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Keywords:Extended Kalman lterGeneralized likelihood ratio methodFault diagnosis and accommodationFailure toleranceNonlinear model predictive control

time progresses, slow drifts in unmeasured disturbances andchanges in process parameters can lead to signicant mismatchin plant and model behavior. Also, NMPC schemes are typicallydeveloped under the assumption that sensors and actuators arefree from faults. However, soft faults, such as biases in sensors oractuators, are frequently encountered in the process industry. Inaddition to this, some sensor(s) and/or actuator(s) may fail during

The conventional approach to deal with the model-plant mis-match in the NMPC formulations is through the introduction ofadditional articial states in the state observer [24]. The main lim-itation of this approach is that the number of extra states intro-duced cannot exceed the number of measurements. This impliesthat it is necessary to have a priori knowledge of which subset offaults are most likely to occur or which parameters are most likelyto drift. In such a formulationwhere the state vector is permanentlyaugmented with subset of parameters to be estimated, the stateestimates can become biased when unanticipated abrupt1. Introduction

The need to operate continuous processes over wide operatingranges and semi-batch/batch processes efciently has motivatedthe development of nonlinear MPC (NMPC) techniques over lasttwo decades. These techniques employ nonlinear models for pre-diction. The prediction model is typically developed once in thebeginning of implementation of an NMPC scheme. However, as

operation, which results in loss of degrees of freedom for control.Occurrences of such parametric changes, soft faults and failuresprogressively result in severe model-plant mismatch. This can leadto a signicant degradation in the closed loop performance of theNMPC scheme and may also lead to instability. Thus, to arrestthe degradation in controller performance, it is extremely impor-tant to isolate the root causes of the plant model mismatch and,if possible, compensate for them on-line.There is growing realization that on-line model maintenance is the key to realizing long term benets of apredictive control scheme. In this work, a novel intelligent nonlinear state estimation strategy is pro-posed, which keeps diagnosing the root cause(s) of the plant model mismatch by isolating the subsetof active faults (abrupt changes in parameters/disturbances, biases in sensors/actuators, actuator/sensorfailures) and auto-corrects the model on-line so as to accommodate the isolated faults/failures. To carryout the task of fault diagnosis in multivariate nonlinear time varying systems, we propose a nonlinearversion of the generalized likelihood ratio (GLR) based fault diagnosis and identication (FDI) scheme(NL-GLR). An active fault tolerant NMPC (FTNMPC) scheme is developed that makes use of the fault/fail-ure location and magnitude estimates generated by NL-GLR to correct the state estimator and predictionmodel used in NMPC formulation. This facilitates application of the fault tolerant scheme to nonlinearand time varying processes including batch and semi-batch processes. The advantages of the proposedintelligent state estimation and FTNMPC schemes are demonstrated by conducting simulation studieson a benchmark CSTR system, which exhibits input multiplicity and change in the sign of steady stategain, and a fed batch bioreactor, which exhibits strongly nonlinear dynamics. By simulating a regulatorycontrol problem associated with an unstable nonlinear system given by Chen and Allgower [H. Chen,F. Allgower, A quasi innite horizon nonlinear model predictive control scheme with guaranteed stability,Automatica 34(10) (1998) 12051217], we also demonstrate that the proposed intelligent state estima-tion strategy can be used to maintain asymptotic closed loop stability in the face of abrupt changes inmodel parameters. Analysis of the simulation results reveals that the proposed approach provides a com-prehensive method for treating both faults (biases/drifts in sensors/actuators/model parameters) andfailures (sensor/ actuator failures) under the unied framework of fault tolerant nonlinear predictivecontrol.a b s t r a c tIntelligent state estimation for fault toler

Anjali P. Deshpande a, Sachin C. Patwardhan b , Shanka Systems and Control Engineering, Indian Institute of Technology, Bombay, Powai, MumbDepartment of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, McDepartment of Chemical Engineering, Indian Institute of Technology, Madras, Chennait nonlinear predictive control

S. Narasimhan c

400076, Indiabai 400076, India36, India

of faults (abrupt changes in unmeasured disturbance, parameterdrifts, sensor/actuator biases) and sensor/actuator failures in a typ-changes/faults occur. Moreover, the permanent state augmentationapproach cannot systematically deal with the difculties arising outof sensor biases or actuator/sensor failures. The difculties encoun-tered while selecting such a subset in design of extended Kalmanlter (EKF) for a complex large dimensional system (TennesseeEastman problem) have been highlighted by Ricker and Lee [2].

Attempts to develop fault-tolerant MPC schemes have mainlyfocused on dealing with sensor or actuator failures [57]. Yu etal. [6] have proposed to develop a failure tolerant cascaded Kalmanlter with online tuning parameters. This approach involves thedesign of main and auxiliary Kalman lter (KF) based on reliableset of measurements and complete set of measurements, respec-tively. The auxiliary KF is used to remove the bias from the esti-mates given by the main KF. The steady state gain of auxiliary KFis modied online based on the failed measurements. Though thisapproach achieves fault tolerance while maintaining the integrityin the estimate of the lost output, the fault detection and isolationaspect does not feature in the formulation. Recently, Prakash et al.[8] have proposed an active fault tolerant linear MPC (FTMPC)scheme, which can systematically deal with soft faults in a uniedframework. The FTMPC scheme is developed by integrating gener-alized likelihood ratio (GLR) method, a model based fault detectionand identication (FDI) scheme, with the state space formulationof MPC based on Kalman lter. The GLR method performs faultidentication using innovation sequence generated by the Kalmanlter over a moving window of data in the past and this facilitatesvery close integration of the FDI and MPC schemes. The main lim-itation of these approaches arises from the use of linear perturba-tion model for performing control and diagnosis tasks. The use oflinear models not only restricts its applicability to a narrow oper-ating range but also limits the diagnostic abilities of fault detectionand identication (FDI) components to only linear additive typefaults. As a consequence, many faults that have a nonlinear effecton the system dynamics, such as abrupt changes in model param-eters or unmeasured disturbances, have to be approximated as lin-ear additive faults. Moreover, the FTMPC scheme does not dealwith failures of sensors or actuators.

Recently, Mhaskar et al. [9,10] have presented an approach thatdeals with control system or actuator failure in nonlinear processessubject to constraints. They have presented an approach for designof robust hybrid predictive candidate controllers, which guaran-tees stability from an explicitly characterized set of initial condi-tions, subject to uncertainty and constraints. Reconguration orcontroller switching is done to activate or deactivate the constitu-ent control conguration in order to achieve fault tolerance. TheFault tolerant controller uses the knowledge of the stability regionsof the back up control congurations to guide the state trajectorywithin the stability regions of the back up control congurationsto enhance the fault tolerance capabilities. Their approach, how-ever, requires nonlinear system under consideration to have inputafne structure. In another article, Mhaskar et al. [11] have pre-sented an integrated fault detection and fault-tolerant controlstructure, for SISO nonlinear systems with input constraints sub-ject to control failures. A bounded Lyapunov based controller hasbeen developed, which depends on construction of control Lyapu-nov function. Upon failure of the primary controller, the faulty con-guration is shut down and a well functioning fall backconguration is switched on. It may be noted that various controlstructures are developed by exploiting specic structural featuresof a nonlinear system, as no standard method is available for con-struction of these control Lyapunov functions. Also, these ap-proaches, as proposed, do not address difculties arising fromabrupt changes in model parameters, mean shift in unmeasured

188disturbances, sensor/actuator biases and failed sensors.Examination of various fault tolerant MPC/NMPC formulations

proposed in literature reveals that the design of state observer isical situation where the number of degrees of freedom available forobserver design (synonymous with the number of measurementsavailable for observer construction) is limited (i.e. far less thanthe number faults and failures to be dealt), then it becomes imper-ative to introduce some degree of intelligence in the state estima-tion to overcome these limitations [12]. In the present work, anintelligent nonlinear state estimation strategy is proposed, whichkeeps diagnosing the root cause(s) of the plant model mismatchby isolating the subset of active faults and auto-corrects the modelon-line so as to accommodate the isolated faults. To carry out thetask of fault diagnosis in multivariate nonlinear time varying sys-tems, we propose a nonlinear version of the generalized likelihoodratio (GLR) based FDI scheme, which is referred to as nonlinear GLR(NL-GLR) in the rest of the text. The NL-GLR scheme, along with thefault location, also generates an estimate of the fault magnitude,which is used to correct the prediction model used in the proposedfault tolerant NMPC (FTNMPC) formulation. As the proposed NL-GLR scheme is computationally demanding, it is further simpliedfor online implementation (SNL-GLR). This simplication is basedon linearization of nonlinear process model around a nominal tra-jectory. The signicant contributions of the work described in thispaper are

Development of an active fault tolerant control scheme for non-linear processes by suitably integrating a nonlinear version ofthe GLR method for FDI with a nonlinear model based controller.

Development of fault/failure isolation strategy when multiplefaults and failures occur simultaneously.

Development of a comprehensive method for treating bothfaults (biases/drifts in sensors/actuators/model parameters)and failures (sensor/actuator failures) in fault diagnosis andaccommodation.

