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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

There are different schemes based on observers to detect and isolate faults in dynamic processes. In the case of fault diagnosis in instruments (FDI) there are different diagnosis schemes based on the number of observers: the Simplified Observer Scheme (SOS) only requires one observer, uses all the inputs and only one output, detecting faults in one detector; the Dedicated Observer Scheme (DOS), which again uses all the inputs and just one output, but this time there is a bank of observers capable of locating multiple faults in sensors, and the Generalized Observer Scheme (GOS) which involves a reduced bank of observers, where each observer uses all the inputs and m-1 outputs, and allows the localization of unique faults. This work proposes a new scheme named Simplified Interval Observer SIOS-FDI, which does not requires the measurement of any input and just with just one output allows the detection of unique faults in sensors and because it does not require any input, it simplifies in an important way the diagnosis of faults in processes in which it is difficult to measure all the inputs, as in the case of biologic reactors.

Processes supervision systems for operators have evolved as new techniques of detection and isolation of faults have appeared. Research in this field has also grown as the complexity of industrial processes has grown and this has motivated the development of different focuses for FDI system design.

The diagnosis of faults can be done using observers. One great advantage of the diagnosis schemes based on observers is that in comparison with other methods they are very large schemes. The high level of complexity in current industrial processes has led to a situation where the amount of information generated by these processes can overcome the capacity of analysis of human operators, which hinders decision making [

As an example of a process that is hard to supervise, in this work the production of biogas in an anaerobic reactor is used as a case study in which faults are diagnosed and isolated using a scheme based on observers. In many publications about non linear observers for the design of FDI systems, the residuals are based in the error of the estimation obtained by the observer [

Diagnosis schemes based on observers can be classified according the type of fault detected: sensor faults (Instrument Fault Detection or IFD), actuator faults (Actuator Fault Detection or AFD), and component faults (Component Fault Detection or CFD). Diagnosis schemes can also be classified according the number of observers that are used. There are schemes with one observer: a

The SIOS-IFD is a scheme with just one interval observator, of reduced order, for faults in sensors. The main advantage of the SIOS-IFD scheme over all the previously presented schemes, is the fact that no input measurements are is required; it is only necessary to have knowledge of the interval of values that the named inputs can reach. SIOS-IFD only allows the localization of faults in one sensor, because it requires the in line measurement of just one output.

_{i}_{3})_{1} e _{2}) and in that way be able to generate the responses: _{1} = _{1} – _{1} y _{2} = _{2} – _{1}.

Next the simplified ADM1 mathematic model of the UASB reactor of the Instituto Tecnológico de Orizaba, Veracruz, México (

Simplified ADM1 Model:
_{1} is the concentration of the anaerobic mass, _{1} is the concentration of organic matter expressed as chemic oxygen demand (COD), _{CH4}_{m1}_{d}_{s1}_{1}_{PH}_{LL}_{UL}

In this section the design of the interval observer designed for the IOS – IFD scheme is presented. The designed interval observer is capable of stimating value _{1}_{CH4}_{1}

The model described by the set of differential non linear ^{n}^{m×n}^{m}^{n×n}^{n}

The asymptotic observer is designed under the assumption that all inputs are known, and _{2}(t) (dim v_{2}(t) = m)_{1}(t) (dim v_{1}(t) = s = n – m)_{11}(^{s×s}, _{12}(^{s×m}, _{21}(^{m×s}, _{22}(^{m×m}, _{1} ∈ ℜ^{s×r}, _{2} ∈ ℜ^{m×r}, _{1} ∈ ℜ^{s}_{2} ∈ ℜ^{m}

In the design of this asymptonic observer, it was assumed that the rate of dilution D, and the inputs to the digestor are considered as knowntherefore A(t) y b(t) are known ∀ _{1}_{2}(t) =_{1}_{2}(t) = 1),_{1}_{1}_{1}(t) =_{1} Q_{CH4}^{T} _{1}(t) = 2)_{1}_{2}_{2}= –_{1}_{1}_{2}^{§}, (where _{2}^{§} is the pseudo reverse generalization of _{2}

