Compa ison be ween wo s a e es ima ion
echniques o linea sys ems ?
D. Me hy ⇤C. S oica Maniu ⇤T. Alamo ⇤⇤ E.F. Camacho ⇤⇤
S. Ben Chabane ⇤⇤⇤
⇤Labo a oi e des Signaux e Sys `emes, Cen aleSup´elec-CNRS-Uni .
Pa is-Sud, Uni e si ´e Pa is Saclay, Gi -su -Y e e, F ance
(e-mail: {do y.me hy,c is ina.s oica}@supelec. ).
⇤⇤ Depa men o Ingenie ´ıa de Sis emas y Au om´a ica, Uni e sidad de
Se illa, Camino de los Descub imien os, 41092 Se illa, Spain
(e-mail: alamo@ca uja.us.es, edua [email p o ec ed].es).
⇤⇤⇤ GIPSA Lab, Con ol Depa men , F ance
(e-mail: so [email protected] enoble-inp. ).
Abs ac : This pape p esen s a compa ison in e ms o accu acy and complexi y be ween wo
app oaches used o s a e es ima ion o linea sys ems: a classic Kalman il e and a gua an eed
se -membe ship s a e es ima ion echnique. The main goal o his pape is o analyze he
ad an ages o hese echniques and o combine hem in he u u e in a new accu a e and simple
ex ension ha handles sys em unce ain ies and chance cons ain s. Two academic examples
illus a e he main di↵e ences be ween he compa ed echniques.
Keywo ds: Se -membe ship es ima ion, Kalman il e , linea sys ems, ellipsoidal se
1. INTRODUCTION
Gene ally, p ocess con ol equi es accu a e in o ma ion
abou he plan . Howe e , he measu ed a iables do no
o ally desc ibe he beha io o he sys em. Pa icula ly,
he en i e sys em s a e is no always accessible. This is why
i is impo an o ge access o he unknown in o ma ion
using a ailable da a/knowledge. Va ious me hods o s a e
es ima ion a e sugges ed in he li e a u e and hey can be
di ided in o wo ca ego ies. S ochas ic app oaches such as
he Kalman Fil e (see Kalman (1960)) assume he p io
knowledge o he dis ibu ion o he pe u ba ions and
he measu emen noises (in gene al Gaussian dis ibu ion)
aking in o accoun ce ain cha ac e is ics like he mean
and he co a iance. This assump ion can be some imes
un ealis ic. Thus de e minis ic app oaches (Be sekas and
Rhodes (1971), Fogel and Huang (1982)) ha conside s
unknown bu bounded pe u ba ions and bounded noises
ha e been elabo a ed. The e a e se e al de e minis ic ap-
p oaches used o s a e es ima ion, like se -membe ship
s a e es ima ion (Schweppe (1968)), in e al obse e s
(Pou asgha e al. (2016)) o obus il e ing me hods (El
Ghaoui and Cala io e (2001)). In he implemen a ion o
se -based de e minis ic es ima ion me hods, a ious se s
a e used: poly opes (Wal e and Pie -Lahanie (1989)),
zono opes (Combas el (2003), Alamo e al. (2005), Le e al.
(2013)), ellipsoids (Ku zhanski and V´alyi (1996), Du ieu
e al. (2001), Polyak e al. (2004), Da yin e al. (2006),
Da yin and Ku zhanski (2012), Che nousko (1994)). The
low complexi y o ellipsoids makes hem widely used com-
pa ed o poly opes which o↵e be e accu acy o he
es ima ion. Combas el (2015) ecen ly p oposed a combi-
?The i s au ho is a i s yea PhD s uden .
na ion be ween s ochas ic and de e minis ic app oaches,
mo e exac ly a zono opic Kalman il e .
In he p esen pape , a compa ison in e ms o accu acy
and compu a ion complexi y is made be ween wo es ima-
ion echniques s udied in he li e a u e: an ellipsoidal se -
membe ship s a e es ima ion (Ben Chabane e al. (2014a),
Ben Chabane e al. (2014b)) and a classical Kalman il e .
