In e na ional Jou nal o Chaos, Con ol, Modelling and Simula ion (IJCCMS) Vol.9, No.1/2/3, Sep embe 2020
DOI:10.5121/ijccms.2020.9301 1
STATE OBSERVER DESIGNS FOR QUARTER-CAR
PASSIVE SUSPENSION
Tasya Y. Ch is nan asa i
Depa men o Enginee ing Physics, Ins i u Teknologi Sepuluh Nopembe , Su abaya,
60111, Indonesia
ABSTRACT
This pape p esen s s a e obse e designs o qua e -ca passi e suspension. P oposed designs
co espond o wo heo ies, ull-o de s a e obse e and obse e on closed-loop sys em. Those obse e s
a e used o s a es and es ima ion e o s obse a ion. Simula ion is done using MATLAB and SIMULINK.
MATLAB is used o calcula e bo h eedback gain ma ix and obse e gain ma ix whe eas SIMULINK is
applied o build s a e space block. Resul s show ha hose obse e s wo k e ec i ely and i obse e
heo ies. This wo k may mo i a e o con inue o o he s eps o obse e designs, obse e designs o hal -
ca and ull ca passi e suspension.
KEYWORDS
Qua e -Ca Passi e Suspension, Obse e , Closed-Loop Sys em
1. INTRODUCTION
Suspension plays an essen ial ole in isola ing ib a ion due o oad su ace and imp o ing human
com o and sa e y. So ha , suspension has been widely adop ed in many esea ches since 1980s.
Recen ly, he e a e h ee ypes o suspension ha a e implemen ed and es ed: passi e, semi-
ac i e, and ac i e. All o hem use ei he pneuma ic o hyd aulic ope a ion [1]. Each ype has i s
unique model, cha ac e is ic, and componen s.
Passi e suspension is one o ypes o suspension ha widely in ol ed because o i s simplici y.
Passi e suspension consis s o dampe and sp ing as i s main componen s. No ene gy is added o
passi e suspension sys em compa ed o ac i e suspension sys em. Al hough i is ha d o achie e
an op imal condi ion o ide com o when using passi e suspension, passi e suspension is
a ou able o a ligh weigh ehicle [2].
The e a e many analyses conce ning passi e suspension sys em model in o de o main ain ide
com o .I. Maciejewski e al. (2009) p esen ed modi ied passi e suspension o minimize ib a ion
[3].Hassaan (2004) has examined ca dynamics o qua e -ca passi e suspension o op imize ide
com o [4]. Ve os e al. (2005) did design op imiza ion o passi e suspension unde andom
oad exci a ion [5].
Many esea che s ha e also ied o achie e op imal condi ion o passi e suspension h ough
se e al me hods. Ani ban. C. Mi a e al. (2016)ha e applied Gene ic Algo i hm o op imize ide
com o and oad holding o passi e suspension [6]. Bha ga Gadh i e al. (2016) ha e used
NSGA-II, SPEA2, and PESA-II o ehicle passi e suspension mul i-objec i e op imiza ion [7].
M. Zehsaz e al. (2011) ha e done op imiza ion based on expe imen al and nume ical me hods
In e na ional Jou nal o Chaos, Con ol, Modelling and Simula ion (IJCCMS) Vol.9, No.1/2/3, Sep embe 2020
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o ac o ’s cabin passi e suspension [8].Smi h and Wang (2004) used ine e in o de o
op imize passi e suspension [9].
Those esea ches ha e no co e ed and esponded all challenges o passi e suspension sys em.
One o hose challenges is s a e obse a ion p ocess. The e a e wo p oblems ega ding s a e
obse a ion p ocess ha should be ackled, s a e es ima ion om he measu emen s such as
accele a ion o eloci y and s a e obse a ion i ac ual s a e is no measu able [10]. Bo h cases
equi e s a e obse e and/o il e ha can gene a e good es ima ed s a e, such as eloci y, mass
displacemen , and suspension de lec ion. Those es ima ions can imp o e ide quali y and u he
can be implemen ed in o he opics.
This pape is ocusing on obse e designs o passi e suspension in qua e -ca model based on
ull-o de s a e obse e and closed-loop sys em heo ies. Two obse e designs will be
p esen ed. The i s design answe s i s condi ion ha es ima ed s a es cons uc om a ailable
measu emen and second design will deal wi h he second condi ion whe e he e is no measu able
s a e.
2. QUARTER-CAR PASSIVE SUSPENSION MODEL
A s uc u e diag am o qua e -ca passi e suspension model can be seen in Figu e 1. The
diag am consis s o a sp ung mass and unsp ung mass. Sp ung mass ep esen s he mass o
qua e o ca chassis while unsp ung mass e lec s he mass o a single wheel assembly. Bo h
masses a e connec ed using a linea sp ing and iscous dampe . Subsc ip o s e e s o sp ung
elemen s, subsc ip u is o unsp ung elemen s, meanwhile subsc ip s ands o oad
displacemen .
