be expressed as
120585 = 119860120585 + 119861 [1198910 (120585) + 120591 (119905 minus 119889)] 119910 = 119862119879120585 (7)
where 120585 = [1205851198791 1205851198792 ]119879 120591(119905minus119889) = 119869(1205851)119872minus1120591(119905minus119889) and1198910(120585) =119891(120585) minus 11986011205851 minus 11986021205852According to the approximation of NN there exists a
bounded reconstruction error 120576(120576 le 120576) and an ideal weight119882 such that system (4) is described by
120585 = 119860120585 + 119861 [119882119879Φ (120585) + 120591 (119905 minus 119889) + 120576] 119910 = 119862119879120585 (8)
where 119882 is the ideal NN weight and 119882 le 119882 (119882 is apositive constant) The sigmoid function Φ(119885) = [Φ1(119885)Φ2(119885) sdot sdot sdot Φ119899(119885)]119879 isin 119877119899 is differentiable with respect to 119909and Φ(sdot) le Φ holds with a positive constantΦ
Journal of Control Science and Engineering 3
Then the NN of system (8) can be written as
120585 = 119860120585 + 119861 [119879Φ(120585) + 120591 (119905 minus 119889)] 119910 = 119862119879120585 (9)
where 120585 is a state vector of NN and is a synaptic weightmatrixThe sigmoid functionΦ(119909) = 119886(1 + 119890minus119887119909) + 119888 forall119886 119887 isin119877+ 119888 isin 119877 where 119886 119887 and the real number 119888 are the boundthe slope and the bias of sigmoidal curvature respectively
Let estimation error 120585 = 120585minus120585 output error 119910 = 119910minus119910 andNN weight error = 119882 minus Using (8) and (9) we have
120585 = 119860120585 + 119861 [119882119879Φ (120585) minus 119879Φ(120585) + 120576] 119910 = 119862119879120585 (10)
Next a main result will be given In the following 120582m(119875)and 120582M(119875) denote the minimum and maximum eigenvalueof corresponding matrix 119875Theorem 2 Consider system (1) with the identification model(9) and conditions (19) Let the NN weight update law beprovided by
119882 = Γ (119910119879Φ(120585) minus 120583) (11)
in which Γ = Γ119879 gt 0 is the learning parameter and 120583 isa constant Then the estimation error 120585 and neural networkweight error are uniformly ultimately bounded (UUB)
Proof Consider a Lyapunov function defined by
119881 = 12120585119879119875120585 + 12 tr (119879Γminus1) (12)
Calculate the derivative (12) along (10) and (11) we have
= minus12120585119879119876120585 + 120585119879119875119861 [119882119879Φ (120585) minus 119879Φ(120585) + 120576]
+ tr (119879Γminus1 119882) (13)
From (11) it follows that
= minus12120585119879119876120585 + 120585119879119875119861 [119882119879Φ (120585) minus 119879Φ(120585) + 120576]
minus 119910119879Φ(120585) + 120583 tr (119879) (14)
From the definition of we obtain the following equation
tr (119879) = tr (119879 (119882 minus )) = 119882119879 1003817100381710038171003817100381710038171003817100381710038171003817 minus 10038171003817100381710038171003817100381710038171003817100381710038172 (15)
Moreover there exist three positive constants 1205721 1205722 and1205723 such that
10038171003817100381710038171003817119882119879Φ (120585) minus 119879Φ(120585) + 12057610038171003817100381710038171003817 le 1205721 1003817100381710038171003817100381712058510038171003817100381710038171003817 + 1205722 1003817100381710038171003817100381710038171003817100381710038171003817 + 1205723 (16)
So
le minus (12120582m (119876) minus 1205721 119875119861) 10038171003817100381710038171003817120585100381710038171003817100381710038172
+ (1205722 119875119861 + Φ) 1003817100381710038171003817100381710038171003817100381710038171003817 1003817100381710038171003817100381712058510038171003817100381710038171003817 + 1205723 119875119861 1003817100381710038171003817100381712058510038171003817100381710038171003817+ 1205831198821003817100381710038171003817100381710038171003817100381710038171003817 minus 120583 10038171003817100381710038171003817100381710038171003817100381710038172
(17)
According to Youngrsquos inequality 119886119887 le (1198862 +12058121198872)2120581 with120581 gt 0 then there are 1205810 gt 0 1205811 gt 0 and 1205812 gt 0 such that
1003817100381710038171003817100381710038171003817100381710038171003817 1003817100381710038171003817100381712058510038171003817100381710038171003817 le100381710038171003817100381710038171205851003817100381710038171003817100381721205810 +
12058102 10038171003817100381710038171003817100381710038171003817100381710038172
1205723 119875119861 1003817100381710038171003817100381712058510038171003817100381710038171003817 le (1205723 119875119861)221205811 + 12058112 10038171003817100381710038171003817120585100381710038171003817100381710038172
119882 1003817100381710038171003817100381710038171003817100381710038171003817 le 119882221205812 + 12058122 10038171003817100381710038171003817100381710038171003817100381710038172
(18)
Assume that
1198771 = 12120582m (119876) minus 1205721 119875119861 minus 121205810 minus121205811 gt 0
1198772 = 120583 minus 121205831205812 minus
121205810 gt 0
(19)
Then
le minus1198771 10038171003817100381710038171003817120585100381710038171003817100381710038172 minus 1198772 10038171003817100381710038171003817100381710038171003817100381710038172 + 1198773 le minus120578119881 + 1198773 (20)
where
120578 = min 1198771120582M (119875) 1198772
1198773 = 12057223 119875119861221205811 + 119882221205812 (21)
Thus estimator error 120585 and NNweight error are UUB
4 Predictive Control
Input delay (measurement delay and computational delaycan be represented by input delay) is a source of instabilitywhich is frequently encountered in the practical systems Forachieving tracking performance a predictive controller isproposed to compensate for the time-delay present in AUVFigure 1 is the control structure diagram of AUV system (1)
In fact the NN weight stores the dynamical systeminformation Based on the structure of NN in (9) anonline predictor is proposed For improving the accuracy ofpath tracking effectively the nonlinear prediction model isemployed here Now let the predictor of system (8) be
120585119901 (119905 + 119889 | 119905) = 119860120585119901 (119905 + 119889 | 119905)+ 119861 [119879Φ(120585119901 (119905 + 119889 | 119905)) + 120591 (119905)]
119910119901 (119905 + 119889 | 119905) = 119862119879120585119901 (119905 + 119889 | 119905) (22)
4 Journal of Control Science and Engineering
Predictive model
RNN
AUV
Delay
y(t)r(t)
W(t)
[0 ΛT]
[ΛT 1]
BT A
Kr eminussd
d(t)
xd(t) (t)
p(t + d | t)
(t) (t minus d)
y (t)minus
minus
minus
minus+
+
minus
minus
+
WT (t) Φ (p (t + )d | t)
Figure 1 Control structure of AUV system (1)
where 120585119901(119905 + 119889 | 119905) and 119910119901(119905 + 119889 | 119905) are the prediction stateand output of system (8) with the initial condition 120585119901(119889 | 0) =120585(0)
If prediction model (22) is precise then 120585119901(119905 + 119889 | 119905) =120585(119905 + 119889) This mean that 120585 ahead of time 119889 can be predictedvia 120585119901(119905+119889 | 119905) in predictionmodelTherefore the difficulty incontrolling time-delay plant can be overcome However dueto the modeling errors in prediction model (22) errors existinevitably Now define a predictor error as 119890(119905) = 120585(119905 + 119889) minus120585119901(119905 + 119889 | 119905) It follows from (8) and (22) that
119890 (119905) = 119860119890 (119905) + 119861 [119882119879Φ (120585 (119905 + 119889))minus 119879 (119905) Φ (120585119901 (119905 + 119889 | 119905)) + 120576 (119905 + 119889)]
119910119890 (119905) = 119862119879119890 (119905) (23)
Next we will prove that the predictor error (23) isbounded Define an error vector as
120575 (119905) = 120585119901 (119905 + 119889 | 119905) minus 120578119889 (119905) (24)
Define a filtered error as
119903 (119905) = Λ119879120575 (119905) = [Λ119879 1] 120575 (119905) (25)
where Λ = [1205821 1205822]119879 is an appropriately chosen coefficientvector such that 120575(119905) rarr 0 exponentially as 119903(119905) rarr 0 Thenusing (22) the filtered error can be written as
119903 (119905) = Λ119879 120575 (119905) = Λ119879 [119860120585119901 (119905 + 119889 | 119905)+ 119861 (119879Φ(120585119901 (119905 + 119889 | 119905)) + 120591 (119905))]= [0 Λ119879] 120575 (119905) minus 120578119889 + 119861119879119860120585119901 (119905 + 119889 | 119905)+ 119879Φ(120585119901 (119905 + 119889 | 119905)) + 120591 (119905)
(26)
Now choose 119870119903 gt 0 and let
120591 (119905) = minus [0 Λ119879] 120575 (119905) + 120578119889 minus 119861119879119860120585119901 (119905 + 119889 | 119905)minus 119879Φ(120585119901 (119905 + 119889 | 119905)) minus 119870119903119903 (119905)
= minus [0 Λ119879] 120575 (119905) + 120578119889 minus 119861119879119860120575 (119905) minus 119861119879119860120578119889 (119905)minus 119879Φ(120585119901 (119905 + 119889 | 119905)) minus 119870119903119903 (119905)
(27)
That is a control input of AUV is
120591 (119905) = 119872119869minus1 (1205851) (minus [0 Λ119879] 120575 (119905) + 120578119889minus 119861119879119860120585119901 (119905 + 119889 | 119905) minus 119879Φ(120585119901 (119905 + 119889 | 119905))minus 119870119903119903 (119905)) = 119872119869minus1 (1205851) (minus [0 Λ119879] 120575 (119905) + 120578119889minus 119861119879119860120575 (119905) minus 119861119879119860119909119889 (119905) minus 119879Φ(120585119901 (119905 + 119889 | 119905))minus 119870119903119903 (119905))
(28)
Note that the predictor state 120585119901(119905 + 119889 | 119905) and theassociated error 119903(119905) are used in AUV controller (28) theNN approximation term 119879(119905)Φ(120585119901(119905 + 119889 | 119905)) from (22)is employed to accommodate the unknown nonlinearityTherefore the stability of the closed-loop system can beguaranteed
From (28) (26) becomes
119903 (119905) = minus119870119903119903 (119905) (29)
5 Stability Analysis
Assume that the parameters are chosen such that
1198774 = 1198772 + 121205813 gt 01198775 = 1
2120582m (119876) 121205813 minus
121205814 minus 1205732 gt 0
1198776 = minus119870119903 gt 0(30)
Journal of Control Science and Engineering 5
where 1198772 is defined as in Theorem 2 and 1205813 1205814 are positiveconstants that can be chosen
Theorem 3 (let Assumption 1 hold) Consider the input delayAUV system (1) under condition (30) the NN weight updatelaw (11) and controller (28) Then(1) all the closed-loop signals are UUB(2) the path tracking error120596(119905) = 119910(119905+119889)minus120578119889(119905) convergesto a neighborhood of the origin whose size can be adjusted bycontrol parameters
Proof Consider a Lyapunov function defined by
119881 = 12120585119879119875120585 + 1
2119879Γminus1 + 12119890119879119875119890 + 1
21199032fl 1198811 + 1198812 + 1198813 + 1198814
(31)
The derivative of 1198811 and 1198812 can be deduced following theproof of Theorem 2 Thus we have
3 = 12119890119879 (119860119879119875 + 119875119860) 119890 + 119890119879119875119861 [119882119879Φ (120585 (119905 + 119889))minus 119879Φ(120585119901 (119905 + 119889 | 119905)) + 120576 (119905 + 119889)]
(32)
In fact via Taylor series expansion there exist positiveconstants 1205731 1205732 and 1205733 such that10038171003817100381710038171003817119882119879Φ (120585 (119905 + 119889)) minus 119879Φ(120585119901 (119905 + 119889 | 119905))10038171003817100381710038171003817
= 10038171003817100381710038171003817Φ (120585119901 (119905 + 119889 | 119905))+ 119882119879 [Φ (120585 (119905 + 119889)) minus Φ (120585119901 (119905 + 119889 | 119905))]10038171003817100381710038171003817le 1205731 1003817100381710038171003817100381710038171003817100381710038171003817 + 1205732 119890 + 1205733
(33)
So
3 le minus12119876m 1198902 + 119890 (1205731 1003817100381710038171003817100381710038171003817100381710038171003817 + 1205732 119890 + 1205733 + 120576) (34)
By using Youngrsquos inequality there exist positive numbers 1205813and 1205814 such that
119890 1003817100381710038171003817100381710038171003817100381710038171003817 le 119890221205813 + 12058132 10038171003817100381710038171003817100381710038171003817100381710038172
(1205733 + 120576) 119890 le (1205733 + 120576)221205814 + 12058142 1198902
(35)
3 le minus(12119876m minus 1
21205813 minus121205814 minus 1205732) 1198902
+ 121205813 1003817100381710038171003817100381710038171003817100381710038171003817 + (1205734 + 119889 + 120576)221205814
(36)
4 = 119903 119903 = minus119870119903 1199032 (37)
Therefore
le minus1198771 10038171003817100381710038171003817120585100381710038171003817100381710038172 minus 1198774 10038171003817100381710038171003817100381710038171003817100381710038172 minus 1198775 1198902 minus 1198776 1199032 + 1198777le minus120578119881 + 1198777
(38)
where 1198771 is defined as in (19) and 120578 is positive constantdefined by
120578 = min 1198771120582max (119875) 11987741198775120582max (119875) 1198776
1198777 = 1198773 + (1205734 + 119889 + 120576)221205814
(39)
Then according to Lyapunov theorem error 120585 NNweighterror predictor error 119890 and filtered error 119903 are all UUBThe control error 120575 is thus bounded based on (24) andAssumption 1 Therefore the NN weights and 120585119901(119905 + 119889 | 119905)are bounded
Finally the boundedness of path tracking error 120596(119905) willbe proved Since
120596 (119905) = 119910 (119905 + 119889) minus 120578119889 (119905)= 1205851 (119905 + 119889) minus 1205851199011 (119905 + 119889 | 119905) + 1205851199011 (119905 + 119889 | 119905)
minus 119910119889 = 1198901 (119905) + 1205751 (119905) (40)
then
lim119905rarrinfin
120596 (119905) = lim119905rarrinfin
10038171003817100381710038171198901 (119905)1003817100381710038171003817 + lim119905rarrinfin
10038171003817100381710038171205751 (119905)1003817100381710038171003817le lim119905rarrinfin
119890 (119905) + lim119905rarrinfin
120575 (119905) (41)
Therefore the tracking error120596(119905) is bounded because 119890(119905)120575(119905) and 120585(119905) are boundedRemark 4 Compared with [20ndash33] there are three advan-tages Firstly output feedback is considered in this paperSecondly the nonlinear prediction model is employed toimprove the accuracy of predictive control Finally time-delay is considered in path tracking control of AUV whichhas more real significance
6 Simulation Analysis
Example 5 The simplified dynamics model of INFANTEAUV [2] in the horizontal plane with input delay is adoptedas follows in this paper
= 119906 cos (120595) minus V sin (120595) 119910 = 119906 sin (120595) + V cos (120595)
= 1199030 = 119898VV + 119898119906119903119906119903 + 119889VΓ = 119898119903 119903 + 119889119903
(42)
where 119909 119910 and 120595 are the surge position sway position andyaw angle in the body-fixed frame and 119906 V and 119903 denotesurge sway and yaw velocities respectively Γ is the yawmomentThe symbol 119868119911 denotes the moment of inertia of theAUV119873sdot is nonlinear hydrodynamic damping and
6 Journal of Control Science and Engineering
119898V = 119898 minus 119884V119898119906119903 = 119898 minus 119884119903119889V = minus119884V119906V minus 119884V|V|V |V| 119898119903 = 119868119911 minus 119873 119903119889119903 = minus119873V119906V minus 119873V|V|V |V| minus 119873119903119906119903
(43)
Since the consideredmodel (42) is in the horizontal planeand has no disturbance we can assume that 119906 = 1119898119904in principle In the following the desired path is 119909119889(119905) =20 sin 2120587119905200 119910119889(119905) = 20 minus 20 cos 2120587119905200 and 120595119889(119905) =2120587119905200
The model parameters of INFANTE AUV are as follows
119898 = 22345119896119892119868119911 = 20001198731198982
119883 = minus142119896119892119873 119903 = minus13501198731198982119884V = minus1715119896119892119884V = minus346119896119892119898119884119903 = 435119896119892119873V = minus686119896119892119873119903 = minus1427119896119892119898
119884V|V| = minus667119896119892119898119873V|V| = 443119896119892
(44)
NN parameters are selected as follows Φ(119909) = 1(1 +119890minus119886119909) where 119886 = 05 120583 = 03 Γ = 6The delay constant 119889 = 2 Other parameters in controller
are 1198601 = minus31198683 1198602 = minus21198683 1205821 = 2 1205822 = 2 and 119870119903 = 8The initial position and the surge speed of the AUV