The above contributions allow application of the fault tolerantscheme to nonlinear and time varying processes including batchand semi-batch processes. The proposed fault tolerant scheme alsoovercomes the limitation on the number of extra states that can beadded to the state space model in NMPC for offset removal and al-lows bias compensation for more variables than the number ofmeasured outputs. The advantages of the proposed state estima-tion and control scheme are demonstrated by conducting simula-tion studies on a benchmark CSTR system, which exhibits inputmultiplicity and change in the sign of steady state gain, and a fedbatch bioreactor, which exhibits strongly nonlinear dynamics. Bysimulating regulatory control problem associated with a unstablenonlinear system given by Chen and Allgower [1], we also demon-strate that the proposed intelligent state estimation strategy can beused to recover closed loop stability in the face of abrupt changesin model parameters.

The rest of this article is organized as follows. To begin with, wedevelop the nonlinear version of GLR method. A fault tolerantNMPC formulation is presented in the subsequent section. We thenproceed to present the results of simulation case studies. The mainconclusions reached based on the analysis of these results are pre-sented in the last section.

2. Fault diagnosisthe key to integration of fault tolerance with predictive control. Ifit is desired to achieve tolerance with respect to a broad spectrumIn this section we develop an FDI method based on a nonlinearversion of GLR scheme for diagnosing faults in nonlinear dynamicsystems. To begin with, the method is described as applied once

when a single fault is detected for the rst time. Modications nec-essary for on-line implementation of the FDI scheme when multi-ple faults occur sequentially are described later.

2.1. Model for normal behavior

Consider a continuous time nonlinear stochastic system de-scribed by the following set of equations:

xk 1 xk Z k1TkT

Fxs;uk;p;dkds 1

dk dwk 2yk Hxk vk 3where x 2 Rn; y 2 Rr and u 2 Rm represent the state variables, mea-sured outputs and manipulated inputs, respectively, and T repre-sents sampling interval. The variables p 2 Rp and d 2 Rd representthe vector of parameters and unmeasured disturbance variables,respectively, which are likely to undergo deterministic changes. Inaddition, the unmeasured disturbances are also assumed to under-go random uctuations. For mathematical tractability, these aresimulated as piecewise constant between each sampling periodand changing randomly from their nominal value at each samplinginstant. Here, vk and wk are zero mean Gaussian white noise se-quences with known covariance matrices. When process is not fullyunderstood or when it is not possible to develop mechanistic mod-els of each component of a system, it is often possible to developgrey box model by combining equations arising from rst principleswith some black box model components.

Eqs. (1) and (3) represent the normal or fault free behavior ofthe process, which can be used to develop a state estimator undernormal operating conditions. In the present work, the state estima-tion is carried out using the standard linearized version of EKF [13]as follows:

x^k 1jk x^kjk Z k1TkT

Fxs;mk; p; dds 4x^kjk x^kjk 1 Lkck 5It may be noted that we distinguish between the controller output,mk, and manipulated input, uk, entering the process. Underideal and fault-free conditions, the controller output equals themanipulated input. The innovation sequence under normal operat-ing condition is computed as

ck yk Hx^kjk 1The state and innovation covariance estimates are updated asfollows:

Pkjk 1 UkPk 1jk 1UkT CdkQCdkT 6Vk CkPkjk 1CkT R 7Lk Pkjk 1CkTVk1 8Pkjk I LkCkPkjk 1 9

where

Uk expAkT; Ak oFox

x^kjk1;mk1;p;d

Ck oHxox

x^kjk1

;

Bdk oFod

x^kjk1;mk1;p;d; Cdk

Z T0

expAksBdkds

Here, p and d are assumed to be the nominal values of parameters

and the unmeasured disturbances, respectively. In remainder of thetext, we refer to this EKF as normal EKF.2.2. Fault detection

When process starts behaving abnormally, the rst task is to de-tect the deviations from the normal operating conditions. To sim-plify the task of fault detection, it is further assumed that, undernormal operating conditions, the innovation sequence fckg is azero mean Gaussian white noise sequence with covariance Vk.The whiteness of innovation sequence generated by the normalEKF is taken as an indicator of absence of plant-model mismatch.A signicant and sustained departure from this behavior is as-sumed to result from model plant mismatch. To detect such depar-tures systematically, a simple statistical test namely fault detectiontest (FDT) as given in Prakash et al. [14] is modied based on theinnovations obtained from the normal EKF. This test is applied ateach time instant to estimate the time of occurrence of a fault.The test statistic for this purpose is given as follows:

k ckTVk1ck 10

Since it is assumed that innovation sequence is a zero mean Gauss-ian white noise process, the above test statistic follows a centralchi-square distribution [15,16] with r degrees of freedom, whichcan be used to x the threshold. Here r is the number of measure-ments. If FDT is rejected, the occurrence of a fault is further con-rmed by examining innovation sequence in the time intervalt; t N. The test statistic given by Eq. (11) is used for this purpose,which follows a central chi-square distribution with rN 1 de-grees of freedom:

t;N XtNkt

ckTVk1ck 11

If this test statistic exceeds the threshold, the occurrence of the faultor failure is conrmed. Window size for FDI computations is thetuning parameter. A large size of window reduces false alarmsand also improves the magnitude estimates of the fault. Howevera very high value of N may result in operating the process in a de-graded mode for a long time, which may have a deteriorating effecton the performance.

2.3. Fault and failure models

Once the occurrence of a fault is conrmed, the next step is toisolate the fault and estimate its magnitude. To identify the fault(s)that might have occurred, it is necessary to develop a model foreach hypothesized fault or failure that describes its effect on theevolution of the process variables. A fault can either develop asan abrupt (step-like) change or as a slow drift from its nominal va-lue. For example, an abrupt change in jth parameter can be mod-elled as

ppj k p bpjepjrk t 12

Here, bpj represents change in the parameter value from its nominalvalue, epj represents parameter fault vector with jth element equalto unity and all other elements equal to zero and rk t representsa unit step function dened as

rk t 0 if k < t; rk t 1 if kP tSimilarly, if bias occurs in jth sensor at instant t, then, subsequent tothe occurrence of bias in the sensor, the behavior of measured out-puts is modeled as follows:

yyj k Hxk byjeyjrk t vk 13

189On the other hand, if jth unmeasured disturbance changes as a slowdrift, then the corresponding fault model

for this fault mode observer computed using equations of the formddj k dwk bdjedj1k t1k t 0 if k < t; 1k t t if kP t

14

Here, bdj represents the magnitude of the unmeasured disturbancevariable change and edj represents the corresponding fault locationvector.

When an actuator or a sensor fails abruptly, then the models forfailure modes have to be developed in a different manner [17]. Forexample, if jth actuator is stuck abruptly at instant t, then plant in-put uk subsequent to the failure (denoted as uuj k) can be repre-sented as

uuj k mk buj eTujmkeujrk t 15

where buj represents constant value at which the jth actuator isstuck. Model given in Eq. (15), indicates that though the controllermanipulatesmk in the usual manner, the signal going to the plantfrom an actuator becomes constant due to some fault in theactuator.

When jth sensor fails abruptly at instant t, it is often observedthat we get a constant reading close to the value measured by dig-ital to analog converter before the failure occurs. Thus, if jth sensorfails at instant t, we propose to model the behavior of the measure-ment vector subsequent to the failure as follows:

yyj k Hxk byj eTyjHxkeyjrk t vk 16

where byj represents constant value at which the jth sensor readingis stuck. According to Eq. (16), the measurement coming from a par-ticular sensor in a plant gives a signal with constant mean when asensor fails, though the true plant output is changing.

2.4. Review of linear GLR method

In this work, a new approach has been proposed for fault iden-tication based on EKF. As motivation for developing the new ap-proach is derived from the version of linear GLR, method proposedby Narasimhan and Mah [18] andWillsky and Jones [19], a brief re-view of their method is presented here. The linear GLR methodmakes use of the innovation sequences generated by the normalKalman lter and Kalman lters obtained under different faultassumptions. Let fct . . . ct Ng represent the sample of innova-tion vectors generated by the normal Kalman lter over a windowfor time t; t N after a fault is detected. This innovation sequenceobtained from the normal Kalman Filter is viewed as a Gaussianrandom process with unknown means lk; t and covariancematrices Vk; t. The hypothesis H0 for absence of a fault in the ob-served data can be written as

H0 : lk; t 0

which is referred to as null hypothesis and the alternate hypothesisH1 for the presence of a fault in the observed data can be written as

H1 : lk; t bfjGfj k; tefj gfj k; tk 2 t; t N and f 2 p;d; y;u

where bfj refers to magnitude of a fault fj;Gfk; t and gfj k; t repre-sent fault signature matrix and fault signature vector, respectively,which describe the effect of fault fj on the innovations. Fault identi-cation (fault location and magnitude estimation) is carried out bymaximizing the log-likelihood function

T supbfj ;fj

Tfj 17

190Tfj XtNkt

cTkVk1ck infbfj

XtNkt

cTfj kVk1cfj k 18(6)(9). For each hypothesized fault, a separate fault mode observeris developed in a similar manner over window t; t N.

