Some examples of time-delaysystems

I. Fluid flow model for a congested router in TCP/AQM controlled network

pTC

tQtR

QCtR

tWtN

QCtR

tWtN

tQ

tRtptRtR

tRtWtW

tRtW

+=

=

−

>−=

−−

−−=

)()(

0,0,)(

)()(max

0)(

)()(

)(

))(())((

))(()(

2

1

)(

1)(

ɺ

ɺ

Hollot et al., IEEE TAC 2002Model of collision-avoidance type:

W: window-sizeQ: queue lengthN: number of TCP sessionsR: round-trip-timeC: link capacityp: probability of packet markTp: propagation delay

Interpretation of AQM as a feedback control problem: )(Qfp=

Sender ReceiverBottleneck router

link c

rtt R

queue Q

acknowledgement

packet marking

We assume: - N constant, R is constant, p=K Q

Normalization of state and time

=

−

>−=

−−−=

0,0,)(

max

0)(

)(

)()()(

2

11)(

QCR

tWN

QCR

tWN

tQ

RtQKR

RtWtW

RtW

ɺ

ɺ

( )

=−>−

=

−−−=

0,0,)(max

0)()(

)1()1()(2

11)(

qctw

qctwtq

tqktwtwtw

ɺ

ɺ

RttN

QqWwoldnew )()(,, ===

KNkN

RCc == ,

4 parameters

2 parameters

CRNK ,,,

( )

=−>−

=

−−−=

0,0,)(max

0)()(

)1()1()(2

11)(

qctw

qctwtq

tqktwtwtw

ɺ

ɺ

)2,(),( 2**

kccqw =

0)1(~2

)1(~1)(~1

)(~2

=−+−++ tqkc

tqc

tqc

tq ɺɺɺɺ

Unique steady state solutionLinearization:

Linearized model

02

1)(

1)(

22 =+++ −− λλ e

kceλ

ctλ

ctλ

II. A car following system

Car following model in a ring configuration

speed vk-1

speed vk

Simplest model:

Refinements:- taking multiple cars into account- distribution of the delay

0 2 4 6 8 100

0.05

0.1

ξ

f(ξ)

gap τ

Possible choice for f: a gamma distribution with a gap

( , , )T nτthree parameters:

k-1

k

Teξ τ−−

∼

System consisting of p agents, each described by an integrator:

Directed, time-invariant communication graph:

Node set 1,…,pSet of vertices E: Weighted adjacency matrixStrongly connected

,( , ) 0k lk l E α∈ ⇔ ≠,: diagonal entries zero, non-diagonal entries k lαA

Interpretation as a consensus protocol

( ) ( ),

( ) ( ), 1, ,k k

k k

v t u t

y t v t k p

== =

ɺ

…

Consensus protocol:

( ),( , ) 0

( ) ( ) ( ) ( ) , 1, ,k k l l kk l E

u t f y t y t d k pα θ θ θ θ∞

∈

= − − − =∑ ∫ …

Successive passage of teeth

⇒ delay Rotation of each tooth

⇒ periodic coefficients

Cutting process Successive passage of the same point of the piece

⇒ delay Orientation of tooth w.r.t.

workpiece is fixed ⇒ constant coefficients

workpiece(fixed / translates)

tool (rotates)

Milling process

0

( ) ( ) ( ) ( ) ( ( ))

( ) ( )

x t A ωt x t B ωt x t τ t

τ t τ δ f t

= + − = + Ω

ɺ

unstable steady state chatter or oscillations of workpiece/tool irregular surface

Both cases: speed determinesdelay

III. Rotating cutting and milling machines

tool(fixed)

Workpiece(rotates)

speed

time

Fast modulation of rotational machine speed, N, around the nominal valueA measure to improve stability and prevent chatter:

Variable speed machines

)(1~)( tNtτsince

Modulating the machine speed= modulating the delay in the model

(see work of Jayaram,Sexton,Stone, etc.)

! Stabilizing effect of delay variation !