are (0 20) and 0119898119904 respectively The simulation resultsare shown in Figures 2ndash4 The path tracking errors in119909 119910 and 120595 are given in Figure 2 The control forces in119909 and 119910 and the control torque of yaw 120595 are given inFigure 3 From these simulation figures we can see thatthe tracking performance is unsatisfying at the beginning ofsimulation this is because the controller performs mainlydepending on the adaptive control The good tracking ofposition is obtained by the proposed adaptive NN predictivecontroller by and by Figure 4 is the path tracking inhorizontal plane From Figure 4 we can see that AUV canrealize tracking control smoothly and converge to the desiredtrajectory
7 Conclusion
This paper investigates the path tracking problem for anAUV with input delay Based on predictive and adaptive NN
minus020
0204
e x(m
)
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
minus05
0
05
e (r
ad)
minus05
0
05
e y(m
)
100 200 300 400 500 600 700 8000time (s)
Figure 2 Path tracking errors
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
0
20
40Fx
(N)
0
100
200
Fy
(N)
minus100
0
100
M
(Nm
)
Figure 3 Control forces in 119909 and 119910 and control torque of yaw
control theory a predictive controller is given The outputfeedback control algorithm is employed here The NN isused to estimate the dynamic uncertain nonlinear functioninduced by hydrodynamic coefficients and coupling of thesurge sway and yaw angular velocity The predictive controlis introduced to compensate the input delay present in AUVThe stability of the controller was analyzed by Lyapunov the-orem Simulation results showed that the proposed controllerperforms well with stability
Data Availability
We are sorry that we cannot share the data in our article nowbecause future works are based on its results The methodsin this paper are effective methods for investigation of pathfollowing for autonomous underwater vehicles with input
Journal of Control Science and Engineering 7
AUV trajectorydesired trajectory
minus5
0
5
10
15
20
25
30
35
40
45
y (m
)
minus20 minus15 minus10 minus5 0 5 10 15 20 25minus25x (m)
Figure 4 Path tracking response in 119909119910 plane
delay We will apply a patent on the relevant studies So wecannot share the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the Hubei Provincial NaturalScience Foundation of China (2016CFB273) and the Researchand Innovation Initiatives of WHPU (2018Y20)
References
[1] P Jantapremjit and P A Wilson ldquoGuidance-control based pathfollowing for homing and docking using an autonomous under-water vehiclerdquo in Proceedings of the OCEANSrsquo08 MTSIEEEKobe-Techno-Oceanrsquo08 - Voyage toward the Future OTOrsquo08 jpnApril 2008
[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007
[3] A Adhami-Mirhosseini M J Yazdanpanah and A P AguiarldquoAutomatic bottom-following for underwater robotic vehiclesrdquoAutomatica vol 50 no 8 pp 2155ndash2162 2014
[4] M Kim H Joe J Kim and S-c Yu ldquoIntegral sliding modecontroller for precise manoeuvring of autonomous underwatervehicle in the presence of unknown environmental distur-bancesrdquo International Journal of Control vol 88 no 10 pp2055ndash2065 2015
[5] R Cui X Zhang and D Cui ldquoAdaptive sliding-mode attitudecontrol for autonomous underwater vehicles with input nonlin-earitiesrdquo Ocean Engineering vol 123 pp 45ndash54 2016
[6] Y Wang L Gu M Gao and K Zhu ldquoMultivariable outputfeedback adaptive terminal sliding mode control for underwa-ter vehiclesrdquoAsian Journal of Control vol 18 no 1 pp 247ndash2652016
[7] X Qi ldquoAdaptive coordinated tracking control of multipleautonomous underwater vehiclesrdquo Ocean Engineering vol 91pp 84ndash90 2014
[8] R Cui C Yang Y Li and S Sharma ldquoAdaptive Neural NetworkControl of AUVs With Control Input Nonlinearities UsingReinforcement Learningrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 47 no 6 pp 1019ndash1029 2017
[9] B B Miao T S Li and W L Luo ldquoA DSC and MLP basedrobust adaptive NN tracking control for underwater vehiclerdquoNeurocomputing vol 111 pp 184ndash189 2013
[10] Y-C Liu S-Y Liu and N Wang ldquoFully-tuned fuzzy neuralnetwork based robust adaptive tracking control of unmannedunderwater vehicle with thruster dynamicsrdquo Neurocomputingvol 196 pp 1ndash13 2016
[11] J Ghommam and M Saad ldquoBackstepping-based cooperativeand adaptive tracking control design for a group of underactu-ated AUVs in horizontal planrdquo International Journal of Controlvol 87 no 5 pp 1076ndash1093 2014
[12] C Shen B Buckham and Y Shi ldquoModified CGMRES Algo-rithm for Fast Nonlinear Model Predictive Tracking Control ofAUVsrdquo IEEE Transactions on Control Systems Technology vol25 no 5 pp 1896ndash1904 2017
[13] J Gao A A Proctor Y Shi and C Bradley ldquoHierarchicalModel Predictive Image-Based Visual Servoing of UnderwaterVehicles with Adaptive Neural Network Dynamic ControlrdquoIEEE Transactions on Cybernetics vol 46 no 10 pp 2323ndash23342016
[14] J Sun and J Chen ldquoNetworked predictive control for systemswith unknown or partially known delayrdquo IET Control Theory ampApplications vol 8 no 18 pp 2282ndash2288 2014
[15] S Li and G-P Liu ldquoNetworked predictive control for nonlinearsystems with stochastic disturbances in the presence of datalossesrdquo Neurocomputing vol 194 pp 56ndash64 2016
[16] J Zhao L Zhang and X Qi ldquoA necessary and sufficientcondition for stabilization of switched descriptor time-delaysystems under arbitrary switchingrdquo Asian Journal of Controlvol 18 no 1 pp 266ndash272 2016
[17] S Sirouspour and A Shahdi ldquoModel predictive control fortransparent teleoperation under communication time delayrdquoIEEE Transactions on Robotics vol 22 no 6 pp 1131ndash1145 2006
[18] Y Sheng Y Shen and M Zhu ldquoDelay-dependent globalexponential stability for delayed recurrent neural networksrdquoIEEE Transactions on Neural Networks and Learning Systems2016
[19] O M Smith ldquoA controller to overcome deadtimerdquo ISA J vol 6p 2833 1959
[20] C F Caruntu and C Lazar ldquoNetwork delay predictive com-pensation based on time-delay modelling as disturbancerdquoInternational Journal of Control vol 87 no 10 pp 2012ndash20262014
[21] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
[22] S H HosseinNia and M Lundh ldquoA general robust MPCdesign for the state-space model application to paper machineprocessrdquo Asian Journal of Control vol 18 no 5 pp 1891ndash19072016
[23] Y Ding Z Xu J Zhao KWang and Z Shao ldquoEmbeddedMPCcontroller based on interior-point method with convergencedepth controlrdquoAsian Journal of Control vol 18 no 6 pp 2064ndash2077 2016
8 Journal of Control Science and Engineering
[24] J-Q Huang and F L Lewis ldquoNeural-network predictive controlfor nonlinear dynamic systems with time-delayrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 14 no 2pp 377ndash389 2003
[25] B An and G Liu ldquoModel-based predictive controller designfor a class of nonlinear networked systemswith communicationdelays and data lossrdquo Chinese Physics B vol 23 no 8 p 0802022014
[26] N Sharma C M Gregory and W E Dixon ldquoPredictor-basedcompensation for electromechanical delay during neuromuscu-lar electrical stimulationrdquo IEEE Transactions on Neural Systemsand Rehabilitation Engineering vol 19 no 6 pp 601ndash611 2011
[27] N Fischer A Dani N Sharma and W E Dixon ldquoSaturatedcontrol of an uncertain nonlinear system with input delayrdquoAutomatica vol 49 no 6 pp 1741ndash1747 2013
[28] T Wang H Gao and J Qiu ldquoA combined adaptive neuralnetwork and nonlinear model predictive control for multiratenetworked industrial process controlrdquo IEEE Transactions onNeural Networks and Learning Systems vol 27 no 2 pp 416ndash425 2016
[29] PDai ZGong andGGuo ldquoPerformance analysis of amodularmultilevel converter drive system with model predictive con-trolrdquo International Journal of Control andAutomation vol 9 no1 pp 387ndash398 2016
[30] C Wei J Luo H Dai Z Yin W Ma and J Yuan ldquoGloballyrobust explicit model predictive control of constrained systemsexploiting SVM-based approximationrdquo International Journal ofRobust and Nonlinear Control vol 27 no 16 pp 3000ndash30272017
[31] M Parvez Akter S Mekhilef N Mei Lin Tan and H AkagildquoModified Model Predictive Control of a Bidirectional AC-DCConverter Based on Lyapunov Function for Energy StorageSystemsrdquo IEEETransactions on Industrial Electronics vol 63 no2 pp 704ndash715 2016
[32] R Prasanth Kumar A Dasgupta and C S Kumar ldquoRobusttrajectory control of underwater vehicles using time delaycontrol lawrdquo Ocean Engineering vol 34 no 5-6 pp 842ndash8492007
[33] K Mukherjee I N Kar and R K P Bhatt ldquoRegion trackingbased control of an autonomous underwater vehicle with inputdelayrdquo Ocean Engineering vol 99 pp 107ndash114 2015
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2 Journal of Control Science and Engineering
nonlinear part of AUV at first Then predictive control algo-rithm is employed to compensate for the delay produced ininput channel The proposed predictive model is a nonlinearmodel Stability of the closed-loop system is guaranteed basedon Lyapunov stability theory Finally a simulation example ispresented to show the effectiveness of the proposed controlstrategy
The remainder of this paper is organized as follows Theproblem of path tracking for AUV is formulated in Section 2Section 3 is devoted to identification of AUV system by NNStability analysis for the boundness of error state and NNweight estimation error are also performed The predictorand the corresponding control are derived in Section 4 Theproblems of dealing with the time-delay and stability analysisare illustrated in Section 5 Section 6 validates the feasibilityand performance of the proposed control law by simulationexperiment Some conclusions are given in Section 7
2 Problem Formulation
In the horizontal plane a 3-DOF AUV with input delay canbe modeled as
120578 = 119869 (120578) ]119872] + 119862 (]) ] + 119863 (]) ] + 119892 (120578) = 120591 (119905 minus 119889)
ℎ = 120578(1)
where 120578 = [119909 119910 120595]119879 denotes the vehicle location andorientation in the earth-fixed frameThe vector ] = [119906 V 119903]119879is the velocities expressed in the body-fixed frame 119872 =119872119877119861+119872119860 is the inertia matrix of rigid body119872119877119861 with addedmass119872119860Thematrix119862(]) is skew symmetrical and it denotedthe Coriolis and centripetal forces Linear and quadraticdamping forces are considered in the total hydrodynamicdamping matrix 119863(]) The vector 119892(120578) is the combinedgravitational and buoyancy forces in the body-fixed frame120591 is the input of the system and the vector of the forcesand moments on AUV induced by the input and fins 119889 isa known constant time-delay ℎ is the output of the systemThe kinematic transformation matrix transformation fromthe body-fixed frame to earth-fixed frame is denoted by 119869(120578)and
119869 (120578) = 119877 (120595) = [[[
cos120595 minus sin120595 0sin120595 cos120595 00 0 1
]]]
(2)
Let1205851 = 1205781205852 = 119869 (120578) ] (3)
Then system (1) changed to1205851 = 12058521205852 = 119891 (1205851 1205852) + 119869 (1205851)119872minus1120591 (119905 minus 119889) 119910 = 1205851
(4)
where nonlinear uncertain function
119891 (1205851 1205852) = 119869 (1205851) 119869minus1 (1205851) 1205852 + 119869 (1205851)sdot 119872minus1 [minus119862 (119869minus1 (1205851) 1205852)minus 119863 (119869minus1 (1205851) 1205852) 119869minus1 (1205851) 1205852 minus 119892 (1205851)]
(5)
The objective of this paper is that the output 119910 of system(4) tracks a desired trajectory 120578119889 with all internal signals andcontrol commands remaining bounded For this purpose wemake the following assumption
Assumption 1 The desired trajectory vector 120577119889 =[120578119889 120578119889 120578119889]119879 is available for measurement and 120578119889 and120578119889 are bounded3 Identification of AUV System
There are two steps to design the output feedback controllerfor AUVwith input delay First an NN is designed to identifysystem (4) Then we will use predictive control algorithmto compensate for the delay that presents in communicationchannel of AUV
Let
119860 = [ 0 11986831198601 1198602]
119861 = [01198683]
119862 = [1198683 0]119879
(6)
where 1198683 denotes the identity matrix matrices1198601 and1198602 areparameters that can be chosen such that matrix 119860 is stableThen there exist symmetric positive definite matrices 119875 and119876 such that Lyapunovmatrix equations119860119879119875+119875119860 = minus119876 hold
Hence system (4) can be expressed as
120585 = 119860120585 + 119861 [1198910 (120585) + 120591 (119905 minus 119889)] 119910 = 119862119879120585 (7)
where 120585 = [1205851198791 1205851198792 ]119879 120591(119905minus119889) = 119869(1205851)119872minus1120591(119905minus119889) and1198910(120585) =119891(120585) minus 11986011205851 minus 11986021205852According to the approximation of NN there exists a
bounded reconstruction error 120576(120576 le 120576) and an ideal weight119882 such that system (4) is described by
120585 = 119860120585 + 119861 [119882119879Φ (120585) + 120591 (119905 minus 119889) + 120576] 119910 = 119862119879120585 (8)
where 119882 is the ideal NN weight and 119882 le 119882 (119882 is apositive constant) The sigmoid function Φ(119885) = [Φ1(119885)Φ2(119885) sdot sdot sdot Φ119899(119885)]119879 isin 119877119899 is differentiable with respect to 119909and Φ(sdot) le Φ holds with a positive constantΦ
Journal of Control Science and Engineering 3
Then the NN of system (8) can be written as
120585 = 119860120585 + 119861 [119879Φ(120585) + 120591 (119905 minus 119889)] 119910 = 119862119879120585 (9)
where 120585 is a state vector of NN and is a synaptic weightmatrixThe sigmoid functionΦ(119909) = 119886(1 + 119890minus119887119909) + 119888 forall119886 119887 isin119877+ 119888 isin 119877 where 119886 119887 and the real number 119888 are the boundthe slope and the bias of sigmoidal curvature respectively
Let estimation error 120585 = 120585minus120585 output error 119910 = 119910minus119910 andNN weight error = 119882 minus Using (8) and (9) we have
120585 = 119860120585 + 119861 [119882119879Φ (120585) minus 119879Φ(120585) + 120576] 119910 = 119862119879120585 (10)
Next a main result will be given In the following 120582m(119875)and 120582M(119875) denote the minimum and maximum eigenvalueof corresponding matrix 119875Theorem 2 Consider system (1) with the identification model(9) and conditions (19) Let the NN weight update law beprovided by
119882 = Γ (119910119879Φ(120585) minus 120583) (11)
in which Γ = Γ119879 gt 0 is the learning parameter and 120583 isa constant Then the estimation error 120585 and neural networkweight error are uniformly ultimately bounded (UUB)
Proof Consider a Lyapunov function defined by
119881 = 12120585119879119875120585 + 12 tr (119879Γminus1) (12)
Calculate the derivative (12) along (10) and (11) we have
= minus12120585119879119876120585 + 120585119879119875119861 [119882119879Φ (120585) minus 119879Φ(120585) + 120576]
+ tr (119879Γminus1 119882) (13)
From (11) it follows that
= minus12120585119879119876120585 + 120585119879119875119861 [119882119879Φ (120585) minus 119879Φ(120585) + 120576]
minus 119910119879Φ(120585) + 120583 tr (119879) (14)
From the definition of we obtain the following equation
tr (119879) = tr (119879 (119882 minus )) = 119882119879 1003817100381710038171003817100381710038171003817100381710038171003817 minus 10038171003817100381710038171003817100381710038171003817100381710038172 (15)
Moreover there exist three positive constants 1205721 1205722 and1205723 such that
10038171003817100381710038171003817119882119879Φ (120585) minus 119879Φ(120585) + 12057610038171003817100381710038171003817 le 1205721 1003817100381710038171003817100381712058510038171003817100381710038171003817 + 1205722 1003817100381710038171003817100381710038171003817100381710038171003817 + 1205723 (16)
So
le minus (12120582m (119876) minus 1205721 119875119861) 10038171003817100381710038171003817120585100381710038171003817100381710038172
+ (1205722 119875119861 + Φ) 1003817100381710038171003817100381710038171003817100381710038171003817 1003817100381710038171003817100381712058510038171003817100381710038171003817 + 1205723 119875119861 1003817100381710038171003817100381712058510038171003817100381710038171003817+ 1205831198821003817100381710038171003817100381710038171003817100381710038171003817 minus 120583 10038171003817100381710038171003817100381710038171003817100381710038172
