This paper is concerned with balanced realization and model reduction for discrete-time nonlinear systems. Singular perturbation type balanced truncation method is proposed. In this procedure, the Hankel singular values and the related controllability and observability properties are preserved, which is a natural generalization of both the linear discrete-time case and the nonlinear continuous-time case.

Heuberger P.

Jacquelien M. A. Scherpen

Kenji Fujimoto

Scherpen J. M. A.

English

This paper discusses balanced realization and model order reduction for both continuous-time and discrete-time general nonlinear systems based on singular value analysis of the corresponding Hankel operators. Singular value analysis clarifies the gain structure of a given nonlinear operator. Here it is proved that singular value analysis of smooth Hankel operators defined on Hilbert spaces can be characterized by simple equations in terms of their states. A balanced realization and model order reduction procedure is derived based on it, and several important properties such as stability, balanced form, Hankel norm, controllability, and observability of the original system are preserved. The work improves the earlier results of [K. Fujimoto and J. M. A. Scherpen, IEEE Trans. Automat. Control, 50 (2005), pp. 2-18] and then continues with new balancing and model reduction results

Fujimoto, Kenji

Scherpen, Jacquelien M. A.

NARCIS 

Balanced Realization and Model Order Reduction for Nonlinear Systems based on Singular Value Analysis

Fujimoto, Kenji,

Scherpen, Jacquelien M.A.,

University of Groningen Digital Archive

Balancing and model reduction for discrete-time nonlinear systems
based on Hankel singular value analysis
Kenji Fujimotoa and Jacquelien M. A. Scherpenb
aDept. of Mechanical Science and Engineering
Graduate School of Engineering
Nagoya University
Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
fujimoto@nuem.nagoya-u.ac.jp
bDelft Center for Systems and Control
Delft University of Technology
Mekelweg 2, 2628 CD Delft
The Netherlands
J.M.A.Scherpen@dcsc.tudelft.nl
AbstractThis paper is concerned with balanced realization
and model reduction for discrete-time nonlinear systems. Singu-
lar perturbation type balanced truncation method is proposed.
In this procedure, the Hankel singular values and the related
controllability and observability properties are preserved, which
is a natural generalization of both the linear discrete-time case
and the nonlinear continuous-time case.
I. INTRODUCTION
In linear control systems theory, balanced realization and
model reduction theory plays an important role in both
theoretical and practical research elds [11]. Motivated by
this, its nonlinear extension was investigated by many authors
[9], [5], [8], [4]. The authors have provided a new balanced
realization method based on singular value analysis of the
Hankel operator of the nonlinear plant [1], [2] as a precise
nonlinear counterpart of the linear case result. In those
former results, balancing and model reduction method for
continuous-time nonlinear systems was obtained, although
its discrete-time version was not investigated.
Balanced realization for discrete-time nonlinear systems
were also investigated by some authors [10], [6], [3]. How-
ever, though there is a strong similarity to the continuous-
time case, those results are not immediately obtained from
the continuous-time results. In particular, model reduction
theory based on balancing for discrete-time nonlinear sys-
tems was not obtained so far.
In this paper, we provide a balancing and model reduction
method for discrete-time nonlinear systems. This method is
a natural nonlinear generalization of the linear case as well
as a discrete-time counterpart of our continuous-time case
result. We prove that there exists a balanced realization for
nonlinear discrete-time systems which is quite similar to the
continuous-time case and that a model reduction method
based on this realization and a singular perturbation based
truncation approach derives a reduced order model which
preserves several important properties of the original system
such as controllability, observability and the gain property.
II. PROBLEM SETTING AND PRELIMINARIES
Consider an `2-stable discrete-time nonlinear system
 :

x(t + 1) = f(x(t);u(t))
y(t) = h(x(t);u(t)) (1)
with x(t) 2 Rn, u(t) 2 Rm and y(t) 2 Rr. It's controlla-
bility operator C : `m
2 (Z+) ! Rn and observability operator
O : Rn ! `
p
2(Z+) are dened by
x0 = C(u) :

x(t   1) = f(x(t);u(t)) x(1) = 0
x0 = x(0)
y = O(x0) :

x(t + 1) = f(x(t);0) x(0) = x0
y(t) = h(x(t);0) :
The Hankel operator is given by their composition
H = O  C:
The corresponding controllability and observability functions
are dened by
Lc(x) =
1
2
kC
y(x)k
2
`2 (2)
Lo(x) =
1
2
kO(x)k2
`2 (3)
where Cy is the norm-minimizing pseudo-inverse of C, that
is,
C
y(x) = arg inf
u2`m
2
C(u)=x
kuk`2:
Balanced realization investigated in this paper (also bal-
anced realization for continuous-time systems in [1], [2]) is
closely related to the solution of singular value analysis of
the Hankel operator H as
(dH(u))
  H(u) =  u;  2 R:
Solutions of this equation are important because they char-
acterize critical points of kH(u)k=kuk, hence the gain max-
imizing input argsupu(kH(u)k=kuk) is also contained in
them.
In the authors' former result, the following theorem was
proved.
Theorem 1: [3] Suppose that C, Cy and O are differen-
tiable, and that there exist  2 R and  2 Rn satisfying
@Lo()
@
= 
@Lo()
@
: (4)
Then v 2 `m
2 (Z+) dened by
v := Cy()satises the equation for singular value analysis of H
(dH(v))
  H(v) =  v: (5)
Suppose moreover that the Jacobian linearization of  has
non-zero and distinct Hankel singular values. Then there exist
n solutions curves  = i(s) 2 Rn, s 2 R satisfying i(0) =
0 for Equation (4) in a neighborhood of the origin.
Here we call the solution v of Equation (5) a singular
vector of H, and the corresponding input-output ratio
 =
kH(v)k
kvk
a singular value of H, respectively. Singular value functions
and singular vector functions corresponding to i(s) are
dened as follows for convenience.
vi(s) := Cy(i(s)) (6)
i(s) :=
kH(vi(s))k
kvi(s)k
(7)
The curves in the state-space i(s) play the role of the
coordinate axes of the balanced realization. Balanced real-
ization and the corresponding model reduction method in the
continuous-time case was derived based on them. See [1], [2]
for the detail.
III. MAIN RESULTS
A. Observability and controllability functions
As a preparation for the model reduction of discrete-
time systems, we need to characterize the observability
and controllability functions Lo(x) and Lc(x) by algebraic
equations which are similar to the Hamilton Jacobi equations
in the continuous-time case.
Lemma 1: Suppose that x = 0 of the system
x(t + 1) = f(x(t);0)
is asymptotically stable. Then a smooth observability function
Lo(x) in (3) exists if and only if
Lo(f(x;0))   Lo(x) +
1
2
h(x;0)
Th(x;0) = 0; Lo(0) = 0
(8)
has a smooth solution Lo(x).
Proof: Sufciency is proved rst. Suppose that the
observability function Lo(x) exists. Then the denition of
the observability function (3) implies that
Lo(x(0)) =
1
2
1 X
t=0
h(x(t);0)
Th(x(t);0)
=
1
2
1 X
t=1
h(x(t);0)Th(x(t);0)
+
1
2
h(x(0);0)
Th(x(0);0)
= Lo(x(1)) +
1
2
h(x(0);0)Th(x(0);0)
= Lo(f(x(0);0)) +
1
2
h(x(0);0)
Th(x(0);0):
This equation has to hold for an arbitrary initial state x(0),
that is, it satises the equation (8) since Lo(0) = 0. This
proves sufciency.
Next, necessity is proved. Suppose that the equation (8)
has a smooth solution  Lo(x). The equation (8) implies that
 Lo(x) =  Lo(F(x)) +
1
2
h(x;0)Th(x;0)
=  Lo(F(F(x)))
+
1
2
h(x;0)Th(x;0) +
1
2
h(F(x);0)Th(F(x);0)
= 
= lim
k!1
 
 Lo(F
k(x))+
1
2
k X
i=0
h(F
i(x);0)
Th(F
i(x);0)
!
= lim
k!1
 Lo(F k(x)) + Lo(x)
= Lo(x)
where F(x) := f(x;0). The last equation holds because
the system x(t + 1) = F(x(t)) is asymptotically stable and
because  Lo(0) = 0. This completes the proof.
This result is a natural nonlinear generalization of the
linear case result. In the linear case, the dynamics (1) reduces
to
 :

x(t + 1) = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t)
with appropriate matrices A, B, C and D. Here the observ-
ability function is in a quadratic form
Lo(x) =
1
2
xTGox:
The algebraic equation (8) reduces down to
A
TGoA   Go + C
TC = 0
which is the Lyapunov equation for the observability Gram-
mian in the linear case.
A similar result for the controllability function is obtained
as follows. Let us consider an optimal control problem
minimizing a cost function
min
v2`2(Z+)
x(1)=0; x(0)=x0
1 X
t=0
ku(t)k2 (9)
for the dynamics of C
x(t + 1) = f 1(x(t);u(t))
where f 1 denotes the inverse of f(x;u) with respect to x,
that is,
f(f 1(x;u);u) = x
holds. Let us denote the input u achieving the minimization
in (9) by u = u?(x). Then the dynamics of Cy : x0 7! v
becomes
Cy :

