ACTIVE FAILURE DETECTION: AUXILIARY SIGNAL DESIGN AND ON-LINE DETECTION

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1 ACTIVE FAILURE DETECTION: AUXILIARY SIGNAL DESIGN AND ON-LINE DETECTION R. Nikoukhah,S.L.Campbell INRIA, Rocquencourt BP 5, 7853 Le Chesna Cedex, France. fax: +( Department of Mathematics, North Carolina State Universit, Raleigh, NC , USA. fax: slc Kewords: Failure detection, Auxiliar Signal, Model Identification. Abstract This paper describes an active approach for model identification and failure detection in the presence of quadraticall bounded uncertaint. After developing the underling geometr, two particular examples of this approach involving static and continuous models are described. Several examples are given. Introduction There are two general approaches to failure detection and isolation. One is a passive approach adetector monitors input and outputs of the sstem and decides whether, and if possible what kind of, a failure has occurred. A passive approach is used for continuous monitoring. The detector has no wa of acting upon the sstem. In contrast, in an active approach the sstem is acted upon on a periodic basis or at critical times using a test signal (auxiliar signal to exhibit abnormal behaviors. The decision of whether or not the sstem has failed should be made b the end of the test period. The active approach has the advantages that it can sometimes detect failures that are not detectable during the normal operation of the sstem. This is especiall important for evaluating subsstem status before the subsstem s performance becomes crucial. An example would be evaluating the brakes Figure : General sstem structure. while moving but before a truck has to stop. An active approach also often permits quicker detection of afailure. Of course, it is usuall important that the test signal be small in some sense in order to not interfere with normal operation. In [2, 3, 4, 5] we have begun the investigation of amulti-model active approach for model identification and failure detection. These earlier papers have focused on the theor and computation for various special cases. As this work has progressed a general framework encompassing all of these cases and several additional ones has begun to become evident. This paper will discuss this general framework for the first time. 2 Geometr of the approach The structure of the active failure detection method considered here is described in Figure. The sstem is acted on b both a control input u and the auxiliar signal v. The sstem is subject to noise ν and there is an output. The information available for

2 A (v A (v Figure 2: No guaranteed detection. Figure 3: Increasing the magnitude of v. the decision is u, v,. The auxiliar signal v is precomputed and known while u, become available in real time during the test period. Thus u, cannot be used in the design of the detection filter. The can onl be used in its application. Failure detection is to be robust with respect to the noise ν. Itisassumed that the noise satisfies a quadratic bound. Depending on the quadratic form we will see that this includes both norm bounded noise and model uncertaint. The approach presented here can be applied to more than two models [2] but we focus here on the two model case. It is assumed that there are two possible models, Model andmodel. For a given auxiliar signal v we define A i (v to be the set of input-output pairs (u, consistent with Model i. That is, those pairs which are realiable b that model. Thus A (v: set of normal input-outputs. A (v: set of input-outputs when failure occurs. Failure detection consists of observing the inputs and outputs and deciding to which set the belong. For guaranteed detection we need that A (v A (v =. ( In this paper we focus on linear models of different tpes. Suppose that a given v does not give guaranteed detection. Then we have the situation in Figure 2. As we appl increasing amounts of v, the A i (v move in the direction of the arrows as indicated in Figure 3. If the A i (v are bounded in the appropriate directions, then we reach a multiple of v the A i (v are tangent as in Figure 4, and after that the are disjoint. Guaranteed detection occurs Figure 4: Guaranteed detection with minimal v. when the sets A i (v are disjoint or tangent. In Figures 2-4 the A i (v staed a constant sie and shape. We will see later that for some choices of noise measure that the A i (v can also grow as v sincreases. The goal then is to have an auxiliar signal v of the smallest possible sie and an easil computed function µ = h(u, so that the value of µ will tell us whether failure has occurred. Sections 3 and 4 develop the needed theor for one model. Section 5 then considers auxiliar signal design when there are two models. 3 Static Case It is instructive to consider first linear sstems in the static case. If one then thinks of the matrices as being operators this motivates a number of later developments. Absence of uncertaint: If we have no uncertaint the model is just = Gu (2

