# Integrated Fault Estimation and Fault-Tolerant Control for Dynamic Positioning of Ships

Authors: Xiaogong Lin; Heng Li; Anzuo Jiang; Juan Li
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

An integrated fault estimation and fault-tolerant control scheme is developed in this paper for dynamic positioning of ships in the presence of an actuator fault. First, an auxiliary derivative output of dynamic positioning ships is constructed in order to satisfy the so-called observer matching condition, and a high-gain observer is designed to exactly estimate the auxiliary derivative outputs. Then, a fault-tolerant controller is developed for dynamic positioning ships based on the iterative learning observer. By means of Lyapunov–Krasovskii stability theory, it is proved that the proposed fault-tolerant controller is able to estimate the total fault effects and states of ships accurately via the iterative learning observer and also to stabilize the closed-loop system. In addition, the parameter design of the proposed fault-tolerant control system can be conveniently solved in terms of linear matrix inequalities. Finally, simulation studies for dynamic positioning ships with actuator faults are carried out, and the results validate the effectivity of the proposed fault-tolerant control scheme.


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## Body

## 1. Introduction

Dynamic positioning systems (DPS) can only rely on their own propulsion to counteract the interference of the external environment and maintain the ship’s position and heading at the fixed location or along the predetermined track [1]. With the increasing focus on the ocean exploitation, dynamic positioning system has become an attractive research topic and has drawn great attention from control communities during the past two decades, with many results reported in the literature, such as hybrid control [2], fuzzy control [3, 4], backstepping [5, 6], and model predictive control [7]. The highly disturbed marine environment will lead to aging of the components of ships, which causes inevitable malfunction in actuators. Once the actuator faults occur, it is not possible that the faulty thrusters can be repaired or replaced by the backup one in time. Therefore, how to design a fault-tolerant controller (FTC) for dynamic positioning (DP) ships is a critical problem.Recently, to increase the safety and reliability of DPS, many researches on fault-tolerant control for DPS have been carried out. The authors in [5] constructed an iterative learning observer to estimate the fault signal and combined the pseudoinverse method to generate a fault-tolerant controller for the dynamic positioning of ships. Andrea and Tor utilized an unknown input observer (UIO) technique to produce a fault detection and isolation mechanism for an overactuated marine vessel [8]. Fault detection and diagnosis mechanism, based on two techniques: the parity space approach and the Luenberger observer, was proposed to guarantee a fault-tolerant robust control for the dynamic positioning of ships [9]. A fault-tolerant supervisory controller was designed based on the certainty equivalence principle, which could solve both sensor and actuator faults of ships [10]. Other remarkable results on FTC for dynamic positioning of ships can be found in [11, 12]. However, most of the current studies of FTC for DPS are carried out as two separate entities, fault estimation and fault-tolerant control, such that it is difficult to determine whether or not fault estimation satisfies high requirement for the fault-tolerant controller. Moreover, there exists the time delay problem in the fault-tolerant control strategies in [8, 9, 12] due to using the fault detection and the isolation (FDI) module.Motivated by the aforementioned considerations, an integrated fault estimation and fault-tolerant control scheme for DPS with actuator faults are presented in this paper. Comparing with the adaptive law, the updating law in this paper is an algebraic equation, which requires less on-line computing power and is easy to implement in practice. Moreover, the novelty of the approach with respect to existing results consists of the following key points:(1)
In [5, 11], the authors made a strict assumption that the velocities of ships are measurable. In fact, the velocities of ships cannot be deduced from the measured signals through differentiation because such a scheme tends to significantly magnify the noise level in velocities [1]. Instead, in the proposed work, such assumption is not needed.(2)
Unlike in [8, 9, 12]where fault detection and isolation and fault-tolerant control are designed separately, in this paper, the fault estimation and the FTC are simultaneously considered, and their coupling problem can be effectively solved.(3)
This paper is concerned with the fault-tolerant control design problem with a general actuator fault mode. Compared with the supervisory control method in [10], the proposed scheme does not need to know the specific fault mode information which is indispensable for the supervisory fault-tolerant controller.The rest of this paper is organized as follows. Several preliminaries and problem formulation are presented in Section2. Section 3 addresses the fault-tolerant controller for DPS based on the iterative learning observer with theoretical stability analysis of the closed-loop system. In Section 4, we illustrate the effectiveness of the proposed method via simulations on the dynamic positioning of ships. Finally, some concluding remarks are provided in Section 5.Throughout the paper,λminQ and λmaxQ are the minimum and maximum eigenvalues of Q, respectively; Q>0Q<0 denotes that Q is a positive (negative) definite matrix, • represents Euclidean norm of the vector or the matrix, and ∗ means symmetric term.


