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Fault diagnosis and fault tolerant control using reduced order models.


During the past two decades there has been an increasing trend towards the design and development of fault; tolerant control system (FTCS), which can maintain an acceptable control performance despite the occurrence of faults. Fault tolerant control schemes proposed in the literature can be classified as active or passive schemes (Patton, 1997). In a passive scheme, both the state estimator (or observer) and controller is designed so that it is robust with respect to pre-specified set of faults including unknown input disturbances (Frank and Wunnenburg, 1989; Chen and Patton, 1999). In contrast, in an active FTCS design, the state estimator and controller is designed for normal operation, and both are reformulated or restructured on-line, as and when a fault is deemed to have occurred. An important component of an active FTCS design is the fault detection and identification (FDI) method. Among the FDI approaches, model based methods employing analytical redundancy are found to be most suitable for control applications, since the information it provides facilitates explicit corrective measures.

Recently, Prakash et al. (2002, 2005) have developed active FTCS schemes, which integrate a FDI methodology based on generalized likelihood ratio (GLR) method (Willsky, 1976; Narasimhan and Mah, 1988) with a controller, through an on-line compensation mechanism. They demonstrated through simulation and experimental studies, the superior performance of the FTCS over the conventional controller in the presence of various types of soft faults. The key component in their FTCS design is the use of a linear model (and a state estimator or Kalman filter) developed from first principles. The first principles modelling approach used in their work enables identification of various types of faults. For systems like multi-component distillation columns, even if we develop the first principles model, the resulting dimension of the model is quite high. This can lead to excessive computational demand in real-time application of the proposed FTCS. In addition to this, system observability is a prime requirement in the Kalman filter design for obtaining unique and unbiased estimates of states. Typically, not all the states are observable from the relatively few measurements that are available in such large scale systems. Thus, there is a need to develop a suitable approach, which will alleviate computational difficulties and improve upon the state estimation particularly when the system dimension is large.

In this paper, we propose the use of reduced order normal and fault models for FDI purposes which are useful for on-line active fault tolerant control of large dimensional systems. Although, methods for obtaining reduced order models for control purposes have been developed earlier, we propose a modified approach so that the reduced order models are useful for both FDI and control purposes. We also integrate these reduced order models with on-line fault tolerant control strategies developed by Prakash et al. (2002, 2005). A third useful contribution of this paper is to show how inferential control schemes can be implemented using reduced order models. The effectiveness of the proposed approach is demonstrated through simulation of an ideal binary distillation column example (Luyben, 1990).

This paper is organized as follows. In the following section, an FDI relevant model order reduction method is described. The formulation of BFTCS and FTMPC schemes based on reduced order models is presented in Fault Tolerant Control Schemes Section. Simulation results are presented in Simulation Studies Section followed by conclusions obtained from this study.


Fault detection and identification (FDI) is concerned with detecting whether a fault has occurred, and if so, identify the cause and provide an estimate of the fault magnitude. We briefly describe the FDI strategy based on the GLR method for identification of various types of faults. Although, in principle, the GLR method can be used to detect and identify both soft faults such as biases in sensors, actuators, and changes in parameters, as well as hard faults such as complete failure of the sensor or actuator, we restrict our focus in this paper only to soft faults. Achieving fault tolerance with respect to hard failures may require reconfiguration of the control structure and is outside the scope of this paper.

Brief Review of GLR Strategy

The FDI strategy makes use of process models under normal and faulty operating conditions. A linear discrete stochastic state space equation is used to model the process as follows:

x(k + 1) = [PHI]x(k) + [[GAMMA].sub.u]u(k) + w(k) (1)

y{k} = Cx(k) + v(k) (2)

where x [member of] [R.sup.n] represents the states, u [member of] [R.sup.m] represents the manipulated inputs, y [member of] [R.sup.r] represents the measured outputs, w [member of] [R.sup.n] and v [member of] [R.sup.r] are assumed to be independent zero mean Gaussian white noise sequences with covariance matrix [R.sub.1] and [R.sub.2], respectively. Under normal operation, the above model can be used to obtain the optimal estimates of the state variables using a Kalman filter (Astrom and Wittenmark, 1994). The innovations (or measurement residuals) generated by the Kalman filter are given as:

y(k) = y(k) - C[??](k|k - 1) (3)

where [??]([k|k - 1) denote the state estimates predicted at time k using all measurements made up to time (k - 1). Simple chi-square statistical tests based on these innovations are employed for fault detection and subsequent fault confirmation over a time window. Fault identification is carried out with the GLR method, where for each hypothesized fault, the characteristic innovations trend termed as fault signature is determined from the corresponding fault model and the normal estimator model. For soft faults caused by biases in sensors and actuators which are assumed to occur as step changes, the corresponding fault models can be easily obtained as follows. In the presence of a sensor bias, the measurement model Equation (2) is given by:

y(k) = Cx(k) + v(k) + [be.sub.y,i] (4)

for k [greater than or equal to] [t.sub.f], where [t.sub.f] is the time of occurrence of a fault. Likewise, in the presence of an actuator bias the state transition Equation (1) is modelled as:

x(k + 1) = [PHI]x(k) + [[GAMMA].sub.u]u(k) + w(k) + b[[GAMMA].sub.u][e.sub.u,i] (5)

In the above equations, b represents the magnitude of the fault, [e.sub.y,i] and [e.sub.u,i] are unit vectors of appropriate dimensions, subscripts y, u signifies the fault type with i as the index of the measurement or actuator where the bias occurs.

The fault models for step changes in process parameters or unmeasured disturbance variables can be described as:

x(k + 1) = [PHI]x(k) + [[GAMMA].sub.u]u(k) + w(k) + b[[GAMMA].sub.f'],[e.sub.f',i] (6)

where the matrix [[GAMMA].sub.f'] depends on the type of fault that has occurred (disturbance or parameter) with f' [member of] {d, p}. It may be noted that deviations in process parameters and or unmeasured disturbance variables from their nominal values affect the coefficients of the matrices [PHI] and [GAMMA] and are known as multiplicative faults. However, a linearized approximation around the nominal values lead to the additive fault model given by Equation (6) (Prakash et al., 2002).