The next step is to generate estimates of the parameters of thefault model for each hypothesized fault. Taking motivation fromlinear GLR method, the fault magnitude estimation problem is for-mulated as a nonlinear optimization problem as follows:

minbfj

Jfj XtNit

cTfj iVfj i1cfj i 27

where cfj i and Vfj i are the innovations and the innovations covari-ance matrices, respectively, computed using the fault mode observerwhere cfj k ck bfjGfj k; tefj gfj k; t represents the innova-tion sequence generated by the fault mode Kalman lter developedunder the assumption that fault fj has occurred. Thus, the fault iso-lation can be viewed as nding the observer that best explains datain window t; t N. It may be noted that the rst term in Eq. (18) issame for all hypothesized faults. Thus, once a fault is detected, thenthe fault type together with its magnitude (i.e. bfj and fj) is deter-mined by solving the following set of optimization problems:

infbfj ;fj

XtNkt

cTfj kVk1cfj k 19

2.5. Nonlinear GLR method

In the GLR method, linearity of the process model/observers andadditive nature of faults can be exploited to develop recursive rela-tionships between the innovation sequences generated by the nor-mal Kalman lter and fault mode Kalman lters developed underdifferent fault assumptions (ref. [14] for details). However, for ageneral nonlinear system governed by Eqs. (1) and (3), most ofthe faults affect system dynamics in a nonlinear manner and sim-ilar recurrence relationships cannot be derived. Thus, to develop anonlinear analog of the GLR method, we formulate a separate EKFfor each hypothesized fault model over the time window t; t N,with the assumption that a fault has occurred at time instant t. Wethen pose the problem of fault isolation as nding the fault modeobserver that best explains the measurement sequencefyt . . . yt Ng collected over a window for time t; t N.

To understand the proposed FDI method, consider an observerdeveloped under the assumption that an actuator has failed.Assuming that actuator j has failed at instant t, the process behav-ior over window t; t N can be described as follows:

xuj i 1 xuj i Z i1TiT

Fxuj s;uuj i;p;dds 20yuj i Hxuj i vi 21where uuj is given by Eq. (15). Assuming that a fault has occurred att, the corresponding fault mode observer can be formulated asfollows:

x^uj iji 1 x^uj i 1ji 1 Z iTi1T

Fxuj s;muj i 1; p; dds

22muj i mi buj eTujmieuj 23x^uj iji x^uj iji 1 Luj icuj i 24cuj i yi Hx^uj iji 1 25x^uj tjt x^tjt 26where i 2 t; t N and Luj i represents the Kalman Gain matricescorresponding to fault fj. The fault mode observer that best explainsthe measurement sequence fyt . . . yt Ng is one for which the va-lue of Jfj is minimum. Thus, the fault fj that corresponds to

minfj2p;d;y;u

Jfj 28

is isolated as the fault that has occurred at time t and its corre-sponding magnitude estimate b^fj is taken as fault magnitude. Thisproposed approach for fault identication, which is motivated bylinear GLR method, is referred to as Nonlinear GLR (NL-GLR) methodin the rest of the text.

2.6. Simplication of NL-GLR method

The NL-GLR method proposed above involves solving multiplenonlinear optimization problems, which are subjected to nonlinearODE constraints. Such NLPs are notoriously difcult to solve andcomputationally demanding from the viewpoint of on-line imple-mentation. To simplify the task of on-line fault isolation, we pro-pose a simplied version of the proposed NL-GLR method. Thissimplication is based on linearization of the nonlinear processmodel along a nominal trajectory dened as follows:

fx^iji;mi 1; d; p : i 2 t 1; t N 1gwhich is generated using the Normal EKF under the assumption thatno fault has occurred over window t; t N. Now, for small magni-tude faults, the system dynamics under faulty conditionsfxfj i : i 2 t; t Ng can be viewed as deviation from the nominaltrajectory generated by Normal EKF. Under the hypothesis of occur-rence of fault fj, let the deviation in the state estimates from thenominal trajectory be represented as

dxfj i xfj i x^iji 29Then, using the Taylor series expansion in the neighborhood of thenominal trajectory and neglecting higher order terms, a time vary-ing linear perturbation model in the neighborhood of the nominaltrajectory can be obtained as follows:

dxfj i 1 ,i Uidxfj i Cuidmi Cdidd Cpidp Cdiwi 30

dyfj i Cidxfj i vi 31dmi mi mi 1 32yfj i Cix^iji dyfj i 33

i 2 t 1; t N 1where dd and dp represent vectors of abrupt changes in unmea-sured disturbances and model parameters from their nominal val-ues, respectively. The time varying vector ,i and matricesUi;Cui;Cdi;Cpi;Ci appearing in the above set of equationsare computed by linearizing the normal process model along thenominal trajectory as follows:

,i Z T0

expAiqFx^iji;mi 1; p; ddq 34

Ui expAiT; Ai oFox

; Ci oHox

35

Cui Z T0

expAiqBuidq; Bui oFom

36

Cpi Z T0

expAiqBpidq; Bpi oFop

37

Cdi Z T0

expAiqBdidq; Bdi oFod

38

x^iji;mi 1; p; d 39Based on the above perturbation model a separate linearized obser-

ver is formulated over a time window t; t N for each hypothe-sized fault. For example, consider a case where jth actuator hasfailed. Let buj denote the magnitude at which the jth input is stuck.Assuming that the fault has occurred at t, the linearized observercan be formulated as follows:

dx^uj iji 1 ,i 1 Ui 1dx^uj i 1ji 1 40 Cui 1dmuj i 1

c^uj i dyi Cidx^uj iji 1 41dx^uj iji dx^uj iji 1 Lic^uj i 42dmuj i muj i mi 1 43

dmi buj eTujmieuj 44dyi yi Cix^iji 45for i 2 t; t N starting from the initial conditiondx^uj t 1 j t 1 0. It may be noted that the Kalman gain matri-ces fLi : i 2 t; t Ng obtained from the normal EKF are used forstate correction. Also, the time varying matrices Ui;Cui, andCi have to be computed only once by this approach, which signif-icantly reduces on-line computational burden. The fault magnitudefor each hypothesized fault is estimated from the following optimi-zation problem:

minbfj Jfj XtNit

c^Tfj iVi1c^fj i 46

f 2 y; p;d; u 47where Vi is the covariance matrix for the innovations from thenormal EKF and cfj i is the innovation sequence from the linearizedobserver under fault hypothesis fj. The fault isolation can now becarried out by nding fault fj that corresponds to minimum valueof Jfj .

Once a fault fj is isolated, a rened estimate of the fault magni-tude is generated by formulating a nonlinear optimization problem(27) as described in the previous sub-section. This simplication ofNL-GLR, referred to as SNL-GLR in the rest of the text, reduces theon-line computational burden signicantly. The nonlinear optimi-zation is carried out only once for renement of fault magnitudeestimate for the fault that has been isolated.

2.7. Multiple simultaneous faults

In the previous section, the proposed FDI method has been de-scribed for the case when a single (root cause) fault occurs. If mul-tiple (root cause) faults occur simultaneously (i.e. at the same timeinstant), the above formulation can be extended to isolate and esti-mate magnitudes of multiple simultaneous faults as follows. In thiscase, we propose to enumerate all possible combinations of multi-ple faults that can occur simultaneously and develop fault modeobservers for each hypothesized combination. For example ifsimultaneous faults are to be hypothesized in jth sensor and lthparameter, then the observer for this combination of faults canbe formulated as follows:

x^plyjiji 1 x^pl;yji 1ji 1 Z iTi1T

Fx^pl;yjt;mi;

p bplepl ; ddt 48x^pl;yjiji x^pl;yjiji 1 Lpl;yjicpl;yji 49cpl;yji yi Hx^pl;yji byjeyj 50for i 2 t; t N with x^pl;yjt 1jt 1 x^t 1jt 1 51Fault magnitude estimation problem for each hypothesized combi-nation is then formulated and solved similar to formulation (27)discussed in Section 2.5. However, when multiple simultaneousfaults are hypothesized together with single faults the fault models

191have unequal number of unknown parameters (i.e. different degreesof freedom). Consequently, the fault isolation step cannot be carriedout using the minimum value of Jfj as described in Section 2.5. To

cation. On the other hand, choosing N to be small reduces delay in

mator and predictor can be modied in analogous manner.such as biases in sensors or actuators results in signicant mis-match in plant and model behavior (behavior mismatch). In addi-tion, hard failures, like failures of actuators and sensors can leadto signicant structural plant model mismatch (structure mis-match). The conventional approach to deal with the behavior mis-match in the NMPC formulations is through the introduction ofadditional articial states in the state observer [24]. The mainlimitation of this permanent augmentation approach is that thenumber of extra states that can be introduced cannot exceed thenumber of measurements. This implies that it is necessary to havea priori knowledge of a subset of faults that are most likely to occuror a subset of parameters that are most likely to drift. In such a for-mulation, the state estimates can become biased when unantici-pated faults and/or parameter drifts occur. When NMPCformulation is used for inferential control of some unmeasuredquality variables, the biased state estimates can have detrimentaleffect on the closed loop performance. The accuracy of the stateestimates, which is the prime concern in the inferential control for-mulation, can be maintained only if identical model is used forfault diagnosis and control and the model is corrected at the cor-rect location when a fault or abrupt change occurs [12]. Moreover,the permanent augmentation of state space model cannot system-fault isolation. However, it can increase false alarms and results inlarger variance errors in fault magnitude estimates. Based onsimulation studies, Prakash et al. [14] have suggested that thewindow length N can be chosen approximately equal to half thetime required for the estimator to converge after a change occurs.

