nonlinear observer design

The Frobenius TheoremState ObservabilityAnalysis and Design

Literature

State Observer Design

Sopasakis Pantelis

October 5, 2012

Sopasakis Pantelis State Observer Design


Literature

Introduction to DistributionsDuality and IntegrabilityThe Frobenius Theorem

What is a Distribution

A distribution is a mapping D : <n −→ V(<n) which maps everyx ∈ <n to a linear subspace of <n according to the formulaD(x) = span{f1 (x) , f2 (x) , . . . , fp (x)}, where fi : <n −→ <n are thegenerator vector fields. The set of all distributions definedU ⊂ <n will be denoted as D(U).



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Some Definitions

I A distribution is called nonsingular or a distribution ofconstant-degree k if dimD(x) = k for every x ∈ <n.

I Let D ∈ D(U). A x0 ∈ U is said to be a regular point of D ifthere exists an open neighborhood U0 of x0 such that D isnonsingular in U0.

I A distribution D is called smooth if there exist vector fields{fi}i∈F that span D.



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A fundamental result on smooth distributions

Let D ∈ D(U) be a nonsingular smooth distribution of constantdegree k and Y : U −→ <n a smooth vector-valued function in D,i.e Y (x) ∈ D(x) for every x ∈ U. Then there exist k smoothreal-valued functions mj : U → < such that

Y (x) =k∑

j=1

mj(x)Xj(x)

where Xj ∈ D.



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Lie Algebras on Mn(<) and C∞(<n).

A Lie Algebra is a linear vector space L over a field F endowedwith a bilinear operation [·, ·] : L× L→ L, such that:

1. [x , x ] = 0 for every x ∈ L

2. [x [yz ]] + [y [zx ]] + [z [xy ]] = 0 for every x , y , z ∈ L

This operator is known as Lie Bracket or Commutator.

1. If L = Mn(<) then the commutator is (usually) defiend to be[A,B] = AB − BA for every A,B ∈ Mn(<).

2. If L = C inf(<n) then [f , g ] = ∇g · f −∇f · g



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Involutive Distributions

A distribution D ∈ D(U) is called involutive if [f , g ] ∈ D for everyf , g ∈ D.



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Duality

A codistribution is a mapping W : <n → V((<n)?) which mapsevery x ∈ <n to a linear subspace of (<n)?. The set ofcodistributions defined on a subset U ⊂ <n is denoted by D?(U).

The annihilator of a distribution D ∈ D(U) is a codistributionD⊥ ∈ D?(U) defined as

D⊥(x) = {w ∈ (<n)? : 〈w , u〉 = 0, ∀u ∈ D(x)}

It is remarkable that dim(D) + dim(D?) = n.



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The Distribution Integrability Problem

Problem Formulation:Given a distribution DF ∈ D(U) of constant degree p which isspanned by the k ≥ p columns of a mapping F : <n → Mn×k(<)specify necessary and sufficient conditions such that there existn − p vector fields λ1, λ2, . . . , λp : U → <n that their derivativesannigilate the vector fields in D, i.e.

dλj · F (x) = 0



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Complete Integrability

Let D be a nonsingular distribution of constant degree d definedon an open set U ⊂ <n and 〈U〉 = <n. The distribution D iscalled completely integrable if for each x0 ∈ U there exists anopen neighborhood U0 of x0 and n − d real-valued functionsλ1, λ2, . . . , λn−d , defined on U0 such that dλ1, dλ2, . . . , dλn−dspan the annihilator of D.



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The Frobenius Theorem

Let D ∈ D(U) be a nonsingular distribution of constant degree dand 〈U〉 = <n. The following are equivalent:

1. D is completely integrable

2. D is involutive



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Local Decompositions of Control SystemsState Observability

f -invariant distributions

Let D ∈ D(U) and f : <n → <n. The distribution D is said to bef -invariant if for every τ ∈ D : [f , τ ] ∈ D.



