Projective State Observers for Large-Scale Linear Systems
Tomonori Sadamoto1,2 , Takayuki Ishizaki1,2 , and Jun-ichi Imura1,2

Abstract— In this paper, towards efficient state estimation
for large-scale linear systems, we propose a novel frameworkof low-dimensional functional state observers, which we calla projective state observer, with the provision of a systematic
design procedure. The projective state observer can be regarded
as a generalization of functional state observers that is definedby means of the orthogonal projection taking into account thesystem controllability/observability in a quantitative manner.
The efficiency of the proposed observer is shown through a
numerical example for a reaction-diffusion system evolving
over a directed complex network. Moreover, through thisobserver design, we discuss a trade-off relation between lowdimensionality and an observation error.

I. I NTRODUCTION
With the recent developments in engineering, the architecture of systems has been more complex and larger inscale. An appropriate state observer is needed for realizing
state feedback control of large-scale systems. However, in theclassical observer design methods, the resultant observer isnecessary to have a dimension comparable with a system tobe observed; see e.g., [1], [2], [3]. Thus, a high-dimensional
observer is inevitably required for large-scale systems. Inview of this, a design method of low-dimensional observers
is crucial to deal with large-scale systems.
As related work, some methods for designing lowdimensional observers can be found in literature; see [4],[5]. In these methods, a state observer is often designed
for a low-dimensional approximate model of systems that is
obtained by a model reduction technique. For example, [5]proposes a design procedure of low-dimensional observers
by using an approximant obtained by the balanced truncation[6]. However, in this method, the construction of approximate
models is rather heuristic because the relation between theapproximation error and observation error is not discussed
theoretically.
To discuss the observation error for low-dimensional observers theoretically, we need to take into account some differences between the classical and low-dimensional observer
design. For example, in the case of classical observers, suchas the Luenberger-type observer, the external input, which
is applied to both system and observer, does not have any
influence on observation errors, i.e., the error dynamics isexactly uncontrollable. On the other hand, in the case of
low-dimensional observers, the external input possibly excites observation errors, since the observer dynamics cannot
1 Department of Mechanical and Environmental Informatics, Graduate
School of Information Science and Engineering, Tokyo Institute of Technology; 2-12-1, Meguro, Tokyo, Japan:
2 Japan Science and Technology Agency, CREST 4-1-8 Honcho,
Kawaguchi, Saitama, 332-0012, Japan{sadamoto@cyb., ishizaki@, imura@}mei.titech.ac.jp

exactly imitate the dynamical system behavior for the input
signals.
Against this background, in the first half of this paper, we
propose a novel framework of low-dimensional observers,
which we call a projective state observer. The projective state
observer can be regarded as a generalization of functional
state observers that is defined by means of the orthogonal projection taking into account the system controllability/observability in a quantitative manner. Furthermore, we
derive a tractable representation of the error system clarifyingthat not only an initial state estimation error but also the input
signal and the initial value response of systems are relevantto the observation error.
In the second half of this paper, based on the error
analysis in the first half, we propose a systematic design
procedure for projective state observers. More specifically,
we propose two types of design procedures for projective
state observers that can estimate a set of specific states given
in advance, and can estimate a kind of average behavior ofsystems, respectively. The former one is effective if we wantto estimate a specific limited number of states. Although
this type of observers is proposed in [7], the approach isdifferent from the design procedure shown in this paper. Thisis because the approach is based on observer reduction wherea reduced order observer approximates a full-state observer
for the original system.
To explain the basic concept of the latter one, let us
imagine the following situation: From a microscopic point ofview, the behavior of fluid is realized by complex interaction
among a huge number of molecules; on the other hand, froma macroscopic point of view, we can observe only a kind of
average behavior of molecules. This fact suggests that theestimation of average behavior is essential to control the fluid
behavior (i.e., large-scale systems) from a macroscopic pointof view.
However, it should be noted that to find a set of averagestates that is suitable for the state estimation is nontrivial ingeneral. To find such a set of average states, we apply theconcept of clustered model reduction developed in [8], [9].
Consequently, we can design a projective state observer thatcan systematically find a set of states that is desirable for
estimating the average system behavior. The efficiency of
the two design procedures for projective state observers isshown through a numerical example for a reaction-diffusion
system evolving over a directed complex network.
This paper is organized as follows: In Section II, we
first give a mathematical formulation of projective state
observers. Furthermore, deriving a tractable representation of
the error system, we clarify differences between the design

