Online Implementation of Inequality Constraints Monitoring in Dynamical Systems

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 by𝑥𝑦̇((𝑡𝑡))==𝐶𝐴𝑥𝑥((𝑡𝑡))++𝐷𝐵𝑢𝑢((𝑡𝑡))++𝐹𝐹₁₂𝑑(𝑡),𝑑(𝑡),(1) where 𝑥(𝑡)∈ℜ^𝑛,𝑢(𝑡)∈ℜ^𝑟, and 𝑦(𝑡)∈ℜ^𝑚 are the state, input, and output vectors, respectively. 𝑑(𝑡)∈ℜ^𝑘 represents a fault signal. In the following sections, this kind of fault, which corrupts the state equations, will be qualified as internal fault. Matrices 𝐴,𝐵,𝐶,𝐷,𝐹₁, and 𝐹₂ 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 𝑦(𝑡) up to order 𝑠, which leads to the following form:𝑌(𝑡,𝑠)−𝐺_𝑠𝑈(𝑡,𝑠)=𝐻_𝑠𝑥(𝑡)+𝐹_𝑠𝐷₁(𝑡,𝑠).(2) With𝑌(𝑡,𝑠)=⎛⎜⎜⎜⎜⎜⎜⎝𝑦⋮𝑦𝑦̇_(𝑠)⎞⎟⎟⎟⎟⎟⎟⎠,𝑈(𝑡,𝑠)=⎛⎜⎜⎜⎜⎜⎜⎝𝑢⋮𝑢𝑢̇_(𝑠)⎞⎟⎟⎟⎟⎟⎟⎠,𝐷₁(𝑡,𝑠)=⎛⎜⎜⎜⎜⎜⎜⎝𝑑⋮𝑑𝑑̇_(𝑠)𝐻_𝑠⎞⎟⎟⎟⎟⎟⎟⎠,=⎛⎜⎜⎜⎜⎜⎜⎝𝐶𝐶⋮𝐶𝐴𝐴_𝑠⎞⎟⎟⎟⎟⎟⎟⎠,𝐺_𝑠=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐶𝐶𝐴𝐶𝐷⋮𝐴𝐵𝐵_𝑠−1𝐶𝐷⋮0𝐵⋱𝐷00⋱⋯⋯⋯⋮000𝐹𝐵⋯𝐶𝐴𝐵𝐶𝐵𝐷_𝑠⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐹²𝐶𝐹₁00⋯0𝐶𝐴𝐹₁𝐹₂0⋯0𝐶𝐴⋮_𝑠−1𝐶⋮𝐹₁⋱𝐹₂⋱⋯⋮0𝐹₁⋯𝐶𝐴𝐹₁𝐶𝐹₁𝐹₂⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(3) The unknown variable 𝑥(𝑡) (state vector) may be eliminated using a projection matrix 𝑊 [5] as follows: 𝑊𝐻_s=0.(4) The rows of 𝑊 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 𝑟(𝑡):𝑟(𝑡)=𝑊𝑌(𝑡,𝑠)−𝐺_s𝑈(𝑡,𝑠),(5)𝑟(𝑡)=𝑊𝐹_s𝐷₁(𝑡,𝑠).(6) The residual 𝑟(𝑡) is computed using online input and output values (5). 𝑟(𝑡) will be different from zero in the presence of an internal fault 𝑑(𝑡) and remains equal to zero in the normal (𝑑(𝑡)=0) 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: 𝐻𝑥(𝑡)≤ℎ,(7) where ℎ∈ℜ^𝑞 is a constant vector (expresses the saturation level), 𝐻 is a real matrix of dimension (𝑞⋅𝑛), and 𝑥(𝑡) is a state vector of dimension 𝑛.

Define the following signal 𝑣(𝑡) as 𝑣(𝑡)=−𝐻𝑥(𝑡)+ℎ,(8) where 𝑣(𝑡)∈ℜ^𝑞 is a real, nonnegative vector. It represents an indicator of the inequality constraint validity (7). 𝑣(𝑡) expresses the distance that separates the inequality constraints from their saturation level. The test diagnosis consists in checking the nonnegativity of the components of 𝑣(𝑡) (i.e., it exist elements 𝑣_𝑖(𝑡)<0), so the constraint 𝑖 is violated, then an alarm signal is generated as a result of a fault. Thus, 𝑣(𝑡) may be considered as an IC-residual.

