Detection Signal Design for Failure Detection and Isolation For Linear Dynamic System
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Detection signal design for failure detection and isolation for linear dynamic systems H´ector E. Rubio Scola. No 3935 Mai 2000. ISSN 0249-6399. ISRN INRIA/RR--3935--FR+ENG. ` THEME 4. apport de recherche.
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(4) Detection signal design for failure detection and isolation for linear dynamic systems Héctor E. Rubio Scola Thème 4 — Simulation et optimisation de systèmes complexes Projet Meta2 Rapport de recherche n˚3935 — Mai 2000 — 51 pages. Abstract: We present a new methodology to calculate a filter that permits failure detection and isolation in a dynamic system. Assuming that the normal and the failed behaviors of a process can be modeled by two linear systems subject to inequality bounded perturbations, a method for on line implementation of a detection signal, guaranteeing detection of failure, is presented. To make the failures detectable, the injection of the test signal that improves the detectability property of failure in the dynamic process is proposed which achieves detectability on line. All the operations needed for our method are implemented by using large linear optimization problems. Examples are shown. Key-words: failure detection, large scale programming, failure isolation, bounded perturbations, active detection.. (Résumé : tsvp). CIUNR - FCEIA - Universidad Nacional de Rosario, Argentina.. Unit´e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T´el´ephone : 01 39 63 55 11 - International : +33 1 39 63 55 11 T´el´ecopie : (33) 01 39 63 53 30 - International : +33 1 39 63 53 30.
(5) Conception de signal de détection et d’isolation de panne pour les systèmes dynamiques linéaires Résumé : Nous présentons une nouvelle méthodologie pour calculer un filtre qui permet la détection et l’isolation de pannes dans un système dynamique. En supposant que le comportement du système normal et du système en panne d’un processus peut être modélisé par deux systèmes linéaires avec des contraintes bornées, une méthode pour trouver le signal de detection est presentée. Cette méthode, sous une condition de détectabilité, peut garantir la détection et l’isolation des pannes. Pour rendre les pannes discernables, on propose l’injection d’un signal de test qui réalise la détectabilité en ligne. Toutes les opérations nécessaires pour notre méthode sont résolues en utilisant la programmation linéaire creuse à grande dimension. Plusieurs exemples sont traités. Mots-clé : détection de pannes, programmation grande échelle, isolement de pannes, perturbations bornées, détection.
(6) Detection signal design for failure detection and isolation. 3. Contents 1 Introduction. 4. 2 System model. 7. 3 Detection filter implementation. 11. 4 Examples 4.1 Example 1 . . 4.2 Example 2 . . 4.2.1 Case 1 4.2.2 Case 2 .. 21 21 26 28 30. 5 Test signal design 6 Examples 6.1 Example 3 6.2 Example 4. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 34 42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44. 7 Appendix. 46. 8 Conclusion. 49. RR n˚3935.
(7) 4. H.E. Rubio Scola. 1 Introduction The conception of failure detection systems entails the consideration of several points. Conceiving a system that will react rapidly when a failure occurs is what usually matters most interesting; yet, in systems with a high performance, one cannot generally tolerate important degradation in performance during normal system operation. That is why these two considerations are often conflicting. In other words, a system which is conceived to react quickly to sudden changes needs to be sensitive to some high frequency effects, which in turn will tend to increase the system’s sensitivity to noise, through the occurrence of false alarms signals by the failure detection system. The difference between these conception issues is best studied if we take a precise example in which we can evaluate the costs of the various differences. For instance, false alarms will be better tolerated in a highly redundant system configuration than in a system deprived of significant backup capacities. There are two ways of tackling the problem of failure detection and isolation. In first place, a passive approach: the detector monitors the inputs and the outputs of a system to know whether a failure has occurred and, if possible, what kind of failure. To achieve this, the measured input-output is compared with the normal behavior of the system. The passive approach is used to continuously monitor the system, particularly when the detector cannot act upon the system for material or security reasons. In the field of failure detection, most of the work is devoted to this type of approach. This approach of detecting changes in dynamical systems has been carefully studied (Willsky, 1976; Mironovrski, 1980; Isermann, 1984; Basseville and Benveniste, 1986; Clark 1986) in many field applications, to achieve failure detection in controlled systems or signal segmentation for recognition. Most of the time domain model-based methods use all the known or estimated model parameters to solve the two fundamental steps of change detection, that is to say residual generation and choice of the statistical decision function (Willsky, 1976). For instance, both filter innovations and parity checks involve all the model parameters, with possible inclusion of parameter uncertainties, and classical coefficients of probability or bayesian test proceed similarly (Basseville et all, 1987) The active approach to failure detection consists in acting upon the system on a periodic basis or at critical times using a test signal so as to show abnor-. INRIA.
(8) Detection signal design for failure detection and isolation. 5. mal behaviors which would otherwise remain undetected during normal operation (Nikoukhah,1998). The detector can act either by taking over the usual inputs of the system or through a special input channel, perhaps modifying the structure of the system. The structure of the failure detection method considered in this paper is depicted in Figure 1. v: test signal noise y: output Dection System. Filter. u: input. Failure decision. Figure 1: Active failure detection.. At some given time during normal operation of the system, a test signal is injected into the system for a finite period of time. This signal exposes the failure modes of the system which are detected by the detection filter. The conception of test signals has been an important question in system identification, but their use to detect failures has been introduced in (Zhang, 1989, Kerestecioˇglu, 1993, Uosaki et al., 1984). The test signals (called auxiliary inputs) in these works are regarded as linear inputs of stochastic models, and their objective is to optimize some statistical properties of the detector. This work is oriented to the detection and isolation of faults in an active approach. We present a methodology to calculate a detector (filter) that allow us to make the detection and isolation of faults. The selection of the test signal, as well as the calculation of the filter is computed off-line. The complexity of filter computation and design of the test signal is important when dealing with medium systems and long test signals. All the operations needed for our method are implemented by solving large linear optimization problems. The mathematical operations that must make the filter to decide the good or bad operation of the system reduce to a multiplication and a sum of each input and each output, which is very easy to implement in real time. Examples are given in Section 4.. RR n˚3935.
(9) 6. H.E. Rubio Scola. Structure of the failure detection and isolation A structure of the failure detection and isolation method is proposed in this work at some given time during normal operation of the system, a test signal is injected into the system for a finite period of time. This signal is supposed to expose the failure modes of the system which are then detected by the detection filter. The problem considered here has two parts: Part 1: Find a signal such that the possible input-output set of the normal system be disjoint to the possible input-output set of the failed system. Part 2: For a found test signal , given an input-output, recognize if this inputoutput is in the set of the normal system or in the set of the failed system. The outline of the paper is as follows. In Section 2, basic assumptions are presented and the model is introduced. The solution to Part 2 is characterized in Section 3. This solution is computed off-line over a finite horizon, see for example Nikoukhah (1999). The complexity of the off-line computation can be important when dealing with very large systems and long test signals and this solution cannot be used for real problems. In this paper, we propose computationally implementable solutions, using algorithms for sparse matrices. We obtain a reduced time of computation for very large systems and long test signals, and we can use the proposed method to real problems. The examples, one proposed by Nikoukhah (1998) and other proposed by Clark (1980) are given in Section 4. The solution to Part 1 is only given in the case where the test signal enters the system linearly, the solution is considered in Section 5 and in Section 6 two examples are presented.. INRIA.
