A nonlinear fault-tolerant thruster allocation architecture for underwater remotely operated vehicles

Transcription

1 Preprints, th IFAC Conference on Control Applications in Marine Systems September 3-6, 26. Trondheim, Norway A nonlinear fault-tolerant thruster allocation architecture for underwater remotely operated vehicles Maria Letizia Corradini Andrea Cristofaro School of Science and Technology Mathematics Division, University of Camerino, 6232 (MC, Italy Abstract: This paper proposes an actuator failure tolerant robust control scheme for underwater Remotely Operated Vehicles (ROVs. A reduced order observer has been introduced for estimating the ROV velocities and a sliding mode control law has been developed using the available position measurements and the velocity estimates provided by the observer to achieve output regulation. A thruster failure is shown to be detectable simply checking the presence of any deviation of the observed sliding surfaces. Moreover, an isolation policy for the failed thruster is proposed. Finally, control reconfiguration is performed exploiting the inherent redundancy of actuators with the implementation of an adaptive input allocation architecture. An extensive simulation study has been performed, supporting the effectiveness of the proposed approach.. INTRODUCTION Unmanned Underwater Vehicles (UUVs are a costeffective solution for performing complex tasks in the underwater environment without risking human life, such as environmental data gathering, transportation of assembling modules for submarine installations, inspection of underwater structures.with increasing mission durations in complex marine applications, one of the primary concerns is the failure occurrence on the actuators (Fossen, 994; Sarkar et al., 22. When actuator failures occur and result in abnormal operations, the only present solution is to abort the mission, and use a damage control to make UUVs surface (Isermann and Ballé, 997. Therefore, the problem of reliability and security of UUVs, especially their ability of actuator fault tolerance, has become a major concern. It is desirable to incorporate a function of actuator fault detection and isolation into the control system, so that it is possible to detect and identify actuator fault and/or failures and estimate their severity, in order to design optimal compensation laws. This paper presents a fault-tolerant control allocation scheme in the framework of underwater robotics, specifically addressing an underwater Remotely Operated Vehicle (ROV (Longhi and Rossolini, 989 used by SNAMprogetti (Fano, Italy in the exploitation of combustible gas deposits at great water depths. The vehicle is equipped with four thrusters, controlling its position and orientation in planes parallel to the sea surface, and is connected with the surface vessel by a supporting cable which controls the vehicle depth and provides power and communication facilities. The control system is composed of two independent parts: the first part, placed on the surface vessel, monitors the vehicle depth and it typically involves a human operator, while the second part is automated and controls the position and orientation of the vehicle in the dive plane. In this paper the attention has been focused on this second part of the control system: the ROV is supposed to move on a plane, with three degrees of freedom. Due to this assumption, the thrusters config- Authors are listed in alphabetical order. Corresponding author: andrea.cristofaro@unicam.it uration is redundant. A typical efficient way to exploit such redundancy is the implementation of a control allocation architecture (Johansen and Fossen, 23, this being a paradigm capable