nonlinear reduced-order observer-based predictive control...

Research ArticleNonlinear Reduced-Order Observer-Based Predictive Controlfor Diving of an Autonomous Underwater Vehicle

Xuliang Yao1 Guangyi Yang1 and Yu Peng2

1College of Automation Harbin Engineering University Harbin 150001 China2Beijing Aerospace Automatic Control Institute Beijing 100854 China

Correspondence should be addressed to Guangyi Yang hahaygyhrbeueducn

Received 15 October 2016 Accepted 14 December 2016 Published 3 January 2017

Academic Editor Juan R Torregrosa

Copyright copy 2017 Xuliang Yao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The attitude control and depth tracking issue of autonomous underwater vehicle (AUV) are addressed in this paper By introducinga nonsingular coordinate transformation a novel nonlinear reduced-order observer (NROO) is presented to achieve an accurateestimation of AUVrsquos state variables A discrete-time model predictive control with nonlinear model online linearization (MPC-NMOL) is applied to enhance the attitude control and depth tracking performance of AUV considering the wave disturbance nearsurface In AUV longitudinal control simulation the comparisons have been presented between NROO and full-order observer(FOO) and also between MPC-NMOL and traditional NMPC Simulation results show the effectiveness of the proposed method

1 Introduction

Autonomous underwater vehicle (AUV) is an important toolfor ocean exploitation Due to its strong autonomous abilityAUV can complete the mission given by human and hasbecome widely used in military and scientific research [1]

Underactuated AUV is a kind of underwater vehicleswhose actual control inputs are less than degrees of freedom(DOF) [2] When AUV navigates near surface diving controlis one of the key research fields a variety of methodshave been proposed for AUV in vertical plane Yan etel [3] proposed backstepping technology and IFTSMC forsurge velocity and depth control respectively However thecoupled interaction between the two controllers was notconsidered An improved dynamic fuzzy SMC is establishedto enhance the depth tracking control performance in [4]Adhami-Mirhosseini et al [5] proposed a nonlinear dynamiccontroller combined with Fourier series expansion and pseu-dospectral ideas to achieve the depth tracking objectiveIn [6] an analytical SMC method is used to obtain thetime optimal trajectory tracking and the effectiveness of theproposed controller was verified via simulations The fuzzyfeedback linearization methods studied in [7] abandonedtwo general assumptions and used the nonlinear dynamicsof depth motion directly However discrete-time model was

not taken into consideration and there is lack of simulationverification of the proposed algorithm Hsu and Liu [8]investigated a steady-state depth errors problem caused bycenter of gravity change and considered the effects of gravityand buoyancy in AUV dynamics By adding a switchingPI controller in external-loop the steady-state depth errorswere eliminated finally A static output feedback controllerwas studied in [9] to complete the diving task however alinearized model was used

Model predictive control can calculate a sequence offuture control signals in advance in order to track a givenfuture reference Moreover predictive control allows theintegration of system constraints in the controller designand minimizes a cost function defined over a predictionhorizon ensuring near-optimal performance of the systemThe predictive control of AUVmotion is one of the promisingresearch areas in marine control systems field In [10] visualservoing MPC is used to bring current-influenced AUVaperiodic algorithm-generated control loop closures A costfunction minimization method is used in [11] with MPCcalled least squares whose advantage is the real-time exe-cution ability to optimize sawtooth paths for an underwaterglider A reduced dynamical model proposed in [12] is usedto control AUV in vertical and horizontal planes respectively

HindawiDiscrete Dynamics in Nature and SocietyVolume 2017 Article ID 4394571 15 pageshttpsdoiorg10115520174394571

2 Discrete Dynamics in Nature and Society

The MPC is formulated as two independent optimizationparts However by using the same initial states for both partsof linearization and neglecting the weakly coupled states thecontrol optimization differs from the actual situation a lotIn [13] a sampling based MPC is proposed to generate thecontrol sequence with constraints effectively These methodshave a good performance in the field ofMPCuse for AUV butthey do not deal with wave disturbances The method in [14]used MPC to predict future disturbances and counteract thewave disturbances A linearwave theory solver is employed byestimator to approximate the fluid dynamics but constraintswere not considered NMPC is an extension of MPC by usingnonlinear systems and constraints applications of NMPCfor AUV are less frequent In [15] NMPC is used to designthe kinematic loop in a hierarchical structure for AUV DPcontrol

Because of fewer actuators and sensors there is anadvantage for underactuated AUV to simplify the design andreduce costs However it brings highly coupled nonlinearmodels which increases the difficulty of controller designIn general some state variables of AUV are unmeasured dueto a lack of sensors or sensor faults So an observer-basedcontroller becomes significant A dual observer in [16] whichcombines a state and a perturbation observer aims to solvethe problem about being sensitive to external disturbanceZhang et al [17] considered the elevator angle constraints andproposed a controller based onMPCwith artificial bee colonyalgorithm and a classical linear state observer is used In [18]a novel discrete-time proportional and integral observer isused to estimate the states output input and disturbancetogether Zhang et al [19] proposed a new reduced-orderobserver with multiconstrained thoughts by using specificsystem decomposition However these results only use linearmodels

In contrast most practical systems are nonlinear andtherefore nonlinear models are required Kinsey et al [20]proposed a nonlinear observer based on dynamic model ofAUVwhich is used to estimate the vehiclersquos velocityHoweverin theirmodel the coupling terms are neglected and therewasonly one degree of freedomMahapatra et al [21] investigateda nonlinear119867infin feedback control algorithm with a nonlinearstate observer but the error dynamics stability was notconsidered in observer design Zhang et al [22] extendedan adaptive state observer to a class of nonlinear systemsHowever due to the selected special Lyapunov matrix thereis a reduction of the accuracy of state estimation

At the same time MPC need state variables to achievedesired output tracking in the minimization of cost functionso the role of observer-based MPC has a prominent positionin practical implementation However there is a situationthat some states variables of underactuated AUV can bemeasured So it is not necessary to estimate all the variablesCompared with FOO reduced-order observer (ROO) hassimplified structure and good estimation performance [2324] which can be designed to estimate only the unmeasurablestates Recently ROO designs used in MPC are exploited inboth linear [25 26] and nonlinear systems [27ndash30] Howeverfew research results are reported on NROO-based MPC forAUV So the motivation of this paper is the aforementioned

problems and deficiencies and the aim is to design a NROO-based MPC with nonlinear AUV kinematics and dynamicsmodel

In this paper amodel predictive controllerwith nonlinearmodel online-linearization based on NROO is proposed tosolve the attitude control and depth tracking issues Thedesign process is as follows First the nonlinear dynamicsin AUV vertical plane are described and Euler approxi-mation is used to discrete the proposed model Secondthe nonlinear reduced-order observer is designed accordingto the dynamics above to estimate the unmeasured statevariables of AUV By solving a convex optimization problemand using LMI technique the estimation performance isenhanced and the wave disturbance influence is restrictedThird a model predictive control based on model online-linearization is presented in which the controller designprocess is independent of the NROO and it is convenientfor the following simulation Finally simulation results showthe good robustness and efficiency of the proposed methodswhich can be applied to AUV attitude control and depthtracking field

The rest of the paper is organized as follows Section 2described the AUV vertical plane dynamics NROO design isproposed in Section 3 Section 4 introduces theMPC-NMOLstrategy Section 5 gives the simulation results of AUV toverify the effectiveness of the proposed approach

2 AUV Vertical Plane Dynamics

The vertical plane dynamic and kinematic model of AUV aredescribed in this section and the following assumptions areconsidered

Assumption 1 (1) AUV navigates at a constant speed 119906 = 119880119888(2) To analyze the diving control sway yaw and rollmotions can be ignored and that means 119910 = 0 V = 0 and120601 = 120595 = 0(3) Center of buoyancy stays constant with no changemade to external hull but the position of center of gravity isvariable(4)The hydrodynamics drag terms whose order is higherthan two are neglected(5) Pitch angle 120579 has small perturbations at zero

Therefore the dynamic and kinematic model of AUV canbe expressed in the nonlinear forms as follows119898[ minus 119906119902 minus 119909119892] = Σ119885ext119868119910 + 119898 [119911119892120596119902 minus 119909119892 + 119909119892119906119902] = Σ119872ext = minus119906120579 + 120596 = 119902

(1)

whereΣ119885ext = 119885119867119878 + 119885120596|120596|120596 |120596| + 119885 + 119885 + 119885119906119902119906119902+ 119885119906120596119906120596 + 1198851205751199062120575 + 119885waveΣ119872ext = 119872119867119878 +119872119902|119902|119902 10038161003816100381610038161199021003816100381610038161003816 + 119872 + 119872 + 119872119906119902119906119902+119872119906120596119906120596 +1198721205751199062120575 +119872wave

Discrete Dynamics in Nature and Society 3

119885119867119878 = (119882 minus 119861) cos 120579119872119867119878 = (119911119887119861 minus 119911119892119882)120579 + (119909119887119861 minus 119909119892119882) (2)119868119910 119898 are the moment of inertia and the mass of AUV

respectively (119909119892 119910119892 119911119892) and (119909119887 119910119887 119911119887) are the coordinatesof AUVrsquos center of gravity and buoyancy 119906 120596 and 119902 arethe surge heave and pitch angle velocity in body-fixedframe respectively 119911 are depth in earth-fixed frame 120575 is theelevator deflection angle 119885120596|120596|120596|120596| 119885 119885 119885119906119902 119885119906120596 119885120575119872119902|119902|119902|119902|119872119872119872119906119902119872119906120596 and119872120575 are the hydrodynamicscoefficients of AUV and 119885wave119872wave are the wave force andmoment

The nonlinear model (1) can be simply expressed as thefollowing system119888 (119905) = 119860119888119909119888 (119905) + 119892119888 (119905 119909 (119905)) + 119861119888119906 (119905) + 119863119888119908 (119905) 119910 (119905) = 119862119909119888 (119905) (3)

where

119860119888 = 1198721minus1[[[[[11989911 11989912 0 011989921 11989922 0 119899241 0 0 minus1199060 1 0 0

]]]]]

119861119888 = 1198721minus1[[[[[11988512057511987212057500]]]]]

119863119888 = 1198721minus1 [1198682times202times2] 119892119888 (119905 119909 (119905))

= 1198721minus1[[[[[

119885120596|120596|120596 |120596| + (119882 minus 119861)minus119898119911119892120596119902 +119872119902|119902|119902 10038161003816100381610038161199021003816100381610038161003816 minus (119909119892119882minus 119909119887119861)00]]]]]

1198721 = [[[[[11989811 11989812 0 011989821 11989822 0 00 0 1 00 0 0 1

]]]]]

119862 = [0 0 1 00 0 0 1] 11989811 = 119898 minus 11988511989812 = minus119885 minus 11989811990911989211989821 = minus119898119909119892 minus11987211989822 = 119868119910 minus11987211989911 = 119885119906120596119880119888

11989912 = (119885119906119902 + 119898119880119888) 11989921 = 11987211990612059611988011988811989922 = (119872119906119902 minus 119898119909119892119880119888) 11989924 = minus (119911119892119882minus 119911119887119861)

(4)

Due to the digital controller implemented by usingzero-order hold a discrete-time vector-form nonlinear stateequation is obtained by Euler approximation [31]

119909 (119896 + 1) = 119860119909 (119896) + 119892 (119896 119909 (119896)) + 119861119906 (119896) + 119863119908 (119896) 119910 (119896) = 119862119909 (119896) (5)

where 119909(119896) = [1199091 1199092 1199093 1199094]119879 isin 119877119899 1199091 = 120596 1199092 = 119902 1199093 = 1199111199094 = 120579 119879119904 is the sampling period 119860 = (119879119904119860119888 + 119868) 119861 = 119879119904119861119888119863 = 119879119904119863119888 119908(119896) = [119885wave 119872wave] isin 119877119889 119906(119896) isin 119877119898 is theinput 120575 119892(119896 119909(119896)) = 119879119904119892119888(119905 119909(119905)) is a nonlinear-term vectorand 119910(119896) isin 119877119901 denotes the output vector In this paper 119899 = 4119889 = 2119898 = 1 119901 = 2 119862 is a full-rank matrix and rank(119862) = 2and we assume that pair (119860 119862) is observableRemark 2 In this paper the disturbance due to surface wavesis considered as the main disturbance type and written as119908(119896) Sea state is used to classify the sea conditions andexpressed by wind speed and wave height Wave height isaffected by wind and the wave profile is affected by waterdepth For surface waves modelling we assume the longcrested and unidirectional sea state The superposition prin-ciple is used to complete an irregular long crested wave seastate simulation The single parameter Pierson-Moskowitzspectrum [32] is used in this model and its spectrum densityis defined as follows

119878 (120596) = 81 times 10minus311989221205965 exp[minus31111986721199041205964] (6)

where 119867119904 is the significant wave height (m) and 120596 iswave frequency Assuming the AUV body is small enoughcomparedwith the incomingwave and thewave speed is largeenough in relation to the hull diameter1198630 the transient forceand moment are found by using Morisonrsquos equation [33] andintegration along the length of the hull 1198710 at each step isobtained below

119885wave = int1198710

(119862119889 12058811986302 (120596120596 minus 120596)2+ 119862119898 1205881205871198632

04 (120596 minus )2)119889119909119872wave = int

1198710

(119862119889 12058811986302 (120596120596 minus 120596)2+ 119862119898 1205881205871198632

04 (120596 minus )2)119909119889119909

(7)


AUV

NROO

MPC

u y

w

Stateestimation

yref

Figure 1 Block diagram of the proposed system

where 120588 is fluid density 119862119889 is the drag coefficient and 119862119898 isthe added mass coefficient

In (7)

120596120596 = 119873sum119894=1

120596119890119894120572119894 exp (minus119896119894119911) sin (120596119890119894119905 + 120579119894) 120596 = minus 119873sum

119894=1

1205961198901198942120572119894 exp (minus119896119894119911) cos (120596119890119894119905 + 120579119894) (8)

where 120596119890119894 is encountering frequency 120572119894 = radic2119878(120596119894)119889120596 119873 isthe equal interval frequency waveband and chosen in thisarticle to discretize the wave spectrum 119896119894 is wave numberand 120579119894 is random phase shift (0 lt 120579119894 lt 2120587) in relation to eachfrequency

Assumption 3 119892(119896 119909(119896)) is a Lipschitz function [34] vector1003817100381710038171003817119892 (119896 1199091) minus 119892 (119896 1199092)10038171003817100381710038172 le 119871119892 10038171003817100381710038171199091 minus 119909210038171003817100381710038172 (9)

which is assumed to satisfy the following conditions

(1) The effect of 119885119867119878 is approximated to zero(2) 119909119892 asymp 119909119887 = 0

where 119871119892 gt 0 is a Lipschitz constant and sdot 2 denotes theEuclidean norm Moreover 119892(0 119896) = 0Remark 4 This paper proposedMPC-NMOLdesignmethodbased on NROO for a nonlinear Lipschitz model of AUVThe NROO is independent of the MPC-NMOL which isconstructed to guarantee the stability and performance ofAUV diving motion Figure 1 shows the structure of theproposed system

