Automatica 45 (2009) 1524–1529

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Automatica

journal homepage: www.elsevier.com/locate/automatica

Brief paper

Elementwise decoupling and convergence of the Riccati equation in theSG algorithmI

Alexander Medvedev ∗, Magnus EvestedtInformation Technology, Uppsala University, SE-751 05 Uppsala, Sweden

a r t i c l e i n f o

Article history:Received 28 July 2008Received in revised form24 November 2008Accepted 13 February 2009Available online 24 March 2009

Keywords:Riccati equationsRecursive estimationDiscrete systemsWindupExponentially stable

a b s t r a c t

It is shown that the difference Riccati equation of the Stenlund–Gustafsson (SG) algorithm for estimationof linear regression models can be solved elementwise. Convergence estimates for the elements of thesolution to the Riccati equation are provided, directly relating convergence rate to the signal-to-noiseratio in the regression model. It is demonstrated that the elements of the solution lying in the directionof excitation exponentially converge to a stationary point while the other elements experience boundedexcursions around their current values.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

One of the recent contributions to a plethora of covarianceanti-windup algorithms for the Kalman filter is a specializationof the Riccati equation suggested in Stenlund and Gustafsson(2002), further referred to as Stenlund–Gustafsson (SG) algorithm.A similar idea is presented somewhat later in Cao and Schwartz(2004).The SG algorithm has been suggested as an ad hoc solution

and without convergence analysis but it appeared to work well insimulations. An important feature of the algorithm that has alreadybeen demonstrated in the seminal publication is that the differenceRiccati equation can be written for this particular case in the formof a linear (Sylvester) matrix equation. This fact has facilitatedfurther results on the dynamics of the SG algorithm and is provenfor a general case in Evestedt and Medvedev (2006b).

I This work has been carried out with financial support by Swedish ResearchCouncil. A conference version of this paper was presented at 17th IFAC WorldCongress, July, 6–11, 2008, Seoul, Korea. This paper was recommended forpublication in revised form by Associate Editor Er-Wei Bai, under the direction ofEditor Torsten Söderström.∗ Corresponding address: Information Technology, Uppsala University, Box 337,SE-751 05 Uppsala, Sweden. Tel.: +46 18 4713064; fax: +46 18 503611.E-mail address: Alexander.Medvedev@it.uu.se (A. Medvedev).

0005-1098/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2009.02.010

In Medvedev (2004), a formal proof for non-divergence of theRiccati equation for the SG algorithm was given using theory ofconverging matrix products. It was also shown in the same paperthat the linear mapping corresponding to the Riccati (Sylvester)equation is not a contraction in most common norms but aparacontraction in some unspecified norm. The latter discoveryposed the question why and under what conditions the SGalgorithm converges.The remarkable robustness of the SG algorithm against lack of

excitation called for its use in a number of engineering applicationssuch as active vibration control (Olsson, 2005), acoustic echocancelation (Evestedt, Medvedev & Wigren, 2008) and changedetection (Evestedt & Medvedev, 2006a, in press).In this paper, convergence properties of the SG algorithm are

studied via an elementwise decomposition of the Riccati equa-tion, clearly revealing the underlying convergencemechanism andits close relationship to the excitation condition of recursive pa-rameter estimation, (Ljung & Gunnarsson, 1990). After some pre-liminaries introducing necessary notions and notation, recursiverelationships for elementwise solution of the Riccati equation arederived. Further, exponential convergence of the Riccati equa-tion in the direction of excitation is proved followed by conver-gence rate bounds. It is also shown that each element outside ofthe current excitation direction remains within a bounded inter-val whose range is defined by the excitation and the parametersof the algorithm. Obtained convergence results are illustrated bysimulation.