x(t + 1) = f 1(x(t);u?(x(t))) x(0) = x0
v(t) = u?(x(t))Lemma 2: Suppose that x = 0 of the feedback system
x(t + 1) = f 1(x(t);u?(x(t)))
is asymptotically stable. Then a smooth controllability func-
tion Lc(x) in (2) exists if and only if
Lc(f 1(x;u?(x)))   Lc(x) +
1
2
u?(x)Tu?(x); Lc(0) = 0
(10)
has a smooth solution Lc(x).
Proof: This lemma can be proved as a corollary of
Lemma 1 by identifying Cy with O.
These results are natural generalization of the continuous-
time case results where the equations (8) and (10) are
Hamilton-Jacobi equations.
B. Balanced realization
As in the continuous-time case [2], we can prove the
existence of balanced realization for discrete-time nonlinear
systems.
Theorem 2: Consider the state-space system  in (1) and
suppose that its Jacobian linearization has non-zero and
distinct Hankel singular values. Then, in a neighborhood of
the origin, there exists a coordinate transformation convert-
ing  into a system whose controllability and observability
functions are described by
Lc(x) =
1
2
n X
i=1
x2
i
i(xi)
Lo(x) =
1
2
n X
i=1
x
2
ii(xi)
with the singular value functions i's dened in (7). In
particular, if the above coordinate transformation is dened
globally, then
sup
u2`2(Z+)
kH(u)k
kuk
= max
i
sup
s2R
i(s):
The proof follows along the same lines as the proof of
Theorem 5 in [2], and it is omitted for the reason of space.
This realization is a natural nonlinear generalization of the
linear case, because the balanced realization in the linear case
has the controllability and observability functions
Lc(x) =
1
2
x
TG
 1
c x; Lo(x) =
1
2
x
TGox
with the controllability and observability Grammians Gc and
Go which are balanced as
Gc = Go = diag(1;:::;n)
with the Hankel singular values of the system. In Theorem
2, we have its nonlinear counterpart
Lc(x) =
1
2
x
TGc(x)
 1x; Lo(x) =
1
2
x
TGo(x)x
with
Gc(x) = Go(x) = diag(1(x1);:::;n(xn))
with the singular value functions i()'s of the Hankel
operator H.
C. Model reduction
This subsection gives a model reduction method based on
the balanced realization given in Theorem 2 with a singular
perturbation type balanced truncation technique.
Consider the system  in (1) and suppose that the system
is balanced in the sense of Theorem 2. Suppose moreover
that the singular value functions satisfy
max
s
i(s) > max
s
i+1(s):
Namely, the coordinate axis xi plays a more important role
than xj in the input-output mapping. Moreover we assume
that
max
s
k(s)  max
s
k+1(s)
holds for a certain k, and divide the state-space according to
k as
x = (xa;xb) 2 Rk  Rn k
f(x;u) =

fa(xa;xb;u)
fb(xa;xb;u)

2 Rk  Rn k:
Then, accordingly, we obtain two reduced order systems by
a singular perturbation based truncation method

a :
8
<
:
xa(t + 1) = fa(xa(t);xb(t);ua(t))
xb(t) = fb(xa(t);xb(t);ua(t))
ya(t) = h(xa(t);xb(t);ua(t))

b :
8
<
:
xa(t) = fa(xa(t);xb(t);ub(t))
xb(t + 1) = fb(xa(t);xb(t);ub(t))
yb(t) = h(xa(t);xb(t);ub(t))
:
Here we suppose that the equation
xa = fa(xa;xb;u) (11)
has a unique solution
xa = ^ fa(xb;u); (12)
and that the equation
xb = fb(xa;xb;u) (13)
has a unique solution
xb = ^ fb(xa;u): (14)
Note that these equations always have solutions at least in
a neighborhood of the origin if the Jacobian linearization ofthe system  is asymptotically stable. Then we obtain explict
forms

a :

xa(t + 1) =  fa(xa(t);ua(t))
ya(t) =  ha(xa(t);ua(t)) (15)
b :

xb(t + 1) =  fb(xb(t);ub(t))
yb(t) =  hb(xb(t);ub(t))
(16)
with
 fa(xa(t);ua(t)) := fa(xa(t); ^ fb(xa(t);ua(t));ua(t))
 ha(xa(t);ua(t)) := h(xa(t); ^ fb(xa(t);ua(t));ua(t))
 f
b(x
b(t);u
b(t)) := f
b( ^ f
a(x
b(t);u
b(t));x
b(t);u
b(t))
 hb(xb(t);ub(t)) := h( ^ fa(xb(t);ub(t));xb(t);ub(t))
by substituting the equations (12) and (14) for . For
those reduced order systems, we can prove the following
properties.
Theorem 3: Consider the system  in (1) and the trun-
cated systems a and b in (15) and (16). Then, in a
neighborhood of the origin, a and b are balanced in the
sense of Theorem 2 and
a
i (xa
i ) = i(xa
i ) i 2 f1;:::;kg

b
i(x
b
i) = k+i(x
b
i) i 2 f1;:::;n   kg
hold with a
i 's and b
i's the singular value functions of
the systems a and b, respectively. In particular, if those
functions are dened globally, then
sup
u2`m
2 (Z+)
kH(u)k
kuk
= sup
s2R
a
1(s):
Proof: Suppose that the system  in (1) is balanced in
the sense of Theorem 2. Then it implies that Lo(x) can be
divided into two parts
Lo(x) = L
a
o(x
a) + L
b
o(x
b) (17)
where
L
a
o(x
a) :=
1
2
k X
i=1
x
2
ii(xi)
Lb
o(xb) :=
1
2
n X
i=k+1
x2
ii(xi):
On the other hand, the equations (11) and (13) imply that
fa( ^ fa(xb;u);xb;u) = ^ fa(xb;u) (18)
fb(xa; ^ fb(xa;u);u) = ^ fb(xa;u): (19)
Let us substitute (14) for (8). Then we obtain
0 =

Lo(f(x;0))   Lo(x) +
1
2
h(x;0)Th(x;0)

 

xb= ^ fb(xa;u)
= Lo(f(xa; ^ fb(xa;0);0))   Lo(xa; ^ fb(xa;0))
+
1
2
h(x
a; ^ f
b(x
a;0);0)
Th(x
a; ^ f
b(x
a;0);0)
=

La
o(fa(xa; ^ fb(xa;0);0)) + Lb
o(fb(xa; ^ fb(xa;0);0))

 

La
o(xa) + Lb
o( ^ fb(xa;0))

+
1
2
h(xa; ^ fb(xa;0);0)Th(xa; ^ fb(xa;0);0)
= La
o(  fa(xa;0))   La
o(xa) +
1
2
 ha(xa;0)T ha(xa;0):
Here the third equation follows from (17), and the last
equation follows from (18) and (19). Then Lemma 1 implies
that La
o(xa) is the observability function of the system
a. Further, it can be easily proved that Lb
o(xb) is the
observability function of b by substituting (12).
In a similar way, as in the proof of Lemma 2, by identify-
ing Cy with O, we can prove that the controllability functions
La
c(xa) and Lb
c(xb) of the systems a and b are given by
La
c(xa) :=
1
2
k X
i=1
x2
i
i(xi)
Lb
c(xb) :=
1
2
n X
i=k+1
x2
i
i(xi)
which prove the former part of the theorem. The latter part
follows immediately. (See [2].) This completes the proof.
This theorem reveals several properties of the proposed
model reduction method:
 This model reduction derives balanced reduced order
models.
 Singular value functions are preserved and, in particular,
the gain of the related Hankel operator (which is called
Hankel norm) is preserved.
 Since singular value functions are preserved, some prop-
erties related to controllability and observability of the
original system is preserved.
This is both a natural nonlinear generalization of the linear
case result [7] and a natural discrete-time counterpart of the
continuous-time nonlinear systems case [1], though that was
based on balanced truncation, where here we use a singular
perturbation model reduction procedure so that we preserve
the structure.
IV. CONCLUSION
This paper was devoted to balanced realizations and model
reduction for discrete-time nonlinear dynamical systems
based on Hankel singular value analysis. Firstly, we proved
the existence of a balanced realization similar to continuous-
time case result. Secondly, a balanced truncation methodbased on a singular perturbation approach was proposed.
In this method, several important properties of the original
system such as controllability, observability and the gain
property are preserved.
V. REFERENCES
[1] K. Fujimoto and J. M. A. Scherpen. Balancing and
model reduction for nonlinear systems based on the
differential eigenstructure of Hankel operators. In Proc.
40th IEEE Conf. on Decision and Control, pages 3252
3257, 2001.
[2] K. Fujimoto and J. M. A. Scherpen. Nonlinear balanced
realization based on singular value analysis of Hankel
operators. In Proc. 42nd IEEE Conf. on Decision and
Control, pages 60726077, 2003.
[3] K. Fujimoto and J. M. A. Scherpen. Singular value
analysis of Hankel operators for general nonlinear sys-
tems. In Proc. European Control Conference, 2003.
[4] J. Hahn and T. F. Edgar. Reduction of nonlinear models
using balancing of empirical gramians and Galerkin
projections. In Proc. American Control Conference,
pages 28642868, 2000.
[5] S. Lall, J. E. Marsden, and G. Glavaski. A subspace
approach to balanced truncation for model reduction
of nonlinear systems. Int. J. Robust and Nonlinear
Control, 12(6):519535, 2002.
[6] R. Lopezlena and J. M. A. Scherpen. Energy functions
and balancing for nonlinear discrete-time systems. In
Proc. of IFAC World Congress, pages 659664, 2002.
[7] B. C. Moore. Principal component analysis in linear
systems: controllability, observability and model reduc-
tion. IEEE Trans. Autom. Contr., AC-26:1732, 1981.
[8] A. Newman and P. S. Krishnaprasad. Computing
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Balancing and model reduction for discrete-time nonlinear systems based on Hankel singular value analysis

This paper discusses balanced realization and model order reduction for both continuous-time and discrete-time general nonlinear systems based on singular value analysis of the corresponding Hankel operators. Singular value analysis clarifies the gain structure of a given nonlinear operator. Here it is proved that singular value analysis of smooth Hankel operators defined on Hilbert spaces can be characterized by simple equations in terms of their states. A balanced realization and model order reduction procedure is derived based on it, and several important properties such as stability, balanced form, Hankel norm, controllability, and observability of the original system are preserved. The work improves the earlier results of [K. Fujimoto and J. M. A. Scherpen, IEEE Trans. Automat. Control, 50 (2005), pp. 2-18] and then continues with new balancing and model reduction results.</p

ARTS repository - University of Groningen

University of Groningen

This paper is concerned with balanced realization and model reduction for discrete-time nonlinear systems. Singular perturbation type balanced truncation method is proposed. In this procedure, the Hankel singular values and the related controllability and observability properties are preserved, which is a natural generalization of both the linear discrete-time case and the nonlinear continuous-time case

Scherpen, Jacquelien M.A.