The esul s illus a ed in his pape a e he main mo i-
a ion o de elop a u u e ex ension ha will combine
he ad an ages o he wo compa ed echniques: be e
accu acy and less complexi y.
The emainde o he pape is o ganized as ollows. Sec-
ion 2 o mula es he s a e es ima ion p oblem o lin-
ea sys ems. Sec ion 3 b ie ly p esen s he ellipsoidal se -
membe ship s a e es ima ion echnique. Sec ion 4 eminds
he s a e es ima ion using he classical Kalman Fil e . Sec-
ion 5 exposes he compa ison be ween he wo echniques.
Sec ion 6 p oposes wo illus a i e examples. Finally, con-
clusions and pe spec i es a e d awn in Sec ion 7.
No a ion. An in e al deno ed [a, b] is he se de ined by
{x2R:axb}.Thus,B= [-1,1] can be deno ed as
uni a y in e al. A box ([a1,b
1],...,[an,b
n])>is an in e al
ec o . A uni a y box in Rnis a box composed by nuni a y
in e als. The iden i y ma ix o size nis de ined by In.
A andom a iable xno mally dis ibu ed wi h mean o ¯x
and wi h a iance o 2is ep esen ed by x⇠N¯x, 2.
2. PRELIMINARIES AND SETUP
Conside he ollowing disc e e- ime Linea Time In a ian
(LTI) sys em
⇢xk+1 =Axk+Buk+Ewwk
yk=Cxk+Duk+F k(1)
whe e xk2Rnxis he s a e ec o o he sys em, uk2Rnu
is he inpu ec o , and yk2Rnyis he measu ed ou pu
ec o a sample ime k. The ma ices A,B,C,D,Ewand
F ha e he app op ia e dimensions. He e, wk2Rnxis a
ec o con aining he s a e pe u ba ions, while k2Rny
con ains he measu emen noises.
Combining he s a e pe u ba ions and he measu emen
noises in one ec o !k=[
wk k]>2Rnx+ny, he sys em
(1) can be ew i en in an equi alen o m
⇢xk+1 =Axk+Buk+E!k
yk=Cxk+Duk+F!k(2)
wi h he ma ices E=⇥Ew0nx,ny⇤and F=⇥0ny,nxF ⇤.
In his wo k, we aim o compa e an es ima e o he s a e
o he sys em (1) p o ided by wo app oaches ha a e
u he de ailed in Sec ions 3 and 4.
3. GUARANTEED ELLIPSOIDAL
SET-MEMBERSHIP STATE ESTIMATION
This sec ion b ie ly desc ibes he gua an eed ellipsoidal
se -membe ship s a e es ima ion p oposed by Ben Cha-
bane e al. (2014a) o he sys em (2).
In his con ex , conside ha he ini ial s a e x0belongs
o he ellipsoid:
E(P0,x
0,⇢
0)={x2Rnx:(xx0)>P0(xx0)⇢0}
wi h he shape ma ix P0=P>
00, he cen e x0and
he so called adius ⇢0.
Gi en an ellipsoidal es ima ion se o xk,wi h¯xk he
nominal es ima ed s a e, he objec i e o his echnique is
o ob ain an ellipsoidal se es ima ion o xk+1. Figu e
1 illus a es he 2-s ep p ocedu e (p edic ion and co -
ec ion) o calcula e he es ima ion se . A each sample
ime k, he g een se ep esen s he p edic ed s a e se .
The yellow s ip ep esen s he se o s a es compa ible
wi h he measu emen s yk+1. The blue ellipsoid (which
con ains he s a e es ima ion se ) o e app oxima es he
in e sec ion o he p edic ed s a e se and he measu emen
s ip. Repea ing he p ocedu e a each ime kleads o a
gua an eed es ima ion se ha con ains he s a e o he
sys em.
