On the Equivalence of Control Systems on the Orthogonal Group SO (4)

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On the Equivalence of Control Systems on the Orthogonal Group SO (4)

Ross M. Adams ∗ , Rory Biggs, Claudiu C. Remsing

Department of Mathematics (Pure and Applied) Rhodes University, Grahamstown 6140, South Africa

WSEAS International Conference on DYNAMICAL SYSTEMS and

CONTROL, Porto, Portugal, July 1–3, 2012

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Outline

1 Introduction

2 The orthogonal group SO (4)

3 Equivalence

4 An illustrative example

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Outline

1 Introduction

3 Equivalence

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Problem statement

Left-invariant control affine systems on Lie groups.

Study the local geometry by introducing a natural equivalence relation.

Detached feedback equivalence and L-equivalence.

Classify, under L-equivalence, all homogeneous systems on

SO (4).

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Left-invariant control affine systems

A left-invariant control affine system Σ = (G, Ξ) g ˙ = g Ξ(1, u) = g(A + u 1 B 1 + . . . u ` B ` ) where g ∈ G , u ∈ R ` and A, B 1 , . . . , B ` ∈ g.

The parameterization map Ξ(1, ·) : R ` → g is an injective affine map (i.e., B 1 , . . . , B ` are linearly independent).

The trace of Σ is Γ = im Ξ(1, ·) = A + hB 1 , . . . , B ` i.

Σ is homogeneous if A ∈ hB 1 , . . . , B ` i.

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Equivalences

Let Σ = (G, Ξ) and Σ 0 = (G, Ξ 0 ).

Detached feedback equivalence

Σ and Σ 0 are (locally) detached feedback equivalent if there exist 1 ∈ N and 1 ∈ N 0 , and

a (local) diffeomorphism Φ = φ × ϕ : N × R ` → N 0 × R ` , φ(1) = 1, such that

T g φ · Ξ (g, u) = Ξ 0 (φ(g), ϕ(u)) for all g ∈ N and u ∈ R `

L-equivalence

Σ and Σ 0 are L-equivalent if there exists a Lie algebra automorphism ψ : g → g such that

ψ · Γ = Γ 0 .

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Outline

1 Introduction

3 Equivalence

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The orthogonal group SO (4)

The orthogonal group SO (4) = n

g ∈ GL (4, R ) : g > g = 1, det g = 1 o is a six-dimensional, non-commutative, semisimple, compact Lie group.

The Lie algebra

so (4) = n

A ∈ R 4×4 : A > + A = 0 o

.

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Decomposition so (4) ∼ = so (3) ⊕ so (3)

Natural Basis

Isomorphism ς : so (3) ⊕ so (3) → so (4).

Induces a natural basis for so (4).

E 1 E 2 E 3 E 4 E 5 E 6

E 1 0 E 3 −E 2 0 0 0

E 2 −E 3 0 E 1 0 0 0

E 3 E 2 −E 1 0 0 0 0

E 4 0 0 0 0 E 6 −E 5

E 5 0 0 0 −E 6 0 E 4

E 6 0 0 0 E 5 −E 4 0

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Automorphisms of so (4)

Lemma

Group of inner automorphisms Int (so (4)) =

ψ 1 0 0 ψ 2

: ψ 1 , ψ 2 ∈ SO (3)

.

Proposition

Aut (so (4)) is generated by Int (so (4)) and the automorphism ζ =

0 1 1 0

.

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Outline

1 Introduction

3 Equivalence

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Basic observations

Proposition

Let Γ, e Γ ⊂ so (4) and ψ ∈ Aut (so (4)). Then ψ · Γ = e Γ ⇐⇒ ψ · Γ ⊥ = e Γ ⊥ .

Any R ∈ SO (3) can be expressed as a product of rotations ρ 1 (θ), ρ 2 (θ) and ρ 3 (θ), denoted respectively





1 0 0

0 cos θ − sin θ 0 sin θ cos θ



 ,





cos θ 0 sin θ

0 1 0

− sin θ 0 cos θ









cos θ − sin θ 0 sin θ cos θ 0

0 0 1



 .

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Example

Let the trace of Σ be Γ =

D E 1 +

√

3E 3 + E 4 , 3E 2 + E 5 + E 6 E . Then

(ρ 2 ( π 3 ), 1) · Γ =

cos π 3 E 1 − sin π 3 E 3 +

√

3 (sin π 3 E 1 + cos π 3 E 3 ) + E 4 , 3E 2 + E 5 + E 6

= h2E 1 + E 4 , 3E 2 + E 5 + E 6 i . Also,

(1, ρ 1 (− π 4 )) · h2E 1 + E 4 , 3E 2 + E 5 + E 6 i

= D

E 1 + 1 2 E 4 , E 2 +

√ 2 3 E 5

E

= e Γ.

Therefore, Σ is L-equivalent to Σ e (with trace e Γ). Note that ψ = ρ 2 ( π 3 ), ρ 1 (− π 4 )

is such that ψ · Γ ⊥ = e Γ ⊥ .

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One-input systems

Theorem

Any one-input system is L-equivalent to a system

Ξ 1 1 (1, u) = u 1 E 1

Ξ 1 2,α (1, u) = u 1 (E 1 + αE 4 )

for some 0 < α ≤ 1.

Proof

Let Σ be a one-input system.

