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Course on Randomized Algorithms, Seville 1© IEIIT – CNR 2005

Randomized Algorithms for Analysis and Control of Uncertain

SystemsRoberto Tempo

IEIIT-CNR, Politecnico di [email protected]

Escuela Superior de Ingenieros, Universidad de Sevilla

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Course Schedule

� Monday, March 14, 11:00 - 14:00

� Tuesday, March 15, 9:30 - 14:00 and 15:30 - 18:30

� Wednesday, March 16, 9:30 - 14:00 and 15:30 - 18:30

� Thursday, March 17, 9:30 - 14:00

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� Additional documents, papers, etc, please consult

http://staff.polito.it/tempo/

� Questions after the course can be sent to

[email protected]

� Some MATLABTM

codes are available at

http://www.polito.it/~dabbene

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Acknowledgments

� This research is supported by IEIIT-CNR of Italy

� Thanks to

Giuseppe Calafiore, Politecnico di Torino, Italy

Fabrizio Dabbene, IEIIT-CNR, Italy

Boris Polyak, Russian Academy of Sciences, Russia

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Overview

1. Preliminaries

2. Robustness Analysis and Design

3. Randomized Algorithms

4. Random Vector Generation

5. Random Matrix Generation

6. Randomized Algorithms for Robust Design

7. Probabilistic LMIs

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Overview

8. Probabilistic LPV Systems

9. Randomized Algorithms for Switched Systems

10. Miscellaneous Topics

11. Mixed Deterministic/Randomized Methods

12. Applications

Conclusions and Discussions

References

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CHAPTER 1

Preliminaries

Keywords: probability, robustness, uncertainty, good and bad sets

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Randomized Algorithms (RAs)

� Randomized algorithms are frequently used in many

areas of engineering, computer science, physics,

finance, optimization,…

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Randomized Algorithms (RAs)

� Combinatorial optimization, computational geometry

� Examples: Data structuring, search trees, graph

algorithms, sorting, …

� Motion and path planning problems

� Mathematics of finance: Computation of path integrals

� Bioinformatics (string matching problems)

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RAs and Uncertain Systems

� Main point of the course: Rigorous study and

development of RAs for uncertain systems and control,

and related areas

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Uncertainty

� Uncertainty has been always a critical issue in control

theory and applications

� First methods to deal with uncertainty were based on a

stochastic approach

� Optimal control: LQG and Kalman filter

� Since early 80’s alternative deterministic approach

(worst-case or robust) has been proposed

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Robustness

� Major stepping stone in 1981: Formulation of the H∞

problem by George Zames

� Various “robust” methods to handle uncertainty now

exist: Structured singular values, Kharitonov,

optimization-based (LMI), l-one optimal control,

quantitative feedback theory (QFT)

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Robustness

� Late 80’s and early 90’s: Robust control theory became

a well-assessed area

� Successful industrial applications in aerospace,

chemical, electrical, mechanical engineering, …

� However, …

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Limitations of Robust Control

� Researchers realized some drawbacks of robust control

� Computational Complexity: Worst case robustness is

often NP-hard (not solvable in polynomial time unless

P==NP))

� Various robustness problems are NP-hard

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Limitations of Robust Control

� Consider uncertainty ∆ bounded in a set B of radius ρ.

Largest value of ρ such that the systems is stable for all

∆ ∈ B is called (worst-case) robustness margin

� Conservatism: Worst case robustness margin may be

small

� Discontinuity: Worst case robustness margin may be

discontinuous wrt problem data

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New Paradigm Proposed

� New paradigm proposed is based on uncertainty

randomization and leads to randomized algorithms for

analysis and synthesis

� Within this setting a different notion of problem

tractability is needed

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Brief History of RAs

� 1980: New notion of probability of instability by Stengel in the

context of flight control[1]

� 1986: Stengel’s book on stochastic optimal control[2]

� 1989: CDC paper[3] titled “Probabilistic Robust Controller Design”

� Basic idea: Use of Monte Carlo methods

� Major criticism: Absence of new mathematical tools[1] R.F. Stengel (1980)

[2] R.F. Stengel (1986)

[3] P. Djavdan and H.J.A.F. Tulleken and M.H. Voetter and H.B. Verbruggen and G.J. Olsder (1989)

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� 1996: CDC (held in Kobe) two papers[1,2] on explicit

bounds on the sample size

� 1998: Development of methods for “average” design

based on statistical learning theory by Vidyasagar[3]

[1] P.P. Khargonekar and A. Tikku (1996) [2] R. Tempo, E. W. Bai and F. Dabbene (1996)[3] M. Vidyasagar (1998)

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� More recently, various research directions:

� Sample generation in various sets

� Bounds on the sample complexity

� Stochastic gradient algorithms for control design

� RAs for specific applications

� Combination of deterministic/randomized techniques

� RAs for nonconvex problems

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New Paradigm Proposed

� New paradigm proposed is based on uncertainty

randomization and leads to randomized algorithms for

analysis and synthesis

� Main motivation: Limitations of robust control

� Computational complexity (NP-hardness), conservatism

and discontinuity of the robustness margin

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Probability andRobustness

� The interplay of Probability and Robustness for control of uncertain systems

� Robustness: Deterministic uncertainty bounded

� Probability: Random uncertainty (pdf is known)

� Computation of the probability of performance

� Controller which stabilizes mostuncertain systems

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Key Features

� We obtain larger robustness margins at the expense of a

small risk

� We study the probability degradation beyond the

robustness margins

� Computational complexity is generally not an issue:

Randomized algorithms are low complexity

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Uncertain Systems

M(s) ∆ UncertaintySystem

� ∆ belongs to a structured set Bˇ

– Parametric uncertainty q

– Nonparametric uncertainty ∆i

– Mixed uncertainty

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Worst Case Model

� Worst case model: Set membership uncertainty

� The uncertainty ∆ is bounded in a set Bˇ

∆ ∈ Bˇ

� Real parametric uncertainty q=[q1,…, ql] ∈ —l

qi ∈ [qi-, qi

+]

� Nonparametric uncertainty

∆i ∈ { ∆i∈ —n,n: ||∆i|| ≤ 1}

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Robustness

� Robustness is to guarantee stability and minimize performance for all ∆ in Bˇ

M(s)

∆

d e

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Objective of Robustness

� Objective of robustness: To guarantee stability and

performance for all

∆ ∈ Bˇ

� Different probabilistic paradigm based on uncertainty

randomization of ∆ within Bˇ

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Example: Flexible Structure

� Mass spring damper model

� Real parametric uncertainty affecting stiffness and

damping

� Complex unmodeled dynamics (nonparametric)

m1

l1

k1

m2

l2

k2

m3

l3

k3

m4

l4

k4

m5

l5

k5

l6

k6

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Flexible Structure

� M-∆ configuration for controlled systems and study stability of

q1, q2 ∈ —

∆1∈¬4,4

∆ ∈ Bˇ(( == {∆ ∈ ˇ:: σ(∆ )) < ρ):} : :

BAsICsM 1)()( −−=

∆=∆

1

52

51

00

00

00

Iq

Iq

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Probability Degradation Function

0 .3 5 0 .4 0 .4 5 0 .5 0 .5 5 0 .6 0 .6 5 0 .7

0 .9 4

0 .9 5

0 .9 6

0 .9 7

0 .9 8

0 .9 9

1

1 .0 1

P ro b a b i li s tic ra d iu s ρ

Est

ima

ted

pro

ba

bili

ty d

eg

rad

atio

n

394.0≈ρ

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Probabilistic Model

� Probability density function associated to Bˇ

� We now assume that ∆ is a random matrix with given

density function f∆(∆) and support Bˇ

� Example: ∆ is uniform in Bˇ

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Uniform Density

� We take f∆(∆)=U[Bˇ] (uniform density within Bˇ). That

is

� In this case, for a subset S ⊆ Bˇ

[ ]

∈∆=

otherwise

ifvol

0)(

1ˇ

ˇˇ

BBBU

{ })()(

)(Pr

ˇˇ BB vol

vol

vol

d SS S =

∆=∈∆ ∫

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Performance Function

� In classical robustness we guarantee that a certain

performance requirement is attained for all ∆∈Bˇ

� This can be stated in terms of a performance function

J = J(∆)

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Example 1: H∞ Performance

� Compute the H∞ norm of the transfer function Fu(M,∆)

between the error e and the disturbance d

J(∆) = || Fu(M, ∆)||∞

Fu(M,∆) is called the upper LFT (Linear Fractional

Transformation)

� For given γ >0, check if

J(∆) < γ

for all ∆ in Bˇ

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Example 1: H∞ Performance - 2

� Continuous time SISO systems with real parametric uncertainty q. Consider the transfer function P(s,q) =

where q1 ∈ [0.2, 0.6] and q2 ∈ [10-5,3·10-5]

� Letting J(q) = || P(s,q)||∞, we choose γ=0.003

� Check if J(q)<γ for all q in these intervals

( ) )5.0102(5.000102.0)05.010(105.0

21

52

22

51

521

qsqsq

qsqq

+⋅+++++

−−

−

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Example 1: H∞ Performance - 3

� The set of q1, q2 for which J(q)<γ is shown below

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.651

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3x 10

-5

q1

q2

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Example 2[1] : Robust Stability

� Robust stability of continuous time systems with real parametric uncertainty q. Consider the closed loop polynomial

Then,

J(q) = max{Reλ1(q), …, Re λn(q)}

where λ1(q),…,λn(q) are the roots of p(s,q)

l

lL sqasqaqaqsp )()()(),( 10 +++=

[1] G. Truxal (1961)

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Example 2: Robust Stability - 2

� Consider the closed loop uncertain polynomial p(s,q) =

where q1 ∈ [0.3, 2.5], q2 ∈ [0,1.7] and r=0.5

� Check stability for all q in these intervals

( ) ( ) ( ) 3221212121

2 132661 ssqqsqqqqqqr +++++++++++

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Example 2: Robust Stability - 3

� Set of unstable polynomials

� Taking r=0 the unstable set reduces to a singleton

q1

q2

0.3 2.5

0

1.7

r

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P1: Performance Verification

� For given performance level γ, check whether

J(∆) ≤ γ

for all ∆ in Bˇ

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P2: Worst-Case Performance

� Find Jmax such that

)(maxmax ∆=∈∆

JJˇB

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Drawbacks: Complexity

� Complexity issue: Worst-case robustness is often NP-hard (not solvable in polynomial time unless P=NP)[1]

� In classical robustness a number of problems are NP-hard

– stability of interval matrices

– structured singular value

– static output feedback

– …

[1] V. Blondel and J.N. Tsitsiklis (2000)

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Conservatism and Complexity: Trade-off

� Uncertain parameters often enter into the system in a

nonlinear fashion

� To avoid complexity issues (or just to find a solution of

the problem) overbounding techniques are used

� Worst-case robustness margins may be very conservative

� Another issue: Discontinuity of the robustness margin

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Good and Bad Sets

� We define two subsets of Bˇ

∆∆∆∆good= {∆: J(∆) ≤ γ } ⊆ Bˇ

∆∆∆∆bad = {∆: J(∆) > γ } ⊆ Bˇ

� ∆∆∆∆good is the set of ∆’s satisfying performance� Measure of robustness is

( ) ∫ ∆=good

dvol good ∆∆∆∆∆∆∆∆

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Example of Good and Bad Sets

q1

q2

0.3 2.5

0

1.7

∆∆∆∆bad

∆∆∆∆good

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Example of Good and Bad Sets - 2

q1

q2

0.3 2.5

0

1.7

∆∆∆∆bad

∆∆∆∆good

Taking smallr

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Probabilistic Robustness Measure

� In worst-case analysis we compute γ such that all ∆satisfy performance. Equivalently, we evaluate γ such that

∆∆∆∆good = Bˇ

� In a probabilistic setting, we are satisfied if the ratio

is close to one

( )( )

ˇBvol

vol good∆∆∆∆

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CHAPTER 2

Nuts and Bolts ofRobustness Analysis and Design

Keywords: M-∆ configuration, structured singular value, stability radii, Kharitonov Theorem

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General Robust Control Framework

� Performance is measured by a characteristic of e.