## 2. Preliminaries and Problem Formulation

### 2.1. Modeling of Ships

For the horizontal motion of a surface vessel, let the Earth-fixed positionx,y and orientation ψ of the vessel relative to an Earth-fixed frame XEYEZE be expressed in vector form by η=xyψT, and let the velocities decomposed in a body-fixed reference frame XYZ be represented by the vector ν=uvrT. These three modes are referred to as the surge, sway, and yaw of a vessel [1]. The Earth-fixed inertial frame and body-fixed frame are depicted by Figure 1.Figure 1
Earth-fixed inertial frame and body-fixed frame.The kinematics and dynamics of ships in 3-DOF can be described as follows:(1a)η˙=Jψν,(1b)Mν˙=τ−Dν+d+τf,where Jψ=cosψ−sinψ0sinψcosψ0001 is the rotation matrix from the body-fixed frame to the Earth-fixed inertial frame; M∈R3∗3 and D∈R3∗3 denote the inertia matrix and damping matrix, respectively; τ,τf represents the control input vector and uncertain fault vector, respectively; and d denotes the time-varying external disturbance caused by winds, waves, and ocean currents.Remark 1.
In practice, most faults are nonlinear functions of the state vector and/or input vector. The formulation given byτf=τfη,ν,τ can capture such practical fault models [13].
By defining the new vessel parallel coordinate position asηp=JTψη, for low-speed assumptions, we can deduce that η˙p≈ν [1]. Then, the linear time-invariant state-space model of ships can be written in the compact form as(2a)x˙=Ax+Bτ+Ef,(2b)y=y1y2y3T=Cx,(2c)yi=Cix,where x=ηTνTT, f=d+τf, A=03∗3I3∗303∗3−M−1D, B=E=03∗3M−1, and C=I0.
The control objective of this paper is to design a fault-tolerant control lawτ for dynamic positioning of ships in the absence of velocity measurements and subject to actuator faults, for the purpose that the DP ships can maintain the desired position ηd.


### 2.2. Assumptions and Lemmas

Throughout this paper, the following assumptions are made.Assumption 1.
For low-speed applications, it can also be assumed thatMT=M>0 and D>0.Remark 2.
Assumption 1 is true if starboard and port symmetries and low speed are assumed [14]. Actually, the nonlinear damping term can be neglected since the linear term dominates at lower speeds. This is a good assumption for dynamic positioning of ships [1].Assumption 2.
The following inequality holdsft−K1ft−T=fdt≤f¯d, where T is a short time delay and we call it as a learning interval later.Definition 1.
(relative degree [15]). The relative degree γi of the systems (1a) and (1b) with respect to the i-th output yi is defined as follows:(3)CiAkE=0,for allk<γi−1,CiAγi−1E≠0.Definition 2.
(Invariant zeros). For a MIMO state-space model, the invariant zeros are the complex values ofs for which the rank of Rosenbrock’s system matrix sI−AEC0 drops from its norm value.Lemma 1 (Schur complement lemma [16]).
SupposeS=S1S2∗S4 is a given symmetric matrix, where S1∈Rm∗m. Then, the following three conditions are equivalent:(4)S<0,S1<0,S4−S2TS1−1S2<0,S4<0,S1−S2S4−1S2T<0.Lemma 2 (Young’s inequality [17]).
LetL,R, and Ft be real matrices of appropriate dimensions with Ft being a matrix function. Then, for any ε>0 and FTtFt≤I, we have(5)LFtR+RTFTtLT≤1εLLT+εRTR.Lemma 3 (UIO existence conditions [18]).
There exist a matrixL, G and a symmetric positive-definite matrix P such that(6)PA+LC+A+LCTP<0,ETP=GC,if and only if the rankCE=rankE and the invariant zeros of A,E,C lie in the open left-hand complex plane.


## 2.1. Modeling of Ships

For the horizontal motion of a surface vessel, let the Earth-fixed positionx,y and orientation ψ of the vessel relative to an Earth-fixed frame XEYEZE be expressed in vector form by η=xyψT, and let the velocities decomposed in a body-fixed reference frame XYZ be represented by the vector ν=uvrT. These three modes are referred to as the surge, sway, and yaw of a vessel [1]. The Earth-fixed inertial frame and body-fixed frame are depicted by Figure 1.Figure 1
Earth-fixed inertial frame and body-fixed frame.The kinematics and dynamics of ships in 3-DOF can be described as follows:(1a)η˙=Jψν,(1b)Mν˙=τ−Dν+d+τf,where Jψ=cosψ−sinψ0sinψcosψ0001 is the rotation matrix from the body-fixed frame to the Earth-fixed inertial frame; M∈R3∗3 and D∈R3∗3 denote the inertia matrix and damping matrix, respectively; τ,τf represents the control input vector and uncertain fault vector, respectively; and d denotes the time-varying external disturbance caused by winds, waves, and ocean currents.Remark 1.
In practice, most faults are nonlinear functions of the state vector and/or input vector. The formulation given byτf=τfη,ν,τ can capture such practical fault models [13].
By defining the new vessel parallel coordinate position asηp=JTψη, for low-speed assumptions, we can deduce that η˙p≈ν [1]. Then, the linear time-invariant state-space model of ships can be written in the compact form as(2a)x˙=Ax+Bτ+Ef,(2b)y=y1y2y3T=Cx,(2c)yi=Cix,where x=ηTνTT, f=d+τf, A=03∗3I3∗303∗3−M−1D, B=E=03∗3M−1, and C=I0.
The control objective of this paper is to design a fault-tolerant control lawτ for dynamic positioning of ships in the absence of velocity measurements and subject to actuator faults, for the purpose that the DP ships can maintain the desired position ηd.


