IEEE Control Systems Society - sct.ieiit.cnr.it · Computational Aspects of Contr. Sys. Control Education Discrete Event Systems Distributed Parameter Systems Hybrid Systems Industrial

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Northeastern University, Shenyang © RT 2010

IEEE Control Systems Society

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IEEE – Institute of Electrical and Electronics Engineers

Formed in 1963 by merger of the American Institute ofElectrical Engineers (founded 1884) and the Institute ofRadio Engineers (founded 1912)

IEEE is the world’s leading association for theadvancement of technology, has more than 375,000members located in 160 countries; 45% of its membersare from outside the US

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CSS – Control Systems Society

CSS is one of 38 Societies of IEEE, with a total numberof 8,338 members, founded in 1956

Number of represented countries: 116 USA members: 3,553 members

R. Tempo, “Internationalizationand Globalization” IEEEControl Systems Magazine,February 2010

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IEEE Control Systems Society

President: Roberto Tempo

VPPA: Christos G. Cassandras VPTA: Shuzhi Sam Ge VPMA: Shinji Hara VPFA: Pradeep Misra VPCA: Maria Elena Valcher

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CSS Transactions and Magazine

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2010 CSS Sponsored Conferences

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CSS Technical Committees Aerospace Controls Automotive Controls Behavioral Systems and Contr. Th. Computational Aspects of Contr. Sys. Control Education Discrete Event Systems Distributed Parameter Systems Hybrid Systems Industrial Process Contr Intelligent Control Manufacturing Autom. Robotic Contr. Networks and Communications Nonlinear Systems and Controls

Systems Biology System Id & Adaptive Control Systems with Uncertainty Variable Structure and SMC

http://www.ieeecss.org/main/technical-committees/aerospace-controls�

http://www.ieeecss.org/main/technical-committees/automotive-controls�

http://www.ieeecss.org/main/technical-committees/behavioral-systems-and-control-theory�

http://www.ieeecss.org/main/technical-committees/computer-aided-control-system-design�

http://www.ieeecss.org/main/technical-committees/control-education�

http://www.ieeecss.org/main/technical-committees/discrete-event-systems�

http://www.ieeecss.org/main/technical-committees/distributed-parameter-systems�

http://www.ieeecss.org/main/technical-committees/hybrid-systems�

http://www.ieeecss.org/main/technical-committees/industrial-process-control�

http://www.ieeecss.org/main/technical-committees/intelligent-control�

http://www.ieeecss.org/main/technical-committees/manufacturing-automation-and-robotic-control�

http://www.ieeecss.org/main/technical-committees/networks-and-communications�

http://www.ieeecss.org/main/technical-committees/nonlinear-systems-and-controls�

http://www.ieeecss.org/main/technical-committees/biosystems-and-control�



http://www.ieeecss.org/main/technical-committees/system-identification-a-adaptive-control�



http://www.ieeecss.org/main/technical-committees/systems-with-uncertainty�





http://www.ieeecss.org/main/technical-committees/variable-structure-and-sliding-mode-control�



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President’s Messages IEEE CSM 2010

1. “Internationalization and Globalization” February

2. “Research, a Never-Ending Story” April

3. “From Hard Copy to Electronic” June

4. “Handwriting, Typing and LaTeX” August

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Randomized Algorithms for Systems and Control: Theory and Applications

Roberto TempoIEIIT-CNR

Politecnico di Torino, [email protected]

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Randomized Algorithms: A Success Story

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A Success Story

Randomized Algorithms (RAs) are successfully usedin various areas outside control

1. CS: Sorting problems (e.g., QuickSort algorithm)

2. CS: Data structuring, search trees, graph algorithms

3. Mathematics of finance: Computation of integrals

4. Genomics: String matching and classification

5. Robotics: Motion and path planning problems

6. Control of Unmanned Aerial Vehicles (UAV)

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Monte Carlo and Las Vegas Algorithms

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Monte Carlo and Las Vegas

Monte Carlo was invented by Metropolis, Ulam, vonNeumann, Fermi, … (Manhattan project)

Las Vegas first appeared in computer science in the lateseventies

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Randomized Algorithm: Definition

Randomized Algorithm (RA): An algorithm that makesrandom choices during its execution to produce a result

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Example of a “random choice” is a coin tossheads or tails

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

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

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Monte Carlo Randomized Algorithm

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

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Monte Carlo Randomized Algorithm

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

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Example of Monte Carlo: Area/Volume Estimation

