A Tutorial on Incremental Stability Analysis using Contraction Theory

Abstract

This paper introduces a methodology for differential nonlinear stability analysis using contraction theory (Lohmiller and Slotine, 1998). The methodology includes four distinct steps: the descriptions of two systems to be compared (the plant and the observer in the case of observer convergence analysis, the plant and the controller in the case of tracking controller analysis), the definition of an abstract system common to the two systems and denoted as the 'virtual system', and the convergence study of the virtual system using its virtual dynamics representation. The approach is illustrated on several simple examples.

Full text

Modeling, Identification and Control, Vol. 31, No. 3, 2010, pp. 93–106, ISSN 1890–1328
A Tutorial on Incremental Stability Analysis
using Contraction Theory
J. Jouffroy 1T.I. Fossen 2,3
1Mads Clausen Institute, University of Southern Denmark, DK-6400 Sønderborg, Denmark. E-mail:
jerome@mci.sdu.dk
2Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim,
Norway. E-mail: fossen@ieee.org
3Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, NO-7491 Trondheim,
Norway.
Abstract
This paper introduces a methodology for differential nonlinear stability analysis using contraction theory
(Lohmiller and Slotine,1998). The methodology includes four distinct steps: the descriptions of two
systems to be compared (the plant and the observer in the case of observer convergence analysis, the plant
and the controller in the case of tracking controller analysis), the definition of an abstract system common
to the two systems and denoted as the “virtual system”, and the convergence study of the virtual system
using its virtual dynamics representation. The approach is illustrated on several simple examples.
Keywords:
Contraction theory, exponential stability, incremental stability, Lyapunov stability, methodology.
Introduction
Stability analysis has long been recognized as a key-
stone in the control systems community, and many
techniques have been proposed to check this important
property. Among them, Lyapunov theory has become
a central tool of the control community, and Lyapunov
functions have proven fundamental in stability analy-
sis and control design of nonlinear and time-varying
systems described in the state-space (see for example
Khalil (1996), Krsti´c et al. (1995), or Slotine and Li
(1991).
One of the main features of Lyapunov-based stabil-
ity analysis is the consideration of systems having an
equilibrium at the origin of the state-space. In more
general cases, such as e.g. trajectory tracking control,
the standard methodology consists in making use of
an appropriate change of coordinates to put the sys-
tem under study in the suitable form.
Contraction theory is a more recent tool for ana-
lyzing the convergence behavior of nonlinear systems
in state-space form; see Lohmiller and Slotine (1998),
Slotine and Wang (2003) and Jouffroy (2003b) for the
explicit incorporation of inputs in the framework of
contraction. 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. The 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 or use the results of
a theory, is now well-established for Lyapunov func-
tions. While it might be tempting to apply directly
these techniques to the world of contraction, a more
fruitful approach may be to take into account the speci-
ficities of contraction theory to see how it applies on
doi:10.4173/mic.2010.3.2 c
2010 Norwegian Society of Automatic Control

Modeling, Identification and Control
several concrete examples. The purpose of this paper is
to contribute in this important issue, as well as suggest
means to compare contraction with Lyapunov stability
analysis. The present tutorial paper is based on Jouf-
froy and Slotine (2004) and Jouffroy (2005).
After this introduction, and in addition to a brief
recall of the main results of contraction theory, we
shortly discuss in Section 1a few simple techni-
calities related to the main criterion of contraction
whose Lyapunov counterpart would be the celebrated
Lyapunov equation. Contraction being also an incre-
mental form of stability, i.e.stability of the system
trajectories with respect to one another (see Angeli
(2002), Fromion et al. (1999) and references therein
for other forms of incremental stability, and Lohmiller
(1999), Lohmiller and Slotine (2000b), Lohmiller and
Slotine (2000a), Egeland et al. (2001), Aghannan
and Rouchon (2003), Jouffroy and Lottin (2002) and
Jouffroy and Opderbecke (2007) for other applications
of contraction), Section 2discusses the importance
of the term “incremental” on the methodological
point-of-view introducing a simple example whose
stable behavior can be easily concluded with a simple
Lyapunov function. In Section 3, we use the remarks
of the previous section to deal more specifically with
the methodological aspects induced by the nature
of contraction theory. As expected, these are quite
different from those of a traditional stability analysis
using the original Lyapunov theory and ideally, would
allow to expect contraction to perform as well as in
the Lyapunov case. The approach is illustrated with
different simple examples throughout the paper and
Section 4deals specifically with several application
examples, namely a robot controller, a ship controller
and the Extended Kalman Filter. Concluding remarks
end the paper.
1 Contraction theory
1.1 Contraction analysis
In the following, consider systems described by a non-
linear deterministic differential equation in the form
˙x=f(x, t) (1)
where xis the n-dimensional vector corresponding the
state of the system, tis the time, and fis a nonlinear
vector field. In addition, we make the further assump-
tion that the system is smooth and that any solution
x(x0, t) initialized in x0of equation (1) exists and is
unique. One of the main features of contraction theory
is to use the concept of virtual displacements of the
state xwhich, roughly speaking, consists of a slight
modification of the state to see the change it produces
on the velocity vector ˙x. The standard notation of a
virtual displacement, introduced by Lagrange (Lanc-
zos,1970, p. 38), is δx.
From there, the so-called virtual dynamics are intro-
duced by computing the first variation of equation (1),
i.e.
δ˙x=δf =∂f
∂x (x, t)δx (2)
If now a state dependent local and virtual change of
coordinates
δz = Θ(x, t)δx (3)
(where Θ(x, t) is a nonsingular transformation matrix)
is performed on expression (2), the virtual dynamics
can be expressed in δz-coordinates as
δ˙z=F(x, t)δz (4)
where the generalized Jacobian Fis given by
F=˙
Θ+Θ∂f
∂x Θ−1(5)
We are now ready to state the main definition of
Lohmiller and Slotine (1998):
Definition 1 Given the system equations ˙x=f(x, t),
a region of the state space is called a contraction re-
gion with respect to a uniformly positive definite met-
ric M(x, t) = Θ>Θ, if there exists a strictly positive
constant βMsuch that
F=˙
Θ+Θ∂f
∂x Θ−1≤ −βMI(6)
or equivalently
∂f
∂x
T
M+˙
M+M∂f
∂x ≤ −2βMM(7)
is verified in that region.
From this definition, Theorem 2 in Lohmiller and
Slotine (1998) is stated as:
Theorem 1 Given the system 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 contained at all times in a contraction
region with respect to M(x, t), remains in that ball and
converges exponentially to that trajectory.
Proof. See Lohmiller and Slotine (1998).
Intuitively, the above result means that if the tem-
poral evolution of a virtual displacement tends to zero
as time goes to infinity, this being true for all states x
and at all time, the whole flow will “shrink” to a point,
hence the term “contraction”.
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J. Jouffroy and T.I. Fossen, “A Tutorial on Incremental Stability Analysis using Contraction Theory”
(a) t= 2s(b) t= 3s(c) t= 4s
(d) t= 5s(e) t= 6s(f) t= 7s
Figure 1: Contracting volume of a nonlinear system
Example 1 To illustrate the above idea, consider the
following simulation of a simple three-dimensional
nonlinear system described by the following equation


