Hybrid modeling based double-granularity fault detection and diagnosis for quadrotor helicopter

Nonlinear Analysis: Hybrid Systems 21 (2016) 22–36

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Hybrid modeling based double-granularity fault detection and diagnosis for quadrotor helicopter Yue Wang a,b , Bin Jiang a,∗ , Ningyun Lu a , Jun Pan b a

College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

b

The 28th Research Institute of China Electronics Technology Group Corporation, Nanjing 210017, China

article

info

Article history: Received 17 July 2014 Accepted 22 December 2015 Keywords: Hybrid modeling Fault detection and diagnosis Quadrotor helicopter Structure fault

abstract Fault detection and diagnosis (FDD) is an effective technology to assure the safety and reliability of quadrotor helicopters. However, there are still some unsolved problems in the existing FDD methods, such as the trade-offs between the accuracy and complexity of system models used for FDD, and the rarely explored structure faults in quadrotor helicopters. In this paper, a double-granularity FDD method is proposed based on the hybrid modeling of a quadrotor helicopter which has been developed in authors’ previous work. The hybrid model consists of a prior model and a set of non-parametric models. The coarse-granularity-level FDD is built on the prior model which can isolate the faulty channel(s); while the fine-granularity-level FDD is built on the nonparametric models which can isolate the faulty components in the faulty channel. In both coarse and fine granularity FDD procedures, principal component analysis (PCA) is adopted for online fault detection. Using such a double-granularity scheme, the proposed FDD method has inherent ability in detecting and diagnosing structure faults or failures in quadrotor helicopters. Experimental results conducted on a 3-DOF hover platform can demonstrate the feasibility and effectiveness of the proposed hybrid modeling technique and the hybrid model based FDD method. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Quadrotor helicopters are typical modern complex systems, and have in general more complicated aerodynamic characteristics and quite special flight attitudes compared with fixed-wing aircrafts [1,2]. On one hand, they have placed greater demands on accurate system model and robust controller design to assure flight quality and safety, which has raised the great interest of research in the field of control theory and application. On the other hand, quadrotor helicopters are prone to various faults or failures along with the growth of running time; therefore, Fault Detection and Diagnosis (FDD) is becoming a new hotspot of quadrotor helicopters to assure their safety and reliability. FDD has been the subject of intensive research in various research fields for more than 40 years, and fruitful results have been reported in the literatures and books [3–9]. Faults in a control system are usually classified into actuator fault, sensor fault and structure fault [10]. For the flight control systems of quadrotor helicopters, Table 1 summarizes the reported faults and the corresponding FDD methods [11–24]. Despite these encouraging achievements, there are still many problems unsolved in the existing FDD methods when applied to quadrotor helicopters, such as:

∗

Corresponding author. E-mail address: [email protected] (B. Jiang).

http://dx.doi.org/10.1016/j.nahs.2015.12.005 1751-570X/© 2016 Elsevier Ltd. All rights reserved.

Y. Wang et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 22–36

23

Table 1 Common faults in quadrotor helicopters and the employed FDD methods. Methods

Luenberger observers

Faults

Advantages

Bias, dead zone, scale factor on accelerometer [11] Bias on velocity and mach measurements [12] Bias elevator or pitch rate sensor [13]

Small false alarm rate Short detection delay

Computational burden

Robustness to model uncertainty

Difficult to distinguish faults from unmodeled disturbances

Control surfaces (loss of effectiveness, locking) [14]

Isolation of simultaneous faults

Loss of control effectiveness [15,16]

Same advantages as Luenberger observers Gaussian measurement noise and state perturbations are taken into account

Kalman filters Failures of sensors in an engine [17]

Drawbacks

Well-established for linearized models only; Gaussian assumptions are not always valid

Bias on sensor in flight control surface [18] Sliding mode observers

Bias in IMU [19] Drift in rudder throttle [20]

Fault estimation Quick convergence Estimation of some disturbances

Collective control stuck [21] Neural networks

Bias/drift of IMU sensors or actuators [22] Tail or wing damage [23]

Require no dynamical models

Computational burden Difficult in parameter tuning Choice of network structure may be difficult Huge on-line learning time Learning convergence not guaranteed

