ROBUST REDUNDANCY RELATIONS FOR FAULT DIAGNOSIS IN NONLINEAR DESCRIPTOR SYSTEMS

Proceedings of te IASTED International Conference Modelling Identification and Control (AsiaMIC 23) April - 2 23 Puket Tailand ROBUST REDUNDANCY RELATIONS FOR FAULT DIAGNOSIS IN NONLINEAR DESCRIPTOR SYSTEMS Alexey Sumsky* Alexey Zirabok* * Far Eastern Federal University Russia e-mail: sumsky@mailprimoryeru zirabok@mailru ABSTRACT Te problem of fault diagnosis in nonlinear descriptor systems is studied witin te scope of analytical redundancy concept Solution of te problem assumes te cecking redundancy relations existing among measured system inputs and outputs A novel metod is proposed for constructing robust redundancy relations on te base of system models described by differential-algebraic equations wit polynomial functions wose coefficients may be unknown KEY WORDS Fault detection and isolation descriptor systems redundancy relations robustness model transformation differential geometric tools Introduction An increasing demand on reliability fault tolerance and safety for critical purpose control systems calls for te use of fault detection and isolation (FDI) Numerous metods ave been proposed for FDI witin te scope of te analytical redundancy concept [4] According to tis concept te FDI process includes residual generation as a result of mismatc between te system beaviour and its reference model beaviour followed by decision making trow evaluation of te residual Different tecniques ave been elaborated for solving residual generation problem see eg [] [4] [8] As soon as te modelling uncertainty and faults bot may act upon te residual te robustness problem arises Te essence of robust FDI is to make te residual insensitive to uncertainty and simultaneously sensitive to faults Tere exist tree main approaces to solve te robustness problem Te first of tem is based on adaptation principle Its use assumes tat te uncertainties represented as unknown constant (or slow varying) coefficients of te reference model are estimated for te reference model tuning [] Realization of tis approac may cause difficulties if te number of unknown coefficients subjected to estimation is considerablete second approac is closely allied to full decoupling problem wic solutions ave been considered in several works [5] [8] [7] All tese solutions put no demands on time beaviour of unknown coefficients but are caracterized by te existence conditions tat impose strict limitation on te acceptable number of above coefficients Te tird approac deals wit optimization principle Different criteria ave been proposed in te framework of multi-objective optimization problem aimed at acieving te trade-off between low sensitivity of te residual to uncertainty and its ig sensitivity to te faults [6] [9] [2] Increasing acceptable number of unknown coefficients tis approac rejects te idea to make te residual insensitive to uncertainty An interesting solution of te robust residual generation problem as been proposed in [5] for sensor FDI in stable systems wit linear structure Te distinguising feature of tis solution is tat it does not use te explicit model: all te matrices of te linear model are assumed to be unknown Tus it allows overcoming te limitation of te conventional full decoupling based metods Using an idea of tis solution te metod was proposed in [7] for constructing te redundancy relations for robust FDI in nonlinear uncertain systems described by differential equations wose rigt-and side is polynomial Tis metod involves nonlinear transformation of te initial system model into strict feedback form and ten its conversion to input-output description Te redundancy relations are immediately found from te last description In [9] and [2] tis metod was extended to time retarded and ybrid systems respectively Tere exists an important class of control systems known as descriptor or singular systems wic are described by so-called differential-algebraic equations Tis class includes suc engineering objects as power and cemical reactors