hierarchical performance analysis of uncertain large scale ... · synchronization specifications...
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Hierarchical Performance Analysis of UncertainLarge Scale Systems
K. LaibA. Korniienko G. Scorletti F. Morel
Laboratoire AmpèreÉcole Centrale de Lyon
GT MOSAR, Grenoble, 6 October, 2015
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 1 / 28
1 IntroductionMotivationProblem formulationProblem analysis
2 Proposed approachRobustness analysis and QC PropagationHierarchical approach
3 Application Example
4 Discussion
5 Conclusion and future work
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 2 / 28
Introduction Motivation
Context : PLL network
Large Scale Systems (LSS) : Phase Locked Loop (PLL) network
PLL network to deliver clock signal to synchronous multi-core processors
How to guarantee synchronization ?
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 3 / 28
Introduction Motivation
Context : PLL network
Large Scale Systems (LSS) : Phase Locked Loop (PLL) network
PLL network to deliver clock signal to synchronous multi-core processors
How to guarantee synchronization ?
4 3 2 1
8 7 6 5
12 11 10 9
16 15 14 13
+
-
Introduce global synchronization error
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 3 / 28
Introduction Motivation
Context : PLL network
Large Scale Systems (LSS) : Phase Locked Loop (PLL) network
PLL network to deliver clock signal to synchronous multi-core processors
How to guarantee synchronization ?
4 3 2 1
8 7 6 5
12 11 10 9
16 15 14 13
+
-
Introduce global synchronization error
Synchronization specifications (performance) are guaranteed ifTrg−→eg satisfies some frequency constraints
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 3 / 28
Introduction Motivation
Context : Performance
Performance is expressed in frequency domain.
100
101
102
103
104
105
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
30
frequency, rad/sec
Mag
nit
ud
e, d
B
S1
ω0
+40 dB/dec
Max peak
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 4 / 28
Introduction Motivation
Context : Performance
Performance is expressed in frequency domain.
100
101
102
103
104
105
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
30
frequency, rad/sec
Mag
nit
ud
e, d
B
S2
S1
ω0
+40 dB/dec
Max peak
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 4 / 28
Introduction Motivation
Context : Performance
Performance is expressed in frequency domain.
100
101
102
103
104
105
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
30
frequency, rad/sec
Mag
nit
ud
e, d
B
S2
S1
ω0
+40 dB/dec
Max peak
0 0.05 0.1 0.15 0.2 0.25-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
time, sec
ε 1(t
), u
.e.
S1
S2
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 4 / 28
Introduction Motivation
Context : Uncertainties
Active clock distribution network
Technological dispersions, modeling errors =⇒ uncertainties (∆)
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 5 / 28
Introduction Motivation
Context : Uncertainties
Active clock distribution network
Technological dispersions, modeling errors =⇒ uncertainties (∆)Uncertain subsystems
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 5 / 28
Introduction Motivation
Context : Uncertainties
Active clock distribution network
Technological dispersions, modeling errors =⇒ uncertainties (∆)Uncertain subsystems
Uncertain Network
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 5 / 28
Introduction Motivation
Context : Uncertainties
Active clock distribution network
Technological dispersions, modeling errors =⇒ uncertainties (∆)Uncertain subsystems
Uncertain Network
Robustness analysis :Perform the worst case robustness analysis for all the uncertainties ∆i
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 5 / 28
Introduction Motivation
Context : Performance
100
101
102
103
104
105
106
-140
-120
-100
-80
-60
-40
-20
0
20
frequency, rad/sec
Mag
nit
ud
e, d
B
nominal
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 6 / 28
Introduction Motivation
Context : Performance
100
101
102
103
104
105
106
-140
-120
-100
-80
-60
-40
-20
0
20
frequency, rad/sec
Mag
nit
ud
e, d
B
nominal
with uncertainties
upper bound
+40 dB/dec
ω0
Max peak
