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Contents Linear time-varying systems and growth rates Constrained switching
The top Lyapunov exponentof switched linear systems with dwell times
Fabian Wirth
Institute of MathematicsUniversity of Wurzburg
The Dynamics of ControlOcotber 1–3, 2010.
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Contents Linear time-varying systems and growth rates Constrained switching
Linear time-varying systems and growth ratesGrowth ratesSwitched linear systemsIrreducibilityA converse Lyapunov theorem
Constrained switchingDwell timesA converse Lyapunov theoremAverage Dwell Time
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Contents Linear time-varying systems and growth rates Constrained switching
Growth rates
Families of Linear Time-Varying Systems
Consider a family of time-varying linear systems
x = A(u(t))x , t ≥ 0
x(0) = x0 ∈ Rn,
u :R+ → U measurable
The evolution operator is denoted by Φu(t, s), t ≥ s ≥ 0.
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Contents Linear time-varying systems and growth rates Constrained switching
Growth rates
Exponential growth rates
x = A(u(t))x , u : R+ → U measurable
The Lyapunov exponent of a solution is
λ(x0, u) = lim supt→∞
1
tlog ‖Φu(t, 0)x0‖,
Lyapunov exponents for periodic u are called Floquet exponents.
The upper Bohl exponent corresponding to u is
β(u) = lim supt,s→∞
1
tlog ‖Φu(t + s, s)‖
The collections of all Lyapunov, Floquet and Bohl exponents are
ΣLy(A,U), ΣFl(A,U), ΣBohl(A,U).
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Contents Linear time-varying systems and growth rates Constrained switching
Growth rates
Exponential growth rates
x = A(u(t))x , u : R+ → U measurable
The Lyapunov exponent of a solution is
λ(x0, u) = lim supt→∞
1
tlog ‖Φu(t, 0)x0‖,
Lyapunov exponents for periodic u are called Floquet exponents.
The upper Bohl exponent corresponding to u is
β(u) = lim supt,s→∞
1
tlog ‖Φu(t + s, s)‖
The collections of all Lyapunov, Floquet and Bohl exponents are
ΣLy(A,U), ΣFl(A,U), ΣBohl(A,U).
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Contents Linear time-varying systems and growth rates Constrained switching
Growth rates
Exponential growth rates
x = A(u(t))x , u : R+ → U measurable
The Lyapunov exponent of a solution is
λ(x0, u) = lim supt→∞
1
tlog ‖Φu(t, 0)x0‖,
Lyapunov exponents for periodic u are called Floquet exponents.
The upper Bohl exponent corresponding to u is
β(u) = lim supt,s→∞
1
tlog ‖Φu(t + s, s)‖
The collections of all Lyapunov, Floquet and Bohl exponents are
ΣLy(A,U), ΣFl(A,U), ΣBohl(A,U).
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Contents Linear time-varying systems and growth rates Constrained switching
Growth rates
Read this
F. Colonius, W. Kliemann. Infinite-time optimal control andperiodicity. Appl. Math. Optim. 1989.F. Colonius, W. Kliemann. Stability radii and Lyapunov exponents.Proc. Workshop Bremen. 1990.F. Colonius, W. Kliemann. Linear control semigroups acting onprojecitve space. J. Dyn. Diff. Equations. 1993.F. Colonius, W. Kliemann. Minimal and maximal Lyapunovexponents of nonlinear control systems. J. Diff. Equations. 1993.F. Colonius, W. Kliemann. The Lyapunov spectrum of families oftime-varying matrices. Trans. Amer. Math. Society. 1996.F. Colonius, W. Kliemann. The Morse spectrum of linear flows onvector bundles. Trans. Amer. Math. Society. 1996.
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Contents Linear time-varying systems and growth rates Constrained switching
Growth rates
The Colonius-Kliemann intuition
Considerx = A1x andx = A2xin R2.
