Bachelo a bei
im S udiengang Ma hema ik
Leh s uhl ü angewand e Ma hema ik
Op imal Senso Placemen o linea Sys ems
einge eich on: Maximilian P is e
einge eich am: 17.08.2012
Be eue : P o . D . Tobias Damm
D .-Ing. Ul ich Münz (Siemens AG)
Abs ac
The aim o senso placemen is o obse e he s a e o a dynamical sys em while using only a
small pa o he a ailable ou pu in o ma ion. Thus, he obse e does no need senso s a e e y
possible node o he sys em. We use senso placemen because i is no p ac ical o la ge-scale
ne wo ks, such as powe g ids, o place senso s a each node. Wi h an op imal senso placemen
we ob ain a subse o senso s which minimizes he obse e e o in compa ison o any o he
subse o he same size. This means we gene a e an op imal obse a ion wi h he gi en numbe
o senso s.
We compu e he obse e e o , o he linea dynamical sys ems we conside , wi h he
H2
-no m
o he obse e e o sys em. In his app oach, we op imize bo h he subse o selec ed senso s
and he obse e gain ma ix in pa allel. The op imiza ion p oblem is non-con ex bo h in a
cons ain , which bounds he
H2
-no m, as well as in he objec i e unc ion which uses a
`0
-no m
o coun he used senso s. To ob ain a semide ini e p og am, we i s elax he
`0
-no m by an
i e a i e eweigh ed
`1
-no m. Second, we use a e o mula ion o he
H2
-no m wi h linea ma ix
inequali ies o eplace an occu ing bilinea and he e o e non-con ex e m.
We use his compu a ionally e icien o mula ion o he senso placemen p oblem o de i e h ee
algo i hms. Fu he mo e, exis ing algo i hms, which do no use he con ex e o mula ion o he
op imiza ion p oblem, we e implemen ed. The algo i hms a e compa ed ex ensi ely ela ing o
execu ion ime, pe o mance o he chosen senso s, and he applicabili y on a p ac ical p oblem.
The p ac ical p oblem is a model o a high- ol age powe g id wi h he aim o measu e he phase
angles and he equencies a e e y node. The esul o he compa ison is ha a algo i hm wi h a
g eedy app oach sol es he op imiza ion p oblem as and usually wi h a good solu ion. Howe e ,
his algo i hm is p oblema ic because he sho sigh ed g eedy app oach canno exclude ha a
wo s case solu ion is gene a ed. The bes esul s in gene al we e p oduced by a no el app oach
made in his hesis. This no el algo i hm i e a i ely sol es he elaxed op imiza ion p oblem and
inds nea -op imal senso subse s.
Con en s
1 In oduc ion 7
2 P oblem Fo mula ion 9
2.1 P oblem Fo mula ion o Disc e e-Time Sys ems . . . . . . . . . . . . . . . . . . 9
2.2 P oblem Fo mula ion o Con inuous-Time Sys ems . . . . . . . . . . . . . . . . . 10
3 Con ex Relaxa ion o he Op imiza ion P oblem 11
3.1 Con ex Relaxa ion o he `0-Objec i e........................ 11
3.2 Re o mula ion o he H2-Pe o mance Cons ain . . . . . . . . . . . . . . . . . . 12
3.3 Algo i hms o Op imal Senso Placemen o Disc e e-Time Sys ems . . . . . . . 14
3.3.1 Algo i hm wi h Subs i u ion . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3.2 Algo i hm wi h Cone Complemen a i y Linea iza ion . . . . . . . . . . . . 17
3.4 Algo i hm o Op imal Senso Placemen o Con inuous-Time Sys ems . . . . . 19
4 Compa ison o Di e en Algo i hms 23
4.1 S a e o he a Algo i hms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1.1 G eedyAlgo i hm ............................... 23
4.1.2 B anch and Bound Algo i hm . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Run imeCompa ison.................................. 24
4.3 Pe o manceCompa ison ............................... 26
4.4 Applica ionExample.................................. 28
4.5 Concluding Compa ison o he Algo i hms . . . . . . . . . . . . . . . . . . . . . . 31
5 Conclusion and Fu u e Di ec ions 33
6 Appendix - Ma hema ical De ini ions and No a ion 35
6.1 `0-Vec o -No m..................................... 35
6.2 H2-Sys em-No m.................................... 35
6.3 Ma icesandLMIs................................... 36
Bibliog aphy 37
5

1 In oduc ion
In his hesis, we conside he p oblem o obse ing he s a e o dynamical sys ems, such as ene gy
ne wo ks, as accu a e as possible wi h a small numbe o senso s. This p oblem is in e es ing
since he e exis la ge ne wo ks, whe e i is no cos -e icien o place a senso a e e y node o
he sys em. Consequen ly, we can no measu e all he s a es o he sys em which esul s in an
obse e e o . The aim o his hesis is o ind an op imal subse o senso s, i.e. a subse , such
ha e e y o he subse o he same size has a bigge obse e e o .
The p oblem o senso selec ion appea s in se e al a eas o applica ion like obo ics, senso
placemen o s uc u es, a ge acking, chemical plan con ol and wi eless ne wo ks as lis ed
in Joshi and Boyd [2009]. The ield o applica ion o his hesis a e ene gy ne wo ks, especially
high- ol age ne wo ks, as seen in he example o Sec ion 4.4.
We conside he p oblem o choosing an op imal subse om among
n
po en ial senso s o a
ime-in a ian linea dynamic sys em o s a e es ima ion subjec o whi e inpu noise. Each
senso can measu e one componen o he ou pu ec o
y
. Thus he p ocess o senso selec ion
educes he a ailable in o ma ion o he obse e . The senso selec ion consequen ly has an
in luence on he obse e e o . The aim o his hesis is o choose an H2-op imal subse , i.e. a
subse ha minimizes he
H2
-no m o he obse e e o sys em. The
H2
-no m desc ibes he
o al ou pu ene gy o he impulse esponse o he e o sys em. A simple app oach o e alua e
he bes
k
senso subse would be o calcula e he
H2
-no m wi h an op imal obse e o all
possible
n
k
combina ions o subse s. Bu since
n
k
g ows apidly wi h inc easing
n
and
k
, his
me hod is no p ac ical. Fo example, wi h
n
= 50 po en ial senso s and
k
= 25 senso s o choose
he e a e o e 1014 possible uples, so a sequen ial e alua ion is ob iously no possible.
