Some examples of time-delaysystems

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

pTCtQ

tR

QCtRtW

tN

QCtRtW

tNtQ

tRtptRtR

tRtWtWtR

tW

+=

=

>

=

=

)()(

0,0,)()()(max

0)()()(

)(

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

21

)(1)(

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)()(

)()()(211)(

QCR

tWN

QCR

tWNtQ

RtQKR

RtWtWR

tW

( )

=

>=

=

0,0,)(max0)()(

)1()1()(211)(

qctwqctw

tq

tqktwtwtw

RttN

QqWw oldnew )()(,, ===

KNkN

RCc == ,

4 parameters

2 parameters

CRNK ,,,

( )

=

>=

=

0,0,)(max0)()(

)1()1()(211)(

qctwqctw

tq

tqktwtwtw

)2,(),( 2** kccqw =

0)1(~2

)1(~1)(~1)(~2

=+++ tqkctqc

tqc

tq

Unique steady state solutionLinearization:

Linearized model

02

1)(1)(2

2=+++ e

kce

ct

ct

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

f

(

)

gap

Possible choice for f: a gamma distribution with a gap

( , , )T nthree parameters:

k-1

k

Te

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

Directed, time-invariant communication graph:

Node set {1,,p}Set of vertices E: Weighted adjacency matrixStrongly connected

,( , ) 0k lk l E

,: diagonal entries zero, non-diagonal entries k lA

Interpretation as a consensus protocol

( ) ( ),( ) ( ), 1, ,

k k

k k

v t u t

y t v t k p=

= =

Consensus protocol:

( ),

( , ) 0( ) ( ) ( ) ( ) , 1, ,k k l l k

k l Eu t f y t y t d k p

= =

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 t x t B t x t t t f t

= +

= +

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~)( tNtsinceModulating 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 tT x t x t K x tx t x t x t

= + +

+ = + +

= +

= +

=

,

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 equation0 max1

( ) ( ) ( ), ( ) , maxm ni i ii ix t A x t A x t x t == + =

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

[ ]max( ,0 , ),n C

Reformulation of the DDE over

mapping abstract ODE

Initial condition is a function segment

[ ]max( ,0 , ),n C

[ ]max , ( )( )t x t Let be the forward solution with initial condition and let[ ]max( ) ( ), ,0tx x t = +

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

A : infinitesimal generator of T(t)

( ){ max max1

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

(0) (0) ( ) ,

, ( ).

m

i i ii

A A

=

=

= +

=

D A C

A D A

010

( ) ( )( ), 0,

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

i ii

t

t t

A s A s ds t

+

=

=

+ +

+ + + >

T

T T

0max

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

( ) ( ( )),te P t A T( )( ( )) exp ( )t t =T A

{ }0

1

\ 0 :

) 0i

n n

m

ii

v

I A A e v

=

=

( ) 0 ( )H = A[ ]max, ,0ve eigenfunction

finite-dimensional nonlinear eigenvalue problem

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

((.): spectrum, P(.): point-spectrum)

01

( ) 0, ( ) : det ,im

ii

H H I A A e =

= =

or equivalently

Properties

( ) ( ),P A =A

eigenfunction [ ]max, ,0ve

Characteristic roots,eigenvalues of A

Eigenvalues of T(1)exp(.)1 0 1 1.5

1

0

1

Real axis

I

m

a

g

i

n

a

r

y

a

x

i

s

3 2 1 0 1100

50

0

50

100

Real axis

I

m

a

g

i

n

a

r

y

a

x

i

s

Mapping is not one-to-one

But: characteristic roots can be obtained from (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, Bhler 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( ) ( ) : ( , ) , = >

A A R AC

1( , ) ( ) :A I = R Aresolvent

-pseudospectrum of an operator A d x xdt

=

A(or system

computable as level sets of resolvent norm

{ }( ) ( ) 0,forsome with = + =

Stability radius- partitionate the complex plane into disjunct sets, d u=C C C- Assume that ( ) d A C

under mild conditions:

uC

x

x

x

x

x

xx

dC

x

dC infinity

general formula:

{ }21

1

11

inf inf 0 : ( ) for some satisfying ,

sup ( )

sup ( )

du

u

Cd

r

I

I

= + C

=

mm wp

wpw

/)(

/)()(

00

,111,,

,111,,

,,

2221 =+==

=+==

==

qpqp

qpqp

pp

(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)( has dimension n x n !