� Closed loop: e=Fu(M,∆)d

P(s)

∆

K(s)

d e

u y

( )KPMM

MMsM ,)(

2221

1211l

F=

=

M(s) strictly proper

[1] K. Zhou and J.C. Doyle (1998)

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Uncertainty

� Parametric uncertainty q

� Nonparametric uncertainty ∆i

� Mixed uncertainty

∆ Uncertainty

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Structured Uncertainty

� Subspace defining uncertainty structure

� Norm-bounded structured uncertainty

[ ]{ }brr IqIq ∆∆=∆∆= ,,,,,blockdiag: 11 1LL

llˇ

{ }ρρ ≤∆∈∆∆= ,:)( ˇˇ

B

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Robust Stability

� Let A,B,C be a realization of M11(s). Define the set

� The feedback connection is robustly stable if and only if every element in Aρ is stable.

� The stability radius

{ })(,: ρρ ˇBA ∈∆∆+== ∆∆ CBAAA

ρ

{ }stablerobustly is :sup ρρρ A=

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Examples of Structures

� Full complex block: ̌ = ¬n,m

� Full real block: ̌ = —n,m

( ) ∞

==)(

1)(sup

1

1111 sMjM ωσρ

ω

( ]

1

11111

111121,0 ))(Re())(Im(

))(Im())(Re(infsup

−

−∈

−=

ωωγωγω

σρ γω jMjM

jMjM

Complex stability radius

Real stability radius

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Mixed Structured Uncertainty

� Mixed uncertainty:

� µ is the structured singular value

))((sup1

ωµρ

ω jM=

{ }0))(det(:)(inf1

))((11 =∆−∆

=∈∆ ωσ

ωµjMI

jMˇ

Structured stability radius


llˇ

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Stability Radii

� We deal with stability radii in state space

{ })(,: ρρ ˇBA ∈∆∆+== ∆∆ CBAAA

{ }stablerobustly is :sup ρρρ A=

10

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Quadratic Stability of Interval Matrices

� Consider A∆=A+∆, where

� A∆ is quadratically stable if and only if ∃ P >0 such that

for all vertex matrices Ai

� Simultaneous solution of Lyapunov inequalities is a convex problem, but number of matrix inequalities grows as

nnA ,, —∈≤∆ ∞ ρ

0<+ iTi PAPA

2

2n

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Rank One µ Problem

� Consider stable t.f. M11(s)=u(s)vT(s) where u(s) and v(s) are l-dimensional vectors of rational functions

� Let ∆=diag(q1,…,ql), then

� This leads to a polytope of polynomials

� Edge Theorem: The polytope is stable if and only if the one-dimensional edges are stable

∑=

+=∆−l

111 )()(1))(det(

iiii svsuqsMI

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Interval Polynomials

� An interval polynomial is of the form

where qi∈[qi-, qi

+]

� Kharitonov Theorem:p(s,q) is stable if and only if fourparticular vertex polynomials (Kharitonov polynomials) are stable

lL ssqsqqqsp ++++= 2

210),(

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Convex Optimization in Control

� Many robust analysis and design problems may be formulated as convex optimization problems[1]

� A unifying framework is that based on SemidefiniteProgramming (SDP)

H2/H∞ control

Computable bounds for µ analysis/synthesis

Multi-model robust design

[1] S. P. Boyd L. El Ghaoui, E. Feron and V. Balakrishnan (1994)

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Robust Optimization

� Many control problems in the presence of uncertainty may be cast as robust SDP

� For generic uncertainty structures, we can only compute relaxations of original robust SDP

subject to ,min xcT

0),( >∆xF)()()(),( 110 ∆++∆+∆=∆ mmFxFxFxF L

)(, ρˇB∈∆= Tii FF

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CHAPTER 3

Randomized Algorithms

Keywords: Monte Carlo methods, law of large numbers, Chernoff bound, worst-case bound

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Recall:Probabilistic Robustness Measure

� In worst-case analysis we compute γ such that all ∆satisfy performance. Equivalently, we evaluate γ such that

∆∆∆∆good = Bˇ

� In a probabilistic setting, we are satisfied if the ratio

is close to one

( )( )

ˇBvol

vol good∆∆∆∆

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Probability of Performance[1]

� We define the probability of performance as

pγ = Pr{J(∆) ≤ γ }

� Notice that, if f∆(∆) is uniform, then

( )( )

ˇBvol

volp good∆∆∆∆

=γ

[1] R.F. Stengel (1980)

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� Let

� Define the set of stable polynomials

� The volume of stable polynomials

Volume of Stable PolynomialsContinuous Time[1]

nnsasaaasp +++= L10),(

{ }0)Re(:0),(,10:1 ≥∀≠≤≤∈= + ssaspaa in

n —S

( )nn volV S=

!1n

Vn ≤

∞→→≈ − neV nn as0

2

[1] A.S. Nemirovskii and B.T. Polyak (1994)

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Volume of Stable PolynomialsDiscrete Time[1]

� Let

� Define the set of stable polynomials

� The volume of stable polynomials

V1=2, V2=4, V3=16/3

nzzaaazp +++= L10),(

{ }10),(: >∀≠∈= zazpa nn —S

( )nn volV S=

odd1

2

1 nVV

Vn

nn

−+ =

∞→→≈ − nnVnn

n as02 22

( ) even1 2

11 n

VnVnV

Vn

nnn

−

−+ +

=

[1] A.T. Fam (1989)

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Bˇ

Bˇ

Volume of Stable PolynomialsDiscrete Time - 2

� If Sn ⊆ Bˇ we can compute in closed form the probability of stability

a0

a1

210),( zzaaazp ++=

( )( )

ˇB

S

vol

volp goodn ∆∆∆∆≡

=γ

Sn

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Randomized Algorithms

� Volume estimate and numerical integration with Monte

Carlo methods

� Monte Carlo and Quasi-Monte Carlo methods[1]

� Breaking the curse of dimensionality

[1] H. Niederreider (1992)

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Ingredients for RAs

� Assume that ∆ is random with pdff∆(∆) with support Bˇ

� Accuracy ε ∈(0,1) and confidenceδ ∈(0,1) be assigned

� Performance function for analysis and level↓ ↓J = J(∆) γ

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Randomized Algorithms for Analysis

� Two classes of randomized algorithms for probabilistic

robust performance analysis

� Probabilistic performance verification (computepγ)� Probabilistic worst case performance (compute worst-

case performance)

� Both are based on Monte Carlo methods, i.e. on

uncertainty randomization

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Randomized Algorithms - 2

� We estimate pγ by means of a randomized algorithm

� First, we generate N i.i.d. samples

∆1, ∆2, …, ∆N ∈ Bˇ

according to the density f∆

� We evaluate J(∆1), J(∆2), …, J(∆N)

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Empirical Probability

� Construct an indicator function

� An estimate of pγ is the empirical probability

where Ngood is the number of samples such that J(∆i) ≤ γ

( )N

NI

Np good

N

i

iN =∆= ∑

=1

1ˆ

( ) ≤∆

=∆otherwise

JifI

ii

0

)(1 γ

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A Reliable Estimate

� The empirical probability is a reliable estimate if

� Find the minimum N such that

where ε ∈(0,1) andδ ∈(0,1)

� ε andδ are called accuracy and confidence

{ } εγγ ≤−≤∆=− NN pJpp ˆ)(Prˆ

{ } δεγ −≥≤− 1ˆPr Npp

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Law of Large Numbers[1]

� For any ε ∈(0,1) andδ ∈(0,1), if

then

δε 241≥N

[1] J. Bernoulli (1713)


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Remarks

� The number of samples computed with the Law of Large Numbers is independent of the number of blocks in ∆, the density function f∆ and the size of Bˇ

� The number of samples N is very large

2.0⋅109

99.5%

0.5%

1.0⋅1010

99.9%

0.5%

5.0⋅1010

99.5%

0.1%

2.5⋅1011N

99.9%1-δ0.1%ε

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Other Bounds

� The Bernoulli bound is based on the Chebyshev

inequality

� Other bounds are also available, such as those based

on the Bienaymé inequality

� A bound that largely improves the previous ones, for

small values of δ and ε, is the Chernoff bound

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Chernoff Bound[1]

� For any ε ∈(0,1) andδ ∈(0,1), if

then

2

2

2

log

εδ≥N


[1] H. Chernoff (1952)

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Comparison Between Bounds

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Chernoff Bound - 2

� We remark that the Chernoff bound greatly improves upon Bernoulli. The dependence on 1/δ is logarithmic

� The dependence on 1/ε is still quadratic

1.2⋅105

99.5%

0.5%

1.6⋅106

99.9%

0.5%

3.0⋅106

99.5%

0.1%

3.9⋅106N

99.9%1-δ0.1%ε

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Computational Complexity of RAs

� RAs are efficient (polynomial-time) because

1. Random sample generation of ∆i can be performed in

polynomial-time

2. Cost of checking stability for fixed ∆i is polynomial-

time

3. Sample size is polynomial in the problem size and

probabilistic levelsε and δ

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Polynomial-Time

� The cost associated with the evaluation of J(∆i) for fixed ∆i is polynomial-time in many cases. For example, when checking stability or H∞ performance

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Cost of Checking Stability

� Consider a polynomial

� To check left half plane stability we can use the Routhtest. The number of multiplications needed is

� The number of divisions and additions is equal to this number

� We conclude that checking stability is O(n2)

odd for 4

1 even for

4

22

nn

nn −

nnsasaaasp +++= L10),(

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Worst-Case Performance

� Recall that

� Generate N i.i.d. samples

∆1, ∆2, …, ∆Ν ∈ Bˇ

according to the density f∆

� Compute

)(maxmax ∆=∈∆

JJˇB

)(maxˆ1

i

NiN JJ ∆=

= K

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Worst-Case Bound[1,2]

� For any ε ∈(0,1) andδ ∈(0,1), if

thenε

δ

−

≥1

1

1

loglog

N

[1] P.P. Khargonekar and A. Tikku (1996)

[2] R. Tempo, E. W. Bai and F. Dabbene (1996)

{ }{ } δε −≥≤>∆ 1ˆ)(PrPr NJJ

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Comparison

� The number of samples required is much smaller than that needed with the Chernoff Bound.