In his hesis, we p o ide se e al algo i hms, pa ly based on con ex op imiza ion, o a nea
op imal solu ion o he senso selec ion p oblem. And we use one combina o ial algo i hm wi h a
b anch and bound echnique o an op imal solu ion in o de o compa e he pe o mance o he
elaxed algo i hms o his op imal solu ion.
When compa ing he execu ion ime o he algo i hms, i can be seen ha he b anch and bound
algo i hm is a NP-ha d p oblem, while he o he algo i hms ha e a polynomial compu a ional
complexi y.
The p oblem we conside consis s o wo componen s. Fi s we ha e o choose a subse o senso s
which ha e o be used. Second, we ha e o calcula e an op imal obse e gain ma ix o hese
se ings in o de o e alua e he objec i e unc ion. In his app oach, we p o ide a se ing ha
sol es hese wo p oblems in pa allel.
Va ious pape s ea ela ed p oblems o he one discussed he e. In Mo e al. [2011], we ind
a gene al app oach o senso selec ion s a egies o wi eless senso ne wo ks. Howe e , he
amewo k does no hold o ou case since he s a egy is chosen o a ini e ime ho izon. This
ini eness is used in he included manipula ion o he op imiza ion p oblem o a ecu si ely
de ined equa ion o he s a e es ima ion. Gi en an in ini e ime ho izon, we could no pe o m
he same e o mula ions.
7
In Schule e al. [2012], we ind an almos dual p oblem o he obse e design in his hesis. This
pape wi h he i le Decen alized S a e Feedback Con ol o In e connec ed P ocess Sys ems
seeks o minimize he numbe o measu emen links be ween senso s and con olle s and c ea es
a con ex op imiza ion p oblem, ha is simila o he p oblem de i ed in his hesis. Ne e heless,
he de i a ion o he con ex op imiza ion p oblem is no applicable o he p oblem discussed
he e, because, on he one hand, he con olle p oblem has sligh ly di e en bilinea i ies which
ha e o be subs i u ed o con ex op imiza ion, and on he o he hand Schule e al. [2012] use
he H∞-no m in con as o he he e applied H2-no m.
The app oach in his hesis s a s wi h desc ibing he sys em o he es ima ion e o . By
in oducing a design ma ix, we c ea e he possibili y o a y he se ing o he p oblem, i.e.
o punish es ima ion e o s in ce ain s a es mo e han in o he s. In he nex s ep, we use
a LMI-cha ac e iza ion o he
H2
-no m o e o mula e his cons ain o a semide ini e one.
The men ioned cons ain is elaxed in wo di e en ways ha a e wa ds esul in di e en
algo i hms. The
`0
-no m o obse e -spa si y in he objec i e is a non-con exi y ha is elaxed
by an i e a i e weigh ed `1-no m.
The es o he hesis is o ganized as ollows. We p esen he p oblem o mula ion in Chap e 2. In
Chap e 3, we desc ibe con ex elaxa ions o he op imiza ion p oblem and he de i ed algo i hms
bo h o disc e e- ime and con inuous- ime sys ems. In Chap e 4, ex ensi e compa ison o he
di e en algo i hms is p esen ed, including a un ime and a pe o mance compa ison as well as
an example o an ene gy-ne wo k model o es he algo i hms in p axis. The hesis concludes
wi h a summa y and an ou look in Chap e 5.
8
2 P oblem Fo mula ion
2.1 P oblem Fo mula ion o Disc e e-Time Sys ems
Conside he ollowing disc e e- ime dynamical sys em
xk+1 =Axk+Bwk(2.1a)
yk=Cxk, (2.1b)
whe e
k∈N+
0
,
xk∈Rn
is he s a e o a sys em wi h
n
s a es and
yk∈Rm
,
m≤n
, is he se
o possible senso s. The unknown inpu
wk∈Rp
is ze o-mean whi e noise wi h uni a iance.
Assuming ha he ma ices
A∈Rn×n
,
B∈Rn×p
, and
C∈Rm×n
a e known, we de ine a
Luenbe ge obse e .
ˆxk+1 =Aˆxk+L(yk−ˆyk)(2.2a)
ˆyk=Cˆxk, (2.2b)
whe e
L∈Rn×m
desc ibes he obse e gain ma ix ha has o be designed. An op imal
L
could be ound by using he p inciple o he Kalman il e , which uses he same se up as he one
desc ibed he e.
We can exp ess he obse e e o e=x−ˆx:
ek+1 = (A−LC)ek+Bwk(2.3a)
zk=Wek. (2.3b)
The ma ix
W∈Rn×n
can be used as a design ma ix when he desi ed accu acy o he
obse a ions o ce ain s a es di e s o i a scaling has o be done. Elsewise
W
could simply be
de ined as an iden i y ma ix.
Le us assume he e exis s a s abilizing
L
. Thus we can gua an ee ha he
H2
-no m o he
sys em exis s and cha ac e ize he obse e e o wi h ha no m, again as in he se up o Kalman
il e . Wi h he aim o minimize he numbe o used senso s wi h a bounded obse e e o we
can de ine he ollowing simpli ied op imiza ion p oblem:
min
L”no. o senso s”(2.4a)
s. . ||Σe(L)||2
2< γ, (2.4b)
whe e Σ
e
(
L
)is a sho o m o he e o sys em depending on
L
as desc ibed in (2.3).
|| ·||2
deno es he
H2
-no m o he sys em and
γ
is a p ede ined uppe bound o he squa ed
H2
-
no m.
The numbe o used senso s in (2.4) is he numbe o non-ze o columns in
L
, i.e. i e e y elemen
o a column o
L
is ze o, he co esponding en y in
yk
has no in luence on he obse e sys em,
9
Algo i hm 1. Algo i hm wi h subs i u ion
: Relaxed senso placemen algo i hm wi h
linea iza ion ia subs i u ion o disc e e- ime sys ems
1.
