Identification for passive robust fault detection using zonotope-based set-membership approaches

-1Identification for Passive Robust Fault Detection using Zonotopes* Joaquim Blesa*, Vicenç Puig*, Jordi Saludes**  *Automatic Control Department, Technical University of Catalonia (UPC) Pau Gargallo 5, 08028 Barcelona, Spain; (e-mail:{ joaquim.blesa,vicenc.puig}@upc.edu) **Applied Mathematics Department (MA2), Technical University of Catalonia (UPC) Carrer Colom 11, 08222 Terrassa, Spain (e-mail:[email protected]) Abstract: In this paper, the problem of identification for passive robust fault detection when parametric modeling uncertainty is considered. In particular, a zonotope is used to bound the model parametric uncertainty. Two identification methods are introduced following, respectively, the worst-case and setmembership approaches. Then, the underlying hypothesis are discussed and performance is compared. These two identification approaches lead to two robust fault detection tests (namely, the direct and inverse tests) that are also discussed. A case study based on a four tanks system is used to exemplify the properties of the two identification and associated fault detection approaches. Keywords: Robust fault detection, identification, interval models, zonotopes, set-membership  *This work was supported in part by the Research Commission of the Generalitat of Catalunya (Grup SAC ref. 2005SGR00537) and by Spanish Ministry of Education (CICYT projects ref. DPI-2005-05415, ref. DPI-2006-11944 and ref. DPI2008-01996). The material of this paper was partially presented at 7th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, July 2009, Barcelona, Spain. 1. INTRODUCTION The principle of model-based fault detection is to test whether the measured system inputs and outputs are consistent with the system behaviour described by a faultless model. If the measurements are inconsistent with the model of the faultless system, the existence of a fault is proved. The residual vector usually describes the consistency check between the predicted and the -2real behaviour. Ideally, the residuals should only be affected by the faults. However, the presence of disturbances, noise and modeling errors causes the residuals to become nonzero and thus interferes with the detection of faults. Therefore, the fault detection procedure must be robust against these undesired effects (Chen and Patton, 1999). In case that parametric uncertainties are taken into account, the healthy system model should include a vector of uncertain parameters bounded by sets that contains all possible parameter values when the system operates normally. In the robust fault detection literature, so far, parameters have been bounded using intervals and the resulting model is known as an interval model (Puig, 2008). In case of modeling a dynamic system using an interval model, the predicted output is also bounded by an interval. Then, fault detection test is based on checking if zero is contained or not in the residual interval after propagating the parameter uncertainty to the residual (Fagarasan et al., 2004; Puig et al., 2008; Sainz et al., 2002, Ploix, 2006). Alternatively, in this paper, parameter uncertainty will be bound using a more complex shape: a zonotope, that will allow to obtain less conservative fault detection models and results. The use of zonotopes in fault detection has already been suggested in the literaure. See for example (Combastel et al, 2008; 2009). One of the key points in model based fault detection is how models and their uncertainty bounds are obtained. Classical system identification methods provide only an estimation of the nominal model but do not provide a reliable means for bounding the uncertainty associated with the model. This problem has been mainly stated in many papers coming from robust control field. Recently some methodologies that provides a model with its uncertainty has been developed but thinking always in its application to control (Reinelt, 2002). In fact in this community, robust