[PPT] - Some examples of time-delay systems I. Fluid flow model for a PowerPoint Presentation

SLIDE 1

Some examples of time-delay systems

SLIDE 2

I. Fluid flow model for a congested router in

TCP/AQM controlled network

p

T C t Q t R Q C t R t W t N Q C t R t W t N t Q t R t p t R t R t R t W t W t R t W + =        =         − > − = − − − − = ) ( ) ( , , ) ( ) ( ) ( max ) ( ) ( ) ( ) ( )) ( ( )) ( ( )) ( ( ) ( 2 1 ) ( 1 ) ( ɺ ɺ Hollot et al., IEEE TAC 2002 Model of collision-avoidance type:

W: window-size Q: queue length N: number of TCP sessions R: round-trip-time C: link capacity p: probability of packet mark Tp: propagation delay

Interpretation of AQM as a feedback control problem:

) (Q f p=

Sender Receiver Bottleneck router link c rtt R queue Q acknowledgement packet marking

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

SLIDE 3

Normalization of state and time

       =       − > − = − − − = , , ) ( max ) ( ) ( ) ( ) ( ) ( 2 1 1 ) ( Q C R t W N Q C R t W N t Q R t Q K R R t W t W R t W ɺ ɺ

( )

   = − > − = − − − = , , ) ( max ) ( ) ( ) 1 ( ) 1 ( ) ( 2 1 1 ) ( q c t w q c t w t q t q k t w t w t w ɺ ɺ

R t t N Q q W w

ld

new ) ( ) (

, , = = =

KN k N RC c = = , 4 parameters 2 parameters

C R N K , , ,

SLIDE 4

( )

   = − > − = − − − = , , ) ( max ) ( ) ( ) 1 ( ) 1 ( ) ( 2 1 1 ) ( q c t w q c t w t q t q k t w t w t w ɺ ɺ

) 2 , ( ) , (

2 * *

kc c q w = ) 1 ( ~ 2 ) 1 ( ~ 1 ) ( ~ 1 ) ( ~

2

= − + − + + t q kc t q c t q c t q ɺ ɺ ɺ ɺ

Unique steady state solution Linearization:

Linearized model

2 1 ) ( 1 ) (

2 2

= + + +

− − λ λ

e kc e λ c t λ c t λ

SLIDE 5

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

2 4 6 8 10 0.05 0.1

ξ f(ξ)

gap τ

Possible choice for f: a gamma distribution with a gap

( , , ) T n τ

three parameters:

k-1 k

T

e

ξ τ − −

∼

SLIDE 6

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 matrix Strongly connected

,

( , )

k l

k l E α ∈ ⇔ ≠

,

: diagonal entries zero, non-diagonal entries

k l

α

Interpretation as a consensus protocol

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

k k k k

v t u t y t v t k p = = = ɺ …

Consensus protocol:

( )

, ( , )

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

k k l l k k l E

u t f y t y t d k p α θ θ θ θ

∞ ∈

= − − − =

∑ ∫

…

SLIDE 7

Successive passage of teeth ⇒ delay Rotation of each tooth ⇒ periodic coefficients

Cutting process

Successive passage of the same point

f the piece

⇒ delay Orientation of tooth w.r.t. workpiece is fixed ⇒ constant coefficients

workpiece (fixed / translates) tool (rotates) Milling process ( ) ( ) ( ) ( ) ( ( )) ( ) ( ) 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 determines delay

III. Rotating cutting and milling machines

tool (fixed) Workpiece (rotates)

SLIDE 8

speed time

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

Variable speed machines

) ( 1 ~ ) ( t N t τ

since Modulating the machine speed = modulating the delay in the model

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

! Stabilizing effect of delay variation !

