delay robustness mirkin

7/26/2019 Delay Robustness Mirkin

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Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks

An Opinionated View on Delay Robustness

Leonid Mirkin

Faculty of Mechanical EngineeringTechnion IIT

May 18, 2006
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Single-delay system with uncertain delay

Consider LTI system

x(t) =A0x(t) + A1x(t h), x() =0, [h, 0]

or, equivalently, in thes-domain:

sx(s) =A0x(s) + A1esh

x(s).

Wed like to be able to check

whether this system isstableh [0,h]for some h > 0.

This clearly requires that

A1: delay-free system is stable, i.e.,A0+ A1is Hurwitz.
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Precise methods

Methods yielding exact stability intervals:

Nyquist criterion(Tsypkin, 1946)

Delay-sweeping arguments

(Cooke & Grossman, 1982; Walton & Marshall, 1987)

Schur-Cohn criterion inspirations(J. Chen, G. Gu, & Nett, 1995)

Common pitfalls:

not suitable for analytic controller design

not readily extendible to multiple-delay systems
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Precise methods

Methods yielding exact stability intervals:

Nyquist criterion(Tsypkin, 1946)

Delay-sweeping arguments

(Cooke & Grossman, 1982; Walton & Marshall, 1987)

Schur-Cohn criterion inspirations(J. Chen, G. Gu, & Nett, 1995)

Common pitfalls:

not suitable for analytic controller design

not readily extendible to multiple-delay systems
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The quest for alternatives

Has intensified during the last decade:

a zillion of papers published

dominated by Lyapunov-Krasovski (LK) methods(LMI solutions derived via state-space LK technique)

d l f ll h fi l d k
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Outline

Lyapunov-Krasovski methods & model transformations

Good ol (scaled) Small Gain Theorem

Some comparisons

Possible refinements

Concluding remarks

M d l f i S ll G i Th S i R fi C l di k
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Outline



Some comparisons


Concluding remarks

M d l t f ti S ll G i Th S i R fi t C l di k
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Lyapunov-Krasovski methods

Analysis based on

1. constructing a Lyapunov-Krasovski functional (storage function), like

V=x(t)P1x(t) + 2x(t)0h

P2()x(t+ )d

+0h

0h

x (t+ )P3(, )x(t+ )dd+ ,

2. calculating its derivative along system trajectory,

3. completing squares via approximating some cross-terms,

4. ending up with LMI conditions guaranteeing that V < 0.



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Lyapunov-Krasovski methods: transformations

Apparent obstacle in the use of this approach is that

equation x(t) =A0x(t) + A1x(t h)is not quite compatible

with Lyapunov-Krasovski techniques (not LK-friendly).

Conventional way to circumvent this obstacle is to

transform this model to more suitable form

by rearranging its terms.

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Lyapunov-Krasovski methods: transformations

Apparent obstacle in the use of this approach is that

equation x(t) =A0x(t) + A1x(t h)is not quite compatible

with Lyapunov-Krasovski techniques (not LK-friendly).

Conventional way to circumvent this obstacle is to

transform this model to more suitable form

by rearranging its terms.



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The first transformation

Rewrite

sx =A0x + A1eshx= (A0+ A1)x A1(1 e

sh)x

= (A0+ A1)x A11esh

s sx

= (A0+ A1)x A11esh

s (A0x + A1e

shx)

or in the time domain:

x(t) = (A0+ A1)x(t) A1

h0

A0x(t ) + A1x(t h )

d.

Turns out to be more LK-friendly, yet introducesadditional dynamics:

(s) =det

sI A0 A1esh

det

sI A1

1esh

s

,

stability of which is hard to check (source of additionalconservatism).

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The first transformation

Rewrite

sx =A0x + A1eshx= (A0+ A1)x A1(1 e

sh)x

= (A0+ A1)x A11esh

s sx

= (A0+ A1)x A11esh

s (A0x + A1e

shx)


x(t) = (A0+ A1)x(t) A1

h0

A0x(t ) + A1x(t h )

d.

