Automatica 44 (2008) 3126–3132

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Automatica

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Brief paper

Partial state observers for linear systems with unknown inputsI

Shreyas Sundaram, Christoforos N. Hadjicostis ∗Department of Electrical and Computer Engineering and Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2307, United States

a r t i c l e i n f o

Article history:Received 31 March 2006Received in revised form14 December 2007Accepted 6 May 2008Available online 4 November 2008

Keywords:Unknown input observersFixed-lag estimationFunctional observersSystem inversionState estimation

a b s t r a c t

We consider the problem of constructing partial state observers for discrete-time linear systems withunknown inputs. Specifically, for any given system, we develop a design procedure that characterizesthe set of all linear functionals of the system state that can be reconstructed through a linear observerwith a given delay. By treating the delay as a design parameter, we allow greater flexibility in estimatingstate functionals, and are able to obtain a procedure that directly produces the corresponding observerparameters. Our technique is also applicable to continuous-time systems by replacing delayed outputswith differentiated outputs.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

When designing control systems, there is frequently somedegree of uncertainty surrounding the plant. For example, someof the plant parameters may not be known, or the plant maybe subject to unmeasurable disturbances and faults (Saif &Guan, 1993). These uncertainties can often be incorporated intothe system model by treating them as unknown inputs, andresearchers have studied ways to reconstruct these unknowninputs by using only the output of the system and (possibly) theinitial system state (e.g., see Moylan (1977), Sain and Massey(1969)). These investigations have revealed that it will generallybe necessary to use delayed outputs (or differentiated outputs inthe continuous-time case) in order to reconstruct the unknowninputs. In other words, estimation of the unknown input at timeinstant kmight require the outputs of the systemup to time instantk+α, for somenonnegative integerα. Analogously, for continuous-time systems, the α-th derivative of the output might be requiredin order to estimate the input at time t . A system is said to beinvertible if it is possible to reconstruct the inputs in the abovemanner. The related problem of state estimation in linear systemswith unknown inputs has also been studied extensively, andvarious design procedures for state observers have been developed

I This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor Masayuki Fujita,under the direction of Editor Ian R. Petersen.∗ Corresponding author. Tel.: +1 217 265 8259; fax: +1 217 244 1653.E-mail addresses: ssundarm@uiuc.edu (S. Sundaram), chadjic@uiuc.edu,

chadjic@control.csl.uiuc.edu (C.N. Hadjicostis).

0005-1098/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2008.05.011

(e.g., see Darouach, Zasadzinski, and Xu (1994), Hou, Pugh, andMuller (1999), Kurek (1983), Valcher (1999) and Yang and Wilde(1988)). It has been shown that the problem of state observationin the presence of unknown inputs is equivalent to the problem ofstable system inversion (Jin, Tahk, & Park, 1997; Saberi, Stoorvogel,& Sannuti, 2000; Sundaram & Hadjicostis, 2007); in other words,estimating the state of a linear system allows one to estimatethe unknown inputs, and vice versa. As a consequence, it wasshown in Jin et al. (1997), Saberi et al. (2000) and Sundaram andHadjicostis (2007) that by considering delayed observers, one canpotentially estimate the entire state of the system even when nozero-delay observer exists (i.e., examining delayed outputs of thesystem allows one to obtain more information about the state).This observation is in line with the fact that systems that can beinverted with some nonzero delay cannot necessarily be invertedwith zero delay (Sain & Massey, 1969).Not all systems will have a stable inverse and, even if such

an inverse exists, the delay required to construct it might beintolerably high. Consequently, it will not be possible to build afull state observer for a large class of systems. In such cases, onecan ask the more general question that we consider in this paper:what kind of information can be obtained about the state within aspecified delay? Various special cases of this problem have beeninvestigated in the literature. For example, for any system, onecan construct a zero-delay partial state observer (i.e., an observerthat does not make use of delayed outputs); this problem wasstudied in Hou and Muller (1994), Hou et al. (1999) and Tsui(1996). The problem of reconstructing a particular function of thestates (possibly with a nonzero delay) was studied in Saberi et al.(2000) through a geometric approach. The work in Jin et al. (1997)and Sundaram and Hadjicostis (2007) considered the problem of