Synchronization specifications (performance) are guaranteed if the upper boundsatisfies the frequency constraints
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 6 / 28
Introduction Problem formulation
PLL network Performance
16 PLLs mutually synchronized
4 3 2 1
8 7 6 5
12 11 10 9
16 15 14 13
+
-
Two uncertain parameters for every PLL =⇒ 32 uncertain parameters
Nowadays networks : 100 PLLs =⇒ 200 uncertain parameters=⇒ classic method is not applicable
16 PLL network to show classic method results
Objective Compute an upper bound on ‖Trg→eg‖ for all the uncertainties
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 7 / 28
Introduction Problem analysis
Problem analysis
Large scale robustness analysis : two aspects problem
1 Robustness analysis : IQC based analysis (input-output description)
2 Large scale : decomposition techniques from graph theory
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 8 / 28
Introduction Problem analysis
Problem analysis
Large scale robustness analysis : two aspects problem
1 Robustness analysis : IQC based analysis (input-output description)
2 Large scale : decomposition techniques from graph theory
Direct application of IQC based analysis =⇒ important computation time
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 8 / 28
Introduction Problem analysis
Problem analysis
Large scale robustness analysis : two aspects problem
1 Robustness analysis : IQC based analysis (input-output description)
2 Large scale : decomposition techniques from graph theory
Direct application of IQC based analysis =⇒ important computation time
Few methods combining the two aspects : [Andersen et al., 2014]
Complex large scale
analysis problem
Complex large scale
optimization problem
Small scale
optimization problems
Modeling Optimization
High complexity
Low complexity
Specific solver
Large scale IQC analysis
Decomposition techniques
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 8 / 28
Introduction Problem analysis
Problem analysis
Large scale robustness analysis : two aspects problem
1 Robustness analysis : IQC based analysis (input-output description)
2 Large scale : decomposition techniques from graph theory
Direct application of IQC based analysis =⇒ important computation time
Few methods combining the two aspects : [Andersen et al., 2014]
Complex large scale
analysis problem
Complex large scale
optimization problem
Small scale
optimization problems
Simplified small scale
analysis problems
(distributed)
Modeling Optimization
High complexity
Low complexity
Specific solver
Large scale IQC analysis
Decomposition techniques
Hierarchical techniques
Small scale IQC analysis Standard
solver
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 8 / 28
Proposed approach Robustness analysis and QC Propagation
Integral Quadratic Constraints (IQC)
Integral Quadratic Constraints (IQC)
∫ +∞
−∞
(z(jω)w(jω)
)∗ΦP(jω)
(z(jω)w(jω)
)dω ≥ 0
Possibility to cover classical characterizations of performance
L2 gain∫ +∞
0
‖z(t)‖2 dt ≤ γ2
∫ +∞
0
‖w(t)‖2 dt⇐⇒∫ +∞
−∞
(z(jω)w(jω)
)∗ (−I 00 γ2
)(z(jω)w(jω)
)dω ≥ 0
Passivity∫ +∞
0
z(t)T w(t)dt ≥ 0⇐⇒∫ +∞
−∞
(z(jω)w(jω)
)∗ (0 II 0
)(z(jω)w(jω)
)dω ≥ 0
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 9 / 28
Proposed approach Robustness analysis and QC Propagation
Proposed approach
Linear Time Invariant Systems
z(jω) = T(jω)w(jω) and QC based analysis
Frequency domain : frequency response at ω0
Performance : compute an upper bound on the frequency response (σ(T) ≤ γ)min
γγ s.t.
(TI
)∗ (−I 00 γ2I
)(TI
)≥ 0
100
101
102
103
104
105
106
-140
-120
-100
-80
-60
-40
-20
0
20
frequency, rad/sec
Mag
nit
ud
e, d
B
nominal
with uncertainties
upper bound
+40 dB/dec
ω0
Max peak
General performance(
TI
)∗ ( X YY∗ Z
)(TI
)≥ 0
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 10 / 28
Proposed approach Robustness analysis and QC Propagation
Proposed approach : Robust Performance Theorem (LTI systems)
QC for performance and uncertainty : Classical interpretation
Theorem (Robust Performance Theorem)
T is X, Y, Z dissipative i.e.(TI
)∗ ( X YY∗ Z
)(TI
)≥ 0 ∀ ∆ ∈ ∆ =⇒ QC of T
if and only if
1)(
∆I
)∗ (Φ11 Φ12Φ∗12 Φ22
)(∆I
)≥ 0 ∀∆ ∈ ∆ =⇒ QC of ∆
2)(
MI
)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0
0 Y∗ 0 Z
(MI
)> 0