˙x
˙y
˙z
=

−0.1 0 0
0−0.1 0
0 0 −0.1−0.01z2


x
y
z

+

100
010
001
u(8)
where (x, y, z)>is the state. Let ube a time-dependent
control input u= (2 sin t, 2 cos t, 5t−2)>.This system
was simulated for many different initial conditions in
a ball of radius R=px2
0+y2
0+z2
0= 10 and centered
about the origin. Figure 1represents the evolution of
this ball in time. Since system (8) fulfils the conditions
of the above criterion, the volume contracts in time to
a point, as predicted by the theory.
It can actually be shown that the condition of defi-
nition 1and theorem 1is not only sufficient but neces-
sary, as stated in the following converse theorem:
Theorem 2 If the system which equations are ˙x=
f(x, t)is exponentially convergent, i.e.its virtual dis-
placements verify the following inequality
δx>δx ≤kδx>
0δx0e−βt
(where δx0=δx(0) and kand βare strictly positive
constants) then it is also contracting with respect to a
uniformly positive definite and initially upper bounded
metric M(x, t).
Proof. See Lohmiller and Slotine (1998, section 3.5).
On the methodological aspect, first note that the as-
sumptions that are used on the metric Mimply that
it can become unbounded as time goes to infinity. Let
us see what simple implication it may have for using
criteria (6) and (7). Indeed, by looking at the uni-
form negative definiteness condition of equation (7),
one could think of checking
∂f
∂x
>
M+˙
M+M∂f
∂x ≤ −βI (9)
where βis a strictly positive constant. This would
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Modeling, Identification and Control
obviously imply
d
dt δx>M δx≤ −βδx>δx (10)
However, in order to be able to conclude exponential
convergence, one would like to have
d
dt δx>M δx≤ −βMδx>M δx (11)
with βMa strictly positive constant. Then, note that
the assumptions on Min Lohmiller and Slotine (1998,
Section 3.5) can be expressed as
σ2
minδx>δx ≤δx>Mδx ≤σ2
max(t)δx>δx (12)
where σmin is a strictly positive constant which stands
for the uniform positive definiteness of M, and σmax(t)
is a strictly positive time-dependent function stating
that Mis bounded for bounded t, but may be un-
bounded as t→+∞. From eq. (10) and eq. (12), we
thus get
−βδx>δx ≤ − β
σ2
max(t)δx>Mδx (13)
which in turn implies that
d
dt δx>M δx≤ − β
σ2
max(t)δx>Mδx (14)
Using eq. (12) once again, we finally transform eq. (14)
into
δx>δx ≤σ2
max(0)
σ2
min
δx>
0δx0e−βRt
0
1
σ2
max(τ)dτ (15)
Hence, because of the form (15), we might not get an
exponential convergence if we read eq. (9) for eq. (7).
As a consequence, this latter condition must be read
as
∂f
∂x
>
M+˙
M+M∂f
∂x ≤ −βMM(16)
which straightforwardly implies
δx> ∂f
∂x
>
M+˙
M+M∂f
∂x !δx ≤ −βMδx>M δx
(17)
and therefore
δx>δx ≤σ2
max(0)
σ2
min
δx>
0δx0e−βMt(18)
which indicates exponential convergence of the trajec-
tories of ˙x=f(x, t).
Note that by using the local transform (3), one can
change eq. (17) into
δz>F>+Fδz ≤ −βMδz >δz (19)
hence the equivalence between the negativity condition
on eq. (6) and (7).
Comparing with the usual Lyapunov quadratic func-
tions that are used to prove stability of linear (time-
varying) systems ˙x=A(t)x, remark that these latter
imply the well-known Lyapunov equation
P(t)A(t) + AT(t)P(t) + ˙
P(t) = −Q(t) (20)
where it is often assumed that both P(t) and Q(t) are
uniformly positive and upper bounded matrices, i.e.
pminI≤P(t)≤pmax I(21)
and
qminI≤Q(t)≤qmax I(22)
While equation (20) together with the bounds
above are very important for computational pur-
poses (Gaji´c and Qureshi,1995), it does not have
the“proportionality” form of (16) induced by the term
βM, leading to the equivalence with (19) which makes
it easy to find the transform Θ under which the system
is contracting.
On the other hand, such a proportional inequality as
eq. (16) might sometimes be difficult to verify without
assuming any upper boundedness of the metric M, as
it will be seen through an example later in this paper.
Finally, since
d
dt δz>δz≤ − 2βmax (x, t)δz >δz (23)
where βmax(x, t) is the largest eigenvalue of the sym-
metric part of F, note that criterion (6) can be re-
laxed to conclude exponential convergence by requir-
ing e.g. that the moving-window time-average of F
be upper bounded, i.e. that for some finite T > 0,
Rt+T
tλmax(x, τ )dτ be uniformly negative definite in
time, as studied in Jouffroy (2003a).
1.2 Partial contraction
In this section we briefly recall the basic principles of an
extension of contraction analysis, the so-called partial
contraction analysis. The reader is referred to Slotine
(2003) for details.
Theorem 3 Consider a nonlinear system which can
be put under the form
˙x=f(x, x, t) (24)
and assume that the auxiliary system
˙y=f(y, x, t) (25)
96