system identification is used to describe the new methodologies of system identification that provide not only a nominal model but also a reliable estimate of the uncertainty associated with the model. See for example the set-membership parameter estimation algorithms proposed by Milanese (1996), that produces a set of parameters that are consistent with the model structure that has been selected and assumed noise bounds. Alternatively, Campi et al. (2009) has suggested an adaptation of classical system identifications methods in order to provide the nominal model plus the uncertainty bounds for parameters that guarantee that all collected data from the system in non-faulty scenarios will be included in the model prediction interval (worst-case parameter estimation). The main contribution of this paper is to provide worst-case/set-membership parameter estimation algorithms using zonotopes to bound the parameter uncertainty and focusing on their application to passive robust fault detection. -3The structure of the paper is the following: Section 2 is dedicated to describe the problem, first focused on LTI systems. In Section 3, two identification approaches for robust fault detection are presented, recalling the underlying hypothesis and existing implementation algorithms. In Section 4, the fault detection tests associated to the models delivered by the proposed identification approaches are introduced. In Section 5, the performance of the two fault detection approaches in case of additive and multiplicative faults are studied. In Section 6, the extension of the indentification algorithms for LTI models to LPV models is introduced. In Section 7, a case study based on a well known control benchmark (the four-tanks system) is used to exemplify and compare the identification methods for robust fault detection presented in this paper. Finally, in Section 8, the conclusions of the paper are provided. 2. MODEL PARAMETRISATION Let us assume that the system can be expressed by an LTI model in regressor form (MA model) with additive noise as follows: ˆ () ()() () () ()yk k k ek yk ekφθ (1) where: - ()kφ is the regressor vector of dimension 1n  which can contain any function of inputs ()uk and outputs† ()yk . - ()kθΘ is the parameter vector of dimension 1n  - Θ is the set that bounds parameter values. - ()ek is the sensor additive noise bounded by a constant ()ek   . In this paper, the uncertain parameter set Θ is described by a zonotope centered in a nominal model:   00 : nn    Θθ HB θHz z B (2) where: - 0n θ  is the nominal model. - nn  H -1nn B is a unitary box composed by n unitary (   1,1B) interval vectors. - denotes the Minkowski sum. Notice that a particular case corresponds to the case the parameter set Θ is bounded by an interval box: 11 [,] [,] [ , ] ii n n         Θ where 0 iii  and 0 iii   with 0 i   and i=1,…,n -4This set can be viewed as a zonotope with Hequal to a nn    diagonal matrix: 12 ( , ,..., ) n diag    H (3) Depending on the consideration of variation of parameters from one instant k-1 to the following instant k two very different approaches, based on model (1), can be defined. 1. No restriction on variation in parameters, this is () ( 1) , kk k  θθ (3) Then the parameter can vary in one time instant to the farthest parameter in the set  , this is the worst-case variation. In the following this approach will be called worst-case approach. This approach was first suggested by Ploix (1999) in the context of fault detection. Further works using this approach are Adrot (2000), Calafiore (2002) and Campi (2009). 2. No variation in parameter is considered, this is () ( 1) 0, kk k  θθ (4) Then, the parameter is unknown but considered constant () ( 1) , kk k   θθ θ (5) Algorithms following this approach are also known as “set-membership parameter estimation” algorithms. In Milanese (1996) there is a survey of such methods. These different approaches will be described in the following two sections. 