From the linearity of the system and filter, the effect of each fault on the expected values of the innovations at any time can be obtained as:


where the subscript f [member of] {y, u, d, p} denotes the fault type. Here, the matrix [G.sub.f] (k; [t.sub.f]) , which is referred to as signature matrix, depends upon the time [t.sub.f] at which the fault has occurred and time k at which the innovations are computed. The vector [g.sub.f,i] which we refer to as the fault signature vector, depends upon the fault type and location. The computational details of these signature matrices and signature vectors for different faults are given in Prakash et al. (2002). The GLR method essentially identifies the fault whose signature best fits the observed innovations pattern.

Model Reduction For FDI Development

Model reduction techniques are widely used in controller design. Among the various approaches, the balanced truncation technique (Moore, 1981), removes after suitable transformations, the states that are difficult to control or observe. Since such states contribute little to the understanding of the process input-output behaviour, their removal does not significantly alter the quality of the model predictions. The distinct advantage of this reduction approach is that one can obtain an observable subsystem for the Kalman filter design to be used in FDI development. For stable systems, controllers designed using reduced order models will perform well. However, for unstable systems, reduced order models based on coprime factorization techniques are recommended for controller synthesis (Obinata and Anderson, 2001).

As explained in the preceding sub-section, the GLR based FDI strategy makes use of process models under normal as well as faulty operating conditions. Therefore, the requirement in fault diagnosis is to adequately capture fault-output dynamics as opposed to a controller design where it is important to capture the effect of manipulated inputs on the outputs. Thus, the standard approach for obtaining reduced order models to design controllers needs to be modified.

A generic state space process model which describes the effect of manipulated inputs, parametric and disturbance faults, can be obtained by combining the normal and fault models (Equations (1),(5) and (6)) as follows:

x(k + 1) = [PHI]x(k) + [GAMMA][u.sub.c], (k) (8)

y(k) = Cx(k) (9)

where x(k) [member of] [R.sup.n], y(k) [member of] [R.sup.r] are variables as defined with Equations (1) and (2). The input vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] defined as:

[u.sub.c] = [[[u.sup.T] [d.sup.T] [p.sup.T].sup.T] (10)

denotes the complete input set, which includes all manipulated inputs as well as fault inputs. The matrix [GAMMA] = [[[GAMMA].sub.u], [[GAMMA].sub.d] [[GAMMA].sub.p]] is the corresponding input coupling matrix with individual elements as explained in Equations (5) and (6), and [m.sub.u] is the total number of manipulated inputs, unmeasured disturbances, and parameters likely to change.

Let [zeta] be a vector of variables related to x by a balancing transformation:

x = T[zeta] (11)

The state space model Equations (8) and (9) in transformed domain is given as:

[zeta](k + 1) = [??][zeta](k) + [??][u.sub.c] (k) (12)

y(k) = [??][zeta](k) (13)

where [??] = [T.sup.-1][PHI]T; [??] = [T.sup.-1] [GAMMA] and [??] = CT. The balanced transformation T, defined for stable systems is computed such that the solutions of the following Lyapunov equations:

[??]P[[??].sup.T] - P + [??],[[??].sup.T] = 0 (14)

[[??].sup.T]Q[??] - Q + [[??].sup.T][??] = 0 (15)

are equal and diagonal, that is P = Q = diag([[sigma].sub.1], [[sigma].sub.2], ..., [[sigma].sub.n]), where [[sigma].sub.1] [greater than or equal to] [[sigma].sub.2] [greater than or equal to] ... [greater than or equal to] [[sigma].sub.n] > 0. P and Q are the discrete-time controllability and observability Gramian matrices, respectively (Skogestad and Postlethwaite, 1996). The [[sigma].sub.i] are the ordered Hankel singular values, more generally defined as the positive square root of the eigenvalues of the Hankel matrix H, which is the product of the Gramians:

[[sigma].sub.i] = [[lambda].sup.1/2.sub.i](H), i = 1, ..., n (16)

H = PQ (17)

In a balanced realization, each state in the transformed domain [[zeta].sub.i] is just as controllable as it is observable. The magnitude of [[sigma].sub.i] is a relative measure of the contribution that state [[zeta].sub.i] makes to the input-output behaviour. The balanced truncation approach removes the states corresponding to the smallest Hankel singular values. The state vector [zeta] is partitioned into two vectors [[zeta.sub.1] and [[zeta.sub.2] where [[zeta.sub.1] corresponds to the first l largest Hankel singular values and [[zeta].sub.2] the remaining n - l smaller singular values. With appropriate partitioning of [??], [??], [??] the state space equations becomes:



Setting [[zeta].sub.2] = 0 gives the following reduced order model:

[[zeta].sub.1](k + 1) = [[??].sub.11[[zeta].sub.1](k) + [[??].sub.1] [u.sub.c](k) (20)

[??](k) = [[??].sub.1] [[zeta].sub.1(k) (21)

which is used for the development of FDI as explained in the following section.

To decide upon the truncation order l, we can look for large changes in the Hankel singular values plot. We define the truncation index (TI) as the fraction of the original system response that is retained in the reduced dimensional model, which is expressed mathematically as:

TI = [[summation].sup.l.sub.i=1] [[sigma].sub.i]/[[summation].sup.n.sub.i=1] [[sigma].sub.i] (22)

The order of reduced dimensional state space model (l) is decided on the basis of TI value chosen.

