active failure detection: auxiliary signal design and

ACTIVE FAILURE DETECTION: AUXILIARY SIGNAL DESIGN AND ON-LINE DETECTION R. Nikoukhah∗ , S. L. Campbell† ∗

†

INRIA, Rocquencourt BP 105, 78153 Le Chesnay Cedex, France. fax: +(33) 1-39-63-57-86 e-mail: [email protected]

Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA. fax: 1-919-515-3798 e-mail: [email protected] http://www.math.ncsu.edu/˜slc

Keywords: Failure detection, Auxiliary Signal, Model Identification.

Abstract This paper describes an active approach for model identification and failure detection in the presence of Figure 1: General system structure. quadratically bounded uncertainty. After developing the underlying geometry, two particular examples of this approach involving static and continuous models are described. Several examples are given. while moving but before a truck has to stop. An active approach also often permits quicker detection of a failure. Of course, it is usually important that the 1 Introduction test signal be small in some sense in order to not inThere are two general approaches to failure detec- terfere with normal operation. tion and isolation. One is a passive approach where a detector monitors input and outputs of the system In [2, 3, 4, 5] we have begun the investigation of and decides whether, and if possible what kind of, a a multi-model active approach for model identificafailure has occurred. A passive approach is used for tion and failure detection. These earlier papers have continuous monitoring. The detector has no way of focused on the theory and computation for various special cases. As this work has progressed a general acting upon the system. framework encompassing all of these cases and sevIn contrast, in an active approach the system is acted eral additional ones has begun to become evident. upon on a periodic basis or at critical times using a This paper will discuss this general framework for test signal (auxiliary signal) to exhibit abnormal be- the first time. haviors. The decision of whether or not the system has failed should be made by the end of the test period. The active approach has the advantages that it 2 Geometry of the approach can sometimes detect failures that are not detectable The structure of the active failure detection method during the normal operation of the system. This is considered here is described in Figure 1. The system especially important for evaluating subsystem status is acted on by both a control input u and the auxilbefore the subsystem’s performance becomes cru- iary signal v. The system is subject to noise ν and cial. An example would be evaluating the brakes there is an output y. The information available for

A (v) 0

A1(v)

Figure 3: Increasing the magnitude of v.

Figure 2: No guaranteed detection.

the decision is u, v, y. The auxiliary signal v is precomputed and known while u, y become available in real time during the test period. Thus u, y cannot be used in the design of the detection filter. They can only be used in its application. Failure detection is to be robust with respect to the noise ν. It is assumed that the noise satisfies a quadratic bound. Depending on the quadratic form we will see that this includes both norm bounded noise and model uncertainty. 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 0 and Model 1. For a given auxiliary signal v we define Ai (v) to be the set of input-output pairs (u, y) consistent with Model i. That is, those pairs which are realizable by that model. Thus

Figure 4: Guaranteed detection with minimal v.

when the sets Ai (v) are disjoint or tangent. In Figures 2-4 the Ai (v) stayed a constant size and shape. We will see later that for some choices of noise measure that the Ai (v) can also grow as v s increases.

The goal then is to have an auxiliary signal v of the smallest possible size and an easily computed funcA0 (v): set of normal input-outputs. tion µ = h(u, y) so that the value of µ will tell us A1 (v): set of input-outputs when failure occurs. whether failure has occurred. Sections 3 and 4 deFailure detection consists of observing the inputs velop the needed theory for one model. Section 5 and outputs and deciding to which set they belong. then considers auxiliary signal design when there are two models. For guaranteed detection we need that A0 (v) ∩ A1 (v) = ∅.

(1) 3

In this paper we focus on linear models of different types. Suppose that a given v does not give guaranteed detection. Then we have the situation in Figure 2. As we apply increasing amounts of v, the Ai (v) move in the direction of the arrows as indicated in Figure 3. If the Ai (v) are bounded in the appropriate directions, then we reach a multiple of v where the Ai (v) are tangent as in Figure 4, and after that they are disjoint. Guaranteed detection occurs

Static Case

It is instructive to consider first linear systems in the static case. If one then thinks of the matrices as being operators this motivates a number of later developments. Absence of uncertainty: If we have no uncertainty the model is just y = Gu

(2)

Figure 6: Realizable set for Example 1.