of easily handling constraints and optimization requirements. The actuator failure diagnosis scheme presented in the paper is composed by the usual modules performing detection and isolation of faults (Gertler, 22. A reduced order observer has been specifically designed for the ROV exploiting the features of the underwater vehicle, permitting to estimate the underwater vehicle velocities, whose direct measurements are typically difficult to be gathered and poorly reliable. These velocity estimations and the available position measurements, have been then used for developing a robust sliding mode control law (Utkin, 992, which is able to solve the regulation problem for the ROV positions, with respect to the reference ones. The developed sliding surfaces have been used both for designing a robust ROV control algorithm ensuring plant regulation, and for detecting the thruster failures. Failure detection is here performed simply checking the presence of any deviation of the observed sliding surfaces, which can be due only to the occurrence of a thruster failure. Exploiting the ROV structure, faulty thruster can be successfully isolated. Once the failed actuator has been identified, control reconfiguration is performed using the redundant healthy actuators. While in previous works, such as (Corradini and Orlando, 24, a supervisor was supposed to be in charge of applying and monitoring the control reconfiguration, the main contribution of the present paper is to present an automated reconfiguration method based on adaptive control allocation. The proposed scheme takes inspiration from well established results on fault-tolerant control allocation (Cristofaro and Johansen, 24b; Cristofaro et al., 25; Tohidi et al., 26 but incorporates also some special features of the particular ROV model. 2. MATHEMATICAL MODEL OF THE ROV 2. ROV nonlinear model The equations describing the ROV dynamics have been obtained from classical mechanics (Longhi and Rossolini, Copyright 26 IFAC 285

2 26 IFAC CAMS Sept 3-6, 26. Trondheim, Norway 989, (Conter et al., 989, (Corradini and Orlando, 997. The ROV considered as a rigid body can be fully described with six degrees of freedom, corresponding to the position and orientation with respect to a given coordinate system. Let us consider the inertial frame R (, x, y, z and the body reference frame R a ( a, x a, y a, z a (Conter et al., 989 shown in Fig.. The ROV position with respect to R is expressed by the origin of the system while its orientation by the roll, pitch, and yaw angles ψ, θ, and φ, respectively. Being the depth controlled by the surface vessel, the ROV is considered to operate on surfaces parallel to the x y plane. Accordingly the controllable variables are x, y, and the yaw angle φ. It should be noticed that the roll and pitch angles ψ and θ will not be considered in the dynamic model: their amplitude, in fact, has been proved to be negligible in a wide range of load conditions, and with different intensities and directions of the underwater current as well (Longhi and Rossolini, 989, (Conter et al., 989. Therefore, the ROV model is described by the following system of differential equations (Longhi and Rossolini, 989, (Conter et al., 989, (Corradini and Orlando, 997, (Corradini et al., 2 : p ẍ + (p 2 cos (φ + p 3 sin (φ V x V + p 4 x p 5 V cx V c = T x p ÿ + (p 2 sin (φ + p 3 cos (φ V y V + p 4 y p 5 V cy V c = T ( y p 6 φ + p7 φ φ ( + p 8 V c 2 φ φc sin + p 9 = M z 2 Fig.. ROV operational Fig.. ROV operational configuration. The ROV position with respect to R is expressed by the origin of the system while its orientation by the roll, pitch, and yaw angles,, and, respectively. Being the depth controlled by the surface