Observers, invariance and autonomy J. Trumpf ANU, Canberra based - - PowerPoint PPT Presentation

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Observers, invariance and autonomy

• J. Trumpf

ANU, Canberra based on joint work with C. Lageman and R. Mahony

July 2008

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Outline

1

The problem

2

Symmetry and projected systems

3

Synchrony and error functions

4

Internal models and innovation terms

5

Observer design

6

Conclusions

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Outline

1

The problem

2

Symmetry and projected systems

3

Synchrony and error functions

4

Internal models and innovation terms

5

Observer design

6

Conclusions

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Left invariant systems

We consider systems of the form ˙ X = Xu where the state X evolves on a finite dimensional, connected Lie group G and the (admissible) input is in the associated Lie algebra g.

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Left invariant systems

We consider systems of the form ˙ X = Xu where the state X evolves on a finite dimensional, connected Lie group G and the (admissible) input is in the associated Lie algebra g. Xu is shorthand notation for TeLXu where TeLX is the derivative

• f the left multiplication map

LX : G − → G, Y → XY at the identity element Y = e of G. This is then a map g ≃ TeG − → TXG, so X → Xu is a left invariant vector field.

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Left invariant systems

This is a special case of a well studied class of systems on Lie groups (Brockett, Jurdjevic, Sussmann, ...) with no drift and a full basis set of control vector fields.

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Left invariant systems

This is a special case of a well studied class of systems on Lie groups (Brockett, Jurdjevic, Sussmann, ...) with no drift and a full basis set of control vector fields. For G = SO(3) and g = so(3) these are the kinematic equations ˙ R = RΩ describing the time evolution of the attitude (=orientation) of the center of mass of a rigid body in 3D space. Here, R is the rotation matrix relating the inertial coordinate frame to the body-fixed frame and Ω contains the angular velocities. Similar for G = SE(3) ≃ SO(3) ⋉ R3 and g = se(3).

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Outputs

˙ X = Xu We consider outputs of the form y = h(X, y0) with a right action h : G × M − → M, where M is a smooth manifold (i.e. a homogeneous space of G).

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Outputs

˙ X = Xu We consider outputs of the form y = h(X, y0) with a right action h : G × M − → M, where M is a smooth manifold (i.e. a homogeneous space of G). For G = SO(3) think of M = S2 and y = RTy0 which describes the direction a fixed landmark is seen in by, say, a camera mounted on board.

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Note that this is different from the case y = h(X, y0) where h is a left action. For G = SO(3) think of M = S2 and y = Ry0 which describes the direction a fixed marking on the rigid body is seen in from the ground.

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Note that this is different from the case y = h(X, y0) where h is a left action. For G = SO(3) think of M = S2 and y = Ry0 which describes the direction a fixed marking on the rigid body is seen in from the ground. Only this latter case has been studied in the literature (in the context of control).

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

The problem

˙ X = Xu y = h(X, y0) Suppose we have measurements of u and measurements of y. We want to construct observers that estimate X, i.e. systems with input (u, y) and state ˆ X where ˆ X is a reasonable estimate

• f X.

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

The problem

˙ X = Xu y = h(X, y0) Suppose we have measurements of u and measurements of y. We want to construct observers that estimate X, i.e. systems with input (u, y) and state ˆ X where ˆ X is a reasonable estimate

• f X.

Think: noisy measurements ...

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Outline

1

The problem

2

Symmetry and projected systems

3

Synchrony and error functions

4

Internal models and innovation terms

5

Observer design

6

Conclusions

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Systems with symmetry

A system that is globally given by ˙ x = f(x, u), x ∈ N can be regarded as a map f : B − → TN where B is a trivial bundle over N. (We allow general bundles here.) Add an output h : N − → M.

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Systems with symmetry

A system that is globally given by ˙ x = f(x, u), x ∈ N can be regarded as a map f : B − → TN where B is a trivial bundle over N. (We allow general bundles here.) Add an output h : N − → M. A Lie group H is called a symmetry of this system if there are left actions SB and SN and a right action SM such that f(SB(X, v)) = TSN

X f(v)

h(SN(X, x)) = SM(X, h(x)) (Cf. Grizzle/Marcus, Nijmeijer/van der Schaft, Tabuda/Pappas)

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Projected systems

˙ X = Xu y = h(X, y0) (S) Proposition: stab(y0) is a symmetry of (S). Theorem: (S) projects to the system on M ˙ y = TXπ(Xu), X ∈ π−1(y) where π : G − → M is the canonical projection.

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Projected systems

˙ X = Xu y = h(X, y0) (S) Proposition: stab(y0) is a symmetry of (S). Theorem: (S) projects to the system on M ˙ y = TXπ(Xu), X ∈ π−1(y) where π : G − → M is the canonical projection. Corollary: Two states X, Y ∈ G are indistinguishable if and

• nly if XY −1 ∈ stab y0.

(Cf. Sussmann)

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

The idea

Construct an observer for the projected system and lift it up.