3.1. Parity Space Approach

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

The constraint equality (8) is added to (2) to obtain𝑌(𝑡,𝑠)−𝐺_𝑠𝑈(𝑡,𝑠)=𝐻_𝑠𝑥(𝑡)+𝐹_𝑠𝐷₁(𝑡,𝑠),(9a)𝑣(𝑡)=−𝐻𝑥(𝑡)+ℎ.(9b)Consider the following projection matrix 𝑊_𝑝: 𝑊_𝑝⎡⎢⎢⎣𝐻_𝑠𝐹_𝑠−𝐻0⎤⎥⎥⎦=0,with𝑊^𝑝=𝑊_𝑝1,𝑊_𝑝2.(10)𝑊_𝑝1 and 𝑊_𝑝2 have appropriate dimensions. Using this partitioning:𝑊𝑊_𝑝1𝐻_𝑠=𝑊_𝑝2𝐻,_𝑝1𝐹_𝑠=0.(11) Projecting (9a), (9b) using 𝑊_𝑝 leads to 𝑊_𝑝2𝑣(𝑡)=−𝑊_𝑝1𝑌(𝑡,𝑠)+𝑊_𝑝1𝐺_𝑠𝑈(𝑡,𝑠)+𝑊_𝑝2ℎ.(12) And one obtains𝑣(𝑡)=𝑊^+𝑝2−𝑊_𝑝1𝑌(𝑡,𝑠)+𝑊_𝑝1𝐺_𝑠𝑈(𝑡,𝑠)+𝑊_𝑝2ℎ,(13) where 𝑊^+𝑝2 is the left pseudoinverse of 𝑊_𝑝2. So, 𝑣(𝑡) 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 𝐷₁(𝑡,𝑠) of the expression (9a) by a projection matrix 𝑊_𝑠 such that𝑊_𝑠𝐹_𝑠=0(14)𝑊_𝑠 exists if 𝑚(𝑠+1)>rank[𝐹_𝑠]. We obtain the following equation:𝑊_𝑠𝑌(𝑡,𝑠)−𝐺_𝑠𝑈(𝑡,𝑠)=𝑊_𝑠𝐻_s𝑥(𝑡).(15) It is assumed that 𝑊_𝑠𝐻_s=Ω.

If Ω is regular matrix, then 𝑥(𝑡)=Ω⁻¹𝑊_𝑠𝑌(𝑡,𝑠)−𝐺_𝑠𝑈(𝑡,𝑠)(16) otherwise𝑥(𝑡)=Ω⁺𝑊_𝑠𝑌(𝑡,𝑠)−𝐺_𝑠𝑈(𝑡,𝑠),(17) where Ω⁺=(Ω^𝑇Ω)⁻¹Ω^𝑇a left pseudoinverse of Ω.

Substituting 𝑥(𝑡) in ((9b),𝑣(𝑡)=−𝐻Ω⁺𝑊_𝑠𝑌(𝑡,𝑠)−𝐺_𝑠𝑈(𝑡,𝑠)+ℎ.(18) The advantage of the substitution approach is the evaluation form of the IC-residual 𝑣(𝑡) which is not possible to obtain by the projection method.

The procedure consists in eliminating the matrix [^𝐼_𝑚(𝑠+1)^−𝐺_𝑠] using a projection matrix 𝑊_𝑑 as follows:𝐼_𝑚(𝑠+1)−𝐺_𝑠⎡⎢⎢⎣𝑈𝑌((𝑡𝑡,,𝑠𝑠))⎤⎥⎥⎦=𝐻^𝑠𝑥(𝑡)+𝐹_𝑠𝐷₁(𝑡,𝑠).(19) We obtain 𝑥(𝑡)=−Ω^+𝑑𝑊_𝑑𝐹_𝑠𝐷₁(𝑡,𝑠), where Ω_𝑑=𝑊_𝑑𝐻_𝑠.