Elevator bias [24]

(1) Most FDD methods use linearized system models, which may pose strong constraints on their application potentials; (2) Most FDD methods are purely model-based, while the measurement data that contain rich information on system behaviors and incipient faults are not fully utilized; (3) Sensor and actuator faults have been extensively studied so far; however, structure faults are rarely mentioned. Focusing on the above unsolved problems, this paper dedicates to improve the modeling accuracy to obtain a reliable and efficient FDD method. Hybrid modeling strategy [25–27] is an effective way to get more accurate models and to make full use of measurement data, which has been used in various industrial processes and aerospace fields [28–30]. A hybrid modeling strategy for a quadrotor helicopter based on physical effect analysis and nonlinearity measure has been proposed by the authors [31,32]. It clarified the scopes of the prior model and the nonparametric model according to the physical effects, and used fuzzy inference to select proper linearization methods in accordance with the nonlinearity degrees. The prior model was used to assure the global generalization performance; while the nonparametric model can explain the un-modeled characteristics so as to increase the accuracy and obtain a good local approximation performance. Based on the hybrid modeling technique, this paper focuses on hybrid model based fault detection and diagnosis. The key contributions can be summarized as follows: (1) A double-granularity fault diagnosis method is presented based on the hybrid model of a quadrotor helicopter. The coarse-granularity-level diagnosis is built on the prior model which can be used to isolate the faulty channel(s). The fine-granularity-level diagnosis is built on the nonparametric model to isolate the faulty component(s) in the faulty channel(s). (2) The proposed FDD method has superior ability in detecting and diagnosing structure faults. As verified in Section 4, two structure faults are simulated on a quadrotor helicopter experimental platform. One is a broken-rotor-blade fault and the other is a motor fault. Experimental results can show the effectiveness and superiority of the proposed FDD method for structure faults. 2. Quadrotor helicopter Quadrotor helicopter is a kind of multi-rotor helicopter that is lifted and propelled by four rotors. Such a four-rotor design allows quadrotor helicopters to be highly reliable and maneuverable. The main mechanical components in a quadrotor helicopter include a frame, four propellers, four electric motors, and electrical components such as electronic speed control module, on-board computer or controller board, and battery. Attitude control of a quadrotor helicopter can be achieved by adjusting the rotor angular speeds. Fig. 1 shows the schematic system of a quadrotor helicopter, where ϕ , θ and φ denote the yaw angle, pitch angle and roll angle, respectively; Ω 1–Ω 4 are angular speeds of the four propellers; F1–F4 are lift forces; and M1–M4 denote the corresponding motors. Adjusting the speeds of M1 and M3 can change the pitch angle; while adjusting the speeds of M2 and M4 can change

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Y. Wang et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 22–36

Fig. 1. Control principle of quadrotor helicopters. Table 2 Physical effects in a quadrotor helicopter. Effects

Sources

Outcome variables

Major effects

Propeller rotation Blades flapping Change in propeller rotation speed Center of mass position

Roll torque, Pitch torque Vertical force, Resistance torque Yaw torque Gravity

Change in orientation of the rigid body Change in orientation of the propeller plane The aircraft near the surface

