2. Lagrangian modelling

There is a huge amount of literature about Lagrange's equations of movement, however, see for example Ref. [8]. The structure of a Planar-Vertical-Take-Off and Landing (PVTOL) is represented in Figure 1.

The absolute linear position to the PVTOL is defined in the inertial frame x � y � z axes with two generalized coordinates <sup>ξ</sup><sup>T</sup> <sup>¼</sup> <sup>Δ</sup> <sup>½</sup> y z �. One additional generalized coordinate, the angular position,

Figure 1. Schema of the PVTOL system.

isolation of faults could also be used. An idea consists in taking advantage of the structure given by the Lagrangian model of a PVTOL in order to develop an algorithm for the detection, isolation

Many research studies dealing with the fault detection and isolation (FDI) problem have been already published, most of them deals with linear systems, see for instance Refs. [1–3]. On the other hand, for non-linear systems, some solutions exist, based on the inherited characteristics, see Refs. [4, 5] for more details. The most common approach used for FDI is the hardware redundancy; however, this approach normally represents an increment in weight and economical cost of the aircraft. In order to avoid this problem, somemathematical relations could be used, the simplest way is to compare two or more internal signals, having as goal to create a residue, which, in fact will be zero if the system is working normally and different from zero if not. In order to create such relations it is common to exploit some intrinsic characteristics of the systems. See for instance Ref. [6]. Diagnosis for the PVTOL system has been considered previously using a Hamiltonian formalism [7]. A Lagrangian formalism is used to model a PVTOL in order to obtain an aircraft model. The Euler-Lagrange model of the PVTOL is used to develop an algorithm for fault diagnosis. Diagnosis implies the detection, isolation and identification of a fault. The considered approach is based on the knowledge of a system model as well as the model of the possible faults. The idea is to use non-linear decoupling approach to derivate a set of subsystems, each related to a specific fault or a set of faults. An observer-based residual generation is designed for each subsystem. Detection and isolation of faults can be reached at this stage, for fault identification a kind of approximated inversion algorithm to meet the different diagnostic levels. The results are obtained taking advantage of the structure given by the Euler-Lagrange modelling of the PVTOL

as well as from recent results related to observer design and fault identification.

approach can be used effectively for the diagnosis of a PVTOL system.

2. Lagrangian modelling

represented in Figure 1.

generalized coordinates <sup>ξ</sup><sup>T</sup> <sup>¼</sup>

Fault diagnosis algorithms can be developed for a more or less general Euler-Lagrange model of a system, which, in fact, also include a PVTOL system. Fault diagnosis includes detection, isolation and identification of faults. In order to meet a diagnosis task, an observer-based residual generator is designed in order to determine whether a fault is present. A decoupling approach is used in order to guarantee also a fault isolation task. As discussed, both steps could be systematically developed for the considered system model. Further, fault isolation is approached using a kind of approximated system inversion to develop approximated fault estimation through dynamic inversion of the corresponding residual equation. The schema is shown using a specific example of a PVTOL. As presented in the results, the proposed

There is a huge amount of literature about Lagrange's equations of movement, however, see for example Ref. [8]. The structure of a Planar-Vertical-Take-Off and Landing (PVTOL) is

The absolute linear position to the PVTOL is defined in the inertial frame x � y � z axes with two

<sup>Δ</sup> <sup>½</sup> y z �. One additional generalized coordinate, the angular position,

and identification of faults.

136 Lagrangian Mechanics

is defined in the inertial frame. Note that the pitch angle θ, i.e. the rotation angle around the yaxis, and yaw angle Ψ, i.e. the rotation of the PVTOL around the z-axis, are zero. The only angular movement is the roll angle φ, i.e. the rotation around the x-axis.

$$\xi = \begin{bmatrix} y \\ z \end{bmatrix}, \quad \eta = \phi, \ \eta = \begin{bmatrix} y \\ z \\ \phi \end{bmatrix} \tag{1}$$

The origin of the body frame (also the origin of the inertial frame) is the centre of mass of the PVTOL system. The PVTOL is assumed to have a symmetric structure with the two arms aligned with the body x-axis. The inertia is represented by Jx.

