A new es ima ion o he lowe e o bound in balanced unca ion me hod I
Ha Binh Minha, Ca les Ba lleb, En ic Fossasc
aSchool o Applied Ma hema ics and In o ma ics, Hanoi Uni e si y o Science and Technology, 1 Dai Co Vie , Hanoi, Vie nam
bDepa men o Applied Ma hema ics IV and he Ins i u e o Indus ial and Con ol Enginee ing, Uni e si a Poli `ecnica de Ca alunya, Spain
cIns i u e o Indus ial and Con ol Enginee ing, Uni e si a Poli `ecnica de Ca alunya, Spain
Abs ac
Fo a single-inpu /single-ou pu (SISO) linea ime-in a ian dynamical sys em, he classical H∞-no m lowe e o bound o bal-
anced unca ion me hod is
kG(s)−G (s)kH∞≥σ +1,
whe e σi,i=1,...,n,a e he Hankel singula alues o sys em in dec easing o de . In his pape we p o ide a new es ima ion o
he lowe e o , acco ding o
kG(s)−G (s)kH∞≥max{σd,2X
i<J
siσi},
whe e siis he sign associa ed wi h he Hankel singula alue σiin Obe ’s canonical o m. The index subse Jand he numbe din
abo e inequali y will be in oduced in he pape . We show by means o an example ha he new bound may be ele an in deciding
which s a es o keep in he balanced unca ion me hod, and ha using he classical esul no always yields he bes esul .
Keywo ds: linea ime-in a ian sys ems, H∞-no m, lowe e o bound, balanced unca ion, model o de educ ion, Obe ’s
canonical o m.
1. In oduc ion
The H∞-no m lowe and uppe e o bounds o he balanced
unca ion me hod a e gi en by
σ +1≤ kG(s)−G (s)kH∞≤2
n
X
i= +1
σi,(1.1)
whe e σi,i=1,...,n, a e he Hankel singula alues o he
sys em (see e.g., [3, 5]). F om hese inequali ies i ollows
ha , in o de o ge he smalles e o o he unca ed sys em,
one should, in any case, dis ega d he s a es associa ed wi h he
smalles Hankel singula alues.
Al hough selec ing he unca ed sys em using he abo e
idea yields gene ally good esul s, and is widely used in p ac-
ice, in some cases, as shown in he example a he end o he
pape , a smalle e o is ob ained i one selec s di e en ly he
s a es o disca d. In his pape we explain his si ua ion by ob-
aining a be e lowe e o bound, and showing how his may
in luence he selec ion o he unca ed sys em.
Ou esul es s on he new lowe bound o he H∞no m o
a sys em p oposed in [6, 8, 11], which is based on compu ing
IThis wo k was suppo ed by Vie namese Na ional Founda ion o Science
and Technology De elopmen (NAFOSTED) unde g an 101.02-2013.18, and
by Spanish CICYT p ojec s DPI2010-15110 and DPI2011-25649.
Email add esses: [email p o ec ed];
[email p o ec ed] (Ha Binh Minh), [email p o ec ed]
(Ca les Ba lle), [email p o ec ed] (En ic Fossas)
he ans e unc ion a ze o equency. In his pape , we use
he same idea o compu e he lowe e o bound o he balanced
unca ion me hod.
Th oughou he pape , we deno e ma ices and ec o s by
bold- ace le e s, o example A, and scala s by no mal le e s,
as in a. The symbols Rand Cdeno e he ields o eal and
