Fault Detection and Isolation

Fault diagnosis of a class of linear multiple-input and multiple-output (MIMO) systems is developed here. An emulator-based scheme is proposed to detect and isolate faults in a system formed by interconnected subsystems. Emulators, which are hardware or software devices, are connected to the input and measurement outputs in cascade with the subsystems whose faults are to be diagnosed. The role of an emulator is to induce variations in cascade combination of the nominal fault-free subsystem so as to mimic the actual perturbations that may occur in the subsystem during the offline identification phase. The emulator-generated data are employed in the reliable identification of the nominal system, the associated Kalman filter, and a map that relates the emulator parameters to the feature vector. In the operational stage, the Kalman filter residual is used to detect a fault in the system; the emulator parameter that has varied is estimated, and using the emulator-feature vector map, the faulty subsystem is isolated. The main contributions of this work are accurate and reliable identification of the system, the fault diagnosis of multivariable systems using feature vector-emulator map fault diagnosis of multivariable systems, and the establishment of the key properties of the Kalman filter for fault detection. The proposed scheme was successfully evaluated on a number of simulated as well as physical systems.


Introduction
Fault detection and isolation (FDI) of physical systems-especially mission critical systems including nuclear reactors, aircraft, automotive systems, spacecraft, autonomous vehicles, and fast rail transportation-is becoming increasingly important in recent times thanks mainly to advances in sensors, computing, and communication technologies. It still poses a challenge in view of the stringent and conflicting requirements, high probability of correct detection and isolation, low false alarm probability, and timely decision on the fault status.
The identification of the system model is crucial to the performance of the fault diagnosis scheme. The more accurate the identified model, the higher is the probability of correct diagnosis and the lower is the false alarm probability. The reliability and accuracy of the identification hinges on ensuring that the identified model is captured completely and what is leftover is the information-less zero-mean white noise process. As the Kalman filter is a zero-mean white noise process if and only if there is no mismatch between the identified model and the model of the system, the identification scheme should minimize the residual of the Kalman filter-instead the equation error, which in general, is a colored noise [1]. The widely popular, consistent, and efficient scheme that meets the above state requirement is the prediction error method (PEM) [2]. The PEM identifies the system by minimizing the residual of the Kalman filter.
A physical system is subject to perturbation resulting from the variations of the parameters and effects nonlinearities resulting in the deviation in the neighborhood of the nominal operating point. A model identified at a nominal operating point will not capture the static and the dynamic behavior of the perturbed system. To overcome this, an emulator, which is a hardware or a software device, is connected to either an accessible input or an accessible output in cascade with a subsystem to mimic its operating scenarios [3][4][5]. The powerful concept of emulators, which is employed to mimic the likely operating scenarios for single-input and single-output (SISO) system, is extended to multiple-input and multiple-output (MIMO) and multiple-input and single-output (MISO) system. The system is identified and the feature vector-emulator map is estimated from the emulator-generated data covering all likely operating scenarios including the normal and the faulty ones similar in spirit to that employed in training the neural network [6]. The identified nominal model, an optimal nominal model, is robust to model perturbation in the neighborhoods of the nominal operating point. It may be worth noting that the conventional scheme uses only the input-output data from the system in the nominal operating scheme.
There are essentially three approaches to the failure detection and isolation problem: the nonparametric approach, the parametric approach, and the combined approach. The non-parametric approach is based on analyzing a residual. The residual is defined as a signal, which is ideally non-zero in a statistical sense when there is a failure present, and zero otherwise. The residual may be generated using Kalman filters, observers, unknown-input observers, other forms of detection filters, and parity equations [7][8][9][10][11][12]. In view of the following key properties of the Kalman filter listed below, the Kalman filter is deemed the most preferable for both fault detection and fault isolation [1]: a. Model matching: The residual is a zero-mean white noise process if and only if there is no mismatch between the actual model of the system and its identified model embodied in the Kalman filter, that is, and its variance is minimum.
b. Optimal estimation: The estimate is optimal in the sense that it is the best estimate that can be obtained by any estimator in the class of all estimators that are constrained by the same assumptions.
c. Robustness: Thanks to the feedback (closed-loop) configuration of the Kalman filter with residual feedback, the Kalman filter provides the highest robustness against the effect of disturbance and model variations.