## 2.2. Assumptions and Lemmas

Throughout this paper, the following assumptions are made.Assumption 1.
For low-speed applications, it can also be assumed thatMT=M>0 and D>0.Remark 2.
Assumption 1 is true if starboard and port symmetries and low speed are assumed [14]. Actually, the nonlinear damping term can be neglected since the linear term dominates at lower speeds. This is a good assumption for dynamic positioning of ships [1].Assumption 2.
The following inequality holdsft−K1ft−T=fdt≤f¯d, where T is a short time delay and we call it as a learning interval later.Definition 1.
(relative degree [15]). The relative degree γi of the systems (1a) and (1b) with respect to the i-th output yi is defined as follows:(3)CiAkE=0,for allk<γi−1,CiAγi−1E≠0.Definition 2.
(Invariant zeros). For a MIMO state-space model, the invariant zeros are the complex values ofs for which the rank of Rosenbrock’s system matrix sI−AEC0 drops from its norm value.Lemma 1 (Schur complement lemma [16]).
SupposeS=S1S2∗S4 is a given symmetric matrix, where S1∈Rm∗m. Then, the following three conditions are equivalent:(4)S<0,S1<0,S4−S2TS1−1S2<0,S4<0,S1−S2S4−1S2T<0.Lemma 2 (Young’s inequality [17]).
LetL,R, and Ft be real matrices of appropriate dimensions with Ft being a matrix function. Then, for any ε>0 and FTtFt≤I, we have(5)LFtR+RTFTtLT≤1εLLT+εRTR.Lemma 3 (UIO existence conditions [18]).
There exist a matrixL, G and a symmetric positive-definite matrix P such that(6)PA+LC+A+LCTP<0,ETP=GC,if and only if the rankCE=rankE and the invariant zeros of A,E,C lie in the open left-hand complex plane.


## 3. Main Results

### 3.1. Construction of Auxiliary Derivative Outputs and the High-Gain Observer

In this subsection, the auxiliary derivative outputs (ADOs) are generated using a high-gain observer (HGO) in order to make the new auxiliary system satisfy the UIO existence condition which is defined in Lemma 3. The ADO estimation error is shown to be uniformly ultimately bounded with respect to a ball whose radius is a function of design parameters.After calculation and analysis, we get thatrankCE≠rankE, and the relative degree of the dynamic positioning of ships is 2. Then, by the definition of the relative degree, we can construct the auxiliary derivative output matrix(7)Ca≜C1C1AC2C2AC3C3A=100000000100010000000010001000000001,such that rankCaE=rankE=3. Then, we construct the HGO to obtain accurately the ADOs:(8)ya=Cax=ya11ya12ya21ya22ya31ya32T,just based on the measurable outputs y=Cx. In order to simplify the analysis process and to highlight important concepts, we only consider the i-th pair of ADOs (8) which is given by(9)yai≜yai1yai2.Then, by taking the first-time derivative of the ADOs (9) along x˙=Ax+Bτ+Ef, one can achieve that(10)y˙ai=A¯iyai+B¯iτ+b¯iϕ,yio=c¯iyai,where ϕ=CiAEf+CiA2x, c¯i=10, A¯i=0100, B¯i=03∗1CiAB, and b¯i=01. We construct the HGO for ADOs (10) as follows:(11)y˙hi=A¯iyhi+B¯iτ+l¯ic¯iyai−yhi,yhio=c¯iyhi,where l¯i=χi1/μχi2/μ2T is a design parameter vector with μ>0. According to (10) and (11), the error dynamics can be described as(12)y˜˙ai=−li11−li20y˜ai+b¯iϕ,where y˜ai=yai−yhi, li1=χi1/μ, and li2=χi2/μ2. Define the new states as ζai≔Dμ−1y˜ai, where Dμ=μ001. Then, we have(13)μζ˙ai=Λiζai+μb¯iϕ,where(14)Λi=−χi11−χi20.Λ
i can be stabilized by selecting the appropriate parameters of χij. Using the arguments in [19, 20], the following lemma can be given that we will use in next subsection.Lemma 4.
For the high-gain observer (11), there exist a positive constant β, arbitrarily small positive number μ¯ and a finite time Tμ¯ such that y˜h=yh−ya≤βμ¯ for t≥t0+Tμ¯, where yh=yh1Tyh2Tyh3TT.


