Complete Statistical Characterization of Discrete-Time LQG and Cumulant Control

As the performance index of the linear-quadratic-Gaussian (LQG) problem is governed by the noncentral generalized Chi-square distribution, solely controlling the expected value of the performance index, as the traditional LQG theory aims at, is insufficient to deliver a satisfactory solution in some situations. While the risk sensitive control does control a specific weighting sum of various moments of the performance index, the single degree of freedom in adjusting the weighting coefficients in this specific weighting sum of various moments of the performance index often prevents the risk sensitive control from generating a desired pattern of high order moment-distribution. We achieve in this note the complete statistical characterization of the performance index for the discrete-time LQG formulation. More specifically, we derive a recursive relationship to obtain cumulants of various orders of the performance index successively. Parameterized in feedback gain, the optimal feedback control law can be computed off-line by solving a static polynomial optimization problem, thus serving two design goals: (i) To shape the probability density function (pdf) of the performance index to attain, at least approximately, a given desired pattern by regulating cumulants of various orders, and (ii) to improve the performance measure of an incumbent solution (generated by the risk sensitive control, for example) by adjusting the levels of cumulants of various orders.

[1]  Larry S. Karp.  Higher moments in the linear-quadratic-gaussian problem , 1985 .

[2]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[3]  Duan Li,et al.  Performance-First Control for Discrete-Time LQG Problems , 2009, IEEE Transactions on Automatic Control.

[4]  M. Sain,et al.  Linear Cumulant Control and Its Relationship to Risk-Sensitive Control , 2000 .

[5]  Jianhua Zhang,et al.  Bounded stochastic distributions control for pseudo-ARMAX stochastic systems , 2001, IEEE Trans. Autom. Control..

[6]  Bernard Hanzon,et al.  REPORT RAPPORT , 2001 .

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

[8]  Zhong-Ping Jiang,et al.  Finite-Time Stabilization of Nonlinear Systems With Parametric and Dynamic Uncertainties , 2006, IEEE Transactions on Automatic Control.

[9]  H. Cramér.  Mathematical methods of statistics , 1947 .

[10]  S. R. Liberty,et al.  Design-performance-measure statistics for stochastic linear control systems , 1978 .

[11]  Robert Engel,et al.  A continuous-time observer which converges in finite time , 2002, IEEE Trans. Autom. Control..

[12]  Hong Wang,et al.  Minimum entropy control of closed-loop tracking errors for dynamic stochastic systems , 2003, IEEE Trans. Autom. Control..

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

[14]  Alexander S. Poznyak,et al.  Lyapunov function design for finite-time convergence analysis: "Twisting" controller for second-order sliding mode realization , 2009, Autom..

[15]  Leonid Fridman,et al.  Uniform Second-Order Sliding Mode Observer for mechanical systems , 2010, 2010 11th International Workshop on Variable Structure Systems (VSS).

[16]  Didier Henrion,et al.  GloptiPoly 3: moments, optimization and semidefinite programming , 2007, Optim. Methods Softw..

[17]  Leonid M. Fridman,et al.  High order sliding mode observer for linear systems with unbounded unknown inputs , 2010, Int. J. Control.

[18]  Dennis S. Bernstein,et al.  Finite-Time Stability of Continuous Autonomous Systems , 2000, SIAM J. Control. Optim..

[19]  P. Whittle.  Risk-Sensitive Optimal Control , 1990 .

[20]  Giorgio Bartolini,et al.  A survey of applications of second-order sliding mode control to mechanical systems , 2003 .

[21]  Lei Guo,et al.  PID controller design for output PDFs of stochastic systems using linear matrix inequalities , 2005, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[22]  Duan Li,et al.  Mean-variance control for discrete-time LQG problems , 2003, Proceedings of the 2003 American Control Conference, 2003..

[23]  Frank Allgöwer,et al.  Analysis and design of polynomial control systems using dissipation inequalities and sum of squares , 2006, Comput. Chem. Eng..

[24]  M. Sain.  Control of linear systems according to the minimal variance criterion--A new approach to the disturbance problem , 1966 .

[25]  M. Berberan-Santos.  Expressing a Probability Density Function in Terms of another PDF: A Generalized Gram-Charlier Expansion , 2007 .

[26]  Hong Wang,et al.  Minimum entropy control of closed loop tracking errors for stochastic systems using Youla parameterization , 2006 .

[27]  Duan Li,et al.  Closed-loop optimal control law for discrete-time LQG problems with a mean-variance objective , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[28]  M. Sain,et al.  Performance-measure densities for a class of LQG control systems , 1971 .

[29]  F. Allgöwer,et al.  An Observer That Converges in Finite Time Due to Measurement-Based State Updates , 2008 .

[30]  Markos V. Koutras,et al.  Quadratic forms in spherical random variables: Generalized noncentral x2 distribution , 1984 .

[31]  D. B. Izosimov,et al.  Block control principle. II , 1990 .