Se he i e a ion coun
µ
o ze o and choose an app op ia e alue o he pa ame e
α
.
Ini ialize he weigh s ec o
ω(0)
o
ω(0)
j
= 1 o
j
= 1
, . . . , n
and choose a su icien ly small
ε > 0.
2. Sol e he op imiza ion p oblem (3.6).
3. Upda e he weigh s:
ω(µ+1)
j=1
||˜
L∗j||1+ε,j= 1, . . . , n.
4.
Te mina e he i e a ion on con e gence wi h
L
=
X−1˜
L
. O he wise inc ease
µ
by 1 and
e u n o S ep 2.
5. Imp o e he choice o Lby sol ing he op imiza ion p oblem:
min
γ, ˜
L,X
γ(3.7a)
s. . X=XT0(3.7b)
ace(BTXB)< γ (3.7c)
−X+WTW ATX−˜
CT˜
LT
XA −˜
L˜
C−X!≺0, (3.7d)
whe e
˜
C
:=
C
wi h p ede ined ze o lines. The
j
h line is se o ze o i he
j
h senso was
emo ed in he p e ious solu ion o S ep 2 (i.e. i ||L∗j||1ε).
The weigh s ec o
ω(0)
could also be ini ialized as a ze o ec o in S ep 2 o he algo i hm. Thus,
he i s i e a ion would yield o a op imal obse e gain ma ix wi h no ze o columns in
L
and
ω(1)
would coun e ac he ac ual size o he column in an op imal non-spa se obse e . Wi h he
cu en implemen a ion i is possible ha a senso is se o ze o in he i s i e a ion because
he column o
L
is smalle han he o he columns. This is in gene al no equi alen o a bad
pe o mance o he senso .
In S ep 3 o Algo i hm 1,
˜
L
o
L
could be used o pe o m he upda e on he weigh s. The eason
why
˜
L
is used he e is ha we wan o coun e ac he size o
˜
L
in he objec i e. As men ioned in
Rema k 1, i is equi alen o educe ei he he non-ze o columns o Lo ˜
L.
In S ep 3 o Algo i hm 1, we in oduced he a iable
ε
o imp o e he eweigh ing wi h
ω(µ)
and
a oid nume ical p oblems, such as di iding by ze o, as p oposed in Candes e al. [2008].
S ep 5 o Algo i hm 1 is necessa y, since he obse e , ha is calcula ed in he op imiza ion
p oblem (3.6), has a subop imal obse e gain ma ix, because e en he eweigh ed
`1
-no m in
he objec i e minimizes he absolu e alue o he en ies o
˜
L
. A e S ep 5, he exac
H2
-no m
o he obse e e o sys em can be calcula ed.
The inequa ion
||L∗j||1ε
in S ep 5 is equi alen o he s a emen ha he
j
h column is
coun e ac ed wi h a maximum weigh . This means ha he co esponding senso is emo ed.
The e o e, he j h senso is no conside ed when compu ing he imp o emen o L.
Rema k 2.
As w i en in equa ion (2.5), i is also possible o choose a cons an
γ
and emo e
γ
om he objec i e unc ion. We choose he o he way because i is on he one hand mo e
16

di icul o es ima e an app op ia e
γ
be o e s a ing he op imiza ion, and on he o he hand he
algo i hm has no incen i e o choose he bes subse o
k
senso s, i ano he subse o
k
senso s
also sa is ies he bound on he H2-no m.
3.3.2 Algo i hm wi h Cone Complemen a i y Linea iza ion
As men ioned be o e, we will conside he ma ix inequali y (3.3) o pe o m a cone complemen-
a i y linea iza ion. He e, you can see he ma ix inequali y again:
−X+WTW AT−CTLT
A−LC −X−1!≺0. (3.8)
The ollowing Lemma is necessa y o o mula ing he algo i hm wi h cone complemen a i y
linea iza ion.
Lemma 3. Wi h X0 he ollowing s a emen s hold:
(i) The ollowing wo exp essions a e equi alen :
a) KX =I
b) ace(KX) = nand K I
I X!0
(ii) K I
I X!0⇒ ace(KX)≥n.
P oo :
Ad (i): Unde condi ion o X0 he ollowing s eps a e alid:
K I
I X!0
⇔Re o mula ion o he LMI wi h he Schu complemen : K−IX−1I0
⇔K−X−10. Pos mul iplying wi h X leads o he nex ma ix inequali y.
⇔KX −I0
Wi h his e o mula ion he p oo o "a)
⇒
b)" is ob ious since
KX
=
I
implies ace(
KX
) =
ace(I) = nand KX −I0holds.
Fo he p oo o "b)
⇒
a)" we will look a a cha ac e is ic o he ma ix inequali y
KX −I
0
and use he p emise ace(KX) = ace(I) = n.
ace(KX −I)=0is alid, because o ace(KX) = ace(I) = n.
⇒KX −I
has all eigen alues equal o ze o. Because assuming ha
KX −I
has a posi i e
eigen alue, which is possible since
KX −I
0, would esul in
KX −I
ha ing a nega i e
eigen alue so ha he sum o all eigen alues is ze o again. Tha leads o a con adic ion o he
s a emen KX −I0.
Ad (ii): Unde condi ion o X0 he ollowing s ep is alid:
K I
I X!0⇔KX −I0 ollows om he i s s eps o he p oo o (i).
Since he ace o a ma ix is he sum o i s eigen alues, i esul s ha ace(
KX
)
≥ ace
(
I
) =
n
.

17
The lemma is i al o ou app oach o he second algo i hm because we now ha e he possibili y
o in oduce a new a iable
K
ha beha es as he in e se o
X
. The LMI men ioned in Lemma
3 can be used as a cons ain , while ace(
KX
)can be pa o he objec i e unc ion. Wi h
minimizing ace(
KX
), we will ecei e
K
as he in e se o
X
because ace(
KX
)
≥n
and
K=X−1a he minimum, whe e ace(KX) = n.