   University of GroningenBalancing and model reduction for discrete-time nonlinear systems based on Hankel singularvalue analysisFujimoto, Kenji; Scherpen, Jacquelien M.A.Published in:Proceedings of the 16th MTNS ConferenceIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2004Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Fujimoto, K., & Scherpen, J. M. A. (2004). Balancing and model reduction for discrete-time nonlinearsystems based on Hankel singular value analysis. In V. Blondel, P. V. Dooren, B. D. Moor, B. Motmans, &J. Willems (Eds.), Proceedings of the 16th MTNS Conference University of Groningen, Research Instituteof Technology and Management.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019Balancing and model reduction for discrete-time nonlinear systemsbased on Hankel singular value analysisKenji Fujimotoa and Jacquelien M. A. ScherpenbaDept. of Mechanical Science and EngineeringGraduate School of EngineeringNagoya UniversityFuro-cho, Chikusa-ku, Nagoya 464-8603, Japanfujimoto@nuem.nagoya-u.ac.jpbDelft Center for Systems and ControlDelft University of TechnologyMekelweg 2, 2628 CD DelftThe NetherlandsJ.M.A.Scherpen@dcsc.tudelft.nlAbstract— This paper is concerned with balanced realizationand model reduction for discrete-time nonlinear systems. Singu-lar perturbation type balanced truncation method is proposed.In this procedure, the Hankel singular values and the relatedcontrollability and observability properties are preserved, whichis a natural generalization of both the linear discrete-time caseand the nonlinear continuous-time case.I. INTRODUCTIONIn linear control systems theory, balanced realization andmodel reduction theory plays an important role in boththeoretical and practical research fields [11]. Motivated bythis, its nonlinear extension was investigated by many authors[9], [5], [8], [4]. The authors have provided a new balancedrealization method based on singular value analysis of theHankel operator of the nonlinear plant [1], [2] as a precisenonlinear counterpart of the linear case result. In thoseformer results, balancing and model reduction method forcontinuous-time nonlinear systems was obtained, althoughits discrete-time version was not investigated.Balanced realization for discrete-time nonlinear systemswere also investigated by some authors [10], [6], [3]. How-ever, though there is a strong similarity to the continuous-time case, those results are not immediately obtained fromthe continuous-time results. In particular, model reductiontheory based on balancing for discrete-time nonlinear sys-tems was not obtained so far.In this paper, we provide a balancing and model reductionmethod for discrete-time nonlinear systems. This method isa natural nonlinear generalization of the linear case as wellas a discrete-time counterpart of our continuous-time caseresult. We prove that there exists a balanced realization fornonlinear discrete-time systems which is quite similar to thecontinuous-time case and that a model reduction methodbased on this realization and a singular perturbation basedtruncation approach derives a reduced order model whichpreserves several important properties of the original systemsuch as controllability, observability and the gain property.II. PROBLEM SETTING AND PRELIMINARIESConsider an `2-stable discrete-time nonlinear systemΣ :{x(t + 1) = f(x(t), u(t))y(t) = h(x(t), u(t))(1)with x(t) ∈ Rn, u(t) ∈ Rm and y(t) ∈ Rr. It’s controlla-bility operator C : `m2 (Z+) → Rn and observability operatorO : Rn → `p2(Z+) are defined byx0 = C(u) :{x(t− 1) = f(x(t), u(t)) x(∞) = 0x0 = x(0)y = O(x0) :{x(t + 1) = f(x(t), 0) x(0) = x0y(t) = h(x(t), 0).The Hankel operator is given by their compositionH = O ◦ C.The corresponding controllability and observability functionsare defined byLc(x) =12‖C†(x)‖2`2 (2)Lo(x) =12‖O(x)‖2`2 (3)where C† is the norm-minimizing pseudo-inverse of C, thatis,C†(x) = arg infu∈`m2C(u)=x‖u‖`2 .Balanced realization investigated in this paper (also bal-anced realization for continuous-time systems in [1], [2]) isclosely related to the solution of singular value analysis ofthe Hankel operator H as(dH(u))∗ ◦ H(u) = λ u, λ ∈ R.Solutions of this equation are important because they char-acterize critical points of ‖H(u)‖/‖u‖, hence the gain max-imizing input arg supu(‖H(u)‖/‖u‖) is also contained inthem.In the authors’ former result, the following theorem wasproved.Theorem 1: [3] Suppose that C, C† and O are differen-tiable, and that there exist λ ∈ R and ξ ∈ Rn satisfying∂Lo(ξ)∂ξ= λ∂Lo(ξ)∂ξ. (4)Then v ∈ `m2 (Z+) dened byv := C†(ξ)satises the equation for singular value analysis of H(dH(v))∗ ◦ H(v) = λ v. (5)Suppose moreover that the Jacobian linearization of Σ hasnon-zero and distinct Hankel singular values. Then there existn solutions curves ξ = ξi(s) ∈ Rn, s ∈ R satisfying ξi(0) =0 for Equation (4) in a neighborhood of the origin.Here we call the solution v of Equation (5) a singularvector of H, and the corresponding input-output ratioσ =‖H(v)‖‖v‖a singular value of H, respectively. Singular value functionsand singular vector functions corresponding to ξi(s) aredefined as follows for convenience.vi(s) := C†(ξi(s)) (6)σi(s) :=‖H(vi(s))‖‖vi(s)‖(7)The curves in the state-space ξi(s) play the role of thecoordinate axes of the balanced realization. Balanced real-ization and the corresponding model reduction method in thecontinuous-time case was derived based on them. See [1], [2]for the detail.III. MAIN RESULTSA. Observability and controllability functionsAs a preparation for the model reduction of discrete-time systems, we need to characterize the observabilityand controllability functions Lo(x) and Lc(x) by algebraicequations which are similar to the Hamilton Jacobi equationsin the continuous-time case.Lemma 1: Suppose that x = 0 of the systemx(t + 1) = f(x(t), 0)is asymptotically stable. Then a smooth observability functionLo(x) in (3) exists if and only ifLo(f(x, 0))− Lo(x) +12h(x, 0)Th(x, 0) = 0, Lo(0) = 0(8)has a smooth solution Lo(x).Proof: Sufficiency is proved first. Suppose that theobservability function Lo(x) exists. Then the definition ofthe observability function (3) implies thatLo(x(0)) =12∞∑t=0h(x(t), 0)Th(x(t), 0)=12∞∑t=1h(x(t), 0)Th(x(t), 0)+12h(x(0), 0)Th(x(0), 0)= Lo(x(1)) +12h(x(0), 0)Th(x(0), 0)= Lo(f(x(0), 0)) +12h(x(0), 0)Th(x(0), 0).This equation has to hold for an arbitrary initial state x(0),that is, it satisfies the equation (8) since Lo(0) = 0. Thisproves sufficiency.Next, necessity is proved. Suppose that the equation (8)has a smooth solution L¯o(x). The equation (8) implies thatL¯o(x) = L¯o(F (x)) +12h(x, 0)Th(x, 0)= L¯o(F (F (x)))+12h(x, 0)Th(x, 0) +12h(F (x), 0)Th(F (x), 0)= · · ·= limk→∞(L¯o(Fk(x))+12k∑i=0h(F i(x), 0)Th(F i(x),0))= limk→∞L¯o(Fk(x)) + Lo(x)= Lo(x)where F (x) := f(x, 0). The last equation holds becausethe system x(t + 1) = F (x(t)) is asymptotically stable andbecause L¯o(0) = 0. This completes the proof.This result is a natural nonlinear generalization of thelinear case result. In the linear case, the dynamics (1) reducestoΣ :{x(t + 1) = Ax(t) + Bu(t)y(t) = Cx(t) + Du(t)with appropriate matrices A, B, C and D. Here the observ-ability function is in a quadratic formLo(x) =12xTGox.The algebraic equation (8) reduces down toATGoA−Go + CTC = 0which is the Lyapunov equation for the observability Gram-mian in the linear case.A similar result for the controllability function is obtainedas follows. Let us consider an optimal control problemminimizing a cost functionminv∈`2(Z+)x(∞)=0, x(0)=x0∞∑t=0‖u(t)‖2 (9)for the dynamics of Cx(t + 1) = f−1(x(t), u(t))where f−1 denotes the inverse of f(x, u) with respect to x,that is,f(f−1(x, u), u) = xholds. Let us denote the input u achieving the minimizationin (9) by u = u?(x). Then the dynamics of C† : x0 7→ vbecomesC† :{x(t + 1) = f−1(x(t), u?(x(t))) x(0) = x0v(t) = u?(x(t))Lemma 2: Suppose that x = 0 of the feedback systemx(t + 1) = f−1(x(t), u?(x(t)))is asymptotically stable. Then a smooth controllability func-tion Lc(x) in (2) exists if and only ifLc(f−1(x, u?(x))) − Lc(x) +12u?(x)Tu?