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 11 / 28
Proposed approach Robustness analysis and QC Propagation
Proposed approach : Robust Performance Theorem (LTI systems)
QC for performance and uncertainty : Classical interpretation
Theorem (Robust Performance Theorem)
T is X, Y, Z dissipative i.e.(TI
)∗ ( X YY∗ Z
)(TI
)≥ 0 ∀ ∆ ∈ ∆ =⇒ QC of T
if and only if
1)(
∆I
)∗ (Φ11 Φ12Φ∗12 Φ22
)(∆I
)≥ 0 ∀∆ ∈ ∆ =⇒ QC of ∆
2)(
MI
)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0
0 Y∗ 0 Z
(MI
)> 0
Condition 1) : infinite dimensional
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 11 / 28
Proposed approach Robustness analysis and QC Propagation
Proposed approach : Robust Performance Theorem (LTI systems)
QC for performance and uncertainty : Classical interpretation
Theorem (Robust Performance Theorem)
T is X, Y, Z dissipative i.e.(TI
)∗ ( X YY∗ Z
)(TI
)≥ 0 ∀ ∆ ∈ ∆ =⇒ QC of T
if and only if
1)(
∆I
)∗ (Φ11 Φ12Φ∗12 Φ22
)(∆I
)≥ 0 ∀∆ ∈ ∆ =⇒ QC of ∆
2)(
MI
)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0
0 Y∗ 0 Z
(MI
)> 0
Condition 1) : infinite dimensional
Parametrize Φ with Φ∆ in 1) and test 2) =⇒ Construct a ’basis’ Φ∆ for Φ
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 11 / 28
Proposed approach Robustness analysis and QC Propagation
Proposed approach : Robust Performance Theorem (LTI systems)
QC for performance and uncertainty : Classical interpretation
Theorem (Robust Performance Theorem)
T is X, Y, Z dissipative i.e.(TI
)∗ ( X YY∗ Z
)(TI
)≥ 0 ∀ ∆ ∈ ∆ =⇒ QC of T
if and only if =⇒ if ( and only if )1) ∃ Φ ∈ Φ∆ =⇒ QC of ∆
2)(
MI
)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0
0 Y∗ 0 Z
(MI
)> 0
Condition 1) : infinite dimensional
Parametrize Φ with Φ∆ in 1) and test 2) =⇒ Construct a ’basis’ Φ∆ for Φ
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 11 / 28
Proposed approach Robustness analysis and QC Propagation
Proposed approach : Robust Performance Theorem (LTI systems)
QC for performance and uncertainty : Classical interpretation
Theorem (Robust Performance Theorem)
T is X, Y, Z dissipative i.e.(TI
)∗ ( X YY∗ Z
)(TI
)≥ 0 ∀ ∆ ∈ ∆ =⇒ QC of T
if and only if =⇒ if ( and only if )1) ∃ Φ ∈ Φ∆ =⇒ QC of ∆
2)(
MI
)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0
0 Y∗ 0 Z
(MI
)> 0
Condition 1) : infinite dimensional
Parametrize Φ with Φ∆ in 1) and test 2) =⇒ Construct a ’basis’ Φ∆ for Φ=⇒ conservative (pessimist) results
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 11 / 28
Proposed approach Robustness analysis and QC Propagation
Proposed approach : Robust Performance Theorem (LTI systems)
QC for performance and uncertainty : Classical interpretation
Theorem (Robust Performance Theorem)
T is X, Y, Z dissipative i.e.(TI
)∗ ( X YY∗ Z
)(TI
)≥ 0 ∀ ∆ ∈ ∆ =⇒ QC of T
if and only if =⇒ if ( and only if )1) ∃ Φ ∈ Φ∆ =⇒ QC of ∆
2)(
MI
)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0
0 Y∗ 0 Z
(MI
)> 0
Condition 1) : infinite dimensional
Parametrize Φ with Φ∆ in 1) and test 2) =⇒ Construct a ’basis’ Φ∆ for Φ=⇒ conservative (pessimist) results
Conservatism depends on Φ∆
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 11 / 28
Proposed approach Robustness analysis and QC Propagation
Proposed approach : Robust Performance Theorem (LTI systems)
QC for performance and uncertainty : New interpretation
Theorem (Robust Performance Theorem)
T is X, Y, Z dissipative i.e.(TI
)∗ ( X YY∗ Z
)(TI
)≥ 0 ∀ ∆ ∈ ∆
if ( and only if )
1) ∃ Φ ∈ Φ∆
2)(
MI
)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0
0 Y∗ 0 Z
(MI
)> 0
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 12 / 28
Proposed approach Robustness analysis and QC Propagation
Proposed approach : Robust Performance Theorem (LTI systems)
QC for performance and uncertainty : New interpretation
Theorem (Robust Performance Theorem)
T is X, Y, Z dissipative i.e.(TI
)∗ ( X YY∗ Z
)(TI
)≥ 0 ∀ ∆ ∈ ∆
if ( and only if )
1) ∃ Φ ∈ Φ∆
2)(
MI
)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0
0 Y∗ 0 Z
(MI
)> 0
Local step : find simple QC for every Ti
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 12 / 28
Proposed approach Robustness analysis and QC Propagation
Proposed approach : Robust Performance Theorem (LTI systems)
QC for performance and uncertainty : New interpretation
Theorem (Robust Performance Theorem)
