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Actuator fault detection and isolation on a quadrotor unmanned aerial vehicle modeled as a linear parameter-varying system

Actuator fault detection and isolation on a quadrotor unmanned aerial vehicle modeled as a linear... This paper presents the design of a fault detection and diagnosis system for a quadrotor unmanned aerial vehicle under partial or total actuator fault. In order to control the quadrotor, the dynamic system is divided in two subsystems driven by the translational and the rotational dynamics, where the rotational subsystem is based on a linear parameter-varying model. A robust linear parameter-varying observer applied to the rotational subsystem is considered to detect actuator faults, which can occur as total failures (loss of a propeller or a motor) or partial faults (degradation). Furthermore, fault diagnosis is done by analyzing the displacements of the roll and pitch angles. Numerical experiments are carried out in order to illustrate the effectiveness of the proposed methodology. Keywords Fault diagnosis, quadrotor unmanned aerial vehicle, linear parameter-varying systems, fault detection, actuator fault, lin- ear parameter-varying observer Date received: 15 August 2018; accepted: 19 December 2018 search for techniques that guarantee stability, control, Introduction and robustness. However, the increase in civil applica- Unmanned aerial vehicles (UAVs) are gaining more tions makes necessary to consider new and difficult and more attention in recent years due to their impor- situations regarding the vehicle’s safety, for example, tant contribution and profitable application in various flying in an urban environment where the security is a tasks such as surveillance, search, rescue, remote sen- critical target to achieve. For such reason, it is neces- sing, geographical studies, as well as various military sary to develop new robust fault detection and isolation and security applications. The most popular UAVs systems which guarantee the security of the UAV dur- are fixed-wing and multirotor UAVs (helicopters, ing the all-time fly envelope. The studies on fault quadrotors, hexacopters, among others). Fixed-wing aircrafts can fly forward at high speed and are suitable for long distances due to their configuration, which TURIX-Dynamics Diagnosis and Control Group, Tecnolo´gico Nacional makes the energy consumption less than a multirotor. de Me´xico/Instituto Tecnolo´gico de Tuxtla Gutie´rrez, Tuxtla Gutie´rrez, Nonetheless, they cannot take-off and land vertically. Mexico Faculty of Engineering and Technology, Autonomous University of However, multirotor vehicles can take-off and land ver- Tlaxcala, Apizaco, Mexico tically and stay in a hover position, which could be very Tecnolo´gico Nacional de Me´xico/Instituto Tecnolo´gico de Hermosillo, convenient for some applications such as surveillance, Hermosillo, Mexico precision farming, power-line autonomous inspection, Mode´lisation, Information et Syste´mes (MIS) Laboratory, University of and delivering. This work is focused on the study of Picardie Jules Verne (UPJV), Amiens, France quadrotors UAV. A quadrotor UAV is a simple, Corresponding author: affordable, and easy-to-fly system that has been widely Guillermo Valencia-Palomo, Tecnolo´gico Nacional de Me´xico/Instituto used to develop and implement guidance methods, Tecnolo´gico de Hermosillo, Av. Tecnolo´gico y Perife´rico Poniente S/N, navigation control, fault diagnosis, fault-tolerant con- 83170 Hermosillo, Mexico. trol, among others. Many research works focus on the Email: gvalencia@ith.mx Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage). Guzma´n-Rabasa et al. 1229 diagnosis and fault-tolerant control for quadrotors method by considering a sliding mode LPV observer 1 20 were reviewed in Zhang et al., where it is shown that was proposed in Chandra et al. A multi-level method these works can be divided into different types accord- combining dynamic control allocation and control ing to faults, testbeds, frameworks, problems consid- reconfiguration was presented in Pe´ ni et al. Tan ered, and tools employed. et al. proposed a robust fault detection method in a Robust and efficient fault diagnosis systems require discrete-time qLPV system for the longitudinal move- mathematical models that represent the dynamic beha- ment of an UAV. Rotondo et al. presents an LPV vior of the UAV subject to aerodynamic and measure- observer for diagnosing sensor faults, and this observer ment disturbances. It is well known that the best has the advantage of not being noise-sensitive. A fault representation of such systems is driven by a set of non- estimation scheme in actuators of a helicopter is pre- linear ordinary differential equations. Nonetheless, it is sented in Liu et al., which reduces the impact of the also known that nonlinear systems are complex and dif- transient phenomenon. The problem of fault detection ficult to study, which reduces their applicability. An and isolation has been addressed mostly in sensors. attractive alternative to represent nonlinear dynamics is Nevertheless, application to actuators faults has not through linear parameter-varying (LPV) systems, which been fully investigated. One of the particular interests is are a class of linear systems whose state space matrices the work of Rotondo et al., which develops a robust depend on a set of time-varying measured or estimated qLPV fault-tolerant control subject to actuator faults parameters. The idea of controlling LPV systems has with very good performance, however; the proposed been introduced in Kamen and Khargonekar, then solution is in the order of thousands of gains, which are further extended during the last two decades to produce difficult to handle in a real-time implementation. This many control methods and for fault detection. implies a challenging and open problem. Therefore, the Examples of these methods include observers, fault- objective here is to develop an effective actuator fault 8 9 tolerant controllers, H controllers, sliding mode detection and isolation method simplifying the com- observers, among others. plexity of the solution. Among the different classes of LPV systems, the This paper presents an actuator fault detection and quasi-LPV (qLPV) systems have become of importance isolation method based on an observer for a quadrotor in the last years due to the fact that the nonlinearities UAV modeled as a qLPV system. The main contribu- are hidden (or embedded) in the varying parameters, in tion of this work is the use of a robust H observer to this case the varying parameters depend on exogenous generate a fault detection system that allows locating signals. This work considers this class of LPV the actuator that presents the inappropriate behavior in systems. a simple but effective manner using a reduced model of The qLPV systems are an alternative for the repre- the quadrotor. The fault detection method considers sentation of nonlinear systems used to address prob- the dynamics of the rotors and their influence on the lems such as fault detection and control. In that same pitch and roll displacements under faults. These displa- sense, some recently highlighted works are as follows: cements are used as an indicator of faults by comparing Dahmani et al. presented a robust observer to esti- the information given by the sensors and the observer. mate the dynamics of a vehicle evaluated through an Then, fault detection and isolation are done from the experimental configuration; Aouaouda et al. pro- evaluation of the generated residual signals. To guaran- posed an active fault-tolerant tracking controller tee the effectiveness of the method, stability analysis is (FTTC) scheme applied to a vehicle dynamics system. done by considering a quadratic Lyapunov equation, The stabilization problem of qLPV systems is addressed which relies on a set of linear matrix inequalities (LMI). in Wag et al., where a controller is designed to guar- Finally, numerical results are given to illustrate the antee the stability of a closed-loop system. Also, Hu effectiveness of the method under partial and total et al. proposed an output-feedback control strategy rotor faults. for the path following of autonomous ground vehicles. This paper is organized as follows: the next section More recently, qLPV has been considered for