STATESPACEMETHODST
whyharecontrolsystems D V F F
n ee ee
ya E d d
n
predictbehaviour of dynamicalsystem f
ta bca
M r
unmodelleddynamics i 0r
c g w
uncertainenvironment disturbances s n ar
measurementnoise
ty d
Actuate Feedback
usedifandonlyif
SIEEIIT.am there
effqdtfq D uncertainty
GPS noise
Flistutrbancestmodellingerror
Controller I
Closedloops Feedbackdrawbacks
aircraft Copenlooponly candestabilisesystem
noon
i
Fn
itne.n
function
m.n.ge n
controller
FILI HK Jt G w
feedback theONLY
way toattenuateunknownsw
statespaceSystems
choiced staterector
XCHEIR states 1excludetimeconstantvariables
x t f xlt ult t
2 eiindudeupb
explicit ultelR inputs faeagehcdaifdfafggthegteddey.ajaiq.be
or measuredoutputs
F Xlt xLH ult t o
implicit DAE
Yale
yltt hlxlthultt.tt u
lI y
linearTime
Invariantuttems
f xlt ua t isalinearfundionotxHlandulHonly
Gene oftensettoloo wt
og
lineanisedaboutequilibrium X 0 flxe.ae 0
Ant Butt
t AER dynamics
matrix CEIR sensormatrix
m
yLH CxLt tDult BelR inputmatrix DEIR directterm often170
, y
Closed
Loopsyst controlledlegulated
statefeedbactm
yariables x Ix
w
tEz
y y o
w
yorzy woffuarnstanteesaswea
OutputFeedback
mmmm
Ie
Plantlsystem
a i
u
state controller
a
hey stat.com iacn.o
w
Tf
ftEzInffI z
ExampleState
mum Space Model
we want x fCxa andyhlx.nl
Inputs ufhi.FI Measuredoutputy fHfFg
x
a
States ftp.t.no
Xy
If
Xs
Iii ii sit at H to
Feedforward Feedback Controller
Iteedn d UffeIsystemT Y
ii s i
i i
i
, Jacobian Linearisaton Equilibriumstateinputpainke Ue
I
CtI fCxTthuLt E
ctt hcxtthut.tt f xe.ae O
y aboutequilibrium cxe.ae L
yiguery
I nearising
we define e
E fcxhitftxe.intfIaeFiexeltfIa
aEieye y
a a we
y yeyehexeme
i x A Bu night
ExampleInvertedPendulum
ehlx.it xeueltffl4iY
hf uejdtATuIfa9ieYe
y
jcxtPumm
Equilibriumpointsatfcxe.ae O
X O and X Intl fan O1,2
chooseue 0
181 axe 8
zcoscx.su flea
zfxztf
fmoffsincxi
0
Ext.EE coscxn easincxiu rIII ifmge rII
B ftaho.o oscx.it
jl
Ey
c
I o t Ou C I 0 and 17 0
MatrixExponential a Ax at e xlo
onlyfmatrix is a diagona
b4 ex 17 12734 3 1,537 fee matrixdoesthisweek
AeAtT
dqett
whyharecontrolsystems D V F F
n ee ee
ya E d d
n
predictbehaviour of dynamicalsystem f
ta bca
M r
unmodelleddynamics i 0r
c g w
uncertainenvironment disturbances s n ar
measurementnoise
ty d
Actuate Feedback
usedifandonlyif
SIEEIIT.am there
effqdtfq D uncertainty
GPS noise
Flistutrbancestmodellingerror
Controller I
Closedloops Feedbackdrawbacks
aircraft Copenlooponly candestabilisesystem
noon
i
Fn
itne.n
function
m.n.ge n
controller
FILI HK Jt G w
feedback theONLY
way toattenuateunknownsw
statespaceSystems
choiced staterector
XCHEIR states 1excludetimeconstantvariables
x t f xlt ult t
2 eiindudeupb
explicit ultelR inputs faeagehcdaifdfafggthegteddey.ajaiq.be
or measuredoutputs
F Xlt xLH ult t o
implicit DAE
Yale
yltt hlxlthultt.tt u
lI y
linearTime
Invariantuttems
f xlt ua t isalinearfundionotxHlandulHonly
Gene oftensettoloo wt
og
lineanisedaboutequilibrium X 0 flxe.ae 0
Ant Butt
t AER dynamics
matrix CEIR sensormatrix
m
yLH CxLt tDult BelR inputmatrix DEIR directterm often170
, y
Closed
Loopsyst controlledlegulated
statefeedbactm
yariables x Ix
w
tEz
y y o
w
yorzy woffuarnstanteesaswea
OutputFeedback
mmmm
Ie
Plantlsystem
a i
u
state controller
a
hey stat.com iacn.o
w
Tf
ftEzInffI z
ExampleState
mum Space Model
we want x fCxa andyhlx.nl
Inputs ufhi.FI Measuredoutputy fHfFg
x
a
States ftp.t.no
Xy
If
Xs
Iii ii sit at H to
Feedforward Feedback Controller
Iteedn d UffeIsystemT Y
ii s i
i i
i
, Jacobian Linearisaton Equilibriumstateinputpainke Ue
I
CtI fCxTthuLt E
ctt hcxtthut.tt f xe.ae O
y aboutequilibrium cxe.ae L
yiguery
I nearising
we define e
E fcxhitftxe.intfIaeFiexeltfIa
aEieye y
a a we
y yeyehexeme
i x A Bu night
ExampleInvertedPendulum
ehlx.it xeueltffl4iY
hf uejdtATuIfa9ieYe
y
jcxtPumm
Equilibriumpointsatfcxe.ae O
X O and X Intl fan O1,2
chooseue 0
181 axe 8
zcoscx.su flea
zfxztf
fmoffsincxi
0
Ext.EE coscxn easincxiu rIII ifmge rII
B ftaho.o oscx.it
jl
Ey
c
I o t Ou C I 0 and 17 0
MatrixExponential a Ax at e xlo
onlyfmatrix is a diagona
b4 ex 17 12734 3 1,537 fee matrixdoesthisweek
AeAtT
dqett