A. Medvedev, M. Evestedt / Automatica 45 (2009) 1524–1529 1525

2. Preliminaries

Consider the following difference Riccati equation typical toparameter estimation of regression models

P(t) = P(t − 1)−P(t − 1)ϕ(t)ϕT(t)P(t − 1)r(t)+ ϕT(t)P(t − 1)ϕ(t)

+ Q (t) (1)

where P(·) ∈ Rn×n, Q (·) ∈ Rn×n,Q (·) = Q T(·),Q (·) ≥ 0,P(0) = PT(0), P(0) ≥ 0, ϕ(·) ∈ Rn is a regressor vector, r(t) isa positive scalar, and t ∈ {1, 2, . . . ,∞}.In Stenlund and Gustafsson (2002), Q (t) is treated as an ‘‘input

signal’’ that can be designed to provide desirable behavior of P(t),for instance convergence to a certain point or manifold.A special case of the difference Riccati equation arises in the SG

algorithm where the following form of the free matrix term

Q (t) =Pdϕ(t)ϕT(t)Pd

r(t)+ ϕT(t)Pdϕ(t)(2)

is introduced for a given constant Pd ∈ Rn×n, Pd > 0. Evidently, thematrix Pd is then a stationary point of (1). Notice that the classicalstability proof for the Kalman filter assuming Q (t) > 0 providede.g. in Jazwinski (1970) does not apply to the SG algorithm sincerank Q (t) = 1, according to (2).As shown in Stenlund and Gustafsson (2002) for non-singular

P(t) and in Evestedt and Medvedev (2006b) for a general case, thedeviation from the stationary point E(t) = P(t) − Pd obeys therecursion

E(t + 1) = A−1t (P(t))E(t)A−Tt (Pd) (3)

where At(X) = I + r−1(t)Xϕ(t)ϕT(t).The persistence of excitation condition plays a significant role

in the dynamic behavior of (1). The condition is that there existc ∈ R+ and an integer N such that for all k

cI ≤k+N∑t=k

ϕ(t)ϕ(t)T. (4)

When (4) is not satisfied, some eigenvalues of P(·) grow linearlyin time. This phenomenon is usually referred to as (covariance)windup. The SG algorithm has been developed in Stenlund andGustafsson (2002) specifically in order to deal with the windupproblem in a systematic manner and Riccati equation (1, 2) isproved to be non-diverging under lack of excitation in Medvedev(2004).Introduce the following notation

ρt =ϕT(t)ϕ(t)r(t)

, ρ∗ = inftρt , Ut =

ϕ(t)ϕT(t)ϕT(t)ϕ(t)

where Ut is a projection matrix specifying the direction ofexcitation and ρt characterizes the relative instantaneous intensityof excitation. The latter can loosely be called signal-to-noise ratiosince ϕT(t)ϕ(t) is the squared Euclidean norm of the regressionvector and r(t) usually stands for the variance of measurementnoise in the underlying regression model.

3. Elementwise solution

It is practical to bring Eq. (3) to a more compact and structure-revealing form, as suggested in Medvedev (2004).Denote the normalized eigenvector of Ut corresponding to

the eigenvalue equal to one as ξ t1 . Eigenvalue–eigenvectordecomposition of Ut immediately yields Ut = ξ t1ξ

t1T. Thus, the

direction of excitation at time t can be characterized by the vector

ξ t1 . Let the sequence {Ut} be persistently exciting on each intervalof n consecutive steps, i.e. for τ = {0, n, 2n, . . .}

det T (τ ) 6= 0, T (τ ) =[ξ τ1 . . . ξ τ+n−11

].

Now the matrix T is constant on each interval t = τ , . . . , τ +

n − 1 and can be used as a Lyapunov transformation (Medvedev,2004; Rugh, 1996). Essential properties of T are that it includes allpossible directions of excitation and it is always non-singular. Thus,the columns of the transformation matrix are denoted as

T (t) =[ξ 11 (t) . . . ξ n1 (t)

]with time variable dropped when appropriate to save space.Introduce a new state matrix Z(t) = T T(t)E(t)T (t). After the

Lyapunov transformation, (3) reads

Z(t + 1) = A−1t (P(t))Z(t)A−Tt (Pd) (5)

where

At(·) = T TAt(·)T−T.

It is essential to keep in mind that T (t) is a time-varyingtransformation and changes after each n steps of the algorithm.This implies that the state elements of Z(t) evolve not only dueto recursion (5) but as well as a function of T (t).Let ξ i1 be the current excitation direction, i = 1, . . . , n. For some

X ∈ Rn×n, X ≥ 0, consider two vectors

dTi (X) =[ξ 11TXξ i1 . . . ξ n1

TXξ i1

]and

DTi (X) =[D1i (X) . . . Dni (X)

]which are related to each other as to

Di(X) =ρidi(X)

1+ ρiξ i1TXξ i1

. (6)

In Medvedev (2004), the matrix function Ai(X) and its inverseare calculated to be

Ai(X) = I +[0n×(i−1) ρidi(X) 0n×(n−i)

],

A−1i (X) = I −[0n×(i−1) Di(X) 0n×(n−i)

].