Substituting _{1}_{2}

Finally, _{e}^{+}

_{e}^{−}

The eigenvalues of _{e}^{+}_{e}^{−}

Now, in order to make the observer asymptotically stable, the following conditions must be fulfilled:

_{e}^{−}_{i,j}

_{e}^{+}_{e}^{−}

Since both conditions are fulfilled, the observer is asymptotically stable. Next the design of an interval observer based on the designed asymptotic observer is presented. Once the asymptotic observer that will work as base observer have been designed, we continued with the design of the interval observer, which is based in the supposition that the values of the input vector ^{+}^{−}^{−}^{+}

For the upper limit:

For the lower limit:

The convergence of the interval observer is based in the principal of cooperation defined by [^{+}(_{1}^{+} – _{1} y ^{−}_{1} – _{1}^{−} be the errors of estimation associated with

It uses ^{+}^{−}_{1}^{−1} ^{+}(_{1}^{−1} ^{−}(

In

This implies that if _{1}^{−}(0) ≤ _{1}(0) ≤ _{1}^{+} (0) then _{e}_{1}^{−}(_{1}(_{1}^{+}(

To experimentally verify the operation of the developed interval observer, the anaerobic digester was fed with wastewater from a brewery, which had maximum values of 3 gCOD/L and minimum values of 2 g COD/L, these values are based to the digester being accustomed to consume 3 gCOD/L. If the concentration of the water is greater than this, it can be diluted to achieve the desired value, but if it is lesser, it will be difficult to remove excess water to achieve the desired value, being in the worst case a concentration of 2 gCOD/L. The dilution rate is bounded by a maximum value of 0.74 d^{−1} and minimum value of 0.26 d^{−1} (these parameters were obtained experimentally by applying to the plant a positive bounded control as explained by Zavala [_{CH4}_{CH4}

The developed SIOS-IFD scheme is able to detect unique, sudden and permanent faults in sensors _{1}_{1}_{CH4}_{1} = _{1} – _{1} and response _{2} = _{2} – _{2}, being _{1}_{1}_{2}_{CH4}

_{1}, on day 45 of experimentation. This figure shows that both responses have faults on day 45, because both _{1} and _{CH4}, have been estimated from _{1}_{1}

The array of symptoms presented in _{1}

_{1}_{1}_{CH4}

Shaded In _{1}_{1}_{1}_{CH4}

In this article experimental and simulation results of a novel system of diagnosis of faults in sensors are presented. It has been named as SIOS-IFD (Simplified Observer Interval Scheme—Instrument Fault Detection). The main advantage of the SIOS-IFD scheme in comparison with all the other schemes presented above is the fact that the measurement does not require any input, only knowledge of the range of values that such targets can achieve is required. The SIOS-IFD scheme only allows the location of a single sensor fault, for which it requires the in-line measurement in a single output.

SIOS-IFD.

UASB reactor.

Estimation of the concentration of anaerobic mass x1.

Estimation of methane biogas flow _{CH4}

Responses to a single sudden and permanent fault, +5% in the sensor _{1}

Developed response assessment scheme.

Model Parameters.

_{m1} |
5.1 gCOD/gCOD d |

_{s1} |
0.5 gCOD/L |

_{d} |
0.02 L/d |

_{1} |
0.1 gCOD/gCOD |

_{CH4} |
0.35 LCH_{4}/gCOD |

0.5 (adimensional) |

Structured Diagnostic Matrix.

_{1} |
_{1} |
_{1} |
_{1} |
_{CH4} |
_{CH4} | |
---|---|---|---|---|---|---|

r_{1} |
1 | −1 | 1 | −1 | 0 | 0 |

r_{2} |
1 | −1 | 0 | 0 | 1 | −1 |