(x), Lc(0) = 0(10)has a smooth solution Lc(x).Proof: This lemma can be proved as a corollary ofLemma 1 by identifying C† with O.These results are natural generalization of the continuous-time case results where the equations (8) and (10) areHamilton-Jacobi equations.B. Balanced realizationAs in the continuous-time case [2], we can prove theexistence of balanced realization for discrete-time nonlinearsystems.Theorem 2: Consider the state-space system Σ in (1) andsuppose that its Jacobian linearization has non-zero anddistinct Hankel singular values. Then, in a neighborhood ofthe origin, there exists a coordinate transformation convert-ing Σ into a system whose controllability and observabilityfunctions are described byLc(x) =12n∑i=1x2iσi(xi)Lo(x) =12n∑i=1x2i σi(xi)with the singular value functions σi’s dened in (7). Inparticular, if the above coordinate transformation is denedglobally, thensupu∈`2(Z+)‖H(u)‖‖u‖= maxisups∈Rσi(s).The proof follows along the same lines as the proof ofTheorem 5 in [2], and it is omitted for the reason of space.This realization is a natural nonlinear generalization of thelinear case, because the balanced realization in the linear casehas the controllability and observability functionsLc(x) =12xTG−1c x, Lo(x) =12xTGoxwith the controllability and observability Grammians Gc andGo which are balanced asGc = Go = diag(σ1, . . . , σn)with the Hankel singular values of the system. In Theorem2, we have its nonlinear counterpartLc(x) =12xTGc(x)−1x, Lo(x) =12xTGo(x)xwithGc(x) = Go(x) = diag(σ1(x1), . . . , σn(xn))with the singular value functions σi(·)’s of the Hankeloperator H.C. Model reductionThis subsection gives a model reduction method based onthe balanced realization given in Theorem 2 with a singularperturbation type balanced truncation technique.Consider the system Σ in (1) and suppose that the systemis balanced in the sense of Theorem 2. Suppose moreoverthat the singular value functions satisfymax±sσi(±s) > max±sσi+1(±s).Namely, the coordinate axis xi plays a more important rolethan xj in the input-output mapping. Moreover we assumethatmax±sσk(±s)  max±sσk+1(±s)holds for a certain k, and divide the state-space according tok asx = (xa, xb) ∈ Rk × Rn−kf(x, u) =(fa(xa, xb, u)f b(xa, xb, u))∈ Rk × Rn−k.Then, accordingly, we obtain two reduced order systems bya singular perturbation based truncation methodΣa :xa(t + 1) = fa(xa(t), xb(t), ua(t))xb(t) = f b(xa(t), xb(t), ua(t))ya(t) = h(xa(t), xb(t), ua(t))Σb :xa(t) = fa(xa(t), xb(t), ub(t))xb(t + 1) = f b(xa(t), xb(t), ub(t))yb(t) = h(xa(t), xb(t), ub(t)).Here we suppose that the equationxa = fa(xa, xb, u) (11)has a unique solutionxa = fˆa(xb, u), (12)and that the equationxb = f b(xa, xb, u) (13)has a unique solutionxb = fˆ b(xa, u). (14)Note that these equations always have solutions at least ina neighborhood of the origin if the Jacobian linearization ofthe system Σ is asymptotically stable. Then we obtain explictformsΣa :{xa(t + 1) = f¯a(xa(t), ua(t))ya(t) = h¯a(xa(t), ua(t))(15)Σb :{xb(t + 1) = f¯ b(xb(t), ub(t))yb(t) = h¯b(xb(t), ub(t))(16)withf¯a(xa(t), ua(t)) := fa(xa(t), fˆ b(xa(t), ua(t)), ua(t))h¯a(xa(t), ua(t)) := h(xa(t), fˆ b(xa(t), ua(t)), ua(t))f¯ b(xb(t), ub(t)) := f b(fˆa(xb(t), ub(t)), xb(t), ub(t))h¯b(xb(t), ub(t)) := h(fˆa(xb(t), ub(t)), xb(t), ub(t))by substituting the equations (12) and (14) for Σ. Forthose reduced order systems, we can prove the followingproperties.Theorem 3: Consider the system Σ in (1) and the trun-cated systems Σa and Σb in (15) and (16). Then, in aneighborhood of the origin, Σa and Σb are balanced in thesense of Theorem 2 andσai (xai ) = σi(xai ) i ∈ {1, . . . , k}σbi (xbi ) = σk+i(xbi ) i ∈ {1, . . . , n− k}hold with σai ’s and σbi ’s the singular value functions ofthe systems Σa and Σb, respectively. In particular, if thosefunctions are dened globally, thensupu∈`m2 (Z+)‖H(u)‖‖u‖= sups∈Rσa1 (s).Proof: Suppose that the system Σ in (1) is balanced inthe sense of Theorem 2. Then it implies that Lo(x) can bedivided into two partsLo(x) = Lao(xa) + Lbo(xb) (17)whereLao(xa) :=12k∑i=1x2i σi(xi)Lbo(xb) :=12n∑i=k+1x2i σi(xi).On the other hand, the equations (11) and (13) imply thatfa(fˆa(xb, u), xb, u) = fˆa(xb, u) (18)f b(xa, fˆ b(xa, u), u) = fˆ b(xa, u). (19)Let us substitute (14) for (8). Then we obtain0 =[Lo(f(x, 0))− Lo(x) +12h(x, 0)Th(x, 0)]∣∣∣∣xb=fˆb(xa,u)= Lo(f(xa, fˆ b(xa, 0), 0))− Lo(xa, fˆ b(xa, 0))+12h(xa, fˆ b(xa, 0), 0)Th(xa, fˆ b(xa, 0), 0)=(Lao(fa(xa, fˆ b(xa, 0), 0)) + Lbo(fb(xa, fˆ b(xa, 0), 0)))−(Lao(xa) + Lbo(fˆb(xa, 0)))+12h(xa, fˆ b(xa, 0), 0)Th(xa, fˆ b(xa, 0), 0)= Lao(f¯a(xa, 0))− Lao(xa) +12h¯a(xa, 0)Th¯a(xa, 0).Here the third equation follows from (17), and the lastequation follows from (18) and (19). Then Lemma 1 impliesthat Lao(xa) is the observability function of the systemΣa. Further, it can be easily proved that Lbo(xb) is theobservability function of Σb by substituting (12).In a similar way, as in the proof of Lemma 2, by identify-ing C† with O, we can prove that the controllability functionsLac (xa) and Lbc(xb) of the systems Σa and Σb are given byLac(xa) :=12k∑i=1x2iσi(xi)Lbc(xb) :=12n∑i=k+1x2iσi(xi)which prove the former part of the theorem. The latter partfollows immediately. (See [2].) This completes the proof.This theorem reveals several properties of the proposedmodel reduction method:• This model reduction derives balanced reduced ordermodels.• Singular value functions are preserved and, in particular,the gain of the related Hankel operator (which is calledHankel norm) is preserved.• Since singular value functions are preserved, some prop-erties related to controllability and observability of theoriginal system is preserved.This is both a natural nonlinear generalization of the linearcase result [7] and a natural discrete-time counterpart of thecontinuous-time nonlinear systems case [1], though that wasbased on balanced truncation, where here we use a singularperturbation model reduction procedure so that we preservethe structure.IV. CONCLUSIONThis paper was devoted to balanced realizations and modelreduction for discrete-time nonlinear dynamical systemsbased on Hankel singular value analysis. Firstly, we provedthe existence of a balanced realization similar to continuous-time case result. Secondly, a balanced truncation methodbased on a singular perturbation approach was proposed.In this method, several important properties of the originalsystem such as controllability, observability and the gainproperty are preserved.V. REFERENCES[1] K. Fujimoto and J. M. A. Scherpen. Balancing andmodel reduction for nonlinear systems based on thedifferential eigenstructure of Hankel operators. In Proc.40th IEEE Conf. on Decision and Control, pages 3252–3257, 2001.[2] K. Fujimoto and J. M. A. Scherpen. Nonlinear balancedrealization based on singular value analysis of Hankeloperators. In Proc. 42nd IEEE Conf. on Decision andControl, pages 6072–6077, 2003.[3] K. Fujimoto and J. M. A. Scherpen. Singular valueanalysis of Hankel operators for general nonlinear sys-tems. In Proc. European Control Conference, 2003.[4] J. Hahn and T. F. Edgar. Reduction of nonlinear modelsusing balancing of empirical gramians and Galerkinprojections. In Proc. American Control Conference,pages 2864–2868, 2000.[5] S. Lall, J. E. Marsden, and G. Glavaski. A subspaceapproach to balanced truncation for model reductionof nonlinear systems. Int. J. Robust and NonlinearControl, 12(6):519–535, 2002.[6] R. Lopezlena and J. M. A. Scherpen. Energy functionsand balancing for nonlinear discrete-time systems. InProc. of IFAC World Congress, pages 659–664, 2002.[7] B. C. Moore. Principal component analysis in linearsystems: controllability, observability and model reduc-tion. IEEE Trans. Autom. Contr., AC-26:17–32, 1981.[8] A. Newman and P. S. Krishnaprasad. Computingbalanced realizations for nonlinear systems. In Proc.Symp. Mathematical Theory of Networks and Systems,2000.[9] J. M. A. Scherpen. Balancing for nonlinear systems.Systems & Control Letters, 21:143–153, 1993.[10] E. I. Verriest and W. S. Gray. Discrete-time nonlinearbalancing. In Proc. 5th IFAC Symp. Nonlinear ControlSystems, pages 515–520, 2001.[11] K. Zhou, J. C. Doyle, and K. Glover. Robust andOptimal Control. Prentice-Hall, Inc., Upper SaddleRiver, N.J., 1996.