Body gyroscopic torque Rotor gyroscopic torque Air velocity

Aerodynamic effect

Minor effects

Inertial counter torques Gravity effect Gyroscopic effect Near surface effect

the roll angle. Besides the clear flight principle, quadrotor helicopters have in general a large amount of data measured on attitude angles, motor voltages, aerodynamic torques and gyroscopic torques, which are stored in on-board computer. The availability of system mechanism and various variables’ measurements offers a solid foundation for obtaining a more accurate system model via hybrid modeling. According to the Ref. [33], we make the following assumptions before modeling: (1) Structure of the quadrotor helicopter is strictly symmetrical and rigid; (2) Thrust and drag are proportional to the output voltages of the motors; 3. Hybrid modeling of quadrotor helicopter Quardrotor helicopter dynamics are in essence nonlinear [31]. However, the original nonlinear models of quardrotor helicopters are rarely directly used for controller design because most of the controller design strategies demand parameterized models. Linearization is a widely used technique for nonlinear systems to obtain parameterized models. For quardrotor helicopters, it is often assumed that quardrotor helicopter dynamics can be accurately modeled as linear for attitude and altitude control. In fact, this assumption is only reasonable in hovering flight regime with slow velocities [32]. This paper focuses on the derivation of a new form of dynamic equations for both control and FDD purposes, which can take the major nonlinearities into consideration in a hybrid modeling framework. Analysis of physical effects is the first step in hybrid modeling. Main physical effects acting on a quadrotor helicopter are listed in Table 2. The aerodynamic effect, inertial counter torques and gravity effect can be considered as the major physical effects, since they determine the global characteristics of a quadrotor helicopter. Description of these major physical effects falls into the scope of prior modeling. Gyroscopic effect and near surface effect can be considered as the minor physical effects, which determine the local characteristics. Nonparametric modeling techniques are more suitable for describing the minor physical effects. 3.1. The prior model Aerodynamic torques (including roll, yaw and pitch torques) are the major torques in a quadrotor helicopter [34], which are caused by the pull forces and the resistances generated by rotors’ rotation. For control or FDD purpose, the prior model of a quadrotor helicopter usually takes only the aerodynamic torques (i.e., the major physical effects) into consideration [35,36], which can be formulated as,

  Jxx φ¨ = Kf (V1 − V3 )l Jyy θ¨ = Kf (V2 − V4 )l   Jzz ϕ¨ = Kt ,c (V1 + V3 − V2 − V4 )

(1)

where ϕ¨ , θ¨ and φ¨ denote the yaw, pitch and roll angular accelerations; Jxx , Jyy and Jzz are the equivalent moments of inertia about roll, pitch and yaw axis, respectively; Kf is the propeller force-thrust constant, Kt ,c is the counter rotation propeller torque-thrust constant, V1 –V4 are motor voltages, l is the distance between the pivot to each motor.

Y. Wang et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 22–36

˙ T and u = [V1 By defining x = [ϕ θ φ ϕ˙ θ˙ φ] a quadrotor helicopter can be written as,

V3

V2

25

V4 ]T as the state and input vectors, the prior model of

x˙ = Ax + Bu

(2)

y = Cx where,

0 0 0 A= 0  0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

 0  0  0  K  t ,c  J B=  zz  lKf  J  yy 

0 0 1 , 0  0 0



0 0 0 K t ,c Jzz

−

0

lKf Jyy 0

0 0 0

−

K t ,c Jzz 0

lKf

0 0 0



   Kt ,c   −  Jzz  ,   0    lKf  −

 C =

1 0 0

0 1 0

0 0 1

0 0 0

0 0 0

0 0 . 0



Jxx Jxx Eq. (1) or (2) is a linearized model of a quadrotor helicopter, which has been widely used in existing literatures. Such a linear model is simple and easy to use in controller design. However, it indeed poses strong constraints in practical applications, as it is valid only for hovering flight regime with low velocities. In order to reduce the conservatism in controller design and improve the robustness of the controlled system, it is always attractive to improve the accuracy of the system model. 3.2. The nonparametric models Besides aerodynamic torques, gyroscopic torques should also be considered in system modeling, since they determine the local characteristics. By doing so, a more accurate model can be obtained as,

 ¨ ˙ ˙  Jxx φ = Kf (V1 − V3 )l + (Jyy − Jzz )ϕ˙ θ + Jr θ (Ω1 + Ω3 − Ω2 − Ω4 ) ¨ ˙ ˙ Ω1 + Ω3 − Ω2 − Ω4 ) (3) Jyy θ = Kf (V2 − V4 )l + (Jzz − Jxx )ϕ˙ φ − Jr φ(   ˙1 +Ω ˙3 −Ω ˙2 −Ω ˙ 4) Jzz ϕ¨ = Kt ,c (V1 + V3 − V2 − V4 ) + (Jxx − Jyy )θ˙ φ˙ + Jr (Ω ˙ 1 –Ω ˙ 4 denote angular where ϕ˙ , θ˙ and φ˙ denote yaw, pitch and roll angular speeds; Jr is the motor rotor moment of inertia; Ω accelerations for the four propellers. Since nonlinear and coupling terms exist in the model Eq. (3), it brings difficulties to system analysis and controller design. In order to reduce the complexity of controller design without scarifying the accuracy of the system model, the authors have proposed to select proper linearization methods by measuring the nonlinearity degree for the nonlinear terms in the hybrid model, where a nonlinear measure method [37–39] is adopted and an approximate lower bound for the nonlinearity U measure ΘN LB is computed by, U ΘN LB

 

≥

sup

A∈A,ω∈Ξ

1 

A

A20 (ω, A) +

∞  A2 (ω, A) k

k=2

2

.

