ROBUST FAULT ISOLATION USING NON-LINEAR INTERVAL OBSERVERS: THE DAMADICS BENCHMARK CASE STUDY. Vicenç Puig, Alexandru Stancu, Joseba Quevedo

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1 ROBUST FAULT ISOLATION USING NON-LINEAR INTERVAL OBSERVERS: THE DAMADICS BENCHMARK CASE STUDY Vicenç Puig, Alexandru Stancu, Joseba Quevedo Autoatic Control Departent (ESAII - Capus de Terrassa Technical University of Catalonia (UPC Rabla Sant Nebridi,. 8 Terrassa (Spain vicenc.puig@upc.es Abstract: This paper presents a passive robust fault detection and isolation approach using non-linear interval observers. In industrial coplex systes there is usually soe uncertainty on odel paraeters that can be bounded using intervals. A odel with paraeters bounded in interval is known as an interval odel. Intervals observers propagate paraeter uncertainty to the residual generating an adaptive threshold that allow to robust detect syste faults. In order to isolate faults, a bank of those observers with a specified fault signature is required. Finally, this approach will be applied to detect and isolate soe of the faults proposed in an industrial actuator used as an FDI benchark in the European project DAMADICS. Copyright 5 IFAC Keywords: Fault Detection, Fault Diagnosis, Robustness, Envelope Generation, Adaptive Threshold, Optiisation, Sliding Window Principle.. INTRODUCTION The proble of robustness in fault detection using observers has been treated basically using the active approach, based on decoupling the effects of the uncertainty fro the effects of the faults on the residual (Chen, 999. On the other hand, the passive approach is based on propagating the effect of the uncertainty to the residuals and then using adaptive thresholds (Horak, 988(Puig,. In this paper, the passive approach based on adaptive thresholds using a odel with uncertain paraeters bounded in intervals, also known as an interval odel, will be presented in the context of observer ethodology, deriving their corresponding interval version (Puig, b. Moreover, soe non-linearities present in the real syste will be included in the structure of the odel in order to iprove the accuracy of the predicted behaviours. This will lead to the design of non-linear interval observers that have been already proposed for robust fault detection in Stancu (. In this paper, the integration of such algoriths with existent fault isolation algoriths will be presented. Finally, a fault isolation case study based on the industrial actuator used as FDI benchark in the European project DAMADICS will be used to show the effectiveness of the fault isolation capabilities of the proposed approach.. ROBUST FAULT DETECTION USING NON- LINEAR INTERVAL OBSERVERS. Adaptive thresholding The proble of adaptive threshold generation in discrete tie-doain using non-linear interval observers with a Luenberger-like structure can be

2 forulated atheatically as follows: at every tieinstant an interval for the predicted output [ ˆ, ˆ( y k ] y (or alternatively, for the residual should be coputed subject to: ( k + = g( ( k, u( k, θ + K( y( k yˆ ( k yˆ ( k = h( ( k, u( k, θ where: R nx and ŷ R ny are estiated state and output vectors of diension nx and ny respectively; g and h are the state space and easureent non-linear function ; θ is the vector of uncertain paraeters of diension p with their values bounded by a copact set ( θ Θ of box type, i.e., p Θ = { θ R θ θ θ } and K is the gain of the observer designed to guarantee observer stability for all θ Θ. In case that [ ˆ y, ˆ ] y y ( holds no fault can be indicated. Otherwise, a fault should be indicated. Fault detection test ( is r = y ˆy. equivalent to checking if [ ] [ ]. Interval observation as an interval siulation In order to deterine [ ˆy, ˆ( y k ], observer equation ( can be reorganised as a syste with one output and two inputs, according to ( k + = g ˆ o ( x, uo( k, θ ˆ y( k = h(, u( k, θ where: [ u( k y( k ] t u o =. Then, interval observation can be forulated as an interval siulation. Existent algoriths can be classified according to if they copute the output interval using (Puig, 5: one step-ahead iteration based on previous approxiations of the set of estiated states (region based approaches or a set of punctual trajectories generated by selecting particular values of θ Θ using heuristics or optiisation (trajectory based approaches. The first approach assues iplicitly that the uncertain paraeters are tievarying. In this paper the second approach is followed since uncertain paraeters are considered tie-invariant.. NON-LINEAR INTERVAL OBSERVATION ALGORITHM In order to preserve uncertain paraeter tieinvariance, a functional relation between states and paraeters at any tie instant k that will relate initial ( and present state is derived. This is possible through forulating an optiisation proble that considers all the transitions fro the initial to the present state as it is usually done in forulating an optial control proble. Using this idea the following algorith is proposed (Stancu, : Algorith. Let y = { y(, y(, y(,... y( k } be a easureent trajectory of syste ( and assuing that the uncertainty on the initial state is such that