(7)

Then Ai(X) is a sum of a unit matrix and a matrix whose non-zero elements are collected in just one column. The position ofthe column is defined by the current excitation direction ξ i1. Thestructure of Ai(X) brought about by the Lyapunov transformationmakes it possible to distinguish between the elements of Z inthe direction of excitation and those outside of it. Formally, theelements zkl, (k = i) ∨ (l = i) are in the excitation direction andall others (e.g. zkl, k = 1, . . . , n; l = 1, . . . , n; k 6= i, l 6= i) areoutside.For X ≥ 0, let λj(X) ≤ λj+1(X), j = 1, . . . , n− 1 be the ordered

eigenvalues of the matrix.

Proposition 1. Let P(t) be a solution to (1) and (2), then thefollowing relationship holds for all i, k ∈ [1, n]

Dki (P(t)) =ρi

(ξ k1TPdξ i1 + zik(t)

)1+ ρi

(ξ i1TPdξ i1 + zii(t)

) .

1526 A. Medvedev, M. Evestedt / Automatica 45 (2009) 1524–1529

Proof. From the definitions of Z and E, it follows that

P(t) = Pd + E(t) = Pd + T−T(t)Z(t)T−1(t).

Then

Dki (P(t)) =ρi

(ξ k1TPdξ i1 + ξ

k1TT−T(t)Z(t)T−1(t)ξ i1

)1+ ρi

(ξ i1TPdξ i1 + ξ

i1TT−T(t)Z(t)T−1(t)ξ i1

) . (8)

Now, taking into account the identity

ξ k1TT−T(t) =

[0 . . . 1 0 . . .

]k

leads to

ξ k1TT−T(t)Z(t)T−1(t)ξ i1 = zik(t).

The above equality together with (8) yields the sought result. �

In Medvedev (2004), it was demonstrated how the originalrecursive scheme (3) can be at each step decomposed into a setof fourth-order linear autonomous systems of triangular structure.Furthermore, at each t , the dynamics of each element zkl ofthe matrix Z is shown to be defined by four numbers, namelyDli(Pd),D

ii(Pd),D

ki (P),D

ii(P). Therefore, the updates of the elements

of Z outside of the current excitation direction are completelydefined by the values of the elements in the excitation direction.Provided (1, 2) is initialized at a symmetric matrix P(0), the

solution P(t) is obviously symmetric. Thus, a further simplificationof the equation structure can be obtained for such practicallyimportant cases by utilizing the result of Proposition 1.

Proposition 2. For symmetric solutions of (3), i.e. P(t) = PT(t), theelements of z(t) = {zkl, k = 1, . . . , n; l = 1, . . . , n} are governedby

zkl(t + 1) = zkl(t)+ ρi

(ξ l1TPdξ i1ξ

k1TPdξ i1

1+ ρiξ i1TPdξ i1

−(ξ l1TPdξ i1 + zil)(ξ

k1TPdξ i1 + zik)

1+ ρi(ξ i1TPdξ i1 + zii(t))

)(9)

or, for the special case of l = i

zki(t + 1) =(1+ ρiξ i1

TPdξ i1)zki(t)− ρiξk1TPdξ i1zii(t)(

1+ ρiξ i1TPdξ i1

) (1+ ρi(ξ i1

TPdξ i1 + zii(t))) (10)

for which the expression reads for k = l = i as

zii(t + 1) =zii(t)(

1+ ρiξ i1TPdξ i1

) (1+ ρi(ξ i1

TPdξ i1 + zii(t))) . (11)