Figure 5: Additive uncertainty.

where G is m × n and 4≥ m. For the system (2) we We must express γ directly in terms of {u, y}. The solution to the optimization problem (6), (7) is can use a residual-based test:

 
T
u  u   −GT1  T −1 µ = T (y − Gu) = H (3) γ(u, y) = u −G1 I . G2 G2 y y I y   where H = T −G I . Realizability of {u, y} is
 u equivalent to µ being zero. Thus if we let µ = H where H satisfies y Additive uncertainty: A more realistic case is when
 we have additive uncertainty  −1  −GT1  T −G1 I , H H= G2 GT2 I y = G1 u + G 2 ν (4) we have γ(u, y) = µ2 . Then the realizability test where G1 and G2 are matrices. We assume G2 is full becomes µ2 < d. If this test does not hold, then row rank. The noise ν is assumed to satisfy (here · we have a failure. is the usual Euclidian norm): ν2 < d.

(5) Example 1 Consider the following simple example

as illustrated in Figure 5. Then {u, y} will be realizable if there exists a ν satisfying (5) and (4).

y = u + ν, ν 2 < 1.

In order to construct a residual-based test to deter- The realizable set {u, y} is given by mine the realizability of a given {u, y}, we consider γ(u, y) = (y − u)2 < 1 the optimization problem: γ(u, y) = min ν2 ν

(6)

and is illustrated in Figure 6.

Slope uncertainty: Of course, in many applications there is uncertainty in the models themselves. To (7) include this type of uncertainty we consider uncery = G1 u + G2 ν. tainty in the following form which follows the forThe function γ(u, y) provides the realizability test. mulation of [6]: If γ(u, y) < d, then {u, y} is realizable. y = (G11 + G12 ∆(I − G22 ∆)−1 G21 )u (8) If γ(u, y) ≥ d, then {u, y} is not realizable.

subject to

where


J=

 I 0 , 0 −I

(17)

and {v, z} satisfy (11) and (12) for given {u, y}. These constraints can be expressed as:

 u ν G1 = G2 (18) y z Figure 7: Model uncertainty.


G1 =

 −G11 I , −G21 0


G2 =

 G12 0 .(19) G22 −I

where ∆ is any matrix whose maximum singular Let A⊥ be a matrix of maximal full column rank such that AA⊥ = 0. That is, A⊥ is a maximal rank value σ ¯ satisfies right annihilator. Then the solution of the above opσ ¯ (∆) ≤ 1. (9) timization problem is as follows. Suppose Note that (8) is a perturbation of

G2⊥ JGT2⊥ > 0.

(20)

y = G11 u.

(10) Then the solution to optimization problem (16) is

T u u This corresponds to modeling an uncertain gain as T T −1 G1 (G2 JG2 ) G1 . (21) γ(u, y) = y y shown in Figure 7. The system of Figure 7 can be re-expressed as Example 2 Suppose that (8) is y = G11 u + G12 ν (11) y = (1 + ∆)u (22) z = G21 u + G22 ν (12) with ν = ∆z. Note that combining (13) with (12) we get

with |∆| ≤ 1. From (8), it is easy to see that we can take G22 = 0 and G11 = G12 = G21 = 1. Thus (13)

 −1 1 1 0 G1 = , G2 = . (23) −1 0 0 −1

(14) H can be obtained using the J-spectral factorization. It is easy to verify that (G2 JGT2 )−1 = J so that we Solving for ν and substituting into (11) gives (8). can let H = G1 , that is,
 Since (9) holds, we get from (13) that ν2 ≤ z2 . y−u µ= . (24) Thus the characterization of realizability is −u ν = ∆(G21 u + G22 ν)

ν2 − z2 ≤ 0.

(15) The realizability test (21) then becomes

To get this in terms of just {u, y} we again set up an (25) γ(u, y) = µT Jµ = y 2 − 2uy ≤ 0. optimization problem. Let To see why this inequality is correct, simply note that
T
 2 2 2 ν ν γ(u, y) = min J (16) y − 2uy = u (∆ − 1). This inequality defines the z z ν,z set of realizable {u, y} and is illustrated in Figure 8.

