bounded-error estimation using dead zone and bounding ellipsoid

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, VOL. 8, 31-42 (1994)

BOUNDED-ERROR ESTIMATION USING DEAD ZONE AND BOUNDING ELLIPSOID

R. J. EVANS, C. ZHANG AND Y. C. SOH* Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, Vic. 305.2,

Australia

SUMMARY The use of a dead zone and a bounding ellipsoid for parameter estimation when measurement errors are bounded is discussed. The size of the dead zone is set to be exactly equal to the assumed noise bound. The algorithm retains the properties of computing parameter point estimates and allows a bounding ellipsoid to be computed at each iterative step.

KEY WORDS Bounding ellipsoid Dead zone Identification Parameter estimation Recursive least squares Unknown but bounded error

1. INTRODUCTION

In system identification problems it is often necessary to process observation data and records which are corrupted by noise. The usual approach is to model the noise as a random variable or stochastic process',' characterized by its mean and covariance and a model of the noise autocorrelation. Statistical inference is then employed to derive an identification scheme. While this approach is satisfactory in some circumstances, it is often severely limited because the prior information may be insufficient to characterize the noise, especially if the records are short. This and other inaccurate assumptions on the statistical nature of the noise often result in biased estimates of the system parameters. Thus in physical processes where the statistical distribution of the noise is not known a different approach is needed. In this paper we consider the so-called unknown but bounded noise model and consider the performance of a modified recursive least-squares algorithm with a dead zone under this noise model.

Many physical problems of interest can be accurately modelled by

yk = do*, 6k- 1) + W k , I W k ( < wk, k = 1 , 2 , ... (1)

wherebk) is an observed output sequence, O* is the unknown system parameter vector, 6k-L is an observed information vector available at time k, g(., .) is a known function and (wk j is an unobservable but bounded noise sequence. The bounds Wk >, 0 are assumed known. The model (1) is commonly referred to as an equation error formulation. Note that apart from the assumption of a known (Wk) sequence there is no other assumption on the statistical nature

* On leave from School of Electrical Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 2263.

CCC 0890-6327/94/01003 1-12 0 1994 by John Wiley & Sons, Ltd.

32 R . J . EVANS, C. ZHANG AND Y. C. SOH

of W k . Thus, for example, the observation noise can be correlated and biased. Under such a bounded assumption on W k , the identification problem usually involves the estimation of a model and a region in the parameter space where the true model 6* may lie.

State estimation with a bounded noise assumption has been investigated by Schweppe3 and Bertsekas and R h ~ d e s , ~ where a simple recursive estimation scheme based on a bounding ellipsoid is proposed. The bounding ellipsoid contains all state values which are consistent with the observations, together with some state values which are not compatible with the observed data. Fogel and Huang' and Dasgupta and Huang6 used similar bounding ellipsoid methods for parameter estimation. Their algorithms resemble weighted recursive least-squares estimation. At each iteration step a parameter estimate and a minimum size ellipsoid (based on measures such as determinant and trace) are computed. The ellipsoid bounds all parameter values which are compatible with the observation records. Other recursive and non-recursive approaches have also been examined. '-" Some of these algorithms derive a bounding orthotope procedure*-" where linear programming is used to compute the minimum size orthotope.

The treatment of bounded noise has also appeared in the adaptive control literature. ''-I6

One strategy in the parameter estimation process is to introduce a dead zone where the parameter update will cease when the prediction error is smaller than the dead zone threshold. Therefore it updates the estimate only when the information content of the new measurement is 'relevant'. This avoids catastrophic behaviour of the system in the presence of under modelling error and poor excitation. The dead zone treatment of bounded noise is very simple and attractive and has been shown to improve performance and robustness in adaptive control schemes. 129'3

Bounding ellipsoid parameter estimation algorithms and parameter estimation algorithms with a dead zone are analysed and compared in Reference 17. It is shown that bounding ellipsoid parameter estimation algorithms can also be written as algorithms with a dead zone. The difference between algorithms in these two families relates to computation and the weighting factors for the dead zone threshold.