The Lagrangian is defined as the sum of kinetic energy minus the potential energy ðEpotÞ. In the case of the PVTOL, the kinetic energy consist of two parts, one related to the translational energy ðEtranÞ and the second related to the rotational energy ðErotÞ:

$$\mathcal{L}(q, \dot{q}) = E\_{tran} + E\_{rot} - E\_{pot} \tag{2}$$

The movement equations of Lagrange are given by

$$\frac{d}{dt}\left[\frac{\partial \mathcal{L}(q,\dot{q})}{\partial \dot{q}}\right] - \frac{\partial \mathcal{L}(q,\dot{q})}{\partial q} = \begin{bmatrix} f\_y \\ f\_z \\ \dot{\ell} \end{bmatrix} \tag{3}$$

where f <sup>y</sup> represent the generalized forces on the y-axis, f <sup>z</sup> represent the generalized forces on the z-axis and ℓ is the torque.

For the PVTOL results:

$$E\_{\rm tran} = \frac{1}{2} m[\dot{y} \quad \dot{z}] \begin{bmatrix} \dot{y} \\ \dot{z} \end{bmatrix} \tag{4}$$

$$E\_{\rm rot} = \frac{1}{2} J\_{\rm x} \omega^2 = \frac{1}{2} J\_{\rm x} \dot{\phi}^2 \tag{5}$$

$$E\_{\rm pot} = \text{mgz} \tag{6}$$

So that the Lagrangian results

$$\mathcal{L}(q, \dot{q}) = \frac{1}{2}m\dot{y}^2 + \frac{1}{2}m\dot{z}^2 + \frac{1}{2}J\_x\dot{\phi}^2 - mgz \tag{7}$$

and the terms

$$\begin{bmatrix} \frac{\partial \mathcal{L}}{\partial \dot{q}} = \begin{bmatrix} m\dot{y} \\ m\dot{\bar{z}} \\ J\_x \dot{\phi} \end{bmatrix} \end{bmatrix} \tag{8}$$

$$
\frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{\boldsymbol{\phi}}} \right) = \begin{bmatrix} m\ddot{\boldsymbol{\chi}} \\ m\ddot{\boldsymbol{z}} \\ J\_x \dot{\boldsymbol{\omega}} \end{bmatrix} = \begin{bmatrix} m\ddot{\boldsymbol{\chi}} \\ m\ddot{\boldsymbol{z}} \\ J\_x \ddot{\boldsymbol{\phi}} \end{bmatrix} \tag{9}
$$

$$
\frac{\partial \mathcal{L}}{\partial q} = \begin{bmatrix} 0 \\ mg \\ 0 \end{bmatrix} \tag{10}
$$

The generalised forces (in the inertial frame) are given by

$$f\_y = \cos\left(\phi\right)\mathcal{U}\_y - \sin\left(\phi\right)\mathcal{U}\_z \tag{11}$$

$$f\_z = \sin\left(\phi\right)\mathcal{U}\_y + \cos\left(\phi\right)\mathcal{U}\_z \tag{12}$$

where Uz represents the total thrust force (the sum of the forces of each rotor), acting on the z-axis of the body frame. Uy corresponds to the side forces on the y-axis of the body frame. The moment acting on the rolling angle is given by ℓ.

The movement equations are given by

$$
\begin{bmatrix} m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & f\_x \end{bmatrix} \begin{bmatrix} \ddot{y} \\ \ddot{z} \\ \ddot{\phi} \end{bmatrix} + \begin{bmatrix} 0 \\ mg \\ 0 \end{bmatrix} = \begin{bmatrix} \cos(\phi) & -\sin(\phi) & 0 \\ \sin(\phi) & \cos(\phi) & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} U\_y \\ U\_z \\ \ell \end{bmatrix} \tag{13}$$

#### 3. Diagnosis approach

Fault diagnosis aim to detect the fault occurrence in the functional units of the system, as well as to classify the different faults and to determine the type, magnitude and cause of faults, which leads to undesired behaviour of the whole system. The fault diagnosis can be achieved by hardware redundancy or software redundancy also called analytical redundancy. One technique of fault diagnosis is the model-based fault diagnosis, which employs software redundancy.

In the model-based fault diagnosis technique, the system behaviour is online reconstructed by a mathematical model, which is implemented in the software form. In this scheme, the system model run in parallel to the system and both of them are driven by the same control inputs. Thus, in the fault-free case, reconstructed system variables by the system model follow the corresponding real system variables and show a derivation in the faulty case.