complex numbe s, espec i ely.
We conside he class o single-inpu /single-ou pu (SISO)
linea dynamical sys ems wi h ime-in a ian s a e-space eal-
iza ion
˙
x( )=Ax( )+bu( ),(1.2)
y( )=cx( ), ∈R,(1.3)
whe e (A,b,c)∈Rn×n×Rn×1×R1×n,x( )∈Rn,u( )∈Rand
y( )∈R. The s a e-space sys em (1.2)-(1.3) gene a es a ans e
unc ion
G(s) :=c(sI−A)−1b,s∈C(1.4)
o which some imes we use he no a ion A b
c!.
The H∞-no m o a linea ime-in a ian sys em is de ined
by
kG(s)kH∞:=max
ω∈R
σmax(G(jω)),
whe e σmax(G(jω)) is he la ges singula alue o G(jω). In
he SISO case, howe e , G(jω) is jus a complex numbe , and
he e o e σmax(G(jω)) =|G(jω)|, which gi es
kG(s)kH∞=max
ω∈R
σmax(G(jω)) =max
ω∈R
|G(jω)|.
P ep in submi ed o Else ie Oc obe 1, 2013
The ou line o his pape is as ollows. In Sec ion 2 we ecall
Obe ’s canonical o m o balanced ealiza ion [8]. This ealiza-
ion is use ul o in es iga e he H∞-no m lowe e o bound o
he balanced unca ion me hod, which will be discussed in Sec-
ion 3. In Sec ion 4, a nume ical example is p esen ed, which
shows ha he classical balanced unca ion me hod does no
always yield he bes esul . Finally, ou conclusions a e p e-
sen ed in Sec ion 5.
2. Obe ’s canonical o m o balanced ealiza ion
2.1. Balanced ealiza ion
Assume ha he sys em G(s)= A b
c!is asymp o ically
s able and is in a minimal ealiza ion, i.e., Ais s able, he pai
(A,b) is con ollable and he pai (A,c) is obse able. The con-
ollabili y and obse abili y G amians Pand Qo he sys em
a e, espec i ely, he solu ions o he algeb aic Lyapuno equa-
ions
AP +PAT+bbT=0,(2.1)
ATQ+QA +cTc=0.(2.2)
The balancing ans o ma ion is a s a e ans o ma ion ha
makes he con ollabili y and obse abili y G amians iden ical
and diagonal, i.e. i he ans o ma ion is gi en by xb( )=
T−1x( ), hen
(Ab,bb,cb)=(T−1AT,T−1b,cT),(2.3)
Pb=T−1PT−T=Σ:=diag(σ1, σ2, . . . , σn)
=TTQT =Qb,(2.4)
whe e σ1≥σ2≥ · · · ≥ σn>0 a e he Hankel singula alues
o he sys em. The ealiza ion (Ab,bb,cb) is called a balanced
ealiza ion o sys em [7].
2.2. Balanced unca ion
In o de o ob ain an o de educed model, we assume ha
(Ab,bb,cb) a e in balanced ealiza ion. Le J:={i1,...,i } ⊂
{1,...,n}be he indexes o he s a es ha we wan o keep in he
educed model. Le IJ:=col{ei1,...,ei }, whe e ejis he j- h
column ec o o iden i y ma ix In. Then, he educed-o de
sys em G (s) is ob ained by unca ing he (n− ) s a es which
do no belong o J, as ollows:
AJ:=IJAbIJ,bJ:=IJbb,cJ:=cbIJ,
G (s) :=cJ(sI−AJ)−1bJ.(2.5)
2.3. Obe ’s canonical o m o SISO balanced ealiza ion
Suppose ha he SISO linea ime-in a ian sys em G(s)=
Abbb
cb!=cb(sI−Ab)−1bbis in balanced ealiza ion. Mo e-
o e , o simplici y eason, we assume ha he Hankel singula
alues o G(s)a e dis inc , i.e., σ1> σ2>· · · > σn.Then in
his case, ollowing [8], G(s)= Abbb
cb!can be w i en in
Obe ’s canonical o m:
G(s)=




