The simplicity and robustness of the dead zone algorithm for identification with bounded noise are the main motivations for this paper. Firstly we reduce the threshold of the dead zone to the smallest possible size so that it is exactly equal to the size of the assumed noise bound. In contrast, the currently available least-squares-type results I 2 , l 3 require that the dead zone threshold be larger than the known noise bound. Secondly we derive a bounding ellipsoid for each iteration step, including the situation where the prediction error falls within the dead zone. Hence we can systematically reduce the size of the bounding ellipsoid to reflect increased knowledge about the parameter estimate. Again in contrast with existing bounding ellipsoid estimation algorithm^,^'^ the dead zone threshold used here relates directly to the system noise bound. While the bounding ellipsoid we derive is not optimal in terms of minimum size as discussed in Reference 5 , our algorithm has the advantage that it is simple and there is no need for parameter update if the new measurement contains no fresh information. It also provides a robust and smoothly convergent point parameter estimate, which is highly desirable in adaptive control.

The remainder of this paper is organized as follows. Section 2 is devoted to formulating the problem of parameter estimation with bounded noise. The modified recursive least-squares algorithm with a dead zone is presented in Section 3. Section 4 presents the bounded error estimation algorithm with a bounding ellipsoid and a dead zone. A simple extension of this algorithm to handle unknown but bounded time-varying parameters is presented in Section 5 , followed by conclusions in Section 6 .

DEAD ZONE AND BOUNDING ELLIPSOID 33

2. PROBLEM FORMULATION

Consider a single-input, single-output system which is modelled by an autoregressive exogenous input (ARX) equation of the form

n m

i = l j = O y k = a i y k - i + b j U k - j + Wk (2)

where ( Y k ] and ( u k ] are measurable output and input sequences respectively and b) is a sequence representing disturbances and noise that corrupt the system.

If we define f?*T = [f?, d2 ... el] as the true system parameter vector and 4;-I = [ Y k - 1 . . . y k - n U k ... U k - m ] as the regression vector, with I = n + rn + 1, then (2) can be rewritten as

Y k = $ : - l e * + W k (3)

The standard system identification problem is to devise a scheme or an algorithm for identifying the system parameter f?* based on the observed sequence ( y k , @ k - I ) . Of course, assumptions about the unobservable sequence ( W k ) are required. Here we shall assume that W k is known to be bounded via

I W k [ < W k , k = 1 , 2 , ... (4)

where ( w k ) is a known sequence. From (3) and (4) it is clear that

I Y k - dT-If?*( < w k ( 5 )

Thus for each measurement (y i , 4 i - 1 ) the set of all possible parameters f? which are consistent with the observation is given by

Sj = (eE R': ( Y i - &f? 1 < Wj) (6)

After k measurements the set of f? which is consistent with ( 5 ) must be given by k

S:= n S i i = l

(7)

Since S t defines a set of f? which are consistent with k measurements, any member of S$ will be a valid estimate of 8 * as far as the k measurements are concerned. Hence, given k measurements, it is always possible to use linear programming to compute orthotopes to bound S $ . 8 9 9 However, in many on-line applications this approach is computationally too demanding.

Our identification problem is thus to design a recursive parameter-updating procedure that will give parameter point estimates and a bounding region in the parameter space containing the true parameter. We also want the algorithm to be simple and robust, since this is crucial in many adaptive signal-processing and adaptive control applications.

3. MODIFIED RECURSIVE LEAST SQUARES WITH DEAD ZONE

The idea of using a dead zone to deal with bounded noise is not new and several results have appeared in the literature. l 2 - I 4 The dead zone is employed to prevent parameter update when the prediction error becomes smaller than a bound, thus eliminating the effect of noise from the measurements. It is desirable that the dead zone threshold coincide with the known noise bound. However, at present, all the available least-squares-type algorithms with a dead

34 R. J . EVANS, C. ZHANG AND Y. C. SOH

2 0 n e ' ~ - ' ~ require the dead zone threshold to be larger than the known noise bound. This problem is addressed here and we show that the exact noise bound can be used. Thus our result retains all the advantages of a recursive scheme with a dead zone while allowing the extreme value of the prediction error to be exactly equal to the known noise bound.

Now let 8 k be an estimate of 8* at time k and define the parameter estimation error as

8k = 8* - 6 k (8)

Then together with (3) the prediction error at time k is given by T -

e k = Y k - 8 Z - l d J k - l = dJ k - i 8 k - 1 + Wk

Note that the estimation error & - I and the unobservable noise Wk are combined in e k .

