New residual generation design for fault detection

Proceedings of the 7th World Congress The International Federation of Automatic Control Seoul Korea July 6-8 New residual generation design for fault detection B Larroque F Noureddine F Rotella Ecole Nationale d Ingénieurs de Tarbes (ENIT) 65 Tarbes France (Tel: +3356447; e-mails: (benoitlarroque faridnoureddine fredericrotella)@enitfr) Abstract: A new design procedure of a reduced order unknown input observer (UIO) is proposed to generate residuals for fault detection isolation (FDI) The originality of this work consists in the adopted approach for the procedure implementation Indeed the kernel of the actuator fault distribution matrix is generated thanks to generalized inverses The Kronecker productis used tosolve asylvesterequation which appears in the equations of an UIO Residuals generated by bank of observers allow on the one hand the detection isolation of every actuator fault and on the other hand the isolation between actuator faults and sensor faults INTRODUCTION Unknown input observers (UIOs) are usually used when limitation disturbance effects are targeted This approach has been carried out as early as the 8s when researchers in controlfield paid greatattentiontouios (Kudvaetal 98 Yang and Wilde 988) Model-based faultdiagnosis techniques mathematical description and definitions are detailed in Chen and Patton 999 Many works in the field of FDI are based on the design of a full order UIO (Chen and Patton 999 Demetriou 5) In fact there are more degrees of freedom available for the design of structured residuals However in mostcases a reduced orderobserveris suitable forafdiapproach (Koenigand Mammar ) Asimple reduced-order UIOdesign procedure isdescribed in this article The observer leads to structured residuals in order to detect and isolate actuator faults from sensor faults In this procedure we use generalized inverses to design the observer and it enables to introduce some arbitrary parameters which are very useful Indeed they offer some opportunitiesin the design Severalworksuse ageneralized inverse to treat the equation which ensures that the residualis insensitive tothe actuatorfaults The arbitrary matrix which appears by solving a linear system with generalized inverse is used when it is possible to assign the observer dynamics See for instance Kudva et al 98 Kurek 983 and Darouach et al 994 who define existence conditions by using this approach too In this article we also solve a linear system by finding the kernel of an application Another particular point of our approach is that we set the structure of the matrix which provides the dynamics of the observer An arbitrary matrix introduced in the resolution allows to ensure the compatibility of the linear system we finally have to solve This work was supported by the "Laboratoire Génie de Production (LGP)" in Tarbes (France) and by the "Réseau de Formation des Jeunes Chercheurs en Automatique (JCA)" to design the UIO Existence conditions are given by several tests in the presented procedure In order to describe our purpose notations are presented A linear time invariant system is considered where r actuator faults and m sensor faults can occur on the system The system model is described as ẋ(t) = Ax(t) + Bu(t) + K a f a () y(t) = Cx(t) + K s f s = I m x(t) + K s f s () where for every time t x(t) R n u(t) R r f a R r y(t) R m and f s R m are respectively the state known input actuatorfaultassimilated as unknown input output of the system and the sensor fault Matrices A R n n B R n r and C R m n are known constant matrices constitutingthe state space model Notice from () thata special structure for the sensormatrixc is assumed which can always be fulfilled under hypothesis of independent sensors K a R n r and K s R m m are respectively the distribution matrices of actuator and sensor faults It is usually admitted that faults are constant additive terms With a FDI approach and following Chen and Patton 999 it is possible to write () as ẋ(t) = Ax(t) + Bu(t) + y(t) = Cx(t) + r K ai f ai (3) m K si f si (4) = I m x(t) + m K s f s (5) which means that the distribution matrix K a can be split in r distribution vectors and that the distribution matrix K s can be split in m distribution vectors So f ai fits the i-th actuator fault and K ai is its distribution vector That is the same for f s which fits the i-th sensor fault and K s the associated distribution vector 978--34-789-/8/$ 8 IFAC 733 38/876-5-KR-374

