Change detection based on algebraical consistency tests 11.1.. 404 Change detection based on algebraical consistency tests That is, we will study a noise free state space model The dif
Trang 1Change detection based on algebraical consistency tests
11.1 B a s i c s 4 0 3
11.2 P a r i t y space change d e t e c t i o n 407
11.2.1 Algorithm derivation 407
11.2.2 Some rank results 409
11.2.3 Sensitivity and robustness 411
11.2.4 Uncontrollable fault states 412
11.2.5 Open problems 413
11.3 An observer approach 4 1 3 11.4 An input-output approach 414
11.5 Applications 4 1 5 11.5.1 Simulated DC motor 415
11.5.2 DC motor 417
11.5.3 Vertical aircraft dynamics 419
11 l Basics
Consider a batch of data over a sliding window, collected in a measurement vector Y and input vector U As in Chapter 6, the idea of a consistency test is to apply a linear transformation to a batch of data, AiY + BiU + ci
The matrices Ai, Bi and vector G are chosen so that the norm of the linear transformation is small when there is no change/fault according to hypothesis
Hi, and large when fault Hi has appeared The approach in this chapter
measures the size of
llAiY + BiU + till
as a distance function in a algebraic meaning (in contrast to the statistical meaning in Chapter 6 ) The distance measure becomes exactly ‘zero’ in the non-faulty case, and any deviation from zero is explained by modeling errors and unmodeled disturbances rather than noise
Adaptive Filtering and Change Detection
Fredrik Gustafsson Copyright © 2000 John Wiley & Sons, Ltd ISBNs: 0-471-49287-6 (Hardback); 0-470-84161-3 (Electronic)
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That is, we will study a noise free state space model
The differences to, for example, the state space model (9.1) are the following:
0 The measurement noise is removed
0 The state noise is replaced by a deterministic disturbance d t , which may have a direct term to yt
0 The state change may have a dynamic profile f t This is more general than the step change in (9.1), which is a special case with
f t = Ot-kV
0 There is an ambiguity over how to split the influence from one fault between B f , D f and f t The convention here is that the fault direction
is included in B f , D f (which are typically time-invariant) That is, B; f:
is the influence of fault i , and the scalar fuult profile f: is the time-varying
size of the fault
The algebraic approach uses the batch model
& = OZt-L+1 -k Huut -k HdDt -k H f F t , (11.3) where
For simplicity, time-invariant matrices are assumed here The Hankel matrix
H is defined identically for all three input signals U , d and f (the subscript
defines which one is meant)
The idea now is to compute a linear combination of data, which is usually referred to as a residual
rt A wT(& - &ut) = wT(Oxt-L+l + H& + H f F t ) (11.5)
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This equation assumes a stable system (Kinnaert et al., 1995) The equation
rt = 0 is called a parity equation For the residual to be useful, we have to
impose the following constraints on the choice of the vector or matrix W :
no fault The columns of wT are therefore vectors in the null space of the
matrix (0, Hd) This can be referred to as decoupling of the residuals
from the state vector and disturbances
Sensitive to faults:
Together with condition 1, this implies rt = wT(Y,-H,Ut) = w T H f F t #
0 whenever Ft # 0
For isolation, we would like the residual to react differently to the differ-
ent faults That is, the residual vectors from different faults f', f 2 , ,
f " f should form a certain pattern, called a residual structure R There are two possible approaches:
a) Transformation of the residuals;
This design assumes stationarity in the fault That is, its magnitude
is constant within the sliding window This implies that there will
be a transient in the residual of length L See Table 11.1 for two common examples on structures R
b) For fault decoupling, a slight modification of the null space above
is needed Let H i be the fault matrix from fault fz There are now n f such matrices Replace (11.6) and (11.7) with the iterative
might excite other residuals causing incorrect isolation, a risk which
is eliminated here On the other hand, it should be remarked that detection should be done faster than isolation, which should be done after transient effects have passed away
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Table 11.1 The residual vector rt = w T ( K - H,Ut) is one of the columns of the residual structure matrix R, of which two common examples are given