3. Intelligent state estimation for fault tolerant NMPC

NMPC techniques use nonlinear model for prediction, which istypically developed once in the beginning of implementation ofan NMPC scheme. However, as time progresses, slow changes inunmeasured disturbances and/or process parameters and faultsalleviate this difculty, we propose to use Akaike Information Crite-rion (AIC) for fault isolation when multiple simultaneous faults arehypothesized together with single faults. Thus, the test statisticused for fault isolation is as follows:

minfj2 all hypothesized faults

AICfj N ln1NJfj

2/ 52

where Jfj represents the value of the prediction error term obtainedafter solving the magnitude estimation problem and / representsthe total number of parameters estimated when fault fj has oc-curred. The fault, i.e. either a single fault or a set of simultaneousfaults, that yields minimum value for AIC is isolated as the fault thathas occurred. It may be noted that

The proposed fault isolation strategy based on AIC can also beemployed when fault models with different number of unknownparameters (e.g. step jump in a parameter and slow drift in theparameter) are hypothesized.

The number of faults that can be hypothesized to occur simulta-neously cannot exceed the number of measurements due toobservability considerations.

Remark 1. It may be noted that the choice of window length Ndetermines the trade-off between delay in diagnosis and accu-racy of diagnosis. A large value of N results in less false alarms andsmaller variance errors in fault magnitude estimate. However,choosing larger N also introduces a longer delay in fault identi-

192atically deal with the difculties arising out of sensor biases andsensor/actuator failures. Sensor faults: If jth sensor bias is isolated, the measured output iscompensated as

yck yk b^yjeyj 56and used in FDI as well as NMPC formulation for computing innova-tion sequence. Compensation for actuator bias:

x^kjk 1 x^k 1jk 1 Z kTk1T

Fx^t;mk 1

b^ujeuj ; p; ddt 57x^kjk x^kjk 1 Lkck 58

x^k l 1jk x^k ljk Z kl1T

Fx^t;mk ljkIn this section, we describe the integration of the conventionalstate space based NMPC formulation with NL-GLR or SNL-GLRscheme, which is capable of generating unbiased state estimatesby intelligently correcting the state estimator. To begin with, wedescribe the modications necessary in the state estimator usedfor FDI a well as NMPC, when a fault is detected for the rst timeby FDI component. Modications necessary for dealing with recur-rence of the fault, occurrence of another fault at subsequent timeinstants and drifting (non-stationary) changes in unmeasured dis-turbances/model parameters are described later. We then proceedto propose NMPC formulation that can deal with behavioral as wellas structural changes in the model and state estimator. A sche-matic representation of the proposed FTNMPC scheme is shownin Fig. 1.

3.1. On-line modications to state estimator and predictor

Consider a situation where FDT has been rejected at time in-stant t and subsequently FCT has been rejected at time t N forthe rst time. Further assume that at instant t N a fault fj hasbeen isolated using NL-GLR/SNL-GLR method and the fault magni-tude has been estimated using data collected in the intervalt; t N. During the interval t; t N, the NMPC formulation isbased on the prediction model given by equations

x^k l 1jk x^k ljk Z kl1TklT

Fx^t;mk

ljk; p; d; tdt 53

However after the identication of the fault at instant t N, themodel for kP t N is modied as follows:

Step jump in unmeasured disturbance/model parameter: When FDIcomponent isolates abrupt change in unmeasured disturbance,the prediction equations in the state estimator and future pre-dictions in NMPC are modied as follows:

x^kjk 1 x^k 1jk 1 Z kTk1T

Fx^t;mk 1; p; d

b^djedj dt 54x^kjk x^kjk 1 Lkck 55

x^k l 1jk x^k ljk Z kl1TklT

Fx^k ljk;mk; p; d

b^djedj dtIf an abrupt change is detected in a parameter, then the state esti-klT

b^ujeuj ; p; ddt 59

Failed actuator: In state estimation and prediction, the failedactuator is treated as constant mjk b^uj for kP t N, whereb^uj is the estimate of stuck actuator signal for jth actuator.

Failed sensor: When the FDI component isolates a sensor failure,the failed sensor measurement is removed from the set of mea-surements used for state estimation.

After a fault, say fj, is diagnosed and corrections are made to thestate estimator, it becomes necessary to correct the state estimateswhile re-starting the state estimator at k t N based on themodied model. The state vector and state error covariance matrixestimated at k t N in the magnitude renement step is used tore-start the state estimation with the modied model and state ob-server by setting

x^t Njt N x^fj t Njt NPt Njt N Pfj t Njt Nand these values are used subsequently for state propagation andcovariance update.

3.2. Correction for drifting disturbances, parameters and multiplesequential faults

The main concern with the above approach is that the magni-tude and the position of the fault may not be accurately estimated.

Disturbance /Parameter Changes Faults/Failures

Process

Intelligent EKF

Optimization

Updated parameters/ structure/constraintsSet point

Outputs

NL-GLR based FDI

FT-NMPC

Prediction Model

Model Update

Disturbance /Parameter Changes Faults/Failures

Process

Intelligent EKF

Optimization

Updated parameters/ structure/constraintsSet point

Outputs

NL-GLR based FDI

FT-NMPC

Prediction Model

Model Update

Fig. 1. Fault tolerant NMPC (FTNMPC): schematic representation.Thus, there is a need to introduce integral action in such a way thatthe errors in estimation of fault magnitude or position can be cor-rected in the course of time. Furthermore, other faults may occur atsubsequent times. Thus, in the on-line implementation of NMPC,application of FDI method resumes at t N 1. The FDI methodmay identify a fault in the previously identied location or a newfault/parameter change/mean shift in unmeasured disturbancemay be identied. In either case, the Eqs. (54)(59) can be modiedusing cumulative estimate of the corresponding biases [14], whichare computed as

~bfj Xnfl1

b^fj l with initial value b^fj 0 0 60

where f 2 u; y denotes the fault type occurring at jth position andnf represents the number of times a fault of type f was conrmedand isolated in the jth position. Similarly, cumulative unmeasureddisturbance vector, ~dtdl , can be dened as follows:

~dtdl ~dtdl1 Xdj1

b^dj tdl edj~d0 d

61where tdl represent the last time instant when unmeasured distur-bance fault was isolated, tdl1 represent the time instant previous totdl when such fault was isolated and d^tdl represents the fault vec-tor (point) estimate at time instant tdl . Cumulative parameter vectorcan be dened in a similar manner as follows:

~ptpl ~ptpl1 Xpj1

b^pj tpl epj

~ptpl p62

The cumulative bias estimates given by Eqs. (60)(62) are used inEqs. (54)(59) in place of the point estimates b^fjefj ;

d b^djedj andp b^pjepj . The use of cumulative bias estimates can be looked uponas a method of introducing integral action to account for plant mod-el mismatch, in which some of the states (cumulative bias esti-mates) are integrated at much slower rate and at irregularsampling intervals. It may be noted that the use of cumulative biasestimates to correct the EKF also implies that the denition of nor-mal behavior model keeps changing as and when faults are detectedand subsequently the model is compensated for the faults. Thus,after sufciently long time, the normal behaviormodel used for stateestimation and fault diagnosis can be represented as follows:

xk 1 xk Z k1TkT

Fxt; ~uk; ~ptpl ; ~dtdl wdkdt 63

~uk mk Xmj1

~bujeuj 64

yk Hxk Xrj1

~byjeyj vk 65

provided no actuator/sensor failures are diagnosed. When a newfault is detected, the on-line diagnosis problem is now formulatedas follows:

infb^fj tfl1;fj

Xtfl1Nktfl1

cTfj kVk1cfj k 66

x^fj iji1 x^fj i1ji1

Z iTi1T

Fxfj s; ~ui1; ~ptpl ; ~dtdl ; b^fj tfl1efj ds 67

x^fj iji x^fj iji1Lfj icfj i 68

cfj i yi Hx^fj iji1Xrj1

~byjeyj

( )69

x^fj tjt x^tjt 70where i 2 tfl ; tfl1 N and b^fj tfl1efj inuence the system dynam-ics through the cumulative bias expressions given by Eqs. (60)(62). In abstract form, if h represents the set of all corrections thatare made to the model subsequent to diagnosis, then the above for-mulation followed by the model correction (i.e. fault accommoda-tion) step is equivalent to a slow rate recursion of the form:

htfl1 Whtfl ;Xtfl1; tfl1 NXtfl1; tfl1 N fyi;ui : i 2 tfl ; tfl1 Ngwhere tfl represent the last time instant when a fault was isolated,tfl1 represent the time instant previous to tfl when a fault was iso-lated and W represents update rule through NL-GLR. This is tanta-mount to using all the data collected after each fault detection forupdating the model. As a consequence, the use of cumulative biasestimate improves parameter/bias estimates and reduces the vari-ance errors if a fault is isolated in the same location multiple times.

193In fact, Eqs. (60)(65) together represent a multi-rate model withexpected values of unmeasured disturbances and parameterschanging at a signicantly slower and irregular sampling rates.