3 Nonlinear Reduced-Order Observer Design

31 NROO Algorithm Design Consider that 119862 is full-rowrank and there always exists a matrix119862perp isin 119877(119899minus119901)times119899 such that

[ 119862perp119862] isin 119877119899times119899 is nonsingular (119862perp can be given as an orthogonal

basis of the null-space of 119862) Then under the nonsingularcoordinate transformation 119911(119896) = 119879119909(119896) in which 119879 = [ 119862perp

119862]

is used the dynamics (5) can be expressed into the form

[1199111 (119896 + 1)1199112 (119896 + 1)] = [11986011 1198601211986021 11986022

][1199111 (119896)1199112 (119896)]+ [119862perp119862 ]119892 (119896 119879minus1119911 (119896))+ [11986111198612] 119906 (119896) + [11986311198632

]119908 (119896) 119910 (119896) = [0119901times(119899minus119901) 119868119901] [1199111 (119896)1199112 (119896)]

(10)

where 1199111(119896) isin 119877119899minus119901 and 1199112(119896) isin 119877119901 are new state vectors 119868119901is identity matrix whose dimension is 119901 times 119901 and

119879119860119879minus1 = [11986011 1198601211986021 11986022

] 119879119861 = [11986111198612] 119879119863 = [11986311198632

] 119862119879minus1 = [0119901times(119899minus119901) 119868119901] 119911 (119896) = 119879119909 (119896) = [1199111 (119896)1199112 (119896)]

(11)

and then

1199112 (119896) = 119910 (119896) 1199111 (119896 + 1) = 119860111199111 (119896) + 11986012119910 (119896)+ 119862perp119892 (119896 119879minus1119911 (119896)) + 1198611119906 (119896)+ 1198631119908 (119896) 119910 (119896 + 1) = 119860211199111 (119896) + 11986022119910 (119896) + 119862119892 (119896 119879minus1119911 (119896))+ 1198612119906 (119896) + 1198632119908 (119896)

(12)

By using the hypothesis input 120583(119896) and output 120588(119896)120583 (119896) = 11986012119910 (119896) + 1198611119906 (119896) 120588 (119896) = 119910 (119896 + 1) minus 11986022119910 (119896) minus 1198612119906 (119896) (13)


we get1199111 (119896 + 1) = 119860111199111 (119896) + 119862perp119892 (119896 119879minus1119911 (119896)) + 120583 (119896)+ 1198631119908 (119896) 120588 (119896) = 119860211199111 (119896) + 119862119892 (119896 119879minus1119911 (119896)) + 1198632119908 (119896) (14)

From the dynamics (14) NROO is constructed as

1 (119896 + 1) = 119860111 (119896) + 119862perp119892 (119896 119879minus1 (119896)) + 120583 (119896)minus 119866 ( (119896) minus 120588 (119896)) (15)

(119896) = 119860211 (119896) + 119862119892 (119896 119879minus1 (119896)) (16)

where 1(119896) isin 119877119899minus119901 is the observer state of the reduced-order system (14) (119896) isin 119877119901 is the output of reduced-orderobserver and 119866 isin 119877(119899minus119901)times119901 is the observer gain matrix

Denote119890 (119896) = 1 (119896) minus 1199111 (119896) 119866 (119896) = 119892 (119896 119879minus1 (119896)) minus 119892 (119896 119879minus1119911 (119896)) (17)

and then from (14)ndash(16) the state estimation error dynamicsare described by the following

119890 (119896 + 1) = (11986011 minus 11986611986021) 119890 (119896) + (119862perp minus 119866119862)119866 (119896)+ (1198661198632 minus 1198631) 119908 (119896) (18)

Equation (18) is rewritten as the following form

119890 (119896 + 1) = (1198601 minus 1198661198602) 119890 (119896) + (1198661198622 minus 1198621) 119866 (119896)+ (1198661198632 minus 1198631) 119908 (119896) (19)

where 1198601 = 11986011 1198602 = 11986021 1198621 = minus119862perp and 1198622 = minus119862Theorem 5 gives a constrainedNROOdesign whose state

estimation performance is specified via a 119867infin performanceindex

Theorem 5 Let a prescribed 119867infin performance level 1205741 gt 0and if there exist a symmetric positive definite matrix 119875 isin119877(119899minus119901)times(119899minus119901) a matrix 119884 isin 119877(119899minus119901)times119901 and a 1205761 gt 0 such thatthe following condition holds

[[[[[[[[

minus119875 1198751198601 minus 1198841198602 1198841198622 minus 1198751198621 1198841198632 minus 1198751198631 0lowast minus119875 + 12057611198712119892119868119899minus119901 0 0 119868119899minus119901lowast lowast minus1205761120582min (119879119879119879) 119868119899 0 0lowast lowast lowast minus1205741119868119889 0lowast lowast lowast lowast minus1205741119868119899minus119901

]]]]]]]]le 0 (20)

where 119884 = 119875119866 and lowast denotes the symmetric elements in amatrix then the error dynamics (18) satisfy 119867infin performanceindex 119890(119896)2 le 1205741119908(119896)2 and the NROO gain matrix can beobtained by 119866 = 119875minus1119884Proof Choose the following Lyapunov function119881 (119896) = 119890119879 (119896) 119875119890 (119896) (21)

According to error dynamics (18) and (21) the differenceΔ119881(119896) isΔ119881 (119896) = 119881 (119896 + 1) minus 119881 (119896)= 119890119879 (119896 + 1) 119875119890 (119896 + 1) minus 119890119879 (119896) 119875119890 (119896)= 119890119879 (119896) (1198601 minus 1198661198602)119879 119875 (1198601 minus 1198661198602) 119890 (119896)+ 119866119879 (119896) (1198661198622 minus 1198621)119879 119875 (1198661198622 minus 1198621) 119866 (119896)+ 119908119879 (119896) (1198661198632 minus 1198631)119879 119875 (1198661198632 minus 1198631) 119908 (119896)+ 2119890119879 (119896) (1198601 minus 1198661198602)119879 119875 (1198661198622 minus 1198621) 119866 (119896)+ 2119890119879 (119896) (1198601 minus 1198661198602)119879 119875 (1198661198632 minus 1198631) 119908 (119896)+ 2119866119879 (119896) (1198661198622 minus 1198621)119879 119875 (1198661198632 minus 1198631) 119908 (119896)minus 119890119879 (119896) 119875119890 (119896)

(22)

Since 119910(119896) is measurable 2(119896) can be substituted by 119910(119896)and the observer state (119896) can be written as

(119896) = 119879minus1 [1 (119896)119910 (119896) ] (23)

One gets

(119896) minus 119909 (119896) = 119879minus1 [1 (119896)119910 (119896) ] minus 119879minus1 [1199111 (119896)119910 (119896)]= 119879minus1 [119890 (119896)0 ] (24)

Since119866(119896) = 119892(119896 119879minus1(119896))minus119892(119896 119879minus1119911(119896)) and 119892(119896 119909(119896))satisfies the Lipschitz condition119866 (119896)2 le 119871119892 (119896) minus 119909 (119896)2 (25)

for a positive scalar 1205761 we have1205761119866119879 (119896) 119866 (119896) le 12057611198712119892 ( (119896) minus 119909 (119896))119879 ( (119896) minus 119909 (119896))= 12057611198712119892 [119890119879 (119896) 0] (119879minus1)119879 119879minus1 [119890 (119896)0 ] (26)


The above inequality is multiplied by 120582min(119879119879119879) on bothsides and we have

1205761120582min (119879119879119879)119866119879 (119896) 119866 (119896)le 12057611198712119892120582min (119879119879119879) [119890119879 (119896) 0] (119879minus1)119879 119879minus1 [119890 (119896)0 ]le 12057611198712119892 [119890119879 (119896) 0] (119879119879)minus1 (119879119879119879)119879minus1 [119890 (119896)0 ]= 12057611198712119892 [119890119879 (119896) 0] [119890 (119896)0 ] = 12057611198712119892119890119879 (119896) 119890 (119896)

(27)

which is

12057611198712119892119890119879 (119896) 119890 (119896) minus 1205761120582min (119879119879119879)119866119879 (119896) 119866 (119896) ge 0 (28)

where 120582min(sdot) is the smallest eigenvalue matrixWe define

1198691 = 119870sum119896=0

[ 11205741 119890119879 (119896) 119890 (119896) minus 1205741119908119879 (119896) 119908 (119896)] (29)

Under zero initial conditions one gets

1198691 le 119870sum119896=0

[ 11205741 119890119879 (119896) 119890 (119896) minus 1205741119908119879 (119896) 119908 (119896) + Δ119881 (119896)] (30)

Substituting (22) and (28) into (30) it follows that

1198691 le 119870sum119896=0

[ 11205741 119890119879 (119896) 119890 (119896) minus 1205741119908119879 (119896) 119908 (119896)+ 12057611198712119892119890119879 (119896) 119890 (119896) minus 1205761120582min (119879119879119879)119866119879 (119896) 119866 (119896)+ Δ119881 (119896)] = 119870sum

119896=0

[119890119879 (119896) 119866119879 (119896) 119908119879 (119896)]sdot Ω[[[

119890 (119896)119866 (119896)119908 (119896)]]] (31)

where

Ω = [[[[11988811 (1198601 minus 1198661198602)119879 119875 (1198661198622 minus 1198621) (1198601 minus 1198661198602)119879 119875 (1198661198632 minus 1198631)lowast 11988822 (1198661198622 minus 1198621)119879 119875 (1198661198632 minus 1198631)lowast lowast 11988833

]]]]

11988811 = (1198601 minus 1198661198602)119879 119875 (1198601 minus 1198661198602) minus 119875 + 12057611198712119892 + 11205741 119868119899minus11990111988822 = (1198661198622 minus 1198621)119879 119875 (1198661198622 minus 1198621) minus 1205761120582min (119879119879119879) 11986811989911988833 = (1198661198632 minus 1198631)119879 119875 (1198661198632 minus 1198631) minus 1205741119868119889

(32)

By using the Schur complement lemma Ω le 0 isequivalent to

[[[[[[[

11988811 minus 11205741 119868119899minus119901 (1198601 minus 1198661198602)119879 119875 (1198661198622 minus 1198621) (1198601 minus 1198661198602)119879 119875 (1198661198632 minus 1198631) 119868119899minus119901lowast 11988822 (1198661198622 minus 1198621)119879 119875 (1198661198632 minus 1198631) 0lowast lowast 11988833 0lowast lowast lowast minus1205741119868119899minus119901]]]]]]]le 0 (33)

Furthermore (33) can be rewritten as

[[[[[[(1198601 minus 1198661198602)119879 119875(1198661198622 minus 1198621)119879 119875(1198661198632 minus 1198631)119879 1198750

]]]]]]119875minus1 [119875 (1198601 minus 1198661198602) 119875 (1198661198622 minus 1198621)

sdot 119875 (1198661198632 minus 1198631) 0]

+ [[[[[[[

minus119875 + 12057611198712119892 0 0 119868119899minus119901lowast minus1205761120582min (119879119879119879) 119868119899 0 0lowast lowast minus1205741119868119889 0lowast lowast lowast minus1205741119868119899minus119901]]]]]]]le 0(34)


Let us use Schur complement lemma again finally weobtain (20) which guarantees 119890(119896)2 le 1205741119908(119896)2 Soif (20) holds then (19) is exponentially stable with a 119867infin

performance index 119890(119896)2 le 1205741119908(119896)2Remark 6 We use the measurable output 119910(119896) = 1199112(119896) andthe inequality 120582min(119875)119886119879119886 le 119886119879119875119886 for any matrix 119875 gt 0 andreal vector 119886 to make the Lipschitz condition transform into1205761120582min(119879119879119879)119866119879(119896)119866(119896) le 12057611198712119892119890119879(119896)119890(119896) which guaranteesthe diagonal element (3 3) is nonzero and negative and helpsto form constraint (20)

Remark 7 In Theorem 5 the LMI constraint (20) estimatesthe states by suppressing the influence of the disturbance term119908(119896) on 119890(119896) By solving the convex optimization problemthe level 1205741 can be obtained in order to minimize 119890(119896)Apparently the NROO order 119899 minus 119901 is less than the FOO[35] order 119899 and the transfer function from 119908(119896) to 119890(119896) issimplified Therefore NROO has the advantages of simpleconstruction and better performance which is compared toFOO

32 NROO Existence Condition On one hand for the exis-tence of NROO the necessity of condition rank(119862119863) =rank(119863) has been proved in [36] It is called observabilitywith unknown input which means the necessity of conditioncan guarantee the observability of systems with unknowndisturbances 119908(119896)

On the other hand for the existence of FOO the necessityof condition requires the pair (119860 119862) to be observable [35]that means

rank(120582119868119899minus119901 minus 11986011 minus11986012minus11986021 120582119868119901 minus 119860220119901times(119899minus119901) 119868119901 )= rank(120582119868119899minus119901 minus 1198601111986021

) + 119901 = 119899(35)

It is not difficult to prove that the formula above isequivalent to the necessity of condition in NROO becausethe nonsingular coordinate transformation which will beintroduced in next section cannot transform the observabilityof the system which means the NROO has the same range ofapplication as FOO in this paper

33 State Estimation The NROO gain matrix 119866 has beenderived in the previous section However hypothesis output120588(119896) includes future output 119910(119896 + 1) which is not available inpractical implementation In this section a novel expressionwill be introduced for state estimation

Substituting (16) and (23) into (15) one gets1 (119896 + 1) = (11986011 minus 11986611986021) 1 (119896)+ (119862perp minus 119866119862) 119892(119896 119879minus1 [1 (119896)119910 (119896) ])+ 119868119899minus119901120583 (119896) + 119866120588 (119896)= (1198601 minus 1198661198602) 1 (119896)

+ (1198661198622 minus 1198621) 119892(119896 119879minus1 [1 (119896)119910 (119896) ])+ 119868119899minus119901120583 (119896) + 119866120588 (119896)

(36)

Then substituting 120583(119896) and 120588(119896) into (36) yields1 (119896 + 1) = (1198601 minus 1198661198602) 1 (119896)+ (1198661198622 minus 1198621) 119892(119896 119879minus1 [1 (119896)119910 (119896) ])+ (119868119899minus11990111986012 minus 11986611986022) 119910 (119896) + 119866119910 (119896 + 1)+ (119868119899minus1199011198611 minus 1198661198612) 119906 (119896)

(37)