robotic manipulators wit olonomic constraints electrical circuits etc Known metods related to FDI in descriptor systems [7] [] [3] [4] and [22] deal wit linear models in te main In [23] an observerbased metod as been proposed for fault estimation in nonlinear descriptor systems described by perfect models Also te useful solutions of te problem ave been considered in te recent papers [2] and [6] In present paper redundancy relations based metod [8] is extended to nonlinear descriptor systems wit parametric uncertainties Te rest of paper is organized as follows Section 2 considers te main aspects of robust DOI: 236/P23799-22 39

residual generator design Section 3 deals wit te problem of nonlinear descriptor system transformation into strict feedback form Solution of tis problem is of principle for robust redundancy relations design An example is given in Section 4 Section 5 concludes te paper 2 Residual Generator Design 2 Descriptor System Specification In present paper redundancy relations based metod is considered for residual generation in descriptor systems specified by differential-algebraic equations of te form E x& = f ( x( ) + g( x( ) u( + were dd ( x( ) wd + d f ( x( ) w f () y = ( x( ) n x X R is te vector of state p u U R is te input vector output vector q wd R and w f l y Y R is te s R are te vectors given for disturbances and faults representation respectively f and g d d d f are nonlinear vector and matrix functions of appropriate dimensions respectively E is te constant matrix of dimension r n rank E = r r n t is time It is assumed tat all te components of functions f g a polynomial form d d σ σ σ P ϑ = i i 2 n d f and ave i 2 i n i ( x) ϑ x x K x (3) wit constant coefficients ϑ i It is also assumed tat some or all tese coefficients are unknown In (3) j n is an appropriate component of te vector x j x and te top index σ j i denotes te degree of tis component in i t term of te polynomial For ealty system equality w f = olds Faults in te system result in No special assumptions w f are made about fault dynamics and disturbances; w f ( and w d ( are considered as unknown bounded functions of time 22 Computational Form for Redundancy Relations Let w d = w f = Introduce coordinate transformation x = α ( x( ) = φ i m (4) y ( y( ) (5) suc tat in transformed coordinates te system () takes strict feedback form wit = x& f ( ξ ) + g ( ξ ) u( i m (6) y = ( x ) (7) ξ () ( t ) = y( ( ( i ) ( ) i ξ ξ = ( i ) 2 i s x* Te components of () x x = x ( M ( m) x α and φ are assumed to be polynomials of te form P P ( ϑ ) ( x) and P P ( ϑ ) ( y) wile te components of f g and take polynomial form PP ( ϑ ) ( ξ ) and P P ( ϑ) ( x ) respectively Te coefficients of above polynomials ave a form of polynomials of ϑ denoted P (ϑ) It is of principle for furter consideration tat in contrast to te functions α f g and te function φ does not contain unknown coefficients For te system (6) and all i i m te corresponding iterated integral is a functional of system inputs and outputs defined at time interval [ t t ] by recurrence t () () () () () I( t t ) = ( f ( ξ ( θ)) + g ( ξ ( θ)) u( θ)) dθ + x t t Ii ( t t ) = ( f ( Ii ( θi )) + g ( Ii ( θi )) u( θi )) dθi + x t wit 2 i m (8) () ξ ( θi ) I ( θ ) I ( θ ) = i i i M Ii ( θi ) It follows immediately from (6) (8) tat 4

x ( t ) = Ii ( t t ) (9) involved instead of te matrix Z q( ) Taking (7) and (9) into account one obtains y ( t ) = ( I( t t ) K Im( t t )) () Because of polynomial form for all functions from (6) (7) equation () can be rewritten as follows y ( t ) = C( W ( t t ) () were te matrix C ( dependent on unknown () ( m) coefficients and unknown vectors x K x is time invariant at time interval [ t t ] wile W ( t t ) is te column vector of functionals dependent only on te system inputs and outputs measured at time interval [ t t ] Let t k = k t were t is some sampling period k = 2 K and introduce te matrix ( W( t t ) W( t t ) W( t t )) Vi ) = k i k i k i k i+ K k i k Denote by q ( minimal integer satisfying condition and let rankv