   University of GroningenBalancing and model reduction for discrete-time nonlinear systems based on Hankel singularvalue analysisFujimoto, Kenji; Scherpen, Jacquelien M.A.Published in:Proceedings of the 16th MTNS ConferenceIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2004Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Fujimoto, K., & Scherpen, J. M. A. (2004). Balancing and model reduction for discrete-time nonlinearsystems based on Hankel singular value analysis. In V. Blondel, P. V. Dooren, B. D. Moor, B. Motmans, &J. Willems (Eds.), Proceedings of the 16th MTNS Conference University of Groningen, Research Instituteof Technology and Management.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Balancing and model reduction for discrete-time nonlinear systemsbased on Hankel singular value analysisKenji Fujimotoa and Jacquelien M. A. ScherpenbaDept. of Mechanical Science and EngineeringGraduate School of EngineeringNagoya UniversityFuro-cho, Chikusa-ku, Nagoya 464-8603, Japanfujimoto@nuem.nagoya-u.ac.jpbDelft Center for Systems and ControlDelft University of TechnologyMekelweg 2, 2628 CD DelftThe NetherlandsJ.M.A.Scherpen@dcsc.tudelft.nlAbstract— This paper is concerned with balanced realizationand model reduction for discrete-time nonlinear systems. Singu-lar perturbation type balanced truncation method is proposed.In this procedure, the Hankel singular values and the relatedcontrollability and observability properties are preserved, whichis a natural generalization of both the linear discrete-time caseand the nonlinear continuous-time case.I. INTRODUCTIONIn linear control systems theory, balanced realization andmodel reduction theory plays an important role in boththeoretical and practical research fields [11]. Motivated bythis, its nonlinear extension was investigated by many authors[9], [5], [8], [4]. The authors have provided a new balancedrealization method based on singular value analysis of theHankel operator of the nonlinear plant [1], [2] as a precisenonlinear counterpart of the linear case result. In thoseformer results, balancing and model reduction method forcontinuous-time nonlinear systems was obtained, althoughits discrete-time version was not investigated.Balanced realization for discrete-time nonlinear systemswere also investigated by some authors [10], [6], [3]. How-ever, though there is a strong similarity to the continuous-time case, those results are not immediately obtained fromthe continuous-time results. In particular, model reductiontheory based on balancing for discrete-time nonlinear sys-tems was not obtained so far.In this paper, we provide a balancing and model reductionmethod for discrete-time nonlinear systems. This method isa natural nonlinear generalization of the linear case as wellas a discrete-time counterpart of our continuous-time caseresult. We prove that there exists a balanced realization fornonlinear discrete-time systems which is quite similar to thecontinuous-time case and that a model reduction methodbased on this realization and a singular perturbation basedtruncation approach derives a reduced order model whichpreserves several important properties of the original systemsuch as controllability, observability and the gain property.II. PROBLEM SETTING AND PRELIMINARIESConsider an `2-stable discrete-time nonlinear systemΣ :{x(t + 1) = f(x(t), u(t))y(t) = h(x(t), u(t))(1)with x(t) ∈ Rn, u(t) ∈ Rm and y(t) ∈ Rr. It’s controlla-bility operator C : `m2 (Z+) → Rn and observability operatorO : Rn → `p2(Z+) are defined byx0 = C(u) :{x(t − 1) = f(x(t), u(t)) x(∞) = 0x0 = x(0)y = O(x0) :{x(t + 1) = f(x(t), 0) x(0) = x0y(t) = h(x(t), 0).The Hankel operator is given by their compositionH = O ◦ C.The corresponding controllability and observability functionsare defined byLc(x) =12‖C†(x)‖2`2 (2)Lo(x) =12‖O(x)‖2`2 (3)where C† is the norm-minimizing pseudo-inverse of C, thatis,C†(x) = arg infu∈`m2C(u)=x‖u‖`2 .Balanced realization investigated in this paper (also bal-anced realization for continuous-time systems in [1], [2]) isclosely related to the solution of singular value analysis ofthe Hankel operator H as(dH(u))∗ ◦ H(u) = λ u, λ ∈ R.Solutions of this equation are important because they char-acterize critical points of ‖H(u)‖/‖u‖, hence the gain max-imizing input arg supu(‖H(u)‖/‖u‖) is also contained inthem.In the authors’ former result, the following theorem wasproved.Theorem 1: [3] Suppose that C, C† and O are differen-tiable, and that there exist λ ∈ R and ξ ∈ Rn satisfying∂Lo(ξ)∂ξ= λ∂Lo(ξ)∂ξ. (4)Then v ∈ `m2 (Z+) defined byv := C†(ξ)satisfies the equation for singular value analysis of H(dH(v))∗ ◦ H(v) = λ v. (5)Suppose moreover that the Jacobian linearization of Σ hasnon-zero and distinct Hankel singular values. Then there existn solutions curves ξ = ξi(s) ∈ Rn, s ∈ R satisfying ξi(0) =0 for Equation (4) in a neighborhood of the origin.Here we call the solution v of Equation (5) a singularvector of H, and the corresponding input-output ratioσ =‖H(v)‖‖v‖a singular value of H, respectively. Singular value functionsand singular vector functions corresponding to ξi(s) aredefined as follows for convenience.vi(s) := C†(ξi(s)) (6)σi(s) :=‖H(vi(s))‖‖vi(s)‖(7)The curves in the state-space ξi(s) play the role of thecoordinate axes of the balanced realization. Balanced real-ization and the corresponding model reduction method in thecontinuous-time case was derived based on them. See [1], [2]for the detail.III. MAIN RESULTSA. Observability and controllability functionsAs a preparation for the model reduction of discrete-time systems, we need to characterize the observabilityand controllability functions Lo(x) and Lc(x) by algebraicequations which are similar to the Hamilton Jacobi equationsin the continuous-time case.Lemma 1: Suppose that x = 0 of the systemx(t + 1) = f(x(t), 0)is asymptotically stable. Then a smooth observability functionLo(x) in (3) exists if and only ifLo(f(x, 0)) − Lo(x) +12h(x, 0)Th(x, 0) = 0, Lo(0) = 0(8)has a smooth solution Lo(x).Proof: Sufficiency is proved first. Suppose that theobservability function Lo(x) exists. Then the definition ofthe observability function (3) implies thatLo(x(0)) =12∞∑t=0h(x(t), 0)Th(x(t), 0)=12∞∑t=1h(x(t), 0)Th(x(t), 0)+12h(x(0), 0)Th(x(0), 0)= Lo(x(1)) +12h(x(0), 0)Th(x(0), 0)= Lo(f(x(0), 0)) +12h(x(0), 0)Th(x(0), 0).This equation has to hold for an arbitrary initial state x(0),that is, it satisfies the equation (8) since Lo(0) = 0. Thisproves sufficiency.Next, necessity is proved. Suppose that the equation (8)has a smooth solution L̄o(x). The equation (8) implies thatL̄o(x) = L̄o(F (x)) +12h(x, 0)Th(x, 0)= L̄o(F (F (x)))+12h(x, 0)Th(x, 0) +12h(F (x), 0)Th(F (x), 0)= · · ·= limk→∞(L̄o(Fk(x))+12k∑i=0h(F i(x), 0)Th(F i(x),0))= limk→∞L̄o(Fk(x)) + Lo(x)= Lo(x)where F (x) := f(x, 0). The last equation holds becausethe system x(t + 1) = F (x(t)) is asymptotically stable andbecause L̄o(0) = 0. This completes the proof.This result is a natural nonlinear generalization of thelinear case result. In the linear case, the dynamics (1) reducestoΣ :{x(t + 1) = Ax(t) + Bu(t)y(t) = Cx(t) + Du(t)with appropriate matrices A, B, C and D. Here the observ-ability function is in a quadratic formLo(x) =12xTGox.The algebraic equation (8) reduces down toATGoA − Go + CTC = 0which is the Lyapunov equation for the observability Gram-mian in the linear case.A similar result for the controllability function is obtainedas follows. Let us consider an optimal control problemminimizing a cost functionminv∈`2(Z+)x(∞)=0, x(0)=x0∞∑t=0‖u(t)‖2 (9)for the dynamics of Cx(t + 1) = f−1(x(t), u(t))where f−1 denotes the inverse of f(x, u) with respect to x,that is,f(f−1(x, u), u) = xholds. Let us denote the input u achieving the minimizationin (9) by u = u?(x). Then the dynamics of C† : x0 7→ vbecomesC† :{x(t + 1) = f−1(x(t), u?(x(t))) x(0) = x0v(t) = u?(x(t))Lemma 2: Suppose that x = 0 of the feedback systemx(t + 1) = f−1(x(t), u?(x(t)))is asymptotically stable. Then a smooth controllability func-tion Lc(x) in (2) exists if and only ifLc(f−1(x, u?(x))) − Lc(x) +12u?(x)Tu?