of classical functional state observers and that of projective
state observers. In Section III-A, we describe a road map
for systematic design of projective state observers. Then, in
Section III-B and Section III-C, we propose two design procedures of projective state observers which can estimate a setof states and average behavior of the system, respectively. In
Section IV, we show the efficiency of the proposed methods
through a numerical example of a networked system. Finally,
concluding remarks are provided in Section V.
Notation The following notation is used:R
set of real numbers
unit matrix of size n × nInnegative (positive) definitenessM ≺ On (M  On )of a symmetric matrix M ∈Rn×n
M  On (M  On )negative (positive) semidefiniteness of a symmetric matrix
M ∈ Rn×n
im(M )
range space spanned by the column vectors of a matrix Mtr(M )
trace of a matrix M
the Frobenius norm of a matrix
M F
MThe L2 -norm of a square integrable function v(t) ∈ Rn isdefined by
 ∞
 12
v T (t)v(t)dt.
v(t)L2 :=0

The H∞ -norm of a stable proper transfer matrix G and theH2 -norm of a stable strictly proper transfer matrix G arerespectively defined by
G(s)H∞ := sup G(jω),ω∈RG(s)H2 :=

1
2π

 ∞

T

 12

tr(G(jω)G (−jω))dω−∞

where  ·  denotes the induced 2-norm.
II. F UNDAMENTALS OF P ROJECTIVE S TATE O BSERVERS
A. Preliminary Review of Functional State Observers
In this paper, we deal with the n-dimensional linear systemdescribed byẋ = Ax + Bu(1)Σ:
, x(0) = x0
y = Cx + Du
where A ∈ Rn×n , B ∈ Rn×mu , C ∈ Rmy ×n , and D ∈Rmy ×mu . To simplify the arguments, we show results forstable systems. Similar results are available also for unstablesystems. In Σ in (1), a measurement output signal is denotedby y ∈ Rmy . Furthermore, we define a signal to be estimatedbyz = Sx(2)where S ∈ Rmz ×n .

In this notation, let us consider the n-dimensional(Luenberger-type) observer described byx̂˙ = Ax̂ + Bu + H(y − ŷ)(3)O:, x̂(0) = x̂0ŷ = C x̂ + Du
where the observer gain H ∈ Rn×my is a design parameter.
As being compatible with (2), we define the estimation of zbyẑ = S x̂.
(4)
In what follows, O is referred to as a functional state observer[2], [3] since z and ẑ are defined as functions of x and x̂.
By denoting the state error as e := x − x̂, we obtain theerror system with this functional state observer asė = (A − HC)eE:
, e(0) = e0(5)
Δ = Sewhere
Δ := z − ẑ,

e0 := x0 − x̂0denote the observation error and the initial state error,
respectively. Note that the observation error is a functionof e0 , namely(6)
Δ = Δ(t; e0 ).
Usually, we design the observer gain H so that this error
system has a desirable behavior. To regulate a convergence
rate of observation errors, for a given constant δ ≥ 0, one
can design H such thatsupe0 =0Δ(t; e0 )L2
≤ δ.
e0 

(7)

In the next subsection, we define a lower-dimensional functional observer as a generalization of this n-dimensional
functional state observer.
B. Error Analysis of Projective State Observers
In this subsection, for Σ in (1), we define an n̂-dimensional
functional state observer as a generalization of O in (3). Morespecifically, by means of the orthogonal projection [6], we
define the n̂-dimensional functional state observer byx̂˙ = P AP T x̂ + P Bu + H(y − ŷ)OP :, x̂(0) = x̂0ŷ = CP T x̂ + Du
(8)where the observer gain H ∈ Rn̂×my and the projection matrix P ∈ Rn̂×n satisfying P P T = In̂ are design parameters.
Without loss of generality, we assume n̂ ≤ n. Similarly to(4), we define the estimation of z in (2) byẑ = SP T x̂.