By substituting 𝑥(𝑡) in the equation 𝑣(𝑡),𝑣(𝑡)=𝐻Ω^+𝑑𝑊_𝑑𝐹_𝑠𝐷₁(𝑡,𝑠)+ℎ.(20) The fault vector 𝐷₁(𝑡,𝑠) can be expressed with the inputs and outputs variables by eliminating the state vector with projection matrix 𝑊. From (9a), find 𝑊 such that 𝑊𝐻_𝑠=0:𝑊𝑌(𝑡,𝑠)−𝐺_𝑠𝑈(𝑡,𝑠)=𝑊𝐹_𝑠𝐷₁(𝑡,𝑠)(21)𝑊 exists if 𝑚(𝑠+1)>rank[𝐻_𝑠],𝐷₁(𝑡,𝑠)=Γ⁺𝑊𝑌(𝑡,𝑠)−𝐺_𝑠𝑈(𝑡,𝑠)(22)Γ=𝑊𝐹_𝑠, Γ⁺ is the left pseudoinverse of Γ.

3.2. Unknown Input Observer Approach

In the literature, many well-known results are available for the design of a state observer [6–10]. 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 𝑥̇(𝑡)=𝑣𝐴(𝑡𝑦)𝑥(=(𝑡𝑡))−=+𝐻𝐶𝐵𝑥𝑥𝑢((𝑡()𝑡𝑡))+,+ℎ𝐹.𝑑(𝑡),(23) It is assumed that the matrix 𝐹 is full column rank and that the pair (𝐴,𝐶) is observable.

3.2.1. Constraint Observer Design

We propose an estimation of the IC-residual 𝑣(𝑡).

To determine 𝑣(̂𝑡), we can estimate the state vector 𝑥(𝑡) as follows: 𝑣̂(𝑡)=−𝐻𝑥̂(𝑡)+ℎ,(24)𝑣(̂𝑡) and 𝑥̂(𝑡) represent the estimates of 𝑣(𝑡) and 𝑥(𝑡), respectively.

The test diagnosis consists in checking the nonnegativity of vector 𝑣(̂𝑡).

As it is shown above, we have estimated the entire state 𝑥(𝑡) to obtain 𝑣(̂𝑡). 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:𝜔̇(𝑡)𝑣=̂𝑁𝜔(𝑡)+𝐿𝑦(𝑡)+𝐺𝑢(𝑡),₁(𝑡)=𝜔(𝑡)+𝐸𝑦(𝑡).(25) We assume that 𝑣₁(𝑡)=𝑣(𝑡)−ℎ and 𝑣₁̂(𝑡)=𝑣(̂𝑡)−ℎ.

𝜔(𝑡): state vector of observer and matrices 𝑁,𝐿,𝐺, and 𝐸 are to be designed so that 𝑣₁̂(𝑡) asymptotically estimates 𝑣₁(𝑡)(𝑣₁̂(𝑡)→𝑣₁(𝑡)).

Let an error vector 𝑒(𝑡) be defined as 𝑒(𝑡)=𝑣₁̂(𝑡)−𝑣₁(𝑡)=𝜔(𝑡)+𝐸𝑦(𝑡)+𝐻𝑥(𝑡).(26) Hence,𝑒̇(𝑡)===+((𝑣𝐻𝐻̇₁̂𝑁(𝑡++𝜔)−𝐸𝐸(𝑡)𝐶𝐶𝑣+̇₁))((𝑥𝐺𝑡𝐴̇)(𝑢𝑡𝑥=)((𝑡+𝑡𝜔))̇+𝜔+(𝑡̇)(𝐿𝐵𝑡+)𝐶𝑢𝑥𝐸(𝑡()𝑦𝑡+)̇(𝑡.)𝐹+𝑑𝐻(𝑡)𝑥)̇(𝑡)(27) We assume 𝑃=𝐻+𝐸𝐶.(28) And replace 𝜔(𝑡)=𝑒(𝑡)−𝑃𝑥(𝑡) into (27), so error dynamic 𝑒(̇𝑡) is equal to𝑒̇(𝑡)=+𝑁𝑃𝑒𝐹(𝑑𝑡)(+𝑡).(𝑃𝐴+𝐿𝐶−𝑁𝑃)𝑥(𝑡)+(𝑃𝐵+𝐺)𝑢(𝑡)(29) To obtain an error dynamic independent of 𝑥(𝑡), 𝑢(𝑡), and 𝑑(𝑡) as the form, 𝑒(̇𝑡)=𝑁𝑒(𝑡) with 𝑁 being a Hurwitz (stability) matrix, the following conditions are to be satisfied:𝑁𝑃𝑃𝑃𝐵𝑃==+𝐹𝑃𝐻𝐺=𝐴+0=+,𝐸0𝐿,,𝐶,((((33330000bdac))))𝑁 is stable.