(x), Lc(0) = 0(10)has a smooth solution Lc(x).Proof: This lemma can be proved as a corollary ofLemma 1 by identifying C† with O.These results are natural generalization of the continuous-time case results where the equations (8) and (10) areHamilton-Jacobi equations.B. Balanced realizationAs in the continuous-time case [2], we can prove theexistence of balanced realization for discrete-time nonlinearsystems.Theorem 2: Consider the state-space system Σ in (1) andsuppose that its Jacobian linearization has non-zero anddistinct Hankel singular values. Then, in a neighborhood ofthe origin, there exists a coordinate transformation convert-ing Σ into a system whose controllability and observabilityfunctions are described byLc(x) =12n∑i=1x2iσi(xi)Lo(x) =12n∑i=1x2i σi(xi)with the singular value functions σi’s defined in (7). Inparticular, if the above coordinate transformation is definedglobally, thensupu∈`2(Z+)‖H(u)‖‖u‖= maxisups∈Rσi(s).The proof follows along the same lines as the proof ofTheorem 5 in [2], and it is omitted for the reason of space.This realization is a natural nonlinear generalization of thelinear case, because the balanced realization in the linear casehas the controllability and observability functionsLc(x) =12xTG−1c x, Lo(x) =12xTGoxwith the controllability and observability Grammians Gc andGo which are balanced asGc = Go = diag(σ1, . . . , σn)with the Hankel singular values of the system. In Theorem2, we have its nonlinear counterpartLc(x) =12xTGc(x)−1x, Lo(x) =12xTGo(x)xwithGc(x) = Go(x) = diag(σ1(x1), . . . , σn(xn))with the singular value functions σi(·)’s of the Hankeloperator H.C. Model reductionThis subsection gives a model reduction method based onthe balanced realization given in Theorem 2 with a singularperturbation type balanced truncation technique.Consider the system Σ in (1) and suppose that the systemis balanced in the sense of Theorem 2. Suppose moreoverthat the singular value functions satisfymax±sσi(±s) > max±sσi+1(±s).Namely, the coordinate axis xi plays a more important rolethan xj in the input-output mapping. Moreover we assumethatmax±sσk(±s)  max±sσk+1(±s)holds for a certain k, and divide the state-space according tok asx = (xa, xb) ∈ Rk × Rn−kf(x, u) =(fa(xa, xb, u)f b(xa, xb, u))∈ Rk × Rn−k.Then, accordingly, we obtain two reduced order systems bya singular perturbation based truncation methodΣa :xa(t + 1) = fa(xa(t), xb(t), ua(t))xb(t) = f b(xa(t), xb(t), ua(t))ya(t) = h(xa(t), xb(t), ua(t))Σb :xa(t) = fa(xa(t), xb(t), ub(t))xb(t + 1) = f b(xa(t), xb(t), ub(t))yb(t) = h(xa(t), xb(t), ub(t)).Here we suppose that the equationxa = fa(xa, xb, u) (11)has a unique solutionxa = f̂a(xb, u), (12)and that the equationxb = f b(xa, xb, u) (13)has a unique solutionxb = f̂ b(xa, u). (14)Note that these equations always have solutions at least ina neighborhood of the origin if the Jacobian linearization ofthe system Σ is asymptotically stable. Then we obtain explictformsΣa :{xa(t + 1) = f̄a(xa(t), ua(t))ya(t) = h̄a(xa(t), ua(t))(15)Σb :{xb(t + 1) = f̄ b(xb(t), ub(t))yb(t) = h̄b(xb(t), ub(t))(16)withf̄a(xa(t), ua(t)) := fa(xa(t), f̂ b(xa(t), ua(t)), ua(t))h̄a(xa(t), ua(t)) := h(xa(t), f̂ b(xa(t), ua(t)), ua(t))f̄ b(xb(t), ub(t)) := f b(f̂a(xb(t), ub(t)), xb(t), ub(t))h̄b(xb(t), ub(t)) := h(f̂a(xb(t), ub(t)), xb(t), ub(t))by substituting the equations (12) and (14) for Σ. Forthose reduced order systems, we can prove the followingproperties.Theorem 3: Consider the system Σ in (1) and the trun-cated systems Σa and Σb in (15) and (16). Then, in aneighborhood of the origin, Σa and Σb are balanced in thesense of Theorem 2 andσai (xai ) = σi(xai ) i ∈ {1, . . . , k}σbi (xbi ) = σk+i(xbi ) i ∈ {1, . . . , n − k}hold with σai ’s and σbi ’s the singular value functions ofthe systems Σa and Σb, respectively. In particular, if thosefunctions are defined globally, thensupu∈`m2 (Z+)‖H(u)‖‖u‖= sups∈Rσa1 (s).Proof: Suppose that the system Σ in (1) is balanced inthe sense of Theorem 2. Then it implies that Lo(x) can bedivided into two partsLo(x) = Lao(xa) + Lbo(xb) (17)whereLao(xa) :=12k∑i=1x2i σi(xi)Lbo(xb) :=12n∑i=k+1x2i σi(xi).On the other hand, the equations (11) and (13) imply thatfa(f̂a(xb, u), xb, u) = f̂a(xb, u) (18)f b(xa, f̂ b(xa, u), u) = f̂ b(xa, u). (19)Let us substitute (14) for (8). Then we obtain0 =[Lo(f(x, 0)) − Lo(x) +12h(x, 0)Th(x, 0)]∣∣∣∣xb=f̂b(xa,u)= Lo(f(xa, f̂ b(xa, 0), 0)) − Lo(xa, f̂ b(xa, 0))+12h(xa, f̂ b(xa, 0), 0)Th(xa, f̂ b(xa, 0), 0)=(Lao(fa(xa, f̂ b(xa, 0), 0)) + Lbo(fb(xa, f̂ b(xa, 0), 0)))−(Lao(xa) + Lbo(f̂b(xa, 0)))+12h(xa, f̂ b(xa, 0), 0)Th(xa, f̂ b(xa, 0), 0)= Lao(f̄a(xa, 0)) − Lao(xa) +12h̄a(xa, 0)Th̄a(xa, 0).Here the third equation follows from (17), and the lastequation follows from (18) and (19). Then Lemma 1 impliesthat Lao(xa) is the observability function of the systemΣa. Further, it can be easily proved that Lbo(xb) is theobservability function of Σb by substituting (12).In a similar way, as in the proof of Lemma 2, by identify-ing C† with O, we can prove that the controllability functionsLac (xa) and Lbc(xb) of the systems Σa and Σb are given byLac(xa) :=12k∑i=1x2iσi(xi)Lbc(xb) :=12n∑i=k+1x2iσi(xi)which prove the former part of the theorem. The latter partfollows immediately. (See [2].) This completes the proof.This theorem reveals several properties of the proposedmodel reduction method:• This model reduction derives balanced reduced ordermodels.• Singular value functions are preserved and, in particular,the gain of the related Hankel operator (which is calledHankel norm) is preserved.• Since singular value functions are preserved, some prop-erties related to controllability and observability of theoriginal system is preserved.This is both a natural nonlinear generalization of the linearcase result [7] and a natural discrete-time counterpart of thecontinuous-time nonlinear systems case [1], though that wasbased on balanced truncation, where here we use a singularperturbation model reduction procedure so that we preservethe structure.IV. CONCLUSIONThis paper was devoted to balanced realizations and modelreduction for discrete-time nonlinear dynamical systemsbased on Hankel singular value analysis. Firstly, we provedthe existence of a balanced realization similar to continuous-time case result. Secondly, a balanced truncation methodbased on a singular perturbation approach was proposed.In this method, several important properties of the originalsystem such as controllability, observability and the gainproperty are preserved.V. REFERENCES[1] K. Fujimoto and J. M. A. Scherpen. Balancing andmodel reduction for nonlinear systems based on thedifferential eigenstructure of Hankel operators. In Proc.40th IEEE Conf. on Decision and Control, pages 3252–3257, 2001.[2] K. Fujimoto and J. M. A. Scherpen. Nonlinear balancedrealization based on singular value analysis of Hankeloperators. In Proc. 42nd IEEE Conf. on Decision andControl, pages 6072–6077, 2003.[3] K. Fujimoto and J. M. A. Scherpen. Singular valueanalysis of Hankel operators for general nonlinear sys-tems. In Proc. European Control Conference, 2003.[4] J. Hahn and T. F. Edgar. Reduction of nonlinear modelsusing balancing of empirical gramians and Galerkinprojections. In Proc. American Control Conference,pages 2864–2868, 2000.[5] S. Lall, J. E. Marsden, and G. Glavaski. A subspaceapproach to balanced truncation for model reductionof nonlinear systems. Int. J. Robust and NonlinearControl, 12(6):519–535, 2002.[6] R. Lopezlena and J. M. A. Scherpen. Energy functionsand balancing for nonlinear discrete-time systems. InProc. of IFAC World Congress, pages 659–664, 2002.[7] B. C. Moore. Principal component analysis in linearsystems: controllability, observability and model reduc-tion. IEEE Trans. Autom. Contr., AC-26:17–32, 1981.[8] A. Newman and P. S. Krishnaprasad. Computingbalanced realizations for nonlinear systems. In Proc.Symp. Mathematical Theory of Networks and Systems,2000.[9] J. M. A. Scherpen. Balancing for nonlinear systems.Systems & Control Letters, 21:143–153, 1993.[10] E. I. Verriest and W. S. Gray. Discrete-time nonlinearbalancing. In Proc. 5th IFAC Symp. Nonlinear ControlSystems, pages 515–520, 2001.[11] K. Zhou, J. C. Doyle, and K. Glover. Robust andOptimal Control. Prentice-Hall, Inc., Upper SaddleRiver, N.J., 1996.