the presents the preliminaries on qLPV systems; the study of UAVs, specifically to represent its dynamics ‘‘Dynamic model and problem definition’’ section pre- and for the construction of fault detection systems. For sents the mathematical model of the quadrotor and its instance, a fault detection based on an unknown input qLPV representation; the ‘‘Main result’’ section pre- LPV observer in discrete-time for a helicopter is pre- sents the linear quadratic regulator (LQR) controller sented in De Oca et al.; in that same sense, a fault used to stabilize the system, the observer design, and detection and isolation method using set-valued observ- fault detection in actuators, where the convergence of ers was proposed in Rosa and Silvestre for a fixed- the robust observer is guaranteed by Lyapunov func- wing aircraft. Aguilar-Sierra et al. proposed a fault tions and a H robustness technique; the ‘‘Numerical estimation method by considering polynomial observ- results’’ section presents the numerical results where ers. The use of the Lyapunov quadratic function in the developments are applied to the quadrotor. Finally, LPV systems to control and stabilize aerial vehicles is the last section of the manuscript presents the presented in Sadeghzadeh et al. A fault reconstruction conclusions. 1230 Measurement and Control 52(9-10) Preliminaries on qLPV systems The sector nonlinearity approach is used to construct the qLPV model for a given nonlinear system. Considering a nonlinear system of the form m m x _(t)= fðÞ x(t), u(t) x(t)+ gðÞ x(t), u(t) u(t); ð1Þ y(t)= hðÞ x(t), u(t) x(t); m m m where f , g , and h are smooth nonlinear matrices, n n x y x 2 R is the state vector, y 2 R is the output vector, and u 2 R is the input vector. The non-constant terms of the nonlinear system are grouped into the scheduling functions r . The schedul- ing functions of the h sub-models satisfy the following convex set sum property Figure 1. The quadrotor UAV configuration. 8i 2½ 1, 2, ... , h , rðÞ z(t) 50 rðÞ z(t) =1, 8t: i i i=1 ð2Þ Dynamic model and problem definition The nonlinearities of the system are considered as Nonlinear representation variant parameters, so that they are involved in weight- The quadrotor UAV configuration is shown in ing functions, given by Figure 1. The state vector of the quadrotor is hl  z (t) T j j j j j = x, y, z, v , v , v , f, u, c, p, q, r ; h ðÞ  = , h ðÞ  =1  h ; x y z z 0 1 hl  hl j j ð3Þ where x, y, z and v , v , v are the positions and veloci- x y z j=1,2, ... , p; ties of the quadrotor relative to the inertial frame fC g, respectively; f, u, and c are the Euler angles where hl and hl are the upper and lower limits that W j j roll, pitch, and yaw, respectively; and p, q, and r reach the nonlinearity, respectively; then, for each p z denote the angular velocity relative to the body frame nonlinearity, m=2 local models are obtained. The fBg. Figure 1 also shows the torques t , t , and t for weighting functions are calculated as the product of the f u c each Euler angle. functions corresponding to each nonlinearity of the sys- 25,26 By considering the Newton–Euler formalism, the tem, that is nonlinear model is obtained as r ðÞ z = r h : ð4Þ x _ = v ; i ij j > x j=1 y_ = v ; Then, by considering equation (4), the nonlinear system > z_ = v ; (equation (1)) is represented by the qLPV model _ ð6Þ v = ðÞ cos c sin u cos f + sin c sin f ; x _(t)= rðÞ z(t) A ; i _ i v = ðÞ cos c sin u cos f  cos c sin f ; > y ð5Þ i=1 m y(t)= Cx(t)+ D d(t); > u v = g ðÞ cos u cos f ; z 0 with A = A x(t)+ B u(t)+ B d(t), where x(t) 2 R , i i i di m q p > f = p + rðÞ tan u cos f + qðÞ tan u sin f ; u(t) 2 R , d(t) 2 R , and y(t) 2 R are the state vector, > _ the control input, the disturbance, and the measured u = q cos f  r sin f; > z output vector, respectively. A , B , B , and D are con- i i di d > c = rðÞ sec u cos f + qðÞ sec u sin f ; > z stant matrices of appropriate dimensions. r (z(t)) are > I  I y z 2 scheduling functions which depend on x(t). Note that p_ = qr + ; ð7Þ I I x x the main advantage of LPV systems is that they are > > I  I u composed by a set of local linear models, and the nonli- z x 3 q_ = pr + ; > z nearities are hidden in the scheduling functions. This is I I > z y a powerful property that makes possible to use tech- I  I u > x y 4 : r_ = qr + ; niques developed for linear systems, but applied to non- z z I I z z linear systems. In the next section, this method will be used to obtain a qLPV representation of a quadrotor where I , I , and I represent the moments of inertia x y z UAV. along the x, y, and z directions, respectively; also, m Guzma´n-Rabasa et al. 1231 Table 1. Quadrotor UAV parameters. qLPV model Two subsystems describe the dynamics of the quadro- Name Description Value tor, one describes the translational part (equation (6)) m Mass 0:65 kg and the other describes the rotational part (equation 3 2 I Inertia on x axis 7:5310 kg s (7)). It is possible to study them separately due to the 3 2 I Inertia on y axis 7:5310 kg s fact that they are decoupled. This property will be used 2 2 I Inertia on z axis 1:3310 kg s to detect an actuator failure because the pitch, roll, and yaw angles change dramatically in the presence of a rotor failure. The state vector of the rotational dynamics (equation (7)) is defined as and g are the system mass and the constant of gravity, respectively. The parameters required for the model are _ _ _ x(t)= ; f, u, c, f, u, c presented in Table 1 and were obtained from 27 T Bouabdallah. =½ x (t), x (t), x (t), x (t), x (t), x (t) : 1 2 3 4 5 6 Subsystem (equation (6)) represents the translational The inputs are u(t)= ½ u (t) u (t) u (t)  . Then, 2 3 4 dynamics and subsystem (equation (7)) represents the an equivalent system representation is obtained as rotational dynamics over the reference frame fC g. 2 3 0 0 010 0 These dynamics are considered as different subsystems 6 7 0 0 001 0 due to the fact that are decoupled with respect to the 6 7 6 7 0 0 000 1 system inputs. These characteristics will be used to 6 7 x _(t)= x(t) 6 7 0 0 000 a x (t) design a fault detection observer by considering the 1 5 6 7 4 5 0 0 000 a x (t) rotational dynamic only. 2 4 000 a x (t)0 0 The control inputs u are defined in terms of propel- 3 5 2 3 2 3 00 0 ler angular speeds. The four inputs comprising the total 6 7 00 0 6 7 thrust, roll, pitch, and yaw torques are 6 7 000 ð10Þ 6 7 6 7 00 0 6 7 6 7 000 6 7 2 2 2 2 +6 7u(t)+ d(t); 6 7 u = b(O +O +O +O ); 1 6 I 7 100 1 2 3 4 x 6 7 6 1 7 4 5 0 0 2 2 4 5 010 u = lb(O  O ); 2 4 ð8Þ 001 2 2 2 3 u = lb(O  O ); 1 3 000 100 2 2 2 2 u = d O +O  O +O ; 4 5 4 y(t)= 000 010 x(t); 1 2 3 4 000 001 where l is the distance from the center of mass to the where a ¼ðI  I Þ=I , a ¼ðI  I Þ=I , and rotors, b and d are the thrust and drag factors, respec- 1 y z x 2 z x y a ¼ðI  I Þ=I . tively, and O is the angular speed of the propeller with 3 x y z As it can be seen from the system (equation (10)), i 2½1, .. .,4. matrix A contains two non-constant terms, that is, It is noteworthy that the control inputs (u , u , u , u ) i 1 2 3 4 x (t) and x (t). These elements are used to construct should be designed to stabilize the aerial vehicle. 5 4 the scheduling functions of the qLPV system, obtaining Moreover, the diagnostic ability of the system is estab- z (t)= x (t) and z (t)= x (t). As mentioned before, lished utilizing the thrusts as a function of the control- 1 5 2 4 the qLPV representation has 2 local models—since for lers. Then, the angular speeds can be computed with each nonlinear term, two scheduling functions are respect to the control inputs u as required. Since each nonlinear term is bounded, the u u u 1 3 4 weighting functions are constructed as follows O = +  ; 4b 2bl 4d 22  z (t) 1 1 1 u u u 1 2 4 z : hðÞ z (t) = , hðÞ z (t) =1  h ; 2 1 1 1 0 1 0 O = + + ; 4b 2bl 4d ð9Þ 22  z (t) 2 2 2 u u u 1 3 4 z : hðÞ z (t) = , hðÞ z (t) =1  h : 2 2 2 0 1 0 O =   ; 4b 2bl 4d u u u 1 2 4 The limits of the weighting functions are O =  + : 4b 2bl 4d z 2½022  and z 2½022 , where z and z are 1 2 1 2 expressed in rad=s. This consideration is necessary for the numerical From the representation of the system (equation experiments in order to induce actuator faults in the (5)), its LPV representation is obtained UAV, which can be observed with respect to changes on the angular speeds for the ith rotor. Furthermore, in x _(t)= rðÞ z(t) A ; ð11Þ order to design the fault diagnosis algorithm, the fol- i=1 lowing section presents the characteristics of a qLPV y(t)= Cx(t)+ D d(t); ð12Þ model and identifies the terms required to obtain a qLPV representation the system (equation (7)). with A = A x(t)+ B u(t)+ B d(t). i i i di 1232 Measurement and Control 52(9-10) The D matrix and the B matrix are the same; the P = P . 