Adaptive Filtering and Change Detection Fredrik Gustafsson Copyright © 2000 John Wiley & Sons, Ltd ISBNs: 0-471-49287-6 (Hardback); 0-470-84161-3 (Electronic) Change detection basedon likelihood ratios 9.1 Basics 9.2.The 343 346 likelihood approach 9.2.1 Notation 346 9.2.2 Likelihood 347 348 9.2.3 Likelihood ratio 9.3.The GLR test 349 9.4.The MLR test 353 9.4.1 RelationbetweenGLRand MLR 353 9.4.2 A two-filter implementation 355 9.4.3 Marginalization of the noiselevel 361 9.4.4 Stateandvariancejump 363 364 9.4.5 Summary 9.5.Simulationstudy 365 9.5.1.A MonteCarlosimulation 365 9.5.2 Complexity 369 9.A Derivation of the GLR test 370 9.A.l Regression model for the jump 370 9.A.2 TheGLRtest 371 9.B LS-based derivation of the MLR test 372 9.1 Basics This chapter is devoted to the problem of detecting additive abrupt changes offset in linear state space models Sensor and actuator faults as a sudden or drift can all be modeled as additive changes In addition disturbances are traditionally modeled as additivestate changes The likelihood ratio formulation provides a general framework for detecting such changes and to isolate the fault/disturbance Chanae detection based on likelihood ratios 344 The state space model studied in this chapter is xt+1 = A m + B,,tut + &,tVt + ut-kBe,tv Y t =Gxt + et + D u p t + ut-rcDe,tv (9.1) (9.2) The additive change (fault) v enters at time L as a step (ut denotes the step function) Here ut, et and 20 are assumed to be independent Gaussian variables: ut Qt) et N O , Rt) zo &)- Furthermore, they are assumedto be mutually independent The state change v occurs at the unknown time instant L, and S(j) is the pulse function that is one if j = and zero otherwise The set of measurements y1, y2, ,YN,each of dimension p , will be denoted yN and y F denotes the set yt, yt+l, ,Y N This formulation of the change detection problem can be interpreted as an input observer or input estimator approach A similar model is used in Chapter 11 To motivate the ideas of this chapter, let us consider the augmented state space model, assuming a change at time t = k (compare with Examples 6.6 and 8.2.4): That is, at time t = k the parameter value is changed as a step from Bt = for t < k to Qt = v for t k It should be noted that v and Q both denote the magnitude of the additive change, but the former is seen as an input and the latter as a state, or parameter The advantageof the re-parameterization is that we can apply the Kalman filter directly, with or without change detection, and we have an explicit fault state that can be used for fault isolation The Kalman filter applied to the augmented state space model gives a parameter estimator &+l,, = et,,-, +m y t - Ct%lt l - D6&1 - Du,tUt), 9.1 Basics 345 High gain (Q: = 6t kaI) I Ut Yt Filter 21, P1 No gain (Q: = 0) I Ut Yt Filter - Hyp test P, P W P O ,PO Figure 9.1 Two parallel filters One is based on the hypothesis no change ( H o ) and the other on a change at time k (H1( k ) ) By including more hypothesis for k , a filter bank is obtained Here we have split the Kalman filter quantities as so the covariance matrix of the change (fault component) is P/' Note that K! = before the change The following alternatives directly appear: Kalman filter-based adaptive filtering, where the state noise covariance is used to track 19 0 Whiteness-based residual test, where the Kalman filter innovations are used and the state covariance block Q," is momentarily increased when a change is detected A parallel filter structure as in Figure 9.1 The hypothesis test can be accomplished by one of the distance measures in Chapter This chapter is devoted to customized approaches for detecting and explicitely estimating the change v The approach is based on Likelihood Ratio ( L R ) tests using Generalized Likelihood Ratio ( G L R ) or Marginalized Likelihood Ratio (MLR).The derivation of LR is straightforward from (9.3), but the special structure of the state space model can be used to derive lower order filters The basic idea is that the residuals from a Kalman filter, assuming no change, can be expressed as a linear regression Chanae detection based on likelihood ratios 346 Linear regression formulation The nominal Kalmanfilter, assumingno abrupt change, is applied, and theadditive change is expressed as a linearregression with the innovations as measurements with the following notation: Kalman filter + Auxiliary recursion Residual regression Compensation Q-l, + Et pt, pt Et = xt ' p ? + et Xtlt-l +P P The third equation indicates that we can use RLS to estimate the change v , and the fourthequation shows how to solve the compensation problem after