It can be observed from Equation (20), that the matrix [[??].sub.1] is the coupling matrix which describes the effect of the manipulated inputs and faults on the reduced transformed states [[zeta].sub.1]. By partitioning the columns of this matrix corresponding to manipulated inputs and fault inputs, respectively the reduced order normal model and fault models for each type of fault can be obtained. The normal process behaviour is described as:

[[zeta].sub.1] (k + 1) = [[??].sub.11] [[zeta].sub.1] (k) + [[??].sub.u1]u(k) (23)

along with the output equation as given by Equation (21). The different fault models are obtained as follows: Sensor bias:

[??](k) = [[??].sub.1][[zeta].sub.1](k) + [b.sub.y,i][e.sub.y,i][sigma](k - t) (24)

Actuator bias:

[[zeta].sub.1] (k + 1) = [[??].sub.11] [[zeta].sub.1] (k) + [[??].sub.u1]u(k) + [b.sub.u,i][[??].sub.u1][e.sub.u,i] (25)

Disturbance change:

[[zeta].sub.1] (k + 1) = [[??].sub.11][[zeta].sub.1](k) + [[??].sub.u1]u(k) + [b.sub.d,i][[??].sub.d1][e.sub.d,i] (26)

Parameter change:

[[zeta].sub.1] (k + 1) = [[??].sub.11][[zeta].sub.1](k) + [[??].sub.u1]u(k) + [b.sub.p,i][[??].sub.p1][e.sub.p,i] (27)

It may be noted that a reduced order model for control purposes is developed by considering only the manipulated inputs as part of the input set. By considering both the manipulated inputs and the fault inputs as the complete inputs to the process in the procedure described above, we obtain a reduced order model which is useful both for control and FDI purposes. It is possible to develop a reduced order model for implementing a controller and a different reduced order model for model based FDI, by appropriate choices of the process inputs. In such a case, the reduced order models so developed will not be identical and does not allow for easy integration of the FDI and controller modules. In order to implement the FTMPC scheme, identical models must be used in FDI and MPC. The above modified balanced truncation procedure ensures this.

In the following sections, we briefly describe the two fault tolerant control schemes developed by Prakash et al. (2002, 2005) and the modifications, if any, required to use the reduced order model in these schemes.


The following two schemes for achieving fault tolerance with respect to soft faults have been proposed earlier.

* Basic fault tolerant control scheme (BFTCS): In this scheme (Prakash et al., 2002), the FDI module is integrated 'externally' with an existing multi-loop or multivariable controller, by a suitable compensation method. An important feature of this scheme is that no modifications are made to the control algorithm or controller parameters and thus this scheme can be applied to interface with any type of controller.

* Fault tolerant model predictive control (FTMPC): This scheme (Prakash et al., 2005) is specifically developed to interface the FDI module with a model predictive controller (MPC). In order to achieve tight integration, a common model is used both in the FDI and controller modules.

The FTCS principally uses the information provided by the FDI scheme to appropriately adapt/modify the controller and the FDI scheme.

Basic Fault Tolerant Control Scheme

As stated earlier, soft faults caused by biases in sensors, actuators and abrupt changes in disturbance and operating parameters are considered in the design of FTCS. Typically, a conventional feedback controller with no input constraints can deal with actuator biases, disturbance and parameter changes. This is due to the fact that integral action is usually incorporated in the controller, which can reject such disturbances. However, in the presence of a sensor bias, it takes corrective action on the biased value thereby introducing an offset in the true value of the process output. The BFTCS was developed to give good control performance despite the occurrence of any type of soft fault, by appropriately integrating the FDI method with controller. In BFTCS (Figure 1), the FDI module is like an external interface to the existing feedback control system. If a sensor bias is identified at time instant t + N, then for all subsequent times the faulty measurements are corrected using the estimated bias magnitude as follows:

[y.sub.c](k) = y(k) - [[??].sub.y,i][e.sub.y,i], k > (t + N) (28)

The corrected measurements are provided to the control and FDI modules for all subsequent time instants.


For all other types of input faults that are identified, the FDI model is compensated and used in the Kalman filter for obtaining all subsequent state estimates as follows:

[??](k + 1|k) = [PHI][??](k|k) + [[GAMMA].sub.u]u(k) + [[GAMMA].sub.w]w(k) + [[??].sub.f,i][[GAMMA].sub.f][e.sub.f,i], k > (t + N) (29)

where f [member of] {u, d, p}. In addition, the state estimates are corrected for all faults as follows:

[[??].sub.c],(k|k) = [??](k|k) - [[??].sub.f,i] [J.sub.f](k,t), k [member of] [t,t + N) (30)

The above compensation is applied to correct for the bias in the state estimates introduced due to delay of N sampling instants in identifying the fault. For on-line implementation, the fault compensation scheme has to be modified further as described below.

On-line implementation strategy

Immediately after compensation of measurements/model and state estimates, we resume application of the fault detection and identification method as before. If a new fault is identified then the compensation scheme as described above can be applied. However, there is always a possibility that the same fault is identified again, in which case we need to make use of cumulative fault estimates in the output and state correction Equations (28) and (29), respectively. This way, an integral action is introduced in the on-line compensation mechanism, which in turn provides a certain degree of self-correcting ability to the FDI. As a result, the errors in estimation of the fault magnitude or position can be corrected in course of time. The cumulative bias estimates [[??].sub.f,i]([n.sub.f,i]) are computed as:

[[??].sub.f,i]([n.sub.f,i]) = [[n.sub.f,i].summation over (j=1)][[??].sub.f,i](j), [[??].sub.f,i](0) = 0 (31)

where [n.sub.f,i] represents number of times a fault of type f is isolated in the ith position. The above expression for the cumulative bias estimates can be rewritten as:

[[??].sub.f,i]([n.sub.f,i]) = [[??].sub.f,i] ([n.sub.f,i] - 1) + [[??].sub.f,i]([n.sub.f,i]) (32)


[[??].sub.f,i](0) = 0 (33)

and [[??].sub.f,i] ([n.sub.f,i]) represents the last estimated magnitude of the fault of type f in the ith position. We can define the following cumulative bias correction vector:

[[??].sub.u] = [m.summation over (i=1)] [[??].sub.u,i]([n.sub.u,i])[e.sub.u,i] (34)

[[??].sub.d] = [d.summation over (i=1)] [[??].sub.d,i]([n.sub.d,i])[e.sub.d,i] (35)

[[??].sub.b] = [p.summation over (i=1)] [[??].sub.p,i]([n.sub.p,i])[e.sub.p,i] (36)

[[??].sub.y] = [r.summation over (i=1)] [[??].sub.y,i]([n.sub.y.sub.i])[e.sub.y,i] (37)

Using the above definitions of cumulative bias vectors, the output and state correction Equations (28) and (29) are modified as:

[y.sub.c](k) = y(k) - [[??].sub.y] (38)

[??](k + 1|k) = [PHI][??](k|k) + [[GAMMA].sub.u](u(k) + [[??].sub.u]) + [[GAMMA].sub.d][[??].sub.d] + [[GAMMA].sub.p][[??].sub.p] (39)

Under ideal conditions, when a fault is correctly identified and its magnitude is exactly estimated, the use of compensated quantities in the FDI model will neutralize the effect of the fault that has occurred. This enables the FDI method to detect and identify any subsequent fault that may occur.