d. Model mismatch:
If there is a model mismatch, the residual will not be a zero-mean white noise process and an additive term termed fault-indicative term. The fault-indicative term is affine in the deviation in the linear regression or the transfer function model.
The feature vector-emulator map relating the deviation of the feature vector and variations of the emulator parameter is used for fault isolation if a fault is detected. The influence vector, which is the partial derivative of the feature vector with respect to an emulator parameter, plays a crucial role in pinpointing the faulty subsystem and tracks its parameter variation.
The main contributions here are the development of emulator-based system identification, and estimation of the feature vector-emulator map and its application to performance monitoring and fault diagnosis of multivariable system. The key properties of the Kalman filter, including model matching, whitening of the equation error, and residual expression for the modelmismatch case, are established for MIMO, MISO, and SISO systems.
The chapter is organized as follows. In Section 2, the mathematical model of the multiple-input and multiple-output system in state-space, frequency-domain, and a linear regression form is developed. The multiple-input and single-output and the single-input, single-output models are derived. Modeling of faults is also given. In Section 3, the concept of emulators, the generation of emulator-perturbed data, and its role in the identification of the system, the estimation of the feature vector-emulator map for fault isolation is developed. In Section 4, the identification of the system and the associated Kalman filter using prediction error method is suggested. The feature vector-emulator map is estimated using the expression of the Kaman filter residual in the model-mismatch case. In Section 5, the model of the Kalman filter, residual model, and the key properties of this filter are given. The key properties of the residual are established including whitening of the equation error, and expressions for the residual for the model-mismatch case. In Section 6, Bayesian approach to fault diagnosis is explained. Finally, in Sections 7 and 8, the successful evaluation of the proposed scheme on both a simulated and physical system is given, respectively.

Mathematical model of the system
The MIMO state-space model of the system denoted ðA, B, CÞ is given by where xðkÞ ¼ ½ x 1 ðkÞ x 2 ðkÞ x 3 ðkÞ … x n ðkÞ T , yðkÞ ¼ ½ y 1 ðkÞ y 2 ðkÞ y 3 ðkÞ … y q ðkÞ T , rðkÞ ¼ ½ r 1 ðkÞ r 2 ðkÞ r 3 ðkÞ … r p ðkÞ T , wðkÞ and vðkÞ , are respectively, nx1 state vector, qx1 output, px1 input to the system, px1 disturbance and qx1 measurement noise; A, B, C, E w are nxn state transition, nxp input, and qxn output and nxp input disturbance matrices; A and C are block diagonal matrices; A ¼ Expressing the frequency-domain model (5) in a linear regression form yields where υ ij ðzÞ ¼ D j ðzÞϑ ij ðzÞ; ψ T ji ðkÞ is 1x2n j regression vector formed of the regression vectors, formed ψ T yji ðkÞ associated with y ji ðkÞ, and ψ T ri ðkÞ associated with input r i ðkÞ: ψ T yji ðkÞ ¼ ½ Ày ji ðk À 1Þ Ày ji ðk À 2Þ : Ày ji ðk À n j Þ ; ψ T ri ðkÞ ¼ ½ r i ðk À 1Þ r i ðk À 2Þ : r i ðk À n j Þ; θ ji is 2n j x1 feature vector formed of the n j coefficients of the denominator polynomial D j ðzÞ and the numerator polynomial N ij ðzÞ: Remarks: In the operational stage, we may not have access to the output y j ðkÞ, termed y ji ðkÞ, generated by the input r i ðkÞ alone when rest of the inputs are set to zero. It is estimated during the identification phase of the multi-input and single-output model relating the accessible output y j ðkÞ generated by all the inputs rðkÞ.

Multi-input and single-output pairing
Using Eq. (5), the output y j ðzÞ is the output due to all the inputs rðkÞ of MISO system, which is Expressing the frequency-domain model (10) in a linear regression form yields where ψ T j ðkÞ is 1xðn j þ n j pÞ regression vector formed of the regression vectors ψ T yj ðkÞ associated with y j ðkÞ, and ψ T r ðkÞ associated with rðkÞ: ψ T yj ðkÞ ¼ ½ Ày j ðk À 1Þ Ày j ðk À 2Þ : Ày j ðk À n j Þ ; ψ T r ðkÞ ¼ ½ ψ T r1 ðkÞ ψ T r2 ðkÞ : ψ T rp ðkÞ ; θ j is ðn j þ n j pÞx1 feature vector formed of the n coefficients of the denominator polynomial D j ðzÞ and the n j p coefficients of the numerator polynomial N j ðzÞ; where θ yj ¼ ½ a j1 a j2 : a jnj T ; θ rj ¼ ½ θ T rj1 θ T rj2 : θ T rjp T .

