114731168-Contraction-Analysis.pdf

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Introduction

Nonlinear system analysis has been very successfully applied to particular classes of

systems and problems, but it still lacks generality, as e.g. in the case of feedback linearization,

or explicitness, as e.g. in the case of Lyapunov theory. A body of new results is derived using

elementary tools from continuum mechanics and differential geometry, leading to what we

shall call contraction analysis.

Intuitively, contraction analysis is based on a slightly different view of what stability is,

inspired by fluid mechanics. Regardless of the exact technical form in which it is defined,

stability is generally viewed relative to some nominal motion or equilibrium point.

Basically, a nonlinear dynamic system is called contracting if initial conditions or

temporary disturbances are forgotten exponentially fast, so that all trajectories converge to a

unique trajectory. After a brief review of key results of contraction theory, we introduce the new concept of partial contraction, which extends contraction analysis to include convergence

to behaviors or to specific properties (such as equality of state components, or convergence to

a manifold). Partial contraction provides a very general analysis tool to investigate the stability

of large‐scale systems. It is especially powerful to study synchronization behaviors, and it

inherits from contraction the feature that convergence and long‐term behavior can be analyzed

separately, leading to significant conceptual simplifications.

Contraction theory is a recent tool for analyzing the convergence behavior of nonlinear

systems in state‐space form. One of the main features of contraction is that, contrary to

traditional Lyapunov‐based analysis, it does not require the explicit knowledge of a specific

attractor. Stability analysis is performed through extensive use of virtual displacements, with

the system dynamics being described in a differential framework.

Methodology, i:e, how to apply the results of a theory, is well‐established for Lyapunov

functions. Rather than just importing these techniques to the world of contraction, a more

fruitful approach is to take into account the specificities of contraction theory and study their

application to concrete examples. Our purpose is to contribute to this important issue, as well

as suggest means to compare contraction with Lyapunov stability analysis.



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1.

Contraction Analysis

Consider systems described by general nonlinear deterministic differential equations of

the form

= f(x, t)

where x is the n‐dimensional vector corresponding the state of the system, t is time, and f is a

nonlinear vector field. The above equation may also represent the closed‐loop dynamics of a

controlled system with state feedback u(x, t) resulting in a smooth nonlinear system of the form

(t) = f (x(t), u(x, t), t)

where x(t) , and f : × × → .

All quantities are assumed to be real and smooth, by which is meant that any required

derivative or partial derivative exists and is continuous.

1.1 A basic convergence result

Considering the local flow at a given point x leads to a convergence analysis between

two neighboring trajectories. If all neighboring trajectories converge to each other (contraction

behavior) global exponential convergence to a single trajectory can then be concluded.

The plant equation = f(x, t) can be thought of as an n‐dimensional fluid flow, where is the n‐dimensional “velocity” vector at the n‐dimensional position x and time t. Assuming as

we do that f(x, t) is continuously differentiable, = f(x, t) yields the exact differential relation

δ = (x, t) δx

where δx is a virtual displacement (a virtual displacement is an infinitesimal displacement at

fixed time). Note that virtual displacements, pervasive in physics and in the calculus of

variations, and extensively used, are also well‐defined mathematical objects. Formally, δx

defines a linear tangent differential form, and δδx the associated quadratic tangent form,

both of which are differentiable with respect to time.

Consider now two neighboring trajectories in the flow field = f(x, t), and the virtual

displacement δx between them as shown in the figure. The squared distance between these

two trajectories can be defined as δδx, leading from δ to the rate of change

(δδx) = 2 δ δ = 2 δ δx



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

(i.e., the largest eigenvalue of (

+

)), we thus have

(δδx) ≤ 2 δδx

and hence,

|| δx || ≤ || δ

||

,

Assume now that (x, t) is uniformly strictly negative (i.e., β > 0, x, t ≥ 0,

(x, t) ≤ −β < 0 ). Then, from (3) any infinitesimal length δx converges exponentially to zero. By

path integration, this immediately implies that the length of any finite path converges

exponentially to zero. This motivates the following definition.

Definition 1: Given the system equations = f(x, t), a region of the state space is called a

contraction region if the Jacobian is uniformly negative definite in that region.

By

uniformly negative definite we mean that

β > 0, x, t ≥ 0, (

+

) ≤ −β I < 0

More generally, by convention all matrix inequalities will refer to the symmetric parts of

the square matrices involved − for instance, we shall write the above as ≤ −β I < 0. By a



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region we mean an open connected set. Extending the above definition, a semi‐contraction

region corresponds to being negative semi‐definite, and an indifferent region to

being

skew‐symmetric.

Consider now a ball of constant radius centered about a given trajectory, such that given this trajectory the ball remains within a contraction region at all times (i.e., t ≥ 0). Because any

length within the ball decreases exponentially, any trajectory starting in the ball remains in the

ball (since by definition the center of the ball is a particular system trajectory) and converges

exponentially to the given trajectory (Figure 2). Thus, as in stable linear time‐invariant (LTI)

systems, the initial conditions are exponentially “forgotten.” This leads to the following

theorem:

Theorem 1: Given the system equations = 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.

This sufficient exponential convergence result may be viewed as a strengthened version

of Krasovskii’s classical theorem on global asymptotic convergence. Note that its proof is very

straight‐forward, even in the non‐autonomous case, and even in the non‐global case, where it

guarantees explicit regions of convergence. Also, note that the ball in the above theorem may

not be replaced by an arbitrary convex region – while radial distances would still decrease,

tangential velocities could let trajectories escape the region.



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Example 1: In the system

= −x +

the Jacobian is uniformly negative definite and exponential convergence to a single trajectory is

guaranteed. This result is of course obvious from linear control theory.

Example 2: Consider the system

= −t ( + x)

For t ≥ to > 0, the Jacobian is again uniformly negative definite and exponential convergence to

the unique equilibrium point x = 0 is guaranteed.

1.2 Generalization of the convergence analysis

Theorem 1 can be vastly extended simply by using a more general definition of differential length. The result may be viewed as a generalization of linear eigenvalue analysis

and of the Lyapunov matrix equation. Furthermore, it leads to a necessary and sufficient

characterization of exponential convergence.

1.2.1 General definition of length

The line vector δx between two neighboring trajectories in Figure 1 can also be

expressed using the differential coordinate transformation

δz = Θδx

where Θ(x, t) is a square matrix. This leads to a generalization of our earlier definition of

squared length

δzδz = δxM δx

where M(x, t) = ΘΘ represents a symmetric and continuously differentiable metric − formally, δzδz defines a Riemann space. Since δz is in general not integrable, we cannot expect to find

explicit new coordinates z(x, t), but δz and δzδz can always be defined, which is all we need.