We denote 120594(119896) = 1(119896) minus 119866119910(119896) in order to eliminate119910(119896 + 1) as follows120594 (119896 + 1)= (1198601 minus 1198661198602) (120594 (119896) + 119866119910 (119896))+ (1198661198622 minus 1198621) 119892(119896 119879minus1 [1 (119896)119910 (119896) ])+ (119868119899minus11990111986012 minus 11986611986022) 119910 (119896)+ (119868119899minus1199011198611 minus 1198661198612) 119906 (119896)

= (1198601 minus 1198661198602) 120594 (119896)+ (1198661198622 minus 1198621) 119892(119896 119879minus1 [1 (119896)119910 (119896) ])+ ((1198601 minus 1198661198602) + (119868119899minus11990111986012 minus 11986611986022)) 119910 (119896)+ (119868119899minus1199011198611 minus 1198661198612) 119906 (119896)

(38)

Finally the state estimator is

(119896) = 119879minus1 [120594 (119896) + 119866119910 (119896)119910 (119896) ] (39)

4 Model Predictive Control with NonlinearModel Online Linearization

41 Nonlinear Predictive Model Online Linearization In thissection MPC-NMOL was proposed to deal with the non-linear state equation (5) Due to the time-consuming andcomputational complex problem nonlinear optimization isconverted into a quadratic optimization by considering anonline linearized method Using gain scheduling technique[12] the online linearized dynamics at current samplinginstance are given as119909 (119896 + 1) = (119860 + 119892 (119896 119909)) 119909 (119896) + 119861119906 (119896) + 119863119908 (119896) 119910 (119896) = 119862119909 (119896) (40)


where the nonlinear-term 119892(119896 119909) is linearized from119892(119896 119909(119896)) and one gets

119892 (119909 119896) = 120597119892 (119896 119909 (119896))120597119909 (119896) 10038161003816100381610038161003816100381610038161003816119909(119896)=(119896) (41)

where the current operating state 119909(119896) is defined by theestimation state (119896) at 119896th sampling instance

We denote the general expression as follows

119892 (119896 119909 (119896)) = [[[[[[

119892111199091 100381610038161003816100381611990911003816100381610038161003816 + 119892121199092 100381610038161003816100381611990921003816100381610038161003816 + 1198921311990911199092119892211199091 100381610038161003816100381611990911003816100381610038161003816 + 119892221199092 100381610038161003816100381611990921003816100381610038161003816 + 119892231199091119909200]]]]]] (42)

where the nonlinear-term coefficient in (42) can be calculatedby1198721 119879119904 and 119892119888(119905 119909(119905))mentioned before

Substitute (42) into (41) and we can get

119892 (119896 119909) = [[[[[[

1198921 1198922 0 01198923 1198924 0 00 0 0 00 0 0 0]]]]]] (43)

where

1198921 = 11989211 100381610038161003816100381611003816100381610038161003816 + 119892111 sgn (1) + 1198921321198922 = 11989212 100381610038161003816100381621003816100381610038161003816 + 119892122 sgn (2) + 1198921311198923 = 11989221 100381610038161003816100381611003816100381610038161003816 + 119892211 sgn (1) + 1198922321198924 = 11989222 100381610038161003816100381621003816100381610038161003816 + 119892222 sgn (2) + 119892231(44)

We assume that

119908 (119896) = 119908 (119896 + 1 | 119896) = sdot sdot sdot = 119908 (119896 + 119873119901 | 119896) (45)

where 119873119901 is the predictive horizon Then considering that119908(119896) is the difference between the estimation state (119896) and119909(119896) (calculated by 119909(119896 minus 1) 119906(119896 minus 1) and 119892(119896 minus 1 119909)) thismeans

119908 (119896)= (119896)minus [(119860 + 119892 (119896 minus 1 119909)) 119909 (119896 minus 1) + 119861119906 (119896 minus 1)] (46)

Here thewave disturbancemodel can be considered as thestate disturbance and we can use it to estimate the externaldisturbance in vertical plane of AUV motion

Denote

119860 = 119860 + 119892 (119896 minus 1 119909) (47)

and then the iterative predictive states over119873119901 at step 119896 is (119896 + 1 | 119896) = 119860 (119896) + 119861 (119896 | 119896) + 119908 (119896) (119896 + 2 | 119896) = 119860 (119896 + 1) + 119861 (119896 + 1 | 119896) + 119908 (119896) (119896 + 119873119901 | 119896) = 119860 (119896 + 119873119901 minus 1)+ 119861 (119896 + 119873119901 minus 1 | 119896) + 119908 (119896)

(48)

where (119896 + 119873119901 | 119896) is the predictive input at step 119896When we implement MPC algorithm an incremental

predictive model is always required and therefore the input(119896 + 119894 | 119896) can be replaced by Δ(119896 + 119894 | 119896) which means (119896 + 119894 | 119896) = (119896 + 119894 minus 1) + Δ (119896 + 119894 | 119896) (49)and here we assume that (119896 + 119894 | 119896) will change just at everystep (119894 lt 119873119906) and remain constant after step 119896 + 119894 (119873119906 le 119894 lt119873119901 minus 1)

With the assumption that the predictive value of 119908(119896)at sample instance 119896 is zero the vector output predictionequation can be calculated and expressed in condensed formwhich predicts the future dynamic behavior of the AUVlongitudinal motion over the horizon119873119901

y = Ψ (119896) + Γ119906 (119896 minus 1) +Θu (50)where

y = [[[[ (119896 + 1 | 119896) (119896 + 119873119901 | 119896)

]]]]

u = [[[[Δ (119896 | 119896)Δ (119896 + 119873119906 minus 1 | 119896)

]]]]

Ψ = [[[[[[[[

1198621198601198621198602

119862119860119873119901

]]]]]]]]

Γ =[[[[[[[[[[

119862119861119862 (119860119861 + 119861)119873119901minus1sum119894=0

119862119860119894119861

]]]]]]]]]]

Θ =[[[[[[[[[[

119862119861 sdot sdot sdot 0119862 (119860119861 + 119861) sdot sdot sdot 0 d

119873119901minus1sum119894=0

119862119860119894119861 sdot sdot sdot 119873119901minus119873119906sum119894=0

119862119860119894119861

]]]]]]]]]]

(51)


42 Consideration of Constraints In order to guarantee thecorrect operation of the AUV the constraints of the inputelevator deflection 120575 isin 119877119873119906 over control horizon 119873119906 andsystem output y isin 1198772119873119901 over predictive horizon 119873119901 areused to define 119872 and 120574 for each time step 119896 consideringthe physical limitations of the driving device in practicalimplementation and one obtains

120575min le 120575 le 120575maxΔ120575min le Δ120575 le Δ120575maxymin le y le ymax

(52)

For simplification of the following discussion and with-out loss of generality we set the control horizon119873119906 = 2 andthen from (52) we can get

[[[119872111987221198723

]]]Δ120575 le[[[119873111987321198733

]]] (53)

where

1198721 = [minusΛ 2Λ 2

] 1198731 = [minus120575min + Λ 1120575 (119896 minus 1)120575max minus Λ 1120575 (119896 minus 1) ]

1198722 = [minus1198682times21198682times2 ] 1198732 = [minusΔ120575minΔ120575max

] 1198723 = [minusΘ

Θ]

1198733 = [minusymin +Ψ (119896) + Γ120575 (119896 minus 1)ymax minusΨ (119896) minus Γ120575 (119896 minus 1) ]

Δ120575 = [ Δ120575 (119896)Δ120575 (119896 + 1)] Λ 1 = [11] Λ 2 = [1 01 1]

(54)

We describe (53) by 119872Δ120575 le 120574 (55)

which is equivalent to the constraints in the next section andthe constraints of the elevator deflection angle 120575 and pitchangle 120579 corresponding to the inputs and outputs of theMPCare characterized by119872 and 120574

43 Optimization with Constraints Conventional MPC per-formance index can be written as

119869 = (y minus yref)119879119876 (y minus yref) + u119879119877u (56)

where yref is a future reference vector and119877 and119876 are positivedefinite weighted matrices

To simplify the expression (56) can be rewritten asfollows

119869 = 12u119879119864u + u119879119865 + f0st 119872u le 120574 (57)

where

119864 = 2 (Θ119879119876Θ + 119877) 119865 = 2Θ119879119876 (Ψ (119896) + Γ119906 (119896 minus 1) minus yref) f0 = (Ψ (119896) + Γ119906 (119896 minus 1) minus yref)119879sdot (Ψ (119896) + Γ119906 (119896 minus 1) minus yref)

(58)

To minimize the quadratic function subject to (55) a QP(Quadratic Programming) problemhas come out Let us con-sider the expression which contains the Lagrangemultipliersthat is a QP problem subject to equality constraints119872u = 120574by the formula below

119869 = 12u119879119864u + u119879119865 + f0 + 120582119879 (119872u minus 120574) (59)

Theminimization of 119869 is to take the first partial derivativeswith respect to u and 120582 and we make them equal to zero andobtain the formula below120597119869120597u = 119864u +119872119879120582 + 119865 = 0 (60)

120597119869120597120582 = 119872u minus 120574 = 0 (61)

Theminimization of 119869 can bemade by finding the optimalu and 120582 via (60) and (61) where

120582 = minus (119872119864minus1119872119879)minus1 (120574 +119872119864minus1119865) (62)

u = minus119864minus1 (119872119879120582 + 119865) = 120578 minus 119864minus1119872119879120582 (63)

where 120578 = minus119864minus1119865 is the global optimal solutionThe inequality constraints may comprise active con-

straints and inactive constraints in (55) We use both119872119894 and120574119894 to form the 119894th inequality constraint If 119872119894u = 120574119894 aninequality constraint 119872119894u le 120574119894 can be considered as activeand if119872119894u lt 120574119894 it is inactive Here we use the Kuhn-Tuckerconditions [37] to define the active and inactive constraints interms of 120582 If the active set were known the original problemcould become equality constrains problem in (59)

In the conventional active setmethod [38] which belongsto the primal methods the solutions are based on u (called


decision variables) If the MIMO system has too manyconstraints the calculations are complex and it is not astraightforward work

A dual method can be used to identify the constraintswhich are inactive systematicallyThe inactive constraints canbe eliminated in the solution and 120582 are called dual variableshere For constrained minimization problem this method isa very simple programming procedure The dual problem isderived from original primal problem as follows Substituting(63) in (59) the dual problem is written as

min120582ge0

(12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574) (64)

where the matrices119867 and119870 are given by119867 = 119872119864minus1119872119879119870 = 120574 +119872119864minus1119865 (65)

Subject to 120582 ge 0 we minimize the dual performanceindex

119869 = 12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574 (66)

where the set of 120582 are denoted as 120582lowast By using Hildrethrsquos QPprocedure [39] the dual problem is solved and the methodcan be written as120582119894 (119896 + 1) = max [0 120573119894 (119896 + 1)] (67)

with120573119894 (119896 + 1)= minus 1ℎ119894119894 [[119896119894 +

119894minus1sum119895=1

ℎ119894119895120582119895 (119896 + 1) + 119873119896sum119895=119894+1

ℎ119894119895120582119895 (119896)]] (68)

where ℎ119894119895 is the 119894119895th element in119867 119896119894 is the 119894th element in 119870and119873119896 is the number of rows of119870 In this method there are120582119895(119896+1) and120582119895(119896) in one iterative cycle Andwe set120582119895(0) = 0at 119896 = 1 and the iterative procedure will converge to 120582lowast as aresult Substitute 120582lowast into (63) and we have

u = 120578 minus 119864minus1119872119879120582lowast (69)

where 120582lowast = [1205731(119896 + 1) 1205732(119896 + 1) 120573119873119896(119896 + 1)]119879According to the receding horizon control in MPC the

first elements (elevator deflection Δ120575(119896)) in u are taken toconstruct Δ119906(119905)Remark 8 Because Hildrethrsquos QP is a search-based point-by-point algorithm there is no matrix inversion calculationHowever if the number of the active constraints is morethan the number of ursquos or the active constraints are linearlydependent then 120582 will not converge to 120582lowast and the iterationwill terminate at the largest value of the iterative counterBut the algorithm will not end because there is no matrixinversion calculation In this case finally the algorithmwill end in a near-optimal solution with the violation ofconstraints This is the reason why we use Hildrethrsquos QPhere for its good ability to automatically recover from adeterioration constrained process

44 Steps of NROO-Based MPC-NMOL with Constraints

(a) Set values of119873119875119873119906 and specify 119876 119877(b) Get the estimation of current state (119896) sample

current depth 119911(119905) and pitch angle 120579(119905)(c) Calculate matrix 119860 by online-linearization at current

operating point which is defined by (119896) to getmatrixesΨ Γ andΘ Update constraints matrixes119872and 120574 by using 120575(119905 minus 1) and then 119864 119865 and f0 can becalculated

(d) Check if the global optimal solution 120578 satisfies theconstraints If so make 120582lowastequal to zero vector and goto (f) If not go to (e)

(e) Calculate matrices 119867 and 119870 and then the dualvariable 120582lowast can be calculated from (61)

(f) Get Δ119906(119905) from optimal solution u(g) Go to step (b)

5 Simulation Results

In this paper simulations are presented to demonstrate theeffectiveness of NROO-based MPC-NMOL The method isused in a given depth control of REMUS AUV which isdeveloped by MIT (Massachusetts Institute of Technology)The values of nonlinear model parameters are shown asfollows

119860 = [[[[[[

09921 00063 0 0000200453 09961 0 minus00071001 0 1 minus001510 001 0 1]]]]]]

119861 = [[[[[[

minus00065minus00400]]]]]]

119862 = [0 0 1 00 0 0 1]

119863 = [[[[[[

00002 00 0001200002 00 00012]]]]]]

119892 (119896 119909 (119896)) = 119872minus1

[[[[[[

minus0021199091 100381610038161003816100381611990911003816100381610038161003816 minus 000661199092 100381610038161003816100381611990921003816100381610038161003816000461199091 100381610038161003816100381611990911003816100381610038161003816 minus 022721199092 10038161003816100381610038161199092100381610038161003816100381600]]]]]]

(70)

Here the physical parameters of REMUS AUV which canbe found from [40] are shown in Table 1


0 50 100 150Sampling instant

minus06

minus04

minus02

0

02

x1

x1

Estimation of x1(a)


x1Estimation of x1

minus06

minus04

minus02

0

02

x1

(b)

Figure 2 Comparison of 1199091 and 1 (a) using the NROO and (b) using the FOO

51 Nonlinear Reduced-Order Observer Design From theresult that rank(119862119863) = rank(119863) = 2 and (119860 119862) areobservable we can easily verify the existence of NROO

By using Schmidt orthogonalization we can get thematrix 119879 from the combination of 1198621 and its standardorthogonal basis1198622 Obviously the nonsingular matrix119879 hasonly one form which is

119879 = [[[[[[

1 0 0 00 1 0 00 0 1 00 0 0 1]]]]]] (71)

We choose 119871119892 = 0001 as the value of the Lipschitzconstant and then the matrices parameter values in (19) canbe calculated as follows