Multi-input and multiple-output system
Extending the results of the time-domain expression to the MIMO (3), we get where ψ T ðkÞ is qxðn þ npqÞ regression matrix formed of the regression vectors fψ T ij ðkÞg, and θ is ðn þ npqÞx1 feature vector formed of θ j , j ¼ 1, 2, …, q is given as follows: The regression model (14) is the time-domain version of the frequency-domain model (3).
Expressing the time-domain model (14) in the frequency domain, we get

Interconnected system
The system is an interconnection of subsystems such as the plant, the actuator, the sensors, and the controllers shown in Figure 1. Subfigure A at the top shows that jth output of the system y ji is given by Eq. (10) where y ij ðzÞ given in Eq. (5) is the output generated by the input r i acting alone.
Subfigure B at the bottom shows that the transfer function G ji ðzÞ in the path from the input r i to the output y ij is formed of subsystems {G ijl ðzÞ}. The subsystem G ijl ðzÞ is driven by the input u jil ðzÞ and its output is corrupted by the disturbance w jil ðzÞ. The input and the output of G ji ðzÞ are r i and y ji , respectively, v ji is the measurement noise, ϑ ji given in Eq. (5) is the combined effect of the disturbances {w jik } and {v ji } on the output y ji ðzÞ.

Modeling of faults
There are two types of fault models, namely the additive and the multiplicative (or parametric) types. In the additive type, a fault is modeled as an additive exogenous input to the system, whereas in the multiplicative type, a fault is modeled as a change in the parameters, which completely characterize the fault behavior of the subsystems. Although the multiplicative and additive perturbation models are equivalent, the multiplicative-type perturbation model is preferable. The multiplicative perturbation model of the cascade combination of subsystems can actually model the particular perturbation in any one of the subsystems under consideration.

Emulators
The emulator-based identification scheme is motivated by the model-free artificial neural network approach to capture the static and the dynamic behavior by presenting neural network data covering likely operating scenarios. An identified model at each operating point characterizes the behavior of the system in the neighborhood of that point. In practice, however, the system model may be perturbed because of variations in the parameters of the system. To overcome this problem, the system model is identified by performing a number of emulator parameter-perturbed experiments proposed in [4][5]. Each experiment consists of perturbing one or more emulator parameters. A linear model, termed optimal model, is identified as a best fit to the input-output data from the set of emulated perturbations. The optimal model thus obtained characterizes the behavior of the system over wider operating regions (in the neighborhood of the operating point), whereas the conventional model characterizes the behavior merely at the nominal operating point (i.e., the conventional approach assumes that the model of the system remains unperturbed at every operating point). The optimal model is more robust, that is, the identification errors resulting from the variations in the emulator parameters are significantly lower compared to those of the conventional one based on performing a single experiment (i.e., without using emulators). Fault Detection and Isolation http://dx.doi.org/10.5772/67870 During the system identification phase, a number of experiments are performed by (a) not perturbing the emulator parameters and (b) perturbing the emulator parameters one at a time, simultaneously perturbing two at a time, three at time, and so on till perturbing all of them. The input-output data collected from all experiments are termed emulator-generated data.
• Nominal system model and the Kalman filter: The emulator-generated data are used to identify the nominal optimal model of the system and the optimal Kalman filter model using the prediction error method.
• Estimation of the influence vectors: Using the least-squares method, the influence vectors are identified recursively using the input-output data obtained from the emulator-perturbed parameter experiments. First, the influence vector for influence vector for the single parameter perturbation is identified, and then using the estimated influence vector, the influence vector for the two simultaneous emulator perturbations is estimated. Generalizing, the influence vector for m simultaneous perturbation is identified, and then using all previous m estimates of the influence vectors, the ðm þ 1Þth influence vector is identified.
The emulators are transfer functions, which are connected in cascade with the subsystems to generate likely operating scenarios including normal and faulty one for reliable and accurate identification of the system, its associated Kalman filter, and the feature vector-emulator map.
Emulators are connected to the system during the identification phase and its parameter is varied to generate likely operating scenarios. During the operational phase, the static emulators are disconnected, as it were, by setting them to unit values. The dynamic emulator, however, is not disconnected. Its gain is set to unity and its phase made a non-zero negligibly small value so that (a) both of these parameters have a negligible effect on the dynamic behavior of the system during the operational phase and (b) the order of the system during the identification and the operational phases remains identical to ensure mathematical tractability without causing performance degradation. The role of the emulator-generated data includes the following: 3.1. Emulator-generated data for MISO system The MISO system is given by Eq. (11) relating all the inputs rðkÞ and the output y j ðkÞ identified by connecting an emulator E j ðzÞ in cascade with rðzÞ. The emulator is a first-order all-pass filter given by where jγ j1 j < 1 to ensure stability. The emulators γ j1 and γ j2 are varied one at a time, and both simultaneously. During the identification, an emulator E j ðzÞ, which is a first-order all-pass filter (17), is connected to the input r j ðkÞ in cascade with nominal model G j0 ðzÞ. A number of experiments are performed by varying the emulator parameters γ j1 , γ j2 one at a time and both simultaneously to acquire emulator-generated data: it is assumed for simplicity that the same input is applied to all the experiments. Using Eq. (10), the MISO model relating r j ðkÞ and y j ðkÞ becomes y el where y e1 j ðzÞ, y e2 j ðzÞ, and y e3 j ðzÞ denote, respectively, the output generated by varying γ j1 , γ j2 and both γ j1 , γ j2 .