7th IFAC World Congress (IFAC'8) Seoul Korea July 6-8 For every matrix X R m n let us denote X X n X = = X X X n X m X mn with X ij R and X i R m we can write vect(x) = X X m X n X mn T = X T X T Xn T T where X T stands for the transpose A generalized inverse of the (r q) matrix X is defined as a (q r) matrix denoted X {} (Ben-Israel and Greville 974) such that XX {} X = X For every matrix X a generalized inverse exists and the set of generalized inverses of the matrix X is given by X {} + Y X {} XY XX {} (6) where X {} is a particular generalized inverse for X and Y is an arbitrary (q r) matrix Moreover if Iρ X = ρq ρ L r ρq ρ where ρ = rank X and L is a given (r ρ) ρ matrix then we can choose X {} I = ρ ρr ρ q ρρ q ρr ρ Two equivalent conditions ensure the existence of a solution for M = NΓ { } { } M rank N = rank or M(I N q N {} N) = When these conditions are fulfilled a general solution for M = NΓ is given by Γ = MN {} + Z(I NN {} ) where Z is an arbitrary matrix The Kroneckerproduct defined in Brewer 978 C(ms nt) of two matrices A(m n) and B(s t) is defined by A B A n B C = A B = A m B A mn B RESIDUAL GENERATION The aim of this section is to provide a procedure to design areduced orderuio The fundamentalissue of this design is the generation of a residual which is insensitive to only one actuator fault (K ai f ai term in (3)) or insensitive to all actuator faults (K a f a term in ()) To simplify the notations we will consider in the following matrix K a However if a residual insensitive to only one fault has to be designed we have to replace K a by K ai Residual and UIO design Using the basic principles of functional observers design (Franck and Wünnenberg 989 Chen and Patton 999) residual r(t) can be estimated by r(t) = G z(t) + G y(t) (7) when r(t) R and z(t) R q are the residual and the observer state vector respectively The observer state vector z(t) is governed by ż(t) = N z(t) + Qu(t) + Ly(t) (8) with G R q G R m N R q q Q R q r L R q m and T R q n The estimation error is defined by e(t) = z(t) Tx(t) (9) Dimensions q and g and matrices namely G G N Q L and T have to be determined to obtain lim t e(t) = in the fault free case Thus the following conditions can be deduced (Tsui 4) N is Hurwitz () Q = TB () LC = TA NT () Byconsideringthese conditions the state estimationerror is governed by ė = Ne TK a f a + LK s f s When the system is subjected tofaultand by substituting y(t) defined in () and z(t) defined in (9) in (7) we get r(t) = G (e(t) + Tx(t)) + G Cx(t) = G e(t) + (G T + G C)x(t) + G K s f s Firstly r(t) must be independent of the state vector so G T + G C = (3) which constitutes a new constraint for the design of the dynamic system observer-residual as constraints () () and () Furthermore in steady state and considering f a constant r(t) verifies the following condition r = lim t r(t) = G N TK a f a (4) ( G N LK s G K s ) fs In order to make the residual r insensitive to fault f a matrix T should be orthogonal to K a It follows that TK a = K T a T T = T T Ker(K T a ) (5) Moreover to obtain the residual r defined in (4) sensitive to sensor faults f s the following necessary condition must be satisfied 7333

7th IFAC World Congress (IFAC'8) Seoul Korea July 6-8 G K s G N LK s (6) Indeed as seen in (?) if this condition is not fulfilled the residual will be insensitive to sensor faults and so as TK a = the residual will always be equal to zero To treat (7) it is interesting to set T T = V X where V R n n and X R n q Matrix V is such that its columns span Ker(Ka T ) So V verifies rank(v ) = dim(ker(k T a ) = n dim(im(k T a )) = n rank(k a ) = q (7) K a and n are known a priori (7) defines the observer order In fact K a reflects the effects of actuator faults on the system so we claim that K a = B To design V equation TK a = is solved Remark In the case where only one fault is considered (K a K ai ) observer order q is such that q = n As said previously T T = V X (8) with V = (I m (Ka T ) {} Ka T ) (9) and X is an arbitrary matrix such that X R n q By studying T T (8) in () we get A T V X V XN T = C T L T () Equation () is a Sylvester one By using the Kronecker product this equation can be written as (I q A T V ) (N V )vect(x) = vect(c T L T ) () To ensure condition () N is chosen diagonal N = q diag{n i } where each n i is nonzero and must have a negative real part In fact each n i represents an eigenvalue of N or a