IV. Heating system

Linear system of dimension 6, 5 delays,,

Goal of feedback: achieving asymptotic stability, and maximizing response time

temperature to be controlledsetpoint

(PhD Thesis Vyhlidal, CTU Prague, 2003)

,

,

( ) ( ) ( ) ( )

1 1( ) ( ) ( ) ( ) ( ) ( )

2 2

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

h h h h b a b u h set u

a a a c e a h a c e

d d d d a d

c c c c c d c

e c set c

T x t x t K x t K x t

q qT x t x t x t K x t x t x t

T x t x t K x t

T x t x t K x t

x t x t x t

η τ τ

τ τ

τη τ

= − − + − + − + − = − + − + − − − = − + − = − − + −

= −

ɺ

ɺ

ɺ

ɺ

ɺ

,

T

h set h a d c ex K x x x x x =

System

Control law (PI+ state feedback)

Computation of characteristic rootsand stability regions

Operators associated to a delay equation

0 max1( ) ( ) ( ), ( ) , max

m ni i ii i

x t A x t A x t x tτ τ τ=

= + − ∈ =∑ɺ ℝ

0, ( )t tx x x= ∈A D A0( ) , 0tx t x t= ≥T

[ ]max( ,0 , ),nϕ τ∈ − ℝC

Reformulation of the DDE over

mapping abstract ‘ODE’

Initial condition is a function segment

[ ]max( ,0 , ),nτ− ℝC

[ ]max, ( )( )t x tτ ϕ∈ − ∞ →Let be the forward solution with initial condition ϕ and let

[ ]max( ) ( ), ,0tx x tϕ θ θ τ= + ∈ −

T(t) : solution (time-integration) operator over interval t

A : infinitesimal generator of T(t)

( ) max max

1

( ) ([ ,0]) : continuous on ,0 and

(0) (0) ( ) ,

, ( ).

m

i i ii

A A

ϕ τ ϕ τ

ϕ ϕ ϕ τ

ϕ ϕ ϕ=

= ∈ − −

= + −

= ∈

∑

D A C

A D A

ɺ

ɺ

ɺ010

( ) ( )

( ), 0,

(0) ( ( ) )(0) ( ( ) )( ) , 0t m

i ii

t

t t

A s A s ds tθ

ϕ θϕ θ θ

ϕ ϕ ϕ τ θ+

=

=+ + ≤

+ + − + >

∑∫

T

T T

0maxτ−

ϕϕϕϕ

Spectral propertiesλ is a characteristic root if and only if it satisfies the characteristic equation

( ) ( ( )),te P tλλ σ σ∈ ⇒ ∈A T

( )( ( )) exp ( )t tσ σ=T A

01

\ 0 :

) 0i

n n

m

ii

v

I A A e vλτλ

×

−

=

∃ ∈

− − =

∑

ℂ

( ) 0 ( )H λ λ σ= ⇔ ∈ A

[ ]max, ,0veλθ θ τ∈ −eigenfunction

finite-dimensional nonlinear eigenvalue problem

infinite-dimensional linear eigenvalue problemsfor A and T(t)

(σ(.): spectrum, Pσ(.): point-spectrum)

01

( ) 0, ( ) : det ,i

m

ii

H H I A A e λτλ λ λ −

=

= = − −

∑

or equivalently

Properties

( ) ( ),P Aσ σ=A

eigenfunction [ ]max, ,0veλθ θ τ∈ −

Characteristic roots,eigenvalues of A

Eigenvalues of T(1)exp(.)

−1 0 1 1.5

−1

0

1

Real axis

Imag

inar

y ax

is

−3 −2 −1 0 1−100

−50

0

50

100

Real axis

Imag

inar

y ax

is

Mapping is not one-to-one

But: characteristic roots can be obtained from σ(T(t)) by computing also the corresponding eigenfunction

Two-stage approach to compute characteristic roots

1a. Discretize A or T(t) , with t fixed, into a matrix

2. Correct the approximate characteristic roots with Newton iterations on the characteristic equation, up to the desired accuracy

Discretizing T(t)- linear multi-step methods (Engelborghs et al.)- subspace iteration (Engelborghs at al)- spectral collocation (Verheyden et al.)- Chebychev expansion (Butcher, Bühler et al.)- semi-discretization (Stepan et al.)

Discretizing A (Breda et al)

1b. Compute the (rightmost or dominant) eigenvaluesof this matrix

Routine in the Matlab package DDE-BIFTOOL

- Linear multi-step method to discretize T(h), combined with Lagrange interpolation to evaluate delayed terms

- Newton correction- Automatic choice of discretization steplength h, to capture all the

characteristic roots in a given half plane, possible

+ uncorrected rootso corrected roots

Pseudospectra and stability radii of nonlinear eigenvalue problems,

with application to time-delay systems

Overview

PseudospectraApproaches to exploit structure of nonlinear

eigenvalue problems

via structured matrix perturbations by redefining pseudospectra

Emphasis on computable expressions

Numerical examplesConcluding remarks

Pseudospectra

1( ) ( ) : ( , ) ,ε λ λ

ε Λ = Σ ∪ ∈ >

A A R AC

1( , ) ( ) :A Iλ λ −= −R A

resolvent

ε-pseudospectrum of an operator Ad

x xdt

=

A(or system

computable as level sets of resolvent norm

( ) ( ) 0,forsome withε λ δ δ δ εΛ = ∈Σ + = <A A A A A

( ) :Σ ⋅ spectrum

−6 −4 −2 0 2 4 6−50

0

50

ℜ(λ)

