cyrus mostajeran rodolphe sepulchre university of...

Invariant Differential Analysis

Cyrus Mostajeran Rodolphe Sepulchre

University of Cambridge

UKACC PhD Presentation Showcase

UKACC PhD Presentation Showcase Slide 2

Introduction Ø  Despite the many great successes of linear modelling in control

engineering, such methods inevitably fail to account for a variety of crucial global behaviours of nonlinear systems.

Ø  Differential analysis is concerned with the derivation of global statements

about the nonlinear dynamics by studying the linearization of the system, also known as the variational system.

Ø  If P1 for the linearization, then P2 for the nonlinear dynamics.

Differential Stability Ø  An illuminating example of this is the study of stability via local

contraction analysis, where local contraction of length by the flow is used to infer stability.

Ø  The global behavior is captured by the local behavior.

Ø  In a linear space, contraction with respect to a local measure of length

that is invariant under translations yields quadratic Lyapunov theory.

UKACC PhD Presentation Showcase Slide 3

V

loc

(x, �x) = �x

TP (x)�x = �x

TP �x , V (x) = x

TPx

[On contraction analysis for non-linear systems, Lohmiller, Slotine, 1998]

[A di↵erential Lyapunov framework for contraction analysis, Forni, Sepulchre, 2014]

V (γ(0), γ(0))

γ(s)

γ(0)γ(0)

γ(1)M

ψt0(t, γ(0))

ψt0(t, γ(s))

ψt0(t, γ(1))

x = f(x)˙δx = ∂f(x)δx

δx(t)

Differential Positivity Ø  Positive systems are defined as linear systems that leave a cone

invariant, and are closely related to the theory of monotone systems. Ø  Differentially positive dynamical systems are defined as systems whose

linearizations along trajectories are positive. A nonlinear map is differentially positive if it leaves a cone field invariant.

UKACC PhD Presentation Showcase Slide 4

Ø  A cooperative system on a linear

space is monotone with respect to the positive orthant, and thus differentially positive with respect to a constant cone field:

Cone Field: KM(x) := Rn+ for all x 2 M = Rn

[Di↵erentially positive systems, Forni, Sepulchre, 2014]

∂ψt(x)KX (x)

ψt(x)

x

KX (x) KX (ψt(x))

Invariant Differential Analysis Ø  In many problems the state space of the nonlinear system of interest is

not a vector space, but a nonlinear manifold. Ø  Construction of suitable geometric structures such as local measures of

length and cone fields is generally intractable. One approach is to make these structures invariant in some sense.

UKACC PhD Presentation Showcase Slide 5

TeG = g

TgG

Lge

gK ⊂ TeG

dLg (K) ⊂ TgG Ø  This notion can be made

precise on Lie groups and homogeneous spaces, where invariance is achieved with respect to a group of transformations.

Case study: Damped Nonlinear Pendulum Ø  The nonlinear pendulum:

Ø  Strict differential positivity of the nonlinear pendulum with respect to an

invariant cone field on the cylinder can be used to infer the limit cycle behavior of the pendulum.

UKACC PhD Presentation Showcase Slide 6

⌃ :

⇢# = vv = � sin(#)� kv + ⌧

#⌧

k

stable fixed point

stable limit cycle

bistability

Nonlinear dynamics and Chaos, Strogatz, 1994

#

v

The pendulum

⌃ :

⇢# = vv = � sin(#)� kv + ⌧

�⌃ :

⇢˙�# = �v˙�v = � cos(#)�#� k�v

#

v

⌧

k

stable fixed point

stable limit cycle

bistability

K(#, v) := �# � 0, �#+ �v � 0�#

�v

Strict Diff+ with respect to the cone field

The pendulum

⌃ :

⇢# = vv = � sin(#)� kv + ⌧

�⌃ :

⇢˙�# = �v˙�v = � cos(#)�#� k�v

#

v

⌧

k

stable fixed point

stable limit cycle

bistability

0 1

� cos(#) �k

�

k=4 k=3 k=2

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

δ v

δ ϑ

The pendulum

⌃ :

⇢# = vv = � sin(#)� kv + ⌧

�⌃ :