(17)
According to Youngrsquos inequality 119886119887 le (1198862 +12058121198872)2120581 with120581 gt 0 then there are 1205810 gt 0 1205811 gt 0 and 1205812 gt 0 such that
1003817100381710038171003817100381710038171003817100381710038171003817 1003817100381710038171003817100381712058510038171003817100381710038171003817 le100381710038171003817100381710038171205851003817100381710038171003817100381721205810 +
12058102 10038171003817100381710038171003817100381710038171003817100381710038172
1205723 119875119861 1003817100381710038171003817100381712058510038171003817100381710038171003817 le (1205723 119875119861)221205811 + 12058112 10038171003817100381710038171003817120585100381710038171003817100381710038172
119882 1003817100381710038171003817100381710038171003817100381710038171003817 le 119882221205812 + 12058122 10038171003817100381710038171003817100381710038171003817100381710038172
(18)
Assume that
1198771 = 12120582m (119876) minus 1205721 119875119861 minus 121205810 minus121205811 gt 0
1198772 = 120583 minus 121205831205812 minus
121205810 gt 0
(19)
Then
le minus1198771 10038171003817100381710038171003817120585100381710038171003817100381710038172 minus 1198772 10038171003817100381710038171003817100381710038171003817100381710038172 + 1198773 le minus120578119881 + 1198773 (20)
where
120578 = min 1198771120582M (119875) 1198772
1198773 = 12057223 119875119861221205811 + 119882221205812 (21)
Thus estimator error 120585 and NNweight error are UUB
4 Predictive Control
Input delay (measurement delay and computational delaycan be represented by input delay) is a source of instabilitywhich is frequently encountered in the practical systems Forachieving tracking performance a predictive controller isproposed to compensate for the time-delay present in AUVFigure 1 is the control structure diagram of AUV system (1)
In fact the NN weight stores the dynamical systeminformation Based on the structure of NN in (9) anonline predictor is proposed For improving the accuracy ofpath tracking effectively the nonlinear prediction model isemployed here Now let the predictor of system (8) be
120585119901 (119905 + 119889 | 119905) = 119860120585119901 (119905 + 119889 | 119905)+ 119861 [119879Φ(120585119901 (119905 + 119889 | 119905)) + 120591 (119905)]
119910119901 (119905 + 119889 | 119905) = 119862119879120585119901 (119905 + 119889 | 119905) (22)
4 Journal of Control Science and Engineering
Predictive model
RNN
AUV
Delay
y(t)r(t)
W(t)
[0 ΛT]
[ΛT 1]
BT A
Kr eminussd
d(t)
xd(t) (t)
p(t + d | t)
(t) (t minus d)
y (t)minus
minus
minus
minus+
+
minus
minus
+
WT (t) Φ (p (t + )d | t)
Figure 1 Control structure of AUV system (1)
where 120585119901(119905 + 119889 | 119905) and 119910119901(119905 + 119889 | 119905) are the prediction stateand output of system (8) with the initial condition 120585119901(119889 | 0) =120585(0)
If prediction model (22) is precise then 120585119901(119905 + 119889 | 119905) =120585(119905 + 119889) This mean that 120585 ahead of time 119889 can be predictedvia 120585119901(119905+119889 | 119905) in predictionmodelTherefore the difficulty incontrolling time-delay plant can be overcome However dueto the modeling errors in prediction model (22) errors existinevitably Now define a predictor error as 119890(119905) = 120585(119905 + 119889) minus120585119901(119905 + 119889 | 119905) It follows from (8) and (22) that
119890 (119905) = 119860119890 (119905) + 119861 [119882119879Φ (120585 (119905 + 119889))minus 119879 (119905) Φ (120585119901 (119905 + 119889 | 119905)) + 120576 (119905 + 119889)]
119910119890 (119905) = 119862119879119890 (119905) (23)
Next we will prove that the predictor error (23) isbounded Define an error vector as
120575 (119905) = 120585119901 (119905 + 119889 | 119905) minus 120578119889 (119905) (24)
Define a filtered error as
119903 (119905) = Λ119879120575 (119905) = [Λ119879 1] 120575 (119905) (25)
where Λ = [1205821 1205822]119879 is an appropriately chosen coefficientvector such that 120575(119905) rarr 0 exponentially as 119903(119905) rarr 0 Thenusing (22) the filtered error can be written as
119903 (119905) = Λ119879 120575 (119905) = Λ119879 [119860120585119901 (119905 + 119889 | 119905)+ 119861 (119879Φ(120585119901 (119905 + 119889 | 119905)) + 120591 (119905))]= [0 Λ119879] 120575 (119905) minus 120578119889 + 119861119879119860120585119901 (119905 + 119889 | 119905)+ 119879Φ(120585119901 (119905 + 119889 | 119905)) + 120591 (119905)
(26)
Now choose 119870119903 gt 0 and let
120591 (119905) = minus [0 Λ119879] 120575 (119905) + 120578119889 minus 119861119879119860120585119901 (119905 + 119889 | 119905)minus 119879Φ(120585119901 (119905 + 119889 | 119905)) minus 119870119903119903 (119905)
= minus [0 Λ119879] 120575 (119905) + 120578119889 minus 119861119879119860120575 (119905) minus 119861119879119860120578119889 (119905)minus 119879Φ(120585119901 (119905 + 119889 | 119905)) minus 119870119903119903 (119905)
(27)
That is a control input of AUV is
120591 (119905) = 119872119869minus1 (1205851) (minus [0 Λ119879] 120575 (119905) + 120578119889minus 119861119879119860120585119901 (119905 + 119889 | 119905) minus 119879Φ(120585119901 (119905 + 119889 | 119905))minus 119870119903119903 (119905)) = 119872119869minus1 (1205851) (minus [0 Λ119879] 120575 (119905) + 120578119889minus 119861119879119860120575 (119905) minus 119861119879119860119909119889 (119905) minus 119879Φ(120585119901 (119905 + 119889 | 119905))minus 119870119903119903 (119905))
(28)
Note that the predictor state 120585119901(119905 + 119889 | 119905) and theassociated error 119903(119905) are used in AUV controller (28) theNN approximation term 119879(119905)Φ(120585119901(119905 + 119889 | 119905)) from (22)is employed to accommodate the unknown nonlinearityTherefore the stability of the closed-loop system can beguaranteed
From (28) (26) becomes
119903 (119905) = minus119870119903119903 (119905) (29)
5 Stability Analysis
Assume that the parameters are chosen such that
1198774 = 1198772 + 121205813 gt 01198775 = 1
2120582m (119876) 121205813 minus
121205814 minus 1205732 gt 0
1198776 = minus119870119903 gt 0(30)
Journal of Control Science and Engineering 5
where 1198772 is defined as in Theorem 2 and 1205813 1205814 are positiveconstants that can be chosen
Theorem 3 (let Assumption 1 hold) Consider the input delayAUV system (1) under condition (30) the NN weight updatelaw (11) and controller (28) Then(1) all the closed-loop signals are UUB(2) the path tracking error120596(119905) = 119910(119905+119889)minus120578119889(119905) convergesto a neighborhood of the origin whose size can be adjusted bycontrol parameters
Proof Consider a Lyapunov function defined by
119881 = 12120585119879119875120585 + 1
2119879Γminus1 + 12119890119879119875119890 + 1
21199032fl 1198811 + 1198812 + 1198813 + 1198814
(31)
The derivative of 1198811 and 1198812 can be deduced following theproof of Theorem 2 Thus we have
3 = 12119890119879 (119860119879119875 + 119875119860) 119890 + 119890119879119875119861 [119882119879Φ (120585 (119905 + 119889))minus 119879Φ(120585119901 (119905 + 119889 | 119905)) + 120576 (119905 + 119889)]
(32)
In fact via Taylor series expansion there exist positiveconstants 1205731 1205732 and 1205733 such that10038171003817100381710038171003817119882119879Φ (120585 (119905 + 119889)) minus 119879Φ(120585119901 (119905 + 119889 | 119905))10038171003817100381710038171003817
= 10038171003817100381710038171003817Φ (120585119901 (119905 + 119889 | 119905))+ 119882119879 [Φ (120585 (119905 + 119889)) minus Φ (120585119901 (119905 + 119889 | 119905))]10038171003817100381710038171003817le 1205731 1003817100381710038171003817100381710038171003817100381710038171003817 + 1205732 119890 + 1205733
(33)
So
3 le minus12119876m 1198902 + 119890 (1205731 1003817100381710038171003817100381710038171003817100381710038171003817 + 1205732 119890 + 1205733 + 120576) (34)
By using Youngrsquos inequality there exist positive numbers 1205813and 1205814 such that
119890 1003817100381710038171003817100381710038171003817100381710038171003817 le 119890221205813 + 12058132 10038171003817100381710038171003817100381710038171003817100381710038172
(1205733 + 120576) 119890 le (1205733 + 120576)221205814 + 12058142 1198902
(35)
3 le minus(12119876m minus 1
21205813 minus121205814 minus 1205732) 1198902
+ 121205813 1003817100381710038171003817100381710038171003817100381710038171003817 + (1205734 + 119889 + 120576)221205814
(36)
4 = 119903 119903 = minus119870119903 1199032 (37)
Therefore
le minus1198771 10038171003817100381710038171003817120585100381710038171003817100381710038172 minus 1198774 10038171003817100381710038171003817100381710038171003817100381710038172 minus 1198775 1198902 minus 1198776 1199032 + 1198777le minus120578119881 + 1198777
(38)
where 1198771 is defined as in (19) and 120578 is positive constantdefined by
120578 = min 1198771120582max (119875) 11987741198775120582max (119875) 1198776
1198777 = 1198773 + (1205734 + 119889 + 120576)221205814
(39)
Then according to Lyapunov theorem error 120585 NNweighterror predictor error 119890 and filtered error 119903 are all UUBThe control error 120575 is thus bounded based on (24) andAssumption 1 Therefore the NN weights and 120585119901(119905 + 119889 | 119905)are bounded
Finally the boundedness of path tracking error 120596(119905) willbe proved Since
120596 (119905) = 119910 (119905 + 119889) minus 120578119889 (119905)= 1205851 (119905 + 119889) minus 1205851199011 (119905 + 119889 | 119905) + 1205851199011 (119905 + 119889 | 119905)
minus 119910119889 = 1198901 (119905) + 1205751 (119905) (40)
then
lim119905rarrinfin
120596 (119905) = lim119905rarrinfin
10038171003817100381710038171198901 (119905)1003817100381710038171003817 + lim119905rarrinfin
10038171003817100381710038171205751 (119905)1003817100381710038171003817le lim119905rarrinfin
119890 (119905) + lim119905rarrinfin
120575 (119905) (41)
Therefore the tracking error120596(119905) is bounded because 119890(119905)120575(119905) and 120585(119905) are boundedRemark 4 Compared with [20ndash33] there are three advan-tages Firstly output feedback is considered in this paperSecondly the nonlinear prediction model is employed toimprove the accuracy of predictive control Finally time-delay is considered in path tracking control of AUV whichhas more real significance
6 Simulation Analysis
Example 5 The simplified dynamics model of INFANTEAUV [2] in the horizontal plane with input delay is adoptedas follows in this paper
= 119906 cos (120595) minus V sin (120595) 119910 = 119906 sin (120595) + V cos (120595)
= 1199030 = 119898VV + 119898119906119903119906119903 + 119889VΓ = 119898119903 119903 + 119889119903
(42)
where 119909 119910 and 120595 are the surge position sway position andyaw angle in the body-fixed frame and 119906 V and 119903 denotesurge sway and yaw velocities respectively Γ is the yawmomentThe symbol 119868119911 denotes the moment of inertia of theAUV119873sdot is nonlinear hydrodynamic damping and
6 Journal of Control Science and Engineering
119898V = 119898 minus 119884V119898119906119903 = 119898 minus 119884119903119889V = minus119884V119906V minus 119884V|V|V |V| 119898119903 = 119868119911 minus 119873 119903119889119903 = minus119873V119906V minus 119873V|V|V |V| minus 119873119903119906119903
(43)
Since the consideredmodel (42) is in the horizontal planeand has no disturbance we can assume that 119906 = 1119898119904in principle In the following the desired path is 119909119889(119905) =20 sin 2120587119905200 119910119889(119905) = 20 minus 20 cos 2120587119905200 and 120595119889(119905) =2120587119905200
The model parameters of INFANTE AUV are as follows
119898 = 22345119896119892119868119911 = 20001198731198982
119883 = minus142119896119892119873 119903 = minus13501198731198982119884V = minus1715119896119892119884V = minus346119896119892119898119884119903 = 435119896119892119873V = minus686119896119892119873119903 = minus1427119896119892119898
119884V|V| = minus667119896119892119898119873V|V| = 443119896119892
(44)
NN parameters are selected as follows Φ(119909) = 1(1 +119890minus119886119909) where 119886 = 05 120583 = 03 Γ = 6The delay constant 119889 = 2 Other parameters in controller
are 1198601 = minus31198683 1198602 = minus21198683 1205821 = 2 1205822 = 2 and 119870119903 = 8The initial position and the surge speed of the AUV
are (0 20) and 0119898119904 respectively The simulation resultsare shown in Figures 2ndash4 The path tracking errors in119909 119910 and 120595 are given in Figure 2 The control forces in119909 and 119910 and the control torque of yaw 120595 are given inFigure 3 From these simulation figures we can see thatthe tracking performance is unsatisfying at the beginning ofsimulation this is because the controller performs mainlydepending on the adaptive control The good tracking ofposition is obtained by the proposed adaptive NN predictivecontroller by and by Figure 4 is the path tracking inhorizontal plane From Figure 4 we can see that AUV canrealize tracking control smoothly and converge to the desiredtrajectory
7 Conclusion
This paper investigates the path tracking problem for anAUV with input delay Based on predictive and adaptive NN
minus020
0204
e x(m
)
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
minus05
0
05
e (r
ad)
minus05
0
05
e y(m
)
100 200 300 400 500 600 700 8000time (s)
Figure 2 Path tracking errors
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
0
20
40Fx
(N)
0
100
200
Fy
(N)
minus100
0
100
M
(Nm
)
Figure 3 Control forces in 119909 and 119910 and control torque of yaw
control theory a predictive controller is given The outputfeedback control algorithm is employed here The NN isused to estimate the dynamic uncertain nonlinear functioninduced by hydrodynamic coefficients and coupling of thesurge sway and yaw angular velocity The predictive controlis introduced to compensate the input delay present in AUVThe stability of the controller was analyzed by Lyapunov the-orem Simulation results showed that the proposed controllerperforms well with stability
Data Availability
We are sorry that we cannot share the data in our article nowbecause future works are based on its results The methodsin this paper are effective methods for investigation of pathfollowing for autonomous underwater vehicles with input
Journal of Control Science and Engineering 7
AUV trajectorydesired trajectory
minus5
0
5
10
15
20
25
30
35
40
45
y (m
)
minus20 minus15 minus10 minus5 0 5 10 15 20 25minus25x (m)
Figure 4 Path tracking response in 119909119910 plane
delay We will apply a patent on the relevant studies So wecannot share the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the Hubei Provincial NaturalScience Foundation of China (2016CFB273) and the Researchand Innovation Initiatives of WHPU (2018Y20)
References
[1] P Jantapremjit and P A Wilson ldquoGuidance-control based pathfollowing for homing and docking using an autonomous under-water vehiclerdquo in Proceedings of the OCEANSrsquo08 MTSIEEEKobe-Techno-Oceanrsquo08 - Voyage toward the Future OTOrsquo08 jpnApril 2008
[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007
[3] A Adhami-Mirhosseini M J Yazdanpanah and A P AguiarldquoAutomatic bottom-following for underwater robotic vehiclesrdquoAutomatica vol 50 no 8 pp 2155ndash2162 2014
[4] M Kim H Joe J Kim and S-c Yu ldquoIntegral sliding modecontroller for precise manoeuvring of autonomous underwatervehicle in the presence of unknown environmental distur-bancesrdquo International Journal of Control vol 88 no 10 pp2055ndash2065 2015
[5] R Cui X Zhang and D Cui ldquoAdaptive sliding-mode attitudecontrol for autonomous underwater vehicles with input nonlin-earitiesrdquo Ocean Engineering vol 123 pp 45ndash54 2016
[6] Y Wang L Gu M Gao and K Zhu ldquoMultivariable outputfeedback adaptive terminal sliding mode control for underwa-ter vehiclesrdquoAsian Journal of Control vol 18 no 1 pp 247ndash2652016
[7] X Qi ldquoAdaptive coordinated tracking control of multipleautonomous underwater vehiclesrdquo Ocean Engineering vol 91pp 84ndash90 2014
[8] R Cui C Yang Y Li and S Sharma ldquoAdaptive Neural NetworkControl of AUVs With Control Input Nonlinearities UsingReinforcement Learningrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 47 no 6 pp 1019ndash1029 2017
[9] B B Miao T S Li and W L Luo ldquoA DSC and MLP basedrobust adaptive NN tracking control for underwater vehiclerdquoNeurocomputing vol 111 pp 184ndash189 2013