0, Q , and W with attenuation level d i i intention is that the perturbation vector affects the g. 0 8i 2½1, 2, .. . , h, such that the following optimi- angular velocities. Matrices A , B , and B are not zation problem has solution i i di included here due to space limitations. However, these min g can be easily obtained by evaluating equation (10) in P, Q , W the upper and lower limits of z and z 1 2 under the following LMI constraints 2 3 T T Main result G PB  Q D C W rd di i d T T T 2 T T 4 5 \ 0; ð16Þ B P  D Q gID W di d d Controller design WC WD I A control strategy for stabilizing the quadrotor while T T T with G = A P  C Q + PA  Q C. Then, the hovering is necessary as this is an open-loop unstable rd i i i i observer parameters are computed by L = P Q . system. The translational dynamics and the yaw angle i i are controlled by an LQR controller. The objective of this controller is to determine control signal so that the Proof. In order to guarantee asymptotic convergence of system to be controlled can meet physical constraints the estimation error to zero and robustness against dis- and minimize/maximize a cost/performance function. turbances d(t), the following H criterion is considered In this case, R is the cost of actuators, which is defined by the designer. The controller is not presented here J :¼ VeðÞ (t) + J \ 0; ð17Þ rd 1 due to space limitations. However, this can be consulted T 2 T J :¼ r (t)r(t)  g d (t)d(t)\ 0; ð18Þ in Reyes-Valeria et al. Then, for the remaining of the paper, it is considered that the UAV is stabilized. where J represents the L gain of system (equation rd 2 (16)) from d(t)to r(t) bounded by g. V(e(t)) is a Lyapunov function defined as V(e(t)) = e (t)Pe(t). The Observer design derivative of the Lyapunov function is synthesized as Assuming that system (equation (12)) is observable, the T T VeðÞ (t) :¼ e_ (t)Pe(t)+ e (t)Pe_(t)\ 0: ð19Þ following fault diagnosis LPV observer is proposed h By substituting equation (14) in equation (19), the x ^(t)= rðÞ z(t)ðÞ A x ^(t)+ B u(t)+ LðÞ y(t)  Cx ^(t) , following is equivalent i i i i=1 VeðÞ (t) :¼ðÞ ðÞ A  LC e(t)+ðÞ B  L D d(t) Pe(t) i di i d ð13Þ + e(t) P((A  L C)e(t) i i where x(t) is the estimated state vector and L are the +(B  L D )d(t))\ 0: di i d gain matrices. The gain matrices of the fault diagnosis observer (equation (13)) must be designed in order to ð20Þ guarantee the convergence of the state estimation error After some algebraic manipulation, the following is and maximize the robustness against disturbances d(t). obtained The estimation error is defined as e(t)= x(t)  x ^(t): e(t) e(t) VeðÞ (t) = F \ 0; ð21Þ d(t) d(t) The error dynamics is computed as with e_(t)= rðÞ z(t) ððÞ A  L C etðÞ i i ð14Þ G PB  PL D rd di i d i=1 F = +ðÞ B  L D dtðÞÞ, di i d T T T and the residual state space error system is given by and G = A P  C L P + PA  PL C. rd i i i i By manipulating equation (18), the following is r(t)=WCðÞ e(t)+ D d(t) , ð15Þ obtained where W is the residual weighting matrix that has to be T T T T C W WC C W WD determined. J :¼ : ð22Þ T T 2 D W WD  g I Then, the problem is reformulated to ensure stability of asymptotic equations (14) and (15) despite the per- By considering equations (21) and (22), the condition turbation vector d(t). The following theorem gives suf- (17) is rewritten as ficient conditions to reach this objective. T T U PB  PL D + C W WD rd di i d d J :¼ \ 0: rd T T 2 D W WD  g I Theorem 1. The robust observer (equation (13)) for sys- ð23Þ tem (equation (5)) exists if there exist matrices Guzma´n-Rabasa et al. 1233 Figure 2. Displacement fault indicator for: (a) actuator 3; (b) actuator 1; (c) actuator 2; (d) actuator 4. Then, Q = PL is defined in order to eliminate the i i Table 2. Diagnosis inference model for one actuator fault. quadratic terms from equation (23), such as Fault a Fault a Fault a Fault a 1 2 3 4 T T G PB  Q D + C W WD rd di i d d J :¼ \ 0: ð24Þ rd T T 2 D W WD  g I r (f) 22 ++ r (u) 2 ++ 2 Finally, the Schur complement implies equation (16). This completes the proof. Table 3. Decisions based on binary logic. Actuator fault detection and diagnosis r (f) r (u) Fault a Fault a Fault a Fault a The faults considered are additive faults; this is because 1 2 1 2 3 4 most of the faults in actuators are deviations of the sig- 0 0 1000 nals (bias) due to a stop or increase/decrease in the 0 1 0100 rotors velocity, damage in the propellers, malfunction 1 1 0010 of the electronic speed controllers, among others. The 1 0 0001 model-based fault detection and diagnosis method designed on this section is based on the approach pre- sented in Saied et al. First, residual signals based on a the roll and pitch angles are zero. However, if a fault robust state-observer, for the rotational subsystem occurs, the roll and pitch angles vary depending on (equation (7)), is generated in order to compare the which actuator fails. If a threshold is predefined in the expected behavior of the system with the measured one. pitch angle, it is easy to identify a fault in actuator 3 These residuals are defined as the differences between for a negative displacement, as shown in Figure 2(a); or the measured angles given by the inertial measurement a fault in actuator 1 for a positive displacement as unit (IMU) of the UAV and the estimated angles, which shown in Figure 2(b). Similarly, for a predefined are the system and the observer outputs, respectively. threshold in the roll angle, a fault in actuator 2 can be The residuals are defined as identified for a negative displacement as shown in Figure 2(c); or a fault in actuator 4 if the displacement r (t)= f(t)  f(t); is positive as shown in Figure 2(d). r (t)= u(t)  u(t): Therefore, given the residuals r and r , a diagnostic 1 2 interference model can be constructed in order to iso- Once the residuals are generated, they are analyzed in order to detect and isolate faults in the actuators. In late the occurrence of an actuator fault as shown in a fault-free case, the residual r value is close to zero, Table 2. while in a faulty case, the residual changes its value. If Decision-making is done through a scheme of binary the residual value is bigger than a predefined threshold, logic by substituting negative and positive displace- then this is an indication of fault occurrence. To ments (+ , ) for values of 0 and 1. Table 3 presents explain this idea further, let us consider a quadrotor in the fault cases according to the residuals, where the a hover flight mode. When the system is operating binary logic indicates which actuator is faulty. This is fault-free in this flying mode, the expected values for represented by the following equations 1234 Measurement and Control 52(9-10) fault a = :r ^:r ; 1 1 2 fault a = :r ^ r ; 2 1 2 ð25Þ fault a = r ^ r ; 3 1 2 fault a = r ^:r : 4 1 2 where fault a represents a failure in the actuator i. Numerical results This section presents numerical tests in order to show the effectiveness of the fault detection and diagnosis method. Observer synthesis and controller test For the synthesis of the proposed LPV observer with H performance (equation (13)), the LMI presented in Figure 3. Reference and tracking entries obtained. equation (16) has been solved using the YALMIP Toolbox. The resulting matrices are The attenuation level obtained is g =1:39. This 2 3 value guarantees an adequate performance of the obser- 0:13840 00:0025 0 0 ver and its robustness against disturbances. 6 7 6 7 00:1384 0 0 0:0025 0 Once the observer is designed, the system’s perfor- 6 7 6 7 6 00 2:0339 0 0 0:04077 mance is tested in closed-loop using the LQR controller 6 7 P = ; 6 7 designed for the stabilization of the quadrotor. The 0:00250 01:0785 0 0 6 7 6 7 numerical experiments were performed in MATLAB / 6 7 00:0025 0 0 1:0785 0 4 5 Simulink . 