Xq1

si-innite horizon formulation, the NMPC objective function isJDm l0

Dmk ljkTWmDmk ljk 73

subject to following constraints:

efk ljk yrk y^k ljkDmk ljk mk ljk mk l 1jkx^k l 1jk x^k ljk

Z kl1TklT

Fx^t;mk ljk

Xmj1

~bujeuj ; ~ptpl ; ~dtdl ; tdt 74

~yk ljk Gx^k ljk 75

mL 6mk ljk Xmj1

~bujeuj 6mU

DmL 6 Dmk ljk 6 DmU

where l 2 0;Np. Here, Np represents prediction horizon, q repre-sents control horizon, yrk represents the future setpoint trajectoryand ~y Gx represents the vector of controlled outputs, which may,in general, differ from the measured outputs y Hx. It may beThe minimum gap between two such changes equals the windowlength used for fault isolation. Thus, this model effectively separatesunmeasured disturbances into two components: (a) stationary col-ored noise modelled through extended Kalman lter; (b) non-sta-tionary low frequency mean changes captured through ~dtdl and~ptpl . The above self-adapting form of model with slowly time vary-ing parameters is used in the proposed NMPC formulation.

Remark 2. It may be noted that Eqs. (61) and (62) slowly modeldrifting disturbances as sequence of step changes. However, it islikely that in some situations drift model given Eq. (14) is moreappropriate and has to be used. The mechanism for modelcorrection has to be suitably modied when a fault is modelledas a drift. If the mean value of some unmeasured disturbance/parameter changes continuously and at a much faster rate, thenthe time scale separation that is implicit in the proposed formu-lation may not be acceptable. In such a situation, that specicparameter or unmeasured disturbance variable can be included inthe state vector and its value can be estimated together with theother states. While such permanent augmentation will reduce adegree of freedom available for diagnosis, the proposed approachcan still be used for diagnosing remaining faults without requiringany signicant modications.

3.3. Fault tolerant NMPC formulation

At any sampling instant k, the nonlinear model predictive con-trol problem is formulated as a constrained optimization problemwhereby the future manipulated input moves denoted as

fmkjk;mk 1jk . . .mk Np 1jkgare determined by minimizing an objective function involving pre-dicted controller errors. Typical objective function used in an NMPCformulation is of the form

minmkjk;mk1jk...mkNp1jk

J Je JDm 71

Je XNpl1

efk ljkTWEefk ljk 72

194noted that constraints on manipulated input are modied to accom-modate bises in manipulated inputs. While the above modiedmodel can deal with faults, actuator failure may require additionalmodied by including the following additional term:

J Je JDm J1 80J1 xk NpjkTW1xk Npjk 81

where W1 represents the terminal state penalty matrix, which iscomputed by solving an appropriate Lyapunov equation (ref.Appendix). In addition, the predicted state xk pjk is constrainedto lie within a terminal set Xx dened as

Xx : fx 2 RnjxTW1x 6 agin the neighborhood of the operating steady state. If the Jacobianlinearization of the nonlinear system to be controlled is stabilizableat the operating steady state, then, it has been shown that feasibilityof the open loop quasi-innite horizon control problem at timet 0 implies nominal asymptotic stability of the closed loopsystem.

It may be noted that, the terminal region Xx and the penaltymatrix W1 are functions of the model parameters. As a conse-quence, if the model parameters/unmeasured disturbances under-go abrupt and large changes during plant operation, the closedloop stability can no longer be guaranteed using (W1;Xx) com-puted initially. We propose a remedy to this problem wherebywe recompute the terminal region Xx and the penalty matrix W1every time the FDI component diagnoses an abrupt change in mod-el parameters/unmeasured disturbance. Under the ideal conditionmeasure such as modication of the control objective to accommo-date the failure. For example, if dimension of the setpoint vectorequals the number of manipulated inputs and an actuator failureis diagnosed, then, the NMPC objective function is modied byrelaxing setpoint on one of the controlled outputs.

In an ideal situation where all the behavioral changes in theplant are detected and isolated by the proposed FDI scheme, NMPCformulated using model (71)(75) can provide offset free control.However, since we are dealing with a stochastic system, all faults/changes that occur in the plant may not get diagnosed correctly. Toachieve offset free control in such a scenario, the prediction equa-tions can be modied as follows:

~xk l 1jk x^k ljk Z kl1TklT

Fx^t;mk ljk

Xmj1

~bujeuj ; ~ptpl ; ~dtdl ; ;tdt 76

x^k l 1jk ~xk l 1jk Lkck 77y^k ljk Gx^k ljk ek 78ek yk y^kjk 79

where y^k=k is computed using Eq. (75). When the sets of mea-sured and controlled outputs are identical, this simple modicationin the prediction equation can eliminate offset without requiringstate augmentation [20].

3.4. Closed loop stability

The above nite horizon formulation of NMPC does not guaran-tee closed loop stability even under nominal conditions. Chen andAllgower [1] have shown that inclusion of terminal weighting inthe NMPC objective function (quasi-innite horizon formulation)can guarantee asymptotic closed loop stability under nominal con-ditions in the absence of any unmeasured disturbances. In the qua-where a fault is correctly isolated and its magnitude is accuratelyestimated, this pro-active measure can ensure nominal closed loopstability under certainty equivalence control.

steady state gain reduces to zero at the peak and changes its signacross the peak [20,21,23].

In this case study, we hypothesize ten different faults consistingof single faults such as (a) biases in two actuators, (b) biases andfailures of two sensors, (c) step jump in inlet concentration Cai,(d) step change in reaction rate parameter and simultaneous faults

195It may be noted that Chen and Allgower [1] assume exactknowledge of the complete state vector in their formulation. Inour formulation, on the other hand, we make use of the state esti-mate x^kjk to initialize prediction in the NMPC formulation. Thestability of NMPC and EKF pair is still an open issue as the separa-tion principle does not hold in the nonlinear case.

4. Simulation case studies

Simulation studies are carried out to evaluate the proposedintelligent state estimation (referred to as intelligent EKF in the restof the text) and FTNMPC schemes by simulating control problemsassociated with the following highly nonlinear systems:

CSTR exhibiting input multiplicity [20,21]. Unstable nonlinear system described in Chen and Allgower [1]. Fed-batch bioreactor [22].

The performance of the conventional NMPC (CNMPC) that em-ploys conventional EKF for state estimation is compared with theperformance of the proposed FTNMPC scheme under different faultscenarios. In all the three case studies the CNMPC formulation isbased on the nominal model given by Eqs. (1)(3) and state estima-tor given by Eqs. (6)(9). The future trajectory predictions inCNMPC formulation are carried out using Eq. (52). In addition, amodel-plant mismatch compensation scheme similar to (77)(79) has been used in CNMPC formulations used in the CSTR andfed-batch bioreactor case studies for eliminating offset.

4.1. CSTR with input multiplicity

The system under consideration consists of a CSTR in which areversible exothermic reaction of type AB is carried out. Thenominal parameters and the optimum operating steady state usedin the simulation studies can be found in Li and Biegler [21] andEconomou [23].

The dynamic model used for simulating the CSTR system is asfollows [20]:

dCadt

FhAc

Cai Ca K1Ca K2Cb 82dCbdt

FihAc

Cb K1Ca K2Cb 83dTdt

1hAc

FiTi T HrqCp K1Ca K2Cb 84dhdt

1Ac

Fi kh

p 85

K1 kf expEf=T; K2 kb expEb=T 86In the present work, the concentration of component By1 Cb andlevel (y2 h) in the CSTR are treated as two controlled outputs ofthe system. The inlet ow rate u1 Fi and inlet feed temperatureu2 Ti are used as manipulated variables. The constraints im-posed on manipulated inputs are as follows:

0 6 Fi 6 2 and 300 6 Ti 6 550

The inlet concentration Cai is treated as unmeasured disturbanceand it is assumed to be corrupted with a zero mean white noise sig-nal of standard deviation 0:05 mol=m3. The sampling interval ischosen as 0.4 min. This system exhibits input multiplicity andchange in the sign of the steady state gain in the operating region.For a xed value of ow rate, the concentration Cb as a functionof inlet ow temperature has a well dened maximum. Thus, the

objective is to control the concentration Cb at the optimum oper-ating point of the system where the conversion is maximum. Regu-lating the CSTR at the optimum point is a challenging task as theas (e) simultaneous occurrence of step changes in Cai and kf (e)simultaneous occurrence of bias in level sensor and inlet concen-tration Cai. The controlled outputs are concentration, Cb, andreactor level. The tuning parameters used in the controller formu-lation and SNL-GLR method are given in Tables 1 and 2respectively.

4.1.1. Optimum seeking control in presence of parametric faultsIn this sub-section, it is assumed that measured outputs are

same as controlled outputs, i.e.

Gx Hx 0 1 0 1 xand measurements of Cb and h are assumed to be corrupted with azero mean white noise signals with standard deviations0:005 mol=m3 and 0.002 m, respectively. The control problem isto regulate the system at the optimum operating point in the faceof abrupt changes in parameters and unmeasured disturbances. Itmay be noted that the location of the maximum conversion pointis a function of model parameters and unmeasured disturbances.Under nominal operating conditions, the optimum operating pointis located at Cb 0:5088 mol=m3 for h 0:16 m. However, whenthere is a signicant shift in the mean value of model parametersor unmeasured disturbances, the maximum concentration of Cbpredicted by the nominal model is different than the maximumachievable output in the plant. Patwardhan and Madhavan [24]have discussed two possible situations arising due to shift in theoptimum point: (a) sub-optimal operation when the maximumattainable conversion in the plant shifts above the nominal maxi-mum; (b) unattainable setpoint when the maximum shifts belowthe nominal maximum. The later situation results in a steady stateoffset and may lead to input saturation and loss of control. In thissection, we demonstrate that the proposed FTNMPC formulation,in combination with on-line steady state optimization, can be usedto track the changing optimum operating point.

