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A robust deconvolution scheme for fault detection and isolation of uncertain linear systems: an LMI approach

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Optimal H∞H∞ deconvolution filter theory is exploited for the design of robust fault detection and isolation (FDI) units for uncertain polytopic linear systems. Such a filter is synthesized under frequency domain conditions which ensure guaranteed
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  AROBUST DECONVOLUTION LMI PROCEDUREFORFAULT DETECTION ANDISOLATION OFUNCERTAINLINEAR SYSTEMS: ANLMI APPROACHA. Casavola ∗ D. Famularo ∗∗ G.Franz`e ∗∗  DEIS - Universit `a degli studi della Calabria, Rende(CS),87036 ITALY,  { casavola, franze } @deis.unical.it ∗∗  DIMET - Universit `a degli studi di Reggio Calabria, ReggioCalabria (RC) 89100, ITALY, domenico.famularo@unirc.it Abstract: A robust Fault Detection and Isolation (FDI) scheme for uncertain polytopiclinear systems based on optimal H  ∞  deconvolution filters is discussed. The filter must becapable to satisfy two sets of  H  ∞  constraints: the first is a disturbance-to-faultdecouplingrequirement, whereas the second expresses the capability of the filter to track the faultsignals in a prescribed frequency range. By means of the Projection Lemma, a quasi-convex formulation of the problem is obtained via LMIs. Finally, a FDI logic consistingof an adaptive thresholds scheme based on the on-line rms evaluation of relevant systemvariables is proposed. The effectiveness of the design technique is illustrated via anumerical example.  Copyright   c  2005 IFAC  Keywords: Fault detection, Fault isolation, Uncertainty, Linear systems, Convexprogramming1. INTRODUCTIONFault Detection andIsolation (FDI) techniquesare im-portant topics in systems engineering from the view-point of improving the system reliability. A fault rep-resents any kind of malfunction in a plant that leads tounacceptable anomalies in the overall system behav-ior. Such a malfunction may occur due to componentfailures inside the main frame of the process, sensorsand/or actuators.The issue of fault detection and isolation has been ad-dressed by many authors in several books and surveyarticles where many different design methodologieshave been exploited (model based approach, param-eter estimation, generalized likelihood ratio etc.). See(Frank, 1990; Patton  et al. , 1989; Qiu  et al. , 1993)andreferences therein for comprehensive and up-to-datetutorials.In this paper a novel robust H  ∞  FDI design procedureis proposedfor polytopicuncertainLTI systems wherethe residual generator is a deconvolution filter whosedynamic does not depend on any nominal plant real-ization. Such a filter will be designed so as to robustlydecouple the residuals (Fault Detection) from the dis-turbances and conversely to enhance the sensitivityto each fault signal by properly separating classesof different faults (Fault Isolation). The