(10) Detection signal design for failure detection and isolation. 7. 2 System model We consider systems which can be modeled as follows:.
(11) %&. ! #"$ '( )* & +, -. /"0. (1). for 1 325467686849-;:< , where = > 49@?BAC2549-D:< E is a test signal , and % are inputs and outputs which are measured on-line. , , '( , )F , ! , -. , and +, are matrices and vectors of appropriate dimensions which depend on and we use the notation G H I > # , etc. The (known) inputs and the (unknown) perturbations "KJ are both supposed LNM M to satisfy. LSR #"$ PO Q R (2) O Q M > ?IT(U0VXW9Y Lh where , M Z? R LNT\ R []V^W_Y , %` ? TSabV^W_Y , "0 Z?T\cV^W9Y , Q Z? TSdV^W9Y , Q e? Tgf9V^W_M Y , and R , are givenLmatrices M LNR of appropriate dimensions. Q . Q . . The vectors , and the matrices , LhM LhM LGM alsoLNR depend on , i.e. S > / , etc. No assumption is made on , and except that (2) are consistent. In the same way, for the failed system, we choose a similar model in the variables i , , % , "0i , and considered the system with failure as follows:. i5<
(12) j hi, ki5 hi5 & il I!il #"0i, %& '(i, ki, m)Fi, +,i, -.i, #"$il for n <2468676849-o:p , Perturbation "0i and input satisfy the inequality constraints: LNM M l i /. $ " ] q P. O Q LSR R il il O Q i,. (3). (4). M i l r. s ? ( T u U t X V 9 W Y 1. v ? \ T ] [ ^ V 9 W Y & % 1. v ? S T b a ^ V _ W Y 0 " , i w. v ? \ T x c x t X V 9 W Y Q , , , i, r? where R LhM LNR , TSdtxVXW9Y , Q i, e?pTyfztxVXW9Y and ilR , il , are given matrices ofLNappropriate diM LM R Q l i . Q , i . , i . , i . mensions. The vectors , , also deLGand M the LNR matrices pend on . No assumption is made on i , i except that (4) are consistent, for n <2468676849-o:p .. RR n˚3935.
(13) 8. H.E. Rubio Scola. The matrices and the vectors have not necessarily the same dimensions in both systems. The systems have in common only the input and the output %` . The basic assumption is that the normal mode, and the failed mode of the system can be modeled as in (1), (2), (3) and (4). But the system matrices can (and hopefully are) different for different modes. Note that unlike most other approaches to uncertainty modeling in dynamical systems for the purpose of failure detection, " is not a stochastic white noise sequence, but rather an arbitrary inequality bounded discrete sequence. A fundamental, and reasonable, assumption here is that, during the test period, the system is either in normal mode or failed mode; no transition occurs during the test period. ` x49 ? AC24_- :I E is given. The test We assume that the test signal signal is a sequence of vectors, as short as possible, such that the constraints on the operating system (1),(2) and the constraints on the failed system (3), (4) are inconsistent. In Section 5, we show how to construct a signal . We introduce the vectors 4Q4 , defined by. . . . . 4#il 4_%` m 4_ m49"0 4 "0i,
(14) if -1 4#il -r
(15) if M x Q R +l Q Q 4 M +, i,i, QQ R i,i, . for n <2468676849-o:p , and the following matrix. . . 0. 2 . .
(16) . . . ? A 249-o:p E n <-. defined by the following scheme:. 0 ... .. . -o:p
(17) . . . (5). INRIA.
(18) Detection signal design for failure detection and isolation. . . 9. . In the matrix , the submatrices and
(19) share the columns corresponding to the variables `Z< u and >ilZ
(20) , and. :Nh 2 :N'( 2 2 :Sil 2 :N'(i, for n <24 46664u-o:p
(21) .. . . 2. . Analogously we define. . . . . 2. : N :N)F :Nil :h)*il. :h! :h-e 2 2. 2. 2 2 :h!il j2 :h-ei5 2. 2 2. 2 .
(22) . . . . 0 ... . .. -o:p
(23) . where. ]. . 2. 2. 2. 2. 2. 2. 2. 2 2. . with a similar scheme:. 0. . . 2. 2. 2 2. LhM. L R2 N L R 2 N i,. 2 2 2. 2. (6). 2. 2. 2. 2. . 2. 2 LhM 2 il 2 2 2. . . for n <24 46664u-o:p
(24) . (1), (2), (3) and (4) can be rewritten as:. . O. Q. . (7). We write in matrix form the equations and inequations for the operating system (1), (2) and the failed system (3), (4) separately:. RR n˚3935.
(25) 10. H.E. Rubio Scola For the operating system :. . . We define and. . . . . . . Q x ]. . O. . Q . with a scheme similar to. :S# 2 :h'( j2. . (8). . and. 2. :h :! j2 :N)* :N-. 2 LhM 2 2 2 LNR 2 j2 2 2 2 2 2 2 2 2 2 Q M R 4 ] 6 +lx Q . , where. 2. 2 2. . For the failed system :. ii O Q ii We define analogously i and i , where 2 :Nhil j2 :Ni, : i5 2 , i ]. 2 :N'(il :N)*i5 :N-.il 2 2 2 2 2 LhM il 2 2 2 i, ] 2 2 2 LNR i, 2 2 2 2 il Q M i, R Q i, y 4 il ] 6 +,i, Q i, We define the following polyhedrons. . ]. . . verifies y4. . i. . (9). 2. 2. 2. . verifies y6. INRIA.
(26) Detection signal design for failure detection and isolation. 11. 3 Detection filter implementation Let ` 94 r? AC2549-D: E be a test signal such that the solution sets of systems x491?pA 249-=E be an observation of (8) and (9) have no intersection and let the state of the system. The detection problem consists now to decide whether this vector is compatible with (8) or with (9). Since inequalities (8) and (9) define two disjoint convex polyhedrons, our problem is reduced to know in which polyhedron the vector stays. If a hyperplane can be found that separates the polyhedrons, it is sufficient to find at which side of the hyperplane the vector lies. Our work is limited to find such an hyperplane. Its existence is guaranteed by the classical convexity theory. The following theorem shows that we can obtain directly constraints involving only inputs and outputs % for testing failure. Therefore we do not need to know the states variables > and >i of the systems (1) and (3).. . . . . Theorem 1 Let defined by. . . . and. . be two non empty disjoint convex cylindrical polyhedrons. 5 m4# 4_%4_m4 "549" . 5 m4# _4 %4_m4 "549" If we note k=
(27) separates and , then . . . * _% " OB . .