In the presence of a sensor bias, a conventional control scheme produces an offset in the true value of a controlled output because its biased value is being used for control. On the other hand, the BFTCS compensation mechanism first reconstructs a plausible true value from the bias estimate provided by the FDI for use in the control law. The compensation scheme will therefore remove the resulting offset once the sensor bias is correctly identified and estimated.

For implementing this scheme, we make use of the reduced order normal and fault models given by Equations (23)-(27) in the FDI module. It may be noted, that in this scheme fault tolerance is achieved by modifying the measurements provided to the controller or the manipulated inputs computed by the controller as the case may be. Thus, it is not necessary to obtain the estimates of the original state variables and the reduced order models can be used in FDI in lieu of the full order model.

Fault Tolerant Model Predictive Control Scheme

The FTMPC scheme was developed as an integrated model based scheme, which can perform the dual task of control and fault compensation simultaneously based on the same model. The MPC was specifically chosen to perform control tasks as it also uses the Kalman filter for state estimation and prediction. We first describe the conventional MPC followed by the FTMPC formulation.

Conventional MPC formulation

The conventional MPC formulation (Ricker, 1990) makes use of the linear dynamic state space model of the form given in Equations (1) and (2) used in the development of FDI scheme.

Prediction model. At each sampling instant, the current state estimates [??](k|k) and the process model Equations (1) and (2) are used to predict the future behaviour of the process over a finite time horizon of length [N.sub.p] as follows:

[??](k + j + 1|k) = [PHI][??](k + j|k) + [[GAMMA].sub.u]u (k + j|k) + K(k)[gamma](k) (40)

[??](k + j + 1|k) = C[??](k + j + 1|k) + [I - CK(k)][gamma](k) (41)

[gamma](k) = y(k) - C[??](k|k) (42)

Objective problem formulation. At any sampling instant k, the model predictive control problem is defined as a constrained optimization problem where the future manipulated input moves u (k|k), u(k + 1|k), ..., u (k + [N.sub.c] - 1|k) are determined by minimizing an objective function:

min {[[N.sub.p].summation over (j=1)] e[(k + j|k).sup.T][W.sub.e]e(k + j|k) + [[N.sub.c]-1].summation over (j=0)] [DELTA]u [(k + j|k).sup.T][W.sub.u][DELTA]u(k + j|k)} (43)

subject to the constraints Equations (40)-(42), and

u(k + [N.sub.p] - 1|k) = ... u (k + [N.sub.c] + 1|k) = u(k + [N.sub.c]|k) = u(k + [N.sub.c] - 1|k) (44)

[x.sup.L] [less than or equal to] [??](k + j|k) [less than or equal to] [x.sup.H], j = 1, ..., [N.sub.p] (45)

[x.sup.L] [less than or equal to] [??](k + j|k) [less than or equal to] [y.sup.H], j = 1, ..., [N.sub.p] (46)

[u.sup.L] [less than or equal to] u(k + j|k) [less than or equal to] [u.sup.H], j = 0, ..., [N.sub.c] - 1 (47)

[DELTA][u.sup.L] [less than or equal to] [DELTA]u (k + j|k) [less than or equal to] [DELTA][u.sup.H], j = 1, ..., [N.sub.c] - 1 (48)


e(k + j|k) = [y.sub.r](k + j|k) - [??](k + j|k) (49)

[DELTA]u(k + j|k) = u(k + j|k) - u(k + j - 1|k) (50)

Here, {[y.sub.r] (k + j|k): j = 1, 2, ..., [N.sub.p]} represents the future set point trajectory, and We, [W.sub.e], [W.sub.u] are symmetric positive semi-definite weighting matrices on the set-point error and manipulated inputs, respectively. The desired closed loop performance is achieved by appropriately selecting prediction horizon [N.sub.p], control horizon [N.sub.c], [W.sub.e] and [W.sub.u]. The controller is implemented in the moving horizon framework, that is after solving the optimization problem, only the first move u(k|k) is implemented on the process and the optimization problem is formulated at the next sampling instant based on the updated information from the process.

FTMPC formulation

Figure 2 shows the schematic of the FTMPC scheme. In the FTMPC scheme, the FDI module and the MPC are tightly coupled by using the same model both in FDI and controller modules. The information provided by the FDI module is used to reformulate the control law, depending upon the type of fault that is identified as described below.

Similar to BFTCS, the on-line FDI method is applied for fault detection and identification. Cumulative bias estimates (as defined by Equations (34)-(37)) of all fault types identified until current time are maintained. These cumulative bias estimates are used to modify the model and, hence, the estimator used in the Kalman filter for state estimation. The state estimates at each time are obtained as:

[??](k|k - 1) = [PHI][??](k - 1|k - 1) + [[GAMMA].sub.u][u(k - 1) + [[??].sub.u]] + [[GAMMA].sub.d][[??].sub.d] + [[GAMMA].sub.p][[??].sub.p] (51)

[??](k|k) = [??](k|k - 1) + K(k)[[gamma].sub.c](k|k) (52)

[[gamma].sub.c](k|k) = [y.sub.c](k) - C[??](k|k - 1) (53)

[y.sub.c] (k) = y(k) - [[??].sub.y] (54)

The cumulative fault magnitude estimates are also used to modify the future predictions used in MPC as follows:


with the constraints on the future manipulated input moves being modified as:

[u.sup.L] [less than or equal to] u(k + j|k) + [[??].sub.u] [less than or equal to] [u.sup.H], j = 0, ..., [N.sub.c] - 1 (56)

As can be seen, this type of an integration mode has a distinct advantage in the presence of input, output or state constraints as they can be suitably reformulated based on the fault estimates provided by FDI. Further, the information provided by the FDI module can be used to generate unbiased output estimates in the presence of various types of faults.


The reduced order normal and fault models can be used, instead of the full order model, in the above FTMPC scheme without any changes provided there are no constraints on the original state variables, and measured outputs are controlled. If constraints on the state variables are imposed or the controlled variables are inferred variables (inferential control), then we need to obtain the estimates of the original variables at each time from the filtered/ predicted estimates of the reduced order model, as described below.