Emulator-generated data for SISO system
The feature vector-emulator map of the SISO system (5) is estimated for the isolation of faults in the subsystems fG ijℓ ðzÞg. The emulators E jiℓ ðzÞ are connected to an accessible input or output fu jiℓ g in cascade with the subsystems fG jiℓ ðzÞg to mimic their variations. In other words, the known emulator parameter variations mimic those of the unknown parameters of the associated subsystems. The accessible inputs include the tracking error, the control input, actuator input, and sensor output.
The emulator E jiℓ ðzÞ may be a dynamic system, a constant gain ðγ jiℓ Þ, a gain, and a pure delay of d time instants ðγ jiℓ z Àd Þ, a first-order all-pass filter or a Blaschke product of all firstorder-pass filters The emulator E ji ðzÞ is chosen to be a product of a static gain and a first-order all-pass filter to mimic the behavior of the subsystem G ji ðzÞ ¼ G jiℓ ðzÞ of the SISO system given by Eqs. (5) and (6) In order to ensure stability of the dynamic emulator, parameter γ ji1 is constrained by jγ jiℓ j < 1.
Connecting the emulator E ji ðzÞ given in Eq. (19) to the nominal SISO model G ji0 ðzÞ using Eqs. (5) and (6), we get where y e1 ji ðzÞ, y e2 ji ðzÞ , and y e3 ji ðzÞ denote, respectively, the output generated by varying γ ji1 , γ ji2 and both γ ji1 and γ ji2 . Figure 2 shows an example of a closed-loop position control system formed of a controller, an actuator, a plant, and a sensor in the path connecting the tracking error e ri ðkÞ ¼ r i ðkÞ À y ji ðkÞ and the output y ji . Only e ri ðkÞ, u ij1 ðkÞ, and u ij3 ðkÞ are the measurement outputs. The emulators 1þγ ji1 z À1 , and E ji2 ¼ γ ji2 are connected to u ji1 , and E ji3 ¼ γ ji3 is connected to u ji3 to mimic Fault Detection and Isolation http://dx.doi.org/10.5772/67870 the perturbations in the dynamic plant G ji1 ðzÞ, the static actuator G ji2 ðzÞ ¼ k A and the static sensor G ji3 ðzÞ ¼ k s , respectively, where E ji1 ðzÞ is dynamic, and E ji2 and E ji3 are static emulators.
The nominal static emulator is set to unit value γ 0 ijk ¼ 1. The variation Δγ jik of an emulator γ ijk may be expressed in terms of its nominal value γ 0 jik as Δγ jik ¼ γ jik À γ 0 jik .

Feature vector-emulator map
The feature vector-emulator map for the SISO and the MISO systems is developed subsequently.

