Page 1
Intr oduction to Linear and Nonlinear Observers
Zoran Gajic, Rutgers University
Part1 — Review Basic Observability(Controllability) Results
Part 2 — Introductionto Full- andReduced-OrderLinear Observers
Part3 — Introductionto Full- andReduced-OrderNonlinearObservers
1
Page 2
PART 1: BASIC OBSERVABILITY (CONTROLLABILITY) RESULTS
Observability Theorem in Discrete-Time
The linear discrete-timesystemwith the correspondingmeasurements
�
�
is observableif and only if the observabilitymatrix
� ��
� �� ��...� ������
� ��� ���
has rank equal to .
2
Page 3
Observability Theorem in Continuous-Time
The linear continuous-timesystemwith the correspondingmeasurements
is observableif and only if the observabilitymatrix
�...�����
� ���������
has full rank equal to .
3
Page 4
Controllability Theorem in Discrete-Time
The linear discrete-timesystem
� �
is controllableif andonly if the controllability matrix defined
� � � ... � � ... ... ��� �� � � � ! �#" $
has full rank equal to .
4
Page 5
Controllability Theorem in Continuous-Time
The linear continuous-timesystem
is controllableif andonly if the controllability matrix definedby
... ... ... %�&�' %)( * %#+ ,
has full rank equal to .
5
Page 6
Similarity Transformation
For a given system-
we can introducea new statevector by a linear coordinatetransformationas
where is somenonsingular matrix. A newstatespacemodelis obtainedas
-
where
.0/ .�/ .�/
6
Page 7
EigenvalueInvariance Under a Similarity Transformation
A new statespacemodel obtainedby the similarity transformationdoesnot
changeinternalstructureof themodel,that is, theeigenvaluesof thesystemremain
the same. This can be shownas follows
1�2 1�21�2
Notethat in this proof thefollowing propertiesof thematrix determinanthavebeen
used 2 3 4 2 3 41�2
7
Page 8
Controllability Invariance Under a Similarity Transformation
The pair is controllable if and only if the pair is controllable.
This theoremcan be provedas follows
... ... ... 5�6�7... 6�7 ... ... 5�6�7 6�7... ... ... 5�6�7
Since is a nonsingularmatrix (it cannotchangethe rank of the product ),
we get
8
Page 9
Observability Invariance Under a Similarity Transformation
The pair is observable if and only if the pair is observable.
The proof of this theoremis as follows
8...9�:�;
:�;:�; :�;:�; 8 :�;
...:�; 9�:�; :�;8
...9�:�;:�;
that is,
:�;
The nonsingularityof implies
9
Page 10
PART 2: INTRODUCTION TO LINEAR OBSERVERS
Sometimesall statespacevariablesarenot availablefor measurements,or it is not
practical to measureall of them, or it is too expensiveto measureall statespace
variables. In order to be able to apply the statefeedbackcontrol to a system,all
of its state space variables must be available at all times. Also, in somecontrol
systemapplications,oneis interestedin havinginformationaboutsystemstatespace
variablesat any time instant. Thus, one is facedwith the problemof estimating
systemstatespacevariables.This canbe doneby constructinganotherdynamical
systemcalledtheobserveror estimator,connectedto thesystemunderconsideration,
whoserole is to producegoodestimatesof the statespacevariablesof the original
system.
10
Page 11
The theory of observersstartedwith the work of Luenberger (1964, 1966,
1971) so that observersare very often called Luenberger observers. According
to Luenberger, any system driven by the output of the given system can serve as an
observer for that system.
Two main techniquesare availablefor observerdesign.
Thefirst oneis usedfor thefull-order observerdesignandproducesanobserver
that hasthe samedimensionas the original system.
The secondtechniqueexploits the knowledgeof some state spacevariables
availablethrough the output algebraicequation(systemmeasurements)so that a
reduced-order observer is constructedonly for estimatingstatespacevariables
that are not directly obtainablefrom the systemmeasurements.