Using the reduced order models in the FDI module, estimates of the reduced dimensional state vector [[zeta].sub.1] in the transformed domain [zeta] are obtained. Since the reduced order model has been derived under the assumption that [[zeta].sub.2] is zero, the estimate of the full transformed vector [zeta](k|k) can be obtained as:

[??](k|k) = [[[[??].sub.1](k|k) 0].sup.T] (57)

The transformed state vector [zeta] is related to original state vector x by a nonsingular transformation. Thus, by applying the inverse transformation we recover an estimate of the complete original state vector as:

[??](k|k) = [T.sup.-1] [??](k|k) (58)

In a similar manner, the predicted estimates of the full state vector in the MPC module for each time instant of the prediction horizon, can be computed from the predicted estimates of the states [[zeta].sub.j|k] for all j > k. The constraint residuals, and the objective function in MPC can be obtained as a function of the full order state estimates. With this modification, the inferential FTMPC scheme can be implemented.


The performance of the proposed scheme is tested through simulation of a 20-tray single feed binary distillation column (Luyben, 1990). A non-linear model of the column derived from first principles under some assumptions is described in Appendix A. The full-order model of the process consists of 42 non-linear ordinary differential equations. The reflux flow rate (R) and vapour boil-up rate (V) are manipulated to control the top and bottom product compositions ([x.sub.d] and [x.sub.b]). The nominal steady state operating data for the column are given in Table 1.

It was first verified that the linearized model of the process around the operating state steady state was open loop stable. A reduced order model of the binary distillation column is developed and used in the BFTCS and FTMPC schemes as described in the preceding section. It should be noted that the reduced order linear normal and fault models are used in FDI (and MPC controller), while the process is simulated using either a full-scale linearized process model or the non-linear model as indicated later. The use of non-linear model for simulating the process will reveal the combined effect of process model mismatch introduced due to linearization and the reduced-order model approximation used in FDI.

Performance Measures

Simulations are carried out for different fault scenarios. Corresponding to each fault scenario (also denoted as a simulation run), several simulation trials each consisting of L measurement samples are carried out. Both the conventional control strategy as well as the fault tolerant control strategy are applied to the simulation trials in each run and the performance of the schemes evaluated using the following measures as defined by Prakash et al. (2002).

* Performance index ([PI.sub.i]) for a controlled variable i is defined as:

[PI.sub.i] = [ISE.sub.i] (fault tolerant control scheme)/ [ISE.sub.i](conventional control scheme) (59)

where [ISE.sub.i] is the sum of squared differences between the true value of the measured variable and the corresponding set point in any simulation trial. The value of the [PI.sub.i] averaged over all the trials in a simulation run is reported.

* Percentage of successfal trials (PST)

PST = number of successful trials ([N.sub.s])/ total number of trials ([N.sub.T]) x 100 (60)

where a successful trial is defined as one in which the introduced fault is identified at least once in that trial.

* False alarm index (FAI)

FAI = total number of false alarms ([N.sub.F])/ number of trials ([N.sub.T]) x length of simulation trial (L)/window length (N) (61)

Results and Discussion

Case 1: BFTCS studies

In this particular case, the two product compositions to be controlled are considered as the measured variables. Seven different faults are hypothesized namely, bias in the two concentration measurements ([x.sub.d] and [x.sub.b]), bias in the two actuators corresponding to the manipulated inputs (reflux R and vapour boil-up [V.sub.B]), step change in the two disturbance variables (feed flow rate F and feed composition [z.sub.f]) and a step change in the tray efficiency parameter (n). We have in total five inputs (R, [V.sub.B], F, [z.sub.f] and [eta]) to be considered for obtaining a reduced order model for fault diagnosis and control. The reduced order linear model of the process is obtained as follows.

Model reduction. A linearized continuous state space model is developed by linearizing the non-linear equations around the normal operating point (reported in Table 1) as follows:

[??](t) = Ax(t) + [B.sub.u]u(t) + [B.sub.d]d(t) + [B.sub.p]p(t) (62)

y(t) = Cx(t) (63)

where the state space matrices [A B C] and fault coupling matrices [[B.sub.u] [B.sub.d] [B.sub.p]] are obtained by numerically evaluating the Jacobians of the differential equations with respect to the state, input and fault variables. A truncation index (TI) of 0.999 was chosen to decide upon the model order of 8 as shown in the plot of Hankel singular values in Figure 3.

Figure 4 compares unit step responses of the distillate product composition obtained with respect to different inputs for the full-scale linearized and reduced order linear model. As can be seen from the Figures 4a-e, the reduced order model is able to capture the system dynamics reasonably well for all manipulated and disturbance inputs that are considered. Figures 5a-e shows a similar close match for the bottom product composition. The reduced 8th order continuous model so obtained is discretized with a sampling time of 1 min for use in FDI and controller development.


An unconstrained DMC control law is developed with tuning parameters as given in Table 2. The reduced model based FDI strategy is incorporated in the BFTCS design with the tuning parameters as given in Table 3. Guidelines for choosing these tuning parameters as recommended by Prakash et al. (2002) are also found useful for this particular application.

In all simulation runs, 50 simulation trials (NT) were performed with each trial lasting for 1000 (L) sampling instants. State noise is simulated through random fluctuations in unmeasured disturbance variables (feed flow rate F and feed composition [z.sub.f]) as Gaussian zero-mean white noise sequences with standard deviations 5 and 1 percent of their nominal values, respectively. Measurement noise is simulated as Gaussian zero-mean white noise sequence with standard deviation 1 percent of their respective nominal values. The Kalman filter for FDI development is designed using a reduced order normal description of the process with these particular noise characteristics. For all the simulation cases reported, the fault is introduced at the 1st sampling instant. The magnitude of the sensor and actuator bias introduced was in proportion to the standard deviation of the output or input, as the case may be under normal operation (reported in Table 1).

A comparative study of FDI and BFTCS control performance is made for the following three subcases namely,

* FDI based on full-scale linear model; process simulated using full-scale linear model (FS-L).

* FDI based on reduced order model; process simulated using full-scale linear model (RO-L).