![Page 9: The top Lyapunov exponent of switched linear systems …num.math.uni-bayreuth.de/en/conferences/irsee_2010/home/download/... · Contents Linear time-varying systems and growth rates](https://reader034.vdocuments.mx/reader034/viewer/2022052516/5aab82b47f8b9aa9488c1583/html5/thumbnails/9.jpg)
Contents Linear time-varying systems and growth rates Constrained switching
Growth rates
The Colonius-Kliemann intuition
Considerx = A1x andx = A2xin R2.
![Page 10: The top Lyapunov exponent of switched linear systems …num.math.uni-bayreuth.de/en/conferences/irsee_2010/home/download/... · Contents Linear time-varying systems and growth rates](https://reader034.vdocuments.mx/reader034/viewer/2022052516/5aab82b47f8b9aa9488c1583/html5/thumbnails/10.jpg)
Contents Linear time-varying systems and growth rates Constrained switching
Growth rates
The Colonius-Kliemann intuition
Project thesystems onto theunit sphereor ratherprojective space.
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Contents Linear time-varying systems and growth rates Constrained switching
Growth rates
The Colonius-Kliemann intuition
Project thesystems onto theunit sphereor ratherprojective space.
![Page 12: The top Lyapunov exponent of switched linear systems …num.math.uni-bayreuth.de/en/conferences/irsee_2010/home/download/... · Contents Linear time-varying systems and growth rates](https://reader034.vdocuments.mx/reader034/viewer/2022052516/5aab82b47f8b9aa9488c1583/html5/thumbnails/12.jpg)
Contents Linear time-varying systems and growth rates Constrained switching
Growth rates
The Colonius-Kliemann intuition
Project thesystems onto theunit sphereor ratherprojective space.
![Page 13: The top Lyapunov exponent of switched linear systems …num.math.uni-bayreuth.de/en/conferences/irsee_2010/home/download/... · Contents Linear time-varying systems and growth rates](https://reader034.vdocuments.mx/reader034/viewer/2022052516/5aab82b47f8b9aa9488c1583/html5/thumbnails/13.jpg)
Contents Linear time-varying systems and growth rates Constrained switching
Growth rates
The Colonius-Kliemann intuition
Project thesystems onto theunit sphereor ratherprojective space.
![Page 14: The top Lyapunov exponent of switched linear systems …num.math.uni-bayreuth.de/en/conferences/irsee_2010/home/download/... · Contents Linear time-varying systems and growth rates](https://reader034.vdocuments.mx/reader034/viewer/2022052516/5aab82b47f8b9aa9488c1583/html5/thumbnails/14.jpg)
Contents Linear time-varying systems and growth rates Constrained switching
Growth rates
The Colonius-Kliemann intuition
Project thesystems onto theunit sphereor ratherprojective space.
DC
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Contents Linear time-varying systems and growth rates Constrained switching
Growth rates
The Colonius-Kliemann spectrum of families oftime-varying matrices
Generically, there are a finite number of control sets. In particular,one invariant control set.To each control set we can associate a set of FLoquet exponentsand Lyapunov exponents.In particular we have the Gelfand formula
sup ΣFL = max ΣLy = max ΣBohl .
Assumptions: cl intU = cl U, local accessibility of the projectedsystem, ...
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Contents Linear time-varying systems and growth rates Constrained switching
Switched linear systems
We consider a finite set of matrices M = {A1, . . . ,Am} ⊂ Rn×n
and the associated switched linear system
x(t) = Aσ(t)x(t) (1)
whereσ : R→M
is the switching signal.
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Contents Linear time-varying systems and growth rates Constrained switching
Switched linear systems
Exponential growth rates
M = {A1, . . . ,Am} ⊂ Rn×n x(t) = Aσ(t)x(t) (1)
Options for defining a uniform exponential growth rate of (1):
I trajectory-wise (Lyapunov exponents):
κ := maxσ∈S,x0
{lim supt→∞
1
tlog ‖ϕ(t, x0, σ)‖
},
I using norms of evolution operators (Bohl exponents):
ρ := limt→∞
1
tlog max
σ∈S‖Φσ(t, 0)‖ ,
It is known by Fenichel’s uniformity lemma thatκ = ρ .