SLIDE 9

IV. Heating system

Linear system of dimension 6, 5 delays,,

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

temperature to be controlled setpoint

(PhD Thesis Vyhlidal, CTU Prague, 2003)

SLIDE 10

, ,

( ) ( ) ( ) ( ) 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 q T 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 τ η τ = − − + − + −   + −    = − + − + − − −       = − + −   = − − + −  = −    ɺ ɺ ɺ ɺ ɺ

, T h set h a d c e

x K x x x x x   =  

System Control law (PI+ state feedback)

SLIDE 11

Computation of characteristic roots and stability regions

SLIDE 12

Operators associated to a delay equation

max 1

( ) ( ) ( ), ( ) , max

m n i i i i i

x t A x t A x t x t τ τ τ

=

= + − ∈ =

∑

ɺ ℝ

, ( )

t t

x x x = ∈

( )

,

t

x t x t = ≥

[

]

max

( ,0 , ),

n

ϕ τ ∈ − ℝ

Reformulation of the DDE over

mapping abstract ‘ODE’ Initial condition is a function segment

[ ]

max

( ,0 , ),

n

τ − ℝ

[

]

max,

( )( ) t x t τ ϕ ∈ − ∞ →

Let be the forward solution with initial condition ϕ and let

[ ]

max

( ) ( ), ,0

t

x x t ϕ θ θ τ = + ∈ −

(t) : solution (time-integration)

perator over interval t

: infinitesimal generator of (t)

( )

{

max max 1

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

m i i i i

A A ϕ τ ϕ τ ϕ ϕ ϕ τ ϕ ϕ ϕ

=

= ∈ − −  = + −   = ∈

∑

ɺ

ɺ ɺ

1

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

t m i i i

t t t A s A s ds t

θ

ϕ θ ϕ θ θ ϕ ϕ ϕ τ θ

+ =

= + + ≤    + + − + >  

∑ ∫

max

τ −

ϕ ϕ ϕ ϕ

SLIDE 13

Spectral properties

λ is a characteristic root if and only if it satisfies the characteristic equation

( ) ( ( )),

t

e P t

λ

λ σ σ ∈ ⇒ ∈

(

)

( ( )) exp ( ) t t σ σ =

{ }

1

\ 0 : )

i

n n m i i

v I A Ae v

λτ

λ

× − =

∃ ∈   − − =    

∑

ℂ

( ) ( ) H λ λ σ = ⇔ ∈

[

]

max

, ,0 veλθ θ τ ∈ −

eigenfunction

finite-dimensional nonlinear eigenvalue problem infinite-dimensional linear eigenvalue problems for and (t) (σ(.): spectrum, Pσ(.): point-spectrum)

1

( ) 0, ( ) : det ,

i

m i i

H H I A Ae λτ λ λ λ

− =

  = = − −    

∑

r equivalently

Properties

( ) ( ), P A σ σ =

eigenfunction

[ ]

max

, ,0 veλθ θ τ ∈ −

SLIDE 14

Characteristic roots, eigenvalues of Eigenvalues of (1) exp(.)

−1 1 1.5 −1 1 Real axis Imaginary axis −3 −2 −1 1 −100 −50 50 100 Real axis Imaginary axis

Mapping is not one-to-one But: characteristic roots can be obtained from σ((t)) by computing also the corresponding eigenfunction

SLIDE 15

Two-stage approach to compute characteristic roots

1a. Discretize or (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)

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 (Breda et al)

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

SLIDE 16

Routine in the Matlab package DDE-BIFTOOL

Linear multi-step method to discretize (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 roots

corrected roots

SLIDE 17

Pseudospectra and stability radii of nonlinear eigenvalue problems, with application to time-delay systems

SLIDE 18

Overview

Pseudospectra Approaches to exploit structure of nonlinear eigenvalue problems

via structured matrix perturbations by redefining pseudospectra

Emphasis on computable expressions

Numerical examples Concluding remarks

SLIDE 19

Pseudospectra

1 ( ) ( ) : ( , ) ,

ε

λ λ ε   Λ = Σ ∪ ∈ >    

1

( , ) ( ) : A I λ λ

−

= −

resolvent

ε-pseudospectrum of an operator

d x x dt  =  

(or system

computable as level sets of resolvent norm

{ }

( ) ( ) 0,forsome with

ε

λ δ δ δ ε Λ = ∈Σ + = <

( ) :