(9)In the rest of this paper, we refer to this functional state
observer OP as a projective state observer.
To analyze the observation error for this projective state
observer, we derive a tractable representation of the error
system as follows:Theorem 1: Let Σ in (1) be given with S in (2). Define
OP as in (8), and let ΔP := z − ẑ with z and ẑ defined asin (2) and (9). Then, it follows thate0ξ˙ = Aξ + BuEP :, ξ(0) =(10)x0
ΔP = Sξwhere
e0 := P x0 − x̂0(11)III. D ESIGN OF P ROJECTIVE S TATE O BSERVERS
A. A Road Map for Systematic Design
In what follows, we aim to find the design parameters P
and H in (8) so that the observation error due to (i), (ii) and(iii) are suppressed as much as possible. Since the dynamics
of the error system is linear, we can represent each error
factor as
ΔP (t; e0 , 0, 0),and

P AP T − HCP T (P A − HC)(In − P T P )
0
A



0
B :=
, S := SP T S(In − P T P ) .
B



A :=

Proof: By taking a state as X̂ := [x̂T xT ]T , we have˙X̂ = ÂX̂ + B̂u , X̂ (0) = x̂0x0
ΔP = Ŝ X̂
where




P AP T − HCP T HC
0
A


PB
B̂ :=
, Ŝ := [−SP T S].
B

T :=

−In̂

P
In



with a given constant δ ≥ 0.
Next, to see the effect of x0 and u more explicitly,
supposing that e0 = 0, we derive the Laplace domain
representation of the observation error due to the second andthird factors asξ(0) =

e0
x0

(14)

ΞP,H (s) := CΞ (sIn̂ − AΞ )−1 BΞ + DΞ

(15)

with
−1

AΞ := P AP T − HCP T ,CΞ := SP T ,, T B̂ andFrom Theorem 1, we notice that EP in (10) corresponds
to a generalized representation of the error system E in (5).
This is because, if P = In , we have





A − HC 00A=, B=, S= S 00ABandΔ̂P (s; x0 , u) := ΞP,H (s)XP (s; x0 , u)where

= T −1 .

From the similarity transformation of T ÂTŜT −1 , the claim follows.

BΞ := (P A − HC)PT
DΞ := SPe0 = x0 − x̂0 .

Furthermore, it should be emphasized that ΔP in (10) is afunction of not only e0 but also x0 and u, namely
ΔP = ΔP (t; e0 , x0 , u),(12)which is clearly contrasted with Δ in (6) for the traditional
functional state observer. An intuitive explanation on theseerror factors is as follows:(i)the error due to the initial state error e0 defined asin (11),(ii)
the error amplified by the initial value response of
Σ, i.e., y = CeAt x0 , and(iii) the error amplified by the dynamical discrepancyof Σ and OP for the external input u.
In conclusion, we see that the error factors (i), (ii) and (iii)should be taken into account for the design of projective state
observers.

TandXP (s; x0 , u) := P (sIn − A)−1 [x0 + Bu(s)](16)

with an orthogonal complement P ∈ R(n−n̂)×n of P ∈Rn̂×n such thatTP T P + P P = In .

,

ΔP (t; 0, 0, u).