We shall assume M to be uniformly positive definite, so that exponential convergence of δz to 0

also implies exponential convergence of δx to 0.

Distance between two points and with respect to the metric M is defined as the

shortest path length (i.e., the smallest path integral ||δz|| ) between these two points.

Accordingly, a ball of center c and radius R is defined as the set of all points whose distance to c

with respect to M is strictly less than R. The two equivalent definitions of length in δzδz lead



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to two formulations of the rate of change of length: using local coordinates δz leads to a

generalization of linear eigenvalue analysis, while using the original system coordinates x leads

to a generalized Lyapunov equation.

1.2.2 Generalized eigenvalue analysis

The time derivative of δz = Θδx can be computed as

δz = Θ δx + Θδx = Θ δx + Θ δx = (Θ + Θ

)δx = (Θ + Θ) Θ δz = Fδz

Formally, the generalized Jacobian

F = (Θ + Θ) Θ

represents the covariant derivative of f in δz coordinates. The rate of change of squared length

can be written (δzδz) = δz.δz + δz. δz = 2δz δz = 2δzδz

Similarly to the reasoning in Theorem 1, exponential convergence of δz (and thus of δx)

to 0 can be determined in regions with uniformly negative definite F. This result may be

regarded as an extension of eigenvalue analysis in LTI systems, as the next example illustrates.

Example 3: Consider first the LTI system

= Ax

and the coordinate transformation z = Θx (where Θ is constant) into a Jordan form

= ΘAΘ z = Λ z

For instance, one may have

Λ =

0 0 0 00 0 0 0

0 0

0 00 0 0 0 0 0

where the ’s are the eigenvalues of the system, and ρ < 2 is the normalization factor of the

Jordan form. The covariant derivative F = Λ is uniformly negative definite if and only if the

system is strictly stable, a result which obviously extends to the general n‐dimensional case.



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Now, consider instead a gain‐scheduled system. Let A(x, t) = be the Jacobian of the

corresponding nonlinear, non‐autonomous closed‐loop system = f (x, t), and define at each

point a coordinate transformation Θ(x, t) as above. Uniform negative definiteness of F = Λ +

Θ Θ (a condition on the “logarithmic” derivative of Θ) then implies exponential convergence

of this design.

This result also allows one to compute an explicit region of exponential convergence for

a controller design based on linearization about an equilibrium point, by using the

corresponding constant Θ.

Note that the above could not have been derived simply by using Krasovskii’s

generalized asymptotic global convergence theorem, even in the global asymptotic case and

even using a state transformation, since an explicit z does not exist in general.

1.2.3 Metric

analysis

δz can equivalently be written in δx coordinates

Θ δz = M δ + ΘΘ δx = (M

+ ΘΘ )δx

using the covariant velocity differential Mδ + ΘΘ δx. The rate of change of length is

(

δxM δx) =

δx (

+

M + M

) δx

so that exponential convergence to a single trajectory can be concluded in regions of ( + M + M

) ≤ −M (where is a strictly positive constant). It is immediate to verify that

these are of course exactly the regions of uniformly negative definite F. If we restrict the metric

M to be constant, this exponential convergence result represents a generalization and

strengthening of Krasovskii’s generalized asymptotic global convergence theorem. It may also

be regarded as an extension of the Lyapunov matrix equation in LTI systems.

1.2.4 Generalized contraction analysis

The above leads to the following generalized definition, superseding Definition 1 (which

corresponds to Θ = I and M = I).

Definition 2: Given the system equations = f(x, t), a region of the state space is called a

contraction region with respect to a uniformly positive definite metric M(x, t) = ΘΘ, if



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equivalently F is uniformly negative definite +

M + M ≤ −M (with constant > 0)

in that region.

As earlier, regions where F or equivalently

+

M + M

are negative semi‐definite

(skew‐symmetric) are called semi‐contracting (indifferent). The generalized convergence result

can be stated as:

Theorem 2: Given the system equations = 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 contained

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 to the metric M(x, t).

We always assume this generalized form when we discuss contraction behavior.

1.3 A converse theorem

Conversely, consider now an exponentially convergent system, which implies that β >

0, k ≥ 1, such that along any system trajectory x(t) and t ≥ 0,

δxδx ≤ k δδ

Defining a metric M(x(t), t) by the ordinary differential equation (Lyapunov equation)

= ‐

M ‐ M – βM M(t = 0) = kI

and using (δxM δx), we can write δxδx as

δxδx ≤ δxMδx = k δδ

Since this holds for any δx, the above shows that M is uniformly positive definite, M ≥ I.

Thus, any exponentially convergent system is contracting with respect to a suitable metric.

Note from the linearity of that M is always bounded for bounded t. Furthermore,

while M may become unbounded as t → +∞, this does not create a technical difficulty, since

the boundedness of δxMδx still implies that δx tends to zero exponentially and also indicates

that the metric could be renormalized by a further coordinate transformation.



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Thus, Theorem 2 actually corresponds to a necessary and sufficient condition for

exponential convergence of a system. In this sense it generalizes and simplifies a number of

previous results in dynamic systems theory.

For instance, note that chaos theory leads at best to sufficient stability results. Lyapunov

exponents, which are computed as numerical integrals of the eigenvalues of the symmetric part

of the Jacobian , depend on the chosen coordinates x and hence do not represent intrinsic

properties.

1.4 A Note On Krasovskii’s Theorem

It should be clear to the reader familiar with the many versions of Krasovskii’s theorem

that by now we have ventured quite far from this classical result. Indeed, Krasovskii’s theorem

provides a sufficient, asymptotic convergence result, corresponding to a constant metric M.

Also, it does not exploit the possibility of a pure differential coordinate change as in δz. It is also interesting to notice that the type of proof used here is very significantly simpler than that

used, say, for the global non‐autonomous version of Krasovskii’s theorem. This in turn allows

many further extensions, as the next sections demonstrate.

1.5 Linear properties of generalized contraction analysis

Introductions to nonlinear control generally start with the warning that the behavior of

general nonlinear non‐autonomous systems is fundamentally different from that of linear

systems. While this is unquestionably the case, contraction analysis extends a number of

desirable properties of linear system analysis to general nonlinear non‐autonomous systems.

I. Solutions in δz(t) can be superimposed, since δz = F(x, t)δz around a specific trajectory

x(t) represents a linear time‐varying (LTV) system in local δz coordinates. Note that the

system needs not be contracting for this result to hold.