S. Sundaram, C.N. Hadjicostis / Automatica 44 (2008) 3126–3132 3127

reconstructing the entire state of the systemvia a delayed observer.In comparison to the works cited above, the main contribution ofthis paper is a characterization of all possible linear functionals ofthe state that can be reconstructed with a given delay through alinear observer of a particular form. Since our observers allow theuse of delayed outputs, they can reconstruct, for some systems, alarger number of linear functionals of the state (as compared to thezero-delay observers commonly considered in the literature). Inaddition to obtaining the set of observable linear functionals of thestates, our approach directly produces the observer parameters.Observers that provide delayed estimates of the system state

can be used in a variety of applications, including feedbackcontrol (Franklin, Powell, & Workman, 1998), communicationsystems based on the principle of chaotic synchronization, (Inoue &Ushio, 2001), and fault detection and identification schemes (Saif& Guan, 1993). In many of these cases, a delay in estimating thestates will generally not pose a problem, although the amount ofdelay that can be tolerated will depend upon the system underconsideration and the particular application. If there is amaximumtolerable delay for a given application, the procedure presented inthis paper can be used to find the set of linear functionals of thestate that can be estimated within that delay. Also note that in thecase of continuous-time systems, our observers require derivativesof the outputs, in place of delayed outputs.

2. Preliminaries

Consider a discrete-time linear time-invariant system of theform

xk+1 = Axk + Bdkyk = Cxk + Ddk, (1)

with state vector x ∈ Rn, unknown input d ∈ Rq, output y ∈Rp, and system matrices (A, B, C,D) of appropriate dimensions.We will make the standard assumption that the pair (A, B) iscontrollable.1 Note that we do not include known inputs in orderto avoid cluttering the development (they can be readily handledby a straightforward — and rather standard — modification of ourobserver design procedure). The response of system (1) over α+ 1(α = 0, 1, 2, . . .) time units is given byykyk+1...yk+α

︸ ︷︷ ︸

yk:k+α

=

CCA...CAα

︸ ︷︷ ︸

Θα

xk

+

D 0 · · · 0CB D · · · 0...

.... . .

...

CAα−1B CAα−2B · · · D

︸ ︷︷ ︸

Mα

dkdk+1...dk+α

︸ ︷︷ ︸

dk:k+α

. (2)

The matrices Θα and Mα in the above expression can also beexpressed recursively as

Θα =

[C

Θα−1A

], Mα =

[D 0

Θα−1B Mα−1

], (3)

1 We will make this technical assumption simply to ensure that the state vectorcan take on arbitrary values during the operation of the system.

with Θ0 = C and M0 = D. Throughout our development, for anygiven matrix X with rank rx, we will seek to find square invertiblematrices U and V satisfying

UXV =[Irx 00 0

]. (4)

Such matrices can be readily obtained by first performingthe singular value decomposition of X , which produces squareorthogonal matrices U and V such that UXV =

[Σ 00 0

], where

Σ = diag(σ1, . . . , σrx) is a diagonal matrix containing the singularvalues of X (Chen, 1984). The matrices U and V in (4) can then beobtained as U =

[Σ−1 00 I

]U, V = V . Other groupings are also

possible, which implies that these matrices are not unique.

3. Partial state observer

Our objective in this paper is to characterize the set of linearfunctionals of the state vector xk that can be reproduced through alinear observer of the form

zk+1 = Ezk + Fyk:k+αχk = zk + Gyk:k+α, (5)

where the nonnegative integer α is the observer delay, and thematrices E, F , and G are chosen such that χk → Txk as k → ∞for some matrix T of maximum possible rank (defined below). Thedelayα is assumed to be a designparameter;we later discusswhy alarger delaymay enlarge (but certainly not reduce) the set of linearfunctionals that can be reconstructed.

Definition 1 (Maximum Possible Rank). We say matrix T is ofmaximum possible rank to indicate that it is of maximum rank for agiven system with a given delay. If this maximum rank is n, then itwill be possible to estimate the entire state vector for the systemwith the given delay but, in general, themaximum rankwill be lessthan n.