3 Figure 5: Additive uncertaint. Figure 6: Realiable set for Example. G is m n and 4 m. Forthesstem (2 we can use a residual-based test: u µ = T ( Gu =H (3 H = T ( G I. Realiabilit of {u, } is equivalent to µ being ero. Additive uncertaint: Amore realistic case is when we have additive uncertaint = G u + G 2 ν (4 G and G 2 are matrices. We assume G 2 is full row rank. The noise ν is assumed to satisf (here is the usual Euclidian norm: ν 2 <d. (5 as illustrated in Figure 5. Then {u, } will be realiable if there exists a ν satisfing (5 and (4. In order to construct a residual-based test to determine the realiabilit of a given {u, }, weconsider the optimiation problem: subject to γ(u, =min ν ν 2 (6 = G u + G 2 ν. (7 The function γ(u, provides the realiabilit test. If γ(u, <d,then{u, } is realiable. If γ(u, d,then{u, } is not realiable. We must express γ directl in terms of {u, }. The solution to the optimiation problem (6, (7 is γ(u, = ( u T ( G T I Thus if we let µ = H H T H = ( G T I (G2 G T 2 ( G I u. u H satisfies (G2 G T 2 ( G I, we have γ(u, = µ 2.Then the realiabilit test becomes µ 2 <d.if this test does not hold, then we have a failure. Example Consider the following simple example = u + ν, ν 2 <. The realiable set {u, } is given b γ(u, =( u 2 < and is illustrated in Figure 6. Slope uncertaint: Of course, in man applications there is uncertaint in the models themselves. To include this tpe of uncertaint we consider uncertaint in the following form which follows the formulation of [6]: =(G + G 2 (I G 22 G 2 u (8

4 J = I, (7 I and {v, } satisf ( and (2 for given {u, }. These constraints can be expressed as: u ν G = G 2 (8 Figure 7: Model uncertaint. is an matrix whose maximum singular value σ satisfies Note that (8 is a perturbation of σ(. (9 = G u. ( This corresponds to modeling an uncertain gain as shown in Figure 7. The sstem of Figure 7 can be re-expressed as with = G u + G 2 ν ( = G 2 u + G 22 ν (2 ν =. (3 Note that combining (3 with (2 we get ν = (G 2 u + G 22 ν (4 Solving for ν and substituting into ( gives (8. Since (9 holds, we get from (3 that ν 2 2. Thus the characteriation of realiabilit is ν 2 2. (5 To get this in terms of just {u, } we again set up an optimiation problem. Let γ(u, =min ν, T ν J ν (6 G = G I G2, G G 2 2 =.(9 G 22 I Let A be a matrix of maximal full column rank such that AA =.Thatis,A is a maximal rank right annihilator. Then the solution of the above optimiation problem is as follows. Suppose G 2 JG T 2 >. (2 Then the solution to optimiation problem (6 is γ(u, = T u u G T (G 2 JG T 2 G. (2 Example 2 Suppose that (8 is =(+ u (22 with. From (8, it is eas to see that we can take G 22 =and G = G 2 = G 2 =.Thus G =, G 2 =. (23 H can be obtained using the J-spectral factoriation. It is eas to verif that (G 2 JG T 2 = J so that we can let H = G,that is, µ = ( u u The realiabilit test (2 then becomes. (24 γ(u, =µ T Jµ = 2 2u. (25 To see wh this inequalit is correct, simpl note that 2 2u = u 2 ( 2. This inequalit defines the set of realiable {u, } and is illustrated in Figure 8.

5 Figure 9: Realiable set for Example 3. Figure 8: Realiable set for Example 2. Note that in Figure 6 the set of s for a given u stas the same sie while the set varies in sie with u in Figure 8. A similar behavior occurs when designing auxiliar signals later in this paper. The A i (v are fixed sie with additive uncertaint and var in sie with model uncertaint. General uncertaint model: In order to capture both additive and slope uncertainties we consider the more general model u ν G = G 2 (26 with G 2 full row rank and with the noise constraint, T ν J ν <d (27 J = diag(i, I. Thisisalinear model with a quadratic criteria on the possible noises and perturbations. Example 3 Consider the following sstem u ν = with ( ν ν 2 T ( ( ν ν 2 ν 2 (28 <. (29 This sstem is closel related to Example 2. The realiable set is defined b γ(u, = 2 2u < and is illustrated in Figure 9. 4 Dnamical sstems Continuous and discrete sstems are of interest but space allows us onl to consider the continuous case. Our approach builds on the formulations in [6]. The static case just discussed leads to the following uncertain dnamical sstem modeled over [,T]: ẋ = Ax + Bu + Mν, (3 = Gx + Hu, (3 = Cx + Du + Nν (32 ν and are respectivel the noise input and noise output representing model uncertaint. N is full row rank. The constraints on the noises are: x( T P x( + s ν 2 2 dt<d, s [,T]. (33 which is the continuous analogue of (5. No weights on ν, are needed since the can be incorporated into the model coefficients. This formulation includes both additive uncertaint and model uncertaint.