Turns out to be more LK-friendly, yet introducesadditional dynamics:

(s) =det

sI A0 A1esh

det

sI A1

1esh

s

,

stability of which is hard to check (source of additionalconservatism).

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p g

The second transformation

Rewrite

sx=A0x + A1eshx

I + A1

1esh

s

sx= (A0+ A1)x


d

dt

x(t) + A1 h0

x(t )d

= (A0+ A1)x(t).

Turns out to be more LK-friendly, yet

requires the stability of

det

I + A11esh

s

=0 or, equiv., of x(t) = A1

h0

x(t )d,

which is hard to verify, so it might be source of additional conservatismtoo.

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p g

The second transformation

Rewrite

sx=A0x + A1eshx

I + A1

1esh

s

sx= (A0+ A1)x


d

dt

x(t) + A1 h0

x(t )d

= (A0+ A1)x(t).

Turns out to be more LK-friendly, yet

requires the stability of

det

I + A11esh

s

=0 or, equiv., of x(t) = A1

h0

x(t )d,

which is hard to verify, so it might be source of additional conservatismtoo.

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The third transformation

Rewrite

sx =A0x + A1eshx= (A0+ A1)x A1

1esh

s sx

(in fact, this is midway toward first transformation) or in the time domain:

x(t) = (A0+ A1)x(t) A1 h0

x(t )d.

Somehow is also LK-friendly, yet claimed to

introduce additional terms to V

i.e., leads to overdesign (might be source of additionalconservatismtoo).

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The third transformation

Rewrite

sx =A0x + A1eshx= (A0+ A1)x A1

1esh

s sx

(in fact, this is midway toward first transformation) or in the time domain:

x(t) = (A0+ A1)x(t) A1 h0

x(t )d.

Somehow is also LK-friendly, yet claimed to

introduce additional terms to V

i.e., leads to overdesign (might be source of additionalconservatismtoo).

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The fourth transformation

Rewrite

sx=A0x + A1eshx

sx= y

y= (A0+ A1)x A11esh

s y

or in the time domain asdescriptorsystem

I 0

0 0

x(t)

y(t)

=

0 I

A0+ A1 I

x(t)

y(t)

0

A1

h0

y(t )d.

Not surprise that it is also considered LK-friendly. Moreover, it is

claimed to be less conservative.

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The fourth transformation

Rewrite

sx=A0x + A1eshx

sx= y

y= (A0+ A1)x A11esh

s y

or in the time domain as descriptor system

I 0

0 0

x(t)

y(t)

=

0 I

A0+ A1 I

x(t)

y(t)

0

A1

h0

y(t )d.

Not surprise that it is also considered LK-friendly. Moreover, it is

claimed to be less conservative.

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What bothers me

conservatism sources are hidden

LK functional choice, cross-terms approximations, model transformation,. . .

rationale behind model transformations is obscure (recondite?)(mysteriously, they all concentrated on 1e

sh

s , yet I found no hint why)

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Outline



Some comparisons


Concluding remarks

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The Small Gain Theorem

G(s)

(s)

TheoremLet G(s)and (s)be stable and such that

1 and G < 1.Then the closed-loop system is internally stable.



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Delay as unstructured uncertaintyI

Straightforward approach is to exploit the facts that

esh 1, h.Then,

sx=

A0x+

A1e

sh

x stableh > 0 if

sx=A0x + A1 x stable

1

We then end up withdelay-independent(sufficient) condition

(sI A0)1A1 < 1,

which is easily solvable.

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Delay as unstructured uncertaintyI

Straightforward approach is to exploit the facts that

esh 1, h.Then,

sx=A0

x + A1e

sh x stable

h > 0 if


1

We then end up withdelay-independent(sufficient) condition

(sI A0)1A1 < 1,

which is easily solvable.



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Delay as unstructured uncertaintyI (contd)

Conservatism can be a bit reduced by noticing that

Mesh =eshM, M Rnn

Then,

sx=A0

x + A1esh x

stableh > 0

if


1such thatM=M

We then end up withdelay-independentcondition

M= M > 0such thatM(sI A0)1A1M

1

< 1,

which is LMI-able.