⇢˙�# = �v˙�v = � cos(#)�#� k�v

#

v

⌧

k

stable fixed point

stable limit cycle

bistability

k=4 k=3 k=2

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ

-4 -2 0 2 4-5

-4

-3

-2

-1

0

1

eig

s

ϑ-4 -2 0 2 4

-5

-4

-3

-2

-1

0

1

eig

s

ϑ-4 -2 0 2 4

-5

-4

-3

-2

-1

0

1

eig

sϑ

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ

The pendulum

damping coe�cient: k

KM(#, v) := {(�#, �v) 2 T(#,v)M : �# � 0, �#+ �v � 0}

⌃ :

⇢# = vv = � sin(#)� kv + ⌧

#⌧

k

stable fixed point

stable limit cycle

bistability

Nonlinear dynamics and Chaos, Strogatz, 1994

#

v

The pendulum

⌃ :

⇢# = vv = � sin(#)� kv + ⌧

�⌃ :

⇢˙�# = �v˙�v = � cos(#)�#� k�v

#

v

⌧

k

stable fixed point

stable limit cycle

bistability

K(#, v) := �# � 0, �#+ �v � 0�#

�v

Strict Diff+ with respect to the cone field

The pendulum

⌃ :

⇢# = vv = � sin(#)� kv + ⌧

�⌃ :

⇢˙�# = �v˙�v = � cos(#)�#� k�v

#

v

⌧

k

stable fixed point

stable limit cycle

bistability

0 1

� cos(#) �k

�

k=4 k=3 k=2

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

δ v

δ ϑ

The pendulum

⌃ :

⇢# = vv = � sin(#)� kv + ⌧

�⌃ :

⇢˙�# = �v˙�v = � cos(#)�#� k�v

#

v

⌧

k

stable fixed point

stable limit cycle

bistability

k=4 k=3 k=2

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ

-4 -2 0 2 4-5

-4

-3

-2

-1

0

1

eig

s

ϑ-4 -2 0 2 4

-5

-4

-3

-2

-1

0

1

eig

s

ϑ-4 -2 0 2 4

-5

-4

-3

-2

-1

0

1

eig

s

ϑ

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ

The pendulum

constant torque input: ⌧

⌃ :

⇢# = vv = � sin(#)� kv + ⌧

#⌧

k

stable fixed point

stable limit cycle

bistability

Nonlinear dynamics and Chaos, Strogatz, 1994

#

v

The pendulum

⌃ :

⇢# = vv = � sin(#)� kv + ⌧

�⌃ :

⇢˙�# = �v˙�v = � cos(#)�#� k�v

#

v

⌧

k

stable fixed point

stable limit cycle

bistability

K(#, v) := �# � 0, �#+ �v � 0�#

�v

Strict Diff+ with respect to the cone field

The pendulum

⌃ :

⇢# = vv = � sin(#)� kv + ⌧

�⌃ :

⇢˙�# = �v˙�v = � cos(#)�#� k�v

#

v

⌧

k

stable fixed point

stable limit cycle

bistability

0 1

� cos(#) �k

�

k=4 k=3 k=2

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

δ v

δ ϑ

The pendulum

⌃ :

⇢# = vv = � sin(#)� kv + ⌧

�⌃ :

⇢˙�# = �v˙�v = � cos(#)�#� k�v

#

v

⌧

k

stable fixed point

stable limit cycle

bistability

k=4 k=3 k=2

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ

-4 -2 0 2 4-5

-4

-3

-2

-1

0

1

eig

s

ϑ-4 -2 0 2 4

-5

-4

-3

-2

-1

0

1

eig

s

ϑ-4 -2 0 2 4

-5

-4

-3

-2

-1

0

1

eig

s

ϑ

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ

The pendulum

UKACC PhD Presentation Showcase Slide 7

Conclusion

Ø Work in 1st year of project: §  A mathematical framework for studying key concepts in differential

analysis, including invariant differential lyapunov theory and invariant cone fields

Ø  Future work (next 2 years): §  Invariant control and design.

§  Investigate the effectiveness of invariant differential positivity in the study of limit cycles and oscillations.

§  Invariant orders and positivity. Extension

of linear consensus theory to nonlinear spaces.

x = γ(t0)

y = γ(t1)

γ′(t) ∈ KM (γ(t))

γ : [t0, t1] → M