Figure 9: Realizable set for Example 3. Figure 8: Realizable set for Example 2. This system is closely related to Example 2. The realizable set is defined by γ(u, y) = y 2 − 2uy < 1 Note that in Figure 6 the set of y’s for a given u stays and is illustrated in Figure 9. the same size while the set varies in size with u in Figure 8. A similar behavior occurs when designing auxiliary signals later in this paper. The Ai (v) are 4 Dynamical systems fixed size with additive uncertainty and vary in size Continuous and discrete systems are of interest but with model uncertainty. space allows us only to consider the continuous case. General uncertainty model: In order to capture Our approach builds on the formulations in [6]. The both additive and slope uncertainties we consider the static case just discussed leads to the following unmore general model certain dynamical system modeled over [0, T ]:

 u ν G1 = G2 (26) x˙ = Ax + Bu + M ν, (30) y z z = Gx + Hu, (31) with G2 full row rank and with the noise constraint, y = Cx + Du + N ν (32)
T
 ν ν J 0, ∀t ∈ [0, T ].

5

Auxiliary signal design

 T

−4v + y = ν0 ν02 < 1 and




 1 1 1 0 − v+ y = ν 1 0 0 −1 1
 1 0 T ν1 ν < 1. 0 −1 1

(38) (39)

(40) (41)

Figure 10 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:

The discussion so far has only addressed how to σ(v) ≥ 1 characterize realizability when there is one model present. We now consider the situation where there are two models and an auxiliary signal v which is where to be constructed to aid in identification. We again σ(v) = inf max(ν0T J0 ν0 , ν1T J1 ν1 ) ν0 ,ν1 ,u,y begin by considering the static case.

(42)

(43)

So the optimization problem (46) can be expressed as follows: φβ (v) = inf ν T Jβ ν

(48)

Gv = Hν

(49)

ν

subject to

where


  X0 G = W0 W1 , X1
  H0 0  , H = W0 W1 0 H1

 βJ0 0 ν0 , Jβ = . ν = 0 (1 − β)J1 ν1 

Figure 10: Realizable sets for Example 4 and minimal v.

subject to (36), i = 0, 1. It can be shown that σ(v) Proper auxiliary signal and the separability inin (43) can be expressed as: dex: An auxiliary signal v is proper if the two realizable sets A0 (v) and A1 (v) are disjoint. Thus an σ(v) = max inf (βν0T J0 ν0 + (1 − β)ν1T J1 ν1 ) auxiliary signal v is proper if and only if σ(v) ≥ 1. β∈[0,1] ν0 ,ν1 ,u,y Let V denote the set of proper auxiliary signals v. Suppose that v is measured in an inner product norm   so there exists a positive definite matrix Q such that

 v 
 T 1/2 H0 0 X0 Y0 Z0   ν0 u = . (44) the norm of v is (v Qv) . Then 0 H1 X1 Y1 Z1 ν1 y
− 12 ∗ T , (50) γ = min v Qv v∈V The optimization problem (43) can then now be expressed as follows: is called the separability index for positive definite subject to

σ(v) = max φβ (v)

(45)

β

matrix Q. The v realizing the min is called an optimal proper auxiliary signal.

Using the fact that σ(v) is quadratic in v, the problem of finding an optimal auxiliary signal can be formuφβ (v) = min (βν0T J0 ν0 + (1 − β)ν1T J1 ν1 ) (46) lated as follows:

where

ν0 ,ν1 ,u,y

v=0

Since u and y appear only in the constraint (44), we can eliminate them from the optimization problem by premultiplying (44) with 

 W0 W1 =

σ(v) φβ (v) = max T v Qv v=0,β∈[0,1] v T Qv v T Vβ v = max v=0,β∈[0,1] v T Qv

λ∗ = max

subject to (44).




Y0 Z 0 Y1 Z 1

⊥ (47)

where A⊥ is a maximal rank left annihilator of A.