### 3.2. Integrated Design of Fault Estimation and FTC via Iterative Learning Observers

It is proved in [15] that the invariant zeros of the triples A,E,C and A,E,Ca are identical. Moreover, the system (2a)–(2c) has no invariant zeros. By Lemma 3, there must exist matrices L,G and a symmetric positive-definite matrix P such that(15)PA−LCa+A−LCaTP<0,GCa=ETP.A fault-tolerant controller based on the iterative learning observer and high-gain observer for dynamic positioning of ships with actuator faults is proposed as follows:(16a)τt=−Kτx^t−f^t,(16b)x^˙t=Ax^t+Bτt+Ef^t+Lyht−Cax^t,(16c)y˙ht=A¯yht+B¯τt+L¯c¯Caxt−yht,(16d)f^t=K1f^t−T+K2yht−Cax^t,where A¯=diagA¯1,A¯2,A¯3, B¯=B¯1,B¯2,B¯3T, L¯=l¯1T,l¯2T,l¯3TT, and c¯=c¯1,c¯2,c¯3; Kτ is the control gain matrix to be designed; f^ is the reconstructed signal of the fault and is used as feed-forward compensation; and K1>0 and K2>0 are the learning rates.Remark 3.
From the formulas (16a)–(16d), it can be observed that the high-gain observer (16c) can be designed independently of the whole fault-tolerant controller, and the function of the high-gain observer is to generate a set of auxiliary signals yh to update the state of the iterative learning observer (16b) and fault estimation (16d). Parameter T in equation (16d) is called the learning interval that can be adjusted to guarantee fault-reconstruction accuracy. It should be selected large if a fault is constant or slow-varying; otherwise, it should be chosen small. Specifically, in sampled-data control systems, it can be taken as the sampling interval or as an integer multiple of the sampling interval. Parameters K1,K2 are the gains of the updating law (16d). Generally speaking, K1>0 should be chosen to be 1 or close to 1 such that fdt=ft−K1ft−T is small enough.Theorem 1.
The dynamics of dynamic positioning ships described in (2a)–(2c) is considered, and Assumptions 1 and 2 are supposed. If the fault-tolerant controller is designed as (16a)–(16d) with the parameters Kτ=N2N1−1 and L=P−1N3, where N1,N2,N3,P are the solutions of the following LMIs:(17)minθ,0<θ,0<N1,0<P,−θIETP−α1K2CaT∗−I≤0,Ω1BN200E∗−2γN1γIn00∗∗Ω21N30∗∗∗−1ε2μ¯0∗∗∗∗−1ε1I<0,−Q+1+ε4α1α1K1TK1+ε3α12μ¯K1TK2K2TK1<0,where μ¯≤λmin−Q1/ε2λmaxPLPLT, Q1=PA−LCa+A−LCaTP, Ω1=AN1−BN2+N1AT−N2TBT, and Ω21=PA−N3Ca+ATP−CaTN3T; then, the state-estimation error, fault-estimation error, and the state vector of the closed-loop system are uniformly ultimately bounded.Proof.
Denotex˜=x−x^ and f˜=f−f^. Consider the Lyapunov–Krasovskii function candidate V for t≥t0+Tμ¯ as(18)V=x˜TPx˜+xTP1x+∫t−Ttf˜TQf˜.
The derivative of the Lyapunov–Krasovskii function (18) with respect to time can be derived as(19)V˙=x˜T2PA−LCax˜+2x˜TPEf˜−2x˜TPLyh−ya−ε0f˜TQf˜+α0f˜TQf˜−f˜Tt−TQf˜t−T+xT2P1A−BKτx+2xTP1BKτx˜+2xTP1Ef˜,where α0=1+ε0,ε0>0. Fault-estimation error can be calculated as follows:(20)f˜=fdt+K1f˜t−T−K2y˜h−K2Cax˜,where y˜h=yh−ya, and fd=ft−K1ft−T. Following Lemma 2, we have(21)2xTP1Ef˜≤ε1xTP1EP1ETx+1ε1f˜Tf˜,where ε1>0. Then, the derivative of the Lyapunov–Krasovskii function (19) can be rewritten as(22)V˙≤x˜T2PA−LCax˜+2x˜TPEf˜−2x˜TPLy˜h−ε0f˜TQf˜+α1f˜Tf˜−f˜Tt−TQf˜t−T+xT2P1A−BKτ+ε1P1EP1ETx+2xTP1BKτx˜≤x˜T2PA−LCax˜+2x˜TPE−α1K2CaTK1f˜t−T+2x˜TPE−α1K2CaTfdt+2x˜Tα12K2CaT−PEK2Cax˜−2x˜TPLy˜h−ε0f˜TQf˜+α1fdTfd+α1f˜Tt−TK1TK1f˜t−T+2α1fdTK1f˜t−T−f˜Tt−TQf˜t−T+xT2P1A−BKτ+ε1P1EP1ETx+2xTP1BKτx˜+α1y˜hTK2TK2y˜h−2α1fdTK2y˜h−2α1f˜Tt−TK1TK2y˜h−2x˜TPE−α1K2CaTK2y˜h,where α1=1+ε0λmaxQ+1/ε1. By Lemma 3, we can always choose K2=1/α1G such that(23)α1K2Ca=ETP.
Then, equation (22) can be rewritten as(24)V˙≤x˜T2PA−LCax˜−ε0f˜TQf˜+2α1fdTK1f˜t−T+α1fdTfd−f˜Tt−TQf˜t−T+α1f˜Tt−TK1TK1f˜t−T+2xTP1BKτx˜+xT2P1A−BKτ+ε1P1EP1ETx−2x˜TPLy˜h+α1y˜2TK2TK2y˜h−2α1fdTK2y˜h−2α1f˜Tt−TK1TK2y˜h−x˜TPEPETα1x˜.
By using Lemma 2 and Lemma 4, we get(25)−2x˜TPLy˜h+α1y˜hTK2TK2y˜h−2α1fdTK2y˜h−2α1f˜Tt−TK1TK2y˜h=−2x˜TPLμ¯1μ¯y˜h+α1y˜hTK2TK2y˜h−2α1fdTK2y˜h−2α1f˜Tt−TK1TK2μ¯1μ¯y˜h≤ε2μ¯x˜TPLPLTx˜+1ε2μ¯β2+α1λmaxK2TK2β2μ¯2+2α1βμ¯K2fd+ε3α12μ¯f˜Tt−TK1TK2K2TK1f˜t−T+1ε3μ¯β2,2α1fdTK1f˜t−T≤ε4α12f˜Tt−TK1TK1f˜t−T+1ε4fd2.
Therefore, the derivative of the Lyapunov–Krasovskii function (24) can be further transformed into the compact form as follows:(26)V˙≤xtx˜f˜T2P1A−BKτ+ε1P1EP1ET︷Π1P1BKτ0∗2PA−LCa+ε2μ¯PLPLT︷f2−1α1PEPET︸Π20∗∗−ε0Q︸−Q0xtx˜f˜+f˜Tt−T−Q+1+ε4α1α1K1TK1+ε3α12μ¯K1TK2K2TK1︷Γf˜t−T+α1+1ε4fd2+1ε2+1ε3+α1λmaxK2TK2μ¯β2μ¯+2α1βμ¯K2fd︸κ2.
If the proper parameters have been chosen such that(27)Γ=−Q+1+ε4α1α1K1TK1+ε3α12μ¯K1TK2K2TK1<0,Q0>0,then (26) can be transformed into V˙≤−κ1X2+κ2, where X=xx˜f˜T, κ1=λminQ0, and κ2=α1+1/ε4f¯d2+1/ε2+1/ε3+α1λmaxK2TK2μ¯β2μ¯+2α1βμ¯K2f¯d. It follows that X2>κ2/κ1 renders V˙<0. It is obviously observed that V is uniformly ultimately bounded. According to the definition of V, the signals of X are bounded. Theorem 1 is thus proved.
Next, we will analyze how to select parameters to satisfy the aforementioned formula. For simplicity of analysis, rewrite−Q0<0:(28)Π1P1BKτ0∗Π20∗∗−ε0Q<0,where(29)Π1=P1A−BKτ+A−BKτTP1+ε1P1EP1ET,Π2=PA−LCa+A−LCaTP+ε2μ¯PLPLT−1α1PEPET.
Furthermore, by using Lemma 1, (28) is equivalent to (30)Π1P1BKτ∗Π2<0.
Postmultiply and premultiply (30) by P1−1, it is obtained that(31)P1−1Π1P1−1BKτP1−1∗P1−1Π2P1−1<0.
Since(32)PEPET=P0M−10M−1TP=000PM−1M−1P≥0,one has(33)P1−1Π2P1−1≤P1−1Ω2P1−1,where(34)Ω2=Π2+1α1PEPET=PA−LCa+A−LCaTP+ε2μ¯PLPLT.
Thus, sufficient conditions for (31) are obtained as follows:(35)P1−1Π1P1−1BKτP1−1∗P1−1Ω2P1−1<0.
From Lemma 3, we conclude thatL can always be chosen such that(36)PA−LCa+A−LCaTP=−Q1<0.
Then, we chooseμ¯≤λmin−Q1/ε2λmaxPLPLT such that Ω2<0. Since P1−1,Ω2 are symmetric matrices, so we obtain(37)P1−1+γΩ2−1TΩ2P1−1+γΩ2−1<0,P1−1Ω2P1−1<−2γP1−1−γ2Ω2−1.
Thus, we obtain the following sufficient condition for (35):(38)P1−1Π1P1−1BKτP1−1∗−2γP1−1−γ2Ω2−1<0.
By using Lemma 1 repeatedly, (38) can be converted into equivalent condition as follows:(39)Ω1BKτP1−100E∗−2γP1−1γIn00∗∗Ω21PL0∗∗∗−1ε2μ¯0∗∗∗∗−1ε1I<0,where Ω1=A−BKτP1−1+P1−1A−BKτT and Ω21=PA−LCa+A−LCaTP.
LetP1−1=N1, KτN1=N2, and PL=N3; then, we can obtain the LMIs in Theorem 1. Besides, using the conclusions in [21], the equation α1K2Ca=ETP can be converted into LMI equivalently as follows:(40)−θIETP−α1K2CaT∗−I≤0,where θ is a small positive scalar.