x( X. At each tie step copute ˆX [ ] =,ˆ x, solving the following optiisation proble to deterine ˆx : = in ( globally subject to : = g ( ˆ x( k, u ( k, θ... ( = g ( (, u (, θ ( [ ( j, ( j ] [ θ, θ] j for j =,...,k θ where: g, u, is the state space ( θ observer function and [ u( k y( k ] t observer input ( u = is the o And solving again the previous optiisation proble substituting in by ax to deterine ˆx. The wrapping effect is avoided since uncertainty is not propagated fro step to step but instead always fro the initial state. This approach yields the accurate tie-invariant interval observation without any conservatis, assuing that the previous optiisation probles could be solved with infinite precision and the global optiu could be deterined (Puig, b. However, in practice it only could be solved with a given precision. On the other hand, one of the ain drawbacks of this approach is the high coputational coplexity of the optiisation algorith since at every iteration an additional restriction is added. So, the aount of coputation needed is increasing with tie being ipossible to operate over a large tie interval. The length increase proble in the previous approach can be solved if the observer is asyptotically stable, as it is presented in Stancu (. In this case, any transients in the observer settle to negligible values in a finite-tie. Therefore, for any tie k, it is possible to approxiate (6 using a sliding window, starting at tie k-l and ending at k, where L is the length of this window. Of course, the paraeter tie-invariance is only guaranteed inside the sliding window. This is why this approach is referred as alost tieinvariant. Finally, if the observer satisfies the isotony property (Gouzé,, the solution of the

3 optiisation probles is located in the vertices of the uncertain initial state and paraeter space. FAULT ISOLATION USING A BANK OF INTERVAL OBSERVERS While a single interval observer (residual is sufficient to detect faults, a bank (or a vector of interval observers (residuals are required for fault isolation. Given a set of residuals r = [ r,,rn ], the theoretical fault signature atrix, FSM, can be defined binary codifying the presence or not of a variable in every residual. This atrix has as any rows as residuals and as any coluns as variables appearing in the residuals. The eleent FSM of this atrix is equal to if its j th variable appears in the expression of the i th residual, otherwise is equal to. This atrix provides the theoretical influence of faults on the residuals in the following way: the j th colun can be associated to fault in the j th variable. If ultiple faults are considered then nuber of coluns of signature fault atrix has to be expanded up to all possible cobinations that are considered. For each residual derived fro the corresponding odel, a decision procedure should be ipleented in order to check the isatch between the corresponding odel and the real observations, using the fault detection procedure described in Section. The result of these tests applied to the whole set of odels will be the experiental or actual fault signature of the syste: s = [ s,,sn ]. Then, fault isolation will consist in looking for the theoretical fault signature in fault signature atrix that atches with the experiental signature. However, in practice to ake ore robust this fault selecting the theoretical fault signature with the least distance fro the actual fault signature refines isolation process. This fault vector will indicate the actual fault signature distance to every theoretical fault signature allowing to isolate faults (Fig.. P (k... P n (k P (k P (k Observer Observer P n (k Observer n Residual Vector r (k r (k... r n (k ij Fault Detection Signature Vector s (k... Fault... Isolation s n (k Fault Vector d (k d n (k Fig.. Fault detection and isolation using a bank of interval observers 5. THE DAMADICS BENCHMARK ISOLATION CASE STUDY The application exaple to test the interval observer approach to robust fault isolation, deals with an industrial sart actuator consisting of a flow servovalve driven by a sart positioner, proposed as an FDI benchark in the European DAMADICS project. The sart actuator consists of a control valve, a pneuatic servootor and a sart positioner (Bartys,. The DAMADICS sart actuator can be decoupled in seven coponents, described by their corresponding eleentary relations, listed in Table. Table. List of coponents of DAMADICS servoactuator Coponent Pneuatic Servootor Eleentary Relation X = e ( P, F s vc Fvc e( X,P,P,T Control Valve Fv e( X,P,P,T Position Controller = e ( SP,PV CVP P e ( X,CVP,P Electro-Pneuatic s = 5 z Transducer Positioner Transducer = e ( X PV 6 P e ( P Chaber Pressure s = 7 s Transducer Flow Transducer Fv = e8 ( Fv Considering the following easured variables: rod displaceent (X, servootor chaber pressure (P s, inlet valve pressure (P, outlet valve pressure (P, fluid flow controlled by the valve (F v and controller output (CVP and using structural analysis (Staroswiecki, four analytical redundancy relations can be obtained: r ( X r ( P,P, X r ( P,P,F = r ( CVP,SP, X s s,cvp = v = = The paraeters and their intervals of uncertainty of such relations are obtained using a set-ebership paraeter estiation approach siilar to that proposed by Ploix (999, that guarantees that data fro free fault scenarios are covered by the interval odel. 5. Dynaic redundancy relations: bank of interval observers Analytical redundancy relations r and r are dynaic relations that will be evaluated through two reduced observers: one for the rod displaceent and the other for the servootor chaber pressure, that correct partially (through observer gain the estiation using easureents. This iplies that estiation will also be affected by the fault. (5