For the first one, similarly to (7), we consider regulating theconvergence rate of the initial state error by introducing theindex of
ΔP (t; e0 , 0, 0)L2
≤δ
(13)
sup
e0 
e0 =0Â :=DefineΔP (t; 0, x0 , 0),(17)

From this expression, we can expect that the observation
error due to the second and third factors will be small if thenorms of ΞP,H and XP are small enough.
However, it should be noted that simultaneous designof P and H is difficult because ΞP,H involves the design
parameters in a bilinear fashion. To overcome this difficulty,
we exploit the following facts:• The parameter H appears in the system ΞP,H , but notin XP .
• The system XP involves the parameter P (or equivalently P ), but not H.
• By a suitable choice of P , we can vanish DΞ in (15),which may directly increase the norm of ΞP,H .
Based on these facts, we first find P that minimizes the normof XP with the constraint of DΞ = 0, and then find H thatminimizes the norm of ΞP,H . Taking this road map for the
projective state observer design, we provide the following
result:Theorem 2: Let Σ in (1) be given with S in (2). For aconstant α ≥ 0, define Φ  On such thatAΦ + ΦAT + BB T + αIn = 0.

(18)

Furthermore, take P ∈ Rn̂×n such thatim S T ⊆ im(P T ),andP P T = In̂tr(Φ) − tr(P ΦP T ) ≤ .

(19)
(20)

If there exist
γ > 0,Y ∈ Rn̂×my

X  On̂ ,such that X ≺ δ 2 In̂ andsym(XP AP T − Y CP T ) + P S T SP T∗
≺ On
P AT P T X − P C T Y T−γIn−n̂(21)
where sym(M ) := M + M T and P ∈ R(n−n̂)×n satisfying(17), then OP in (8) with
H = X −1 Y(22)satisfies (13) for any x0 ∈ Rn and x̂0 ∈ Rn̂ , andΔP (t; 0, 0, u)2L2 + αΔP (t; 0, x0 , 0)2L2 ≤ γ

(23)

n

for the unit impulse input u and x0 ∈ R such that x0  = 1,
where e0 and ΔP are defined as in (11) and (12).
Proof: First, we evaluate ΔP (t; 0, 0, u)2L2 for theimpulse input u. Based on the error system expression in(14), we haveΔP (t; 0, 0, u)2L2 ≤ ΞP,H (s)2H∞ P (sI − A)−1 B2H2
where ΞP,H is defined as in (15). Substituting (22) into (21),
we havesym(XP AP T − XHCP T ) + P S T SP T∗
≺ On .
P AT P T X − P C T H T X−γIn−n̂(24)
Note that the first condition in (19) guarantees
TDΞ = SP = 0.
Thus, from the bounded-real lemma [10], (24) guarantees
ΞP,H (s)H∞ < γ. Note that there exists Φ(1)  On
satisfyingAΦ(1) + Φ(1) AT + BB T = 0
for the stable system Σ. Using P (sI − A)−1 B2H2 =T
tr(P Φ(1) P ), we haveT

ΔP (t; 0, 0, u)2L2 ≤ γtr(P Φ(1) P ).
Next, we evaluate ΔP (t; 0, x0 , 0)2L2 . Note that there exists
Φ(2)  On satisfying AΦ(2) +Φ(2) AT +In = 0. Thus, usingx0 xT
0  In(25)

for any x0 ∈ Rn such that x0  = 1, we have P (sI −TA)−1 x0 2H2 ≤ tr(P Φ(2) P ). ThereforeΔP (t; 0, 0, u)2L2 + αΔP (t; 0, x0 , 0)2L2

T≤ γtr P (Φ(1) + αΦ(2) )P .

