Indra Narayan Kar

Department of Electrical Engineering,Indian Institute of Technology Delhi,

New Delhi –110016E.mail: ink@ee.iitd.ac.in

Contraction Theory : A Tool for Design and Analysis of Nonlinear Systems

Introduction

Concepts, Definition

Applications to example problems

Design analysis of frequency estimator

Synchronisation of a network of nonlinear systems

Contraction Analysis/Theory or simply Contraction

( , ), 1,2,...... .= =x f x& i t i r

• Given a system or multiple systems• Questions: Does the solution or trajectory converges

• Analyze the convergence of nonlinear systems trajectories with respect to each other–Property regarding the convergence between two arbitrary system trajectories.– If initial conditions or disturbances are forgotten exponentially fast.

),(.

txfx =

where x � Rm ; f : (m×1) vector function.

),(.x

xxfx δδ∂

∂=

t

Contraction analysis/Theory or simply contraction

•First variation in x

•Nonlinear system

Virtual dynamics of two neighboring trajectories

xxxxxfxxxxx δδλδδδδδδ T

mTTT t

dtd ),(222)(

.≤

∂∂

==⇒

If is negative definite in nature, then all solution trajectories of system converge together exponentially

),( tm xλ

•Contraction RegionFor system ,a region of state space is called a contracting region if the

Jacobian is uniformly negative definite (UND) in that region.) , (

.tx xf=

xf

∂∂

•UND of a Jacobian means( , )t∂∂

f xx

0 - s.t. 0 , ,0 <≤∂∂

≥∀∀>∃ It ααxfx

0I- 21

<≤⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+∂∂

⇒ αT

xf

xf

•Time derivative of squared distance

Important featuresUtilizes differential framework. A type of

incremental stability methodNo issues like selection of energy functionEstablishes exponential stability of systemsHandles synchronization problems effectively

• Coordinate transformation

θ(x, t) : uniformly invertible matrix. Squared distance between trajectories is

where is a uniformly positive definite matrix.

zxfzzzzz δθθθδδδδδ 1)

.(2

.2)( −

∂∂

+==⇒ TTT

dd

xMxzz δδδδ TT =

=z xδ θδ

• Generalized Jacobian matrix F is defined as

1.

)( −

∂∂+= θθθxfF

If matrix F is U.N.D., exponential convergence of δZ to zero is guaranteed.

( )ttt T ,),(),( xxxM θθ=

• Time derivative of weighted square length

Generalisation of the convergence analysis

Results:For the system in (1), If there exists a uniformly positive definite matrix M(x, t) s.t. the associated generalized Jacobian matrix

1.

)( −

∂∂+= θθθxfF

is UND, then all system trajectories converge exponentially to a single trajectory with convergence rate where is the largest eigenvalue of the symmetric part of F. Then the system is said to be contracting.

),( tm xλ ),( tm xλ

•Intutively, if the temporal evolution of a virtual displacement tends to 0 as time goes to infinity, this being true for all state x and at all time, the whole flow will shrink to a point, hence the term contraction.

Results

Given system 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 the given trajectory. Further, global exponential convergence to this given trajectory is guaranteed if the whole state space region is contracting.

= f( , )x x t&

Contraction of two neighboring trajectories.

Relative tangential velocity General

trajectory

Relative radial velocity

Given trajectory

Contraction region

Contraction theory and Krasovaskii’s Stability Analysis

• For n-dimensional continuous system having dynamics as

• Transformation is δz = Θ(x(t), t), where Θ(x(t), t) : (n×n) invertible matrix.

• Contraction of differential system means exponential stability of actual system.• Select of Lyapunov function

• it yields contraction condition F ≤ αI with α < 0.

• If function f is autonomous & M is constant, then Krasovaskii’s sufficientcondition for asymptotic stability is equivalent to contraction of such systems

)),(()(.

ttt xfx =

)()),(()()),()( tttttttdtd xxJx(x

xfx δδδ =

∂∂

=⇒

)()),(()( ttttdtd zxFz δδ =

)()),(()()),(( tttttt T xxMxxV δδδ =

)),((2)),(( If.

tttt xVxV δαδ ≤

Contraction applied to stability analysis

1 1 1

2 1 2

11

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦ ⎣ ⎦

&

&

x x xx x x

1 1 1

2 1 2

1 1 0 Jacobian J= 0

1 0 1− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= <⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − −⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

&

&

y x yy x y

• The y-system is contracting and has two particular solutions, namely

1 1 1

2 2 2

0 and

0⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

y x yy x y

1 2 and will both tend to 0 exponentially ⇒ x x

[ ] [ ]