SISO system
Consider the emulator-perturbed SISO system (20) relating the inputs r i ðkÞ and y ji ðkÞ and the associated linear regression model (7). The feature vector θ ji is a nonlinear function of the emulator parameter γ ji ¼ ½ γ ji1 γ ji2 . Assuming that the feature vector θ ji is a continuous function of γ ji , then using Weierstrass approximation theorem, the feature vector-emulator map becomes where Δθ ji ¼ θ ji À θ 0 ji ;Δγ jℓ ¼ γ jℓ À γ 0 jℓ is the parameter variation;θ 0 ji is the nominal feature vector; Ω ji1 is a 2n j x1 vector of partial derivatives of the feature vector θ ji with respect to γ ji1 evaluated at the unperturbed nominal emulator value γ 0 ji1 . Similarly, Ω ji2 is a 2n j x1 vector of partial derivatives of the feature vector θ ji with respect to γ ji2 evaluated at the unperturbed nominal emulator value γ 0 ji2 , Ω ji12 is the second partial derivatives with respect to γ ji1 and γ ji2 evaluated at the unperturbed nominal emulator value γ 0 ji1 and γ 0 ji2 . The partial derivative terms Ω ji1 , Ω ji2 Ω ji12 , which are the Jacobean of the feature vector θ ji with respect to the emulator parameters fγ jik g, are termed influence vectors. The influence vectors play a crucial role in isolating a fault occurring in any subsystem. The influence vectors Ω ji1 , Ω ji2, and Ω ji12 track the degree of variations in the parameters of the subsystem perturbations.
Substituting for θ ji in (7), the variation Δy ji ðkÞ ¼ y ji ðkÞ À y 0 ji ðkÞ between the actual output y ji ðkÞ and the nominal fault-free output y 0 ji ðkÞ becomes Let Ω ji be an influence matrix associated with the emulators located at the path ij A number of emulator parameter-perturbed experiments are performed by perturbing the parameters of the emulators (20). For each experiment, N input-output data ðy e j ðkÞ, rðkÞÞ are obtained, k ¼ 1, 2, …, N. The input rðkÞ for each experiment is chosen to be persistently exciting. The regression models associated with the experiments and Eq. (22) are given as follows:

MISO system
Consider the emulator-perturbed MISO system (18) relating the inputs rðkÞ and y j ðkÞ, and the associated linear regression model (11). Similar to Eqs. (21) and (24), we get

Identification
The prediction error method can be derived from the residual model of the Kalman filter, which is presented in the next section. It is used to identify both the nominal system and the Kalman filter associated with the system without the need for a priori knowledge of the covariances of the noise and the disturbance statistics. Prediction error method is consistent, efficient, and a gold standard for system identification, and can identify open-loop and closedloop systems. The variance the parameter estimates asymptotically approaches the Cramer-Rao lower bound.
Optimal models: The optimal system and the associated Kalman filter are identified using the prediction error method using computationally efficient scheme. First, the MISO system is identified and then the SISO system is derived from the estimate of feature vector associated with the MISO system. The emulator-generated data generated using Eq. (18) are used to identify MISO system (10) and the nominal feature vector θ 0 j for Eq. (11), which is the best least-squared fit to set all perturbed feature vector θ j , and the Kalman gain K j0 are estimated. Let the optimal state-space model of the MISO system be ðA j0 , B j0 , C j0 Þ and associated Kalman filter be ðA j0 À K j0 C j0 , ½K j0 B j0 , C j0 Þ. Let the optimal transfer matrix of the MISO system and the optimal estimate of the output be G Then, the best estimate of the feature vector θ ji of the SISO system (7), denoted θ 0 ji, and the Kalman gain are estimated from θ 0 j .

Model of the Kalman filter
The Kalman filter forms the backbone of the MISO and the SISO systems fault detection and for fault isolation, respectively. The Kalman filter is a closed-loop system, which is (a) an exact copy of the identified nominal of the system driven by the residual, which is the error between the output and its estimate, and (b) is stabilized by the Kalman gain.
MISO system: Using the state-space model ðA j0 , B j0 , C j0 Þ derived from the identified nominal feature vector θ 0 j . The Kalman filter ðA j0 À K j0 C j0 , ½K j0 B j0 , C j0 Þ associated with the MISO system (10) isx j ðk þ 1Þ ¼ ðA j0 À K j0 C j0 Þx j ðkÞ þ K j0 y j ðkÞ þ B j0 rðkÞ y j ðkÞ ¼ C j0xj ðkÞ e j ðkÞ ¼ y j ðkÞ Àŷ j ðkÞ ð29Þ wherex j ðkÞ andŷ j ðkÞ are, respectively, the minimum variance estimates of the state and the output. Figure 3 shows the nominal fault-free system and the Kalman filter. The structure of the Kalman filter is based on the internal model principle, which embodies the nominal system model ðA j0 , B j0 , C j0 Þ. The inputs to the Kalman filter are the input rðkÞ and the output y j ðkÞ which is corrupted by the disturbance w j ðkÞ and the measurement noise v j ðkÞ.