T is X, Y, Z dissipative i.e.(TI
)∗ ( X YY∗ Z
)(TI
)≥ 0 ∀ ∆ ∈ ∆
if ( and only if )
1) ∃ Φ ∈ Φ∆
2)(
MI
)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0
0 Y∗ 0 Z
(MI
)> 0
Local step : find simple QC for every Ti =⇒ reduce the complexity
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 12 / 28
Proposed approach Robustness analysis and QC Propagation
Proposed approach : Robust Performance Theorem (LTI systems)
QC for performance and uncertainty : New interpretation
Theorem (Robust Performance Theorem)
T is X, Y, Z dissipative i.e.(TI
)∗ ( X YY∗ Z
)(TI
)≥ 0 ∀ ∆ ∈ ∆
if ( and only if )
1) ∃ Φ ∈ Φ∆
2)(
MI
)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0
0 Y∗ 0 Z
(MI
)> 0
Local step : find simple QC for every Ti =⇒ reduce the complexity
Ti are seen as uncertainty ∆i
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 12 / 28
Proposed approach Robustness analysis and QC Propagation
Proposed approach : Robust Performance Theorem (LTI systems)
QC for performance and uncertainty : New interpretation
Theorem (Robust Performance Theorem)
T is X, Y, Z dissipative i.e.(TI
)∗ ( X YY∗ Z
)(TI
)≥ 0 ∀ ∆ ∈ ∆
if ( and only if )
1) ∃ Φ ∈ Φ∆
2)(
MI
)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0
0 Y∗ 0 Z
(MI
)> 0
Local step : find simple QC for every Ti =⇒ reduce the complexity
Ti are seen as uncertainty ∆i
Global step : use local QC to find global QC
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 12 / 28
Proposed approach Robustness analysis and QC Propagation
Proposed approach : Robust Performance Theorem (LTI systems)
QC for performance and uncertainty : New interpretation
Theorem (Robust Performance Theorem)
T is X, Y, Z dissipative i.e.(TI
)∗ ( X YY∗ Z
)(TI
)≥ 0 ∀ ∆ ∈ ∆
if ( and only if )
1) ∃ Φ ∈ Φ∆
2)(
MI
)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0
0 Y∗ 0 Z
(MI
)> 0
Local step : find simple QC for every Ti =⇒ reduce the complexity
Ti are seen as uncertainty ∆i
Global step : use local QC to find global QC =⇒ conservative results
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 12 / 28
Proposed approach Robustness analysis and QC Propagation
Proposed approach : Robust Performance Theorem (LTI systems)
QC for performance and uncertainty : New interpretation
Theorem (Robust Performance Theorem)
T is X, Y, Z dissipative i.e.(TI
)∗ ( X YY∗ Z
)(TI
)≥ 0 ∀ ∆ ∈ ∆
if ( and only if )
1) ∃ Φ ∈ Φ∆
2)(
MI
)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0
0 Y∗ 0 Z
(MI
)> 0
Local step : find simple QC for every Ti =⇒ reduce the complexity
Ti are seen as uncertainty ∆i
Global step : use local QC to find global QC =⇒ conservative results
=⇒ create a basis for QC of Ti (to use as Φ∆ in global step)
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 12 / 28
Proposed approach Robustness analysis and QC Propagation
Proposed approach : Robust Performance Theorem (LTI systems)
Classical interpretation :For given X, Y and Z find Φ from basis Φ∆
New interpretation :
Find basis for X, Y and Z from given Φ ∈ Φ∆
Propagate the old basis into the new basis
=⇒ QC propagation
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 13 / 28
Proposed approach Robustness analysis and QC Propagation
Proposed approach : Robust Performance Theorem (LTI systems)
Classical interpretation :For given X, Y and Z find Φ from basis Φ∆
New interpretation :
Find basis for X, Y and Z from given Φ ∈ Φ∆
Propagate the old basis into the new basis
=⇒ QC propagation
Difficulties
Size : not too big/small
Quality : describes the best the uncertain system
Efficient computation : convex
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 13 / 28
Proposed approach Robustness analysis and QC Propagation
Robustness Analysis : QC classes
Some classes of QC with geometric interpretationsdisc [Dinh et al., 2013]band [Dinh et al., 2014]cone [Laib et al., 2015]
Real(w*z)0.98 0.99 1 1.01 1.02 1.03
Imag
inar
y (w
*z)
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
Real(w*z)0.98 0.99 1 1.01 1.02 1.03
Imag
inar
y (w
*z)
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
Real(w*z)0.98 0.99 1 1.01 1.02 1.03
Imag
inar
y (w
*z)
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
Formulate as convex optimization (no graphical computation)
Some physical interests : gain, phase, . . .