� The dependence on 1/ε is basically linear

1.16⋅106

99.999%

0.001%

1.06⋅103

99.5%

0.5%

1.38⋅103

99.9%

0.5%

5.30⋅103

99.5%

0.1%

9.21⋅104

99.99%

0.01%

6.91⋅103N

99.9%1-δ0.1%ε

≈−

εε1

1log

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Interpretation of Worst Case Bound

vol(∆∆∆∆bad)< ε

J(∆)

∆

NJJ ˆmax −

J(∆)

∆vol(∆∆∆∆bad)< ε

NJJ ˆmax −

15

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Volumetric Interpretation

� In the case of f∆(∆) uniform, we have

� Therefore

is equivalent to

{ } ( )( )

ˇBvol

volJJ bad

N

∆∆∆∆=>∆ ˆ)(Pr

{ }{ } δε −≥≤>∆ 1ˆ)(PrPr NJJ

( ) ( ){ } δε −≥≤ 1Pr ˇBvolvol bad∆∆∆∆

IEIIT-CNRIEIIT-CNR


Confidence Intervals

� The Chernoff and worst-case bounds can be computed a-priori and are explicit

� The sample size obtained with the confidence intervals is not explicit

� Given δ ∈(0,1), upper and lower confidence intervals pLand pU are such that

{ } δγ −=≤≤ 1Pr UL ppp

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Confidence Intervals - 2

� The probabilities pL and pU can be computed a posteriori when the value of Ngood is known, solving equations of the type

with δL+δU=δ

( )

( ) UkN

UkU

N

k

LkN

LkL

N

Nk

ppk

N

ppk

N

good

good

δ

δ

=−

=−

−

=

−

=

∑

∑

1

1

0

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Confidence Intervals - 3

γp̂

Up

Lp

δ = 0.05

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� Let J(q) be the maximum real part of the roots of the closed loop polynomial

with q varying in the box Bq={q∈—n,||q||∞≤1}

� Suppose that p(s,0) is stable. Then, a box

Bq(ρ)={q∈—n,||q||∞≤ρ}

of radius ρ>0 can be determined so that

J(q)<0 for all q ∈ Bq(ρ)

nn sqasqaqaqsp )()()(),( 10 +++= L

ExampleIEIIT-CNRIEIIT-CNR


� Can we estimate a box bigger than Bq(ρ) so that only a small number of plants in this box are unstable?

� We solve such a problem by applying the previous results

Example - 2

16

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� Consider the polynomial

where

44

3310 )()()()(),( sqasqasqaqaqsp +++=

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

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

( ) ( )[ ]( )( ) ( ) ( )35.0501100)(

35.0501100

5.050110021)(

35.0501100

215.0501100)(

215.0501100)(

432

22

13

432

22

1

22

21432

432

22

1

432

22

11

432

22

10

++++−=++++−+

+−+++=+++−+

++++−=+++−=

qqqqqa

qqqq

qqqqqa

qqqq

qqqqqa

qqqqqa

Example -3 IEIIT-CNRIEIIT-CNR


� The parameters q=[q1,q2,q3,q4] vary in the box [-1/2,1/2]4

� Since a0(q)=0 for q1=0.01, the box containing only stable plant is surely smaller than [-0.01,0.01]4

� Taking ε=0.001 and δ=0.01 we compute

� We generate q1,q2,…,q4603 i.i.d. samples using the uniform density function

ε

δ

−

>=1

1

1

log

log4603N

Example - 4

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Example - 5

� We define the performance function J(q) as the maximum real part of the roots of p(s,q) and compute

� With probability at least 1-δ=0.99, the volume of the bad set ∆bad is not greater than ε vol(Bˇ) = ε = 0.001

� We obtain an increase in size which is at least

6,250,000

00000042395.0)(maxˆ4603,,1

<−=∆==

i

iN JJ

K

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Computational Learning Theory

� The Law of Large Numbers studies the problem

where pγ = Pr{J(∆) ≤ γ }

� The function J(∆) is fixed

� Computational Learning Theory computes bounds on the sample size for the problem

where J is a class of functions


( ){ } δεγ −≥∈∀≤−≤∆ 1,ˆ)(PrPr JJpJ N

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VC and P-dimension[1,2]

� The bounds obtained depend on quantities called VC-dimension (if J is a binary valued function), or P-dimension (if J is a continuous valued function)

� VC and P-dimension are measures of the problem complexity

� The bounds obtained are very conservative

[1] M. Vidyasagar (1997)

[2] E.D. Sontag (1998)

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Systems and Control Application

� The class of functions can take into account control parameters

� We replace J(∆) with J(∆,κ) where κ are the controller parameters

� With this approach, we generate random samples for both ∆ and κ, i.e.

∆1, ∆2, …, ∆N and κ1, κ2, …, κN

� We will discuss this approach in the broader contest of control system design with randomized algorithms

17

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Choice of the Distribution

� The probability Pr{∆ ∈S}

depends on f∆(∆)

� It may vary between 0

and 1 depending on the

pdf f∆(∆)

0.5

1

1.5

2

2.5

0

0.5

1

1.5

0

0.5

1

1.5

2

2.5

3

3.5

0.5

1

1.5

2

2.5

0

0.5

1

1.5

0

0.05

0.1

0.15

0.2

0.25

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� The bounds discussed are independent on the choice of the distribution. To compute the estimates we need to know the distribution

� Some research has been done in order to find the worst-case distribution in a certain class[1]

� Uniform distribution is the worst-case if a certain target is convex and centrally symmetric

[1] B. R. Barmish and C. M. Lagoa (1997)

IEIIT-CNRIEIIT-CNR



� Minimax properties of the uniform distribution have been shown in[1]

[1] E. W. Bai, R. Tempo and M. Fu (1998)

IEIIT-CNRIEIIT-CNR


CHAPTER 4

Random Vector Generation

Keywords: radial distributions, inversion method, generalizedGamma density, uniform distribution in norm balls

IEIIT-CNRIEIIT-CNR


RN and Univariate Generation

� Random number generation (RNG): Various methods

available for uniform generation in the interval [0,1)

� Linear and nonlinear RNGs, Fibonacci, feedback shift

register, BBS, MT, …

� Non-uniform univariate random variables: Suitable

functional transformations (e.g., inversion method)

IEIIT-CNRIEIIT-CNR


Multivariate Sample Generation

� Emphasis on finite sample size (Chernoff bound)

� Development of techniques not based on asymptotic

methods such as Metropolis (random walk) or Hit-or-

Miss

� Multivariate random variables: Rejection and conditional

density methods

� Conditional density method requires computation of

multiple integrals and can be used in some special cases

18

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Uniform Sample Generation in lp Balls

� We study parametric uncertainty q in lp norm balls

� Objective: Uniform sample generation in the ball

B— = { q∈—n : ||q||p ≤ 1}

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1step 3

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Rejection Method

� Rejection method: Curse of dimensionality

� Rejection rate for generation of uniform samples in the

Euclidean sphere using an hypercube as bounding set

)12/( )/2()( +Γ= nn nπη

4. 107

n=20

5. 1013401.543.24231.90991.27321

n=30n=10n=4n=3n=2n=1

IEIIT-CNRIEIIT-CNR


Gamma Density

� Need to introduce a technical tool

� A random variable x has (unilateral) Gamma densitywith parameters (a,b) if

0,)(

1)( /1 ≥

Γ= −− xex

baxf bxa

ax

� We write x ∼ G(a,b)

� There exist standard and efficient methods for random generation according to G(a,b)

IEIIT-CNRIEIIT-CNR


Non-uniform Distributions:Inversion Method

� A standard tool for univariate random variable generation is the inversion method

� Let w∈— be a r.v. with uniform distribution in [0, 1]. Let F be a continuous distribution function on — with inverse

� Then, the r.v. z=F-1(w) has distribution F

� The generation is obtained by passing a uniform r.v. through an appropriate transformation F-1

{ }10,)(:inf)(1 ≤≤==− yyxFxyF

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Change of Variables

� Let x be a r.v. with pdffx(x)

� Let x=g(y), g invertible

� The pdf of y is

y

ygygfyf xy ∂

∂= )())(()(

� This method also has multivariate extensions

IEIIT-CNRIEIIT-CNR


Gamma Density

� A random variable x∈— has Gamma density with parameters (a,b) if

0,)(

1)( /1 ≥

Γ= −− xex

baxf bxa

ax

� We denote x ∼ G(a,b)

� There exist standard and efficient methods for random generation according to G(a,b)

19

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Generalized Gamma Density

� A random variable x∈— has Generalized Gamma density with parameters (a,c) if

0,)(

)( 1 ≥Γ

= −− xexa

cxf

cxcax

� We denote x ∼ Gg(a,c)

IEIIT-CNRIEIIT-CNR


Generalized Gamma Density - 2

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

( ) px

ppg epG −

Γ=

)(1

,1

1

p=1p=2p=4p=10p=100

IEIIT-CNRIEIIT-CNR


Comments

� If z ~ G(a,1), taking x = z1/c we have x ~ Gg(a,c)

� Samples distributed according to a bilateral density x ~ fx(x) can be obtained from a unilateral density z ~ fz(z) taking

x = sz

wheres is an independent random sign taking values+1 and-1 with equal probability

IEIIT-CNRIEIIT-CNR


Joint Density

� Let x=[x1,…,xn] be a r.v. with components xi

independently distributed according to the (bilateral) Generalized Gamma density with parameters 1/p,p

� The joint density of x is

pp

pi x

nn

nx

n

i

x ep

pe

p

pxf

||||

1

)/1(2

)/1(2

)( −−

=Γ

=Γ

= ∏

IEIIT-CNRIEIIT-CNR


Examples

� Multivariate normal N(0,I)

is Gg(1/2,2)

� Multivariate Laplace density

is Gg(1,1)

( )xx

nx

T

exf 2

1

2

1)(

−=

π

∑=

−

i

ix

nx exf21

)(

IEIIT-CNRIEIIT-CNR


Radial Densities

� A random vector x∈≈n has {p-radial density if

� g(r) is called the defining functionof x

px xrrgxf == ),()(

p

i

p

ipxx

/1

where

= ∑

20

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Examples of Radial Densities

� The multivariate NormalN(0,I)

is l2-radial

� The multivariate Laplace density

is l1-radial

( )xTx

nx exf 21

2

1)(

−=

π

∑−

= iix

nx exf21

)(

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Radial Density, p=2

22||||)( x

x exf ακ −=

IEIIT-CNRIEIIT-CNR


Radial Density, p=1

1||||)( xx exf ακ −=

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Radial Densities and Uniformity

� The key connection between lp-radial densities and uniform distributions in lp norm balls is given by the following fact

� Let x∈—n be lp-radial, and let w be uniform in [0, 1],

then the random vector

has uniform distribution on the norm ball

n

p

wxx

y /1=

{ }1: ≤p

xx

IEIIT-CNRIEIIT-CNR


Real Random Vectors

� Theorem[1]: Let ≈≡—. If ξi are real r.v. distributed according to the Generalized Gamma density

and w∈[0,1] is a uniformly distributed r.v., then the

vector

is uniformly distributed in B—

[1] G. Calafiore, F. Dabbene and R. Tempo (1998)

( )pG pgi ,~ 1ξ

[ ]Tn

p

xxx

wy n ξξ ,,, 1

1

K==

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Algorithm for Real Uniform Generation

� Generate n independent real scalars ξi ~Gg(1/p,p)

� Construct vector x of components xi=si ξi,where si are random signs

� Generate z=w1/n, where w is uniform in [0, 1]

� Return

� Similar algorithm exists for complex vectors

zxx

yp

=

21

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Real RandomVectors– Step 1

-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4step 1

IEIIT-CNRIEIIT-CNR



-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1step 2

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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1step 3

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Real Random Vectors p=1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

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Real Random Vectors p=0.7

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

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Real Random Vectors p=4

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

22

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Complex Random Vectors

� Theorem: Let ≈≡¬. If ηi are complex r.v. uniformly distributed on the unit circle, ξi are real r.v. distributed according to the Generalized Gamma density

and w∈[0,1] is a uniformly distributed r.v., then the vector

is uniformly distributed in B¬

( )pG pgi ,~ 1ξ

[ ]Tnn

p

xxx

wy n ηξηξ ,,, 1121

K==

IEIIT-CNRIEIIT-CNR


Non-uniform Distributions

� lp-radial densities may be used to obtain generic lp-

radial distributions (other than uniform) on the lp-norm

ball

� Let x∈—n be lp-radial, and let z~fza scalar r.v., then the

random vector has density

where v is a normalization constant

zxx

yp

=

pzny yrrfnr

rgyf === − ),(1

)()( 1ν

IEIIT-CNRIEIIT-CNR


Matrix Extensions

� Matrix Hilbert-Schmidt norms

� For 1-induced and ∞-induced matrix norms, the problem reduces to vector case

� Matrix spectral (max singular value) norm does not reduce to vector case

IEIIT-CNRIEIIT-CNR


CHAPTER 5

Matrix Sample Generation

Keywords: singular value decomposition, spectral norm, Haar density, conditional density method, Selberg integral, examples

IEIIT-CNRIEIIT-CNR


Matrix Sample Generation

� How to generate efficiently uniform matrix samples?