We define the dead zone function as

e k - w k if ek > w k

if - w k f e k < w k f ( e k , w k ) = 0 I l?k + w k if e k < - w k

The above function is shown in Figure 1. Furthermore, we define

( 0 otherwise

(9)

0 otherwise

(13) P i Pk =

1 + 4 z - l p k - I d J k - I

where P k - l is a matrix of the parameter estimation scheme to be defined later. Although CYk is an unmeasurable function, it is clear that CYk, p i and Pk satisfy the inequality

1 > Q k a @ A > P k > O (14)

Furthermore, (Yk, @I and Pk are strictly positive if f ( e k , w k ) # 0. With the above preliminaries we propose the following recursive estimator with a dead zone:

Figure 1.


p i 1 = pi21 f l k $ k - I d ; - l , Po=POT>O (16)

We shall now establish several desirable properties of the algorithm (15), (16). Loosely speaking, the dead zone algorithm has two key properties: first, the estimation error decreases at each step; second, the prediction error decreases until it reaches the size of the dead zone (and hence the size of the known noise bound). These results are made precise below.

Theorem I

has the following properties assuming that ( 5 ) holds: The estimator (15), (16) with e k , f ( . , .) and flk defined as in (9), (10) and (13) respectively

(i) v k = 8 Z ~ i V k , where 8k = e* - Ok, is a non-increasing function and thus

where

36

with

Note that cf > 0 if f(ek, W k ) # 0. Thus v k < V k - I with strict inequality when f(ek, w k ) # 0. Hence we obtain (i). Note that from (17) we also have

A v k < - Pk-lf(ek, w k ) 2 (19)

Summing both sides of (19), we obtain (ii). Then (a) immediately follows. To prove (b), we note that from (a) we have

ft = 0 P i lim k-w 1 + G k - 1

Hence if ) is bounded, fk --+ 0 and this implies 1 ek I < w k . Part (c) follows from the result of part (a).

4. ESTIMATION USING DEAD ZONE AND BOUNDING ELLIPSOID

In the previous section we were mainly concerned with an estimate 6% of the system parameter. While this estimate has many desirable properties, we need to know how far this estimate is from the true parameter at each iteration. This can be achieved by defining a bounding ellipsoid E k with & as its centre, i.e.

(20)

where p k is a positive definite matrix which, together with the scalar ~ k , defines the size and shape of the bounding ellipsoid. Obviously we want E k to contain the true parameter 8* and we also want ( E k ) to be a decreasing sequence, since this will reflect the increased knowledge about the estimate 8k. This can be done by initializing the recursive algorithm with a sufficiently large ellipsoid EO that contains the true system parameter 8*. Then at each iteration of the estimation we systematically reduce the size of E k while ensuring 8* E Ek.

One difficulty with the existing dead zone algorithms is that once the prediction error is within the dead zone, the algorithm no longer updates. Here we still update the bounding ellipsoid even though there is no parameter update. This is highly desirable, since a small prediction error should enable us to gain more knowledge about the current estimate and to reduce the size of the bounding ellipsoid.

In order to achieve these desirable properties, we propose the following recursive estimation algorithm with a dead zone and a bounding ellipsoid.

E k = (6: (6 - e k ) T P k 1 ( 6 - ek) < 7;)

DEAD ZONE AND BOUNDING ELLIPSOID 3 1

Algorithm I

1. Set k = 0 and initialize 80, PO = Pa > 0 such that

2. k = k + l . 3 . If I ej I > wk (i.e. f ( e k , wk) f 0), then

2 T

(1 ek 1 -k wk)2

P i 1 = X*P/2I+ (1 - x * ) T k - l 6 k - l 6 k - 1

= ri-1 Ek = (8: (6 - 8k)TPi'(6 - g k ) < T k j 2

where A * € (0,1], which maximizes de tP i ' in the sense that detPi'(X*) 2 detPiI(A) for all X c (0, 11.

4. Go to step 2.

Remark 1

The above estimation algorithm is different from the algorithm (15) and (16) in that it provides a bounding ellipsoid centred at the point estimate 8k. The matrix Pk has a new interpretation in the sense that it characterizes the size and shape of the bounding ellipsoid. While the point estimation algorithm (15), (16) ceases the updating of & and Pk when the prediction error is small, the above algorithm may still update the matrix Pk to reduce the size of the ellipsoid. In general the point estimate provided by the algorithm with a bounding ellipsoid will be different from that of the algorithm (15) and (16).