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 14 / 28
Proposed approach Robustness analysis and QC Propagation
Robustness Analysis : QC classes
Some classes of QC with geometric interpretationsdisc [Dinh et al., 2013]band [Dinh et al., 2014]cone [Laib et al., 2015]
Real(w*z)0.98 0.99 1 1.01 1.02 1.03
Imag
inar
y (w
*z)
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
Real(w*z)0.98 0.99 1 1.01 1.02 1.03
Imag
inar
y (w
*z)
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
Real(w*z)0.98 0.99 1 1.01 1.02 1.03
Imag
inar
y (w
*z)
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
Formulate as convex optimization (no graphical computation)
Some physical interests : gain, phase, . . .
Cone : Phase uncertainty information
The phase notion for Single-Input Single-Output (SISO) systems is well defined
For Multi-Input Multi-Output (MIMO) systems ? ?
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 14 / 28
Proposed approach Robustness analysis and QC Propagation
Robustness Analysis : QC classes
Some classes of QC with geometric interpretationsdisc [Dinh et al., 2013]band [Dinh et al., 2014]cone [Laib et al., 2015]
Real(w*z)0.98 0.99 1 1.01 1.02 1.03
Imag
inar
y (w
*z)
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
Real(w*z)0.98 0.99 1 1.01 1.02 1.03
Imag
inar
y (w
*z)
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
Real(w*z)0.98 0.99 1 1.01 1.02 1.03
Imag
inar
y (w
*z)
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
Formulate as convex optimization (no graphical computation)
Some physical interests : gain, phase, . . .
Cone : Phase uncertainty information
The phase notion for Single-Input Single-Output (SISO) systems is well defined
For Multi-Input Multi-Output (MIMO) systems ? ?=⇒ Numerical range
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 14 / 28
Proposed approach Robustness analysis and QC Propagation
Robustness Analysis : Numerical Range
For a given a frequency response Γ of a system T
The numerical range N (Γ)
N (Γ) = w∗z | z = Γw,w ∈ Cnw and ‖w‖ = 1
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 15 / 28
Proposed approach Robustness analysis and QC Propagation
Robustness Analysis : Numerical Range
For a given a frequency response Γ of a system T
The numerical range N (Γ)
N (Γ) = w∗z | z = Γw,w ∈ Cnw and ‖w‖ = 1
Certain numerical range
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 15 / 28
Proposed approach Robustness analysis and QC Propagation
Robustness Analysis : Numerical Range
For a given a frequency response Γ of a system T
The numerical range N (Γ)
N (Γ) = w∗z | z = Γw,w ∈ Cnw and ‖w‖ = 1
Certain numerical range Uncertain numerical range
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 15 / 28
Proposed approach Robustness analysis and QC Propagation
Robustness Analysis : Cone QC [Laib et al., 2015]
Theorem
Given the frequency responseof an uncertain system T
Finding the smallest α :
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 16 / 28
Proposed approach Robustness analysis and QC Propagation
Robustness Analysis : Cone QC [Laib et al., 2015]
Theorem
Given the frequency responseof an uncertain system T
Finding the smallest α :
Quasiconvex optimisation problem
LMI constraints
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 16 / 28
Proposed approach Robustness analysis and QC Propagation
Robustness Analysis : Cone QC [Laib et al., 2015]
Theorem
Given T = ∆ ?M, let λ = cotα
2
minλ,Ω
D1, G1, D1, G1
D2, G2, D2, G2
λ
s.t :
λ
(D1 0
0 D2
)+
(D1 0
0 −D2
)> 0
λ
(M 0I 00 M0 I
)∗ (B1 00 B2
)(M 0I 00 M0 I
)+
(M 0I 00 M0 I
)∗ (A1 00 A2
)(M 0I 00 M0 I
)> 0(
D1 0
0 D2
)> 0 and
(M 0I 00 M0 I
)∗ (B1 00 B2
)(M 0I 00 M0 I
)> 0
=⇒ Efficient tools to solve the problem
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 16 / 28
Proposed approach Hierarchical approach
Resume
Level 1
pq
M
T
T1
z w
Ti
TN
1
1 1
11
1
1
1
1
1 Consider hierarchical structure of the systemFind basis (QC description) for Ti with Robust Performance TheoremPropagate this basis to the global level