To begin with, we demonstrate performance of our schemewhen two faults occur sequentially. Initially, the process is con-trolled at the nominal operating point. At t 26:4 min, the reac-tion rate parameter kf is changed from 1 to 1.3. This increasesthe maximum attainable concentration form Cb 0:5088 mol=m3to Cb 0:5738 mol=m3. The proposed SNL-GLR method correctlyisolates this fault and magnitude estimated is k^f 1:2263 whichis further rened using NL-GLR to k^f 1:2945. The optimum con-centration operating point computed based on the rened estimateof kf is Cb 0:5728 mol=m3. Thus, the concentration setpoint ischanged to Cb 0:5728 mol=m3 subsequent to fault diagnosis.Subsequent to this fault a step jump of 0.2 is given in inlet concen-tration (Cai) at k 306. This increases the maximum attainableconcentration from Cb 0:5738 mol=m3 to Cb 0:6886 mol=m3.The proposed SNL-GLR method correctly isolates this fault andmagnitude estimated after renement is 0.2018. The optimumconcentration operating point computed based on this estimate

Table 1CSTR example: controller tuning parameters

Prediction horizon 12Control horizon 3

10 0 Error weighting matrix 0 1

Set-point [0.5088 0.16]

is Cb 0:6883 mol=m3. Thus, the concentration setpoint is chan-ged to Cb 0:6883 mol=m3 subsequent to fault diagnosis. TheCNMPC, however, is unaware of the nature and type of unmea-sured disturbance and attempts to achieve the original setpointof Cb 0:5088 mol=m3. This results in suboptimal operation ofthe CSTR. Fig. 2 compares the performances of both the controllersin the presence of multiple sequential faults. A better insight intotheir behavior is obtained when we compare the state estimationerrors generated by conventional EKF and the proposed intelligentEKF (see Fig. 3). It can be seen from Fig. 3 that both the conven-tional EKF and Intelligent EKF generate biased estimates of Caand Cb immediately after the fault occurs. However, as soon asthe fault is correctly diagnosed and compensated, the states esti-

addition, the biased level sensor gives rise to an offset in true valueof reactor level and the setpoint in the case of CNMPC. Figs. 4 and 5compare the performances of both the controllers and state esti-mators, respectively. Similar to the sequential fault case, the statesestimated using intelligent EKF move close to their true values andbias in state estimation is eliminated soon after the fault is cor-rectly diagnosed and compensated.

4.1.2. Estimator reconguration on sensor failureIn this subsection, we assume that the reactor temperature

measurements are also available together with measurements ofCb and h, i.e.

Hx 0 1 1 1 x 87while the controlled outputs are

Gx 0 1 0 1 xi.e. Cb and h. The temperature measurements are assumed to be cor-rupted with a zero mean white noise signal with standard deviation0.02 C. We begin simulations under the scenario that all three sen-

Table 2CSTR example: SNL-GLR tuning parameters

Window for fault conrmation 60Level of signicance for fault detection 0.5Level of signicance for fault conrmation 0.01

196mated using intelligent EKF are close to their true values and biasin state estimation is eliminated. On the other hand, the bias in theestimates of Cb persists in case of conventional EKF even when Cb isdirectly measured. This can be attributed to persistent plant modelmismatch that develops subsequent to occurrence of abruptchanges in parameters.

In another simulation run, to evaluate the performance of theproposed state estimation and control scheme when multiplefaults occur simultaneously, we introduce a bias of (0.02 m) in le-vel sensor and step jump of magnitude +0.1 in the inlet concentra-tion (Cai) at t 26:4 min. The proposed SNL-GLR method correctlyisolates this simultaneous fault and rened fault magnitudes esti-mated are 0.0196 m and 0.0955 mol/m3. The optimum concen-tration operating point computed based on these estimates isCb 0:5574 mol=m3 (true optimum point under the changed con-ditions is Cb 0:5597 mol=m3). Thus, the concentration setpoint ischanged to Cb 0:5574 mol=m3 subsequent to fault diagnosis andFTNMPC shifts the average steady state concentration to 0.552. TheCNMPC that employs conventional EKF for state estimation,however, attempts to reject these abrupt changes as an unmea-sured disturbance to achieve the original setpoint ofCb 0:5088 mol=m3, which results in suboptimal operation. In

0 50 100

0.5

0.6

0.7

0.8

C B(m

ol/m

3 )

Setpoint FTNMPCFTNMPC OutputSetpoint CNMPCCNMPC Output0 50 1000.1

0.12

0.14

0.16

0.18

Tim

Rea

cto

r Lev

el (m

)

Set-pointFTNMPC OutputCNMPC Output

Fig. 2. CSTR example: comparison of controlled outputs ofsors are functioning well with reactor operating at a suboptimaloperating point 0:4088 0:16 . At sampling instant k 71(t 28:4 min) setpoint is changed to a new value 0:5088 0:16 and just prior to this at k 66 (t 26:4 min), a failure in the sensorfor Cb is simulated by holding sensor output constant at subsequenttime instants. Fig. 6 shows the state estimation errors for the pro-posed intelligent EKF before and after fault compensation. As canbe observed from this gure, the state estimates become biasedwhen concentration sensor fails. However, in the Intelligent EKFscheme, after the failure has been identied by the SNL-GLR meth-od, the state estimator is recongured using only temperature andlevel measurements, i.e. by setting

Hx 0 0 1 1 x 88This measure eliminates the bias in the estimate of Cb and enablesFTNMPC to track this setpoint change using correctly estimatedconcentration Cb (see Fig. 7) from the available level and tempera-ture measurements. Before the failure is isolated and compensated,the controller attempts to increase reactor concentration byincreasing the throughput and thereby increasing the reactor level.However, FTNMPC is able to recover the level to the desired set-point subsequent to the fault accommodation.

150 200 250

150 200 250

e(min)FTNMPC and CNMPC under multiple sequential faults.

0 50 100 150 200 250-0.2

-0.1

0

0.1

0.2

Est.

Erro

r (CA

) (mo

l/m3 )

0 50 100 150 200 250-0.1

0

0.1

Time (min)Est

. Err

or (C

B) (m

ol/m

3 )

Intelligent EKFConventional EKF

Intelligent EKFConventional EKF

Fig. 3. CSTR example: comparison of estimation errors of intelligent EKF and conventional EKF under multiple sequential fault.

0 20 40 60 80 100 1200.45

0.5

0.55

0.6

0.65

Tru

e

CB

(mo

l/m3)

0 20 40 60 80 100 1200.1

0.15

0.2

Time(min)

Tru

e Le

ve

l (m)

Set-pointFTNMPC OutputCNMPC Output

FTNMPC SetpointFTNMPC OutputCNMPC SetpointCNMPC Output

Fig. 4. CSTR example: behavior of true values of outputs of FTNMPC and CNMPC under multiple simultaneous fault.

0 20 40 60 80 100 120-0.05

0

0.05

Est.

Erro

r (C A

) (mo

l/m3 )

0 20 40 60 80 100 120-0.02

0

0.02

0.04

Time(min)

Est.

Erro

r (C B

) (mo

l/m3 )

Intelligent EKFConventional EKF

Int. EKFConv. EKF

Fig. 5. CSTR example: comparison of estimation errors of intelligent EKF and conventional EKF under multiple simultaneous fault.

197

0 10 20 30 40 50 60 70 80-0.05

0

0.05

Est.

Erro

r (CA

) (mo

l/m3 )

0 10 20 30 40 50 60 70 80-0.06

-0.04

-0.02

0

0.02

Time(min)

Est.

Erro

r (CB

) (mo

l/m3 )

Fig. 6. CSTR example: state estimation errors for intelligent EKF under sensor failure.

0 10 20 30 40 50 60 70 800.35

0.4

0.45

0.5

0.55

C B(m

ol/m

3 )

0 10 20 30 40 50 60 70 800.1

0.15

0.2

0.25

Time(min)

Rea

cto

r Le

ve

l (m)

SetpointTrue Output

SetpointTrue Output

Failure Introduced

Failure compensated

Fig. 7. CSTR example: true values of outputs of FTNMPC under concentration sensor failure.

0 20 40 60 80 100 120 140 1600.9

1

1.1

1.2

1.3

Inle

t Con

c. (m

ol/m3

)

0 20 40 60 80 100 120 140 1600.45

0.5

0.55

0.6

0.65

Time (min)

Conc

. B (m

ol/m3

)

TrueEstimated

Set-pointTrue-output

Fig. 8. CSTR example: estimation of drifting disturbance using SNL-GLR and optimum peak seeking using FTNMPC.

198

4.1.3. Isolation and estimation of drifting disturbanceIn this sub-section, we demonstrate that the proposed approach

is capable of modeling drifting disturbances through cumulative

bias vectors dened in Section 3.2. When a continuously driftingdisturbance is present in the process, it will get modelled (approx-imated) as a sequence of step changes. To test the performance ofSNL-GLR in identifying such drifting faults, a drifting disturbance isadded in the inlet concentration Cai as shown in Fig. 8. As can beexpected, the SNL-GLR identies the fault at the same locationrepeatedly. As the concentration drifts, the magnitude estimategets rened each time the fault is isolated till it matches the chang-ing mean value of the disturbance. Fig. 8 compares the actual dis-turbance added and its estimated value as given by SNL-GLR. Eachtime a change in Cai is isolated, the setpoint for Cb is changed to thenew optimum value as shown in this gure.