11
Page 12
Full-Order Observer Design
Considera linear time invariant continuoussystem
< =
where > , ? , @ with constantmatrices having
appropriatedimensions. Since the systemoutput variables, , are available
at all times,we may constructanotherartificial dynamicsystemof order (built,
for example,of capacitorsandresistors)having the samematrices
< =
and comparethe outputs and .
12
Page 13
Thesetwo outputswill be different since in the first casethe systeminitial
condition is unknown,and in the secondcaseit hasbeenchosenarbitrarily.
The differencebetweenthesetwo outputswill generatean error signal
which can be usedas the feedbacksignal to the artificial systemsuch that the
estimation(observation)error is reducedasmuchaspossible,
hopefully to zero (at least at steadystate). This can be physically realized by
proposingthe system-observerstructureasgiven in the next figure.
13
Page 14
uAB
FB K
C +D-
y=CxESystem
ObserverF
Ce
y=CxEx xG
System-observer structure
In this structure representsthe observergain andhasto be chosensuchthat
the observationerror is minimized. The observeraloneis given by
14
Page 15
Remark 1:
Note that the observer has the same structure as the system plus the driving
feedback term that contain information about the observation error
Therole of thefeedbacktermis to reducetheobservationerror
to zero (at steadystate).
Remark 2:
The observeris usually implementedon line as a dynamicsystemdriven by the
sameinput as the original systemandthe measurementscomingfrom the original
systems,that is (note )
15
Page 16
It is easyto derivean expressionfor dynamicsof the observationerror as
If the observergain is chosensuch that the feedbackmatrix is
asymptotically stable, then the estimationerror will decay to zero for any
initial condition H . This can be achieved if the pair is observable.
More precisely,by takingthe transposeof theestimationerror feedbackmatrix, i.e.I I I
, we seethat if thepairI I
is controllable,thenwe canlocate
its poles in arbitrarily asymptoticallystablepositions. Note that controllability of
thepairI I
is equalto observabilityof thepair , seeexpressionsfor
the observabilityand controllability matrices.
16
Page 17
In practicethe observerpoles should be chosento be about ten times faster
than the systempoles. This can be achievedby setting the minimal real part of
observereigenvaluesto be ten times bigger than the maximal real part of system
eigenvalues,that is
JLKNM OQPSRSTVUXWYT�U J[Z]\ RV^�R`_aT J
(in practice10 canbe replaceby 5 or 6). Theoretically,the observercanbe made
arbitrarily fast by pushingits eigenvaluesfar to the left in the complexplane,but
very fast observersgeneratenoise in the system.
17
Page 18
System-ObserverConfiguration
Wewill showthatthesystem-observerstructurepreservestheclosed-loopsystem
poles that would have beenobtainedif the linear perfect statefeedbackcontrol
had been used. The system under the perfect state feedbackcontrol, that is
has the closed-loopform as
so that the eigenvaluesof the matrix are the closed-loopsystempoles
under perfect statefeedback.
In thecaseof thesystem-observerstructure,asgivenin thegivenblock diagram,
weseethattheactualcontrolappliedto boththesystemandtheobserveris givenby
18
Page 19
By eliminating , and from the
augmentedsystem-observerconfiguration,weobtainthefollowing closed-loopform
What are the eigenvaluesof this augmentedsystem?
If we write the system-errorequation,we have
Sincethe statematrix of this systemis upperblock triangular,its eigenvaluesare
equalto theeigenvaluesof matrices and . A very simplerelation
exists among and
19
Page 20
Note that the matrix is nonsingular.In order to go from -coordinatesto -
coordinateswe haveto usethe similarity transformation,which preservesthe same
eigenvalues,that is and , arealsotheeigenvaluesin the
-coordinates.
Separation Principle
This important observationthat the system-observerconfigurationhas closed-
loop poles separatedinto the original systemclosed-looppoles obtainedunder
perfect state feedback, , and the actual observerclosed-looppoles,
, is known as the separation principle.
Hence,we canindependentlydesignthesystempolesusingthesystemfeedback
gain and independentlydesignthe observerpolesusing the observerfeedback
gain .