Remark 2

The rationale for computing A * when lekl < wk is as follows. When 8 k = 8 k - l ,

+ Wk and hence cpT-16k-l = ek - W k . Thus a set of 6 which is centred at 13k and ek = is consistent with the current measurement is given by

$ k = ( 6 : ( 6 - 8 k ) T ~ k - I # T - l ( e - 8 k ) < ( I e k I + wk)2] Therefore for any X E (0, I ]

38 R. J. EVANS, C. ZHANG AND Y. C. SOH

defines an ellipsoid enclosing the intersection of Ek-1 and Sk. The smallest ellipsoid is thus obtained by maximizing det Pk'(X) with respect to X.

The computation of X * to maximize det Pk'(X) is straightforward and is given by the following.

Lemma 1

Suppose that f(ek, wk) = 0. Then there exists a X* € (0, 1) such that det Pk'(X*) 2 det Pk'(X) for all X E (0,l) if and only if

where I = n + m + 1 (i.e. the dimension of the system parameter). Furthermore,

if the above inequality holds.

Proof. Let D(X) = det Pkl(X). Then Tk-l4k-1 2 4 T k-1

D(X) = det ( h P i j ~ + (1 - (, ek + wk)2

Thus

However,

Also,

Hence the result.


Remark 3

The above lemma indicates that if the signal is sufficiently large, we can reduce the size of the bounding ellipsoid even though the prediction error is small and there is no parameter updating.

We can now establish that with Algorithm 1 the sequence of bounding ellipsoids is decreasing.

Theorem 2

Consider the bounding ellipsoid estimator with a dead zone (Algorithm 1) and suppose that 8 * c Eo. Then O * € Ek for all subsequent k. Furthermore, (EkJ is a decreasing sequence with respect to the determinant measure.

Proof. Define v k = @PL%, where I% = (8* - & ) . From the algorithm we have

v k - T : < v k - l - T i - 1

However, Vk-1 < T ; - 1 iff O*€Ek-I. Thus

v k < T: whence 8 * € Ek

Hence it is clear that if O*€Eo, then 8*€Ek for all k. To prove that (Ek) is a decreasing sequence, we first note that when f(ek, Wk) # 0,

p-1 k = p-1 k-1 -k @k4k-I4:-1, @k > 0

d = 4 - 1 - c i < d-1 since E: > o Thus det P i 1 > det Pi21. Also,

Hence Ek is a smaller bounding ellipsoid than Ek-1. If f (ek, Wk) = 0, then r i = 72-1 and det Pi'(X*) 2 det PL'(1) = det Pp-!1, where equality holds if X* = 1. Hence the size of Ek is

The bounding ellipsoid estimation algorithm updates the matrix Pk even though there is no updating of the point estimate I!?&. Thus in general the point estimate 6 k provided by the algorithm with a bounding ellipsoid will be different from that of the algorithm (15) and (16). However, the bounding ellipsoid estimation algorithm retains a number of properties of the estimation algorithm with a dead zone in (15) and (16). These are given in the following theorem.

no larger than Ek-I. 0

Theorem 3

(Algorithm 1) has the following properties: If ( 5 ) is satisfied and 8 * € Eo, then the bounding ellipsoid estimator with a dead zone

m - (i)

(ii) lim 1 ek I < wk provided that (r#~k- 11 is bounded

(iii) Iim 1) Bk - 6 k - 1 1) = 0.

P k f ,$ < co, where f k = f(ek, Wk), and this implies that lim @ k f i = 0 k = l k-+W

k - + m

&- fo r ,


Progf. (i) With E: as defined in (18) and according to Algorithm 1, the bounding ellipsoid E k centred at is characterized by

k

i = l (6 - 6 k ) T P k 1 ( 6 - 6 k ) < 7; = 7; - E f

W W

Then we have C @k f ' k < C &'k < 7'0 and lim f i k f ' k = o follows. k = l k = l k - m

(ii) Lemma 2 in the Appendix shows that P k is bounded for all k . This together with limk-, P k f 2 = 0 and boundedness of C$k gives limk.+,f(ek, w k ) = 0. Hence we establish (ii).

0 (iii) The result follows from (i) and the boundedness of P k .

