ON DESIGNING LEAST ORDER RESIDUAL GENERATORS FOR FAULT DETECTION AND ISOLATION. Andras Varga

ON DESIGNING LEAST ORDER RESIDUAL GENERATORS FOR FAULT DETECTION AND ISOLATION Andras Varga German Aerospace Center, DLR-Oberpaenhoen Institute o Robotics and Mechatronics D-82234 Wessling, Germany Email: andras.varga@dlr.de Abstract: This paper addresses the problem o designing residual generators with least dynamical order to solve a class o ault detection and isolation problems. The main computational ingredients are discussed or the determination o least order residual generators achieving a desired ault-to-residual inluence matrix. An application o the proposed techniques to the pitch axis actuator ault monitoring or a Boeing 747 aircrat is presented. Keywords: Fault detection, minimum degree design, dynamic covers, aircrat ault monitoring 1. INTRODUCTION The design o ault detectors with least dynamical orders is an important aspect rom practical point o view. Although there are many approaches proposed in the literature to design ault detectors, as or example (Ding and Frank, 199; Gertler, 1998; Chen and Patton, 1999) and literature cited therein, only very ew o them address explicitly the design o least order detectors (Yuan et al., 1997; Frisk and Nyberg, 21; Varga, 23a). An apparently open problem is to determine a least order residual generator which achieves a desired ault-to-residual inluence matrix. This problem has as main application the solution o various ault isolation problems (Patton and Hou, 1998; Gertler, 2) and is traditionally addressed by designing a bank o single-output residual generation ilters. However, the reduction o the resulting total order o the bank o ilters was considered only in (Yuan et al., 1997) and at present it appears that no systematic approach exists to solve this problem. In this paper we present an enhanced approach, along the lines o dynamic covers based technique o Varga (23a), to design single-output residual generation ilters. A basic computational ingredient is a reliable numerical algorithm to compute least order rational nullspaces o rational matrices using state-space methods (Varga, 23a). The main computation in this algorithm is the orthogonal reduction o the system pencil matrix to a Kronecker-like orm, which allows to obtain, practically without any additional computation, a least order rational nullspace basis. This approach can be combined with coprime actorization techniques to determine stable rational bases which are candidate solutions to the ault detection problem. The existence conditions o the solution can be easily checked using the outcome o the nullspace computation algorithm. Least order ault detectors can be obtained by selecting an appropriate linear combination o the basis vectors by eliminating non-essential dynamics using dynamic covers based manipulations. We also discuss the least order design problem or a given

ault inluence speciication. For this purpose, a bank o least order residual ilters is determined by computing nullspace bases related to each row o the ault inluence matrix. The eectiveness o this approach in conjunction with the assignment o a uniorm dynamics to all detectors is illustrated via an application to the pitch axis actuator ault monitoring or a Boeing 747 1/2 aircrat. 2. DESIGN OF LEAST ORDER DETECTORS Consider the linear time-invariant system described by the input-output relations y(λ) = G u (λ)u(λ)+g d (λ)d(λ)+g (λ)(λ), (1) where y(λ), u(λ), (λ), and d(λ) are Laplace- or Z-transormed vectors o the p-dimensional system output vector y(t), m u -dimensional control input vector u(t), m -dimensional ault signal vector (t), and m d -dimensional disturbance vector d(t), respectively, and where G u (λ), G (λ) and G d (λ) are the transer-unction matrices (TFMs) rom the control inputs to outputs, ault signals to outputs, and disturbances to outputs, respectively. According to the system type, λ = s in the case o a continuous-time system or λ = z in the case o a discrete-time system. A residual generator or ault detection must ulill two basic requirements: (1) to generate zero residuals in the ault-ree case, or arbitrary control and disturbance inputs; (2) to generate nonzero residuals when any ault occurs in the system. These requirements can be made precise by looking or a linear residual generator (or detector) o least dynamical order having the general orm y(λ) r(λ) = R(λ) (2) u(λ) such that: (i) r(t) = when (t) = or all u(t) and d(t); and (ii) r(t) when i (t), or i = 1,..., m. Besides