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Contents Linear time-varying systems and growth rates Constrained switching
Switched linear systems
Exponential growth rates
M = {A1, . . . ,Am} ⊂ Rn×n x(t) = Aσ(t)x(t) (1)
Options for defining a uniform exponential growth rate of (1):I trajectory-wise (Lyapunov exponents):
κ := maxσ∈S,x0
{lim supt→∞
1
tlog ‖ϕ(t, x0, σ)‖
},
I using norms of evolution operators (Bohl exponents):
ρ := limt→∞
1
tlog max
σ∈S‖Φσ(t, 0)‖ ,
It is known by Fenichel’s uniformity lemma thatκ = ρ .
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Contents Linear time-varying systems and growth rates Constrained switching
Switched linear systems
Exponential growth rates
M = {A1, . . . ,Am} ⊂ Rn×n x(t) = Aσ(t)x(t) (1)
Options for defining a uniform exponential growth rate of (1):I trajectory-wise (Lyapunov exponents):
κ := maxσ∈S,x0
{lim supt→∞
1
tlog ‖ϕ(t, x0, σ)‖
},
I using norms of evolution operators (Bohl exponents):
ρ := limt→∞
1
tlog max
σ∈S‖Φσ(t, 0)‖ ,
It is known by Fenichel’s uniformity lemma thatκ = ρ .
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Contents Linear time-varying systems and growth rates Constrained switching
Switched linear systems
Exponential growth rates
M = {A1, . . . ,Am} ⊂ Rn×n x(t) = Aσ(t)x(t) (1)
Options for defining a uniform exponential growth rate of (1):I trajectory-wise (Lyapunov exponents):
κ := maxσ∈S,x0
{lim supt→∞
1
tlog ‖ϕ(t, x0, σ)‖
},
I using norms of evolution operators (Bohl exponents):
ρ := limt→∞
1
tlog max
σ∈S‖Φσ(t, 0)‖ ,
It is known by Fenichel’s uniformity lemma thatκ = ρ .
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Contents Linear time-varying systems and growth rates Constrained switching
Switched linear systems
Arbitrary switching
We are considering
M = {A1, . . . ,Am} ⊂ Rn×n x(t) = Aσ(t)x(t) (1)
where the switching signal is any measurable function.An equivalent formulation considers the linear inclusion
x(t) ∈ {Ax(t) | A ∈M} .
Denote the exponential growth rate of this inclusion by ρ.
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Contents Linear time-varying systems and growth rates Constrained switching
Switched linear systems
Arbitrary switching
We are considering
M = {A1, . . . ,Am} ⊂ Rn×n x(t) = Aσ(t)x(t) (1)
where the switching signal is any measurable function.An equivalent formulation considers the linear inclusion
x(t) ∈ {Ax(t) | A ∈M} .
Denote the exponential growth rate of this inclusion by ρ.
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Contents Linear time-varying systems and growth rates Constrained switching
Irreducibility
IrreducibilityM is called irreducible, if only the trivial subspaces {0} and Kn areinvariant under all A ∈M and otherwise reducible.
Modulo a similarity transformation for reducible M all A ∈M areof the form
A11 A12 . . . . . . A1d
0 A22 A23 . . . A2d
0 0 A33...
.... . .
. . ....
0 . . . 0 Add
,
whereMii := {Aii ; A ∈M}
is irreducible or Aii = 0 for all A ∈M.
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Contents Linear time-varying systems and growth rates Constrained switching
Irreducibility
IrreducibilityM is called irreducible, if only the trivial subspaces {0} and Kn areinvariant under all A ∈M and otherwise reducible.Modulo a similarity transformation for reducible M all A ∈M areof the form
A11 A12 . . . . . . A1d
0 A22 A23 . . . A2d
0 0 A33...
.... . .
. . ....
0 . . . 0 Add
,
whereMii := {Aii ; A ∈M}
is irreducible or Aii = 0 for all A ∈M.