[ ] [ ]

1 2 1 2

1 2 1 2

1Lyapunov function: 2

0 Easy to conclude

=

= − < ⇒&

T

T

V x x x x

V x x x x

1 1

2

2 1

21

Difficult to conclud12

e1

⎡ ⎤ ⎡ ⎤ ⎡ ⎤= ⇒⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

−

⎣

+− ⎦−

&

&

x xx

x xx x

δ δδ δ

Actual system

Virtual system

Actual & Virtual System Concept

xpxx ),,(.

tρ−=where : state vector, p: parameter vector, ρ (x,p, t) ≥ α I > 0.1×∈ mRx

ypxy ),,(.

tρ−=

ypxy δρδ ),,(.

t−=

• Virtual System

To show exponential stability, define new system as

Let δy is virtual differential increments in y, then

As ρ (x,p, t) ≥ α I > 0, virtual system is contracting. Hence, actual system states also converge together, exponentially

(b)

•Define virtual system •Establish contracting nature of virtual system.•It leads to exponential convergence of virtual system states to actual system states.

• Actual System (a)

Simple Example

= −s sz z p

0

.

.

. ( )

s s s

s s ss

s s s s u

=− +=− −

=− − − +

x px pyy x z yz x y z R

Lorenz chaotic system is

Where . With coordinate change0 ,0 >+= puRR

0

0 01 0

0 1

s s

s s s

s s s R u

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= − − − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

x p p xy p x yz x z p

&

&

&

Taking simple controller system becomes)( 0Ru +−= p

01

0 1

s s

s s s

s s s

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= − − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

x p p xy p x yz x z

&

&

&

Selecting virtual system as

01

0 1s

s

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

−= − − −

−

x p p xy p x yz x z

&&&

01

0 1s

s

δ δδ δδ δ

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= − − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

x p p xy p x yz x z

&

&

&

Dynamics in differential framework will be

The symmetric part of Jacobian matrix comes out to be

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=

10001000p

J s

It is UND, so system is exponentially stabilized to T) 0 0( p

01

0 1

s s

s s s

s s s

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= − − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

x p p xy p x yz x z

&

&

&

Actual system

s

s

f f

∫ ∫g yx +- +

+

System-I System-II

Contraction Theory for Observers

Contraction Theory for ObserversContraction theory based analysis is used for:

•Design of observers/estimators.

• Actual System s sx x u= +&

• Observer for the Systemsˆ ˆ ˆx x u L(x x)= + + −&

u =control input, L =gain selected s.t. L > 1.

•Define virtual system as

•Establish contracting nature of virtual system.•It leads to exponential convergence of observer states to actual system states.

sL(x )x x u x= + −+&

Numerical Example

1 2

2 1 22 3x xx x x u== − − +

&

&

Actual system Observer

1 2 1 1

2 1 2

( )2 3

y y L x yy y y u

= + −= − − +

&

&

• Virtual Systemuzzz

zxLzz

+−−=

−+=

212

1121

32

)(.

.

1

2 3

LJ

−=

− −

⎡ ⎤⎢ ⎥⎣ ⎦

• Jacobian of virtual system

⎥⎦

⎤⎢⎣

⎡−−−−

=+=35.05.0

)(21 L

JJJ Ts

• Symmetric part of Jacobian :

• Js to be UND, if L > 1/12

Figure: (a)-(b) Convergence of actual and observer system states and (c) Estimation error variation.

0 5 10 15 200

0.1

0.2

0.3

0.4

Estim

atio

n of

1st

sta

te

Time (t)

(a)

0 5 10 15 20-0.2

-0.1

0

0.1

0.2

Estim

atio

n of

2nd

sta

te

Time (t)

(b)

0 5 10 15 20-0.4

-0.2

0

0.2

0.4

Est

imat

ion

erro

r

Time (t)

(c)

x2

y2z2

x1

y1z1

[ ] [ ]

[ ]

(0) 0.1 0.2 , (0) 0.4 0.2

(0) 0 0.1

T Ty

T

= = −

=

x

z

0.10.5 0.1 sin( ) & L= 1/6tu e t−= +

A measurable sinusoidal signal involving multiple frequencies

: Non-zero amplitudes; : phases and : Non-zero frequencies s.t.

• Signals are mostly nonlinear functions of frequency.• Dynamic estimator is proposed using contraction theory.• Provides simultaneous online estimation of frequencies.• Designing asymptotic frequency estimators.

jiji ≠∀≠αα

∑=

+++=n

iiiiiii tAtAt

12211 )]cos()sin([)( θαθαy

ii 21 ,θθ iαii AA 21 ,

Frequency Estimation in Signal Theory

–Design Steps for Estimator:

– Select a dynamical system to generate given signal.