satisfactionof requirements on disturbances decoupling and faultsensitivity enhancement leads to the minimization of standard H  ∞ -norm optimization problems. In particu-lar, the first consists in minimizing the  H  ∞ -norm of the disturbance-to-residual map whereas the secondcorresponds to solve an optimal  H  ∞  tracking prob-lem where the objective is that the residual optimallytracks the fault signal over a prescribed frequencyrange.The design methodology presented in this paper is atwo steps procedure. In the first step, the synthesis of a robust FDI filter via LMI optimization techniques isdescribed. Via the Projection Lemma and Congruencetransformations (see (Tuan  et al. , 2003) for details),  the H  ∞  norm constraints can be converted into quasi-LMIs feasibility conditions and efficiently solved bystandard semidefinite programming solvers. The sec-ond step consists in equipping the residual generatorwith the capacity of discriminating between real andfalse alarms. This is done by resorting to decision log-ics based on adaptive thresholds, computed on-line onthe basis of time-windowed rms-norms of the residualresponses. It worth pointing out that the use of stan-dard Luenberger observers based on nominal models(Casavola et al. ,2003),insteadofdeconvolutionfiltersas here proposed, would have led to nonlinear ma-trix conditions and nonconvexoptimization problems,much more difficult to be solved.2. PROBLEM FORMULATIONConsider the following uncertain continuous-time lin-ear system described by the following state-spacemodel P  :  ˙  x ( t  ) =  Ax ( t  )+  B u u ( t  )+[  B  f   B d  ]   f  ( t  ) d  ( t  )   y ( t  ) =  Cx ( t  )+  D u u ( t  )+[  D  f   D d  ]   f  ( t  ) d  ( t  )  (1)where  x ( t  )  ∈  R n represents the state,  y ( t  )  ∈  R  p themeasured output,  f  ( t  )  ∈ R m a set of detectable faultsignals,  d  ( t  )  ∈  R d  bounded energy sensors/actuatordisturbances and  u ( t  )  ∈  R r  an external manipulablecommand. The plant matrices belong to the polytopicfamily   A B u  BC D u  D   ∈   A ( α )  B ( α ) C  ( α )  D ( α )  = s ∑ i = 1 α i   A i  B u , i  B i C  i  D u , i  D i  , α ∈ Γ    (2)where  B  : = [  B  f   B d  ] ,  D  : = [  D  f   D d  ]  and Γ  is the unitarysimplex Γ  : =  ( α 1 ,... , α s )  : s ∑ i = 1 α i  =  1 , α i ≥ 0  W.l.o.g. we can assume that the polytope system isquadratically stable (see (Boyd  et al. , 1994)). Thisis the case e.g. when the system is pre-compensated.Notice that such a condition is necessary in orderto satisfy the Bounded Real Lemma (see inequalities(19)-(20) in Section III). In this case,  u ( t  )  plays therole of a tracking reference signal.Fault detection and isolation (FDI) relies on the gen-eration of a signal, referred to as residual, which mustbe sensitive to failures, viz. capable to distinguishfail-ures from disturbances and discriminate failures eachother. Specifically, the design must ensure that resid-uals are “close” to zero in fault-free situations