3. WORST-CASE APPROACH †In this paper we will focus on the single output case. -53.1 Problem definition Given a sequence of M regressor vector values ()kφ and measurement values ()yk in a fault free scenario and rich enough from the identifiability point of view. The aim is to estimate parameters and their uncertainty of a model parameterised as in Eq. (1) that can describe all the measurements ()yk considering the worst-case approach, this is no restriction in parameter variation. In this case, the set of uncertain parameters Θ should be obtained in such a way that all measured data in a fault free scenario will be covered by the worst-case predicted output produced by using model (1) and the uncertainty parameter set (“worst-case model”), that is ˆˆ () () , ()yk yk yk    that can be rewritten as the following equations ˆ() ()yk yk   and ˆ() ()yk yk    1,...,kM   (6) where:   ˆ( ) max ( ) ( ) with ( )yk k k k φθ θ Θ (7a)   ˆ( ) min ( ) ( ) with ( )yk k k k φθ θ Θ (7b) Then at every instant k, the regressor vector φ(k) and the measured output y(k) define two half-spaces k Θ and k Θ in n   with the worst-case conditions (6) as follows:   :() () n kkyk     Θθ φθ and   :() () n kkyk     Θθ φθ . Then, the sets Θ and Θ that fulfil respectively () ()kykφθ  and () ()kyk  φθ  1,...,kM   are defined by 1 M k k  ΘΘ and 1 M k k  ΘΘ And finally, a set Θ that fulfils both worst-case conditions (6) 1,...,kM   satisfies and    ΘΘ ΘΘ (8) The following two figures show graphically one example with 2n  θ. -61 θ 2 θ k Θ k Θ () ()kykφθ  () ()kykφθ  1k 1 θ 2 θ 123 ΘΘΘΘ 123 ΘΘ ΘΘ 1k 2k 3k Figure 1. a) Half spaces k Θ and k Θwith regressor vector φ(k) and measured output y(k) for k=1. b) Sets Θ and Θ with data from k=1 to k=3. 3.1 Worst-case parameter estimation Considering that the parameter set Θ can be described as the zonotope (2), the maximum (7a) and minimum prediction (7b) provided by model (1), considering worst-case variation in parameters, are given by (see Proposition 1 in Appendix) 0 1 ˆˆ () () ()yk y k kφH (9a) 0 1 ˆˆ () () ()yk y k kφH (9b) where 0 ˆ() y k is the model output prediction with nominal parameters: 00 ˆ() ()yk kφθ where 000 1 ( ,..., ) n   θ (10) Notice that in the particular case of interval parameters follows 11 () φ() n ii i kk    φH (11) according to (3). Replacing equations (9a) and (9b) in inclusion conditions (6), the optimal zonotope that fulfills the “worstcase condition” can be computed by solving the Problem 1. Problem 1: “Worst-case Parameter Estimation “ (general case) min ( ( ))f H ΘH subject to: 0 1ˆ () () ()kykyk φH  1,...,kM -7In this problem, the cost function f is usually the interval prediction thickness that can be calculated as 1 11 ˆˆ (() ()) 2 () MM kk yk yk k    φH (12) Problem 1 can be in general very hard to solve (Campi, 2009). In order to reduce the complexity, the zonotope that bounds Θ can be parameterised such that 0  HH, that corresponds with a zonotope with predefined shape (determined by 0 H) and a scalar . Then, in this case, the interval prediction thickness (12) is given by 01 11 ˆˆ (() ()) 2 () ( ) MM kk yk yk k f     φH (13) and restrictions can be expressed as follows: 0 0 0101 ˆ () () ˆ () () () () yk y k kykyk k   φHφH such that Problem 1 can be rewritten as the Problem 2. Problem 2: “Worst-case Parameter Estimation” (particular case) 01 1 min 2 ( ) N k k  φH subject to: 0 01 ˆ () () () yk y k k   φH 1,...,kM The optimal solution of this problem is given by:  0 1,..., 01 ˆ () () sup () kM yk y k k         φH (14) Remark: From Eq. (14) follows: If 0 , defines a parameter set Θparameterised as the zonotope in (2) with 0  HH that fulfils conditions (6) for all available data. On the other hand, if 0  then 0 ˆ() ()yk yk   0and 0 ˆ () ()yk y k   0 1,...,kM . This means 00 ˆˆ () () , ()yk yk yk    1,...,kM  . i.e. the only source of uncertainty is the sensor additive noise. 