IEIIT-CNR SWAN, ARRI ©RT 2006 1 Las Vegas and Monte Carlo Randomized Algorithms for Systems and...

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SWAN, ARRI SWAN, ARRI ©RT 2006©RT 2006

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Las Vegas and Monte Carlo Randomized Algorithms for Systems and Control

Roberto Tempo

IEIIT-CNR

Politecnico di Torino

[email protected]

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Overview

A Success Story Randomized Algorithms, Monte Carlo and Las Vegas Some Recent Research Directions Applications: High Speed Networks and UAV

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A Success Story

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A Success Story

Randomized AlgorithmsRandomized Algorithms (RAs) are successfully used in

various areas, including computer science, numerical

analysis, optimization, …

… but in systems and control their use is often limited

to Monte Carlo simulations

Example: Example: Sorting problem

Algorithm:Algorithm: RandQuickSort (RQS)

RQS is implemented in the Linux sorting command

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RandQuickSort (RQS)

given n real x1 x2 x3 need to sort them

numbers x4 x5 x6 in increasing order RQS is an iterative algorithm consisting of two phases 1. randomly select a number xi (e.g. x4)

2. perform deterministic comparisons between xi and (n-1) remaining numbers

x2 x3 x4 x1 x5

x6

numbers smaller than x4 numbers larger than x4

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Running Time of RQS

Because of randomization, running time may be different from one execution of the algorithm to the next one

RQS is very fast: average running time is O(n log (n)) This is a major improvement compared to brute force

approach for example when n = 2m

Average running time is also a highly probable running time (Chernoff bound)

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Randomized Algorithms, Monte Carlo and Las Vegas

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Randomized Algorithm: Definition

Randomized AlgorithmRandomized Algorithm (RA): An algorithm that makes random choices during execution to produce a result

For hybrid systems, “random choices” could be switching between different states or logical operations

For uncertain systems, “random choices” require (vector or matrix) random sample generation

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Monte Carlo and Las Vegas RA

Monte Carlo Randomized AlgorithmMonte Carlo Randomized Algorithm (MCRA): A randomized algorithm that may produce incorrect results, but with bounded error probability

Las Vegas Randomized AlgorithmLas Vegas Randomized Algorithm (LVRA): A randomized algorithm that always produces correct results, the only variation from one run to another is the running time

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Uncertain Systems

Consider random uncertainty and a bounding set B is a (real or complex) random vector (parametric

uncertainty) or matrix (nonparametric uncertainty) Consider a performance function

J( Rn,m → R

and level > 0 Define worst case and average performance

Jmax max J(JaveEJ(B

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Example - H Performance

H performance of sensitivity function

S(s,) = 1/(1 + P(s,) C(s))

J() = ||S(s,)||

Objective:Objective: Check if

Jmax andJave

These are uncertain decision problems

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Two Problem Instances

We have two problem instances for worst case performance

Jmax and Jmax >

and two problem instances for average case performance

Jave and Jave >

This leads to one-sidedone-sided and two-sidedtwo-sided MC randomized algorithms

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One-Sided MCRA

One-sided MCRA:One-sided MCRA: Always provide a correct solution in one of the instances (they may provide a wrong solution in the other instance)

Consider the empirical maximum

JJmax max J(i

where N is the sample size

Check if JJmax or JJmax >

i1,…,N

^

^ ^

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One-Sided MCRA: Case 1

Jmax

JJmax

JJmaxJmax <

algorithm provides a correct solution

1 2 3 4 56

J(1)

J(2)

J(3)

J(4)

J(5)

J(6)

J

^

^

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One-Sided MCRA: Case 2

JmaxJJmaxmax

Jmax >

JJmax<

algorithm may provide a wrong solution

1 2 3 4 56

J(1)

J(2)

J(5)

J(4)

J(6)

J(3)

J

^

^

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Two-Sided MCRA

Two-sided MCRA:Two-sided MCRA: They may provide a wrong solution in both instances

Consider the empirical average

JJave ave J(i

where N is the sample size

Check if JJave orJJave >

i1,…,N

^

^ ^

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Two-Sided MCRA

Jave

1 2 3 4 56

J(1)

J(2)

J(3)

J(4)

J(5)

J(6)

J

JJave^

Jave >

JJave< ^

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Two-Sided MCRA

Jave

1 2 3 4 56

J(1)

J(2)

J(3)

J(4)

J(5)

J(6)

J

JJave^

^Jave <

JJave>

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Las Vegas Randomized Algorithms

We also have zero-sidedzero-sided (Las Vegas) randomized algorithms

Las Vegas Randomized Algorithm (LVRA):Las Vegas Randomized Algorithm (LVRA): Always give the correct solution

Running time may be different from one run to another LVRA has more limited applicability than MCRA

Example:Example: RandQuickSort

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Current Research on LVRA

Switched systems:Switched systems:

- design a common Lyapunov function for systems

x(t) = A x(t)

where A is an interval matrix with entries ranging between upper/lower bounds

Consensus control:Consensus control:

- design randomized algorithms achieving finite-time average consensus for connected networks

.

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Uncertain Systems, Optimization, System Identification

From common to piecewise Lyapunov functions[1]

Ellipsoidal randomized algorithm[2] and stopping rules[3]

RAs for semi-infinite programming[4]

MRAS methods for global optimization[5]

Estimation via MCMC[6]

RAs for model validation[7] and system identification[8]

…

[1] H. Ishii, T. Basar and R. Tempo (2005) [2] S. Kanev, B. De Schutter and M. Verhaegen (2002) [3] Y. Oishi and H. Kimura (2003) [4] V. B. Tadic, S. P. Meyn and R. Tempo (2006) [5] J. Hu, M.C. Fu and S.I. Marcus (2005) [6] J.C. Spall (2004)

[7] M. Sznaier, C. M. Lagoa and M.C. Mazzaro (2005) [8] X. Bombois, G. Scorletti, M. Gevers, P. Van den Hof and R. Hildebrand (2006)

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Applications of RAs

RAs have been developed for many control applications

Control of flexible structures

Robustness of high speed networks

Stability of quantized sampled-data systems

Control design for brushless DC motors

Synthesis of real time embedded controllers

Mini-UAV control design

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Applications of RAs

RAs have been developed for many control applications

Control of flexible structures

Robustness of high speed networks

Stability of quantized sampled-data systems

Control design for brushless DC motors

Synthesis of real time embedded controllers

Mini-UAV control design

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Mini-UAV Control Design

Study and development of a

real-time land control and

monitoring system for fire

prevention in Sicily

Uncertainty description

Development of three RAs for

gain synthesis and robustness

analysis (according to flying

quality military specs)

[1] L. Lorefice, B. Pralio and R. Tempo (2006)

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References

“Randomized Algorithms for Analysis and Control of

Uncertain Systems” by R. Tempo, G. Calafiore and F.

Dabbene, Springer-Verlag, 2005

Additional documents, papers, MATLABTM

codes, etc, please consult

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

IEIIT-CNR SWAN, ARRI ©RT 2006 1 Las Vegas and Monte Carlo Randomized Algorithms for Systems and...

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