[10] Y-C Liu S-Y Liu and N Wang ldquoFully-tuned fuzzy neuralnetwork based robust adaptive tracking control of unmannedunderwater vehicle with thruster dynamicsrdquo Neurocomputingvol 196 pp 1ndash13 2016
[11] J Ghommam and M Saad ldquoBackstepping-based cooperativeand adaptive tracking control design for a group of underactu-ated AUVs in horizontal planrdquo International Journal of Controlvol 87 no 5 pp 1076ndash1093 2014
[12] C Shen B Buckham and Y Shi ldquoModified CGMRES Algo-rithm for Fast Nonlinear Model Predictive Tracking Control ofAUVsrdquo IEEE Transactions on Control Systems Technology vol25 no 5 pp 1896ndash1904 2017
[13] J Gao A A Proctor Y Shi and C Bradley ldquoHierarchicalModel Predictive Image-Based Visual Servoing of UnderwaterVehicles with Adaptive Neural Network Dynamic ControlrdquoIEEE Transactions on Cybernetics vol 46 no 10 pp 2323ndash23342016
[14] J Sun and J Chen ldquoNetworked predictive control for systemswith unknown or partially known delayrdquo IET Control Theory ampApplications vol 8 no 18 pp 2282ndash2288 2014
[15] S Li and G-P Liu ldquoNetworked predictive control for nonlinearsystems with stochastic disturbances in the presence of datalossesrdquo Neurocomputing vol 194 pp 56ndash64 2016
[16] J Zhao L Zhang and X Qi ldquoA necessary and sufficientcondition for stabilization of switched descriptor time-delaysystems under arbitrary switchingrdquo Asian Journal of Controlvol 18 no 1 pp 266ndash272 2016
[17] S Sirouspour and A Shahdi ldquoModel predictive control fortransparent teleoperation under communication time delayrdquoIEEE Transactions on Robotics vol 22 no 6 pp 1131ndash1145 2006
[18] Y Sheng Y Shen and M Zhu ldquoDelay-dependent globalexponential stability for delayed recurrent neural networksrdquoIEEE Transactions on Neural Networks and Learning Systems2016
[19] O M Smith ldquoA controller to overcome deadtimerdquo ISA J vol 6p 2833 1959
[20] C F Caruntu and C Lazar ldquoNetwork delay predictive com-pensation based on time-delay modelling as disturbancerdquoInternational Journal of Control vol 87 no 10 pp 2012ndash20262014
[21] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
[22] S H HosseinNia and M Lundh ldquoA general robust MPCdesign for the state-space model application to paper machineprocessrdquo Asian Journal of Control vol 18 no 5 pp 1891ndash19072016
[23] Y Ding Z Xu J Zhao KWang and Z Shao ldquoEmbeddedMPCcontroller based on interior-point method with convergencedepth controlrdquoAsian Journal of Control vol 18 no 6 pp 2064ndash2077 2016
8 Journal of Control Science and Engineering
[24] J-Q Huang and F L Lewis ldquoNeural-network predictive controlfor nonlinear dynamic systems with time-delayrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 14 no 2pp 377ndash389 2003
[25] B An and G Liu ldquoModel-based predictive controller designfor a class of nonlinear networked systemswith communicationdelays and data lossrdquo Chinese Physics B vol 23 no 8 p 0802022014
[26] N Sharma C M Gregory and W E Dixon ldquoPredictor-basedcompensation for electromechanical delay during neuromuscu-lar electrical stimulationrdquo IEEE Transactions on Neural Systemsand Rehabilitation Engineering vol 19 no 6 pp 601ndash611 2011
[27] N Fischer A Dani N Sharma and W E Dixon ldquoSaturatedcontrol of an uncertain nonlinear system with input delayrdquoAutomatica vol 49 no 6 pp 1741ndash1747 2013
[28] T Wang H Gao and J Qiu ldquoA combined adaptive neuralnetwork and nonlinear model predictive control for multiratenetworked industrial process controlrdquo IEEE Transactions onNeural Networks and Learning Systems vol 27 no 2 pp 416ndash425 2016
[29] PDai ZGong andGGuo ldquoPerformance analysis of amodularmultilevel converter drive system with model predictive con-trolrdquo International Journal of Control andAutomation vol 9 no1 pp 387ndash398 2016
[30] C Wei J Luo H Dai Z Yin W Ma and J Yuan ldquoGloballyrobust explicit model predictive control of constrained systemsexploiting SVM-based approximationrdquo International Journal ofRobust and Nonlinear Control vol 27 no 16 pp 3000ndash30272017
[31] M Parvez Akter S Mekhilef N Mei Lin Tan and H AkagildquoModified Model Predictive Control of a Bidirectional AC-DCConverter Based on Lyapunov Function for Energy StorageSystemsrdquo IEEETransactions on Industrial Electronics vol 63 no2 pp 704ndash715 2016
[32] R Prasanth Kumar A Dasgupta and C S Kumar ldquoRobusttrajectory control of underwater vehicles using time delaycontrol lawrdquo Ocean Engineering vol 34 no 5-6 pp 842ndash8492007
[33] K Mukherjee I N Kar and R K P Bhatt ldquoRegion trackingbased control of an autonomous underwater vehicle with inputdelayrdquo Ocean Engineering vol 99 pp 107ndash114 2015
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Journal of Control Science and Engineering 3
Then the NN of system (8) can be written as
120585 = 119860120585 + 119861 [119879Φ(120585) + 120591 (119905 minus 119889)] 119910 = 119862119879120585 (9)
where 120585 is a state vector of NN and is a synaptic weightmatrixThe sigmoid functionΦ(119909) = 119886(1 + 119890minus119887119909) + 119888 forall119886 119887 isin119877+ 119888 isin 119877 where 119886 119887 and the real number 119888 are the boundthe slope and the bias of sigmoidal curvature respectively
Let estimation error 120585 = 120585minus120585 output error 119910 = 119910minus119910 andNN weight error = 119882 minus Using (8) and (9) we have
120585 = 119860120585 + 119861 [119882119879Φ (120585) minus 119879Φ(120585) + 120576] 119910 = 119862119879120585 (10)
Next a main result will be given In the following 120582m(119875)and 120582M(119875) denote the minimum and maximum eigenvalueof corresponding matrix 119875Theorem 2 Consider system (1) with the identification model(9) and conditions (19) Let the NN weight update law beprovided by
119882 = Γ (119910119879Φ(120585) minus 120583) (11)
in which Γ = Γ119879 gt 0 is the learning parameter and 120583 isa constant Then the estimation error 120585 and neural networkweight error are uniformly ultimately bounded (UUB)
Proof Consider a Lyapunov function defined by
119881 = 12120585119879119875120585 + 12 tr (119879Γminus1) (12)
Calculate the derivative (12) along (10) and (11) we have
= minus12120585119879119876120585 + 120585119879119875119861 [119882119879Φ (120585) minus 119879Φ(120585) + 120576]
+ tr (119879Γminus1 119882) (13)
From (11) it follows that
= minus12120585119879119876120585 + 120585119879119875119861 [119882119879Φ (120585) minus 119879Φ(120585) + 120576]
minus 119910119879Φ(120585) + 120583 tr (119879) (14)
From the definition of we obtain the following equation
tr (119879) = tr (119879 (119882 minus )) = 119882119879 1003817100381710038171003817100381710038171003817100381710038171003817 minus 10038171003817100381710038171003817100381710038171003817100381710038172 (15)
Moreover there exist three positive constants 1205721 1205722 and1205723 such that
10038171003817100381710038171003817119882119879Φ (120585) minus 119879Φ(120585) + 12057610038171003817100381710038171003817 le 1205721 1003817100381710038171003817100381712058510038171003817100381710038171003817 + 1205722 1003817100381710038171003817100381710038171003817100381710038171003817 + 1205723 (16)
So
le minus (12120582m (119876) minus 1205721 119875119861) 10038171003817100381710038171003817120585100381710038171003817100381710038172
+ (1205722 119875119861 + Φ) 1003817100381710038171003817100381710038171003817100381710038171003817 1003817100381710038171003817100381712058510038171003817100381710038171003817 + 1205723 119875119861 1003817100381710038171003817100381712058510038171003817100381710038171003817+ 1205831198821003817100381710038171003817100381710038171003817100381710038171003817 minus 120583 10038171003817100381710038171003817100381710038171003817100381710038172
(17)
According to Youngrsquos inequality 119886119887 le (1198862 +12058121198872)2120581 with120581 gt 0 then there are 1205810 gt 0 1205811 gt 0 and 1205812 gt 0 such that
1003817100381710038171003817100381710038171003817100381710038171003817 1003817100381710038171003817100381712058510038171003817100381710038171003817 le100381710038171003817100381710038171205851003817100381710038171003817100381721205810 +
12058102 10038171003817100381710038171003817100381710038171003817100381710038172
1205723 119875119861 1003817100381710038171003817100381712058510038171003817100381710038171003817 le (1205723 119875119861)221205811 + 12058112 10038171003817100381710038171003817120585100381710038171003817100381710038172
119882 1003817100381710038171003817100381710038171003817100381710038171003817 le 119882221205812 + 12058122 10038171003817100381710038171003817100381710038171003817100381710038172
(18)
Assume that
1198771 = 12120582m (119876) minus 1205721 119875119861 minus 121205810 minus121205811 gt 0
1198772 = 120583 minus 121205831205812 minus
121205810 gt 0
(19)
Then
le minus1198771 10038171003817100381710038171003817120585100381710038171003817100381710038172 minus 1198772 10038171003817100381710038171003817100381710038171003817100381710038172 + 1198773 le minus120578119881 + 1198773 (20)
where
120578 = min 1198771120582M (119875) 1198772
1198773 = 12057223 119875119861221205811 + 119882221205812 (21)
Thus estimator error 120585 and NNweight error are UUB
4 Predictive Control
Input delay (measurement delay and computational delaycan be represented by input delay) is a source of instabilitywhich is frequently encountered in the practical systems Forachieving tracking performance a predictive controller isproposed to compensate for the time-delay present in AUVFigure 1 is the control structure diagram of AUV system (1)
In fact the NN weight stores the dynamical systeminformation Based on the structure of NN in (9) anonline predictor is proposed For improving the accuracy ofpath tracking effectively the nonlinear prediction model isemployed here Now let the predictor of system (8) be
120585119901 (119905 + 119889 | 119905) = 119860120585119901 (119905 + 119889 | 119905)+ 119861 [119879Φ(120585119901 (119905 + 119889 | 119905)) + 120591 (119905)]
119910119901 (119905 + 119889 | 119905) = 119862119879120585119901 (119905 + 119889 | 119905) (22)
4 Journal of Control Science and Engineering
Predictive model
RNN
AUV
Delay
y(t)r(t)
W(t)
[0 ΛT]
[ΛT 1]
BT A
Kr eminussd
d(t)
xd(t) (t)
p(t + d | t)
(t) (t minus d)
y (t)minus
minus
minus
minus+
+
minus
minus
+
WT (t) Φ (p (t + )d | t)
Figure 1 Control structure of AUV system (1)
where 120585119901(119905 + 119889 | 119905) and 119910119901(119905 + 119889 | 119905) are the prediction stateand output of system (8) with the initial condition 120585119901(119889 | 0) =120585(0)
If prediction model (22) is precise then 120585119901(119905 + 119889 | 119905) =120585(119905 + 119889) This mean that 120585 ahead of time 119889 can be predictedvia 120585119901(119905+119889 | 119905) in predictionmodelTherefore the difficulty incontrolling time-delay plant can be overcome However dueto the modeling errors in prediction model (22) errors existinevitably Now define a predictor error as 119890(119905) = 120585(119905 + 119889) minus120585119901(119905 + 119889 | 119905) It follows from (8) and (22) that
119890 (119905) = 119860119890 (119905) + 119861 [119882119879Φ (120585 (119905 + 119889))minus 119879 (119905) Φ (120585119901 (119905 + 119889 | 119905)) + 120576 (119905 + 119889)]
119910119890 (119905) = 119862119879119890 (119905) (23)
Next we will prove that the predictor error (23) isbounded Define an error vector as
120575 (119905) = 120585119901 (119905 + 119889 | 119905) minus 120578119889 (119905) (24)
Define a filtered error as
119903 (119905) = Λ119879120575 (119905) = [Λ119879 1] 120575 (119905) (25)
where Λ = [1205821 1205822]119879 is an appropriately chosen coefficientvector such that 120575(119905) rarr 0 exponentially as 119903(119905) rarr 0 Thenusing (22) the filtered error can be written as
119903 (119905) = Λ119879 120575 (119905) = Λ119879 [119860120585119901 (119905 + 119889 | 119905)+ 119861 (119879Φ(120585119901 (119905 + 119889 | 119905)) + 120591 (119905))]= [0 Λ119879] 120575 (119905) minus 120578119889 + 119861119879119860120585119901 (119905 + 119889 | 119905)+ 119879Φ(120585119901 (119905 + 119889 | 119905)) + 120591 (119905)
(26)
Now choose 119870119903 gt 0 and let
120591 (119905) = minus [0 Λ119879] 120575 (119905) + 120578119889 minus 119861119879119860120585119901 (119905 + 119889 | 119905)minus 119879Φ(120585119901 (119905 + 119889 | 119905)) minus 119870119903119903 (119905)
= minus [0 Λ119879] 120575 (119905) + 120578119889 minus 119861119879119860120575 (119905) minus 119861119879119860120578119889 (119905)minus 119879Φ(120585119901 (119905 + 119889 | 119905)) minus 119870119903119903 (119905)
(27)
That is a control input of AUV is
120591 (119905) = 119872119869minus1 (1205851) (minus [0 Λ119879] 120575 (119905) + 120578119889minus 119861119879119860120585119901 (119905 + 119889 | 119905) minus 119879Φ(120585119901 (119905 + 119889 | 119905))minus 119870119903119903 (119905)) = 119872119869minus1 (1205851) (minus [0 Λ119879] 120575 (119905) + 120578119889minus 119861119879119860120575 (119905) minus 119861119879119860119909119889 (119905) minus 119879Φ(120585119901 (119905 + 119889 | 119905))minus 119870119903119903 (119905))
(28)
Note that the predictor state 120585119901(119905 + 119889 | 119905) and theassociated error 119903(119905) are used in AUV controller (28) theNN approximation term 119879(119905)Φ(120585119901(119905 + 119889 | 119905)) from (22)is employed to accommodate the unknown nonlinearityTherefore the stability of the closed-loop system can beguaranteed
From (28) (26) becomes
119903 (119905) = minus119870119903119903 (119905) (29)
5 Stability Analysis
Assume that the parameters are chosen such that
1198774 = 1198772 + 121205813 gt 01198775 = 1
2120582m (119876) 121205813 minus
121205814 minus 1205732 gt 0
1198776 = minus119870119903 gt 0(30)
Journal of Control Science and Engineering 5
where 1198772 is defined as in Theorem 2 and 1205813 1205814 are positiveconstants that can be chosen
Theorem 3 (let Assumption 1 hold) Consider the input delayAUV system (1) under condition (30) the NN weight updatelaw (11) and controller (28) Then(1) all the closed-loop signals are UUB(2) the path tracking error120596(119905) = 119910(119905+119889)minus120578119889(119905) convergesto a neighborhood of the origin whose size can be adjusted bycontrol parameters
Proof Consider a Lyapunov function defined by
119881 = 12120585119879119875120585 + 1
2119879Γminus1 + 12119890119879119875119890 + 1
21199032fl 1198811 + 1198812 + 1198813 + 1198814
(31)
The derivative of 1198811 and 1198812 can be deduced following theproof of Theorem 2 Thus we have
3 = 12119890119879 (119860119879119875 + 119875119860) 119890 + 119890119879119875119861 [119882119879Φ (120585 (119905 + 119889))minus 119879Φ(120585119901 (119905 + 119889 | 119905)) + 120576 (119905 + 119889)]
(32)
In fact via Taylor series expansion there exist positiveconstants 1205731 1205732 and 1205733 such that10038171003817100381710038171003817119882119879Φ (120585 (119905 + 119889)) minus 119879Φ(120585119901 (119905 + 119889 | 119905))10038171003817100381710038171003817
= 10038171003817100381710038171003817Φ (120585119901 (119905 + 119889 | 119905))+ 119882119879 [Φ (120585 (119905 + 119889)) minus Φ (120585119901 (119905 + 119889 | 119905))]10038171003817100381710038171003817le 1205731 1003817100381710038171003817100381710038171003817100381710038171003817 + 1205732 119890 + 1205733
(33)
So
3 le minus12119876m 1198902 + 119890 (1205731 1003817100381710038171003817100381710038171003817100381710038171003817 + 1205732 119890 + 1205733 + 120576) (34)
By using Youngrsquos inequality there exist positive numbers 1205813and 1205814 such that
119890 1003817100381710038171003817100381710038171003817100381710038171003817 le 119890221205813 + 12058132 10038171003817100381710038171003817100381710038171003817100381710038172
(1205733 + 120576) 119890 le (1205733 + 120576)221205814 + 12058142 1198902
(35)
3 le minus(12119876m minus 1
21205813 minus121205814 minus 1205732) 1198902
+ 121205813 1003817100381710038171003817100381710038171003817100381710038171003817 + (1205734 + 119889 + 120576)221205814
(36)
4 = 119903 119903 = minus119870119903 1199032 (37)
Therefore
le minus1198771 10038171003817100381710038171003817120585100381710038171003817100381710038172 minus 1198774 10038171003817100381710038171003817100381710038171003817100381710038172 minus 1198775 1198902 minus 1198776 1199032 + 1198777le minus120578119881 + 1198777