– Design a auxilary system by evolving unknown frequencies as new states.Design a suitable estimator for new system considering the measurement ofsignal.

– Apply contraction theory to select the gains of estimator so as to achieveasymptotic stability.

– Compute the frequency estimates from the steady state value of theadditional states

Frequency Estimator Design

Sinusoid with two Unknown Frequencies

Amplitudes and phases are known. :Unknown frequencies.

A dynamical system generating given sinusoid

Output of dynamical system

2

1 1 2 21

sin cosi i i i i ii

y(t) [A (α t θ ) A (α t θ )]=

= + + +∑

ii AA 21 , ii 21 ,θθ 21 &αα

1 2

2 3

3 4

4 1 1 2 2c c

ζ ζ

ζ ζ

ζ ζ

ζ ζ ζ

=

=

=

=− −

&

&

&

&

44332211 ζζζζ kkkky +++=

Parameters are the coefficients of the

characteristic polynomial

where1

22

4)( cscssq ++=ic

∑∏==

==2

1

22

2

1`

21 ;

ii

ii cc αα

Question: Estimate 1 2& c c

Frequency Estimator Design

Output of estimator

1 2

2 3

3 4

4 5 1 6 3

5

6

0

0

ζ ζ

ζ ζ

ζ ζ

ζ ζ ζ ζ ζ

ζ

ζ

=

=

=

= − −

=

=

&

&

&

&

&

&

1 1 2 2 3 3 4 4= + + +y k x k x k x k x

1 2

2 3

3 4

4 5 1 6 3

5 1

6 3

ˆ( )ˆ( )

ˆ( )ˆ( )

=== + −

= − − + −= − −= − −

&

&

&

&

&

&

x xx xx x y yx x x x x y yx x y yx x y y

1

2

3

4

g

g

g

g

Proposed estimator structureDynamical system

B.B. Sharma and I. N. Kar, “Design of Asymptotically Convergent Frequency Estimator Using Contraction Theory”, IEEE Tran. on Automatic Control, pp. 1932-1937, 2008.

Simulations

Two frequency sinusoid

Amplitude ; unknown frequencies

• Initial conditions • Gains

• Select for s.t. polynomial isstable.• Select s.t. and

1 2[sin( ) sin( )]y A t tα α= +

10=A 1 21 & 2 α α= =

.]0015.05.01[)0( T=x.2,2 ==

21gg

ik 3,2,1=i 322

13)( kskskssp

111ggg +++=

4k 1,1,1 ===144434 ggg kkk 0>

24gk

Simulations

-50

510

-5

0

5-4

-2

0

2

4

x1 state

(a)

x2 state

x 3 sta

te

0 10 20 30 40-10

-5

0

5

10

t (time in seconds)

x 4 sta

te

(b)

0 10 20 30 40-2

0

2

4

6

8

10

t (time in seconds)

x5 a

nd x

6 sta

tes

(c)

0 10 20 30 40-8

-6

-4

-2

0

2

4

t (time in seconds)

sign

al e

stim

atio

n er

ror

(d)

x5x6

Figure: Two frequencies estimation case: (a) Phase portrait in three dimensions, (b)Variation of state x4, (c) Variation of states x5 & x6 and (d) Signal estimation error.

Synchronization of Networks

• A network of N identical nonlinear systems ( may be oscillators)• Copupling can be either linear or nonlinear function of all or some of the

system states• Potential application running from coordination problems in robotics to

mechanisms for secure communications. Important in the area of biology, robotics, complex systems in nature and technology etc.

• Synchronization in the context of networks means states of each and every node should match to each other.

• The network is said to be synchronised if all the systems converge toward the same synchronous state.

• Find conditions to guarantee the synchronisation of a network of identical systems

Different Configurations

s

s

f f

∫ ∫g yx +- +

+

System-I System-II

• One way coupling : signal flow is permitted in one direction only.• Master- Slave synchronisation• Transmitter – receiver synchronisation (secure communication)

Different Configurations

(a) (b)

4S

3S

2S

1S NS

5S

6S

1NS −

1S

2S

3S

4S5S

6S

1NS −

NS

• One way coupling : signal flow is permitted in one direction only.• Two way (Bidirectional coupling means signal flow is permitted in both the directions)• Chain network have systems (one way or two way) connected in cascade.• Ring network have systems (one way or two way) connected in cascade and forming a

closed loop.• Networks with all-all topology• Networks with arbitrary topology

Partial Contraction

For a nonlinear system with dynamics

let an auxiliary system (Observer-like system) is defined as

If this auxiliary system is contracting w.r.t. y, and verifies a smooth specific property, then all trajectories of the original x-system verify this property, exponentially. The original system is said to be partially contracting

),,(.

txxfx =

),,(.

txyfy =

•Specific Properties can be matching of states, convergence to equilibrium points,convergence to manifolds etc.

