#### Abstract

This paper deals with fault detection in dynamical systems where the state variables evolutions are constrained by inequality constraints. The latter corresponds either to physical limitations or to safety specification. Two classical residual generation approaches are studied, namely, parity space and unknown input observer approaches, and are extended to monitor the inequality constraints. A practical implementation on a real process is performed and permits to validate the relevance of the proposed methods.

#### 1. Introduction

Fault diagnosis is playing a crucial role in modern industrial processes to enhance the dependability of such systems. This has led to the development of a wide variety of model-based approaches such as parity space, observer-based, or parameter identification methods [1, 2].

Dynamical systems are usually modelled by state and measurement equations that represent a set of equality constraints. In many practical applications, the state evolution is constrained by physical laws to stay in a given subspace. Moreover, it is generally required, for safety reasons, that the system works in particular conditions. As a consequence, a set of inequalities that constrain the evolution of the state variables in normal (no-fault) situation may be added to the state equations.

Monitoring a practical application requires not only the detection and isolation of the faults that corrupt the state evolution, but also the checking of the inequality constraints fulfilment. Even if the inequality constraints monitoring is very relevant for practical applications, it has not been extensively studied. The parity space approach has been extended in [3, 4], to this aim. The authors proposed to rewrite the inequalities as added equality constraints, thus to project the extended set of equality constraints in the so-called parity space. This leads to generating new residual signals that may be called *inequality constraints residuals (IC-residuals)*, which should stay positive under a normal situation and become negative when inequalities are corrupted. It has been pointed out that faults that corrupt the inequality constraints do not necessarily corrupt the state equations and thus cannot be detected by classical residuals. In this paper, we came back first on the parity space approach and propose a substitution method to generating the IC-residuals. Then, we propose to use an unknown input observer whose objective is to detect internal and external faults. The conditions to design such full or reduced-order observer are established.

This paper is organised as follows: Section 2 briefly recalls the principle of the parity space approaches, and its extension to inequality constraints. In Section 3, we propose to design unknown input observers to estimate the inequality constraint residuals. In Section 4, a reduced-order observer is proposed. Finally, we show the results of a practical implementation of the two IC-residuals generation approaches on a real process. We conclude the paper with some remarks and orientations of our future work.

#### 2. Preliminaries

Consider a dynamic linear system described by
where , and are the state, input, and output vectors, respectively. represents a fault signal. In the following sections, this kind of fault, which corrupts the state equations, will be qualified as *internal *fault. Matrices , and are known constant matrices of appropriate dimensions.

The purpose of the parity space approach is to generate a residual that will inform us about the state of the studied system. The principle of this approach is to derive the output up to order , which leads to the following form: With The unknown variable (state vector) may be eliminated using a projection matrix [5] as follows: The rows of define the so-called parity space.

The projection of (2) in the parity space leads to the computation form (5) and evaluation form (6) of the residual : The residual is computed using online input and output values (5). will be different from zero in the presence of an internal fault and remains equal to zero in the normal situation.

#### 3. Extension to Dynamical Systems Subject to Inequality Constraints

In practical systems, the system states must stay in a given safe domain during a normal operation. Thus, a set of inequality constraints may be added as follows: where is a constant vector (expresses the saturation level), is a real matrix of dimension , and is a state vector of dimension .

Define the following signal as where is a real, nonnegative vector. It represents an indicator of the inequality constraint validity (7). expresses the distance that separates the inequality constraints from their saturation level. The test diagnosis consists in checking the nonnegativity of the components of (i.e., it exist elements ), so the constraint is violated, then an alarm signal is generated as a result of a fault. Thus, may be considered as an IC-residual.

##### 3.1. Parity Space Approach

In [3, 4], the authors have shown how may be computed using the parity space approach.

The constraint equality (8) is added to (2) to obtainConsider the following projection matrix : and have appropriate dimensions. Using this partitioning: Projecting (9a), (9b) using leads to And one obtains where is the left pseudoinverse of . So, is expressed as a function of the known variables (inputs and outputs).