J. Jouffroy and T.I. Fossen, “A Tutorial on Incremental Stability Analysis using Contraction Theory”
is contracting with respect to y. If a particular solu-
tion 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) for all t≥0
and the solution with the specific property. This im-
plies that x(t) verifies the specific property exponen-
tially.
Note that contraction may be trivially regarded as
a particular case of partial contraction. Also, consider
for instance an original system in the form
˙x=c(x, t) + d(x, t) (26)
where function cis contracting in a constant metric.
The auxiliary contracting system may then be con-
structed as
˙y=c(y, t) + d(x, t) (27)
and the specific property of interest may consist e.g. of
a relationship between state variables.
2 Incremental and non-incremental
exponential stability
In Jouffroy (2002) and Jouffroy and Slotine (2004) it
was remarked that some particular examples could be
quite difficult to analyze at first glance using contrac-
tion analysis, whereas their stable behavior was easily
proven with Lyapunov functions, as illustrated in the
following example.
Example 2 Consider the system:
d
dt xs
ys=−1xs
−xs−1 xs
ys(28)
This system is very easily proven to be GES (Glob-
ally Exponentially Stable) using the Lyapunov func-
tion V=1
2(xs, ys)>(xs, ys). Note the skew-symmetric
structure that one often encounters e.g. using back-
stepping techniques. The stability analysis is easy
mainly because the cross-terms neutralize each other
in the expression of the time derivative of V. Indeed,
˙
V=−(xs, ys)>(xs, ys)<0,(xs, ys)6= (0,0). Now
using contraction, the virtual dynamics are expressed
as
d
dt δxs
δys=−1 + ysxs
−2xs−1 δxs
δys(29)
Clearly, the skew-symmetric structure is destroyed in
the derivation process leading to the expression of the
Figure 2: Incremental and non-incremental norms of
system (28) with (xs1(0), ys1(0)) = (−20,10)
and (xs2(0), ys2(0)) = (10,20).
Jacobian (29). Hence the difficulty to conclude to
contracting behavior whereas it was straightforward to
prove GES using a simple Lyapunov function.
Why such a difference? First note that one of the
main differences between Lyapunov and contraction is
that the latter enables to conclude exponential conver-
gence of any couple of trajectories, while the former
simply lead to GES with respect to the origin, which
is a weaker form of stability than incremental stabil-
ity. Indeed, take another form of incremental stability
than contraction, i.e. an incremental Lyapunov func-
tion (the concept was introduced in Angeli (2002))
V(xs1, xs2, ys1, ys2) =
1
2(xs1−xs2, ys1−ys2)>(xs1−xs2, ys1−ys2) (30)
where (xs1, ys1) and (xs2, ys2) are two particles of the
state space (xs, ys). By computing the time derivative
of eq. (30), the reader will notice that verifying the in-
cremental stability of system (28) is actually quite diffi-
cult. This is because incremental stability is a stronger
notion than stability with respect to the origin.
Also, and in more down-to-earth considerations, if
we simulate the above system for two different initial
conditions, and trace the norm of the difference be-
tween the two corresponding particles, we might get a
curve as the one of Figure 2. The overshoot of this
curve shows that if the system is contracting, its Ja-
cobian will be uniformly definite negative under a par-
ticular metric M(xs, t)6=Ibecause an identity metric
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Modeling, Identification and Control
would correspond to a curve bounded by an exponen-
tial whose starting point is the norm of the initial value
of the difference vector (xs1−xs2, ys1−ys2).
Hence contraction should therefore be compared to
an incremental form of stability. Also, contraction im-
plying an exponential form of convergence, the form
of incremental stability under study should be expo-
nential. As a consequence, let us give the following
incremental version of GES:
Definition 2 The system ˙x=f(x, t)is said to be in-
crementally Globally Exponentially Stable (Incremen-
tally GES) if there exist two strictly positive constants
kand λsuch that the following inequality is verified
kx(x10, t)−x(x20 , t)k ≤ kkx10 −x20ke−λt (31)
(where k•k is the Euclidian norm) for all x10 and x20
in n, all t≥0.
Once incremental GES is defined, the question is how
to relate contraction with the former. The following
lemma answers this question.
Lemma 1 Assume that the system
˙x=f(x, t) (32)
is globally contracting with the contraction rate λand
with respect to the uniformly positive definite and
bounded metric M(x, t),i.e.
σ2
minI≤M(x, t)≤σ2
maxI(33)
where σmin and σmax are two strictly positive constants.
Then system (32) is also incrementally GES, with i.e.
k=σmax
σmin
.(34)
Proof. The first part of the proof is based on Opial
(1960) (see also Jouffroy (2005)). Consider a straight
line segment s(α) between x10 and x20 defined by
s(α) = αx10 + (1 −α)x20, α ∈[0,1], (35)
whose length is kx10 −x20k. Consider then the curve
”generated” by s(α), defined by x(s(α), t), α ∈[0,1].
The length L(t) of this curve is given by
L(t) = Zα=1
α=0 