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Contents Linear time-varying systems and growth rates Constrained switching
A converse Lyapunov theorem
A converse Lyapunov theorem
Theorem (Barabanov 1988)If M is irreducible there exists a norm v on Kn such that for allx ∈ Kn, t ≥ 0:
∀σ : v(Φσ(t, 0)x) ≤ eρtv(x)
∃σ : v(Φσ(t, 0)x) = eρtv(x)
The proof relies in many ways on the fact that the set of evolutionoperators
{Φσ(t, s) | σ measurable , t ≥ s ≥ 0 }
is a semigroup.
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Contents Linear time-varying systems and growth rates Constrained switching
A converse Lyapunov theorem
A converse Lyapunov theorem
Theorem (Barabanov 1988)If M is irreducible there exists a norm v on Kn such that for allx ∈ Kn, t ≥ 0:
∀σ : v(Φσ(t, 0)x) ≤ eρtv(x)
∃σ : v(Φσ(t, 0)x) = eρtv(x)
The proof relies in many ways on the fact that the set of evolutionoperators
{Φσ(t, s) | σ measurable , t ≥ s ≥ 0 }
is a semigroup.
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Contents Linear time-varying systems and growth rates Constrained switching
A converse Lyapunov theorem
Three proofs of the Gelfand formula
I Berger, Wang, Linear Algebra and its Applications, 1992 -algebraic
I Elsner, Linear Algebra and its Applications, 1995 - usingBarabanov norms
I Shi, Wu, Pang, Linear Algebra and its Applications, 1997 -using Barabanov norms
note: continua of matrices not needed, acessibility not needed.
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Contents Linear time-varying systems and growth rates Constrained switching
Constrained Switching
So far we have dealt with unrestricted switching.There are many suggestions in the literature for restrictedswitching with some type of dwell-time condition
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Contents Linear time-varying systems and growth rates Constrained switching
Dwell TimeWe consider a finite set of matrices M = {A1, . . . ,Am} ⊂ Rn×n
and the associated switched linear system
x(t) = Aσ(t)x(t) (2)
whereσ : R→M
is the switching signal.
Now switching signals satisfying a dwell-time condition areconsidered, i.e. for h > 0 the set of admissible switching signals is
Sdwell(h) := {σ | σ is piecewise constant and
its discontinuities are at least h apart.}
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Contents Linear time-varying systems and growth rates Constrained switching
Dwell TimeWe consider a finite set of matrices M = {A1, . . . ,Am} ⊂ Rn×n
and the associated switched linear system
x(t) = Aσ(t)x(t) (2)
whereσ : R→M
is the switching signal.Now switching signals satisfying a dwell-time condition areconsidered, i.e. for h > 0 the set of admissible switching signals is
Sdwell(h) := {σ | σ is piecewise constant and
its discontinuities are at least h apart.}
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Contents Linear time-varying systems and growth rates Constrained switching
M = {A1, . . . ,Am} ⊂ Rn×n
x(t) = Aσ(t)x(t)
Switching signals look like this
≥ h
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Contents Linear time-varying systems and growth rates Constrained switching
Average Dwell Time
For t < T let Nσ(T , t) denote the number of discontinuities of σin [t,T ].Switching signals are said to have an average dwell time h if for allt < T it holds that
Nσ(T , t) ≤ N0 +T − t
h.
Sav (h,N0) := {σ : R→M | σ is piecewise constant and
satisfies the average dwell time condition.}
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Contents Linear time-varying systems and growth rates Constrained switching
Average Dwell Time
For t < T let Nσ(T , t) denote the number of discontinuities of σin [t,T ].Switching signals are said to have an average dwell time h if for allt < T it holds that
Nσ(T , t) ≤ N0 +T − t
h.
Here h is called the average dwell time andN0 is the chatter bound.
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Contents Linear time-varying systems and growth rates Constrained switching
Average Dwell Time - Switching Signals
M = {A1, . . . ,Am} ⊂ Rn×n, S = Sav (h,N0)
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Contents Linear time-varying systems and growth rates Constrained switching
A remark on topologyPropositionThe sets Sdwell(h) and Sav (h,N0) are compact in the weak∗
topology of L∞, resp. `∞.In particular, the systems
x(t) = Aσ(t)x(t)
together with the shift on Sdwell(h) resp. Sav (h,N0) are linearflows with compact base space.We therefore have ρ = eκ for these flows by Fenichel’s uniformitylemma.Conley’s theorem says that if 0 is uniformly exponentially stable,there exist a Lyapunov function for the set {0} × S.