Proof. See Appendix. �

4. Convergence

For an arbitrary element of Z , elementwise recursion (9) ateach step is a sum of the previous value of the element and anupdate that is independent of the element in question. This alsoexplains why the mapping P(t) → P(t + 1) is not a contraction,(see Medvedev (2004) for a detailed discussion) and significantlycomplicates the choice of tools for convergence analysis.Decomposition (9) is valid for any element of Z nomatterwhere

it is situated in the matrix with respect to the current excitation. Itcan be intuitively expected that the elements of Z in the direction ofexcitation converge while all other elements do not. Experimental

studies in Evestedt andMedvedev (2006a), Evestedt andMedvedev(2008) and Olsson (2005), indicate that the solution of Riccatiequation (1, 2) converges to Pd whenever excitation condition (4)is satisfied. However, no formal proof of this fact can be found inthe literature.Regarding stationary behavior of the SG algorithm under lack of

excitation studied in Evestedt and Medvedev (2006b), it is shownthat solutions of (3) converge to Pd in the elements that lie inthe persistent directions of excitation. However, the mechanismof convergence has so far not been revealed.The result of Proposition 2 gives an insight into why and

how the solutions of (3) converge. Indeed, (11) implies veryfast exponential convergence of the diagonal elements of Z inthe excited directions. Therefore, the following approximation isjustified

zii(t + 1) ≈zii(t)(

1+ ρiξ i1TPdξ i1

)2 .Clearly, the rate of convergence increases for higher signal-to-noiseratios. Now, it is safe to assume that zii is small which yields thefollowing approximation for the off-diagonal elements of Z in thedirections of excitation, cf. (10)

zki(t + 1) ≈zki(t)

1+ ρiξ i1TPdξ i1

.

The elements zki, k 6= i also exhibit exponential convergence but ata lower rate than for the diagonal elements. Thus, the assumptionthat zki, k = 1, . . . , n are small is as well justified. This turns (9)intozkl(t + 1) ≈ zkl(t), k 6= i, l 6= i.Naturally, all the approximations above are valid in the vicinityof the stationary point Pd and good excitation conditions, i.e.reasonably high values of ρi.As demonstrated in Evestedt and Medvedev (2008), the

approximate relations following from elementwise form (9) areuseful in deriving recursive parameter estimation algorithms withreduced computational load, as well as preserved anti-windupproperty and exponential convergence under sufficient excitation.Recapitulating for the simplifications above, it can be concluded

that all elements of Z in the directions of excitation converge ex-ponentially to zero while all other elements remain approximatelyconstant.At this point, it is instructive to recall that the Lyapunov

transformation T (t) changes each n steps therefore altering thevalues of generally all elements of Z . After such transformationchange, any zkl can either increase or decrease. This however doesnot influence the type of convergence due to the properties ofLyapunov transformations (Rugh, 1996).In the sequel of the section, formal convergence proofs for

solutions of (5) are provided. First the convergence in the currentdirection of excitation is examined.

Proposition 3. The mapping[zki(t + 1)zii(t + 1)

]= Fki

[zki(t)zii(t)

](12)

where

Fki =E(ρiξ

k1TPdξ i1, ρiξ

i1TPdξ i1)+ ρiξ

i1TPdξ i1I

(1+ ρiξ i1TPdξ i1)

(1+ ρi

(ξ i1TPdξ i1 + zii(t)

))and

E(a, b) =[1 −a0 1− b

]is a contraction for all k = 1, . . . , n; k 6= i.

A. Medvedev, M. Evestedt / Automatica 45 (2009) 1524–1529 1527

Proof. Indeed, a direct evaluation gives

λ2(F TkiFki) =1

2(1+ ρiξ i1TPdξ i1)2

(1+ ρi

(ξ i1TPdξ i1 + zii(t)

))2×

((1+ ρiξ i1

TPdξ i1)

2+ ρ2i (ξ

k1TPdξ i1)

2+ 1

+ ρi

√((ξ i1TPdξ i1)2 + (ξ

k1TPdξ i1)2

)fki

)(13)

where fki = (2+ ρiξ i1TPdξ i1)

2+ ρ2i (ξ

k1TPdξ i1)

2. Noticing that

(ξ k1TPdξ i1)

2≤ λ2n(Pd)

and that λ2(F TkiFki) is monotonically decreasing in ρi and ξi1TPdξ i1

with

limρi→0

λ2(F TkiFki) = 1, limξ i1TPdξ i1→0

λ2(F TkiFki) < 1

completes the proof. �

Proposition 3 proves that all the elements of Z(t) in the currentexcitation direction exponentially converge.The results below guarantee that the increase in the elements

outside of the current excitation direction is bounded at each stepfrom above and below.