(4)

Remark 1. Nonlinearity measure of an input–output stable causal system N : Ua → Y for input signals u ∈ U ⊆ Ua is ∥G[u]−N [u]∥ defined by the nonnegative number φNU = infG∈Y supu∈U ∥N [u]∥ VT , where G : Ua → Y is a linearization operator, and the VT

norm ∥·∥VT is defined as ∥α∥VT =

  T 1 T

0

α 2 (t )dt.

Remark 2. The nonlinearity measure in Eq. (4) can be obtained by taking the input set ULB to be a series of sinusoids parameterized by A and Ξ , where ULB = ⟨u|u(t ) = A sin(ωt ), A ∈ A, ω ∈ Ξ ⟩. The steady state output can be formulated ∞ as y(t ) = A0 + k=1 Ak sin(kωt + φk ). Remark 3. A mapping from nonlinearity degrees to linearization methods can be implemented by fuzzy inference since the U range of nonlinearity measure ΘN LB is from 0 to 1 which is in accordance with the universe of fuzzy set. Four linearization methods are taken into consideration [40]: Linear least square fit (LLSF), Taylor series expansion at equilibrium points (TSE), T-S fuzzy inference (TSFI) [41], and subspace system identification (SSI) [42]. According to the nonlinearity degree and the characteristics of these linearization methods, Mamdani-type fuzzy inference model is set up to select proper linearization methods. Table 3 shows the linearization methods used to obtain the nonparametric models, where aij and bij stand for the term in the itch row and jet column in the matrices A and B, respectively. More details can be referred in Ref. [31].

26

Y. Wang et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 22–36 Table 3 Linearization methods used to obtain the nonparametric models for quadrotor helicopter. Physical effect

Channel

Mathematical expression

Elements affected in matrices A and B in Eq. (2)

Linearization method

Body gyroscopic effect

Roll Pitch Yaw

(Jyy − Jzz )ϕθ (Jzz − Jxx )ϕφ (Jxx − Jyy )θφ

a64 , a65 a54 , a56 a45 , a46

TSE TSE TSE

Rotor gyroscopic effect

Roll Pitch Yaw

˙ Ω1 + Ω3 − Ω2 − Ω4 ) Jr θ( −Jr φ(Ω1 + Ω3 − Ω2 − Ω4 ) Jr (Ω1 + Ω3 − Ω2 − Ω4 )

b61 , b62 , b63 , b64 , a65 b51 , b52 , b53 , b54 , a56 b41 , b42 , b43 , b44

TSFI TSFI SSI

3.3. The hybrid model With the prior model Eq. (2) and the nonparametric models in Table 3, the hybrid model of a quadrotor helicopter is given by

     ¨ ˙ ˙  Jxx φ = Kf (V1 − V3 )l + fTSE  (Jyy − Jzz )ϕ˙ θ  + fTSFI Jr θ (Ω1 + Ω3 − Ω2 − Ω4 )  ˙ Ω1 + Ω3 − Ω2 − Ω4 ) Jyy θ¨ = Kf (V2 − V4 )l + fTSE (Jzz − Jxx )ϕ˙ φ˙ − fTSFI Jr φ(       ˙ ˙ ˙1 +Ω ˙3 −Ω ˙2 −Ω ˙ 4) Jzz ϕ¨ = Kt ,c (V1 + V3 − V2 − V4 ) + fTSE (Jxx − Jyy )θ φ + fSSI Jr (Ω

(5)

or in the compact form, x˙ = Ahybrid x + Bhybrid u y = C x.