(51)

for some symmetric matrix Vβ . We are assuming here that H⊥T Jβ H⊥ > 0. Let λβ = max v=0

v T Vβ v . v T Qv

(52)

Then clearly λ∗ = maxβ λβ . But λβ is the largest λ a proper auxiliary signal exists if and only if λ∗ > 0. Then an optimal auxiliary signal is such that v T Vβ v − λv T Qv = 0.

v ∗ = Kζ ∗

(53)

(58)

where K = Gl HJβ−1 H T , Gl is any left-inverse of G, So if λ > 0, then the set of proper auxiliary signals ζ ∗ = αζ where ζ is any non-zero vector satisfying is not empty and any optimal proper auxiliary signal T is a solution of (53) satisfying (λ∗ HJβ−1 − GGT )ζ = 0 (59) ∗ H ∗

∗ (54) where β is a value of β maximizing in (57), and 1 . (60) α=

If λ∗ ≤ 0, then the set of proper auxiliary signals is ∗ ζ T K T QKζ λ empty. If λ∗ > 0, then the separability index is given by 5.2 Continuous-time case

v T Qv =

1 . λ∗

1 γ∗ = √ . λ∗

(55) By using ν to represent ν and z, and by breaking up the input u into two inputs u and v, we obtain the following models:

Example 5 For Example 4, φβ (v) is: 



T 

¯i u + Mi νi , x˙ i = Ai xi + Bi v + B ¯ i u + Ni νi Ei y = Ci xi + Di v + D



ν0 ν0 β      1−β ν10 ν10  inf ν0 ,ν10 ,ν11 −(1 − β) ν11 ν11 subject to      ν0 1 0 0 −4 1
 −1 1 v = 0 1 0  ν10  . y 0 0 −1 −1 0 ν11

(61) (62)

where i = 0, 1 correspond to normal and failed systems. Ni ’s are assumed to have full row rank. The noise constraints are Si (v, s) = xi (0)T Pi0−1 xi (0)  s + νiT Ji νi dt < 1, ∀s ∈ [0, T ], (63) 0

We can show that φβ (v) is defined for all β ∈ [0, 1] where the Ji ’s are signature matrices. and is given by φβ (v) = (9β − 1)(1 − β)v 2 . So Non-existence of a solution to (61), (62) and (63) is σ(v) = maxβ∈[0,1] φβ (v) can easily be computed equivalent to: in this case by setting the derivative of φβ (v) with σ(v, s) ≥ 1 (64) respect to β to zero to get β ∗ = 5/9. This gives 16 2 ∗ σ(v) = 9 v and thus v = 3/4. where Alternative solution: Instead of performing two opσ(v, s) = ν ,νinf,u,y max(S0 (v, s), S1 (v, s)), (65) 0 1 x0 ,x1 timizations, we can combine them to have a one step T solution. Suppose that H⊥ Jβ H⊥ > 0. Then λβ corsubject to (61)-(62), i = 0, 1. As in the static case, responds to the largest λ such that we can reformulate (65) as: det(λHJβ−1 H T − GGT ) = 0.

(56)

σ(v, s) = max φβ (v, s) β∈[0,1]

Computing

(66)

where λ∗ = max λβ β∈[a,b]

(57)

φβ (v, s) = ν ,νinf,u,y βS0 (v, s) + (1 − β)S1 (v, s) 0

1

x0 ,x1

subject to (61)-(62), i = 0, 1.

As we did in the static case, we reformulate the optimization problem as follows

We consider here only the case where there is no u. That is, all the inputs are used as an auxiliary signal φβ (v, s) λβ,s = max . (68) for failure detection. Let v=0 |v|2


 x0 ν0 A0 0 x= , ν= , A= , Suppose for some β ∈ [0, 1], that we have 0 A1 x1 ν1 N⊥ T Jβ N⊥ > 0, ∀ t ∈ [0, T ], and that the Riccati 
  M0 0 equation , N = F0 N0 F1 N1 , M= 0 M1 P˙ = (A − SR−1 C)P + P (A − SR−1 C)T   −P C T R−1 CP + Q − SR−1 S T , P (0) = Pβ D = F0 D0 + F1 D1 , C = F0 C0 F1 C1 ,
B=


−1   0 βP0,0 B0 −1 , Pβ = −1 , B1 0 (1 − β)P1,0
 0 βJ0 , Jβ = 0 (1 − β)J1 






F0 F1 =

E0 E1

⊥ .