Proposition 4. For each element zkl(t), k 6= i, l 6= i of Z(t) in (5),the following inequality applies

|zkl(t + 1)− zkl(t)| < zii(t)+2ρi(1+ ρiξ i1

TPdξ i1) (14)

Proof. Consider the second term on the right-hand side of (9).Recalling from Proposition 2 in Medvedev (2004) that

|Dki (P)| =ρi|ξ

k1TPdξ i1 + zik|

1+ ρi(ξ i1TPdξ i1 + zii(t))

< 1

the following inequalities follow

−|ξ l1TPdξ i1| − |ξ

l1TPdξ i1 + zil|

< ρi

(ξ l1TPdξ i1ξ

k1TPdξ i1

1+ ρiξ i1TPdξ i1

−(ξ l1TPdξ i1 + zil)(ξ

k1TPdξ i1 + zik)

1+ ρi(ξ i1TPdξ i1 + zii(t))

)< |ξ l1

TPdξ i1| + |ξ

l1TPdξ i1 + zil|.

Therefore

−1ρi(1+ ρiξ i1

TPdξ i1)−

1ρi

(1+ ρi(ξ i1

TPdξ i1 + zii(t))

)< zkl(t + 1)− zkl(t)

<1ρi(1+ ρiξ i1

TPdξ i1)+

1ρi

(1+ ρi(ξ i1

TPdξ i1 + zii(t))

)which immediately gives (14). �

Notice also that regarding bounds on the updates of theelements of Z outside of the direction of excitation, the lowerbound is always negative and the upper bound is always positivedespite indefinite sign of zii. Indeed, since Dii(·) is bounded 0 <Dii(P(t)) < 1 and taking into account (A.5), one obtains

0 <1ρi(1+ ρiξ i1

TPdξ i1) < zii(t)+

2ρi(1+ ρiξ i1

TPdξ i1). (15)

5. Bounds

The bounds provided by (14) capture a phenomenon observedin simulation of (5), namely that the excursions in the elementsof Z(t) outside of the current excitation direction decrease withtime. This is explained by exponential convergence of zii. Anotherproperty of (5) is that the excursions lessen with higher excitationintensity. This can be as well deduced from (14).Constant and therefore less sharp bounds can be easily obtained

around the stationary solution, i.e. assuming that zii is small

−2ρ∗(1+ ρ∗λn(Pd)) < zkl(t + 1)− zkl(t)

<2ρ∗(1+ ρ∗λn(Pd)).

For bounds valid even near the initial conditions on (5), moreinsight into evolutions of zii is needed.As mentioned before, zii(t) can take both positive and negative

values. Inequality (15) implies that the negative values of zii(t) arefor any t bounded from below

zii(t) > −1ρi(1+ ρiξ i1

TPdξ i1) ≥ −

1ρi(1+ ρiλn(Pd)) .

This means that the range of negative zii is basically defined byPd for large signal-to-noise ratios and by the ratio itself when it ispoor.Furthermore, by comparing (A.6) and (11), one obtains

zii(t + 1) =1ρi

(Dii(P(t))− D

ii(Pd)

).

Since both Dii(P(t)) and Dii(Pd) are positive and less than one

|Dii(P(t))− Dii(Pd)| < 1

and, for all t

|zii(t + 1)| <1ρi. (16)

The inequality above shows that, for good excitation conditions,zii should take a small value already after the first step of thealgorithm in the excitation direction in question. Taking intoaccount the exponential convergence due to (11), all zii are quitesmall for t > n, provided the excitation is good and the noisevariance r in the regression model is low.The convergence of (1) and (2) is fast for high values of ρt and

a properly chosen Pd. One can also conclude that increasing theconvergence rate through the choice of Pd also increases the rangeof variation for zii.The result of Proposition 3 gives rise to the following inequality∥∥∥∥zki(t + 1)zii(t + 1)

∥∥∥∥22≤ λ2(F TkiFki)

∥∥∥∥zki(t)zii(t)

∥∥∥∥22, k 6= i. (17)

Taking into account (11) yields

z2ki(t + 1) ≤ λ2(FTkiFki)z

2ki(t) +

λ2(F TkiFki)−

1(1+ ρiξ i1

TPdξ i1)2 (1+ ρi(ξ i1

TPdξ i1 + zii))2 z2ii (t)

< λ2(F TkiFki)(z2ki(t)+ z

2ii (t)

).