(38)
where 1198771 is defined as in (19) and 120578 is positive constantdefined by
120578 = min 1198771120582max (119875) 11987741198775120582max (119875) 1198776
1198777 = 1198773 + (1205734 + 119889 + 120576)221205814
(39)
Then according to Lyapunov theorem error 120585 NNweighterror predictor error 119890 and filtered error 119903 are all UUBThe control error 120575 is thus bounded based on (24) andAssumption 1 Therefore the NN weights and 120585119901(119905 + 119889 | 119905)are bounded
Finally the boundedness of path tracking error 120596(119905) willbe proved Since
120596 (119905) = 119910 (119905 + 119889) minus 120578119889 (119905)= 1205851 (119905 + 119889) minus 1205851199011 (119905 + 119889 | 119905) + 1205851199011 (119905 + 119889 | 119905)
minus 119910119889 = 1198901 (119905) + 1205751 (119905) (40)
then
lim119905rarrinfin
120596 (119905) = lim119905rarrinfin
10038171003817100381710038171198901 (119905)1003817100381710038171003817 + lim119905rarrinfin
10038171003817100381710038171205751 (119905)1003817100381710038171003817le lim119905rarrinfin
119890 (119905) + lim119905rarrinfin
120575 (119905) (41)
Therefore the tracking error120596(119905) is bounded because 119890(119905)120575(119905) and 120585(119905) are boundedRemark 4 Compared with [20ndash33] there are three advan-tages Firstly output feedback is considered in this paperSecondly the nonlinear prediction model is employed toimprove the accuracy of predictive control Finally time-delay is considered in path tracking control of AUV whichhas more real significance
6 Simulation Analysis
Example 5 The simplified dynamics model of INFANTEAUV [2] in the horizontal plane with input delay is adoptedas follows in this paper
= 119906 cos (120595) minus V sin (120595) 119910 = 119906 sin (120595) + V cos (120595)
= 1199030 = 119898VV + 119898119906119903119906119903 + 119889VΓ = 119898119903 119903 + 119889119903
(42)
where 119909 119910 and 120595 are the surge position sway position andyaw angle in the body-fixed frame and 119906 V and 119903 denotesurge sway and yaw velocities respectively Γ is the yawmomentThe symbol 119868119911 denotes the moment of inertia of theAUV119873sdot is nonlinear hydrodynamic damping and
6 Journal of Control Science and Engineering
119898V = 119898 minus 119884V119898119906119903 = 119898 minus 119884119903119889V = minus119884V119906V minus 119884V|V|V |V| 119898119903 = 119868119911 minus 119873 119903119889119903 = minus119873V119906V minus 119873V|V|V |V| minus 119873119903119906119903
(43)
Since the consideredmodel (42) is in the horizontal planeand has no disturbance we can assume that 119906 = 1119898119904in principle In the following the desired path is 119909119889(119905) =20 sin 2120587119905200 119910119889(119905) = 20 minus 20 cos 2120587119905200 and 120595119889(119905) =2120587119905200
The model parameters of INFANTE AUV are as follows
119898 = 22345119896119892119868119911 = 20001198731198982
119883 = minus142119896119892119873 119903 = minus13501198731198982119884V = minus1715119896119892119884V = minus346119896119892119898119884119903 = 435119896119892119873V = minus686119896119892119873119903 = minus1427119896119892119898
119884V|V| = minus667119896119892119898119873V|V| = 443119896119892
(44)
NN parameters are selected as follows Φ(119909) = 1(1 +119890minus119886119909) where 119886 = 05 120583 = 03 Γ = 6The delay constant 119889 = 2 Other parameters in controller
are 1198601 = minus31198683 1198602 = minus21198683 1205821 = 2 1205822 = 2 and 119870119903 = 8The initial position and the surge speed of the AUV
are (0 20) and 0119898119904 respectively The simulation resultsare shown in Figures 2ndash4 The path tracking errors in119909 119910 and 120595 are given in Figure 2 The control forces in119909 and 119910 and the control torque of yaw 120595 are given inFigure 3 From these simulation figures we can see thatthe tracking performance is unsatisfying at the beginning ofsimulation this is because the controller performs mainlydepending on the adaptive control The good tracking ofposition is obtained by the proposed adaptive NN predictivecontroller by and by Figure 4 is the path tracking inhorizontal plane From Figure 4 we can see that AUV canrealize tracking control smoothly and converge to the desiredtrajectory
7 Conclusion
This paper investigates the path tracking problem for anAUV with input delay Based on predictive and adaptive NN
minus020
0204
e x(m
)
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
minus05
0
05
e (r
ad)
minus05
0
05
e y(m
)
100 200 300 400 500 600 700 8000time (s)
Figure 2 Path tracking errors
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
0
20
40Fx
(N)
0
100
200
Fy
(N)
minus100
0
100
M
(Nm
)
Figure 3 Control forces in 119909 and 119910 and control torque of yaw
control theory a predictive controller is given The outputfeedback control algorithm is employed here The NN isused to estimate the dynamic uncertain nonlinear functioninduced by hydrodynamic coefficients and coupling of thesurge sway and yaw angular velocity The predictive controlis introduced to compensate the input delay present in AUVThe stability of the controller was analyzed by Lyapunov the-orem Simulation results showed that the proposed controllerperforms well with stability
Data Availability
We are sorry that we cannot share the data in our article nowbecause future works are based on its results The methodsin this paper are effective methods for investigation of pathfollowing for autonomous underwater vehicles with input
Journal of Control Science and Engineering 7
AUV trajectorydesired trajectory
minus5
0
5
10
15
20
25
30
35
40
45
y (m
)
minus20 minus15 minus10 minus5 0 5 10 15 20 25minus25x (m)
Figure 4 Path tracking response in 119909119910 plane
delay We will apply a patent on the relevant studies So wecannot share the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the Hubei Provincial NaturalScience Foundation of China (2016CFB273) and the Researchand Innovation Initiatives of WHPU (2018Y20)
References
[1] P Jantapremjit and P A Wilson ldquoGuidance-control based pathfollowing for homing and docking using an autonomous under-water vehiclerdquo in Proceedings of the OCEANSrsquo08 MTSIEEEKobe-Techno-Oceanrsquo08 - Voyage toward the Future OTOrsquo08 jpnApril 2008
[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007
[3] A Adhami-Mirhosseini M J Yazdanpanah and A P AguiarldquoAutomatic bottom-following for underwater robotic vehiclesrdquoAutomatica vol 50 no 8 pp 2155ndash2162 2014
[4] M Kim H Joe J Kim and S-c Yu ldquoIntegral sliding modecontroller for precise manoeuvring of autonomous underwatervehicle in the presence of unknown environmental distur-bancesrdquo International Journal of Control vol 88 no 10 pp2055ndash2065 2015
[5] R Cui X Zhang and D Cui ldquoAdaptive sliding-mode attitudecontrol for autonomous underwater vehicles with input nonlin-earitiesrdquo Ocean Engineering vol 123 pp 45ndash54 2016
[6] Y Wang L Gu M Gao and K Zhu ldquoMultivariable outputfeedback adaptive terminal sliding mode control for underwa-ter vehiclesrdquoAsian Journal of Control vol 18 no 1 pp 247ndash2652016
[7] X Qi ldquoAdaptive coordinated tracking control of multipleautonomous underwater vehiclesrdquo Ocean Engineering vol 91pp 84ndash90 2014
[8] R Cui C Yang Y Li and S Sharma ldquoAdaptive Neural NetworkControl of AUVs With Control Input Nonlinearities UsingReinforcement Learningrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 47 no 6 pp 1019ndash1029 2017
[9] B B Miao T S Li and W L Luo ldquoA DSC and MLP basedrobust adaptive NN tracking control for underwater vehiclerdquoNeurocomputing vol 111 pp 184ndash189 2013
[10] Y-C Liu S-Y Liu and N Wang ldquoFully-tuned fuzzy neuralnetwork based robust adaptive tracking control of unmannedunderwater vehicle with thruster dynamicsrdquo Neurocomputingvol 196 pp 1ndash13 2016
[11] J Ghommam and M Saad ldquoBackstepping-based cooperativeand adaptive tracking control design for a group of underactu-ated AUVs in horizontal planrdquo International Journal of Controlvol 87 no 5 pp 1076ndash1093 2014
[12] C Shen B Buckham and Y Shi ldquoModified CGMRES Algo-rithm for Fast Nonlinear Model Predictive Tracking Control ofAUVsrdquo IEEE Transactions on Control Systems Technology vol25 no 5 pp 1896ndash1904 2017
[13] J Gao A A Proctor Y Shi and C Bradley ldquoHierarchicalModel Predictive Image-Based Visual Servoing of UnderwaterVehicles with Adaptive Neural Network Dynamic ControlrdquoIEEE Transactions on Cybernetics vol 46 no 10 pp 2323ndash23342016
[14] J Sun and J Chen ldquoNetworked predictive control for systemswith unknown or partially known delayrdquo IET Control Theory ampApplications vol 8 no 18 pp 2282ndash2288 2014
[15] S Li and G-P Liu ldquoNetworked predictive control for nonlinearsystems with stochastic disturbances in the presence of datalossesrdquo Neurocomputing vol 194 pp 56ndash64 2016
[16] J Zhao L Zhang and X Qi ldquoA necessary and sufficientcondition for stabilization of switched descriptor time-delaysystems under arbitrary switchingrdquo Asian Journal of Controlvol 18 no 1 pp 266ndash272 2016
[17] S Sirouspour and A Shahdi ldquoModel predictive control fortransparent teleoperation under communication time delayrdquoIEEE Transactions on Robotics vol 22 no 6 pp 1131ndash1145 2006
[18] Y Sheng Y Shen and M Zhu ldquoDelay-dependent globalexponential stability for delayed recurrent neural networksrdquoIEEE Transactions on Neural Networks and Learning Systems2016
[19] O M Smith ldquoA controller to overcome deadtimerdquo ISA J vol 6p 2833 1959
[20] C F Caruntu and C Lazar ldquoNetwork delay predictive com-pensation based on time-delay modelling as disturbancerdquoInternational Journal of Control vol 87 no 10 pp 2012ndash20262014
[21] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
[22] S H HosseinNia and M Lundh ldquoA general robust MPCdesign for the state-space model application to paper machineprocessrdquo Asian Journal of Control vol 18 no 5 pp 1891ndash19072016
[23] Y Ding Z Xu J Zhao KWang and Z Shao ldquoEmbeddedMPCcontroller based on interior-point method with convergencedepth controlrdquoAsian Journal of Control vol 18 no 6 pp 2064ndash2077 2016
8 Journal of Control Science and Engineering
[24] J-Q Huang and F L Lewis ldquoNeural-network predictive controlfor nonlinear dynamic systems with time-delayrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 14 no 2pp 377ndash389 2003
[25] B An and G Liu ldquoModel-based predictive controller designfor a class of nonlinear networked systemswith communicationdelays and data lossrdquo Chinese Physics B vol 23 no 8 p 0802022014
[26] N Sharma C M Gregory and W E Dixon ldquoPredictor-basedcompensation for electromechanical delay during neuromuscu-lar electrical stimulationrdquo IEEE Transactions on Neural Systemsand Rehabilitation Engineering vol 19 no 6 pp 601ndash611 2011
[27] N Fischer A Dani N Sharma and W E Dixon ldquoSaturatedcontrol of an uncertain nonlinear system with input delayrdquoAutomatica vol 49 no 6 pp 1741ndash1747 2013
[28] T Wang H Gao and J Qiu ldquoA combined adaptive neuralnetwork and nonlinear model predictive control for multiratenetworked industrial process controlrdquo IEEE Transactions onNeural Networks and Learning Systems vol 27 no 2 pp 416ndash425 2016
[29] PDai ZGong andGGuo ldquoPerformance analysis of amodularmultilevel converter drive system with model predictive con-trolrdquo International Journal of Control andAutomation vol 9 no1 pp 387ndash398 2016
[30] C Wei J Luo H Dai Z Yin W Ma and J Yuan ldquoGloballyrobust explicit model predictive control of constrained systemsexploiting SVM-based approximationrdquo International Journal ofRobust and Nonlinear Control vol 27 no 16 pp 3000ndash30272017
[31] M Parvez Akter S Mekhilef N Mei Lin Tan and H AkagildquoModified Model Predictive Control of a Bidirectional AC-DCConverter Based on Lyapunov Function for Energy StorageSystemsrdquo IEEETransactions on Industrial Electronics vol 63 no2 pp 704ndash715 2016
[32] R Prasanth Kumar A Dasgupta and C S Kumar ldquoRobusttrajectory control of underwater vehicles using time delaycontrol lawrdquo Ocean Engineering vol 34 no 5-6 pp 842ndash8492007
[33] K Mukherjee I N Kar and R K P Bhatt ldquoRegion trackingbased control of an autonomous underwater vehicle with inputdelayrdquo Ocean Engineering vol 99 pp 107ndash114 2015
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4 Journal of Control Science and Engineering
Predictive model
RNN
AUV
Delay
y(t)r(t)
W(t)
[0 ΛT]
[ΛT 1]
BT A
Kr eminussd
d(t)
xd(t) (t)
p(t + d | t)
(t) (t minus d)
y (t)minus
minus
minus
minus+
+
minus
minus
+
WT (t) Φ (p (t + )d | t)
Figure 1 Control structure of AUV system (1)
where 120585119901(119905 + 119889 | 119905) and 119910119901(119905 + 119889 | 119905) are the prediction stateand output of system (8) with the initial condition 120585119901(119889 | 0) =120585(0)
If prediction model (22) is precise then 120585119901(119905 + 119889 | 119905) =120585(119905 + 119889) This mean that 120585 ahead of time 119889 can be predictedvia 120585119901(119905+119889 | 119905) in predictionmodelTherefore the difficulty incontrolling time-delay plant can be overcome However dueto the modeling errors in prediction model (22) errors existinevitably Now define a predictor error as 119890(119905) = 120585(119905 + 119889) minus120585119901(119905 + 119889 | 119905) It follows from (8) and (22) that
119890 (119905) = 119860119890 (119905) + 119861 [119882119879Φ (120585 (119905 + 119889))minus 119879 (119905) Φ (120585119901 (119905 + 119889 | 119905)) + 120576 (119905 + 119889)]
119910119890 (119905) = 119862119879119890 (119905) (23)
Next we will prove that the predictor error (23) isbounded Define an error vector as
120575 (119905) = 120585119901 (119905 + 119889 | 119905) minus 120578119889 (119905) (24)
Define a filtered error as
119903 (119905) = Λ119879120575 (119905) = [Λ119879 1] 120575 (119905) (25)
where Λ = [1205821 1205822]119879 is an appropriately chosen coefficientvector such that 120575(119905) rarr 0 exponentially as 119903(119905) rarr 0 Thenusing (22) the filtered error can be written as
119903 (119905) = Λ119879 120575 (119905) = Λ119879 [119860120585119901 (119905 + 119889 | 119905)+ 119861 (119879Φ(120585119901 (119905 + 119889 | 119905)) + 120591 (119905))]= [0 Λ119879] 120575 (119905) minus 120578119889 + 119861119879119860120585119901 (119905 + 119889 | 119905)+ 119879Φ(120585119901 (119905 + 119889 | 119905)) + 120591 (119905)
(26)
Now choose 119870119903 gt 0 and let
120591 (119905) = minus [0 Λ119879] 120575 (119905) + 120578119889 minus 119861119879119860120585119901 (119905 + 119889 | 119905)minus 119879Φ(120585119901 (119905 + 119889 | 119905)) minus 119870119903119903 (119905)
= minus [0 Λ119879] 120575 (119905) + 120578119889 minus 119861119879119860120575 (119905) minus 119861119879119860120578119889 (119905)minus 119879Φ(120585119901 (119905 + 119889 | 119905)) minus 119870119903119903 (119905)
(27)
That is a control input of AUV is
120591 (119905) = 119872119869minus1 (1205851) (minus [0 Λ119879] 120575 (119905) + 120578119889minus 119861119879119860120585119901 (119905 + 119889 | 119905) minus 119879Φ(120585119901 (119905 + 119889 | 119905))minus 119870119903119903 (119905)) = 119872119869minus1 (1205851) (minus [0 Λ119879] 120575 (119905) + 120578119889minus 119861119879119860120575 (119905) minus 119861119879119860119909119889 (119905) minus 119879Φ(120585119901 (119905 + 119889 | 119905))minus 119870119903119903 (119905))
(28)
Note that the predictor state 120585119901(119905 + 119889 | 119905) and theassociated error 119903(119905) are used in AUV controller (28) theNN approximation term 119879(119905)Φ(120585119901(119905 + 119889 | 119905)) from (22)is employed to accommodate the unknown nonlinearityTherefore the stability of the closed-loop system can beguaranteed
From (28) (26) becomes
119903 (119905) = minus119870119903119903 (119905) (29)
5 Stability Analysis
Assume that the parameters are chosen such that
1198774 = 1198772 + 121205813 gt 01198775 = 1
2120582m (119876) 121205813 minus
121205814 minus 1205732 gt 0
1198776 = minus119870119903 gt 0(30)
Journal of Control Science and Engineering 5
where 1198772 is defined as in Theorem 2 and 1205813 1205814 are positiveconstants that can be chosen