00 0:0407 0 0 0:9623 Figure 3 shows the references in dashed line and the 2 3 0:9733 0 0:1825 tracking trajectory in solid line. The tracking trajec- 6 7 tories were obtained thanks to the LQR controller. A 6 00:9733 0:18257 6 7 three-dimensional (3D) representation is illustrated in 6 7 6 0:2271 0:2271 0:9676 7 6 7 Figure 4, where the three graphs presented in Figure 3 L = ; 6 7 1:4639 0 5:9739 6 7 are combined. The trajectory obtained follows the ref- 6 7 6 7 01:4639 5:9739 erence, so the LQR controller fulfills its purpose. The 4 5 resulting trajectory is described below with the help of 11:3568 11:3568 1:6225 2 3 points in the path: 0:9733 0 0:1825 6 7 At t = 0 s, the quadrotor is at the origin, that is, 6 7 00:9733 0:1825 6 7 at point A = ( 0, 0, 0 ). At t = 5 s, the posi- 6 7 6 0:2271 0:2271 0:9676 7 6 7 tion of the quadrotor is 5 m on the z axis, while L = ; 6 7 1:4639 0 5:9739 6 7 on the other two axes remains at zero, and this 6 7 6 7 location corresponds to point B = ( 0, 0, 5 ). 01:4639 5:9739 4 5 The path continues to the point 11:3568 11:3568 1:6225 2 3 C=(5, 0, 5) and D=(5, 5, 5) in a 0:9733 0 0:1825 time of t = 10 s and t = 20 s, respectively. 6 7 6 7 00:9733 0:1825 At t = 30 s, the trajectory comes to the point 6 7 6 7 located in E = ( 0, 5, 5 ). 6 0:2271 0:2271 0:96767 6 7 L = ; 6 7 The trajectory of the quadrotor comes back to B 1:4639 0 5:9739 6 7 6 7 by drawing a square on the xy-plane and finally 6 7 01:4639 5:9739 4 5 return to the origin A. 11:3568 11:3568 1:6225 2 3 Fault detection and isolation 0:9733 0 0:1825 6 7 The numerical experiments begin once the quadrotor 6 00:9733 0:1825 7 6 7 has been stabilized in hover flight mode. In order to test 6 7 0:2271 0:2271 0:9676 6 7 the effectiveness of the proposed method, two different 6 7 L = : 6 7 6 1:4639 0 5:97397 scenarios are considered, partial and total faults. In all 6 7 6 7 cases, the observer is fed with noisy measurements of 01:4639 5:9739 4 5 the system input/output. The following describes each 11:3568 11:3568 1:6225 case. Guzma´n-Rabasa et al. 1235 Figure 4. Tracking and reference in 3D space. Partial fault. In this scenario, three different types of faults are induced to actuator 3. For the first fault, the magnitude of the control signal of the actuator required to stabilize the UAV in hover flight mode is reduced by 20% in 40\ t\ 50 s; in the second fault, the actuator is affected by sinusoidal variations of the speed in 70\ t\ 80 s; and finally, in 100\ t\ 110 s, an incipi- ent fault, which reduces the control signal of the actua- tor from 100% to 80% in 10 s, is induced. All of the induced faults provoke changes on the residual signals. The faults and their effects on the residuals are shown in Figure 5. The faults are isolated from the evaluation of the residuals as shown in Figure 6. Also, note that due to the robust approach, disturbances are well atte- nuated and do not affect the performance of the observers. Total fault. A total fault is induced by turning off the actuator 3 at t = 20 s. By considering the residual eva- luation, the fault is detected and achieved by the eva- luation of the residual as can be observed in Figure 7. Once the fault is detected, fault diagnosis is performed considering the dynamic behavior of roll and pitch as shown in Figure 2. The residuals exceed the thresholds, giving positive values, that is, r = 1 and r = 1, indi- 1 2 Figure 5. Residuals with partial fault and faults induced to cating the total fault in actuator 3 as shown in Figure 8 actuator 3. and Table 2. Similar analysis can be done to detect faults in the remaining actuators, and in each case, the actuator from 100% to 70%. It can be seen that both method can detect and isolate the fault, which demon- approaches have the same performance as they activate strates its effectiveness and applicability. the alarm when the fault occurs. The difference between the approaches is mainly Comparisons. To complement the results, Figure 9 that Rotondo et al.’s work considers the complete shows the comparison between the approach of quadrotor model, while in this work, the fault detection Rotondo et al. and the approach proposed in this and isolation are only applied to the rotational paper. Specifically, the figure shows the alarms of dynamics. Considering the complete model also implies actuator 1 for a partial fault scenario. The induced that more residues must be taken into account for the fault consists in a reduction of the control signal of the fault detection and isolation, and therefore, the 1236 Measurement and Control 52(9-10) Figure 6. Partial fault diagnosis on actuator 3. number of vertices is reduced (four vertices) since the method has been applied to the rotational dynamics of the quadrotor only, while in Rotondo et al., the num- ber of vertices exceeds 2000, leading to an equal num- ber of gains L ’s that have to be considered online. Clearly, the method presented here is easier to handle and implement in a real application. Comment on stability in case of an actuator failure Quadrotors are open-loop unstable systems and depend on a controller for their stabilization. Therefore, the stability analysis depends mainly in the controller which is out of the scope of this paper. Nevertheless, it is important to remark that, in a faulty situation, the con- troller may not be able to stabilize the system, depend- Figure 7. Residuals with total fault on actuator 3 at t =20 s. ing on the severity or intensity of the affectation. If the fault is partial and with very low intensity, a conven- inference matrix is larger and its online analysis is more tional controller may be able to stabilize the system complex; although, it is also acknowledged that faults regarding the fault as a model mismatch or disturbance. can be also detected and isolated while the UAV is For a more severe affectation, there are also passive/ moving in the space, not only in hover flight mode. active fault-tolerant control strategies reported in the Moreover, although the computation of the optimiza- literature that increase the actuator affectation toler- 31–33 tion problem (equation (16)) is performed offline, it is ance. If the fault in one actuator is total, an emer- always important to analyze its complexity since the gency protocol has to be activated such as the use of a time needed to solve the LMI grows exponentially as parachute; another option may be to give up control- the problem dimension grows. In this case, the LMI ling the quadrotor’s yaw angle, and use the remaining dimension is smaller than the one presented in actuators to achieve a horizontal spin. Although with Rotondo et al. Furthermore, it is well known that the a very low probability, two (opposed) or three actuators LMI problem could become unfeasible as the number can fail simultaneously and a specific control law is 35,36 of vertices grow. In the approach presented here, the required for those scenarios. Guzma´n-Rabasa et al. 1237 Figure 8. Total fault diagnosis on actuator 3. of a total actuator fault, and also, a mixed strategy H=H will be considered to increase fault sensitivity. Moreover, experimental validation of the developed fault detection and isolation algorithm in a device that allows a safe indoor test is also considered for future work. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Figure 9. Alarms for actuator 1 in partial fault scenario. Funding This work was supported by the Tecnolo´ gico Nacional de Conclusion Me´ xico (TecNM) under the program Apoyo a la Investigacio´n Cientı´fica y Tecnolo´gica (grant numbers 6210.17-P and This paper presented a robust fault diagnosis and isola- 6723.18-P). Additional support was provided by the Consejo tion scheme for actuator faults of a quadrotor UAV Nacional de Ciencia y Tecnologı´a under the program modeled as an LPV system. In order to detect partial Ca´tedras Conacyt (project number 88). or total actuator faults, a residual was generated using a H observer applied to the rotational dynamics. The ORCID iD observer was designed to be robust against distur- bances. Sufficient conditions were obtained to compute Guillermo Valencia-Palomo https://orcid.org/0000-0002- 3382-8213 the observer gains in order to guarantee its conver- gence. Simulation results show the effectiveness of the proposed method in order to detect and isolate partial References and total actuator faults. Future work will include the 1. Zhang Y, Chamseddine A, Rabbath C, et al. Develop- design of a fault-tolerant control strategy in order to ment of advanced FDD and FTC techniques with guarantee the safety of the UAV even in the presence 1238 Measurement and Control 52(9-10) application to an unmanned quadrotor helicopter a fixed-wing aircraft. Control Eng Pract 2013; 21(3): 242– testbed. 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Actuator fault detection and isolation on a quadrotor unmanned aerial vehicle modeled as a linear parameter-varying system

