Probabilistic Design

1

• Las Vegas and Monte Carlo Randomized Algorithms
  for Systems and Control
• Roberto Tempo
• IEIIT-CNR
• Politecnico di Torino
• roberto.tempo_at_polito.it

2
Overview

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

3
A Success Story
4
A Success Story

• Randomized 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 Sorting problem
• Algorithm RandQuickSort (RQS)
• RQS is implemented in the Linux sorting command

5
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

6
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)

7

• Randomized Algorithms, Monte Carlo and
• Las Vegas

8
Randomized Algorithm Definition

• Randomized 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

9
Monte Carlo and Las Vegas RA

• Monte Carlo Randomized Algorithm (MCRA) A
  randomized algorithm that may produce incorrect
  results, but with bounded error probability
• Las Vegas Randomized Algorithm (LVRA) A
  randomized algorithm that always produces correct
  results, the only variation from one run to
  another is the running time

10
Uncertain Systems

• Consider random uncertainty D and a bounding set
  B
• D is a (real or complex) random vector
  (parametric uncertainty) or matrix (nonparametric
  uncertainty)
• Consider a performance function
• J(D) Rn,m ? R
• and level g 0
• Define worst case and average performance
• Jmax max J(D) Jave ED(J(D))

D?B
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Example - H? Performance

• H? performance of sensitivity function
• S(s,D) 1/(1 P(s,D) C(s))
• J(D) S(s,D)?
• Objective Check if
• Jmax ? g and Jave ? g
• These are uncertain decision problems

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Two Problem Instances

• We have two problem instances for worst case
  performance
• Jmax ? g and Jmax g
• and two problem instances for average case
  performance
• Jave ? g and Jave g
• This leads to one-sided and two-sided MC
  randomized algorithms

13
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
• Jmax
  max J(Di)
• where N is the sample size
• Check if Jmax ? g or Jmax g

i1,,N


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One-Sided MCRA Case 1
algorithm provides a correct solution
J(D)
?
Jmax

Jmax
J(D3)
J(D2)
J(D4)

Jmax J(D1)
J(D5)
J(D6)
D
D1 D2 D3 D4
D5 D6
15
One-Sided MCRA Case 2
algorithm may provide a wrong solution
J(D)
Jmax
?

Jmax
J(D3)
J(D2)
J(D4)
Jmax ? Jmax J(D1)

J(D5)
J(D6)
D
D1 D2 D3 D4
D5 D6
16
Two-Sided MCRA

• Two-sided MCRA They may provide a wrong solution
  in both instances
• Consider the empirical average
• Jave
  ave J(Di)
• where N is the sample size
• Check if Jave ? g or Jave g

i1,,N


17
Two-Sided MCRA
J(D)
Jave ? Jave
J(D3)
J(D2)
J(D4)
Jave
?
J(D1)

J(D5)
Jave
J(D6)
D
D1 D2 D3 D4
D5 D6
18
Two-Sided MCRA
J(D)
Jave ?

J(D3)

Jave
J(D2)
J(D4)
?
Jave
J(D1)
J(D5)
J(D6)
D
D1 D2 D3 D4
D5 D6
19
Las Vegas Randomized Algorithms

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

20
Current Research on LVRA

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

.
21
Uncertain Systems, Optimization, System
Identification

• From common to piecewise Lyapunov functions1
• Ellipsoidal randomized algorithm2 and stopping
  rules3
• RAs for semi-infinite programming4
• MRAS methods for global optimization5
• Estimation via MCMC6
• RAs for model validation7 and system
  identification8
• 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)

22
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

23
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

24
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/rober
  to.tempo