−b2
1
2σ1
−b1b2
s1s2σ1+σ2· · · −b1bn
s1snσ1+σnb1
−b2b1
s2s1σ2+σ1
−b2
2
2σ2· · · −b2bn
s2snσ2+σnb2
.
.
.
.
.
.
...
.
.
.
.
.
.
−bnb1
sns1σn+σ1
−bnb2
sns2σn+σ2· · · −b2
n
2σnbn
s1b1s2b2· · · snbn




























,(2.6)
whe e si=1 o −1 is he sign associa ed wi h he Hankel singu-
la alue σi. No ice ha i all he signs sia e equal, ei he 1 o
−1, hen he ma ix Abis symme ic, and cb=bbo cb=−bb.
These special cases will be conside ed la e on.
3. A new lowe e o bound
The H∞-no m o sys em is always bigge han |G(0)|. Mo e-
o e , i all he Hankel singula alues a e dis inc hen G(0) can
be compu ed, in e ms o Hankel’s singula alues, as ollows
G(0) =2
n
X
i=1
siσi.(3.1)
The s a emen and he p oo o his esul can be ound in [6,
8, 11]. By combining his wi h he classical lowe bound o
sys em [5], we ge he ollowing esul :
Theo em 3.1. Assume ha G(s)is s able SISO sys em and all
he Hankel singula alues a e dis inc . Then,
kG(s)kH∞≥max{σ1,2
n
X
i=1
siσi}.(3.2)
Rema k 3.2. One can conside wo cases whe e he new lowe
bound (3.2) eaches he uppe bound o sys em (see Theo em
4.1 in [10], Theo em 4.1 in[9], o Rema k 2.3 in[8]).
(a) In he case ha si=1 o all i =1,...,n, i.e. he
case ha G(s)has s a e-space symme ic ealiza ion A=
AT,b=cT, hen
kG(s)kH∞≥2(σ1+· · · +σn).
I ollows ha kG(s)kH∞=2(σ1+· · ·+σn)since he lowe
bound is equal o he uppe bound.
(b) In he case ha si=−1 o all i =1,...,n, i.e. he case
ha G(s)has ealiza ion A=AT,b=−cT, hen we ge
he same esul as abo e
kG(s)kH∞=2(σ1+· · · +σn).
Now using he same idea as in Theo em 3.1, we a e in a
posi ion o o mula e he main esul o his pape .
Theo em 3.3. Assume ha G(s)is a s able SISO sys em and
ha all he Hankel singula alues a e dis inc . Le J ⊂ {1,...,n}
2
be gi en, and d :=max{j|j<J}. I G (s)is educed model ob-
ained by he balanced unca ion me hod om G(s), hen he
lowe bound o kG(s)−G (s)kH∞is
kG(s)−G (s)kH∞≥max{σd,2X
i<J
siσi}.(3.3)
P oo o Theo em 3.3. Since he balanced unca ion me hod e-
ains he Hankel singula alues σi,i∈ J, as well as he signs
si,∈ J, associa ed wi h hem, o he educed sys em G (s), we
ge ha
G(0) −G (0) =2
n
X
i=1
siσi−2X
i∈J
siσi=2X
i<J
siσi,
which implies ha
kG(s)−G (s)kH∞=max
ω∈R
σmax(G(jω)−G (jω))
=max
ω∈R
|G(jω)−G (jω)|
≥ |G(0) −G (0)|
=2X
i<J
siσi.
Now we conside wo special cases in Theo em 3.3. In hese
cases, he uppe bound and he lowe bound o kG(s)−G (s)kH∞
a e he same, and he e o e he exac e o o balanced unca-
ion me hod can be compu ed. The p oo is omi ed since i is
ob ious.
Co olla y 3.4. Wi h he assump ions as in Theo em 3.3, we ge
ha :
(a) In he case ha all unca ed s a es ha e he sign equal o
1, i.e. si=1 o all i <J, he H∞-no m o (G(s)−G (s))
is
kG(s)−G (s)kH∞=2X
i<J
σi.
(b) In he case ha all unca ed s a es ha e he sign equal
o −1, i.e. si=−1 o all i <J, we ge he same esul as
abo e
kG(s)−G (s)kH∞=2X
i<J
σi.
We conside now wo special cases, namely when G(s) has
s a e-space symme ic ealiza ion A=AT,b=cT, o when
A=AT,b=−cT.
Co olla y 3.5. Wi h he same assump ions as in Theo em 3.3,
one has ha
(a) (Theo em 4.4, [9]) In he case ha G(s)has s a e-space
symme ic ealiza ion A=AT,b=cT, one has
kG(s)−G (s)kH∞=2(σ +1+· · · +σn).
(b) In he case ha G(s)has ealiza ion A=AT,b=−cT,
one ge s also
kG(s)−G (s)kH∞=2(σ +1+· · · +σn).
P oo . The esul s in Co olla y 3.5 co espond o he special
cases in Co olla y 3.4 since all unca ed s a es ha e he sign 1
o −1.
4. A nume ical example
In classical balanced unca ion me hod one unca es he
s a es ha ing he smalles Hankel singula alues. The example
ha we p esen shows ha aking in o accoun he new igh e
lowe bound may educe he e o on he educed o de model
sys em wi hou inc easing i s o de .
Conside he ollowing linea sys em, gi en by Obe ’s eal-
iza ion o m by
b=
1
1
1
1
1