the requirement that the TFM o the detector R(λ) has least possible McMillan degree, it is also necessary, or physical realizability, that R(λ) is a proper and stable TFM. As detector, we can always choose R(λ) as a rational row vector. The ulillment o requirement (ii) ensures that aults produce non-zero residual responses. When designing ault detectors this requirement or ault detectability is usually replaced by the stronger request that persistent (constant) aults produce asymptotically persistent (constant) residuals. This requirement is known as strong ault detectability and has a special importance or practical applications. The requirements (i) and (ii) can be easily transcribed in equivalent algebraic conditions. The condition (i) is equivalent to where G(λ) = R(λ)G(λ) = (3) Gu (λ) G d (λ), (4) I mu while the detectability condition (ii) is equivalent to R (i) (λ), i = 1,..., m (5) where R (i) (λ) is the i-th column o G (λ) R (λ) := R(λ) (6) Enorcing the strong detectability o constant aults is equivalent to ensure inite non-zero DCgains or each column o R (λ), thus to ask < R (i) (λ s) <, i = 1,..., m (7) where λ s = or λ s = 1 depending on the system type (continuous or discrete). From (3) it appears that R(λ) is a let annihilator o G(λ), thus one possibility to determine R(λ) is to compute irst a minimal basis N l (λ) or the let nullspace o G(λ), and then to build a stable scalar output detector as R(λ) = h(λ)n l (λ), (8) representing a linear combination o the rows o N l (λ), such that conditions (5) or (7) are ulilled. The above expression o R(λ) represents a parametrization o all possible detectors and is the basis or the class o nullspace methods introduced in (Frisk and Nyberg, 21). While this work relies on using polynomial nullspace bases or N l (λ), an alternative approach relying on proper rational bases has been proposed by the author (Varga, 23a). The main advantage o this latter method is to rely exclusively on reliable numerical techniques based on state-space computations. In what ollows, we discuss more in details the two main design steps, namely, the determination o N l (λ) and the choice o appropriate h(λ) and we discuss reliable numerical methods to perorm these steps. 3. COMPUTATION OF LEAST ORDER RATIONAL NULLSPACE BASES The approach to compute least order detectors proposed in this paper is strongly connected to the details o the computations o the let nullspace basis N l (λ). Thereore, in this section we summarize the computational approach o (Varga, 23a) to determine least order rational nullspace bases

o a rational matrix which underlies the proposed procedure. This method is also equivalent to the pencil approach to design ault detectors proposed in (Patton and Hou, 1998). We assume the system (1) has the ollowing descriptor state space realization Eλx(t)=Ax(t) + B u u(t) + B d d(t) + B (t) y(t)=cx(t) + D u u(t) + D d d(t) + D (t) (9) with the n-dimensional state vector x(t) and where λx(t) = ẋ(t) or λx(t) = x(t + 1) depending on the type o the system, continuous or discrete, respectively. In general, the square matrix E can be singular, but we will assume that the linear pencil A λe is regular. For proper TFMs in (1), we can choose a standard state space realization where E = I. In general we can assume that the representation (9) is minimal, that is, the pair (A λe, C) is observable and the pair (A λe, [ B u B d B ]) is controllable. The corresponding TFMs o the model in (1) are G u (λ) = C(λE A) 1 B u + D u G d (λ) = C(λE A) 1 B d + D d G (λ) = C(λE A) 1 B + D The (p + m u ) (m u + m d ) TFM G(λ) in (4) can be realized in state space orm as A λe B u B d G(λ) = C D u D d I mu Assume that the rational matrix G d (λ) has rank r d min(p, m d ). It ollows that a let nullspace basis N l (λ) satisying N l (λ)g(λ) = (1) is a (p r d ) (p+m u ) rational matrix whose rows are rational vectors which orm a basis or the let nullspace o G(λ). The method described in (Varga, 23a) exploits the simple act that N l (λ) is a let nullspace basis o G(λ) i [ M l (λ) N l (λ) ] is a let nullspace basis o the system matrix A λe B u S(λ) = B d C D u D d I mu Thus to compute N l (λ) we can determine equivalently a let nullspace basis Y l (λ) or S(λ) and then N l (λ) simply results as N l (λ) = Y l (λ). I p+mu. N l (λ) can be computed (see bellow) by employing linear pencil reduction algorithms based on orthogonal transormations. Bases with special properties (e.g., stable) can be obtained by postprocessing N l (λ), using the act that i N l (λ) = M 