## 3.1. Construction of Auxiliary Derivative Outputs and the High-Gain Observer

In this subsection, the auxiliary derivative outputs (ADOs) are generated using a high-gain observer (HGO) in order to make the new auxiliary system satisfy the UIO existence condition which is defined in Lemma 3. The ADO estimation error is shown to be uniformly ultimately bounded with respect to a ball whose radius is a function of design parameters.After calculation and analysis, we get thatrankCE≠rankE, and the relative degree of the dynamic positioning of ships is 2. Then, by the definition of the relative degree, we can construct the auxiliary derivative output matrix(7)Ca≜C1C1AC2C2AC3C3A=100000000100010000000010001000000001,such that rankCaE=rankE=3. Then, we construct the HGO to obtain accurately the ADOs:(8)ya=Cax=ya11ya12ya21ya22ya31ya32T,just based on the measurable outputs y=Cx. In order to simplify the analysis process and to highlight important concepts, we only consider the i-th pair of ADOs (8) which is given by(9)yai≜yai1yai2.Then, by taking the first-time derivative of the ADOs (9) along x˙=Ax+Bτ+Ef, one can achieve that(10)y˙ai=A¯iyai+B¯iτ+b¯iϕ,yio=c¯iyai,where ϕ=CiAEf+CiA2x, c¯i=10, A¯i=0100, B¯i=03∗1CiAB, and b¯i=01. We construct the HGO for ADOs (10) as follows:(11)y˙hi=A¯iyhi+B¯iτ+l¯ic¯iyai−yhi,yhio=c¯iyhi,where l¯i=χi1/μχi2/μ2T is a design parameter vector with μ>0. According to (10) and (11), the error dynamics can be described as(12)y˜˙ai=−li11−li20y˜ai+b¯iϕ,where y˜ai=yai−yhi, li1=χi1/μ, and li2=χi2/μ2. Define the new states as ζai≔Dμ−1y˜ai, where Dμ=μ001. Then, we have(13)μζ˙ai=Λiζai+μb¯iϕ,where(14)Λi=−χi11−χi20.Λ
i can be stabilized by selecting the appropriate parameters of χij. Using the arguments in [19, 20], the following lemma can be given that we will use in next subsection.Lemma 4.
For the high-gain observer (11), there exist a positive constant β, arbitrarily small positive number μ¯ and a finite time Tμ¯ such that y˜h=yh−ya≤βμ¯ for t≥t0+Tμ¯, where yh=yh1Tyh2Tyh3TT.


