observers for non-linear systems

Observers for non-linear systemsManipulation Lab talk

Siddhartha Srinivasa

The Robotics Institute

Carnegie Mellon University

Observers for non-linear systems p.1/35

http://www.cs.cmu.edu/~siddh

Outline

General frameworkAn exampleNon-linear systemsSome differential geometryReview of linear observabilityObservability rank conditionLinear observersObserver design : Lie-algebraic methodConclusionsThings I did not doReferences


General frameworkSystem

Identify forces and torques

Equations of motion, constraint equations

Pick suitable state variables

Write the equations (implicit) in state-space form(explicit)

Analyze the state-space equations for system behaviour


An example


An example

Inverted pendulum with DC motor control

DC motor armature controlled

motor inertia pendulum inertia

For the motor

For the pendulum

"! ! # $ %'&


An example

Choose state variables

() ( (* State space form

+,.-

(/)(

(/*0

1.2

+

,.-

( 34 $ %& () )5 6874 9 (*

)5 6;:4 (


Non-linear systems

( ? (A@ >

B C (

( ? (

DFE )> D # D (

( system state in a state-space manifold G H I?@ # D smooth vector fields on M?

drift vector field#) @ # @J J J # input vector fields>) @ > @J J J > scalar controlsC

output map


Review of linear observability

A SISO system

( K ( L >

B (

Kalman Rank Condition for observability

M NOPOPOPOQOPO

K

K

... K ISR )TPTPTPTQTPT


Linear observers

System

( = K ( L >B = (Observer

VU ( = K U ( L > B U B U B = U (

> B

+-

U B

The error dynamics is given by :

K W

Eigenvalues of (A-LC) arbitrarily placed by a proper choiceof L.


A first pass : output injection

Consider the nonlinear system

( ? ( ( X H I

B C ( B X H Create an observer with linear output injection

YU ( ? U ( B U B

U B C U ( where

X H I Z is the observer gain matrix we have controlover


A first pass : error dynamics

( U (

? ( [ ? U ( B U B \Error dynamics is nonlinear and stability is unclear. But ...

The stability of a linearized system about its fixedpoint implies the local stability of the correspondingnonlinear system about that fixed point

Why? I dont have a good answer, but here are some pic-

tures of Lyapunov


The many moods of Aleksandr Mikhailovich Lyapunov

angry happy


A first pass : linearizing error

Linearizing the error dynamics about the fixed point

? ( [ ? U ( C ( C U ( \

U ( (

? ( [ ? ( C ( C ( \

? ( ? ( ] ?

] (

C ( C ( ] C

] (

] ?

] ( ] C

] (


A first pass : conclusion

] ?

] ( ] C

] (

The linearization is a function of the true state ( which isnot a fixed quantity

unknown to us ... its what were trying to estimate!

Also, the linearization is valid only for a smallneighbourhood about the fixed point.


Some differential geometry

C_^ H I_` a H - a smooth function? ^ H I_` a H I - a vector fieldb'c @ c d - the standard dot product on H Ie C f C - the gradient of C with respect to (

The Lie derivative of

C

w.r.t.

?

is given by :

hg C b e C@ ? d f Cc ?

Also, e g C ig e C


Example

C ( B

? (( B

e C

jk jmljk jmn

o (

g C b e C@ ? d o ( ( B

e g C

p ( B

(


Observability rank condition

The observation space O is the linear space of thefunctions

C) @ C @J J J @ C and all repeated Lie derivatives

q_r q 9J J J qts Cvu

w @ o@J J J @ N @ o@J J Jx D X [ ?@ #) @ # @J J J @ # \

Intuitively, O comprises of the output functions and themagnitude of their derivatives along all possible systemtrajectories (in infinitesimal time).


Observability rank condition

The observability codistribution dO(x) is defined as

e y ( z M [ e{ ( | { X y\The system is locally observable at state (5 if

e ! [ e y (5 \


Must do example...

+,-

()(

(*0

12

+

,-

(

$ %& (/) (/*

( (*0

12 +

,-

012 >

B (/) C (


Taking Lie-derivatives

e C [ \

ig C e Cc ? (

e g C [ \

g g C e ig C c ? $ %& () (*

e g g C [h} ~ $ () \

y (

+

,-

e Ce g C

e g g C 0

12

+

,-

} ~ $ (/) 0

12


Observer design : transformation

( ? ( ( X H I

B C ( B X HWe wish to find a smooth one-to-one onto global nonlineartransformation ( which gives us

K # B

B W

Why do we care?