(6)

The hybrid model Eq. (5) or Eq. (6) is essentially a linearized and parameterized model, so it inherits all advantages of the linear system model like Eq. (1) or Eq. (2). It is simple, straightforward and easy to use. But the hybrid model Eq. (5) is superior to the model Eq. (1) because it can describe the system’s behavior more precisely. More details on the hybrid model can refer to Ref. [31]. 4. A double-granularity FDD method 4.1. Key idea Clarification of modeling scopes based on analysis of physical effects enables the hybrid model to have clear structure, which offers potential conveniences for developing an efficient FDD method. Beyond that, FDD will also benefit from process data that have been involved in the modeling phase. Thus, a hybrid model based double-granularity FDD method is proposed here. A coarse-granularity-level FDD is implemented on the basis of the prior model, which dedicates to detect and diagnose the faults that change the global behavior of a quadrotor helicopter. A fine-granularity-level FDD is based on the nonparametric models to focus on the change of local behavior of a quadrotor helicopter. In both the coarse and fine granularity levels, principal component analysis (PCA) is adopted for fault detection and diagnosis. That is, in the proposed double-granularity FDD method, the hybrid model is used as a skeleton to provide different levels of fault information, based on which, PCA is used to perform fault detection and diagnosis. The proposed double-granularity FDD method can be depicted by Fig. 2. Details are discussed as follows. 4.2. Why using PCA? The reasons are listed as below. (1) Measurements on most process variables (input, output or state variables) are available on a quadrotor helicopter. Process measurements contain rich information on system’s behavior. They can be more informative than the observed output residuals in the traditional observer based FDD methods [43], because changes in correlation of all measurable variables can be very useful for FDD. (2) Applying the traditional observer based FDD methods on the basis of the hybrid model is not proper because it cannot make full use of the hybrid model. Process measurements and the deep understanding of the physical effects will be completely ignored. (3) PCA is indeed an effective and efficient data-driven FDD method, because it can focus on the change of correlation structure among process measurements and reveal more faulty information in the system.

Y. Wang et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 22–36

27

Fig. 2. Key steps of the proposed FDD method.

4.3. Why combining PCA and the hybrid model? The main disadvantage of applying PCA for FDD is that, PCA merges everything together, making it difficult to isolate and interpret the occurred faults. Fortunately, the developed hybrid model can offer a clear hierarchical structure. As mentioned above, a hybrid model contains prior model(s) and non-parametric model(s). Prior model is coarse since it focuses on the global performance of the system, from which we can grasp key information at the coarse granularity level. Non-parametric model is fine because it pays close attention to the local approximation performance, from which we can obtain detailed information at the fine granularity level. Combination of PCA and the hybrid model can enhance the performance of fault detection and diagnosis. 4.4. FDD based on the combination of the hybrid model and PCA Within the framework of hybrid modeling, PCA is applied several times in different levels of FDD procedures. As shown in Fig. 2, PCA is firstly applied for coarse-granularity-level fault detection based on the attitude angles and motor voltages. PCA is used a second time for coarse-granularity-level FDD based on the attitude angles, motor voltages and the aerodynamic torques. PCA is used a third time for fine-granularity-level FDD based on all data information (including three attitude angles, four motor voltages, three aerodynamic torques, three body gyroscopic torques and three rotor gyroscopic torques). Attitude angles and motor voltages can be directly measured by sensors, which are the primary variables. Aerodynamic torques, body gyroscopic torques and rotor gyroscopic torques are usually calculated by the model (i.e., the hybrid model Eq. (3)), which are the secondary variables.

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Y. Wang et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 22–36

4.4.1. Coarse-granularity-level fault detection In the coarse-granularity-level fault detection phase, only the primary variables (i.e., the three attitude angles and the four motor voltages) are monitored using PCA, as shown in Fig. 2. Let X denote the data matrix consisting of n samples of the seven primary variables {ϕ, θ , φ, V1 , V2 , V3 , V4 } at healthy status. PCA decomposes X as 7 