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SAGE
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© The Author(s) 2019
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0020-2940
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10.1177/0020294018824764
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Abstract

This paper presents the design of a fault detection and diagnosis system for a quadrotor unmanned aerial vehicle under partial or total actuator fault. In order to control the quadrotor, the dynamic system is divided in two subsystems driven by the translational and the rotational dynamics, where the rotational subsystem is based on a linear parameter-varying model. A robust linear parameter-varying observer applied to the rotational subsystem is considered to detect actuator faults, which can occur as total failures (loss of a propeller or a motor) or partial faults (degradation). Furthermore, fault diagnosis is done by analyzing the displacements of the roll and pitch angles. Numerical experiments are carried out in order to illustrate the effectiveness of the proposed methodology. Keywords Fault diagnosis, quadrotor unmanned aerial vehicle, linear parameter-varying systems, fault detection, actuator fault, lin- ear parameter-varying observer Date received: 15 August 2018; accepted: 19 December 2018 search for techniques that guarantee stability, control, Introduction and robustness. However, the increase in civil applica- Unmanned aerial vehicles (UAVs) are gaining more tions makes necessary to consider new and difficult and more attention in recent years due to their impor- situations regarding the vehicle’s safety, for example, tant contribution and profitable application in various flying in an urban environment where the security is a tasks such as surveillance, search, rescue, remote sen- critical target to achieve. For such reason, it is neces- sing, geographical studies, as well as various military sary to develop new robust fault detection and isolation and security applications. The most popular UAVs systems which guarantee the security of the UAV dur- are fixed-wing and multirotor UAVs (helicopters, ing the all-time fly envelope. The studies on fault quadrotors, hexacopters, among others). Fixed-wing aircrafts can fly forward at high speed and are suitable for long distances due to their configuration, which TURIX-Dynamics Diagnosis and Control Group, Tecnolo´gico Nacional makes the energy consumption less than a multirotor. de Me´xico/Instituto Tecnolo´gico de Tuxtla Gutie´rrez, Tuxtla Gutie´rrez, Nonetheless, they cannot take-off and land vertically. Mexico Faculty of Engineering and Technology, Autonomous University of However, multirotor vehicles can take-off and land ver- Tlaxcala, Apizaco, Mexico tically and stay in a hover position, which could be very Tecnolo´gico Nacional de Me´xico/Instituto Tecnolo´gico de Hermosillo, convenient for some applications such as surveillance, Hermosillo, Mexico precision farming, power-line autonomous inspection, Mode´lisation, Information et Syste´mes (MIS) Laboratory, University of and delivering. This work is focused on the study of Picardie Jules Verne (UPJV), Amiens, France quadrotors UAV. A quadrotor UAV is a simple, Corresponding author: affordable, and easy-to-fly system that has been widely Guillermo Valencia-Palomo, Tecnolo´gico Nacional de Me´xico/Instituto used to develop and implement guidance methods, Tecnolo´gico de Hermosillo, Av. Tecnolo´gico y Perife´rico Poniente S/N, navigation control, fault diagnosis, fault-tolerant con- 83170 Hermosillo, Mexico. trol, among others. Many research works focus on the Email: gvalencia@ith.mx Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage). Guzma´n-Rabasa et al. 1229 diagnosis and fault-tolerant control for quadrotors method by considering a sliding mode LPV observer 1 20 were reviewed in Zhang et al., where it is shown that was proposed in Chandra et al. A multi-level method these works can be divided into different types accord- combining dynamic control allocation and control ing to faults, testbeds, frameworks, problems consid- reconfiguration was presented in Pe´ ni et al. Tan ered, and tools employed. et al. proposed a robust fault detection method in a Robust and efficient fault diagnosis systems require discrete-time qLPV system for the longitudinal move- mathematical models that represent the dynamic beha- ment of an UAV. Rotondo et al. presents an LPV vior of the UAV subject to aerodynamic and measure- observer for diagnosing sensor faults, and this observer ment disturbances. It is well known that the best has the advantage of not being noise-sensitive. A fault representation of such systems is driven by a set of non- estimation scheme in actuators of a helicopter is pre- linear ordinary differential equations. Nonetheless, it is sented in Liu et al., which reduces the impact of the also known that nonlinear systems are complex and dif- transient phenomenon. The problem of fault detection ficult to study, which reduces their applicability. An and isolation has been addressed mostly in sensors. attractive alternative to represent nonlinear dynamics is Nevertheless, application to actuators faults has not through linear parameter-varying (LPV) systems, which been fully investigated. One of the particular interests is are a class of linear systems whose state space matrices the work of Rotondo et al., which develops a robust depend on a set of time-varying measured or estimated qLPV fault-tolerant control subject to actuator faults parameters. The idea of controlling LPV systems has with very good performance, however; the proposed been introduced in Kamen and Khargonekar, then solution is in the order of thousands of gains, which are further extended during the last two decades to produce difficult to handle in a real-time implementation. This many control methods and for fault detection. implies a challenging and open problem. Therefore, the Examples of these methods include observers, fault- objective here is to develop an effective actuator fault 8 9 tolerant controllers, H controllers, sliding mode detection and isolation method simplifying the com- observers, among others. plexity of the solution. Among the different classes of LPV systems, the This paper presents an actuator fault detection and quasi-LPV (qLPV) systems have become of importance isolation method based on an observer for a quadrotor in the last years due to the fact that the nonlinearities UAV modeled as a qLPV system. The main contribu- are hidden (or embedded) in the varying parameters, in tion of this work is the use of a robust H observer to this case the varying parameters depend on exogenous generate a fault detection system that allows locating signals. This work considers this class of LPV the actuator that presents the inappropriate behavior in systems. a simple but effective manner using a reduced model of The qLPV systems are an alternative for the repre- the quadrotor. The fault detection method considers sentation of nonlinear systems used to address prob- the dynamics of the rotors and their influence on the lems such as fault detection and control. In that same pitch and roll displacements under faults. These displa- sense, some recently highlighted works are as follows: cements are used as an indicator of faults by comparing Dahmani et al. presented a robust observer to esti- the information given by the sensors and the observer. mate the dynamics of a vehicle evaluated through an Then, fault detection and isolation are done from the experimental configuration; Aouaouda et al. pro- evaluation of the generated residual signals. To guaran- posed an active fault-tolerant tracking controller tee the effectiveness of the method, stability analysis is (FTTC) scheme applied to a vehicle dynamics system. done by considering a quadratic Lyapunov equation, The stabilization problem of qLPV systems is addressed which relies on a set of linear matrix inequalities (LMI). in Wag