Fig. 1. S a e es ima ion using ellipsoids
Mo e p ecisely, a each ime k, he adius o he ellipsoidal
se is minimized by sol ing a Linea Ma ix Inequali y
(LMI) p oblem (see Ben Chabane e al. (2014a) o mo e
de ails)
min
,Yk,⇢k+1
⇢k+1
subjec o
8
>
>
>
<
>
>
>
:
"P⇤⇤
0⇢k+1 ⇢k⇤
PAYkC(PE YkF)!kP#0
⇢k+1 ⇢k+
0<<1
(3)
o all !k2Bnx+ny,wi hYk=PLkand he nominal
es ima ed s a e ¯xk+1 =A¯xk+Buk+Lk(ykC¯xkDuk).
The symbol ”*” deno es symme ical e ms.
In ac , exp ession (3) gua an ees ha he sys em s a e
xk+1 belongs o he ellipsoid E(P, ¯xk+1,⇢
k+1).
An imp o emen o his me hod in e ms o accu acy
is p oposed in Ben Chabane e al. (2014b), wi h he
ad an age ha i can deal wi h in e al unce ain ies
on he e olu ion ma ix A. The main di↵e ence wi h
espec o Ben Chabane e al. (2014a) is he use o
he measu emen yk+1 oge he wi h addi ional quad a ic
cons ain s on he pe u ba ions !k.Thisimp o emen
allows us o educe e en mo e he size o he ellipsoidal
es ima ed s a e se by sol ing an addi ional op imiza ion
p oblem
min
⇢0
k+1,P 0,¯x0
k+1,H,⌧,µi
⇢0
k+1
subjec o
8
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>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
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>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
2
6
6
6
4
⌧P+C>HC ⇤⇤
⌘1⌧¯x>
k+1P⌘
2
nx+ny
X
i=1
µi⇤
P0P0¯x0
k+1 P0
3
7
7
7
5
0,
P00,
F>HF <
nx+ny
X
i=1
µiTi,
⌧0,
⌧<1,
⇢0
k+1 >⌧⇢
k+1,
µi0,i=1,...,n
x+ny
(4)
wi h ⌘1=(yk+1 +Duk+1)>HC and ⌘2=⇢0
k+1 ⌧⇢k+1 +
⌧¯x>
k+1P¯xk+1 +(yk+1 +Duk+1)>H(yk+1 +Duk+1).
Supposing ha xk+1 2E(P, ¯xk+1,⇢
k+1), he exp ession
(4) o↵e s an imp o ed ellipsoidal s a e es ima ion se
E0(P0,¯x0
k+1,⇢
0
k+1).
4. KALMAN FILTER
Recall he LTI sys em (1) aking in o conside a ion ha
wkand ka e andom, independen whi e Gaussian noises,
wi h he co a iance ma ices deno ed by Gwand G ,
espec i ely. No ice ha he s a e is a andom Gaussian
ec o deno ed by x⇠N(¯x, ) and pa icula ly he ini ial
s a e is ep esen ed by x0⇠Nx0|1,G
0|1.
The Kalman il e design is di ided in o wo s eps:
•P edic ion. A p e iously es ima ed s a e ˆxk1|k1and
he linea nominal model (wi hou any pe u ba ion)
a e used o p edic he alue o he nex es ima ed
s a e ˆxk|k1as well as he s a e es ima e co a iance
Gk|k1
ˆxk|k1=Aˆxk1|k1+Buk1(5)
Gk|k1=AGk1|k1A>+EwGwE>
w(6)
•Co ec ion. The cu en ou pu measu emen s and
he s a is ical p ope ies o he model a e used o
co ec he s a e es ima ion, leading o compu e he
s a e es ima e co a iance
Sk=CGk|k1C>+F G F>
(7)
Kk=Gk|k1C>S1
k(8)
ˆxk|k=ˆxk|k1+Kk(ykCˆxk|k1) (9)
Gk|k=(IKkC)Gk|k1(10)
wi h Kk he Kalman gain and Sk he inno a ion co a iance
a he sample ime k.
5. COMPARISON
The main di↵e ence be ween he app oaches p esen ed
in Sec ions 3 and 4 can be mainly spo ed in e ms o
sys em modeling. The ellipsoidal se -membe ship s a e
es ima ion Ben Chabane e al. (2014a) gua an ees he
s a e es ima ion bounds wi hin an ellipsoid o any LTI
sys em (1) o (2), while ce ain equi emen s should be
me in o de o e icien ly un he classical Kalman il e .