Figu e 1. Qua e -ca Passi e Suspension Model
2.1. Ma hema ical Model
Acco ding o New on’s law, equa ions o mo ion o qua e -ca passi e suspension depic ed in
Figu e 1 a e
(1)
(2)
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Whe e a iable desc ip ions a e de ailed in Table 1.
Table 1. Va iables and Desc ip ions
Va iable
Desc ip ion
Sp ung mass
Unsp ung mass
Displacemen o sp ung mass
Displacemen o unsp ung mass
Road displacemen
Sp ing s i ness
Ti e s i ness
Damping coe icien
2.2. S a e Space Model
This pape uses s a e space app oach o iden i y inpu and ou pu o sys em. S a e space o m is
de ined as
(3)
whe e a e cons an ma ices. S a e space o m equi es a iables o de ine he dynamic sys em.
Those a iables a e s a e a iable ( , ou pu a iable , and inpu a iable .
Qua e -ca passi e suspension s a e space o m in ol es ou s a e a iables, wo ou pu
a iables, and an inpu a iable. De ails o a iables is desc ibed in Table 2.
Table 2. S a e Space Va iables
Va iable
Ma ix
S a e a iables
Ou pu a iables
Inpu a iable
The e o e, we can ew i e (1) and (2) in s a e space o m as ollows
(4)
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3. STATE OBSERVER
Obse a ion is me hod o unmeasu able s a e a iable es ima ion. I can be ackled wi h se e al
compu e p og ams (o de ices) and one o he is s a e obse e . The heo y o obse e is
o igina ed om Luenbe ge ’s wo ks in 1960s o 1970s [11,12,13].Acco ding o Luenbe ge , he
sys em inpu and ou pu can cons uc an es ima e o sys em s a e a iables. The e o e, S a e
obse e uses da a om con ol a iables and measu emen s o he ou pu o es ima e s a e
a iables. The e a e wo majo kinds o s a e obse e , ull-o de s a e obse e and educed-o de
s a e obse e . Full-o de s a e obse e obse es all s a e a iables whe eas educed-o de s a e
obse e es ima es ewe han o al numbe o s a e a iables. S a e obse e o simply obse e
can be designed o ei he con inuous- ime sys em o disc e e- ime sys em. This pape uses ull-
o de obse e o con inuous- ime sys em.
3.1. Full-o de S a e Obse e Model
Obse e has he same s uc u e wi h he eal sys em plus eedback e m ha b ings in o ma ion
abou obse a ion e o [10,14]. The e o e, ull-o de obse e can be ob ained om ac ual
sys emplus a co ec ion. The co ec ion ac s as eedback signal o obse e . The co ec ion is
based on he sys em ou pu and es ima ed ou pu . Ac ual sys em is desc ibed by
(5)
he obse e is designed as
(6)
whe e L is obse e gain. The obse e equa ion (6) is de i ed om ac ual sys em wi h ue s a e
eplaced by es ima ed s a e . Ou pu equa ion is also de i ed om ac ual sys em’s ou pu wi h
eplaced by . As men ioned in a o emen ioned explana ion, obse e inpu s a e and and
as ou pu . Disc epancy be ween and ac s as co ec ion and educes he e ec o ac ual s a e
a iables and es ima ed s a e a iables gap.
To ob ain he s a e space model o obse e , we subs i u e o obse e ’s s a e equa ion.
(7)
i can be ew i en as
(8)
Al hough D ma ix appea s in (8), i has no in luence on es ima ed s a e p oduced by obse e . I
is because co ec ion cancels ou D. I is p oo ed by subs i u ing and o co ec ion.
(9)
Figu e 2 ep esen s eal sys em wi h ull-o de s a e obse e wi h D = 0.
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3.2. Es ima ion E o and Obse e Gain
The aim o obse e is o p oduce es ima ion o ue s a e . I is easonable o assume he e will
be some gap be ween ue s a e and es ima ed s a e bu i is hoped ha he gap will dec ease o e
ime. The gap is called es ima ion e o . Es ima ion e o can be de ined as
(10)
hen he equa ion (5) and (6) becomes
(11)
Dynamic beha iou o es ima ion e o is de e mined by eigen alues o ma ix. I
is s able, e o will end o ze o and will con e ge wi h o any ini ial alue o and .