vessel, the ROV is considered to operate on surfaces parallel to the x y plane. Accordingly the controllable variables are x, y, and the yaw angle. It should be noticed that the roll and pitch angles and will area of rotation, r is the equivalent arm of action, d i (i =, 2, 3 are the vehicle dimensions along the x a, y a and z a axes, respectively, and φ c is the angle between the x axis and the velocity direction of the current. This model is in agreement with models usually proposed in literature for underwater ROV s moving in the dive plane (Fossen, 994. The quantities T x, T y and M z appearing in ( are the decomposition of the thrust and the torque provided by the four vehicle thusters along the axes of R: B. State Space ROV model Define the vectors z p =[z z 2 z 3 ] T = x y 286 z v =[z 4 z 5 z 6 ] T = ẋ ẏ T, the state vector: z = z T p T x = cos (φ (τ + τ 2 + τ 3 + τ 4 cos (α sin (φ ( τ τ 2 + τ 3 + τ 4 sin (α T y = sin (φ (τ + τ 2 + τ 3 + τ 4 cos (α + cos (φ ( τ τ 2 + τ 3 + τ 4 sin (α M z = ( τ + τ 2 τ 3 + τ 4 d a with α = π 4, d a = (d x sin (α + d y cos (α (see Fig State Space ROV model z T T T v = z z 2 z 3 z 4 z 5 z 6 T, (2 Define the vectors z p = [z z 2 z 3 ] T = [ x y φ ] T, z v = [z 4 z 5 z 6 ] T = [ ẋ ẏ φ ] T, the state vector: z = [ z T ] T p = [ z z 2 z 3 z 4 z 5 z 6 ] T and introduce the input vector u = [ u u 2 u 3 ] T = [ T x T y M z ] T. Moreover, since model parameters and submarine current are not exactly known, bounded uncertainties are taken into account as follows: p i = ˆp i + p i, where V c = [V cx V cy ] T is the submarine current velocity, V = [V x V y ] T = [(ẋ V cx (ẏ V cy ] T, and the expressions of coefficients p i (i =,..., 9 are reported in V cy = ˆV cy + V cy, V cy ρ Vcy, being ˆp i, ˆVcx, ˆVcy the p i ρ pi, i =,..., 9, V cx = ˆV cx + V cx, V cx ρ Vcx, (Corradini and Orlando, 997, (Corradini et al., 2, nominal values of the parameter and of the submarine current components, respectively, and p i, V cx, V cy where M is the vehicle mass, m is the addition mass, I z is the vehicle inertia moment around the z axis, i z is the the corresponding uncertainties, bounded by ρ pi, ρ Vcx, addition inertia moment, M c is the resistance moment of ρ Vcy, respectively. Considering the above definitions and the cable, L is the cable length, T v the vehicle weight in equations (, the following state space model is obtained: the water, W the weighor length unit of the cable, ρ w ż(t = f(z(t + f(z(t, u(t + gu(t (3 the water density, C dc is the drag coefficient of the cable, and D c is the cable diameter, C di is the drag coefficient where of the i th side wall (i =, 2, C ri the packing coefficient z 4 (depending on the geometrical characteristics of the i th z 5 side wall (i =, 2, S i is the area of the i th side wall (i =, 2, C d is the drag coefficient of rotation, C r is z 6 f(z = f the packing us considercoefficient the inertial frame of rotation, R (,x,y,z S and is the body equivalent side wall (i =, 2, S i is the area 4 (z 3 z 4 N z + ϕ 4 (z of the i th side wall f 5 (z 3 z 5 N z + ϕ 5 (z 2 reference frame R a ( a,x a,y a,z a [7] shown in Fig.. (i =, 2, C d is the drag coefficient of rotation, C r is the packing coefficient of rotation, S is the equivalent ˆp 7 (4 z 6 z 6 area + ϕof 6 (z 3 rotation, r is the equivalent arm of action, ˆp 6 d i (i =, 2, 3 are the vehicle dimensions along the x a 3 3, y a and z a axes, [ ] { respectively, and c isg the = angle between the x axis and the diag, ˆp, ˆp } 3 3 =: g velocity direction of the current. This model is in agreement ˆp 6 with models usually proposed in literature for underwater ROV s moving beingin the dive plane []. The quantities T x, T y and M z appearing in ( are the decomposition of the thrus 4 (z and 3 the = ˆp torque (ˆp provided 2 c 3 + by ˆp 3 the s 3 four vehicle propellers along the axes of R: 8 f 5 (z 3 = ˆp (ˆp 2 s 3 + ˆp 3 c 3 T x = cos ( (T + T 2 + T 3 + T 4 cos ( >< sin ( ( T T 2 + T 3 + T 4 sin( T y =sin( (T + ϕt 42 (z+ T 3 = ˆp + T 4 ( ˆp cos ( + 4 z + ˆp 5 (2 ˆVcx ˆV c cos ( ( T T 2 + T 3 + T 4 sin( >: M z =( T + T 2 T 3 + T 4 d a ϕ 5 (z 2 = ˆp ( ˆp 4 z 2 + ˆp 5 ˆVcy ˆV c with = 4, d a =(d x sin ( +d y cos ( (see Fig. (. + ˆp 9 z T v ϕ 6 (z 3 = ˆp 6 (ˆp 8 ˆV c 2 sin z 3 ˆφ c 2