It can be easily checked that the factor z2ii in the first inequality isnon-negative and strictly less than one.

1528 A. Medvedev, M. Evestedt / Automatica 45 (2009) 1524–1529

Fig. 1. Bound (16) on the diagonal element in the direction of excitation. Directionof excitation is changed periodically, i = 1, 2, 3 . . ..

6. Simulation results

The bounds on the elements of Z(t) are illustrated using asimulation example with n = 3. Regressor vectors are chosen tovary in norm but be periodic in the direction of excitation whichgives a constant transformation matrix T .Letting r(t) = 1 yields ρt = ϕT(t)ϕ(t). In this simulation,

ρi is taken as a random sequence of numbers between 0 and10. This choice is only for the purpose of illustration since it isnot what typically occurs when the Riccati equation is used in aparameter estimation algorithm. In the latter case, the regressorvectors include input and output signal values and possess certainstructure. The properties of the parameter estimates in the SGalgorithm are investigated in Evestedt and Medvedev (2008).Furthermore, the matrices P(0) and Pd are selected as random

positive definite matrices to make the example as generalas possible. The Riccati equation in the SG-algorithm wasimplemented elementwise according to Proposition 2 and theevaluated elements were utilized to illustrate (17) and (16).In Fig. 1, the actual values of zii are given and comparedwith the

upper bound provided by (16). As anticipated, being independentof the equation convergence, the bound is useful during initialiteration steps of the Riccati equation but becomes conservativefurther on.The contraction property proved in Proposition 3 is illustrated

in Fig. 2. The three subplots correspond to three elements of Z(t)in the current excitation direction at the time instances when theyare outside of themain diagonal. Since the direction of excitation isperiodic, each of the elements appears on the main diagonal whilethe vector Di(·) propagates through the columns of the secondterm on the right-hand side of (7). Clearly, this bound captures theconvergence and is not excessively conservative.

7. Conclusion

It is shown that the Riccati equation of the SG algorithm canbe solved elementwise. The Riccati equation in question has afree term that is of rank 1 and its convergence properties aretherefore not guaranteed by the standard Kalman filter theory. Theexpressions for the elementwise solution reveal the underlyingconvergence mechanism and are instrumental in the derivationof the upper and lower bounds on the elements of the Riccatiequation solution.

Fig. 2. The stepwise contraction property of the SG-algorithm in the excitationdirection. The plots show the right-hand side (∗) and the left-hand side (o) of (17).The instances when zki ≡ zii are omitted here and provided in Fig. 1 instead.

Appendix. Proof of Proposition 2

Let ξ t1 = ξ i1. Then the dynamics of the elements of z(t) ={zkl, k = 1, . . . , n; l = 1, . . . , n} are governed by

zkl(t + 1) = E(Dki (Pd),D

ii(Pd)

)⊗ E

(Dli(P(t)),D

ii(P(t))

)zkl(t) (A.1)

wherezTkl =

[zkl zki zil zii

].

Notice that for symmetric solutions, zkl and zlk are equivalent up topermutation of elements. Rewriting (A.1) elementwise giveszkl(t + 1) = zkl(t)− Dli(P)zki(t)− D

ki (Pd)zil(t)

+Dli(P)Dki (Pd)zii(t) (A.2)

zki(t + 1) = (1− Dii(P))zki(t)− Dki (Pd)(1− D

ii(P))zii(t)

zil(t + 1) = (1− Dii(Pd))zil(t)− Dli(P)(1− D

ii(Pd))zii(t)

zii(t + 1) = (1− Dii(Pd))(1− Dii(P))zii(t).

For the case of symmetric P , Z is also symmetric sinceZ(t) = T T(t)(P(t)− Pd)T (t)and Pd = PTd .A relationship corresponding to (A.1) but written for zlk defines

elementwise equations similar to (A.2) but for zlk, zik, zli. Takinginto account that Z is symmetric leads to the following identitiesvalid for all t(Dki (P)− D

ki (Pd)

)zil =

(Dli(P)− D

li(Pd)

)zki

+(Dli(Pd)D

ki (P)− D

li(P)D

ki (Pd)

)zii(

Dii(Pd)− Dii(P)

)zki =

(Dki (Pd)(1− D

ii(P))

−Dki (P)(1− Dii(Pd))

)zii(

Dii(P)− Dii(Pd)

)zil =

(Dli(P)(1− D

ii(Pd))

− Dli(Pd)(1− Dii(P))

)zii.