Example

( )( )

( ) ( )( )

7 4 21 2 1 1 2

3 5 4 22 1

4 2 10 6 4 21 2 1 2 1

2 1 2

22 10 4 12

2 10

3 2 10

2 10

= − + −

= − − + −

⇒ + − = − + + −

&

&

x x x x x

x x x x

d x x x x x xdt

x

( )10 61 2y-system: 4 12= − +&y x x y

4 21 2 The -system has two particular solutions 2 10 and 0• = + − =y y x x y

1 24 21 2

The system stays at the origin if (0) (0) 0. Otherwise, it tends to reach

( ) 2 ( ) 10.

⇒ = =

+ =

x x

x t x t

1 2 The y-system is contracting as long as not both ( ) and ( ) equal to 0.• x t x t

Construction of auxiliary / virtual system

• Depends on state variables of nonlinear system and some virtual state variables• The substitution of the i-th system state variables into the virtual state variables returns the dynamics of the i-th system of the network• Contraction property with respect to these virtual state variables immediately implies synchronisation

( ) ( ) ( )( ) ( ) ( )

= + −= + −

&

&

z f z h w h zw f w h z h w

•Two coupled nonlinear systems

•Virtual systems ( ) 2 ( ) ( ) ( ) ,( , )+= =+−& h zx f x h x xh w z wφ

• Trajectories of the nodes are particular solutions (x-solutions) of the virtual system.

• If the virtual system is contracting with respect to the x state variable, the two particular solutions , namely z and w, will converge to each other. Synchronisation is then attained

( , , ) ( ) ( ) ( ), ( , , ) ( ) ( ) ( )= + − = + −z z w f z h w h z w z w f w h z h wφ φ

( ) ( )2∂ ∂= −

∂ ∂f x h xJ

x x•Jacobian of virtual system

Important Features

•New method to study dynamic behavior of coupled nonlinear oscillators/systems.•Extends contraction analysis to include the convergence to specific properties.•Specific Properties can be matching of states, convergence to equilibrium points,convergence to manifolds etc.•Provides general stability analysis framework for large complex systems.•Powerful tool to study synchronization behavior.•Synchronization results are global in nature.

• Master System ( , )=&x f x t

• Slave System= coupling matrix of dimensionC

selected be gain toscalar =k

( , ) ( )

Virtual System

= + −&y f y t kC x y

( , ) ( )z f z t z kC x z= + −&

• Selection of suitable gain k ensures UND nature of Js.• As virtual system in (44) is particular solution of (45), so UND nature of (45)ensures

exponential convergence of system in (45).• As both master and slave are particular solutions of the virtual system (44), their

corresponding states will converge to each other, exponentially.

Two systems in one way coupled form

Two Systems with Bidirectional Coupling

Dynamics of two bidirectional coupled systems

1 1 2 1

2 2 1 2

x f (x ,t) kC[(x x )]x f (x ,t) kC[(x x )]&

&

= + −

= + −1 2

nx , x ;C : (n n) coupling matrixk : coupling gain to be designed

∈ℜ ×}

1 22virtual system

f ( ,t) kC kC(x x )&ϕ = ϕ − ϕ + +

Network with all-to-all coupling topology

Dynamics of network with all-to-all coupling

1 1 2 1 3 1 1

2 2 1 2 3 2 2

1 2 1

= + + + += + + + +

= + + + +

n

n

n n n n nn-

x f (x ,t) kC[(x - x ) (x - x ) ... (x - x )]x f (x ,t) kC[(x - x ) (x - x ) ... (x - x )]

x f (x ,t) kC[(x - x ) (x - x ) ... (x - x )]

&

&

M

&

; ::∈ℜ

×

nix k common coupling gain ;

C (n n) coupling matrix.