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Local Inner Triangular Decomposition ∗

Let D be a distribution posessing the following properties:

1. D is nonsingular, of constant degree d and involutive.

2. D is f -invariant for some f : <n → <n

Then for every x0 ∈ U there exists a neighborhood U0 3 x0 and acoordinate transormation z = Φ(x); x ∈ U, such that:

f (z) = f (Φ−1(x)) =

f1 (z1, z2, . . . , zd , zd+1, . . . , zn)f2 (z1, z2, . . . , zd , zd+1, . . . , zn)

. . .

fd (zd , zd+1, . . . , zn). . .

fn (zd , zd+1, . . . , zn)

A.J.Kerner, Normal Forms for linear and nonlinear systems, Contemp. Math. 68, 157-189, 1987



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How much state can we know? ∗∗

For a dynamic system

x = f (x) +∑m

i=1 gi (x)uiy = h(x), y ∈ <p

Let its triangular representation be:

ζ1 = θ1 (ζ1, ζ2) +∑m

i=1 γ1,i (ζ1, ζ2) uiζ2 = θ2 (ζ2) +

∑mi=1 γ2,i (ζ2) ui

yi = hi (ζ2)

for x ∈ U0 3 x0. The “unobservable” manifold of the system is theslice:

Sx = {υ ∈ U0 : ζ2 (υ) = ζ2 (x)}

S.R.Kou, D.L.Eliot and T.J.Tarn, Observability of Nonlinear Systems, Information and Control 22(1), 89-99, 1972



Literature

State ObserversObserver Linearization ProblemObserver Canonical FormObserver DesignExtended Linearization Design Method

Observers

An observer is a dynamic system such that its output converges tothe state of a given system as t →∞, that is

‖ξ (t)− x (t) ‖ t→∞−−−→ 0



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The Observer Linearization Problem ∗∗∗

Given a dynamical system x = f (x), y = h(x) with scalar output yand an initial state x0, specify a neighborhood U0 3 x0 and a localcoord. transf. z = Φ(x) and a mapping k : h(U0)→ <n such that:[

∂Φ∂x f (x)

]x=Φ−1(z)

= Az + k(Cz)

h(Φ−1(z)) = Cz

for z ∈ Φ(U0) and A ∈ Mn(<) and CT ∈ <nsuch that:

rank

CCA

...CAn−1

= n⇔ (C ,A) is observable



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Solvability of the Observer Linearization Problem ∗∗∗∗

The OLP is solvable at x0 only if the following conditions holds:

dim span{dh|x0 , d (fh) |x0 , d

(f 2h)|x0 , . . . , d

(f n−1h

)|x0

}= n

where fh = f (h) = 〈dh, f 〉 = Lf h and f kh = Lkf h

A. Isidori, Nonlinear Control Systems - An Introduction, Springer-Verlag editions, 2ns ed, 1989, pp.217-226



Literature


Solvability of the Observer Linearization Problem

The OLP is solvable at x0 iff the following conditions hold:

1. dim span{dh|x0 , d (fh) |x0 , d

(f 2h)|x0 , . . . , d

(f n−1h

)|x0

}= n

2. The unique vector field τ which satisfiesdh|x0

d (fh) |x0

d(f 2h)|x0

...d(f n−1h

)|x0

· τ =

00...01

is such that

[ad i

f τ, adjf τ]

= 0 for every 0 ≤ i and j ≤ n − 1



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Observer Canonical Form

Given an observable system x = f (x) with x ∈ <n and y = h(x)with y ∈ < find a local coord. trans x = T (x∗) s.t.

x∗ =

0 · · · 0

1...

. . ....

1 0

x∗ −

f ∗0 (x∗n )f ∗1 (x∗n )

...f ∗n−1 (x∗n )

= f ∗ (x∗)

y =[

0 . . . 0 1]x∗ = ηT x∗

X.H. Xia, W.B. Gao, Nonlinear observer design by observer canonical forms, Int. J. contr. 47(4) 2988, 1081-1100



Literature


Inverted Pendulum - OCF ∗∗∗∗∗

Suppose of the system: x1

x2

x3

=

x2

sinx1 + x3

x2 + x3

= f (x)

y = x1 = h (x)

Hint:

O(x) · ∂T∂x∗1= η and ∂T

∂x∗ =[ad0

f∂T∂x∗1

· · · adn−1f

∂T∂x∗1

]



Literature


Inverted Pendulum - OCF

The OCF of the inverted pendulum nonlinear system is:

x∗ =

0 0 01 0 00 1 0

x∗ +

sinx∗3x∗3 + sinx∗3

x∗3

= f ∗(x∗)

y = x∗3



Literature


Observer Design Based on the OCF

With every observable system in the OCF we associate thefollowing observer:

ξ =

0 0 0 · · · 01 0 0 · · · 00 1 0 · · · 0...