Remark 1. Typically, functional observers are used to estimate aparticular functional Txk (i.e., the matrix T is fixed) Chen (1984). Incontrast, in this paper, we fix the delay α and try to find the matrixT ofmaximumpossible rank forwhich a functional observer can beconstructed (a similar problemwas considered in (Hou et al., 1999)for zero-delay observers). Suppose the observer (5) is constructedfor amatrix T of maximum possible rank, and onewishes to obtainthe functional T̄ xk for some matrix T̄ . It is possible to obtain thisfunctional through an observer of the form (5) if and only if T̄ = RTfor some matrix R; the functional T̄ xk is then given by Rχk, whereχk is the estimate of Txk obtained from (5).

Remark 2. Note that we are attempting to maximize the set ofstate functionals that can be estimated through an observer of theform (5). This form is commonly considered when constructingobservers to estimate the entire system state (e.g., Darouach et al.(1994), Sundaram and Hadjicostis (2007), Valcher (1999) and Yangand Wilde (1988)), and our contribution is (i) to extend the useof such observers to systems where the entire state cannot beestimated, and (ii) to allow the use of delays. A potentially moregeneral observer structure can be obtained by replacing the outputequation in (5) byχk = Hzk+Gyk:k+α , for somematrixH . However,it is not apparent whether such observers would allow a larger setof linear functionals to be reproduced (as compared to the originalobserver (5)), and we will leave this question to be answered infuture research.While the observer (5) might appear to be non-causal at first

glance, it is, in reality, a delayed observer. In other words, oneshould view this observer as producing an estimate of the stateTxk−α using the outputs of the system up to time-step k.

3128 S. Sundaram, C.N. Hadjicostis / Automatica 44 (2008) 3126–3132

To obtain the observer parameters, we use (1), (2) and (5) toexamine the estimation error

ek+1 ≡ χk+1 − Txk+1= Ezk + Fyk:k+α + Gyk+1:k+α+1 − TAxk − TBdk= Eek + (F − EG) yk:k+α + Gyk+1:k+α+1+ (ET − TA) xk − TBdk.

Define the matrix K ≡[(F − EG) 0

]+[0 G

], where the zero

matrices have p columns each. If we partition the matrices F and Gas

F =[F0 F1 · · · Fα

], G =

[G0 G1 · · · Gα

],

where Fi and Gi have p columns each, the matrix K can be writtenas

K ≡[F0 − EG0 F1 − EG1 + G0 · · · Fα − EGα + Gα−1 Gα

].

(6)

Note that since we are free to choose F and G, the matrix K can bechosen to have any value we require. Using (2), the error can thenbe expressed as

ek+1 = Eek + Kyk:k+α+1 + (ET − TA) xk − TBdk= Eek + (ET − TA+ KΘα+1) xk+ KMα+1dk:k+α+1 − TBdk. (7)

In order to force the error to go to zero (asymptotically) regardlessof the values of xk (recall that xk can take on arbitrary values dueto the controllability assumption on the system) and the inputs,matrices E, K and T must simultaneously satisfy

KMα+1 =[TB 0 · · · 0

], (8)

ET − TA+ KΘα+1 = 0, (9)

and E must be a stable matrix. We will refer to Eq. (8) as theinput decoupling condition, and to Eq. (9) as the state decouplingcondition. After a few brief remarks, we will describe how to findmatrices E, K and T (of maximum possible rank) satisfying (8)and (9).

Remark 3. Note that Eqs. (8) and (9) encompass all linearfunctionals of the state that can be directly obtained from theoutput yk:k+α . Specifically, if there is a matrix Q that satisfiesQMα = 0, we can directly obtain the linear functional QΘαxkfrom the output as QΘαxk = Qyk:k+α . One can readily use theidentities in (3) to verify that Eqs. (8) and (9) are satisfied bysetting T = QΘα , K =

[0 Q

](where the zero matrix has

p columns), and E = 0. Since our design procedure will findthe matrix T of maximum possible rank in (8) and (9), it alsocaptures all directly obtainable functionals, and so we do not needto devote any special attention to these functionals. Note that thismeans that the constructed observer may be of larger dimensionthan necessary (since one does not strictly require an observer toreconstruct the directly observable functionals), but the intent ofthis paper is not to construct a minimum-dimension observer, butrather to characterize (and reconstruct) the set of all possible linearfunctionals of the state that can be estimated for a given systemwith a given delay.

