4. Application results

The method presented in the previous section is applied in a PVTOL. Only additive faults are taken into account, the faults could affect sensors (y, z and φ) and the control inputs (uz, uy and ℓ), the faulty case is restricted to one fault at a time, meaning that it is assured that if a fault appears, it is impossible that another fault occurs. Once the fault has occurred, it still presents until the end of the simulation.

The faulty PVTOL system is defined as

$$
\begin{bmatrix} m\cos\left(\phi\right) & m\sin\left(\phi\right) & 0\\ -m\sin\left(\phi\right) & m\cos\left(\phi\right) & 0\\ 0 & 0 & J\_z \end{bmatrix} \begin{bmatrix} \ddot{y} \\ \ddot{z} \\ \ddot{\phi} \end{bmatrix} + \begin{bmatrix} mg\sin\left(\phi\right) \\ mg\cos\left(\phi\right) \\ 0 \end{bmatrix} = \begin{bmatrix} U\_y \\ U\_z \\ \ell \end{bmatrix} + \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} f\_{uy} \\ f\_{uz} \\ f\_\ell \end{bmatrix},\tag{17}
$$

$$y\_o = \begin{bmatrix} y \\ z \\ \phi \end{bmatrix} + \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} f\_{sy} \\ f\_{sz} \\ f\_{s\phi} \end{bmatrix},\tag{18}$$

where f uy is the fault in the control input Uy, f uz is the fault in the control input Uz, f <sup>ℓ</sup> is the fault in the control input <sup>ℓ</sup>, <sup>f</sup> sz is the fault in the sensor of position in the vertical movement, <sup>f</sup> sy is the fault in the sensor of position in the horizontal movement and f <sup>s</sup><sup>φ</sup> is the fault in the sensor of angle φ.

Among the different detection methods available in the literature [2], the threshold-based approach is one of the most common thanks to its simplicity and accuracy. The principle of this approach is based on the idea that the parameters of the system (e.g. mass and dimensions) could vary because of the measurement or estimation errors. Those data are used to determine a threshold, which is computed by varying the internal parameters of the system in a certain � percentage. This is carried out in order to avoid false alarms caused by the difference between the mathematical model and the real system.

The internal parameters of the PVTOL that may vary are the mass (m) and the inertia (Jx), in order to cover the worst case scenario, both parameters vary at the same time þ10 and �10%. As a result, the thresholds are fixed as depicted in Figure 3.

Figure 3. Amplitudes of the detection thresholds. - -, detection threshold; —, residues.

d dt

vector of sensor faults and N ∈ R<sup>n</sup>�<sup>n</sup> is a constant matrix.

4. Application results

The faulty PVTOL system is defined as

m cos ðφÞ m sin ðφÞ 0 �m sin ðφÞ m cos ðφÞ 0 0 0 Jx

of the simulation.

140 Lagrangian Mechanics

2 6 4

of angle φ.

presented in the system Eq. (15) can be detected by the residual generator

3 7 5

2 4

yo ¼

difference between the mathematical model and the real system.

y€ z€ φ€ 3 5 þ

> y z φ

2 6 4

where f uy is the fault in the control input Uy, f uz is the fault in the control input Uz, f <sup>ℓ</sup> is the fault in the control input <sup>ℓ</sup>, <sup>f</sup> sz is the fault in the sensor of position in the vertical movement, <sup>f</sup> sy is the fault in the sensor of position in the horizontal movement and f <sup>s</sup><sup>φ</sup> is the fault in the sensor

Among the different detection methods available in the literature [2], the threshold-based approach is one of the most common thanks to its simplicity and accuracy. The principle of this approach is based on the idea that the parameters of the system (e.g. mass and dimensions) could vary because of the measurement or estimation errors. Those data are used to determine a threshold, which is computed by varying the internal parameters of the system in a certain � percentage. This is carried out in order to avoid false alarms caused by the

2 6 4 2 6 4

∂Lðq;q\_Þ ∂q\_ � �

� <sup>∂</sup>Lðq;q\_<sup>Þ</sup>

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

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

The method presented in the previous section is applied in a PVTOL. Only additive faults are taken into account, the faults could affect sensors (y, z and φ) and the control inputs (uz, uy and ℓ), the faulty case is restricted to one fault at a time, meaning that it is assured that if a fault appears, it is impossible that another fault occurs. Once the fault has occurred, it still presents until the end

> mg sin ðφÞ mg cos ðφÞ 0

> > 100 010 001

> 3 7 5

Uy Uz ℓ

3 5 þ 2 4

100 010 001 3 5

5; (18)

2 4

f uy f uz f ℓ

3

5, (17)

2 4

f sy f sz f sφ 3 7 7

<sup>∂</sup><sup>q</sup> ¼ ð<sup>τ</sup> <sup>þ</sup> QFaÞ;

yf ¼ ðq þ NFsÞ;

rðtÞ ¼ yfðtÞ � yoðtÞ (16)

(15)

As explained in the beginning of this section only additive faults are taken into account, since the controller is designed to stabilize the system in hover flight, the fault amplitude is defined as a percentage of the initial value for sensors and a percentage of the maximum amplitude of the input control. This percentage is fixed �10% for sensors and �5% for control inputs. The faults are triggered 7 seconds after the beginning of the simulation and it is persistent until the end.