* FDI based on reduced order model; process simulated using full-scale non-linear model (RO-NL).

The three subcases correspond to different degrees of process-model mismatch. While in FS-L there is no process-model mismatch since the same model is used both for simulation and in FDI, there may be a minor mismatch in RO-L due to the use of a reduced order model in FDI, and a more significant mismatch in RO-NL since the process is simulated using the non-linear differential equations.

For the no fault case, Table 4 reports the number of trials that are successful out of the total 50 trials that were carried out in a simulation run, the number of false alarms committed in the entire simulation run and the PI values for the two controlled outputs ([x.sub.d], [x.sub.b]). In the tables, both the average values as well as the standard deviation in a simulation run are reported. For linear process simulation (FS-L, RO-L), the performance with full-scale and reduced order FDI model is almost identical with respect to FDI as well as FTCS performance. With non-linear process simulation (RO-NL), the FDI performance deteriorates because of the resulting process model mismatch. Nevertheless, the PI values indicate that the control performance of BFTCS and conventional control scheme are comparable despite the false alarms that are committed. The reason is that all the false alarms that are committed are well within the noise band as can be seen from the cumulative bias plots in Figure 6. We next compare the performances measures when different faults are introduced. Along with the performance measures as defined in Performance Measures Section, the average value of the cumulative bias estimate and its standard deviation in a simulation run are given in the tables for different fault scenarios.


For sensor and parametric faults (as reported in Tables 5 and 6), the FDI and BFTCS control performance with the two different models for FDI and linear process simulation (FS-L and RO-L) are comparable in terms of number of successful detections, number of false alarms committed, cumulative bias estimated and the PI values. As has been pointed out by Prakash et al. (2002) a conventional control scheme causes an offset in the presence of a sensor bias. However, the proposed FTCS scheme is capable of eliminating this offset. This is also clearly brought out in the results of the PI values for the biased variable in Table 5, which are significantly smaller than unity. Figure 7 shows the response of the distillate concentration for the BFTCS and conventional control scheme using the reduced order model in a typical simulation trial. This figure clearly shows that the conventional control scheme produces an offset between the true value and the set point while the fault tolerant scheme does not. In the case of parametric faults, the BFTCS does not further improve the control performance as the DMC conventional controller can itself handle such small magnitude process parameter changes.

For actuator and disturbance fault with the full-scale model (i.e., for FS-L as reported in Tables 7 and 8, respectively), there is a significant deterioration in FDI performance (in terms of number of false alarms committed) and in control performance with PI values of 1.36 and 1.79 (due to fault accommodation for false alarms). In fact, these are quite unacceptable results from control performance point of view. However, the reduced dimensional model gives satisfactory performance for all the fault cases considered (Tables 5-8). We find that the system observability property, a prime requirement in the design of Kalman filter for obtaining unique estimates of the states is the cause of the deteriorated FDI performance with full-scale model. For the full-scale model, with the state dimension as 42, the rank of observability matrix is 14. For the reduced order model, with the state dimension as 8, the rank of observability matrix is 8. When compared to the reduced order observable system description, the Kalman filter implementation with full-scale unobservable model yields non-unique solution. The impact of these suboptimal state estimates on the on-line FDI performance is brought out through these simulation studies.


In all the fault cases reported here (Tables 5-8), the non-linear process simulation deteriorates the FDI and BFTCS control performance when compared to the linear case because of the resulting process model mismatch. However, the PI values indicate that the resulting FTCS control performance is superior to the conventional control scheme in the presence of a sensor bias. Furthermore, the performance of the two schemes is comparable for the different input faults considered namely, actuator, disturbance and parameter fault.


The computation time requirements on a 3 GHz Pentium IV machine for state estimation and for fault identification using reduced order and full order models were compared. The computing time for state estimation for each sampling instant was obtained by averaging over the 1000 sampling instants in a simulation trial, and was found to be 8.2 x [10.sup.-5] s using reduced order model and 9.3 x [10.sup.-5] s using full order model. The computation time for fault identification is the time required each time a fault is detected, and is obtained by averaging over all fault detection events in a simulation run. This was found to be 1.9 x [10.sup.-2] s using reduced order model and 2.4 x [10.sup.-2] s using full order model. Thus, for this process, using a reduced order model decreases the computing time by about 12 percent for state estimation and 25 percent for fault identification.


Case 11: FTMPC studies

We have assumed in Case I that the product compositions (controlled outputs) are the measured variables. In most practical applications, these outputs are estimated or inferred from tray temperature measurements, since the latter are far more frequently available and less expensive when compared to concentration measurements (Mejdell and Skogestad, 1993).

In this case study, we demonstrate the use of FTMPC using reduced order model(s) for inferential control. For this purpose, six temperature measurements are used corresponding to the tray locations 3, 8, 10, 17, reboiler and condenser. A linear empirical relationship is considered between the column tray temperatures and the corresponding tray compositions (as given in Appendix A).

Model reduction. We consider a linear state space model with state equation given by Equation (62) and output equation expressed as:

[y.sub.[theta]](t) = [C.sub.[theta]]x(t) + [v.sub.[theta]](t) (64)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] represents the measured output vector for developing a reduced order model between the temperature output measurements and the different fault inputs. As in Case I, a TI value of 0.999 was chosen as the cut off limit which corresponds to a reduced order model of 15 states. The step responses of the tray temperatures obtained using the full-order model and reduced order model matched well for step changes in the different inputs.

Inferential control. The inferential control implementation essentially requires estimates of the two original state elements, that is the product compositions [x.sub.d] and [x.sub.b] for control. In the FTMPC scheme we had described how the complete original state estimates can be obtained from the reduced order state estimates. The estimates of the two product compositions can be extracted from the complete state estimates.

The FTMPC scheme is developed with FDI and controller implementation based on the reduced order models obtained above (with tuning parameters as reported in Tables 9 and 10, respectively). In all simulation runs, 50 simulation trials ([N.sub.T]) were performed with each trial lasting for 1000 (L) sampling instants. Eleven different faults are hypothesized namely, bias in the six temperature measurements, bias in the two actuators for reflux and vapour boil-up, step change in the two disturbance variables (feed flow rate and feed composition) and a step change in the tray efficiency parameter. For all the simulation cases reported, the fault is introduced at the 1st sampling instant. State noise is simulated similar to Case I. Measurement noise is simulated as Gaussian zero mean white noise sequence with standard deviation of 1[degrees]C on the measured temperatures.