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 University of Groningen
Balancing and model reduction for discrete-time nonlinear systems based on Hankel singular
value analysis
Fujimoto, Kenji; Scherpen, Jacquelien M.A.
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Balancing and model reduction for discrete-time nonlinear systems
based on Hankel singular value analysis
Kenji Fujimotoa and Jacquelien M. A. Scherpenb
aDept. of Mechanical Science and Engineering
Graduate School of Engineering
Nagoya University
Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
fujimoto@nuem.nagoya-u.ac.jp
bDelft Center for Systems and Control
Delft University of Technology
Mekelweg 2, 2628 CD Delft
The Netherlands
J.M.A.Scherpen@dcsc.tudelft.nl
Abstract— This paper is concerned with balanced realization
and model reduction for discrete-time nonlinear systems. Singu-
lar perturbation type balanced truncation method is proposed.
In this procedure, the Hankel singular values and the related
controllability and observability properties are preserved, which
is a natural generalization of both the linear discrete-time case
and the nonlinear continuous-time case.
I. INTRODUCTION
In linear control systems theory, balanced realization and
model reduction theory plays an important role in both
theoretical and practical research fields [11]. Motivated by
this, its nonlinear extension was investigated by many authors
[9], [5], [8], [4]. The authors have provided a new balanced
realization method based on singular value analysis of the
Hankel operator of the nonlinear plant [1], [2] as a precise
nonlinear counterpart of the linear case result. In those
former results, balancing and model reduction method for
continuous-time nonlinear systems was obtained, although
its discrete-time version was not investigated.
Balanced realization for discrete-time nonlinear systems
were also investigated by some authors [10], [6], [3]. How-
ever, though there is a strong similarity to the continuous-
time case, those results are not immediately obtained from
the continuous-time results. In particular, model reduction
theory based on balancing for discrete-time nonlinear sys-
tems was not obtained so far.
In this paper, we provide a balancing and model reduction
method for discrete-time nonlinear systems. This method is
a natural nonlinear generalization of the linear case as well
as a discrete-time counterpart of our continuous-time case
result. We prove that there exists a balanced realization for
nonlinear discrete-time systems which is quite similar to the
continuous-time case and that a model reduction method
based on this realization and a singular perturbation based
truncation approach derives a reduced order model which
preserves several important properties of the original system
such as controllability, observability and the gain property.
II. PROBLEM SETTING AND PRELIMINARIES
Consider an `2-stable discrete-time nonlinear system
Σ :
{
x(t + 1) = f(x(t), u(t))
y(t) = h(x(t), u(t))
(1)
with x(t) ∈ Rn, u(t) ∈ Rm and y(t) ∈ Rr. It’s controlla-
bility operator C : `m2 (Z+) → R
n and observability operator
O : Rn → `p2(Z+) are defined by
x0 = C(u) :
{
x(t− 1) = f(x(t), u(t)) x(∞) = 0
x0 = x(0)
y = O(x0) :
{
x(t + 1) = f(x(t), 0) x(0) = x0
y(t) = h(x(t), 0)
.
The Hankel operator is given by their composition
H = O ◦ C.
The corresponding controllability and observability functions
are defined by
Lc(x) =
1
2
‖C†(x)‖2`2 (2)
Lo(x) =
1
2
‖O(x)‖2`2 (3)
where C† is the norm-minimizing pseudo-inverse of C, that
is,
C†(x) = arg inf
u∈`m2
C(u)=x
‖u‖`2 .
Balanced realization investigated in this paper (also bal-
anced realization for continuous-time systems in [1], [2]) is
closely related to the solution of singular value analysis of
the Hankel operator H as
(dH(u))∗ ◦ H(u) = λ u, λ ∈ R.
Solutions of this equation are important because they char-
acterize critical points of ‖H(u)‖/‖u‖, hence the gain max-
imizing input arg supu(‖H(u)‖/‖u‖) is also contained in
them.
In the authors’ former result, the following theorem was
proved.
Theorem 1: [3] Suppose that C, C† and O are differen-
tiable, and that there exist λ ∈ R and ξ ∈ Rn satisfying
∂Lo(ξ)
∂ξ
= λ
∂Lo(ξ)
∂ξ
. (4)
Then v ∈ `m2 (Z+) dened by
v := C†(ξ)
satises the equation for singular value analysis of H
(dH(v))∗ ◦ H(v) = λ v. (5)
Suppose moreover that the Jacobian linearization of Σ has
non-zero and distinct Hankel singular values. Then there exist
n solutions curves ξ = ξi(s) ∈ Rn, s ∈ R satisfying ξi(0) =
0 for Equation (4) in a neighborhood of the origin.
Here we call the solution v of Equation (5) a singular
vector of H, and the corresponding input-output ratio
σ =
‖H(v)‖
‖v‖
a singular value of H, respectively. Singular value functions
and singular vector functions corresponding to ξi(s) are
defined as follows for convenience.
vi(s) := C
†(ξi(s)) (6)
σi(s) :=
‖H(vi(s))‖
‖vi(s)‖
(7)
The curves in the state-space ξi(s) play the role of the
coordinate axes of the balanced realization. Balanced real-
ization and the corresponding model reduction method in the
continuous-time case was derived based on them. See [1], [2]
for the detail.
III. MAIN RESULTS
A. Observability and controllability functions
As a preparation for the model reduction of discrete-
time systems, we need to characterize the observability
and controllability functions Lo(x) and Lc(x) by algebraic
equations which are similar to the Hamilton Jacobi equations
in the continuous-time case.
Lemma 1: Suppose that x = 0 of the system
x(t + 1) = f(x(t), 0)
is asymptotically stable. Then a smooth observability function
Lo(x) in (3) exists if and only if
Lo(f(x, 0))− Lo(x) +
1
2
h(x, 0)Th(x, 0) = 0, Lo(0) = 0
(8)
has a smooth solution Lo(x).
Proof: Sufficiency is proved first. Suppose that the
observability function Lo(x) exists. Then the definition of
the observability function (3) implies that
Lo(x(0)) =
1
2
∞∑
t=0
h(x(t), 0)Th(x(t), 0)
=
1
2
∞∑
t=1
h(x(t), 0)Th(x(t), 0)
+
1
2
h(x(0), 0)Th(x(0), 0)
= Lo(x(1)) +
1
2
h(x(0), 0)Th(x(0), 0)
= Lo(f(x(0), 0)) +
1
2
h(x(0), 0)Th(x(0), 0).
This equation has to hold for an arbitrary initial state x(0),
that is, it satisfies the equation (8) since Lo(0) = 0. This
proves sufficiency.
Next, necessity is proved. Suppose that the equation (8)
has a smooth solution L¯o(x). The equation (8) implies that
L¯o(x) = L¯o(F (x)) +
1
2
h(x, 0)Th(x, 0)
= L¯o(F (F (x)))
+
1
2
h(x, 0)Th(x, 0) +
1
2
h(F (x), 0)Th(F (x), 0)
= · · ·
= lim
k→∞
(
L¯o(F
k(x))+
1
2
k∑
i=0
h(F i(x), 0)Th(F i(x),0)
)
= lim
k→∞
L¯o(F
k(x)) + Lo(x)
= Lo(x)
where F (x) := f(x, 0). The last equation holds because
the system x(t + 1) = F (x(t)) is asymptotically stable and
because L¯o(0) = 0. This completes the proof.
This result is a natural nonlinear generalization of the
linear case result. In the linear case, the dynamics (1) reduces
to
Σ :
{
x(t + 1) = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t)
with appropriate matrices A, B, C and D. Here the observ-
ability function is in a quadratic form
Lo(x) =
1
2
xTGox.
The algebraic equation (8) reduces down to
ATGoA−Go + C
TC = 0
which is the Lyapunov equation for the observability Gram-
mian in the linear case.
A similar result for the controllability function is obtained
as follows. Let us consider an optimal control problem
minimizing a cost function
min
v∈`2(Z+)
x(∞)=0, x(0)=x0
∞∑
t=0
‖u(t)‖2 (9)
for the dynamics of C
x(t + 1) = f−1(x(t), u(t))
where f−1 denotes the inverse of f(x, u) with respect to x,
that is,
f(f−1(x, u), u) = x
holds. Let us denote the input u achieving the minimization
in (9) by u = u?(x). Then the dynamics of C† : x0 7→ v
becomes
C† :
{
x(t + 1) = f−1(x(t), u?(x(t))) x(0) = x0
v(t) = u?(x(t))
Lemma 2: Suppose that x = 0 of the feedback system
x(t + 1) = f−1(x(t), u?(x(t)))
is asymptotically stable. Then a smooth controllability func-
tion Lc(x) in (2) exists if and only if
Lc(f
−1(x, u?(x))) − Lc(x) +
1
2
u?(x)Tu?(x), Lc(0) = 0
(10)
has a smooth solution Lc(x).
Proof: This lemma can be proved as a corollary of
Lemma 1 by identifying C† with O.
These results are natural generalization of the continuous-
time case results where the equations (8) and (10) are
Hamilton-Jacobi equations.
B. Balanced realization
As in the continuous-time case [2], we can prove the
existence of balanced realization for discrete-time nonlinear
systems.
Theorem 2: Consider the state-space system Σ in (1) and
suppose that its Jacobian linearization has non-zero and
distinct Hankel singular values. Then, in a neighborhood of
the origin, there exists a coordinate transformation convert-
ing Σ into a system whose controllability and observability
functions are described by
Lc(x) =
1
2
n∑
i=1
x2i
σi(xi)
Lo(x) =
1
2
n∑
i=1
x2i σi(xi)
with the singular value functions σi’s dened in (7). In
particular, if the above coordinate transformation is dened
globally, then
sup
u∈`2(Z+)
‖H(u)‖
‖u‖
= max
i
sup
s∈R
σi(s).
The proof follows along the same lines as the proof of
Theorem 5 in [2], and it is omitted for the reason of space.
This realization is a natural nonlinear generalization of the
linear case, because the balanced realization in the linear case
has the controllability and observability functions
Lc(x) =
1
2
xTG−1c x, Lo(x) =
1
2
xTGox
with the controllability and observability Grammians Gc and
Go which are balanced as
Gc = Go = diag(σ1, . . . , σn)
with the Hankel singular values of the system. In Theorem
2, we have its nonlinear counterpart
Lc(x) =
1
2
xTGc(x)
−1x, Lo(x) =
1
2
xTGo(x)x
with
Gc(x) = Go(x) = diag(σ1(x1), . . . , σn(xn))
with the singular value functions σi(·)’s of the Hankel
operator H.
C. Model reduction
This subsection gives a model reduction method based on
the balanced realization given in Theorem 2 with a singular
perturbation type balanced truncation technique.
Consider the system Σ in (1) and suppose that the system
is balanced in the sense of Theorem 2. Suppose moreover
that the singular value functions satisfy
max
±s
σi(±s) > max
±s
σi+1(±s).
Namely, the coordinate axis xi plays a more important role
than xj in the input-output mapping. Moreover we assume
that
max
±s
σk(±s)  max
±s
σk+1(±s)
holds for a certain k, and divide the state-space according to
k as
x = (xa, xb) ∈ Rk × Rn−k
f(x, u) =
(
fa(xa, xb, u)
f b(xa, xb, u)
)
∈ Rk × Rn−k.
Then, accordingly, we obtain two reduced order systems by
a singular perturbation based truncation method
Σa :