et al., where a controller is designed to guar- Finally, numerical results are given to illustrate the antee the stability of a closed-loop system. Also, Hu effectiveness of the method under partial and total et al. proposed an output-feedback control strategy rotor faults. for the path following of autonomous ground vehicles. This paper is organized as follows: the next section More recently, qLPV has been considered for the presents the preliminaries on qLPV systems; the study of UAVs, specifically to represent its dynamics ‘‘Dynamic model and problem definition’’ section pre- and for the construction of fault detection systems. For sents the mathematical model of the quadrotor and its instance, a fault detection based on an unknown input qLPV representation; the ‘‘Main result’’ section pre- LPV observer in discrete-time for a helicopter is pre- sents the linear quadratic regulator (LQR) controller sented in De Oca et al.; in that same sense, a fault used to stabilize the system, the observer design, and detection and isolation method using set-valued observ- fault detection in actuators, where the convergence of ers was proposed in Rosa and Silvestre for a fixed- the robust observer is guaranteed by Lyapunov func- wing aircraft. Aguilar-Sierra et al. proposed a fault tions and a H robustness technique; the ‘‘Numerical estimation method by considering polynomial observ- results’’ section presents the numerical results where ers. The use of the Lyapunov quadratic function in the developments are applied to the quadrotor. Finally, LPV systems to control and stabilize aerial vehicles is the last section of the manuscript presents the presented in Sadeghzadeh et al. A fault reconstruction conclusions. 1230 Measurement and Control 52(9-10) Preliminaries on qLPV systems The sector nonlinearity approach is used to construct the qLPV model for a given nonlinear system. Considering a nonlinear system of the form m m x _(t)= fðÞ x(t), u(t) x(t)+ gðÞ x(t), u(t) u(t); ð1Þ y(t)= hðÞ x(t), u(t) x(t); m m m where f , g , and h are smooth nonlinear matrices, n n x y x 2 R is the state vector, y 2 R is the output vector, and u 2 R is the input vector. The non-constant terms of the nonlinear system are grouped into the scheduling functions r . The schedul- ing functions of the h sub-models satisfy the following convex set sum property Figure 1. The quadrotor UAV configuration. 8i 2½ 1, 2, ... , h , rðÞ z(t) 50 rðÞ z(t) =1, 8t: i i i=1 ð2Þ Dynamic model and problem definition The nonlinearities of the system are considered as Nonlinear representation variant parameters, so that they are involved in weight- The quadrotor UAV configuration is shown in ing functions, given by Figure 1. The state vector of the quadrotor is hl  z (t) T j j j j j = x, y, z, v , v , v , f, u, c, p, q, r ; h ðÞ  = , h ðÞ  =1  h ; x y z z 0 1 hl  hl j j ð3Þ where x, y, z and v , v , v are the positions and veloci- x y z j=1,2, ... , p; ties of the quadrotor relative to the inertial frame fC g, respectively; f, u, and c are the Euler angles where hl and hl are the upper and lower limits that W j j roll, pitch, and yaw, respectively; and p, q, and r reach the nonlinearity, respectively; then, for each p z denote the angular velocity relative to the body frame nonlinearity, m=2 local models are obtained. The fBg. Figure 1 also shows the torques t , t , and t for weighting functions are calculated as the product of the f u c each Euler angle. functions corresponding to each nonlinearity of the sys- 25,26 By considering the Newton–Euler formalism, the tem, that is nonlinear model is obtained as r ðÞ z = r h : ð4Þ x _ = v ; i ij j > x j=1 y_ = v ; Then, by considering equation (4), the nonlinear system > z_ = v ; (equation (1)) is represented by the qLPV model _ ð6Þ v = ðÞ cos c sin u cos f + sin c sin f ; x _(t)= rðÞ z(t) A ; i _ i v = ðÞ cos c sin u cos f  cos c sin f ; > y ð5Þ i=1 m y(t)= Cx(t)+ D d(t); > u v = g ðÞ cos u cos f ; z 0 with A = A x(t)+ B u(t)+ B d(t), where x(t) 2 R , i i i di m q p > f = p + rðÞ tan u cos f + qðÞ tan u sin f ; u(t) 2 R , d(t) 2 R , and y(t) 2 R are the state vector, > _ the control input, the disturbance, and the measured u = q cos f  r sin f; > z output vector, respectively. A , B , B , and D are con- i i di d > c = rðÞ sec u cos f + qðÞ sec u sin f ; > z stant matrices of appropriate dimensions. r (z(t)) are > I  I y z 2 scheduling functions which depend on x(t). Note that p_ = qr + ; ð7Þ I I x x the main advantage of LPV systems is that they are > > I  I u composed by a set of local linear models, and the nonli- z x 3 q_ = pr + ; > z nearities are hidden in the scheduling functions. This is I I > z y a powerful property that makes possible to use tech- I  I u > x y 4 : r_ = qr + ; niques developed for linear systems, but applied to non- z z I I z z linear systems. In the next section, this method will be used to obtain a qLPV representation of a quadrotor where I , I , and I represent the moments of inertia x y z UAV. along the x, y, and z directions, respectively; also, m Guzma´n-Rabasa et al. 1231 Table 1. Quadrotor UAV parameters. qLPV model Two subsystems describe the dynamics of the quadro- Name Description Value tor, one describes the translational part (equation (6)) m Mass 0:65 kg and the other describes the rotational part (equation 3 2 I Inertia on x axis 7:5310 kg s (7)). It is possible to study them separately due to the 3 2 I Inertia on y axis 7:5310 kg s fact that they are decoupled. This property will be used 2 2 I Inertia on z axis 1:3310 kg s to detect an actuator failure because the pitch, roll, and yaw angles change dramatically in the presence of a rotor failure. The state vector of the rotational dynamics (equation (7)) is defined as and g are the system mass and the constant of gravity, respectively. The parameters required for the model are _ _ _ x(t)= ; f, u, c, f, u, c presented in Table 1 and were obtained from 27 T Bouabdallah. =½ x (t), x (t), x (t), x (t), x (t), x (t) : 1 2 3 4 5 6 Subsystem (equation (6)) represents the translational The inputs are u(t)= ½ u (t) u (t) u (t)  . Then, 2 3 4 dynamics and subsystem (equation (7)) represents the an equivalent system representation is obtained as rotational dynamics over the reference frame fC g. 2 3 0 0 010 0 These dynamics are considered as different subsystems 6 7 0 0 001 0 due to the fact that are decoupled with respect to the 6 7 6 7 0 0 000 1 system inputs. These characteristics will be used to 6 7 x _(t)= x(t) 6 7 0 0 000 a x (t) design a fault detection observer by considering the 1 5 6 7 4 5 0 0 000 a x (t) rotational dynamic only. 2 4 000 a x (t)0 0 The control inputs u are defined in terms of propel- 3 5 2 3 2 3 00 0 ler angular speeds. The four inputs comprising the total 6 7 00 0 6 7 thrust, roll, pitch, and yaw torques are 6 7 000 ð10Þ 6 7 6 7 00 0 6 7 6 7 000 6 7 2 2 2 2 +6 7u(t)+ d(t); 6 7 u = b(O +O +O +O ); 1 6 I 7 100 1 2 3 4 x 6 7 6 1 7 4 5 0 0 2 2 4 5 010 u = lb(O  O ); 2 4 ð8Þ 001 2 2 2 3 u = lb(O  O ); 1 3 000 100 2 2 2 2 u = d O +O  O +O ; 4 5 4 y(t)= 000 010 x(t); 1 2 3 4 000 001 where l is the distance from the center of mass to the where a ¼ðI  I Þ=I , a ¼ðI  I Þ=I , and rotors, b and d are the thrust and drag factors, respec- 1 y z x 2 z x y a ¼ðI  I Þ=I . tively, and O is the angular speed of the propeller with 3 x y z As it can be seen from the system (equation (10)), i 2½1, .. .,4. matrix A contains two non-constant terms, that is, It is noteworthy that the control inputs (u , u , u , u ) i 1 2 3 4 x (t) and x (t). These elements are used to construct should be designed to stabilize the aerial vehicle. 