Applying Lemma 3 o he ma ix inequali y (3.8), he op imiza ion p oblem (2.6) esul s in:
min
γ,L,X,K γ+α
n
X
j=1
ω(µ)
j||L∗j||1+β ace(XK)(3.9a)
s. . X=XT0(3.9b)
ace(BTXB)< γ (3.9c)
−X+WTW AT−LTC
A−LC −K!≺0(3.9d)
K I
I X!0, (3.9e)
whe e
β
is ano he pa ame e o adjus ing he objec i e unc ion. As we can see, linea izing
he ma ix inequali y esul ed in a bilinea e m in he objec i e unc ion. Ou aim is o ha e
semide ini e p og am so we need o eplace he bilinea pa . We could use a linea iza ion me hod
as seen in El Ghaoui e al. [1997] o sol e such a p oblem. A a gi en poin (
Xold, Kold
), a linea
app oxima ion o ace(XK)is
φlin(X, K) = cons an + ace(XKold +XoldK).
The op imiza ion p oblem o he algo i hm wi h cone complemen a i y linea iza ion (CCL) is
hen:
min
γ,L,X,K γ+α
n
X
j=1
ω(µ)
j||L∗j||1+β ace(XKold +XoldK)(3.10a)
s. . X=XT0(3.10b)
ace(BTXB)< γ (3.10c)
−X+WTW AT−LTC
A−LC −K!≺0(3.10d)
K I
I X!0. (3.10e)
Algo i hm 2. Algo i hm wi h CCL
: Relaxed senso placemen algo i hm wi h cone comple-
men a i y linea iza ion o disc e e- ime sys ems
1.
Find a easible solu ion o
X
in (3.4) and se
Xold
=
X
. I he e a e none, exi . Se he
ou e i e a ion coun µand he inne i e a ion coun k o ze o.
2.
Se
Kold
=
X−1
old
and choose app op ia e alues o he pa ame e s
α
,
β
,
γ
and
δ
. Ini ialize
he weigh s ec o ω(0) o ω(0)
j= 1 o j= 1, . . . , n and choose a su icien ly small ε > 0.
3. Sol e he op imiza ion p oblem (3.10).
18
4.
I
| ace
(
XK
)
−n|< δ
, se
k
= 0 and go o S ep 5, else se
k
=
k
+ 1,
Xold
=
X
and
Kold =Kand go o S ep 3.
5. Upda e he weigh s:
ω(µ+1)
j=1
||L∗j||1+ε,j= 1, . . . , n.
6.
Te mina e he ou e i e a ion on con e gence. O he wise inc ease
µ
by 1 and e u n o
S ep 3.
7.
Imp o e he choice o
L
by sol ing he op imiza ion p oblem (3.7). De ine
˜
C
:=
C
wi h
p ede ined cons an ze o lines. The
j
h line is se o ze o i he
j
h senso was emo ed in
he p e ious solu ion o S ep 3 (i.e. i ||L∗j||1ε).
Rema k 3.
In S ep 1 o Algo i hm 2, he e a e se e al possibili ies o ind a easible solu ion in
(3.4). You could pe o m one i e a ion o (3.6) and use he gene a ed
X
. In his case you could
make he i s upda e o he weigh s ec o a e wa ds.
Whe eas a as e me hod is o ind
X
by calcula ing a non-spa se Kalman il e and he
co esponding Lyapuno ma ix o he obse e e o sys em. Bu in his case he ma ix
X
has
o change much du ing he nex i e a ions o emo e senso s. This means ha a lo o i e a ions
a e needed because he linea iza ion me hod punishes any changes in
X
and
K
. The e o e, he
algo i hm emo es senso s only e y slowly.
The objec i e unc ion o his algo i hm is di ided up in h ee di e en pa s, which a e he bound
on he
H2
-no m
γ
, he
`1
- elaxa ion o he numbe o senso s, and he punishing e m o he
in e se ma ix o
X
. These h ee e ms ha e o ally di e en unc ions in he op imiza ion p oblem
and ha e a di e se scale o each e m. The e o e, i is ha d o ind he igh pa ame e iza ion
o he e ms.
Depending on he applica ion o he algo i hm, i is possible o educe he complexi y o he
objec i e unc ion. I only a educed accu acy is needed o he esul , one could ake he
e m
β ace
(
XKold
+
XoldK
) om he objec i e unc ion and use i as a cons ain . Wi h an
app op ia e cons an ζ he cons ain could look like:
ace(XKold +XoldK)<2n+ζ.
As men ioned in Rema k 3, a e y small
ace
(
XKold
+
XoldK
)slows down he emo al o senso s.
The e o e, i could e en be an ad an age o pu his e m in o he cons ain s and allow g ea e
inaccu acy in o de o ha e a esul wi h less senso s.
3.4 Algo i hm o Op imal Senso Placemen o Con inuous-Time
Sys ems
In his sec ion, we de i e he app oach o he algo i hm wi h subs i u ion o con inuous-
ime sys ems. The algo i hm is e y simila o he one o disc e e- ime sys ems bu he LMI
cha ac e iza ion o he H2-no m is di e en since he Lyapuno equa ion is o ano he o m.
The ollowing Lemma is like Lemma 2, bu i is no necessa y o show he second equi alence
since he subs i u ion e en wo ks when applied on he a iables in he Lyapuno inequali y.
19
Fu he mo e, in his hesis he cone complemen a i y linea iza ion is used only o disc e e- ime
sys ems as in he li e a u e. Consequen ly, he augmen ed LMI o con inuous- ime sys ems is o
no use o ou app oach.
Lemma 4. (As in Riebe [2006]) Conside a con inuous- ime sys em wi h ans e unc ion
G(s) = "A B
C0#:= C(sI −A)−1B.
The ollowing wo s a emen s a e equi alen :
(i) ||G(s)||2
2< γ and Ais asymp o ically s able.
(ii)
The e exis s a posi i e de ini e symme ic solu ion
P
o he Lyapuno equa ion
ATP
+
PA
=
−CTCand ace(BTPB)< γ.