3 26 IFAC CAMS Sept 3-6, 26. Trondheim, Norway with z 4 = z 4 ˆV cx, z 5 = z 5 ˆV cy, N z = z z2 5, c 3 = cos(z 3, s 3 = sin(z 3, ˆV c = ˆV 2 cx + ˆV cy, 2 ˆφc = arctan ( ˆVcy ˆV cx, and the term f(z, u = [,,, 4 (z, z 3, z 4, T x, 5 (z 2, z 3, z 5, T y, 6 (z 3, z 6, M z ] T can be easily computed considering the uncertainties p i (i =... 9, V cx, V cy. 3. LOW-LEVEL ROBUST CONTROL DESIGN 3. Robust Design of the Reduced Order Observer In this section, a reduced order observer, specifically designed for the ROV, will be proposed. In fact, while position measurements are usually available and are in general sufficiently reliable, velocity measurements are either difficult to be gathered and poorly reliable. The observer design will be carried out exploiting the following features of the ROV: in the model (3, (4, positive scalars γ 4, γ 24, γ 5, γ 25 can be found such that: < γ 4 f 4 (z 3 γ 24, < γ 5 f 5 (z 3 γ 25 ; there is a maximal (known velocity z4 max, z5 max, z6 max achievable by the vehicle, as a consequence of the existence of a maximal power supply by thrusters. Therefore it exists a bound M = (z4 max V cx 2 + (z5 max V cy 2 such that N z M. as a consequence of the existence of a maximal power supply by thrusters, control variables T x, T y, M z are bounded, too, and bounds can be easily computed also for the uncertain terms. Therefore, it can be assumed that: 4 (z, z 3, z 4, T x ρ 4 ; 5 (z 2, z 3, z 5, T y ρ 5 ; 6 (z 3, z 6, M z ρ 6. Define ξ = [ξ ξ 2 ξ 3 ] T, and consider the following reduced state observer: ξ = f 4 (z 3 ˆN z ξ + ϕ 4 (z + u + v ξ 2 = f 5 (z 3 ˆN z ξ2 + ϕ 5 (z 2 + u 2 + v 2 ξ 3 = α 6 Mξ3 + ϕ 6 (z 3 + u 3 + v 3 (5 with ξ = ξ V cx, ξ2 = ξ 2 V cy, α 6 = p 7 /p 6, M = sup{ z6 max, M}, ˆNz = (ξ V cx 2 + (ξ 2 V cy 2 and v, v 2, v 3 are auxiliary parameters to be specified. A robust control law, coupled with the above observer, will be here presented aimed at solving the regulation problem for the variables z, z 2, z 3 with respect to reference variable z d = [z d z 2d z 3d ] T. Define the following sliding surface: s = [ s s 2 s 3 ] T = (ξ ż d + Λ(z p z d =. (6 with Λ = diag{λ i }, λ i >, i =,..., 3, and being ɛ = z p z d the tracking error. Lemma. Consider the uncertain ROV model (3. The control law u = u eq + u n, with: u eq = g f 4(z 3 ξ ˆNz ϕ 4 (z λ (z 4 ż d + z d v f 5 (z 3 ξ 2 ˆNz ϕ 5 (z 2 λ 2 (z 5 ż 2d + z 2d v 2 α 6 Mξ3 ϕ 6 (z 3 λ 3 (z 6 ż 3d + z 3d v 3 [ ] (ρ4 + ϖsign( s u n = g (ρ 5 + ϖsign( s 2, ϖ > (7 (ρ 6 + ϖsign( s 3 guarantees the asymptotical achievement of a sliding motion on (6 with a finite reaching phase. Proof. The achievement of a sliding motion on (6 is guaranteed by the following condition: ( s T s = s ξ z d + λ (z 4 ż d + ( ( s 2 ξ2 z 2d + λ 2 (z 5 ż 2d + s 3 ξ3 z 3d + λ 3 (z 6 ż 3d < (8 which can ( be fulfilled imposing separately three inequalities, i.e. s i ξi z id + λ i (z 3+i ż id <, i =, 2, 3. The first inequality gives, e.g.: s ( f 4 (z 3 ˆN z ξ + ϕ 4 (z + v z d + λ (z 4 ż d < and one gets immediately the controller (7. In particular, selecting ϖ arbitrarily small but positive guarantees a finite reaching phase. Define the observation error as e = [e e 2 e 3 ] T = z v ξ. The