� Vector case is solved for the real and complex case, for

any lp norm ball

� Matrix case is solved for the real and complex case, for

any Hilbert-Schmidt p-norm (reduces to the vector case)

� For 1 and ∞ -induced matrix norms, the problem reduces

to the vector case

� Hard problem for the spectral norm

IEIIT-CNRIEIIT-CNR


A First Attempt: Rejection

� Methods based on rejection of samples generated from an outer-bounding set fail, due to dimensionality issues

� Let

then

� Uniform generation in BF and B∞ is easy

{ }nnF

nnF ≤∆∈∆= :)( ,¬B

{ }1:)1( , ≤∆∈∆= ∞∞ nn¬B

)1()1();()1( ()BB()()BB() ∞⊂⊂ nF

{ }1)(:)1( , ≤∆∈∆= σnn¬B

23

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Rejection Rates

� Let η be the average number of samples that one needs to generate in the outer set, to find one sample in the good set

1e372e236e124e81.8e54688ηF

1e955e542e262e168.7e88,64012η∞

10865432n =

IEIIT-CNRIEIIT-CNR


Singular Value Decomposition

� Consider ∆∈≈n,m, m≥ n.

� Singular Value Decomposition

∆ = UΣ V*

where U∈≈n,n and V∈≈m,n have orthonormal columns,

and

Σ = diag{σ1,σ2,…,σn}

where σ1>σ2>…>σn>0

IEIIT-CNRIEIIT-CNR


A Class of Matrix pdfs

� Unitarily Invariant densities: Depend only on s.v. of ∆

� Radial Symmetric densities: Depend only on norm of ∆

� Uniform distribution in B

is a special case of radial density

{ })()( Σ=∆= ∆∆ ffIF

{ }))(()( ∆=∆= ∆∆ σffRF

[ ]BU=∆∆ )(f

IEIIT-CNRIEIIT-CNR


The pdf of theSingular Values – Real Case

� Theorem[1]: Let ∆∈—n,m. The following statements are equivalent

� The pdff∆(∆) is unitarily invariant

� The joint pdf of U, Σ and V is )()()(),,(,, VffUfVUf VUVU Σ=Σ ΣΣ

{ }[ ]{ }[ ]

( )∏∏≤<≤=

−∆Σ −Σ=Σ

>==

==

nkiki

n

k

nmk

iT

V

TU

ff

VIVVVVf

IUUUUf

1

22

1

,1

)()(

0][,:)(

:)(

σσσ—

U

U

U


IEIIT-CNRIEIIT-CNR


Real Matrices

� U— is a normalization constant

� Proof of previous theorem is based on the determination of the Jacobian of the mapping between ∆ and its SVD factors U,Σ,V. Details are highly technical

( )( )

( )( )∏∏

+−==+

−

−Γ−Γ

−Γ−Γ=

m

nmk

n

kmn

mn

kk

kk

112/)1(

2/)1(

12/)1(

12/)1(

2)8( π

—U

IEIIT-CNRIEIIT-CNR


Uniform Matrices – Real Case

� For particular case of uniform matrices

with σ1>σ2>…>σn>0

� The value of K— is given by the Selberg Integral

∏−

= +−+Γ+Γ+Γ++Γ=

1

0

2/

)2/)1(()12/()2/2/3()2/)1((

!n

i

n

nmiiiim

nK π—

( )∏∏≤<≤=

−Σ −=Σ

nkiki

n

k

nmkKf

1

22

1

)( σσσ—

24

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The pdf of theSingular Values – Complex Case

� Theorem[1]: Let ∆∈¬n,m. The following statements are equivalent

� The pdff∆(∆) is unitarily invariant

� The joint pdf of U, Σ and V is )()()(),,(,, VffUfVUf VUVU Σ=Σ ΣΣ

{ }[ ]{ }[ ]

( )∏∏≤<≤=

+−∆Σ −Σ=Σ

>==

==

nkiki

n

k

nmk

iV

U

ff

VIVVVVf

IUUUUf

1

222

1

1)(2

,1*

*

)()(

0][,:)(

:)(

σσσ¬

U

U

U


IEIIT-CNRIEIIT-CNR


Complex Matrices

� U¬ is a normalization constant

� Proof of previous theorem is based on the determination of the Jacobian of the mapping between ∆ and its s.v.d. factors U,Σ,V

∏=

−−= n

k

mnn

kmkn1

)!()!(

2 π¬U

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Uniform Matrices – Complex Case

� Consider particular case of uniform matrices, and change of variables xi=σi

2 , with ordering condition removed

� The value of Kx is given by the Selberg Integral

∏ ∏= ≤<≤

− −=n

i nkiki

nmixx xxxKxf

1 1

2)()(

∏−

= +−+Γ+Γ++Γ=

1

02 )1()1(

)1(!

1 n

ix nmii

im

nK

IEIIT-CNRIEIIT-CNR


Outline of Sample Generation Method

� For uniform ∆, its SVD factors are independently distributed

1. Generate the samples of U and V (easy problem)

2. Generate the samples of Σ (hard problem)

3. Build matrix sample ∆=UΣVT

IEIIT-CNRIEIIT-CNR


Generation of Haar Samples

� Uniform distribution over orthogonal (or unitary) group

is known as the Haar invariant distribution

� Fundamental property: If U is Haar, then QU has same distribution as U, for any fixed orthogonal (unitary) matrix Q

IEIIT-CNRIEIIT-CNR


Generation of Haar Samples

� Haar matrix U∈—n,n may be generated by means of QR decomposition as follows1. X =randn(n,n);

2. [Q,R]=QR(X);

3. U=Q;

� Complex case works similarly

� Rectangular Haar matrices work similarly

25

IEIIT-CNRIEIIT-CNR

Course on Randomized Algorithms, Seville © IEIIT – CNR 2005

Generation of the Singular Valuesfor Complex Matrices

IEIIT-CNRIEIIT-CNR


Conditional Density Method

� Is a general method that reduces generation according to

one n-dimensional distribution to n one-dimensional

sample generation problems

� Drawback: Requires computation of marginal densities

� The previous is a very hard problem in general:

Computing multiple integrals

IEIIT-CNRIEIIT-CNR


Conditional Densities Method - 2

� Write as

where

and

),...,( 1 nx xxf

)|()|()(),...,( 11122111 −= nnnnx xxxfxxfxfxxf LL

)()(

)|(111

111

−−− =

ii

iiiii xxf

xxfxxxf

L

LL

∫∫∫ += ninxii dxdxxxfxxf LL 111 ),...,(...)(

IEIIT-CNRIEIIT-CNR


Conditional Density Method - 3

� A vector x∈—n with density fx(x) can be obtained

generating sequentially the xi, i=1,…,n

� Each xi is generated independently according to the univariatedistribution

)|( 11 −iii xxxf L

IEIIT-CNRIEIIT-CNR


Computing the Marginal Densities: Complex Matrices

Let

be a partial Vandermonde matrix

[ ])()()(

111

21

112

11

222

21

21

i

ni

nn

i

i

i xxx

xxx

xxx

xxx

XXXV L

L

MLMM

L

L

L

=

=

−−−

∫∫∫ += ninxii dxdxxxfxxf LL 111 ),...,(...)(

IEIIT-CNRIEIIT-CNR


Key Result

� The marginal density is equal to

� Where M=R-1,

� Proof of result based on Dyson-Mehta Theorem

),...,( 1 nx xxf

∏=

−−=i

k

nmki

Tixnx xM

M

inKxxf

11

)!(),...,( VV

[ ] nlrnmlr

R lr ,,1,;1

1, K=

−−++=

26

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Dyson-Mehta Theorem

� Let Zn∈—n,n be a symmetric matrix such that

1. [Zn] i,j=ψ(xi,xj)

2.

3.

where dµ is a suitable measure, and c is a constant. Then

where Zn-1 is the submatrix obtained from Zn removing the row

and column containing xn

∫ = cxdxx )(),( µψ

∫ = ),()(),(),( zxydzyyx ψµψψ

∫ −+−= )det()1()()det( 1nnn ZncxdZ µ

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Key Result - 2

� Given x1, x2,…, xi-1, the marginal density is expressed as a polynomial in xi

� The constants Ci and the coefficients bi are computed by means of appropriate recursions

� We have efficient way to compute conditional densities

∑−

=

−−=

)1(2

01)(

n

k

kik

nmiiii xbxCxp

∑−

=

−− =

)1(2

011 ),,|(

n

k

kik

nmiiiii xbxKxxxf K

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Generation of the Singular Valuesfor Real Matrices

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Computing the Marginal Densities: Real Matrices

� Use again the conditional method

� Mathematical details are different from the complex case

� We again obtain marginal densities in “closed form”

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Key Result

� The marginal density fi (σ1,…, σi) may be computed as

where

being

∏=

−Ψ= −

i

k

nmki

Ki n

Rf1

2212

),,()( 1 σσσσ K

∫∫∫ +=ΨiD innii xdxdxxx )()(|)(|...),,( 11 µµ L&K V

{ }10 1 <<<<= xxD ni L&

kkk dxxxd υµ =)(

)1(21 −−= nmυ

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Computation of ψ

� Theorem: Ψi(x1,…,xi) is equal to

where αi=(υ+1)(n-1)

− −

−−

00),,(0

),,()(

det

11

111

22

1

iTi

iiiK

i

xx

xxxM

x nRi

K

K

V

Vα

−

−

−

−=

odd, for

00)(

00)(

)()()(

even for 0)(

)()(

)(in

xF

x

xFxxS

inx

xxS

xM

iT

iT

iii

iT

ii

i

X

X

X

X

( )( )( )( )

21

21

21

21

1

)(;)(1 −−+−−

−−−+

==j

xijkjkj

xkjijk

j

ikj

i xFxS

27

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Marginal Densities IEIIT-CNRIEIIT-CNR