q( ) = rank Vq( ( ) (3) ( y ( t ) y ( t ) y ( t )) Zq( ) = k q( k q( + K k From () equality Zq( ) = C( q( ) Vq( ) olds According to (3) te matrix V q( ) is singular Using nonzero vector from above v( t k ) ker V q( ( ) one obtains Z q( ( ) v( ) = (4) independently of te matrix C( t k q( ) and terefore independently of te values of unknown coefficients ϑ i Computational form for redundancy relations is given by r ) = Φ q( ) v( ) v( ) ker ( Vq( )) (5) 23 Decoupling and Fault Detection Conditions Consider firstly te condition of decoupling wit respect to disturbances Let w d w f = In tis case one can write y ( t ) = C( W ( t t ) + F( t t ) (7) were F ( t t ) is some functional dependent on te system inputs and outputs as well as on unknown values of ϑ x ( w d ( at time interval [ t t ] Due to v( t k ) ker V q( ( ) wit r ) = Ψq( ) v( ) (8) ( F( t t ) F( t t ) F( t t )) Ψi ) = k i k i k i k i+ K k i k Sufficient condition for decoupling follows immediately from (8) in te form Condition (9) olds if Ψq( ( t k ) = (9) ( α x) d ( x) = i m d Indeed if is true te functions α i m give perfect transformation of () to (6) (7) ie F ( t t ) = even if w d Reasoning along similar lines it is easy to sow tat ( α x) d f ( x) for some i i m is te necessary condition for detection of te fault caused by w f 24 Residual Generation Assuming tat analytical expressions for te vector of functionals W ( t t ) and te vector function φ are known consider an algoritm for te residual r( t k ) calculating Involving tis algoritm te residual is calculated at every instant of time t k wit te matrix ( φ( y( t )) φ( y( t )) φ( y( t ))) Φ q( ) = k q( k q( + K k (6) t k q( t (22) 4

according to basic relation (5) by andling system inputs and outputs measured at time interval [ q( ] Steps 2 and 8 of algoritm provide zero value of te residual at te initial stage of diagnosis wen (22) is violated Algoritm Take i = 2 If t k < i t go to step 8 3 Calculate te matrix V i ( t k ) according to 4 If i = take i = 2 and go step 2 5 If te matrix V i ( t k ) is of full column rank ten take i = i + and go to step 2 oterwise take q ( = i 6 Calculate nonzero vector v ) kervq( ) 7 Calculate te matrix Φ q( ) according to (6) and te residual according to (5) End 8 Take r ( t k ) = End 3 Model Transformation Nonlinear transformation of te model () into strict feedback form (6) (7) makes a basis for redundancy relations constructing Solution of te model transformation task is split into two stages At te first stage initial model is reduced to te model described by ordinary differential equations wile at te second stage strict feedback form is found 3 Preliminary Constructions Let w f = w d = For te matrix E introduce equivalent matrix of rank r E ( ) r = I r r according to factorization r ( n r) wit unit and zero matrices denoted E = Q E P (23) r I r r and r ( n r) respectively and non-singular square matrices Q and P of appropriate dimensions Using coordinate transformation x = P x( (24) equations () are reduced to te form f ) = Q f ( x) x= P g x ) = Q g( x) x x= P x ) = ( x) x= P x ( Under assumption about bounded time derivatives x& i ( for i > r one can write x& i = wu i r r + i n (27) were w u ( is unknown input vector of dimension n r 32 Transformation into Strict Feedback Form Taking into account (24)-(27) consider transformation of te system () to (6) (7) Let te functions α i m are given in te form α ( x) = ( ρ( x)) (28) were te functions ) i m specify coordinate transformation of te model (25)-(27) to (6) (7) and ρ ( x ) = P x In tis case solution of te model transformation task is reduced to te problem of te functions ( i ) ) i m and φ (y) determination Denote () ) = ) ) = ( i ) ( i ) ) )) i 2 Following [8] it easily may be sown tat te system (25)-(27) admits transformation into strict feedback form (6) (7) if and only if te functions i m and φ satisfy conditions f ( )) = ( x ) f ( x ) g ( )) = ( x ) g ) (29) ( ( ) ) r n r x = (3) I( n r) ( n r) () ( m ) ( ) K )) = φ( ( x )) (3) To take into account additional conditions it is useful to rewrite tem in te form wit x& i = fi ) + gi ) u( i r (25) y = ) (26) ( d x= P x x ) P d ( x) = (32) ( f x= P x x ) P d ( x) (33) 42

