Introduction to Contraction Theory.pdf

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Introduction to Contraction Theory

Seminar 2008

Helmut Hauser

Institute for Theoretical Computer Science

31.Jan, 2008

Helmut Hauser Institute for Theoretical Computer Science


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Overview

1 Introduction and Basic Theorems



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Overview


2 Connecting Contractive Systems



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Overview


2 Connecting Contractive Systems3 Applications of Contraction Theory



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Overview


2 Connecting Contractive Systems3 Applications of Contraction Theory

4 Summary



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Introductions

Some DefinitionsWe are considering n-dimensional deterministic nonlinear systems of the form

x = f (x, t )

with x ∈ Rn being the the state vector



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Introductions


x = f (x, t )

with x ∈ Rn being the the state vectorand f being a nonlinear vector function



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Introductions


x = f (x, t )

with x ∈ Rn

being the the state vectorand f being a nonlinear vector function

f is assumed to be smooth



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Introductions


x = f (x, t )

with x ∈ Rn



all quantities assumed to be real and smooth (any required derivative orpartial derivative exists)



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Introductions


x = f (x, t )

with x ∈ Rn




Note: system can be in general time-variant!



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Introductions


x = f (x, t )

with x ∈ Rn




Note: system can be in general time-variant!

Note: may also represent closed-loop dynamics of system with statefeedback u(x, t )



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Introduction - Basic Idea

Basic Idea

Classical stability analysis is relative to some nominal motion orequilibrium point



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Basic Idea


Contraction Theory states ”A system is stable if in some region any initialconditions or temporary disturbances are somehow forgotten”



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Basic Idea



Do not care about the nominal motion itself, just show that all trajectoriesconverge



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Basic Idea




Analysis is inspired by fluid mechanics



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Basic Idea





Stability can be therefor analyzed differentially: Do nearby trajectoriesconverge?



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Basic Idea








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Basic Idea






Fluid Mechanics Interpretation

x = f (x, t ) can be seen as an n-dimensional fluid flow, where x is then-dimensional ”velocity” vector at the n-dimensional position x and time t .



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The path to Contraction Theory

With δ x being a virtual displacement

(= infinitesimal displacement atfixed time) we define a well defineddifferential relation:

δ x =∂ f

∂ x(x, t ) δ x

Virtual dynamics of neighboring

trajectories

δ x

δ x



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δ x =∂ f

∂ x(x, t ) δ x

the associated quadratic tangentform of δ x is δ xT δ x. Looking at rateof change of the quadratic distancebetween to neighboring trajectories:

d

dt (δ

x

T δ

x) = 2δ

x

T δ

x

from above

= 2δ

x

T ∂ f

∂ xδ

x


trajectories

δ x

δ x



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δ x =∂ f

∂ x(x, t ) δ x

the associated quadratic tangentform of δ x is δ xT δ x. Looking at rateof change of the quadratic distancebetween to neighboring trajectories:

d

dt (δ

x

T δ

x) = 2δ

x

T δ

x

from above

= 2δ

x

T ∂ f

∂ xδ

x

Now we want to have a negative rateof change = trajectories converge


trajectories

δ x

δ x



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The path to Contraction Theory - cont.

d

dt (δ xT

δ x) = 2δ xT ∂ f

∂ x |{z} Jacobian

δ x



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d

dt (δ xT

δ x) = 2δ xT ∂ f

∂ x |{z} Jacobian

δ x

Denoting λmax (x, t ) the largest eigenvalue of the symmetric part of the

Jacobian ∂ f ∂ x

d

dt (δ xT

δ x) ≤ 2λmax δ xT δ x



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d

dt (δ xT

δ x) = 2δ xT ∂ f

∂ x |{z} Jacobian

δ x

Denoting λmax (x, t ) the largest eigenvalue of the symmetric part of the

Jacobian ∂ f ∂ x

d

dt (δ xT

δ x) ≤ 2λmax δ xT δ x

and hence,δ x ≤ δ x0e

R t

0 λmax (x,t )dt

if λmax (x, t ) is uniformly strictly negative then δ x converges exponentially tozero.



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DefinitionGiven the systems equations x = f (x, t ), a region of the state space is called acontraction region if the Jacobian ∂ f

∂ xis uniformly negative definite in that

region.