4. Set-Membership approach -84.1 Problem definition As in the worst-case approach (section 3), the aim is to estimate parameters and their uncertainty of a model parameterised as in Eq. (1), Given a sequence of data in a fault free scenario and rich enough from the identifiability point of view, that can describe all the measurements ()yk considering now the set-membership approach, this is no variation in parameters. In this case, the model is called a “consistent model” since the predicted behaviour is always inside the interval of possible measurements. That is   ˆ() () ,()yk yk yk or alternatively, it can be rewritten as follows ˆ() ()yk yk   and ˆ() ()yk yk    1,...,kM   (15) Where ˆ() ()yk k  φθ and  θΘ (15) As θΘ conditions (15) are satisfied and ˆ()yk can be bounded by ˆˆˆ () () ()yk yk yk where ˆ() y kand ˆ() y k are defined as in (7), set-membership conditions (15) can be rewritten as ˆ() ()yk yk  and ˆ() ()yk yk    1,...,kM   (17) In this case, the model is called a “consistent model” since the predicted behaviour is always inside the interval of possible measurements. At every instant k the regressor vector φ(k) and the measured output y(k) with the set-membership conditions (15) define a strip  :()() n kyk k       Fθφθ (18) Finally, a set Θ that fulfils both set-membership conditions (15) 1,...,kM   satisfies  1 M k k  ΘF. The following two figures show graphically one example with 2n θ. -91 θ 2 θ  1 :(1)(1) n y   Fθφθ    1 θ 2 θ F1 F2 F3 Figure 2. a) Strip with regressor vector and φ(k) and measured output ()yk for k=1. b) Intersection of the three stripes with data from k=1 to k=3. 4.2 Set-Membership parameter estimation Using this approach, the parameter set Θ that contains all models consistent with data, known as Feasible Parameter Set (FPS), is defined as follows:   |() () () , 1, , nyk k yk k M     FPS θφθ (19) The exact description of FPS is in general not simple, and existing algorithms usually approximate the FPS using an inner/outer simpler shapes as boxes, parallelotopes, ellipsoids or zonotopes (Milanese 1996). The approximation set is called Approximated Feasible Parameter Set ( AFPS ). In this paper, algorithms that provided inner/outer AFPS using zonotopes in case of using model (1) are used. Outer approximations Outer approximation algorithms find the parameter set Θof minimum volume such that FPS Θ. This kind of algorithms usually implies an excessive computational cost and recursive forms have been proposed in Vicino and Zappa (1996), in case of using parallelotopes, and in Bravo et al. (2006), in case of using zonotopes. These recursive approaches are based in computing iteratively the outer AFPS using parallelotopes/zonotopes and related operations as follows 211 1kkk M M      AFPS AFPS F AFPS AFPS F ΘAFPS AFPS F   (20) -16a) Fault k F b) No Fault revealed k FPS out A FPS in AFPS k FPS out A FPS in A FPS k F c) Undecided k F d) Undecided k FPS out A FPS in AFPS k FPS out A FPS in A FPS k F Figure 4. Different situations that can be distinguished in fault detection when using inner and outer approximations Another way to understand the inverse test is to consider it as a parameter estimation (Puig, 2006), where for every time instant k, the parameter vector ()kθ can be estimated from input and output measurements over a N samples temporal window as follows ˆ() ( ())()kkkgYθ§ (35) where () () () kN k k        φ φ  and () () () yk N k yk       Y Then, the inverse test, in a similar way than the direct test, is based on evaluating the following residual (Gertler, 1998) -17ˆ ˆ () ( ()) () ( ())(() ()) () ()kkkkkkkk rgRg YY θθθ (36) with () () () rk N k rk       R and where ()kθis the normal value obtained from system modelling in a non-faulty situation. This, the detection test (30) can be rewritten as follows 0()k  θ  (37) where   ˆ () ()| () () (),() k kkkkkkrr θθθ θθθAFPS (38) Remark: The implementation of (37) instead of (30), that always can be implemented, is subject to the condition of permanent excitation ( ()k is invertible). 