∂x(s(α), t)
∂α 



dα. (36)
Defining now vas
v=∂x(s(α), t)
∂α , (37)
it can be seen that vverifies
d
dt v=∂f (x, t)
∂x v(38)
Then, introducing the local transform Θ(x,t) corre-
sponding the metric M(x, t) under which the system
is contracting
w= Θ(x,t)v, (39)
we get
d
dt w=F(x,t)w. (40)
Assuming global contraction with rate λmeans that F
is uniformly negative definite, and that
w>w≤w>
0w0e−2λt (41)
which in turn leads to
v>v≤σ2
max
σ2
min
v>
0v0e−2λt (42)
due to the bounds on M(x, t). Finally, we have
kv(α, t)k ≤ σmax(0)
σmin
kv(α, 0)ke−λt (43)
for all α∈[0,1]. After integration on α, we finally have
kx(x10, t)−x(x20 , t)k ≤ L(t)
≤σmax
σmin
e−λtL(0) = σmax
σmin
kx10 −x20ke−λt (44)
which concludes the proof of the lemma.
The other way is even simpler, as can be seen in the
following lemma.
Lemma 2 Assume that the system
˙x=f(x, t) (45)
is incrementally GES. Then system (45) is also globally
contracting.
Proof. Since eq. (31) is valid for all x1=x(x10, t)
and x2=x(x20, t), then it is also valid for x1=x+δx
and x2=x. Therefore eq. (31) implies
kδxk ≤ kkδx0ke−λt. (46)
Then, using the converse theorem of the last section,
the above inequality implies that (45) is globally con-
tracting.
The above lemmas give therefore an equivalence be-
tween contraction and incremental exponential stabil-
ity. This can be summarized with the following theo-
rem:
Theorem 4 The system
˙x=f(x, t) (47)
is incrementally GES if and only if it is globally con-
tracting.
Proof. Immediate from Lemma 1and Lemma 2.
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J. Jouffroy and T.I. Fossen, “A Tutorial on Incremental Stability Analysis using Contraction Theory”
3 Contraction as a flow-oriented
approach to stability analysis
3.1 Virtual system / actual systems
In order to be able to compare Lyapunov theory with
contraction in terms of applications, one would have
to take into account their differences by requiring the
verification of the same stability property. Hence the
following question arises: how to prove that system
(28) is GES using contraction?
To answer this question, which is the starting point
of the methodology proposed in this paper, let us first
consider the following elementary generalization.
Example 3 Consider the system
˙xs=−D(xs)xs(48)
where xs∈Rn,D(xs) + D>(xs)≥αI > 0. Since
the time-derivative of the quadratic Lyapunov function
V=1
2x>
sxsis
˙
V=−x>
sD(xs)xs≤ −αx>
sxs(49)
the equilibrium point xs= 0 is GES.
To link the above result with contraction theory and
incremental stability, let us first go back to the proof
of Lemma 1, which implies the definition of a path be-
tween two particles x1(t) and x2(t), the state xs(t) of
eq. (48) would represent one end of the path (i.e. for
example x1(t) = xs(t)), while the origin of the state-
space would be the other end of the path (x2(t) = 0).
In terms of systems and differential equations, it means
that these two signals are particular solutions of a sin-
gle system. However, to one particular solution can
correspond several different systems, meaning there is
generally some freedom in choosing the system gener-
ating these two solutions. Such a perspective was first
noticed by Polish mathematician Z. Opial that used a
very similar criterion to the one of contraction theory,
to then apply it to compare different systems (see the
historical review Jouffroy (2005)).
This is also the viewpoint that is adopted in Section
1.2 to describe partial contraction analysis (Slotine and
Wang,2003), where the choice of the so-called auxiliary
system gives this freedom.
Additionally, note that the definition of a differential
equation is quite abstract and general if no particular
initial value is specified. Specifically, consider a virtual
system
˙x=f(x, t) (50)
which can be seen as an auxiliary system in the frame-
work of partial contraction. Then, a particular solution
can be specified for example as
xs=x(xs0, t) (51)
in explicit form, or, in implicit form
˙xs=f(xs, t) (52)
which in the following will be called an actual system.
Note that this clear separation of the abstract level of
the virtual system from the more concrete level of ac-
tual systems is also close in spirit to object-oriented
programming, where classes and objects defined in an
abstract way have to be instantiated to be fully mate-
rialized.
In the example above, a possible way of defining the
virtual system corresponding to eq. (48) would be the
following equation
˙x=−D(xs)x(53)
which is possible since, if we choose x=xsas a par-
ticular solution, we find actual system (48). On the
other hand, note that the origin of the state-space is
also a particular solution of (53). This fits well with
a methodology for using contraction theory since, if
system (53) (and more generally system (50)) is con-
tracting, then, as stated by Theorem 1, any couple of
trajectories, and particularly the ones of interest, will
converge to each other.
Coming back to virtual system (53) and calculating
its virtual dynamics
δ˙x=−D(xs)δx (54)
it is easy to conclude to contracting behavior of (53)
and hence of exponential convergence of its two par-
ticular solutions x=xsand x= 0. This allows us to
conclude that system (48) is GES.
Note that if we would have worked directly on eq.
(48) using contraction, we would have searched for an
incremental form of GES, which is difficult to check, as
we saw in Example 2.
3.2 From observers to controllers
It is worth noting that contraction was first developed
in the context of observers; see Lohmiller and Slotine
(1996b) for the main principle and Lohmiller and Slo-
tine (1996a) for application examples, for which the
virtual system corresponded exactly to the observer
equation, as shown by the following example.
Example 4 Define the following observer
˙
ˆx= ˆx+u+k(xs−ˆx) (55)
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Modeling, Identification and Control
where k > 1and uis the control input. The observer
estimates the state of the system
˙xs=xs+u+k(xs−xs).(56)
Using the virtual system/actual systems description is
quite straightforward, since eq. (55) and eq. (56) are
particular systems of the virtual system
˙x=x+u+k(xs−x) (57)
which is contracting.
However, virtual system (57) corresponds exactly
to the observer (55) itself. Indeed, it was noted in
Lohmiller and Slotine (1998) that for observer conver-
gence analysis, one simply had to verify that the system
to be estimated is a particular solution of the observer
to ensure that ˆxwill converge exponentially to the ac-
tual state xsof the system. By duality, it was also
stated that one would have an exponentially conver-
gent tracking controller provided that the system to
be controlled is a particular solution of the contracting
controller. This last statement is true for many con-
trollers, in particular for linear static state feedback
controllers. But it can be vastly extended using the
virtual system/actual systems description, as seen e.g.
in Example 3. Let us discuss this point further through
the continuation of Example 3using a control input.
Example 5 Consider the system
˙xs=−D(xs)xs+u(58)
where xsRn,D(xs) + D>(xs)≥αI > 0and u is the
control. Define the controller
˙xd=−D(xs)xd+u+K(xd)(xs−xd) (59)
where the n-dimensional square matrix Kis positive
semi-definite. This controller makes xsand xdcon-
verge exponentially to one another since the virtual sys-
tem
˙x=−D(xs)x+u+K(xd)(xs−x) (60)
whose particular solutions x=xsand x=xdare re-
spectively syst. (58) and syst. (59), is contracting.
The reader has certainly noticed that the result is
quite obvious since the chosen virtual system is actually
linear. However, note that such controllers as eq. (59)
are often used with Lyapunov-based techniques (see for
example Skjetne et al. (2004)), precisely because they
make easier the analysis of the time derivative of the
Lyapunov function V(˜x), where ˜x=xs−xd.
This interpretation of partial contraction is of course
useful for larger classes of systems than eq. (58). Con-
sider for instance a nonlinear system of the form
˙xs=f(xs, xs, xd, u, t) (61)
and assume that the controller equation is such that
˙xd=f(xd, xs, xd, u, t) (62)
where xd(t) is the desired state. Consider now the
virtual system
˙x=f(x, xs, xd, u, t) (63)
If the virtual system is contracting, then xtends to xd
exponentially, since both are particular solutions of the
x-system.
Note that in the analysis of the controller that was
carried out above, controller (62) is represented in an
implicit form, contrary to the usual u=c(xs, xd,˙xd, t)
form. In our opinion, it clarifies the reading and the
comparison of system and controller, as well as brings
a unified view of both observers and controllers con-
vergence analysis by adopting an observer perspective.
This last point can also be related to the concept of
dual observers due to Brasch that are alluded to in Lu-
enberger (1971, Section 6) where if an observer could
be seen as a system S2tracking another system S1, the
corresponding controller would be S1that the system-
to-be-controlled S2would have to follow. Finally, note
that this point-of-view also allows to go back and forth
between observer and controller design, as shown by
Example 5in which we can design a “tricky” but sim-
ple observer for system (58) by replacing xdwith ˆx
in (59) if xsis measured (see also Jouffroy and Lottin
(2002) for an application to observer design for Dy-
namic Positioning of marine vessels).
Hence, we can summarize the above discussion by
introducing a methodology for controller stability anal-
ysis using contraction theory which could be sketched
as follows.
•write the “target” system equation ( ˙xs=f(xs, t)),
•write the controller equation in implicit form,
•define the virtual system whose particular solu-
tions or actual systems are the target system and
the controller,
•analyze the virtual dynamics of the virtual system
to conclude to contracting behavior.
One might wonder about several types of controllers
when related to the above methodology, like for exam-
ple PID controllers. In this case, the dimension of the
100