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Advantages:

easilyunderstandable easilytractable

easily extendible tomultiple-delayproblems

easily incorporable into controllerdesign(Hoptimization)

Disadvantages:

delay independent, hence too conservative

(not so many, if any, problems, where delays can become arbitrarily large) all phase information about delay is neglected

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Advantages:

easily understandable easily tractable

easily extendible to multiple-delay problems

easily incorporable into controller design (Hoptimization)

Disadvantages:

delay independent, hence tooconservative

(not so many, if any, problems, where delays can become arbitrarily large) allphase informationabout delay isneglected

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Delay as unstructured uncertaintyII

Rewrite

sx=A0x + A1eshx= (A0+ A1)x A1(1 e

sh)x.

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Rewrite


sh)x.

Term1 esh is a better candidate for approximations because its

size (norm) does depend on the phase lag ofejh.

Re

Im

1

1 ej1h

Re

Im

1

1 ej2h

Re

Im

1

1 ej3h

Here1

< 2

< 3

.

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Rewrite


sh)x.

Term1 esh is a better candidate for approximations because its

size (norm) does depend on the phase lag ofejh.

Re

Im

1

1 ej1h

Re

Im

1

1 ej2h

Re

Im

1

1 ej3h

Here1

< 2

< 3

.

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Covering1 esh

Re

Im

1

h

lh(

)

Simple geometry yields that

lh() =

2 sin h

2 ifh

2 ifh >



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Delay as unstructured uncertaintyII (contd)

Then,

sx= (A0+ A1)x A1(1 esh)x stableh [0,h]

if

sx= (A0+ A1)x + A1 x stable/lh 1

We then have delay-dependent (sufficient) condition

(sI A0 A1)1

A1 lh(s) < 1,which might not be easy to check though, because

lh()is not rational.

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Then,

sx= (A0+ A1)x A1(1 esh)x stableh [0,h]

if

sx= (A0+ A1)x + A1 x stable/lh 1

We then havedelay-dependent(sufficient) condition

(sI A0 A1)1

A1 lh(s) < 1,which might not be easy to check though, because

lh()is not rational.

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Rational approximations oflh

We need to construct stable and rationalW(s)such that

|W(j)| lh(), .

Some examples:

W0(s) = hs,

W1(s) = 2

3hs

hs+2

3

W3(s) = 2.007hs

hs+2s2+1.567s+2

s2+1.283s+2 with =2.358/h.

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|W(j)| lh(), .

Some examples:

W0(s) = hs,

W1(s) = 23hshs+2

3

(note that|W1(j)| 0),

W3(s) = 2.007hs

hs+2s2+1.567s+2

s2+1.283s+2 with =2.358/h.


l f
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|W(j)| lh(), .

Some examples:

W0(s) = hs,

W1(s) = 23hshs+2

3

W3(s) = 2.007hs

hs+2s2+1.567s+2

s2+1.283s+2 with =2.358/h.

We then end up with delay-dependent (sufficient) condition

(sI A0 A1)1A1W(s) < 1,

which is easily calculable. . .


l f l
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|W(j)| lh(), .

Some examples:

W0(s) = hs,

W1(s) = 23hshs+2

3

W3(s) = 2.007hs

hs+2s2+1.567s+2

s2+1.283s+2 with =2.358/h.

. . . or, exploitingM(1 esh) = (1 esh)M, with (sufficient) condition

M=M > 0such thatM(sI A0 A1)1A1W(s)M

1 < 1,which is LMI-able.


D l d i II ( d)
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The idea is rather old (pre-LK), it

can be traced back to (Owens & Raya, 82; Morari & Zafiriou, 89) and extensively exposed in (Wang, Lundstrom & Skogestad, 94).

Advantages:

easily understandable

easily tractable


easily incorporable into controller design (Hoptimization) conservatism sources, unlike LK approach, clearly seen.

Disadvantages:

seems to be too conservative in general

conservatism sources, unlike LK approach, clearly seen1.

1This appears to harm the acceptance of the method.