X = TP T =

tj pTj

(7)

j =1

tj = X p j where tj (n×1) is the principal component (PC) vector or score vector; pj (7×1) is the loading vector; and T and P are the score matrix and loading matrix, respectively. According to the mechanism analysis of quadrotor helicopters, the seven variables {ϕ, θ, φ, V1 , V2 , V3 , V4 } are not independent and there exists an inherent correlation structure among them. This indicates that the majority of variance information can be extracted by retaining the first few orthogonal PCs, and the dimension of original variables can be largely reduced accordingly. By retaining only the first A PCs, X can be approximated by A 

T

Xˆ = TP =

tj pTj

j =1

(8)

T = XP X = Xˆ + E where T (n × A) is the score matrix in PC subspace P (7 × A) is the corresponding loading matrix and Xˆ (n × 7) is the reconstruction of the original data matrix X ; E is the residual matrix. For online implementation, when a new sample xnew of {ϕ, θ , φ, V1 , V2 , V3 , V4 } is measured, it is projected onto the score subspace using the loading matrix P to get its score vector tnew and the residual vector enew by tnew = xnew P

(9)

T

enew = xnew − xˆ new = xnew · (I − P P ).

The following two statistics, SPE and T 2 , which subject to χ 2 and F distributions, respectively [44,45], are used for fault detection where δα2 and Tα2 are statistical control limits and α is the confidence level. Once SPE or T 2 exceeds the control limit, a fault is alarmed. SPE = e eT =

j =1 −1 T

T = x PΛ 2

m  (xj − xˆ j )2 ≤ δα2

T

P x ≤ Tα

  δα2 = θ1  Tα2 ∼

(10)

2

cα

2θ2 h20

θ1

A(n − 1) n−A

1/h0 θ2 h0 (h0 − 1)  +1+ θ12

(11)

FA,n−A,α .

4.4.2. Coarse-granularity-level FDD The objective of coarse-granularity-level fault diagnosis is to isolate the faulty channel(s). PCA is performed on the different groups of variables shown in Table 4. The variables are collected from the three channels (pitch, roll and yaw), which can be divided into three types: attitude angles, aerodynamic torques and body gyroscopic torques. With such structural information, it is easy to isolate different channel faults. Firstly, the SPE and T 2 values are calculated following the steps in Section 4.4.2 after a fault is detected. Then, contributions of the original variables to the out-of-control statistics are calculated for isolating the faulty channels. The contribution of xnew (j) to the ith PC score ti is calculated by [45], Cti ,xnew (j) =

/,

xnew (j)pi,j t i

(12)

where pi,j is the ith element in the jth column in the loading matrix P. For SPE, the contribution of xnew (j) is calculated by, CSPE ,xnew (j) = sign(xnew (j) − xˆ new (j)) ·

(xnew (j) − xˆ new (j))2 SPE

.

(13)

Y. Wang et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 22–36

29

Table 4 Variables considered in the coarse-granularity level fault diagnosis. Channel

Physical variables

Pitch

Pitch angle Pitch torque Body gyroscopic torque around pitch channel

Roll

Roll angle Roll torque Body gyroscopic torque around roll channel

Yaw

Yaw angle Yaw torque Body gyroscopic torque around yaw channel

Fig. 3. Experimental platform. (a) Quanser 3 DOF hover plant; (b) Four motors.

4.4.3. Fine-granularity-level FDD The objective of the fine-granularity-level fault diagnosis is to isolate the specific variable(s) in the faulty channel(s), based on the results of the coarse-granularity-level fault diagnosis. Implementation steps are the same as in Section 4.4.3. The four motor voltages and the three rotor gyroscopic torques are analyzed using PCA respectively, because they have close relationships with the physical components of a quadrotor helicopter. Therefore, in terms of such clear causality between the variables and the faulty components, precise fault information on structure failure can be obtained at the fine granularity level fault diagnosis. 5. Experimental results and discussion 5.1. Experimental platform The 3-DOF hover platform, as shown in Fig. 3, is used to verify the feasibility and effectiveness of the proposed hybrid modeling method and the hybrid model based FDD strategy. Table 5 lists the main parameters, so the control matrix in the prior model Eq. (2) is

 0  0  0   K t ,c   B =  Jzz  lK  f   Jyy  0

0 0 0 K t ,c Jzz lKf

−

Jyy 0

0 0 0

−

K t ,c Jzz 0

lKf Jxx

0 0 0



   0  0 Kt ,c     − 0   Jzz  = − 0.0036    0.042447 0   0  lKf − Jxx

5.2. Experimental setup The operating condition is set up as: (1) The desired pitch, roll and yaw angle are all 0.3°; (2) The basic voltage is 3.5 V.