1 l (λ)ñl(λ) is a stable let coprime actorization (LCF), then the numerator matrix Ñl(λ) can be equally employed as a let nullspace basis o G(λ). Let Q and Z be orthogonal matrices (or instance, determined by using the algorithms o (Beelen, 1987; Varga, 1996) such that the transormed pencil S(λ) := QS(λ)Z is in the Kronecker-like staircase orm A r,reg λe r,reg A r,l λe r,l S(λ) = A l λe l C l (11) where the descriptor pair (A l λe l, C l ) is observable, E l is non-singular, and A r,reg λe r,reg has ull row rank excepting possibly a inite set o values o λ (i.e, the invariant zeros o S(λ)). It ollows that we can choose the nullspace Ỹl(λ) o S(λ) in the orm Ỹ l (λ) = [ C l (λe l A l ) 1 I ]. Then the nullspace o G(λ) is [ N l (λ) = Ỹl(λ)Q and i we partition [ Q I p+mu I p+mu ] B r,l = B l D l in accordance with the column partitioning o Ỹ l (λ), we obtain N l (λ) = C l (λe l A l ) 1 B l + D l (12) Thus, (A l λe l, B l, C l, D l ), with E l nonsingular, is a descriptor system representation or N l (λ). Note that, to obtain this nullspace basis, we perormed exclusively orthogonal transormations on the system matrices. We can prove that all computed matrices are exact or a slightly perturbed original system. It ollows that the algorithm to compute the nullspace basis is numerically backward stable. Consider now the detailed [ structure ] o the ull column rank subpencil l Al λe. We can assume that this subpencil is in the ollowing observability staircase orm A l,l+1 A l,l λe l,l A l,1 λe l,1. A..... l 1,l... A1,1 λe 1,1 (13) C l A,1 where A i,i+1 IR µi µi+1, with µ l+1 =, are ull column rank upper triangular matrices, or i =,..., l. Note that this orm is automatically obtained by using the pencil reduction algorithms described in (Beelen, 1987; Varga, 1996). The let (or row) Kronecker indices o G(λ) result as ]

ollows: there are µ i 1 µ i Kronecker blocks o size i (i 1), or i = 1,..., l + 1. The row dimension o N l (λ) is given by the total number o Kronecker indices, thus l+1 (µ i 1 µ i ) = µ i=1 It ollows that µ := p r d. The order o the realization (12) is n l := l i=1 µ i and it was shown in (Varga, 23a) that this order is equal to the least possible one. 4. CHECKING FAULT DETECTABILITY We discuss shortly how conditions (5) or (7) can be eiciently checked or a candidate detector corresponding to the computed let nullspace basis N l (λ). For this, we use the outcome o the let nullspace algorithm to compute Q B D = B D, where the row partitioning o the right hand side corresponds to the column partitioning o Ỹl(λ). It easy to show that R (λ) = Ỹl(λ)Q B D = C l (λe l A l ) 1 B + D Since the pair (A l λe l, C l ) is observable, checking the condition (5) is equivalent to veriy that B(i) D (i), i = 1,..., m B (i) D (i) where and denote the i-th columns o B and D, respectively. To check the strong detectability condition (7), λ s must not be a zero o R (i) (λ). Direct evaluation o R (i) (λ s) may be not possible i N l (λ), and thus presumably also R (i) (λ), has poles in λ s. However, such poles can be easily perturbed by using a randomly generated output injection K, to work, instead o R (λ), with the perturbed R (λ) = C l (λe l A l KC l ) 1 ( B + K D ) + D 5. DETECTOR ORDER REDUCTION Let N l (λ) be a let proper nullspace basis o G(λ) (i.e., satisying N l (λ)g(λ) = ). In this section we will address the choice o a vector h(λ) such that the scalar output detector R(λ) := h(λ)n l (λ) has least McMillan degree. Assume N l (λ) has the standard state space realization Al λi N l (λ) = nl B l (14) where we replaced A l and B l by E 1 l A l and E 1 l B l, respectively. It is easy to show that instead N l (λ) we can equally use Al + KC Ñ l (λ) = l λi B l + KD l C l where K is an arbitrary output injection matrix. Moreover, all solutions o order at most n l can be generated in the orm Al + KC R(λ) = l λi nl B l + KD l (15) hc l hd l C l D l D l where h is chosen such that (5) (or (7)) is ulilled. Thus the least order design problem can be reormulated as ollows: or a given h determine the injection matrix K such that a maximum number o eigenvalues o A l + KC l becomes unobservable or the pair (A l + KC l, hc l ). In a dual setting in terms o transposed matrices, the least order solution o the above problem can be recast as a minimal order dynamic cover design problem. Consider the set J = {V : A T l V Im C T l + V, Im ( C T l h T ) V} and let J denote the set o subspaces J o least dimension. I V J then a matrix K T can be determined such that ( A T l + Cl T K T ) V + Im ( Cl T h