7. FAULT SENSITIVITY OF THE PROPOSED FAULT DETECTION TESTS In fault detection, two kinds of faults are typically considered: additive faults (in input/output variables) and multiplicative faults (in parameters) (Gertler, 1998). Thus, including both type of faults in the system (1), the residual can be written as () ()() ()() () () () () y y kkkfkkfkkfkek   φθ φθ θ φ (39) where ()k φ f and () y f k represent the input and output sensor faults (additive) and ()k θ f represent the parametric faults (multiplicative). Eq. (1) can be rewritten as 0 () () () () f y kyk kek    (40) where 0()yk is the non-fault response and () fk is the effect of the faults 0() ()()yk k k  φθ (41) () ()() () () () fy kfkkfk kfk   φθ θφ (42) § In the case of a linear in the parameters models, Eq. (30) corresponds to the non-recursive least squares formula: 1 (()) ( ()()) () tt kkkk  g -18According to Gertler (1998), the minimum detectable fault corresponds to a fault that brings a residual to its threshold (“triggering limit”), assuming that no other faults and nuisance inputs are present. In this section it will be determined the minimum effect of a fault () fk that will guarantee the fault detection in the fault detection methods presented in section 6. Direct test In case of using model (1), the set of possible residual defined in (27) can be bounded by _ () [(),()]krkrk (43) where ˆ () () ()rk yk yk and _ˆ () () ()rk yk yk , with ˆ()yk and ˆ()yk as defined in (7), then test (26) detects an inconsistency when () 0rk or _() 0rk  (44) that, considering (40), implies 0 () () () ˆ() fkkekyk y  (45a) 0 () () () ˆ() fkkekyk y   (45b) In order to bound the inequalities the smallest values of 0()ky and ()ek are considered in (45a) and the biggest values in (45b), then () 2 ˆˆ () () fkyk yk  and () 2 ˆˆ () () fkyk yk    This is () 2 ˆˆ () () fkyk yk    (46) That in the case of bounding the parameters by a Zonotope leads to 1 () 2 () 2 fkk   φH (47) -19Inverse test In case of using model (1), and the test based on (34) where the Aproximation of the feasible parameter set at instant k can be bounded by a zonotope as 0n kk k AFPS θBX (48) and the strip k F as defined in (18) can be implemented using the zonotope support strip (Bravo 2006) defined by   :() n kdu qkq  Fθφθ  (49) where 0 1 () () uk k qk kφθ φ X (50a) 0 1 () () dk k qk kφθ φX (50b) Then the test (34) is equivalent to check if () d qyk or () u qyk   (51) For more details see Vicino & Zappa (1996). Inequalities (50) taking into account (40) lead to 0 () () () fd kq keky    (52a) 0 () () () fu kq kyek     (52b) In order to bound the inequalities the biggest values of 0()ky and ()ek are considered in (52a) and the smallest values in (52b), then 1 () 2 () 2 fk kk   φX and 1 () 2 () 2 fk kk   φX This is 1 () 2 () 2 fk kk   φX (53) As can be extracted from (47) and (53) the minimum effect detectable of a fault have the same structure from the direct and inverse tests. () (()) fr kkφ (54) where () rk is the interval prediction thickness considering the additive noise. This is () max(()() ()) min(()() ()) rk kkek kkek   φθ φθ (55) -20with () k kθΘ and ()ek  the difference is that the set of parameters k Θ and additive noise may be different. In this work we will focus only in the instant output sensor faults (additive) and parametric faults (multiplicative). In case of instant output sensor faults the effect of the fault is () () fy kfk   (56) while in the case of parametric fault is () () () fkkfk θ φ (57) Remark that the effect of parametric faults depends on the regressor vector. 8. CASE STUDY: FOUR TANKS SYSTEM 8.1 Description of the system A quadruple-tank process, proposed by Johansson (2000), will be used to illustrate the results presented in this paper. A schematic diagram of the system is shown in Fig. 4a. The process inputs are 1 vand 2 v (input voltages to the pumps) and the outputs are the tank levels 1 hand 2 h. The experiments presented in this section just consider the residual coming from the first tank, assuming that levels 1 hand 3 h and voltage 1 v are measured: 3 11 11 1311 11 1 22 () a dh a k g hghvek dt A A A      (45) where 1()ekis the measured noise( 1( ) 0.05ek cm  ) and 2 128 A cm, 3 13.33 /kcmVs, 2 981 / g cm s and 10.7   assumed constants. Eq. (45) can be discretized by the Euler method with sampling time 1ts  : -213 111 11 1 3 1 1 11 1 () (1) 2(1) 2(1) (1) () a ak hk hk ghk ghk vk ek AA A      (46) and can be transformed in a LPV model as 111123211 () ( ()) ( 1) ( ()) ( 1) ( 1) ()hk apkhk bpkhk bvk ek  (47) where: 11 () ( 1)pk hk and 23 () ( 1)pk hk are the scheduling variables; 1 1 11 2 (())1 () ag apk A pk  and 3 12 12 2 (()) () a g bpk A pk  are the LPV parameters; 11 2 1 k b A   is a LTI parameter. 