II. Using this point of view, Theorem 2 can also be applied to other norms, such as ||||

= || and |||| = ∑||, with associated balls defined accordingly. Using the

same reasoning as in standard matrix measure results, the corresponding

convergenceresults are

|||| ≤ ( + ∑ || ) |||| |||| ≤ ( + ∑ || ) ||||

III. Global contraction precludes finite escape, under the very mild assumption x* , c ≥ 0, t ≥ 0, ||Θf(x*, t)|| ≤ c

Indeed, no trajectory can diverge faster from x* than bounded Θf(x*, t) and thus cannot

become unbounded in finite time. The result can be extended to the case where x* may

itself depend on time, as long as it remains in an a priori bounded region.



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IV. A convex contraction region contains at most one equilibrium point, since any length

between two trajectories is shrinking exponentially in that region.

V. This further implies that, in a globally contracting autonomous system, all trajectories

converge exponentially to a unique equilibrium point. Indeed, using V(x) = M(x, t)

f(x) as a Lyapunov‐like function (an extension of the standard proof of Krasovskii’s

Theorem for autonomous systems) yields = ( + M + M ) f(x) ≤ −V

which shows that = f(x) tends to 0 exponentially, and thus that x tends towards a finite

equilibrium point.

VI. The output of any time‐invariant contracting system driven by a periodic input tends

exponentially to a periodic signal with the same period. Indeed, consider a time‐

invariant nonlinear system driven by a periodic input ω(t) of period T > 0, = f(x, ω(t))

Let (t) be the system trajectory corresponding to the initial condition = , and let

(t) be the system trajectory corresponding to the system being initialized instead at

(T) = . Since f is time‐invariant and ω(t) has period T , (t) is simply as shifted version of (t), t ≥ T, (t) = (t − T)

Furthermore, if we now assume that the dynamics is contracting, then initial

conditions are exponentially forgotten, and thus (t) tends to (t) exponentially.

Therefore, from (t), (t − T) tends towards (t) exponentially. By recursion, this

implies that t, 0 ≤ t < T , the sequence (t + nT) is a Cauchy sequence, and therefore

the limiting function lim t nT exists, which completes the proof.

VII. Consider the distance R = ||δz|| between two trajectories and , contained at all

times in a contraction region characterized by maximal eigenvalues

λ(x, t) ≤ −β < 0 of

F. The relative velocity between these trajectories verifies + | λ| R ≤ 0

Assume now, instead, that represents a desired system trajectory and the actual

system trajectory in a disturbed flow field = f(x, t) + d(x, t). Then + | λ| R ≤ || Θd ||

For bounded disturbance || Θd || any trajectory remains in a boundary ball around the

desired trajectory. Since initial conditions R(t = 0) are exponentially forgotten, we can

also state that any trajectory converges exponentially to a ball of radius R with arbitrary

initial condition R(t = 0).

VIII. The above can be used to describe a contracting dynamics at multiple resolutions using

multiscale approximation of the dynamics with bounded basis functions, as e.g. in wavelet analysis. The radius R with respect to the metric M of the boundary ball to

which all trajectories converge exponentially becomes smaller as resolution is increased,

making precise the usual “coarse grain” to “fine grain” terminology.

1.6 The key advantages of contraction analysis



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• Contraction theory is a differential‐based analysis. Therefore, it is much easier to

combine multiple systems.

• Contraction theory is more generalized. For instance, all examples and theorems in

[158] are for autonomous systems whereas contraction theory can be easily applied to

general time‐varying systems including systems with time‐varying desired trajectories.

•

Convergence of contraction theory is stronger: exponential and global. Asymptotic convergence of a PD controller is not enough for tracking demanding time‐varying

reference inputs of nonlinear systems.

• The proof is intuitive and much simpler. This truly sets contraction analysis aside from

other methodologies. For example, one can recall the proof of global and time‐varying

version of Krasovskii’s theorem.

• Contraction theory also permits a non‐constant metric and a pure differential

coordinate change, which is not explicit in [158]. This generalization of the metric

(generalized Jacobian, not just ) is one of the differences with other methodologies.



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2.

Partial Contraction

Theorem 3: Consider a nonlinear system of the form

x = f (x, x, t)

and assume that the auxiliary system

= f (y, x, t)

is contracting with respect to y. If a particular solution of the auxiliary y‐system 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.

Proof: The virtual, observer‐like y‐system has two particular solutions, namely y(t) = x(t), t ≥ 0

and the solution with the specific property. Since all trajectories of the y‐system converge

exponentially to a single trajectory, this implies that x(t) verifies the specific property

exponentially.

Example 4: Consider a system of the form

= c(x, t) + d(x, t)

where function c is contracting in a constant metric1 . The auxiliary contracting system may

then be constructed as

= c(y, t) + d(x, t)

The specific property may consist e.g. a relationship between state variables, or simply of a

particular trajectory.

Example 5: Consider a convex combination or interpolation between contracting dynamics

= ∑ x, t x, t

where the individual systems = x, t) are contracting in a common metric M(x) = Θx Θ(x)

and have a common trajectory

(t) (for instance an equilibrium), with all

x,t ≥ 0 and

∑ x,t = 1. Then all trajectories of the system globally exponentially converge to the trajectory (t). Indeed, the auxiliary system

= ∑ x, t y, t

is contracting (with metric M(y) ) and has x(t) and (t) as particular solutions.

Note that contraction may be trivially regarded as a particular case of partial contraction.



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3.

Combinations of Contracting Systems

Combinations of contracting systems satisfy simple closure properties, subsets of which

are reminiscent of the passivity formalism.

3.1 Parallel

combination

Consider two systems of the same dimension

= ( , t)

= ( , t)

with virtual dynamics

δ

=

δz

δ = δz

and connect them in a parallel combination. If both systems are contracting in the the same

metric, so is any uniformly positive superposition

(t) δ + (t) δ where α > 0, t ≥ 0, (t) ≥ α

Theorem 4: Consider two time‐varying nonlinear systems, contracting in the same metric

function which does not explicitly depend on time such that M(x) =

Θx Θ(x) and

Θ =

x :

x = (x, t) i = 1, 2

Then, any positive superposition with α > 0

x = α (x, t) + α (x, t)

is contracting in the same metric. By recursion, this parallel combination can be extended to

any number of systems.

