Incremental stability of bimodal Filippov systems in Rn
Davide Fiore
Supervisor: Mario di Bernardo
University of Naples "Federico II"
Department of Electrical Engineering and Information Technology
IEEE International Meeting on
Analysis and Applications of Nonsmooth Systems
10-12 September 2014, Como, Italy
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Outline
1
Introduction and Motivation
2
Incremental stability of bimodal Filippov systems
3
An example
4
Conclusions
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Introduction
A dynamical system is said to be incrementally stable if its solutions asymptotically
converge towards each other
In turn it implies that initial conditions or temporary disturbances are
asymptotically forgotten in time
This allows us to study convergence properties of a system without the need of
having an exact knowledge of some attractors or nominal solutions
Incremental stability plays an important role in many nonlinear control problems
including
tracking
design of nonlinear state
observers
synchronization in networks
of dynamical systems
output regulation problems
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Incremental stability and Contraction Theory
Contraction theory provides sufficient conditions for incremental exponential
stability of a smooth dynamical system.
Definition
Consider the dynamical system
x˙ = f (x)
defined on a forward invariant and path-connected set C ⊆ Rn and let x(t) and z(t) be
any two of its solutions with initial conditions x(t0 ) = x0 ∈ C and z(t0 ) = z0 ∈ C.
The system is said to be incrementally exponentially stable (IES) on C if there exist
constants K ≥ 1 and λ > 0 such that
|x(t) − z(t)| ≤ K e−λ(t−t0 ) |x0 − z0 |
D. Fiore and M. di Bernardo
∀t ≥ t0 ,
∀x0 , z0 ∈ C
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Incremental stability and Contraction Theory
Definition
The smooth dynamical system x˙ = f (x) is said to be contracting on a set C ⊆ Rn if
there exists some norm in C, with associated matrix measure µ, such that, for some
constant c > 0, it holds
∂f
µ
(x) ≤ −c
∀x ∈ C, ∀t ≥ t0
∂x
Note: For example, the matrix measure µ induced by Euclidean norm is the maximum
∂f
.
eigenvalue of the symmetric part of ∂x
The basic result of nonlinear contraction analysis states that
If a system is contracting, then all of its
trajectories exponentially converge towards
each other, i.e. the system is IES.
∂f
µ
(x) ≤ −c ⇒ ∆x → 0
∂x
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Motivation
Still, most of the results available in the literature on both incremental stability and
contraction assume that the system vector field is continuously differentiable.
Only relatively few results deal with the problem of investigating incremental
stability of piecewise-smooth and switched dynamical systems.
See for example
Slotine et al. (1998) for constrained systems
Pavlov et al. (2005) for PWA systems
di Bernardo et al. (2011) for Carathéodory systems
di Bernardo and Liuzza (2013) for planar Filippov systems
The aim of our research is to extend the existing results to the case of
n-dimensional Filippov systems.
These theoretical results are useful in applications like
design PWS observers
proving convergence in networks of PWS systems
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Bimodal Filippov systems
A common class of PWS dynamical systems particularly important in many
applications is that of bimodal Filippov systems
A bimodal Filippov system is given by
(
f + (x) if h(x) > 0
x˙ =
f − (x) if h(x) < 0
where
f + (x) and f − (x) are two smooth vector fields
h(x) is a smooth scalar function whose zero set defines the smooth discontinuity
manifold Σ := {x ∈ Rn : h(x) = 0}
We want to find sufficient conditions for a such class of systems able to guarantee
incremental exponential stability on a subset C ⊂ Rn .
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The key idea
We want that every pairs of trajectories converge
towards each other in C independently of where
they start from, on the switching manifold Σ or
outside of it.
To achieve this we require that:
1
2
ˆ is globally attractive for all system solutions evolving in C\Σ
ˆ
The sliding region Σ
The sliding vector field is contracting, and therefore such that every pair of solutions
within the sliding region will exponentially converge towards each other
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ˆ to be attractive
1. Conditions for Σ
The switching manifold Σ is guaranteed to be attractive in C if the two vector fields
f + (x) and f − (x) point towards it in every point on C.
For example, assume Σ be a hyperplane with h(x) = hT (x − xh ) for some
constant vectors h, xh ∈ Rn .