## 3.2. Integrated Design of Fault Estimation and FTC via Iterative Learning Observers

It is proved in [15] that the invariant zeros of the triples A,E,C and A,E,Ca are identical. Moreover, the system (2a)–(2c) has no invariant zeros. By Lemma 3, there must exist matrices L,G and a symmetric positive-definite matrix P such that(15)PA−LCa+A−LCaTP<0,GCa=ETP.A fault-tolerant controller based on the iterative learning observer and high-gain observer for dynamic positioning of ships with actuator faults is proposed as follows:(16a)τt=−Kτx^t−f^t,(16b)x^˙t=Ax^t+Bτt+Ef^t+Lyht−Cax^t,(16c)y˙ht=A¯yht+B¯τt+L¯c¯Caxt−yht,(16d)f^t=K1f^t−T+K2yht−Cax^t,where A¯=diagA¯1,A¯2,A¯3, B¯=B¯1,B¯2,B¯3T, L¯=l¯1T,l¯2T,l¯3TT, and c¯=c¯1,c¯2,c¯3; Kτ is the control gain matrix to be designed; f^ is the reconstructed signal of the fault and is used as feed-forward compensation; and K1>0 and K2>0 are the learning rates.Remark 3.
From the formulas (16a)–(16d), it can be observed that the high-gain observer (16c) can be designed independently of the whole fault-tolerant controller, and the function of the high-gain observer is to generate a set of auxiliary signals yh to update the state of the iterative learning observer (16b) and fault estimation (16d). Parameter T in equation (16d) is called the learning interval that can be adjusted to guarantee fault-reconstruction accuracy. It should be selected large if a fault is constant or slow-varying; otherwise, it should be chosen small. Specifically, in sampled-data control systems, it can be taken as the sampling interval or as an integer multiple of the sampling interval. Parameters K1,K2 are the gains of the updating law (16d). Generally speaking, K1>0 should be chosen to be 1 or close to 1 such that fdt=ft−K1ft−T is small enough.Theorem 1.
The dynamics of dynamic positioning ships described in (2a)–(2c) is considered, and Assumptions 1 and 2 are supposed. If the fault-tolerant controller is designed as (16a)–(16d) with the parameters Kτ=N2N1−1 and L=P−1N3, where N1,N2,N3,P are the solutions of the following LMIs:(17)minθ,0<θ,0<N1,0<P,−θIETP−α1K2CaT∗−I≤0,Ω1BN200E∗−2γN1γIn00∗∗Ω21N30∗∗∗−1ε2μ¯0∗∗∗∗−1ε1I<0,−Q+1+ε4α1α1K1TK1+ε3α12μ¯K1TK2K2TK1<0,where μ¯≤λmin−Q1/ε2λmaxPLPLT, Q1=PA−LCa+A−LCaTP, Ω1=AN1−BN2+N1AT−N2TBT, and Ω21=PA−N3Ca+ATP−CaTN3T; then, the state-estimation error, fault-estimation error, and the state vector of the closed-loop system are uniformly ultimately bounded.Proof.
Denotex˜=x−x^ and f˜=f−f^. Consider the Lyapunov–Krasovskii function candidate V for t≥t0+Tμ¯ as(18)V=x˜TPx˜+xTP1x+∫t−Ttf˜TQf˜.
The derivative of the Lyapunov–Krasovskii function (18) with respect to time can be derived as(19)V˙=x˜T2PA−LCax˜+2x˜TPEf˜−2x˜TPLyh−ya−ε0f˜TQf˜+α0f˜TQf˜−f˜Tt−TQf˜t−T+xT2P1A−BKτx+2xTP1BKτx˜+2xTP1Ef˜,where α0=1+ε0,ε0>0. Fault-estimation error can be calculated as follows:(20)f˜=fdt+K1f˜t−T−K2y˜h−K2Cax˜,where y˜h=yh−ya, and fd=ft−K1ft−T. Following Lemma 2, we have(21)2xTP1Ef˜≤ε1xTP1EP1ETx+1ε1f˜Tf˜,where ε1>0. Then, the derivative of the Lyapunov–Krasovskii function (19) can be rewritten as(22)V˙≤x˜T2PA−LCax˜+2x˜TPEf˜−2x˜TPLy˜h−ε0f˜TQf˜+α1f˜Tf˜−f˜Tt−TQf˜t−T+xT2P1A−BKτ+ε1P1EP1ETx+2xTP1BKτx˜≤x˜T2PA−LCax˜+2x˜TPE−α1K2CaTK1f˜t−T+2x˜TPE−α1K2CaTfdt+2x˜Tα12K2CaT−PEK2Cax˜−2x˜TPLy˜h−ε0f˜TQf˜+α1fdTfd+α1f˜Tt−TK1TK1f˜t−T+2α1fdTK1f˜t−T−f˜Tt−TQf˜t−T+xT2P1A−BKτ+ε1P1EP1ETx+2xTP1BKτx˜+α1y˜hTK2TK2y˜h−2α1fdTK2y˜h−2α1f˜Tt−TK1TK2y˜h−2x˜TPE−α1K2CaTK2y˜h,where α1=1+ε0λmaxQ+1/ε1. By Lemma 3, we can always choose K2=1/α1G such that(23)α1K2Ca=ETP.