Proof: Since the original dynamics are contracting with a time‐independent metric, the

generalized Jacobian, F becomes

F = (Θ (x) + Θ(x) ) Θx = (

+ Θ(x) ) Θx

Since the individual systems are contracting,

(δzδz) = δz( + Θ

) Θδz ≤ −2λδzδz, i = 1,2



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Hence, the δz dynamics of the combined system result in

(δzδz) = δz( α α + Θ (α α )) Θδz

=

αδz(

+ Θ

)

Θδz +

αδz(

+ Θ

)

Θδz

≤ −2 (αλ + αλ) δzδz

This parallel combination is extremely important if the system possesses a certain

geometrical symmetry. For example, the M and C matrices from the Lagrangian equation might

satisfy = + and = + .

3.2 Hierarchical combination

Theorem 5: Consider two contracting systems, of possibly different dimensions and metrics,

and connect them in series, leading to a smooth virtual dynamics of the form

δzδz = F 0F F δzδz

Then the combined system is contracting if F21 is bounded.

Proof: Intuitively, since δz tends to zero, δz tends to zero as well for a bounded F . By using

a smooth transformation Θ:

Θ = (

00 )

Then, the new transformed Jacobian matrix is given as

ΘFΘ = (F 0F F),

which is negative definite for a sufficiently small > 0.

Composite variables, s = q − q + Λ (q−q) are an excellent example of a hierarchical

combination. Its usage for nonlinear tracking control and robust sliding control shall be

discussed in the subsequent sections.

Another interesting combination that occurs in the nonlinear control synthesis is

feedback combination.

3.3 Feedback combination

Similarly, connect instead two systems of possibly different dimensions



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= (, , t)

= (, , t)

in the feedback combination

δzδz = F G F δzδz

The augmented system is contracting if and only if the separated plants are contracting.

Theorem 6: The overall dynamics of the generalized virtual displacements might be

represented as

δz

δz= F δz

δz= F G

F δz

δz

where F and F are symmetric negative definite matrices.

In order to prove contraction of the combined system, one needs to check if F is

negative definite. From the standard linear algebra, F is uniformly negative definite if and only if

F < G FG

A sufficient condition for the above inequality can be given as

λ(

F )λ(

F ) >

σ (G)

where λ(∙) is the contraction rate, which is the absolute value of the largest eigenvalue (least

negative) of the contracting system. σ(∙) denotes the largest singular value.

It should be noted that the overall contraction rate is no greater than the individual

contraction rate of F and F by the eigenvalue interlacing theorem. Indeed, one can find the

explicit lower bound of the contraction rate as

λ(F) = ‐ σ



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4.

Coupling Configuration of Oscillators

4.1 One‐Way Coupling Configuration

Consider a pair of one‐way (unidirectional) coupled identical oscillators

, t , t u u

where , are the state vectors, f (, t) the dynamics of the uncoupled oscillators,

and u u the coupling force.

Theorem 7: If the function f − u is contracting in (3), two systems and will reach synchrony

exponentially regardless of the initial conditions.

Proof: The second subsystem, with u(

) as input, is contracting, and

(t) =

(t) is a

particular solution.

Example 6: Consider two coupled identical Van der Pol oscillators

α 1 0 α 1 ακ

where α, ω and κ are strictly positive constants (this assumption will apply to all our Van der Pol

examples). Since the system

+ α(

+ κ − 1)

+

x = u(t)

is semi‐contracting for κ > 1, will synchronize to asymptotically.

Note that a typical engineering application with a one‐way coupling configuration is

observer design, in which case u() represents the measurement. The result of Theorem 2 can

be easily extended to a network containing n oscillators with a chain structure (or more

generally, a tree structure)

, t

, t u u… , t u u

where the synchronization condition is the same as that for system.

4.2 Two‐Way Coupling Configuration



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The meaning of synchronization may vary in different contexts. In this paper, we define

synchronization of two (or more) oscillators , as corresponding to a complete match, i.e., = . Similarly, we define anti‐synchronization as = − . These two behaviors are called

in‐phase synchronization and anti‐phase synchronization in many communities.

4.2.1 Synchronization

Theorem 8: Consider two coupled systems. If the dynamics equations verify

− h ( , t) = − h ( , t)

where the function h is contracting, then and will converge to each other exponentially,

regardless of the initial conditions.

Proof: Given initial conditions

(0) and

(0), denote by

(t) and

(t) the solutions of the two

coupled systems. Define

g ( , , t) = − h ( , t) = − h ( , t)

and construct the auxiliary system

= h(y, t) + g ((t), (t), t)

This system is contracting since the function h is contracting, and therefore all solutions

of y converge together exponentially. Since y = (t) and y = (t) are two particular solutions,

this implies that

(t) and

(t) converge together exponentially.

Remarks

• Theorem 8 can also be proved by constructing another auxiliary system

h, t g, , t h, t g, , t

which has a particular solution verifying the specific property = . Since this auxiliary system

is composed of two independent subsystems driven by the same inputs, the proof can be

simplified as above by using a auxiliary system of reduced dimension.

• Theorem 7 is a particular case of Theorem 8. So is, for instance, a system of two‐way coupled

identical oscillators of the form

, t u u , t u u



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In such a system tends to exponentially if f − 2u is contracting. Furthermore, because the

coupling forces vanish exponentially, both oscillators tend to their original limit cycle behavior,

but with a common phase.

• Although contraction properties are central to the analysis, the overall system itself in general

is not contracting, and the common phase of the steady states is determined by the initial

conditions (0) and (0). This stresses the difference between contraction and partial

contraction.

• Theorem 8 can be easily extended to coupled discrete‐time systems, using discrete versions

[40] of contraction analysis, to coupled hybrid systems, and to coupled systems expressed by

partial‐differential‐equations.

• Contraction of the auxiliary system also implies that bounded variations in subsystem

dynamics lead to bounded synchronisation errors.

Example 7: Consider again two coupled identical Van der Pol oscillators

α 1 ακ α 1 ακ

One has

α κ κ 1 α κ κ 1

From Theorem 3 and the result in the Appendix, we know that these two oscillators will reach

synchrony asymptotically if κ κ > 1 for non‐zero initial conditions.

4.2.2 Anti‐Synchronization

In a seminal paper inspired by Turing’s work, Smale describes a mathematical model

where two identical biological cells, inert by themselves, can be excited into oscillations

through diffusion interaction across their membranes. Using Theorem 8, we can build a coupled

system

, t u, u, , t u, u, t

to describe Smale’s model.

Theorem 9: If the uncoupled dynamics h in above is contracting and odd in x, + will

converge to zero exponentially regardless of the initial conditions. Moreover, for non‐zero

initial conditions, and will oscillate and reach anti‐synchrony if the system



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= h(z, t) − 2u(z, t)

has a stable limit‐cycle.

Proof: Replace

by −

in Theorem 8.

