Randomized Algorithms and Probabilistic Methods for Robustness

The main objective of this research area is to study probabilistic and randomized methods for analysis and design of uncertain systems. This area is fairly recent, even though its roots lie in the robustness techniques for handling complex control systems developed in the 1980s. In contrast to these previous deterministic techniques, its main ingredient is the use of probabilistic concepts. One of the goals of this research endeavor is to provide a reapprochement between the classical stochastic and robust paradigms, combining worst-case bounds with probabilistic information, thus potentially reducing the conservatism inherent in the worst-case design. In this way, the control engineer gains additional insight that may help bridging the gap between theory and applications.

The algorithms derived in the probabilistic context are based on uncertainty randomization and are usually called randomized algorithms. For control systems analysis, these algorithms have low complexity and are associated with robustness bounds that are generally less conservative than the classical ones, obviously at the expense of a probabilistic risk of failure.

In the classical robustness analysis framework, the main objective is to guarantee that a certain system property is attained for all uncertainties Δ bounded within a specified set B. To this end, it is useful to define a performance function:
J (Δ):B → R

and an associated performance level γ . In general, the function J (Δ) can take into account the simultaneous attainment of various performance requirements. In the framework of robust synthesis, the performance function depends also on some "design" parameters θ in a bounding set " and takes the form
J (Δ,θ ): (B,Θ )→R

When dealing with probabilistic methods for robustness, we assume that the uncertainty Δ is a random matrix distributed according to a probability density function f(Δ) with support B. Then, we formulate various problems. For example, the Probabilistic Performance Problem can be stated as: Given a performance function J(Δ) with associated level γ and a density function f(Δ) with support B, compute the probability of performance

P = ProbΔ{J(Δ)≤ γ}.

The probability of performance measures the probability that a level of performance is achieved when Δ is a random matrix distributed according to f (Δ ). We remark that this probability is in general difficult to compute either analytically or numerically, since it basically requires the evaluation of a multidimensional integral. In some cases, it can be evaluated in closed form, in other cases randomized algorithms may be used to this end. The ultimate objective of this research is to develop low complexity randomized algorithms for various analysis and synthesis problems, i.e. for specific performance functions.

References:

  1. H. Ishii and R. Tempo, "Probabilistic Sorting and Stabilization of Switched Systems," Automatica, Vol. 45, pp. 776-782, 2009.


  2. Y. Fujisaki, Y. Oishi and R. Tempo, "Mixed Deterministic/Randomized Methods for Fixed Order Controller Design," IEEE Transactions on Automatic Control, Vol. 53, pp. 2033-2047, 2008.


  3. C. Lagoa, F. Dabbene and R. Tempo, "Hard Bounds on the Probability of Performance with Application to Circuit Analysis," IEEE Transactions on Circuits and Systems I, Vol. 55, pp.3178-3187, 2008.


  4. R. Tempo and H. Ishii, "Monte Carlo and Las Vegas Randomized Algorithms for Systems and Control: An Introduction," European Journal of Control, Vol. 13, pp. 189-203, 2007.


  5. G. Calafiore, F. Dabbene and R. Tempo, "A Survey of Randomized Algorithms for Control Synthesis and Performance Verification," Journal of Complexity, Vol. 23, pp. 301-316, 2007.


  6. H. Ishii, T. Basar and R. Tempo, "Randomized Algorithms for Synthesis of Switching Rules for Multimodal Systems," IEEE Transactions on Automatic Control, Vol. AC-50, pp. 754-767, 2005.


  7. R. Tempo, G. Calafiore and R. Tempo, "Randomized Algorithms for Analysis and Control of Uncertain Systems," Springer-Verlag,London,2005.


  8. Y. Fujisaki, F. Dabbene and R. Tempo, "Probabilistic Robust Design of LPV Control Systems," Automatica, Vol. 39, pp. 1323-1337, 2003.


  9. B. T. Polyak and R. Tempo, "Probabilistic Robust Design with Linear Quadratic Regulators," Systems and Control Letters, Vol. 43, pp. 343-353, 2001.


  10. G. Calafiore, F. Dabbene and R. Tempo, "Randomized Algorithms for Probabilistic Robustness with Real and Complex Structured Uncertainty," IEEE Transactions on Automatic Control, Vol. AC-45, pp. 2218-2235, 2000.


  11. E. W. Bai, R. Tempo and M. Fu, "Worst-Case Properties of the Uniform Distribution and Randomized Algorithms for Robustness Analysis," Mathematics of Control, Signals and Systems, Vol. 11, pp. 183-196, 1998.


  12. R. Tempo, E. W. Bai and F. Dabbene, "Probabilistic Robustness Analysis: Explicit Bounds for the Minimum Number of Samples," Systems and Control Letters, Vol. 30, pp. 237-242, 1997.


back