20
Page 21
System-ObserverConfiguration in SIMULINK
y(t)
xhat
(t)
Mux
y
syst
em o
utpu
t y(t
)
yhat
obse
rver
out
put y
hat(
t)
C*
u
mat
rix C
−F
* u
feed
back
gai
n −
F
t
To
Wor
kspa
ce2
x’ =
Ax+
Bu
y =
Cx+
Du
Obs
erve
r (s
tate
spa
ce fo
rm)
x’ =
Ax+
Bu
y =
Cx+
Du
Line
ar S
yste
m (
stat
e sp
ace
form
)
Clo
ck
21
Page 22
Since b , c , d , the statespaceform for the systemmatrices
shouldbe set (by clicking on andopeningthe observerstatespaceblock) as
>> A=A; B=B; C=C; D=zeros(p,r);% assumingD=0
>> % to be ableto run simulationyou mustassignany valueto the systeminitial
>> % conditionsincein practicethis value is given, but unknown,that is
>> x0 = “any vector of dimensionn”
Since the observeris implementedas
the observerstatespacematricesin SIMULINK shouldbe specified(by clicking
on and openingthe observerstatespaceblock) as
>> Aobs=A-K*C; Bobs=[B K]; Cobs=eye(n);Dobs=zeros(n,r+p);
>> xobs=’anyn-dimensionalvector’
22
Page 23
Discrete-Time Full-Order Observer
The sameprocedurecan be applied to in the discrete-timedomain producing
the analogousresults.
Discrete-timesystem:
e fe
Discrete-time observer:
e f e ge
Observation error dynamics ( ) :
e e e e e e h
23
Page 24
i is chosento makethe observerstable, i i i , andmuch
faster than the system,which requires
i i i i i i
In practice,the observershouldbe six to ten times fasterthan the system.
Closed-loopsystem-observerconfiguration
i i ii i i i i
The system-errordynamic
i i i i ii i i
The separationprinciple holds also.
24
Page 25
Reduced-Order Observer (Estimator)
Considerthe linear systemwith the correspondingmeasurements
j k
We will show how to derive an observerof reduceddimensionsby exploiting
knowledgeof the outputmeasurementequation.Assumethat the outputmatrix
hasrank , which meansthat theoutputequationrepresents linearly independent
algebraicequations. Thus, equation
produces algebraicequationsfor unknownsof . Our goal is to constructan
observerof order for estimationof theremaining statespacevariables.
25
Page 26
In orderto simplify derivationsandwithout lossof generality,we will considerthe
linear systemwith the correspondingmeasurementsdefinedby
l mn
This is possiblesinceit is knownfrom linearalgebrathat ifn�o�p
thenit existsa nonsingularmatrixp o�p
suchthat n , which implies
q0r n q�r
Hence,mappingthe systemin the new coordinatesvia the similarity transfor-
mation,we obtain the given structurefor the measurementmatrix.
r rs r s
26
Page 27
Partitioningcompatibly the systemequation,we have
tu
tXt tSuuvt uXu
tu
tu
t
The statevariables t are directly measured(observed)at all times, so that
t . To constructan observerfor u , we use the knowledgethat
an observer has the same structure as the system plus the driving feedback term
whose role is to reduce the estimation error to zero. Hence,anobserverfor u is
u uvt t uXu u u u
Since doesnot carry informationabout u , this observerwill not be able
to reducethe correspondingobservationerror to zero, u u u .
27
Page 28
However,if we differentiatethe output variablewe get
w wXw w wyx x w
that is carriesinformationabout x . The reduced-orderobserverwith the
feedbackinformation coming from is
x xvw w xXx x x xwXw w wyx x w
The observationerror dynamicscanbe obtainedfrom x x x as
x xzx x wSx x
To placethereduced-observerpolesarbitrarily (the reduced-orderobservermustbe
stableandmuch fasterthan the system),we need { xXx { wSx controllable.
28
Page 29
By duality between controllability and observability, controllability of| }X} | ~S}
is dual to observabilityof}X} ~S}
.
It is easyto showusing the Popov-Belevitchobservabilitytest
� � �
that observableimplies}X} ~S}
.
Hence,if the original systemis observable,we canconstructthe reduced-order
observerwhoseobservationerror will decayquickly to zero.