to be found However, it should be clear from the previously described design approaches that it is the window size L which is the main design parameter
to trade off sensitivity (and robustness) t o decreased detection performance Equation (11.5) can be expressed in filter form as
rt = A(q)Yt - B(&,
where A(q) and B(q) are polynomials of order L There are approaches de-
scribed in Section 11.4, that design the FIR filters in the frequency domain There is also a close link to observer design presented in Section 11.3 Ba- sically, (11.5) is a dead-beat observer of the faults Other observer designs correspond to pole placements different from origin, which can be achieved by filtering the residuals in (11.5) by an observer polynomial C(q)
Another approach is also based on observers The idea is to run a bank of filters, each one using only one output This is called an dedicated observer in
Clark (1979) The observer outputs are then compared and by a simple voting
strategy faulty sensors are detected The residual structure is the left one in Table 11.1 A variant of this is to include all but one of the measurements in the observer This is an efficient solution for, for example, navigation systems, since the recovery after a detected change is simple; use the output from the observer not using the faulty sensor A fault in one sensor will then affect all but one observer, and voting can be applied according to the right-hand struc- ture in Table 11.1 An extension of this idea is t o design observers that use all but a subset of inputs, corresponding to the hypothesized faulty actuators This is called unknown input observer (Wiinnenberg, 1990) A further alter- native is the generalized observer, see Patton (1994) and Wiinnenberg (1990),
which is outlined in Section 11.3 Finally, it will be argued that all observer
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and frequency domain approaches are equivalent to, or special cases of, the parity space approach detailed here
Literature
For ten years, the collection by Patton et al (1989) was the main reference in this area Now, there are three single-authored monographs in Gertler (1998), Chen and Patton (1999) and Mangoubi (1998) There are also several survey papers, of which we mention Isermann and Balle (1997), Isermann (1997) and Gertler (1997)
The approaches to design suitable residuals are: parity space design (Chow and Willsky, 1984; Ding et al., 1999; Gertler, 1997), unknown input observer (Hong et al., 1997; Hou and Patton, 1998; Wang and Daley, 1996; Wunnenberg, 1990; Yang and Saif, 1998) and in the frequency domain (Frank and Ding, 199413; Sauter and Hamelin, 1999) A completely different approach is based on
reasoning and computer science, and examples here are A r z h (1996), Blanke
et al (1997), Larsson (1994) and Larsson (1999) In the latter approach, Boolean logics and object-orientation are keywords
A logical approach to merge the deterministic modeling of this chapter with the stochastic models used by the Kalman filter appears in Keller (1999)
elements equal to the singular values of (0 H d ) , and its left eigenvectors are
the rows of U The null space N of (0 Hd) is spanned by the last columns
of U , corresponding to eigenvalues zero In the following, we will use the same notation for the null space N , as for a basis represented by the rows of a matrix
N In MAT LAB^^ notation, we can take
[UyD,V1=svd( CO Hdl) ;
n=rank(D) ;
N=U(:,n+l:end)';
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The MATLABTM function null computes the null space directly and gives a
slightly different basis Condition (11.6) is satisfied for any linear combination
of N ,
wT = T N ,
where T is an arbitrary (square or thick) matrix To satisfy condition (11.7),
we just have to check that no rows of N are orthogonal t o H f If this is the case, these are then deleted and we save N In MATLABTM notation, we take
The last thing to do is t o choose T t o facilitate isolation Assume there
are n f 5 nN faults, in directions f ' , f 2 , , f " f Isolation design is done by first choosing a residual structure R The two popular choices in Table 11.1 are
Here 1~ is a vector of L ones and @ denotes the Kronecker product That
is, 1~ @ f i is another way of writing F: when the fault magnitude is constant
during the sliding window In MATLABTM notation for n f = 3, this is done
by
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To summarize, we have the following algorithm:
Algorithm 7 7.7 Parity space change detection
Given: a state space model (11.1)
Design parameters: sliding window size L and residual structure R