(28) _%G 9 " O 9% x"F " D+ an hyperplane which ( 2 , i.e. the hyperplane equation is
(29). 9%G + (10). Proof: Let us suppose that the hyperplane is defined by. k
(30). 9%G "Z " < + then m4 " where x4
(31) 4 4 are not all zero. Let 4# 4_% 4_ 4 " 49" ? . the point m4/ 4_% 4_ 4 "549" ? It follows that:. z*
(32). RR n˚3935. 9% . x" " +. (11).
(33) 12. H.E. Rubio Scola. . . . . . . Since
(34) _% 9 " is fixed, and g x" can take any value, because m4 " are arbitrary, the expression (11) can take values + , which contradicts the assumption that the hyperplane separates the two convex polyhedrons and . Thus ] 2 and B2 . Analogously
(35) and must be zero.. . . . Construction of separating hyperplane The following lemma and its corollary show that it is possible to convert the problem separating two polyhedrons into an equivalent, problem separating of a polyhedron from the origin of coordinates.. . . . . . Lemma 1 Let and be two non-empty convex polyhedrons. There exists an hyperplane separating and , if and only if the convex polyhedron : does 2 2 not contain , i.e.,if and only if there exists an hyperplane separating and the convex polyhedron : . Proof: See Rockafellar [18] p.98 (Theorem 11.4). . . . Corollary 1 Let and be two non-empty convex polyhedrons. Then, the hyper + + plane separates and if and only if the hyperplane separates 2 and the convex polyhedron : . i.e. a hyperplane separating and can be 2 : chosen parallel to a hyperplane separating and .. . . . . . . Thanks to corollary 1, the problem that we are going to solve is to find an hyperplane that separates a polyhedron from the origin of coordinates. We will solve it by linear programming and taking into account the geometric property of the convex polyhedrons given in theorem 1.. . The solution sets of systems (8) and (9) have no intersection then and ie? i , where Let and i such that ?. . . . . . . . i .. INRIA.
(36) Detection signal design for failure detection and isolation. ]. 13. 4#i, 4_%Kx 4_ 49"0 4 "0i,
(37) -1 4#i, -r
(38) . if ? A 249-o:p E <n if. 4#i, 4#% i, 4_i 4 "$ 49"0i,
(39) -r 4#il-r
(40) . if ? A 249- : E * if. and. i, ]. We note % 94_/ as operating system output-input pair, i.e. in the equation (1) (of the operating system) we change the pair %4_> by %, 4#> obtaining :. <
(41) j %K. h >x I #"$ '(x k I)*x > +,x -. /"0. (12). for n <2468676849-o:p . Analogously % i 4_iu as failed system output-input pair, i.e. in the equation (3) (of the failed system) we change the pair %>4_> by %,i 4_i
(42) obtaining :. i,Z<
(43) % i,. hil ki, Ihi, >i, il I!il #"0i, '(i, il )*i, >i, +,i, -ei5 #"$i,. (13). for n <2468676849-o:p . We also introduce the deviation between input-output pairs for normal and failed systems as follows :. , ]. . R . . We introduce the vector. . . . . defined by. . . . and the following matrices and defined by where diag 2 x4
(44) 46768684 -o:p
(45) y4 diag 2 4
(46) x46768674 -o:p
(47) . . RR n˚3935. . . . . % x : >x . . . . K% i, i, . . A 4 . . and. E 4. (14). 3A 4 . . E.
(48) 14. H.E. Rubio Scola. .
(49) . . . . . . Figure 2: and i properties. INRIA.
(50) Detection signal design for failure detection and isolation. . . ]. 2. . 2. 2. 2 hi, 2 2. :. . 4. y. . . LhM 2 2. LNR 2 :. 15. il. . 2 2 2. . Using now (12), (2), (13), (4) and (14) we obtain :. . . . We define. . . . Q O. . (15). . verifies . . 2. We assume that the solution sets of systems (8) and (9) have no intersection, i.e. system (7) has no solution. Then (15) has no solution of the form (15) becomes (7) if. . . . 2. because. .. We introduce a relaxation parameter . . 3A 2 4
(51) 2 4
(52) 2 x4666x4 -. :p
(53) x4
(54) -o:p u x4
(55) -o:p
(56) kE. (16). in the equations (15), we obtain :. : : . O. . O. . O. . Q. :]Q
(57) \ . . We define the following polyhedrons : . with following properties :. I2? for B2 .. RR n˚3935. . verifies . . (17).
(58) 16. H.E. Rubio Scola If. . .
(59) then. . . . The projection of H:. . .. i on the coordinate is for <2 .. . Chosing an adequate we can enlarge the polyhedron it. See figure 3.. . until <2 belongs to. . .
(60) . t. . hiperplane of contact. Figure 3: linear programming problem. INRIA.
(61) Detection signal design for failure detection and isolation. 17. Taking into account the polyhedron (17) we solve the following linear programming problem:.
(62) f W
(63) subject to. : : . . O . 4 4
(64)
(65) . . O. . . . (19) (20) (21) (22) (23). . . . (18). Q. :]Q
(66) \ 2 2 O. . . f . and a solution to (18). Then permuting the rows Let of (19), (20) and (21) (evaluated at 492 ), in such a way that equality constraints appear first. We introduce the permutation matrix x4
(67) 4 such that. . &A 4. . . E . dJ . d J . . 4. . . Q . . Q d QJ . . . . 4. . d J . . (24). We rewrite the equation (19) using B2 as follows :. dJ . . . . . d 2 J 2. Q d j QJ . . d J . . (25). Analogously, we define
(68) and for the equations (20) and (21):. .
(69) A 4. . E :. RR n˚3935. . d
(70) : J
(71) dJ . dJ
(72)
(73). . d
(74) J
(75). . . . A 4. . E>. . dJ . d
(76) 2 Q d
(77) d
(78) QJ
(79) J
(80) J
(81) 2 2 d d d J2 xJ J . : :. . 4. . . . . . d J. . (26) (27) (28).
(82) 18. H.E. Rubio Scola We write the active constraints of (19), (20) and (21). : d d
(83) d. . . 2. :. Q d :]Q d
(84) d. d 2 d
(85) 2 d. . . . d d
(86) d . . (29). The general equation for hyperplanes defined by these active constraints are :. : d d
(87) d. . . d d
(88) d. :. We rewrite (30) as:. where. : d dd
(89). . Q d :]Q d
(90) d. N. . . . 4. . . . . : d dd
(91). r 4. . d d
(92) d . . (30). d. (. (31). Q d :]Q d
(93) d. . . . 4. . d. . d d
(94) d. . As we said before, the variables m4#`i 4 "54 "$i do not appear in the hyperplane definition, because it is defined by m4_%4_m4 " and `i 4_%4_m4 "0iu , so the variables which participate in the two sets are %4_> (Theorem 1). The following theorem will show how transforming (31) to obtain an expression of the hyperplanes according to Theorem 1.. . . . ` . If and Lemma 2 Let be a full rank matrix whose columns span < h B2 , where satisfy the constraints & . d , then satisfies. .