Isnt that asking for a bit too much?Observers for non-linear systems p.21/35

Why do we care?

Remember, we can measure B.Create an observer of the form

U K U # B B U B

U B W U

Error dynamics

U

K # B [ K U # B B U B \

K W Error dynamics is exactly like the linear observer!

We can place poles wherever we wish.


Can we really do this?

Comparative study of nonlinear state-observationtechniquesWalcott BL, Corless MJ, Zak SHInternational Journal of Control, 1987, Vol. 45, No. 6,2109-2132

The derivation is long and complicated

I dont fully understand it.

But, I can implement their algorithm


Back to the example

+,-

()(

(*0

12

+

,-

(

$ %& (/) (/*

( (*0

12 +

,-

012 >

B (/) C ( The observability matrix is given by

y (

+

,-

e Ce g C

e g g C 0

12

+

,-

} ~ $ () 0

12


The Algorithm

Compute

y R )

+,P,P,.-

} ~ $ 0

1P1P1.2The starting vector

jjmr is the last column of y R )

] ] )

+,P,P,-

0

1P1P12


The Algorithm

Build the Jacobian matrix of T

] ] M e

5 ?@]

] ) @ Me) ?@

] ] ) @ M

e ?@]

] )

M e) ?@ # [ ?@ #\

] ?

] ( # ] #

] ( ?

M e ?@ # [ ?@ M e R ) ?@ # \

M e5 ?@ # #


The Algorithm

Build the Jacobian matrix of T

] ]

+,Q,P,-

o0

1Q1P12

Solving for T

+

,P,P,.-

o0

1P1P1.2


The Algorithm

Apply the transformation

+

,P,Q,-

0

1P1Q12

+,P,Q,-

$ %'& *

$ %& * * *

01P1Q12

B *


The Algorithm

The observer

U

+

,P,P,.-

0

1P1P1.2U

+,P,P,.-

$ %'& B

$ %'& B BB

01P1P1.2

+,P,P,.-

N)N

N*0

1P1P1.2 B U B

U B U


The Algorithm

Error dynamics U

+

,P,P,- N)

N

N*0

1P1P12 K

Error in original coordinates

_ D 3 K R ) D 3


Why was this so easy?

The tricky bit is to go from

jjmr to T. For more complexsystems, you will have to integrate coupled partialdifferential equations.Bestle and Zeitz(1983) provide a hack for that.

But Im not going to go into it.


Conclusions

Given a nonlinear system, M N y ( tells us if thesystem is observable or not.

If the system is observable, the Lie-algebraic approachgives a smooth nonlinear transformation T that convertsthe state space equation into a form that we like.

Once this is done, we can use good old Luenbergeroutput injection to stabilize the error dynamics.


Things I did not do

The icky derivations.

Other kinds of observers : Extended linearization, Thauobserver, VSS technique, GHO observer.

Extended Kalman filters.


References

1. Nonlinear control systems, Alberto Isidori, Third Edition,Springer

2. Geometric and dynamic sensing, PhD Thesis, Yan-BinJia, CMU-RI-TR-97-50

3. Comparative study of nonlinear state-observationtechniques, Walcott BL, Corless MJ, Zak SH, Int. J.Control, 1987, Vol. 45, No. 6, 2109-2132

4. Canonical form observer design for non-lineartime-variable systems, D. Bestle and M. Zeitz., Int. J.Control, 38(2):419-431, 1983

5. Manifolds : Calculus on curved surfaces, Lyle Noakes,http://www.maths.uwa.edu.au/ rkeal-ley/mf3/manifolds/manifolds.htm


References


OutlineGeneral frameworkAn exampleAn exampleAn exampleNon-linear systemsReview of linear observabilityLinear observersA first pass : output injectionA first pass : error dynamicssmall {The many moods of Aleksandr Mikhailovich Lyapunov}A first pass : linearizing errorA first pass : conclusionSome differential geometryExampleObservability rank conditionObservability rank conditionMust do example...Taking Lie-derivativesObserver design : transformationWhy do we care?Can we really do this?Back to the exampleThe AlgorithmThe AlgorithmThe AlgorithmThe AlgorithmThe AlgorithmThe AlgorithmWhy was this so easy?ConclusionsThings I did not doReferencesReferences

observers for non-linear systems

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