D l t t d t i t II ( td)
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Advantages:

easilyunderstandable

easilytractable

easily extendible tomultiple-delayproblems

easily incorporable into controllerdesign(Hoptimization) conservatism sources, unlike LK approach, clearly seen.

Disadvantages:

seems to be too conservative in general

conservatism sources, unlike LK approach, clearly seen1.



D l t t d t i t II ( td)
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Advantages:

easily understandable

easily tractable


easily incorporable into controller design (Hoptimization) conservatism sources, unlike LK approach,clearly seen.

Disadvantages:

seems to be tooconservativein general

conservatism sources, unlike LK approach,clearly seen1.



O tli
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Outline



Some comparisons


Concluding remarks


SG isnt more conservative than LK
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Two papers in the early 2000s showed that in many cases

LK conditions might actually be more conservative than SG conditions.These are

(Huang & Zhou, 00), who cast delay robustness problem as -problem andclaimed that LK-based solutions are much more conservative

(LK derivations mostly useW0-bound and static scaling);(Zhang, Knospe, & Tsiotras, 01), who proved that several LK-based results

are equivalent to (statically scaled) SG-based results, which

useW0-bound on|1 ejh|.


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SG isn t more conservative than LK




(LK derivations mostly useW0-bound and static scaling);(Zhang, Knospe, & Tsiotras, 01), who proved that several LK-based results




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SG isn t more conservative than LK




(LK derivations mostly useW

0-bound and static scaling);(Zhang, Knospe, & Tsiotras, 01), who proved that several LK-based results




Descriptor approach vs Small Gain Theorem
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Descriptor approach vs. Small Gain Theorem

Consider covering of|1 ejh|withW0 =sh. We have that our system is

stableh [0,

h]if

sx(s) = (A0+ A1)x(s) A1(s)hsx(s)

is stable for all

1such thatM=Mfor allM. In principle, this

is guaranteed ifM= M> 0such that to

M

I + (A0+ A1)(sI A0 A1)1

A1M1 < 1h ,

yet we may want to rewrite it as

0 M

s

I 0

0 0

0 I

A0+ A1 I

1

0

A1M1

< 1

h

and end up with the problem of

calculatingHnorm of adescriptorsystem.


Descriptor approach vs Small Gain Theorem
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Descriptor approach vs. Small Gain Theorem

Consider covering of|1 ejh|withW0 =sh. We have that our system is

stableh [0,

h]if

sx(s) = (A0+ A1)x(s) A1(s)hsx(s)

is stable for all

1such thatM=Mfor allM. In principle, this

is guaranteed ifM= M> 0such that to

M

I + (A0+ A1)(sI A0 A1)1

A1M1 < 1h ,

yet we may want to rewrite it as

0 M

s

I 0

0 0

0 I

A0+ A1 I

1

0

A1M1

< 1

h

and end up with the problem of

calculatingHnorm of adescriptorsystem.


H norm of descriptor systems
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Theorem (Rehm & Allgover, 99)

Letdet(sE A)0, thenC(sE A)1B < iff

X such that E X=X E 0and

A X + X A X B C

B X I 0C 0 I

< 0.

In our case,

E X= X E 0 I 0

0 0 X11 X12

X21 X22

=X 11 X 21

X 12 X 22 I 0

0 0 0.

HenceX12 =0 and X11 =X

11 0.


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A X + X A X B C

B X I 0C 0 I

< 0.

In our case,

E X= X E 0 I 0

0 0 X11 X12

X21 X22

=X 11 X 21

X 12 X 22 I 0

0 0 0.


11 0.


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A X + X A X B C

B X I 0C 0 I

< 0.

In our case,

E X= X E 0 I 0

0 0 X11 0

X21 X22

=X 11 X 21

0 X 22 I 0

0 0 0.


11 0.