Theorem

Given the systems equation x = f (x, t ), any trajectory, which starts in a ball of

constant radius centered about a given trajectory and contained at all times in

a contraction region, remains in that ball and converges exponentially to this

trajectory.

Furthermore, global exponential convergence to the given trajectory is

guaranteed if the whole state space is a contraction region.



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Examples

Example

Consider the systemx = −x + e

t

and the Jacobian ∂ f ∂ x

= −1 which is globally negative definite.



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Examples

Example

Consider the systemx = −x + e

t

and the Jacobian ∂ f ∂ x

= −1 which is globally negative definite.

Example

Consider the systemx = −t (x

3 + x )

and the Jacobian ∂ f

∂ x

= −t (3x 2 + 1) which is globally negative definite fort ≥ t 0 ≥ 0.



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Generalization - Contraction Theory

Basic Idea

Instead of using standard differential length we can use a more generaldefinition of differential length



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Basic Idea


A line vector δ x can be expressed by using a differential coordinatetransformation

δ z = Θδ x



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Basic Idea



δ z = Θδ x

where Θ(x, t ) is a square matrix and uniformly positive definite.



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Basic Idea



δ z = Θδ x


a quadratic distance is then

δ zT δ z = δ x

T M δ x

Helmut Hauser Institute for Theoretical Computer ScienceIntroduction to Contraction Theory

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Basic Idea



δ z = Θδ x


a quadratic distance is then

δ zT δ z = δ x

T M δ x

with M(x, t ) = ΘT Θ representing a symmetric and continuouslydifferentiable metric.


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Generalization - cont.

Same steps as before:

Calculating the time derivative of δ z

d

dt

δ z = „Θ + Θ∂ f

∂ x«

Θ−1

δ z = Fδ z


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d

dt δ z = „Θ + Θ

∂ f

∂ x«Θ−1

δ z = Fδ z

with F =“

Θ + Θ ∂ f ∂ x

”Θ−1 is called the generalized Jacobian


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d

dt δ z = „Θ + Θ

∂ f

∂ x«Θ−1

δ z = Fδ z

with F =“

Θ + Θ ∂ f ∂ x

”Θ−1 is called the generalized Jacobian

the rate of change of the squared length

d

dt

(δ zT δ z) = 2δ zT

Fδ z


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Definition

Given the systems equations x = f (x, t ), a region of the state space is called acontraction region with respect to a uniformly positive definite metric

M(x, t ) = ΘT Θ if the generalized Jacobian F =“

Θ + Θ ∂ f ∂ x

”Θ−1 is uniformly

negative definite in that region.

TheoremGiven the systems equations x = f (x, t ), any trajectory, which starts in a ball of

constant radius with respect to the metric M(x, t ), centered at a given

trajectory and containend at all times in a contraction region with respect to

M(x, t ), remains in that ball and converges exponentially to this trajectory.

Furthermore, global exponential convergence to the given trajectory is

guaranteed if the whole state space is a contraction region with respect the metric M(x, t ).


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Example

Example

For a linear systemx = Ax

the coordinate transformation z = Θx (constant!) into a Jordan form.

F =

„Θ + Θ

∂ f

∂ x

«Θ−1 = (0 + ΘA) Θ

−1

and therefore ΘAΘ−1 has to be uniformly negative definite. This is true if andonly if the system is strictly stable.


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Example

Example

FitzHugh-Nagumo model (simplification of the Hodgkin-Huxley model):

v = c

„v + w −

1

3v

3 + I

«

w = −1

c

(v − a + bw )

with c ,a and b being some constants, I the input, v the membrane voltage andw the recovery variable. With

Θ =

»1 00 c

–

we get following generalized Jacobian:

F =

»c (1− v 2) 1−1 − b

c

–


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Additional Possibilities

Time varying systems can be analyzed


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Approach extensible to MIMO (Multiple Input Multiple Output) systems


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Switching networks can be analyzed too


Addi i l P ibili i

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Hybrid Systems (discrete and continuous mixture)


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Hybrid Systems (discrete and continuous mixture)System (plant & controller) can be designed to be contractive


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Observer can be designed to be contractive in conjunction with the plant


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The rate of convergence is bounded by λmax



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The rate of convergence is bounded by λmax

Contractive systems are robust, temporal disturbances vanish exponentially


Connecting Contractive Systems

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Interesting questions raises:Does the property of contraction hold for bigger systems built out of contractive systems? → the answer is yes!