Figu e 2. Ac ual Sys em wi h S a e Obse e
akes c ucial pa in designing ull-s a e obse e . I in luences he speed o e o
con e gence. I is dependen o obse abili y o he sys em. I sys em (A,C) is comple ely
obse able, hen he eigen alues o can be chosen a bi a ily [10,15,16]. The chosen
eigen alues o should comply s ic ule ha complex eigen alue mus be pai ed wi h i s
conjuga e. can be de e mined using a ious me hods, such as Acke mann’s o mula, ma ix
ans o ma ion, and place in MATLAB [10,17].
3.3. E ec o Addendum o S a e Obse e on Closed-Loop Sys em
I we design a closed-loop sys em, x( ) is assumed o be a ailable o eedback. In ac , ac ual
s a e may no be measu able. So ha , s a e obse e is needed and is used o eedback.
The e will be wo-s ages p ocess. Fi s s age will de e mine eedback gain ma ix ( ) o p oduce
desi ed cha ac e is ic equa ion and second s age will se obse e gain ma ix ( ) o yield desi ed
obse e cha ac e is ic equa ion [10]. Figu e 3 ep esen s s a e obse e on closed-loop sys em.
In e na ional Jou nal o Chaos, Con ol, Modelling and Simula ion (IJCCMS) Vol.9, No.1/2/3, Sep embe 2020
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Inpu o his sys em is
(12)
Subs i u ing ha inpu , s a e equa ion becomes
(13)
Using de ini ion o es ima ion e o , s a e equa ion u ns o
(14)
Es ima ed-s a e equa ion becomes
(15)
Figu e 3. S a e Obse e on Closed-Loop Sys em
Tha sys em has wo sepa a e eigen alues, eigen alues o and eigen alues o .
Hence, bo h eigen alues can be independen ly and sepa a ely placed in he desi ed loca ions. I
he sys em (A,B)is con ollable, eigen alues o can be a bi a ily placed in he plane bu
i should ollow a ule ha complex eigen alues mus appea in pai s wi h i s complex conjuga e.
Feedback gain ma ix can be calcula ed wi h Acke mann’s o mula and place in MATLAB.
4. OBSERVER DESIGNS FOR QUARTER-CAR PASSIVE SUSPENSION
The pa ame e s used o he obse e designs a e de ailed in Table 3.
Table 3. Simula ion Pa ame e s
Pa ame e
Value
Uni
100
kg
10
kg
5
kN/m
20
kN/m
0.1
kN.s/m
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De ails o coe icien s o s a e space o m a e desc ibed in Table 4.
Table 4. De ails o Coe icien s o S a e Space Fo m
Coe icien
Value
A
B
C
D
0
This pape ep esen s wo designs, ull s a e obse e and obse e on closed-loop sys em. S ep
inpu as ep esen a i e o oad displacemen will be applied in i s design. Bo h designs equi e
obse e gain ma ix ( ) which is compu ed in MATLAB. Ano he design also needs eedback
gain ma ix ( ) which is calcula ed in MATLAB. These sys ems a e simula ed in SIMULINK.
Ini ial alues o ac ual sys em and obse e and o he simula ion de ails a e explained in Table 5.
Table 5. Simula ion De ails
Simula ion De ail
Value
Ini ial alues o ac ual sys em
Ini ial alues o obse e
Eigen alues o
Eigen alues o
These obse e designs use block diag ams depic ed in Figu e 3 and 4. Figu e 5 and 6 show
SIMULINK con igu a ions o ull-o de s a e obse e design and ull s a e obse e on closed-
loop sys em espec i ely.
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Figu e 5. Simulink Con igu a ion o Full-O de S a e Obse e
Figu e 6. Simulink Con igu a ion o Full S a e Obse e on Closed-Loop Sys em
5. SIMULATION RESULT
Simulink simula ion yielded he ollowing esul s. Each s a e p oduces di e en esul al hough
all o hem i obse e heo y. Figu e 4 and 6 p esen compa ison o ac ual s a e and es ima ed
s a e o ull-o de s a e obse e and obse e on closed-loop sys em espec i ely. Figu e 5 and 7
show he es ima ion e o s o ull-o de s a e obse e and obse e on closed loop sys em.
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Figu e 7. Ac ual and Es ima ed S a es o Full-O de S a e Obse e
Figu e 8. Es ima ion E o s o Full-O de S a e Obse e
Resul s show ha es ima ed s a es con e ge o ac ual s a es wi h ce ain speed and es ima ion
e o s u n o ze o o e ime. Fi s es ima ed s a e ( akes 1.2s. Second es ima ed s a e ( )
coincides wi h ac ual s a e a 1.6s. Thi d es ima ed s a e ( ) needs 1.2s o coincide wi h i s eal
s a e. Fou h es ima ed s a e ( ) con e ges o ac ual s a e a 1.4s. Es ima ion e o s need
iden ical ime wi h s a e con e gence o con e ge o ze o. This speed depends on eigen alues o
.