6 For example, as shown in [6] the uncertain sstem (3, (3 can be used to model ẋ = (A + M Gx +(B + M Hu, = (C + N Gx +(D + N Hu σ( (t x( T P x( <d B formulating as an optimiation problem, we obtain the realiabilit test γ s (u, <d, for all s [,T] (34 γ s (u, = s µ T R µdt (35 and µ is the output of the following sstem ˆx = (A SR C PC T R Cˆx +(S + P C Du T R + Bu Hu µ = C Du ˆx Hu with ˆx( = x. P is the solution of the Riccati equation P = (A SR CP + P (A SR C T P C T R CP + Q SR S T,P( = P Q = MM T,S= ( MN T MJ T NN T NJ R = T C JN T, C =. I G We also assume that N T (I J T JN >, t [,T]. 5 Auxiliar signal design The discussion so far has onl addressed how to characterie realiabilit when there is one model present. We now consider the situation there are two models and an auxiliar signal v which is to be constructed to aid in identification. We again begin b considering the static case. 5. Static case Let v be the auxiliar input. Suppose that there are two models. B using ν to represent ν and,andb breaking up the input u into two inputs u and v, we get v Xi Y i Z i u = H i ν i (36 with the constraints, ν T i J i ν i <, (37 for i =,. Thecase i =corresponds to the unfailed sstem and i =,the failed sstem. Here J is asignature matrix (diagonal matrix with + and entries. Vectors u and are the on-line measured input and output as before. We wantatestsignalv that not onl permits perfect identification ( but also is small in some sense. If the minimiation is on the norm of v, our problem becomes that of finding a v of smallest norm for which there exist no solution to (36 and (37, for i =and simultaneousl. Example 4 Consider the following simple sstems: and 4v + = ν (38 ν 2 < (39 v + = ν (4 ν T ν <. (4 Figure shows the input-output sets and the minimal v that provides perfect identification. Non-existence of a solution to (36 and (37 is equivalent to: σ(v = σ(v (42 inf ν,ν,u, max(νt J ν,ν T J ν (43

7 So the optimiation problem (46 can be expressed as follows: subject to φ (v =inf ν νt J ν (48 Gv = Hν (49 Figure : Realiable sets for Example 4 and minimal v. subject to (36, i =,. Itcan be shown that σ(v in (43 can be expressed as: σ(v = max inf ν,ν,u, (νt J ν +( ν T J ν [,] subject to ( X Y Z v u H = X Y Z H ( ν ν. (44 The optimiation problem (43 can then now be expressed as follows: φ (v = σ(v =max φ (v (45 min ν,ν,u, (νt J ν +( ν T J ν (46 subject to (44. Since u and appear onl in the constraint (44, we can eliminate them from the optimiation problem b premultipling (44 with Y Z W W = (47 Y Z A is a maximal rank left annihilator of A. G = X W W, X H = ( H W W ν = ( ν ν,j =, H ( J ( J. Proper auxiliar signal and the separabilit index: An auxiliar signal v is proper if the two realiable sets A (v and A (v are disjoint. Thus an auxiliar signal v is proper if and onl if σ(v. Let V denote the set of proper auxiliar signals v. Suppose that v is measured in an inner product norm so there exists a positive definite matrix Q such that the norm of v is (v T Qv /2.Then γ = 2 min v V vt Qv, (5 is called the separabilit index for positive definite matrix Q. Thev realiing the min is called an optimal proper auxiliar signal. Using the fact that σ(v is quadratic in v,the problem of finding an optimal auxiliar signal can be formulated as follows: = max v σ(v v T Qv = = max v, [,] max φ (v v, [,] v T Qv v T V v v T Qv for some smmetric matrix V. here that H T J H >. Let = max v (5 We are assuming v T V v v T Qv. (52

8 Then clearl = max. But is the largest such that v T V v v T Qv =. (53 So if >, then the set of proper auxiliar signals is not empt and an optimal proper auxiliar signal is a solution of (53 satisfing v T Qv =. (54 If, then the set of proper auxiliar signals is empt. If >,then the separabilit index is given b γ =. (55 Example 5 For Example 4, φ (v is: inf ν,ν,ν ν ν ν T subject to 4 v = ν ν ( ν ν ν. ν We can show that φ (v is defined for all [, ] and is given b φ (v = (9 ( v 2. So σ(v = max [,] φ (v can easil be computed in this case b setting the derivative of φ (v with respect to to ero to get = 5/9. This gives σ(v = 6 9 v2 and thus v =3/4. Alternative solution: Instead of performing two optimiations, we can combine them to have a one step solution. Suppose that H T J H >. Then corresponds to the largest such that Computing det(hj HT GG T =. (56 = max (57 [a,b] a proper auxiliar signal exists if and onl if >. Then an optimal auxiliar signal is v = Kζ (58 K = G l HJ HT,G l is an left-inverse of G, ζ = αζ ζ is an non-ero vector satisfing ( HJ H T GG T ζ = (59 is a value of maximiing in (57, and α = 5.2 Continuous-time case ζ T K T QKζ. (6 B using ν to represent ν and, andb breaking up the input u into two inputs u and v, weobtain the following models: ẋ i = A i x i + B i v + B i u + M i ν i, (6 E i = C i x i + D i v + D i u + N i ν i (62 i =, correspond to normal and failed sstems. N i s are assumed to have full row rank. The noise constraints are S i (v, s = x i ( T P i x i( s + νi T J i ν i dt <, s [,T], (63 the J i s are signature matrices. Non-existence of a solution to (6, (62 and (63 is equivalent to: σ(v, s (64 σ(v, s = inf max(s ν,ν,u, (v, s, S (v, s, (65 x,x subject to (6-(62, i =,. Asinthe static case, we can reformulate (65 as: φ (v, s = σ(v, s = max [,] φ (v, s (66 inf ν,ν,u, x,x S (v, s+( S (v, s