whilesuitably deviating from zero in the presence of faults.With these premises, the idea is to consider a residualgenerator based on a deconvolution filter having thefollowing general structure. F   :   ˙  x F  ( t  ) =  A F   x F  ( t  )+  B F  s ( t  )  z ( t  ) =  L F   x F  ( t  )+  H  F  s ( t  )  (3)where  x F  ( t  ) ∈ R n F  ,  z ( t  ) ∈ R  p and s ( t  )  : = [  y T  ( t  )  u T  ( t  )] T  ∈ R  p + r  (4)Note that in the above filter structure the informationcoming from the reference input  u ( t  )  is directly usedfor estimation purposes.Let r  ( t  )  : =  z ( t  ) −  y ( t  )  (5)be the residual vector. Accordingly, the augmentedsystem of becomes G  :  ˙  x cl ( t  ) =  A cl  x cl ( t  )+ B  cl , u u ( t  )+ B  cl   f  ( t  ) d  ( t  )  r  ( t  ) =  C  cl  x cl ( t  )+ D  cl , u u ( t  )+ D  cl   f  ( t  ) d  ( t  )  (6)  x cl ( t  )  : =   x ( t  )  x F  ( t  )  , A cl  : =   A  0  B F   C  0   A F   , B  cl  =   B B F    D  f  0  D d  0  , B  cl , u  : =   B u  B F    D u  I  r   , C  cl  : =   H  F   C  0  − C L F   , D  cl  : =  H  F    D  f  0  D d  0  −   D  f   D d   , D  cl , u  : =  H  F    D u  I  r   −  D u and the residual vector, depending on disturbances,fault signals and command inputs, can be rewritten as r  ( s ) =  G rf   ( s )  f  ( s )+ G rd  ( s ) d  ( s )+ G ru ( s ) u ( s )  (7)in terms of uncertain transfer functions  G rd  ( s ) ,  G rf   ( s ) and  G ru ( s ) .  We shall assume hereafterthat the numberof faults to be isolated is less than or equal to thenumber of outputs. Such an assumption is necessarybecause we want to consider simultaneous fault oc-currences.The objectives of robust residual generation are par-tially conflicting each other. In fact, there exists atrade-off between the minimization of the effects of the disturbanceandreferenceinputonthe residualandthe maximization of the residual sensitivity to faults.The first leads to the minimization of the  H  ∞ -normsof   G rd   and  G ru .  Notice that  G ru   =  0 here becauseof the model uncertainty, whereas it would be zeroin the uncertainty-free case thanks to the separationprinciple. The fault sensitivity enhancement wouldcorrespond to the maximization of the minimum sin-gular values of   G rf   ,  which is a nonconvex functionof the convolution filter matrices. This complicationcan be gone around via the Smallest Gain Lemma(Rank, 1998),which allows the replacementof a max-imization problem regarding the minimum singularvalue with the minimization of a standard  H  ∞ -normmodel-matching problem, specifically the minimiza-tion of the  H  ∞ -norm of the difference between theresiduals  r  ( s )  and the faults  f  ( s ) . The latter transfor-mation involves a certain degree of conservativenessup an extent that depends on  γ   f   in (10), the lowerthe best. For solvability reasons, the above problemonly makes sense over a prescribed frequency range, Ω : = [ ω  i , ω  s ] . Frequency weighting is also importantfrom practical points of view if the disturbances and  faults have known spectra. Then, for prescribed suit-able levels  γ  d  ,  γ  u  and  γ   f  , the above problems can berecast in the simultaneoussatisfaction ofthe followingconditions:max α ∈ Γ  σ ω  ∈ Ω 1  ( G rd  (  j ω  )) ≤ γ  d  , γ  d   >  0 .  (8)max α ∈ Γ  σ ω  ∈ Ω 1  ( G ru (  j ω  )) ≤ γ  u ,  γ  u  >  0 .  (9)max α ∈ Γ  σ ω  ∈ Ω 2  ( W   f   (  j ω  ) − G rf   (  j ω  )) ≤ γ   f   ,  γ   f   >  0 . (10)Note that conditions (8) and (9) translate into therobust decoupling of the residual w.r.t. disturbancesand reference inputs in the frequency interval  Ω 1 ,whereas condition (10) means that, in the frequencyinterval  Ω 2 ,  r  ( s )  robustly tracks a filtered version of the fault signal,  W   f   ( s )  f  ( s ) , with  W   f   ( s )  stable appro-priately chosen. The detection and isolation problemscan be recast into the following multi-objective  H  ∞ optimization problem: Optimal FDI design problem (OFDP) -  Given posi-tive reals  a ,  b  and  c , find a filter realization  F  ( s )  suchthatmin F  ( s ) a γ  d   + b γ   f   + c γ  u subject tomax α ∈ Γ   F  ( s ) P sd  ( s ) − P  yd  ( s )  ∞ ≤ γ  d  ,  (11)max α ∈ Γ   F  ( s ) P su ( s ) − P  yu ( s )  ∞ ≤ γ  u ,  (12)max α ∈ Γ   W   f   ( s ) − ( F  ( s ) P sf   ( s ) − P  yf  ( s ))  ∞ ≤ γ   f   (13)where  P sd  ( s )  : =  P  yd  ( s ) 0  ,  P su ( s )  : =  P  yu ( s )  I  r   , P sf   ( s )  : =  P  yf  ( s ) 0  . Constants  a ,  b  and  c  are used to trade-off between theconflicting requirements (11), (12) and (13).ThesecondimportanttaskforFDIconsists intheeval-uation of the generated residuals. One widely adoptedapproach is to choose a threshold  J  th  >  0 and use thelogic  J  r  ( t  )  >  J  th  ⇒  faults  J  r  ( t  ) ≤  J  th  ⇒  no faults (14)for fault detection and  J  r  , i ( t  )  >  J  th , i  ⇒  i-th fault  J  r  , i ( t  ) ≤  J  th , i  ⇒  without i-th fault (15)for fault isolation where  J  r  ( t  ) =   1 t  Z   t  0 r  T  ( τ ) r  ( τ ) d  τ  (16)See (Frank   et al. , 1997) for a detailed discussionabout this index and similarly for  J  r  i ( t  ) . Details andproperties of the detection and isolation logic used inthis paper will be given in next Section 4.3. LMI FORMULATIONThe design of the residual observer (3) is accom-plished by recurring to the above  OFDP  optimizationproblem.Hereitwill beshown,followingsimilarlinesas in (Tuan  et al. , 2003), that  OFDP  can be refor-mulated as an LMI optimization problem when thefrequency constraints (11) and (13) are replaced byan equivalentset of   µ -parameterizedLMI (quasi-LMI)feasibility conditionsTo this end,let a minimalstate-space realization ofthetracking filter W   f   ( s )  be given by W   f   ( s )  : =   A r   B r  ,  f  C  r   D r  ,  f   while P sd  ( s ) : =   A B d    C  0    D d  0  , P su ( s ) : =   A B u   C  0    D u  I  r   , P sf   ( s ) : =   A B  f    C  0    D  f  0  are the state-space realizations of   P sd  ( s ) ,  P su ( s )  and P sf   ( s )  in (11), (12) and (13). It follows that  W   f   ( s ) − ( F  ( s ) P sf   ( s )  − P sf  ( s ))  can be realized as   Ar   0 0  Br  ,  f  0  A  0  B f  0  BF   C  0   AF  BF    D f  0  C r   −  H F   C  0  + C   −  LF   − (  H F    D f  0  −  D f   )  = :  ˜  A  0 ˜  B f  BF   ˜ C AF  BF    D f  0  − ˜  L f   −  LF   − ˜  H  f   (17)for some matrices  A F  ,  B F  ,  L F   and  H  F   to be deter-mined with  A F   asymptotically stable. Coherently, byadding the unobservable/uncontrollable modes of   A r  ,non-minimal state-space realizations of   F  ( s ) P sd  ( s ) − P sd  ( s )  and  F  ( s ) P su ( s ) − P su ( s )  assume the followingform   Ar   0 0 00  A  0  Bd  0  BF   C  0   AF  BF    Dd  0  0  H F   C  0  − C LF  H F    Dd  0  −  Dd   = :  ˜  A  0 ˜  Bd  BF   ˜ C AF  BF    Dd  0  ˜  Ld  LF   ˜  H d   (18)   Ar   0 0 00  A  0  Bu 0  BF   C  0   AF  BF    Du I r   0  H F   C  0  − C LF  H F    Du I r   −  Du  = :  ˜  A  0 ˜  Bu BF   ˜ C AF  BF    Du I r   ˜  Lu LF   ˜  H u  (19)Theadditionoftheunobservable/uncontrollablemodesof   A r   in(18)and(19)makesit possibleto usethesamematrix vertices ˜  A i  for the tracking and the decouplingobjectives.Thischoiceis alsonecessaryinordertoap-ply the Projection Lemma and obtain a convexdesign.For the same reason,note that the filter dimension(thedimension of   A F  ) must satisfy n F   =  n r   (tracking)  + n  (plant) (Full Order Filter)Then, by exploiting the Bounded Real Lemma, con-ditions (8), (9) and (10) are jointly satisfied iff thereexist filter matrices  A F  ,  B F  ,  L F   and  H  F  , with  A F   asymptotically stable, and an auxiliary matrix  X   =  X  T  ∈ R 2 n F  × 2 n F  ,  X   > 0, such that the following matrixinequalities   ˜  A  0  BF   ˜ C  f  AF   T   X  +  X   ˜  A  0  BF   ˜ C  f  AF    X   ˜  B f  BF    D f  0   − ˆ  LT  f  −  LT F    ˜  BT  f    D f  0  T  BT F    X   − γ   f  I   − ˜  H T  f   − ˜  L f   −  LF    − ˜  H  f   − γ   f  I   (20)   ˜  A  0  BF   ˜ C d  AF   T   X  +  X   ˜  A  0  BF   ˜ C d  AF    X   ˜  Bd  BF    Dd  0   ˆ  LT d  LT F    ˜  BT d    Dd  0  T  BT F    X   − γ  d  I   ˜  H T d   ˜  Ld  LF    ˜  H d   − γ  d  I   (21)   ˜  A  0  BF   ˜ C u AF   T   X  +  X   ˜  A  0  BF   ˜ C u AF    X   ˜  Bu BF    Du I r    ˆ  LT u LT F    ˜  BT u   Du I r   T   BT F    X   − γ  u I   ˜  H T u  ˜  Lu LF    ˜  H u  − γ   f  I   (22)will be negative definite. Notice that, for any givenquadruple (  A F  ,  B F  ,  L F  ,  H  F  ) with  A F   stable, the aboveinequalities are jointly solvable for some symmetricalmatrix  X   >  0 and for sufficiently large  γ   f  ,  γ  d   and  γ  u .As a usual in the multiobjective optimization, a singlematrix  X   is used in both the LMI conditions (20), (21)and (22).By using the Projection Lemma (PL) and followingsimilar lines of (Tuan  et al. , 2003), inequalities (20),(21) and (22) are satisfied iff (see (Tuan  et al. , 2003)for details) the following matrix inequalities  −  µ ( V  + V T  )  V T  A cl ,  f   +  X V T  B  cl ,  f   0  µV T  A T cl ,  f  V  +  X   −  X   0  − ˜  LT  f  −  LT F    0 B  T cl ,  f  V   0  − γ   f  I   − ˜  H T  f   00   − ˜  L f   −  LF    − ˜  H  f   − γ   f  I   0  µV   0 0 0  −  X   (23)  −  µ ( V  + V T  )  V T  A cl , d  +  X V T  B  cl , d   0  µV T  A T cl , d V  +  X   −  X   0   ˜  LT d  LT F    0 B  T cl , d V   0  − γ  d  I   ˜  H T d   00   ˜  Ld  LF    ˜  H d   − γ  d  I   0  µV   0 0 0  −  X   (24)  −  µ ( V  + V T  )  V T  A cl , u +  X V T  B  cl , U   0  µV T  A T cl , uV  +  X   −  X   0   ˜  LT u LT F    0 B  T cl , uV   0  − γ  u I   ˜  H T u  00   ˜  Lu  −  LF    ˜  H u  − γ  u I   0  µV   0 0 0  −  X   (25)are negative definite. Here,  V   is a slack variable of proper dimensions and  µ ≥ 0 is a scalar that can beselectedtobesufficientlylargetorender(23),(24)and(25) feasible A cl ,  f   : =   ˜  A  0  B F   ˜ C   f   A F   , B  cl ,  f   : =  ˜  B  f   B F    D  f  0  , A cl , d   : =   ˜  A  0  B F   ˜ C  d   A F   , B  cl , d   : =  ˜  B d   B F    D d  0  . A cl , u  : =   ˜  A  0  B F   ˜ C  u  A F   , B  cl , u  : =  ˜  B u  B F    D u  I  r   . By partitioning V   and  X   as 2 × 2 block-matrices V   =  V  11  V  12 V  21  V  22  ,  X   =   X  1  X  T  3  X  3  X  2  and by taking into account the structure of  A cl ,  f  , B  cl ,  f  , A cl , d   and B  cl , d  , (23)-(25) become  −  µ ( V  11  + V  T  11 )  −  µ ( S  2  + S  T  2  )  V  T  11  ˜  A +  ˆ  B F   ˜ C   f   +  ˆ  X  1  ˆ  A F   +  ˆ  X  T  3 −  µ ( S  2  + S  T  2  )  −  µ ( S  1  + S  T  1  )  V  T  12  ˜  A +  ˆ  B F   ˜ C   f   +  X  3  ˆ  A F   +  X  2 ( ∗ ) ( ∗ )  − ˆ  X  1  − ˆ  X  T  3 ( ∗ ) ( ∗ )  − ˆ  X  3  − ˆ  X  2 ( ∗ ) ( ∗ ) ( ∗ ) ( ∗ )( ∗ ) ( ∗ ) ( ∗ ) ( ∗ )( ∗ ) ( ∗ ) ( ∗ ) ( ∗ )( ∗ ) ( ∗ ) ( ∗ ) ( ∗ ) V  T  11  ˜  B  f   +  ˆ  B F   D  f   0  µV  T  11  µS  2 V  T  12  B +  ˆ  B F   D  f   0  µS  T  2  µS  1 0   C  T  f  − C  T  f   H  T F   + C  T  f    0 00  − ˆ  L T F   0 0 − γ   f   I   − (  H  F    D  f  0  −  D  f   ) T  0 0 ( ∗ )  − γ   f   I   0 0 ( ∗ ) ( ∗ )  − ˆ  X  1  − ˆ  X  T  3 ( ∗ ) ( ∗ )  − ˆ  X  3  − ˆ  X  2  <  0  −  µ ( V  11  + V  T  11 )  −  µ ( S  2  + S  T  2  )  V  T  11  ˜  A +  ˆ  B F   ˜ C  d   +  ˆ  X  1  ˆ  A F   +  ˆ  X  T  3 −  µ ( S  2  + S  T  2  )  −  µ ( S  1  + S  T  1  )  V  T  12  ˜  A +  ˆ  B F   ˜ C  d   +  X  3  ˆ  A F   +  X  2 ( ∗ ) ( ∗ )  − ˆ  X  1  − ˆ  X  T  3 ( ∗ ) ( ∗ )  − ˆ  X  3  − ˆ  X  2 ( ∗ ) ( ∗ ) ( ∗ ) ( ∗ )( ∗ ) ( ∗ ) ( ∗ ) ( ∗ )( ∗ ) ( ∗ ) ( ∗ ) ( ∗ )( ∗ ) ( ∗ ) ( ∗ ) ( ∗ ) V  T  11  ˜  B d   +  ˆ  B F   D d   0  µV  T  11  µS  2 V  T  12  B +  ˆ  B F   D d   0  µS  T  2  µS  1 0   0 C  T d   H  T F   − C  T d    0 00 ˆ  L T F   0 0 − γ  d   I   (  H  F    D d  0  −  D d  ) T  0 0 ( ∗ )  − γ  d   I   0 0 ( ∗ ) ( ∗ )  − ˆ  X  1  − ˆ  X  T  3 ( ∗ ) ( ∗ )  − ˆ  X  3  − ˆ  X  2  <  0  −  µ ( V  11  + V  T  11 )  −  µ ( S  2  + S  T  2  )  V  T  11  ˜  A +  ˆ  B F   ˜ C  d   +  ˆ  X  1  ˆ  A F   +  ˆ  X  T  3 −  µ ( S  2  + S  T  2  )  −  µ ( S  1  + S  T  1  )  V  T  12  ˜  A +  ˆ  B F   ˜ C  d   +  X  3  ˆ  A F   +  X  2 ( ∗ ) ( ∗ )  − ˆ  X  1  − ˆ  X  T  3 ( ∗ ) ( ∗ )  − ˆ  X  3  − ˆ  X  2 ( ∗ ) ( ∗ ) ( ∗ ) ( ∗ )( ∗ ) ( ∗ ) ( ∗ ) ( ∗ )( ∗ ) ( ∗ ) ( ∗ ) ( ∗ )( ∗ ) ( ∗ ) ( ∗ ) ( ∗ )  V  T  11  ˜  B u  +  ˆ  B F   D u  0  µV  T  11  µS  2 V  T  12  B +  ˆ  B F   D u  0  µS  T  2  µS  1 0   0 C  T u  H  T F   − C  T u   0 00 ˆ  L T F   0 0 − γ  u  I   (  H  F    D u  I  r   −  D u ) T  0 0 ( ∗ )  − γ  u  I   0 0 ( ∗ ) ( ∗ )  − ˆ  X  1  − ˆ  X  T  3 ( ∗ ) ( ∗ )  − ˆ  X  3  − ˆ  X  2  <  0 where  ˆ  A F   =  V  T  21  A F  V  − 122  V  21 ,  ˆ  B F   =  V  T  21  B F  ,  ˆ  L F   =  L F  V  − 122  V  21 ,  S  1  = V  T  21 V  − T  22  V  21 ,  S  2  = V  T  21 V  − T  22  V  12 ˆ  X   =   ˆ  X  1  ˆ  X  T  3 ˆ  X  3  ˆ  X  2  =   I   00  V  T  21 V  − T  22   X    I   00  V  − 122  V  21  The previousthree inequalities are indeed quasi-linear(  µ -parameterized LMIs) in ˆ  X  ,  S  1 ,  S  2 ,  ˆ  A F  ,  ˆ  B F  , V  11 ,  ˆ  L F  ,  H  F  . The next main result summarizes theabove discussion and providesa procedurefor solving OFDP : Theorem 1.  - A feasible solution to the OFDP prob-lem is obtained by solving a sequence of   µ -para-meterized optimization problemsmin ˆ  X  , S  1 , S  2 ,  ˆ  A F  ,  ˆ  B F  , V  11 , ˆ  L F  ,  H  F  a γ  d   + b γ   f   + c γ  u subject toˆ  X   >  0 ( 23 ) − ( 25 ) , evaluated over the polytope vertices (2) . (26)Foranychoiceof   µ > 0,ifsolvable,the aboveproblemis convex and admits a unique solution. Remark1.  The matrices  A F  ,  B F  ,  L F  ,  H  F  , defining theresidual generator can be derived by means of the fol-lowingprocedure.Letusdenote ˆ  X  ,  S  1 ,  S  2 ,  ˆ  A F  ,  ˆ  B F  , V  11 , ˆ  L F   a solution of (26):(1) compute V  22 , V  21  ( n F  × n F  )bysolvingthefollow-ing factorization problem S  1  = V  T  21 V  − 122  V  21 , (2) compute  A F  ,  B F  ,  L F  A F  = V  − T  21 ˆ  A F  V  − 121  V  22 ,  B F  = V  − T  21  ˆ  B F  ,  L F  = ˆ  L F  V  − 121  V  22 . 4. THRESHOLDS COMPUTATIONThe detection and isolation decision logic is based onthe adaptive residual thresholds evaluation proposedby (Frank   et al. , 1997). Consider first the detectionproblem and let the time-windowed rms-norm  J  r  ( t  ) =   r   rms , t   =   1 t  Z   t  0 r  T  ( τ ) r  ( τ ) d  τ ,  J  r  i ( t  ) =   r  i  rms , t   =   1 t  Z   t  0 r  T i  ( τ ) r  i ( τ ) d  τ be a convenient residual measure. Under fault-freeconditions,(7)becomes r  ( s ) = G rd  d  ( s )+ G ru u ( s )  andvia the Perseval’s Theorem one has that  r   rms , t  ,  f  = 0  =  r  d   + r  u  rms , t  ≤ G rd   ∞   d   rms +  G ru  ∞   u  rms , t  =  γ  d   ν + γ  u   u  rms , t  where  γ   f   is the solution of (10) and  ν  is a convenientupper-boundto the rms-norm of the worst disturbanceacting on the plant. As a consequence, the followingthreshold results  J  th ( t  )  : =  γ  d   ν + γ  u   u  rms , t   .  (27)Isolation thresholds can be derived in a similar way.Let us consider the  i -th component of residual vector r  ( t  ) . When the  i -th fault signal is equal to zero in  [ 0 , t  ] ,one has that  r  i  rms , t  ,  f i = 0  =  r  i , d   + m ∑  j = 1 ,  j  = i r  i ,  f  j  + r  i , u  rms , t  ,  f i = 0 ≤ G rd   ∞  d   rms + m ∑  j = 1 ,  j  = i  e T i  G rf  e  j  ∞   f   j  rms , t   + γ  u  u  rms , t  where  e i  is the canonical basis of   R m and  r  i , d   and r  i , u  denote the distinct amounts of   r  i  depending ondisturbances and, respectively, on the external input u ( t  ) .  By denoting with  ξ ij  : =  e T i  G rf   e  j  ∞  the  H  ∞ -norm of the map between the  j -th fault to the  i -thresidual, a convenient isolation threshold is given by  J  th , i ( t  )  : =  γ  d   ν + m ∑  j = 1 ,  j  = i ξ ij β  j  + γ  u   u  rms , t   (28)where  β  j  denotes an upper bound to the rms-norm of the  j − th  fault class.5. AN ISOLATION EXAMPLEThis exampleaims at illustrating the isolation capabil-ity of filters designed by the proposed method. To thisend, consider the following uncertain LTI system P 0   y 1 ( s ) =  k  1 s 2 + θ 1 s + θ 2 ( u ( s )+  f  1 ( s ))+  k  1 ( sT  1  + 1 )( s 2 + θ 1 s + θ 2 ) d  1 ( s )  y 2 ( s ) =  k  3 k  1 s 2 + θ 1 s + θ 2 ( u ( s )+  f  1 ( s ))+  k  4 sT  4  + 1  f  2 ( s )+  k  1 ( sT  1  + 1 )( s 2 + θ 1 s + θ 2 ) d  1 ( s )+  k  2 ( sT  2  + 1 ) sT  3  + 1 d  2 ( s ) where:  T  1  =  0 . 1,  T  2  =  10,  T  3  =  0 . 2,  T  4  =  1,  k  1  = 1,  k  2  =  0 . 2,  k  3  =  k  4  =  10 and the parameters  θ 1 , θ 2  belonging to the intervals 0 . 5  ≤  θ 1  ≤  1 . 2 ,  1  ≤ θ 2  ≤  1 . 5 The signals  d  1 ( t  )  and  d  2 ( t  )  are assumedto be unitary variance white noises. We are inter-ested here to isolate faults over the frequency inter-val  Ω  = [ 0 ,  1 ] rads . Here, we consider the followingoutput filter ( P ( s ) =  H  ( s ) P 0 ( s ) ) and tracking filter  H  ( s ) = diag ([  1 s + 1 ,  1 s + 1 ]) , W   f   ( s ) = diag ([  1 s 2 + s + 1 ,  1 s + 1 ]) . Note that a polytopic state-space realization of   P ( s ) consisting of four vertices results. By solving thequasi-convex problem (26) for  a  =  1 ,  b  =  1 , c  =  1 ,  itresults that the lowest value of   µ  ensuring the feasi-bility of (26) equals  µ  =  1 . 2 ,  and the correspondingoptimalvaluesof theobjectivefunctiontermsare γ  d   = 0 . 0230 , γ   f   = 0 . 7749and γ  u  = 1 . 3351 . Inthis example,thesimulationshavebeencarriedoutbytakingtheun-certain plant parameters constant at one of its vertices( θ 1  =  0 . 5,  θ 2  =  1). The detection and isolation capa-bility of the filter can be observed in Fig. 1, where thetime responses  r  i ( t  ) ,  i  =  1 , 2 for  u ( t  ) =  0 are reportedunder superimposed unitary variance white noises  d  1 and  d  2  and for the following faults occurrence  f  1 ( t  ) =  0  t   <  5s.1 5s. ≤ t   ≤ 50s.0 50s.  <  t   ≤ 100s. ,  f  2 ( t  ) =  0  t   ≤ 50s.1 50s.  <  t   ≤ 100s.
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