T ) V Thus, determining a minimal dimension subspace V is equivalent to a Type I minimal order cover design problem, and a computational approach to solve it has been proposed in (Varga, 23b). The outcome o this method is, besides V, the matrix K which achieves a maximal order reduction by pole-zero cancellations. The details o this procedure are presented in (Varga, 23b). The general idea is to perorm a preliminary orthogonal similarity transormation on the system matrices in (14) by applying a special version o the controllability staircase orm algorithm (see or example (Varga, 1981)) to the pair ( ) [ Cl T ht Cl T ], A T l. With additional block permutations and block row/column transormations, we can bring the transormed system matrices in the ollowing special orm Â11 V 1 Â 12 B1 A l V =, V 1 B l =, Â 21 Â 22 B 2 Cl Ĉ11 Ĉ 12 V =, hc l Ĉ 22 where the pairs (Ĉ11, Â11) and (Ĉ22, Â22) are observable, and the submatrices Ĉ11 and Â21 have the particular structure

[ Â21 Ĉ 11 ] A21 = C 11 with C 11 having ull column rank. Thus, by taking K = V K 2 with K 2 satisying K 2 C 11 + A 21 =, we achieve the cancellation o the maximum number o unobservable eigenvalues. The resulting detector o least McMillan degree, obtained by deleting the unobservable part, is [ Â22 + K 2 Ĉ 12 λi R(λ) B ] 2 + K 2 D l = hd l Ĉ 22 The main aspect which remains to be discussed is the choice o matrix h. Generally with a randomly generated h one achieves a detector whose order corresponds to the maximum degree o a minimal polynomial basis. This result can be justiied by looking at the details o determining polynomial basis vectors. Using the Kronecker-like orm based method to determine a minimal polynomial basis N l (λ) proposed by Beelen (1987) (see also (Varga, 23a)), the let polynomial basis is determined using the resulting staircase orm o the observable pair (A l λe l, C l ) in (13). The let Kronecker indices µ i, i =, 1,..., l+1, (recall that µ l+1 = ) provide the complete inormation on the number o polynomial basis vectors o a given degree. The main result o (Beelen, 1987, Section 4.6) in this respect states that there are µ i 1 µ i polynomial vectors o degree i 1. The maximum degree is thus l and this is precisely the order which can be achieved applying the cover algorithm. In the case when no disturbance inputs are present, this is a well know result in designing unctional observers (Luenberger, 1966). Lower orders detectors can be obtained using particular choices o the row vector h. By exploiting the details o the staircase orm (13) o the pair (A l λe l, C l ) (see (Beelen, 1987)) we can show that i only the last j elements o h are nonzero (e.g., chosen randomly), then the corresponding least order solution corresponds to a combination o the irst j vectors o the polynomial basis. In this way, a systematic search can be perormed by generating successive choices o h with increasing number o nonzero elements and checking or the resulting residual generator the conditions (5) (or (7)). The resulting detectors have non-decreasing orders and thus the irst detector satisying these conditions represents a satisactory least order design. For more details see (Varga, 27). 6. RELATION TO OTHER APPROACHES We discuss shortly the relationships o the proposed method with other methods proposed in the literature. The method o (Yuan et al., 1997) achieves the order reduction o the detector by assigning the eigenstructure o the matrix A l + KC l with multiple eigenvalues in a maximum number o Jordan blocks. The dimensions o these blocks are constrained by the observability indices o the pair (A l, C l ) by the well-known undamental theorem o linear state variable eedback due to Rosenbrock (197, p.19). In single-output only one Jordan block can exist or each distinct eigenvalue, and thereore this technique can be used to cancel a maximum number o unobservable eigenvalues. However, the computational procedure sketched in (Yuan et al., 1997) relies on a delicate eigenstructure assignment and is not numerically reliable. Note that assigning multiple eigenvalues leads in general to an increased sensitivity o the eigenvalue problem. In a second computational step, the non-observable part o the resulting system must be eliminated. Perorming irst an eigenstructure assignment to produce unobservable multiple eigenvalues appears to be just a computational trick and thus an unnecessary computational detour. In contrast, the method based on dynamic covers is a direct technique and involves only reliable numerical