Remark 4. Even though Eq. (8) was obtained in a purely algebraicmanner (in order to decouple the unknown inputs from theestimation error), it is instructive to examine the significanceof this equation from the perspective of input reconstruction.Specifically, since Mα+1 is the input-to-output matrix in Eq. (2),Eq. (8) has the interpretation that one must obtain the functionalTBdk from the output yk:k+α+1 in order to estimate the state

functional Txk+1. The intuition behind this requirement is obtainedby examining system (1), and noting that Txk+1 is a function ofTAxk and TBdk. Since dk is unknown and completely arbitrary,one must estimate TBdk from the outputs in order to correctlyupdate the estimate of the state functional. This reveals the tightcoupling between state estimation and unknown input estimation.In particular, one can potentially estimate a larger number of statefunctionals if one obtains more information about the unknowninputs from the output, and this was the initial motivation forconsidering delayed estimators (recall from the introduction ofthe paper that for some systems, delayed outputs actually enablethe estimation of the state and unknown input vectors in theirentirety (Sain & Massey, 1969; Sundaram & Hadjicostis, 2007)).The determination of an explicit formula relating the delay α tothe rank of T is more challenging, and will be an avenue for futureresearch.

3.1. Input decoupling

We start by analyzing the input decoupling condition given by(8). Using (3), we can write this condition as

K[D 0ΘαB Mα

]=[TB 0

]. (10)

LetN be amatrix whose rows form a basis for the left null space ofthe last (α+ 1)q columns ofMα+1 (i.e.,N

[0Mα

]= 0). We see from

the above expression that the matrix K must be of the form

K = K̄N , (11)

(for some matrix K̄ ), so that Eq. (10) becomes

K̄N

[DΘαB

]= TB. (12)

Let the rank ofN[DΘαB

]be denoted by r . Then there exists a pair of

nonsingular matrices U and V such that

UN

[DΘαB

]V =

[Ir 00 0

]. (13)

Since U is invertible, we can define

K̂ = K̄U−1 (14)

for some K̂ ≡[K̂1 K̂2

], where K̂1 has r columns. Right-

multiplying Eq. (12) by V , we get

[K̂1 K̂2

] [Ir 00 0

]= TBV ≡ T

[Γ1,1 Γ1,2

], (15)

whereΓ1,1 represents the first r columns of BV , andΓ1,2 representsthe last q − r columns. From the above expression, we see that Tmust satisfy TΓ1,2 = 0. LetN1 be amatrixwhose rows form a basisfor the left null space of Γ1,2. This implies that T must be of theform

T = T1N1 (16)

for some matrix T1, and thus we have to maximize the rank ofT1 in order to maximize the rank of T . Eq. (15) also shows thatK̂1 = TΓ1,1 = T1N1Γ1,1, and K̂2 remains a free matrix. Next, weuse the above parameterization of K and T to address the statedecoupling condition (9).

S. Sundaram, C.N. Hadjicostis / Automatica 44 (2008) 3126–3132 3129

3.2. State decoupling (following input decoupling)

Using the parameterization of K and T (given by (11), (14) and(16)) in (9), we obtain

ET1N1 − T1N1A+[T1N1Γ1,1 K̂2

]UNΘα+1 = 0. (17)

Defining[Φ1Φ2

]≡ UNΘα+1, (18)

whereΦ1 has r rows, Eq. (17) becomes

ET1N1 −[T1 K̂2

] [N1 (A− Γ1,1Φ1)−Φ2

]= 0. (19)

Denote the rank ofN1 by r1. SinceN1 is of full row rank, there existsa nonsingular matrix V1 such that

N1V1 =[Ir1 0

]. (20)

Right-multiplying (19) by V1, we get[ET1 0

]−[T1 K̂2

] [Γ2,1 Γ2,2

]= 0, (21)

where[Γ2,1 Γ2,2

]≡

[N1(A− Γ1,1Φ1

)−Φ2

]V1 (22)

and Γ2,1 has r1 columns. Letting N2 denote a matrix whose rowsform a basis for the left null space of Γ2,2, we see from (21) that[T1 K̂2

]= T2N2, (23)

for somematrix T2. IfwepartitionN2 asN2 =[N2,1 N2,2

], where

N2,1 has r1 columns, we obtain

T1 = T2N2,1. (24)

