Cost distribution shaping: the relation between Bode integral, entropy, risk-sensitivity, and cost cumulant control

The cost function in stochastic optimal control is viewed as a random variable. Then the classical linear-quadratic-Gaussian control, entropy control, risk-sensitive control, and cost cumulant control can be viewed as the cost distribution shaping methods. We would survey the existing relations between entropy, Bode integral, and risk-sensitive cost function. Furthermore, we would relate the cost cumulants with information theoretic entropy, and Bode integral. The interpretation of cost cumulant control is given in terms of the control entropy minimization. The paper also relates information theoretic entropy with exponential-of-integral cost function using a Lagrange multiplier and calculus of variations. Finally, the logarithmic-exponential-of-integral cost function is related to the information theoretic entropy using large deviation theory.

[1]  H. Saunders,et al.  Probability, Random Variables and Stochastic Processes (2nd Edition) , 1989 .

[2]  Pablo A. Iglesias,et al.  Tradeoffs in linear time-varying systems: an analogue of Bode's sensitivity integral , 2001, Autom..

[3]  J. Lynch,et al.  A weak convergence approach to the theory of large deviations , 1997 .

[4]  P. Iglesias.  An analogue of Bode's integral for stable nonlinear systems: relations to entropy , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[5]  M. Kendall,et al.  The Advanced Theory of Statistics: Volume 1, Distribution Theory , 1978 .

[6]  M. Sain,et al.  CUMULANT MINIMIZATION AND ROBUST CONTROL , 1992 .

[7]  P. Davies,et al.  Kendall's Advanced Theory of Statistics. Volume 1. Distribution Theory , 1988 .

[8]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[9]  Keith Glover.  Minimum entropy and risk-sensitive control: the continuous time case , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[10]  R. Ellis,et al.  Large deviations and statistical mechanics , 1985 .

[11]  M. A. Peters,et al.  Minimum Entropy Control for Time-Varying Systems , 1997 .

[12]  J. H. van Schuppen,et al.  System identification with information theoretic criteria , 1995 .

[13]  T. Runolfsson.  The equivalence between infinite-horizon optimal control of stochastic systems with exponential-of-integral performance index and stochastic differential games , 1994, IEEE Trans. Autom. Control..

[14]  Mark Bartlett,et al.  An Introduction to Stochastic Processes with Special Reference to Methods and Applications. , 1955 .

[15]  K. Uchida,et al.  On the central controller: Characterizations via differential games and LEQG control problems , 1989 .

[16]  K. Glover,et al.  State-space formulae for all stabilizing controllers that satisfy and H ∞ norm bound and relations to risk sensitivity , 1988 .

[17]  G. Saridis.  Entropy formulation of optimal and adaptive control , 1988 .

[18]  H. W. Bode,et al.  Network analysis and feedback amplifier design , 1945 .

[19]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

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

[21]  J. Freudenberg,et al.  Right half plane poles and zeros and design tradeoffs in feedback systems , 1985 .

[22]  M. Kendall,et al.  Kendall's advanced theory of statistics , 1995 .

[23]  S. Varadhan,et al.  Asymptotic evaluation of certain Markov process expectations for large time , 1975 .