Descriptor approach vs. Small Gain Theorem (contd)
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Proceedings further, we get LMI solvability condition

A X21+ X 21A X11 X21+ AX22 X

21A1M

1 0

X11 X21+ X22A

X22 X 22 X22A1M

1 M

M1A 1X21 M1A 1X22

1h

I 0

0 M 0 1h

I

< 0

or, equivalently (via Schur complement of the (4, 4)term), LMI

A X21+ X 21A X11 X

21+

AX22 X21A1h

X11 X21+ X22A

hY X22 X 22 X22A1h

hA 1X21

hA 1X22

hY

< 0,

where A .=A0+ A1andY

.=M2.

This is

exactly the condition of (Fridman, 01) derived via LK technique.


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Proceedings further, we get LMI solvability condition

A X21+ X 21A X11 X21+ AX22 X

21A1M

1 0

X11 X21+ X22A

X22 X 22 X22A1M

1 M

M1A 1X21 M1A 1X22

1h

I 0

0 M 0 1h

I

< 0

or, equivalently (via Schur complement of the (4, 4)term), LMI

A X21+ X 21A X11 X

21+

AX22 X21A1h

X11 X21+ X22A

hY X22 X 22 X22A1h

hA 1X21

hA 1X22

hY

< 0,

where A .=A0+ A1andY

.=M2.

This is

exactly the condition of (Fridman, 01) derived via LK technique.


Descriptor approach is a version of SGT too
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p pp

Thus, we have that

descriptor transformation leads to conditions, which are equivalent toapplication of scaled Small Gain Theorem under covering jh >|1 ejh|(in a sense, the weakest covering) bringing in some redundancy into state-space realization via

A1+ (A0+ A1)sI A0 A11A1

0 I

s

I 0

0 0

0 I

A0+ A1 I

1 0

A1

,

which does not introduce any additional dynamics.

In other words, descriptor transformation appears to be

smart solution to problem one should not have gotten into in the first

place.


Descriptor approach is a version of SGT too
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p pp

Thus, we have that

descriptor transformation leads to conditions, which are equivalent toapplication of scaled Small Gain Theorem under covering jh >|1 ejh|(in a sense, the weakest covering) bringing in some redundancy into state-space realization via

A1+ (A0+ A1)sI A0 A11A1

0 I

s

I 0

0 0

0 I

A0+ A1 I

1 0

A1

,

which does not introduce any additional dynamics.

In other words, descriptor transformation appears to be

smart solution to problem one should not have gotten into in the first

place.


Example
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p

Consider the system (Kolmanovski & Richard, 99):

x(t) =

1 0.5

0.5 1

x(t) +

2 2

2 2

x(t h)

The following stability bounds are available:

Method IV SGT+W0 SGT+W1 SGT+lhhmax 0.271 0.2716 0.3042 0.3047

whereunscaled SGT was used.


Some arguments in favor of SGT.
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g

Lyapunov-Krasovski functional:

1. Pros and cons obscure2. Hinges uponW0(s) = hs(?)

3. Hinges upon static scalings

4. Design limited to FD controllers(hence nominal delay has to be h

0=0)

Small Gain Theorem:

1. Pros and cons transparent2. Can use tighter coverings

3. Can use dynamic scalings ()

4. Design can use Smith predictors(hence nominal delay may be h

0=

h

2)




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g




4. Design limited to FD controllers(hence nominal delay has to be h0 =0)

Small Gain Theorem:



4. Design can use Smith predictors(hence nominal delay may be h0 =

h

2

)


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Small Gain Theorem:




h

2

)


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Small Gain Theorem:




h

2

)


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Small Gain Theorem:




h

2

)


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Small Gain Theorem:




h

2

)

The question is

what makes LK methods so dominating ?


Outline
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Some comparisons


Concluding remarks


Trading off DIS-DDS
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We can also rewrite equationsx=A0x + A1eshxas

sx= (A0+ A1)x A1(1 esh)x A1( I)eshx,

whereis arbitrary. If

=0, delay-independent conditions recovered

=I, delay-dependent conditions recoveredIn general,brings more freedom and this freedom is LMI-able.