4 5.. Analytical redundancy relation r Analytical redundancy relation r is derived using a reduced observer for the rod displaceent and can be forulated as ( k + = + + K ( x ( k + = a + ( a Ae + Ps + 58 r = x x dx where: x = X (the rod displaceent, x = dt (the rod velocity, K x is the displaceent observer gain, x is the rod displaceent easureent and is the sapling tie being equal to s. 5.. Analytical redundancy relation r Analytical redundancy relation r is derived a reduced observer for the chaber pressure can be forulated as + Pa ( k + = ( k Ae xɺ ( k V + Ae x( k ( k + r + K ( k + = ( k where: = P + ( P + k CVP + k CVP s Ps s Ps P z ( + P f f p p a (6 (7 x = (the pressure in the servootor s chaber and x = a (the air ass K Ps is the observer gain and P s is the chaber pressure easureent coing fro the sensors. 5. Static analytical interval redundancy relations Two additional static residuals (r and r can be generated fro fluid ass flow and controller output equations. 5.. Analytical redundancy relation r Analytical redundancy relation r is derived using - fluid ass flow: r = P P ρ f Fv K v( x (8 where: P is the easureent of the inlet pressure, P is the easureent of the outlet pressure and F v is the easureent of the fluid ass flow 5.. Analytical redundancy relation r Analytical redundancy relation r is derived using controller output: r = ( CVP K ( SP x + arcsin( π.8 p where: CVP is the easureent of the output of the controller, SP is the set-point and x is the easureent of the rod displaceent. 5. Theoretical fault signature atrix Considering residuals generated by interval observers (6 and (7, and by the static analytical redundancy relations (8 and (9 the theoretical fault signature atrix presented in Table considering the 9 faults defined in the benchark can be deduced. Looking at Table, it can be noticed that with the considered odels and easureents not all fault could be isolated or even detected. For exaple, f 7, f 8, f and f 9 will not be detected and f can not be isolated fro f, aong other. 5.5 Isolation results Four fault scenarios are considered: a servootor's diaphrag perforation (f, unexpected pressure change across valve (f 7, pressure sensor fault (f and positioner supply pressure drop (f 6. Looking at Table, fault f and f present the sae fault signature f and f. So, in practice they could not be isolated. Fault f 6 presents the sae fault signature with the considered residuals than f 9. And, finally, fault f 7 can be isolated since it presents a unique fault signature. In the following fault scenarios only those residuals that are affected by the corresponding fault will be presented because of the lack of space Fault f (servootor's diaphrag perforation Fro theoretical fault signature atrix presented in Table, fault f should only affect residual r and r. Fault f is introduced at tie instant 5 s. Fro Figure and, it can be observed that this fault is detected by residuals r and r. However, residual r is not as persistent as residual r since it only indicates the fault presence between 5 and 7 s. So, fault f only could be isolated in this tie interval. (9

5 Fig. Residual r evaluation under fault f Fig.5 Residual r evaluation under fault f 7 Fig. Residual r evaluation under fault f 5.5. Fault f 7 (unexpected pressure change across valve Fault f 7 should only affect residuals r, r and r according to theoretical fault signature atrix presented in Table. It is introduced at tie instant 5 s. Fro Figure, 5 and 6, it can be observed that this fault is detected by residuals r, r and r. However, residual r is not as persistent as residuals r and r since it only indicate the fault presence between 5 and 8 s. Therefore, fault f 7 only could be isolated in this tie interval. Fig.6 Residual r evaluation under fault f Fault f (pressure sensor fault Fault f should only affect residuals r and r according to theoretical fault signature atrix presented in Table. Fault f is introduced at tie instant 5 s. Fro Figure 7 and 8, it can be observed that this fault is clearly detected by residuals r and r. As conclusion, fault f only could be isolated. Fig.7 Residual r evaluation under fault f 5.5. Fault f 6 (Positioner supply pressure drop Fig. Residual r evaluation under fault f 7 Finally, fault f 6 should affect only residuals r according to theoretical fault signature atrix presented in Table. Residuals r, r and r do not detect the fault and for the lack of space are not included. Fault f 6 is introduced at tie instant 5 s. Fro Figure 9, it can be observed that is detected by residual r. So fault f 6 only could be isolated.