Expressions of the residual
The expression for the residuals for the MISO system e j ðzÞ and the SISO system e ji ðzÞ is derived from the Kalman filter (29).
MISO model: The frequency-domain expression, relating the n u x1 input rðzÞ and output y j ðzÞ to the residual e j ðzÞ is given by the following model, termed residual model: where F j0 ðzÞ ¼ jzI À A j0 þ K j0 C j0 j is the characteristic polynomial termed Kalman polynomial; where N 0ji ðzÞ is the i th element of N j0 ðzÞ.

Key properties of the Kalman filter residual
The Kalman filter forms the backbone of the proposed scheme in view of its key properties proved in [1]. These properties exploited in developing the system identification using the

Propositions
We establish important results, in the form of lemmas that are crucial to the development of the proposed fault diagnosis scheme. In Lemma 1, it is shown that (a) the system transfer function can be estimated from the residual model and (b) Kalman filter whitens the output error ϑ j ðzÞ given in Eq. (10). Lemma 2 shows that the residual will not be a zero-mean white noise process if there is a model mismatch, and there will be an additive fault indicating term, which is a function of the deviation between the actual feature vector θ j of the system model ðA j , B j , C j Þ and the nominal fault-free feature vector θ 0 j of nominal fault-free model ðA j0 , B j0 , C j0 Þ.
Case 1: The system and the nominal models are identical Lemma 1: where G j0 ðzÞ is the transfer function of the nominal fault-free model ðA j0 , B j0 , C j0 Þ. Adding and subtracting y j ðzÞ inside the bracket on the right-hand side yields e j ðzÞ ¼ D j0 ðzÞ D j0 ðzÞF j0 ðzÞ y j ðzÞ À 1 À D j0 ðzÞ y j ðzÞ À N j0 ðzÞrðzÞ ð41Þ Using the expression for the regression model (11) and substituting for the actual and the nominal fault-free cases, we get Remarks: If there is a model mismatch because of variations in the subsystem parameters, the residual is no longer zero-mean white noise process. The residual has an additive term, which is affine in the deviation in the system transfer function ΔG j ðzÞ or equivalently affine in the feature vector ψ T jf ðzÞΔθ j . The additive terms are termed fault indicators. This shows that the Kalman filter provides a unifying approach to handle both fault detection and fault isolation. In view of the key properties, the Kalman filter is employed for identification and the fault diagnosis. In system identification, the criterion for determining whether the identified model has captured completely the dynamic behavior of the system is that the residual (error between the output and its estimate obtained using the identified model) is a zero-mean white noise process. Consider the problem of identification of the system. Since the equation error υðkÞ is a colored noise process, the parameter estimates will be biased and inefficient. To overcome this, the input and the output are whitened using the Kalman filter as shown in Eq. (35) of Corollary 1. The Kalman filter model (29) may be interpreted as an inverse system generating the innovation sequence eðkÞ , or alternatively as a whitening-filter realization of a state-space model that is driven by both the disturbance and measurement noise.

Bayesian approach fault diagnosis
The objective of fault detection is to assert whether the given residual belongs to a set of faultfree data or faulty residual data, while fault isolation is determined to which class of emulatorperturbed residual the given data belong. The problem of fault detection and fault isolation is formulated by a pattern classification problem. Fault detection is a binary pattern classification, while the fault isolation is a multi-class pattern classification. The Bayesian decision strategy is employed to assert appropriate class label. The Bayesian decision strategy is based on the a posteriori conditional probability of deciding a hypothesis given the data, a priori probability of the hypothesis, and a performance measure. The decision strategy is determined from the minimization of the performance measure with respect to all hypotheses.
The Nx1 residual eðkÞ is located in a different region of the N-dimensional plane depending upon the fault type. In the ideal case regions, there will not be overlaps between regions associated with different fault types. However, due to noise, disturbances, and other measurement artifacts there will be overlap between the various regions. Hence, Bayesian strategy is employed to asset an appropriate class label to ensure a high-probability correct decision, and a low probability of false alarms.