This is to guarantee the convergence of 𝑒(𝑡)(𝑒(𝑡)→0as𝑡→∞).

The well-known necessary and sufficient conditions [8] for the existence of this unknown input observer are rank[𝐶𝐹]=rank[𝐹],(31)Pair((𝑃𝐴),𝐶)isobservable.(32) Condition (31) is deduced from (30c) by replacing 𝑃 by its expression: 𝑃=𝐻+𝐸𝐶. Condition (32) is deduced from (30a). A similar proof is given in detail in the next section.

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

From (30a)–(30d), the matrix 𝐿 is determined:𝐿=−𝐾+𝑁𝑃𝐶⁺−𝐶⁺.(33)

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 𝐶 takes the form 𝐶=[^𝐼_𝑚⁰]. 𝐶 is partitioned into an 𝑚∗𝑚 identity matrix and 𝑚∗(𝑛−𝑚) zero matrix. The proposed reduced-constraint observer has a lower-order 𝑞, where 𝑞≤𝑛−𝑚. Its design requires to decompose the matrices 𝐴,𝐹, and 𝐻 as follows:𝐴=⎡⎢⎢⎣𝐴¹¹𝐴₁₂𝐴₂₁𝐴₂₂⎤⎥⎥⎦,𝐹=⎡⎢⎢⎣𝐹₁𝐹₂⎤⎥⎥⎦,𝐻=𝐻₁𝐻₂(34) with 𝐴𝐴₁₁₂₂∈ℜ^𝑚∗𝑚,𝐴₁₂∈ℜ^{𝑚∗(𝑛−𝑚)},𝐴₂₁∈ℜ^{(𝑛−𝑚)∗𝑚},∈ℜ𝐻^{(𝑛−𝑚)∗(𝑛−𝑚)}₁,𝐹₁∈ℜ^𝑚∗𝑘,𝐹₂∈ℜ^{(𝑛−𝑚)∗𝑘},∈ℜ^𝑞∗𝑚,𝐻₂∈ℜ^{𝑞∗(𝑛−𝑚)}.(35) By substituting these matrices in (30a)–(30d) and decomposing the matrix 𝑃 as follows: 𝑃=[^𝑃₁^𝑃₂] with 𝑃₁∈ℜ^𝑞∗𝑚 and 𝑃₂∈ℜ^{𝑞∗(𝑛−𝑚)}, we obtain𝐿𝑁+𝑃𝑃₁₂𝐴₁₁+𝑃₂𝐴₂₁=𝑁𝑃₁,(36a)𝑃=𝑃𝑃𝐵₁𝐴=−₁₂+𝐺,𝑃₂𝐴₂₂,((3366bc)₁)𝐹₁𝐻+𝑃₂𝐹₂=0,(36d)₂𝐻=𝑃₂,(36e)₁+𝐸=𝑃₁.(36f)The necessary and sufficient conditions for the existence of such reduced observer are given by the following proposition:rank−𝐻₂𝐹₂=rank𝐹₁,(37)Pair𝐾₁,𝐾₂isobservable.(38)

Proof. From (36e) and (36d), we obtain 𝑃₁𝐹₁=−𝐻₂𝐹₂.(39)𝑃₁ exists if rank[−𝐻₂𝐹₂]=rank[𝐹₁].
Now, we can write 𝑃₁ as follows [13]: 𝑃₁=−𝐻₂𝐹₂𝐹⁺¹+𝑍𝐼_𝑚−𝐹₁𝐹⁺¹,(40) where 𝑍 is an arbitrary matrix of appropriate dimension and 𝐹⁺¹ is the generalized inverse of 𝐹₁(𝐹⁺¹=(𝐹^𝑇1𝐹₁)⁻¹𝐹^𝑇1).
From (36b), we obtain 𝑁=𝑃₁𝐴₁₂𝐻⁺²+𝐻₂𝐴₂₂𝐻⁺²,(41)𝐻⁺² is the generalized inverse of 𝐻₂. Substitute (40) into (41); it gives 𝑁=𝐾₁−𝑍𝐾₂.(42) With 𝐾₁𝐾=₂𝐻₂−𝐹₂𝐹⁺¹𝐴₁₂+𝐴₂₂𝐻⁺²,=𝐹₁𝐹⁺¹−𝐼_𝑚𝐴₁₂𝐻⁺².(43) Matrix 𝑍 is chosen so that matrix 𝑁 is stable. This condition is fulfilled if the pair (𝐾₁,𝐾₂) is observable [8, 10].
The matrices 𝐺,𝐿, and 𝐸 are obtained from (36a)–(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 𝑅₁,𝑅₂,𝑅₃, and 𝑅_𝑠 are connecting by pipes. A pump 𝑃 (nova 180) distributes water from the tank 𝑅_𝑠 in both reservoirs 𝑅₁ and 𝑅₃ through two electrovalves 𝑉₁ and 𝑉₂. Three level sensors (vega 61) measure the water levels ℎ_𝑖 of the three tanks.