FTMPC performance with non-linear process simulation. For the FTMPC scheme with reduced order models, we only present the results of simulation for the most difficult case when the process is simulated using the complete non-linear model while the FTMPC scheme uses the reduced order linear model. For the no fault case, 40 out of 50 trials were successful with no fault detections. In the entire simulation run, 12 false alarms were detected, but they did not adversely affect the FTMPC performance.

The performance measures using FTMPC for different types of faults are given in Table 11. A sensor bias of 1 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for the third tray temperature measurement is considered. Different input faults considered are namely, bias in the actuator corresponding to the reflux flow rate (-3[[sigma].sub.R]), change in the feed flow rate disturbance input of 10 percent and a change in the tray efficiency parameter of 17 percent from their respective nominal values.

It is interesting to note that a bias in one of the temperature leads to biased estimates of the product concentration in a conventional MPC scheme, which in turn leads to an offset between the true value and set point of both the controlled variables (Table 11). The FTMPC scheme on the other hand, gives unbiased estimates of the controlled variables. This results in significantly improving control performance, as indicated by the lower PI values.

For actuator biases, or changes in unmeasured disturbances or parameters again a conventional MPC scheme leads to biased estimates of concentrations. Figure 8 shows the true values of the controlled outputs with conventional and FTMPC scheme in the presence of a parametric fault. When compared to the conventional control scheme which gives an offset, the FTMPC removes this offset subsequently once the fault is identified and the unbiased estimates provided by FDI are used in the control law. This clearly indicates that the conventional approach to account for such faults in model predictive control schemes will not work satisfactorily when used for inferential control, although they may be able to reject such faults in non-inferential schemes.


The computing time requirements for state estimation (for each sampling instant) was found to be 4.6 x [10.sup.-4] and 8.4 x [10.sup.-4] s, respectively, using a reduced order and full order model in FTMPC. For each fault identification, the computing time required using reduced order and full order models was found to be 3.6 x [10.sup.-2] and 4.4 x [10.sup.-2] s, respectively. Thus, for this example, the use of reduced order model decreases the computing time requirements for state estimation and fault identification by 48 and 25 percent, respectively.


In this work, we have developed a FDI relevant reduced order model which enables efficient real-time implementation of fault tolerant control schemes for large dimensional processes. State estimation using reduced order observable models considerably improves the diagnostic performance when compared to the use of unobservable full-scale model. In the case of inferential control, the FTMPC scheme performance is superior to the conventional control scheme when various soft faults occur. In particular, for distillation operations, where it is a common practice to infer product compositions from the temperature measurements, the results highlight the significant merit of incorporating tolerance to soft faults.


Financial support for carrying out this work was provided by the Department of Science and Technology through the project CHE/00-01/051/DST/SACHX and Council of Scientific and Industrial Research (CSIR SRF No. 9/84(303)/98-EMR-I), India.


The following assumptions are made in deriving a non-linear model from first principles: constant overflows, constant relative volatility, linear liquid flow dynamics, constant pressure, no vapour hold-up, total condenser, perfect level control in condenser and reboiler. The only modification made to the column simulation by Luyben (1990) is that the Murphree vapour phase stage efficiency [eta] is considered (assumed to be the same for all trays) in order to demonstrate parametric faults. The resulting model equations consists of two differential (material balance and component balance) and three algebraic equations (vapour liquid equilibrium, tray efficiency and liquid hydraulic relation) for each stage i modelled as follows:

d[M.sub.i]/dt = [L.sub.i+1] - [L.sub.i] (A-1)

[M.sub.i] d[X.sub.i]/dt = [L.sub.i+1] ([X.sub.i+1] - [X.sub.i]) + [V.sub.B] ([Y.sub.i-1] - [Y.sub.i]) (A-2)

[Y.sup.*.sub.i] = [alpha][X.sub.i]/1 + ([alpha] - 1)[X.sub.i] (A-3)

[[eta].sub.i] = [Y.sub.i] - [Y.sub.i-1]/[Y.sup.*.sub.i] - [Y.sub.i-1] (A-4)

[L.sub.i] = [[bar.L].sub.i] + [[bar.M].sub.i] - [[bar.M].sub.i]/[beta] (A-5)

For feed plate (i = 10) with saturated liquid feed, there is an additional term F in the material balance and F([Z.sub.f] - [X.sub.i]) in the component balance. It should also be noted that [L.sub.N+1] = R (trays numbered from bottom to top) is the reflux flow rate which is manipulated along with [V.sub.B] to control the product compositions. From the above modelling equations, the distillation column can be described by the following set of non-linear differential equations:

[??](t) = g(X, U, D, P, t) (A-6)

where the state vector X(t) represents the hold-up ([M.sub.i]) and liquid concentration ([X.sub.i]) on each tray, U = [R, [V.sub.B]] represents the manipulated input vector, D = [F, [z.sub.f]] represents the disturbance vector and P represents the model parameter vector [eta]. For inferential control we assume that tray compositions are estimated from tray temperature measurements which are related to the liquid compositions as follows:

[T.sub.i] = [X.sub.i][T.sub.bL] + (1 - [X.sub.i]) [T.sub.bH] (A-7)

where [T.sub.i] is the temperature of tray i and [X.sub.i] is the liquid mole fraction. [T.sub.bL] and [T.sub.bH] are the boiling points of the two pure components. For the column considered, we take:

[T.sub.bL] = 341.9 K; [T.sub.bH] = 355.4 K (A-8)

Manuscript received November 17, 2006; revised manuscript received October 10, 2007; accepted for publication October 14, 2007.