5 4 the scheduling functions of the qLPV system, obtaining Moreover, the diagnostic ability of the system is estab- z (t)= x (t) and z (t)= x (t). As mentioned before, lished utilizing the thrusts as a function of the control- 1 5 2 4 the qLPV representation has 2 local models—since for lers. Then, the angular speeds can be computed with each nonlinear term, two scheduling functions are respect to the control inputs u as required. Since each nonlinear term is bounded, the u u u 1 3 4 weighting functions are constructed as follows O = +  ; 4b 2bl 4d 22  z (t) 1 1 1 u u u 1 2 4 z : hðÞ z (t) = , hðÞ z (t) =1  h ; 2 1 1 1 0 1 0 O = + + ; 4b 2bl 4d ð9Þ 22  z (t) 2 2 2 u u u 1 3 4 z : hðÞ z (t) = , hðÞ z (t) =1  h : 2 2 2 0 1 0 O =   ; 4b 2bl 4d u u u 1 2 4 The limits of the weighting functions are O =  + : 4b 2bl 4d z 2½022  and z 2½022 , where z and z are 1 2 1 2 expressed in rad=s. This consideration is necessary for the numerical From the representation of the system (equation experiments in order to induce actuator faults in the (5)), its LPV representation is obtained UAV, which can be observed with respect to changes on the angular speeds for the ith rotor. Furthermore, in x _(t)= rðÞ z(t) A ; ð11Þ order to design the fault diagnosis algorithm, the fol- i=1 lowing section presents the characteristics of a qLPV y(t)= Cx(t)+ D d(t); ð12Þ model and identifies the terms required to obtain a qLPV representation the system (equation (7)). with A = A x(t)+ B u(t)+ B d(t). i i i di 1232 Measurement and Control 52(9-10) The D matrix and the B matrix are the same; the P = P . 0, Q , and W with attenuation level d i i intention is that the perturbation vector affects the g. 0 8i 2½1, 2, .. . , h, such that the following optimi- angular velocities. Matrices A , B , and B are not zation problem has solution i i di included here due to space limitations. However, these min g can be easily obtained by evaluating equation (10) in P, Q , W the upper and lower limits of z and z 1 2 under the following LMI constraints 2 3 T T Main result G PB  Q D C W rd di i d T T T 2 T T 4 5 \ 0; ð16Þ B P  D Q gID W di d d Controller design WC WD I A control strategy for stabilizing the quadrotor while T T T with G = A P  C Q + PA  Q C. Then, the hovering is necessary as this is an open-loop unstable rd i i i i observer parameters are computed by L = P Q . system. The translational dynamics and the yaw angle i i are controlled by an LQR controller. The objective of this controller is to determine control signal so that the Proof. In order to guarantee asymptotic convergence of system to be controlled can meet physical constraints the estimation error to zero and robustness against dis- and minimize/maximize a cost/performance function. turbances d(t), the following H criterion is considered In this case, R is the cost of actuators, which is defined by the designer. The controller is not presented here J :¼ VeðÞ (t) + J \ 0; ð17Þ rd 1 due to space limitations. However, this can be consulted T 2 T J :¼ r (t)r(t)  g d (t)d(t)\ 0; ð18Þ in Reyes-Valeria et al. Then, for the remaining of the paper, it is considered that the UAV is stabilized. where J represents the L gain of system (equation rd 2 (16)) from d(t)to r(t) bounded by g. V(e(t)) is a Lyapunov function defined as V(e(t)) = e (t)Pe(t). The Observer design derivative of the Lyapunov function is synthesized as Assuming that system (equation (12)) is observable, the T T VeðÞ (t) :¼ e_ (t)Pe(t)+ e (t)Pe_(t)\ 0: ð19Þ following fault diagnosis LPV observer is proposed h By substituting equation (14) in equation (19), the x ^(t)= rðÞ z(t)ðÞ A x ^(t)+ B u(t)+ LðÞ y(t)  Cx ^(t) , following is equivalent i i i i=1 VeðÞ (t) :¼ðÞ ðÞ A  LC e(t)+ðÞ B  L D d(t) Pe(t) i di i d ð13Þ + e(t) P((A  L C)e(t) i i where x(t) is the estimated state vector and L are the +(B  L D )d(t))\ 0: di i d gain matrices. The gain matrices of the fault diagnosis observer (equation (13)) must be designed in order to ð20Þ guarantee the convergence of the state estimation error After some algebraic manipulation, the following is and maximize the robustness against disturbances d(t). obtained The estimation error is defined as e(t)= x(t)  x ^(t): e(t) e(t) VeðÞ (t) = F \ 0; ð21Þ d(t) d(t) The error dynamics is computed as with e_(t)= rðÞ z(t) ððÞ A  L C etðÞ i i ð14Þ G PB  PL D rd di i d i=1 F = +ðÞ B  L D dtðÞÞ, di i d T T T and the residual state space error system is given by and G = A P  C L P + PA  PL C. rd i i i i By manipulating equation (18), the following is r(t)=WCðÞ e(t)+ D d(t) , ð15Þ obtained where W is the residual weighting matrix that has to be T T T T C W WC C W WD determined. J :¼ : ð22Þ T T 2 D W WD  g I Then, the problem is reformulated to ensure stability of asymptotic equations (14) and (15) despite the per- By considering equations (21) and (22), the condition turbation vector d(t). The following theorem gives suf- (17) is rewritten as ficient conditions to reach this objective. T T U PB  PL D + C W WD rd di i d d J :¼ \ 0: rd T T 2 D W WD  g I Theorem 1. The robust observer (equation (13)) for sys- ð23Þ tem (equation (5)) exists if there exist matrices Guzma´n-Rabasa et al. 1233 Figure 2. Displacement fault indicator for: (a) actuator 3; (b) actuator 1; (c) actuator 2; (d) actuator 4. Then, Q = PL is defined in order to eliminate the i i Table 2. Diagnosis inference model for one actuator fault. quadratic terms from equation (23), such as Fault a Fault a Fault a Fault a 1 2 3 4 T T G PB  Q D + C W WD rd di i d d J :¼ \ 0: ð24Þ rd T T 2 D W WD  g I r (f) 22 ++ r (u) 2 ++ 2 Finally, the Schur complement implies equation (16). This completes the proof. Table 3. Decisions based on binary logic. Actuator fault detection and diagnosis r (f) r (u) Fault a Fault a Fault a Fault a The faults considered are additive faults; this is because 1 2 1 2 3 4 most of the faults in actuators are deviations of the sig- 0 0 1000 nals (bias) due to a stop or increase/decrease in the 0 1 0100 rotors velocity, damage in the propellers, malfunction 1 1 0010 of the electronic speed controllers, among others. The 1 0 0001 model-based fault detection and diagnosis method designed on this section is based on the approach pre- sented in Saied et al. First, residual signals based on a the roll and pitch angles are zero. However, if a fault robust state-observer, for the rotational subsystem occurs, the roll and pitch angles vary depending on (equation (7)), is generated in order to compare the which actuator fails. If a threshold is predefined in the expected behavior of the system with the measured one. pitch angle, it is easy to identify a fault in actuator 3 These residuals are defined as the differences between for a negative displacement, as shown in Figure 2(a); or the measured angles given by the inertial measurement a fault in actuator 1 for a positive displacement as unit (IMU) of the UAV and the estimated angles, which shown in Figure 2(b). Similarly, for a predefined are the system and the observer outputs, respectively. threshold in the roll angle, a fault in actuator 2 can be The residuals are defined as identified for a negative displacement as shown in Figure 2(c); or a fault in actuator 4 if the displacement r (t)= f(t)  f(t); is positive as shown in Figure 2(d). r (t)= u(t)  u(t): Therefore, given the residuals r and r , a diagnostic 1 2 interference model can be constructed in order to iso- Once the residuals are generated, they are analyzed in order to detect and isolate faults in the actuators. In late the occurrence of an actuator fault as shown in a fault-free case, the residual r value is close to zero, Table 2. while in a faulty case, the residual changes its value. If Decision-making is done through a scheme of binary the residual value is bigger than a predefined threshold, logic by substituting negative and positive displace- then this is an indication of fault occurrence. To ments (+ , ) for values of 0 and 1. Table 3 presents explain this idea further, let us consider a quadrotor in the fault cases according to the residuals, where the a hover flight mode. When the system is operating binary logic indicates which actuator is faulty. This is fault-free in this flying mode, the expected values for represented by the following equations 1234 Measurement and Control 52(9-10) fault a = :r ^:r ; 1 1 2 fault a = :r ^ r ; 2 1 2 ð25Þ fault a = r ^ r ; 3 1 2 fault a = r ^:r : 4 1 2 where fault a represents a failure in the actuator i. Numerical results This section presents numerical tests in order to show the effectiveness of the fault detection and diagnosis method. Observer synthesis and controller test For the synthesis of the proposed LPV observer with H performance (equation (13)), the LMI presented in Figure 3. Reference and tracking entries obtained. equation (16) has been solved using the YALMIP Toolbox. The resulting matrices are The attenuation level obtained is g =1:39. This 2 3 value guarantees an adequate performance of the obser- 0:13840 00:0025 0 0 ver and its robustness against disturbances. 