P oo : (pa ly as in Smi h [2010])
||G(s)||2
2< γ
⇔1
2πZ∞
−∞
ace(G(jω)G(jω)∗)dω < γ, which is he de ini ion o he H2-no m on he
equency domain. In he nex s ep we use an equi alen de ini ion on he ime domain.
⇔Z∞
0
ace(g( )Tg( ))d < γ, whe e g( )is he impulse esponse o he sys em G(s).
⇔ ace[BT(Z∞
0
eAT CTCA d )B]< γ
⇔ ace[BTPB]< γ, whe e Pis he obse abili y G amian (possible, since Ais s able)
which is de ined as P=Z∞
=0
eAT CTCeA d .
⇔ ace[BTPB]< γ, P 0and ATP+PA =−CTC

We now conside he op imiza ion p oblem (p e iously desc ibed in (2.6))
min
L,γ γ+α
n
X
j=1
||
n
X
i=1
|Lij| ||`0(3.11a)
s. . ||Σe(L)||2
2< γ (3.11b)
and apply he op imiza ion p oblem o a con inuous- ime sys em (as desc ibed in (2.8))
˙e= (A−LC)e+Bw (3.12a)
z=We. (3.12b)
20
Using Lemma 4, he esul ing op imiza ion p oblem is he ollowing:
min
γ,L,P γ+α
n
X
j=1
ω(µ)
j||L∗j||`1(3.13a)
s. . P=PT0(3.13b)
ace(BTPB)< γ (3.13c)
ATP+PA −CTLTP−PLC +I= 0, (3.13d)
whe e
P
is he solu ion o he Lyapuno equa ion as in Lemma 4 (ii). The cons ain (3.13d)
could also be w i en as a LMI, i he ma ix
P
is eplaced by a ma ix
Xcon
which is de ined as
he con inuous- ime equi alen o he ma ix Xmen ioned in he p oo o Lemma 2.
We could now pe o m he subs i u ion
˜
L
=
PL
as in Sec ion 3.3.1. Consequen ly, Rema k 1
holds also o his case and we eplace
L
in he objec i e unc ion wi h
˜
L
. This esul s in he
ollowing op imiza ion p oblem:
min
γ, ˜
L,P
γ+α
n
X
j=1
ω(µ)
j||˜
L∗j||`1(3.14a)
s. . P=PT0(3.14b)
ace(BTPB)< γ (3.14c)
ATP+PA −CT˜
LT−˜
LC +I= 0. (3.14d)
The algo i hm wi h subs i u ion is hen o mula ed as:
Algo i hm 3. Algo i hm wi h subs i u ion
: Relaxed senso placemen algo i hm wi h
linea iza ion ia subs i u ion o con inuous- ime sys ems
1.
Se he i e a ion coun
µ
o ze o and choose an app op ia e alue o he pa ame e
α
.
Ini ialize he weigh s ec o
ω(0)
o
ω(0)
j
= 1 o
j
= 1
, . . . , n
and choose a su icien ly small
ε > 0.
2. Sol e he op imiza ion p oblem (3.14).
3. Upda e he weigh s:
ω(µ+1)
j=1
||˜
L∗j||1+ε,j= 1, . . . , n.
4.
Te mina e he i e a ion on con e gence wi h
L
=
P−1˜
L
. O he wise inc ease
µ
by 1 and
e u n o S ep 2.
5. Imp o e he choice o Lby sol ing he op imiza ion p oblem:
min
γ, ˜
L,P
γ(3.15a)
s. . P=PT0(3.15b)
ace(BTPB)< γ (3.15c)
ATP+PA −˜
CT˜
LT−˜
L˜
C+I= 0, (3.15d)
whe e
˜
C
:=
C
wi h p ede ined ze o lines. The
j
h line is se o ze o i he
j
h senso was
emo ed in he p e ious solu ion o S ep 2 (i.e. i ||L∗j||1ε).
21

The commen s conce ning
˜
L
,
L
,
ε
, and
ω
made abou he algo i hm wi h subs i u ion o
disc e e- ime sys ems a e also applicable o his algo i hm.
22
4 Compa ison o Di e en Algo i hms
In his chap e , he algo i hms o Chap e 3 a e compa ed wi h s a e o he a algo i hms. One
is a g eedy app oach in wo di e en e sions as seen in Bach e al. [2010]. The o he algo i hm
uses a b anch and bound echnique o p o ide an op imal solu ion which is also p esen ed in
he i s pa o his chap e . In he second pa o he chap e , he algo i hms a e compa ed in
e ms o compu a ion ime and
H2
-pe o mance o he chosen subse o senso s. Addi ionally,
he algo i hms ha e o sol e an applica ion example consis ing o a model o a powe g id.
4.1 S a e o he a Algo i hms
In he ollowing, we discuss h ee algo i hms ha use he o iginal op imiza ion p oblem wi hou
any elaxa ion. Hence, hese algo i hms canno use in e io poin me hods as he wo algo i hms
men ioned be o e.
These algo i hms do no op imize he chosen subse o senso s and he obse e gain ma ix in
pa allel. In ac , he algo i hms compu e he
H2
-no m o di e en subse s and compa e his
H2
-pe o mance. Hence, he e is no change in di icul y o he compu a ion when he algo i hms
a e implemen ed o disc e e- ime o con inuous- ime models.
4.1.1 G eedy Algo i hm
The i s s a e o he a algo i hm, which is desc ibed, is a g eedy app oach ha is e y in ui i e.
The algo i hm s a s wi h all senso s and compa es all subse s wi h
n−
1senso s. A e u ning-o
he wo s senso he algo i hm consequen ly sea ches o he wo s senso o he emaining subse .
Consequen ly, he g eedy algo i hm inds subse s o e e y size.
This g eedy algo i hm gene a es a as solu ion and is able o gi e an o e iew o how many
senso s a e necessa y o which
H2
pe o mance. Bu as he algo i hm emo es all he senso s
sepa a ely, i could no accoun o co ela ion be ween ce ain senso subse s. This could lead
o a e y poo pe o mance.
The algo i hm sea ches o a subse o
k
senso s, whe e
k
is a p ede ined numbe wi h
k < n
.
This means ha he algo i hm has o emo e n−ksenso s.