A comparison of the measured system variables with their estimates by the system model is called residual. Thus, a residual signal includes the fault effect, and ideally if the residual signal is different from zero, then a fault has occurred otherwise the system is fault free. The residual generation process is carried out in two stages, first, the system outputs have to be estimated, then, the difference between those signals and the signal coming from sensors is computed [9].

Figure 2 shows the general scheme for residual generation using a model-based fault diagnosis technique.

In this contribution, a fault diagnosis for systems with model Euler-Lagrange is presented. A model-based fault diagnosis technique with analytical redundancy is used to obtain a residual generation.

Consider a dynamic system without faults described by the following Euler-Lagrange equations

$$
\frac{d}{dt} \left[ \frac{\partial L(q, \dot{q})}{\partial \dot{q}} \right] - \frac{\partial L(q, \dot{q})}{\partial q} = \tau,\\
$$

$$
y\_o = q,\tag{14}
$$

where τ∈ R<sup>n</sup> is the vector of generalized forces, q∈ R<sup>n</sup> is the vector of generalized coordinates, L is the Lagrangian and y is the vector output.

In this work, additive faults in control input and sensor are considered. The Euler-Lagrange model of the faulty system is defined as

Figure 2. General scheme for residual generation.

For the PVTOL results:

138 Lagrangian Mechanics

So that the Lagrangian results

and the terms

<sup>E</sup>tran <sup>¼</sup> <sup>1</sup> 2

<sup>E</sup>rot <sup>¼</sup> <sup>1</sup> 2

<sup>L</sup>ðq;q\_Þ ¼ <sup>1</sup>

d dt

The generalised forces (in the inertial frame) are given by

moment acting on the rolling angle is given by ℓ.

3 5 y€ z€ φ€ 3 5 þ

0 mg 0

3 5 ¼

2 4

2 4

The movement equations are given by

2 4

3. Diagnosis approach

m 0 0 0 m 0 0 0 Jx

∂L ∂q\_ � �

2 my\_ <sup>2</sup> <sup>þ</sup> 1 2 mz\_ 2 þ 1 2

> ∂L ∂q\_ ¼

> > ¼

∂L ∂q ¼

<sup>m</sup><sup>½</sup> <sup>y</sup>\_ <sup>z</sup>\_ � <sup>y</sup>\_

Jxω<sup>2</sup> <sup>¼</sup> <sup>1</sup> 2

> my\_ mz\_ Jxφ\_

> > 3 5 ¼

0 mg 0

3

3

my€ mz€ Jxφ€ 3

f <sup>y</sup> ¼ cos ðφÞUy � sin ðφÞUz (11)

f <sup>z</sup> ¼ sin ðφÞUy þ cos ðφÞUz (12)

cos ðφÞ � sin ðφÞ 0 sin ðφÞ cos ðφÞ 0 0 01

2 4

2 4

my€ mz€ Jxω\_

> 2 4

where Uz represents the total thrust force (the sum of the forces of each rotor), acting on the z-axis of the body frame. Uy corresponds to the side forces on the y-axis of the body frame. The

> 2 4

Fault diagnosis aim to detect the fault occurrence in the functional units of the system, as well as to classify the different faults and to determine the type, magnitude and cause of faults, which leads to undesired behaviour of the whole system. The fault diagnosis can be achieved by

2 4

z\_ � �

Jxφ\_ <sup>2</sup> (5)

Jxφ\_ <sup>2</sup> � mgz (7)

5 (8)

5 (10)

3 5 2 4

Uy Uz ℓ

3

5 (13)

5 (9)

Epot ¼ mgz (6)

(4)

$$\begin{split} \frac{d}{dt} \left[ \frac{\partial L(q, \dot{q})}{\partial \dot{q}} \right] - \frac{\partial L(q, \dot{q})}{\partial q} = (\tau + QF\_a), \\ y\_f = (q + NF\_s), \end{split} \tag{15}$$

where Fa ∈ R<sup>n</sup> is the vector of control input faults, Q ∈ R<sup>n</sup>�<sup>n</sup> is a constant matrix, Fs ∈ R<sup>n</sup> is the vector of sensor faults and N ∈ R<sup>n</sup>�<sup>n</sup> is a constant matrix.

Assumption 1. Consider an Euler-Lagrange system with faults described by Eq. (15) and the system behaviour is on line reconstructed by the Euler-Lagrange system without faults Eq. (14), then the faults presented in the system Eq. (15) can be detected by the residual generator

$$r(t) = y\_f(t) - y\_o(t) \tag{16}$$