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Possible Connections are:

One-way coupling



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One-way couplingTwo-way coupling



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Parallel



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Parallel

Hierarchies



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Interesting questions raises:

Does the property of contraction hold for bigger systems built out of contractive systems? → the answer is yes!



Parallel

Hierarchies

Feedback




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C g C Sy





Parallel

Hierarchies

Feedback

and others




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g y





Parallel

Hierarchies

Feedback

and others




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g y





Parallel

Hierarchies

Feedback

and others

Note: They can be combined and applied recursively!



One-Way Coupling

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We have two systems:

x1 = f 1(x1, t )

x2 = f 2(x1, t ) + u(x1)− u(x2)

f 1 and f 1 are the dynamics of uncoupled oscillators.

u(x1)− u(x2) is the coupling force.



One-Way Coupling

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x1 = f 1(x1, t )

x2 = f 2(x1, t ) + u(x1)− u(x2)



if f − u is contracting then x1 → x2 exponentially regardless of initial condition.This is interesting when the two systems are two (or more) oscillators → theysynchronize!



One-Way Coupling

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x1 = f 1(x1, t )

x2 = f 2(x1, t ) + u(x1)− u(x2)




Simple proof:The second subsystem, with u(x1) as input, is contracting, and x1(t ) = x2(t ) isa particular solution.



One-Way Coupling

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x1 = f 1(x1, t )

x2 = f 2(x1, t ) + u(x1)− u(x2)




Simple proof:The second subsystem, with u(x1) as input, is contracting, and x1(t ) = x2(t ) isa particular solution.

This can be used to extent networks with chain or tree structures.



Two-Way Coupling

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We have two systems coupled like:

x1 − h(x1, t ) = x2 − h(x2, t )

if h is contracting then x1 and x2 will converge exponentially regardless of initialcondition. Again interesting for two (or more) oscillators → they synchronize!



Two-Way Coupling

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x1 − h(x1, t ) = x2 − h(x2, t )


More Oscillator Couplings and Nonlinear Networks are possible:

It is possible to design the coupling to have contractive behavior and thereforesynchronization.

”Oscillator Death”



Two-Way Coupling

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x1 − h(x1, t ) = x2 − h(x2, t )





Networks with special symmetry



Two-Way Coupling

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x1 − h(x1, t ) = x2 − h(x2, t )






Shutoff of synchrony by just one inhibitory link (fast inhibition)



Two-Way Coupling

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x1 − h(x1, t ) = x2 − h(x2, t )







Leader Following (and even different leaders of arbitrary dynamics candefine different group primitives)



Two-Way Coupling

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x1 − h(x1, t ) = x2 − h(x2, t )







Leader Following (and even different leaders of arbitrary dynamics candefine different group primitives)

and many other case



Parallel Connection

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Two systems

x1 = f 1(x1, t )

x2 = f 2(x2, t )



Parallel Connection

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Two systems

x1 = f 1(x1, t )

x2 = f 2(x2, t )

and virtual dynamics

δ z1 = F1δ z

δ z2 = F2δ z



Parallel Connection

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Two systems

x1 = f 1(x1, t )

x2 = f 2(x2, t )

and virtual dynamics

δ z1 = F1δ z

δ z2 = F2δ z

So we have a linear combination d dt δ z =

Pi αi (t ) d

dt δ zi and combined system is

contractive again with αi > 0 and same metric.



Parallel Connection Example

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Example

Control Primitives with biological control inputs:

x = f (x, t ) +Xi

αi (t )φi (x, t )




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Example


x = f (x, t ) +Xi

αi (t )φi (x, t )

Dynamics f and primitives φi all contracting in the same Θ(x) and αi (t ) > 0then the whole system is contractive.




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Example


x = f (x, t ) +Xi

αi (t )φi (x, t )

Dynamics f and primitives φi all contracting in the same Θ(x) and αi (t ) > 0then the whole system is contractive.

Note: in general a time-varying combination of stable systems does not have tobe stable!