Theorem 3 (let Assumption 1 hold) Consider the input delayAUV system (1) under condition (30) the NN weight updatelaw (11) and controller (28) Then(1) all the closed-loop signals are UUB(2) the path tracking error120596(119905) = 119910(119905+119889)minus120578119889(119905) convergesto a neighborhood of the origin whose size can be adjusted bycontrol parameters
Proof Consider a Lyapunov function defined by
119881 = 12120585119879119875120585 + 1
2119879Γminus1 + 12119890119879119875119890 + 1
21199032fl 1198811 + 1198812 + 1198813 + 1198814
(31)
The derivative of 1198811 and 1198812 can be deduced following theproof of Theorem 2 Thus we have
3 = 12119890119879 (119860119879119875 + 119875119860) 119890 + 119890119879119875119861 [119882119879Φ (120585 (119905 + 119889))minus 119879Φ(120585119901 (119905 + 119889 | 119905)) + 120576 (119905 + 119889)]
(32)
In fact via Taylor series expansion there exist positiveconstants 1205731 1205732 and 1205733 such that10038171003817100381710038171003817119882119879Φ (120585 (119905 + 119889)) minus 119879Φ(120585119901 (119905 + 119889 | 119905))10038171003817100381710038171003817
= 10038171003817100381710038171003817Φ (120585119901 (119905 + 119889 | 119905))+ 119882119879 [Φ (120585 (119905 + 119889)) minus Φ (120585119901 (119905 + 119889 | 119905))]10038171003817100381710038171003817le 1205731 1003817100381710038171003817100381710038171003817100381710038171003817 + 1205732 119890 + 1205733
(33)
So
3 le minus12119876m 1198902 + 119890 (1205731 1003817100381710038171003817100381710038171003817100381710038171003817 + 1205732 119890 + 1205733 + 120576) (34)
By using Youngrsquos inequality there exist positive numbers 1205813and 1205814 such that
119890 1003817100381710038171003817100381710038171003817100381710038171003817 le 119890221205813 + 12058132 10038171003817100381710038171003817100381710038171003817100381710038172
(1205733 + 120576) 119890 le (1205733 + 120576)221205814 + 12058142 1198902
(35)
3 le minus(12119876m minus 1
21205813 minus121205814 minus 1205732) 1198902
+ 121205813 1003817100381710038171003817100381710038171003817100381710038171003817 + (1205734 + 119889 + 120576)221205814
(36)
4 = 119903 119903 = minus119870119903 1199032 (37)
Therefore
le minus1198771 10038171003817100381710038171003817120585100381710038171003817100381710038172 minus 1198774 10038171003817100381710038171003817100381710038171003817100381710038172 minus 1198775 1198902 minus 1198776 1199032 + 1198777le minus120578119881 + 1198777
(38)
where 1198771 is defined as in (19) and 120578 is positive constantdefined by
120578 = min 1198771120582max (119875) 11987741198775120582max (119875) 1198776
1198777 = 1198773 + (1205734 + 119889 + 120576)221205814
(39)
Then according to Lyapunov theorem error 120585 NNweighterror predictor error 119890 and filtered error 119903 are all UUBThe control error 120575 is thus bounded based on (24) andAssumption 1 Therefore the NN weights and 120585119901(119905 + 119889 | 119905)are bounded
Finally the boundedness of path tracking error 120596(119905) willbe proved Since
120596 (119905) = 119910 (119905 + 119889) minus 120578119889 (119905)= 1205851 (119905 + 119889) minus 1205851199011 (119905 + 119889 | 119905) + 1205851199011 (119905 + 119889 | 119905)
minus 119910119889 = 1198901 (119905) + 1205751 (119905) (40)
then
lim119905rarrinfin
120596 (119905) = lim119905rarrinfin
10038171003817100381710038171198901 (119905)1003817100381710038171003817 + lim119905rarrinfin
10038171003817100381710038171205751 (119905)1003817100381710038171003817le lim119905rarrinfin
119890 (119905) + lim119905rarrinfin
120575 (119905) (41)
Therefore the tracking error120596(119905) is bounded because 119890(119905)120575(119905) and 120585(119905) are boundedRemark 4 Compared with [20ndash33] there are three advan-tages Firstly output feedback is considered in this paperSecondly the nonlinear prediction model is employed toimprove the accuracy of predictive control Finally time-delay is considered in path tracking control of AUV whichhas more real significance
6 Simulation Analysis
Example 5 The simplified dynamics model of INFANTEAUV [2] in the horizontal plane with input delay is adoptedas follows in this paper
= 119906 cos (120595) minus V sin (120595) 119910 = 119906 sin (120595) + V cos (120595)
= 1199030 = 119898VV + 119898119906119903119906119903 + 119889VΓ = 119898119903 119903 + 119889119903
(42)
where 119909 119910 and 120595 are the surge position sway position andyaw angle in the body-fixed frame and 119906 V and 119903 denotesurge sway and yaw velocities respectively Γ is the yawmomentThe symbol 119868119911 denotes the moment of inertia of theAUV119873sdot is nonlinear hydrodynamic damping and
6 Journal of Control Science and Engineering
119898V = 119898 minus 119884V119898119906119903 = 119898 minus 119884119903119889V = minus119884V119906V minus 119884V|V|V |V| 119898119903 = 119868119911 minus 119873 119903119889119903 = minus119873V119906V minus 119873V|V|V |V| minus 119873119903119906119903
(43)
Since the consideredmodel (42) is in the horizontal planeand has no disturbance we can assume that 119906 = 1119898119904in principle In the following the desired path is 119909119889(119905) =20 sin 2120587119905200 119910119889(119905) = 20 minus 20 cos 2120587119905200 and 120595119889(119905) =2120587119905200
The model parameters of INFANTE AUV are as follows
119898 = 22345119896119892119868119911 = 20001198731198982
119883 = minus142119896119892119873 119903 = minus13501198731198982119884V = minus1715119896119892119884V = minus346119896119892119898119884119903 = 435119896119892119873V = minus686119896119892119873119903 = minus1427119896119892119898
119884V|V| = minus667119896119892119898119873V|V| = 443119896119892
(44)
NN parameters are selected as follows Φ(119909) = 1(1 +119890minus119886119909) where 119886 = 05 120583 = 03 Γ = 6The delay constant 119889 = 2 Other parameters in controller
are 1198601 = minus31198683 1198602 = minus21198683 1205821 = 2 1205822 = 2 and 119870119903 = 8The initial position and the surge speed of the AUV
are (0 20) and 0119898119904 respectively The simulation resultsare shown in Figures 2ndash4 The path tracking errors in119909 119910 and 120595 are given in Figure 2 The control forces in119909 and 119910 and the control torque of yaw 120595 are given inFigure 3 From these simulation figures we can see thatthe tracking performance is unsatisfying at the beginning ofsimulation this is because the controller performs mainlydepending on the adaptive control The good tracking ofposition is obtained by the proposed adaptive NN predictivecontroller by and by Figure 4 is the path tracking inhorizontal plane From Figure 4 we can see that AUV canrealize tracking control smoothly and converge to the desiredtrajectory
7 Conclusion
This paper investigates the path tracking problem for anAUV with input delay Based on predictive and adaptive NN
minus020
0204
e x(m
)
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
minus05
0
05
e (r
ad)
minus05
0
05
e y(m
)
100 200 300 400 500 600 700 8000time (s)
Figure 2 Path tracking errors
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
0
20
40Fx
(N)
0
100
200
Fy
(N)
minus100
0
100
M
(Nm
)
Figure 3 Control forces in 119909 and 119910 and control torque of yaw
control theory a predictive controller is given The outputfeedback control algorithm is employed here The NN isused to estimate the dynamic uncertain nonlinear functioninduced by hydrodynamic coefficients and coupling of thesurge sway and yaw angular velocity The predictive controlis introduced to compensate the input delay present in AUVThe stability of the controller was analyzed by Lyapunov the-orem Simulation results showed that the proposed controllerperforms well with stability
Data Availability
We are sorry that we cannot share the data in our article nowbecause future works are based on its results The methodsin this paper are effective methods for investigation of pathfollowing for autonomous underwater vehicles with input
Journal of Control Science and Engineering 7
AUV trajectorydesired trajectory
minus5
0
5
10
15
20
25
30
35
40
45
y (m
)
minus20 minus15 minus10 minus5 0 5 10 15 20 25minus25x (m)
Figure 4 Path tracking response in 119909119910 plane
delay We will apply a patent on the relevant studies So wecannot share the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the Hubei Provincial NaturalScience Foundation of China (2016CFB273) and the Researchand Innovation Initiatives of WHPU (2018Y20)
References
[1] P Jantapremjit and P A Wilson ldquoGuidance-control based pathfollowing for homing and docking using an autonomous under-water vehiclerdquo in Proceedings of the OCEANSrsquo08 MTSIEEEKobe-Techno-Oceanrsquo08 - Voyage toward the Future OTOrsquo08 jpnApril 2008
[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007
[3] A Adhami-Mirhosseini M J Yazdanpanah and A P AguiarldquoAutomatic bottom-following for underwater robotic vehiclesrdquoAutomatica vol 50 no 8 pp 2155ndash2162 2014
[4] M Kim H Joe J Kim and S-c Yu ldquoIntegral sliding modecontroller for precise manoeuvring of autonomous underwatervehicle in the presence of unknown environmental distur-bancesrdquo International Journal of Control vol 88 no 10 pp2055ndash2065 2015
[5] R Cui X Zhang and D Cui ldquoAdaptive sliding-mode attitudecontrol for autonomous underwater vehicles with input nonlin-earitiesrdquo Ocean Engineering vol 123 pp 45ndash54 2016
[6] Y Wang L Gu M Gao and K Zhu ldquoMultivariable outputfeedback adaptive terminal sliding mode control for underwa-ter vehiclesrdquoAsian Journal of Control vol 18 no 1 pp 247ndash2652016
[7] X Qi ldquoAdaptive coordinated tracking control of multipleautonomous underwater vehiclesrdquo Ocean Engineering vol 91pp 84ndash90 2014
[8] R Cui C Yang Y Li and S Sharma ldquoAdaptive Neural NetworkControl of AUVs With Control Input Nonlinearities UsingReinforcement Learningrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 47 no 6 pp 1019ndash1029 2017
[9] B B Miao T S Li and W L Luo ldquoA DSC and MLP basedrobust adaptive NN tracking control for underwater vehiclerdquoNeurocomputing vol 111 pp 184ndash189 2013
[10] Y-C Liu S-Y Liu and N Wang ldquoFully-tuned fuzzy neuralnetwork based robust adaptive tracking control of unmannedunderwater vehicle with thruster dynamicsrdquo Neurocomputingvol 196 pp 1ndash13 2016
[11] J Ghommam and M Saad ldquoBackstepping-based cooperativeand adaptive tracking control design for a group of underactu-ated AUVs in horizontal planrdquo International Journal of Controlvol 87 no 5 pp 1076ndash1093 2014
[12] C Shen B Buckham and Y Shi ldquoModified CGMRES Algo-rithm for Fast Nonlinear Model Predictive Tracking Control ofAUVsrdquo IEEE Transactions on Control Systems Technology vol25 no 5 pp 1896ndash1904 2017
[13] J Gao A A Proctor Y Shi and C Bradley ldquoHierarchicalModel Predictive Image-Based Visual Servoing of UnderwaterVehicles with Adaptive Neural Network Dynamic ControlrdquoIEEE Transactions on Cybernetics vol 46 no 10 pp 2323ndash23342016
[14] J Sun and J Chen ldquoNetworked predictive control for systemswith unknown or partially known delayrdquo IET Control Theory ampApplications vol 8 no 18 pp 2282ndash2288 2014
[15] S Li and G-P Liu ldquoNetworked predictive control for nonlinearsystems with stochastic disturbances in the presence of datalossesrdquo Neurocomputing vol 194 pp 56ndash64 2016
[16] J Zhao L Zhang and X Qi ldquoA necessary and sufficientcondition for stabilization of switched descriptor time-delaysystems under arbitrary switchingrdquo Asian Journal of Controlvol 18 no 1 pp 266ndash272 2016
[17] S Sirouspour and A Shahdi ldquoModel predictive control fortransparent teleoperation under communication time delayrdquoIEEE Transactions on Robotics vol 22 no 6 pp 1131ndash1145 2006
[18] Y Sheng Y Shen and M Zhu ldquoDelay-dependent globalexponential stability for delayed recurrent neural networksrdquoIEEE Transactions on Neural Networks and Learning Systems2016
[19] O M Smith ldquoA controller to overcome deadtimerdquo ISA J vol 6p 2833 1959
[20] C F Caruntu and C Lazar ldquoNetwork delay predictive com-pensation based on time-delay modelling as disturbancerdquoInternational Journal of Control vol 87 no 10 pp 2012ndash20262014
[21] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
[22] S H HosseinNia and M Lundh ldquoA general robust MPCdesign for the state-space model application to paper machineprocessrdquo Asian Journal of Control vol 18 no 5 pp 1891ndash19072016
[23] Y Ding Z Xu J Zhao KWang and Z Shao ldquoEmbeddedMPCcontroller based on interior-point method with convergencedepth controlrdquoAsian Journal of Control vol 18 no 6 pp 2064ndash2077 2016
8 Journal of Control Science and Engineering
[24] J-Q Huang and F L Lewis ldquoNeural-network predictive controlfor nonlinear dynamic systems with time-delayrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 14 no 2pp 377ndash389 2003
[25] B An and G Liu ldquoModel-based predictive controller designfor a class of nonlinear networked systemswith communicationdelays and data lossrdquo Chinese Physics B vol 23 no 8 p 0802022014
[26] N Sharma C M Gregory and W E Dixon ldquoPredictor-basedcompensation for electromechanical delay during neuromuscu-lar electrical stimulationrdquo IEEE Transactions on Neural Systemsand Rehabilitation Engineering vol 19 no 6 pp 601ndash611 2011
[27] N Fischer A Dani N Sharma and W E Dixon ldquoSaturatedcontrol of an uncertain nonlinear system with input delayrdquoAutomatica vol 49 no 6 pp 1741ndash1747 2013
[28] T Wang H Gao and J Qiu ldquoA combined adaptive neuralnetwork and nonlinear model predictive control for multiratenetworked industrial process controlrdquo IEEE Transactions onNeural Networks and Learning Systems vol 27 no 2 pp 416ndash425 2016
[29] PDai ZGong andGGuo ldquoPerformance analysis of amodularmultilevel converter drive system with model predictive con-trolrdquo International Journal of Control andAutomation vol 9 no1 pp 387ndash398 2016
[30] C Wei J Luo H Dai Z Yin W Ma and J Yuan ldquoGloballyrobust explicit model predictive control of constrained systemsexploiting SVM-based approximationrdquo International Journal ofRobust and Nonlinear Control vol 27 no 16 pp 3000ndash30272017
[31] M Parvez Akter S Mekhilef N Mei Lin Tan and H AkagildquoModified Model Predictive Control of a Bidirectional AC-DCConverter Based on Lyapunov Function for Energy StorageSystemsrdquo IEEETransactions on Industrial Electronics vol 63 no2 pp 704ndash715 2016
[32] R Prasanth Kumar A Dasgupta and C S Kumar ldquoRobusttrajectory control of underwater vehicles using time delaycontrol lawrdquo Ocean Engineering vol 34 no 5-6 pp 842ndash8492007
[33] K Mukherjee I N Kar and R K P Bhatt ldquoRegion trackingbased control of an autonomous underwater vehicle with inputdelayrdquo Ocean Engineering vol 99 pp 107ndash114 2015
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Journal of Control Science and Engineering 5
where 1198772 is defined as in Theorem 2 and 1205813 1205814 are positiveconstants that can be chosen
Theorem 3 (let Assumption 1 hold) Consider the input delayAUV system (1) under condition (30) the NN weight updatelaw (11) and controller (28) Then(1) all the closed-loop signals are UUB(2) the path tracking error120596(119905) = 119910(119905+119889)minus120578119889(119905) convergesto a neighborhood of the origin whose size can be adjusted bycontrol parameters
Proof Consider a Lyapunov function defined by
119881 = 12120585119879119875120585 + 1
2119879Γminus1 + 12119890119879119875119890 + 1
21199032fl 1198811 + 1198812 + 1198813 + 1198814
(31)
The derivative of 1198811 and 1198812 can be deduced following theproof of Theorem 2 Thus we have
3 = 12119890119879 (119860119879119875 + 119875119860) 119890 + 119890119879119875119861 [119882119879Φ (120585 (119905 + 119889))minus 119879Φ(120585119901 (119905 + 119889 | 119905)) + 120576 (119905 + 119889)]
(32)
In fact via Taylor series expansion there exist positiveconstants 1205731 1205732 and 1205733 such that10038171003817100381710038171003817119882119879Φ (120585 (119905 + 119889)) minus 119879Φ(120585119901 (119905 + 119889 | 119905))10038171003817100381710038171003817
= 10038171003817100381710038171003817Φ (120585119901 (119905 + 119889 | 119905))+ 119882119879 [Φ (120585 (119905 + 119889)) minus Φ (120585119901 (119905 + 119889 | 119905))]10038171003817100381710038171003817le 1205731 1003817100381710038171003817100381710038171003817100381710038171003817 + 1205732 119890 + 1205733