1198601 = [09921 0006300453 09961] 1198602 = [001 00 001] 1198621 = [minus1 0 0 00 minus1 0 0] 1198622 = [0 0 minus1 00 0 0 minus1] 1198631 = [00002 00 00012] 1198632 = [00002 00 00012]

(72)

Table 1 Physical Parameters of REMUS AUV

Description Values Units119898 3048 kgLength 133 mBeam 026 m119868119909119909 0177 Kgsdotm2119868119910119910 345 Kgsdotm2119868119911119911 345 Kgsdotm2

Location of CG (0000196) mLocation of BG (minus061100) m

With the help of MATLAB LMI toolbox condition (20)is solved to obtain 1205741 = 23561 times 103 at the same time otherresults are found as follows

119875 = [1446253 minus547977minus547977 335797 ] 119884 = [426923 minus1275404957 418309] 1205761 = 43871 times 103

(73)

One obtains

119866 = 119875minus1119884 = [07880 1213513007 32259] (74)

Figures 2 and 3 show the estimation of 1199091(119896) and 1199092(119896) byusing NROO and FOO Although both observers can makethe state estimation error converge asymptotically comparedto FOO NROO has a better performance of state estimation



x2

x2

Estimation of x2

minus2

minus1

0

1

(a)


x2Estimation of x2

x2

minus2

minus1

0

1

(b)


0 50 100 150

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

NMPCMPC-NMOL

minus1

Figure 4 Comparison of depth with no disturbance

52 Implementation ofMPC-NMOL Theparameters ofMPCare chosen as 119873119901 = 30 119873119906 = 2 119876 = 119868119873119901times119873119901 and 119877 = 10The initial values of state variables are all zero The wavesdisturbance exposed on REMUS is assumed at a level 3 seastate119867119904 = 088 (m) 120573 = 45∘ 119862119889 = 065 119862119898 = 195 and119873 =271 The surge speed is 119880119888 = 151 (ms) the desired depth is5 (m) and pitch angle is 0∘ Input and output constraints are

120575max = minus120575min = 30∘Δ120575max = minusΔ120575min = 5∘sminus90∘ le 120579 le 90∘(75)


NMPCMPC-NMOL

Pitc

h an

gle (

degr

ee)

minus120

minus100

minus80

minus60

minus40

minus20

0

20

Figure 5 Comparison of pitch angle with no disturbance

Case 1 First we assume that there is no disturbance insimulation process and NMPC is used to compare with theproposed method Both of the two methods (MPC-NMOLand NMPC) have all state variables measurable Figure 4compares MPC-NMOL and NMPC simulation results ofdepth output Figure 5 compares the simulation results ofpitch output Figure 6 compares the elevator deflection angleinput

Case 2 Next it is assumed that wave disturbance affects thestate process and the other condition is the same as Case 1Figures 7ndash9 show the comparison of depths pitch angles andelevator deflection angles with wave disturbance



NMPCMPC-NMOL

Elev

ator

(deg

ree)

minus60

minus40

minus20

0

20

40

Figure 6 Comparison of elevator deflection angle with no distur-bance

0 50 100 150 200

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

MPC-NOMPC-NMOL

minus1

Figure 7 Comparison of depth with disturbance

Figure 10 shows the wave force and moment which canbe seen as state process disturbance and assumed to be zero-mean white noise sequence The wave force and momentare calculated from (7) so they have the same form butdifferent amplitude Furthermore whenwe simulate thewaveforce andmoment we choosemultiple influential frequencieswhich are near the given main frequency of P-M spectrum tosuperimpose the irregular waves

All these results in Cases 1 and 2 demonstrate that AUVcould achieve the desired depth and pitch angle under thewave disturbance In addition the input signals in MPC-NMOL are smooth and without control signal saturation

0 50 100 150 200Sampling instant

Pitc

h an

gle (

degr

ee)

MPC-NOMPC-NMOL

minus100

minus80

minus60

minus40

minus20

0

20

Figure 8 Comparison of pitch angle with disturbance

0 50 100 150 200

0

20

40

60

Sampling instant

Elev

ator

(deg

ree)

MPC-NOMPC-NMOL

minus60

minus40

minus20

Figure 9 Comparison of elevator deflection angle with disturbance

6 Conclusions

In this paper a NROO-based model predictive controllerwith nonlinearmodel online linearization forAUV in verticalplane is presented which controls the depth and pitch angleThis design uses the NROO to estimate the states usedin MPC The design process of the controller also takesinto account the practical elevator deflection constraints andoutput constraints By using a Hildrethrsquos QP procedure theconstraints can be simply handled Making use of the pro-posed MPCmethods the AUV can navigate in vertical planewith desired depth and pitch angle It is robust against roughwave disturbance near surface The simulations carried outprovide the validation of the proposed methods presenting


0

20

0 50 100 150 200

0

10

Sampling instant


minus10

minus20

Mwave (Nm)

Zwave (N)

Figure 10 Wave force and moment

fast dynamical response and strong robustness to externaldisturbances Accurate control and state estimation can alsobe achieved

Competing Interests

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

Acknowledgments

The project is financially supported by the NNSF (NationalNatural Science Foundation) of China underGrant 51279039

References

[1] X Xiang L Lapierre and B Jouvencel ldquoSmooth transition ofAUV motion control from fully-actuated to under-actuatedconfigurationrdquo Robotics amp Autonomous Systems vol 67 pp 14ndash22 2015

[2] W Chen YWei J Zeng H Han and X Jia ldquoAdaptive terminalsliding mode NDO-based control of underactuated AUV invertical planerdquo Discrete Dynamics in Nature and Society vol2016 Article ID 6590517 9 pages 2016

[3] Z Yan H Yu and S Hou ldquoDiving control of underactuatedunmanned undersea vehicle using integral-fast terminal slidingmode controlrdquo Journal of Central South University vol 23 no5 pp 1085ndash1094 2016

[4] G V Lakhekar L M Waghmare and P S Londhe ldquoEnhanceddynamic fuzzy sliding mode controller for autonomous under-water vehiclesrdquo in Proceedings of the IEEE Underwater Technol-ogy (UT rsquo15) IEEE Chennai India February 2015

[5] A Adhami-Mirhosseini M J Yazdanpanah and A P AguiarldquoAutomatic bottom-following for underwater robotic vehiclesrdquoAutomatica vol 50 no 8 pp 2155ndash2162 2014

[6] M B Loc H-S Choi S-S You and T N Huy ldquoTime optimaltrajectory design for unmanned underwater vehiclerdquo OceanEngineering vol 89 pp 69ndash81 2014

[7] Y-H Tseng C-C Chen C-H Lin and Y-S Hwang ldquoTrackingcontroller design for diving behavior of an unmanned under-water vehiclerdquoMathematical Problems in Engineering vol 2013Article ID 504541 10 pages 2013

[8] S-P Hsu and T-S Liu ldquoModifications of control loop toimprove the depth response of autonomous underwater vehi-clesrdquo Mathematical Problems in Engineering vol 2014 ArticleID 324813 12 pages 2014

[9] B Subudhi K Mukherjee and S Ghosh ldquoA static outputfeedback control design for path following of autonomousunderwater vehicle in vertical planerdquo Ocean Engineering vol63 pp 72ndash76 2013

[10] S Heshmati-Alamdari A Eqtami G C Karras D V Dimarog-onas and K J Kyriakopoulos ldquoA self-triggered visual servoingmodel predictive control scheme for under-actuated underwa-ter robotic vehiclesrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo14) pp 3826ndash3831 Hong Kong China June 2014

[11] L Medagoda and S B Williams ldquoModel predictive control ofan autonomous underwater vehicle in an in situ estimatedwatercurrent profilerdquo in Proceedings of the (OCEANS rsquo12) MTSIEEEYeosu Conference The Living Ocean and CoastmdashDiversity ofResources and Sustainable Activities Yeosu Korea May 2012

[12] P Jagtap P Raut P Kumar A Gupta N Singh and F KazildquoControl of autonomous underwater vehicle using reducedorder model predictive control in three dimensional spacerdquoIFAC-PapersOnLine vol 49 no 1 pp 772ndash777 2016

[13] C V Caldwell D D Dunlap and E G Collins Jr ldquoMotionplanning for an autonomous underwater vehicle via samplingbasedmodel predictive controlrdquo inProceedings of theMTSIEEESeattle (OCEANS rsquo10) pp 1ndash6 Seattle Wash USA September2010

[14] D C Fernandez and G A Hollinger ldquoModel predictive controlfor underwater robots in ocean wavesrdquo IEEE Robotics ampAutomation Letters vol 2 no 1 pp 88ndash95 2017

[15] J Gao C Liu and A Proctor ldquoNonlinear model predictivedynamic positioning control of an underwater vehicle with anonboardUSBL systemrdquo Journal ofMarine ScienceampTechnologyvol 21 no 1 pp 57ndash69 2016

[16] P Hamelin P Bigras J Beaudry P-L Richard and M BlainldquoDiscrete-time state feedback with velocity estimation usinga dual observer application to an underwater direct-drivegrinding robotrdquo IEEEASME Transactions on Mechatronics vol17 no 1 pp 187ndash191 2012

[17] W Zhang Y Guo D Meng Z Liang and T Chen ldquoResearchon diving control of underactuated UUV based on modelpredictive control with artificial bee colony algorithmrdquo inProceedings of the 34th Chinese Control Conference (CCC rsquo15)pp 4073ndash4078 IEEE Hangzhou China July 2015

[18] Z Gao T Breikin and H Wang ldquoDiscrete-time proportionaland integral observer and observer-based controller for systemswith both unknown input and output disturbancesrdquo OptimalControl Applications ampMethods vol 29 no 3 pp 171ndash189 2008

[19] K Zhang B Jiang P Shi and A Shumsky ldquoReduced-orderfault estimation observer design for discrete-time systemsrdquo inProceedings of the 10thWorld Congress on Intelligent Control andAutomation (WCICA rsquo12) pp 2959ndash2964 Beijing China July2012


[20] J C Kinsey Q Yang and J C Howland ldquoNonlinear dynamicmodel-based state estimators for underwater navigation ofremotely operated vehiclesrdquo IEEE Transactions on ControlSystems Technology vol 22 pp 1845ndash1854 2014

[21] S Mahapatra B Subudhi and R Rout ldquoDiving control of anAutonomous Underwater Vehicle using nonlinear Hinfin mea-surement feedback techniquerdquo in Proceedings of the OCEANS2016mdashShanghai April 2016

[22] K Zhang B Jiang and P Shi ldquoFast fault estimation andaccommodation for dynamical systemsrdquo IET Control Theoryand Applications vol 3 no 2 pp 189ndash199 2009

[23] A Mustafa K Munawar F M Malik M B Malik M Salmanand S Amin ldquoReduced order observer design with DMPC andLQR for systemwith backlash nonlinearityrdquoArabian Journal forScience amp Engineering vol 39 no 8 pp 6521ndash6530 2014

[24] K Zhang M Staroswiecki and B Jiang ldquoReduced-orderobserver-based fault estimation design for multiple input-multiple output discrete-time systemsrdquo Proceedings of the Insti-tution of Mechanical Engineers Part I Journal of Systems ampControl Engineering vol 226 no 1 pp 101ndash110 2011

[25] S A Davari D A Khaburi F Wang and R M KennelldquoUsing full order and reduced order observers for robustsensorless predictive torque control of induction motorsrdquo IEEETransactions on Power Electronics vol 27 no 7 pp 3424ndash34332012

[26] J Rodas R Gregor M Rivera Y Takase and M ArzamendialdquoEfficiency analysis of reduced-order observers applied to thepredictive current control of asymmetrical dual three-phaseinduction machinesrdquo in Proceedings of the IEEE InternationalSymposium on Sensorless Control for Electrical Drives andPredictive Control of Electrical Drives and Power Electronics(SLEDPRECEDE rsquo13) pp 1ndash7 Munchen Germany October2013

[27] S H Saıd N B Nasr M F Mimouni and F MSahli ldquoOutputfeedback predictive controller for a class of nonlinear systemsrdquoin Proceedings of the American Control Conference (ACC rsquo10)July 2010

[28] R Gregor J Balsevich and B Bogado ldquoReduced-orderobserver for rotor current estimation in speed control of dual-three phase induction machinerdquo in Proceedings of the 3rd IEEEInternational Conference on Power Engineering Energy andElectrical Drives (PowerEng rsquo11) pp 1ndash6 Malaga Spain May2011

[29] V Sundarapandian ldquoReduced order observer design for non-linear systemsrdquo Applied Mathematics Letters vol 19 no 9 pp936ndash941 2006

[30] V Sundarapandian ldquoReduced order observer design fordiscrete-time nonlinear systemsrdquo Applied Mathematics Lettersvol 19 no 10 pp 1013ndash1018 2006

[31] Z Mao B Jiang and P Shi ldquoFault-tolerant control for a classof nonlinear sampled-data systems via a Euler approximateobserverrdquo Automatica vol 46 no 11 pp 1852ndash1859 2010

[32] M Calasan N Soc V Vujicic et al ldquoReview of marinecurrent speed and power coefficienmdashmathematical modelsrdquo inProceedings of the 4th Mediterranean Conference on EmbeddedComputing (MECO rsquo15) pp 427ndash431 Budva Montenegro June2015

[33] J P J Avila and J C Adamowski ldquoExperimental evaluationof the hydrodynamic coefficients of a ROV through Morisonrsquosequationrdquo Ocean Engineering vol 38 no 17-18 pp 2162ndash21702011

[34] D Nesic A R Teel and P V Kokotovic ldquoSufficient conditionsfor stabilization of sampled-data nonlinear systems via discrete-time approximationsrdquo Systems and Control Letters vol 38 no4-5 pp 259ndash270 1999

[35] K Zhang B Jiang and P Shi ldquoObserver-based integratedrobust fault estimation and accommodation design for discrete-time systemsrdquo International Journal of Control vol 83 no 6 pp1167ndash1181 2010

[36] W Kratz ldquoCharacterization of strong observability and con-struction of an observerrdquo Linear Algebra amp Its Applications vol221 pp 31ndash40 1995

[37] D P Bertsekas Nonlinear Programming 1999[38] E F Camacho andC BAlbaModel Predictive Control Springer

Science amp Business Media 2013[39] LWangModel Predictive Control SystemDesign and Implemen-

tation Using MATLAB Springer Science amp Business MediaBerlin Germany 2009

[40] T Prestero ldquoVerification of a 6-degree of freedom simulationmodel for the REMUS AUVrdquo 2001

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The MPC is formulated as two independent optimizationparts However by using the same initial states for both partsof linearization and neglecting the weakly coupled states thecontrol optimization differs from the actual situation a lotIn [13] a sampling based MPC is proposed to generate thecontrol sequence with constraints effectively These methodshave a good performance in the field ofMPCuse for AUV butthey do not deal with wave disturbances The method in [14]used MPC to predict future disturbances and counteract thewave disturbances A linearwave theory solver is employed byestimator to approximate the fluid dynamics but constraintswere not considered NMPC is an extension of MPC by usingnonlinear systems and constraints applications of NMPCfor AUV are less frequent In [15] NMPC is used to designthe kinematic loop in a hierarchical structure for AUV DPcontrol