The Kalman il e wo ks p ope ly when he LTI model
ma ices a e ixed and do no p esen pa ame ic unce -
ain ies. The imp o ed se -membe ship s a e es ima ion
Ben Chabane e al. (2014b) o↵e s gua an eed bounds o
he s a e es ima ion despi e he p esence o possible in e -
al unce ain ies on he e olu ion ma ix Ao he sys em
(1) o (2). Howe e , he Kalman il e o↵e s a educed
compu a ion complexi y wi h espec o he conside ed
se -membe ship es ima ion me hod. In ac , he Kalman
equa ions a e based on basic ma ix ope a ions and he
compu a ional complexi y can be app oxima ed by he
numbe o mul iplica ions pe loop. Using he exp essions
(5)-(10) and conside ing he wo s case scena io (i.e. ull
ma ices) we can app oxima e he il e compu a ional
complexi y o O(N3), wi h N=max(nx,n
y).
The compu a ional complexi y o he ellipsoidal s a e
es ima ion me hod elies on sol ing a LMI op imiza ion
p oblem. The mincx sol e o he Ma lab Robus Con ol
Toolbox is based on he in e io poin me hod Nes e o
and Nemi o ski (1994) which is an i e a i e echnique
sol ing a leas squa e p oblem a each i e a ion. The
complexi y o he me hod in he wo s case scena io can
be app oxima ed o O(m2.75l1.5)wi hm he numbe
o decision a iables and l he numbe o cons ain s
Vandenbe ghe and Boyd (1994). No ice ha m=(nx+
ny)2+nxny+ 2 and l=2
nx+ny + 3 o he op imiza ion
p oblem (3) and m=0.5(n2
x+n2
y)+2.5nx+1.5ny+2 and
l=nx+ny+ 6 o he op imiza ion p oblem (4).
The compa ison allows us o conclude ha he Kalman
il e o↵e s us a be e esul in e ms o complexi y, hus
as e compu a ions. In e ms o accu acy, and o each
i e a ion, he ellipsoidal me hod compu es an ellipsoidal
se o which he eal s a e is gua an eed o belong.
The se -membe ship es ima ion se up (Sec ion 3) o↵e s
he possibili y o use co ela ed/unco ela ed pe u ba-
ions and measu emen noises, howe e he choice o he
pe u ba ion bounds needs good knowledge o he plan .
The Kalman il e uses he assump ion o Gaussian noises,
which can be di icul o e i y o some eal plan s.
S a ing om his esul s, he aim o ou u u e esea ch
wo k is o combine he ad an ages o he wo p esen ed
echniques in o de o p opose an ex en ed me hod ha
handles sys ems unce ain ies (i.e. in e al unce ain ies in
he sys em ma ices) and chance cons ain s.
6. ILLUSTRATIVE EXAMPLES
Two nume ical examples a e conside ed in his sec ion o
illus a e he compa ison o he p esen ed s a e es ima ion
echniques.
Example 1. Conside he ollowing s able LTI sys em
8
<
:
xk+1 =0.80.2
0.30.1xk+0.12
0.02 wk
yk=[
21
]xk+0.2 k
(11)
In his example, we p esen he esul s ob ained by he
imp o ed gua an eeed elipsoidal se -membe ship s a e es-
ima ion (4) compa ed o he esul s ob ained by Kalman
il e .
In o de o make a alid compa ison be ween hese wo
echniques, app op ia e assump ions should be aken e-
ga ding he ini ial s a e, and noises. In ac , we con-
side ha x0⇠Nx0|1,G
0|1and wk⇠N(0,1),
k⇠N(0,1) o he Kalman il e . Fo he ellipsoidal
se -membe ship app oach, he ini ial s a e x0belongs o
E(P0,x
0|1,⇢
0), and he pe u ba ions and measu emen
noises a e bounded, i.e. |wk|1 and | k|1. No ice
ha x0|1=[
55
]>,G0|1=I2,P0= 109I2and
⇢0=2·108.