following result can be given omitting the proof for brevity: Corollary 2. Consider the uncertain plant model (3 driven by the control law (7. The reduced order observer (5 ensures the robust asymptotical vanishing both of the observation error and of the tracking error designing v as follows: [ [f4 (z 3 M( z max ] 4 + ξ + ρ 4 ] v = θ [f [ 5 (z 3 M( z 5 max + ξ 2 + ρ 5 ] α6 ( z 6 max 2 ] + M ξ 3 + ρ 6 [ sign( ξ + ξ z d ż d sign( ξ 2 + ξ 2 z 2d ż 2d sign( ξ 3 + ξ 3 z 3d ż 3d ], θ >. (9 4. FAULT-TOLERANT CONTROL ALLOCATION In the scenario considered in this paper, each thruster is an actuator potentially affected by faults. The basic idea is that, whenever a failure is detected and identified, a supervisor performs a control reconfiguration exploiting thrusters redundancy (three thursters are enough to control the ROV trajectory. In this framework, it is convenient to rewrite the model (3 as follows: ( 3 3 ż = f(z + f(z, u + diag, ˆp, ˆp Gτ ḡ(z p ˆp 6 ( with [ ] τ τ G =, τ = 2 τ, 3 ḡ(z p = [ ] ḡ (z 3 ḡ 2 (z 3 = ḡ 3 (z 3 τ 4 c 3 cos(α s 3 sin(α p p s 3 cos(α c 3 sin(α. p p d a According to the low-level control design in Section 3, a control allocation scheme can be defined in order to optimally distribute the control effect among the thrusters. Specifically, one can consider the high-level control scheme τ all = arg min τ R 4(ω τ 2 + ω 2 τ ω 3 τ ω 4 τ 4 4 subject to ḡ(z p Gτ = u eq + u n, where the weights ω j are nonnegative. The solution can be expressed by means of p 6 287

4 26 IFAC CAMS Sept 3-6, 26. Trondheim, Norway the weighted pseudo inverse τ all = [ḡ(z p G] Ω (u eq + u n, where [ḡ(z p G] Ω = ΩGT (ḡ(z p T [ḡ(z p GΩG T (ḡ(z p T ], 4. Fault diagnosis Ω = diag(ω, ω 2, ω 3, ω 4. Class of faults: The faults are modeled as actuator loss of efficiency. In this regard, let us introduce the efficiency matrix = diag(δ, δ 2, δ 3, δ 4, δ j [, ] j =,..., 4, such that the actual actuator state is τ eff = τ alloc. In particular in a fault-free scenario δ = δ 2 = δ 3 = δ 4 =, or equivalently = I, and hence the identity τ eff = τ alloc holds. On the other hand, in the case of a partial loss of efficiency, e.g. δ j (,, or total failure, e.g. δ j =, the control allocation scheme ceases to produce the desired thrust effect. A wider class might include also faults affecting the actuator own dynamics, leading to potential faulty steady-states (Cristofaro and Johansen, 24a. Assumption 3. Only one of the four thrusters can undergo a fault, i.e. multiple thruster faults cannot be admitted. Moreover, it is assumed that any fault does not compromise controllability of the plant driven by the remaining healthy thrusters. Assumption 4. In view of the fact that the reaching phase can be made arbitrarily short, it is assumed thaaults can occur only after a sliding motion has been achieved on (6. Fault Detection: The detection of a fault and the identification of the failed thruster can be performed by means of simple considerations exploiting the ROV model. To this purpose, it is important to notice that the control law is computed imposing the achievement of a sliding motion of the observed surface (6. Therefore, once the sliding motion is established (i.e. when s i = after the reaching phase, it is straightforward to verify that any deviation of the sliding surface is due to the occurrence of a fault. For instance, if a fault occurs at the time on the thruster τ such that the control input actually supplied undergoes a deviation τ (t for t > with respect to its theoretical value, for the first sliding surface it holds: s (t s (t r = s (t = τ (σ cos(z 3 αdσ p having denoted by t r the time when the sliding motion is achieved (therefore s (t = for t t r. Proposition 5. Consider