Equations (29) and (3) can be used for finding te functions f g i m and of te strict feedback form (6) (7) under known functions ( ) ( ) i i i m φ f g and Consider solvability conditions for (29) (3) and (32) For te functions and introduce distributions ( ) Λ = ker ( / x ) and Λ = ker ( i / x ) It follows from construction of te function tat Λ = Λ ( i ) Λ ( i ) 2 i m (34) Equations (29) (3) and (32) are solvable if for every i i m relations Λ span {[ µ λ] µ { } λ Λ } + f g K g p span{ d d } (35) L { f g g µ < ω λ >= µ K p } ω Λ λ Λ old and Λ is involutive distribution were r ( n r) d = d ( x ) P d d ( x) x= P x I( n r) ( n r) (36) In (35) (36) te symbols [] and L µ denote Lie brackets and Lie derivative of te function along vector field µ respectively Consider a way for finding distribution Λ under given distribution Λ Introduce a set of vector functions λ λ2 K λ λ : R R suc tat = z span { λ λ2 K λ z } Introduce also te series i n n Λ µ S = {} S = {() K( z)} S 2 = {( i j) i z i + j z} S d = {( 2 K z)} of dimension one two z and let S = U z S i Te i = distributions Λi i = 2 K are assigned to appropriate elements of te set S according to te following rule: Λ = { n } so on Λ i = span{ λi} i z Λ z + = span{ λ λ2} Λ z + 2 = span{ λ λ3} Algoritm 2 Take k = 2 Find Λ as minimal involutive distribution including Λ k + span {[ µ λ] µ { f g K g p } λ Λ } + span{ d d } 3 If Λ satisfies (36) ten end oterwise take k = k + and go to step 2 Tis algoritm is based on finite exaustion and results in distribution Λ Λ + span{[ µ λ] µ { f g K g p} λ Λ ) } + span{ d µ d (i } in limiting case Tis distribution satisfies (35) by construction and < ω λ >= ω Λ λ Λ ( i ) ; µ equality (36) follows immediately from te last relation According to Frobenius teorem [2] involutive distribution Λ is integrable ie omogeneous partial differential equation ( / x) λ = λ Λ (37) is solvable for te function Te way for step-by- step calculating te functions i m is given by te next algoritm Algoritm 3 () Take i = Λ ( ) = ker ( / x) 2 Find distribution Λ from Algoritm 2 3 Find te function from (37) 4 If i 2 and Λ ( i ) Λ ( i ) go to step 6 5 Take i = i + find distribution Λ from (34) and go to step 2 6 Take m = i End Remark In order to minimize te dimension of transformed model te function sould contain only te components wic are functionally independent of te () ( i ) function K components and oter 43

components of It requires deleting te redundant components of 2 i m under obtaining tem at step 3 of algoritm Consider now a way for te function φ determination Introduce te function δ (x) = φ ( ( x)) and let Λ δ = ker( δ/ x) be appropriate distribution Clearly Λ δ ker ( / x) On te oter and inclusion Λ I m δ Λ α i = is true Taking bot inclusions into account one obtains Λ I m δ ker ( / x) + Λ α (38) i = Te function δ is found by distribution and ten te function φ may be found 4 Illustrative Example Λ δ integrating Let te system under diagnosis is described by equations x& = ϑ 2 ϑ x ( x + x ) +ϑ ( x 2 ϑ ( x + x ) 4 3 3 6 2 2 4 3 + x x 4 ) + x3( ϑ 3x3( u( + wf + x2( + x3( wd ϑ5 x + x4 y = x2 + x3( All te coefficients ϑi i 6 are unknown Te task is to construct te redundancy relations for obtaining te residual wic is sensitive to w f and insensitive to w d Firstly to solve te task factorization (23) is found wit matrices Q = Er = P = Ten involving coordinate transformation (24) for w f = = one obtains equations of te w d form (25) - (27) respectively x & = ϑx3 x & = ϑ x + ϑ x u + ϑ x ( ) 2 ( 2 2 3 3 6 4 t 2 x& 3 = ϑ4x x3 + ϑ5u2 x& 4 = wu ( After tis te functions x y = x2 () 3 ( x ) =ϑ4x 3ϑ x3 ( x ) = x are obtained from Algoritms 2 3 According to (28) one () 3 as α ( x) =ϑ4( x + x4) 3ϑ x3 α ( x ) = x + x4 Te function δ ( x ) = φ( ( x)) = x + x4 is found from (38) tat gives φ ( y ) = y It is easy to ceck tat olds for i = 2 Terefore necessary condition for