detection of change and estimation (isolation) of v Chapter 10 gives an alternative approach to this problem, where the change is not explicitely parameterized 9.2 Thelikelihood approach Some modifications of the Kalmanfilter equationsare given, and thelikelihood ratio is defined for the problem at hand 9.2.1 Notation The Kalmanfilter equations for a change v E N(0, Pv) at a given time k follows by considering v as an extra state noise component directly from (8.34)-(8.37) V t = 6t-,p, with Qt = 6t-kPv The addressed problem is to modify these equations to the case where k and v are unknown The change instant k is of primary interest, but good state estimates may also be desired 9.2 The likelihood amroach 347 In GLR, v is an unknown constant, while it is considered as a stochastic variable in the MLR test To start with, the change will be assumed to have aGaussianprior.Lateron,anon-informativeprior will be usedwhichis sometimes called aprior of ignorance; see Lehmann (1991) This prior is characterized by a constant density function, p(v) = C Example 9.1 Modeling a change in the mean We can use Eq (9.1) to detect abrupt changes in the mean of a sequence of stochastic variables by letting At = 1,C, = 1,Qt = 0, Bu,t = Furthermore, if the mean before the change is supposed to be 0, a case often considered in the literature (see Basseville and Nikiforov (1993)), we have zo = and no = It is worthmentioning, framework as well that parametric models from Part I11 can fit this Example 9.2 Modeling a change in an ARX model By letting At = I and Ct = (yt-1, yt-2, ,ut-1, ut-2, ), a special case of equation (9.1) is obtained We then have a linear regression description of an ARX model, where xt is the (time-varying) parameter vector and C, the regressors In this way, we can detect abrupt changes in the transfer function of ARX models Note that the change occurs in the dynamics of the system in this case, and not in the system’s state 9.2.2 likelihood The likelihood for the measurements up to time N given the change v at time k is denoted p ( y N I k ,v) The same notationis used for the conditional density function for yN, given k , v For simplicity, k = N is agreed to mean no change There are two principally different possibilities to estimate the change time L: Joint ML estimate of k and v, Here arg m a ~ k ~ [ ~ , ~ l , ~vp) means ( y ~ I the k , maximizing arguments of the likelihood p ( y N I k ,v ) where k is restricted to [l,NI Chanae detection based on likelihood ratios 348 The ML estimate of just k using marginalization of the conditional density function p ( y N I k ,v): The likelihood for data given just k in (9.6) is the starting point in this approach A tool in the derivations is the so-called flat prior, of the form p(v) = C, which is not a proper density function See Section 7.3.3 for a discussion and two examples for the parametric case, whose conclusions are applicable here as well 9.2.3 likelihood ratio In thecontext of hypothesis testing, the likelihood ratios rather than the likelihoods are used The LR test is a multiple hypotheses test, where the different change hypotheses are compared to the nochangehypothesis pairwise In the LR test, the change magnitude is assumed to be known The hypotheses under consideration are H0 : H l ( k ,v) : nochange a change of magnitude v at time k The test is as follows Introduce the log likelihood ratio for the hypotheses as the test statistic: The factor is just for notational convenience We use the convention that H1(N,v) = Ho, so again, k = N means no change Then the LR estimate can be expressed as when v is known Exactly as in (9.5) and (9.7), we have two possibilities of how to eliminate the unknown nuisance parameter v Double maximization gives the GLR test, proposed for change detection inWillsky and Jones(1976), and marginalization the MLR test, proposed in Gustafsson (1996) 9.3 The GLR test 349 9.3 The GLR test Why not just use the augmented statespace model (9.3)and the Kalmanfilter equations in (9.4)? It would be straightforward to evaluate the likelihood ratios in (9.8) for each possible k The answer is as follows: The GLR algorithm is mainlya computational tool that splits the Kalman filter for the full order model (9.3) into a low order Kalman filter (which is perhaps already designed and running) and a cascade coupled filter bank with least squares filters I The GLR test proposed inWillsky and Jones (1976) utilizes this approach GLR’s general applicability has contributed to it now being a standard tool in change detection As summarized in Kerr (1987), GLR has an appealing analytic framework, is widely understood