(95) . . Proof: We have ? @ . (31) we get The equation (31) for e = 0 is. . . . 2 . Multiplying by. d 0 , then d. . . both sides of. 0 B2 , i.e. d. lF2e B2 . INRIA.
(96) Detection signal design for failure detection and isolation. . 19. Multiplying by both members of equation (31) defines a set of hyperplanes s2 (Lemma 2). Considering (14) and Corollary 1, %>4_> \ s+ defines a set of hyperplanes. It exists at least one that separates the two polyhedrons (8) and (9), which we note J %>4_> g + J . Simply we have left to determine +,J . To do that, we calculate the tangent hyperplanes to the two convex polyhedrons and i ..
(97). . . J .
(98). J
(99) . J
(100) then + Jg 46867684
(101) .. If J with . 6 6 6 6 . J . J %>4#` J . 6 6 6 6 J . J
(102) . %>4_>.
(103) J
(104) . or If. . J %4_> %4_>. (32). J
(105) then + Jg. J
(106) .
(107) J . Then the hyperplane equation will be. . J %>4#` H + J6.
(108) . Theorem 2 There exists ? A K4 E such that convex the polyhedrons and i .. . Proof: We remark that. . . if Indeed, let us suppose that. . If. . . ?. Let such that. Multiplying by. RR n˚3935. . then g. . . . ?. . then. ?. and. (33). J %>4_> . 2. <2. + J separates the two. (34). (35). O . g. . < (O . both sides of (36), we get. (36).
(109) 20. H.E. Rubio Scola. It follows that. . n H. . . . 2 i.e. (? Im . Then. B O< . . . > > such that (37). 2 problem (18), i.e. d is the minimal d is solution of the linear programming O 2 since (7) has no solution. It norm such that has solution. d d O< has no solution, i.e. (37) does not hold, i.e. (35) follows that the inequality . . . is false. We can conclude that (34) holds.. It follows that ? A 4 E such that Jk %Kx:N% i 4_x:Ni
(110) < 2 and Jk %Kx4_ . i.e.. J % i 4_iu % 4_ ?. . , % i 4_i
(111) ?. . i,. J J i
(112) ] Since and i are convex, Jk / and Jk i
(113) are convex, and we have Jk % 94_ J % iK4_i
(114) or J % 94_ J %Ki 4_i
(115)
(116) J %K i 4_Jiu % then If J % 4_ J % 9 _ 4 . O 4_/ + J
(117) J % i 4_ i
(118) PO J % i 4_i
(119) . %K94_/ ? % i 4#>i (? i J J% %94_ 4_
(120) J %K i 4_Jiu %> 4_then. %K 4# >g? > . If. . . + J.
(121) J % i 4_i
(122) % i 4_i
(123) ? i. It follows that the hyperplane defined by polyhedrons and i .. . J % i 4_i
(124) . J %>4_> B+ J , separates the two convex . INRIA.
(125) Detection signal design for failure detection and isolation. 21. 4 Examples 4.1 Example 1 The model considered here represents a gas chamber [15]. We will indicate with J the gas flow in , and with the gas flow out. The chamber is supposed to have a uniform pressure J . A sensor is placed inside the chamber for measuring this pressure. Based on this measurement, a central computer may decide to open the emergency valve which releases gas into the atmosphere. This happens in emergency situation, for example if J goes above a critical threshold. The problem here is that emergency situations are rare and the emergency valve which stays closed over long periods of time may not be functioning when needed. For this reason, on a periodic basis, the emergency valve should be tested. The central computer does that by periodically sending a signal to open the emergency valve and monitoring the pressure J to decide whether or not the valve has effectively been opened. The following assumptions are made. Every seconds, the central computer can either read the output of the pressure sensor, or send a signal to open (or keep open if already open) to the valve mechanism. During the test period the valve mechanism is either broken or functional; we exclude the case were the mechanism can brake down or be repaired in middle of the test. The gas temperature is constant during the test period and the gas can be modeled as an ideal gas. The flow J is J , and the output flow is given by bounded above and below, i.e., 2. Jm: , where (the pressure outsize the chamber) is supposed to be constant but unknown during the test period.. . The state vector is. ]. . .
(126) . and the output vector is. RR n˚3935. . . . . J . . / / . . pressure inside the chamber pressure outside the chamber. .
(127) 22. . %` . pressure in chamber
(128). AxE. 1. H.E. Rubio Scola if > 2 if > 2. > denotes the test signal wich can take the values 2 and . When > \ 2 , the pressure is measured and the valve is closed. When > ( , a signal to open the emergency valve (or to keep open in case it is already open) is sent but pressure is not measured. The sample period is h v2567 s. The resulting matrices and vectors of equations (1) and (2) are given by. . . . H \: . 2. . . ! . . 2. . . 2. 2. 2. 1. 3A E. ]. if > 2 if > 2. AC24 E if > ] B2 AxE 2 if > B. -. . LNM. if > <2 2 if > <. 2
(129) '(x ] AxE LNR. /. . We note with. :e 2. 2. 2. :Z. 2. :e. . . . x ] 3AxE. x . 2. 2. )Fx ] AE +lx ] R Q ] AE. 2. AE. 2 if ` B 2 if ` B. if > B2 2 if > B. the empty matrix. INRIA.
(130) Detection signal design for failure detection and isolation. K: 2 ( \: : # / # # :. M Q K: 2 // `> ( (: : # # / / :e The function is defined by: Q :. > / and 56 54 ( 4 r u2 .. if > . 23. 2. . if > 2. 2 (pressure cannot be negaThe constraints on initial conditions are
(131) 2 J [ (corresponds to the lowest steady state pressure J with the tive) and 2 emergency valve closed). This clearly corresponds to the situation where J( , > J. and can be obtained by noting that in steady state , which implies that. . J [. :. 2 H :
(132) 2 . .
(133) 2 . . . 2 . For the system with fault whose equations are (3)and (4), the matrices and vectors are given by. RR n˚3935.
(134) 24. H.E. Rubio Scola. ih. . 2 2. . !i, ] '(i, . . -.i, ] LNR. LNM. 2. . \:. 2 2. 2 #. . 2
(135) AxE. 2. if if. 2. A 254 E if > B 2 AE 2 if > B. . 2. 2. 2. :e 2. . il . i, y AxE. il . hi, ] AE. . :Z 2. :e. . 2. 2. 2. )*i, ] AE. . +,i, ]. R Q ,i ] AxE. 2. AE. if > ] B2 2 if > B. if > ] B2 2 if > B. . K: 2 S: : # # / # :. if > y <2 M Q i, ] K: 2 // >` H ( (: : / / # # 2 if > < :e The constraints on initial conditions if the valve has failed are : i 2 H: i
(136) 2 i
(137) 2 . . 2. INRIA.