Fault detection
Fault detection is posed as a binary hypothesis-testing problem. The criterion to choose between the two hypotheses, namely the presence or an absence of a fault, is based on minimizing the Bayes risk, which quantifies the costs associated with correct and incorrect decisions. The Nx1 Kalman filter residual data eðkÞ generated by Eq. (29) is employed. The minimization of the Bayes risk yields the likelihood ratio test. The decision between the two hypotheses is based on comparing the likelihood ratio, which is the ratio of the conditional probabilities under the two hypotheses, to a threshold value. The resulting binary composite hypothesis-testing problem compares the test statistics of residual eðkÞ with a threshold value η: The test statistics depends upon the input rðkÞ that generates the residual eðkÞ [4]: rðkÞ is an arbitrary signal

Computationally efficient scheme
A computationally efficient scheme is employed here for the detection: • The status of each of the MISO systems G j ðzÞ relating all the inputs rðzÞ and all the outputs y j ðzÞ is evaluated for all j ¼ 1, 2, …, q using the binary hypothesis scheme (44). Using the test statistics of the residuals e j ðkÞ given by Eq. (30) yields • If a fault is asserted in G j ðzÞ, then the status of each of the p subsystems G ji ðzÞ of the SISO system is asserted using the test statistics of the residuals e ji ðkÞ (31): Fault accommodation: If a fault is asserted, then the Kalman gain is adapted online, the system re-identified, and the Kalman filter redesigned accordingly, thus the fault is accommodated and, in the extreme case, the system is shut down for safety reasons.

Evaluation on simulated system
The proposed emulator-based system identification of the system, the associated Kalman filter, feature vector-emulator map, and finally the fault diagnosis are illustrated using an example of a position control system formed of an actuator, a sensor, and a plant.
The poles of the MISO transfer functions G 2 ðzÞ of y 2 ðkÞ and G 1 ðzÞ of y 1 ðkÞ were, respectively, 0:8500 AE j0:3122 and 0:7500 AE j0:3708. The same emulator was used for inducing phase shift to the MISO models. G 2 ðzÞ with poles close to the unit circle was affected more than G 1 ðzÞ with poles well inside. In view of the difference in the perturbations induced in the two models, the mean-squared errors mse 2 and mse c 2 are higher than mse 1 and mse c 1 .

Fault diagnosis
Detection of a fault: Various types of faults include (a) actuator, (b) sensor, and (c) plant, we introduced by varying the columns of B 0 , the rows of C 0, and the diagonal matrices of A 0 . A fault is detected using appropriate test statistics depending upon the reference input waveform from Eq. (45). Since the reference input rðkÞ is a constant waveform, the test statistics for the MISO and the SISO system using Eqs. (46) and (47) are A visual picture of the faulty and the normal subsystems may be deduced from the autocorrelations of the residuals associated with the fault-free, sensor fault, actuator fault, and the plant faults shown in Figure 5. Subfigures A, and B, subfigures C and D, subfigures E, and F, and subfigures G and H show respectively autocorrelations of the residual for the ideal no fault, the sensor fault, the actuator fault, and the plant fault.
Remarks: The maximum value of the autocorrelation of the residual (i.e., its variance) provides an indication of the presence or an absence of the fault. In the case of the sensor fault introduced by perturbing C 20 , it affects only the residual e 2 ðkÞ. The variance of the autocorrelation e 2 ðkÞ is large while that of e 1 ðkÞ indicating a fault in C 2 . However, a fault in either the actuator or the plant, depending upon which elements of B 0 or A 0 are perturbed, may affect both residuals, and hence would be difficult to isolate.

Fault isolation
If a fault is asserted, and the path where the fault is located, then it is isolated using Bayesian multiple hypotheses testing scheme. The size of the fault is also estimated. The objective of fault isolation is to determine which of the emulator parameter has varied using the residual data generated or parameters using the expression for the Kalman filter residual for the modelmismatch case given in Eq. (43). The residual e ji ðkÞ is affine in the unknown emulator parameter variations fΔγ ijk g. The emulator parameter variation that is most likely to fit the perturbed residual with additive term ψ T jif ðzÞΔθ ji is determined sequentially by first hypothesizing single faults. If the estimates thus obtained do not fit the residual, then two simultaneous faults are hypothesized. If again the estimates do not fit the residual model, then hypothesize triple faults, and so on until the estimates fit the residual model. The maximum likelihood method, which is efficient and unbiased, is employed herein to estimate the variation Δγ. The maximum likelihood estimates of the emulator parameters are obtained by minimizing the log likelihood function [13].