(33)
So
3 le minus12119876m 1198902 + 119890 (1205731 1003817100381710038171003817100381710038171003817100381710038171003817 + 1205732 119890 + 1205733 + 120576) (34)
By using Youngrsquos inequality there exist positive numbers 1205813and 1205814 such that
119890 1003817100381710038171003817100381710038171003817100381710038171003817 le 119890221205813 + 12058132 10038171003817100381710038171003817100381710038171003817100381710038172
(1205733 + 120576) 119890 le (1205733 + 120576)221205814 + 12058142 1198902
(35)
3 le minus(12119876m minus 1
21205813 minus121205814 minus 1205732) 1198902
+ 121205813 1003817100381710038171003817100381710038171003817100381710038171003817 + (1205734 + 119889 + 120576)221205814
(36)
4 = 119903 119903 = minus119870119903 1199032 (37)
Therefore
le minus1198771 10038171003817100381710038171003817120585100381710038171003817100381710038172 minus 1198774 10038171003817100381710038171003817100381710038171003817100381710038172 minus 1198775 1198902 minus 1198776 1199032 + 1198777le minus120578119881 + 1198777
(38)
where 1198771 is defined as in (19) and 120578 is positive constantdefined by
120578 = min 1198771120582max (119875) 11987741198775120582max (119875) 1198776
1198777 = 1198773 + (1205734 + 119889 + 120576)221205814
(39)
Then according to Lyapunov theorem error 120585 NNweighterror predictor error 119890 and filtered error 119903 are all UUBThe control error 120575 is thus bounded based on (24) andAssumption 1 Therefore the NN weights and 120585119901(119905 + 119889 | 119905)are bounded
Finally the boundedness of path tracking error 120596(119905) willbe proved Since
120596 (119905) = 119910 (119905 + 119889) minus 120578119889 (119905)= 1205851 (119905 + 119889) minus 1205851199011 (119905 + 119889 | 119905) + 1205851199011 (119905 + 119889 | 119905)
minus 119910119889 = 1198901 (119905) + 1205751 (119905) (40)
then
lim119905rarrinfin
120596 (119905) = lim119905rarrinfin
10038171003817100381710038171198901 (119905)1003817100381710038171003817 + lim119905rarrinfin
10038171003817100381710038171205751 (119905)1003817100381710038171003817le lim119905rarrinfin
119890 (119905) + lim119905rarrinfin
120575 (119905) (41)
Therefore the tracking error120596(119905) is bounded because 119890(119905)120575(119905) and 120585(119905) are boundedRemark 4 Compared with [20ndash33] there are three advan-tages Firstly output feedback is considered in this paperSecondly the nonlinear prediction model is employed toimprove the accuracy of predictive control Finally time-delay is considered in path tracking control of AUV whichhas more real significance
6 Simulation Analysis
Example 5 The simplified dynamics model of INFANTEAUV [2] in the horizontal plane with input delay is adoptedas follows in this paper
= 119906 cos (120595) minus V sin (120595) 119910 = 119906 sin (120595) + V cos (120595)
= 1199030 = 119898VV + 119898119906119903119906119903 + 119889VΓ = 119898119903 119903 + 119889119903
(42)
where 119909 119910 and 120595 are the surge position sway position andyaw angle in the body-fixed frame and 119906 V and 119903 denotesurge sway and yaw velocities respectively Γ is the yawmomentThe symbol 119868119911 denotes the moment of inertia of theAUV119873sdot is nonlinear hydrodynamic damping and
6 Journal of Control Science and Engineering
119898V = 119898 minus 119884V119898119906119903 = 119898 minus 119884119903119889V = minus119884V119906V minus 119884V|V|V |V| 119898119903 = 119868119911 minus 119873 119903119889119903 = minus119873V119906V minus 119873V|V|V |V| minus 119873119903119906119903
(43)
Since the consideredmodel (42) is in the horizontal planeand has no disturbance we can assume that 119906 = 1119898119904in principle In the following the desired path is 119909119889(119905) =20 sin 2120587119905200 119910119889(119905) = 20 minus 20 cos 2120587119905200 and 120595119889(119905) =2120587119905200
The model parameters of INFANTE AUV are as follows
119898 = 22345119896119892119868119911 = 20001198731198982
119883 = minus142119896119892119873 119903 = minus13501198731198982119884V = minus1715119896119892119884V = minus346119896119892119898119884119903 = 435119896119892119873V = minus686119896119892119873119903 = minus1427119896119892119898
119884V|V| = minus667119896119892119898119873V|V| = 443119896119892
(44)
NN parameters are selected as follows Φ(119909) = 1(1 +119890minus119886119909) where 119886 = 05 120583 = 03 Γ = 6The delay constant 119889 = 2 Other parameters in controller
are 1198601 = minus31198683 1198602 = minus21198683 1205821 = 2 1205822 = 2 and 119870119903 = 8The initial position and the surge speed of the AUV
are (0 20) and 0119898119904 respectively The simulation resultsare shown in Figures 2ndash4 The path tracking errors in119909 119910 and 120595 are given in Figure 2 The control forces in119909 and 119910 and the control torque of yaw 120595 are given inFigure 3 From these simulation figures we can see thatthe tracking performance is unsatisfying at the beginning ofsimulation this is because the controller performs mainlydepending on the adaptive control The good tracking ofposition is obtained by the proposed adaptive NN predictivecontroller by and by Figure 4 is the path tracking inhorizontal plane From Figure 4 we can see that AUV canrealize tracking control smoothly and converge to the desiredtrajectory
7 Conclusion
This paper investigates the path tracking problem for anAUV with input delay Based on predictive and adaptive NN
minus020
0204
e x(m
)
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
minus05
0
05
e (r
ad)
minus05
0
05
e y(m
)
100 200 300 400 500 600 700 8000time (s)
Figure 2 Path tracking errors
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
0
20
40Fx
(N)
0
100
200
Fy
(N)
minus100
0
100
M
(Nm
)
Figure 3 Control forces in 119909 and 119910 and control torque of yaw
control theory a predictive controller is given The outputfeedback control algorithm is employed here The NN isused to estimate the dynamic uncertain nonlinear functioninduced by hydrodynamic coefficients and coupling of thesurge sway and yaw angular velocity The predictive controlis introduced to compensate the input delay present in AUVThe stability of the controller was analyzed by Lyapunov the-orem Simulation results showed that the proposed controllerperforms well with stability
Data Availability
We are sorry that we cannot share the data in our article nowbecause future works are based on its results The methodsin this paper are effective methods for investigation of pathfollowing for autonomous underwater vehicles with input
Journal of Control Science and Engineering 7
AUV trajectorydesired trajectory
minus5
0
5
10
15
20
25
30
35
40
45
y (m
)
minus20 minus15 minus10 minus5 0 5 10 15 20 25minus25x (m)
Figure 4 Path tracking response in 119909119910 plane
delay We will apply a patent on the relevant studies So wecannot share the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the Hubei Provincial NaturalScience Foundation of China (2016CFB273) and the Researchand Innovation Initiatives of WHPU (2018Y20)
References
[1] P Jantapremjit and P A Wilson ldquoGuidance-control based pathfollowing for homing and docking using an autonomous under-water vehiclerdquo in Proceedings of the OCEANSrsquo08 MTSIEEEKobe-Techno-Oceanrsquo08 - Voyage toward the Future OTOrsquo08 jpnApril 2008
[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007
[3] A Adhami-Mirhosseini M J Yazdanpanah and A P AguiarldquoAutomatic bottom-following for underwater robotic vehiclesrdquoAutomatica vol 50 no 8 pp 2155ndash2162 2014
[4] M Kim H Joe J Kim and S-c Yu ldquoIntegral sliding modecontroller for precise manoeuvring of autonomous underwatervehicle in the presence of unknown environmental distur-bancesrdquo International Journal of Control vol 88 no 10 pp2055ndash2065 2015
[5] R Cui X Zhang and D Cui ldquoAdaptive sliding-mode attitudecontrol for autonomous underwater vehicles with input nonlin-earitiesrdquo Ocean Engineering vol 123 pp 45ndash54 2016
[6] Y Wang L Gu M Gao and K Zhu ldquoMultivariable outputfeedback adaptive terminal sliding mode control for underwa-ter vehiclesrdquoAsian Journal of Control vol 18 no 1 pp 247ndash2652016
[7] X Qi ldquoAdaptive coordinated tracking control of multipleautonomous underwater vehiclesrdquo Ocean Engineering vol 91pp 84ndash90 2014
[8] R Cui C Yang Y Li and S Sharma ldquoAdaptive Neural NetworkControl of AUVs With Control Input Nonlinearities UsingReinforcement Learningrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 47 no 6 pp 1019ndash1029 2017
[9] B B Miao T S Li and W L Luo ldquoA DSC and MLP basedrobust adaptive NN tracking control for underwater vehiclerdquoNeurocomputing vol 111 pp 184ndash189 2013
[10] Y-C Liu S-Y Liu and N Wang ldquoFully-tuned fuzzy neuralnetwork based robust adaptive tracking control of unmannedunderwater vehicle with thruster dynamicsrdquo Neurocomputingvol 196 pp 1ndash13 2016
[11] J Ghommam and M Saad ldquoBackstepping-based cooperativeand adaptive tracking control design for a group of underactu-ated AUVs in horizontal planrdquo International Journal of Controlvol 87 no 5 pp 1076ndash1093 2014
[12] C Shen B Buckham and Y Shi ldquoModified CGMRES Algo-rithm for Fast Nonlinear Model Predictive Tracking Control ofAUVsrdquo IEEE Transactions on Control Systems Technology vol25 no 5 pp 1896ndash1904 2017
[13] J Gao A A Proctor Y Shi and C Bradley ldquoHierarchicalModel Predictive Image-Based Visual Servoing of UnderwaterVehicles with Adaptive Neural Network Dynamic ControlrdquoIEEE Transactions on Cybernetics vol 46 no 10 pp 2323ndash23342016
[14] J Sun and J Chen ldquoNetworked predictive control for systemswith unknown or partially known delayrdquo IET Control Theory ampApplications vol 8 no 18 pp 2282ndash2288 2014
[15] S Li and G-P Liu ldquoNetworked predictive control for nonlinearsystems with stochastic disturbances in the presence of datalossesrdquo Neurocomputing vol 194 pp 56ndash64 2016
[16] J Zhao L Zhang and X Qi ldquoA necessary and sufficientcondition for stabilization of switched descriptor time-delaysystems under arbitrary switchingrdquo Asian Journal of Controlvol 18 no 1 pp 266ndash272 2016
[17] S Sirouspour and A Shahdi ldquoModel predictive control fortransparent teleoperation under communication time delayrdquoIEEE Transactions on Robotics vol 22 no 6 pp 1131ndash1145 2006
[18] Y Sheng Y Shen and M Zhu ldquoDelay-dependent globalexponential stability for delayed recurrent neural networksrdquoIEEE Transactions on Neural Networks and Learning Systems2016
[19] O M Smith ldquoA controller to overcome deadtimerdquo ISA J vol 6p 2833 1959
[20] C F Caruntu and C Lazar ldquoNetwork delay predictive com-pensation based on time-delay modelling as disturbancerdquoInternational Journal of Control vol 87 no 10 pp 2012ndash20262014
[21] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
[22] S H HosseinNia and M Lundh ldquoA general robust MPCdesign for the state-space model application to paper machineprocessrdquo Asian Journal of Control vol 18 no 5 pp 1891ndash19072016
[23] Y Ding Z Xu J Zhao KWang and Z Shao ldquoEmbeddedMPCcontroller based on interior-point method with convergencedepth controlrdquoAsian Journal of Control vol 18 no 6 pp 2064ndash2077 2016
8 Journal of Control Science and Engineering
[24] J-Q Huang and F L Lewis ldquoNeural-network predictive controlfor nonlinear dynamic systems with time-delayrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 14 no 2pp 377ndash389 2003
[25] B An and G Liu ldquoModel-based predictive controller designfor a class of nonlinear networked systemswith communicationdelays and data lossrdquo Chinese Physics B vol 23 no 8 p 0802022014
[26] N Sharma C M Gregory and W E Dixon ldquoPredictor-basedcompensation for electromechanical delay during neuromuscu-lar electrical stimulationrdquo IEEE Transactions on Neural Systemsand Rehabilitation Engineering vol 19 no 6 pp 601ndash611 2011
[27] N Fischer A Dani N Sharma and W E Dixon ldquoSaturatedcontrol of an uncertain nonlinear system with input delayrdquoAutomatica vol 49 no 6 pp 1741ndash1747 2013
[28] T Wang H Gao and J Qiu ldquoA combined adaptive neuralnetwork and nonlinear model predictive control for multiratenetworked industrial process controlrdquo IEEE Transactions onNeural Networks and Learning Systems vol 27 no 2 pp 416ndash425 2016
[29] PDai ZGong andGGuo ldquoPerformance analysis of amodularmultilevel converter drive system with model predictive con-trolrdquo International Journal of Control andAutomation vol 9 no1 pp 387ndash398 2016
[30] C Wei J Luo H Dai Z Yin W Ma and J Yuan ldquoGloballyrobust explicit model predictive control of constrained systemsexploiting SVM-based approximationrdquo International Journal ofRobust and Nonlinear Control vol 27 no 16 pp 3000ndash30272017
[31] M Parvez Akter S Mekhilef N Mei Lin Tan and H AkagildquoModified Model Predictive Control of a Bidirectional AC-DCConverter Based on Lyapunov Function for Energy StorageSystemsrdquo IEEETransactions on Industrial Electronics vol 63 no2 pp 704ndash715 2016
[32] R Prasanth Kumar A Dasgupta and C S Kumar ldquoRobusttrajectory control of underwater vehicles using time delaycontrol lawrdquo Ocean Engineering vol 34 no 5-6 pp 842ndash8492007
[33] K Mukherjee I N Kar and R K P Bhatt ldquoRegion trackingbased control of an autonomous underwater vehicle with inputdelayrdquo Ocean Engineering vol 99 pp 107ndash114 2015
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
6 Journal of Control Science and Engineering
119898V = 119898 minus 119884V119898119906119903 = 119898 minus 119884119903119889V = minus119884V119906V minus 119884V|V|V |V| 119898119903 = 119868119911 minus 119873 119903119889119903 = minus119873V119906V minus 119873V|V|V |V| minus 119873119903119906119903
(43)
Since the consideredmodel (42) is in the horizontal planeand has no disturbance we can assume that 119906 = 1119898119904in principle In the following the desired path is 119909119889(119905) =20 sin 2120587119905200 119910119889(119905) = 20 minus 20 cos 2120587119905200 and 120595119889(119905) =2120587119905200
The model parameters of INFANTE AUV are as follows
119898 = 22345119896119892119868119911 = 20001198731198982
119883 = minus142119896119892119873 119903 = minus13501198731198982119884V = minus1715119896119892119884V = minus346119896119892119898119884119903 = 435119896119892119873V = minus686119896119892119873119903 = minus1427119896119892119898
119884V|V| = minus667119896119892119898119873V|V| = 443119896119892
(44)
NN parameters are selected as follows Φ(119909) = 1(1 +119890minus119886119909) where 119886 = 05 120583 = 03 Γ = 6The delay constant 119889 = 2 Other parameters in controller
are 1198601 = minus31198683 1198602 = minus21198683 1205821 = 2 1205822 = 2 and 119870119903 = 8The initial position and the surge speed of the AUV
are (0 20) and 0119898119904 respectively The simulation resultsare shown in Figures 2ndash4 The path tracking errors in119909 119910 and 120595 are given in Figure 2 The control forces in119909 and 119910 and the control torque of yaw 120595 are given inFigure 3 From these simulation figures we can see thatthe tracking performance is unsatisfying at the beginning ofsimulation this is because the controller performs mainlydepending on the adaptive control The good tracking ofposition is obtained by the proposed adaptive NN predictivecontroller by and by Figure 4 is the path tracking inhorizontal plane From Figure 4 we can see that AUV canrealize tracking control smoothly and converge to the desiredtrajectory
7 Conclusion
This paper investigates the path tracking problem for anAUV with input delay Based on predictive and adaptive NN
minus020
0204
e x(m
)
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
minus05
0
05
e (r
ad)
minus05
0
05
e y(m
)
100 200 300 400 500 600 700 8000time (s)
Figure 2 Path tracking errors
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
100 200 300 400 500 600 700 8000time (s)
0
20
40Fx
(N)
0
100
200
Fy
(N)
minus100
0
100
M
(Nm
)
Figure 3 Control forces in 119909 and 119910 and control torque of yaw
control theory a predictive controller is given The outputfeedback control algorithm is employed here The NN isused to estimate the dynamic uncertain nonlinear functioninduced by hydrodynamic coefficients and coupling of thesurge sway and yaw angular velocity The predictive controlis introduced to compensate the input delay present in AUVThe stability of the controller was analyzed by Lyapunov the-orem Simulation results showed that the proposed controllerperforms well with stability