Figu e 2 and 3 show he bounds o x1and x2 espec-
i ely a e 10 i e a ions ob ained by he ellipsoidal se -
membe ship s a e es ima ion me hod (4) and he Kalman
il e . The eal s a e x( ed as e ix) is always inside he
gua an eed bounds (in dashed blue) calcula ed by he
ellipsoidal se -membe ship me hod (4). I can be no iced
ha , in his example, he s a e es ima ed wi h he Kalman
il e (black as e ix) has a slowe con e gence and i is
no always inside he gua an eed bounds ob ained wi h
he imp o ed se -membe ship me hod Ben Chabane e al.
(2014b).
Conce ning he compu a ional complexi y, he classic
Kalman il e akes a ound 0.21ms pe i e a ion, while he
se -membe ship es ima ion echnique (LMIs (3) and (4))
spends a ound 9ms o de e mine he es ima ion bounds.
Example 2. The aim o using an uns able sys em, in
which he s a es do no con e ge o 0 is o p o e he
e iciency o he ellipsoidal me hod. In his con ex , we
conside he p e ious example wi h a di↵e en e olu ion
ma ix A, while keeping he same pe u ba ions, noises
and ini ial condi ions
8
<
:
xk+1 =1.50.2
0.30.1xk+0.12
0.02 !k
yk=[
21
]xk+0.2 k
(12)
0246810
Sample ime k
-6
-4
-2
0
2
4
6
Bounds o x1
Ellipsoidal me hod
Real s a e
Kalman il e
Fig. 2. Bounds o x1
0246810
Sample ime k
-2
-1
0
1
2
3
4
5
Bounds o x2
Ellipsoidal me hod
Real s a e
Kalman il e
Fig. 3. Bounds o x2
Figu e 4 and i s zoom (Fig. 6) show he bounds o x1
(in dashed blue) ob ained wi h he ellipsoidal es ima ion
me hod. The bounds o x2a e illus a ed in Fig. 5 and i s
zoom (Fig. 7). I can be no iced ha he eal s a e ( ed
as e ix) is gua an eed o be inside hese bounds, while i is
no he case o he classical Kalman il e (black as e ix).
7. CONCLUSION
In his pape , a b ie compa ison has been made be ween
wo me hodogies used o he s a e es ima ion o disc e e-
ime linea ime in a ian sys ems, subjec o pe u ba-
ions and measu emen noises. The gua an eed ellipsoidal
se -membe ship es ima ion me hod Ben Chabane e al.
(2014b) is compa ed o he classic Kalman Fil e , in e ms
o accu acy and complexi y. The bes es ima ion esul s
(i.e. gua an eed bounds) a e ob ained wi h he imp o ed
es ima ion me hod Ben Chabane e al. (2014b). The main
ad an age o he Kalman il e is i s lowe compu a ional
complexi y. In o de o ake ad an age o he bene i s o
he wo p oposed me hodologies, u u e wo k will consis
on inding a new es ima ion echnique ha gua an ees high
accu acy, wi h a small compu a ional cha ge. Addi ionally,
0246810
Sample ime k
-150
-100
-50
0
50
100
150
200
250
Bounds o x1
Ellipsoidal me hod
Real s a e
Kalman il e
Fig. 4. Bounds o x1
0246810
Sample ime k
-30
-20
-10
0
10
20
30
40
Bounds o x2
Ellipsoidal me hod
Real s a e
Kalman il e
Fig. 5. Bounds o x2
5 5.5 6 6.5 7
Sample ime k
-60
-40
-20
0
20
40
Bounds o x1
Ellipsoidal me hod
Real s a e
Kalman il e
Fig. 6. Zoom on he bounds o x1
pa ame ic unce ain ies and chance cons ain s will be
conside ed in u u e de elopmen s.
7 7.2 7.4 7.6 7.8 8 8.2 8.4 8.6
Sample ime k
-10
-5
0
5
10
15
Bounds o x2
Ellipsoidal me hod
Real s a e
Kalman il e
Fig. 7. Zoom on he bounds o x2
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