fault detection is valid Te model of strict feedback form is found from (29) and (3): () ( 5 2 t 3 () = ( ϑ4 y( x ( t y ( = x x& = 3ϑ ϑ u ( ) x& )) / 3 Using formulas (8) above model is converted to inputoutput description () wit 4 and 4 matrices () ( x x /3 ϑ4 /3 ϑ 5) C ϑ = 3 ( ) t W t t = t t ( θ2) θ2 t y d T t 2 2( θ ) θ θ2 θ u d d t t Te matrix W ( t t ) and te function φ are used for residual generation according to Algoritm Simulation results are given in Figs-3 It was taken under simulation ϑi = i 6 u ( = sin (t ) u 2( t ) = sin (2 t ) In Figure time beaviour of te fault ( w f at time interval 3s 4 s) is given wile Figure 2 illustrates time beaviour of te disturbance ( w d at time interval s 2 s) It is clear from Figure 3 tat te residual is equal to zero for ealty system even if te disturbance presents But for a case of te fault te residual differs from zero 44

References Fig Time beaviour of te fault w f ( Fig2 Time beaviour of te disturbance w d ( Fig3 Time beaviour of te residual r ( 5 Conclusion In tis paper a novel metod as been proposed for constructing te redundancy relations for robust FDI in nonlinear uncertain descriptor systems Te metod involves nonlinear transformation of te initial system model into strict feedback form Existence conditions for transformation ave been formulated and designing algoritms ave been developed Realization of te algoritms needs in fulfilling linear algebra operations functions differentiation and distributions integration All tese calculations may be supported by symbolic programming systems suc as Reduce Maple so on Acknowledgement Tis work was supported by Russian Foundation of Basic Researces grants -8-95a -8-33a [] M Blanke M Kinnaert J Lunze & M Staroswiecki Diagnosis and Fault Tolerant Control (Berlin Heidelberg: Springer-Verlag 23) [2] B Boulkrone & A Zemouce Robust fault diagnosis for a class of nonlinear descriptor systems Proc Int Conference on Control and Fault-Tolerant Systems SysTool Nice France 2 335-34 [3] J Cen & R Patton Robust Model-Based Fault Diagnosis for Dynamic Systems (Boston: Kluwer 999) [4] EY Cow & AS Willsky Analytical redundancy and te design of robust failure detection systems IEEE Trans Automat Contr 29(7) 984 63-64 [5] C De Persis & A Isidori A geometric approac to nonlinear fault detection and isolation IEEE Trans Automat Contr 46(6) 2 853-865 [6] C De Persis & A Isidori Fault detection filters Int J Robust nonlinear control 2 22 729-747 [7] GR Duan D Howe & RJ Patton Robust fault detection in descriptor linear systems via generalized unknown input observers International Journal of Systems Science 33(5) 22 pp 369 377 [8] PM Frank & X Ding Survey of robust residual generation and evaluation metods in observer-based fault detection systems Journal of Process Control 7(3) 997 43-424 [9] J Gertler & M M Kunwer Optimal residual decoupling for robust fault diagnosis Proc Int Conf Tooldiag 93 Toulouse France 993 436-452 [] H Hamdi M Rodrigues C Mecmece D Teilliol & N Braiek Fault detection and isolation in linear parameter-varying descriptor systems via proportional integral observer International Journal of Adaptive Control and Signal Processing 26(3) 22 224-24 [] R Isermann Process fault detection based on modeling and estimation metods: a survey Automatica 2(4) 984 387-44 [2] A Isidori Nonlinear control systems (London: Springer Verlag 989) [3] B Marx D Koenig & D Georges Robust fault diagnosis for linear descriptor systems using proportional integral observers Proc42 nd IEEE Conference on Decision and Control Maui Hawaii USA 23 457 462 [4] M Nyberg & E Frisk Residual generation for fault diagnosis of systems described by linear differential algebraic equations IEEE Transactions on Automatic Control 5 26 995 2 [5] KM Pekpe G Mourot & J Ragot Subspace metod for sensors fault detection and isolation application to grinding circuit monitoring Proc t IFAC Symposium on automation in mining mineral and metal processing Nancy France 24 45

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