Sample Generation: Summary

� Details are highly technical

� Computation of the pdf of the singular values

� Computation of the pdf of U,V (Haar distribution)

� Conditional density method

� Closed-form solution of a multiple integral

� Dyson-Mehta Theorem

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Polynomial-Time Algorithms

� Polynomial-time algorithms for the recursive generation of the singular values have been developed

� Algorithms require at each step only additions and multiplications of polynomial matrices

� Technical details and MATLABTM

software can be found at the URL http://www.polito.it/~dabbene

� Method becomes ill-conditioned for large n (n>20)

� Uniform matrices concentrate on the boundary of the norm-ball

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Quasi-Monte Carlo Methods

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Quasi-Monte Carlo Method[1]

� RAs presented are based on Monte Carlo (MC)

� Monte Carlo used for integration and optimization

� Quasi-Monte Carlo (QMC) is a deterministic version of

MC

[1] H. Niederreiter (1992)

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Quasi-Monte Carlo Method

� For integration, discrepancy is a measure of how the

sample set is “evenly distributed” within the integration

domain (unit cube)

� For optimization, the measure is the dispersion

28

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Discrepancy

� Let S be a non-empty family of subsets of [0,1]n and let

∆1,…, ∆N ∈ [0,1]n be a (deterministic) point set

� The discrepancy is defined as

where supremum is taken wrtS ∈ S, Vol(S) is the

volume of Sand I is the indicator function of the set S

|| )(

)(

sup),...,( 1 SVolN

I

D

N

i

i

NN −

∆

=∆∆

∑

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Integration Error

� Integration error is given by the Koksma-Hlawka

inequality

where V(g) is the variation of g

� To minimize integration error, we need to minimize

discrepancy

),...,( )V( )()/1( )( 1 || NN

iN

iDggNdg ∆∆≤∆−∆∆ ∑∫

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Optimal Sequences

� Need to find low discrepancy (optimal) sequences

which minimize discrepancy

� There are various low discrepancy sequences including

van der Corput (for n=1), Halton, Sobol, Nieterreiter

(for n>1)

� For Halton sequence we have

DN (∆1, ∆2, …, ∆N) = O(N-1 (log N)n)

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MC and QMC

� The average absolute error of MC is O(N-1/2)

� Main result of QMC: The (deterministic) error is

O(N-1 (log N)n)

where n is the problem dimension

� This error is asymptotically smaller than O(N-1/2) but it depends on n

� Huge sample size is required before QMC is superior to MC

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CHAPTER 6

Randomized Algorithms forRobust Design

Keywords: LQ regulators, probabilistic quadratic stabilization,gradient based algorithms, probability-one controller

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Analysis vs Design with Uncertainty

� Starting point: worst-case analysis versus design

� Consider an interval family p(s,q), q∈Bq={q∈—n,||q||∞≤1}

� Analysis problem:

– Check if p(s,q) is stable for all q∈Bq

Answer: Kharitonov Theorem

� Design Problem: – Does there exist a q∈Bq such that p(s,q) is stable?

Answer: Unknown in general

29

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Example

� Example:

with qi∈[-0.5,0.5]� Probability of stability is 0.5n

� For n large the probability goes to zero� Nevertheless, it is easy to design a stable polynomial in

the family, e.g. p(s,q*) for

� Notice that this is not a fragile solution

)())((),( 110 −−−−= nqsqsqsqsp L

25.0*1

*1

*0 −==== −nqqq L

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Synthesis Paradigm

� Design the parameterized controller Kθ to guarantee stability and performance

P

∆

Kθ

d e

u y

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Probabilistic Synthesis Paradigm

� Uncertainty randomization of ∆ in Bˇ

� Convex optimization to design the controller K(s)

P(s)

∆

K(s)

d e

u y

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Synthesis Performance Function

� Recall that the parameterized controller is Kθ

� We replace J(∆) with a synthesis performance function

J = J(∆,θ)

where θ ∈ Θ represents the controller parameters to be

determined and its bounding set

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Randomized Algorithms for Synthesis

� Two classes of RAs for probabilistic synthesis

� Average performance synthesis[1]

� Robust performance synthesis[2]


[2] B. Polyak and R. Tempo (2001)

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Average Performance Synthesis

30

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Statistical Learning Theory for Design

� Statistical learning theory for probabilistic design[1]

� Controller randomization and uncertainty randomization� Bounds on the sample size: issue of conservatism[2]

[1] M. Vidyasagar (1997)[2] V. Kolchinskii, C.T. Abdallah, M. Ariola, P. Dorato and D. Panchenko (2000)

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Average Performance Synthesis

� Denote the average performance of the plant byφ (θ) = E∆ {J(∆,θ)}where E∆ is expectation wrt∆ and optimal average performance is φ * = inf φ (θ)

θ∈Θ� RA returns with probability 1-δ a design vector θθθθΝ such

that φ (θθθθΝ) - φ * ≤ ε

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RA for Average Performance Synthesis

1. Determine finite sample size N=N(ε,δ)

2. Draw N samples ∆1,…, ∆Ν according to the given pdf

3. Return the controller parameter

θθθθΝ = arg inf Ngood(θθθθ)/N θ∈Θ

where Ngood (θθθθ) is the number of samples such that J(∆i,θθθθ)≤ γ

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Comments

� This RA is based upon Statistical Learning Theory[1]

� Sample complexity may be determined using the bound

N ≥ (128/ε 2)(log(8/δ) + d (log (32e/ε ) + log log (32e/ε )))

for any ε ∈(0,1) andδ ∈(0,1) and where d is an upper bound on the P-dimension of the function family

J= = {J(·,θθθθ), for all θθθθ}


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Comments

� An upper bound for d can be computed for several control problems. However, this introduces an extra level of conservatism.

� First issue: The sample complexity is huge

� Second issue: How to compute “inf”

� Randomization may be used also in this case, but this implies to generate controllers at random

� Advantage: Convexity of J(∆, θθθθ) is not required

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Robust Performance Synthesis

31

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Robust Performance Synthesis

� RA returns with probability 1-δ a design vector θθθθΝ such

that

Pr{J(∆, θθθθΝ) ≤ γ } ≥ 1 - ε� Controller parameter θθθθΝ is constructed using a finite

number N of random samples of ∆

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RA for Robust Performance Synthesis

1. Initialization2. Determine sample size function N(k)=N(ε ,δ,k)3. Set k=0, i=0 and choose θθθθ0

4. Feasibility loop5. Set l=0 and feas=true6. While l <N(k) and feas=true7. Set i=i+1, l=l+18. Draw ∆ι according to f∆9. If J(∆ι, θθθθκ) > γ set feas = false10. End While

11. Exit condition12. If feas = true13. Set N=I14. Return θθθθΝ

= θθθθκ+1 and Exit15. End If

16. Update17. Update θθθθκ+1 = ψ upd(∆ι, θθθθκ)18. Set k=k+1 and goto 4

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Comments

� This RA is based on convexity of the performance function J(∆,θ) in θ

� Key point: Update rule ψ upd(∆ι, θθθθκ) in the update step

� Specific implementations of the update rule are a gradient step or a step of the ellipsoidal method

� For any ε, δ, number of steps N(k) of the algorithm is given by[1]

ε

δπ−

−+≤1

1log)6log()1( log2

)(k

kN

[1] Y. Oishi (2003)

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RAs for Optimal Control (LQR)

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Uncertain Systems in State Space

� We consider a state space description of the uncertain

system

with x(0)=x0; x∈—n; u∈—m, ∆∈ Bˇ

� For example, A(∆) is an interval matrix with bounded

entries

)()()()( tButxAtx +∆=&

+− ≤≤ ijijij aaa

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LQ regulator

� The performance index is the quadratic cost function

where S >0 and R >0 are given weights

� The state feedback controller is

where Q=QT >0 is solution of a QMI

( )∫∞

+=0

dtRuuSxxJ TT

xQBRu T 11 −−−=

32

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Quadratic Matrix Inequality

� Find Q=QT >0 such that, for given 0 ≤ γ ≤1,

for all ∆∈ Bˇ

� Then, the cost

is guaranteed for all ∆∈ Bˇ

01

0

1xQxJ T −≤

γ

( ) 02)()( 11 ≤++−∆+∆ −− TTT BBRQSQBBRQAQA γ

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Robust Quadratic Stabilization

� Quadratic stabilization and guaranteed cost can be reduced to checking a finite number of matrix inequalities

� Example: If A(∆) is an interval matrix and R=I, for quadratic stabilization we take γ = 0 and we need to find a solution Q=QT >0 of

AiQ + Q(Ai)T – 2 BBT ≤ 0

for all vertex matrices Ai

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Probabilistic Quadratic Stabilization

� Critical problem: The number of LMIs may be too large and not tractable with classical interior point methods

� Example: If A(∆) is an interval matrix the number of LMIs is equal to the number of vertex matrices

� Probabilistic version of quadratic stabilization and LQ regulator

2

2nVN =

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Probabilistic Solution

� Randomly generate ∆1,…, ∆N ∈ Bˇ. Then, check if the

LMI

AiQ + Q(Ai)T – 2 BBT ≤ 0

is feasible for i= 1,…,N and find a common solution

Q=QT >0

� Critical problem: Even if N is relatively small, this is a

hard computational problem

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Sequential Algorithm

� Key point: Sequential algorithm which deals with one constraint at each step

� At step k we have

Phase 1: Uncertainty randomization of ∆Phase 2: Gradient algorithm and projection

� Final result: Find a solution Q=QT >0 with probability one in a finite number of steps

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Definitions

� Let En be an Euclidean space

and C be the cone of positive semi-definite matrices

=∈== ∑=

n

kik

nTn aAAA

1,

21,—E

{ }0: ≥∈= AA nEC

33

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Projection on a Cone

� For any real symmetric matrix A we define the projection [A]+∈C as

� The projection can be computed through the eigenvaluedecomposition A=TΛTT

� Then

where λi+=max {λi ,0}

[ ] XAAX

−=∈

+

Cminarg

TTTA ++ Λ=][

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Phase 1: Uncertainty Randomization

� Uncertainty randomization: Generate ∆k ∈ Bˇ

� Then, for guaranteed cost we obtain the QMI

( ) 02)()( 11 ≤++−∆+∆ −− TTkTk BBRQSQBBRQAQA γ

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Matrix Valued Function

� Define a matrix valued function

and a scalar function

where || ⋅ || is the Frobenius norm

� We can also take the maximum eigenvalue of V(Q ,∆k)

( )TTkTk

k

BBRQSQBBRQAQA

QV112)()(

),(−− ++−∆+∆

=∆

γ

[ ]+∆=∆ ),(),( kk QVQv

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Phase 2: Gradient Algorithm

� We write

where ∂Q is the subgradient and the stepsizeµk is

and r>0 is a parameter

( ){ }[ ] ( ) >∆∆∂−=

++

otherwise

0, if ,1

k

kkkkQ

kkk

Q

QvQvQQ

µ

( ) ( ){ }( ){ } 2

,

,,kk

Q

kkQ

kkk

Qv

QvrQv

∆∂

∆∂+∆=µ

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Closed-form Gradient Computation

� The function v(Q,∆k) is convex in Q and its subgradientis given by

if v(Q,∆k)≠0, and it is zero otherwise

( ){ }( )[ ] ( ) ( ) ( )[ ]

( )k

kTkkk

kQ

Qv

QVQSAQSAQV

Qv

∆∆+∆++∆∆

=∆∂++

,,)()(,

,

γγ

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Theorem[1]