Because of fewer actuators and sensors there is anadvantage for underactuated AUV to simplify the design andreduce costs However it brings highly coupled nonlinearmodels which increases the difficulty of controller designIn general some state variables of AUV are unmeasured dueto a lack of sensors or sensor faults So an observer-basedcontroller becomes significant A dual observer in [16] whichcombines a state and a perturbation observer aims to solvethe problem about being sensitive to external disturbanceZhang et al [17] considered the elevator angle constraints andproposed a controller based onMPCwith artificial bee colonyalgorithm and a classical linear state observer is used In [18]a novel discrete-time proportional and integral observer isused to estimate the states output input and disturbancetogether Zhang et al [19] proposed a new reduced-orderobserver with multiconstrained thoughts by using specificsystem decomposition However these results only use linearmodels

In contrast most practical systems are nonlinear andtherefore nonlinear models are required Kinsey et al [20]proposed a nonlinear observer based on dynamic model ofAUVwhich is used to estimate the vehiclersquos velocityHoweverin theirmodel the coupling terms are neglected and therewasonly one degree of freedomMahapatra et al [21] investigateda nonlinear119867infin feedback control algorithm with a nonlinearstate observer but the error dynamics stability was notconsidered in observer design Zhang et al [22] extendedan adaptive state observer to a class of nonlinear systemsHowever due to the selected special Lyapunov matrix thereis a reduction of the accuracy of state estimation

At the same time MPC need state variables to achievedesired output tracking in the minimization of cost functionso the role of observer-based MPC has a prominent positionin practical implementation However there is a situationthat some states variables of underactuated AUV can bemeasured So it is not necessary to estimate all the variablesCompared with FOO reduced-order observer (ROO) hassimplified structure and good estimation performance [2324] which can be designed to estimate only the unmeasurablestates Recently ROO designs used in MPC are exploited inboth linear [25 26] and nonlinear systems [27ndash30] Howeverfew research results are reported on NROO-based MPC forAUV So the motivation of this paper is the aforementioned

problems and deficiencies and the aim is to design a NROO-based MPC with nonlinear AUV kinematics and dynamicsmodel

In this paper amodel predictive controllerwith nonlinearmodel online-linearization based on NROO is proposed tosolve the attitude control and depth tracking issues Thedesign process is as follows First the nonlinear dynamicsin AUV vertical plane are described and Euler approxi-mation is used to discrete the proposed model Secondthe nonlinear reduced-order observer is designed accordingto the dynamics above to estimate the unmeasured statevariables of AUV By solving a convex optimization problemand using LMI technique the estimation performance isenhanced and the wave disturbance influence is restrictedThird a model predictive control based on model online-linearization is presented in which the controller designprocess is independent of the NROO and it is convenientfor the following simulation Finally simulation results showthe good robustness and efficiency of the proposed methodswhich can be applied to AUV attitude control and depthtracking field

The rest of the paper is organized as follows Section 2described the AUV vertical plane dynamics NROO design isproposed in Section 3 Section 4 introduces theMPC-NMOLstrategy Section 5 gives the simulation results of AUV toverify the effectiveness of the proposed approach

2 AUV Vertical Plane Dynamics

The vertical plane dynamic and kinematic model of AUV aredescribed in this section and the following assumptions areconsidered

Assumption 1 (1) AUV navigates at a constant speed 119906 = 119880119888(2) To analyze the diving control sway yaw and rollmotions can be ignored and that means 119910 = 0 V = 0 and120601 = 120595 = 0(3) Center of buoyancy stays constant with no changemade to external hull but the position of center of gravity isvariable(4)The hydrodynamics drag terms whose order is higherthan two are neglected(5) Pitch angle 120579 has small perturbations at zero

Therefore the dynamic and kinematic model of AUV canbe expressed in the nonlinear forms as follows119898[ minus 119906119902 minus 119909119892] = Σ119885ext119868119910 + 119898 [119911119892120596119902 minus 119909119892 + 119909119892119906119902] = Σ119872ext = minus119906120579 + 120596 = 119902

(1)

whereΣ119885ext = 119885119867119878 + 119885120596|120596|120596 |120596| + 119885 + 119885 + 119885119906119902119906119902+ 119885119906120596119906120596 + 1198851205751199062120575 + 119885waveΣ119872ext = 119872119867119878 +119872119902|119902|119902 10038161003816100381610038161199021003816100381610038161003816 + 119872 + 119872 + 119872119906119902119906119902+119872119906120596119906120596 +1198721205751199062120575 +119872wave





where

119860119888 = 1198721minus1[[[[[11989911 11989912 0 011989921 11989922 0 119899241 0 0 minus1199060 1 0 0

]]]]]

119861119888 = 1198721minus1[[[[[11988512057511987212057500]]]]]


= 1198721minus1[[[[[


1198721 = [[[[[11989811 11989812 0 011989821 11989822 0 00 0 1 00 0 0 1

]]]]]



(4)


119909 (119896 + 1) = 119860119909 (119896) + 119892 (119896 119909 (119896)) + 119861119906 (119896) + 119863119908 (119896) 119910 (119896) = 119862119909 (119896) (5)




119885wave = int1198710

(119862119889 12058811986302 (120596120596 minus 120596)2+ 119862119898 1205881205871198632

04 (120596 minus )2)119889119909119872wave = int

1198710

(119862119889 12058811986302 (120596120596 minus 120596)2+ 119862119898 1205881205871198632

04 (120596 minus )2)119909119889119909

(7)


AUV

NROO

MPC

u y

w

Stateestimation

yref



In (7)

120596120596 = 119873sum119894=1


119894=1

1205961198901198942120572119894 exp (minus119896119894119911) cos (120596119890119894119905 + 120579119894) (8)










119862]


[1199111 (119896 + 1)1199112 (119896 + 1)] = [11986011 1198601211986021 11986022

][1199111 (119896)1199112 (119896)]+ [119862perp119862 ]119892 (119896 119879minus1119911 (119896))+ [11986111198612] 119906 (119896) + [11986311198632

]119908 (119896) 119910 (119896) = [0119901times(119899minus119901) 119868119901] [1199111 (119896)1199112 (119896)]

(10)


119879119860119879minus1 = [11986011 1198601211986021 11986022

] 119879119861 = [11986111198612] 119879119863 = [11986311198632


(11)

and then

1199112 (119896) = 119910 (119896) 1199111 (119896 + 1) = 119860111199111 (119896) + 11986012119910 (119896)+ 119862perp119892 (119896 119879minus1119911 (119896)) + 1198611119906 (119896)+ 1198631119908 (119896) 119910 (119896 + 1) = 119860211199111 (119896) + 11986022119910 (119896) + 119862119892 (119896 119879minus1119911 (119896))+ 1198612119906 (119896) + 1198632119908 (119896)

(12)






(119896) = 119860211 (119896) + 119862119892 (119896 119879minus1 (119896)) (16)










[[[[[[[[


]]]]]]]]le 0 (20)



(22)


(119896) = 119879minus1 [1 (119896)119910 (119896) ] (23)

One gets







(27)

which is

12057611198712119892119890119879 (119896) 119890 (119896) minus 1205761120582min (119879119879119879)119866119879 (119896) 119866 (119896) ge 0 (28)


1198691 = 119870sum119896=0

[ 11205741 119890119879 (119896) 119890 (119896) minus 1205741119908119879 (119896) 119908 (119896)] (29)


1198691 le 119870sum119896=0

[ 11205741 119890119879 (119896) 119890 (119896) minus 1205741119908119879 (119896) 119908 (119896) + Δ119881 (119896)] (30)


1198691 le 119870sum119896=0


119896=0

[119890119879 (119896) 119866119879 (119896) 119908119879 (119896)]sdot Ω[[[

119890 (119896)119866 (119896)119908 (119896)]]] (31)

where


]]]]


(32)


[[[[[[[





sdot 119875 (1198661198632 minus 1198631) 0]

+ [[[[[[[









) + 119901 = 119899(35)





(36)


(37)



(38)


(119896) = 119879minus1 [120594 (119896) + 119866119910 (119896)119910 (119896) ] (39)





119892 (119909 119896) = 120597119892 (119896 119909 (119896))120597119909 (119896) 10038161003816100381610038161003816100381610038161003816119909(119896)=(119896) (41)



119892 (119896 119909 (119896)) = [[[[[[

119892111199091 100381610038161003816100381611990911003816100381610038161003816 + 119892121199092 100381610038161003816100381611990921003816100381610038161003816 + 1198921311990911199092119892211199091 100381610038161003816100381611990911003816100381610038161003816 + 119892221199092 100381610038161003816100381611990921003816100381610038161003816 + 119892231199091119909200]]]]]] (42)



119892 (119896 119909) = [[[[[[

1198921 1198922 0 01198923 1198924 0 00 0 0 00 0 0 0]]]]]] (43)

where

1198921 = 11989211 100381610038161003816100381611003816100381610038161003816 + 119892111 sgn (1) + 1198921321198922 = 11989212 100381610038161003816100381621003816100381610038161003816 + 119892122 sgn (2) + 1198921311198923 = 11989221 100381610038161003816100381611003816100381610038161003816 + 119892211 sgn (1) + 1198922321198924 = 11989222 100381610038161003816100381621003816100381610038161003816 + 119892222 sgn (2) + 119892231(44)

We assume that

119908 (119896) = 119908 (119896 + 1 | 119896) = sdot sdot sdot = 119908 (119896 + 119873119901 | 119896) (45)




Denote

119860 = 119860 + 119892 (119896 minus 1 119909) (47)


(48)





y = [[[[ (119896 + 1 | 119896) (119896 + 119873119901 | 119896)

]]]]

u = [[[[Δ (119896 | 119896)Δ (119896 + 119873119906 minus 1 | 119896)

]]]]

Ψ = [[[[[[[[

1198621198601198621198602

119862119860119873119901

]]]]]]]]

Γ =[[[[[[[[[[

119862119861119862 (119860119861 + 119861)119873119901minus1sum119894=0

119862119860119894119861

]]]]]]]]]]

Θ =[[[[[[[[[[


119873119901minus1sum119894=0


119862119860119894119861

]]]]]]]]]]

(51)




(52)


[[[119872111987221198723

]]]Δ120575 le[[[119873111987321198733

]]] (53)

where

1198721 = [minusΛ 2Λ 2



] 1198723 = [minusΘ

Θ]


Δ120575 = [ Δ120575 (119896)Δ120575 (119896 + 1)] Λ 1 = [11] Λ 2 = [1 01 1]

(54)







119869 = 12u119879119864u + u119879119865 + f0st 119872u le 120574 (57)

where


(58)


119869 = 12u119879119864u + u119879119865 + f0 + 120582119879 (119872u minus 120574) (59)


120597119869120597120582 = 119872u minus 120574 = 0 (61)










min120582ge0

(12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574) (64)



119869 = 12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574 (66)


with120573119894 (119896 + 1)= minus 1ℎ119894119894 [[119896119894 +

119894minus1sum119895=1

ℎ119894119895120582119895 (119896 + 1) + 119873119896sum119895=119894+1

ℎ119894119895120582119895 (119896)]] (68)














119860 = [[[[[[

09921 00063 0 0000200453 09961 0 minus00071001 0 1 minus001510 001 0 1]]]]]]

119861 = [[[[[[

minus00065minus00400]]]]]]

119862 = [0 0 1 00 0 0 1]

119863 = [[[[[[

00002 00 0001200002 00 00012]]]]]]

119892 (119896 119909 (119896)) = 119872minus1

[[[[[[


(70)




minus06

minus04

minus02

0

02

x1

x1

Estimation of x1(a)


x1Estimation of x1

minus06

minus04

minus02

0

02

x1

(b)




119879 = [[[[[[

1 0 0 00 1 0 00 0 1 00 0 0 1]]]]]] (71)



(72)






(73)

One obtains

119866 = 119875minus1119884 = [07880 1213513007 32259] (74)




x2

x2

Estimation of x2

minus2

minus1

0

1

(a)


x2Estimation of x2

x2

minus2

minus1

0

1

(b)


0 50 100 150

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

NMPCMPC-NMOL

minus1





NMPCMPC-NMOL

Pitc

h an

gle (

degr

ee)

minus120

minus100

minus80

minus60

minus40

minus20

0

20






NMPCMPC-NMOL

Elev

ator

(deg

ree)

minus60

minus40

minus20

0

20

40


0 50 100 150 200

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

MPC-NOMPC-NMOL

minus1





Pitc

h an

gle (

degr

ee)

MPC-NOMPC-NMOL

minus100

minus80

minus60

minus40

minus20

0

20


0 50 100 150 200

0

20

40

60

Sampling instant

Elev

ator

(deg

ree)

MPC-NOMPC-NMOL

minus60

minus40

minus20


6 Conclusions



0

20

0 50 100 150 200

0

10

Sampling instant


minus10

minus20

Mwave (Nm)

Zwave (N)



Competing Interests


Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













where

119860119888 = 1198721minus1[[[[[11989911 11989912 0 011989921 11989922 0 119899241 0 0 minus1199060 1 0 0

]]]]]

119861119888 = 1198721minus1[[[[[11988512057511987212057500]]]]]


= 1198721minus1[[[[[


1198721 = [[[[[11989811 11989812 0 011989821 11989822 0 00 0 1 00 0 0 1

]]]]]



(4)


119909 (119896 + 1) = 119860119909 (119896) + 119892 (119896 119909 (119896)) + 119861119906 (119896) + 119863119908 (119896) 119910 (119896) = 119862119909 (119896) (5)




119885wave = int1198710

(119862119889 12058811986302 (120596120596 minus 120596)2+ 119862119898 1205881205871198632

04 (120596 minus )2)119889119909119872wave = int

1198710

(119862119889 12058811986302 (120596120596 minus 120596)2+ 119862119898 1205881205871198632

04 (120596 minus )2)119909119889119909

(7)


AUV

NROO

MPC

u y

w

Stateestimation

yref



In (7)

120596120596 = 119873sum119894=1


119894=1

1205961198901198942120572119894 exp (minus119896119894119911) cos (120596119890119894119905 + 120579119894) (8)










119862]


[1199111 (119896 + 1)1199112 (119896 + 1)] = [11986011 1198601211986021 11986022

][1199111 (119896)1199112 (119896)]+ [119862perp119862 ]119892 (119896 119879minus1119911 (119896))+ [11986111198612] 119906 (119896) + [11986311198632

]119908 (119896) 119910 (119896) = [0119901times(119899minus119901) 119868119901] [1199111 (119896)1199112 (119896)]

(10)