Compute recursively:
1 The data vectors yt and Ut in (11.3) and the model matrices 0 , Hd, H f
2 The null space N of (0 Hd) is spanned by the last columns of U , cor- responding to eigenvalues zero In MATLABTM formalism, the transfor- mation matrix giving residual structure R is computed by:
4 Change detection if rTr > 0, or rTr > h considering model uncertainties
5 Change isolation Fault i in direction f 2 where i = arg maxi rTRi Ri
denotes column i of R
[dum , il =max (r ’ *R) ;
It should be noted that the residual structure R is no real design parameter, but rather a tool for interpreting and illustrating the result
11.2.2 Some rank results
When does a parity equation exist? That is, when is the size of the null space
N different from O? We can do some quick calculations t o find the rank of N
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It is assumed that the ranks are determined by the column spaces, which is the case if the number of outputs is larger than the number of disturbances (otherwise condition (11.6) can never be satisfied) In (11.9) it is assumed
that the column space of 0 and H d do not overlap, otherwise the rank of
N will be larger That is, a lower bound on the rank of the null space is rank(N) 2 L(nU - n d ) - n,
The calculations above give a condition for detectability For isolability, compute Ni for each fault fi Isolability is implied by the two conditions
Ni # 0 for all i and that Ni is not parallel to Nj for all i # j
Requirement for isolation
In the case of the same number of residuals as faults, we simply take f t =
(TNHf)-'rt The design of the transformation matrix to get the required fault structure then also has a unique solution,
It should be remarked, however, that the residual structure is cosmetics which may be useful for monitoring purposes only The information is available in the residuals with or without transformation
Minimal order residual filters
We discuss why window sizes larger than L = n, do not need to be considered
The simplest explanation is that a state observer does not need to be of a higher dimension (the state comprises all of the information about the system, thus also detection and isolation information) Now so-called Luenberger observers
can be used to further decrease the order of the observer The idea here is that the known part in the state from each measurement can be updated exactly
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A similar result exists here as well, of course A simple derivation goes as
follows
Take a QR factorization of the basis for the null space,
N = &R
It is clear that by using T = QT we get residuals
rt = Tni(yt - &Ut) = QTQR(& - &Ut) = R(& - &Ut)
The matrix R looks like the following:
nd) - nx
The matrix has zeros below the diagonal, and the numbers indicate dimen- sions We just need to use nf residuals for isolation, which can be taken as
the last nf rows of rt above When forming these n f residuals, a number of
elements in the last rows in R are zero Using geometry in the figure above, the filter order is given by
11.2.3 Sensitivity and robustness
The design methods presented here are based on purely algebraic relations
It turns out, from examples, that the residual filters are extremely sensitive
to measurement noise and lack robustness to modeling errors Consider, for example, the case of measurement noise only, and the case of no fault:
rt = wT(% +Et - H,Ut) = wTEt
Here Et is the stacked vector of measurement noises, which has covariance matrix
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Here @ denotes Kronecker product The covariance matrix of the residuals is given by
Cov(rt) = WT Cov(Et)w = W T ( I L 8 R)w (11.14) The reason this might blow up is that the components of W might become sev- eral order of magnitudes larger than one, and thus magnifying the noise See Section 11.5.3 for an example In this section, examples of both measurement noise and modeling error are given, which show that small measurement noise
or a small system change can give very ‘noisy’ residuals, due to large elements
in W
A generalization of this approach in the case the state space model has states
(‘fault states’) that are not controllable from the input ut and disturbance dt
is presented in Nyberg and Nielsen (1997) Suppose the state space model can
and secondly, it is less likely that the null space is orthogonal to (02 H f ) than
to H f in the original design Intuitively, the ‘fault states’ x2 are unaffected by input and disturbance decoupling and are observable from the output, which facilitates detection and isolation