Then, equation (22) can be rewritten as(24)V˙≤x˜T2PA−LCax˜−ε0f˜TQf˜+2α1fdTK1f˜t−T+α1fdTfd−f˜Tt−TQf˜t−T+α1f˜Tt−TK1TK1f˜t−T+2xTP1BKτx˜+xT2P1A−BKτ+ε1P1EP1ETx−2x˜TPLy˜h+α1y˜2TK2TK2y˜h−2α1fdTK2y˜h−2α1f˜Tt−TK1TK2y˜h−x˜TPEPETα1x˜.
By using Lemma 2 and Lemma 4, we get(25)−2x˜TPLy˜h+α1y˜hTK2TK2y˜h−2α1fdTK2y˜h−2α1f˜Tt−TK1TK2y˜h=−2x˜TPLμ¯1μ¯y˜h+α1y˜hTK2TK2y˜h−2α1fdTK2y˜h−2α1f˜Tt−TK1TK2μ¯1μ¯y˜h≤ε2μ¯x˜TPLPLTx˜+1ε2μ¯β2+α1λmaxK2TK2β2μ¯2+2α1βμ¯K2fd+ε3α12μ¯f˜Tt−TK1TK2K2TK1f˜t−T+1ε3μ¯β2,2α1fdTK1f˜t−T≤ε4α12f˜Tt−TK1TK1f˜t−T+1ε4fd2.
Therefore, the derivative of the Lyapunov–Krasovskii function (24) can be further transformed into the compact form as follows:(26)V˙≤xtx˜f˜T2P1A−BKτ+ε1P1EP1ET︷Π1P1BKτ0∗2PA−LCa+ε2μ¯PLPLT︷f2−1α1PEPET︸Π20∗∗−ε0Q︸−Q0xtx˜f˜+f˜Tt−T−Q+1+ε4α1α1K1TK1+ε3α12μ¯K1TK2K2TK1︷Γf˜t−T+α1+1ε4fd2+1ε2+1ε3+α1λmaxK2TK2μ¯β2μ¯+2α1βμ¯K2fd︸κ2.
If the proper parameters have been chosen such that(27)Γ=−Q+1+ε4α1α1K1TK1+ε3α12μ¯K1TK2K2TK1<0,Q0>0,then (26) can be transformed into V˙≤−κ1X2+κ2, where X=xx˜f˜T, κ1=λminQ0, and κ2=α1+1/ε4f¯d2+1/ε2+1/ε3+α1λmaxK2TK2μ¯β2μ¯+2α1βμ¯K2f¯d. It follows that X2>κ2/κ1 renders V˙<0. It is obviously observed that V is uniformly ultimately bounded. According to the definition of V, the signals of X are bounded. Theorem 1 is thus proved.
Next, we will analyze how to select parameters to satisfy the aforementioned formula. For simplicity of analysis, rewrite−Q0<0:(28)Π1P1BKτ0∗Π20∗∗−ε0Q<0,where(29)Π1=P1A−BKτ+A−BKτTP1+ε1P1EP1ET,Π2=PA−LCa+A−LCaTP+ε2μ¯PLPLT−1α1PEPET.
Furthermore, by using Lemma 1, (28) is equivalent to (30)Π1P1BKτ∗Π2<0.
Postmultiply and premultiply (30) by P1−1, it is obtained that(31)P1−1Π1P1−1BKτP1−1∗P1−1Π2P1−1<0.
Since(32)PEPET=P0M−10M−1TP=000PM−1M−1P≥0,one has(33)P1−1Π2P1−1≤P1−1Ω2P1−1,where(34)Ω2=Π2+1α1PEPET=PA−LCa+A−LCaTP+ε2μ¯PLPLT.
Thus, sufficient conditions for (31) are obtained as follows:(35)P1−1Π1P1−1BKτP1−1∗P1−1Ω2P1−1<0.
From Lemma 3, we conclude thatL can always be chosen such that(36)PA−LCa+A−LCaTP=−Q1<0.
Then, we chooseμ¯≤λmin−Q1/ε2λmaxPLPLT such that Ω2<0. Since P1−1,Ω2 are symmetric matrices, so we obtain(37)P1−1+γΩ2−1TΩ2P1−1+γΩ2−1<0,P1−1Ω2P1−1<−2γP1−1−γ2Ω2−1.
Thus, we obtain the following sufficient condition for (35):(38)P1−1Π1P1−1BKτP1−1∗−2γP1−1−γ2Ω2−1<0.
By using Lemma 1 repeatedly, (38) can be converted into equivalent condition as follows:(39)Ω1BKτP1−100E∗−2γP1−1γIn00∗∗Ω21PL0∗∗∗−1ε2μ¯0∗∗∗∗−1ε1I<0,where Ω1=A−BKτP1−1+P1−1A−BKτT and Ω21=PA−LCa+A−LCaTP.
LetP1−1=N1, KτN1=N2, and PL=N3; then, we can obtain the LMIs in Theorem 1. Besides, using the conclusions in [21], the equation α1K2Ca=ETP can be converted into LMI equivalently as follows:(40)−θIETP−α1K2CaT∗−I≤0,where θ is a small positive scalar.