Let A 1 = P 3

i=1 a i E i and A 2 = P 6

i=4 a i E i .

Then Γ 1 = hA 1 i, Γ 2 = hA 2 i, or Γ 3 = hA 1 + A 2 i.

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Proof (cont.)

For Γ 1 , ∃ ψ = (ψ 1 , 1) ∈ Int(so (4)) such that ψ · Γ 1 = hE 1 i = Γ 1 1 . Similarly, there exists a ψ = (1, ψ 2 ) such that ψ · Γ 2 = hE 4 i.

Hence ζ · ψ · Γ 2 = hE 1 i = Γ 1 1 .

For Γ 3 , there exists a ψ = (ψ 1 , ψ 2 ) such that ψ · Γ 3 = hE 1 + αE 4 i for some α > 0.

If α ≤ 1, then ψ · Γ 3 = Γ 1 2,α .

If α > 1, then ζ · ψ · Γ 3 = hE 4 + αE 1 i =

E 1 + α 1 E 4

= Γ 1 2,

1 α

.

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Five-input systems

Corollary

Any five-input system is L-equivalent to a system

Ξ 5 1 (1, u) = u 1 E 2 + u 2 E 3 + u 3 E 4 + u 4 E 5 + u 6 E 6 Ξ 5 2,α (1, u) = u 1 E 2 + u 2 E 3 + u 3 E 5

+ u 4 E 6 + u 5 (αE 1 − E 4 )

for some 0 < α ≤ 1.

Any five-input system has full rank.

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Two-input systems

Theorem

Any two-input system is L-equivalent to a system

Ξ 2 1 (1, u) = u 1 E 1 + u 2 E 2 Ξ 2 2 (1, u) = u 1 E 1 + u 2 E 4

Ξ 2 3,α (1, u) = u 1 E 1 + u 2 (E 2 + αE 5 ) Ξ 2 4,αβ (1, u) = u 1 (E 1 + αE 4 + βE 5 )

+ u 2 (E 2 + αE 5 )

for some α > 0, β ≥ 0.

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Four-input systems

Corollary

Any four-input system is L-equivalent to a system

Ξ 4 1 (1, u) = u 1 E 3 + u 2 E 4 + u 3 E 5 + u 4 E 6 Ξ 4 2 (1, u) = u 1 E 2 + u 2 E 3 + u 5 E 5 + u 6 E 6 Ξ 4 3,α (1, u) = u 1 E 3 + u 2 E 4 + u 3 E 6

+ u 4 (αE 2 − E 5 )

Ξ 4 4,αβ (1, u) = u 1 E 3 + u 2 E 6 + u 3 βE 1 + αE 2

− E 5

+ u 4 (αE 1 − E 4 )

for some α > 0 and β ≥ 0.

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Three-input systems

Theorem

Any three-input system is L-equivalent to a system

Ξ 3 1,αβ (1, u) = u 1 (E 1 + α 1 E 4 ) + u 2 (E 2 + α 2 E 5 ) + u 3 (E 3 + βE 6 )

Ξ 3 2,γ (1, u) = u 1 (E 1 + γE 4 ) + u 2 (E 2 + γ E 5 ) + u 3 (E 3 ± γE 6 )

Ξ 3 3,γ (1, u) = u 1 (E 1 + γE 4 ) + u 2 E 2 + u 3 E 6

for some α 1 ≥ α 2 ≥ |β| ≥ 0 and 0 ≤ γ ≤ 1. Here α 1 6= α 2 or

α 2 6= |β|.

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Outline

1 Introduction

3 Equivalence

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An illustrative example

Any system

Ξ γ (1, u) = γ 1 E 4 + u 1 (γ 2 E 1 + γ 3 E 2 + γ 4 E 3 )

+ u 2 (γ 5 E 3 + γ 6 E 4 ) + u 3 (γ 7 E 4 ) + u 4 (γ 8 E 5 )

is L-equivalent to the system

Ξ(1, e u) = u 1 E 2 + u 2 E 3 + u 3 E 4 + u 4 E 5 .

Here γ i > 0, i = 1, . . . , 8. The automorphism relating the traces of these systems is given by ψ = (ψ 1 , 1) , where

ψ 1 =







γ

3

√

γ

₂²

+γ

²₃

− √ γ

²

γ

²₂

+γ

₃²

0

γ

2

√

γ

₂²

+γ

²₃

γ

3

√

γ

₂²

+γ

₃²

0

0 0 1







.

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An illustrative example

The corresponding feedback transformation ϕ, defined by ψ · Ξ γ (1, u) = Ξ (1, ϕ(u)) e

is given by

ϕ : u 7→





 q

γ 2 2 + γ 3 2 0 0 0

γ 4 γ 5 0 0

0 γ 6 γ 7 0

0 0 0 γ 8





 u +





 0 0 γ 1

0





 .

There exists a φ ∈ Aut(SO (4)) such that T 1 φ = ψ.

φ establishes a one-to-one correspondence between trajectories

of Ξ γ and Ξ. e

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Concluding remark

Obtained a list of equivalence representatives for homogeneous systems on SO (4).

Attempt to obtain a classification of systems.

A classification can be obtained for the one-input case.

For the two-input and three-input cases the calculations become

On the Equivalence of Control Systems on the Orthogonal Group SO (4)