Is robustness of stochastic uncertain systems related to information theory and statistical mechanics? (Keynote Address)

Fifty years ago, when Claude Shannon was developing the Mathematical Theory of Communications, for reliable data transmission, which evolved into the subject of information theory, another discipline was developing dealing with Feedback Control of Dynamical System, which evolved into a scientific subject dealing with decision, stability, and optimization. More recently, a separate discipline dealing with robustness of uncertain systems was born in response to the codification of high performance and reliability in the presence of modeling uncertainties. In principle, robustness in dynamical systems is captured through power dissipation via induced norms and dynamic games, while reliable data transmission is captured through measures of information via entropy, relative entropy, and certain laws of Large Deviations theory. The main ingredient in Large Deviations is the rate functional (or action functional in the classical mechanics terminology), often identified through the Cramer or Legendre-Fenchel Transform. On the other hand, robustness of stochastic uncertain systems is currently under development, using information theoretic as well as statistical mechanics concepts, such as, partition functions, free energy, relative entropy, and entropy rate functional. This lecture will summarize certain connections between fundamental concepts of robustness, information theory, and statistical mechanics, and possibly make future projections into the convergence of these disciplines.

[1]  D. Stroock.  An Introduction to the Theory of Large Deviations , 1984 .

[2]  C. Charalambous,et al.  Optimization of stochastic uncertain systems: large deviations and robustness , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[3]  G. Zames.  Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses , 1981 .

[4]  A. Kyprianou,et al.  Relations between information theory, robustness, and statistical mechanics of stochastic systems , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[5]  A. Bensoussan,et al.  Optimal control of partially observable stochastic systems with an exponential-of-integral performance index , 1985 .

[6]  A. Isidori,et al.  Disturbance attenuation and H/sub infinity /-control via measurement feedback in nonlinear systems , 1992 .

[7]  C. Charalambous,et al.  Minimum principle for partially observable nonlinear risk-sensitive control problems using measure-valued decompositions , 1996 .

[8]  P. Whittle.  A risk-sensitive maximum principle: the case of imperfect state observation , 1991 .

[9]  Charalambos D. Charalambous,et al.  Stochastic Uncertain Systems Subject to Relative Entropy Constraints: Induced Norms and Monotonicity Properties of Minimax Games , 2007, IEEE Transactions on Automatic Control.

[10]  A. Astolfi.  Disturbance Attenuation and H,-Control Via Measurement Feedback in , 1992 .

[11]  M. James,et al.  Risk-sensitive control and dynamic games for partially observed discrete-time nonlinear systems , 1994, IEEE Trans. Autom. Control..

[12]  Wolfgang J. Runggaldier,et al.  Connections between stochastic control and dynamic games , 1996, Math. Control. Signals Syst..

[13]  C. Charalambous.  The role of information state and adjoint in relating nonlinear output feedback risk-sensitive control and dynamic games , 1997, IEEE Trans. Autom. Control..

[14]  C. D. Charalambous,et al.  Optimization of stochastic uncertain systems: large deviations and robustness for partially observable diffusions , 2003, Proceedings of the 2004 American Control Conference.

[15]  P. Whittle.  A risk-sensitive maximum principle , 1990 .

[16]  Ian R. Petersen,et al.  Minimax optimal control of stochastic uncertain systems with relative entropy constraints , 2000, IEEE Trans. Autom. Control..

[17]  J. Willems.  Dissipative dynamical systems part I: General theory , 1972 .

[18]  P. Dupuis,et al.  Minimax optimal control of stochastic uncertain systems with relative entropy constraints , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[19]  Charalambos D. Charalambous,et al.  Characterization of the optimal disturbance attenuation for nonlinear stochastic uncertain systems , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[20]  S.,et al.  Risk-Sensitive Control and Dynamic Games for Partially Observed Discrete-Time Nonlinear Systems , 1994 .

[21]  Rhodes,et al.  Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games , 1973 .

[22]  S. Varadhan.  Large Deviations and Applications , 1984 .