2 For global hierarchical level, investigate the performance with RobustPerformance Theorem
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 17 / 28
Proposed approach Hierarchical approach
Resume
Level 1
pq
M
T
T1
z w
Ti
TN
1
1 1
11
1
1
1
1
1 Consider hierarchical structure of the system
Find basis (QC description) for Ti with Robust Performance TheoremPropagate this basis to the global level
2 For global hierarchical level, investigate the performance with RobustPerformance Theorem
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 17 / 28
Proposed approach Hierarchical approach
Resume
Level 1
pq
M
T
T1
z w
Ti
TN
1
1 1
11
1
1
1
1
1 Consider hierarchical structure of the system
Find basis (QC description) for Ti with Robust Performance TheoremPropagate this basis to the global level
2 For global hierarchical level, investigate the performance with RobustPerformance Theorem
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 17 / 28
Proposed approach Hierarchical approach
Resume
Level 1
pq
M
T
T1
z w
Ti
TN
1
1 1
11
1
1
1
1
sTi
uiyi
qipi
Level 2
∆i
M11
M21 M22
M12i i
ii
1 Consider hierarchical structure of the system
Find basis (QC description) for Ti with Robust Performance TheoremPropagate this basis to the global level
2 For global hierarchical level, investigate the performance with RobustPerformance Theorem
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 17 / 28
Proposed approach Hierarchical approach
Resume
Level 1
pq
M
T
T1
z w
Ti
TN
1
1 1
11
1
1
1
1
sTi
uiyi
qipi
Level 2
∆i
M11
M21 M22
M12i i
ii
1 Consider hierarchical structure of the systemFind basis (QC description) for Ti with Robust Performance Theorem
Propagate this basis to the global level
2 For global hierarchical level, investigate the performance with RobustPerformance Theorem
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 17 / 28
Proposed approach Hierarchical approach
Resume
Level 1
pq
M
T
T1
z w
Ti
TN
1
1 1
11
1
1
1
1
sTi
uiyi
qipi
Level 2
∆i
M11
M21 M22
M12i i
ii
1 Consider hierarchical structure of the systemFind basis (QC description) for Ti with Robust Performance TheoremPropagate this basis to the global level
2 For global hierarchical level, investigate the performance with RobustPerformance Theorem
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 17 / 28
Proposed approach Hierarchical approach
Resume
Level 1
pq
M
∆
∆1
z w
∆i
∆N
1
1 1
11
u1
u1
u1
u1
sTi
uiyi
qipi
Level 2
∆i
M11
M21 M22
M12i i
ii
1 Consider hierarchical structure of the systemFind basis (QC description) for Ti with Robust Performance TheoremPropagate this basis to the global level
2 For global hierarchical level, investigate the performance with RobustPerformance Theorem
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 17 / 28
Proposed approach Hierarchical approach
Resume
Level 1
pq
M
∆
∆1
z w
∆i
∆N
1
1 1
11
u1
u1
u1
u1
sTi
uiyi
qipi
Level 2
∆i
M11
M21 M22
M12i i
ii
Computation time is reduced however conservatism may appear
robustness of feedbacks loops =⇒ simple set may be sufficient
combination of several simple sets =⇒ increase of the computation time
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 18 / 28
Proposed approach Hierarchical approach
Resume
Level 1
pq
M
∆
∆1
z w
∆i
∆N
1
1 1
11
u1
u1
u1
u1
sTi
uiyi
qipi
Level 2
∆i
M11
M21 M22
M12i i
ii
Computation time is reduced however conservatism may appear
robustness of feedbacks loops =⇒ simple set may be sufficient
combination of several simple sets =⇒ increase of the computation time
=⇒ trade-off conservatism/computation time
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 18 / 28
Application Example
PLL network : Local Step
Characterize each PLL with QC with : disc, band and cone
Real(w*z)0.9998 0.9999 1 1.0001 1.0002 1.0003 1.0004
Imag
inar
y (w
*z)
×10-4
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
ω = 15
Real(w*z)0.999 0.9995 1 1.0005 1.001 1.0015 1.002
Imag
inar
y (w
*z)
×10-3
-1.5
-1
-0.5
0
0.5
1
ω = 37
Real(w*z)0.99 0.995 1 1.005 1.01 1.015
Imag
inar
y (w
*z)
×10-3
-16
-14
-12
-10
-8
-6
-4
-2
0
2
4
ω = 136
Real(w*z)0.95 1 1.05
Imag