From the Lyapunov theorem [11], Φ given in (18) satisfiesΦ = Φ(1) + αΦ(2) . In addition, the condition in (20) yields
T≤ .
tr P (Φ(1) + αΦ(2) )PThus, (23) follows. Finally, we show (13). To describe thetime evolution of η(t) := ΔP (t; e0 , 0, 0), we consider thesystem given byė = (P AP T − HCP T )e
, e(0) = e0 .
η = SP T e
From (1, 1) block of (21), this system admits a Lyapnov function V (e) := eT Xe such that V̇ (e(t)) <−η T (t)η(t). Integrating this inequality over [0, ∞) and utilizing V (e(∞)) = 0, we haveη(t)2L2 < V (e(0)) ≤ e0 2 X2
for any e0 ∈ Rn̂ . Hence, (13) follows from X ≺ δ 2 In̂ .
Theorem 2 provides an explicit error bound for projective
state observers. As shown in the proof of this theorem, we
measure the effect of u in terms of the H2 -norm, with similar
results available for the case of the H∞ -norm. One possible
approach to find P in the H∞ -norm evaluation is system
tridiagonalization; see [8] for details.
B. Design of Projective State Observers for Specific State
Estimation
In this subsection, we propose a procedure to constructa projective state observer with S given in advance. This
projective state observer estimates a subset of the state-spaceof Σ, which is specified by z in (2). Such a projective state
observer is efficient when we want to estimate a limited
number of states, as in the use of traditional functional state
observers.
In this situation, we can find P ∈ Rn̂×n such that (19)and (20) in the following manner: First, we find the set{(λi , vi )}i∈{1,...,n} of all eigenpairs of Φ, supposing thatλi ≥ λi+1 and vi  = 1 without loss of generality. Next,
we find minimum k ∈ {1, . . . , n} such that
λk+1 + · · · + λn ≤ (26)

and construct V = [v1 , . . . , vk ] ∈ Rn×k . Finally, by theGram-Schmidt process, we derive P such that
im(P T ) = im([V, S T ]).

(27)

Based on this, supposing that Σ in (1) with S in (2) is given,
we summarize the design procedure of an n̂-dimensional
projective state observer OP such that (13) andΔP (t; 0, 0, u)2L2 + αΔP (t; 0, x0 , 0)2L2 ≤ ρ(28)

where ρ ≥ 0 is a design parameter.
1) Fix an initial value of  ≥ 0.
2) Find P ∈ Rn̂×n such that (19) and (20) by theprocedure above.
3) For a given δ, solve (21) while minimizing γ where P
is constructed by Gram-Schmitt process.

4) If (21) is infeasible or ρ < γ, then take larger δ and ρ
and back to 2).
5) Compute H by (22) and construct a low-dimensional
functional observer OP in (8).
It should be noted that since the number of decision variablesof LMI given by (21) is 12 n̂(n̂ − 1) + n̂my , this design
procedure is computationally tractable if n̂ is small.
C. Design of Projective State Observers for Average State
Estimation
In this subsection, we construct a projective state observerwith S in (2) given by
S = P.

(29)

This implies that S is also a design parameter of projective
state observers. In this setting, the condition in (19) is
automatically satisfied. Furthermore, z and ẑ in (2) and (9)are clearly given by z = P x and ẑ = x̂. It should beemphasized that, unless we impose a meaningful structure
on P , the estimated signal ẑ has no physical meaning.
To accomplish the state estimation in a suitable sense, we
impose the following block-diagonal structure on P :Definition 1: The family of an index set {I[l] }l∈L for L :={1, . . . , L} is called a cluster set, whose element is referred
to as a cluster, if each element I[l] is a disjoint subset of{1, . . . , n} and it satisfies
I[l] = {1, . . . , n}.
l∈LThen, an aggregation matrix compatible with {I[l] }l∈L isdefined by
P := diag(p[1] , . . . , p[L] )Π ∈ RL×n

(30)

average behavior of systems. To this end, we can use thefollowing fact:
Lemma 1: Let Σ in (1) be given, and define Φ  On suchthat (18) for a constant α ≥ 0. If, for each l ∈ L, there exists
a row vector φ[l] ∈ R1×n such that


1
 n T

φ
(34)
(eI[l] ) Φ 12 − pT ≤ |I[l] | 2 θ
[l]
[l]
F

1×|I[l] |satisfying p[l]  = 1 and Φ 12 denotes
where p[l] ∈ R
a Cholesky factor such that Φ = Φ 12 ΦT1 , then it follows that2

tr(Φ) − tr(P ΦP T ) ≤ θ2

L

l=1 |I[l] |(|I[l] | − 1)where P is defined as in (30).
Proof: See [9].
This lemma shows that, if we find a cluster set {I[l] }l∈Lsuch that (34), then the value of  in Theorem 2 can be taken
as