## 4. Simulation Results

In this section, a simulation based on the MATLAB-Simulink platform is presented to confirm the effectiveness of the proposed fault-tolerant control scheme. We consider a DP ship equipped with three azimuth thrustersT1,T2 and T3 and two transverse tunnel thrusters T4 and T5. A schematic diagram of the actuator distribution is shown in Figure 2.Figure 2
Schematic diagram of actuator distribution.For the DP ships, the actual actuator force is related to the control input through the equationτ=Gu with(41)G=⋯cosαi⋯⋯sinαi⋯⋯lxisinαi−lyicosαi⋯,where αi,i=1,…,m denotes the angle between the force of the i-th actuator and surge direction, lxi,lyi is the location of the i-th actuator in the body-fixed frame, and u∈Rm×1 is the actual actuator commands. However, the orientation angles of the thrusters T1,T2 and T3 are time-varying, which increases the complexity of the control design. To reduce the complexity of the simulation, the thruster commands are decomposed into the surge force and sway force [8, 11]. Then, the configuration matrix G can be transformed into the linear form. According to [8], the matrix G is chosen as(42)G=1010100001010111−5.91−19.15.91−19.1018.53035,and the numerical values for the involved matrices are(43)M=107∗0.0275210000.076348−0.0738030−0.0738036.690963,D=106∗0.0000250000.0098650.00137500.0007112.813355.The expected position and heading vectors are taken asηd=0m0m0radT, and the initial states are taken as ηt0=10m10m1radT. The parameters of the fault-tolerant control system are selected as μ¯=10−5, l¯i=50500T, ∀i∈1,2,3, ε0=0.1, ε1=100, ε2=2, ε3=10−6, ε4=0.1, and γ=1. Furthermore, the gain matrices can be calculated by MATLAB LMI toolbox as follows:(44)θ=4.4860×10−9,K1=0.90000.90000.9,K2=−0.00040090.3333000−0.0020.000032.91320.36300−0.000−0.00000.36300.3756,Kτ=108∗0.0033000.00550000.0091−0.008800.0153−0.01480−0.00700.80870−0.01241.3454,L=0.500000.50000000.5000−0.000000.5000−0.00000−0.0000.50000−0.0000.50000.5000000.49990000.5000−0.000000.4870−0.02150−0.0000.50000−0.02150.4575.Two fault scenarios of all thrusters are summarized in Figures3 and 4, in which Case 1 and Case 2 represent the single faults and multiple faults, respectively. In addition, the UIO-based control allocation (CA) method in [8] is taken to show the advantages of the proposed scheme.Figure 3
Fault scenarios of thrusters (Case 1).Figure 4
Fault scenarios of thrusters (Case 2).Simulation results are depicted in Figures5–8. It is shown from Figure 5 that the proposed scheme and the UIO-based CA method both can force the ship to the desired target position and heading in Case 1. Due to the time delay effect caused by FDI, the CA method will deviate slightly when the actuator faults occur, but it can still ensure the stability of the system in Case 1. Furthermore, since the fault signal is estimated and compensated by feedforward in the proposed scheme, the deviations from the equilibrium point are also much smaller than those in the CA method. Also, it can be observed from Figure 7 that the velocity estimation errors can converge to a sufficiently small neighborhood of zero. It is worth noting that the proposed approach can be applied to systems with multiple faults (i.e., Case 2). To verify this point, simulation results on the case of multiple actuator faults are given in Figure 6, demonstrating that the proposed scheme is also effective for the cases of multiple actuator faults. As a comparative method, the CA method is not able to control the system well in Case 2, and we can see that the states of the system have a large deviation in around 110 seconds and 350 seconds whenever multiple faults occur simultaneously.Figure 5
Position and heading of dynamic positioning ships (Case 1).Figure 6
Position and heading of dynamic positioning ships (Case 2).Figure 7
Velocity estimation and actual velocity (Case 1).Figure 8
Velocity estimation and actual velocity (Case 2).In order to verify the effectiveness of the proposed fault reconstruction method in this paper, we will compare it with the adaptive fault observer (AFO) in reference [22]. First, we consider constant fault situations for actuators in dynamic positioning ships, which are created as(45)ft=0,0,0T,0≤t≤100,105,105,4×105T,otherwise.Simulation results for the constant actuator faults are depicted in Figure9. Furthermore, it is assumed that a time-varying fault occurs in the actuator, i.e.,(46)ft=0,0,0T,0≤t≤100,105∗sin0.02tcos20.02t105∗cos0.02tsin20.02t4×105sin0.02tcos20.02t,otherwise.Figure 9
Reconstructed constant fault.Simulation results for the time-varying faults are depicted in Figure10.Figure 10
Reconstructed time-varying fault.It can be observed from the aforementioned simulation results that for a constant fault in Figure9, the convergence of the fault-estimation error can be achieved using the two methods, but the proposed scheme can improve the rapidity of fault reconstruction evidently. As for the time-varying fault in Figure 10, the estimation accuracy of the proposed scheme is significantly better than that of the AFO method. This is because the estimation performance of AFO depends on the first time derivative of the fault signal. By the way, it is worth noting that from Figure 9, although the boundedness of errors can be only guaranteed of the proposed scheme, the upper bound can be made small enough by choosing appropriate parameters.In practical application, the marine environment is complex and the disturbance forces caused by wind, wave, and current cannot be ignored. A frequently used disturbance model for marine control applications is the first-order Markov process(47)d˙=−Ad−1d+Bdωd,where d∈R3 is the vector representing the changing disturbance forces and moment, ωd∈R3 is the vector of the Gaussian white noise with zero mean value, and Ad and Bd are the diagonal matrix of positive time constants and the diagonal matrix scaling the amplitude of Gaussian white noise. For the simulation purposes, the parameters are selected as Ad=diag3000,3000,3000 and Bd=diag1000,1000,1000. Figure 11 displays the total fault estimation when the aforementioned disturbance (47) is taken into account and shows that the proposed method, compared with the AFO method, provides much better performances. Therefore, it is concluded that the proposed scheme is effective in fault reconstruction and fault-tolerant control for the dynamic positioning of ships.Figure 11
Reconstructed time-varying fault and disturbance.


## 5. Conclusion

The paper presented a fault-tolerant control method for dynamic positioning of ships with actuator faults based on iterative learning observers and high-gain observers. The proposed fault-tolerant control scheme does not need the fault detection and isolation mechanism which may cause time delay problem. It has been proved that, by constructing the auxiliary derivative outputs, the UIO existence condition is guaranteed for dynamic positioning ships. Simulation results demonstrate the effectiveness of the proposed scheme in fault-tolerant control and fault reconstruction. It is worth noting that the proposed approach can be applied to systems with multiple faults. In the future, we will consider how to isolate faults from disturbance signals. Furthermore, we will study fault-tolerant control methods for dynamic positioning ships with nonlinear dynamics and other constraints.



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