ℑ(λ

)

(a)

ℜ(λ)

ℑ(λ)

−6 −4 −2 0 2 4 6−50

0

50(b)

spectrum pseudospectra

Stability radius

- partitionate the complex plane into disjunct sets, d u=C C C∪- Assume that ( ) dΣ ⊆A C

under mild conditions:

uC

x

x

x

x

xxx

dC

x

dΓC infinity

general formula:

2

1

1

1

1

inf inf 0 : ( ) for some satisfying ,

sup ( )

sup ( )

du

u

Cd

r

I

I

λ

λ

λ

ε λ δ δ δ ε

λ

λ

∈

−−

∈

−

−

∈Γ

= ≥ ∈ Σ + <

= −

= −

A A A A

A

A

CC

C

desired regioncf. stability

: sufficient to scan boundary

• vibrating system

• time-delay system

• …

Application of above definition to systems goverened by lineardifferential equations requires a formulation in a first order form:

( ) 0det

0)()()(2 =++→

=++

KCM

txKtxCtxM

λλɺɺɺ

1 11 1

2 2

( ) ( )0

( ) ( )

x t x tI

x t x tM K M C− −

= − −

ɺ

ɺ

invertibleM

: ( ), [ , 0]

t t

t

dx x

dt

x x t θ θ τ

=

= + ∈ −

A

Relation with perturbations of coefficient matrices ???

A: infinitesimal generator of solution operator

( ) 0det

)()()(

=−−→−+=

−λτλτ

BeAI

tBxtAxtxɺ

Approaches for exploiting structure

2. Redefine ε-pseudospectra of nonlinear eigenvalue problems (Michiels et al, inspired by Tisseur et al.)

:

0

( ) ( ),m

n ni i i

i

F A p Aλ λ ×

=

= ∈∑ C

entire functions

1. Structured perturbations (Hinrichsen & Kelb,…)

1 11 1

2 2

( ) ( )0

( ) ( )A

x t x tI

x t x tM K M C− −

= − −

ɺ

ɺ

[ ] 1 11

2 2

( ) ( )0

( ) ( )EA

D

x t x tA K C I

x t x tMδ

δ δ−

= + −

ɺ

ɺ

( ; , ) : ( ) 0,forsome withD E Dε λ δ δ δ εΛ = ∈ Σ + Ε = <A A A A AC

1( ; , ) ( ) : ( , ) ,D E E Dε λ λ

ε Λ = Σ ∪ ∈ >

A A R AC

0Mδ =

( ) : det( ( ) 0F Fλ λΣ ∪ ∈ =C

0)()(det0

=

+∑=

m

iiii pAA λδ

- perturbation class

),,(: 0 mAA δδ …=∆- measure on the combined perturbation

, 0, ,n niA i mδ ×∈ =C …

[ ]pmm AwAw δδ ⋯00glob

=∆

pmm Aw

Aw

=∆δ

δ⋮

00

glob

21

100

glob

ppmm

p

Aw

Aw

=∆δ

δ⋮

(1)

(2)

(3)

1 2, ,

: weightsi

p p p

w

+

+

∈ +∞

∈ +∞

R

R

∪

∪

=<⇔<∆∞=

miAw

p

pii ,,0,

:

1glob

2

…εδε

0

( ) ( 1)

glob

:det ( ) ( ) 0,for some

with

m

i i ii

n n m

A A pε λ δ λ

ε

=

× × +

Λ = ∈ + =

∆∈ ∆ <

∑C

C

1

0

1( ) : ( ) ( )

m

i ii

F A p wε βα

λ λ λε

−

=

Λ = Σ ∪ ∈ > ∑C

=

mm wp

wp

w

/)(

/)(

)(00

λ

λλ ⋮

,111

,,

,111

,,

,,

2221 =+==

=+==

==

qpqp

qpqp

pp

βα

βα

βα (1)measureonperturbati(2)measureonperturbati

(3)measureonperturbati

where

Computable expressions

- computation of pseudospectra contours as level sets of function f

- structure is fully exploited !!

∑=

m

iii pA

0

)(λ has dimension n x n !