computations. A second approach addressing the least order design has been proposed by Frisk and Nyberg (21) and relies on a polynomial approach. The nullspace N l (λ) is determined as a polynomial matrix and a least order detector is determined by building linear combinations o several basis vectors. Thus, the least order results as the largest degree o the involved polynomials. Although this method is not generally recommended as a computational method due to the intrinsic ill-conditioning o polynomial representations (see (Varga, 23a)) especially or higher order systems, still the approach provides insight into the structure o the problem and allows to guide the choice o least order candidates. Additional diiculties are expected when converting the polynomial basis into a rational basis and checking conditions (5) (or (7)). The last method we mention was proposed by the author in (Varga, 23a). The order reduction can be achieved by selecting two disjoint components N l,1 (λ) and N l,2 (λ) o the nullspace N l (λ) and computing appropriate Y (λ) such that N l,1 (λ) + Y (λ)n l,2 (λ) has least order. The underlying computational approach relies on employing Type II dynamic covers. For each design, the conditions (5) must be checked. The main problem with this approach is that there are cases when the least order can not be directly achieved. A typical case is when a linear combination o the ull nullspace basis N l (λ) is necessary.

7. SOLVING FAULT ISOLATION PROBLEMS The more advanced unctionality o ault isolation (i.e., exact location o aults) can be oten achieved by designing a bank o ault detectors (Gertler, 1998) or by direct design o ault isolation ilters (Varga, 24b). Designing detectors which are sensitive to some ault and insensitive to others can be reormulated as a standard ault detection problem, by ormally redeining the aults to be rejected in the residual as ictive disturbances. For a given detector R(λ) the corresponding ault inluence matrix S can be coded as S ij = 1 i R,ij (λ s ) S ij = 1 i R,ij (λ s ) = but R,ij (λ) S ij = i R,ij (λ) = I S ij = 1 then we say that ault j is strongly detected in residual i. I S ij = 1 then the ault j is only weekly detected in residual i. The ault j is not detected in residual i i S ij =. To solve a ault isolation problem, a given ault inluence matrix must be achieved by using a bank o single output detectors. Each single output detector achieves an inluence matrix representing a single row o the desired ault inluence matrix. For example, to achieve the complete isolation o maximum k simultaneous aults the choice S = I k is necessary. In many practical applications such a speciication can not be achieved due to the lack o suicient number o measurements. Provided we can assume that the aults occur one at time, a socalled week isolation o k aults is possible using a speciication matrix whose i-th row contains all ones excepting the element in column i which is zero. For example, or 3 aults S is chosen as 1 1 S = 1 1 1 1 I this ault inluence speciication can be achieved, then the occurrence o ault i can be detected i all residuals (excepting the i-th residual) are non-zero. More insight on how to speciy ault inluence matrices can be ound in (Gertler, 2). For an eicient implementation and operation o a residual generator (or a bank o such devices), it is generally desirable to keep the order o the ault detector(s) as low as possible. Thus, minimizing the total order o the detector achieving a given speciication is important or practical applications. Determining a detector o least order or a given speciication S is to the best o our knowledge still an open problem. The only paper addressing this problem is apparently (Yuan et al., 1997). Unortunately, no systematic approach can be derived on basis o the results o (Yuan et al., 1997) and the possible approaches are rather ad-hoc and problem speciic. In what ollows we suggest an alternative approach which usually leads to detectors o quite low orders. Although this approach does not provide a solution to the least order design problem, its main advantage lies in that all computational aspects can be addressed using numerically reliable techniques. The proposed approach is very simple and relies on the least order design techniques developed in this paper. For each row o S a single output detector o least order can be determined, leading to a global detector whose order is the sum o the orders o the component detectors. For practical reasons it is reasonable to assign approximately