De ine:
J
(
k1, k2, . . . , km
)is he
H2
-no m o he e o sys em wi h an op imal educed obse e
which uses he senso s
k1, k2, . . . , km
wi h
k1, k2, . . . , km∈ {
1
,
2
, . . . , n}
. The e alua ion o his
unc ion could be done wi h he p ede ined sys em ma ix
C
, in which one eplaces e e y column
excep he columns
k1, . . . , km
wi h ze o columns in o de o emo e he co esponding senso s.
One way o e alua e he unc ion is o sol e he op imiza ion p oblem (4.1) ha is p esen ed
23
below.
min
γ, ˜
L,X
γ(4.1a)
s. . X=XT0(4.1b)
ace(BTXB)< γ (4.1c)
−X+WTW ATX−CT˜
LT
XA −˜
LC −X!≺0(4.1d)
Algo i hm 4. G eedy algo i hm
: G eedy app oach o he senso selec ion p oblem as in
Bach e al. [2010].
1. Ini ialize M:= {1,2, . . . , n}and i e a ion a iable i:= 1.
2. Compu e m=a g min
lJ(M {l})wi h l∈M.
3. De ine M:= M m.
4.
I
i
=
n−k
, end he algo i hm wi h he solu ion
M
. I no , inc ease
i
by one and go back
o S ep 2.
Mis he de e mined subse o senso s and J(M) he co esponding alue o he H2no m.
The g eedy algo i hm could also be implemen ed backwa ds. This means, he algo i hm s a s
wi h ze o senso s and hen i e a i ely adds he bes senso . This implemen a ion o he algo i hm
is called e e se g eedy algo i hm in he ollowing.
4.1.2 B anch and Bound Algo i hm
The nex algo i hm is a combina o ial algo i hm, which inds he op imal subse o
k
senso s,
whe e
k
is a p ede ined numbe wi h
k < n
. The algo i hm is called b anch and bound algo i hm,
because i a anges he senso s in b anches and looks o an uppe bound a he beginning o
he algo i hm. I he algo i hm inds a good uppe bound, i is possible ha a lo o b anches,
i.e. uples o senso s, will no ha e o be e alua ed. The exac algo i hm as i was implemen ed
du ing he w i ing o his hesis and mo e de ails abou he algo i hm can be ound in Na end a
and Fukunaga [1977].
4.2 Run ime Compa ison
Fo compa ison disc e e- ime dynamical sys ems a e gene a ed andomly wi h
B, C,
and,
W
as iden i y ma ices. The dynamic ma ix
A
is chosen andomly wi h each en y uni o mly,
independen ly, and iden ically dis ibu ed on [0
,
1] and 80% ze o elemen s o ha e a spa se
s uc u e as i is common in p axis o example when conside ing powe g ids.
The con inuous- ime sys ems we e gene a ed by con e ing he andomly gene a ed disc e e- ime
sys ems o a con inuous- ime model. The loss o he spa se s uc u e du ing he con e ing
p ocess was no conside ed in his compa ison.
Algo i hms 1, 2 and 3 ha e objec i e unc ions in which he numbe o chosen senso s is con olled
only indi ec ly by chosing he pa ame e s
α
and
β
. Fo he es , he pa ame e s a e se so ha
24
abou one ou h o he senso s we e emo ed. Howe e o hese algo i hms, he pa am e iza ion
has li le in luence on he execu ion ime i we assume con e gence o he algo i hms. Al hough
a bad se o pa ame e s could easily esul in di e gence, especially when handling he algo i hm
wi h CCL, and hus o no use ul solu ion.
The b anch and bound algo i hm and he g eedy algo i hms had o choose subse s o
n
2
senso s.
In con as o he o he algo i hms, he execu ion ime is dependen on he numbe o chosen
senso s, since bo h s a wi h all senso s and consequen ly emo e senso s. When choosing e y
ew senso s, he b anch and bound algo i hm has a e y la ge execu ion ime. The algo i hm
could e en be ou pe o med by an algo i hm, which ies all possible uples, because he uppe
bound used in he b anch and bound algo i hm would be oo big and he e o e nea ly useless.
The o mula ion o he op imiza ion p oblem o he algo i hms wi h subs i u ion and he
algo i hm wi h CCL is a semide ini e p og amming p oblem, since he cons ain s a e linea
ma ix inequali ies (see also in Vandenbe ghe and Boyd [1999]) and he objec i e is a con ex
unc ion and can e en be ecas as a linea unc ion. This allows us o use in e io poin me hods
which ha e polynomial complexi y as can be seen in S u m [1999]. The eweigh ed
`1
elaxa ion
usually needs only ew s eps o con e ge when he pa ame e
ε
is chosen app op ia e (as in
Candes e al. [2008]).
In he ( e e se) g eedy algo i hm, he numbe o compu a ions o an op imal obse e gain ma ix
and he co esponding
H2
-no m inc eases wi h
n2
, whe e
n
is he numbe o s a es, assuming
ha i has o educe a ce ain pe cen age o he seno s. The op imal obse e gain ma ix could
be e alua ed as a Kalman il e , which has he complexi y o
n3
. So he g eedy algo i hm also
has polynomial complexi y.
The e e se g eedy algo i hm compu es he op imal obse e gain ma ix as o en as he g eedy
algo i hm when choosing a subse o
n
2
senso s. Hence, he execu ion ime o hese wo g eedy
algo i hms is he same. In he ollowing, we do no dis inguish be ween hese wo algo i hms
when conside ing he un ime.
The b anch and bound algo i hm is NP-ha d because a a wo s -case beha io i has o compu e
mo e han all possible subse s o
k
senso s, which a e
n
k
. Fo his eason he b anch and bound
algo i hm was mainly used he e o ha e an op imal subse in o de o compa e he pe o mance
o he o he algo i hms. This compa ison can be seen in he nex sec ion.
Fo each size, we gene a ed 10 andom exampla y sys ems as desc ibed abo e. E e y algo i hm
had o ind he op imal senso s o all hese sys ems. In Figu e 4.1 we see he execu ion ime in
seconds. The displayed execu ion ime is he a e age o he di e en compu a ions.