( ) ( )12 2

2

1: 1M C Kε λ λ λ λ λ

ε− Λ = ∈ + + + + >

C

( ) ( ) ( ) ( ) ( ) 0M M x t C C x t K K xδ δ δ+ + + + + =ɺɺ ɺ

n-by-n matrix

( )1

01

2

1: 1i

m

ii

I A A e eλτ λτε λ λ

ε

−− −

=

Λ = ∈ − − + >

∑C

( ) ( ) ( ) ( ) ( )x t A A x t B B x tδ δ τ= + + + −ɺ

Based on combining the above approaches

0

det ( ) 0m

i ii

A p λ=

= ∑- exploiting the structure of the nonlinear eigenvalue problem,

- imposing structure on perturbations of the coefficient matrices

Examples (in both cases: ):

glob 2max i

iAδ⋅ =

3. Structered pseudospectra of nonlinear eigenvalue problems

What type of structure do we need?

1.) Structural dynamics application (mass-spring system)

1 1 4 6 4 6

2 4 2 4 5 5

3 6 5 3 5 6

0 0

0 0 ( ) ( ) 0;

0 0M K

m k k k k k

m x t k k k k k x t

m k k k k k

+ + − − + − + + − = − − + +

ɺɺ

2.) Laser physics application:

1

0

0 0

( ) ( ) 0 0 ( ) 0;

0 0 0A

g

x t A x g x t τ = + − − =

ɺ

[ ] [ ] [ ]

2

2

1 1 4

( )

1 1

( ) 0 1 0 0 0 1 0 0 1 1 1 0

0 0 0

F M K

F m k k

λ λλ

δ λ δ δ δ

= +

= + + − − +

⋯ ⋯ ⋯

det( ( )) 0 )F λ =( nominal char. eqn.:

0 1

0

( )

1 000

( ) 0 100

0 0

F I A A e

eF A g

e

λτ

λτ

λτ

λ λ

δ λ δ δ

−

−

−

= − −

= − − −

rank 2scalar

, uncertaini im k

0 , uncertainiA g

3.) Systems with multiplicative uncertainty:principle: ( ) ( ) ( ) ( )( ) ( )x t A A x t B B C C x tδ δ δ τ= + + + + −ɺ

( ) ( ) ( ) ( ) ( )

0 ( ) ( ) ( )

x t A A x t B B y t

C C x t y t

δ δδ τ

= + + + = + − −

ɺ

[ ] [ ]

( )

0( ) 0 0 0

0 0

I A BF

Ce I

I IF A I B I C e I

I

λτ

λτ

λλ

δ λ δ δ δ

−

−

− − = −

= − − −

det( ( )) 0F λ =

full blockuncertainy

11

( ) ( ) ( ) ( ) ( ) (1)s

f

j j j j j jjj

F D E d G Hδ λ λ λ λ λ=

=

= ∆ +∑ ∑scalaruncertainty

2

( ) : det( ( ) ( )) 0 for some ( ) of the form (1)

with , 1, , and , 1, ,

s

j j

F F F F

j f d j s

ε λ λ δ λ δ λ

ε ε

Λ = ∈ + =

∆ < = < =

C

… …

Definition of structured ε-pseudospectrum:

In many cases (including the above):

Nominal system:

pseudospectra boundaries computable as level sets of thefunction

to some extent reformulation of problem: efficiency depends on computation / approximation of structured singular value associated with the uncertainty structure.

Computational expressions

11

det( ( )) 0,

( ) ( ) ( ) ( ) ( ), ,j

sf l

j j j j j j j jjj

F

F D E d G H d

λ

δ λ λ λ λ λ ×

==

=

= ∆ + ∆ ∈ ∈∑ ∑ k

C C

1( ) : ( ( )) , wheres F C Tε λ µ λ

ε∆ Λ = ∈ > 1

11 1

1

( )

( )( ) ( ) [ ( ) ( ) ( ) ( )],

( )

( )

f

f s

s

E

ET F D D G G

H

H

λ

λλ λ λ λ λ λ

λ

λ

−

=

⋮

⋯ ⋯

⋮

1 1 s idiag( , , ,d I, ,d I): , ,

1 , 1 .

i ilf jd

i f j f

×∆ = ∆ ∆ ∆ ∈ ∈

≤ ≤ ≤ ≤

kC C… …

General formula:

( ( ))Tλ µ λ∆→

T(λ)

∆

Proof:

Special cases:

1( ) ( ) ( ) ( ), , 1, , : entire functions

f

j j jjF D E q q j fδ λ λ λ λ

== ∆ =∑ …

( ) ( )1

12

1( ) : ( ) ( ) ( ) ( )

fsjj

F E F D qε λ λ λ λ λε

−=

Λ = ∈ >

∑C

structured singular value reduces to 2-norm small dimension of 1( ) ( ) ( )E F Dλ λ λ−

This illustrates the typical trade-off between ‘realism’ of chosen perturbation structure and computational efficiency

1 1,real

2 1 1 10

`

( ( ) ( ) ( )) ( ( ) ( ) ( ))( ) : inf

( ( ) ( ) ( )) ( ( ) ( ) ( ))

1( )

s

f

jj

E F D E F Dj j

E F D E F D

q

ε γ

λ λ λ γ λ λ λλ λ σγ λ λ λ λ λ λ

λε

− −

− − −>

=

ℜ ℑΛ ∩ = ∈ ⋅ ℑ ℜ

>

∑

R R

In addition: qj even, j=1,…,f:

Example:0 0

( ) ( ), ( ) ( )m m

i i i ii i

F A p F A pλ λ δ λ δ λ= =

= =∑ ∑

−0.4 0 0.4−3.5

0

3.5

−0.4 0 0.4−3.5

0

3.5

ℜ(λ)

ℑ(λ)

ℜ(λ)

ℑ(λ)

(a) (b)

Examples

Mass spring system1 1 4 6 4 6

22 4 2 4 5 5

3 6 5 3 5 6

0 0

( ) 0 0

0 0M K

m k k k k k

F m k k k k k

m k k k k k

λ λ+ + − −

= + − + + − − − + +

unstructured pseudospectra

−0.4 0 0.4−3.5

0

3.5

ℜ(λ)

ℑ(λ)

structured pseudospectra

eigenvalues of 2000 simulations of associated random eigenvalue problem

structure of F exploitedstructure of M and K not exploited

−20 −5 100

50

100

ℜ(λ)

ℑ(λ)

−20 −5 100

50

100

−20 −5 100

50

100

ℜ(λ)

ℑ(λ)

ℜ(λ)

ℑ(λ)

(a) (b)

Laser problem

eigenvalues of unperturbed system

structured pseudospectra unstructured pseudospectra

1

0

0 0

( ) 0 0

0 0 0A

g

F I A g e λτλ λ −

= − − −

decay dueto rank increase of A

1

f=s=1:ssv computable viaconvex optimization

Extension to time-varying perturbations

Underlying ideas: L2 gain analysis and Parceval’s theorem

( ) 11

20

( ) ( )( ( )

( )

( )

max ( )

x t A A x t

F I A

F A

r j I Aω

δλ λ

δ λ δ

ω−

−−

≥

= += −

= −

= −C

ɺ2

0( ) ( ( )) ( ( ), sup ( )

t

x t A A t x t A t Mδ δ≥

= + =ɺ

frequency domain

1

1

( ) ( ) ( )

( ) ( )

x t Ax t u t

y t x t

= +=

ɺ

2 2( ) ( ) ( )y t A t u tδ= −

1u

2u

1y

2y

feedback system interconnection is stable if

( ) ( )( )

1 2

1 22 2

11 1

2 20 0

11

200

1

max ( ) 1 max ( )

sup ( ) max ( )

y yu u

it

j I A M M j I A

A t j I A

ω ω

ω

ω ω

δ ω

−− −

≥ ≥

−−

≥≥

<

− < ⇔ < −

⇔ < −

L LG G

feedback interconnection interpretation:

time domain

Extension to systems with time-varying delays

0 0( ) ( ( )) ( ( )) ( ( ))i i i ix t A A t A A t x tδ δ τ δτ= + + + − +∑ɺ

+ weighted combined measure of perturbations,globi

Lower bounds on stability radii can be derived using the following principles:

- exploiting structure of nonlinear eigenvalue problem- linearizing the uncertainty (transformation to a descriptor system / feedback

interconnection interpretation)- embedding the uncertainty due to delay perturbations in a larger class

time domain frequency domain(time-invariant perturbations)

( ) ( ( ( ))) ( )z t x t t x tτ δτ τ= − + − −( ( ))

( ) , ( ) ( )t t

t

y s ds y t x tτ δτ

τ

− +

−

= =∫ ɺ

7( ) ( )

4z t y tµ≤

2 2L L

| ( ) |tδτ µ≤

( 1)( ) ( )

e eZ Y

λτ λδτ

λ λλ

− − −=

( 1)( ) ( ) ( )

j je ez t y t y t

j

ωτ ωδτ

µω ∞

− − −≤ ≤2 2 2L L L

H