(138) Detection signal design for failure detection and isolation. 25. The simple test signal ! jA 24 4 4 492$E , is such that the above constraints are mutually exclusive. Solving the linear programming problem (18) and applying the theorem 2 we obtain:. . . + :N26 %`2 > 56 K2 %`,. (38). The value of + can be determined by solving the linear programming problems:. + [ . s.t.(9) and. + [ JU . s.t. (8). . . and + [ J U. Then the hyperplane equation is:. . . . . . . ,:h256 K%& 2 > 56 2 %`,. . We obtain + [ 6 K2 + [ J U O +FO + [ can be used.. . ,:h256 K%& 2 > 56 2 %`,. . . . 6 5
(139) . So any + satisfying. :N26 %`2 > 56 K2 %`, 6 5. . . . (39). Solving (32), with the output % of a system that works correctly we obtain from the equation (38) the values + between : 6 5 and 6 5
(140) . For the abnormal system the values of + will be placed between 6 K2 and 56 .. RR n˚3935. .
(141) 26. H.E. Rubio Scola. 4.2 Example 2 In this example we report the automatic control system (autopilot) for a hydrofoil boat. A sketch of the basic boat, which employs submerged hydrofoils, is given in Figure 4. A detailed description of this boat and its autopilot appears in [4], [5] and [6]. . :yaw angle. v: lateral velocity Differential Flaps. : roll angle. Rudder. Figure 4: Boat with submerged hydrofoils. The model considered here is the hydrofoil boat in its nominal cruise condition of 45 knots speed and 6 foot foil depth. The state vector is.
(142) . . L . where / . . . L / , / y. . . roll rate, + roll angle, + yaw rate, + lateral velocity,. . . (40). / and > / is the component of the velocity of the. center of mass (with respect to the Earth) along the lateral axis. The yaw, roll, and sway motions of the boat are controlled by the forward rudder and the differentially. INRIA.
(143) Detection signal design for failure detection and isolation. 27. . . operated flaps at the rear ones. We assume for simplicity that we have only single flaps on each side with deflection l / for the port flap and / for the starboard flap. We further assume that the actuators for these flaps are exactly coordinated so that l / y : / . The control vector is:. . . . . . .
(144). . . . . . horizontal flag deflection, + rudder deflection, + . . (41). There are four sensors providing feedback signals and they are monitored. The four instrument signals constitute the instrument output vector. % . %*. . %%
(145) % . . . . . L. . . indicated port acceleration 4 indicated roll angle 49+ indicated yaw rate 49+ indicated lateral acceleration 4. .
(146)
(147). . . (42). The difference equations for the system model are:. < u %`. N . "0 ' ) & -1"0 LhM M #"$ PO Q . (43) (44). In (43) the terms "K and -r"0 represents the disturbance bounded by (44). The autopilot (control laws and actuators) generates the control input from the. four feedback signals and the input command c (helm command) in a conventional manner [5]. It is assumed that the autopilot and instrument failure detection will be realized with digital computers, so a discrete time model of autopilot system is used here. The difference equation for the autopilot model are:. . . <
(148) j where the matrices , , ' , ) , sample period is ] <26X2 s.. . RR n˚3935. . - L &M . Z%` .
(149). ,. ,. and Q. M. c . (45). are listed in the Appendix. The.
(150) 28. H.E. Rubio Scola. It is assumed that two sensors consisting of roll axis gyro and yaw rate gyro are used to measure the state of the system, and the initial state of the system is taken to be. H 2 ] 26. 26 26 26.
(151). 4.2.1 Case 1 We shall consider failed systems with can be modeled as follows:. Z
(152) j %&. N . I "$ 'G )nH -1"0 LhM M #"$ PO Q . A fault vectorE. (46) (47). The “fault vector” represents the effect of the instrument failures which are to be detected and identified. We choice a test signal the control vector , therefore in (46) if then > # ] > , +, > / + > and +li, > / y ) ` & A fault vectorE . Now “fault vector” was chosen to system with failure:. A fault vectorE>. 26. 26 26. 26.
(153) . The respective two candidate models are: Failed system model. ilZ
(154) j %&. Nil $ > # I "0il ' i5 +,i, ` # ` -r"$i G LhM M "$i5 PO Q. (48) (49). Operating system model. <
(155) %`. $ > / I "$ N 'G +,x > # -r"0 LhM M "$ PO Q. (50) (51). INRIA.
(156) Detection signal design for failure detection and isolation. The fault is detected in. . B26 2. 29. 6 , the simple test signal would be. . 26 2 6. . (52). So to decide whether or not a failure has occured, a possible test is. . + :N26 2 K % 2 %
(157) 2 26 K 2 0% 2 . (53). The value of + can be determined by solving the linear programming problems (32), it follows that: If If. :h26 2 2 K 26 2 . . 256X2 2 256 2 . O +nO O +nO. Then the hyperplane equation will be. The system works normally The system works with anomalies. . :h26 2
(158) K % x 2 %
(159) 2 26 K 2 0% 2 26 2 2 . . (54). The autopilot generates the control input (figure 5) from the four feedback. signals and the input command c . The figure 6 shows the dynamic response of. the boat (four output), curve A is the response to c deg. step input and the disturbance null <2 and no instruments faults. The curve B is same as A, but 2 , the curve C is for instruments faults and disturbance with disturbance ; B 2 . The random sequence was chosen to represent Gaussian white stationary noise with zero mean value and of such a intensity as to cause noticeable random. fluctuations in the state variable responses to the standard c K+ command.. . RR n˚3935. . .
(160) 30. H.E. Rubio Scola. u(t) 1.1. 0.9 u2(t) 0.7. 0.5. 0.3 u1(t) 0.1. -0.1 t -0.3 0. 1. 2. 3. 4. 5. 6. 7. 8. A B C. Figure 5: Control variables to a step of 17 deg. 4.2.2 Case 2 Failed system model. <
(161) j %`. 25l66 . where. . '(ih. N $ > / I "$ ' i + > # -1"0 ( LhM M "$i5 PO Q. 256 52 6. 268 6 26 26. 6 2 26 6 26. : 56 2 256 256 256. . . (55) (56). . The null line of '\i represents the effect of disconnect of the lateral indicated acceleration. 26 2 6 , the simple test signal would be The fault is detected in. . . . INRIA.
(162) Detection signal design for failure detection and isolation. 0.5. y1(t). 3.5. 31. y2(t). 3.1 0.3 2.7 2.3. 0.1. 1.9 -0.1 1.5 -0.3. 1.1 0.7. -0.5 t -0.7. t. -0.1. A 0 B C. 1.9. 0.3. 1. 2. 3. 4. 5. 6. 7. 8. A 0 B C. y4(t). 1.9. 1. 2. 3. 4. 5. 6. 7. 8. y3(t). 1.7 1.5. 1.5 1.3. 1.1. 1.1 0.7. 0.9 0.7. 0.3. 0.5 0.3. -0.1 t -0.5 A 0 B C. 0.1. t. -0.1 1. 2. 3. 4. 5. 6. 7. 8. A 0 B C. 1. 2. 3. 4. 5. 6. Figure 6: Dynamic responses of the boat to a step of 17 deg.. RR n˚3935. 7. 8.