J. Jouffroy and T.I. Fossen, “A Tutorial on Incremental Stability Analysis using Contraction Theory”
controller equation can be different from the system un-
der consideration, and one has just to make sure that
the chosen virtual system contains both system and
controller. Switching again to the observer world, this
last remark can be used to reformulate, in a very sim-
ple way, the interesting concept of dynamic observers
as introduced by Park et al. (2002).
The problem of analyzing systems synchronization
can also be studied as in the following example, taken
from Slotine and Wang (2003).
Example 6 Consider two systems ˙x1=f(x1, t)and
˙x2=f(x2, t)coupled in the following manner:
˙x1=f(x1, t) + k(x2, t)−k(x1, t)
˙x2=f(x2, t) + k(x1, t)−k(x2, t)(64)
where k(xi, t)represent the coupling forces. Assuming
that the virtual system
˙x=f(x, t)−2k(x, t) + k(x1, t) + k(x2, t) (65)
is contracting leads to conclude that x1and x2converge
exponentially to each other.
3.3 Incorporating input signals
Another question that might arise when using the
above methodology is how to express in an explicit
manner the impact of different inputs on the behav-
ior of a system. Hence, we will now have to consider
systems described by the following differential equation
˙x=f(x, u, t) (66)
where x∈Rnand u(t)∈Rp. From there, the first
variation of eq. (66) can now be expressed as
δ˙x=∂f
∂x (x, u, t)δx +∂f
∂u (x, u, t)δu (67)
and the local coordinate transform Θ is now control
dependent, i.e.
δz = Θ (x, u, t)δx (68)
and gives the virtual dynamics in the δz-coordinates
δ˙z=F δ z + Θ ∂f
∂u δu (69)
where Fis the generalized Jacobian (6) except for the
dependence on the control input.
From expression (69), it can be seen that provided
that Fis uniformly negative definite for all input, then
the impact of different inputs on the convergent behav-
ior will be bounded if ∂f
∂u and Θ are uniformly bounded.
Thus, as in the ISS framework of Sontag (1989), expres-
sion (69) leads to convergence of a ball around a tra-
jectory. As described in Sontag (2000) in the context
of ISS and in Jouffroy (2003b) in the context of con-
traction, such a point-of-view helps to consider many
different important issues such as robustness, but also
detectability, combination properties such as cascades
and small-gain theorem in a simple way.
In terms of the above-described methodology, the
notation used in this paper indicates that eq. (66) is
the virtual system whose particular solutions need to
be specified to study incremental stability properties
of particular examples. However, since we are dealing
with inputs in addition to the state, we will consider
the couple (x=x1, u =u1) and (x=x2, u =u2) as the
particular solutions describing two systems generated
by eq. (66).
Such a point-of-view happens in particular in the
context of output-feedback where the unavailable state
xsof a plant which should be the input to the feedback
controller is replaced by its estimate ˆxobtained by an
observer.
The following example shows how to reframe the
well-known separation principle in the context of con-
traction for the linear case and a simple nonlinear one.
Example 7 Consider the linear time-invariant system
˙xs=Axs+Bus(70)
ys=Cxs(71)
where us∈Rpand ys∈Rq, and A,Band Care
matrices of appropriate dimensions. Equations (70)-
(71) are assumed to be both controllable and observable.
A linear full-state observer for the plant (70)-(71) takes
the form
˙
ˆx=Aˆx+Bus+LC (xs−ˆx) (72)
where the matrix Lis the observer gain. A state-
feedback controller for (70)-(71) could take the form
˙xd=Axd+Bus+BK (xs−xd) (73)
Remark first that either (72) or (73) can be used to-
gether with (70) to define the following virtual system
˙x=Ax +Bus+F(xs−x) (74)
where Fcan be LC or BK, depending on what action
is chosen, i .e.observation or control. Say now that
instead of controller (73), we want to use the output-
feedback controller
˙xh=Axh+Bus+BK (ˆx−xh) (75)
101