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Contents Linear time-varying systems and growth rates Constrained switching
Construction of Lyapunov functions
x(t) = Aσ(t)x(t) , σ ∈ Sdwell(h),Sav (N0, h).
Remark 1: It is not reasonable to look for a single Lyapunovfunction V for this type of system:If one Lyapunov function exists such that
∇V (x)Ax < −α(‖x‖) , ∀A ∈M
and a positive definite function α, then the system is exponentiallystable with unrestricted switching. But we are interested in thecase that the switching is restricted.Remark 2: By imposing dwell-time conditions the stabilityproperties of a switched system can change.
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Contents Linear time-varying systems and growth rates Constrained switching
Long Dwell Times
PropositionLet M be a compact set of Hurwitz stable matrices, then thereexists a T > 0 such that for all h ≥ T the switched system
x(t) = Aσ(t)x(t) , σ ∈ Sdwell(h)
is uniformly exponentially stable.(easy consequence of results on slowly varying systems: Cesari1967, Desoer 1969)
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Contents Linear time-varying systems and growth rates Constrained switching
Converse Lyapunov Theorems with Dwell-Time
There is one fundamental problem in the construction of Lyapunovfunctions for systems with dwell time: The concatenation ofswitching signals.
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Contents Linear time-varying systems and growth rates Constrained switching
Concatenation
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Contents Linear time-varying systems and growth rates Constrained switching
Concatenation
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Contents Linear time-varying systems and growth rates Constrained switching
Concatenation
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Contents Linear time-varying systems and growth rates Constrained switching
Dwell times
Consider a finite set of matrices M = {A1, . . . ,Am} ⊂ Rn×n andthe associated switched linear system
x(t) = Aσ(t)x(t)
whereσ : R→M
is the switching signal.
We consider switching signals satisfying a dwell-time condition, i.e.for h > 0 the set of admissible switching signals is
S(h) := {σ | σ is piecewise constant and
its discontinuities are at least h apart.}
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Contents Linear time-varying systems and growth rates Constrained switching
Dwell times
Consider a finite set of matrices M = {A1, . . . ,Am} ⊂ Rn×n andthe associated switched linear system
x(t) = Aσ(t)x(t)
whereσ : R→M
is the switching signal.We consider switching signals satisfying a dwell-time condition, i.e.for h > 0 the set of admissible switching signals is
S(h) := {σ | σ is piecewise constant and
its discontinuities are at least h apart.}
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Contents Linear time-varying systems and growth rates Constrained switching
Dwell times
Reprise: The converse Lyapunov theorem for unrestrictedswitching
TheoremIf M is irreducible there exists a norm v on Rn such that for allx ∈ Rn, t ≥ 0:
∀σ : v(Φσ(t, 0)x) ≤ eρtv(x)
∃σ : v(Φσ(t, 0)x) = eρtv(x)
The proof relies in many ways on the fact that the set of evolutionoperators
{Φσ(t, s) | σ measurable , t ≥ s ≥ 0 }
is a semigroup.
This is not the case if a dwell time condition holds.
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Contents Linear time-varying systems and growth rates Constrained switching
Dwell times
Reprise: The converse Lyapunov theorem for unrestrictedswitching
TheoremIf M is irreducible there exists a norm v on Rn such that for allx ∈ Rn, t ≥ 0:
∀σ : v(Φσ(t, 0)x) ≤ eρtv(x)
∃σ : v(Φσ(t, 0)x) = eρtv(x)
The proof relies in many ways on the fact that the set of evolutionoperators
{Φσ(t, s) | σ measurable , t ≥ s ≥ 0 }
is a semigroup. This is not the case if a dwell time condition holds.