4.2. Nonlinear system by Chen and Allgower

The nonlinear system is described by following set of equations[1]:

_x1 x2 ul 1 l _x1_x2 x1 ul 41 lx2

Table 3Chen and Allgower example: controller tuning parameters

Prediction horizon 15Control horizon 15Input weighting R 1State weighting matrix Q diag[0.5 0.5]

Table 4Chen and Allgower example: SNL-GLR tuning parameters

Window for fault conrmation 50Level of signicance for fault detection 0.1Level of signicance for fault conrmation 0.001

0 5 10 15 20 25 30-1

-0.5

0

0.5

Y 1

0 5 10 15 20 25 30-1

-0.5

0

0.5

Y 2

0 5 10 15 20 25 30-1

0

1

2

Time(min)

U

SetpointCNMPC

SetpointCNMPC

Fig. 9. Chen and Allgower example: CNMPC perfor

0 5 10-1

-0.5

0

0.5

y 1

0 5 10-1

012

y 2

2

Fault IntroducedFault Compensated

Setpoint

FTNMPCFTNMPCFTNMPCFTNMPC

1990 5 10-2

0uTim

Fig. 10. ChenAllgower example: comparison of CNMPC and FTNMmance under normal and nominal conditions.

15 20 25 30

15 20 25 30

15 20 25 30

CNMPCCNMPCCNMPC

CNMPC

FTNMPC

CNMPC

CNMPCCNMPCCNMPC

CNMPC

FTNMPCe(min)PC performances in presence of abrupt change in parameter.

11.5

x

Mu=0.9 (original)

Mu=0.5 (True)

d)

ns C

200The nonlinearity of the system is a function of l;which also decidesthe terminal region X and the terminal penalty matrix W1 whichare used in the NMPC objective function for ensuring nominalclosed loop stability (Refer Appendix). Here four different faultshave been hypothesized namely, actuator bias, biases in two sen-sors and parametric fault which is change in l. Keeping the initial(nominal) value of l 0:9, the above process is simulated using asampling time of 0.1 units and with following constraints on input:

2 6 u 6 2The tuning parameters used in controller formulation and SNL-GLRmethod are listed in Tables 3 and 4 respectively.

For l 0:9, the terminal region and penalty matrix are as fol-lows (see Fig. 11):

W1 4:96 0:0367

0:0367 4:96

Xx : fx 2 RnjxTPx 6 5:806g

-1.5 -1 -0.5-1.5

-1

-0.5

0

0.5x

2

Mu=0.5309(Estimate

Fig. 11. ChenAllgower example: stability regioThe control problem is to regulate the process at the origin startingfrom some arbitrary initial state (0.685, 0.685). It is assumedthat the measurements are corrupted with a noise having a varianceof 1 104 and a state noise of standard deviation 1 103 enteringeach state. Fig. 9 shows the controller performance under nominalconditions.

At t 5 min, a step change is introduced in l by changing it to0.5. The CNMPC becomes unstable after occurrence of the paramet-ric change (see Fig. 10). In the FTNMPC formulation, on the otherhand, after the fault is correctly isolated, the estimated value of l(l 0:5309) is used to correct the model used in the state estima-tor as well as prediction model. Along with these modications, theterminal region and the penalty matrix are also recomputed as

W1 14:253 9:2539:253 14:253

Xx : fx 2 RnjxTPx 6 11:66g

Table 5Fed-batch bioreactor example: controller tuning parameters

Prediction horizon 30Control horizon 1

Error weighting matrix 1 00 150

Set-point [90 5]the objective function and the constraint set is modied accord-ingly. Fig. 11 compares the original and modied terminal regions.It may be noted that the terminal region and the penalty matrix cor-responding to l 0:5 are as follows:

W1 16:59 11:5911:59 16:59

Xx : fx 2 RnjxTPx 6 12:57gFig. 11 also shows that the new terminal region computed based onthe estimated value of l is reasonably close to the terminal regioncorresponding to l 0:5. Fig. 10 compares the closed loop behaviorobtained using FTNMPC and CNMPC. As evident form these gures,the proposed FTNMPC formulation is able to retain the closed loopstability under the changed conditions while controlled outputs be-come unbounded in the case of CNMPC.

4.3. Fed-batch bioreactor

0 0.5 1 1.51

NMPC and FTNMPC under parametric change.The unstructured model for penicillin production in a fed-batchfermentor given by Birol et al. [22] has been used for simulationstudies. The model consists of nine differential equations relatingbioreactor states (biomass, substrate, product, dissolved oxygen,CO2 and hydrogen ion concentrations, bioreactor temperatureand bioreactor volume) with six inputs (substrate feed and coolantows, acid and base ows, agitation rate and aeration). The modelequations, initial conditions and nominal values of the model

Table 6Fed-batch bioreactor example: constraints on manipulated inputs

Manipulated variable Input constraints

Aeration rate (l/h) 043Agitation power (W) 0100Acid ow rate (l/h) 00.5e3Base ow rate (l/h) 00.5e3

Table 7Fed-batch bioreactor example: SNL-GLR tuning parameters

Window for fault conrmation 100Level of signicance for fault detection 0.5Level of signicance for fault conrmation 0.01

90

95

isso

lved

Oxy

genparameters are given in Birol et al. [22]. While carrying out thesimulations, we assume that the bioreactor operates isothermallyand the fermentor temperature is maintained at the optimum va-lue (25 C) by a perfect temperature controller. We also assumethat the mean value of substrate feed rate is xed at a constantoptimum value (0.042 l/h), which reproduces a pre-decidedgrowth pattern. The control objective in this case study is to main-tain the dissolved oxygen concentration and pH of the fermenta-tion medium at the set-point by manipulating aeration, agitation

0 50 100 150 2085

0 50 100 150 200

0.5

1

1.5

Time

Peni

cilli

nCon

c. (g

ms/lit

)%

D

Fig. 12. Fed-batch bioreactor example: true values of outputs

0 50 100 150 200

20

40

60

80

Agi

tato

r Po

we

r (W

)

0 50 100 150 200

10

20

30

40

Aer

atio

n R

ate(

lit/h

r)

Time

Fig. 13. Fed-batch bioreactor: manipulated input behavi

Table 8Fed-batch bioreactor example (sensor bias case): comparison of CNMPC and FTNMPCperformances based on ISE values

Output FTNMPC CNMPC

Dissolved oxygen 0.0184 4.6276e+005pH 4.0325 1.90630 250 300 350 400

Set-pointFTNMPCCNMPC

201and acid/base ows. As the culture makes transitions from onegrowth phase to the other (lag phase? exponential phase? sta-tionary phase), the oxygen uptake rate and acid secretion rateschange substantially. With an increase in the batch time, the bio-mass accumulation rate increases and this results in higher oxygenconsumption. Similarly, as time progresses, the acid secretion rateincreases. Thus, the manipulated inputs have to be changed contin-uously to keep pH and DO at desired setpoints. As the inputoutputdynamics changes signicantly with time, isolating faults such aspH controller failure or sudden decrease in agitation power arechallenging problems from the viewpoint of fault diagnosis [22].

Simulations are carried out for a batch time of 400 h, with asampling time of 1.2 min. Measurements of pH and DO are as-sumed to be corrupted by zero mean white noise with standarddeviations of 0.01 and 0.2, respectively. Also, a zero mean whitenoise disturbance with standard deviation of 0.001 has been intro-duced in the substrate addition rate. A total of eleven single faultshave been hypothesized, namely biases and failures in the four

0 250 300 350 400(hrs)

FTNMPCCNMPC

of FTNMPC and CNMPC in presence of bias in DO sensor.

0 250 300 350 400

0 250 300 350 400(hrs)

FTNMPCCNMPC

FTNMPCCNMPC

or of FTNMPC and CNMPC under bias in DO sensor.

actuators and biases in sensors and step jump in substrate additionrate. The tuning parameters and input constraints used in conven-tional NMPC and FTNMPC are listed in Tables 5 and 6, respectively.Also, tuning parameters used in SNL-GLR are listed in Table 7.

4.3.1. Bias in DO sensorFig. 12 presents the comparison of CNMPC and FTNMPC behav-

ior when a bias of 5% occurs in DO measurement at time t 30 h.Fig. 13 compares the corresponding manipulated input behaviorfor both the controllers. During the time window used for faultdiagnosis, both the controllers develop offset between the true va-lue of % DO and the setpoint. However, after fault is correctly iso-lated by the FDI component (estimated fault magnitude 4.854%),FTNMPC moves the true DO percentage close to the desired set-point. On the other hand, the large offset between the true % DOvalue and the setpoint persists in the case of CNMPC. When DOmeasurements become biased, the CNMPC attempts to maintainmeasured % DO at the specied setpoint by increasing both the in-puts, which increases in true values of % DO in the plant. Fig. 12also presents the effect of bias in DO sensor on the Penicillin con-

centration over 400 h of operation. From this gure, it may be ob-served that the Penicillin concentration reduces by 5.94% as aresult of the biased DO measurements when CNMPC formulationis employed. On the other hand, as a consequence of accurate faultaccommodation, the FTNMPC formulation is aware of the bias inDO measurements and manages to maintain the true value of %DO close to the desired setpoint (see Table 8). This results in im-proved Penicillin concentration at the end of the batch.