Modeling, Identification and Control
whose input is the estimate ˆxgiven by observer (72).
Then, the difference in terms of behavior with con-
troller (73) can be seen by writing the following virtual
system
˙xc=Axc+Bus+BK (xo−xc) (76)
where the particular solutions (xc=xd, xo=xs)and
(xc=xh, xo= ˆx)are respectively eq. (73) and eq.
(75), and where xois in turn the state of the virtual
system
˙xo=Axo+Bus+LC(xs−xo) (77)
obtained from the actual systems defined by plant (70)
and observer (72). Finally, putting together the virtual
dynamics of virtual systems (76) and (77), we have
d
dt δxo
δxc=A−LC 0
BK A −BK  δxo
δxc
(78)
which is contracting provided that each element of the
cascade (observer and controller parts) is contracting
and that BK is bounded.
Example 8 Take now, as in Lohmiller and Slotine
(2000b), the following nonlinear closed-loop system
˙zs=f(zs, t) + G(zs, t)u(ˆz, t) (79)
with its corresponding observer
˙
ˆz=f( ˆz, t) + G(zs, t)u(ˆz , t) + e(zs, t)−e(ˆz , t) (80)
and define the virtual observer system
˙zo=f(zo, t) + G(zs, t)u(ˆz, t) + e(zs, t)−e(zo, t) (81)
Like the linear system, define also the virtual controller
system
˙zc=f(zc, t) + G(zc, t)u(zo, t) (82)
which, together with eq. (81) give the virtual dynamics
d
dt δzo
δzc= ∂(f−e)
∂zo0
G∂u
∂zo
∂(f+Gu)
∂zc!δzo
δzc(83)
which is again contracting provided ∂(f−e)
∂zoand ∂(f+Gu)
∂zc
are uniformly negative definite and G∂u
∂zois uniformly
bounded.
4 Applications
4.1 Robot manipulator control design
Consider the nonlinear robot model Asada and Slotine
(1986):
˙qs=vs(84)
H(qs) ˙vs+C(qs, vs)vs+g(qs) = τ(85)
where qs∈Rnis a vector of joint angles, H(qs) =
H>(qs)>0 is the inertia matrix, the matrix C(qs, vs)
defines Coriolis and centripetal terms, g(qs) is a vector
of gravitational torques, and τ∈Rnis a vector of
control torques. Using a control design technique such
as vectorial backstepping gives, for system (84)-(85),
the following nonlinear controller (see Fossen (2002))
τ=H(qs) ˙vr+C(qs, vs)vr+g(qs)−Kds−Kq(qs−qd),
(86)
where qdis a smooth desired trajectory, vd= ˙qd,
and Kdand Kqare strictly positive constant matrices.
Variable sis defined as s= (vs−vd)+Λ(qs−qd), with Λ
is a constant Hurwitz matrix, while vr=qd−Λ(qs−qd).
Controller (86) can easily be rewritten as
H(qs) ˙vd+C(qs, vs) ˙qd+g(qs) = τ
+ [C(qs, vs)Λ + KdΛ + Kq](qs−qd)
+ [H(qs)Λ + Kd](vs−vd) (87)
By comparing this controller with robot model (85),
one can now write the virtual system equation
H(qs) ˙v+C(qs, vs) ˙q+g(qs) = τ
+ [C(qs, vs)Λ + KdΛ + Kq](qs−q)
+ [H(qs)Λ + Kd](vs−v) (88)
and compute its virtual dynamics
Hδ ˙v+C δ ˙q=−[CΛ + KdΛ + Kq]δq
−[HΛ + Kd]δv (89)
which, in matrix form, gives
I0
0H δ˙q
δ˙v=
0I
−[CΛ + KdΛ + Kq]−[HΛ + C+Kd] δq
δv 
(90)
Introducing now the local transform
δq
δs =I0
ΛI δq
δv (91)
gives the generalized Jacobian dynamics
δ˙q
δ˙s=−ΛI
−H−1Kq−H−1(C+Kd) δq
δs 
(92)
Then, we use yet another change of local coordinates
induced by the metric
M=Kq0
0H(93)
102

