The top Lyapunov exponent of switched linear systems...

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.


Linear time-varying systems and growth ratesGrowth ratesSwitched linear systemsIrreducibilityA converse Lyapunov theorem

Constrained switchingDwell timesA converse Lyapunov theoremAverage Dwell Time


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.


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).


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.


Growth rates

The Colonius-Kliemann intuition

Considerx = A1x andx = A2xin R2.


Growth rates


Project thesystems onto theunit sphereor ratherprojective space.


Growth rates


Project thesystems onto theunit sphereor ratherprojective space.

DC


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, ...


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.




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 .




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 .



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.


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.


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.


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.



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.


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


Dwell TimeWe consider a finite set of matrices M = {A1, . . . ,Am} ⊂ Rn×n


x(t) = Aσ(t)x(t) (2)

whereσ : R→M


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.}


Dwell TimeWe consider a finite set of matrices M = {A1, . . . ,Am} ⊂ Rn×n


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



M = {A1, . . . ,Am} ⊂ Rn×n

x(t) = Aσ(t)x(t)

Switching signals look like this

≥ h


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.}


Average Dwell Time


Nσ(T , t) ≤ N0 +T − t

h.

Here h is called the average dwell time andN0 is the chatter bound.


Average Dwell Time - Switching Signals

M = {A1, . . . ,Am} ⊂ Rn×n, S = Sav (h,N0)


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.


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.


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)


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.


Concatenation


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


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



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



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:





is a semigroup.

This is not the case if a dwell time condition holds.


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:





is a semigroup. This is not the case if a dwell time condition holds.



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 .



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.



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.



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.



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.



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.



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.


Average dwell time

Average Dwell Time


Nσ(T , t) ≤ N0 +T − t

h.

Sav (h,N0) := {σ : R→M | σ is piecewise constant and

satisfies the average dwell time condition.}


Average dwell time

Average Dwell Time


Nσ(T , t) ≤ N0 +T − t

h.

Here h is called the average dwell time andN0 is the chatter bound.


Average dwell time

Average Dwell Time - Switching Signals

M = {A1, . . . ,Am} ⊂ Rn×n, S = Sav (h,N0)


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


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


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.


Average dwell time

Gelfand formula

A converse Lyapunov theorem for the class Sav(h,N0) is notknown.But:

The Gelfand formulasup ΣFl = ρ



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.


Average dwell time

Gelfand formula

A converse Lyapunov theorem for the class Sav(h,N0) is notknown.But:The Gelfand formula

sup ΣFl = ρ



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.


Average dwell time

The top Lyapunov exponent of switched linear systems...

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