� Assumption: Every open subset of Bˇ has positive measure

� Theorem: A solution Q, if it exists, is found in a finite number of steps with probability one

� Idea of proof: The distance of Qk from the solution set decreases at each correction step

[1] B.T. Polyak and R. Tempo (2001)

34

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Worst-Case Solution

� The sequential algorithm provides a candidate solution for the set of QMIs

� We can check if this candidate solution satisfies all QMIs and it is a worst-case solution, otherwise we run the algorithm again

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Extensions

� Minimization of a measure of violation for problems

that are not strictly feasible[1,2]

� Uncertainty in the control matrix, B=B(∆), ∆∈ Bˇ

We take the feedback law

where Y and Q=QT >0 are design variables

xYQu 1−=

[1] B.R. Barmish and P. Shcherbakov (1999)

[2] G. Calafiore and B.T. Polyak (2001)

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Further Extensions

� Nonlinear constrained systems

� Disturbance rejection and relations with game theory[1]

� Improvements from the numerical point of view

[1] T. Basar and P. Bernhard (1995)

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Related Literature

� Related literature on optimization and adaptive control with linear constraints[1,2,3,4]

� Stochastic approximation algorithms have been widely studied in the stochastic control and optimization literature[6,7]

[1] S. Agmon (1954)[2] T.S. Motzkin and I.J. Schoenberg (1954)[3] B.T. Polyak (1964)[4] V.A. Bondarko and V.A. Yakubovich (1992)

[6] H.J. Kushner and G.G. Yin (2003)[7] J.C. Spall (2003)

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Subsequent Research

� Design of common Lyapunov functions for switched system[1]

� From common to piecewise Lyapunov functions[2]

� Ellipsoidal algorithm instead of gradient algorithm[3]

� Stopping rule which provides the number of steps[4]

� Nonlinear constrained systems

[1] D. Liberzon and R. Tempo (2004)

[2] H. Ishii, T. Basar and R. Tempo (2004)

[3] S. Kanev, B. De Schutter and M. Verhaegen (2002)

[4] Y. Oishi and H. Kimura (2003)

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Example[1]

� We study a multivariable example for the design of a controller for the lateral motion of an aircraft.

� The model consists of four states and two inputs

( ))(

31.053.2

0035.0

91.30

00

)(10

0

0010

)( tutx

NNYNNNN

Y

LLLtx

rpVg

Vg

rp

−

−+

−+−

=

βββββ

β

β

&&&

&

[1] B.D.O. Anderson and J.B. Moore (1971)

35

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Example - 2

� The state variables are

– x1 bank angle

– x2 derivative of bank angle

– x3 sideslip angle

– x4 jaw rate

� The control inputs are

– u1 rudder deflection

– u2 aileron deflection

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Example - 3

� Nominal values: Lp=-2.93, Lβ=-4.75, Lr=0.78, g/V=0.086, Yβ=-0.11, Nβ=0.1, Np=-0.042, Nβ=2.601, Nr=-0.29

� Perturbed matrix A(∆): each parameter can take values in a range of ±15% of the nominal value

� Quadratic stability (γ=0): take R=I and S=0.01I

� Remark: A(∆) is multiaffine in the uncertain parameters: quadratic stability can be ascertained solving simultaneously 29=512 LMIs

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Example - 4

� Sequential algorithm:

– Initial point Q0 randomly selected

– 800 random matrices ∆k

– The algorithm converged to

=

1.21620.73820.44520.7338

0.73820.77980.70200.1645

0.44520.70201.09270.0843-

0.73380.16450.0843-0.7560

Q

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Example - 5

� The corresponding controller

satisfies all the 512 vertex LMIs and therefore it is also a quadratic stabilizing controller in a deterministic sense

� The optimal LQ controller computed on the nominal plant satisfies only 240 vertex LMIs

== −

0.4954-10.237010.1758-2.8814-

49.9587-43.12844.3731-38.61911QBK T

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CHAPTER 7

Probabilistic Algorithms forRobust LMIs

Keywords: robust LMI, robust and approximate feasibility, stochastic gradient methods, quadratic stability

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Robust LMIs

� We discuss stochastic optimization methods for determining admissible solutions to robust Linear Matrix Inequalities (LMIs)

� Robust LMI

where

mxxF —B ∈∈∆∀<∆ ,;0),(ˇ

nnTiii

m

ii FFFxFxF ,

10 );()(),( —∈=∆+∆=∆ ∑

=

36

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Robust LMIs - 2

� The set Bˇ indicates a structured norm bounded uncertainty set


llˇ

{ }ρρ ≤∆∈∆∆= ,:)( ˇˇ

B

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Robust Feasibility

� Let be a non-empty convex and closed set

� A point is a robustly feasible solution if and only if

� Computing robust solution to LMIs is in general numerically difficult

� Robust SDP techniques can handle relaxations of the problem

m—X ⊆

x~

ˇBX ∈∆∀<∆∈ ;0),~( and ,~ xFx

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Feasibility Indicator Function

� Introduce a scalar function

where F+ indicates the projection of F onto the cone of positive semi-definite matrices

� The function is convex in x

),(),( ∆=∆ + xFxϕ

),( ∆xϕ

FWF W −= ≥+

0minarg

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Robust and Approximate Feasibility

� A point is a robustly feasible solution if and only if

� In case no robustly feasible solution exists, we assume a probability density on ∆, and say that x is an approximately feasible solution if

x~

ˇBX ∈∆∀<∆∈ ;0),~( and ,~ xx ϕ

{ }),(minarg ∆= ∆∈xEx

xϕ

X

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Recall: Projections

� Let F+ be the projection of a symmetric matrix F onto the positive semi-definite cone, then

� F+=RD+RT, where F=RDRT is the eigen-decomposition of F, with D=diag(d1,d2,…,dn), and

D+= diag(d1+,d2

+,…,dn+), where di

+=max(di,0)

� The (matrix) function F+(x) is convex in x

� The (scalar) function is convex in x),(),( ∆=∆ + xFxϕ

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Subgradients of ϕ

� A subgradient of ϕ(x,∆) at x is computed as follows. Let

Then

∆

∆

∆=∆∇

+

+

),(Tr

),(Tr

),(1

),(1

xFF

xFF

xx

m

Mϕ

ϕ

≠∆∆∇

=∆∂otherwise 0

0),( if ),(),(

xxxx

ϕϕϕ

37

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Algorithm for Approximate Feasibility

� Assume ∆ is a random matrix with pdff∆

� Let g(x)=E{ϕ(x,∆)}� Let x be the minimum of g(x) over X

� Let

� Given an initial vector x0 consider the recursion

where [·]X denotes the projection onto the set X, and ∆k

are i.i.d. samples drawn from f∆

{ } ∆∈∀≤∆∂ ,;),( Xxxx µϕ

{ }[ ]X

),(1k

kxkkk xxx ∆∂−=+ ϕλ

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Approximate Feasibility - 2

� Let further

� Theorem[1]:

kkkkk

kk

k

kk mmmxx

mx

m

mx λλ +===+= −−

−1001

1 ,0;

{ } )()()( kCxgxgE k ≤−

∑

∑−

=

−

=+−

= 1

0

1

0

222

0

2)( k

ii

k

iixx

kCλ

λµ

[1] G. Calafiore and B. Polyak (2001)

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Approximate Feasibility - 3

� Moreover, if stepsizes are chosen so that

Then

� The proof of this result is based on convexity (Jensen inequality) and properties of projections

{ }∑∞

=∞=

→

0

0

ii

i

λ

λ

{ } )()(lim xgxgE kk

=∞→

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Algorithm for Robust Feasibility

� Assume that a strong feasibility condition holds: There exist

� Assume that, if x is not a robustly feasible solution, then it must hold that

Pr{F(x,∆) > 0} > 0

such that ,0,* >∈ εXx

ˇBX ∈∆∀≤−∈∀≤∆ ,: ,0),( * εϕ xxxx

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Robust Feasibility - 2

� Consider the recursion

where ∆k are i.i.d. random samples drawn from f∆

� Define the stepsizes

� For any the recursion finds a robustly feasible solution in a finite number of steps, w.p. 1

{ }[ ]X

),(1k

kxkkk xxx ∆∂−=+ ϕλ

{ }{ }

≠∆∆∂

∆∂+∆=

otherwise ,0

0),( if ,),(

),(),(2 kk

kkx

kkxkk

k

xx

xx ϕϕ

ϕεϕηλ

X∈0x

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Example: Quadratic Stability[1]

� Consider a linear system with

� System is quadratically stable if and only if there exists P>0 that satisfies simultaneously 512 Lyapunovinequalities for the vertex matrices

0,,)( 0 >≤∆∆+=∆ rrSAA ijij

=

−

−−=

3141.4014.7004.

1457.4727.2451.

5691.9394.1651.

,

201

001

022

0 SA

[1] B. R. Barmish and P. S. Shcherbakov (2002)

38

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Example - 2

� Let r= 0.5. The stochastic algorithm (robust feasibility) converged after N=50 iterations to the solution

� This solution may be checked to actually satisfy allLyapunov inequalities

=5371.02425.03177.0

2425.00443.28155.0

3177.08155.02487.1

P

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Example - 3

� For r= 1, we know that no actual robustly feasible solution exists

� After N=250 iterations of the approximate feasibility algorithm, we found the solution

� A posteriori probability of feasibility (computed with Monte Carlo using 100,000 samples) is pfeas=0.996

−−−−

=5577.00967.02649.0

0967.07455.19899.0

2649.09899.02042.1

P

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Example - 4

� We recall that to assess quadratic stability form a deterministic viewpoint one needs to solve simultaneous Lyapunov equations

� For n=4, 65,536 vertex matrices

� For n=10, 1,26 1030 vertex matrices

� n=4 is already beyond the capabilities of many current SDP solvers

22n

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Example - 5

� Consider an example with n=10

� Nominally stable

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Example - 6

� Check quadratic stability for element-wise independent uncertainty with r =0.5

� N =2000 iterations yielded the solution

� The estimated a posteriori probability of stability is pfeas=1.0

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Comments

� Two randomized algorithms for determining robustly feasible or approximately feasiblesolutions to robust LMIs

� For the latter algorithms convergence is only asymptotic

� These techniques are useful for problems which are intractable by means of standard (exact) LMI methods

� These solutions may serve as a good initial guess for a deterministic algorithm

39

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Optimization Problems[1]

� Extensions to optimization problems

� Consider convex function f(x) and function g(x,∆) convex in x for fixed ∆

� Semi-infinite (nonlinear) programming problem

min f(x)

g(x,∆) for all ∆ ∈ B

� Reformulation as stochastic optimization

� Drawback: Convergence results are only asymptotic

[1] V. B. Tadic, S. P. Meyn and R. Tempo (2003)

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Scenario Approach[1]

� The scenario approach for convex problems

� Non-sequential method which provides a one-shot solution for general convex problems

� Randomization of ∆ ∈ B and solution of a single convex optimization problem

� Derivation of a new bound on the sample size

� Applications to control systems design

[1] G. Calafiore and M. Campi (2004)

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CHAPTER 8

Probabilistic LPV Systems

Keywords: gain scheduling, Linear-Parameter Varying systems, parameter discretization