.... . .

...0 0 · · · 1 0

−

f ∗0 (x∗n )f ∗1 (x∗n )

...f ∗n−1 (x∗n )

− KT e

where e = ξ − x∗ and KT =[k0 k1 · · · kn−1

]∈ <n



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Error Dynamics

The error e∗ = ξ − x∗ evolves with respect to the linear dynamics:

dedt =

0 0 0 · · · −k0

1 0 0 · · · −k1

0 1 0 · · · −k2...

.... . .

...0 0 · · · 1 −kn−1

e = Y · e

The characteristic polynomial of the matrix Y is

χ (Y ) (s) = k0 + k1s + . . .+ kn−1sn−1 + sn



Literature


Introduction to Extended Linearization

Extended Linearization is a method to tackle the observer designproblem for control systems

x = f (x , u)y = h (x)

x ∈ <n, y ∈ <p, u ∈ <f (0, 0) = 0, h (0) = 0

Let {u = ε, x = xε, f (xε, ε) = 0} be a collection of equillibriumpoints. We assume that the observer admits the representation:

ξ = f (ξ, u) + g (y)− g (y)

B.Walcott et al., Comparative study of nonlinear state observation techniques, Int. J. Contr. 45(6) 1987, 2109-32

F.Thau, Observing the state of nonlinear dynamic systems,Int. J. Contrl 17(3), 1973, 471-9



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Observer Error Dynamics

Let us define e = x − ξ, with ξ = f (ξ, u) + g (y)− g (y). Then:

e = x − ξ = f (x , u)− f (x − e, u)− g (y) + g (y)

for (x , u) close to (xε, ε) we have:

e = [D1f (xε, ε)− Dg (yε)Dh (xε)] e = Ye

The aim of the design consists in determining proper analyticfunction g such that Y perserves constant stable eigenvalues -independent of ε!



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Assumptions

We assume that the following hold:

1. D1f (0, 0)−1 exists

2. (D1f (0, 0) ,Dh (0)) is observable

3. ∂yε∂ε |ε=0 = Dyε|ε=0 = Dh (0)Dxε (0) =

−Dh (0) [D1f (0, 0)]−1 6= 0



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Since (D1f (0, 0) ,Dh (0)) is observable,(D1f (0, 0)T ,Dh (0)T

)is

controllable. Hence we may use the Ackermann’s synthesis formulato determine a C : < → <n such that

D1f (xε, ε)T − Dh (xε)T C (ε)

has a prespecified desired spectrum. Have we found C , g is givenby:

Dg (yε)T = C (ε)

More Details on the whiteboard...



Literature


Fin!

Thank you for your attention!



Literature

Important References

1. A. Isidori, Nonlinear Control Systems - An Introduction, Springer-Verlag editions, 2ns ed, 1989, pp.217-226

2. A.J.Kerner, Normal Forms for linear and nonlinear systems, Contemp. Math. 68, 157-189, 1987

3. S.R.Kou, D.L.Eliot and T.J.Tarn, Observability of Nonlinear Systems, Information and Control 22(1),89-99, 1972

4. X.H. Xia, W.B. Gao, Nonlinear observer design by observer canonical forms, Int. J. contr. 47(4) 2988,1081-1100

5. B.Walcott et al., Comparative study of nonlinear state observation techniques, Int. J. Contr. 45(6) 1987,2109-32

6. F.Thau, Observing the state of nonlinear dynamic systems,Int. J. Contrl 17(3), 1973, 471-9


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