4.3.2. Behavior under pH controller failureTo simulate pH controller failure, in the base addition actuator

failure (stuck control valve) is simulated after initial 10 h of failurefree operation. Fig. 14 compares behavior of: (a) CNMPC under nofailure; (b) CNMPC under the actuator failure; (c) FTNMPC underthe actuator failure. Fig. 14 also compares the acid addition ratesfor cases (b) and (c). In addition, Fig. 15 compares difference be-tween the true and the computed values of base addition rates inFTNMPC and CNMPC. Under normal operating conditions (no fail-ure scenario), CNMPC is able to maintain pH at the desired setpointthroughout the batch. When the base addition actuator fails,

0 50 100 150 200 250 300 350 4004

4.5

5

pH

0 50 100 150 200 250 300 350 4000

2

4

6x 10-4

Time(hrs)

Aci

d Ad

ditio

n Ra

te (l/

hr)

Set-pointNormal OperationCNMPCFTNMPC

FTNMPCCNMPC

roll

20

ted

ev(T

202Fig. 14. Fed-batch bioreactor example: comparison of cont

0 50 100 150-6

-4

-2

0

2x 10-4

Dev

. Bas

e A

dd. R

ate

(l/hr)

-6

-4

-2

0

2x 10-4

v. B

ase

Add

. Rat

e (l/h

r)

FTNMPC: Dev(True-Compu

CNMPC: D0 50 100 150 20De

Fig. 15. Fed-batch bioreactor example: deviation (true computed) bed outputs of FTNMPC and CNMPC under actuator failure.

0 250 300 350 400

) Base Addition Rate

rue-Computed) Base Addition Rate0 250 300 350 400

ase addition rate of FTNMPC and CNMPC under actuator failure.

CNMPC initially recovers from the upset in pH by manipulatingacid addition rate. However, as evident from Fig. 15 CNMPC, con-tinues to calculate changes in base addition rate even after thevalve is stuck. As a consequence, the constraint on the acid addi-tion rate becomes active for t > 50 h. and CNMPC is unable to elim-inate the offset that develops subsequently. Thus, failure of thebase addition actuator results in an offset in pH output in case ofCNMPC.

In the case of FTNMPC, after the failure of actuator is correctlydiagnosed by SNL-GLR, Intelligent EKF makes necessary modica-tions in the estimator as well as prediction model used in NMPC.Thus, subsequent to the failure accommodation, the base additionrate in the state estimator and prediction model is set equal to itsvalue estimated by FDI module and this reduces the difference be-tween the true (i.e. stuck actuator) and computed value of baseaddition rate practically to zero (see Fig. 15). The FTNMPC formu-lation, which is now aware of loss of degree of freedom due toactuator failure, is able to restore pH to the setpoint by t 170 h.and subsequently maintain it at the desired setpoint by manipulat-ing the acid addition rate alone. Thus, as FTNMPC is aware of stuckbase valve, it is able to manage the additional degree of freedomavailable (i.e. acid addition rate) much better when compared toCNMPC and meet the desired control objective. This results inimprovement in Penicillin concentration by 0.13% after 170 h ofoperation. The comparison of CNMPC and FTNMPC performancesin terms of ISE values is presented in Table 9.

5. Conclusions

In this work, a novel intelligent nonlinear state estimation strat-egy has been proposed, which keeps diagnosing the root cause(s)of the plant model mismatch by isolating the subset of active faults(abrupt changes in parameters/disturbances, biases in sensors/actuators, actuator/sensor failures) and auto-corrects the modelon-line so as to accommodate the isolated faults/failures. To facil-itate the diagnosis of faults and failures in nonlinear and time vary-ing processes, we develop a nonlinear version of the generalizedlikelihood ratio (GLR) based fault identication scheme (NL-GLR).Since NL-GLR is computationally demanding, it has been furthersimplied for online implementation using linearization of thenonlinear model around a nominal trajectory (SNL-GLR). The pro-posed approach can deal with sequential as well as simultaneousoccurrences of multiple faults/failures. An active fault tolerantNMPC (FTNMPC) scheme is developed that makes use of thefault/failure location and magnitude estimates generated by SNL-

Table 9Fed-batch bioreactor example (actuator failure case): comparison of CNMPC andFTNMPC performances based on ISE values

Output FTNMPC NMPC

Dissolved oxygen 676.9423 730.0787pH 1.2345 327.4919GLR to correct the state estimator and the prediction model on-line. The proposed model correction strategy overcomes the limita-tion on the number of extra states that can be added to the statespace model in NMPC for offset removal and allows bias compen-sation for more variables than the number of measured outputs.The proposed approach can also be used for achieving fault andfailure tolerance in a semi-batch operation. These advantages ofthe proposed intelligent state estimation and FTNMPC schemeshave been demonstrated by conducting simulation studies on abenchmark CSTR system, which exhibit input multiplicity andchange in the sign of steady state gain, and a fed batch bioreactor,which exhibits strongly nonlinear dynamics. We demonstrate thatthe proposed intelligent state estimator and FTNMPC scheme canbe used for achieving on-line optimizing control of the CSTR sys-tem. By simulating a regulatory control problem associated withan unstable nonlinear system given by Chen and Allgower [1],we have also demonstrated that the proposed intelligent state esti-mation strategy can be used to maintain asymptotic closed loopstability in the face of abrupt changes in model parameters. Anal-ysis of the simulation results reveals that the proposed approachprovides a comprehensive method for treating both faults(biases/drifts in sensors/actuators/model parameters) and failures(sensor/actuator failures) under the unied framework of fault tol-erant predictive control.

It may be noted that, in many applications, the nonlinearity islocalized in a smaller section of a plant and an NMPC scheme basedon mechanistic model is typically developed only for this sub-sec-tion. The proposed FTNMPC approach is expected to be useful foraddressing the problems of on-line model maintenance associatedwith such an NMPC implementation. Also, the proposed approachdoes not address the problems associated with change in the var-iance structure/magnitudes.

Appendix

In this Appendix, we briey summarize the method given byChen and Allgower [1] for computing terminal region X for inclu-sion in the objective function to guarantee nominal stability. Theclass of systems to be controlled is described by the following gen-eral nonlinear set of ordinary differential equations:

dxdt

Fxt;ut 89x0 x0yt Gxt 90such that dF0; 0 0. Here, x 2 Rn; y 2 Rr and u 2 Rm representingthe state variables, measured outputs and manipulated inputs,respectively. The system is subject to input constraints

ut 2 U 8t P 0where U denotes the constraint set on manipulated inputs. Theoptimal control problem over interval t;1 is reformulated asfollows:

minus

J Z tNpt

xsTWxxs usTWUusds

xt NpTW1xt Npsubject to

dxdt

Fxt;utxk xkus 2 U; s 2 k; k Npxk Np 2 X

where Q 2 Rnn and R 2 Rnm denote positive-denite, symmetricweighting matrices and Np is a nite prediction horizon and X rep-resents terminal region. To evaluate terminal region X, consider theJacobian linearization of the system at the origin

dxdt

Ax Bu 91

If Eq. (91) is stabilizable, then a linear state feedback u Kx canbe determined such that Ak A BK is asymptotically stable. Then,

203the following Lyapunov equation:

Ak bITW1 W1Ak bI Q 92

admits a unique positive-denite and symmetric solution W1,where Q Wx KTWuK 2 Rnn is positive denite and symmetric,b 2 0;1 satisesb < kmaxAk 93There exists a constant a 2 0;1 specifying a neighborhood X ofthe region in the form of

Xx : fx 2 RnjxTW1x 6 agsuch that

Kx 2 U; for all x 2 Xx, the linear feedback controller respects theinput constraints in Xx.

Xx is invariant for the nonlinear system (89) and (90) controlledby the local linear feedback u 2 Kx.

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J1x1;u Z 1t1

xtTWxxt mtTWumtdt

subject to nonlinear system (89) and (90) starting from xt1 x1and controlled by local linear state feedback u 2 Kx, is boundedfrom above as follows:

J1x1;u 6 xT1W1x1

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Step 3: Find the largest possible a1 such that Kx 2 U, for allx 2 Xa1 .

Step 4: Making iterations of a simple optimization problem

maxx

fxTW1/x xTW1xjxTW1x 6 ag 94

for the chosen b by reducing a from a1 until the optimum value gi-ven by (94) is nonpositive. Here

/x Fx;Kx Akx

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Intelligent state estimation for fault tolerant nonlinear predictive controlIntroductionFault diagnosisModel for normal behaviorFault detectionFault and failure modelsReview of linear GLR methodNonlinear GLR methodSimplification of NL-GLR methodMultiple simultaneous faults

Intelligent state estimation for fault tolerant NMPCOn-line modifications to state estimator and predictorCorrection for drifting disturbances, parameters and multiple sequential faultsFault tolerant NMPC formulationClosed loop stability

Simulation case studiesCSTR with input multiplicityOptimum seeking control in presence of parametric faultsEstimator reconfiguration on sensor failureIsolation and estimation of drifting disturbance

Nonlinear system by Chen and AllgowerFed-batch bioreactorBias in DO sensorBehavior under pH controller failure

ConclusionsAppendixReferences