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7/26/2019 Delay Robustness Mirkin
1/74
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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Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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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Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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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Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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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Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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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Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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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Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
Outline
Lyapunov-Krasovski methods & model transformations
Good ol (scaled) Small Gain Theorem
Some comparisons
Possible refinements
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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Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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).
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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).
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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p g
The second transformation
Rewrite
sx=A0x + A1eshx
I + A1
1esh
s
sx= (A0+ A1)x
or in the time domain:
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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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p g
The second transformation
Rewrite
sx=A0x + A1eshx
I + A1
1esh
s
sx= (A0+ A1)x
or in the time domain:
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).
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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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).
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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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)
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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Outline
Lyapunov-Krasovski methods & model transformations
Good ol (scaled) Small Gain Theorem
Some comparisons
Possible refinements
Concluding remarks
Model transformations Small Gain Theorem Some comparisons Refinements 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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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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
sx=A0x + A1 x stable
1
We then end up withdelay-independent(sufficient) condition
(sI A0)1A1 < 1,
which is easily solvable.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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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
sx=A0x + A1 x stable
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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Delay as unstructured uncertaintyI (contd)
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
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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Delay as unstructured uncertaintyI (contd)
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
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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Delay as unstructured uncertaintyII
Rewrite
sx=A0x + A1eshx= (A0+ A1)x A1(1 e
sh)x.
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Delay as unstructured uncertaintyII
Rewrite
sx=A0x + A1eshx= (A0+ A1)x A1(1 e
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
.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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Delay as unstructured uncertaintyII
Rewrite
sx=A0x + A1eshx= (A0+ A1)x A1(1 e
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
.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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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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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 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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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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) = 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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
l f
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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) = 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. . .
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
l f l
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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) = 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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
D l d i II ( d)
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Delay as unstructured uncertaintyII (contd)
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 extendible to multiple-delay problems
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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
D l t t d t i t II ( td)
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Delay as unstructured uncertaintyII (contd)
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:
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.
1This appears to harm the acceptance of the method.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
D l t t d t i t II ( td)
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Delay as unstructured uncertaintyII (contd)
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 extendible to multiple-delay problems
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.
1This appears to harm the acceptance of the method.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
O tli
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Outline
Lyapunov-Krasovski methods & model transformations
Good ol (scaled) Small Gain Theorem
Some comparisons
Possible refinements
Concluding remarks
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
SG isnt more conservative than LK
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SG isnt more conservative than LK
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|.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
SG isnt more conservative than LK
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SG isn t more conservative than LK
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|.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
SG isnt more conservative than LK
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SG isn t more conservative than LK
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 useW
0-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|.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
H norm of descriptor systems
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H norm of descriptor systems
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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
H norm of descriptor systems
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H norm of descriptor systems
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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
H norm of descriptor systems
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H norm of descriptor systems
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 0
X21 X22
=X 11 X 21
0 X 22 I 0
0 0 0.
HenceX12 =0 and X11 =X
11 0.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
Descriptor approach vs. Small Gain Theorem (contd)
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Descriptor approach vs. Small Gain Theorem (contd)
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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
Descriptor approach vs. Small Gain Theorem (contd)
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Descriptor approach vs. Small Gain Theorem (contd)
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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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)
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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 h0 =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 h0 =
h
2
)
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
Some arguments in favor of SGT.
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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 h0 =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 h0 =
h
2
)
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
Some arguments in favor of SGT.
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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 h0 =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 h0 =
h
2
)
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
Some arguments in favor of SGT.
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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 h0 =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 h0 =
h
2
)
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
Some arguments in favor of SGT.
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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 h0 =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 h0 =
h
2
)
The question is
what makes LK methods so dominating ?
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
Outline
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Lyapunov-Krasovski methods & model transformations
Good ol (scaled) Small Gain Theorem
Some comparisons
Possible refinements
Concluding remarks
Model transformations Small Gain Theorem Some comparisons Refinements 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)
Model transformations Small Gain Theorem Some comparisons Refinements 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)
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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
Delay-independent stableDelay-dependent
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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
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
.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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).
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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).
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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|.
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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().
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
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().
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
Shifted covering: rational approximationh j
h
b l d b
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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
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
Outline
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Lyapunov-Krasovski methods & model transformations
Good ol (scaled) Small Gain Theorem
Some comparisons
Possible refinements
Concluding remarks
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
Concluding remarks
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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)
Model transformations Small Gain Theorem Some comparisons Refinements Concluding remarks
Concluding remarks
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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)
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