Example 8: Consider specifically Smale’s example [69], where

h(x, t) = A x + 000 u(x) = K

with

A = 1 1 0

1 0 0 2 00 0 2 K = 0 00 0 0 2 00 0 2

For a < −1, h has a negative definite Jacobian and thus is contracting, and h − 2u yields a

stable limit‐cycle, so that the two originally stable cells are excited into oscillations for non‐zero

initial conditions. Requiring in addition that √ 2 < γ < 3/2 guarantees that all the eigenvalues of

K are distinct, real and strictly positive, so that K can be transformed into a diagonal diffusion

matrix through a linear change of coordinates.

4.2.3 Oscillator‐Death

Inverse to Smale’s effect, in the phenomenon called oscillator‐death (or amplitude‐

death) oscillators stop oscillating and stabilize at constant steady states once they are coupled.

Oscillator‐death happens if the overall dynamics is contracting and autonomous, since this

implies that the system tends exponentially to a unique equilibrium.

Example 9: Couple two Van der Pol oscillators with asymmetric forces

α 1 ακ α

1

ακ

where κ > 1. By introducing new variables and as in the Appendix, we get a generalized

Jacobian

F = α κ 1 ω ακ 0ω 0 0 0ακ 0 α κ 1 ω0 0 ω 2 ≤0



20

whose symmetric part is simply that of two uncoupled damped Van der Pol oscillators. Thus

both systems will tend to zero asymptotically.

Example 10: Chaotic synchronization

Much attention has been devoted to synchronization in chaotic systems. A central characateristic of a chaotic system like the Lorenz Attractor is that its long‐term behavior is

often impossible to predict but its short‐term behavior is relatively easy to anticipate, so its

immediate propagation can be controlled. In particular, Pecora et al. reported that this

phenomenon can be used for communications by means of chaotic synchronization. Recently,

Argyris implemented the fist long‐distance demonstration of chaotic synchronization using a

pair of laser diodes, amplifiers, a mirror and more than 120 kilometers of underground fiber.

Consider the Lorenz system

= σ (y − x) = ρx − y – xz

= −βz + xy

with strictly positive constants σ, ρ, β, and, as in [151], a reduced‐order identity observer for

this system based on an available measurement of the variable x,

= ρx − – x

= −β + x

The symmetric part of the observer’s Jacobian is − diag(1, β), and thus the observer is

contracting with an identity metric. Since by construction (,) = (y, z) is a particular solution,

the estimated state converges exponentially to the actual state. This can also be described as

an active‐passive decomposition.



21

5.

High‐Order Contraction Theory

Consider the following second‐order time‐varying nonlinear differential equation

+

(x)

x +

(x, t) = u(t)

where (x) is a smooth function of x, and (x, t) is a smooth function of x and t.

Proving contraction of (2.98) is difficult, since the original contraction analysis is

originally stated for first‐order systems. Finding some smooth coordinate transformation of the

virtual displacement, δz = Θδx, is key such that the generalized Jacobian F associated with the

augmented system of x, x is uniformly negative definite. In general, finding such a

transformation, Θ is not trivial. Even without the use of contraction theory, proving exponential

and global stability of a general time‐varying nonlinear second‐order system can be a very

difficult problem.

We present the main theorem of this section, which provides a sufficient condition for

contraction of (2.98)

Theorem 10: The nonlinear system in (2.98) is contracting (all solutions converge to a single

trajectory exponentially fast from any initial conditions), if

(x) > 0, x

and if

, > 0, ( , ) > 0, t

Proof: The virtual displacement equation of (2.98) is

δ + (x)δx + [ + , ] δx = 0

Equation (2.101) can be written as a first‐order differential equation by defining the

vector state x = x, x :

δ

x =

δx

(δxδx ) = [ 0 1 ] (δxδx )

Let us take the following transformation matrix



22

Θ(x, t) = (

1 0 1 )

It is straightforward to calculate the associated Jacobian as

F = (Θ (x, t) + Θ(x, t)) Θx,t

= [

x 2 ]

From the contraction theory, (2.98) is contracting if the symmetric part of F is uniformly

negative definite, which corresponds to the conditions in Theorem

We illustrate the use of this theorem in a few examples.

Example 11: Driven damped Van der Pol oscillator

Consider the second‐order system

+ (β + α) + x = u(t)

driven by an external input u(t), where α, β, ω are strictly positive constants. Introducing a new

variable y, this system can be written

ωy 3 – βx ωx utω The corresponding Jacobian matrix

F = β α ωω 0

is negative semi‐definite. Therefore,

(δδz) = 2 δ F δz ≤ 0

where δz = δx,δy is the generalized virtual displacement. Thus δδz tends to a lower limit

asymptotically. Now check its higher‐order Taylor expansion:

if δx 0,



23

δδz(t + dt) − δδz(t) = −2 (β + α)δx dt + O(dt)

while if δx = 0,

δ

δz(t + dt) − δ

δz(t) = −4 (β + α

)

δ

! + O(

dt)

So the fact that δδz tends to a lower limit implies that δx and δ both tend to 0.

System is called semi‐contracting, and for any external input all its solutions converge

asymptotically to a single trajectory, independent of the initial conditions. Note that if β < 0 and

u(t) = 0, system has a unique, stable limit cycle.

The first condition in Theorem 10 holds if (x) = (β + α) > 0 uniformly (i.e. α and β are

strictly positive). Also, note that = ω. This is a semi‐contracting system. Indeed, the

generalized Jacobian yields the same F.



24

6.

Nonlinear Networked Systems

Most coupled oscillators in the natural world are organized in large networks, such as

pacemaker cells in heart, neural networks in brain, fireflies with synchronized flashes, crickets

with synchronized chirping, etc. System (4) represents such an instance with a chain structure.

There are many other possible structures, such as e.g. the three symmetric ones illustrated in

Figure 1.

In this section, we show that partial contraction analysis can be used to study

synchronization in networks of nonlinear dynamic systems of various structures and arbitrary

size. Coupling forces can be nonlinear as well.

Figure 1: Networks with different symmetric structures (n = 4)

6.1 Networks with All‐To‐All Symmetry

Consider first a network with all‐to‐all symmetry, that is, with each element coupled to

all the others. Such a network can be analyzed using an immediate extension of Theorem 8.

Theorem 11: Consider n coupled systems. If a contracting function h(

, t) exists such that

− h(, t) = ∙ ∙ ∙ = − h(, t)

then all the systems will synchronize exponentially regardless of the initial conditions.

For instance, for identical oscillators coupled with diffusion‐type force

= f (, t) + ∑ u u i = 1, 2, . . . , n

contraction of f − nu guarantees synchronization of the whole network.