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Contents Linear time-varying systems and growth rates Constrained switching
A converse Lyapunov theorem
A Converse Lyapunov Theorem - Dwell TimesTheoremLet M = {A1, . . . ,Am} ⊂ Rn×n be irreducible and consider a dwelltime h > 0. The following two statements are equivalent
(i) ρ(M, h) = ρ,
(ii) there are norms v1, . . . , vm on Rn with the followingproperties:
vi (eAi tx) ≤ eρtvi (x) for all t ≥ 0, x ∈ Rn, i = 1, . . . ,m,
vj(eAj tx) ≤ eρtvi (x) for all t ≥ h, x ∈ Rn, i , j = 1, . . . ,m.
and for all x0 ∈ Rn there exists a σ such that
vσ(ti )(φ(t, x0, σ)) = eρtvσ(0)(x0) , ∀t ∈ [ti + h, ti+1], i .
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Contents Linear time-varying systems and growth rates Constrained switching
A converse Lyapunov theorem
A Converse Lyapunov Theorem - Dwell TimesTheoremLet M = {A1, . . . ,Am} ⊂ Rn×n be irreducible and consider a dwelltime h > 0. The following two statements are equivalent
(i) ρ(M, h) = ρ,
(ii) there are norms v1, . . . , vm on Rn with the followingproperties:
vi (eAi tx) ≤ eρtvi (x) for all t ≥ 0, x ∈ Rn, i = 1, . . . ,m,
vj(eAj tx) ≤ eρtvi (x) for all t ≥ h, x ∈ Rn, i , j = 1, . . . ,m.
and for all x0 ∈ Rn there exists a σ such that
vσ(ti )(φ(t, x0, σ)) = eρtvσ(0)(x0) , ∀t ∈ [ti + h, ti+1], i .
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Contents Linear time-varying systems and growth rates Constrained switching
A converse Lyapunov theorem
A Converse Lyapunov Theorem - Dwell TimesTheoremLet M = {A1, . . . ,Am} ⊂ Rn×n be irreducible and consider a dwelltime h > 0. The following two statements are equivalent
(i) ρ(M, h) = ρ,
(ii) there are norms v1, . . . , vm on Rn with the followingproperties:
vi (eAi tx) ≤ eρtvi (x) for all t ≥ 0, x ∈ Rn, i = 1, . . . ,m,
vj(eAj tx) ≤ eρtvi (x) for all t ≥ h, x ∈ Rn, i , j = 1, . . . ,m.
and for all x0 ∈ Rn there exists a σ such that
vσ(ti )(φ(t, x0, σ)) = eρtvσ(0)(x0) , ∀t ∈ [ti + h, ti+1], i .
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Contents Linear time-varying systems and growth rates Constrained switching
A converse Lyapunov theorem
A Converse Lyapunov Theorem - Dwell TimesTheoremLet M = {A1, . . . ,Am} ⊂ Rn×n be irreducible and consider a dwelltime h > 0. The following two statements are equivalent
(i) ρ(M, h) = ρ,
(ii) there are norms v1, . . . , vm on Rn with the followingproperties:
vi (eAi tx) ≤ eρtvi (x) for all t ≥ 0, x ∈ Rn, i = 1, . . . ,m,
vj(eAj tx) ≤ eρtvi (x) for all t ≥ h, x ∈ Rn, i , j = 1, . . . ,m.
and for all x0 ∈ Rn there exists a σ such that
vσ(ti )(φ(t, x0, σ)) = eρtvσ(0)(x0) , ∀t ∈ [ti + h, ti+1], i .
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Contents Linear time-varying systems and growth rates Constrained switching
A converse Lyapunov theorem
Outline of proof
Concatenation:
(u �t w)(s) :=
{u(s) , s < tw(s − t) , t ≤ s
.
For each i define the set of switching signals that can beconcatenated at time t to a switching signal that has been equalto i on (t − h, t).
S(i) := {σ ∈ Sh | σ(0) = i or t0(σ) ≥ h} .
Tt(i) := {Φσ(t, 0) | σ ∈ S (i)}.
Note Ti := ∪t≥0Tt(i) is not a semigroup.