8.2 Experiment definition In order to apply identification techniques presented in Section 3, a fault free scenario has been recorded (Fig. 4b). 020 40 60 80 100 120 140 0 5 10 15 Time (s) cm Levels y1 and y3 y1 y3 020 40 60 80 100 120 140 0 0.5 1 Time (s) Volts Pump 1 (v1) (a) (b) Figure 4: (a) Quadruple-tank process. (b) Fault free scenario To illustrate and compare the different behaviour of the two proposed identification methods, two cases will be considered: Case 1. Exact nominal model based in (47), that can be expressed in regressor form as follows: 1() ()( ) () k yk k ekφθp   (48) 1 h2 h 3 h4 h -22with: 11 (1) (1)yk hk , 1 1 2 (1) () ( 1) (1) T yk kuk uk φ         and 1 12 2 (()) () (()) k apk bpk b θp      where 13 (1) (1)uk hk , 21 (1) (1)uk vk  , 11 () ( 1)pk yk   and 21 () ( 1)pk uk  . To apply the methods of identification and fault detection described in Sections 3 and 4 to the LPV model, Eq. (48) can be rewritten as: 11 ˆ '() () () '() ()yk k ek yk ekφω with: 0 111 ˆ '( ) ( ) y kyky   and 123 0 (, , ) Tn     ωΩωHB where 00 1 ˆ() () ( ) k yk kφθp. Nominal LPV parameters come from Eq. (47) 1 0 2 0.1123 1(1) 0.1123 () (1) 0.0833 k pk pk              θp (49) Case 2. Approximate nominal model based on approximating LPV parameters 1 (())apk and 12 (())bpk coming from Eq. (47) by their mean values: 00.9461aa , 0 11 0.0680bb . 020 40 60 80 100 120 140 0.91 0.92 0.93 0.94 0.95 0.96 a(p 1 (k)) and a 0 a 0 a(p 1 (k)) 020 40 60 80 100 120 140 0.02 0.04 0.06 0.08 0.1 b 1 (p 2 (k)) and b 1 0 Time (s) b 1 0 b 1 (p 2 (k)) Figure 5: LPV parameter of Case 1 and approximate parameter of Case 2 evolutions in the fault free scenario Remark: Notice that in Case 1 the only source of uncertainty is the additive noise in the measurements while in Case 2 additionally there is some parametric uncertainty because of the approximation of the LPV parameters by a constant value. 7.3 Worst-case identification results -23A worst-case model has been obtained solving Problem 2 using a diagonal square matrix  000 0123 ,,diagH where 0 10.0385 , 0 20.0781 , 0 30 (no uncertainty in parameter 2 b has been considered) are obtained from the maximum variations in parameters 1 (())apk and 12 (())bpk in the considered operating range:   12.34,10.54 y cm and   11,15ucm. Case 1 Results. In this case since there is not parametric uncertainty, all the data could be explained by the worst-case model just adding the noise bounds to the predicted output provided by the nominal model. This is confirmed when applying Eq. (14), solution of Problem 2, since -0.0063 0  , implying no uncertainty in parameters is necessary to cover the data with the worst-case prediction as discussed in Section 3.4 Case 2 Results. In this case, the solution of Problem 2 using Eq. (14) leads to 0.4523  . Taking into account the considered matrix H0 obtained at the beginning of this section results in   00 12 ,,0diag  H. This means that the parametric uncertainty corresponding to a and b1 is 0 1  and 0 2  , respectively (See Fig. 6a). From Fig. 6b, it can be noticed that all measurements are covered by the bounds of the worst-case prediction. 