Substituting Eqs. (23) and (24) into (21), we get ET2N2,1 =T2N2Γ2,1. If the matrix N2,1 does not have full column rank,this equation places further constraints on matrix T2. Specifically,T2 must be chosen to eliminate the portions of N2Γ2,1 that arenot in the row space of N2,1. In such cases, we must furtherparameterize T2 (and, correspondingly, T and K ) in order to satisfythis constraint. We can use the following iterative algorithm toperform this parameterization; note that the iteration is indexedby variable i, which is initialized with i = 2 in order to maintainconsistency with the notation used so far. We will adopt theconvention that the empty matrix (with zero columns) is of fullcolumn rank.

(1) The equation at this point is

ETiNi,1 − TiNiΓi,1 = 0. (25)

(2) Terminal condition: If Ni,1 is of full column rank, all rowsof NiΓi,1 will be in the row space of Ni,1, and so we donot have to further parameterize Ti in order to satisfy thisconstraint. The algorithm terminates at this point, and we willlater describe how to use the parameterization of Ti producedby the algorithm to solve Eq. (25) with stable E and matrix Ti(and hence T ) of maximum possible rank.

(3) If the terminal condition is not satisfied, let ri denote the rankofNi,1. Find nonsingular matrices Ui and Vi such that

UiNi,1Vi =[Iri 00 0

].

(4) Define

T̄i = TiU−1i , (26)

and right-multiply (25) by Vi to obtain

ET̄i

[Iri 00 0

]− T̄i

[Γi+1,1 Γi+1,2

]= 0, (27)

where[Γi+1,1 Γi+1,2

]≡ UiNiΓi,1Vi, (28)

and Γi+1,1 has ri columns.(5) From (27), T̄i must be in the left null space of Γi+1,2. Let Ni+1be a matrix whose rows form a basis for the left null space ofΓi+1,2, and denote the first ri columns ofNi+1 byNi+1,1. Thus,

T̄i = Ti+1Ni+1 (29)for somematrix Ti+1, and substituting this into (27), we obtainEq. (25) with i replaced by i + 1. Return to step 1 of thealgorithm.

Note that the terminal condition of the algorithm (given in step 2)is guaranteed to be satisfied after a sufficient number of iterations(characterized below). This is because the rows of Ni form a basisfor the left null space of Γi,2 (by definition). Since Γi,2 has the samenumber of rows asNi−1 (from the definition in (28)), and since thedimension of the left null space of a matrix is always less than orequal to the number of rows in that matrix, the rank of Ni will beless than or equal to the rank ofNi−1. The rank ofNiwill be equal tothe rank ofNi−1 if and only ifNi is a square nonsingular matrix, inwhich case Ni,1 will have full column rank, thereby satisfying theterminal condition. Suppose, instead, that the rank ofNi is less thanthe rank ofNi−1. IfNi,1 does not have full column rank, we performone more iteration of the algorithm to obtain the matrixNi+1, andrepeat the above analysis. The key point is that either the first ricolumns of Ni+1 will satisfy the terminal condition, or Ni+1 willhave a smaller rank thanNi. Thus, the terminal condition can onlyfail for a finite number of iterations (say, until iteration j), at whichpoint the matrix Nj becomes the empty matrix (with zero rowsand rank rj = 0). This will cause the matrixNj+1,1 to be the emptymatrix with zero columns (since Nj+1,1 is the matrix obtained bytaking the first rj = 0 columns from Nj+1), thereby satisfyingthe terminal condition. Specifically, since the index variable of thealgorithm starts with i = 2, it will require at most rank[N2] + 1iterations for the terminal condition to be satisfied.Since Ni,1 is of full column rank at the conclusion of the

algorithm (with rank ri), there exists a nonsingular matrix Ui suchthat

UiNi,1 =[Iri0

]. (30)

If we define T̄i = TiU−1i and partition the matrix T̄i as T̄i =[J L

],

where J has ri columns, then by substituting (30) into (25), weobtain EJ −

[J L

]UiNiΓi,1 = 0. Denoting[

AC

]≡ UiNiΓi,1, (31)

whereA has ri rows, we getEJ − JA− LC = 0. (32)We must now choose J , L and E in the above equation to obtain

the matrix T of maximum possible rank satisfying (8) and (9), withE stable. To see how J and L relate to matrices T and K , we use (24),(26) and (29) to obtain matrix T1 asT1 = T2N2,1= T̄2U2N2,1= T3N3U2N2,1...