the same dynamics to all detectors, to ensure that the minimum detection times o all aults are almost the same. Assigning identical dynamics to all detectors has the ollowing interesting consequence: the resulting global dynamics is usually non-minimal and thus an order reduction can be achieved by eliminating the uncontrollable/unobservable eigenvalues. This simple trick works well in many cases leading to detectors o surprisingly low orders. We used this technique or the example o Yuan et al. (1997, Table 2), where a 18 9 ault inluence matrix S served as speciication. Each line o S can be realized by a detector o order 1 or 2 with eigenvalues { 1} or { 1, 2}. The total order o the resulting bank o detectors is 32, but ater perorming the minimal realization procedure a detector o total order 6 has been obtained which achieves the same speciication matrix. Recall that the detector computed in (Yuan et al., 1997) has order 14. 8. MONITORING PITCH AXIS ACTUATOR FAILURES FOR A BOEING 747 A linearized nominal longitudinal model o a Boeing 747 aircrat has the orm (9), where the state, input and output variables are deined as ollows: δq δv T AS x = δα δθ δh e δ eir δ eil δ eor δ eol u = δ ih δep R 1 δep R 2 δep R 3 δep R 4 pitch rate [rad/s] true airspeed [m/s] angle o attack [rad] pitch angle [rad] altitude [m], right inner elevator[rad] le t inner elevator[rad] right outer elevator[rad] le t outer elevator[rad] stabilizer trim angle[rad], thrust engine #1[rad] thrust engine #2[rad] thrust engine #3[rad] thrust engine #4[rad]

δα δ V T AS y = δθ δq δv z δh e angle o attack [rad] acceleration [m/s 2 ] pitch angle [rad] pitch rate [rad/s] vertical velocity [m/s] altitude [m] There is no the disturbance input d(t) in this model. The state space model matrices are given in Appendix A and correspond to a given set o parameter values (mass = 317, kg, center o gravity coordinates: X cg = 25%, Y cg =, Z cg = ) and a speciic light condition straightand-level light at altitude 6 m and speed o 92.6 m/s with a lap setting at 2 o and landing gear up. For more details on the employed model and or additional reerences see (Varga, 27). The ault inputs are deined as 1 right inner elevator ault[rad] 2 le t inner elevator ault[rad] = 3 right outer elevator ault[rad] 4 le t outer elevator ault[rad] 5 stabilizer ault[rad] and thus B = B u (:, 1 : 5) and D = D u (:, 1 : 5). The maximally achievable ault inluence structure is 1 1 1 1 1 1 1 1 1 1 1 S = 1 1 1 1 1 1 1 1 1 From the last three lines o S it can be observed that the isolation o aults grouped in three groups ( 1, 2 ), ( 3, 4 ) and 5 is achievable, although all aults are only weekly detectable. The simplest ault detection task is to determine i any actuator ault in the pitch axis occurred. This comes down to design a ault detector achieving the trivial inluence structure S 1 = [ 1 1 1 1 1 ] by using the lowest order dynamics. To design such a detector, the unction d rom the Fault Detection Toolbox described in (Varga, 26) has been used. Using the least order design option implemented using the proposed approach, a irst order residual generator can be used or this task. Provided we assume that the groups o aults ( 1, 2 ), ( 3, 4 ) and 5 do not simultaneously occur, the achievable speciication 1 1 1 S 2 = 1 1 1 1 1 1 1 can be used or week isolation using the ollowing decision logic: inner elevator ault occurred i r 1 =, r 2, and r 3 ; outer elevator ault occurred i r 1, r 2 =, and r 3 ; stabilizer ault occurred i r 1, r 2, and r 3 =. Using the least order design option, three irst order detectors can be obtained using the unction dbank leading to a detector o total order 3. Note that without the least order design option, a detector o total order 1 results, while using the standard observer based approach (Patton and Hou, 1998), a detector o total order 15 is to be expected. A more realistic setting is to add actuator dynamics to each input actuator-surace channel. The elevator dynamics can be approximated by transer unctions o the orm 37/(s+37), while or the stabilizer dynamics we can take.5/(s +.5). The resulting model has now order 1 and we can achieve the same inluence structure with a bank o three detectors o total order 6. Further enhancement o ault isolation is possible by employing direct measurements o surace positions. For example, with a single additional measurement o the stabilizer surace angle it is possible to achieve the speciication 1 1 S 3 = 1 1 1 and thus to isolate inner elevator, outer elevator and stabilizer aults. The above speciication can be achieved using a bank o three detectors o total order 5. Finally, a complete ault isolation (i.e., S 4 = I 5 ) can be achieved by adding a minimal number o three surace angle measurements rom the two let elevators and the stabilizer. The resulting bank o ive detectors has a total order o 6. 