inar
y (w
*z)
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
ω = 501
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 19 / 28
Application Example
PLL network : Global Step
Compute an upper bound on Trg→eg for all the uncertainties
ω, rad/sec101 102 103 104
Mag
nit
ud
e, d
B
-40
-30
-20
-10
0
10
20PLL network performance analysis
Direct approach: 346.7 sec
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 20 / 28
Application Example
PLL network : Global Step
Compute an upper bound on Trg→eg for all the uncertainties
ω, rad/sec101 102 103 104
Mag
nit
ud
e, d
B
-40
-30
-20
-10
0
10
20PLL network performance analysis
Direct approach: 346.7 sec
Hierarchical approach disc: 16 sec
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 20 / 28
Application Example
PLL network : Global Step
Compute an upper bound on Trg→eg for all the uncertainties
ω, rad/sec101 102 103 104
Mag
nit
ud
e, d
B
-40
-30
-20
-10
0
10
20PLL network performance analysis
Direct approach: 346.7 sec
Hierarchical approach disc: 16 sec
Hierarchical approach disc+band: 46.4 sec
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 20 / 28
Application Example
PLL network : Global Step
Compute an upper bound on Trg→eg for all the uncertainties
ω, rad/sec101 102 103 104
Mag
nit
ud
e, d
B
-40
-30
-20
-10
0
10
20PLL network performance analysis
Direct approach: 346.7 sec
Hierarchical approach disc: 16 sec
Hierarchical approach disc+band: 46.4 sec
Hierarchical approach disc+cone: 43.1 sec
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 20 / 28
Discussion
General Hierarchical Approach
Hierarchical approach
Level 1
pq
M
∆
∆1
z w
∆i
∆N
1
1 1
11
u1
u1
u1
u1
sTi
uiyi
qipi
Level 2
∆i
M11
M21 M22
M12i i
ii
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 21 / 28
Discussion
General Hierarchical Approach
Hierarchical approach
Level 1
pq
M
∆
∆1
z w
∆i
∆N
1
1 1
11
u1
u1
u1
u1
sTi
uiyi
qipi
Level 2
∆i
M11
M21 M22
M12i i
ii
Special case : Direct approach
Level 1
pq
M
∆
∆1
z w
∆i
∆N
1
1 1
11
u1
u1
u1
u1
sTi
uiyi
qipi
Level 2
∆i
M11
M21 M22
M12i i
ii
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 21 / 28
Discussion
General Hierarchical Approach
pq
M
T
T1
z w
Ti
TN
Level 1
1
1 1
11
1
1
1
1
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 22 / 28
Discussion
General Hierarchical Approach
pq
M
T
T1
z w
Ti
TN
Level 1
1
1 1
11
1
1
1
1
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 22 / 28
Discussion
General Hierarchical Approach
pq
M
T
T1
z w
Ti
TN
Level 1
1
1 1
11
1
1
1
1
Level 2
pq
M
T
z w
Ti
TN
2
2 2
22
2
2
2
2
T1
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 22 / 28
Discussion
General Hierarchical Approach
pq
M
T
T1
z w
Ti
TN
Level 1
1
1 1
11
1
1
1
1
Level 2
pq
M
T
z w
Ti
TN
2
2 2
22
2
2
2
2
T1
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 22 / 28
Discussion
General Hierarchical Approach
pq
M
T
T1
z w
Ti
TN
Level 1
1
1 1
11
1
1
1
1
Level 2
pq
M
T
z w
Ti
TN
2
2 2
22
2
2
2
2
T1s
Ti
uiyi
qipi
l
Level l
∆i
M11
M21 M22
M12i i
ii
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 22 / 28
Discussion
General Hierarchical Approach
pq
M
T
T1
z w
Ti
TN
Level 1
1
1 1
11
1
1
1
1
Level 2
pq
M
T
z w
Ti
TN
2
2 2
22
2
2
2
2
T1s
Ti
uiyi
qipi
l
Level l
∆i
M11
M21 M22
M12i i
ii
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 22 / 28
Discussion
General Hierarchical Approach
pq
M
T
T1
z w
Ti
TN
Level 1
1
1 1
11
1
1
1
1
Level 2
pq
Mz w2
2 2
22
∆
∆1
∆i
∆Nu2
u2
u2
u2
sTi
uiyi
qipi
l
Level l
∆i
M11
M21 M22
M12i i
ii
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 22 / 28
Discussion
General Hierarchical Approach
pq
M
T
T1
z w
Ti
TN
Level 1
1
1 1
11
1
1
1
1
Level 2
pq
Mz w2
2 2
22
∆
∆1
∆i
∆Nu2
u2
u2
u2
sTi
uiyi
qipi
l
Level l
∆i
M11
M21 M22
M12i i
ii
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 22 / 28
Discussion
General Hierarchical Approach
pq
M
Δ
Δ 1
z w
Δ
Δ N
Level 1
1
1 1
11
u1
u1
u1
u1
Level 2
pq
Mz w2
2 2
22
∆
∆1
∆i
∆Nu2
u2
u2
u2
sTi
uiyi
qipi
l
Level l
∆i
M11
M21 M22
M12i i
ii
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 22 / 28
Discussion