xa(t + 1) = fa(xa(t), xb(t), ua(t))
xb(t) = f b(xa(t), xb(t), ua(t))
ya(t) = h(xa(t), xb(t), ua(t))
Σb :


xa(t) = fa(xa(t), xb(t), ub(t))
xb(t + 1) = f b(xa(t), xb(t), ub(t))
yb(t) = h(xa(t), xb(t), ub(t))
.
Here we suppose that the equation
xa = fa(xa, xb, u) (11)
has a unique solution
xa = fˆa(xb, u), (12)
and that the equation
xb = f b(xa, xb, u) (13)
has a unique solution
xb = fˆ b(xa, u). (14)
Note that these equations always have solutions at least in
a neighborhood of the origin if the Jacobian linearization of
the system Σ is asymptotically stable. Then we obtain explict
forms
Σa :
{
xa(t + 1) = f¯a(xa(t), ua(t))
ya(t) = h¯a(xa(t), ua(t))
(15)
Σb :
{
xb(t + 1) = f¯ b(xb(t), ub(t))
yb(t) = h¯b(xb(t), ub(t))
(16)
with
f¯a(xa(t), ua(t)) := fa(xa(t), fˆ b(xa(t), ua(t)), ua(t))
h¯a(xa(t), ua(t)) := h(xa(t), fˆ b(xa(t), ua(t)), ua(t))
f¯ b(xb(t), ub(t)) := f b(fˆa(xb(t), ub(t)), xb(t), ub(t))
h¯b(xb(t), ub(t)) := h(fˆa(xb(t), ub(t)), xb(t), ub(t))
by substituting the equations (12) and (14) for Σ. For
those reduced order systems, we can prove the following
properties.
Theorem 3: Consider the system Σ in (1) and the trun-
cated systems Σa and Σb in (15) and (16). Then, in a
neighborhood of the origin, Σa and Σb are balanced in the
sense of Theorem 2 and
σai (x
a
i ) = σi(x
a
i ) i ∈ {1, . . . , k}
σbi (x
b
i ) = σk+i(x
b
i ) i ∈ {1, . . . , n− k}
hold with σai ’s and σbi ’s the singular value functions of
the systems Σa and Σb, respectively. In particular, if those
functions are dened globally, then
sup
u∈`m2 (Z+)
‖H(u)‖
‖u‖
= sup
s∈R
σa1 (s).
Proof: Suppose that the system Σ in (1) is balanced in
the sense of Theorem 2. Then it implies that Lo(x) can be
divided into two parts
Lo(x) = L
a
o(x
a) + Lbo(x
b) (17)
where
Lao(x
a) :=
1
2
k∑
i=1
x2i σi(xi)
Lbo(x
b) :=
1
2
n∑
i=k+1
x2i σi(xi).
On the other hand, the equations (11) and (13) imply that
fa(fˆa(xb, u), xb, u) = fˆa(xb, u) (18)
f b(xa, fˆ b(xa, u), u) = fˆ b(xa, u). (19)
Let us substitute (14) for (8). Then we obtain
0 =
[
Lo(f(x, 0))− Lo(x) +
1
2
h(x, 0)Th(x, 0)
]∣∣∣∣
xb=fˆb(xa,u)
= Lo(f(x
a, fˆ b(xa, 0), 0))− Lo(x
a, fˆ b(xa, 0))
+
1
2
h(xa, fˆ b(xa, 0), 0)Th(xa, fˆ b(xa, 0), 0)
=
(
Lao(f
a(xa, fˆ b(xa, 0), 0)) + Lbo(f
b(xa, fˆ b(xa, 0), 0))
)
−
(
Lao(x
a) + Lbo(fˆ
b(xa, 0))
)
+
1
2
h(xa, fˆ b(xa, 0), 0)Th(xa, fˆ b(xa, 0), 0)
= Lao(f¯
a(xa, 0))− Lao(x
a) +
1
2
h¯a(xa, 0)Th¯a(xa, 0).
Here the third equation follows from (17), and the last
equation follows from (18) and (19). Then Lemma 1 implies
that Lao(xa) is the observability function of the system
Σa. Further, it can be easily proved that Lbo(x
b) is the
observability function of Σb by substituting (12).
In a similar way, as in the proof of Lemma 2, by identify-
ing C† with O, we can prove that the controllability functions
Lac (x
a) and Lbc(x
b) of the systems Σa and Σb are given by
Lac(x
a) :=
1
2
k∑
i=1
x2i
σi(xi)
Lbc(x
b) :=
1
2
n∑
i=k+1
x2i
σi(xi)
which prove the former part of the theorem. The latter part
follows immediately. (See [2].) This completes the proof.
This theorem reveals several properties of the proposed
model reduction method:
• This model reduction derives balanced reduced order
models.
• Singular value functions are preserved and, in particular,
the gain of the related Hankel operator (which is called
Hankel norm) is preserved.
• Since singular value functions are preserved, some prop-
erties related to controllability and observability of the
original system is preserved.
This is both a natural nonlinear generalization of the linear
case result [7] and a natural discrete-time counterpart of the
continuous-time nonlinear systems case [1], though that was
based on balanced truncation, where here we use a singular
perturbation model reduction procedure so that we preserve
the structure.
IV. CONCLUSION
This paper was devoted to balanced realizations and model
reduction for discrete-time nonlinear dynamical systems
based on Hankel singular value analysis. Firstly, we proved
the existence of a balanced realization similar to continuous-
time case result. Secondly, a balanced truncation method
based on a singular perturbation approach was proposed.
In this method, several important properties of the original
system such as controllability, observability and the gain
property are preserved.
V. REFERENCES
[1] K. Fujimoto and J. M. A. Scherpen. Balancing and
model reduction for nonlinear systems based on the
differential eigenstructure of Hankel operators. In Proc.
40th IEEE Conf. on Decision and Control, pages 3252–
3257, 2001.
[2] K. Fujimoto and J. M. A. Scherpen. Nonlinear balanced
realization based on singular value analysis of Hankel
operators. In Proc. 42nd IEEE Conf. on Decision and
Control, pages 6072–6077, 2003.
[3] K. Fujimoto and J. M. A. Scherpen. Singular value
analysis of Hankel operators for general nonlinear sys-
tems. In Proc. European Control Conference, 2003.
[4] J. Hahn and T. F. Edgar. Reduction of nonlinear models
using balancing of empirical gramians and Galerkin
projections. In Proc. American Control Conference,
pages 2864–2868, 2000.
[5] S. Lall, J. E. Marsden, and G. Glavaski. A subspace
approach to balanced truncation for model reduction
of nonlinear systems. Int. J. Robust and Nonlinear
Control, 12(6):519–535, 2002.
[6] R. Lopezlena and J. M. A. Scherpen. Energy functions
and balancing for nonlinear discrete-time systems. In
Proc. of IFAC World Congress, pages 659–664, 2002.
[7] B. C. Moore. Principal component analysis in linear
systems: controllability, observability and model reduc-
tion. IEEE Trans. Autom. Contr., AC-26:17–32, 1981.
[8] A. Newman and P. S. Krishnaprasad. Computing
balanced realizations for nonlinear systems. In Proc.
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2000.
[9] J. M. A. Scherpen. Balancing for nonlinear systems.
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[10] E. I. Verriest and W. S. Gray. Discrete-time nonlinear
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Balancing and model reduction for discrete-time nonlinear systems based on Hankel singular value analysis

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