= TiNiUi−1Ni−1 · · ·U3N3U2N2,1=[J L

]UiNiUi−1Ni−1 · · ·U3N3U2N2,1.

3130 S. Sundaram, C.N. Hadjicostis / Automatica 44 (2008) 3126–3132

From (16) and (23), matrices T and K̂2 (which is parameterized asK̂2 = T2N2,2) are given by

T =[J L

]UiNiUi−1Ni−1 · · ·U3N3U2N2,1N1, (33)

K̂2 =[J L

]UiNiUi−1Ni−1 · · ·U3N3U2N2,2, (34)

and from (11) and (14), we have

K =[TΓ1,1 K̂2

]UN . (35)

Thematrices J and L represent the freedom in thematrix T after wehave satisfied the input and state decoupling conditions, and mustbe chosen tomaximize the rank of T , subject to the constraint givenby (32). Since the matrices Vi are nonsingular, we can repeatedlymultiply (33) on the right by V1, V2, V3, . . . , Vi−1 and use the factsthatN1V1 =

[Ir1 0

]and UjNj,1Vj =

[Irj 00 0

]for 2 ≤ j < i to get

rank[T ] = rank[TV1]= rank

[[J L

]UiNiUi−1Ni−1 · · ·U2N2,1V2

]= rank

[[J L

]UiNiUi−1Ni−1 · · ·U3N3,1V3

]...

= rank[[J L

]UiNi,1

]= rank

[[J L

] [Iri0

]]= rank [J].

Therefore, maximizing the rank of T is equivalent to maximizingthe rank of J . Note that J has ri columns, and so no linear functionalof the state can be reconstructed through the observer (5) if ri = 0(i.e., ifNi,1 is the empty matrix at the conclusion of the algorithm).Next, we address the issue of choosing J (of maximum possiblerank) and L in (32) while ensuring that the error dynamics (givenby E) are stable.

3.3. Stability of the error dynamics

To find the matrix J of maximum possible rank satisfying (32)while ensuring that E is stable, we first note that there is anonsingular matrix P that transforms the pair (A,C) into the form

PAP−1 =[

As 0A21 As̄

], CP−1 =

[Cs 0

], (36)

where the pair (As,Cs) is detectable, and all modes of As̄ areunobservable and unstable. Note that the above form is simplya slightly modified version of the Kalman observable canonicalform (Chen, 1984). Assume that the dimensions of As and As̄ arens×ns and ns̄×ns̄, respectively. Right-multiplying (32) by P−1 anddefining

J̄ = JP−1 (37)

for some J̄ =[J̄1 J̄2

](where J̄1 has ns columns), we get[

EJ̄1 EJ̄2]−[J̄1 J̄2

] [As 0A21 As̄

]−[LCs 0

]= 0. (38)

Note that the matrix J̄ has ns + ns̄ columns, and therefore has atheoretical maximum rank of ns + ns̄. However, examining theabove equation, we see that EJ̄2 = J̄2As̄. For E to be stable,it must not share any eigenvalues with the matrix As̄. This factimplies that J̄2 must be the zero matrix in the above equation(e.g., see Gantmacher (1959)). Substituting J̄2 = 0 into (38), weget EJ̄1 − J̄1As − LCs = 0. Since (As,Cs) is detectable, choosingJ̄1 = Ins produces E = As + LCs, and standard pole-placementtechniques can be used to select the matrix L so that all observable

eigenvalues of the pair (As,Cs) are placed at any desired locationsin the stable region. Furthermore, the unobservable eigenvaluesof the pair (As,Cs) will be at fixed (but stable) locations Brogan(1991). In other words, we choose J̄ (and, correspondingly, J) toselect all the detectable modes of the pair (A,C), and none of theundetectable modes. From (37), we have J =

[Ins 0

]P , and this

allows us to obtain the matrices T and K from (33)–(35). We havenow completed the analysis of the input decoupling condition,state decoupling condition, and the stability condition. With theabove choices of matrices, we see that J (and therefore T ) has rankns. We can now use (6) to map the K matrix to suitable values forF and G in (5). Note that this mapping is not unique. For example,one can choose G0 = G1 = · · · = Gα−1 = 0, thereby gettingK =

[F0 F1 · · · Fα − EGα Gα

], which would correspond to

using only the most delayed measurement in the output of theobserver. Another option is to choose F1 = F2 = · · · = Fα = 0,which corresponds to using only the earliest measurement in thedynamic portion of the observer. Regardless of the actual mappingfrom K to F and G, the observer in (5) will asymptotically estimatethe functional Txk.