If there exist two scalar l1 > 0 and l2 < 0 such that hT f − (x) ≥ l1 for all x ∈ C ∩ G −
and hT f + (x) ≤ l2 for all x ∈ C ∩ G + , then Σ is attractive for every solution starting
in C.
Indeed for ξ ∈ C ∩ G −
d
h(x) = hT f − (x) ≥ l1
dt
integrating we have that the flow reaches Σ at a time instant
ts ≤ −h(ξ)/l1
Similarly for ξ ∈ C ∩ G + .
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2. Contracting sliding vector field
In order for the sliding vector field f sl to be contracting we need to study the
Jacobian of
( sl
f (x) = α f + (x) + (1 − α) f − (x)
with α ∈ [0, 1]
f sl (x) · ∇h(x) = 0
As in general α(x) can have cumbersome expressions, we treat this system as a
constrained system.
We derive conditions for contraction in a
constrained coordinate set z ∈ Rn−1
such that
x = ϕ(z)
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Constrained coordinates
The idea is to:
1
consider a projector P(x) on the hyperplane Tx Σ tangent to the switching manifold Σ
2
move to the set of n − 1 constrained coordinates on Σ, x = ϕ(z).
We then require
∂f sl
µ
(z)
∂z
where
!
≤ −c
∂f sl (z)
∂ sl
=
P(x) · f (x) ∂z
∂x
·
x=ϕ(z)
∂ϕ
∂z
The derivation requires lots of algebra and can be found in di Bernardo, Fiore "Incremental stability of bimodal Filippov systems in Rn ", accepted to 53rd IEEE
Conference on Decision and Control, 2014.
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Incremental stability of bimodal Filippov systems
Theorem
Assume h(x) = hT (x − xh ), for some constant vectors h, xh ∈ Rn .
If there exists a forward invariant and path-connected subset C ⊂ Rn such that
ˆ 6= ∅ and the following conditions hold:
ΣC = C ∩ Σ
1
2
there exists two scalar l1 > 0 and l2 < 0 such that hT f − (x) ≥ l1 for all x ∈ C ∩ G −
and hT f + (x) ≤ l2 for all x ∈ C ∩ G + ;
sl sl
µ ∂f∂z ≤ −c for all z such that ϕ(z) ∈ ΣC , with c > 0, where ∂f∂z is the Jacobian
matrix of the sliding vector field restricted on Σ, and z ∈ Rn−1 are a set of
constrained coordinates on Σ;
then all trajectories of the system globally converge towards each other in C.
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Example
Let us consider the relay feedback system in R2
(
f − (x) = Ax + B if h(x) = Cx < 0
x˙ =
f + (x) = Ax − B if h(x) = Cx > 0
with
0
A=
1
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−1
,
−1
1
B=
,
1
C= 0
1
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Example - Region of incremental stability
Applying the Theorem to this system we can obtain a region C of incremental
stability.
This region is in fact such that
every trajectory starting inside will stay in it for all t > t0 (forward invariance)
both vector fields are directed towards Σ for all x
ˆ
the region is connected, bounded and it intersects the sliding region Σ
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Example - Simulation 1
Let us consider two initial conditions inside the region of incremental stability, one
in G + and the other in G − .
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Example - Simulation 1
Evolution in time of the distance between the trajectories.
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Example - Simulation 2
Let us consider two initial conditions both in G − or in G + .
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Example - Simulation 2
Evolution in time of the distance between the trajectories.
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Example - Remarks
As can been seen from the previous numerical simulations, the system is
incrementally exponentially stable in the green subset C.
It is worth to notice that the constants K ≥ 1 and λ in the definition of IES
|x(t) − z(t)| ≤ K e−λ(t−t0 ) |ξ − ζ|
∀t ≥ t0 ,
∀ξ, ζ ∈ C
appear to be related to the geometry of the region of incremental stability and to
the contraction rate c of the sliding vector field.
An open problem is how to analytically estimate the value of K and λ for a generic
system.
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Conclusions
Contraction theory is a powerful tool to assess incremental stability of a dynamical
system.
Unfortunately there are few results on this topic for PWS systems and, in
particular, for Filippov systems.
We have presented here a sufficient condition for incremental stability of Filippov
systems based on treating the system as a constrained system.
This condition has been defined in terms of the properties of
the subset C
the vector fields f + , f − and f sl
the switching manifold Σ
More work needs to be done to extend the theory to other classes of PWS
systems, like constrained systems, in a more general framework.
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