Estimate the volume of the red area: Generate N sampleuniformly in the rectangle; count how many fall withinthe red area (M), estimated area = M/N

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Las Vegas Randomized Algorithm

Las Vegas Randomized Algorithm (LVRA): Arandomized algorithm that always produces correctresults, the only variation from one run to another is therunning time

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Las Vegas Randomized Algorithm

Las Vegas Randomized Algorithm (LVRA): Arandomized algorithm that always produces correctresults, the only variation from one run to another is therunning time

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Example of Las Vegas

Example of Las Vegas Randomized Algorithm: RandomSurfer Model

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Random Surfer Model

Web representation with incoming and outgoing links

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Random Surfer Model

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Random Surfer Model

Pick an outgoing link at random

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Random Surfer Model

Arriving at a new web page

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Random Surfer Model

Pick another outgoing link at random

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Random Surfer Model

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Random Surfer Model

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Random Surfer Model

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Example of Las Vegas: RQS

Problem: Sorting N real numbers

Algorithm: RandQuickSort (RQS)

RQS is implemented in a C library of Linux for sortingnumbers[1-2]

[1] C.A.R. Hoare (1962) – [2] D.E. Knuth (1998)

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A Success Story in CS

Problem: Sorting N real numbers

Algorithm: RandQuickSort (RQS)

RQS is implemented in a C library of Linux for sortingnumbers

Sorting Problem

given N real x1 x2 x3 sort them in

numbers x4 x5 x6 increasing order

S1

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RandQuickSort (RQS)

The idea is to divide the original set S1 into two setshaving (approximately) the same cardinality

This requires finding the median of S1 (which may bedifficult)

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RandQuickSort (RQS)

RQS is a recursive algorithm consisting of two phases

1. randomly select a number xi (e.g. x4)2. deterministic comparisons between xi and other (N-1) numbers

x2 x3 < x4 < x1 x5 x6

numbers smaller than x4 numbers larger than x4

S3S2

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RQS: Binary Tree Structure

We use randomization at each step of the (binary) tree

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Running Time of RQS

Because of randomization, running time may bedifferent from one run 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 Average running time is highly probable…

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RQS: The Las Vegas Viewpoint

Average running time is O(N log(N)) This running time holds for every input The average running time holds with probability at least

1-1/N Hint: Use the so-called Chernoff bound to prove this Improvements for RQS to avoid achieving the worst

case running time O(N2)

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Find Algorithm

Find Algorithm: Find the k-th smallest number in a set Basically it is a RQS but it terminates when the number

is found Average running time of Find is O(N)

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Sorting of Switched Systems

Motivations: Deciding which systems are more stablethan others is useful information for the controller

This requires finding a LVRA which provides amatrix sorting for the N Lyapunov equations

Matrix version of RandQuickSort is developed[1]

Technical difficulty: The equations may be notcompletely sortable because of sign indefiniteness

[1] H. Ishii, R. Tempo (2007)

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RQS for Matrices: Trinary Tree

We use randomization at each step of the (trinary) tree

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Unmanned Aerial Vehicles (UAVs)

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Italian National Project for Fire Prevention

This activity is supported by the Italian Ministry forResearch within the National Project

Study and development of a real-time land control and monitoring system for fire prevention

Five research groups are involved together with agovernment agency for fire surveillance and patrollocated in Sicily

The aerial platform is based on the MicroHawkconfiguration, developed at the Aerospace EngineeringDepartment, Politecnico di Torino, Italy

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MH1000 Platform - 1

Platform features- wingspan 3.28 ft (1 m)- total weight 3.3 lb (1.5 kg)

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MH1000 Platform - 2

Main on-board equipment- various sensors and two cameras (color and infrared)

DC motor Remote piloting and autonomous flight Flight endurance of about 40 min Flight envelope

- min/max velocity: 33 ft/s (10 m/s) – 66 ft/s (17 m/s)- average velocity: 43 ft/s (14 m/s)

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Flight Envelope (Limits)

Wing loading effect total weight

Propeller sizing effectPropulsive constraint (blu) maximum flight speed

Aerodynamic constraint (red) minimum flight speed (stall effect)

velocity: 33 ft/s (10 m/s) – 66 ft/s (17 m/s)