In order to detect the fault, six different residues are computed, for this, it is assumed that the entire state is available, according to the previous section as follows:

$$R\_1 = y\_f - y \tag{19}$$

$$R\_2 = z\_f - z \tag{20}$$

$$R\_3 = \phi\_f - \phi \tag{21}$$

$$R\_4 = \dot{y}\_f - \dot{y} \tag{22}$$

$$R\_{\mathfrak{F}} = \dot{z}\_f - \dot{z} \tag{23}$$

$$R\_{\mathsf{G}} = \dot{\phi}\_f - \dot{\phi} \tag{24}$$

where the suffix<sup>f</sup> denotes the signal coming from the sensor.

#### 4.1. Sensor faults

The sensor faults considered in this work affect the vertical measurement (z), the horizontal (y) and the inclination angle (φ), as explained before the amplitudes of the faults are 0.2 m, 0.1 m and 0.2, respectively. The controller is designed to decouple the sensor faults, as a result each fault is independent of the others and by consequence all of them are detectable and isolable, thanks to their different fault signatures. Figures 4–6 depict the sensor faults.

Figure 4. Fault affecting y sensor. - -, detection threshold; —, residues.

Figure 5. Fault affecting z sensor. - -, detection threshold; —, residues.

Figure 6. Fault affecting φ sensor. - -, detection threshold; —, residues.

Once the residue exceeds the detection threshold, the fault is considered detected and it will be isolable if and only if the fault signature is different among the others. As expected, and thanks to the controller design every fault affecting the sensors is isolable. The fault signatures are presented in Table 1. X means that the residue exceeds the threshold; O means that even if the residue is affected, it does not surpass the threshold and by consequence this residue is not triggered.


Table 1. Fault signatures of sensors.

4.1. Sensor faults

142 Lagrangian Mechanics

R1

R4

R1

R4 -0.06 -0.04 -0.02 0 0.02 0.04 0.06





0

0.05

The sensor faults considered in this work affect the vertical measurement (z), the horizontal (y) and the inclination angle (φ), as explained before the amplitudes of the faults are 0.2 m, 0.1 m and 0.2, respectively. The controller is designed to decouple the sensor faults, as a result each fault is independent of the others and by consequence all of them are detectable and isolable,

> Time 0 5 10

> > **Residue 5**

Time 0 5 10

**Residue 2**

Time 0 5 10

**Residue 5**

Time 0 5 10

Time 0 5 10

**Residue 6**

Time 0 5 10

**Residue 3**

Time 0 5 10

**Residue 6**

Time 0 5 10

**Residue 3**

**Residue 2**

R3

R6

R3

R6 -0.06 -0.04 -0.02 0 0.02 0.04 0.06




thanks to their different fault signatures. Figures 4–6 depict the sensor faults.

R2

R5

R2

R5 -0.06 -0.04 -0.02 0 0.02 0.04 0.06




Time 0 5 10

**Residue 4**

Time 0 5 10

**Residue 1**

Time 0 5 10

**Residue 4**

Time 0 5 10

Figure 5. Fault affecting z sensor. - -, detection threshold; —, residues.

Figure 4. Fault affecting y sensor. - -, detection threshold; —, residues.

**Residue 1**

#### 4.2. Control inputs faults

The fault amplitudes of the control inputs are fixed by obtaining the 5% of the maximum size of them during an unfaulty simulation, after this processes the amplitudes are fixed to 0.5, 0.425 and 0.008 for uy, uz and ℓ, respectively. Thanks to the controller design, the faults affecting the control inputs are detectable and isolable. Figures 7–9 shows that the detection threshold is exceeded once a fault occurs, by consequence, fault detection is accomplished. Table 2 depicts the fault signatures, it is straightforward to see that they are all different, this behavior confirms that every single fault is detected and isolated.

Figure 7. Fault affecting uz. - -, detection threshold; —, residues.

Figure 8. Fault affecting uy. - -, detection threshold; —, residues.

Lagrangian Model‐Based Fault Diagnosis in a PVTOL http://dx.doi.org/10.5772/66395 145

Figure 9. Fault affecting ℓ. - -, detection threshold; —, residues.


Table 2. Fault signatures of control inputs.