###### 3.1.1. Substitution Method Design

From (9a), we express the unknown state with the known input and output variables. First, we eliminate the vector of the expression (9a) by a projection matrix such that exists if . We obtain the following equation: It is assumed that .

If is regular matrix, then otherwise where a left pseudoinverse of .

Substituting in ((9b), The advantage of the substitution approach is the evaluation form of the IC-residual which is not possible to obtain by the projection method.

The procedure consists in eliminating the matrix using a projection matrix as follows: We obtain , where .

By substituting in the equation , The fault vector can be expressed with the inputs and outputs variables by eliminating the state vector with projection matrix . From (9a), find such that : exists if ,, is the left pseudoinverse of ฮ.

##### 3.2. Unknown Input Observer Approach

In the literature, many well-known results are available for the design of a state observer [6โ10]. The simplicity of its design and the resolution of the difficulty imposed by missing measurements make the observer an attractive general design component [11]. We are interested here in designing an unknown input inequality constraint observer for linear systems subject to inequality constraints. Few works are interested in this problem and are not used in diagnosis. So, we propose an original design based on a constraint observer approach.

Consider a linear system described by It is assumed that the matrix is full column rank and that the pair () is observable.

###### 3.2.1. Constraint Observer Design

We propose an estimation of the IC-residual .

To determine , we can estimate the state vector as follows: and represent the estimates of and , respectively.

The test diagnosis consists in checking the nonnegativity of vector .

As it is shown above, we have estimated the entire state to obtain . This is not always necessary as sometimes there are constraints that do not use all the states. Therefore, we construct an observer that estimates directly the constraints. This observer is called *constraint observer*. It is written in the following form:
We assume that and .

: state vector of observer and matrices , and are to be designed so that asymptotically estimates .

Let an error vector be defined as Hence, We assume And replace into (27), so error dynamic is equal to To obtain an error dynamic independent of , , and as the form, with being a Hurwitz (stability) matrix, the following conditions are to be satisfied: is stable.

This is to guarantee the convergence of .

The well-known necessary and sufficient conditions [8] for the existence of this unknown input observer are Condition (31) is deduced from (30c) by replacing by its expression: . Condition (32) is deduced from (30a). A similar proof is given in detail in the next section.

If conditions (30a)โ(30d), (31), and (32) are fulfilled, the matrix is found: with being a gain matrix chosen so that the pair is observable. The gain can be chosen by a variety of optimisation or pole placement technique; see [12].

From (30a)โ(30d), the matrix is determined:

###### 3.2.2. Reduced-Order Constraint Observers

Now, we construct an observer which takes into account that a part of the state is given by system outputs and so it is already available by direct measurement. An observer of lower dimension called reduced-order observer [11] can be designed.

It can also be assumed that the matrix takes the form . is partitioned into an identity matrix and zero matrix. The proposed reduced-constraint observer has a lower-order , where . Its design requires to decompose the matrices , and as follows: with By substituting these matrices in (30a)โ(30d) and decomposing the matrix as follows: with and , we obtainThe necessary and sufficient conditions for the existence of such reduced observer are given by the following proposition:

*Proof. *From (36e) and (36d), we obtain
exists if .

Now, we can write as follows [13]:
where is an arbitrary matrix of appropriate dimension and is the generalized inverse of .

From (36b), we obtain
is the generalized inverse of . Substitute (40) into (41); it gives
With
Matrix is chosen so that matrix is stable. This condition is fulfilled if the pair is observable [8, 10].

The matrices , and are obtained from (36a)โ(36f).

*Remark 1. *A continuous time model in state space form is used, but all the results are easily transcribed in discrete time.

#### 4. Application to Tank System

Consider the tank system of laboratory ACS depicted in Figure 1. The cylindrical tanks , and are connecting by pipes. A pump (nova 180) distributes water from the tank in both reservoirs and through two electrovalves and . Three level sensors (vega 61) measure the water levels of the three tanks.