9 subject to (6-(62, i =,. We consider here onl the case there is no u. That is, all the inputs are used as an auxiliar signal for failure detection. Let x ν A x =,ν=,a=, x ν A M M =,N= F M N F N, D = F D + F D,C = ( F C F C, B = ( B B P,P, = ( P,, J J =, ( J E F F =. E Then we can reformulate the problem (65 as: T φ (v, s =inf ν,x x(t P x( + ν T J νdt subject to ẋ = Ax + Bv + Mν = Cx + Dv + Nν. Construction of an optimal proper auxiliar signal: For the sake of simplicit of presentation, we start b considering that the optimalit criterion used for the auxiliar signal is just that of minimiing its L 2 norm. Thus the problem to solve is: min v v 2, subject to max v 2 = T [,] s [,T ] v 2 dt. φ (v, s (67 The reason we take the max over s is that the maximum value of φ (v, s does not alwas occur at s = T.Itdoes in most cases though. As we did in the static case, we reformulate the optimiation problem as follows,s = max v φ (v, s v 2. (68 Suppose for some [, ], that we have N T J N >, t [,T], and that the Riccati equation P = (A SR CP + P (A SR C T PC T R CP + Q SR S T, P( = P Q S S T = R M J N T M (69 N has a solution on [,T]. The values of Q, S, R depend on. Then,s is the smallest for which the Riccati equation P = (A S R CP + P (A S R CT PC T R CP + Q S R ST,P( = P ( Q S M B J M B = N D I N D S T R has a solution on [,s]. The values of Q,S,R depend on as well as. This result allows us to compute = max,s s which can be used to compute = max. This gives the separabilit index as γ =. and is used to compute the optimal auxiliar signal v. Construction of optimal auxiliar signal: Assume (an optimal and are computed. Let (x, ζ T

10 specifictpes of problems. In this approach the auxiliar signal design is done off line. It eas to implement the algorithms in Scilab [] or other CAD packages. The on-line detection test can be implemented efficientl using Kalman tpe filters or hperplane tests but space does not permit discussing this here. Acknowledgements Figure : Tpical minimum proper v. be an non-ero solution of the two-point boundarvalue sstem: d x Ω Ω = 2 x dt ζ Ω 2 Ω 22 ζ with boundar conditions: x( = P ζ( ζ(t =. Ω = Ω T 22 = A S R C Ω 2 = Q S R S T Ω 2 = C T R C This boundar value problem is not well-posed and has non trivial solutions (x,ζ.anoptimal auxiliar signal is (for simplicit suppose D = v = αb T ζ α is a constant such that v =/γ. A tpical auxiliar signal is given in Figure. The shape of the v can var depending on the models, the sie of the weight on the initial condition perturbation, and the length of the interval. Several more examples are in [2] and the other references. Research supported in part b the National Science Foundation under DMS-82, ECS-495 and INT References [] C. Bunks, J. P. Chancelier, F. Delebecque, C. Gome (Editor, M. Goursat, R. Nikoukhah, and S. Steer. Engineering and scientific computing with Scilab, Birkhauser, 999. [2] S. L. Campbell, K. Horton, R. Nikoukhah, F. Delebecque. Auxiliar signal design for rapid multimodel identification using optimiation, Automatica, toappear. [3] R. Nikoukhah. Guaranteed active failure detection and isolation for linear dnamical sstems, Automatica, 34, pp , (998. [4] R. Nikoukhah, S. L. Campbell, F. Delebecque. Detection signal design for failure detection: a robust approach, Int. J. Adaptive Control and Signal Processing, 4, pp , (2. [5] R. Nikoukhah, S. L. Campbell, K. Horton, F. Delebecque. Auxiliar signal design for robust multi-model identification, IEEE Trans. Automatic Control, 47, pp , (22. [6] I.R. Petersen, A.V. Savkin, Robust Kalman filtering for signals and sstems with large uncertainties, Birkhauser, Conclusion We have described a framework for multi-model identification and shown the form it takes for two