Then we can reformulate the problem (65) as:  T T −1 ν T Jβ ν dt φβ (v, s) = inf x(0) Pβ x(0) + ν,x

0

subject to x˙ = Ax + Bv + M ν 0 = Cx + Dv + N ν.

where


Q S ST R




=

M N




 Jβ−1

M N

T (69)

has a solution on [0,T]. The values of Q, S, R depend on β. Then λβ,s is the smallest λ for which the Riccati equation P˙ = (A − Sλ Rλ−1 C)P + P (A − Sλ Rλ−1 C)T −P C T Rλ−1 CP + Qλ − Sλ Rλ−1 SλT , P (0) = Pβ where 

T −1

M B Jβ M B 0 Qλ Sλ = N D N D 0 −λI SλT Rλ has a solution on [0, s]. The values of Qλ , Sλ , Rλ depend on β as well as λ.

This result allows us to compute Construction of an optimal proper auxiliary sigλβ = max λβ,s nal: For the sake of simplicity of presentation, we s start by considering that the optimality criterion used for the auxiliary signal is just that of minimizing its which can be used to compute L2 norm. Thus the problem to solve is: λ∗ = max λβ . β min |v|2 , subject to max φβ (v, s) ≥ 1 (67) v β∈[0,1] This gives the separability index as s∈[0,T ] 1 γ∗ = √ . λ∗

where  |v| = 2

T

v2 dt.

and is used to compute the optimal auxiliary signal ∗ The reason we take the max over s is that the max- v . imum value of φβ (v, s) does not always occur at Construction of optimal auxiliary signal: Assume s = T . It does in most cases though. β ∗ (an optimal β) and λ∗ are computed. Let (x, ζ) 0

specific types of problems. In this approach the auxiliary signal design is done off line. It easy to implement the algorithms in Scilab [1] or other CAD packages. The on-line detection test can be implemented efficiently using “Kalman type” filters or hyperplane tests but space does not permit discussing this here.

Acknowledgements Research supported in part by the National Science Foundation under DMS-0101802, ECS-0114095 and INT-9605114. Figure 11: Typical minimum proper v.

References be any non-zero solution of the two-point boundaryvalue system: 


d x x Ω11 Ω12 = ζ Ω21 Ω22 dt ζ

[1] C. Bunks, J. P. Chancelier, F. Delebecque, C. Gomez (Editor), M. Goursat, R. Nikoukhah, and S. Steer. Engineering and scientific computing with Scilab, Birkhauser, 1999.

with boundary conditions:

[2] S. L. Campbell, K. Horton, R. Nikoukhah, F. Delebecque. “Auxiliary signal design for rapid multimodel identification using optimization”, Automatica, to appear.

x(0) = Pβ ∗ ζ(0) ζ(T ) = 0. where Ω11 = −ΩT22 = A − Sλ∗ Rλ−1∗ C T Ω12 = Qλ∗ − Sλ∗ Rλ−1 ∗ Sλ∗ Ω21 = C T Rλ−1∗ C This boundary value problem is not well-posed and has non trivial solutions (x∗ , ζ ∗ ). An optimal auxiliary signal is (for simplicity suppose D = 0) v ∗ = αB T ζ ∗ where α is a constant such that |v ∗ | = 1/γ ∗ . A typical auxiliary signal is given in Figure 11. The shape of the v can vary depending on the models, the size of the weight on the initial condition perturbation, and the length of the interval. Several more examples are in [2] and the other references.

6

Conclusion

We have described a framework for multi-model identification and shown the form it takes for two

[3] R. Nikoukhah. “Guaranteed active failure detection and isolation for linear dynamical systems”, Automatica, 34, pp. 1345–1358, (1998). [4] R. Nikoukhah, S. L. Campbell, F. Delebecque. “Detection signal design for failure detection: a robust approach”, Int. J. Adaptive Control and Signal Processing, 14, pp. 701–724, (2000). [5] R. Nikoukhah, S. L. Campbell, K. Horton, F. Delebecque. “Auxiliary signal design for robust multi-model identification”, IEEE Trans. Automatic Control, 47, pp. 158–163, (2002). [6] I.R. Petersen, A.V. Savkin, Robust Kalman filtering for signals and systems with large uncertainties, Birkhauser, 1999.