Since there is a hardware problem with the valve connect to , only the tanks and are used (Figure 2). The purpose of the tank system is to provide a continuous water flow to a consumer.

The dynamic process of the water tank system can be illustrated by the model (44). It is written using the โmass balanceโ equations: where represents the pump flow. represents the water flow evacuated from the tank , is the fault โclogging valve, โ otherwise 0 (healthy system), represents the water flow between tanks and , which according to Torricelliโs rule is given by We additionally assume that the model of the system is not exactly known. Indeed, the output flow coefficients and are regarded as uncertain constant coefficients.

The values parameters of the tanks system are defined in Table 1.

The system (44) is linearised around its operating point and discretised for a sampling period .

The water level in the supply tank and the middle tank R_{2} has to be maintained at a level:
We get
We consider a window observation . Both substitution and projection methods lead to the same IC-residual . Thus, we have
Using (5), we obtain the residual : Now, we assume that the level sensor of tank is broken, so the only water level in the tank is measured. Consequently, the output variable is written in the following form:
Using (30a)โ(30d), the constraint observer system is
and the existence conditions of the observer are fulfilled:
The pair is observable. It can be tested by using the smith form [14, 15]:
where is the operator variable of -transform.

Now, a reduced-order observer is designed. The purpose is to estimate only the nonmeasurable constraint that monitors the water level in the second tank . It requires to decompose the matrices , and as follows: By applying (36a)โ(36f), the matrices of the observer are So, the dynamical observer is

##### 4.1. Experimental Results

The monitoring algorithms are implemented on the real process described in Figure 1. Two kinds of faults are introduced in the process:(i)Fault 1: clogging of the output valve,(ii)Fault 2: pump flow.

###### 4.1.1. Fault 1

For the flow pump โ=โ4.94โร 10^{โ5}โm^{3}ยทs^{โ1}, the system stabilizes to water level โm for the tank and โm for the tank . The clogging of the output valve between time 1530โs and 2761โs increases the water levels and and shows in Figure 3 an abnormal situation.

Also, the equality residuals become sensitive to this fault. Figure 4 shows that the residuals are disturbed between time 1530โs and 2761โs. Since they are noisy, an analysis signal technique must be used to improve the fault diagnosis [16].

It is seen that the IC-residuals (Figure 5) generated by parity and observer approaches become negative in the time window [1530โs,2761โs]. Hence, the inequality constraints are violated. Thus, the fault is detected.

###### 4.1.2. Fault 2

In normal operation, the flow pump is 10^{โ5}โm^{3}ยทs^{โ1}. We have introduced a fault in the pump; it consists in increasing its flow 10^{โ4}โm^{3}ยทs^{โ1} between time 1705โs and 2382โs. The water levels and increase and show in Figure 6 an abnormal situation.

We remark (Figure 7) that the equality residuals are not very sensitive to this kind of fault. An analysis signal technique must be used to provide a good decision.

However, Figure 8 shows that the IC-residuals become negative. This implies the violation of the inequality constraints which are due to the presence of a fault. So, the IC-residuals provide a good fault detection.

#### 5. Conclusion

In this paper, the advantages provided by the IC-residual to fault detection have been shown. A substitution parity space technique has been proposed in order to express the IC-residual in the evaluation form. Full and reduced-order unknown input constraint observers are designed for the monitoring system. The obtained results are encouraging and similar to those obtained by parity space approach. It has been noted that the IC-residual detects the external faults not considered in the system model. An online implementation of IC-residual on a real process is performed and permits to validate the relevance of the proposed methods.

Our future work will involve the improvement of the fault diagnosis and the fault tolerant control by reducing the fault detection delay.

#### Acknowledgments

The authors would like to thank Professor Vincent Cocquempot at the University of Lille for the discussion and helpful comments. They also thank the valuable comments and suggestions received from the reviewers of this paper.