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Outline

1

The problem

2

Symmetry and projected systems

3

Synchrony and error functions

4

Internal models and innovation terms

5

Observer design

6

Conclusions

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

How to measure errors?

(Smooth) error functions E : M × M − → N Definition: Two systems ˙ y = fy(u, t), ˙ ˆ y = fˆ

y(u, t)

with common input are called E-synchronous if E is constant along corresponding trajectory pairs.

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

The canonical error function

There is a canonical error function Er(ˆ y, y) = h(ˆ XX −1, y0) where π(ˆ X) = ˆ y and π(X) = y.

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

The canonical error function

There is a canonical error function Er(ˆ y, y) = h(ˆ XX −1, y0) where π(ˆ X) = ˆ y and π(X) = y. Theorem: Only ˙ ˆ y = Tˆ

Xπ(ˆ

Xu) is Er-synchronous to ˙ y = TXπ(Xu). Theorem: If ˙ ˆ y = Tˆ

Xπ(ˆ

Xu) is E-synchronous to ˙ y = TXπ(Xu) then E = g ◦ Er.

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Outline

1

The problem

2

Symmetry and projected systems

3

Synchrony and error functions

4

Internal models and innovation terms

5

Observer design

6

Conclusions

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Internal models

Consider system pairs ˙ y = fx(u, t), (S) ˙ ˆ y = fˆ

y(u, y, t).

(O) Definition: (O) has an internal model of (S) if ˆ y = y for corresponding trajectories. Definition: We call a map α : M × M × TM × R − → TM an innovation term if α(ˆ y, y, u, t) ∈ Tˆ

yM and α is zero along

corresponding trajectories.

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

A nice little result

˙ y = TXπ(Xu), (S) ˙ ˆ y = fˆ

y(ˆ

y, y, u, t). (O) Theorem: (O) has an internal model of (S) if and only if it has the form ˙ ˆ y = Tˆ

Xπ(ˆ

Xu) + α(ˆ y, y, u, t), i.e. if and only if it splits into an Er-synchronous term and an innovation term.

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Outline

1

The problem

2

Symmetry and projected systems

3

Synchrony and error functions

4

Internal models and innovation terms

5

Observer design

6

Conclusions

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Good innovation terms

˙ y = TXπ(Xu), (S) ˙ ˆ y = Tˆ

Xπ(ˆ

Xu) + α(ˆ y, y, u, t). (O) Observer design in this framework amounts to finding a good choice for α.

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Good innovation terms

˙ y = TXπ(Xu), (S) ˙ ˆ y = Tˆ

Xπ(ˆ

Xu) + α(ˆ y, y, u, t). (O) Observer design in this framework amounts to finding a good choice for α. Theorem: α yields autonomous Er dynamics if and only if it is equivariant, i.e. Tˆ

yhSα(ˆ

y, y, u, t) = α(hS(ˆ y), hS(y), u, t), and independent of u and t. Then ˙ Er = α(Er, y0).

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Gradient innovations

Introduce a smooth, non-negative cost function f : M × M − → R and choose a Riemannian metric on M. We propose observers of the form ˙ ˆ y = Tˆ

Xπ(ˆ

Xu) − grad1f(ˆ y, y)

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Gradient innovations

Introduce a smooth, non-negative cost function f : M × M − → R and choose a Riemannian metric on M. We propose observers of the form ˙ ˆ y = Tˆ

Xπ(ˆ

Xu) − grad1f(ˆ y, y) Theorem: Assume now that M is reductive. If f is constant along the diagonal and both f and the Riemannian metric are invariant then ˙ Er = −grad1f(Er, y0).

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

A convergence result

˙ y = TXπ(Xu), (S) ˙ ˆ y = Tˆ

Xπ(ˆ

Xu) − grad1f(ˆ y, y) (O) where f and the Riemannian metric are both invariant. Corollary: Let additionally ˆ y → f(ˆ y, y0) be a Morse-Bott function with a global minimum at y0 and no other local minima. Then Er converges to y0 for generic initial conditions.

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

The lift

˙ X = Xu y = h(X, y0) The lifted observer takes the form ˙ ˆ X = ˆ Xu −

• grad1f(π(ˆ

X), y) H = ˆ Xu − grad1˜ f(ˆ X, X) where ˜ f(ˆ X, X) = f(π(ˆ X), π(X)). We get ˆ XX −1 → stab(y0).

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

Outline

1

The problem

2

Symmetry and projected systems

3

Synchrony and error functions

4

Internal models and innovation terms

5

Observer design

6

Conclusions

Trumpf Observers, invariance and autonomy

The problem Symmetry and projected systems Synchrony and error functions Internal models and innovation terms Observer design Conclusions

A first conclusion

Brockett is right: Our main point is that this class of systems is in many ways not more difficult than linear systems of the usual type in Rn. [System theory on group manifolds and coset spaces, SIAM J. Control, 10(2), 1972, p.265]

Trumpf Observers, invariance and autonomy