Hierarchical Combination

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Consider following virtual dynamics

d

dt

„δ z1

δ z2

«=

„F11 0

F21 F22

«„δ z1

δ z2

«

and assume the F21 is bounded and F11 and F22 are uniformly negative definite.



Hierarchical Combination

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Consider following virtual dynamics

d

dt

„δ z1

δ z2

«=

„F11 0

F21 F22

«„δ z1

δ z2

«

and assume the F21 is bounded and F11 and F22 are uniformly negative definite.

Simple Proof:

The first equation does not depend on the second one and is contractive.F21δ (z )2 represents an exponentially decaying disturbance for the secondequation. Thus the whole system converges to a single trajectory.



Examples Hierarchies

Example

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Example

Again Motion Primitives:

x = f (x, t ) +Xi

αi (t )φi (x, t )

the αi (t ) could be outputs of contracting systems of higher up. Again we canguarantee contraction.




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Example


x = f (x, t ) +Xi

αi (t )φi (x, t )


Example

Typical hierarchical processes are chemical chain reactions.

x = q (t )(xf − x) + Nr

withN

the reaction rate coefficients,x

= (c

1. . . c

n−

1T

) withc i the chemicalconcentrations and temperature T , xf the corresponding feed vector,q (t ) the

specific volume flow and r i the normalized reaction rates.




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Example


x = f (x, t ) +Xi

αi (t )φi (x, t )


Example

Typical hierarchical processes are chemical chain reactions.

x = q (t )(xf − x) + Nr

with N the reaction rate coefficients, x = (c 1. . . c n−

1T ) with c i the chemical

concentrations and temperature T , xf the corresponding feed vector,q (t ) thespecific volume flow and r i the normalized reaction rates.Following linear matrix inequalities have to be solved for M > 0

∀i , j NT ij M + MNij ≤ 0



Feedback Connection

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Two systems

x1 = f 1(x1, x2, t )

x2 = f 2(x1, x2, t )

in the feedback combination

d dt

„ δ z1

δ z2

«=„ F1 G−GT F2

«The augmented system is contracting if and only if the separated plants arecontracting and under the rather mild assumption:

F2 < GT

F−11 G



Feedback Connection

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Two systems

x1 = f 1(x1, x2, t )

x2 = f 2(x1, x2, t )

in the feedback combination

d dt

„ δ z1

δ z2

« = „ F1 G−GT F2

«The augmented system is contracting if and only if the separated plants arecontracting and under the rather mild assumption:

F2 < GT

F−11 G

Note: We can usually choose the connection matrix G!



Summary

Pros

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Pros

Contraction theory is applicable for a wide range of nonlinear systems



Summary

Pros

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Contraction theory can be used to design contractive systems



Summary

Pros

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Synchronization of complex networks can be assured by contraction theory



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Pros

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The property of contraction holds over a wide range of possiblecombinations (under some assumption) - Modularity!



Summary

Pros

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One can mix different classes of nonlinear systems and still assurecontraction



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Pros

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Time delays can be incorporated



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Pros

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Pros

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Cons

It is not always easy to find the right metric M.



Summary

Pros

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Cons

It is not always easy to find the right metric M.It is not trivial to prove negative definiteness of big matrices .



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Pros

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Cons

It is not always easy to find the right metric M.It is not trivial to prove negative definiteness of big matrices .

Lohmiller and Slotine do not show the problems - hard to see what elsecould be a problem.



For Further Reading

Winfried Lohmiller and Jean-Jacques E. SlotineO i l i f li

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On contraction analysis for non-linear systems .

Automatica Vol.34, p683-696, 1998.J. J. Slotine and W. LohmillerModularity, evolution, and the binding problem: a view from stability

theory .Neural Networks, Vol 14, p137-145, 2001

Wei Wang and Jean-Jacques E SlotineOn partial contraction analysis for coupled nonlinear oscillators .Biol Cyber, Vol 92, p38-53, 2005

Jean-Jacques E SlotineTalk about Contraction Theory hold at FIAS Summer School, Theoretical

Neuroscience & Complex Systems, Frankfurt, D, August 2007 .can be found in our pdf archive

Winfried Lohmiller and Jean-Jacques E. SlotineNonlinear Proces Control using Contraction Theory .AIChE Journal, Vol 46,Nr:3, p588-596, 2000



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Documents

Introduction to Contraction Theory.pdf