dynamic of the observer With this consideration on N () can be written as ( q diag {A T V } diag q ) {n i V } vect(x) = vect(c T L T ) () By using the notations established before (X) i denotes the i th column of X So () leads to solve q independent linear equations They are expressed for i = q as (A T V n i V )X i = (C T L T ) i (3) Due to the form of C described in () and bysubstituting it in (3) we obtain (A T V n i V )X i = (L T ) i (4) Matrices L and X are determined by solving these q systems In order to generate a residual with (7) G and G have to be determined By multiplying (3) with K a we find G CK a = K T a C T G T = G T Ker(K T a C T ) To solve equation K T a C T G T = with generalized inverses we get G = W T V T (5) with V = (I m (Ka T C T ) {} Ka T C T ) (6) where V R m m and W is an arbitrary matrix such that W R g m Matrix G is thus determined and (7) allows to write G as a solution of G T = V WC (7) Procedure The presented design can be summarized in the following procedure: Step : With (7) n rang(k a ) = q calculate the order of the observer (q > ) If q = go to step Step : Calculate matrix V with (9) V = (I m (K T a ) {} K T a ) Step 3 : Choose q observer dynamics denoted n i for i = q N is defined by n N = n q Step 4 : Solve q linear systems (4) for i = q (A T V n i V )X i = to find X and L Step 5 : Calculate matrix V with (6) Step 6 : With (5) V = (I m (K T a C T ) + K T a C T ) G = W T V T express G according to W Step 7 : Solve (7) G T = V WC to determine G and W with constraint (6) W T V T K s G N LK s and from W find out G Step 8 : Verify (6) G K s G N LK s If (6) is not fulfilled then go to step (L T ) i 7334

7th IFAC World Congress (IFAC'8) Seoul Korea July 6-8 Step 9 : Compute residual generator r(t) End Step : No UIO can be designed with this procedure End 3 EXAMPLE Letus considerthe followingfaultymachine papersystem presented in Kailath 98 This system is more precisely by (-) where A = K a = 5 7 and K s = B = 7 C = (8) (A T V n V ) (A T V n V ) X X X 3 X Each sub-system is defined by and X X 3 = (C T L T ) = (C T L T ) () 8 5 X X = X 3 () 8 5 X X = X 3 L L L L (9) (3) Both actuators and sensors can be subjected to actuator faults (f a and f a ) and sensor faults (f s and f s ) respectively A bank of observers allows to detect and locate f a or f a and detect f s and f s To design the bank of observers we have to: design an UIO which generates a residual insensitive to f a ; design an UIO which generates a residual insensitive to f a ; design an observer which generates a residual sensitive to all faults 3 Residual insensitive to f a We consider : K a = K a = and by applying the described procedure the residual is designed by: Step : As rankk a = then rankt = q = Step : Matrix V is determined with (8) V = Step 3 : As q = we have to choose arbitrary dynamics for the observer namely in and N can be written as N = Step 4 : As q = matrix X is defined by X X X = X X X 3 X 3 So (4) can be written as follows { (A T V n V )X = (C T L T ) (A T V n V )X = (C T L T ) and becomes: By solving (9) and (3) we get X = X = L = 5X L = X L = 5X L = X Therefore by choosing X = and X = we obtain L = 5 L = L = 5 L = Finally N X and L are written as N = X = X 3 X 3 5 L = 4 Step 5 : With (6) V is defined by V = Step 6 : By setting So G is W = W W G = W W Step 7 : From (7) and keeping in mind that T = (V X) T = we obtain G G = W W 7335

7th IFAC World Congress (IFAC'8) Seoul Korea July 6-8 G + G = W W which directly gives G = G + G Equation (6) becomes 5G 5G By setting G G = the previous condition is fulfilled Step 8 : Equation (6) is fulfilled due to G K s G N LK s = 5 Step 9 : Residual is then defined by r = lim r(t) = lim ( z(t) + 5 y(t)) t t and is insensitive to f a 3 Residual insensitive to f a We now consider that K a = K a = 7 Todesign this residual we proceed in the same way we presented before We choose observer dynamics in and and by choosing G = we get the following observer: N = Q = L = 97 47 T = 49 G = G = 49 In orderto complete the FDIproblem aresidualsensitive to all faults (f a f a f s and f s ) has to be designed This residual could be generated thanks to a Luenberger observer 33 Residual sensitive to all faults To generate this residual we design the following Luenberger observer (Luenberger 97) { ˆx(t) = (A KC) ˆx(t) + Bu(t) + Ky(t) ŷ(t) = Cˆx(t) The state estimation error is then x(t) = x(t) ˆx(t) x(t) = ẋ(t) ˆx(t) = (A KC) ˆx(t) + K a f a (t) KK s f s (t) In a fault free case in order that the state estimation error will approach zero asymptotically matrix A KC must be stable K will be determined as a consequencein order to arbitrarily choose eigenvalues of A KC pair (AC) has to be observable (Borne et al ) As this condition is fulfilled in this example we choose to set the eigenvalues which are the observer dynamics in and 3 Eventually K is such that K = 38 95 The residual generated with this observer is the output estimation error defined by ỹ(t) = y(t) ŷ(t) 34 Simulation results fa(t) fs(t) = Cx(t) Cˆx(t) + K s f s (t) = C x(t) + K s f s (t) 4 6 8 4 6 8 fs(t) fs(t) 4 6 8 4 6 8 4 r(t) r(t) r(t) 4 6 8 4 6 8 4 6 8 4 6 8 fa(t) fa(t) 4 6 8 4 6 8 Time(s) Fig Residuals r (t) r (t) and r(t) The simulation which results are given in fig is defined by: system () where constitutive matrices are given in (8) ; actuators faults f a and f a f a appears when t 6;8 and f a appears when t ;4 ; sensor faults f s and f s f s appears when t ; and f s appears when t 5;6 ; residual generator called r (t) insensitive to f a ; residual generator called r (t) insensitive to f a ; residual generator called r(t) sensitive to all faults (f a f a f s and f s ) Thus as shown in fig faults f a and f a can be detected and located In fact an analysis of the residual r(t) allows to detect the time when a fault (f a f a f s or f s ) occurs in the system Therefore after this detection it becomes necessary to analyze: 7336

7th IFAC World Congress (IFAC'8) Seoul Korea July 6-8 r (t) signal: if r (t) then a fault occurs on actuator (f a ) else the fault is either f a or f s or f s ; r (t) signal: if r (t) then a fault occurs on actuator (f a ) else the fault is either f a or f s or f s ; if r (t) and r (t) then a sensor fault occurs: f s or f s In order to be complete in the FDI problem sensor fault can be isolated as seen in Park et al 994 J Park G Rizzoni and W B Ribbens On the representation of sensor faults in fault detection filters Automatica 3():793 795 994 CC Tsui An overview of the applications and solutions of a fundamental matrix equation pair Journal of the Franklin Institute 34:465 475 4 F Yang and RW Wilde Observers for linear systems with unknown inputs IEEE Trans on Automatic Control 33(7):677 68 988 4 CONCLUSION From a FDI perspective a new residual design procedure based on the design reduced-order unknown input observer is presented in this work The starting point of the proposed procedure is the study of the kernel of the fault distribution vector This property allows us to determine at first the observer order Then thanks to Kronecker product we can propose a design procedure to generate a residual insensitive to one or several faults In order to complete the FDI problem we have briefly reminded the method togenerate aresidualsensitive to all faults Thus thanks to a basic detection logic we can distinguish the actuatorfaults amongthem and categorizeactuatorfaults and sensor faults REFERENCES A Ben-Israel and TNE Greville Generalized inverses : theory and applications John Wiley & Sons 974 P Borne G Dauphin-Tanguy JP Richard F Rotella and I Zambettakis Commande et optimisation des processus Editions Technip J Brewer Kronecker products and matrix calculus in system theory IEEE Trans Aut Control AC-5:77 78 978 J Chen and RJ Patton Robust model-based fault diagnosis for dynamic systems Kluwer academic 999 M Darouach M Zasadinsky and SJ Xu Full-order observers forlinearsystems with unknown inputs IEEE Trans Automat Contr 39:66 69 994 MA Demetriou Using unknown input observers for robust adaptive fault detection in vector second-order systems Mechanical SystemsandSignal Processing 9: 9 39 5 PM Franck and J Wünnenberg Fault Diagnosis in Dynamic Systems: Theory and Applications chapter 3: Robust fault diagnosis using unknown input observer schemes pages 47 98 Prentice Hall 989 T Kailath Linear Systems Prentice Hallinformationand system sciences series 98 D Koenig and S Mammar Design of a class of reduced orderunknown inputs nonlinearobserverforfaultdiagnosis In IEEE ACC American Control Conference Arlington Virginia (USA) P Kudva N Viswanadham and A Ramakrishna Observers for linear systems with unknown inputs IEEE Trans on Automatic Control 5():3 5 98 Jerzy E Kurek The state vector reconstruction for linear systems with unkown inputs IEEE Trans Automat Contr 8: 983 DG Luenberger An introduction to observers IEEE Trans Aut Control 6:596 6 97 7337