0 0 0 −0.0036 −0.042447 0

0 0 0 0.0036 0 0.0424447

0 0   0 . 0.0036   0 −0.042447



30

Y. Wang et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 22–36 Table 5 Main parameters of the Quanser hover experimental platform. Symbol

Description

Value/unit

Kt ,c Kf Jxx Jyy Jzz l Jr

Rotation propeller torque-thrust constant Propeller force-thrust constant Equivalent moment of inertia about pitch axis Equivalent moment of inertia about pitch axis Equivalent moment of inertia about yaw axis Distance between pivot to each motor Motor rotor moment of inertia

−0.003 6 N m/V 0.118 8 N/V 0.0552 kg m2 0.0552 kg m2 0.110 kg m2 0.197 m 1.91E − 006 kg m2

Table 6 Simulation of structure faults in the quadrotor helicopter. Fault

Realization

Fault description

A B

Break off the front and back rotor blades Change voltages of the front and left motor

Rotor fault in the roll channel Motor faults in the yaw channel

Table 7 The coarse and fine-granularity-level variables. Granularity level

Variables

Coarse

Attitude angle Aerodynamic torque

Fine

Body gyroscopic torque Rotor gyroscopic torque Motor voltage

Fig. 4. Coarse-granularity-level fault detection results (SPE chart, Fault A).

Under this operating condition, 15 groups of normal test data are collected with 155 sampling points in each group to build the PCA model. As for faulty cases, 12 groups of test data are collected with the same number of sampling points in each group. Two structure faults are simulated, as shown in Table 6. Faults are injected at the very beginning of the experiments because it is difficult to simulate the structural faults in the middle of experiment. According to the key idea of double-granularity FDD method, the coarse and fine-granularity-level variables are summarized in Table 7. 5.3. Experimental results 5.3.1. For fault A Figs. 4 and 5 show the fault detection results. The fault can be detected by the SPE and T 2 monitoring charts at the beginning. Figs. 6–8 are the contribution plots for the three types of variables (attitude angles, aerodynamic torques and the body gyroscopic torques) at the coarse-granularity level. Apparently, the contribution rates of the roll attitude angle (in Fig. 6), the roll torque (in Fig. 7) and the body gyroscopic torque in roll channel (in Fig. 8) are the most significant, inferring that a fault has occurred in the roll channel. Focusing on the roll channel, the motors work well sincethe voltages are within the control limits in the SPE and T 2 monitoring as shown in Figs. 9 and 10. However, the two statistics exceed the control limits for the rotor gyroscopic torques as shown in Figs. 11 and 12, indicating that the faulty component should be the rotor in the roll channel. By the

Y. Wang et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 22–36

31

Fig. 5. Coarse-granularity-level fault detection results (T 2 chart, Fault A).

Fig. 6. Coarse-granularity-level contribution plot for attitude angles (Fault A).

Fig. 7. Coarse-granularity-level contribution plot for aerodynamic torques (Fault A).

Fig. 8. Coarse-granularity-level contribution plot for body gyroscopic torques (Fault A).

fine-granularity-level contribution plot shown in Fig. 13, it is easy to observe that the faulty components are the front and back rotors. It is in accordance with the fault information in Table 6. 5.3.2. For fault B As shown in Figs. 14 and 15, fault B can be detected by the SPE and T 2 monitoring charts. Figs. 16–18 are the coarsegranularity level contribution plots for the attitude angles, aerodynamic torques and the body gyroscopic torques. From Figs. 16 and 17, contributions of the roll and pitch channels are more significant than the yaw channel; however, the yaw channel is the most significant in Fig. 18. It implies that the coarse granularity level FDD may be not accurate because the

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Fig. 9. Fine-granularity-level fault detection for motor voltages (SPE chart, Fault A).

Fig. 10. Fine-granularity-level fault detection results for motor voltages (T 2 chart, Fault A).

Fig. 11. Fine-granularity-level fault detection for rotor gyroscopic torques (SPE chart, Fault A).