Remark 4

Theorems 2 and 3 show that the new bounding ellipsoid estimator with a dead zone takes both the size and centre of the bounding ellipsoid into account and ensures convergence of the bounding ellipsoid as well as its centre estimate. The algorithm is also robust in the sense that the dead zone eliminates noise and disturbances and guarantees that only relevant measurements will be used for estimation. This robustness can be further seen by observing that for some k the estimate 6 k may coincide with the true parameter, i.e. 6 k = 6*. In such cases the algorithm can still reduce the size of the bounding ellipsoid but will not change the estimate &. By comparison, for the same situation the existing bounding ellipsoid estimation algorithm^'.^ may update the centre e k away from the true system parameter when reducing the size of the ellipsoid. We note that the bounding ellipsoid estimation algorithm in Reference 6 has similar estimation convergence properties to those in Theorem 3 . However, these properties require a strong persistent excitation condition. Therefore, compared with the existing bounding ellipsoid estimation algorithms, the proposed algorithm provides a robust and smoothly convergent centre estimate. This is highly desirable, since it is natural that the centre estimate be used for analysis and design in applications such as real-time adaptive control.

5 . UNKNOWN BUT BOUNDED TIME-VARYING PARAMETERS

In many physical systems the system parameters will change with respect to time. Thus the system identification scheme must be able to track this time variation or at least be able to give a bound on the parameters. For a discussion on tracking time-varying systems, see for example the survey papers by Ljung and Gunnarsson" and Norton and Mo. l9

Here we consider a very general model with time-varying parameters. We assume the parameter variations to be bounded with respect to a nominal time-invariant parameter, i.e.

e k = e * + A 6 k , k = 1 , 2 , ... (22)

where O * is time-invariant, 1 1 A 8 k I( < a! and a! is a known positive constant. In this case the dead zone algorithm discussed in the earlier sections can be readily modified to handle both the bounded noise and the bounded time-varying parameters.

With (22) the underlying input-output equation is given by T *

y k = 4 B - l e k + W k = # ' E - l 6 * + d ' Z - l A e k + O k = 4 k - 1 8 + y k

where yk = 4 E - l A e k + O k . Hence if I wk I < w k and 11 A& 11 < a, then I y k I < rk , where

r k = ( 1 d'k-1 I( a! -I- w k


Thus all the results in the preceding sections can be extended in a straightforward manner with wk replaced by rk.

6 . CONCLUSIONS

We have extended the modified recursive least-squares estimation with a dead zone to handle the problem of bounded-error measurements. There are two main contributions. First we have reduced the dead zone to the size of the assumed bounded error. Second we have established the link between a parameter-bounding ellipsoid and the estimation algorithm with a dead zone. We have been able to reduce the size of the bounding ellipsoid even when the prediction error is smaller than the dead zone threshold and there is no updating of the point estimate.

ACKNOWLEDGEMENTS

This work was supported by the Australian Research Council, the Tewksbury Fund and the Rowden-White Foundation.

APPENDIX

Lemma 2

If ( & ] is a bounded sequence, then the matrix Pk satisfies Pk < 00 for all k.

Proof. Let C be a constant such that 11 r#Ik 1 1 < C for all k and let hi'), i = 1 , ..., n + m + 1 , be eigenvalues of P i ' . At time k there exists an integer j , 1 < j < n + m + 1 , such that

Xi') = min ( A f." , i = 1 , . . . , n + rn + 1 )

Then Pk is bounded if and only if A i l ) is bounded away from zero. According to (21) in Algorithm 1 for f ( e k , W k ) = O and for any eigenvalue Xi?, 2 17&z/Wz, the updating of PgJi by (21) gives ? h i ) <Xi?,. This indicates that the updating of Pi21 by (21) does not increase the value of A i ' ) . Consequently, if ni+j X i " 2 I(1- 1)7aC2/ W 2 , the updating of Pk=l1 by (21) does not increase the value of niz, A&') . In the above analysis we have used the fact that the largest possible value of A * in (21) is (I- I)//.

The bounding ellipsoid estimator (Algorithm 1) guarantees Ai' ) n X i "

det P i ' i # j 21 -- det PO1 - PO'

It follows that A & j ) < W z det PO'/( / - 1)7iCz, then I I , ; L ~ A & ~ ) must satisfy

n A&" 2 [(I - 1)7ac2/ wz (23) , # j

Note that the value of X i " is reduced only when (21) is applied to update Pi' and that any decrement in will lead to an increment in nizj A$') to ensure that ,Pk' is increased. Then A i j ) , the minimum eigenvalue of P i 1 , will not be further reduced whenever A i J ) < W z det PO'/(/- 1)7$C2, since (21) can no longer increase the value of I I i # j X i " which satisfies (23). Hence X i " is bounded away from zero and Pk is bounded. 0

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bounded-error estimation using dead zone and bounding ellipsoid

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