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DC motor: Hacker B20-15L (4:1)

controller: Hacker Master Series 18-B-Flight

battery: Kokam 2000HD (3x)

receiver: Schulze Alpha840W

servo: Graupner C1081 (2x)

weight: 58 g

dimensions: Ø 20 x 40 mm

Kv: 3700 rpm/volt

weight: 21 g

dimensions: 33 X 23 X 7 mm

current drain: 18 A

weight: 160 g


capacity: 2000 mAh

weight: 13.5 g


8 channels

weight: 13 g


torque: 12 Ncm

Basic on-board Systems

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Prototype Manufacturing - 1

raw materialpolistyrene

kevlar

fiberglass

carbon fiber

epoxy resinplywood

balsa wood

glue

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working instrumentshot wire foam cutting machine

lifting surfaces outline

slide outlinefuselage reference


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prototype

easy constructionrapid manufacturing

bad model reproducibilityinaccurate geometry


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State Space Model

State space formulation obtained by linearization of thefull (12 coupled nonlinear ODE) model

x(t) = A(∆) x(t) + B(∆) u(t)

u(t) = - K x(t)where x = [V, α, q, θ]T (V flight speed, α angle ofattack, q and θ pitch rate and angle), ∆ uncertainty

Consider longitudinal plane dynamics stabilization Control u is the symmetrical elevon deflection

.

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Uncertainty Description - 1

We consider structured parameter uncertainties affectingplant and flight conditions, and aerodynamic database

Uncertainty vector ∆ = [δ1,..., δ16] where δi∈ [δi-, δi

+] Key point: There is no explicit relation between state

space matrices A and B and uncertainty ∆ This is due to the fact that state space system is obtained

through linearization and off-line flight simulator The only techniques which could be used in this case are

simulation-based which lead to randomized algorithms

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Uncertainty Description - 2

We consider random uncertainty ∆ = [δ1,..., δ16]T

The pdf is either uniform (for plant and flightconditions) or truncated Gaussian (for aerodynamicdatabase uncertainties)

Flight conditions uncertainties need to take into accountlarge variations on physical parameters

Uncertainties for aerodynamic data are related toexperimental measurement or round-off errors

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Plant and Flight Condition Uncertainties

parameter pdf δi % δi- δi

+ #

flight speed [ft/s] U 42.65 ± 15 36.25 49.05 1

altitude [ft] U 164.04 ± 100 0 328.08 2

mass [lb] U 3.31 ± 10 2.98 3.64 3

wingspan [ft] U 3.28 ± 5 3.12 3.44 4

mean aero chord [ft] U 1.75 ± 5 1.67 1.85 5

wing surface [ft2] U 5.61 ± 10 5.06 6.18 6

moment of inertia [lb ft2] U 1.34 ± 10 1.21 1.48 7

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Aerodynamic Database Uncertainties

parameter pdf δi σi #CX [-] G -0.01215 0.00040 8CZ [-] G -0.30651 0.00500 9Cm [-] G -0.02401 0.00040 10CXq [rad-1] G -0.20435 0.00650 11CZq [rad-1] G -1.49462 0.05000 12Cmq [rad-1] G -0.76882 0.01000 13CX [rad-1] G -0.17072 0.00540 14CZ [rad-1] G -1.41136 0.02200 15Cm [rad-1] G -0.94853 0.01500 16

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Standard Deviation and Velocity

Standard deviation is experimentally computed from the velocity

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Random Gain Synthesis (RGS)

Specs in the complex plane

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Randomized Algorithm 1 (RGS) Uniform pdf for controller

gains K in given intervals Accuracy and confidence

ε =4 ·10-5 and η = 3 · 10-4

Number of random samples is computed with “Log-over-Log” Bound obtaining N = 200,000

We obtained s = 5 gains Ki

satisfying specification property S1

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Random Gain Set

gain set KV Kα Kq Kθ

K1 0.00044023 0.09465000 0.01577400 -0.00473510

K2 0.00021450 0.09581200 0.01555500 -0.00323510

K3 0.00054999 0.09430800 0.01548200 -0.00486340

K4 0.00010855 0.09183200 0.01530000 -0.00404380

K5 0.00039238 0.09482700 0.01609300 -0.00417340

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Randomized Algorithm 2 (RSRA) Take Krand from Phase 1 Accuracy and confidence

ε = η = 0.0145 Number of random

samples is computed with Chernoff Bound obtaining N =5,000

Empirical probability is defined using an indicator function

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Empirical Probability of Stability for Phase 2

gain set empirical probability

K1 88.56%

K2 90.60%

K3 89.31%

K4 93.86%

K5 85.14%

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Probability Degradation Function

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Root Locus Plot

Root locus for K2 (left) and K4 (right)

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Bandwidth Criterion

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Randomized Algorithm 3 (RPRA)