Fig. 12. Fine-granularity-level fault detection for rotor gyroscopic torques (T 2 chart, Fault A).

yaw, pitch and roll channels are highly coupled in quadrotor helicopters. Faults occurred in the yaw channel indeed influence the roll and pitch channel.

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Fig. 13. Fine-granularity-level contribution plot for rotor gyroscopic torques (Fault A).

Fig. 14. Coarse-granularity-level fault detection results (SPE chart, Fault B).

Fig. 15. Coarse-granularity-level fault detection results (T 2 chart, Fault B).

Fig. 16. Coarse-granularity-level contribution plot for attitude angles (Fault B).

Focusing on the yaw channel, the motor voltages exceed the control limits as shown in Figs. 19 and 20; while the rotors work well as shown in Figs. 21 and 22. This indicates that the fault occurs in the motor(s) of the yaw channel. According to the contribution plot in Fig. 23, the front and left motors have large contributions to this fault, which agrees well with the fault description in Table 6.

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Fig. 17. Coarse-granularity-level contribution plot for aerodynamic torques (Fault B).

Fig. 18. Coarse-granularity-level contribution plot for body gyroscopic torques (Fault B).

Fig. 19. Fine-granularity-level fault detection for motor voltages (SPE chart, Fault B).

Fig. 20. Fine-granularity-level fault detection for motor voltages (T 2 chart, Fault B).

6. Conclusion A hybrid modeling based double-granularity FDD method has been developed for quadrotor helicopters in this paper. In the hybrid modeling phase, the modeling scopes of the prior model and the nonparametric model are clarified based on analysis of physical effects; fuzzy inference is used to select the proper linearization method in accordance with nonlinearity degree. Then, a double-granularity FDD method is proposed based on the developed hybrid model. The coarse-granularitylevel FDD is implemented on the basis of the prior model, where the objective is to isolate the faulty channel(s). The

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Fig. 21. Fine-granularity-level fault detection for rotors (SPE chart, Fault B).

Fig. 22. Fine-granularity-level fault detection for rotors (T 2 chart, Fault B).

Fig. 23. Fine-granularity-level contribution plots for motor voltages (Fault B).

fine-granularity-level FDD based on the nonparametric models will focus on the change of local behavior of a quadrotor helicopter, where the objective is to isolate the specific variable(s) in the faulty channel(s). The experimental results on a 3-DOF quadrotor helicopter platform have shown the effectiveness of the proposed hybrid modeling technique and the hybrid model based FDD method. Acknowledgment This work is supported by National Natural Science Foundations of China (61490703, 61374141, 61533008). References [1] C. Pedro, L. Rogelio, E.D. Alejandro, Modeling and Control of Mini-Flying Machine, Springer, 2005. [2] O. Shakernia, Y. Ma, T.J. Koo, S.S. Sastry, Landing and unmanned air vehicles: vision-based motion estimation and nonlinear control, Asian J. Control 1 (3) (1999) 128–145. [3] J.J. Gertler, Survey of model-based failure detection and isolation in complex plants, IEEE Control Syst. Mag. 8 (6) (1988) 3–11. [4] M. Basseville, I.V. Nikiforov, Detection of Abrupt Changes: Theory and Application, Prentice-Hall, Englewood Cliffs, 1993. [5] K.M. Zhou, Z. Ren, A new controller architecture for high performance, robust, and fault-tolerant control, IEEE Trans. Automat. Control 46 (10) (2001) 1613–1618. [6] H. Yang, B. Jiang, V. Cocquempot, Qualitative fault tolerance analysis for a class of hybrid systems, Nonlinear Anal. Hybrid Syst. 2 (3) (2008) 846–861. [7] B. Jiang, M. Staroswiecki, V. Cocquempot, Fault accommodation for nonlinear dynamic systems, IEEE Trans. Automat. Control 51 (9) (2006) 1578–1583. [8] W.H. Wang, L.L. Li, D.H. Zhou, K.D. Liu, Robust state estimation and fault diagnosis for uncertain hybrid nonlinear systems, Nonlinear Anal. Hybrid Syst. 1 (1) (2007) 2–15.

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