6 to the anageent and staff of the Lublin sugar factory, Cukrownia Lublin SA, Poland for their collaboration and provision of anpower and access to their sugar plant.the authors wish also to thank the support received by the Research Coission of the Generalitat of Catalunya (ref. SGR6 and by Spanish CICYT (ref. DPI-5. REFERENCES Fig.8 Residual r evaluation under fault f Fig.9 Residual r evaluation under fault f 6 6. CONCLUSIONS In this paper, non-linear interval observers have been presented as an approach to passive robust fault detection and isolation when the onitored process is odelled using a non-linear interval odel. Interval observation has been translated to a couple of optiisation proble that provide the interval for the observed output (one for the axiu and another one for the iniu. Then, the detection test consists in checking if the easured output is or not inside this interval. A bank of such of observers is necessary with a given fault signature atrix in order to isolate faults. Interval observers have been applied to detect and isolate soe of the faults proposed in the DAMADICS benchark providing good results. ACKNOWLEDGMENTS Bartyś, M., de las Heras, S. (. Actuator siulation of the DAMADICS benchark actuator syste. In Proceedings of IFAC SAFEPROCESS, pp , Washington DC. USA. Chen J., Patton, R.J. (999. Robust Model-Based Fault Diagnosis for Dynaic Systes. Kluwer Acadeic Publishers Gouzé, J.L., Rapaport, A, Hadj-Sadok, M.Z. ( Interval observers for uncertain biological systes. Ecological Modelling, No., pp. 5-56,. Horak, D.T. (998 Failure detection in dynaic systes with odelling errors. Journal of Guidance, Control and Dynaics, (6, Ploix, S., Adrot, O., Ragot, J. (999. Paraeter Uncertainty Coputation in Static Linear Models. 8th IEEE Conference on Decision and Control. Phoenix. Arizona. USA. Puig, V., Quevedo, J., Escobet, T., De las Heras, S. (. Robust Fault Detection Approaches using Interval Models. IFAC World Congress (b. Barcelona. Spain. Puig, V., Saludes, J., Quevedo, J. (a Worst-Case Siulation of Discrete Linear Tie-Invariant Interval Dynaic Systes. Reliable Coputing, 9(, pp Puig, V., Quevedo, J., Escobet, T., Stancu, A. (Puig, b Passive Robust Fault Detection using Linear Interval Observers. IFAC SafeProcess,. Washington. USA.. Puig, V., Stancu, A., Quevedo, J. (5 Siulation of uncertain dynaic systes described by interval odels: a survey. IFAC World Congress, 5. Prague. Czech Republic. Stancu, A., Puig, V., Quevedo, J., Patton, R.J. (. Passive robust fault detection using nonlinear interval observers: Application to the DAMADICS benchark proble. In Proceedings of IFAC SAFEPROCESS, pp 97-, Washington DC. USA. (Accepted to be published in Control Engineering Practice Staroswiecki, M., J. P. Cassar & P. Declerk (. A structural fraework for the design of FDI syste in large scale industrial plants. In Issues of Fault Diagnosis for Dynaic Systes. Patton, R.J., Frank, P.M., Clark, R.N. (eds. Springer-Verlag. The authors acknowledge funding support under the EC RTN contract (RTN DAMADICS. Thanks are expressed Table. Fault signature atrix f f f f f 5 f 6 f 7 f 8 f 9 f f f f f f 5 f 6 f 7 f 8 f 9 r r r r