,s=
1
1
−1
1
1

, σ =
5
4
3
2.5
2

,
c=hs1b1s2b2· · · snbni
=h1 1 −111i,
A=

−b2
1
2σ1
−b1b2
s1s2σ1+σ2· · · −b1bn
s1snσ1+σn
−b2b1
s2s1σ2+σ1
−b2
2
2σ2· · · −b2bn
s2snσ2+σn
.
.
.
.
.
.
...
.
.
.
−bnb1
sns1σn+σ1
−bnb2
sns2σn+σ2· · · −b2
n
2σn

=

−0.100 −0.111 0.500 −0.133 −0.143
−0.111 −0.125 1.000 −0.154 −0.167
−0.500 −1.000 −0.167 2.000 1.000
−0.133 −0.154 −2.000 −0.200 −0.222
−0.143 −0.167 −1.000 −0.222 −0.250

.
Le G(s)= A b
c!.G(s) is in balanced ealiza ion and i s
Hankel singula alues a e
σ1=5, σ2=4, σ3=3, σ4=2.5, σ5=2.
I we unca e he 2 s a es ha ing he smalles Hankel singu-
la alues, σ4and σ5, as in he classical balanced unca ion
me hod, we ob ain he ollowing educed-o de sys em.
G1(s)=













−0.100 −0.111 0.500 1
−0.111 −0.125 1.000 1
−0.500 −1.000 −0.167 1
1 1 −1














.
The H∞-no m o e o in his case is
kG(s)−G1(s)kH∞=9,
which is a ained a equency ω=0.
Howe e , i we unca e he 2 s a es co esponding o σ2
and σ3we ob ain he ollowing educed-o de sys em.
G2(s)=













−0.100 −0.133 −0.143 1
−0.133 −0.200 −0.222 1
−0.143 −0.222 −0.250 1
111














.
3
The H∞-no m o e o in his case is
kG(s)−G2(s)kH∞=5.6421,
which is now a ained a equency ω=2.4814.
Hence, unca ing he s a es ha ing he smalles Hankel sin-
gula alues does no always gi e he bes educed sys em. One
can ind an explana ion o his esul in he ligh o he new
lowe e o bound gi en in Theo em 3.3. One has ha
2(σ4+σ5)≥ kG(s)−G1(s)kH∞≥max{σ4,2(σ4+σ5)},
and, he e o e, 9 ≥ kG(s)−G1(s)kH∞≥9, o kG(s)−G1(s)kH∞=
9. On he o he hand,
2(σ2+σ3)≥ kG(s)−G2(s)kH∞≥max{σ2,2|σ2−σ3|},
which leads o 14 ≥ kG(s)−G2(s)kH∞≥4. The lowe bound o
kG(s)−G2(s)kH∞is smalle han he one o kG(s)−G1(s)kH∞,
so i may yield a be e esul , and in ac i does in ou example.
5. Conclusions
This pape has shown ha he H∞-no m lowe bound o
SISO linea sys ems as well as he H∞-no m lowe e o bound
o balanced unca ion me hod can be imp o ed. The echnique
is based on he compu a ion o ans e unc ion a ze o e-
quency. The key poin in his pape is he balanced ealiza ion
ob ained by Obe [8] o he SISO linea sys ems. Whe he he
new bound imp o es he classical one depends on he de ailed
nume ical alues o he Hankel singula alues o he sys em,
as well as o hei signs, bu we ha e shown an explici example
whe e he new bound is ele an .
We should commen ha hese esul s can no be ex ended
o he case o mul i-inpu /mul i-ou pu (MIMO) linea ime-
in a ian sys ems. This is due o he ac ha equali y (3.1)
does no hold o he MIMO case. The ques ion o de ining
a sys ema ic way o imp o e he balanced unca ion gi en he
spec um o Hankel’s singula alues and hei associa ed signs
is an open one.
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4