Σ ⋅

spectrum

−6 −4 −2 2 4 6 −50 50 ℜ(λ) ℑ(λ) (a)

ℜ(λ) ℑ(λ) −6 −4 −2 2 4 6 −50 50 (b)

spectrum pseudospectra

SLIDE 20

Stability radius

partitionate the complex plane into disjunct sets,

d u

=

∪
Assume that

( )

d

Σ ⊆

under mild conditions:

u

x

x x x x x x

d

x

d

Γ

infinity

general formula:

{

}

2 1 1 1 1

inf inf 0 : ( )for some satisfying , sup ( ) sup ( )

d u u Cd

r I I

λ λ λ

ε λ δ δ δ ε λ λ

∈ − − ∈ − − ∈Γ

= ≥ ∈ Σ + <   = −       = −    

desired region
cf. stability

: sufficient to scan boundary

SLIDE 21

vibrating system
time-delay system
…

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

( )

det ) ( ) ( ) (

2

= + + → = + + K C M t x K t x C t x M λ λ ɺ ɺ ɺ

1 1 1 1 2 2

( ) ( ) ( ) ( ) x t x t I x t x t M K M C

− −

      =       − −       ɺ ɺ

invertible M

: ( ), [ , 0]

t t t

d x x dt x x t θ θ τ = = + ∈ −

Relation with perturbations of coefficient matrices ??? : infinitesimal generator of solution operator

( )

det ) ( ) ( ) ( = − − → − + =

−λτ

λ τ Be A I t Bx t Ax t x ɺ

SLIDE 22

Approaches for exploiting structure

2. Redefine ε-pseudospectra of nonlinear eigenvalue problems (Michiels et al,

inspired by Tisseur et al.)

:

( ) ( ),

m n n i i i i

F A p A λ λ

× =

= ∈

∑

entire functions
1. Structured perturbations (Hinrichsen & Kelb,…)

1 1 1 1 2 2

( ) ( ) ( ) ( )

A

x t x t I x t x t M K M C

− −

      =       − −       ɺ ɺ

[

]

1 1 1 2 2

( ) ( ) ( ) ( )

E A D

x t x t A K C I x t x t M

δ

δ δ

−

          = +         −            ɺ

ɺ
{

}

( ; , ) : ( ) 0,forsome with D E D

ε

λ δ δ δ ε Λ = ∈ Σ + Ε = <

1

( ; , ) ( ) : ( , ) , D E E D

ε

λ λ ε   Λ = Σ ∪ ∈ >    

M

δ =

{ }

( ) : det( ( ) F F λ λ Σ ∪ ∈ =

SLIDE 23

) ( ) ( det =       +

∑

= m i i i i

p A A λ δ

perturbation class

) , , ( :

m

A A δ δ … = ∆

measure on the combined perturbation

, 0, ,

n n i

A i m δ

×

∈ =

…

[ ] p

m m A

w A w δ δ ⋯

glob =

∆

p m m A

w A w           = ∆ δ δ ⋮

glob

2 1 1

glob p p m m p

A w A w           = ∆ δ δ ⋮ (1) (2) (3)

{ } { }

1 2

, , : weights

i

p p p w

+ +

∈ +∞ ∈ +∞

∪

∪

        = < ⇔ < ∆ ∞ = m i A w p

p i i

, , , :

1

glob 2

… ε δ ε

}

( ) ( 1) glob

:det ( ) ( ) 0,for some with

m i i i i n n m

A A p

ε

λ δ λ ε

= × × +

   Λ = ∈ + =       ∆∈ ∆ <

∑

SLIDE 24

1

1 ( ) : ( ) ( )

m i i i

F A p w

ε β α

λ λ λ ε

− =

      Λ = Σ ∪ ∈ >          

∑



         =

m m

w p w p w / ) ( / ) ( ) ( λ λ λ ⋮

, 1 1 1 , , , 1 1 1 , , , ,

2 2 2 1

= + = = = + = = = = q p q p q p q p p p β α β α β α

(1) measure

n

perturbati (2) measure

n

perturbati (3) measure

n

perturbati

where

Computable expressions

computation of pseudospectra contours as level sets of

function f

structure is fully exploited !!