General Hierarchical Approach
pq
M
Δ
Δ 1
z w
Δ
Δ N
Level 1
1
1 1
11
u1
u1
u1
u1
Level 2
pq
Mz w2
2 2
22
∆
∆1
∆i
∆Nu2
u2
u2
u2
sTi
uiyi
qipi
l
Level l
∆i
M11
M21 M22
M12i i
ii
Many degrees of freedom to handle the trade-off conservatism/computation time
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 22 / 28
Discussion
General Hierarchical Approach
pq
M
Δ
Δ 1
z w
Δ
Δ N
Level 1
1
1 1
11
u1
u1
u1
u1
Level 2
pq
Mz w2
2 2
22
∆
∆1
∆i
∆Nu2
u2
u2
u2
sTi
uiyi
qipi
l
Level l
∆i
M11
M21 M22
M12i i
ii
Many degrees of freedom to handle the trade-off conservatism/computation time
Number of levels
Number of Ti in each level
Basis for ∆i
Basis for Ti in each level
Parallel computing
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 22 / 28
Discussion
Robust stability
Network with N systems randomly generated [Andersen et al., 2014].
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 23 / 28
Discussion
Robust stability
Network with N systems randomly generated [Andersen et al., 2014].
Complex large scale
analysis problem
Complex large scale
optimization problem
Small scale
optimization problems
Modeling Optimization
High complexity
Low complexity
Specific solver
Large scale IQC analysis
Decomposition techniques
Direct method computation timeProposed method computation time
= 10 for N = 200
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 23 / 28
Discussion
Robust stability
Network with N systems randomly generated [Andersen et al., 2014].
Complex large scale
analysis problem
Complex large scale
optimization problem
Small scale
optimization problems
Simplified small scale
analysis problems
(distributed)
Modeling Optimization
High complexity
Low complexity
Specific solver
Large scale IQC analysis
Decomposition techniques
Hierarchical techniques
Small scale IQC analysis Standard
solver
N0 50 100 150 200
CP
U t
ime
(sec
on
ds)
10-2
10-1
100
101
102
Direct methodDistributed hierarchical method
Direct method computation timeProposed method computation time
= 113 for N = 200
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 24 / 28
Conclusion and future work
Conclusion
Performance analysis of uncertain large scale systems
Important computation time with direct method
Exploit hierarchical structure using basis (QC) propagation
General approach with degrees of freedom
Reduce computation time with possible conservatism
Trade-off conservatism/computation time
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 25 / 28
Conclusion and future work
Perspectives
Perspectives
Systematic decomposition technique using Graph Theory
Combine hierarchical method with specific solvers
Complex large scale
analysis problem
Complex large scale
optimization problem
Small scale
optimization problems
Simplified small scale
analysis problems
(distributed)
Modeling Optimization
High complexity
Low complexity
Specific solver
Large scale IQC analysis
Decomposition techniques
Hierarchical techniques
Small scale IQC analysis Standard
solver
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 26 / 28
Conclusion and future work
Thank you for your attention
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 27 / 28
Conclusion and future work
References
Andersen, M., Pakazad, S., Hanson, A., and Rantzer, A. (2014).
Robust stability analysis of sparsely interconnected uncertain systems.IEEE Transactions on Automatic Control, 59(8) :2151–2156.
Dinh, M., Korniienko, A., and Scorletti, G. (2013).
Embedding of uncertainty propagation : application to hierarchical performance analysis.IFAC Symposium on System, Structure and Control, 5(1) :190–195.
Dinh, M., Korniienko, A., and Scorletti, G. (2014).
Convex hierarchicalrchical analysis for the performance of uncertain large scale systems.IEEE Conference on Decision and Control, pages 5979– 5984.
Laib, K., Korniienko, A., Scorletti, G., and Morel, F. (2015).
Phase IQC for the hierarchical performance analysis of uncertain large scale systems.IEEE Conference on Decision and Control (to appear).
K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 28 / 28