J. Jouffroy and T.I. Fossen, “A Tutorial on Incremental Stability Analysis using Contraction Theory”
to check the quadratic criterion of contraction on eq.
(92), i.e.
d
dt δq>δs>Kq0
0H δq
δs 
=−2δq>δs>KqΛ−Kq
KqKd δq
δs 
=−2δq>δs>KqΛ 0
0Kd δq
δs (94)
leading to contracting behavior of system (88).
From the above computations, one can notice that
we have introduced two different changes of coordinates
into two different forms, namely a local transform Θ
and a metric M. Interestingly, one can see that Θ
is induced by the virtual control law process of the
backstepping procedure, or of the sliding variable s,
while the metric Mis the counterpart of the quadratic
Lyapunov function that is typically used for such a
problem.
Alternatively, consider the energy-based controller
Slotine and Li (1991)
H(qs) ˙vr+C(qs, vs)vr+g(qs)−K(vs−vr) = τ(95)
with Ka constant s.p.d. matrix. The virtual x-system
H(qs) ˙v+C(qs, vs)v+g(qs)−K(vs−v) = τ(96)
has vsand vras particular solutions, and furthermore
is contracting, since the skew-symmetry of the matrix
˙
H−2Cimplies
d
dt δv>H δv =−2δv>(C+K)δv+δv>˙
Hδv =−2δv>Kδv
(97)
Thus vstends to vrexponentially. Making then the
usual choice vr=vd−Λ(qs−qd), where Λ a constant
Hurwitz matrix, implies in turn that qstends to qd
exponentially.
4.2 Ship maneuvering control design
In Fossen (2002), a MIMO nonlinear backstepping
technique for ship maneuvering is presented. Con-
sider a marine vessel moving in the horizontal plane
described by the following model class:
˙ηs=R(ψs)νs
H˙νs+C(νs)νs+D(νs)νs+g(ηs) = τ
where ηs= (xs, ys, ψs)>is the vector of earth-
fixed coordinates and yaw angle of the ship, νs=
(us, vs, rs=˙
ψs)>represent the body-fixed coordinates
(surge, sway, yaw). His the inertia matrix includ-
ing hydrodynamics and added mass, Cis the coriolis
and centripetal matrix, Dthe linear and nonlinear dis-
sipative terms, and gthe vector of gravitational and
buoyancy forces and moments. τis the vector of con-
trol forces and moments. The rotation matrix in yaw
is written as
R(ψs) = 

cos(ψs)−sin(ψs) 0
sin(ψs) cos(ψs) 0
0 0 1 
(98)
The different quantities are defined in Fossen (2002).
Assume that the reference trajectory given by
η(3)
d,¨ηd,˙ηd,and ηdis smooth and bounded. Using vec-
torial backstepping and similarly to section 4.1, the
nonlinear ship controller from Fossen (2002) can be de-
scribed as
˙ηd=R(ψs)νd
H˙νd+C(νs)νd+D(νs)νd+g(ηs) = τ
+ [HR>Λ + R>Kd]R(νs−νd)
+ [H˙
R>Λ+(C+D)R>Λ
+R>(Kp+KdΛ)](ηs−ηd) (99)
where Λ is a constant Hurwitz matrix, Kdand Kpare
strictly positive constant matrix of the feedback part
of the nonlinear PD-controller.
From there, and as in the previous subsection, one
can define the following virtual system
˙η=R(ψs)ν
H˙ν+C(νs)ν+D(νs)ν+g(ηs) = τ
+ [HR>Λ + R>Kd]R(νs−ν)
+ [H˙
R>Λ+(C+D)R>Λ
+R>(Kp+KdΛ)](ηs−η) (100)
whose virtual dynamics can be put into matrix form
(102), which can be shown to be contracting after the
use of the local transform
δη
δs =I0
ΛR δη
δν (101)
I0
0H δ˙η
δ˙ν=0R
−[H˙
R>Λ+(C+D)R>Λ + R>(KdΛ + Kq)] −[HR>ΛR+ (C+D) + R>KdR] δη
δν 
(102)
103