( ) ( )12 22

1: 1M C K

= + + + + >

C

( ) ( ) ( ) ( ) ( ) 0M M x t C C x t K K x + + + + + =

n-by-n matrix

( )101

2

1: 1i

m

ii

I A A e e

=

= + >

C

( ) ( ) ( ) ( ) ( )x t A A x t B B x t = + + +

Based on combining the above approaches

0det ( ) 0

m

i ii

A p =

=

- exploiting the structure of the nonlinear eigenvalue problem,

- imposing structure on perturbations of the coefficient matrices

Examples (in both cases: ):

glob 2max iiA =

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 00 0 ( ) ( ) 0;0 0

M K

m k k k k km x t k k k k k x t

m k k k k k

+ + + + + = + +

2.) Laser physics application:

1

0

0 0( ) ( ) 0 0 ( ) 0;

0 0 0A

gx t A x g x t

= + =

[ ] [ ] [ ]

2

2

1 1 4

( )1 1

( ) 0 1 0 0 0 1 0 0 1 1 1 00 0 0

F M K

F m k k

= +

= + + +

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

0 1

0

( )1 0

00( ) 0 1000 0

F I A A e

eF A g

e

=

=

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 t = + + + +

( ) ( ) ( ) ( ) ( )0 ( ) ( ) ( )x t A A x t B B y t

C C x t y t

= + + +

= +

[ ] [ ]

( )

0( ) 0 0 00 0

I A BF

Ce II I

F A I B I C e II

=

=

det( ( )) 0F =

full blockuncertainy

11

( ) ( ) ( ) ( ) ( ) (1)sf

j j j j j jjj

F D E d G H =

=

= + 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

= + =

< = < =

C

Definition of structured -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,

( ) ( ) ( ) ( ) ( ), ,jsf l

j j j j j j j jjj

F

F D E d G H d

=

=

=

= + kC C

1( ) : ( ( )) , wheres F C T = > 1

11 1

1

( )

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

( )

ff s

s

E

ET F D D G G

H

H

=

{}

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

i ilf jd

i f j f

=

kC C

General formula:

( ( ))T

T()

Proof:

Special cases:

1( ) ( ) ( ) ( ), , 1, , : entire functionsf j j jjF D E q q j f == =

( ) ( )1 12 1( ) : ( ) ( ) ( ) ( )fs jjF E F D q = = > C structured singular value reduces to 2-norm small dimension of 1( ) ( ) ( )E F D

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 jE F D E F D

q

>

=

=

>

R R

In addition: qj even, j=1,,f:

Example:0 0

( ) ( ), ( ) ( )m m

i i i ii i

F A p F A p = =

= =

0.4 0 0.43.5

0

3.5

0.4 0 0.43.5

0

3.5

()

()

()

()

(a) (b)

ExamplesMass spring system

1 1 4 6 4 62

2 4 2 4 5 5

3 6 5 3 5 6

0 0( ) 0 0

0 0M K

m k k k k kF m k k k k k

m k k k k k

+ +

= + + + + +

unstructured pseudospectra

0.4 0 0.43.5

0

3.5

()

()

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

()

()

20 5 100

50

100

20 5 100

50

100

()

()

()

()

(a) (b)

Laser problem

eigenvalues of unperturbed system

structured pseudospectra unstructured pseudospectra

1

0

0 0( ) 0 0

0 0 0A

gF I A g e

=

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 Parcevals theorem

( ) 11 20

( ) ( )( ( )( )

( )

max ( )

x t A A x tF I A

F A

r j I A

= +

=

=

= C

20

( ) ( ( )) ( ( ), sup ( )t

x t A A t x t A t M

= + =

frequency domain

1

1

( ) ( ) ( )( ) ( )

x t Ax t u ty t x t

= +

=

2 2( ) ( ) ( )y t A t u t=

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