119879119860119879minus1 = [11986011 1198601211986021 11986022

] 119879119861 = [11986111198612] 119879119863 = [11986311198632


(11)

and then

1199112 (119896) = 119910 (119896) 1199111 (119896 + 1) = 119860111199111 (119896) + 11986012119910 (119896)+ 119862perp119892 (119896 119879minus1119911 (119896)) + 1198611119906 (119896)+ 1198631119908 (119896) 119910 (119896 + 1) = 119860211199111 (119896) + 11986022119910 (119896) + 119862119892 (119896 119879minus1119911 (119896))+ 1198612119906 (119896) + 1198632119908 (119896)

(12)






(119896) = 119860211 (119896) + 119862119892 (119896 119879minus1 (119896)) (16)










[[[[[[[[


]]]]]]]]le 0 (20)



(22)


(119896) = 119879minus1 [1 (119896)119910 (119896) ] (23)

One gets







(27)

which is

12057611198712119892119890119879 (119896) 119890 (119896) minus 1205761120582min (119879119879119879)119866119879 (119896) 119866 (119896) ge 0 (28)


1198691 = 119870sum119896=0

[ 11205741 119890119879 (119896) 119890 (119896) minus 1205741119908119879 (119896) 119908 (119896)] (29)


1198691 le 119870sum119896=0

[ 11205741 119890119879 (119896) 119890 (119896) minus 1205741119908119879 (119896) 119908 (119896) + Δ119881 (119896)] (30)


1198691 le 119870sum119896=0


119896=0

[119890119879 (119896) 119866119879 (119896) 119908119879 (119896)]sdot Ω[[[

119890 (119896)119866 (119896)119908 (119896)]]] (31)

where


]]]]


(32)


[[[[[[[





sdot 119875 (1198661198632 minus 1198631) 0]

+ [[[[[[[









) + 119901 = 119899(35)





(36)


(37)



(38)


(119896) = 119879minus1 [120594 (119896) + 119866119910 (119896)119910 (119896) ] (39)





119892 (119909 119896) = 120597119892 (119896 119909 (119896))120597119909 (119896) 10038161003816100381610038161003816100381610038161003816119909(119896)=(119896) (41)



119892 (119896 119909 (119896)) = [[[[[[

119892111199091 100381610038161003816100381611990911003816100381610038161003816 + 119892121199092 100381610038161003816100381611990921003816100381610038161003816 + 1198921311990911199092119892211199091 100381610038161003816100381611990911003816100381610038161003816 + 119892221199092 100381610038161003816100381611990921003816100381610038161003816 + 119892231199091119909200]]]]]] (42)



119892 (119896 119909) = [[[[[[

1198921 1198922 0 01198923 1198924 0 00 0 0 00 0 0 0]]]]]] (43)

where

1198921 = 11989211 100381610038161003816100381611003816100381610038161003816 + 119892111 sgn (1) + 1198921321198922 = 11989212 100381610038161003816100381621003816100381610038161003816 + 119892122 sgn (2) + 1198921311198923 = 11989221 100381610038161003816100381611003816100381610038161003816 + 119892211 sgn (1) + 1198922321198924 = 11989222 100381610038161003816100381621003816100381610038161003816 + 119892222 sgn (2) + 119892231(44)

We assume that

119908 (119896) = 119908 (119896 + 1 | 119896) = sdot sdot sdot = 119908 (119896 + 119873119901 | 119896) (45)




Denote

119860 = 119860 + 119892 (119896 minus 1 119909) (47)


(48)





y = [[[[ (119896 + 1 | 119896) (119896 + 119873119901 | 119896)

]]]]

u = [[[[Δ (119896 | 119896)Δ (119896 + 119873119906 minus 1 | 119896)

]]]]

Ψ = [[[[[[[[

1198621198601198621198602

119862119860119873119901

]]]]]]]]

Γ =[[[[[[[[[[

119862119861119862 (119860119861 + 119861)119873119901minus1sum119894=0

119862119860119894119861

]]]]]]]]]]

Θ =[[[[[[[[[[


119873119901minus1sum119894=0


119862119860119894119861

]]]]]]]]]]

(51)




(52)


[[[119872111987221198723

]]]Δ120575 le[[[119873111987321198733

]]] (53)

where

1198721 = [minusΛ 2Λ 2



] 1198723 = [minusΘ

Θ]


Δ120575 = [ Δ120575 (119896)Δ120575 (119896 + 1)] Λ 1 = [11] Λ 2 = [1 01 1]

(54)







119869 = 12u119879119864u + u119879119865 + f0st 119872u le 120574 (57)

where


(58)


119869 = 12u119879119864u + u119879119865 + f0 + 120582119879 (119872u minus 120574) (59)


120597119869120597120582 = 119872u minus 120574 = 0 (61)










min120582ge0

(12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574) (64)



119869 = 12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574 (66)


with120573119894 (119896 + 1)= minus 1ℎ119894119894 [[119896119894 +

119894minus1sum119895=1

ℎ119894119895120582119895 (119896 + 1) + 119873119896sum119895=119894+1

ℎ119894119895120582119895 (119896)]] (68)














119860 = [[[[[[

09921 00063 0 0000200453 09961 0 minus00071001 0 1 minus001510 001 0 1]]]]]]

119861 = [[[[[[

minus00065minus00400]]]]]]

119862 = [0 0 1 00 0 0 1]

119863 = [[[[[[

00002 00 0001200002 00 00012]]]]]]

119892 (119896 119909 (119896)) = 119872minus1

[[[[[[


(70)




minus06

minus04

minus02

0

02

x1

x1

Estimation of x1(a)


x1Estimation of x1

minus06

minus04

minus02

0

02

x1

(b)




119879 = [[[[[[

1 0 0 00 1 0 00 0 1 00 0 0 1]]]]]] (71)



(72)






(73)

One obtains

119866 = 119875minus1119884 = [07880 1213513007 32259] (74)




x2

x2

Estimation of x2

minus2

minus1

0

1

(a)


x2Estimation of x2

x2

minus2

minus1

0

1

(b)


0 50 100 150

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

NMPCMPC-NMOL

minus1





NMPCMPC-NMOL

Pitc

h an

gle (

degr

ee)

minus120

minus100

minus80

minus60

minus40

minus20

0

20






NMPCMPC-NMOL

Elev

ator

(deg

ree)

minus60

minus40

minus20

0

20

40


0 50 100 150 200

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

MPC-NOMPC-NMOL

minus1





Pitc

h an

gle (

degr

ee)

MPC-NOMPC-NMOL

minus100

minus80

minus60

minus40

minus20

0

20


0 50 100 150 200

0

20

40

60

Sampling instant

Elev

ator

(deg

ree)

MPC-NOMPC-NMOL

minus60

minus40

minus20


6 Conclusions



0

20

0 50 100 150 200

0

10

Sampling instant


minus10

minus20

Mwave (Nm)

Zwave (N)



Competing Interests


Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










AUV

NROO

MPC

u y

w

Stateestimation

yref



In (7)

120596120596 = 119873sum119894=1


119894=1

1205961198901198942120572119894 exp (minus119896119894119911) cos (120596119890119894119905 + 120579119894) (8)










119862]


[1199111 (119896 + 1)1199112 (119896 + 1)] = [11986011 1198601211986021 11986022

][1199111 (119896)1199112 (119896)]+ [119862perp119862 ]119892 (119896 119879minus1119911 (119896))+ [11986111198612] 119906 (119896) + [11986311198632

]119908 (119896) 119910 (119896) = [0119901times(119899minus119901) 119868119901] [1199111 (119896)1199112 (119896)]

(10)


119879119860119879minus1 = [11986011 1198601211986021 11986022

] 119879119861 = [11986111198612] 119879119863 = [11986311198632


(11)

and then

1199112 (119896) = 119910 (119896) 1199111 (119896 + 1) = 119860111199111 (119896) + 11986012119910 (119896)+ 119862perp119892 (119896 119879minus1119911 (119896)) + 1198611119906 (119896)+ 1198631119908 (119896) 119910 (119896 + 1) = 119860211199111 (119896) + 11986022119910 (119896) + 119862119892 (119896 119879minus1119911 (119896))+ 1198612119906 (119896) + 1198632119908 (119896)

(12)






(119896) = 119860211 (119896) + 119862119892 (119896 119879minus1 (119896)) (16)










[[[[[[[[


]]]]]]]]le 0 (20)



(22)


(119896) = 119879minus1 [1 (119896)119910 (119896) ] (23)

One gets







(27)

which is

12057611198712119892119890119879 (119896) 119890 (119896) minus 1205761120582min (119879119879119879)119866119879 (119896) 119866 (119896) ge 0 (28)


1198691 = 119870sum119896=0

[ 11205741 119890119879 (119896) 119890 (119896) minus 1205741119908119879 (119896) 119908 (119896)] (29)


1198691 le 119870sum119896=0

[ 11205741 119890119879 (119896) 119890 (119896) minus 1205741119908119879 (119896) 119908 (119896) + Δ119881 (119896)] (30)


1198691 le 119870sum119896=0


119896=0

[119890119879 (119896) 119866119879 (119896) 119908119879 (119896)]sdot Ω[[[

119890 (119896)119866 (119896)119908 (119896)]]] (31)

where


]]]]


(32)


[[[[[[[





sdot 119875 (1198661198632 minus 1198631) 0]

+ [[[[[[[









) + 119901 = 119899(35)





(36)


(37)



(38)


(119896) = 119879minus1 [120594 (119896) + 119866119910 (119896)119910 (119896) ] (39)





119892 (119909 119896) = 120597119892 (119896 119909 (119896))120597119909 (119896) 10038161003816100381610038161003816100381610038161003816119909(119896)=(119896) (41)



119892 (119896 119909 (119896)) = [[[[[[

119892111199091 100381610038161003816100381611990911003816100381610038161003816 + 119892121199092 100381610038161003816100381611990921003816100381610038161003816 + 1198921311990911199092119892211199091 100381610038161003816100381611990911003816100381610038161003816 + 119892221199092 100381610038161003816100381611990921003816100381610038161003816 + 119892231199091119909200]]]]]] (42)



119892 (119896 119909) = [[[[[[

1198921 1198922 0 01198923 1198924 0 00 0 0 00 0 0 0]]]]]] (43)

where

1198921 = 11989211 100381610038161003816100381611003816100381610038161003816 + 119892111 sgn (1) + 1198921321198922 = 11989212 100381610038161003816100381621003816100381610038161003816 + 119892122 sgn (2) + 1198921311198923 = 11989221 100381610038161003816100381611003816100381610038161003816 + 119892211 sgn (1) + 1198922321198924 = 11989222 100381610038161003816100381621003816100381610038161003816 + 119892222 sgn (2) + 119892231(44)

We assume that

119908 (119896) = 119908 (119896 + 1 | 119896) = sdot sdot sdot = 119908 (119896 + 119873119901 | 119896) (45)




Denote

119860 = 119860 + 119892 (119896 minus 1 119909) (47)


(48)





y = [[[[ (119896 + 1 | 119896) (119896 + 119873119901 | 119896)

]]]]

u = [[[[Δ (119896 | 119896)Δ (119896 + 119873119906 minus 1 | 119896)

]]]]

Ψ = [[[[[[[[

1198621198601198621198602

119862119860119873119901

]]]]]]]]

Γ =[[[[[[[[[[

119862119861119862 (119860119861 + 119861)119873119901minus1sum119894=0

119862119860119894119861

]]]]]]]]]]

Θ =[[[[[[[[[[


119873119901minus1sum119894=0


119862119860119894119861

]]]]]]]]]]

(51)




(52)


[[[119872111987221198723

]]]Δ120575 le[[[119873111987321198733

]]] (53)

where

1198721 = [minusΛ 2Λ 2



] 1198723 = [minusΘ

Θ]


Δ120575 = [ Δ120575 (119896)Δ120575 (119896 + 1)] Λ 1 = [11] Λ 2 = [1 01 1]

(54)







119869 = 12u119879119864u + u119879119865 + f0st 119872u le 120574 (57)

where


(58)


119869 = 12u119879119864u + u119879119865 + f0 + 120582119879 (119872u minus 120574) (59)


120597119869120597120582 = 119872u minus 120574 = 0 (61)










min120582ge0

(12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574) (64)



119869 = 12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574 (66)


with120573119894 (119896 + 1)= minus 1ℎ119894119894 [[119896119894 +

119894minus1sum119895=1

ℎ119894119895120582119895 (119896 + 1) + 119873119896sum119895=119894+1

ℎ119894119895120582119895 (119896)]] (68)














119860 = [[[[[[

09921 00063 0 0000200453 09961 0 minus00071001 0 1 minus001510 001 0 1]]]]]]

119861 = [[[[[[

minus00065minus00400]]]]]]

119862 = [0 0 1 00 0 0 1]

119863 = [[[[[[

00002 00 0001200002 00 00012]]]]]]

119892 (119896 119909 (119896)) = 119872minus1

[[[[[[


(70)




minus06

minus04

minus02

0

02

x1

x1

Estimation of x1(a)


x1Estimation of x1

minus06

minus04

minus02

0

02

x1

(b)




119879 = [[[[[[

1 0 0 00 1 0 00 0 1 00 0 0 1]]]]]] (71)



(72)






(73)

One obtains

119866 = 119875minus1119884 = [07880 1213513007 32259] (74)




x2

x2

Estimation of x2

minus2

minus1

0

1

(a)


x2Estimation of x2

x2

minus2

minus1

0

1

(b)


0 50 100 150

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

NMPCMPC-NMOL

minus1





NMPCMPC-NMOL

Pitc

h an

gle (

degr

ee)

minus120

minus100

minus80

minus60

minus40

minus20

0

20






NMPCMPC-NMOL

Elev

ator

(deg

ree)

minus60

minus40

minus20

0

20

40


0 50 100 150 200

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

MPC-NOMPC-NMOL

minus1





Pitc

h an

gle (

degr

ee)

MPC-NOMPC-NMOL

minus100

minus80

minus60

minus40

minus20

0

20


0 50 100 150 200

0

20

40

60

Sampling instant

Elev

ator

(deg

ree)

MPC-NOMPC-NMOL

minus60

minus40

minus20


6 Conclusions



0

20

0 50 100 150 200

0

10

Sampling instant


minus10

minus20

Mwave (Nm)

Zwave (N)



Competing Interests


Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













(119896) = 119860211 (119896) + 119862119892 (119896 119879minus1 (119896)) (16)










[[[[[[[[


]]]]]]]]le 0 (20)



(22)


(119896) = 119879minus1 [1 (119896)119910 (119896) ] (23)

One gets







(27)

which is

12057611198712119892119890119879 (119896) 119890 (119896) minus 1205761120582min (119879119879119879)119866119879 (119896) 119866 (119896) ge 0 (28)


1198691 = 119870sum119896=0

[ 11205741 119890119879 (119896) 119890 (119896) minus 1205741119908119879 (119896) 119908 (119896)] (29)