Mirollo and Strogatz study an all‐to‐all network of pulse‐coupled integrate‐and‐fire oscillators, and derive a similar result on global synchronization.

6.2 Networks with Less Symmetry

Besides its direct application to all‐to‐all networks, Theorem 11 may also be used to

study networks with less symmetry.



25

Example 12: Consider an n = 4 network with two‐way‐ring symmetry (as illustrated in Figure

1(b))

= f (, t) + (u() − u()) + (u() − u()) i = 1, 2, 3, 4

where the subscripts i − 1 and i + 1 are computed circularly. Combining these four oscillators into two groups (, ) and (, ), we find

f , t 2u f , t 2u = f , t 2u f , t 2u = u uu u

Thus, if the function f − 2u is contracting, (, ) converges to (, ) exponentially,

and hence

f , t 2u 2u f , t 2u 2 u

so that in turn converges to exponentially if the function f − 4u is contracting. The four

oscillators then reach synchrony exponentially regardless of the initial conditions.

An extended partial contraction analysis can be used to study the example below, the

idea of which will be generalized in the following section.

6.3 Networks with General Structure

Let us now move to networked systems under a very general coupling structure. For

notational simplicity, we first assume that coupling forces are linear diffusive with gains (associated with coupling from node i to j) positive definite, i.e., = > 0. We further

assume that coupling links are bidirectional and symmetric in different directions, i.e., = .

All these assumptions can be relaxed as we will show later.

Theorem 12: Regardless of initial conditions, all the elements within a generally coupled

network will reach synchrony or group agreement exponentially if

• the network is connected

•

(

) is upper bounded

• the couplings are strong enough

Specifically, the auxiliary system is contracting if

() > max uniformly



26

6.4 Extensions

Besides the properties discussed above, let us make a few more extensions to Theorem

6, and relax assumptions made earlier.

6.4.1 Nonlinear

Couplings

The analysis carries on straightforwardly to nonlinear couplings. For instance,

= f (, t) + ∑ , ,x ,tЄ

where the couplings are of the form

= ( ‐ , x, t)

with

(0, x, t) = 0

i, j, x, t. The whole proof is the same except that we define

= ,, > 0 uniformly

and assume = .

For instance, one may have

= ( (t) + (t) || ||)

with

=

> 0 uniformly and

=

≥ 0, in which case we can construct a simplified

auxiliary system as = f (, t) + ∑ t t || || Є ‐ ∑ + ∑

Note that if the network is all‐to‐all coupled, the coupling forces can be even more

general.

6.4.2 One‐way Couplings

The bidirectional coupling assumption on each link is not always necessary. Consider a

coupled network with one‐way‐ring structure and linear diffusion coupling force

= f (, t) + K ( − ) i = 1, . . . , n

where by convention i − 1 = n when i = 1. We assume that the coupling gain K = > 0 is

identical to all links. Hence,

= − −



27

is negative definite with

= ∑ , 1

Since

( ∑ , 1 ) = (K) (∑ , 1 ) = (K) (1 ‐ cos ),

the threshold to reach synchrony exponentially is

(K) (1 ‐ cos ) > max uniformly

Thus, Theorem 12 can be extended to networks whose links are either bidirectional with = or unidirectional but formed as rings with = K (where K is identical within the same

ring but may differ between different rings). Synchronized groups with increasing complexity

can be generated through accumulation of smaller groups.

Throughout the remainder of the paper, all results on bidirectionally coupled networks

will apply to unidirectional rings as well.

6.4.3 Positive Semi‐Definite Couplings

Theorem 12 requires definite coupling gains. If the are only positive semi‐definite,

additional conditions must be added to the uncoupled system dynamics to guarantee globally

stable synchronization.

Without loss of generality, we assume

= 00 0

where is positive definite and has a common dimension for all links. Thus, we can divide

the uncoupled dynamics into the form

=

with each component having the same dimension as that of the corresponding one in . A

sufficient condition to guarantee globally stable synchronization behavior in the region beyond

a coupling strength threshold is that, i, is contracting and both ( ) and σ( )

are upper bounded.

Example 13: The FitzHugh‐Nagumo (FN) model is a well‐known spiking‐neuron model. Consider

a diffusion‐coupled network with n identical FitzHugh‐Nagumo neurons



28

c 13 I Є 1c

a b i 1, . . . , n

where a, b, c are strictly positive constants. Defining a transformation matrix Θ = 1 00 , which

leaves the coupling gain unchanged, yields the generalized Jacobian of the uncoupled dynamics

= 1 11

Thus the whole network will synchronize exponentially if

(

∑

,ЄN

) =

(

) > c

Note that the model can be generalized using a linear state transformation to a

dimensionless system, with partial contraction analysis yielding a similar result.

6.5 Algebraic Connectivity

For a coupled network with given structure, increasing the coupling gain for a link or

adding an extra link will both improve the synchronization process. In fact, these two

operations are the same in a general understanding by adding an extra term − to the

matrix

. According to Weyl’s Theorem, if square matrices A and B are Hermitian and the

eigenvalues (A), (B) and (A + B) are arranged in increasing order, for each k = 1, 2, . . ., n, we have

(A) + (B) ≤ (A + B) ≤ (A) + (B)

which means immediately

( − ) ≤ ( )

since

(−

) = 0.

Example 14: Kopell and Ermentrout show that closed rings of oscillators will reliably

synchronize with nearest‐neighbor coupling, while open chains require nearest and next‐

nearest neighbour coupling. This result can be explained by assuming all gains are identical and

expressing the synchronization condition as

K > uniformly



29

Assuming n extremely large, for a graph with two‐way‐chain structure

= 2 ( 1 − cos( ) ) ≈ 2

while for a graph with two‐way‐ring structure

= 2 ( 1 − cos( ) ) ≈ 8

As illustrated in Figure 2, although the number of links only differs by one in these two cases,

the effort to synchronize an open chain network is four times of that to a closed one.

Figure 2: Comparison of a chain network and a ring.

Example 15: Consider a ring network, a star network and an all‐to‐all network (Figure 3) as

network size n tends to ∞. For the ring network, the coupling strength threshold for

synchronization tends to infinity. For the star network it only needs to be of order 1, and for the

all‐to‐all network it actually tends to 0.

Thus, predictably, it is much easier to synchronize the star network than the ring. This is

because the central node in the star network performs a global role, which keeps the graph

diameter constant no matter how big the network size is. Such a star‐liked structure is

common. The internet, for instance, is composed of many connected subnetworks with star

structures.

Figure 3: Comparison of three different kinds of networks.

