Dr. Roberto Tempo
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Transcript Dr. Roberto Tempo
IEIIT-CNR
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
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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
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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
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RandQuickSort (RQS)
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given n real
numbers
x1
x4
x2
x5
x3
x6
need to sort them
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
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numbers larger than x4
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Running Time of RQS
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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 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 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
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Uncertain Systems
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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)
DB
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Jave = ED(J(D))
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Example - H Performance
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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
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One-Sided MCRA
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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
J^max = max J(Di)
i=1,…,N
where N is the sample size
Check if J^max g or J^max > g
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One-Sided MCRA: Case 1
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algorithm provides a correct solution
J(D)
g
Jmax
J^
J(D3)
max
J(D2)
J(D4)
^
J(D1)
J(D5)
J(D6)
Jmax < Jmax < g
D
D1
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D2
D3
D4 D5 D6
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One-Sided MCRA: Case 2
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algorithm may provide a wrong solution
J(D)
g
Jmax
J^
J(D3)
max
J(D2)
J(D4)
J(D1)
J(D5)
J(D6)
Jmax > g
^
Jmax < g
D
D1
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D2
D3
D4 D5 D6
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Two-Sided MCRA
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Two-sided MCRA: They may provide a wrong solution
in both instances
Consider the empirical average
J^ave = ave J(Di)
i=1,…,N
where N is the sample size
^
^
Check if Jave g or Jave > g
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Two-Sided MCRA
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Jave > g
^
Jave < g
J(D)
J(D3)
g
J(D2)
J(D4)
J(D1)
Jave
J(D5)
J(D6)
^J
ave
D
D1
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D2
D3
D4 D5 D6
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Two-Sided MCRA
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Jave < g
^
Jave > g
J(D)
J(D3)
g
^J
ave
J(D2)
J(D4)
Jave
J(D1)
J(D5)
J(D6)
D
D1
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D2
D3
D4 D5 D6
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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
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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
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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
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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
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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
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“Randomized Algorithms for Analysis and Control of
Uncertain Systems” by R. Tempo, G. Calafiore and F.
Dabbene, Springer-Verlag, 2005
TM
Additional documents, papers, MATLAB codes, etc,
please consult
http://staff.polito.it/roberto.tempo
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