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LPV

� LPV applications: Aircraft control, automated lane

guidance, communication networks

� Parameters q=q(t) are unknown but bounded in setBq

� They can be measured on-line by the controller

� Critical issue: Parameter discretization (complexity)

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Quadratic LPV

� LPV plant

with q ∈ Bq

� Assumption: Orthogonality conditions are satisfied

=

)(

)(

)(

0))(())((

))((0))((

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

)(

)(

)(

212

121

21

tu

td

tx

tqDtqC

tqDtqC

tqBtqBtqA

ty

te

tx&

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Quadratic LPV - 2

� The main goal is to design an LPV controller of the type

such that quadratic performance of the closed-loop system is guaranteed and

( )( ) q

T

T

dq

dttdtd

dtteteB∈∀<

∫

∫∞

∞

γ21

21

0

0

)()(

)()(sup

=

)(

)(

0))((

))(())((

)(

)(

ty

tx

tqC

tqBtqA

tu

tx c

c

ccc&

40

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Structured QMI Solution[1]

� The LPV problem is solvable if and only if there exist X=XT > 0 and Y=YT > 0 such that for ε > 0

0)()()()(

)()()()(),(

22112

11

≤+−

+++=− IqBqBqBqB

XqCqXCqXAXqAqXPTT

TT

εγ

0)()()()(

)()()()(),(

22112

11

≤+−

+++=− IqCqCqCqC

YqBqYBqYAYqAqYQTT

TT

εγ

0),(1

1

≤

−= −

−

YI

IXYXR

γγ

[1] G. Becker and A. Packard (1994)

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Matrix Valued Function

� Define a matrix valued function

and a scalar function

� Remark: The gradients ∂X{ v(X,Y,q)} and ∂Y{ v(X,Y,q)} can be computed in closed form

=),(00

0),(0

00),(

),,(

YXR

qYQ

qXP

qYXV

[ ]+= ),,(),,( qYXVqYXv

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Gradient-based Algorithm

� We write

� Here µk is a stepsize and

( ){ }( )

( ){ }( )

( ) otherwise and or ,0,, if

,,,,

,,,,

11

1

1

kkkkkkk

kkk

kkkY

kkk

kkk

kkkX

kkk

YYXXqYXv

qYXw

qYXvYY

qYXw

qYXvXX

==>

∂−=

∂−=

++

++

++

µ

µ

( ) ( ){ } ( ){ }( ) 21

22,,,,,, kkk

Ykkk

Xkkk qYXvqYXvqYXw ∂+∂=

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Probabilistic LPV[1]

� Assume that q is random vector with support Bq

� Consider positive measure within Bq

� Theorem: The algorithm converges with probability one

in a finite number of iterations

� Remark: No assumption on the dependence on q of

matrices A, B1, B2, C1, C2, D12, D21

[1] Y. Fujisaki, F. Dabbene and R. Tempo (2003)

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Application Example

� Multivariable example for the design of a controller for

the lateral motion of an aircraft (2 inputs, 3 outputs, 4

state variables)

� Nine uncertain parameters entering multiaffinely into the

state matrix

� Each parameter is perturbed by relative uncertainty ρ

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Numerical Results

� Sequential algorithm

– Initial conditions randomly selected

– 800 iterations

– Computation of Xk and Yk

– Construction of the controller

� We set γ =3

� With ρ = 5% X800 and Y800 satisfy all 29=512 vertex QMIs

41

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Numerical Results

� With ρ = 8% X800 and Y800 still satisfy all 512 vertex QMIs

0.46110.01870.12460.1296

0.01870.60730.13350.2158

0.12460.13350.41660.1546

0.12960.21580.15460.4814

0.50020.14080.02990.1461

0.14080.33950.06870.0035-

0.02990.06870.48410.0715-

0.14610.0035-0.0715-0.6874

800

800

=

=

Y

X

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Algorithm Updates

0 100 200 300 400 500 600 700 800

update

iteration

� The plot shows the algorithm updates

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Fault Tolerant Control (FTC)

� Definition: Fault is an unpermitted deviation of at least one characteristic property or parameter of the system from the acceptable/usual/standard condition[1]

� Example (dramatic): Chernobyl plant[2]

[1] R. Isermann and P. Ballè (1997)

[2] G. Stein, Bode Lecture “Respect the Unstable” (1989)

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Robust Fault Tolerant Control[1]

� Discrete-time plant with time-varying faults f

x(k+1)= A(f) x(k) + B1(f) d(k) + B2(f) u(k)e(k) = C1(f) x(k) + D11(f) d(k) + D12(f) u(k)y(k)= C2(f) x(k) +D21(f) d(k) + D22(f) u(k)

The vector of faults is f = [f1,…,fn] fi(k) = (1+wf,i (k) δ i) gi(k)

where gi(k) is the nominal value and the fault estimate uncertainty δ i is bounded by

| δ i | ≤ 1

[1] S. Kanev and M. Verhaegen (2004)

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Robust Fault Tolerant Control

� Reformulation of the problem as LPV

� Fault estimate uncertainty δ is taken as a random variable with given pdf

� Development of randomized algorithms for computing controller parameters[1,2]

� Random generation of δ and use of ellipsoidal method

� Application: Brushless DC motor

[1] S. Kanev and M. Verhaegen (2004)

[2] S. Kanev (2004)

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Model Predictive Control (MPC)[1]

� Development of randomized algorithms for MPC is an open area

� The separation principle does not hold when parametric uncertainty is present

� Difficulty of deriving output feedback controller

� BMI solution is NP-hard

� Approaches based on finite-horizon output feedback MPC do not always guarantee robust stability of the closed-loop system

[1] E. F. Camacho and F. Bordons (2004)

42

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CHAPTER 9

Randomized Algorithms for Switched Systems

Keywords: common Lyapunov functions, piecewise quadratic function, switching rules

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Common Lyapunov Functions for Finite Families

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Switched Systems Problem

Given Hurwitz matrices and matrix ,

find matrix

We deal with a finite set of matrices

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Common Lyapunov Function

� If P > 0 exists, then the quadratic function

V(x)= xTPx

is a common Lyapunov function for the family of asymptotically stable linear systems

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Special Case[1]

...

quadratic common Lyapunov function∃

In the special case when matrices commute

[1] K. S. Narendra and J. Balakrishnan (1994)

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Function f

� Function f is convex differentiable real-valued functional on the space of symmetric matrices with the property

f(R) ≤ 0 if and only if R ≤ 0

43

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Two Examples of Functions f

1. If λ max is a simple eigenvalue we take

f(R) = λ max(R)

2. f(R) = || R+ ||2

where || . || is Frobenius norm, + is projection into the cone of non-negative definite matrices

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Gradient Algorithms: Preliminaries

Gradient:

( is unit eigenvector of with eigenvalue )

1.

2.

– convex in

given

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Gradient Iteration

– finite family of Hurwitz matrices

– arbitrary symmetric matrix

Gradient iteration:

– visits each index times

( – suitably chosen stepsize)

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Theorem[1]

� Theorem: Solution P >0, if it exists, is found in finite number of steps

� Idea of proof: Distance of Pk from solution set decreases at each correction step

[1] D. Liberzon and R. Tempo (2004)

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Example: Interval Family

� We study interval family of triangular Hurwitzmatrices

� We have vertices

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Numerical Results

� Results with deterministic gradient

� n = 4 (210 inequalities): 10,000 iterations (a few seconds)

� n = 5 (215 inequalities): 10,000,000 iterations (a few hours)

� Randomized gradients gives faster convergence� For comparison: quadstabMATLAB

TMstacks

when n = 5

44

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Comments

� Contribution

� Gradient iteration algorithms for solving simultaneously Lyapunov matrix inequalities

� Deterministic convergence for finite families

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Open Computational Issues

� Performance comparison of different methods

� Comparison with advanced LMI algorithms

� Addressing existence of solutions

� Optimal schedule of iterations

� Additional requirements on solution

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Piecewise Quadratic LyapunovFunctions

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■ Consider the switched system

dx(t)/dt = Aσ(t) x(t)

where x(t) is the state and σ(t) = 1, 2 is the switch■ Beyond common quadratic Lyapunov functions: Piecewise quadratic Lyapunov functions ■ Less conservative but harder to construct

Switched Systems: Another Approach

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■ Given γ > 0 and P1, P2 > 0, suppose that for any x: ||x||=1

xT(P1A1 + A1T P1) x ≤ -γ if xT(P1-P2)x ≥ 0

and

xT(P2A2 + A2T P2) x ≤ -γ if xT(P1-P2)x ≤ 0

Piecewise Lyapunov Functions[1,2]

[1] M. Wicks and R. DeCarlo (1997)

[2] J. Malmborg, B. Bernhardsson and K. J. Astrom (1996)

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■ Then, we construct a switching rule

σ(t) = arg max{i=1,2} x(t)T Pi x(t)■ The piecewise quadratic Lyapunov function

V(x) = max{i=1,2} x(t)T Pi x(t)

decreases along trajectories of the system■ GAS is achieved

Switching Rule

45

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■ This switched system problem is not convexSolution: Combination of randomized algorithms ■ with branch and bound methods■ Design a piecewise quadratic Lyapunov function ■ with probability one in a finite number of steps■ Specific RA and proof techniques are technical

Solution via RA[1]

[1] H. Ishii, T. Basar and R. Tempo (2005)

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CHAPTER 10

Miscellaneous Topics

Keywords:robustness in statistics, value set predictors

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Robustness in Statistics

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Robustness in Statistics

� Parameter estimation

� If ξ ∼ N(0,I) i.i.d., then is the best estimate

� However, if there are outliers, that can be modeled as

ξ ∼ (1-ε)N(0,I)+ε N(0,σ2I) with σ2>>1

then may be very bad

lK,,1* =+= iy ii ξθ

∑=

=−=l

l 1

1minargˆ

iii yy θθ

θ

θ̂

θ̂

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Robust Estimates

� Consider

� ξ is i.i.d. with some density fξ(ξ) belonging to a class P

� Then

� where G(·) is a given function, may be better than

lK,,1* =+= iy ii ξθ

( )∑=

−=l

1

minargi

iG yG θθθ

θ̂

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Robust Estimates – Choice of G

� For instance

is good for normal contaminated distributions

� θG is an intermediate between and the median

quadratic

linear

G(·)

θ̂

46

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Robust Estimates – Optimal G

� Fisher information

� The least favorable distribution is given by

� The best G in the sense of asymptotic variance of θG is

∫

= dttf

tffI

)(

)('

)(

2

ξ

ξ

ξ

)(minarg*ξξ

ξ

fIff P∈

=

)(ln)( ** tftG ξ−=

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Value Set Predictors

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Robust Stability – Value Set

� Consider the uncertain closed-loop polynomial

with q varying in the box Bq={q∈—n,||q||∞≤1}

� Zero exclusion principle: define the value set as

� If p(s,0) is stable and

then robust stability holds

nn sqasqaqaqsp )()()(),( 10 +++= L

{ }qqqjpV B∈= :),()( ωω &

∞≤≤∉ ωω 0 allfor )(0 V

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Probabilistic Predictors of Value Set

� Let q be a random vector uniformly distributed on Bq

� Define the risk-adjusted value set

� Vγ(ω) may be much smaller than V(ω)

{ } γωωωγ −≥∈ 1)(),(Pr:)( VqjpV

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Example – Interval Polynomial

� Consider an interval polynomial

where |ai –ai0|≤ rαi , i=1,…,n-1

� For fixed frequency, the value set is a rectangle (Kharitonov theorem)