(163) 32. H.E. Rubio Scola. . . 26s6 6 2 6 26s K6s 6 2 6. . (57). Then the hyperplane equation will be. 26 2 2 2
(164)
(165) % u 2 H: 26 . % u 26 5 0% u . : 67. . . (58). The autopilot generates the control input (figure 7) from the four feedback. K+ >6 . The figure 8 shows the dynamic signals and the input command c response of the boat % 4_%
(166) 4_% 4#% . The curves A are response with disturbance B2 and no instruments faults. the curves B are same as A, but with B 2 , the curves C are the response with disturbance no null and instrument faults. The random sequence was chosen to represent Gaussian white stationary noise with zero mean value and of such an intensity as to cause noticeable random. fluctuations in the state variable responses to the standard c K+ command.. . . . u(t) 1.2. 1.0. 0.8. u2(t). 0.6. 0.4. u1(t). 0.2. 0. -0.2 t -0.4 0. 1. 2. 3. 4. 5. 6. 7. 8. A B C. Figure 7: Control variables to a step of 17 deg.. INRIA.
(167) Detection signal design for failure detection and isolation. 0.5. y1(t). 3.5. 33. y2(t). 3.1 0.3 2.7 2.3. 0.1. 1.9 -0.1 1.5 -0.3. 1.1 0.7. -0.5 t -0.7. t. -0.1. A 0 B C. 2.7. 0.3. 1. 2. 3. 4. 5. 6. 7. 8. A 0 B C. y4(t). 1.5. 1. 2. 3. 4. 5. 6. 7. 8. y3(t). 1.3. 2.3. 1.1. 1.9. 0.9 1.5 0.7 1.1 0.5 0.7. 0.3. 0.3. 0.1 t. -0.1 A 0 B C. t -0.1. 1. 2. 3. 4. 5. 6. 7. 8. A 0 B C. 1. 2. 3. 4. 5. 6. Figure 8: Dynamic responses of the boat to a step of 17 deg.. RR n˚3935. 7. 8.
(168) 34. H.E. Rubio Scola. 5 Test signal design. . In this section, we show how can we find a detection horizon N and construct a test ` x49 ? A 249-o:I E , as short as possible, such that (1), (2) and (3), signal (4) are mutually exclusive, i.e. i . The solution to this problem is only given in the case where the test signal enters the system linearly. This problem can be considered to be the counter-part of the off-line auxiliary signal design problem of Zhang [27]. We show how a test signal can be designed for a special class of Model (1), (2), (3) and (4). We assume that the 4 do not depend matrices NJ , NJ , ' J , ) J , @J and -J , for on and that. . . > & > & +, i > i +,i > +,i . > / +, > / . +,. i5 > / +,i, > / j. . . (59). . (60). where 9 J x and + J x are matrices of appropriate dimensions and xJ and + J are vectors, for 4 . Then (1), (2), (3) and (4) can be rewritten as:. . where L. . . L. . . diag 2 x4
(169) x46868674- :
(170) y4 :h x :h+, +l Q ] :N i i :N+,i +,i for n <2468676849-o:p .. . . . The problem is then to find the classical convexity theory. Consider the polyhedron:. . c . Q O. . (61). . . such that (61) is not satisfied. To solve it, we use. . 4 5 verifies
(171) . INRIA.
(172) Detection signal design for failure detection and isolation. . 35. c is a convex polyhedron and can be expressed by means of inequality con-. straints in (this results from the fact that the projection of a convex polyhedron is a convex polyhedron). We rewrite (61) as. . L. . . 2 Q. . . . (62). . be a full rank matrix whose columns span Lemma 3 Let and satisfy (62) then and satisfy :
(173) . where. If. . . L. 2. 4. 4
(174) ] B2 then .
(175). . . . . . 4
(176) . If. . 2. (63). . 4. Q . 4. (61) has solution.. The problem that we are going to solve is to find a that does not satisfy (63). Let J be the -th row of matrix , we introduce the following linear programming problem :. . . J. (64). subject to. Let .
(177) . . . 2. 4 be a solution to (64) and we note J J * J. RR n˚3935. J. (65) , then.
(178) 36. H.E. Rubio Scola. . is a equation of the tangent hyperplane to the polyhedron c . Then we can chose such that. J. J. It is possible to consider many criteria for choosing a detection signal , among the ’s for which (61) has no solution : for example, we can choose a minimal norm . We will consider here the following constraints :. >. O. . . (66). for n <2468676849-o:p , which can be rewritten as :. O where. diag. 2 4.
(179) 46768684. . . -o:
(180) .. We define the polyhedron :. . d . We note that there exists no if. Using the convexity of. (67) . . . verifies K y6. satisfying (67) and not satisfying (61) if and only. d d , we have :. c 6. . Lemma 4 Let d be the set of vertices of d . There exists no not satisfying (61) if and only if. d. . c 6. (68). satisfying (67) and. (69). INRIA.
(181) Detection signal design for failure detection and isolation. . 37. . . Figure 9: Test signal design. RR n˚3935.
(182) 38. H.E. Rubio Scola. It follows that the test signal that we are looking for should be such that ? d and ? c . Our construction of the test signal is equivalent to find , a particular vertex in d , which does not belong to c . To test (69) is easier than to test (68) because d is a finite set and we can verify element by element, if there exits some element such that S? d and ? c . We can use this vertex like a test signal. Even though verifies the conditions that we are searching for our test signal, it is an extreme solution, therefore we will try to do better, choosing a point near ? d c , (without loss of generality we can assume the boundary of c . Let 32 ) and the intersection point of the segment with the boundary of that , where is the solution of the following problem: c then . . . . . . . . . subject to. . To find , over interval AC24_algorithm:. (70). L . . Q O. . (71). :1 E , as short as possible, we propose the following. Algorithm 1 Step 1 : Let N = 1 Step 2 : Find d Step 3 : If exits ? to step 1 Step 4 : Calculate. d such that ? . . c go to step 4, else let N = N + 1 and go. . according to (70), (71), *. . INRIA.
(183) Detection signal design for failure detection and isolation. 39. Remark 1 To calculate the set of vertices d , we exploit the property of the block diagonal matrix . We will find the polyhedron vertices defined for each block, then we do the corresponding permutations to find all the elements of d . Let. ?. d , if 2F?. . d , then *. ?. . d , 1? A 24 E. . . (72). subject to
(184) . . . . . . . . 2 2. Let 4 a solution to (72), (73), if and 1?A 24 E , then F not satisfy (63), i.e. ? d is such that ? c . The algorithm 1 can be rewritten as follows :. (73) does. Algorithm 2 Step 1 : Let N = 1 Step 2 : Find d Step 3 : If ? d , such that exist N = N + 1 and go to step 1 Step 4 :. , with ZO. 4 solution of (72), (73), goto set 4, else let. . Remark 2 The number of linear programming problem variables (72), (73), is lower than the number of variables in the problem (70), (71) : in (72), (73) the number of variables is equal to the number of inequality constraints in (71) plus one. There is some flexibility in the choice of the signal test and in some particular cases, it can be interesting to select specific ’s.. RR n˚3935.