Figure 1 

Tank system.

Since there is a hardware problem with the valve 𝑉₂₃ connect 𝑅₂ to 𝑅₃, only the tanks 𝑅₁ and 𝑅₂ are used (Figure 2). The purpose of the tank system is to provide a continuous water flow 𝑞(𝑡) to a consumer.

Figure 2 

A simplified schematic diagram of a part tank process.

The dynamic process of the water tank system can be illustrated by the model (44). It is written using the “mass balance” equations: 𝑆₁⋅𝑑ℎ¹(𝑡)𝑑𝑡=𝑞^𝑢𝑆(𝑡)−𝑞₁₂(𝑡)−𝑞₁(𝑡),₂⋅𝑑ℎ²(𝑡)𝑑𝑡=𝑞¹²𝑦(𝑡)−𝑞₂(𝑡),₁𝑦(𝑡)=ℎ₁(𝑡),₂(𝑡)=ℎ₂(𝑡),(44) where 𝑞_𝑢(𝑡)=𝑢(𝑡),(45) represents the pump flow.𝑞_𝑖(𝑡)=𝜇_𝑖𝑆_𝑝⋅√2⋅𝑔⋅ℎ_𝑖(𝑡)⋅(1−𝑑(𝑡)),(46) represents the water flow evacuated from the tank 𝑅_𝑖(𝑖=1,2), 𝑑(𝑡) is the fault “clogging valve, 𝑑(𝑡)=1” otherwise 0 (healthy system), 𝑞₁₂(𝑡) represents the water flow between tanks 𝑅₁ and 𝑅₂, which according to Torricelli’s rule is given by𝑞₁₂(𝑡)=𝜇₁₂𝑆_𝑝⋅signℎ₁(𝑡)−ℎ₂(𝑡)2⋅𝑔⋅||ℎ₁(𝑡)−ℎ₂(𝑡)||.(47) We additionally assume that the model of the system is not exactly known. Indeed, the output flow coefficients 𝜇_𝑖 and 𝜇₁₂ are regarded as uncertain constant coefficients.

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

Table 1 

Parameters of the tanks system.

The system (44) is linearised around its operating point and discretised for a sampling period 𝑇=1s.