This result is closely related to the Small World problem. Strogatz and Watts showed that the average distance between nodes decreases with the increasing of the probability of adding

short paths to each node. They also conjectured that synchronizability will be enhanced if the

node is endowed with dynamics, which Barahona and Pecora showed numerically.

6.6 Fast Inhibition



30

The dynamics of a large network of synchronized elements can be completely

transformed by the addition of a single inhibitory coupling link. Start for instance with the

synchronized network and add a single inhibitory link between two arbitrary elements a and b

= f (

, t) +

∑ Є + K (

)

= f (, t) + ∑ Є + K ( )

6.7 Switching Networks

Closely related to oscillator synchronization, topics of collective behavior and group

cooperation have also been the object of extensive recent research. A fundamental

understanding of aggregate motions in the natural world, such as bird flocks, fish schools,

animal herds, or bee swarms, for instance, would greatly help in achieving desired collective

behaviors of artificial multi‐agent systems, such as vehicles with distributed cooperative control

rules. Since such networks are composed of moving units and each moving unit can only couple

to its current neighbors, the network structure may change abruptly and asynchronously.

Consider such a network

= f (, t) + ∑ Є i = 1, . . . , n

where ) denotes the set of the active links associated with element i at time t. Apply partial

contraction analysis to each time interval during which the network structure N(t) is fixed. If

(

) >

max uniformly

the auxiliary system is always contracting, since δδy with δy = δ, . . . , δ is continuous

in time and upper bounded by a vanishing exponential (though its time‐derivative can be

discontinuous at discrete instants). Since the particular solution of the auxiliary system in each

time interval is = ∙ ∙ ∙ = = , these n elements will reach synchrony exponentially as they

tend to = ∙ ∙ ∙ = which is a constant region in the state‐space. The threshold phenomenon

described by inequality is also reminiscent of phase transitions in physics and of Bose‐Einstein

condensation.

6.8 Leader

‐Followers

Network

In a network composed of peers, the phase of the collective behavior is hard to predict,

since it depends on the initial conditions of all the coupled elements. To let the whole network

converge to a specific trajectory, a “leader” can be added [27, 36].

Consider the dynamics of a coupled network



31

= f (, t)

= f (, t) + ∑ Є + i = 1, . . . , n

where

is the state of the leader, whose dynamics is independent, and

the state of the ith

follower. is equal to either 0 or 1 and represents the connection from the leader to the followers. denotes the set of peer‐neighbors of element i, i.e., it does not include the

possible link from to . Example: Extension

Suppose that ω is now time‐varying such that

+ (x) x + x = u(t)

where

(x) > 0,

> 0, and

> 0.

The above dynamics is contracting since

( , ) = > 0.



32

7.

Applications of Contraction Analysis

This section illustrates the discussion with some immediate applications of contraction

analysis to specific control and estimation problems.

7.1 PD

observers

Observer design using contraction analysis can be simplified by prior coordinate

transformations similar to those used in linear reduced‐order observer design. Consider the

system

= f(x, t)

y = h(x, t)

where x is the state vector and y the measurement vector. Define a state observer with

= f(, t) − P ( − y) − D

= + Dy

where = h(, t) and = f(,) +

. By differentiation, this leads to the observer dynamics

= f(, t) − P ( − y) − D ( – )

Thus the dynamics of

contains

, although the actual computation is done using

equation and hence is not explicitly used.

7.2 Constrained Systems

Many physical systems, such as mechanical systems with kinematic constraints or

chemical systems in partial equilibrium, are described by an original n‐dimensional dynamics of

the form

= f(z, t)

constrained to an explicit m‐dimensional submanifold (m ≤ n)

z = z(x, t)

The constrained dynamic equations are then of the form

= f(z, t) + n



33

where n represents a superimposed flow normal to the manifold,

n = 0 − the components of

n are Lagrange parameters. In a mechanical system, z represents unconstrained positions and

velocities, x generalized coordinates and associated velocities, and n constraint forces.

Multiplying from the left with

results in

=

( +

) =

f(z)

so that a uniformly positive definite metric M = allows one to compute with =

f(z, t) −

n = − f = +

(f ‐ ) ‐ f

The variation is

δ = ‐

n δx + δz

so that

(δxMδx) = δx

δx ‐ δx

n δx

Consider now a specific trajectory

(t) of the unconstrained flow field which naturally

remains on the manifold z(x, t). Then the normal flow n around this trajectory vanishes, so that

locally the contraction behavior is determined by the projection of the original Jacobian

=

Contraction of the original unconstrained flow thus implies local contraction of the

constrained flow, and the contraction region around the trajectory can be computed.

This result can be used to study the contraction behavior of mechanical systems with

linear external forces, such as PD terms or gravity − Newton’s law in the original unconstrained

state space is then linear, and z = z(x, t) are kinematic constraints. Exponential convergence around one trajectory z(t) at which the constraint forces vanish can then be concluded in the

region where the projection of the original constant Jacobian is uniformly negative definite.

Exponential convergence of a controller or can thus be achieved by stabilizing the

unconstrained dynamics with a PD part, and adding an open‐loop control input to guarantee



34

that a desired trajectory consistent with the kinematic constraints is indeed contained in the

unconstrained flow field.

7.3 Linear Time‐Varying Systems

As another illustration, this section shows that contraction analysis can be used systematically to control and observe linear time‐varying (LTV) systems.

7.3.1 LTV controllers

Consider a general smooth linear time‐varying system

= A(t)x + b(t)u

with control input u = K(t)x + (t). We focus on choosing the gain K(t) so as to achieve

contraction behavior; whereas the open‐loop term

(t) then guarantees that the desired

trajectory, if feasible, is indeed contained in the flow field (this guarantee and a similar

computation is required of any controller design).

We need to find a smooth coordinate transformation δz = Θ(t)δx that leads to the

generalized Jacobian F

F = ( + Θ (A + bK)) =

0 1 0 … 00 0 1 … 0 0 0 0 0 1 …

with the desired (Hurwitz) constant characteristic coefficients . The above equation can be

rewritten in terms of the row vectors θ (j = 1, ∙ ∙ ∙, n) of Θ as

θ + θ (A + bK) = θ j = 1, ∙ ∙ ∙, n − 1

θ + θ (A + bK) = ‐a θθ

In order to make the coordinate transformation Θ independent of the control input, let

us impose recursively, t ≥ 0, the following constraints on the θ

0 = θb = θb

0 = θb = (θ + θA) b − (θb) = θb



35

0 = θb = (θ + θA) b − (θb) = θb

= (θ + θA) b − (θb) = θb

D = θb = θb

where the b are generalized Lie derivatives (Lovelock and Rund, 1989)

b = b

b = A b − b j = 0, ..., n − 1

Choosing D = det |

b ...

b| the above can always be solved algebraically for a

smooth θ, and from (22) the remaining smooth θ j can then be computed algebraically using the recursion

θ = θ + θ A j = 1, ∙ ∙ ∙ , n − 1

This leads to a smooth bounded metric, and the feedback gain K(t) can then be

computed from

D K(t) = ‐a θθ

‐ θ ‐ θ A

Exponential convergence of δx is then guaranteed for uniformly positive definite metric

M = Θ. The uniform positive definiteness of M hence represents a sufficient “contractibility”

condition.