Modeling, Identification and Control
and the metric
M=Kp0
0RHR>(103)
giving indeed
d
dt δη>δs>Kp0
0RHR> δη
δs 
=−2δη>δs>KpΛ 0
0RHR>+Kd δη
δs .
4.3 Extended Kalman Filtering
Despite extensive use of the celebrated Extended
Kalman Filter (EKF) for many practical applications,
its proof of convergence as an observer has been ad-
dressed only recently, using mainly the framework
of the second method of Lyapunov in the determin-
istic case; see for example Reif et al. (1998), for
the continuous-time case and Boutayeb and Darouach
(1997) for the discrete-time case, as well as references
therein. We present, under specific assumptions, a sim-
ple proof of exponential convergence of the EKF based
on contraction theory.
Consider a plant represented by the following non-
linear equations
˙xs=f(xs, t) (104)
ys=h(xs, t) (105)
where xs∈Rnis the state of the system to be esti-
mated, ys∈Rmis the measured output, and where
fand hare smooth vector fields. The EKF observer
structure is
˙
ˆx=f( ˆx, t) + K(ˆx, t) [y−h(ˆx, t)] (106)
where the gain matrix
K(ˆx, t) = P(t)C(ˆx, t)>R−1(107)
is computed using the Riccati matrix differential equa-
tion
˙
P(t) = A(ˆx, t)P(t) + P(t)A>(ˆx, t) + Q
−P(t)C>(ˆx, t)R−1C(ˆx, t)P(t) (108)
where
A(ˆx, t) = ∂ f (x, t)
∂x x=ˆx
, C( ˆx, t) = ∂ h(x, t)
∂x x=ˆx
(109)
The covariance matrices Q=Q>>0 and R=R>>0
for simplicity are assumed to be constant.
We make the highly non-trivial but standard follow-
ing assumption (Reif et al.,1998).
Assumption 1 The Pmatrix of the Riccati equation
(108) is uniformly positive definite and upper bounded,
i.e.there exist two strictly positive constants pmin and
pmax such that
pminI≤P(t)≤pmax I(110)
Taking into account the definitions as well as the as-
sumptions for the EKF described above in (104)-(110),
we are ready to state the following result.
Theorem 5 Under Assumption 1, the estimate ˆxof
the EKF converges exponentially to the actual state xs
of the system ˙xs=f(xs, t).
Proof. The proof starts by using the methodology
described in the previous section. Indeed, examining
(104) and (106), we can define the following virtual
system:
˙x=f(x, t) + K(ˆx, t) [ys−h(x, t)] (111)
which particular solution x=xsgives the state equa-
tion of the plant (104), while the other particular so-
lution x= ˆxgives observer equation (106). It remains
to prove that syst. (111) is contracting. Kand ysare
external functions of time, so that its virtual dynamics
can be written
δ˙x= (A−K C)δx (112)
Consider now the square length defined by the metric
M=P−1
δz>δz =δx>P−1δx (113)
and compute its time-derivative as
d
dt (δx>P−1δx)
=δ˙x>P−1δx+δx>d
dt P−1δx+δx>P−1δ˙x
=δx>[(A−KC)>P−1+d
dt P−1+P−1(A−KC )]δx
=δx>P−1hP(A−KC)>−˙
P+ (A−KC )PiP−1δx
(114)
using the fact that d
dt P−1=−P−1˙
P P −1. Using Ric-
cati matrix differential equation (108) and the defini-
tion of gain matrix (107), this gives
d
dt (δx>P−1δx) = −δx>C>R−1Cδx−δx>P−1QP −1δx
(115)
Since R=R>>0, using the coordinate transform
δy =Cδx on the first term of the right hand side fur-
ther implies
d
dt (δx>P−1δx)≤ −δx>P−1QP −1δx (116)
104

J. Jouffroy and T.I. Fossen, “A Tutorial on Incremental Stability Analysis using Contraction Theory”
Under Assumption 1and using the lower bound qmin
on Q, this in turn implies
d
dt (δx>P−1δx)≤ − qmin
pmax
δx>P−1δx (117)
which shows that virtual system (111) is contracting.
Hence the estimate ˆxconverges exponentially to the ac-
tual state xs.
Note the similarity of the proof with that of the con-
tinuous Kalman filter for linear systems. This is due
to the differential framework in which contraction the-
ory is defined, as well as the appropriate definition of
the virtual system for the stability analysis using con-
traction which would have been much more difficult
working directly on actual system (106). Additionally,
following the discussion at the end of Section 1.1, note
that the EKF proof requires an upper bounded metric
M=P−1to allow conclusion of exponential conver-
gence.
5 Concluding remarks
By taking advantage of the way contraction theory is
defined, we have presented a methodology for incre-
mental stability analysis which depart quite far from
the one that is usually applied in the context of Lya-
punov theory. One of its main features is to consider
two different levels of system description, namely the
virtual system, which can be seen as an abstract def-
inition of a differential equation since no initial value
or particular solution is specified, and the actual sys-
tems or particular solutions that are the result of an
instanciation of the above virtual system.
The other important feature, which is another fun-
damental aspect of contraction theory, is the extensive
use of virtual displacements that help to eliminate in
a rigorous and efficient way the terms that are not di-
rectly 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.
Using this methodology, it seems that it could be ap-
pealing for both linear and nonlinear designs. Indeed,
it makes appear in an explicit way different kinds of
linearities hidden behind an observer or a controller
design, whether these linearities come from a pure lin-
ear system, a state-affine representation, or a Lipschitz
condition.
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