The dashed lines show he g eedy algo i hm and he b anch and bound algo i hm wi h a sligh ly
changed implemen a ion. While he solid lines we e p oduced wi h algo i hms ha used an
implemen a ion o he
kalman
unc ion in Ma lab, he dashed lines sol ed he op imiza ion
p oblem (4.1), which was implemen ed wi h SeDuMi (S u m [1999]) and YALMIP (Lo be g
[2004]). Bo h o he implemen a ions achie e he same esul s, bu he implemen a ion wi h
SeDuMi and YALMIP needs mo e execu ion ime and allows a be e compa ison wi h he o he
h ee algo i hms ha also use SeDuMi and YALMIP.
As we can see, he g eedy algo i hm, which uses he
kalman
unc ion me hod o Ma lab, is he
as es algo i hm. The algo i hm wi h subs i u ion and he algo i hm wi h CCL a e e y simila ,
bu he algo i hm wi h CCL needs mo e execu ion ime, which could be he e ec o he second
loop in he algo i hm. Mo eo e , he algo i hm wi h CCL qui e o en di e ged which was no
conside ed o he execu ion ime he e.
25
The algo i hm wi h cone complemen a i y linea iza ion has he same bene i s as he algo i hm
wi h subs i u ion, excep o he good execu ion- ime and he easy pa am e iza ion. The algo i hm
wi h CCL has a second loop inside o he i e a i e eweigh ed
`1
-minimiza ion which slows he
algo i hm down. The g ea disad an age o his algo i hm is ha i is e y ha d o pa ame e ize
he algo i hm co ec ly. In he applica ion example no app op ia e pa ame e iza ion was ound
al hough o e 50 di e en se s o pa ame e s we e ied. Ne e heless, he algo i hm only lead o
useless esul s.
The b anch and bound algo i hm gene a es he op imal solu ion and is easy o pa ame e ize.
In con as o he o he algo i hms, he b anch and bound algo i hm has a non-polynomial
complexi y and he e o e an execu ion- ime ha is no usable in p axis.
The g eedy algo i hm is an in ui i e and as app oach which gene a es usually a good solu ion.
Howe e , he sho sigh ed app oach canno conside he impo ance o ce ain senso s in small
subse s. Ano he applica ion example was gene a ed whe e he g eedy algo i hm i s emo es
he senso a he highes b anched node, al hough his senso would belong o he op imal subse
a he end. I seems ha highly b anched nodes a e easy o obse e wi h a lo ac i e senso s so
he senso a his node is emo ed a he beginning o he algo i hm. Howe e , wi h ew ac i e
senso s, he highly b anched senso s a e impo an o obse e he s a es a o he nodes. I is
possible ha his p ope y is mo e impo an when conside ing la ge ne wo ks and did no
a ec he esul s in Sec ion 4.3. Howe e , in ou examples he g eedy algo i hm shows a good
pe o mance and gene a es an o e iew o he H2-pe o mance o di e en sizes o subse s.
The e e se g eedy algo i hm is also a as algo i hm wi h a good solu ion o ou examples. In
con as o ou examples, mo e complex g ids wi h ins able pa s can no be sol ed by he e e se
g eedy algo i hm because i needs an obse e wi h only one senso ha has a s able obse e
e o sys em. I such a senso could no be ound, he algo i hm has no s a ing poin and ends
wi h no esul . As he o he g eedy app oach, he e e se g eedy algo i hm can only pe o m
sho sigh ed choices and canno conside co ela ion o he senso s. The e o e, i is possible ha
a e y poo solu ion is p o ided.
When sol ing a senso placemen p oblem in p axis, he algo i hm wi h subs i u ion and he
( e e se) g eedy algo i hm should be chosen. The ( e e se) g eedy algo i hm can p o ide a as
o e iew o he pe o mance o he desi ed numbe o senso s and a i s subse . A e ha , he
algo i hm wi h subs i u ion can be used o imp o e he chosen subse o con i m i s pe o mance.
32

5 Conclusion and Fu u e Di ec ions
In his hesis, we conside ed a
H2
-op imal senso placemen o linea dynamical sys ems o an
in ini e ime ho izon. The model o he sys em can ei he be a disc e e- ime o a con inuous- ime
model. Fu he mo e, he in luence o e o s on ce ain s a es o he sys em can be pa ame e ized
as well as he subse o all possible senso s.
The p oblem o senso placemen a ises in a ious ields o applica ion, o example a obse ing
powe g ids as desc ibed in chap e 4. The aim is he e o educe cos s o expensi e senso s and
s ill ha e he bes possible obse e sys em o a gi en numbe o senso s.
The o iginal p oblem o senso placemen is a non-con ex p oblem, which can ei he be app oached
by combina o ial algo i hms like he b anch and bound algo i hm o wi h g eedy algo i hms in
di e en a ia ions, as desc ibed in Chap e 4. In con as o hese me hods, he main algo i hms
in his hesis use ano he app oach. The p esen ed algo i hms op imize he s uc u e, i.e. he
senso placemen , and he obse e gain ma ix in pa allel. Due o non-con exi ies, which a ise
when coun ing he used senso s and when calcula ing he
H2
-no m wi h an unknown obse e , we
had o e o mula e he op imiza ion p oblem. The disc e e
`0
-no m was elaxed wi h an i e a i e
eweigh ed
`1
-no m. The
H2
-no m was w i en wi h a LMI cha ac e iza ion, which simpli ied
eplacing he non-con ex bilinea pa ei he wi h a subs i u ion o wi h a cone complemen a i y
linea iza ion. The p oposed app oaches we e hen de eloped in o algo i hms and implemen ed o
e icien ly and accu a ely sol e he senso placemen p oblem.
The men ioned algo i hms we e es ed in Chap e 4. The i s es was a compa ison o he
execu ion ime depending on he sys em size. The second es checked on he
H2
-pe o mance o
he chosen senso subse s o he di e en algo i hms, while compa ing he abili y o pa ame e ize
he algo i hms so ha di e en numbe s o senso s we e selec ed. Addi ionally, a p ac ical example
was sol ed. He e, he algo i hms had o de e mine an op imal senso subse . The pe o mance
o he chosen subse s we e compa ed. The example had sligh ly o he equi emen s o he
algo i hms since he p oblem was mo e ill-condi ioned han he andomly gene a ed examples.