∑

= m i i i p

A ) (λ

has dimension n x n !

SLIDE 25

( )

1 2 2 2

1 : 1 M C K

ε

λ λ λ λ λ ε

−

  Λ = ∈ + + + + >    

(

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

n-by-n matrix

( )

1 1 2

1 : 1

i

m i i

I A A e e

λτ λτ ε

λ λ ε

− − − =

      Λ = ∈ − − + >          

∑

( )

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

Based on combining the above approaches

det ( )

m i i i

A p λ

=

  =    

∑

exploiting the structure of the nonlinear eigenvalue problem,
imposing structure on perturbations of the coefficient matrices

Examples (in both cases: ):

glob 2

max

i i

A δ ⋅ =

3. Structered pseudospectra of nonlinear eigenvalue problems

SLIDE 26

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;

M 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;

A

g x t A x g x t τ     = + − − =       ɺ

[

] [ ] [ ]

2 2 1 1 4

( ) 1 1 ( ) 1 1 1 1 1 F M K F m k k λ λ λ δ λ δ δ δ = +             = + + − − +                   ⋯ ⋯ ⋯

det( ( )) ) F λ =

( nominal char. eqn.:

1

( ) 1 0 ( ) 0 1 0 0 F I A A e e F A g e

λτ λτ λτ

λ λ δ λ δ δ

− − −

= − −       = − −     −      

rank 2 scalar

, uncertain

i i

m k

0 ,

uncertain

i

A g

SLIDE 27

3.) Systems with multiplicative uncertainty:

principle:

( ) ( ) ( ) ( )( ) ( ) x t A A x t B B C C x t δ δ δ τ = + + + + − ɺ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x t A A x t B B y t C C x t y t δ δ δ τ = + + +   = + − −  ɺ

[ ] [ ]

( ) ( ) I A B F Ce I I I F A I B I C e I I

λτ λτ

λ λ δ λ δ δ δ

− −

− −   =   −           = − − −              

det( ( )) F λ =

full block uncertainy

1 1

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

s f j j j j j j j j

F D E d G H δ λ λ λ λ λ

= =

= ∆ +

∑ ∑

scalar uncertainty

{

}

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 = ∈ + = ∆ < = < =

…

… Definition of structured ε-pseudospectrum:

In many cases (including the above): Nominal system:

SLIDE 28

pseudospectra boundaries computable as level sets of the function to some extent reformulation of problem: efficiency depends on computation / approximation of structured singular value associated with the uncertainty structure. Computational expressions

1 1

det( ( )) 0, ( ) ( ) ( ) ( ) ( ), ,

j

s f l j j j j j j j j j j

F F D E d G H d λ δ λ λ λ λ λ

× = =

= = ∆ + ∆ ∈ ∈

∑ ∑

1

( ) : ( ( )) , where

s F

C T

ε

λ µ λ ε

∆

  Λ = ∈ >    

1 1 1 1 1

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

f f s s

E E T F D D G G H H λ λ λ λ λ λ λ λ λ λ

−

        =             ⋮ ⋯ ⋯ ⋮

{

}

1 1 s i

diag( , , ,d I, ,d I): , , 1 , 1 .

i i

l f j

d i f j f

×

∆ = ∆ ∆ ∆ ∈ ∈ ≤ ≤ ≤ ≤

…

… General formula:

( ( )) T λ µ λ

∆

→

T(λ)

∆

Proof:

SLIDE 29

Special cases:

1

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

f j j j j

F D E q q j f δ λ λ λ λ

=

= ∆ =

∑

…

( ) (

)

1 1 2

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

f s j j

F E F D q

ε

λ λ λ λ λ ε

− =

  Λ = ∈ >    

∑

structured singular value reduces to 2-norm

small dimension of

1

( ) ( ) ( ) E F D λ λ λ

−

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

1 1 ,real 2 1 1 1 `

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

s f j j

E F D E F D j j E F D E F D q

ε γ

λ λ λ γ λ λ λ λ λ σ γ λ λ λ λ λ λ λ ε

− − − − − > =

    ℜ ℑ  Λ ∩ = ∈ ⋅        ℑ ℜ           >       

∑

In addition: qj even, j=1,…,f:

Example:

( ) ( ), ( ) ( )

m m i i i i i i

F A p F A p λ λ δ λ δ λ

= =

∑ ∑

SLIDE 30

−0.4 0.4 −3.5 3.5 −0.4 0.4 −3.5 3.5

ℜ(λ) ℑ(λ) ℜ(λ) ℑ(λ) (a) (b)

Examples

Mass spring system

1 1 4 6 4 6 2 2 4 2 4 5 5 3 6 5 3 5 6

( )

M 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 0.4 −3.5 3.5 ℜ(λ) ℑ(λ)

structured pseudospectra eigenvalues of 2000 simulations of associated random eigenvalue problem

structure of F exploited structure of M and K not exploited

SLIDE 31

−20 −5 10 50 100

ℜ(λ) ℑ(λ) −20 −5 10 50 100 −20 −5 10 50 100 ℜ(λ) ℑ(λ) ℜ(λ) ℑ(λ) (a) (b)

Laser problem

eigenvalues of unperturbed system structured pseudospectra unstructured pseudospectra

1

( )

A

g F I A g e

λτ

λ λ

−

    = − − −      

decay due

to rank increase of A

1

f=s=1: ssv computable via convex optimization

SLIDE 32

Extension to time-varying perturbations Underlying ideas: 2 gain analysis and Parceval’s theorem

( )

1 1 2

( ) ( )( ( ) ( ) ( ) max ( ) x t A A x t F I A F A r j I A

ω

δ λ λ δ λ δ ω

−

− − ≥

= + = − = − = −

ɺ

2

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

t

x t A A t x t A t M δ δ

≥

= + = ɺ

frequency domain

1 1

( ) ( ) ( ) ( ) ( ) x t Ax t u t y t x t = + = ɺ

2 2

( ) ( ) ( ) y t A t u t δ = −

1

u

2

u

1

y

2

y

feedback system interconnection is stable if

( ) ( ) ( )

1 2 1 2 2 2

1 1 1 2 2 1 1 2

1 max ( ) 1 max ( ) sup ( ) max ( )

y y u u i t

j I A M M j I A A t j I A

ω ω ω

ω ω δ ω

− − − ≥ ≥ − − ≥ ≥

< − < ⇔ < − ⇔ < −

feedback interconnection interpretation:

time domain

SLIDE 33

Extension to systems with time-varying delays

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

i i i i

x t A A t A A t x t δ δ τ δτ = + + + − +

∑

ɺ

+ weighted combined measure of perturbations,

glob

i

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 t τ δτ τ = − + − −

( ( ))

( ) , ( ) ( )

t t t

y s ds y t x t

τ δτ τ − + −

= =

∫

ɺ 7 ( ) ( ) 4 z t y t µ ≤

|

( ) | t δτ µ ≤

( 1) ( ) ( ) e e Z Y

λτ λδτ

λ λ λ

− −

− =

( 1) ( ) ( ) ( )

j j

e e z t y t y t j

ωτ ωδτ

µ ω

∞

− −

− ≤ ≤