6 7 6 7 00:1384 0 0 0:0025 0 Once the observer is designed, the system’s perfor- 6 7 6 7 6 00 2:0339 0 0 0:04077 mance is tested in closed-loop using the LQR controller 6 7 P = ; 6 7 designed for the stabilization of the quadrotor. The 0:00250 01:0785 0 0 6 7 6 7 numerical experiments were performed in MATLAB / 6 7 00:0025 0 0 1:0785 0 4 5 Simulink . 00 0:0407 0 0 0:9623 Figure 3 shows the references in dashed line and the 2 3 0:9733 0 0:1825 tracking trajectory in solid line. The tracking trajec- 6 7 tories were obtained thanks to the LQR controller. A 6 00:9733 0:18257 6 7 three-dimensional (3D) representation is illustrated in 6 7 6 0:2271 0:2271 0:9676 7 6 7 Figure 4, where the three graphs presented in Figure 3 L = ; 6 7 1:4639 0 5:9739 6 7 are combined. The trajectory obtained follows the ref- 6 7 6 7 01:4639 5:9739 erence, so the LQR controller fulfills its purpose. The 4 5 resulting trajectory is described below with the help of 11:3568 11:3568 1:6225 2 3 points in the path: 0:9733 0 0:1825 6 7 At t = 0 s, the quadrotor is at the origin, that is, 6 7 00:9733 0:1825 6 7 at point A = ( 0, 0, 0 ). At t = 5 s, the posi- 6 7 6 0:2271 0:2271 0:9676 7 6 7 tion of the quadrotor is 5 m on the z axis, while L = ; 6 7 1:4639 0 5:9739 6 7 on the other two axes remains at zero, and this 6 7 6 7 location corresponds to point B = ( 0, 0, 5 ). 01:4639 5:9739 4 5 The path continues to the point 11:3568 11:3568 1:6225 2 3 C=(5, 0, 5) and D=(5, 5, 5) in a 0:9733 0 0:1825 time of t = 10 s and t = 20 s, respectively. 6 7 6 7 00:9733 0:1825 At t = 30 s, the trajectory comes to the point 6 7 6 7 located in E = ( 0, 5, 5 ). 6 0:2271 0:2271 0:96767 6 7 L = ; 6 7 The trajectory of the quadrotor comes back to B 1:4639 0 5:9739 6 7 6 7 by drawing a square on the xy-plane and finally 6 7 01:4639 5:9739 4 5 return to the origin A. 11:3568 11:3568 1:6225 2 3 Fault detection and isolation 0:9733 0 0:1825 6 7 The numerical experiments begin once the quadrotor 6 00:9733 0:1825 7 6 7 has been stabilized in hover flight mode. In order to test 6 7 0:2271 0:2271 0:9676 6 7 the effectiveness of the proposed method, two different 6 7 L = : 6 7 6 1:4639 0 5:97397 scenarios are considered, partial and total faults. In all 6 7 6 7 cases, the observer is fed with noisy measurements of 01:4639 5:9739 4 5 the system input/output. The following describes each 11:3568 11:3568 1:6225 case. Guzma´n-Rabasa et al. 1235 Figure 4. Tracking and reference in 3D space. Partial fault. In this scenario, three different types of faults are induced to actuator 3. For the first fault, the magnitude of the control signal of the actuator required to stabilize the UAV in hover flight mode is reduced by 20% in 40\ t\ 50 s; in the second fault, the actuator is affected by sinusoidal variations of the speed in 70\ t\ 80 s; and finally, in 100\ t\ 110 s, an incipi- ent fault, which reduces the control signal of the actua- tor from 100% to 80% in 10 s, is induced. All of the induced faults provoke changes on the residual signals. The faults and their effects on the residuals are shown in Figure 5. The faults are isolated from the evaluation of the residuals as shown in Figure 6. Also, note that due to the robust approach, disturbances are well atte- nuated and do not affect the performance of the observers. Total fault. A total fault is induced by turning off the actuator 3 at t = 20 s. By considering the residual eva- luation, the fault is detected and achieved by the eva- luation of the residual as can be observed in Figure 7. Once the fault is detected, fault diagnosis is performed considering the dynamic behavior of roll and pitch as shown in Figure 2. The residuals exceed the thresholds, giving positive values, that is, r = 1 and r = 1, indi- 1 2 Figure 5. Residuals with partial fault and faults induced to cating the total fault in actuator 3 as shown in Figure 8 actuator 3. and Table 2. Similar analysis can be done to detect faults in the remaining actuators, and in each case, the actuator from 100% to 70%. It can be seen that both method can detect and isolate the fault, which demon- approaches have the same performance as they activate strates its effectiveness and applicability. the alarm when the fault occurs. The difference between the approaches is mainly Comparisons. To complement the results, Figure 9 that Rotondo et al.’s work considers the complete shows the comparison between the approach of quadrotor model, while in this work, the fault detection Rotondo et al. and the approach proposed in this and isolation are only applied to the rotational paper. Specifically, the figure shows the alarms of dynamics. Considering the complete model also implies actuator 1 for a partial fault scenario. The induced that more residues must be taken into account for the fault consists in a reduction of the control signal of the fault detection and isolation, and therefore, the 1236 Measurement and Control 52(9-10) Figure 6. Partial fault diagnosis on actuator 3. number of vertices is reduced (four vertices) since the method has been applied to the rotational dynamics of the quadrotor only, while in Rotondo et al., the num- ber of vertices exceeds 2000, leading to an equal num- ber of gains L ’s that have to be considered online. Clearly, the method presented here is easier to handle and implement in a real application. Comment on stability in case of an actuator failure Quadrotors are open-loop unstable systems and depend on a controller for their stabilization. Therefore, the stability analysis depends mainly in the controller which is out of the scope of this paper. Nevertheless, it is important to remark that, in a faulty situation, the con- troller may not be able to stabilize the system, depend- Figure 7. Residuals with total fault on actuator 3 at t =20 s. ing on the severity or intensity of the affectation. If the fault is partial and with very low intensity, a conven- inference matrix is larger and its online analysis is more tional controller may be able to stabilize the system complex; although, it is also acknowledged that faults regarding the fault as a model mismatch or disturbance. can be also detected and isolated while the UAV is For a more severe affectation, there are also passive/ moving in the space, not only in hover flight mode. active fault-tolerant control strategies reported in the Moreover, although the computation of the optimiza- literature that increase the actuator affectation toler- 31–33 tion problem (equation (16)) is performed offline, it is ance. If the fault in one actuator is total, an emer- always important to analyze its complexity since the gency protocol has to be activated such as the use of a time needed to solve the LMI grows exponentially as parachute; another option may be to give up control- the problem dimension grows. In this case, the LMI ling the quadrotor’s yaw angle, and use the remaining dimension is smaller than the one presented in actuators to achieve a horizontal spin. Although with Rotondo et al. Furthermore, it is well known that the a very low probability, two (opposed) or three actuators LMI problem could become unfeasible as the number can fail simultaneously and a specific control law is 35,36 of vertices grow. In the approach presented here, the required for those scenarios. Guzma´n-Rabasa et al. 1237 Figure 8. Total fault diagnosis on actuator 3. of a total actuator fault, and also, a mixed strategy H=H will be considered to increase fault sensitivity. Moreover, experimental validation of the developed fault detection and isolation algorithm in a device that allows a safe indoor test is also considered for future work. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Figure 9. Alarms for actuator 1 in partial fault scenario. Funding This work was supported by the Tecnolo´ gico Nacional de Conclusion Me´ xico (TecNM) under the program Apoyo a la Investigacio´n Cientı´fica y Tecnolo´gica (grant numbers 6210.17-P and This paper presented a robust fault diagnosis and isola- 6723.18-P). Additional support was provided by the Consejo tion scheme for actuator faults of a quadrotor UAV Nacional de Ciencia y Tecnologı´a under the program modeled as an LPV system. 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Journal

Measurement and ControlSAGE

Published: Nov 1, 2019

Keywords: Fault diagnosis; quadrotor unmanned aerial vehicle; linear parameter-varying systems; fault detection; actuator fault; linear parameter-varying observer

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