Remark 5. Note that our design procedure is a series of increas-ingly refined parameterizations of the design matrices T , E and K(which encapsulate all of the observer parameters), subject to theconstraints imposed by Eqs. (8) and (9).More specifically, each stepof our procedure essentially involves multiplying both sides of anappropriate matrix equation by some nonsingular matrix (whichclearly does not incur any loss of generality). The key point is thatthese nonsingular matrices are chosen so that the resulting equa-tion immediately reveals that the designmatricesmust necessarilylie in certain null spaces. In other words, each step of the algorithmsimply identifies the largest possible subspace that the design ma-trices can lie in, and the size of this subspace is repeatedly reduced(according to constraints imposedby equations that the designma-trices must necessarily satisfy) until all constraints are met. At theend of the algorithm, we are left with the largest possible subspacethat thematrix T can lie in while simultaneously satisfying Eqs. (8)and (9); thus, the design procedure inherently yields a matrix T ofmaximum possible rank.

4. Example

We now provide an example to demonstrate that delayedestimators may be able to reconstruct a larger number of statefunctionals, in comparison to zero-delay estimators. Specifically,consider a system of the form (1) with

A =

0 1 −1 00 1 0 00 0 0 0

0 0 012

, B =0 01 00 11 1

,C =

[1 0 0 0

],D =

[0 0

]. (39)

Wewish to determine the set of functionals of the state that can beobtained through an observer of the form (5)with a delay ofα = 1.To derive the observer parameters, we will require the followingmatrices:

Mα+1 ≡ M2 =

[ D 0 0CB D 0CAB CB D

]=

[0 0 0 0 0 00 0 0 0 0 01 −1 0 0 0 0

],

Θα+1 ≡ Θ2 =

CCACA2

= [1 0 0 00 1 −1 00 1 0 0

].

We start by parameterizing the gains in order to perform inputdecoupling (as described in Section 3.1). A basis for the left null

S. Sundaram, C.N. Hadjicostis / Automatica 44 (2008) 3126–3132 3131

space of the last (α + 1)q = 2q = 4 columns of M2 is given

by N = I3, and the matrix N

[DCBCAB

]in Eq. (12) has a rank of 1.

The matrices U and V from (13) are found to be U =[0 0 11 0 00 1 0

],

V =[1 10 1

]. From (15), we get Γ1,1 =

[0 1 0 1

]′, Γ1,2 =[0 1 1 2

]′. A basis for the left null space of Γ1,2 is taken to beN1 =

[1 0 0 00 1 −1 00 1 1 −1

]. This produces theparameterization of T in

(16). Next,we turn our attention to the problemof state decoupling

(as in Section 3.2). ThematrixV1 in (20) isV1 =

[1 0 0 00 1 1 10 0 1 10 1 1 2

], and

from (18) and (22) we get

Γ2,1 =

0 0 0 −1 0

1 012

0 −1

0 012

0 0

′

,

Γ2,2 =[0 0 0 0 0

]′.

A basis for the left null space of Γ2,2 is taken to be N2 = I5, which

means that the matrix N2,1 in (24) is N2,1 =[I30

]. This leads us

to Eq. (25). Since N2,1 is of full column rank, we do not have tofurther parameterize T2 in (25) in order to ensure thatN2Γ2,1 willbe in the row space of N2,1 (i.e., the terminal condition in step 2of the algorithm is satisfied). To complete the analysis of the statedecoupling condition, we choose U2 = I5 in (30), and from (31)

we getA =[0 1 00 0 00 0.5 0.5

], C =

[−1 0 00 −1 0

]. This leads us to Eq.

