1. IEIIT-CNR SWAN, ARRI ©RT 2006 1 Las Vegas and Monte Carlo Randomized Algorithms for Systems and Control Roberto Tempo IEIIT-CNR Politecnico di Torino roberto.tempo@polito.it

2. IEIIT-CNR SWAN, ARRI ©RT 2006 2 Overview n A Success Story n Randomized Algorithms, Monte Carlo and Las Vegas n Some Recent Research Directions n Applications: High Speed Networks and UAV

3. IEIIT-CNR SWAN, ARRI ©RT 2006 3 A Success Story

4. IEIIT-CNR SWAN, ARRI ©RT 2006 4 A Success Story n Randomized Algorithms n 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 n Example: n Example: Sorting problem n Algorithm: n Algorithm: RandQuickSort (RQS) n RQS is implemented in the Linux sorting command

5. IEIIT-CNR SWAN, ARRI ©RT 2006 5 RandQuickSort (RQS) given n real x 1 x 2 x 3 need to sort them numbers x 4 x 5 x 6 in increasing order n RQS is an iterative algorithm consisting of two phases 1. randomly select a number x i (e.g. x 4 ) 2. perform deterministic comparisons between x i and (n-1) remaining numbers x 2 x 3  x 4  x 1 x 5 x 6 numbers smaller than x 4 numbers larger than x 4

6. IEIIT-CNR SWAN, ARRI ©RT 2006 6 Running Time of RQS n Because of randomization, running time may be different from one execution of the algorithm to the next one n RQS is very fast: average running time is O(n log (n)) n This is a major improvement compared to brute force approach for example when n = 2 m n Average running time is also a highly probable running time (Chernoff bound)

7. IEIIT-CNR SWAN, ARRI ©RT 2006 7 Randomized Algorithms, Monte Carlo and Las Vegas

8. IEIIT-CNR SWAN, ARRI ©RT 2006 8 Randomized Algorithm: Definition n Randomized Algorithm n Randomized Algorithm (RA): An algorithm that makes random choices during execution to produce a result n For hybrid systems, “random choices” could be switching between different states or logical operations n For uncertain systems, “random choices” require (vector or matrix) random sample generation

9. IEIIT-CNR SWAN, ARRI ©RT 2006 9 Monte Carlo and Las Vegas RA n Monte Carlo Randomized Algorithm n Monte Carlo Randomized Algorithm (MCRA): A randomized algorithm that may produce incorrect results, but with bounded error probability n Las Vegas Randomized Algorithm n 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. IEIIT-CNR SWAN, ARRI ©RT 2006 10 Uncertain Systems Consider random uncertainty  and a bounding set B  is a (real or complex) random vector (parametric uncertainty) or matrix (nonparametric uncertainty) n Consider a performance function J(  R n,m → R and level  > 0 n Define worst case and average performance J max  max J(  J ave  E   J(  BB

11. IEIIT-CNR SWAN, ARRI ©RT 2006 11 Example - H  Performance H  performance of sensitivity function S(s,  ) = 1/(1 + P(s,  ) C(s)) J(  ) = ||S(s,  )||  n Objective: n Objective: Check if J max    and  J ave    n These are uncertain decision problems

12. IEIIT-CNR SWAN, ARRI ©RT 2006 12 Two Problem Instances n We have two problem instances for worst case performance J max   and  J max >  and two problem instances for average case performance J ave   and  J ave >  one-sidedtwo-sided n This leads to one-sided and two-sided MC randomized algorithms

13. IEIIT-CNR SWAN, ARRI ©RT 2006 13 One-Sided MCRA n One-sided MCRA: n One-sided MCRA: Always provide a correct solution in one of the instances (they may provide a wrong solution in the other instance) n Consider the empirical maximum J J max  max J(  i   where N is the sample size JJ Check if J max   or  J max >  i  1,…,N ^ ^^

14. IEIIT-CNR SWAN, ARRI ©RT 2006 14 One-Sided MCRA: Case 1 J max  J J max J J max    J max <  algorithm provides a correct solution  1  2  3  4  5  6 J(  1 ) J(  2 ) J(  3 ) J(  4 ) J(  5 ) J(  6 )  J  ^ ^

15. IEIIT-CNR SWAN, ARRI ©RT 2006 15 One-Sided MCRA: Case 2 J max  J max >  J J max  <  algorithm may provide a wrong solution  1  2  3  4  5  6 J(  1 ) J(  2 ) J(  5 ) J(  4 ) J(  6 ) J(  3 )  J  ^ ^

16. IEIIT-CNR SWAN, ARRI ©RT 2006 16 Two-Sided MCRA n Two-sided MCRA: n Two-sided MCRA: They may provide a wrong solution in both instances n Consider the empirical average J J ave  ave J(  i   where N is the sample size JJ Check if J ave   or  J ave >  i  1,…,N ^ ^^

17. IEIIT-CNR SWAN, ARRI ©RT 2006 17 Two-Sided MCRA J ave  1  2  3  4  5  6 J(  1 ) J(  2 ) J(  3 ) J(  4 ) J(  5 ) J(  6 )  J  J J ave ^  J ave >  J J ave  <  ^

18. IEIIT-CNR SWAN, ARRI ©RT 2006 18 Two-Sided MCRA J ave  1  2  3  4  5  6 J(  1 ) J(  2 ) J(  3 ) J(  4 ) J(  5 ) J(  6 )  J  J J ave ^  ^ J ave <  J J ave  > 

19. IEIIT-CNR SWAN, ARRI ©RT 2006 19 Las Vegas Randomized Algorithms zero-sided n We also have zero-sided (Las Vegas) randomized algorithms n Las Vegas Randomized Algorithm (LVRA): n Las Vegas Randomized Algorithm (LVRA): Always give the correct solution n Running time may be different from one run to another n LVRA has more limited applicability than MCRA n Example: n Example: RandQuickSort

20. IEIIT-CNR SWAN, ARRI ©RT 2006 20 Current Research on LVRA n 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 n Consensus control: - design randomized algorithms achieving finite-time average consensus for connected networks.

21. IEIIT-CNR SWAN, ARRI ©RT 2006 21 Uncertain Systems, Optimization, System Identification n From common to piecewise Lyapunov functions [1] Ellipsoidal randomized algorithm [2] and stopping rules [3] RAs for semi-infinite programming [4] n MRAS methods for global optimization [5] n Estimation via MCMC [6] n RAs for model validation [7] and system identification [8] n … [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. IEIIT-CNR SWAN, ARRI ©RT 2006 22 Applications of RAs n RAs have been developed for many control applications n Control of flexible structures n Robustness of high speed networks n Stability of quantized sampled-data systems n Control design for brushless DC motors n Synthesis of real time embedded controllers n Mini-UAV control design

23. IEIIT-CNR SWAN, ARRI ©RT 2006 23 Applications of RAs n RAs have been developed for many control applications n Control of flexible structures n Robustness of high speed networks n Stability of quantized sampled-data systems n Control design for brushless DC motors n Synthesis of real time embedded controllers n Mini-UAV control design

24. IEIIT-CNR SWAN, ARRI ©RT 2006 24 Mini-UAV Control Design n Study and development of a real-time land control and monitoring system for fire prevention in Sicily n Uncertainty description n 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)

25. IEIIT-CNR SWAN, ARRI ©RT 2006 25 References n “Randomized Algorithms for Analysis and Control of Uncertain Systems” by R. Tempo, G. Calafiore and F. Dabbene, Springer-Verlag, 2005 n Additional documents, papers, MATLAB TM codes, etc, please consult http://staff.polito.it/roberto.tempo