29
Page 30
Proof of the claim observable implies �X� �S� :
� �X� � �S��v� �X� ��
�
�S��X� � �
�X� �S�
30
Page 31
The needfor in the reduced-orderobserverequation
� �v� � �X� � � ��X� � �y� � �
canbe eliminatedby introducingthe changeof variables � � � , which
leads to� � � � �
� �X� � �S�� � � �
� �v� � �X� �X� � � �S� �
31
Page 32
Reduced-Order Observer Derivation without a Changeof Coordinates
Considerthe linear systemwith the correspondingmeasurements
� �
Assumethat theoutputmatrix hasrank , which meansthat theoutputequation
represents linearly independentalgebraicequations.Thus,equation
produces algebraicequationsfor unknownsof . Our goal is to constructan
observerof order for estimationof theremaining statespacevariables.
32
Page 33
The procedurefor obtainingthis observeris not unique,which is obviousfrom
the next step. Assumethat a matrix � existssuchthat
�
and introducea vector � as
�
Now, we have
�� �
Sincethe vector is unknown,we will constructan observerto estimateit.
33
Page 34
Introducethe notation
�� � � �
so that
� �
An observerfor canbe constructedby finding first a differentialequationfor
, that is
� � � � � � � �
Note that from this systemwe arenot ableto constructan observerfor since
doesnot containexplicit information aboutthe vector .
34
Page 35
To seethis, we first observethat
�� � � � � �
� �� � �
� �� � � �
The measurements are given by
� �
35
Page 36
If we differentiatethe output variablewe get
� �
i.e. carriesinformation about . An observerfor is obtainedfrom
the last two equationsas
� � � � � �
where � is the observergain. If in the differential equationfor we replace
by its estimate,we will have
� �
36
Page 37
This producesthe following observerfor
� � � � � � � �
Since it is impractical and undesirableto differentiate in order to get
(this operationintroducesnoisein practice),we takethe changeof variables
�
This leadsto an observerfor of the form
� � �
where � � � � � � � �� � � � � � � � � � �
37
Page 38
The estimatesof the original systemstatespacevariablesarenow obtainedas
� � � � � �
The obtainedsystem-reduced-observerstructureis presentedin the next figure.
uAB�
F Bq
KC
q
L2
++
yESystem
Reduced�observer
L�
1+L2K1
q�
xG xG
System-reduced-observer structure
38
Page 39
Setting Reduced-Order-ObserverEigenvaluesin the Desired Location
We needthat the eigenvaluesof the reduced-orderobserver
� � �� � � � �
be roughly ten times faster than the closed-loopsystemeigenvaluesdetermined
by . This can be done if the pair � � � is observable
(analogousresult to the requirement �X� �S� observablefor the casewhen
the first statevariablesare directly measured).This is dual to the requirement
� � � is controllable.
Notethatit canbeshownthat observable implies � � � and
provedsimilarly to the proof of the claim observable implies �z� �y� .
39
Page 40
We can set the reduced-observereigenvaluesusing the following MATLAB
statements:
>> % checkingthe observabilitycondition
>> O=obsv(C1*A*L2,C*A*L2);
>> rank(O); % must be equal to p
>> % finding the closed-loopsystempoles
>> lamsys=eig(A-B*F);maglamsys=abs(real(lamsys))
>> % finding the closed-loopreduced-orderobserverpoles
>> % input desiredlamobs(reduced-orderobservereigenvalues)
>> K1T=place((C1*A*L2)’,(C*A*L2)’,lamobs);
>> K1=K1T’
40
Page 41
PART 3 — INTRODUCTION TO NONLINEAR OBSERVERS
We haveseenthat to observethe stateof the linear systemdefinedby¡ ¢
we constructa linear observer that has the same structure as the system plus the
driving feedback term whose role is to reduce the observation error to zero
Studyingobserversfor nonlinearsystemsis theoreticallymuchharder.However,
we can usethe samelogic to constructa nonlinearobserver.
41
Page 42
Considera nonlinearcontrolledsystemwith measurements
£, ¤ , ¥ , and arenonlinearvector functions,respectively,
of dimensions and .