Data Availability
We are sorry that we cannot share the data in our article nowbecause future works are based on its results The methodsin this paper are effective methods for investigation of pathfollowing for autonomous underwater vehicles with input
Journal of Control Science and Engineering 7
AUV trajectorydesired trajectory
minus5
0
5
10
15
20
25
30
35
40
45
y (m
)
minus20 minus15 minus10 minus5 0 5 10 15 20 25minus25x (m)
Figure 4 Path tracking response in 119909119910 plane
delay We will apply a patent on the relevant studies So wecannot share the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the Hubei Provincial NaturalScience Foundation of China (2016CFB273) and the Researchand Innovation Initiatives of WHPU (2018Y20)
References
[1] P Jantapremjit and P A Wilson ldquoGuidance-control based pathfollowing for homing and docking using an autonomous under-water vehiclerdquo in Proceedings of the OCEANSrsquo08 MTSIEEEKobe-Techno-Oceanrsquo08 - Voyage toward the Future OTOrsquo08 jpnApril 2008
[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007
[3] A Adhami-Mirhosseini M J Yazdanpanah and A P AguiarldquoAutomatic bottom-following for underwater robotic vehiclesrdquoAutomatica vol 50 no 8 pp 2155ndash2162 2014
[4] M Kim H Joe J Kim and S-c Yu ldquoIntegral sliding modecontroller for precise manoeuvring of autonomous underwatervehicle in the presence of unknown environmental distur-bancesrdquo International Journal of Control vol 88 no 10 pp2055ndash2065 2015
[5] R Cui X Zhang and D Cui ldquoAdaptive sliding-mode attitudecontrol for autonomous underwater vehicles with input nonlin-earitiesrdquo Ocean Engineering vol 123 pp 45ndash54 2016
[6] Y Wang L Gu M Gao and K Zhu ldquoMultivariable outputfeedback adaptive terminal sliding mode control for underwa-ter vehiclesrdquoAsian Journal of Control vol 18 no 1 pp 247ndash2652016
[7] X Qi ldquoAdaptive coordinated tracking control of multipleautonomous underwater vehiclesrdquo Ocean Engineering vol 91pp 84ndash90 2014
[8] R Cui C Yang Y Li and S Sharma ldquoAdaptive Neural NetworkControl of AUVs With Control Input Nonlinearities UsingReinforcement Learningrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 47 no 6 pp 1019ndash1029 2017
[9] B B Miao T S Li and W L Luo ldquoA DSC and MLP basedrobust adaptive NN tracking control for underwater vehiclerdquoNeurocomputing vol 111 pp 184ndash189 2013
[10] Y-C Liu S-Y Liu and N Wang ldquoFully-tuned fuzzy neuralnetwork based robust adaptive tracking control of unmannedunderwater vehicle with thruster dynamicsrdquo Neurocomputingvol 196 pp 1ndash13 2016
[11] J Ghommam and M Saad ldquoBackstepping-based cooperativeand adaptive tracking control design for a group of underactu-ated AUVs in horizontal planrdquo International Journal of Controlvol 87 no 5 pp 1076ndash1093 2014
[12] C Shen B Buckham and Y Shi ldquoModified CGMRES Algo-rithm for Fast Nonlinear Model Predictive Tracking Control ofAUVsrdquo IEEE Transactions on Control Systems Technology vol25 no 5 pp 1896ndash1904 2017
[13] J Gao A A Proctor Y Shi and C Bradley ldquoHierarchicalModel Predictive Image-Based Visual Servoing of UnderwaterVehicles with Adaptive Neural Network Dynamic ControlrdquoIEEE Transactions on Cybernetics vol 46 no 10 pp 2323ndash23342016
[14] J Sun and J Chen ldquoNetworked predictive control for systemswith unknown or partially known delayrdquo IET Control Theory ampApplications vol 8 no 18 pp 2282ndash2288 2014
[15] S Li and G-P Liu ldquoNetworked predictive control for nonlinearsystems with stochastic disturbances in the presence of datalossesrdquo Neurocomputing vol 194 pp 56ndash64 2016
[16] J Zhao L Zhang and X Qi ldquoA necessary and sufficientcondition for stabilization of switched descriptor time-delaysystems under arbitrary switchingrdquo Asian Journal of Controlvol 18 no 1 pp 266ndash272 2016
[17] S Sirouspour and A Shahdi ldquoModel predictive control fortransparent teleoperation under communication time delayrdquoIEEE Transactions on Robotics vol 22 no 6 pp 1131ndash1145 2006
[18] Y Sheng Y Shen and M Zhu ldquoDelay-dependent globalexponential stability for delayed recurrent neural networksrdquoIEEE Transactions on Neural Networks and Learning Systems2016
[19] O M Smith ldquoA controller to overcome deadtimerdquo ISA J vol 6p 2833 1959
[20] C F Caruntu and C Lazar ldquoNetwork delay predictive com-pensation based on time-delay modelling as disturbancerdquoInternational Journal of Control vol 87 no 10 pp 2012ndash20262014
[21] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
[22] S H HosseinNia and M Lundh ldquoA general robust MPCdesign for the state-space model application to paper machineprocessrdquo Asian Journal of Control vol 18 no 5 pp 1891ndash19072016
[23] Y Ding Z Xu J Zhao KWang and Z Shao ldquoEmbeddedMPCcontroller based on interior-point method with convergencedepth controlrdquoAsian Journal of Control vol 18 no 6 pp 2064ndash2077 2016
8 Journal of Control Science and Engineering
[24] J-Q Huang and F L Lewis ldquoNeural-network predictive controlfor nonlinear dynamic systems with time-delayrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 14 no 2pp 377ndash389 2003
[25] B An and G Liu ldquoModel-based predictive controller designfor a class of nonlinear networked systemswith communicationdelays and data lossrdquo Chinese Physics B vol 23 no 8 p 0802022014
[26] N Sharma C M Gregory and W E Dixon ldquoPredictor-basedcompensation for electromechanical delay during neuromuscu-lar electrical stimulationrdquo IEEE Transactions on Neural Systemsand Rehabilitation Engineering vol 19 no 6 pp 601ndash611 2011
[27] N Fischer A Dani N Sharma and W E Dixon ldquoSaturatedcontrol of an uncertain nonlinear system with input delayrdquoAutomatica vol 49 no 6 pp 1741ndash1747 2013
[28] T Wang H Gao and J Qiu ldquoA combined adaptive neuralnetwork and nonlinear model predictive control for multiratenetworked industrial process controlrdquo IEEE Transactions onNeural Networks and Learning Systems vol 27 no 2 pp 416ndash425 2016
[29] PDai ZGong andGGuo ldquoPerformance analysis of amodularmultilevel converter drive system with model predictive con-trolrdquo International Journal of Control andAutomation vol 9 no1 pp 387ndash398 2016
[30] C Wei J Luo H Dai Z Yin W Ma and J Yuan ldquoGloballyrobust explicit model predictive control of constrained systemsexploiting SVM-based approximationrdquo International Journal ofRobust and Nonlinear Control vol 27 no 16 pp 3000ndash30272017
[31] M Parvez Akter S Mekhilef N Mei Lin Tan and H AkagildquoModified Model Predictive Control of a Bidirectional AC-DCConverter Based on Lyapunov Function for Energy StorageSystemsrdquo IEEETransactions on Industrial Electronics vol 63 no2 pp 704ndash715 2016
[32] R Prasanth Kumar A Dasgupta and C S Kumar ldquoRobusttrajectory control of underwater vehicles using time delaycontrol lawrdquo Ocean Engineering vol 34 no 5-6 pp 842ndash8492007
[33] K Mukherjee I N Kar and R K P Bhatt ldquoRegion trackingbased control of an autonomous underwater vehicle with inputdelayrdquo Ocean Engineering vol 99 pp 107ndash114 2015
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Journal of Control Science and Engineering 7
AUV trajectorydesired trajectory
minus5
0
5
10
15
20
25
30
35
40
45
y (m
)
minus20 minus15 minus10 minus5 0 5 10 15 20 25minus25x (m)
Figure 4 Path tracking response in 119909119910 plane
delay We will apply a patent on the relevant studies So wecannot share the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work is supported by the Hubei Provincial NaturalScience Foundation of China (2016CFB273) and the Researchand Innovation Initiatives of WHPU (2018Y20)
References
[1] P Jantapremjit and P A Wilson ldquoGuidance-control based pathfollowing for homing and docking using an autonomous under-water vehiclerdquo in Proceedings of the OCEANSrsquo08 MTSIEEEKobe-Techno-Oceanrsquo08 - Voyage toward the Future OTOrsquo08 jpnApril 2008
[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007
[3] A Adhami-Mirhosseini M J Yazdanpanah and A P AguiarldquoAutomatic bottom-following for underwater robotic vehiclesrdquoAutomatica vol 50 no 8 pp 2155ndash2162 2014
[4] M Kim H Joe J Kim and S-c Yu ldquoIntegral sliding modecontroller for precise manoeuvring of autonomous underwatervehicle in the presence of unknown environmental distur-bancesrdquo International Journal of Control vol 88 no 10 pp2055ndash2065 2015
[5] R Cui X Zhang and D Cui ldquoAdaptive sliding-mode attitudecontrol for autonomous underwater vehicles with input nonlin-earitiesrdquo Ocean Engineering vol 123 pp 45ndash54 2016
[6] Y Wang L Gu M Gao and K Zhu ldquoMultivariable outputfeedback adaptive terminal sliding mode control for underwa-ter vehiclesrdquoAsian Journal of Control vol 18 no 1 pp 247ndash2652016
[7] X Qi ldquoAdaptive coordinated tracking control of multipleautonomous underwater vehiclesrdquo Ocean Engineering vol 91pp 84ndash90 2014
[8] R Cui C Yang Y Li and S Sharma ldquoAdaptive Neural NetworkControl of AUVs With Control Input Nonlinearities UsingReinforcement Learningrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 47 no 6 pp 1019ndash1029 2017
[9] B B Miao T S Li and W L Luo ldquoA DSC and MLP basedrobust adaptive NN tracking control for underwater vehiclerdquoNeurocomputing vol 111 pp 184ndash189 2013
[10] Y-C Liu S-Y Liu and N Wang ldquoFully-tuned fuzzy neuralnetwork based robust adaptive tracking control of unmannedunderwater vehicle with thruster dynamicsrdquo Neurocomputingvol 196 pp 1ndash13 2016
[11] J Ghommam and M Saad ldquoBackstepping-based cooperativeand adaptive tracking control design for a group of underactu-ated AUVs in horizontal planrdquo International Journal of Controlvol 87 no 5 pp 1076ndash1093 2014
[12] C Shen B Buckham and Y Shi ldquoModified CGMRES Algo-rithm for Fast Nonlinear Model Predictive Tracking Control ofAUVsrdquo IEEE Transactions on Control Systems Technology vol25 no 5 pp 1896ndash1904 2017
[13] J Gao A A Proctor Y Shi and C Bradley ldquoHierarchicalModel Predictive Image-Based Visual Servoing of UnderwaterVehicles with Adaptive Neural Network Dynamic ControlrdquoIEEE Transactions on Cybernetics vol 46 no 10 pp 2323ndash23342016
[14] J Sun and J Chen ldquoNetworked predictive control for systemswith unknown or partially known delayrdquo IET Control Theory ampApplications vol 8 no 18 pp 2282ndash2288 2014
[15] S Li and G-P Liu ldquoNetworked predictive control for nonlinearsystems with stochastic disturbances in the presence of datalossesrdquo Neurocomputing vol 194 pp 56ndash64 2016
[16] J Zhao L Zhang and X Qi ldquoA necessary and sufficientcondition for stabilization of switched descriptor time-delaysystems under arbitrary switchingrdquo Asian Journal of Controlvol 18 no 1 pp 266ndash272 2016
[17] S Sirouspour and A Shahdi ldquoModel predictive control fortransparent teleoperation under communication time delayrdquoIEEE Transactions on Robotics vol 22 no 6 pp 1131ndash1145 2006
[18] Y Sheng Y Shen and M Zhu ldquoDelay-dependent globalexponential stability for delayed recurrent neural networksrdquoIEEE Transactions on Neural Networks and Learning Systems2016
[19] O M Smith ldquoA controller to overcome deadtimerdquo ISA J vol 6p 2833 1959
[20] C F Caruntu and C Lazar ldquoNetwork delay predictive com-pensation based on time-delay modelling as disturbancerdquoInternational Journal of Control vol 87 no 10 pp 2012ndash20262014
[21] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
[22] S H HosseinNia and M Lundh ldquoA general robust MPCdesign for the state-space model application to paper machineprocessrdquo Asian Journal of Control vol 18 no 5 pp 1891ndash19072016
[23] Y Ding Z Xu J Zhao KWang and Z Shao ldquoEmbeddedMPCcontroller based on interior-point method with convergencedepth controlrdquoAsian Journal of Control vol 18 no 6 pp 2064ndash2077 2016
8 Journal of Control Science and Engineering
[24] J-Q Huang and F L Lewis ldquoNeural-network predictive controlfor nonlinear dynamic systems with time-delayrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 14 no 2pp 377ndash389 2003
[25] B An and G Liu ldquoModel-based predictive controller designfor a class of nonlinear networked systemswith communicationdelays and data lossrdquo Chinese Physics B vol 23 no 8 p 0802022014
[26] N Sharma C M Gregory and W E Dixon ldquoPredictor-basedcompensation for electromechanical delay during neuromuscu-lar electrical stimulationrdquo IEEE Transactions on Neural Systemsand Rehabilitation Engineering vol 19 no 6 pp 601ndash611 2011
[27] N Fischer A Dani N Sharma and W E Dixon ldquoSaturatedcontrol of an uncertain nonlinear system with input delayrdquoAutomatica vol 49 no 6 pp 1741ndash1747 2013
[28] T Wang H Gao and J Qiu ldquoA combined adaptive neuralnetwork and nonlinear model predictive control for multiratenetworked industrial process controlrdquo IEEE Transactions onNeural Networks and Learning Systems vol 27 no 2 pp 416ndash425 2016
[29] PDai ZGong andGGuo ldquoPerformance analysis of amodularmultilevel converter drive system with model predictive con-trolrdquo International Journal of Control andAutomation vol 9 no1 pp 387ndash398 2016
[30] C Wei J Luo H Dai Z Yin W Ma and J Yuan ldquoGloballyrobust explicit model predictive control of constrained systemsexploiting SVM-based approximationrdquo International Journal ofRobust and Nonlinear Control vol 27 no 16 pp 3000ndash30272017
[31] M Parvez Akter S Mekhilef N Mei Lin Tan and H AkagildquoModified Model Predictive Control of a Bidirectional AC-DCConverter Based on Lyapunov Function for Energy StorageSystemsrdquo IEEETransactions on Industrial Electronics vol 63 no2 pp 704ndash715 2016
[32] R Prasanth Kumar A Dasgupta and C S Kumar ldquoRobusttrajectory control of underwater vehicles using time delaycontrol lawrdquo Ocean Engineering vol 34 no 5-6 pp 842ndash8492007
[33] K Mukherjee I N Kar and R K P Bhatt ldquoRegion trackingbased control of an autonomous underwater vehicle with inputdelayrdquo Ocean Engineering vol 99 pp 107ndash114 2015
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
8 Journal of Control Science and Engineering
[24] J-Q Huang and F L Lewis ldquoNeural-network predictive controlfor nonlinear dynamic systems with time-delayrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 14 no 2pp 377ndash389 2003
[25] B An and G Liu ldquoModel-based predictive controller designfor a class of nonlinear networked systemswith communicationdelays and data lossrdquo Chinese Physics B vol 23 no 8 p 0802022014
[26] N Sharma C M Gregory and W E Dixon ldquoPredictor-basedcompensation for electromechanical delay during neuromuscu-lar electrical stimulationrdquo IEEE Transactions on Neural Systemsand Rehabilitation Engineering vol 19 no 6 pp 601ndash611 2011
[27] N Fischer A Dani N Sharma and W E Dixon ldquoSaturatedcontrol of an uncertain nonlinear system with input delayrdquoAutomatica vol 49 no 6 pp 1741ndash1747 2013
[28] T Wang H Gao and J Qiu ldquoA combined adaptive neuralnetwork and nonlinear model predictive control for multiratenetworked industrial process controlrdquo IEEE Transactions onNeural Networks and Learning Systems vol 27 no 2 pp 416ndash425 2016
[29] PDai ZGong andGGuo ldquoPerformance analysis of amodularmultilevel converter drive system with model predictive con-trolrdquo International Journal of Control andAutomation vol 9 no1 pp 387ndash398 2016
[30] C Wei J Luo H Dai Z Yin W Ma and J Yuan ldquoGloballyrobust explicit model predictive control of constrained systemsexploiting SVM-based approximationrdquo International Journal ofRobust and Nonlinear Control vol 27 no 16 pp 3000ndash30272017
[31] M Parvez Akter S Mekhilef N Mei Lin Tan and H AkagildquoModified Model Predictive Control of a Bidirectional AC-DCConverter Based on Lyapunov Function for Energy StorageSystemsrdquo IEEETransactions on Industrial Electronics vol 63 no2 pp 704ndash715 2016
[32] R Prasanth Kumar A Dasgupta and C S Kumar ldquoRobusttrajectory control of underwater vehicles using time delaycontrol lawrdquo Ocean Engineering vol 34 no 5-6 pp 842ndash8492007
[33] K Mukherjee I N Kar and R K P Bhatt ldquoRegion trackingbased control of an autonomous underwater vehicle with inputdelayrdquo Ocean Engineering vol 99 pp 107ndash114 2015
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
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