Therefore, zki and zil in the first equation of (A.2) can be expressedin terms of zii using the last two identities above, thus producing

zkl(t + 1) = zkl(t)+Dki (P)D

li(P)

(1− Dii(Pd)

)Dii(Pd)− D

ii(P)

zii(t)

−Dki (Pd)D

li(Pd)

(1− Dii(P)

)Dii(Pd)− D

ii(P)

zii(t) (A.3)

A. Medvedev, M. Evestedt / Automatica 45 (2009) 1524–1529 1529

for which the expression reads for k = l = i as

zii(t + 1) = (1− Dii(Pd))(1− Dii(P))zii(t). (A.4)

Now the proof is straightforward by substituting in (A.3) thefollowing equalities obtained in virtue of Proposition 1

1− Dii(P) =1

1+ ρi(ξ i1TPdξ i1 + zii(t)

) (A.5)

1− Dii(Pd) =1

1+ ρiξ i1TPdξ i1

Dii(P)− Dii(Pd) =

ρizii

(1+ ρiξ i1TPdξ i1)

(1+ ρi

(ξ i1TPdξ i1 + zii(t)

)) .(A.6)

Exactly as before, (10) and (11) are just the specializations of (9)for l = i and k = l = i.

References

Cao, L., & Schwartz, H. M. (2004). Analysis of the Kalman filter based estimationalgorithm: An orthogonal decomposition approach. Automatica, 5–19.

Evestedt, M., & Medvedev, A. (2006a). Model-based slopping monitoring by changedetection. In Proceedings of IEEE conference on control applications.

Evestedt, M., & Medvedev, A. (2006b). Stationary behavior of an anti-windupscheme for recursive parameter estimation under lack of excitation.Automatica,42(1), 151–157.

Evestedt, M., & Medvedev, A. (2008). Recursive parameter estimation by means ofthe SG-algorithm. In 17th IFAC world congress (available on CD).

Evestedt, M., & Medvedev, A. (2009). Model-based slopping warning inthe LD steel converter process. Journal of Process Control, in press(doi:10.1016/j.jprocont.2009.01.002).

Evestedt, M., Medvedev, A., & Wigren, T. (2008). Windup properties of recursiveparameter estimation algorithms in acoustic echo cancellation. ControlEngineering Practice, 16(11), 1372–1378.

Jazwinski, A. H. (1970). Stochastic processes and filtering theory. Academic Press.Ljung, L., & Gunnarsson, S. (1990). Adaptation and tracking in system identification– a survey. Automatica, 26(1), 7–21.

Medvedev, A. (2004). Stability of a Riccati equation arising in recursive parameterestimation under lack of excitation. IEEE Transactions on Automatic Control, 49,2275–2280.

Olsson, C. (2005). Active vibration control of multibody systems. Ph.D. thesis.Uppsala University.

Rugh, W. J. (1996). Linear system theory. Prentice Hall.Stenlund, B., & Gustafsson, F. (2002). Avoiding windup in recursive parameterestimation. In Preprints of Reglermöte 2002. Linköping, Sweden (pp. 148–153).

Alexander Medvedev received his M.Sc. (honours) andPh.D. degrees in Control Engineering from LeningradElectrical Engineering Institute (LEEI) in 1981 and 1987,respectively. He obtained the docent degree in ControlEngineering from LEEI in 1991 and from Luleå Universityof Technology (LTU) in 1996. He was assigned professorof Control Engineering at LTU in 1997 and at UppsalaUniversity in 2002. His main research interests are controland estimation in time-varying, nonlinear and timedelay systems, as well as control applications in processindustry, electro-mechanics, avionics and biomedicine.

Magnus Evestedt was born in Stockholm, Sweden, in1978. He received his M.Sc. in Engineering Physics andPh.D. degree from the Uppsala University in 2003 and2007, respectively. Since 2008 he works as a technicalconsultant at Prevas AB, Sweden. His interests includeoptimization and control of reheating and annealingfurnaces in the steel industry.