Virtual System

1 2( , ) ( ... )= − + + + + nf t knC kC x x x&φ φ φ

Representative example: Lorenz chaotic system

1 1 2

2 1 1 3 2

1 1 2 3

Chaotic behaviour for 30, 10 & 8/3.

x sx sxx rx x x xx x x bx

r s b

= − += − −

= −

= = =

&

&

&

• Five such systems are interconnected through single variable

• Initial conditions of are different

• At t=2 sec, coupling gains were switched

0 2 4 6 8-40

-20

0

20

t (time in seconds)

1st s

tate

var

iatio

n(a)

x1x4x7x10x13

0 2 4 6 8-40

-20

0

20

40

t (time in seconds)

2nd

stat

e va

riatio

n

(b)

x2x5x8x11x14

0 2 4 6 8-50

0

50

100

t (time in seconds)

3rd

stat

e va

riatio

n

(c)

x3x6x9x12x15

controller switchedon at t= 2 sec.

Figure : Synchronization of five chaotic Lorenz systems with with all to all bidirectional coupling through single variable only (c11 = 1; c22 = 0; c33 = 0): (a), (b) & (c) Variation of state trajectories of different system.

0 2 4 6 8-40

-20

0

20

t (time in seconds)

1st s

tate

var

iatio

n(a)

x1x4x7x10x13

0 2 4 6 8-40

-20

0

20

40

t (time in seconds)

2nd

stat

e va

riatio

n

(b)

x2x5x8x11x14

0 2 4 6 8-20

0

20

40

60

t (time in seconds)

3rd

stat

e va

riatio

n

(c)

x3x6x9x12x15controller switched

on at t= 2 sec.

Figure: Synchronization of five chaotic Lorenz systems with nearest neighbour bidirectional coupling through single variable only (c11 = 1; c22 = 0; c33 = 0): (a), (b) & (c) Variation of state trajectories of different system.

Synchronization of Networks

Key Observations

• Synchronization of networks can be achieved quite easily using virtual system concept.

• Stability results are global in nature as compared to the traditional MSF based stability.

• Provides a common platform to address synchronization of networks in different configurations.

Applications• Observers/Estimators in under actuated surface vessels, robotic,applications

• Cooperative control and target tracking in multi-agent systems.• Analysis of networked systems with group leaders • Group cooperation analysis in animal aggregation applications like birdflocks, fish schools etc.

• Synchronization in complex networks like pacemaker cells in heart, neural networks in brain, synchronized chirping etc.

References:[1] A. Pavlov, A. Pogromvsky, N. Van de Wouv and H. Nijmeijer, “ Convergent dynamics, a tribute to Boris Pavlovich Demidovich”, Systems control Letters, 2004.

[2] Lohmiller, W., and Slotine, J.J.E. “On Contraction Analysis for Nonlinear Systems”, Automatica,1998[3] Lohmiller, W., and Slotine, J.J.E. “ Control system design for mechanical systems using Contraction Theory”, IEEE Trans. Automatic Control, 2000. [4] Wang W. and Slotine J. J. E., “On partial contraction analysis for coupled nonlinear oscillators,” Biol. Cybern., vol. 92, no. 1, 2004.[5] Rifai K and Slotine J. J. E., “Compositional contraction analysis of resetting hybrid systems” , IEEE Trans. Automatic Control, 2006.

[6] Jouffroy J., Some ancestors of contraction analysis, Proc. Of IEEE Conference on Decision and Control, pp.5450-5455, Seville, Spain, 2005. [7] Jouffroy J. and Slotine J.J. E., Methodological remarks on contraction theory. IEEE Conference on Decision and Control Atlantis, Paradise Island, Bahamas, pp.2537-2543, 2004.

[8] Pecora L.M. and Carroll T.L., Synchronization in chaotic systems. Phys Rev Lett, vol. 64, pp.821-24, 1990.[9] Russo G. and M. Bernardo, Contraction theory and master stability function: Linking two approaches to study synchronisation of complex network, IEEE Trans on circuits and systems, 2009.

References:

[10] B.B. Sharma and I. N. Kar, “Design of Asymptotically Convergent Frequency Estimator Using Contraction Theory”, IEEE Tran. on Automatic Control, 2008.[11] B.B. Sharma and I. N. Kar, “Contraction theory based recursive controller design for a class of nonlinear systems”, IET Control theory & applications, (In press), 2009. [12] B.B. Sharma and I. N. Kar, “Contraction based adaptive control of a class of nonlinear systems ”, Proc. of American control control conference, 2009.[13] B.B. Sharma and I. N. Kar, “Synchronisation of a class of chaotic systems in different configurations” (under review)

[14] Pham Q.C, Tabareau N. and Slotine J. J. E., “ A contraction theory approach to stochaisticincremental stability” IEEE Trans. Automatic Control, 2009. [15] B. Girard, Tabareau N., Pham Q.C, and Slotine J. J. E., “ Where neuroscience and dynamic system theory meet autonomous robotics: A contracting basal ganglia model for action selection”, Neural Networks, 2008.

Thank you