The water level in the supply tank 𝑅₁ and the middle tank R₂ has to be maintained at a level:ℎℎ₁<0.23m,₂<0.17m.(48) We get ℎ(𝑘+1)=+⎡⎢⎢⎣010.09.090₋₃22900..09092751⎤⎥⎥⎦ℎ(𝑘)+⎡⎢⎢⎣20.20429⎤⎥⎥⎦𝑢(𝑘)𝑣(𝑘)=𝑦⎡⎢⎢⎣(⎡⎢⎢⎣−𝑘00)1.8=02−0⎡⎢⎢⎣31109⎤⎥⎥⎦⎤⎥⎥⎦ℎ01𝑑⎤⎥⎥⎦((𝑘ℎ𝑘))(+𝑘,)⎡⎢⎢⎣,00..2137⎤⎥⎥⎦.(49) We consider a window observation [𝑘,𝑘+2]. Both substitution and projection methods lead to the same IC-residual 𝑣_𝑠(𝑘). Thus, we have 𝑣_𝑠(𝑘)=×−𝑌⎡⎢⎢⎣−(𝑘0,.0120)1+31−00.03.3₋₁₇05133−0.0000700..030304730000..03030373⎤⎥⎥⎦+⎡⎢⎢⎣00..2137⎤⎥⎥⎦.⎡⎢⎢⎣−−00..05020074−−00..0104083600⎤⎥⎥⎦𝑈(𝑘,2)(50) Using (5), we obtain the residual 𝑟(𝑘): 𝑟(𝑘)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−−000000....6000..33600033111053960373−−−000000...006...033006033014043767333−−−−000000..06....3300063300061500770707−−−−000000..06....0033060033160142070034−−−−−000000.6.....33000733000113011403306−−−−000000..06....0033060033080132367634⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠𝑌(𝑘,2)−⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−−−−661000..387...000.4940313596397244156−−−−−1300066......500067901394069297837941000000⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠𝑈(𝑘,(25)1.)Now, we assume that the level sensor of tank 𝑅₂ is broken, so the only water level ℎ₁ in the tank 𝑅₁ is measured. Consequently, the output variable is written in the following form:𝑦(𝑘)=10ℎ(𝑘);𝐶₁=10.(52) Using (30a)–(30d), the constraint observer system is 𝜔(𝑘+1)=𝑣⎡⎢⎢⎣−̂00.30.90971⎤⎥⎥⎦𝜔(𝑘)+⎡⎢⎢⎣−0.000200⎤⎥⎥⎦𝑦(𝑘),₁(𝑘)=𝜔(𝑘)+⎡⎢⎢⎣−0100⎤⎥⎥⎦𝑦(𝑘)(53) and the existence conditions of the observer are fulfilled:rank𝐶₁𝐹=rank[𝐹]=1.(54) The pair ((𝑃𝐴),𝐶1) is observable. It can be tested by using the smith form [14, 15]: rank⎡⎢⎢⎣𝑧𝐼²𝐶−𝑃𝐴₁⎤⎥⎥⎦=2,(55) where 𝑧 is the operator variable of 𝑧-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 𝑅₂. It requires to decompose the matrices 𝐻,𝑃,𝐴, and 𝐹 as follows:𝐻=𝐻₁𝐻₂=0−1,𝐹=⎡⎢⎢⎣𝐹¹𝐹₂𝐴=⎡⎢⎢⎣𝐴₁₁⎤⎥⎥⎦=10₋₃⎡⎢⎢⎣0,80239⎤⎥⎥⎦,𝐴₁₂𝐴₂₁𝐴₂₂⎤⎥⎥⎦=⎡⎢⎢⎣00.9.09022900..90907215⎤⎥⎥⎦.(56) By applying (36a)–(36f), the matrices of the observer are𝑃=0−1,𝑁=0.9𝐺97=1,0.𝐿=0.002,𝐸=0,(57) So, the dynamical observer is𝜔(𝑘+1)=𝑣0̂.9971𝜔(𝑘)+0.002𝑦(𝑘),₁(𝑘)=𝜔(𝑘).(58)

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 𝑞_𝑢(𝑡) = 4.94 × 10⁻⁵ m³·s⁻¹, the system stabilizes to water level ℎ₁=0.197 m for the tank 𝑅₁ and ℎ₂=0.136 m for the tank 𝑅₂. The clogging of the output valve between time 1530 s and 2761 s increases the water levels ℎ₁ and ℎ₂ and shows in Figure 3 an abnormal situation.

Figure 3 

Water levels ℎ1 and ℎ2 in tanks 𝑅1 and 𝑅2.

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

Figure 4 

Residuals 𝑟𝑖(𝑘).

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

Figure 5 

IC-residual 𝑣(𝑘).

4.1.2. Fault 2

In normal operation, the flow pump is 𝑞_𝑢(𝑘)=4.94× 10⁻⁵ m³·s⁻¹. We have introduced a fault in the pump; it consists in increasing its flow 𝑞_𝑢1(𝑘)=1.027× 10⁻⁴ m³·s⁻¹ between time 1705 s and 2382 s. The water levels ℎ₁ and ℎ₂ increase and show in Figure 6 an abnormal situation.

Figure 6 

Water levels ℎ1 and ℎ2 in tanks 𝑅1 and 𝑅2.

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.

Figure 7 

Residuals 𝑟𝑖(𝑘).

However, Figure 8 shows that the IC-residuals 𝑣(𝑘) 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.

Figure 8 

IC-residual 𝑣(𝑘).

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.