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Contents Linear time-varying systems and growth rates Constrained switching
A converse Lyapunov theorem
Outline of proof
Concatenation:
(u �t w)(s) :=
{u(s) , s < tw(s − t) , t ≤ s
.
For each i define the set of switching signals that can beconcatenated at time t to a switching signal that has been equalto i on (t − h, t).
S(i) := {σ ∈ Sh | σ(0) = i or t0(σ) ≥ h} .
Tt(i) := {Φσ(t, 0) | σ ∈ S (i)}.
Note Ti := ∪t≥0Tt(i) is not a semigroup.
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Contents Linear time-varying systems and growth rates Constrained switching
A converse Lyapunov theorem
Outline of proof
Concatenation:
(u �t w)(s) :=
{u(s) , s < tw(s − t) , t ≤ s
.
For each i define the set of switching signals that can beconcatenated at time t to a switching signal that has been equalto i on (t − h, t).
S(i) := {σ ∈ Sh | σ(0) = i or t0(σ) ≥ h} .
Tt(i) := {Φσ(t, 0) | σ ∈ S (i)}.
Note Ti := ∪t≥0Tt(i) is not a semigroup.
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Contents Linear time-varying systems and growth rates Constrained switching
A converse Lyapunov theorem
Outline of proof II
Tt(i) := {Φσ(t, 0) | σ ∈ S (i)}.If M is irreducible, then
Vi :=⋃t≥0
ρ−tTt(i)
is bounded and irreducible.
Thenwi (x) = sup{‖Φx‖ | Φ ∈ Vi}
defines Lyapunov functions with the property
wi (eAi tx) ≤ eρtwi (x) for all t ≥ 0, x ∈ Rn, i = 1, . . . ,m,
wj(eAj tx) ≤ eρtwi (x) for all t ≥ h, x ∈ Rn, i , j = 1, . . . ,m.
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Contents Linear time-varying systems and growth rates Constrained switching
A converse Lyapunov theorem
Outline of proof II
Tt(i) := {Φσ(t, 0) | σ ∈ S (i)}.If M is irreducible, then
Vi :=⋃t≥0
ρ−tTt(i)
is bounded and irreducible.Then
wi (x) = sup{‖Φx‖ | Φ ∈ Vi}defines Lyapunov functions with the property
wi (eAi tx) ≤ eρtwi (x) for all t ≥ 0, x ∈ Rn, i = 1, . . . ,m,
wj(eAj tx) ≤ eρtwi (x) for all t ≥ h, x ∈ Rn, i , j = 1, . . . ,m.
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Contents Linear time-varying systems and growth rates Constrained switching
A converse Lyapunov theorem
Outline of proof III
This is not enough!!
Tt(i) := {Φσ(t, 0) | σ ∈ S (i)}.
If M is irreducible, then
T∞(i) := lim supt→∞
e−ρtTt(i)
is compact and irreducible.The norms
vi (x) = max{‖Φx‖ | Φ ∈ T∞(i)}
define the norms for which all assertions hold.
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Contents Linear time-varying systems and growth rates Constrained switching
A converse Lyapunov theorem
Outline of proof III
This is not enough!!
Tt(i) := {Φσ(t, 0) | σ ∈ S (i)}.
If M is irreducible, then
T∞(i) := lim supt→∞
e−ρtTt(i)
is compact and irreducible.The norms
vi (x) = max{‖Φx‖ | Φ ∈ T∞(i)}
define the norms for which all assertions hold.
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Contents Linear time-varying systems and growth rates Constrained switching
A converse Lyapunov theorem
A Common Quadratic Lyapunov VersionConsider a finite set of matrices M = {A1, . . . ,Am} ⊂ Rn×n andthe associated switched linear system
x(t) = Aσ(t)x(t) , σ ∈ Sdwell(h) .
Theorem(Colaneri, Geromel, 2005)If there are positive definite matrices Pi > 0, i = 1, . . . ,m suchthat
(i) ATi Pi + PiAi < 0, i = 1, . . . ,m,
(ii)
eATj hPje
Ajh < Pi , i , j = 1, . . . ,m .
then the switched system is exponentially stable.Remark For Hurwitz matrices, the condition of the theorem canalways be satisfied if h is large. So again large enough dwell timesensure stability.