1198691 le 119870sum119896=0

[ 11205741 119890119879 (119896) 119890 (119896) minus 1205741119908119879 (119896) 119908 (119896) + Δ119881 (119896)] (30)


1198691 le 119870sum119896=0


119896=0

[119890119879 (119896) 119866119879 (119896) 119908119879 (119896)]sdot Ω[[[

119890 (119896)119866 (119896)119908 (119896)]]] (31)

where


]]]]


(32)


[[[[[[[





sdot 119875 (1198661198632 minus 1198631) 0]

+ [[[[[[[









) + 119901 = 119899(35)





(36)


(37)



(38)


(119896) = 119879minus1 [120594 (119896) + 119866119910 (119896)119910 (119896) ] (39)





119892 (119909 119896) = 120597119892 (119896 119909 (119896))120597119909 (119896) 10038161003816100381610038161003816100381610038161003816119909(119896)=(119896) (41)



119892 (119896 119909 (119896)) = [[[[[[

119892111199091 100381610038161003816100381611990911003816100381610038161003816 + 119892121199092 100381610038161003816100381611990921003816100381610038161003816 + 1198921311990911199092119892211199091 100381610038161003816100381611990911003816100381610038161003816 + 119892221199092 100381610038161003816100381611990921003816100381610038161003816 + 119892231199091119909200]]]]]] (42)



119892 (119896 119909) = [[[[[[

1198921 1198922 0 01198923 1198924 0 00 0 0 00 0 0 0]]]]]] (43)

where

1198921 = 11989211 100381610038161003816100381611003816100381610038161003816 + 119892111 sgn (1) + 1198921321198922 = 11989212 100381610038161003816100381621003816100381610038161003816 + 119892122 sgn (2) + 1198921311198923 = 11989221 100381610038161003816100381611003816100381610038161003816 + 119892211 sgn (1) + 1198922321198924 = 11989222 100381610038161003816100381621003816100381610038161003816 + 119892222 sgn (2) + 119892231(44)

We assume that

119908 (119896) = 119908 (119896 + 1 | 119896) = sdot sdot sdot = 119908 (119896 + 119873119901 | 119896) (45)




Denote

119860 = 119860 + 119892 (119896 minus 1 119909) (47)


(48)





y = [[[[ (119896 + 1 | 119896) (119896 + 119873119901 | 119896)

]]]]

u = [[[[Δ (119896 | 119896)Δ (119896 + 119873119906 minus 1 | 119896)

]]]]

Ψ = [[[[[[[[

1198621198601198621198602

119862119860119873119901

]]]]]]]]

Γ =[[[[[[[[[[

119862119861119862 (119860119861 + 119861)119873119901minus1sum119894=0

119862119860119894119861

]]]]]]]]]]

Θ =[[[[[[[[[[


119873119901minus1sum119894=0


119862119860119894119861

]]]]]]]]]]

(51)




(52)


[[[119872111987221198723

]]]Δ120575 le[[[119873111987321198733

]]] (53)

where

1198721 = [minusΛ 2Λ 2



] 1198723 = [minusΘ

Θ]


Δ120575 = [ Δ120575 (119896)Δ120575 (119896 + 1)] Λ 1 = [11] Λ 2 = [1 01 1]

(54)







119869 = 12u119879119864u + u119879119865 + f0st 119872u le 120574 (57)

where


(58)


119869 = 12u119879119864u + u119879119865 + f0 + 120582119879 (119872u minus 120574) (59)


120597119869120597120582 = 119872u minus 120574 = 0 (61)










min120582ge0

(12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574) (64)



119869 = 12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574 (66)


with120573119894 (119896 + 1)= minus 1ℎ119894119894 [[119896119894 +

119894minus1sum119895=1

ℎ119894119895120582119895 (119896 + 1) + 119873119896sum119895=119894+1

ℎ119894119895120582119895 (119896)]] (68)














119860 = [[[[[[

09921 00063 0 0000200453 09961 0 minus00071001 0 1 minus001510 001 0 1]]]]]]

119861 = [[[[[[

minus00065minus00400]]]]]]

119862 = [0 0 1 00 0 0 1]

119863 = [[[[[[

00002 00 0001200002 00 00012]]]]]]

119892 (119896 119909 (119896)) = 119872minus1

[[[[[[


(70)




minus06

minus04

minus02

0

02

x1

x1

Estimation of x1(a)


x1Estimation of x1

minus06

minus04

minus02

0

02

x1

(b)




119879 = [[[[[[

1 0 0 00 1 0 00 0 1 00 0 0 1]]]]]] (71)



(72)






(73)

One obtains

119866 = 119875minus1119884 = [07880 1213513007 32259] (74)




x2

x2

Estimation of x2

minus2

minus1

0

1

(a)


x2Estimation of x2

x2

minus2

minus1

0

1

(b)


0 50 100 150

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

NMPCMPC-NMOL

minus1





NMPCMPC-NMOL

Pitc

h an

gle (

degr

ee)

minus120

minus100

minus80

minus60

minus40

minus20

0

20






NMPCMPC-NMOL

Elev

ator

(deg

ree)

minus60

minus40

minus20

0

20

40


0 50 100 150 200

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

MPC-NOMPC-NMOL

minus1





Pitc

h an

gle (

degr

ee)

MPC-NOMPC-NMOL

minus100

minus80

minus60

minus40

minus20

0

20


0 50 100 150 200

0

20

40

60

Sampling instant

Elev

ator

(deg

ree)

MPC-NOMPC-NMOL

minus60

minus40

minus20


6 Conclusions



0

20

0 50 100 150 200

0

10

Sampling instant


minus10

minus20

Mwave (Nm)

Zwave (N)



Competing Interests


Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(27)

which is

12057611198712119892119890119879 (119896) 119890 (119896) minus 1205761120582min (119879119879119879)119866119879 (119896) 119866 (119896) ge 0 (28)


1198691 = 119870sum119896=0

[ 11205741 119890119879 (119896) 119890 (119896) minus 1205741119908119879 (119896) 119908 (119896)] (29)


1198691 le 119870sum119896=0

[ 11205741 119890119879 (119896) 119890 (119896) minus 1205741119908119879 (119896) 119908 (119896) + Δ119881 (119896)] (30)


1198691 le 119870sum119896=0


119896=0

[119890119879 (119896) 119866119879 (119896) 119908119879 (119896)]sdot Ω[[[

119890 (119896)119866 (119896)119908 (119896)]]] (31)

where


]]]]


(32)


[[[[[[[





sdot 119875 (1198661198632 minus 1198631) 0]

+ [[[[[[[









) + 119901 = 119899(35)





(36)


(37)



(38)


(119896) = 119879minus1 [120594 (119896) + 119866119910 (119896)119910 (119896) ] (39)





119892 (119909 119896) = 120597119892 (119896 119909 (119896))120597119909 (119896) 10038161003816100381610038161003816100381610038161003816119909(119896)=(119896) (41)



119892 (119896 119909 (119896)) = [[[[[[

119892111199091 100381610038161003816100381611990911003816100381610038161003816 + 119892121199092 100381610038161003816100381611990921003816100381610038161003816 + 1198921311990911199092119892211199091 100381610038161003816100381611990911003816100381610038161003816 + 119892221199092 100381610038161003816100381611990921003816100381610038161003816 + 119892231199091119909200]]]]]] (42)



119892 (119896 119909) = [[[[[[

1198921 1198922 0 01198923 1198924 0 00 0 0 00 0 0 0]]]]]] (43)

where

1198921 = 11989211 100381610038161003816100381611003816100381610038161003816 + 119892111 sgn (1) + 1198921321198922 = 11989212 100381610038161003816100381621003816100381610038161003816 + 119892122 sgn (2) + 1198921311198923 = 11989221 100381610038161003816100381611003816100381610038161003816 + 119892211 sgn (1) + 1198922321198924 = 11989222 100381610038161003816100381621003816100381610038161003816 + 119892222 sgn (2) + 119892231(44)

We assume that

119908 (119896) = 119908 (119896 + 1 | 119896) = sdot sdot sdot = 119908 (119896 + 119873119901 | 119896) (45)




Denote

119860 = 119860 + 119892 (119896 minus 1 119909) (47)


(48)





y = [[[[ (119896 + 1 | 119896) (119896 + 119873119901 | 119896)

]]]]

u = [[[[Δ (119896 | 119896)Δ (119896 + 119873119906 minus 1 | 119896)

]]]]

Ψ = [[[[[[[[

1198621198601198621198602

119862119860119873119901

]]]]]]]]

Γ =[[[[[[[[[[

119862119861119862 (119860119861 + 119861)119873119901minus1sum119894=0

119862119860119894119861

]]]]]]]]]]

Θ =[[[[[[[[[[


119873119901minus1sum119894=0


119862119860119894119861

]]]]]]]]]]

(51)




(52)


[[[119872111987221198723

]]]Δ120575 le[[[119873111987321198733

]]] (53)

where

1198721 = [minusΛ 2Λ 2



] 1198723 = [minusΘ

Θ]


Δ120575 = [ Δ120575 (119896)Δ120575 (119896 + 1)] Λ 1 = [11] Λ 2 = [1 01 1]

(54)







119869 = 12u119879119864u + u119879119865 + f0st 119872u le 120574 (57)

where


(58)


119869 = 12u119879119864u + u119879119865 + f0 + 120582119879 (119872u minus 120574) (59)


120597119869120597120582 = 119872u minus 120574 = 0 (61)










min120582ge0

(12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574) (64)



119869 = 12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574 (66)


with120573119894 (119896 + 1)= minus 1ℎ119894119894 [[119896119894 +

119894minus1sum119895=1

ℎ119894119895120582119895 (119896 + 1) + 119873119896sum119895=119894+1

ℎ119894119895120582119895 (119896)]] (68)














119860 = [[[[[[

09921 00063 0 0000200453 09961 0 minus00071001 0 1 minus001510 001 0 1]]]]]]

119861 = [[[[[[

minus00065minus00400]]]]]]

119862 = [0 0 1 00 0 0 1]

119863 = [[[[[[

00002 00 0001200002 00 00012]]]]]]

119892 (119896 119909 (119896)) = 119872minus1

[[[[[[


(70)




minus06

minus04

minus02

0

02

x1

x1

Estimation of x1(a)


x1Estimation of x1

minus06

minus04

minus02

0

02

x1

(b)




119879 = [[[[[[

1 0 0 00 1 0 00 0 1 00 0 0 1]]]]]] (71)



(72)






(73)

One obtains

119866 = 119875minus1119884 = [07880 1213513007 32259] (74)




x2

x2

Estimation of x2

minus2

minus1

0

1

(a)


x2Estimation of x2

x2

minus2

minus1

0

1

(b)


0 50 100 150

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

NMPCMPC-NMOL

minus1





NMPCMPC-NMOL

Pitc

h an

gle (

degr

ee)

minus120

minus100

minus80

minus60

minus40

minus20

0

20






NMPCMPC-NMOL

Elev

ator

(deg

ree)

minus60

minus40

minus20

0

20

40


0 50 100 150 200

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

MPC-NOMPC-NMOL

minus1





Pitc

h an

gle (

degr

ee)

MPC-NOMPC-NMOL

minus100

minus80

minus60

minus40

minus20

0

20


0 50 100 150 200

0

20

40

60

Sampling instant

Elev

ator

(deg

ree)

MPC-NOMPC-NMOL

minus60

minus40

minus20


6 Conclusions



0

20

0 50 100 150 200

0

10

Sampling instant


minus10

minus20

Mwave (Nm)

Zwave (N)



Competing Interests


Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















) + 119901 = 119899(35)





(36)


(37)



(38)


(119896) = 119879minus1 [120594 (119896) + 119866119910 (119896)119910 (119896) ] (39)





119892 (119909 119896) = 120597119892 (119896 119909 (119896))120597119909 (119896) 10038161003816100381610038161003816100381610038161003816119909(119896)=(119896) (41)



119892 (119896 119909 (119896)) = [[[[[[

119892111199091 100381610038161003816100381611990911003816100381610038161003816 + 119892121199092 100381610038161003816100381611990921003816100381610038161003816 + 1198921311990911199092119892211199091 100381610038161003816100381611990911003816100381610038161003816 + 119892221199092 100381610038161003816100381611990921003816100381610038161003816 + 119892231199091119909200]]]]]] (42)



119892 (119896 119909) = [[[[[[

1198921 1198922 0 01198923 1198924 0 00 0 0 00 0 0 0]]]]]] (43)

where

1198921 = 11989211 100381610038161003816100381611003816100381610038161003816 + 119892111 sgn (1) + 1198921321198922 = 11989212 100381610038161003816100381621003816100381610038161003816 + 119892122 sgn (2) + 1198921311198923 = 11989221 100381610038161003816100381611003816100381610038161003816 + 119892211 sgn (1) + 1198922321198924 = 11989222 100381610038161003816100381621003816100381610038161003816 + 119892222 sgn (2) + 119892231(44)

We assume that

119908 (119896) = 119908 (119896 + 1 | 119896) = sdot sdot sdot = 119908 (119896 + 119873119901 | 119896) (45)




Denote

119860 = 119860 + 119892 (119896 minus 1 119909) (47)


(48)





y = [[[[ (119896 + 1 | 119896) (119896 + 119873119901 | 119896)

]]]]

u = [[[[Δ (119896 | 119896)Δ (119896 + 119873119906 minus 1 | 119896)

]]]]

Ψ = [[[[[[[[

1198621198601198621198602

119862119860119873119901

]]]]]]]]

Γ =[[[[[[[[[[

119862119861119862 (119860119861 + 119861)119873119901minus1sum119894=0

119862119860119894119861

]]]]]]]]]]

Θ =[[[[[[[[[[


119873119901minus1sum119894=0


119862119860119894119861

]]]]]]]]]]

(51)




(52)


[[[119872111987221198723

]]]Δ120575 le[[[119873111987321198733

]]] (53)

where

1198721 = [minusΛ 2Λ 2



] 1198723 = [minusΘ

Θ]


Δ120575 = [ Δ120575 (119896)Δ120575 (119896 + 1)] Λ 1 = [11] Λ 2 = [1 01 1]

(54)







119869 = 12u119879119864u + u119879119865 + f0st 119872u le 120574 (57)

where


(58)


119869 = 12u119879119864u + u119879119865 + f0 + 120582119879 (119872u minus 120574) (59)


120597119869120597120582 = 119872u minus 120574 = 0 (61)










min120582ge0

(12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574) (64)



119869 = 12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574 (66)


with120573119894 (119896 + 1)= minus 1ℎ119894119894 [[119896119894 +

119894minus1sum119895=1

ℎ119894119895120582119895 (119896 + 1) + 119873119896sum119895=119894+1

ℎ119894119895120582119895 (119896)]] (68)














119860 = [[[[[[

09921 00063 0 0000200453 09961 0 minus00071001 0 1 minus001510 001 0 1]]]]]]

