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

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

Observability Theorem in Continuous-Time

The linear continuous-timesystemwith the correspondingmeasurements

is observableif and only if the observabilitymatrix

�...�����

� ���������

has full rank equal to .

3

Controllability Theorem in Discrete-Time

The linear discrete-timesystem

� �

is controllableif andonly if the controllability matrix defined

� � � ... � � ... ... ��� �� � � � ! �#" $

has full rank equal to .

4

Controllability Theorem in Continuous-Time

The linear continuous-timesystem

is controllableif andonly if the controllability matrix definedby

... ... ... %�&�' %)( * %#+ ,

has full rank equal to .

5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Proof of the claim observable implies �X� �S� :

� �X� � �S��v� �X� ��

�

�S��X� � �

�X� �S�

30

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

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

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

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

To seethis, we first observethat

�� � � � � �

� �� � �

� �� � � �

The measurements are given by

� �

35

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

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

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

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

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

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

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

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

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

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

Nonlinearobserverblock diagramis presentedin the next figure

46

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

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

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

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

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