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Contents Linear time-varying systems and growth rates Constrained switching
A converse Lyapunov theorem
Consequences of the Converse Lyapunov Theorem
With the existence of the Lyapunov functions for systems withdwell time, the following results may be proved.
(i) ρ(M) = ρ(cl convM) is no longer true.
(ii) The maximal exponential growth rate may be approximatedby periodic switching signals – Gelfand formula
(iii) ρ is jointly continuous in M and h.
(iv) ρ is locally Lipschitz continuous in M and h, for Mirreducible.
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Contents Linear time-varying systems and growth rates Constrained switching
Average dwell time
Average Dwell Time
For t < T let Nσ(T , t) denote the number of discontinuities of σin [t,T ].Switching signals are said to have an average dwell time h if for allt < T it holds that
Nσ(T , t) ≤ N0 +T − t
h.
Sav (h,N0) := {σ : R→M | σ is piecewise constant and
satisfies the average dwell time condition.}
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Contents Linear time-varying systems and growth rates Constrained switching
Average dwell time
Average Dwell Time
For t < T let Nσ(T , t) denote the number of discontinuities of σin [t,T ].Switching signals are said to have an average dwell time h if for allt < T it holds that
Nσ(T , t) ≤ N0 +T − t
h.
Here h is called the average dwell time andN0 is the chatter bound.
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Contents Linear time-varying systems and growth rates Constrained switching
Average dwell time
Average Dwell Time - Switching Signals
M = {A1, . . . ,Am} ⊂ Rn×n, S = Sav (h,N0)
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Contents Linear time-varying systems and growth rates Constrained switching
Average dwell time
Is this really more general ?
A system stable with dwell time h may be destabilized byincreasing the chatter bound:Consider switching between two Hurwitz matrices. There areexamples where such systems are just unstable ρ = 0 and thedestabilizing switching is periodic with two switches.
hh
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Contents Linear time-varying systems and growth rates Constrained switching
Average dwell time
Is this really more general ?
In this situation the system is stable with respect to the dwell timeh, but not stable with respect to the set Sav(h, 2).
hh
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Contents Linear time-varying systems and growth rates Constrained switching
Average dwell time
Gelfand formula
A converse Lyapunov theorem for the class Sav(h,N0) is notknown.
But:The Gelfand formula
sup ΣFl = ρ
may still be shown, using new techniques from ergodic theory.I’s still to be dotted and t’s to be crossed.
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Contents Linear time-varying systems and growth rates Constrained switching
Average dwell time
Gelfand formula
A converse Lyapunov theorem for the class Sav(h,N0) is notknown.But:
The Gelfand formulasup ΣFl = ρ
may still be shown, using new techniques from ergodic theory.I’s still to be dotted and t’s to be crossed.
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Contents Linear time-varying systems and growth rates Constrained switching
Average dwell time
Gelfand formula
A converse Lyapunov theorem for the class Sav(h,N0) is notknown.But:The Gelfand formula
sup ΣFl = ρ
may still be shown, using new techniques from ergodic theory.
I’s still to be dotted and t’s to be crossed.
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Contents Linear time-varying systems and growth rates Constrained switching
Average dwell time
Gelfand formula
A converse Lyapunov theorem for the class Sav(h,N0) is notknown.But:The Gelfand formula
sup ΣFl = ρ
may still be shown, using new techniques from ergodic theory.I’s still to be dotted and t’s to be crossed.
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Contents Linear time-varying systems and growth rates Constrained switching
Average dwell time
Conclusions
I In stability theory of linear time-varying systems puzzles stillabound – and new ones are invented all the time.
I Fritz and Didi have put me on the tracks of a subject thatcontinues to fascinate me.
I At a certain point stop followong your teachers as far astechniques are concerned.
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Contents Linear time-varying systems and growth rates Constrained switching
Average dwell time