(185) 40. H.E. Rubio Scola. Particular cases of the signal test Case 1 If in (67). . . y and y . . . for ? AC24_-!:. E , we define the polyhedron. . . O. (74). x4
(186) 46664 U be the set of vertices of polyhedron . Then the and let vertices of d are permutations of elements of . Among all the vertices of d we can restrict the selection of in the following vertices of d subsets of d . . . . . . is the set of such that. f J. > with . . ? AC249 E if ? 7 A
(187) x4u-o:p
(188) kE if. 25467686849-o:p and J4 f ?. (75). ..
(189) is the set of such that ` with 4 ? AC24_-. J f a . ? A 24 E if ? A7 <
(190) 4 kE if if ? 8A &< u x4
(191) -o:p
(192) E. . :I E and Jz4 f 4 a ?. (76). .. Case 2 Another type of signal test of interest can defined by :. >
(193) ] > .
(194). ? AC2549-o:p E. (77). O > O @. with ? A 249-o:p E and where ?. d.. INRIA.
(195) Detection signal design for failure detection and isolation. 41. We can see this problem as a way of aggregating one state variable in our system. Equations (1), (4) are transformed as :. . J e
(196) `<
(197) . . . SJ 9 J 2. %` . . . J >. ' J +J
(198). . . . J >. . SJ 2. . . 0. 2. . K. . @J 2. . "uJ. ) J -ZJ "
(199) Jk. 4 with . Given , a test signal constructed for the extended systems, the signal for the original systems is completely defined by (77) once > 2 (or any > ) is fixed. Then to design the signal test, we can set initial conditions or linear programming problem (70), (71) with additional bounds on some components of . It is possible to consider many criteria for choosing a detection signal , for ? instance that d minimizes the following function :. . .. RR n˚3935.
(200). W
(201) . . . (78).
(202) 42. H.E. Rubio Scola. 6 Examples 6.1 Example 3 The following results are taken from a random system having five state variables and two outputs. For choosing a detection signal we will consider here the following criteria :. : 2K2FO nO. . 2 2. The values of and are determined by Algorithm 2 and typical signal tests obtained by the method proposed in (75) which gives : 3A : 2 254 2 24 2 24 K2 24 2K20E and 26 5 . Then we choose for detection signal :. . B26 . ] A :. 24 K24 24 24 254 20E. Based on the algorithm described in the section 3, the failure detection test can be based on the hyperplane:. . . +. 2 6 2 2 0% x 2 H: 2 6 2 2 0%
(203) 2 26X2 K%
(204) H: 256X2 0%
(205)
(206) H: 256X2 K2K% 256X2 %
(207) 26X2 % 26 2 0%
(208) H: 26 0% x, &268 % x H: 256X2 K%
(209) The value of + is determined by solving the linear program (32) which gives +r u26
(210) and we obtain here the following test: For 6
(211) 2 O +nO 567 2
(212) the system works normally 7 6.
(213) 2. Z . O + O . 6 . . For the system works with anomalies. . . . The following figures show typical signal tests obtained and dynamic responses of system.The curve A are for the operating system and curve B for the failed system.. INRIA.
(214) Detection signal design for failure detection and isolation. v = signal test. v(k). 110. 43. 90 70 50 30 10 -10 -30 -50 -70. k. -90 0. 1. 2. 3. 4. 5. 6. Figure 10: Signal test. y1 = out put. y1(k). 0 -200 -400 -600 -800 -1000. k -1200 0. A1 B. 2. 3. 4. Figure 11: Output % . RR n˚3935. 5. 6.
(215) 44. H.E. Rubio Scola. y2 = out put. y2(k). 100 -100 -300 -500 -700 -900 -1100. k -1300 0. A1 B. 2. 3. 4. 5. 6. Figure 12: Output %
(216). 6.2 Example 4 This example show typical signal tests obtained by the method proposed in (77). The results are taken from a random system having four state variables and two outputs. Test signal:. . . AC24S 68 4 56 54\ 67 K4 l6 54\ 68 4y24\ 68 4y20E The hyperplane equation found by the algorithm is:. 56 K%
(217) H: 56 $ % x H: . . 56 K2K%
(218)
(219) H: 268 0%
(220) H: 256 K%
(221) 6 5 $%
(222) & l 6 K% : 5 6 0%
(223) 256 0% :N26 0%
(224) ] :e 56 5. 6X2 K% .
(225). . . . . . . .
(226) . . . The figures display the signal test and the output vector % 4_%
(227) as a function of time. The curve A are for the operating system and curve B for the failed system.. INRIA.
(228) Detection signal design for failure detection and isolation. 45. v = signal test. v(k) 2.4 2.0 1.6 1.2 0.8 0.4. k 0 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. Figure 13: Signal test. y1 = out put. y1(k). 36 32 28 24 20 16 12 8 4. k. 0 0. 1A B. 2. 3. 4. 5. 6. Figure 14: Output % . RR n˚3935. 7. 8. 9.
(229) 46. H.E. Rubio Scola. y2 = out put. y2(k). 37 33 29 25 21 17 13 9 5 1. k. -3 0. 1A B. 2. 3. 4. 5. 6. 7. 8. 9. Figure 15: Output %
(230). 7 Appendix LM. and Q The matrices , , ' , ) , , - , dynamics of both the plant in discrete time from.. . . B. . . . 26 5 :h256X2 2 26 2 26 :N26X25 26 2 2 u2 26 2 26 2 u2 268 :h268 26 2 2 :h26 2 2
(231) 2 :h256X2 26X2 26 2 2 26X25 26X25 26 256 26 26 2 2 2 256 26 26 256X2 26 26 256. . . :N26X2
(232) :N26X2K2 2
(233) 26 5 :N26X25
(234) 2. . M from (43) and (44) define the. 26 02 26X2K2
(235) 26X2
(236) 26 . . . . 2
(237) . 26 2 6 2 6 2 6 2 2 K2 . . 256 52 6 52 6 52 6. 26 2 6 2 6 2 6. 26 2 6 2 6 2 6. 26 2 6 2 6 2 6. . INRIA.
(238) Detection signal design for failure detection and isolation. 56 26. . ' . 268 6 26 26 . : 56 2 26 26 :Z 68 . 6 2 26 6 : 56. . . 26 2 6 2 . : l6 56X2 2 26 ) 2 26 :h256X2
(239)
(240) 6 26 26 26 26 26 26 2 2 2 26 26 26 26 26 26 26 26 2 2 25 -D 26 26 26 26 26 26 26 26 26 26 26 26 2 2 2 26 26 6 26 26 26 26 256 :Z 6 26 26 26 26 256 26 26 26 26 26 256 26 :h256 26 26 26 256 26 26 26 26 26 256 26 26 :h256 26 26 256 26 26 26 26 26 256 L M N 26 26 26 :h256 26 256. 26 26 26 26 6 256 26 26 26 26 :e 6 256 26 26 26 26 26 256 26 26 26 26 26 :N26 26 26 26 26 26 256 26 26 26 26 26 256 26 26 26 26 26 256 26 26 26 26 26 256 M Q A 54 4 54 4 54 4 54 4 54 4 54 4 l4 54. . . . . . The matrices , discrete-time form. 1. . RR n˚3935. . ,
(241). and. 47. 256 52 6 52 6X2 2K2 256. 256 52 6 52 6 52 6 52 6 52 6 52 6 52 6 52 6 52 6 52 6 52 6 52 6 :N26 256 256 54 0E. . 256 52 6 52 6 52 6 52 6 52 6 52 6 52 6 52 6 52 6 52 6 52 6 52 6 52 6 52 6 :N26 . . . from (45) define the dynamics of autopilot in. 26 2 6 2 6 52 6 2 6 26s 6 2 6 2 6 2 6 52 6 2 6s 6 26. .