the uncertain ROV model (3 driven by the robust controller (7. Suppose that the thruster T k, k {,..., 4} undergoes a fault at time, thus causing a deviation of τ k (t of the control input supplied with respect to its theoretical value. Then one has, for t > : s (t = τ k (σ cos(z 3 ( k 3 αdσ p s 2 (t = τ k (σ sin(z 3 ( k 3 αdσ p s 3 (t = d a ( k τ k (τ dσ ( p 6 where the symbol denotes the operator of division between integers. Proof. The statemenollows directly from Assumption 4 and from the observers (5. From the previous proposition, a sufficienault detection rule immediately follows. Proposition 6. Consider the uncertain ROV model (3 driven by the robust controller (7 under Assumptions 3,4. Suppose that the thruster T k, k {,..., 4} underwent a fault at time. The fault can be detected checking the variables s i (t, i =, 2, 3, at t <, i.e. if ( s (t OR ( s 2 (t OR ( s 3 (t then a fault has occurred. Proof. The proof is straightforward. It simply consists in checking the eventual violation of the sliding mode existence condition, according to (. It is worth recalling that, according to Assumption 4, the sliding motion has been established, and the sliding surface (6 should be zero in the absence of a fault affecting the actuators. Fault Isolation: The identification of the thruster which underwent the fault can be performed by means of simple considerations exploiting the ROV structure. Assume a fault has occurred at time, and define for t > : I c (t = cos(z 3 αdσ; I c2 (t = cos(z 3 + αdσ; I c3 (t = sin(z 3 αdσ; I c4 (t = sin(z 3 + αdσ. Proposition 7. Consider the uncertain ROV model (3 driven by the robust controller (7, under Assumptions 3,4. Suppose that the thruster T k, k {,..., 4} underwent a fault at time, then detected at time t d >. Compute the quantities for t > t d : µ (t = s (t I c (t p µ 3 (t = s (t I c3 (t p µ 22 (t = s 2(t I c2 (t p µ 24 (t = s 2(t I c4 (t p µ 3 (t = s 3 (t p 6 (2 d a The failed thruster τ f can be isolated according to the following rule. Fix a time t > t d. If sign(µ ( t sign(µ 3 ( t then if µ 22 ( t = µ ( t then τ f = τ else τ f = τ 3 ; else (if sign(µ ( t = sign(µ 3 ( t then if µ 22 ( t = µ ( t then τ f = τ 2 else τ f = τ 4 ; Proof. The statemenollows directly from Proposition 5. In facor a short interval (, t, assuming that the loss of effectiveness is occurring slowly enough, the term τ k (σ in ( can be moved outside the integral signs. Iollows that comparing signs of the quantities (2 one can identify the failed actuator, exploiting the model (. Just as an example, suppose that the thruster τ 2 underwent a fault of intensity τ 2 (τ. According to the model (, one has s (t = p τ 2 cos(z 3 αdσ s 2 (t = p τ 2 sin(z 3 αdσ s 3 (t = d a p 6 τ 2 (t (3 288

5 26 IFAC CAMS Sept 3-6, 26. Trondheim, Norway therefore µ = τ 2, µ 3 = τ 2 (t, and µ, µ 3 have the same sign. The same would anyway have occurred if the fault had undergone in the thruster τ 4, in view of the structure of the matrix ḡ(z 3 of the model (. To discriminate between τ 2 and τ 4, it is enough to consider that, from the first two equalities of (3: s (t τ 2 = p t cos(z 3 αdσ = s 2 (t sin(z 3 αdσ This approach can be generalized to the remaining possible cases. In particular, in the case when sign(µ ( t sign(µ 3 ( t, only thrusters τ or τ 3 could have experienced a fault. Moreover, if µ 22 ( t = µ ( t, then the failed actuator is τ, otherwise is τ 3. An analogous argument holds for the case when sign(µ ( t = sign(µ 3 ( t, for which the candidate failed actuators are τ 2 or τ Adaptive control re-allocation After a failure has been detected and isolated by the FD and FI module respectively, control reconfiguration is applied to preserve the desired performances