This e ealed u he di icul ies when pa ame e izing he algo i hm wi h cone complemen a i y
linea iza ion.
The conclusion o all he es s is ha he algo i hm wi h subs i u ion is one o he bes o he
p esen ed algo i hms, since i has no se e e lacks, e.g. execu ion ime o pa ame e iza ion, and
a sa is ying pe o mance. Simila o he g eedy algo i hm, he algo i hm wi h subs i u ion o
con inuous- ime sys ems has a e y sho execu ion ime despi e o i s no ad anced implemen-
a ion. The g eedy app oach and he e e se g eedy app oach bo h lead o good solu ions a
he conside ed examples and a e he as es algo i hms. The sho sigh ed p ocedu e does no
in luence he pe o mance a he desc ibed examples.
Fu u e esea ch conce ning his opic could expand his app oach in o he ollowing di ec ions.
The assump ion ha he sys em ma ix Ais comple ely known may no be ul illed. I is mo e
ealis ic o assume ha one knows
A
wi h ce ain inaccu acies. Ano he possibili y is ha a
non-linea sys em can be es ima ed in a de ined sec o and ha he algo i hms use bounds o
his sec o o de i e an op imal subse o senso s.
33
Ano he di ec ion o u u e esea ch could be he expansion o he model o ac ua o placemen
which is he dual p oblem o he one desc ibed he e. Mo eo e , o ac ua o placemen usually
he
H∞
-no m is used. Consequen ly, he app oach would ha e o be adap ed o he LMI
cha ac e iza ion o his no m.
34
6 Appendix - Ma hema ical De ini ions and
No a ion
6.1 `0-Vec o -No m
The
`0
-no m o ec o s has mo e han one commonly used de ini ion like
||x||0
:=
limp→0||xi||p
p
o
p∈R+
and
x∈Rn
. P ecisely said, he
`0
-no m e e s o he numbe o non-ze o elemen s in
a ec o . The e o e i is s aigh o wa d o de ine also:
||x||0:=
n
X
i=1
|sign(xi)|
wi h
sign(xi) := 




−1 o xi<0
0 o xi= 0
1 o xi>0
Since he posi i e scalabili y (i.e.
|α| ||x||0
=
||αx||0
o
α∈R+
) is no sa is ied in gene al, he
`0-no m is in ac no a eal no m, howe e i is o en e e ed o as he 0-no m.
The `0-no m can be elaxed using a `1-no m, which is de ined as ollows:
||x||1:=
n
X
i=1
|xi|.
6.2 H2-Sys em-No m
De ine
G
(
s
) =
"A B
C0#
:=
C
(
sI −A
)
−1B
as he s a e-space ealiza ion o he ans e ma ix,
QC
:=
R∞
0eA BBTeAT d
as he con ollabili y G amian and
QO
:=
R∞
0eAT CCTeA d
as he
obse abili y G amian o con inuous- ime sys ems.
The H2-no m o a sys em G(s)is de ined as (acco ding o Sche e and Weiland [2000]):
||G(s)||2:= s1
2πZ∞
−∞
ace(G(jω)G(jω)∗)dω. (6.1)
This sys em no m is used o speci y he pe o mance o he sys ems in his hesis, because i is
he ypical no m o obse e s o dynamical sys ems, e.g. he well-es ablished Kalman il e (see
He nbe ge [2010]). A ypical in e p e a ion o he no m is o see
||G
(
s
)
||2
as he o al ou pu
ene gy o an impulse esponse o he sys em(as in Sche e and Weiland [2000]).
35
The ela ion o he H2-no m o he G amians o a sys em G(s) = "A B
C0#is
||G||2
2= ace(CQCCT) = ace(BTQOB).
The de ini ion o he
H2
-no m o disc e e- ime sys ems is e y simila . De ine
G
(
z
) =
"A B
C0#
:=
C
(
zI −A
)
−1B
as he s a e-space ealiza ion o he ans e ma ix o a disc e e- ime sys em.
Consequen ly, he H2-no m o disc e e- ime sys ems is de ined as:
||G(z)||2:= s1
2πZπ
−π
ace(G(ejω)G(ejω)∗)dω. (6.2)
6.3 Ma ices and LMIs
Nex , we will gi e some no a ional speci ica ions o his hesis. Gi en a ma ix
L∈Rn×n
we
will deno e
L−1
and
LT
as he in e se and he anspose o he ma ix.
L∗j
wi h
j∈1,2, . . . , m
de ines he j h column o he ma ix L.
A posi i e o nega i e de ini e (semide ini e) ma ix
L
is w i en as
L
0(
L
0) o
L≺
0
(L0) espec i ely, while >, <, ≤,≥deno e an elemen wise compa ison.
36
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37

Danksagung
An diese S elle möch e ich mich ech he zlich bei He n D
.
Ul ich Münz ü die in ensi e
Be euung bedanken, die eilweise iel Zei in Ansp uch nahm. Ohne die zahl eichen achlichen
Diskussionen und in e essan en An egungen wä e diese A bei so nich zus ande gekommen.
Auße dem bedanke ich mich auch bei de Siemens AG ü die E möglichung de A bei und die
inanzielle Un e s ü zung wäh enddessen.
He n P o
.
D
.
Tobias Damm danke ich ü die seh angenehme Koope a ion und die seh
hil sbe ei e Bean wo ung bei allen achlichen ode o ganisa o ischen F ages ellungen.
Schließlich bedanke ich mich noch bei He n P o
.
D
.
La s G üne ü die Be ei scha , die A bei
des Zwei gu ach e s zu übe nehmen.
E klä ung
Hie mi e klä e ich, dass ich diese Bachelo a bei selbs s ändig e ass und keine ande en als
die on mi angegebenen Quellen und Hil smi el benu z habe und dass ich diese A bei nich
be ei s zu E langung eines akademischen G ades einge eich habe.
Bay eu h, 17. Augus 2012