119861 = [[[[[[

minus00065minus00400]]]]]]

119862 = [0 0 1 00 0 0 1]

119863 = [[[[[[

00002 00 0001200002 00 00012]]]]]]

119892 (119896 119909 (119896)) = 119872minus1

[[[[[[


(70)




minus06

minus04

minus02

0

02

x1

x1

Estimation of x1(a)


x1Estimation of x1

minus06

minus04

minus02

0

02

x1

(b)




119879 = [[[[[[

1 0 0 00 1 0 00 0 1 00 0 0 1]]]]]] (71)



(72)






(73)

One obtains

119866 = 119875minus1119884 = [07880 1213513007 32259] (74)




x2

x2

Estimation of x2

minus2

minus1

0

1

(a)


x2Estimation of x2

x2

minus2

minus1

0

1

(b)


0 50 100 150

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

NMPCMPC-NMOL

minus1





NMPCMPC-NMOL

Pitc

h an

gle (

degr

ee)

minus120

minus100

minus80

minus60

minus40

minus20

0

20






NMPCMPC-NMOL

Elev

ator

(deg

ree)

minus60

minus40

minus20

0

20

40


0 50 100 150 200

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

MPC-NOMPC-NMOL

minus1





Pitc

h an

gle (

degr

ee)

MPC-NOMPC-NMOL

minus100

minus80

minus60

minus40

minus20

0

20


0 50 100 150 200

0

20

40

60

Sampling instant

Elev

ator

(deg

ree)

MPC-NOMPC-NMOL

minus60

minus40

minus20


6 Conclusions



0

20

0 50 100 150 200

0

10

Sampling instant


minus10

minus20

Mwave (Nm)

Zwave (N)



Competing Interests


Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119892 (119909 119896) = 120597119892 (119896 119909 (119896))120597119909 (119896) 10038161003816100381610038161003816100381610038161003816119909(119896)=(119896) (41)



119892 (119896 119909 (119896)) = [[[[[[

119892111199091 100381610038161003816100381611990911003816100381610038161003816 + 119892121199092 100381610038161003816100381611990921003816100381610038161003816 + 1198921311990911199092119892211199091 100381610038161003816100381611990911003816100381610038161003816 + 119892221199092 100381610038161003816100381611990921003816100381610038161003816 + 119892231199091119909200]]]]]] (42)



119892 (119896 119909) = [[[[[[

1198921 1198922 0 01198923 1198924 0 00 0 0 00 0 0 0]]]]]] (43)

where

1198921 = 11989211 100381610038161003816100381611003816100381610038161003816 + 119892111 sgn (1) + 1198921321198922 = 11989212 100381610038161003816100381621003816100381610038161003816 + 119892122 sgn (2) + 1198921311198923 = 11989221 100381610038161003816100381611003816100381610038161003816 + 119892211 sgn (1) + 1198922321198924 = 11989222 100381610038161003816100381621003816100381610038161003816 + 119892222 sgn (2) + 119892231(44)

We assume that

119908 (119896) = 119908 (119896 + 1 | 119896) = sdot sdot sdot = 119908 (119896 + 119873119901 | 119896) (45)




Denote

119860 = 119860 + 119892 (119896 minus 1 119909) (47)


(48)





y = [[[[ (119896 + 1 | 119896) (119896 + 119873119901 | 119896)

]]]]

u = [[[[Δ (119896 | 119896)Δ (119896 + 119873119906 minus 1 | 119896)

]]]]

Ψ = [[[[[[[[

1198621198601198621198602

119862119860119873119901

]]]]]]]]

Γ =[[[[[[[[[[

119862119861119862 (119860119861 + 119861)119873119901minus1sum119894=0

119862119860119894119861

]]]]]]]]]]

Θ =[[[[[[[[[[


119873119901minus1sum119894=0


119862119860119894119861

]]]]]]]]]]

(51)




(52)


[[[119872111987221198723

]]]Δ120575 le[[[119873111987321198733

]]] (53)

where

1198721 = [minusΛ 2Λ 2



] 1198723 = [minusΘ

Θ]


Δ120575 = [ Δ120575 (119896)Δ120575 (119896 + 1)] Λ 1 = [11] Λ 2 = [1 01 1]

(54)







119869 = 12u119879119864u + u119879119865 + f0st 119872u le 120574 (57)

where


(58)


119869 = 12u119879119864u + u119879119865 + f0 + 120582119879 (119872u minus 120574) (59)


120597119869120597120582 = 119872u minus 120574 = 0 (61)










min120582ge0

(12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574) (64)



119869 = 12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574 (66)


with120573119894 (119896 + 1)= minus 1ℎ119894119894 [[119896119894 +

119894minus1sum119895=1

ℎ119894119895120582119895 (119896 + 1) + 119873119896sum119895=119894+1

ℎ119894119895120582119895 (119896)]] (68)














119860 = [[[[[[

09921 00063 0 0000200453 09961 0 minus00071001 0 1 minus001510 001 0 1]]]]]]

119861 = [[[[[[

minus00065minus00400]]]]]]

119862 = [0 0 1 00 0 0 1]

119863 = [[[[[[

00002 00 0001200002 00 00012]]]]]]

119892 (119896 119909 (119896)) = 119872minus1

[[[[[[


(70)




minus06

minus04

minus02

0

02

x1

x1

Estimation of x1(a)


x1Estimation of x1

minus06

minus04

minus02

0

02

x1

(b)




119879 = [[[[[[

1 0 0 00 1 0 00 0 1 00 0 0 1]]]]]] (71)



(72)






(73)

One obtains

119866 = 119875minus1119884 = [07880 1213513007 32259] (74)




x2

x2

Estimation of x2

minus2

minus1

0

1

(a)


x2Estimation of x2

x2

minus2

minus1

0

1

(b)


0 50 100 150

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

NMPCMPC-NMOL

minus1





NMPCMPC-NMOL

Pitc

h an

gle (

degr

ee)

minus120

minus100

minus80

minus60

minus40

minus20

0

20






NMPCMPC-NMOL

Elev

ator

(deg

ree)

minus60

minus40

minus20

0

20

40


0 50 100 150 200

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

MPC-NOMPC-NMOL

minus1





Pitc

h an

gle (

degr

ee)

MPC-NOMPC-NMOL

minus100

minus80

minus60

minus40

minus20

0

20


0 50 100 150 200

0

20

40

60

Sampling instant

Elev

ator

(deg

ree)

MPC-NOMPC-NMOL

minus60

minus40

minus20


6 Conclusions



0

20

0 50 100 150 200

0

10

Sampling instant


minus10

minus20

Mwave (Nm)

Zwave (N)



Competing Interests


Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(52)


[[[119872111987221198723

]]]Δ120575 le[[[119873111987321198733

]]] (53)

where

1198721 = [minusΛ 2Λ 2



] 1198723 = [minusΘ

Θ]


Δ120575 = [ Δ120575 (119896)Δ120575 (119896 + 1)] Λ 1 = [11] Λ 2 = [1 01 1]

(54)







119869 = 12u119879119864u + u119879119865 + f0st 119872u le 120574 (57)

where


(58)


119869 = 12u119879119864u + u119879119865 + f0 + 120582119879 (119872u minus 120574) (59)


120597119869120597120582 = 119872u minus 120574 = 0 (61)










min120582ge0

(12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574) (64)



119869 = 12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574 (66)


with120573119894 (119896 + 1)= minus 1ℎ119894119894 [[119896119894 +

119894minus1sum119895=1

ℎ119894119895120582119895 (119896 + 1) + 119873119896sum119895=119894+1

ℎ119894119895120582119895 (119896)]] (68)














119860 = [[[[[[

09921 00063 0 0000200453 09961 0 minus00071001 0 1 minus001510 001 0 1]]]]]]

119861 = [[[[[[

minus00065minus00400]]]]]]

119862 = [0 0 1 00 0 0 1]

119863 = [[[[[[

00002 00 0001200002 00 00012]]]]]]

119892 (119896 119909 (119896)) = 119872minus1

[[[[[[


(70)




minus06

minus04

minus02

0

02

x1

x1

Estimation of x1(a)


x1Estimation of x1

minus06

minus04

minus02

0

02

x1

(b)




119879 = [[[[[[

1 0 0 00 1 0 00 0 1 00 0 0 1]]]]]] (71)



(72)






(73)

One obtains

119866 = 119875minus1119884 = [07880 1213513007 32259] (74)




x2

x2

Estimation of x2

minus2

minus1

0

1

(a)


x2Estimation of x2

x2

minus2

minus1

0

1

(b)


0 50 100 150

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

NMPCMPC-NMOL

minus1





NMPCMPC-NMOL

Pitc

h an

gle (

degr

ee)

minus120

minus100

minus80

minus60

minus40

minus20

0

20






NMPCMPC-NMOL

Elev

ator

(deg

ree)

minus60

minus40

minus20

0

20

40


0 50 100 150 200

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

MPC-NOMPC-NMOL

minus1





Pitc

h an

gle (

degr

ee)

MPC-NOMPC-NMOL

minus100

minus80

minus60

minus40

minus20

0

20


0 50 100 150 200

0

20

40

60

Sampling instant

Elev

ator

(deg

ree)

MPC-NOMPC-NMOL

minus60

minus40

minus20


6 Conclusions



0

20

0 50 100 150 200

0

10

Sampling instant


minus10

minus20

Mwave (Nm)

Zwave (N)



Competing Interests


Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












min120582ge0

(12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574) (64)



119869 = 12120582119879119867120582 + 120582119879119870 + 12120574119879119864minus1120574 (66)


with120573119894 (119896 + 1)= minus 1ℎ119894119894 [[119896119894 +

119894minus1sum119895=1

ℎ119894119895120582119895 (119896 + 1) + 119873119896sum119895=119894+1

ℎ119894119895120582119895 (119896)]] (68)














119860 = [[[[[[

09921 00063 0 0000200453 09961 0 minus00071001 0 1 minus001510 001 0 1]]]]]]

119861 = [[[[[[

minus00065minus00400]]]]]]

119862 = [0 0 1 00 0 0 1]

119863 = [[[[[[

00002 00 0001200002 00 00012]]]]]]

119892 (119896 119909 (119896)) = 119872minus1

[[[[[[


(70)




minus06

minus04

minus02

0

02

x1

x1

Estimation of x1(a)


x1Estimation of x1

minus06

minus04

minus02

0

02

x1

(b)




119879 = [[[[[[

1 0 0 00 1 0 00 0 1 00 0 0 1]]]]]] (71)



(72)






(73)

One obtains

119866 = 119875minus1119884 = [07880 1213513007 32259] (74)




x2

x2

Estimation of x2

minus2

minus1

0

1

(a)


x2Estimation of x2

x2

minus2

minus1

0

1

(b)


0 50 100 150

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

NMPCMPC-NMOL

minus1





NMPCMPC-NMOL

Pitc

h an

gle (

degr

ee)

minus120

minus100

minus80

minus60

minus40

minus20

0

20






NMPCMPC-NMOL

Elev

ator

(deg

ree)

minus60

minus40

minus20

0

20

40


0 50 100 150 200

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

MPC-NOMPC-NMOL

minus1





Pitc

h an

gle (

degr

ee)

MPC-NOMPC-NMOL

minus100

minus80

minus60

minus40

minus20

0

20


0 50 100 150 200

0

20

40

60

Sampling instant

Elev

ator

(deg

ree)

MPC-NOMPC-NMOL

minus60

minus40

minus20


6 Conclusions



0

20

0 50 100 150 200

0

10

Sampling instant


minus10

minus20

Mwave (Nm)

Zwave (N)



Competing Interests


Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus06

minus04

minus02

0

02

x1

x1

Estimation of x1(a)


x1Estimation of x1

minus06

minus04

minus02

0

02

x1

(b)




119879 = [[[[[[

1 0 0 00 1 0 00 0 1 00 0 0 1]]]]]] (71)



(72)






(73)

One obtains

119866 = 119875minus1119884 = [07880 1213513007 32259] (74)




x2

x2

Estimation of x2

minus2

minus1

0

1

(a)


x2Estimation of x2

x2

minus2

minus1

0

1

(b)


0 50 100 150

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

NMPCMPC-NMOL

minus1





NMPCMPC-NMOL

Pitc

h an

gle (

degr

ee)

minus120

minus100

minus80

minus60

minus40

minus20

0

20






NMPCMPC-NMOL

Elev

ator

(deg

ree)

minus60

minus40

minus20

0

20

40


0 50 100 150 200

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

MPC-NOMPC-NMOL

minus1





Pitc

h an

gle (

degr

ee)

MPC-NOMPC-NMOL

minus100

minus80

minus60

minus40

minus20

0

20


0 50 100 150 200

0

20

40

60

Sampling instant

Elev

ator

(deg

ree)

MPC-NOMPC-NMOL

minus60

minus40

minus20


6 Conclusions



0

20

0 50 100 150 200

0

10

Sampling instant


minus10

minus20

Mwave (Nm)

Zwave (N)



Competing Interests


Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











x2

x2

Estimation of x2

minus2

minus1

0

1

(a)


x2Estimation of x2

x2

minus2

minus1

0

1

(b)


0 50 100 150

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

NMPCMPC-NMOL

minus1





NMPCMPC-NMOL

Pitc

h an

gle (

degr

ee)

minus120

minus100

minus80

minus60

minus40

minus20

0

20






NMPCMPC-NMOL

Elev

ator

(deg

ree)

minus60

minus40

minus20

0

20

40


0 50 100 150 200

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

MPC-NOMPC-NMOL

minus1





Pitc

h an

gle (

degr

ee)

MPC-NOMPC-NMOL

minus100

minus80

minus60

minus40

minus20

0

20


0 50 100 150 200

0

20

40

60

Sampling instant

Elev

ator

(deg

ree)

MPC-NOMPC-NMOL

minus60

minus40

minus20


6 Conclusions



0

20

0 50 100 150 200

0

10

Sampling instant


minus10

minus20

Mwave (Nm)

Zwave (N)



Competing Interests


Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











NMPCMPC-NMOL

Elev

ator

(deg

ree)

minus60

minus40

minus20

0

20

40


0 50 100 150 200

0

1

2

3

4

5

6

Sampling instant

Dep

th (m

)

MPC-NOMPC-NMOL

minus1





Pitc

h an

gle (

degr

ee)

MPC-NOMPC-NMOL

minus100

minus80

minus60

minus40

minus20

0

20


0 50 100 150 200

0

20

40

60

Sampling instant

Elev

ator

(deg

ree)

MPC-NOMPC-NMOL

minus60

minus40

minus20


6 Conclusions



0

20

0 50 100 150 200

0

10

Sampling instant


minus10

minus20

Mwave (Nm)

Zwave (N)



Competing Interests


Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

20

0 50 100 150 200

0

10

Sampling instant


minus10

minus20

Mwave (Nm)

Zwave (N)



Competing Interests


Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









nonlinear reduced-order observer-based predictive control...

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