9. CONCLUSION We proposed a numerically sound computational approach to design least order ault detectors. The least order detector is obtained using a systematic search over single output detectors o increasing orders built as linear combinations o rational nullspace basis vectors. The order reduction is achieved using Type I minimal dynamic covers or which reliable computational techniques are available (Varga, 23b). The proposed method can be eective also in obtaining low order detectors which achieve given ault inluence matrices. A practical actuator ault monitoring example illustrates the potential o this approach to solve ault detection and isolation problems or a Boeing

747 aircrat. For the proposed new method sotware tools have been implemented in the recently developed Fault Detection Toolbox or MATLAB (Varga, 26). The proposed least order design method is particularly eective in determining least order residual generation ilter speciications achieving given ault-to-residual inluence matrices. These speciications can be then used or more realistic designs, where robustness against noisy inputs (e.g., wind gusts) and noisy measurements, as well as robustness against parametric uncertainties must achieved. The robustness aspects in conjunction with least order design will be addressed in a separate work. REFERENCES Beelen, Th.G.J. (1987). New Algorithms or Computing the Kronecker structure o a Pencil with Applications to Systems and Control Theory. Ph. D. Thesis. Eindhoven University o Technology. Chen, J. and R. J. Patton (1999). Robust Model- Based Fault Diagnosis or Dynamic Systems. Kluwer Academic Publishers, London. Ding, X. and P. M. Frank (199). Fault detection via actorization. Systems & Control Lett. 14, 431 436. Frisk, E. and M. Nyberg (21). A minimal polynomial basis solution to residual generation or ault diagnosis in linear systems. Automatica 37, 1417 1424. Gertler, J. (1998). Fault Detection and Diagnosis in Engineering Systems. Marcel Dekker. New York. Gertler, J. (2). Designing dynamic consistency relation or ault detection and isolation. Int. J. Control 73, 72 732. Luenberger, D. G. (1966). Observers or multivariable systems. IEEE Trans. Automat. Control 11, 19 197. Patton, R. J. and M. Hou (1998). Design o ault detection and isolation observers: a matrix pencil approach. Automatica 34(9), 1135 114. Rosenbrock, H. H. (197). State-Space and Multivariable Theory. Wiley, New York. Varga, A. (1981). Numerically stable algorithm or standard controllability orm determination. Electron. Lett. 17, 74 75. Varga, A. (1996). Computation o Kronecker-like orms o a system pencil: Applications, algorithms and sotware. Proc. CACSD 96 Symposium, Dearborn, MI. pp. 77 82. Varga, A. (1998). Computation o coprime actorizations o rational matrices. Lin. Alg. & Appl. 271, 83 115. Varga, A. (23a). On computing least order ault detectors using rational nullspace bases. Proc. o IFAC Symp. SAFEPROCESS 23, Washington D.C. Varga, A. (23b). Reliable algorithms or computing minimal dynamic covers. Proc. o CDC 23, Maui, Hawaii. Varga, A. (24a). Computation o least order solutions o linear rational equations. Proc. o MTNS 4, Leuven, Belgium. Varga, A. (24b). New computational approach or the design o ault detection and isolation ilters. In: Advances in Automatic Control (M. Voicu, Ed.). Vol. 754, The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers. pp. 367 381. Varga, A. (26). A ault detection toolbox or Matlab. Proc. o CACSD 6, Munich, Germany. Varga, A. (27). Design o least order residual generators or ault detection and isolation with application to monitoring actuator/surace aults or a Boeing 747 1/2 aircrat, Technical report, GARTEUR AG- 16 TP-27-1, DLR Oberpaenhoen, Inst. o Robotics and Mechatronics. Yuan, Z., G. C. Vansteenkiste and C. Y. Wen (1997). Improving the observer-based FDI design or eicient ault isolation. Int. J. Control 68(1), 197 218. Appendix A. LINEARIZED BOEING 747 MODEL.4861.317.5588 2.4 1 6.199 3.796 9.848 8.98 1 5 A = 1.53.21.5211 9.3 1 6 1 92.6 92.6.1455.1455.1494.1494 1.286.3122 B u =.71.71.74.74.676.13.35.35.13.1999.1999.1999.1999.4.4.4.4 1.199 3.796 9.848 8.98 1 5 C = 1 1 92.6 92.6 1.3122.1999.1999.1999.1999 D u =