Take Krand from Phase 1 Numer of random samples

is computed with the Chernoff Bound obtaining N =5,000

Empirical probability is defined using an indicator function

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Empirical Probability of Performance for Phase 3

gain set empirical probability

K1 93.58%

K2 95.16%

K3 90.80%

K4 84.78%

K5 96.06%

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Probability Degradation Function

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Bandwidth Criterion

Bandwidth criterion for K1 (left) and K3 (right)

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Gain Selection

Multi-objective criterion as a compromise between different specifications

Finally we selected gain K1 as the best compromise between all the specs and criteria

Detailed results available in[1]

[1] L. Lorefice, B. Pralio and R. Tempo (2009)

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Conclusions: Flight Tests in Sicily - 1

Evaluation of the payload carrying capabilities andautonomous flight performance

Mission test involving altitude, velocity and headingchanging was performed in Sicily

Checking effectiveness of the control laws forlongitudinal and lateral-directional dynamics

Flight control design based on RAs for stabilization andguidance

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Conclusions: Flight Tests in Sicily - 2

Satisfactory response of MH1000

Possible improvements by iterative design procedure

Stability of the platform is crucial for the video qualityand in the effectiveness of the surveillance andmonitoring tasks

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Color Camera: Right Turn

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Color Camera: Landing Phase

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RACTRandomized Algorithms Control Toolbox

http://ract.sourceforge.net

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RACT

RACT: Randomized Algorithms Control Toolbox forMatlab

RACT has been developed at IEIIT-CNR and at theInstitute for Control Sciences-RAS, based on a bilateralinternational project

Members of the projectAndrey Tremba (Main Developer and Maintainer) Giuseppe Calafiore Fabrizio Dabbene Elena Gryazina Boris Polyak (Co-Principal Investigator) Pavel Shcherbakov Roberto Tempo (Co-Principal Investigator)

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RACT

Main features Define a variety of uncertain objects: scalar, vector and

matrix uncertainties, with different pdfs Easy and fast sampling of uncertain objects of almost

any type Sequential randomized algorithms for feasibility of

uncertain LMIs using stochastic gradient and localizationmethods (ellipsoid or cutting plane)

Non-sequential randomized algorithms for optimizationof convex problems

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RACT

Under construction: Non-sequential randomizedalgorithms for feasibility and optimization of non-convex problems

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RACT

RACT: Randomized Algorithms Control Toolbox forMatlab

http://ract.sourceforge.net

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Randomized Algorithms for Systems and Control Applications

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CCC Beijing ©RT 2010

Randomized Algorithms for Systems and Control Applications - 1

Aerospace control andunmanned aerial vehicles(UAVs)[1,2,3]

Multi-agent systemsand consensus[4,5]

[1] C.I. Marrison and R.F. Stengel (1998)[2] B. Lu and F. Wu (2006)[3] L. Lorefice, B. Pralio and R. Tempo (2009)[4] H. Ishii and R. Tempo (2010)[5] L. Pallottino, V.G. Scordio, E. Frazzoli and A. Bicchi (2007)

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Network congestion control[1]

Quantized and switchedsystems[2-3]

Fault detection, isolation,vision-based control[4-5]

[1] T. Alpcan, T. Basar and R. Tempo (2005)[2] H. Ishii, T. Basar and R. Tempo (2004)[3] H. Ishii, T. Basar and R. Tempo (2005)[4] S. Kanev and M. Vehaegen (2006)[5] W. Ma, M. Sznaier and C.M. Lagoa (2007)

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Embedded and electriccircuits[1,2]

Advanced driver assistancesystems[3]

[1] C. Alippi (2002)[2] C.M. Lagoa, F. Dabbene and R. Tempo (2008) [3] O.J. Gietelink, B. De Schutter, and M. Verhaegen (2005)

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Main References

R. Tempo, G. Calafiore and F. Dabbene, “Randomized Algorithms for Analysis and Control of Uncertain Systems,” Springer-Verlag, London, 2005

F. Dabbene and R. Tempo, “Probabilistic and Randomized Tools for Control Design,” The Control Handbook, Taylor & Francis, 2010 (to appear)

Additional documents, papers, etc, please consulthttp://staff.polito.it/roberto.tempo/

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Conclusions

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Randomized Algorithms: A Success Story

Randomized algorithms are a success story for systems and control

Documents

IEEE Control Systems Society - sct.ieiit.cnr.it · Computational Aspects of Contr. Sys. Control Education Discrete Event Systems Distributed Parameter Systems Hybrid Systems Industrial