This freedom is exploited in LK methods too,

either explicitly (parametrized model transformation) or implicitly (via so-called Parks inequality for bounding cross-terms)

(connection not quite transparent, though Zhang, Knospe & Tsiotras (01) showed it via

equivalence of resulting LMIs)


Trading off DIS-DDS
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We can also rewrite equationsx=A0x + A1eshxas

sx= (A0+ A1)x A1(1 esh)x A1( I)eshx,

whereis arbitrary. If

=0, delay-independent conditions recovered

=I, delay-dependent conditions recoveredIn general,brings more freedom and this freedom is LMI-able.

This freedom is exploited in LK methods too,

either explicitly (parametrized model transformation) or implicitly (via so-called Parks inequality for bounding cross-terms)

(connection not quite transparent, though Zhang, Knospe & Tsiotras (01) showed it via

equivalence of resulting LMIs)


Trading off DIS-DDS: example


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An (almost classical) example of (Li & de Souza, 97) considers the system

x(t) =

2 00 0.9

x(t)

1 01 1

x(t h).

This system can be presented as the cascade

1s+2+esh

1s+0.9+esh

esh x1x2


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x(t) =

2 00 0.9

x(t)

1 01 1

x(t h).


1s+2+esh

1s+0.9+esh

esh x1x2

Delay-independent stableDelay-dependent


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x(t) =

2 00 0.9

x(t)

1 01 1

x(t h).


1s+2+esh

1s+0.9+esh

esh x1x2

Delay-independent stableDelay-dependent

This means that the choice = 0 00 I

yieldssx=

2 0

0 1.9

x +

0 0

0 1

(1 esh)x

1 0

1 0

eshx

and effectively reduces this system to sx2

= 1.9x2

+(1 esh)x2

.


Trading off DIS-DDS: example (contd)
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Thus

sx=

2 00 0.9

x

1 01 1

eshx stability sx2 = 0.9x2e

shx2.

It then becomes clear why some methods are less conservative than otherson this particular example (and alike).

Method I II III+PI I+ II+ IV+PI SGT+W3 Exact

hmax .99 .99 4.36 4.35 4.35 4.47 4.84 6.17

More successful methods get rid ofx1, either explicitly (via) or implicitly

(via Parks inequality).


Beyond SGT
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Small Gain Theorem isnottheonlyfrequency-domain robustness tool. Wemay try to

combine small gain and passivity arguments

to end up with less conservative results. This can be done in the

IQC framework

for example, as shown in (Megretski & Rantzer, 97).


Shifted covering
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Clearly,

sx=A0x + A1eshx= (A0+ A1V(s))x A1(V(s)esh)x.

We may then try to

chooseV(s)to reduce conservatism of covering |V(j) ejh|.


Shifted covering: how to chooseV(s)


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Re

Im

1

h

lh(

)

Re

Im

V(j)

h

Simple geometry yields then:

V(j) =

cos h

2 ej

h2 ifh

0 ifh > and lh,V() =

12

lh().


Shifted covering: how to chooseV(s)
http://find/


70/74

Re

Im

1

h

lh(

)

Re

Im

V(j)

h

Simple geometry yields then:

V(j) =

cos h

2 ej

h2 ifh

0 ifh > and lh,V() =

12

lh().


Shifted covering: rational approximationh j

h

b l d b
http://find/


71/74

Frequency responsecos h2 ej

h2 can be quite accurately approximated by

V1(s) = 2hs + 2

.

Covering radii are then:

102

101

100

25

20

15

10

5

0

5

lh

lh,V1

lh,V


Outline
http://find/


72/74



Some comparisons


Concluding remarks


Concluding remarks


73/74

Advantages of LK-methods are in no proportion to their popularity

Relations between LK and SGT are yet to be understood(it looks like that all we can show is that they result in the same LMIs; it would be of

great value to have clear correspondence between intermediate steps of each method)


Concluding remarks
http://find/


74/74

Advantages of LK-methods are in no proportion to their popularity

Relations between LK and SGT are yet to be understood(it looks like that all we can show is that they result in the same LMIs; it would be of

great value to have clear correspondence between intermediate steps of each method)
http://find/

delay robustness mirkin

Documents