Basedon the knowledgeof linear observers,we can proposethe following
structurefor a nonlinearobserver
Hence,the nonlinearobserveris definedby
42
Page 43
Theobservergain is a nonlinearmatrix functionthat in generaldependson
and , that is, . It hasto be chosensuchthat the observationerror,
tendsto zero (at leastat steadystate).
The observationerror dynamicsis determinedby
By eliminating from the error equation,we obtain
At the steadystatewe have
43
Page 44
It is obvious that is the solution of this algebraicequation,which indi-
catesthat the constructedobservermay have at steadystate. The gain
must be chosensuch that the observerand error dynamicsare
asymptoticallystable(to force the error at steadystateto ).
The asymptoticstability will be examinedusing the first stability methodof
Lyapunov. The Jacobinmatrix for the error equationis given by
¦
By the first stability method of Lyapunov, the Jacobianmatrix must have all
eigenvaluesin the left half planefor all working conditions,that is for all
and , where and arethe setsof admissiblestateandcontrol variables.
44
Page 45
The error dynamicsasymptoticstability condition is
§ ¨ª©«¨ ¬�¯®±°²#³´®¶µ·²#¸ §
Similarly, for the observerwe have
¹°
and it is requiredthat the observeris also asymptoticallystable
§ ¹°#©º¨S¬�¯®±°·²»³¼®¶µ·²#¸ §
45
Page 46
Nonlinearobserverblock diagramis presentedin the next figure
46
Page 47
Reduced-Order Nonlinear Observers
Assumethat ½ statevariablesare directly measuredand we needto
constructa nonlinearobserverto estimatethe remaining ¾ ½ ¾ state
variables
½ ½ ½
Let us partition compatiblethe stateequations
½ ½ ½ ¾¾ ¾ ½ ¾
½The estimatefor the statevariablescan be obtainedas
½¾ ¾ ¾
47
Page 48
Let us assumethat the dynamicsystem(observer)for hasthe following form
¿
We haveto find thereduced-orderobservergain ¿ andthereduced-orderobserver
structuredefined by suchthat the observationerror ¿ ¿ ¿ tendsto zero
at steadystate.
The dynamicequationfor the error is obtainedas follows
¿ ¿ ¿ À ¿ ¿ À¿ ¿ ¿ ¿ ¿ ¿ À ¿ ¿
Sinceour goal is that at steadystate ¿ , we have
¿ ¿ ¿ À ¿ ¿
48
Page 49
Hence,the reduced-orderobserverstructureis given by
Á Á Á Á Â Á
The error dynamicmust be asymptoticallystable
Á Á Á Á Â Á ÁÁ Á Á Á
whichmeansthatby thefirst methodof LyapunovtheJacobianmatrix musthaveall
eigenvaluesin the left half planefor all working conditions,that is for all Á Áand , where Á and arethesetsof admissiblestateandcontrol variables.
ÃVÄ ÁÁ
Á ÁÁ Á
Á ÁÁ
Á ÁÁ Á
49
Page 50
The error dynamicsasymptoticstability requirethat
Å ÆVÇzÈ«Æ ÇÊÉ�˯̱Í]ÇSÎ�Ï�ÇV̶зÎÒÑ Å
Similarly, the reduced-orderobserverdynamicsmustbe asymptoticallystable.
The block diagramof the reduced-ordernonlinearobserveris given below
50
Page 51
This lectureon observersis preparedusing the following literature:
[1] Z. Gajic andM. Lelic, Modern Control Systems Engineering, PrenticeHall
International,London,1996,(pages241–247on full- andreduced-orderobservers).
[2] Stefani, Shahian,Savantand Hostetter,Design of Feedback Systems, Ox-
ford University Press,New York, 2002, (pages650–652on reducedorder linear
observer).
[3] B. Friedland,Advanced Control System Design, PrenticeHall, Englewood
Clif fs, 1996 (pages164–166and 174–175,183–187 on full- and reduced-order
nonlinearobservers).
Basic resultson observability(controllability) are reviewedfrom [1].
51