in face of the failure occurrence. While in previous works this has been supposed as part of the human supervisor duty (see for instance (Corradini and Orlando, 24, in this paper we propose an automated reconfiguration method. In particular, the inherent redundancy of the considered ROV can be exploited for fault accommodation using an adaptive scheme for the allocation weight matrix Ω. As output of the FDI module one gets two indices, namely i and j, such that s i (t = max i=,2,3 s i(t >, τ j is the faulty actuator. The aim is to adjust the weights ω, ω 2, ω 3, ω 4 in order to progressively shift the control load on the safe actuators τ j, j j. The signal s i (t will play the role of an indicator of the fault severity and will be used accordingly to tune the adaptation learning rate. Let us formalize these ideas. Denote by t the control reconfiguration initialization time and set Ω = diag(ω,, ω,2, ω,3, ω,4 such that { Ω t < t Ω = (4 Ω(t t t where Ω(t = diag( ω (t, ω 2 (t, ω 3 (t, ω 4 (t is defined as ω j = s i (t ω j ω j =, j j (5 with Ω(t = Ω. According to such adaptation law, the larger is the deviation from the sliding surface due to the fault in the actuator τ j, the larger is the rate of decay of the corresponding weight ω j. The following statement is therefore straightforward. Proposition 8. Consider the ROV model (, with the low-level control law (7 and the high-level re-configured allocation scheme (4-(5. Then, two scenarios are admissible: Actuator performance recovery: there exists t > t such that s i (t = for t t, and hence ω j (t = ω j (t > for t t. Actuator shutdown: one has s i (t > for any t t, with the asymptotic conditions lim s i (t =, lim ω j (t =. t t Remark 9. The first case does correspond to the most likely scenario where, reducing the faulty actuator contribution in the overall thrust generation but keeping it still running, is sufficient to recover the desired tracking performances. Conversely, the actuator shutdown case can be regarded to as a worst-case scenario, where the fault is too severe to allow a partial redistribution of the thrust forces, and therefore a total ban of the faulty device is needed to accomplish the control task. 5. SIMULATION RESULTS The proposed actuator fault tolerant control scheme has been validated by simulation. Tests have been performed in the following operative condition: The measurements of position and orientation are subject to white noise; 2 Parameters variations of % with respect to their nominal value (Corradini and Orlando, 997, (Corradini et al., 2; 3 In the simulation tests, the plant initial condition has been chosen as x ( =, y ( =, φ ( =, ẋ ( =, ẏ ( =, φ ( =, and the set point as y d = [ m m 3 ] T. 4 Favorable submarine current has been considered, with V c = [..] T m/s. Notice that such marine currents have been considered constant since they are very slowly time-varying due to the fact that they model submarine currents at great sea depth. 5 The sliding mode controller has been implemented including boundary layers with the aim of reducing chattering. Accordingly, the fault detection logic does correspond to deviation from boundary layer thresholds. 