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Seema Manuja, (1) Shankar Narasimhanl (1) * and Sachin Patwardhan (2)

(1.) Department of Chemical Engineering, IIT Madras, Chennoi 600036, India

(2.) Department of Chemical Engineering, IIT Bombay, Mumboi 400 076, India

* Author to whom correspondence may be addressed. E-mail address:
Table 1. Steady state parameter values for distillation column

Variable description Value

Number of trays 20

Feed tray 10

Relative volatility ([alpha]) 2

Hydraulic parameter ([beta]) 0.10

Distillate concentration [x.sub.d] 0.904 (0.009)

Bottom concentration [x.sub.b] 0.0272 (0.0002)

Reflux R ([[sigma].sub.R]) 124.08 (5.09) mol/min

Vapour boil-up [V.sub.B] 178.01 (5.52) mol/min

Feed flow rate F ([[sigma].sub.F]) 100.00 (5) mol/min

Feed composition [z.sub.f] 0.5 (0.005)

Each tray efficiency parameter 0.7

Condenser hold-up ([M.sub.D]) 100 mol

Reboiler hold-up ([M.sub.B]) 100 mol

Hold-up on each tray ([M.sup.i]) 10 mol

Table 2. Parameter values for DMC in Case I

Parameter Value

Prediction horizon ([N.sub.P]) 20
Control horizon ([N.sub.c]) 1
Error weighting matrix ([W.sub.e]) diag{1, 37}

Table 3. Parameter values for FDI in Case I

Parameter Value

Window length (N) 60
Level of significance for FDT ([[alpha].sub.FDT]) 0.75
Level of significance for FCT ([[alpha].sub.FCT]) 0.01

Table 4. Comparison of FDI and BFTCS control performance for the
no fault case
 (PI)[X.sub.d] (PI)[X.sub.b]
Model Successful No. of false ([sigma](PI) ([sigma](PI)
 trials alarms [X.sub.d]) [X.sub.b])

FS-L 46 8 1.02 1.001
 (0.08) (0.001)

RO-L 47 4 1.002 1.00
 (0.016) (0.00002)

RC-NL 39 14 1.006 1.015
 (0.032) (0.053)

Table 5. FDI and BFTCS control performance for a bias in

Model [??]([[sigma]. PST FAI

FS-L 0.0265 (0.0017) 100 0.008
RO-L 0.0266 (0.0009) 100 0.006
RO-NL 0.024 (0.001) 100 0.014

Model (PI)[x.sub.d] ([sigma] (PI)[x.sub.b] ([sigma]
 (PI)[X.sub.d]) (PI)[X.sub.b])

FS-L 0.08 (0.008) 1.007 (0.004)
RO-L 0.076 (0.0041) 1.006 (0.0036)
RO-NL 0.090 (0.008) 0.964 (0.009)

Table 6. FDI and BFTCS control performance for a fault in [eta]
(~10% = - 0.075)

Model [??]([[sigma]. PST FAI

FS-L -0.076 (0.003) 100 0.014
RO-L -0.077 (0.003) 100 0.007
RO-NL -0.1 (0.004) 100 0.0024

Model (PI)[x.sub.d] (PI)[x.sub.b]
 ([sigma](PI)[X.sub.d]) ([sigma](PI)[X.sub.b])

FS-L 1.028 (0.095) 1.001 (0.002)
RO-L 1.004 (0.02) 1.0002 (0.002)
RO-NL 1.013 (0.089) 1.0002 (0.0013)

Table 7. FDI and BFTCS control performance for a bias in R
(-1[sigma] = -5)

Model [??]([[sigma]. PST FAI

FS-L -4.87 (0.69) 66 0.096
RO-L -5.07 (0.19) 74 0.03
RO-NL -5.332 (0.188) 96 0.0036

Model (PI)[x.sub.d] ([sigma] (PI)[x.sub.b] ([sigma]
 (PI)[X.sub.d]) (PI)[X.sub.b])

FS-L 1.36 (0.95) 1.01 (0.03)
RO-L 1.013 (0.066) 1.004 (0.016)
RO-NL 1.0002 (0.0013) 1.0002 (0.0007)

Table 8. FDI and BFTCS control performance for a step change in
F (10% = 10)

Model [??]([[sigma]. PST FAI

FS-L 9.9 (0.68) 100 0.14
RO-L 10.001 (0.486) 100 0.07
RO-NL 10.18 (0.479) 100 0.056

Model (PI)[x.sub.d] (PI)[x.sub.b]
 ([sigma](PI)[X.sub.d]) ([sigma](PI)[X.sub.b])

FS-L 1.79 (1.44) 1.01 (0.01)
RO-L 1.007 (0.025) 1.06 (0.066)
RO-NL 1.032 (0.094) 1.001 (0.008)

Table 9. Parameter values for DMC in Case II

Parameter Value

Prediction horizon ([N.sub.p]) 30
Control horizon ([N.sub.c]) 1
Error weighting matrix ([W.sub.e]) diag[11 368]

Table 10. Parameter values for FDI in Case II

Parameter Value

Window length (N) 40
Level of significance for FDT ([[alpha].sub.FDT]) 0.75
Level of significance for FCT ([[alpha].sub.FCT]) 0.01

Table 11. FDI and FTMPC control performance for different faults

Fault type Fault magnitude [??] ([[??].

Sensor bias [MATHEMATICAL 1 1.04 (0.19)

Sensor bias [MATHEMATICAL 3 2.93 (0.16)

Actuator bias (R) -2.5 -2.9 (0.43)

Disturbance change (F) 10 9.65 (0.90)

Parameter change ([eta]) -0.125 -0.18 (0.04)

Fault type PST FAI

Sensor bias [MATHEMATICAL 100 0.0080

Sensor bias [MATHEMATICAL 100 0.006

Actuator bias (R) 84 0.03

Disturbance change (F) 100 0.01

Parameter change ([eta]) 184 10.006

Fault type (PI)[x.sub.d] (PI)[x.sub.b]
 ([sigma] ([sigma]
 (PI)[X.sub.d]) (PI)[X.sub.b])

Sensor bias [MATHEMATICAL 0.84 (0.20) 0.82 (0.11)

Sensor bias [MATHEMATICAL 0.22 (0.05) 0.35 (0.025)

Actuator bias (R) 0.85 (0.24) 1.02 (0.07)

Disturbance change (F) 0.133 (0.033) 0.069 (0.011)

Parameter change ([eta]) 0.53 (0.29) 0.49 (0.29)
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Author:Manuja, Seema; Narasimhan, Shankar; Patwardhan, Sachin
Publication:Canadian Journal of Chemical Engineering
Date:Aug 1, 2008
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