by many researchers and is readily applicable to systems already utilizing a Kalman filter Another advantage with GLR is that it partially solves the isolation problem in fault detection, i.e to locate the physical cause of the change In Kerr (1987), a number of drawbacks with GLR is pointed out as well Among these, we mention problems with choosing decision thresholds, and for some applicationsan untenable computational burden The use of likelihood ratiosinhypothesistesting is motivated by the Neyrnan-Pearson Lemma; see, for instance, Theorem 3.1 in Lehmann (1991) In the application considered here, it says that the likelihood ratio is the optimal test statistic when the change magnitude is known and just one change time is considered This is not the case here, but a sub-optimal extension is immediate: the test is computed for each possible change time, or a restriction to a sliding window, and if several tests indicate a change the most significant is taken as the estimated change time In GLR, the actualchange in the state of a linear system is estimated from data and thenused in the likelihood ratio Starting with the likelihood ratio in (9.8), the GLR test is a double maximization over k and v , where D ( k ) is the maximum likelihood estimate of v, given a change at time k The change candidate iin the GLR test is accepted if Z~(i,fi(i)) > h (9.10) Chanae detection likelihood ratios onbased 350 The threshold h characterizes a hypothesis test and distinguishes the GLR that (9.5) is a special case of (9.10), test from the ML method (9.5) Note where h = If the zero-change hypothesis is rejected, the state estimate can easily be compensated for the detected change The idea in the implementation of GLR in Willsky and Jones (1976) is to make the dependence on v explicit This task is solved in Appendix 9.A The key point is that the innovations from the Kalman filter (9.4) with k = N can be expressed as a linear regression in v , where Et(k) are the innovations from the Kalman filter if v and k were known Here and in the sequel, non-indexed quantitiesas E t are the output from the nominal Kalman filter, assuming no change The GLR algorithm can be implemented as follows Algorithm 9.7 GLR Given the signal model (9.1): Calculate theinnovations from the Kalman filter (9.4) assuming no change Compute the regressors cpt(k) using initialized by zeros at time t = k ; see Lemma 9.7 Here pt is n, is n, X n, Compute the linear regression quantities for each k , k t At time t = N , the test statistic is given by X and 9.3 The GLR test 351 A change candidate is given by k = arg max ~ ( kC,( k ) ) It is accepted is greater than some threshold h (otherwise k = N ) and if Z~(i,fi(i)) the corresponding estimate of the change magnitude is given by C N ( ~=) n,l(i)fN(i) We now make some comments on the algorithm: 0 0 It can be shown that the test statisticZN(L, (L)) under the null hypothesis is x2 distributed Thus, given the confidence level on the test, the threshold h can be found from standard statistical tables Note that this is a multiple hypothesis test performed for each k = 1,2, , N - 1, so nothing can be said about the total confidence level The regressor pt(k) is called a failure signature matrix Jones (1976) in Willsky and The regressors are pre-computable Furthermore, if the system and the Kalman filter are time-invariant, the regressor is only a function of t - k , which simplifies the calculations The formulationinAlgorithm 9.1 is off-line Since theteststatistic involves a matrix inversion of R N , a more efficient on-line method is as follows From (9.34) and (9.37) we get W ) ) = f?(k)Ct(k), where t is used as time index instead of N The Recursive Least Squares (RLS) scheme (see Algorithm5.3),can now be used to update Ct(k) recursively, eliminating the matrix inversion of Rt(k) Thus, the best implementation requires t parallel RLS schemes and one Kalman filter The choice of threshold is difficult It depends not only upon the system's signal-to-noise ratio, but also on the actualnoise levels, as will be pointed out in Section 9.4.3 Example 9.3 DC motor: the GLR test Consider the DC motor in Example 8.4 Assume impulsive additive state changes at times 60, 80, 100 and 120 First the angle is increased byfive units, and then decreased again Then the same fault is simulated on angular velocity That is, = (3 ,v2 = (;5) ,v3 = (3 ,v4 = (_os) Chanae detection based on likelihood ratios 352 Test statistic and threshold 30 20 - 10- ‘0 60 “-50 40 d o G 20 , I i % k l ; G 160 \ l a , 80 100 120 140 160 80 100 120 140 160 KF estimate GLR estimate Q 60 ‘0 40 20 Figure 9.2 Test statistic max-L