V(ω)

Vγ(ω)

nssasaaasp ++++= L2

210),(

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Example – Spherical Polynomial - 2

� Consider a spherical polynomial

where

� V(ω) and Vγ(ω) are ellipses, Vγ(ω) can be rigorously calculated

nssasaaasp ++++= L2

210),(

∑−

=≤−1

02

20)(n

i i

ii raa

α

47

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Example – Spherical Polynomial - 2

� Then

Vγ(ω)

V(ω)

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Linear Functions of Uniform Vectors

� Let q be uniformly distributed on a ball

� Then, for A∈ —m,n

has beta distribution with density

[ ] nq qq —BU ∈~

( )( )AqAqAAT ,1−

=&τ

( )( ) ( ) 10)1(

11

)( 221

22

2 ≤≤−+ΓΓ

+Γ=−−

−τττττ

mnm

mnm

n

f

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CHAPTER 11

Mixed Deterministic/Randomized Methods

Keywords: deterministically tractable and intractable parameters, stability and performance with fixed order controllers

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Randomized vs Deterministic Solutions

� Development of RAs motivated by intractability of various problems in systems and control

� RAs provide (efficiently) a probabilistic solution� The drawback is that this solution is given with a certain

accuracyε and confidenceδ

� Deterministic algorithms provide a strong solution, butcan be used for a narrow class of problems

� Randomized algorithms give a weaker solution, but can be utilized for a broader set of problems

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Mixed Randomized and DeterministicMethods

� Key idea: Development of algorithms which combine randomized and deterministic techniques[1,2]

� ProblemP(η ,θ) with parameters η and

θ� We divide these parameters in two sets consisting of

(deterministically) tractable and intractable

�η are tractable parameters→ deterministic methods

�

θare intractable parameters→ randomized techniques

[1] Y. Fujisaki, Y. Oishi and R. Tempo (2004)[2] Y. Fujisaki, Y. Oishi and R. Tempo (2005)

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Stabilization with FixedOrder Controller

� Specific problemΠ(θ,η) � Strictly proper SISO plantP(s) = NP(s)/DP(s)� Fixed order controller

whereX(s2), Y(s2), Z(s2) and V(s2) are even polynomialsin s

� Objective: Find a stabilizing controller placing the rootsof the closed-loop polynomial in the open LHP

48

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Tractable vs Intractable Parameters

� Definition: The (deterministically) tractable parametersare the coefficientsθ of X(s2) and the intractableparameters are the coefficientsη of Y(s2), Z(s2) and V(s2)

Step 1: We use RAs to compute the coefficientsηStep 2:We use deterministic methods to determine

the coefficients θ

The set of all tractable parameters θ is denoted byθθθθ

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Characterization of TractableParameter Set

� Lemma: Suppose that the parametersη are selectedaccording to a RA. Then, the set θθθθ of all (tractable) stabilizing controller parameters is either empty or is a union of a finite number of polyhedral sets

� Exploitation of this result to determine a stabilizingcontroller

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Polynomial-Time Algorithm

� We construct an algorithm which proceeds in twophases:

1. Computation of a marginal stabilizing controller bymeans of a method based on matrix inversion

2. Computation of a stabilizing controller using a sensitivity approach

The resulting algorithm is polynomial-time

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Extensions toH∞ Performance

� Consider a performance problem with sensitivityfunction

and a (stable) weighting functionW(s)

� The objective is to find a controller C(s) satisfying

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Characterization forH∞ Performance

� Theorem: Suppose that the parametersη are chosen with a RA. Then, the set θ θ θ θ of all tractable controller parameters satisfying

is either empty or is given by

Θ Θ Θ Θ =

where Γi (φ) is a (possibly unbounded) polyhedral set for fixedφand i

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Further Results and Extensions

� Development of polynomial-time algorithms forH∞performance based on matrix inversion and sensitivity methods, adding an extra parameter φ ∈ [0, 2π]

� Extensions to uncertain plants where P(s) is replaced byan interval plantP(s,q,r)

� This extension can be carried on invoking some existing results of parametric stability and by means of a additional parameterλ ∈ [0, 1]

49

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CHAPTER 12

Applications of Randomized Algorithms

Keywords: flexible structures, high-speed networks, brushless DC motors, quantized systems

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Application of RAs

� Randomized algorithms have been developed for

various specific applications

� Control of flexible structures

� Stability and robustness of high speed networks

� Stability of quantized sampled-data systems

� Brushless DC motors

� …

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Application 1

Control of Flexible Structures

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Flexible Structure

� Mass spring damper model

� Real parametric uncertainty affecting stiffness and

damping

� Complex unmodeled dynamics (nonparametric)

m1

l1

k1

m2

l2

k2

m3

l3

k3

m4

l4

k4

m5

l5

k5

l6

k6

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Flexible Structure

� M-∆ configuration for controlled systems

q1, q2 ∈—

∆1∈¬4,4

BAsICsM 1)()( −−=

∆=∆

1

52

51

00

00

00

Iq

Iq

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Probabilistic Stability Margin

� For fixed ρ, we let

� For given p*∈[0,1] we define the probabilistic stability margin

� Clearly

{ }stable is Pr)( CBAp ∆+=&ρ

{ }*)(:sup*)( ppp ≥= ρρρ &

µρ 1

*)( ≥p

50

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Probability Degradation Function

0 .3 5 0 .4 0 .4 5 0 .5 0 .5 5 0 .6 0 .6 5 0 .7

0 .9 4

0 .9 5

0 .9 6

0 .9 7

0 .9 8

0 .9 9

1

1 .0 1

P ro b a b i li s tic ra d iu s ρ

Est

ima

ted

pro

ba

bili

ty d

eg

rad

atio

n

394.01 ≈µ

ρ

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Application 2

Stability and Performance of High-Speed Networks

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Stability and Robustness

� Network topology

� Source and destination

nodes, links (with buffer

and capacity)

� Bottleneck link

� Stability and robustness

C_1

C_2

C_3

C_4

C_5

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Symmetric Single Bottleneck

� Parametric stability (discrete

time) with real uncertain

parameters

� Stability and robustness can

be studied in closed form

� Case study with 20 users

� Roots of the closed loop

polynomial (discrete-time)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Imag

Real

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Non-Symmetric Single Bottleneck

� Closed form analysis is not

possible

� We use RAs based on MC

and QMC

0 0.5 1 1.5 2 2.5 3

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Net

wor

k st

abili

ty

Capacity

Halton Sobol Uniform pdf Optimal gridNiederreiter

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Additional Simulations

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

x 104

0.75

0.8

0.85

0.9

0.95

Net

wor

k st

abili

ty

Capacity

Halton Sobol Uniform pdf Optimal gridNiederreiter

51

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Application 3

Probabilistic StructuredReal Stability Radius

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Structured Real Stability Radius

� Let A∈¬n,n be a stable matrix, and consider the perturbed matrix

with B, Cof appropriate dimensions

� The real stability radius is the size of the smallest destabilizing perturbation

ˇB∈∆∆+=∆ ,)( CBAA

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Probabilistic Stability Radius

� We assume ∆ random, and estimate the probability of stability as a function of the uncertainty radiusρ

� For given p*, the probabilistic real stability radius is defined as

� We estimate the probabilistic stability radius using randomized algorithms

{ }** )(:sup),( pppAR ≥= ρρρ &

{ }ˇ

B∈∆∆= stable, is )(Pr)( Ap ρ

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Numerical Example

� We studied the example

ICB

A

==

−−−−−−

−−−−−−

−−−−−−

=

9809.28238.11812.44435.10764.28445.2

7190.01026.09139.18681.03964.02169.1

6973.02107.04244.00580.06813.01946.0

8705.20364.13362.36874.11677.14202.1

2641.48212.22311.53962.24700.15667.3

3271.15852.18166.21021.19633.09319.0

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Numerical Example - 2

� We computed p(ρ) for ρ∈[0.01 0.05] with two different structures:

� ∆ composed by three 2x2 full real blocks

� ∆ composed by a 4x4 and a 2x2 full real blocks

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Numerical Example - 3

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050.75

0.8

0.85

0.9

0.95

1

1.05

Uncertainty radius ρρρρ

Pro

bab

ility

of

stab

ility

Uncertainty structure 1

Uncertainty structure 2

52

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Application 4

Probabilistic RobustLeast Squares

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Uncertain Least Squares

� Consider the problem

with x ∈ —n, b ∈ —m, ∆∈ Bˇ

� Robust least squares

� Probabilistic least squares

)()( ∆=∆ bxA

2)()(maxminargˆ ∆−∆=∈∆

bxAxx

RLSˇB

{ }2)()(minargˆ ∆−∆= bxAExx

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Probabilistic Robust Least Squares

� Generate N samples of ∆ and compute

with

� We can compute recursively

∑=

∆−∆=N

i

iiN bA

NxE

1

2)()(

1)(ˆ &

)(ˆminargˆ xEx Nx

N =

Nx̂

)()(

),ˆ)()()((ˆˆ11

1

111111

+++

+++−++

∆∆+=

∆−∆∆+=kkT

kk

kkkkT

kkk

AAQQ

xAbAQxx

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PRLS Example

� We consider an uncertain matrix A(∆)=A+∆, ||∆||<1

=

−=

3

1

2

0

2.541

352

110

413

BA

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PRLS Example - 2

� We compute the estimates obtaining

TRLS

TLS

TN

x

x

Nx

]2055.02073.00312.0[ˆ

]9834.97285.90000.10[ˆ

000,10;]3448.01009.01768.0[ˆ

−=

−−=

=−=

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PRLS Example - 3

� To compare the previous estimates we computed the worst-case and the sample residuals

� Worst-case residuals are defined as

we obtained

2** )()(max ∆−∆=

∆bxAr wc

5720.29394.186532.2 === wcRLS

wcLS

wcN rrr

53

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PRLS Example - 4

� Statistics for the sample residuals

0107.05515.22848.2

4090.95164.188962.11

0198.06344.22650.2

Covariance SamplePeakAverage

RLS

LS

N

r

r

r

2

** )()( iii bxAr ∆−∆=

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Conclusions and Discussion

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Conclusions

� A lot of research remains to be done� Applications: Randomized algorithms and sample

generation for specific structures of uncertainty� Theory: Beautiful and nonstandard math behind

randomized algorithms� Complexity: Studying computational complexity of

these algorithms� Computation: High dimensional problems are ill

conditioned� Other Directions: Randomized algorithms for MPC

and FTC

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(Controversial) Conclusions

� A number of years ago, closed-form meant writing

down a “good looking” equation on a piece of paper

� Notion of closed-form solution has changed

substantially

� Various experts (rightfully) convinced us that a new

notion of closed-form is to re-write (if possible) a

control problem as convex optimization

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(Controversial) Conclusions

� Unfortunately, many important control problems are

not convex

� In such cases, we can either introduce relaxation or

use randomization

� Perhaps the control community should pay more

attention to randomization accepting a different (and

weaker) notion of problem solution

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Discussion

54

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Randomized Algorithmsfor Analysis and Control of Uncertain Systems

by RT, GC, FD with a foreword by M. Vidyasagar, Springer-Verlag, London, 2005

http://staff.polito.it/tempo/

randomized algorithms for analysis and control of ...eduardo/cursos/seville6.pdf · 1 ieiit-cnr...

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