(32), where wemust choose J (of maximum possible rank) and L inorder to ensure that the matrix E is stable. To accomplish this, weuse the results in Section 3.3. Since thepair (A,C) is detectable, thematrix P in (36) is simply P = I3, and J = J̄ = I3. The pair (A,C)has two observable modes at 0, and an unobservable (but stable)mode at 0.5. Since these eigenvalues are already stable, we canchoose L to be the zero matrix, and this produces E = A. We havenow satisfied all of the conditions for partial state reconstruction(namely, input decoupling, state decoupling, and stability of theerror dynamics). To determine the set of linear functionals that canbe reproduced and the corresponding observer gain, we use (33)and (34) to get

T =[J L

]U2N2,1N1 =

[1 0 0 00 1 −1 00 1 1 −1

], (40)

K̂2 =[J L

]U2N2,2 =

[0 0 00 0 0

]′,

which produces (from (35)) K =[0 0 00 0 10 0 0

]. Finally, we use (6) to

map thismatrix to F andG. For this example,we choose F1 = 0, and

this gives us F =[0 0 0.250 0 0

]′, G =

[1 0 0.50 1 0

]′. The final state

observer is given by (5). To see why delayed observers are usefulfor this system, one can verify (e.g., by using our design procedure)that the row-space of the matrix

[1 0 0 00 1 1 −1

]characterizes

the set of all functionals that can be estimated by a zero-delayobserver, whereas the above example demonstrates that the setof functionals that can be estimated by a unit-delay estimator ischaracterized by the row space of the rank-3 matrix T in (40).

5. Conclusions

We have provided a characterization of partial state observersfor linear systems with unknown inputs. Our approach developsa parameterization of the observer gain matrices that decouplesthe estimation error from the unknown inputs and the value of thestate. The remaining freedom (after performing this decoupling)is used to reconstruct the maximum possible information aboutthe state, subject to the constraint that the observer dynamicsbe stable. The fact that our procedure allows the use of delays(or differentiated outputs in the continuous-time case) allowsour observers to estimate a larger set of functionals (for certainsystems) than existing zero-delay observers. Interesting avenuesfor future research would be to find sufficient conditions for theexistence of delayed partial observers, and to find an explicitrelationship between the delay α and the rank of matrix T(including an upper bound on the delay required to estimate anyfunctional with a stable observer, and potentially establishing aconnection between thismaximumdelay and theMcMillan degreeof the system).

Acknowledgements

This material is based upon work supported in part by theNational Science Foundation under NSF Career Award 0092696,NSF ITR Award 0085917 and NSF EPNES Award 0224729, and inpart by the Air Force Office of Scientific Research DoD under AFOSRURI Award No F49620-01-1-0365URI. Any opinions, findings, andconclusions or recommendations expressed in this publication arethose of the authors and do not necessarily reflect the views of NSFor AFOSR.

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Shreyas Sundaram is a doctoral student in the Depart-ment of Electrical and Computer Engineering and the Co-ordinated Science Laboratory at the University of Illinoisat Urbana-Champaign. He received the BASc. degree inComputer Engineering from the University of Waterloo in2003, and the MS degree in Electrical Engineering fromthe University of Illinois at Urbana-Champaign in 2005.His research is in the areas of fault-tolerant estimationand control, dealing with topics such as controller design,unknown input observers, and distributed algorithms forcalculating functions and disseminating information over

networks. He was a finalist for the Best Student Paper Award at the 2007 AmericanControl Conference and at the 2008 American Control Conference.

Christoforos N. Hadjicostis received S.B. degrees in Elec-trical Engineering, in Computer Science and Engineering,and in Mathematics, the M.Eng. degree in Electrical En-gineering and Computer Science in 1995, and the Ph.D.degree in Electrical Engineering and Computer Science in1999, all from the Massachusetts Institute of Technology,Cambridge, MA. In August 1999 he joined the Faculty atthe University of Illinois at Urbana-Champaign where heis currently an Associate Professor with the Department ofElectrical and Computer Engineering and a Research Asso-ciate Professor with the Coordinated Science Laboratory.

His current research focuses on fault diagnosis and tolerance in distributed dynamicsystems; error control coding; monitoring, diagnosis and control of large-scale dis-crete event systems; and applications to network security, anomaly detection, en-ergy distribution systems,medical diagnosis, biosequencing, and genetic regulatorymodels.