L
|I
|(|I
|
−
1)
.
 = θ2[l][l]l=1Remark 1: The estimated signal z given by S and P in(29) and (30) with (32) implies an average state in the senseof (33). Alternatively, if we take11,..., ∈ RL×LS = DP, D := diag 
|I1 |
|IL |
(35)
then z implies an average state in the sense of1 xi , l ∈ L.
(36)
zl =
|I[l] |∀i∈I[l]IV. N UMERICAL S IMULATION

with the permutation matrix
Π := [enI[1] , . . . , enI[L] ]T ∈ Rn×n , enI[l] ∈ Rn×|I[l] |(31)

and p[l] ∈ R1×|I[l] | such that p[l]  = 1.
Based on this notion of clustering, the authors have solveda structured model reduction problem for networked systems
in [8], [9]. Note that, if P has the specific structure shownin (30), then the estimated signal ẑ can be interpreted as aweighted average of states of Σ. In particular, if p[l] is in theform of[1, . . . , 1]
∈ R1×|I[l] | ,(32)p[l] =
[1, . . . , 1]
then ẑ corresponds to an average state in the sense of1zl = 
xi , l ∈ L(33)|I[l] | ∀i∈I[l]where zl (resp. xi ) is the lth element of z (resp. the ithelement of x). A method to achieve usual averaging is
described in Remark 1. In what follows, for simplicity, weonly consider the case of (32).
By the design procedure same as in Section III-B, let
us construct a projective state observer that estimates the

In this section, we show the efficiency of the proposed projective state observer through a numerical example. We deal
with a 1000-dimensional reaction-diffusion system evolving
over a complex network model, called the Dorogovtsev
model [12].
First, we show the design result of a projected state
observer for the specific state estimation proposed in Section III-B. Let the input signal u ∈ R5 be applied tothe states of some five nodes, and the measurement output
signal y ∈ R10 is obtained as the states of some 10 nodes.
Moreover, the evaluation output z ∈ R20 is given as the statesof some 20 nodes, which are different from the measurement
output nodes. Based on the procedure shown in Section IIIB, P and H in (8) are constructed for several values of ρ
while taking α = 0 and δ = 2.0. Note that we are required
to take at least n̂ ≥ 20 to satisfy (19) in this setting.
In Fig. 1, we plot the resultant dimension of projective
state observers versus the value of ρ in (28). Furthermore,
Fig. 2 shows the resultant performance of the input response
ΔP (t; 0, 0, u) versus the value of ρ in (28). In addition,
the resultant performance of the initial state error are about
2.0 for all ρ. These results show that there is a trade-off

40

0.1

35
30

0.05

25

Fig. 1.

Evaluation Outputs

0.15

-2

10

-1

10

0

0

10

Dimension of observer versus ρ.

Fig. 2.

-2

10

-1

10

Performance of input response versus ρ.