6 The actuator τ 2 has been supposed to undergo a fault at t = 7 s, with a consequent loss of effectiveness. Before fault occurrence, the control action applied to the ROV is subdivided among the four thrusters (one of which is redundant. Results have been reported in Figs It can be verified that, before the fault occurrence on τ 2, actuators τ 2, τ 3 and τ 4 are able to effectively control the ROV (see Fig Moreover, simulations results show that satisfactory performances are maintained also in the faulty situation, since the ROV controlled outputs effectively follow the reference values (see Fig. 2 also after fault occurrence, and that observation and tracking errors are bounded (see Figs It is interesting to verify that detection of the fault is correctly performed by the sliding surfaces at t = 7 s (see Fig. 6, since the sliding surfaces noticeably deviate from the boundary layer thresholds. After fault isolation, control re-allocation is performed according to the adaptation law (5, with a progressive deactivation of the faulty thruster τ 2 and activation of τ that, along with τ 3 and τ 4, does ensure output regulation and maintain the required control performances. The changes in actuator states are depicted in Fig. 5 where, in particular, the gradual reduction of the τ 2 control effect and the gradual fading in of the actuator τ are clearly visible. REFERENCES Conter, A., Longhi, S., and Tirabassi, C. (989. Dynamic model and self-tuning adaptive control of an underwater vehicle. Proceedings Eighth International Conference on Offshore Mechanics and Arctic Engineering, The Hague, The Netherlands, Corradini, M.L. and Orlando, G. (24. A robust observer-based fault tolerant control scheme for underwater vehicles. Journal of Dynamic Systems, Measurement, and Control, 36(3, Corradini, M., Monteriú, A., and Orlando, G. (2. 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6 26 IFAC CAMS Sept 3-6, 26. Trondheim, Norway x y position [m].5 x y yaw angle [deg] Thruster T Thruster T2 Thruster T3 Thruster T Fig. 2. Outputs. Obs Error e Obs Error e2 Obs Error e Fig. 3. Observation Errors. Tracking Error e Tracking Error e2 Tracking Error e Fig. 4. Tracking Errors. compensation. IEEE Transactions on Control systems Technology, 9(5, Corradini, M. and Orlando, G. (997. A discrete adaptive variable structure controller for mimo systems, and its application to an underwater ROV. IEEE Transactions on Control System Technology, 5(3, Cristofaro, A. and Johansen, T.A. (24a. Fault-tolerant control allocation with actuator dynamics: Finite-time control reconfiguration. In 53rd IEEE Conference on Decision and Control, Cristofaro, A., Polycarpou, M.M., and Johansen, T.A. (25. Fault diagnosis and fault-tolerant control al- Fig. 5. Thrusters. Sliding Surface s Sliding Surface s2 Sliding Surface s Fig. 6. Sliding Surfaces. location for a class of nonlinear systems with redundant inputs. In 25 54th IEEE Conference on Decision and Control (CDC, Cristofaro, A. and Johansen, T.A. (24b. Fault tolerant control allocation using unknown input observers. Automatica, 5(7, Fossen, T. (994. Guidance and control of ocean vehicles. J.Wiley & Sons, New York. Gertler, J. (22. Survey of model-based failure detection and isolation in complex plants. Control Systems Magazine, IEEE, 8(6, 3. Isermann, R. and Ballé, P. (997. Trends in the application of model-based fault detection and diagnosis of technical processes. Control Engineering Practice, 5(5, Johansen, T.A. and Fossen, T.I. (23. Control allocation - a survey. Automatica, 49(5, Longhi, S. and Rossolini, A. (989. Adaptive control for an underwater vehicle: Simulation studies and implementation details. Proc. IFAC Wkshp. Expert Syst. Signal Processing Marine Automa, Denmark, Sarkar, N., Podder, T., and Antonelli, G. (22. Faultaccommodating thruster force allocation of an AUV considering thruster redundancy and saturation. IEEE Transactions on Robotics and Automation, 8(2, Tohidi, S., Sedigh, A., and Buzorgnia, D. (26. Fault tolerant control design using adaptive control allocation based on the pseudo inverse along the null space. International Journal of Robust and Nonlinear Control. Utkin, V. (992. Sliding modes in control optimization. Springer Verlag, Berlin. 29