Criteria for asserting the hypothesis
The most likely hypotheses is determined by verifying which of the emulator parameter or parameters have varied by comparing the deviation with some threshold value Assert Assert where η 1 , η 2 , and η 3 are threshold values. The subsystem associated with the subsystem is asserted to be faulty if the criterion is met.

Evaluation on physical process control system
A laboratory-scale two-tank physical system is formed of a controller, a DC motor, a pump, two tanks connected by a pipe, a flow rate sensor, and a liquid level sensor. The system is interfaced to a PC with the National Instruments LABVIEW for data acquisition and implementing the controller and the soft sensor [14]. The actuator, namely the pump driven by the DC motor, sends the fluid to the first tank to maintain a specified fluid level in the second tank. An evaluation of the proposed scheme for fault diagnosis was performed on a benchmark laboratory-scale process control system using the National Instruments LABVIEW as shown below in Figure 6. The sampling period is T s ¼ 0:05.
Emulator-generated height and flow rate profiles under various types of faults are shown in under the caption Height/Flow rate Profiles for PI controller with Consumer in Fig. 7. Fault Detection and Isolation http://dx.doi.org/10.5772/67870 emulator parameters to 0.25, 0.5, and 0.75 time the nominal values, in order to represent "small," "medium," and "large" faults. However, by virtue of its control design objective, the closed-loop PI controller will hide any fault that may occur in the system and hence will make it difficult to detect it. In addition, the physical system exhibits a highly nonlinear behavior. The flow rate saturates at 4.5 ml/s. The dead-band effect in the actuator exhibits itself as a delay in the output response: when a step reference input is applied, the height output responds after some delay, as a minimum force is required to drive the actuator. These nonlinearities affect the steady-state value of the height: even though there is an integral action in the closed-loop control system, the steady-state error is non-zero for a constant reference input.
The system is modeled as a single-input, multi-output system where r is the reference input, and the outputs are the control input u, the flow rate f and the height h. Faults were induced in the height sensor, the flow sensor, the actuator, and also as a leakage. The proposed fault diagnosis successfully detected and isolated all the faults compared to SISO scheme [14], where all the faults were detected and isolated using the reference input and the height output.

Conclusions
Fault detection and isolation of a class of linear multiple-input and multiple-output system based on the Kalman residual and the emulators were presented. The key properties of the Kalman filter, namely the residual, is a zero-mean white noise process if and only if there is no model mismatch, drive the prediction error identification of the nominal system model, and the Kalman filter. In view of the closed-loop configuration, the noise and the disturbance are attenuated at the estimated output. The Kalman filter is the best minimum variance estimator in the class of all linear estimators.
To handle fault isolation, the powerful and effective concept of emulators was introduced. Similar in spirit to the training of the artificial neural network, a number of emulator parameter-perturbed experiments were performed to capture the perturbation model of the subsystems to help with fault isolation. The influence vectors of the emulator parameters, which are indirectly the associated subsystems, were estimated. The influence vectors captured the emulator perturbation model and hence that of the subsystem.
The residual of the Kalman filter was shown to have an additive fault indicating term when there is a model mismatch due to emulator perturbations. The model-mismatch term is affine in the emulator parameter variations. Using the expression for the fault indicating term, the fault was isolated using the influence vectors and its size was estimated. The residual, being affine in the emulator parameter variation, easily lends itself to the widely used and successful composite Bayes hypothesis-testing scheme for fault isolation.
The future work generated from this work includes its extension to a class of nonlinear multiple-input and multiple-output systems, and the development of a computationally efficient identification of the Kalman filter directly from the input data even for unstable systems. Although a gold standard for system identification, the prediction error method involves a nonlinear optimization problem and hence can suffer from the existence of local minima. Unlike the least-squares approach, it does not offer a closed-form solution to the parameter estimation problem. Instead, it relies on a recursive solution that may be time-consuming (slow convergence rate), computationally complex, and which may also suffer from initialization problems.