relation between the estimation performance and the lowdimensionality of observers. Namely, the estimation performance is improved in compensation for increasing thedimension of observers.
Next, we verify the estimation performance of projective
state observers by comparing z(t) with ẑ(t). In Fig. 3, weplot only three elements of z and ẑ since the other elements
behave similarly. To obtain this figure, a random input signal
is applied over t ∈ [20, 80] and x0 is given randomly andξˆ0 = 0. The resultant dimension of the projective state
observer is n̂ = 47. From this figure, we can confirm thatthe observation error relevant to both input and initial state
response are small.
Finally, we show the design result of projective state
observers for average state estimation proposed in Section
III-C. The observer is constructed by the procedure shownin Section III-B while taking α = 0, δ = 5.0 and ρ = 4.3.
In addition, S is constructed by (35), which implies that z
estimates an average state in the sense of (36). The resultant
dimension of the obtained observer is n̂ = 5, which implies
that the resultant number of clusters is L = 5. In Fig. 4, weplot all trajectories of x and ẑ where a random input signal
is applied over t ∈ [0, 140]. The legends are as follows:
the trajectory of ẑ is depicted by the dotted lines with thecircles where ẑl is color-coded according to l ∈ {1, . . . , 5}.
In addition, xi for each i ∈ I[l] is also color-coded according
to its cluster index l ∈ {1, . . . , 5}. From this figure, we canconfirm that each trajectory of five elements of ẑ is aroundthe center of colored trajectory sets of x. Moreover, theresultant estimation error is z − ẑ2L2 = 1.3 × 10−2 , which
implies that ẑ estimates an average state in the sense of (36)efficiently. As demonstrated in this numerical example, theproposed projective state observer can efficiently find andestimate the average behavior of networked systems.
V. C ONCLUSION
In this paper, we proposed a novel framework of lowdimensional functional state observers, called projective state
observers. In addition, deriving a tractable representation of
the error system, we clarified differences between the designof classical functional state observers and that of projective
state observers. Based on the error analysis, we provided asystematic design procedure. For two types of projective state
observers: the first can estimate a set of states of the systemand the second can find and estimate the average behavior

1
0.5
0
−0.5
−1
0

0

10

20

Fig. 3.

40
Time[s]60

80

Trajectories of z and ẑ.

3Evaluation Outputs

Dimension of Observer

45

20

1.5

0.2

50

2
1
0

−1
−2
−30

50
Fig. 4.

Time[s]100

Trajectories of ẑ and x.

of systems. The efficiency of the proposed design procedureis shown through a numerical simulation.
R EFERENCES[1] D. Luenberger, “An introduction to observers,” Automatic Control,
IEEE Transactions on, vol. 16, no. 6, pp. 596–602, 1971.
[2] M. Darouach, “Existence and design of functional observers for linear
systems,” Automatic Control, IEEE Transactions on, vol. 45, no. 5,
pp. 940–943, 2000.
[3] C.-C. Tsui, “On the order reduction of linear function observers,”
Automatic Control, IEEE Transactions on, vol. 31, no. 5, pp. 447–
449, 1986.
[4] P. Vermeulen and A. Heemink, “Model-reduced variational data assimilation,” Monthly Weather Review, vol. 134, no. 10, pp. 2888–2899,
2006.
[5] A. Lawless, N. Nichols, C. Boess, and A. Bunse-Gerstner, “Using
model reduction methods within incremental four-dimensional variational data assimilation,” Monthly Weather Review, vol. 136, no. 4, pp.
1511–1522, 2008.
[6] A. C. Antoulas, Approximation of Large-Scale Dynamical Systems.
Philadelphia, PA, USA: Society for Industrial and Applied Mathematics, 2005.
[7] T. Sadamoto, T. Ishizaki, and J. Imura, “Low-dimensional functional
observer design for linear systems via observer reduction approach,”in Proc. Conference on Decision and Control. IEEE, 2013, pp. 776–
781.
[8] T. Ishizaki, K. Kashima, J. Imura, and K. Aihara, “Model reduction
and clusterization of large-scale bidirectional networks,” AutomaticControl, IEEE Transactions on, vol. 59, no. 1, pp. 48–63, 2014.
[9] T. Ishizaki, K. Kashima, A. Girard, J. Imura, L. Chen, and K. Aihara,“Clustering-based H2 -state aggregation of positive networks and its
application to reduction of chemical master equations,” in Proc.
Conference on Decision and Control. IEEE, 2012, pp. 4175–4180.
[10] S. P. Boyd, Linear matrix inequalities in system and control theory.
Siam, 1994, vol. 15.
[11] S. Sastry, Nonlinear systems: analysis, stability, and control. Springer
New York, 1999, vol. 10.
[12] S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin, “Structure of
growing networks with preferential linking,” Physical Review Letters,
vol. 85, no. 21, pp. 4633–4636, 2000.