0.92 0.925 0.93 0.935 0.94 0.945 0.95 0.955 0.96 0.965 0.97 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 a b 1 020 40 60 80 100 120 140 2 3 4 5 6 7 8 9 10 11 12 Output y 1 and bounds (cm) Time (s) (a) (b) Figure 6: (a) Uncertain parameters a and 1 b. (b) Measured output and bounds 7.4 Set-membership identification results -24Using the set-membership estimation methods proposed in Section 3.2, the FPS has been approximated by an outer zonotope. Additionally, the inner zonotope has been obtained solving Problem 4 with Eq. (20) and using as H0 the H matrix of the outer zonotope. Case 1 Results. In this case the matrix H of the set-membership outer zonotope calculated is 30.4 1.3 10 10.8      H and the solution of Problem 4 using (14) leads to 0.59 . Both inner and outer zonotopes calculated are showed in Fig. 7, where all strips obtained from every measurement are presented. -0.02 -0.015 -0.01 -0.005 00.005 0.01 0.015 0.02 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02  1  2 -4 -3 -2 -1 0 1 2 3 4 x 10 -3 -4 -3 -2 -1 0 1 2 3 4x 10 -3  1  2 (a) (b) Figure 7: (a) Set-membership inner and outer zonotopes corresponding to Case 1. (b) Detail Case 2 Results. In this case since there is parametric uncertainty and the considered set-membership model only considers additive noise, the FPS is empty. That is 1 M k k  F , implying that does not exist a set-membership model with the assumed structure consistent with all the identification data. But increasing the noise bound ( 0.05cm  ) by the “extra” term 1 2()kφH (with 0.4523 (0.0385,0.0781,0)diagH obtained from the worst-case model corresponding to Case 2) , a setmembership model can be obtained with a FPS equal to the one of the worst-case model presented in Figure 6a. 7.5 Fault detection results In order to show the behaviour of the two tests in front of different faults, two different kinds of faults have been simulated: additive (in sensors) and a multiplicative (in components) faults. -25The model that has been used in the direct test is the identified using the worst-case approach with the approximate nominal model (Case 2) as described in Section 7.3. On the other hand, the model that has been used in the inverse test is the identified using the set-membership approach with nominal LPV model (Case 1) as described in Section 7.4. Fault scenario 1: “Additive fault in input u1 112 Volts u f at t=60s” Figure 8 shows the behaviour of the direct test. In this figure, it can be seen that as the measurement at t=61s is out of the bounds, calculated by the worst-case model, the fault is detected at this instant. On the other hand the residual for t>60s tends to a constant value. 020 40 60 80 100 120 140 2 4 6 8 10 12 14 Time (s) cm Output y 1 and Bounds Output y 1 Bounds 020 40 60 80 100 120 140 No Fault Fault Additive Fault 020 40 60 80 100 120 140 No Fault Fault Direct Test 020 40 60 80 100 120 140 -2 -1 0Residual r Time (s) (a) (b) Figure 8. (a) Measured output and bounds. (b) Direct test Fault detection in fault scenario 1 Figure 9 shows the behaviour of the inverse test in this scenario. In this figure, it can be seen that as the stripe at t=61s does not intersect the outer zonotope, of the set-membership model, the fault is detected at this instant. On the other hand the intersection of the stripes for t>60s is empty and then does not define a new set of parameters consistent with the fault data. This fact can be explained because the fault is additive. -0.1 -0.08 -0.06 -0.04 -0.02 00.02 0.04 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04  1  2 -0.2 -0.1 00.1 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1  1  2 consistent sets consistent sets -32Both conditions (55) and (56) are equivalent when ˆˆ () () wc sm yk yk     and ˆˆ () () wc sm yk yk     (57) that leads to the single condition ˆˆ () () s mwc yk yk    (58) Finally, (54) follows from (58) taking into account that 1 ˆˆ () () 2 ()yk yk k φH from Proposition 1. This completes the proof. Outer approximation of the Feasible Parameter Set (FPS) using zonotopes (Bravo et al. 2006)   |() () () n kyk k yk     Fθφθ 0n kk k AFPS θBX with 12 kk k kn      xx x X () () nkk jjv T B AFPS F Where 0 0 0 () () , if 1 and ( ) 0 () () , otherwise k k kj k j k yk k jn k jk          θ θx vx θ   2,if 1 and () 0 () , otherwise j jj k tn j k jn k j       tt t x T X  () ,if () ,if () k kk i ij k j j i k j k j kij k ij k               x xx x T x x    Then 1kkkAFPS AFPS F with 0 11 1 n kk k   AFPS θBX, where 0* 1() kj θv, * 1() kj TX,   * 0 arg min ( ) ( ) n jn jvoljj  vTB