Note that the terms θ and b distinguish this derivation from the usual pole‐

placement in an LTI system, as well as from related gain‐scheduling. The method guarantees

global exponential convergence, and can be extended straightforwardly to multi‐input systems.

7.3.2 LTV observers

Similarly, consider again the plant above, but assume that only the measurement y =

C(t)x + d(t) is available. Define the observer

= A(t) + E(t) (y − c(t) + d(t)) + b(t)u



36

Since by definition the actual state is contained in the flow field, no “open‐loop” term is

needed, but we need to find a smooth coordinate transformation δ = Σ(t)δ that leads to the

generalized Jacobian F

F = (− + (A − Ec) Σ ) =

0 0 0 1 0 0 0 1 0 0 0 0 1

with the desired (Hurwitz) constant characteristic coefficients . The above equation can be

rewritten in term of the column vectors σ of Σ as

‐ + (A − Ec) σ = σ j = 1, ∙ ∙ ∙ , n − 1

‐

+ (A − Ec)

σ = − (

σ ∙ ∙ ∙

σ)a

Proceeding as before by imposing the constraints

c σ = 0 j = 0, ..., n − 2

c σ = det|c ... c|

where the c are now defined as

c = c

c = c A + c j = 0, ..., n − 1

allows one to solve algebraically for a smooth σ , and then compute the remaining smooth σ

recursively

σ = ‐ + A σ j = 1, ∙ ∙ ∙ , n − 1

leading to a uniformly positive definite metric. The feedback gain E(t) can then be computed

from (28)

ED = (σ ∙ ∙ ∙ σ) a − + A σ

Exponential convergence of δ to zero and thus of to x is then guaranteed for a

bounded metric M = Σ. The boundedness of M hence represents a sufficient observability

condition.



37

Again, the terms and c distinguish this derivation from pole‐placement in LTI

systems, or from extended Kalman filter‐like designs (Bar‐Shalom and Fortmann, 1988). The

method guarantees global exponential convergence, and can be extended straightforwardly to

multi‐output systems.

7.3.3 Separation principle

These LTV designs satisfy a separation principle. Indeed, let us combine the above

controller and observer (perhaps with different coefficient vectors a)

u = K(t) + (t)

Subtracting the plant dynamics from the observer dynamics leads with = − x to

= (A(t) − E(t)C(t))

so that the Jacobian of the error‐dynamics of the observer is unchanged. Since the observer

error‐dynamics and the controller dynamics represent a hierarchical system, they can be

designed separately as long as the control gain K(t) is bounded.

7.3.4 Nonlinear controllers and observers

Consider now the nonlinear system

δ

=

δx + δu δy =

δx

Let

A(t) = ((t), t) B(t) =

((t), t) c(t) = ((t), t)

where (t) is the desired trajectory. Applying the controller and observer designs above to the

LTV system defined by A(t), b(t), and c(t) then guarantees contraction behavior in regions of

uniformly negative definite

= (

(

(t), t) + Θ(

(t), t)(

(x, t) +

(x, t)K(

(t), t)) Θ

t,t

And

= Σ (t (‐ Σ ((t), t) + ( (x, t) ‐ E ((t), t) (x, t)) Σ ((t), t))

similarly to Example. Thus exponential convergence is guaranteed explicitly in a given finite

region around the desired trajectory.



38

Concluding Remarks

By taking advantage of the way the theory is constructed in [12], [20], the methodology

for contraction‐based stability analysis can depart quite far from the one usually applied in the

context of Lyapunov theory. One of its main features, exploiting the incremental nature of the

theory, is to consider two different levels of system description, namely the virtual system,

which can be seen as an abstract definition of a differential equation since no initial value or

particular solution is specified, and the actual systems or particular solutions that are the result

of an instantiation of the above virtual system. As we saw, such an approach can be quite

natural in both a controller context and an observer context.

The other important feature, which is another fundamental aspect of contraction

theory, is the extensive use of virtual displacements which help eliminate in a rigorous and ef

cient way the terms that are not directly responsible for the convergent behavior of the system.

This variational approach was seen to be quite effective at simplifying computations in a variety

of cases.

Such a methodology could be appealing for both linear and nonlinear designs. Indeed, it

makes explicit different kinds of linearities hidden behind an observer or a controller design,

whether these linearities come from a pure linear system, a state‐af ne representation, or a

Lipschitz‐like condition.

Another feature of contraction is that exponential convergence of a system, whether

linear or nonlinear, can always be expressed in terms of a quadratic form of virtual

displacements, thus emphazising the linear‐like character of exponential incremental stability.



References

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Automatica, 34(6)

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Using Contraction Theory”, IEEE Trans. Aut. Control, 45(5)

[3] Lohmiller, W., and Slotine, J.J.E. (2000) “Nonlinear Process Control Using Contraction

Theory”, A.I.Ch.E. Journal, March 2000.

[4] J. J. Slotine and W. Lohmiller, “Modularity, evolution, and the binding problem: a view from

stability theory”, Neural Networks, Vol 14, p137‐145, 2001

[5] W. Wang and J.‐J. E. Slotine, “On partial contraction analysis for coupled nonlinear

oscillators,” Technical report, MIT Nonlinear Systems Laboratory, 2003.

[6] Jérôme Jouffroy, Jean‐Jacques E. Slotine, “Methodological remarks on contraction theory”,

Technical report, MIT Nonlinear Systems Laboratory, 2004.

[7] Lohmiller, W., and Slotine, J.J.E. (2000) “Contraction Analysis of Nonlinear Distributed

Systems”, International Journal of Control, 78(9), 2005.

[8] Lohmiller, W., and Slotine, J.J.E. (2000) “High‐Order Nonlinear Contraction Analysis”, MIT

NSL Report, NSL‐050901, September 2005.

[9] Soon‐Jo Chung, Thesis on “Nonlinear Control and Synchronization of Multiple Langragian

Systems with Applications to Tethered Formation Flight Spacecraft”, MIT 2007.

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