(242) 48. 26. .
(243). H.E. Rubio Scola. 26 26 52 6 26 2 26X2K2 26 26X2K2 256 :Z 56 26X2 5 256 26 26 256 26 26 256 26 26 256 26 26 :N268 :N268 26 256 26 56X2 26 26 26 :Z 56 26 26 26 26 26 26 26 5 26 26 26 26 26 26 26 26 268 :h268 26 26 26 26 26. . . . . . . . 26. 26 2 6 2 6 2 6 26 26 268 . . 5. .
(244) . 26 2 6 2 6 2 6 2 6 26 26 2 . . 26 2 6 2 6 2 6 2 6 2 6 26. 26 2 6 2 6 2 6 2 6 2 6 2 6
(245). . . . . INRIA.
(246) Detection signal design for failure detection and isolation. 49. 8 Conclusion The problem of filtering approach for active detection in linear systems subject to inequality bounded perturbations has been considered. Under certain conditions, there exist test signals that can completely expose various failure modes of the system. A method to design the filter for detecting and isolating failures in systems excited by such test signals has been presented. The complexity of filter computation and design of the test signal can be important , all the operations needed for our method are implemented by solving large linear optimization problems. The method presented here can sometimes be applied to continuous-time linear dynamical systems with discrete-time measurements. This method does not have difficulties with very large systems because it works with sparse matrices and solving large linear optimization problem taking advantage of sparse matrix properties. The efficiency of the proposed approach has been shown by two worked out examples.. References [1] Basseville, M. and Benveniste, A., (eds) (1985) Detection of abrupt changes in signals and dynamical systems. Lecture notes in Control and Information Science, Vol 77, Sprinter Verlag. [2] Basseville, M. and all (1987) Detection and Diagnosis of Changes in the Eigenstructure of Nonstationary Multivariable System, Automatica, Vol 23, No 4, pp. 479-489. [3] Basseville, M. (1988) Detecting Changes in Signals and Systems - A Survey Automatica, Vol. 24 No 3, pp. 309-326. [4] Clark, R. N. and V.M. Walton (1975) Detecting instrument malfunctions in control systems. IEEE Trans. Aerospace Electron. Syst., AES-11, pp. 465473.. RR n˚3935.
(247) 50. H.E. Rubio Scola. [5] Clark, R. N. (1978) Instrument fault detection, IEEE Trans. Aerospace Electron. Syst., AES-14, pp. 456-465. [6] Clark, R. N. and W. Setzer (1980) Sensor fault detection in a system with random disturbances. IEEE Trans. Aerospace Electron. Syst., AES-16, pp. 468-473. [7] Dakin, R. J. (1965) A Tree Search Algorithm for MILP Problems, Comput. J., 8 pp. 250-255. [8] Gondzio, J. (1996) Multiple centrality corrections in a primal-dual method for lineal programming, Computational Optimization and Application, 6, pp. 137-156. [9] Isermann, R. (1984) Process fault detection based on modeling and estimation methods. A survey. Automatica, Vol 20, pp. 387-404. [10] Kerestecioˇglu, F. (1993) Change detection and input design in dynamical systems. Research Studies Press, Taunton, U.K. [11] Kerestecioˇglu, F. and Zarrop(1994) M. B. Input design for detection of abrupt changes in dynamical systems. Int. J. Control 59, no. 4, pp. 1063–1084. [12] Mehrotha, S. (1992) On the implementation of a primal-dual interior point method, SIAM J. Optimization 2., pp. 575-601. [13] Milanese, M. and Vicino, A. (1991) Optimal estimation theory for dynamic systems with set membership uncertainty: An overview. Automatica, Vol 27, No 6, pp. 997-1009 [14] Mironovki, L. A. (1979) Functional diagnosis of linear dynamic systems. Automn Remote Control, 40, pp. 120-128 [15] Nikoukhah, R. (1998) Guaranteed Active Failure Detection and Isolation for Linear Dynamical Systems. Automatica, Vol 34, No 11, pp. 1345-1358 [16] Orchard-Hays, W. (1968) Advanced Linear Programming. Computing Techniques, McGraw-Hill. INRIA.
(248) Detection signal design for failure detection and isolation. 51. [17] Piet-Lahanier, H. and Walter, E. (1990) Exact recursive characterization of feasible parameter sets in the linear case. Math. and Computers in Simulation, 32, pp. 495-504. [18] Rockafellar, R. T. (1972) Convex Analysis. Princeton, New Jersey. Princeton University Press. [19] Rubin, D. (1975) Vertex generation and cardinality constrained linear programs. Operations Research, 23(3), PP. 555-565. [20] Rubio Scola, H. (1997) Implementation of Lipsol in Scilab Rapport INRIA No 0215. [21] Rubio Scola, H. (1999) Solving large linear optimization problems with Scilab: application to multicommodity problems. Rapport Technique No 0227, INRIA. [22] Thie, P. R. (1988) An Introduction to Linear Programming and Game Theory, second edition, John Wiley and Sons. [23] Uosaki, K., Tanaka, I. and Sugiyama, H. (1984) Optimal input design for autoregressive model discrimination with constrained output variance. IEEE Trans. Auto. Contr. , vol. AC-29, no 4, pp. 384-350. [24] Watanabe, K. and M. Iwasaki (1983) A fast computational approach in optimal distributed-parameter state estimation. Trans. ASME J. Dynam. Syst. Meas. Control 105, 1-10. [25] Wilde, E. (1993) A library for doing polyhedral operations, IRSA Report, no 785. [26] Willsky, A. S. (1976) A survey of design methods for failure detection in dynamics systems. Automatica, Vol 12, No 4, pp. 601-611 [27] Zhang, X. J. (1989) Auxiliary signal design in fault detection and diagnosis. Spring-Verlag, Heidelberg. [28] Zhang, Y. (1995) User’s Guide to LIPSOL, Department of Mathematics and Statistics. University of Maryland Baltimore County. USA.. RR n˚3935.
(249) Unit´e de recherche INRIA Lorraine, Technopˆole de Nancy-Brabois, Campus scientifique, ` NANCY 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rhˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN Unit´e de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex Unit´e de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex. ´ Editeur INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) http://www.inria.fr ISSN 0249-6399.
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