» Download Article

Authors
Mavelli, Gabriella; Palumbo, Pasquale

Identifier
10.3182/20080706-5-KR-1001.01355

Index Terms
Stochastic control; Estimation and filtering; Nonlinear systems

Abstract
The paper investigates the optimal control problem for a stochastic linear differential system, driven by a persistent disturbance generated by a nonlinear stochastic exogenous system. The assumption of incomplete information has been assumed, that is neither the state of the system, nor the state of the exosystem are directly measurable. The standard quadratic cost functional has been considered. The approach followed consists of applying the Carleman approximation scheme to the exosystem, which provides a stochastic bilinear system. Then, the optimal regulator is obtained (i.e. the solution to the minimum control problem among all the affine transformations of the measurements). Better performances of the regulator are expected using higher order system approximations.

References

\bibitem[Betts (1999)]{Betts:99}
J.T. Betts.
\newblock Survey of numerical methods for trajectory optimization.
\newblock \emph {J. Guid. Control, Dyna.}, 21:\penalty0 193--207, 1999.

\bibitem[Bryson et al. (1995)]{Bryson:95}
A.E. Bryson, Y.C. Ho.
\newblock \emph{Applied optimal control}.
\newblock Wiley, New York, 1995.

\bibitem[Carravetta et al.(1997)]{Carra:97}
F. Carravetta, A. Germani and M. Raimondi.
\newblock Polynomial filtering of discrete-time stochastic linear systems with multiplicative state noise.
\newblock \emph{IEEE Trans. Autom. Control}, 42:\penalty0 1106--1126, 1997.

\bibitem[Carravetta {\em et al.}, 2000]{Carra:00}
F. Carravetta, A. Germani, M.K. Shuakayev.
A New Suboptimal Approach to the Filtering Problem for
Bilinear Stochastic Differential Systems.
{\it SIAM J. Control Optim. 38}, 1171--1203, 2000.

\bibitem[Carravetta et al.(2007)]{Carra:07}
F. Carravetta and G. Mavelli.
\newblock Suboptimal stochastic linear feedback control of linear systems with state- and control-dependent noise: the incomplete information case.
\newblock \emph{Automatica}, 43:\penalty0 751--757, 2007.

\bibitem[Charalambous et al. (1998)]{ChEl:98}
C. Charalambous and R.J. Elliot.
\newblock Classes of nonlinear partially observable stochastic optimal
control problems with explicit optimal control laws.
\newblock \emph{SIAM J. Control Optim.}, 36:\penalty0 542--578, 1998.

\bibitem[Francis et al. (1976)]{Francis:76}
B.A. Francis, W.M. Wonham.
\newblock The internal model principle of control theory.
\newblock \emph{Automatica}, 12:\penalty0 457--465, 1976.

\bibitem[Germani et al.(2005a)]{GermaPEKF:05}
A. Germani, C. Manes and P. Palumbo.
\newblock Polynomial Extended Kalman Filter.
\newblock \emph{IEEE Trans. Aut. Contr.}, 50:\penalty0 2059--2064, 2005a.

\bibitem[Germani et al.(2005b)]{GermaFamily:05}
A. Germani, C. Manes and P. Palumbo.
\newblock A Family of Polynomial Filters for Discrete-Time Nonlinear Stochastic Systems. In Proc. of 
\newblock \emph{IFAC World Congress on Autom. Contr.}, Prague, 2005b.

\bibitem[Germani et al.(2007)]{Germa:07}
A. Germani, C. Manes and P. Palumbo.
\newblock Filtering of stochastic nonlinear differential systems via a Carleman approximation approach,
\newblock \emph{IEEE Trans. Autom. Control}, 52:\penalty0 2166--2172, 2007.

\bibitem[Higham (2001)]{High:01} 
D.J. Higham.
\newblock An algorithmic introduction to numerical simulation of stochastic differential equations.
\newblock \emph{SIAM Review}, 43:\penalty0 525--546, 2001.

\bibitem[Isidori et al. (1990)]{Isidori:90}
A. Isidori, C.I. Byrnes.
\newblock Output regulation of nonlinear systems.
\newblock \emph{IEEE Trans. Autom. Contr.}, 35:\penalty0 131--140, 1990.

\bibitem[Liptser et al.(1977)]{Lipt:77}
R.S. Liptser and A.N. Shiryayev
\newblock \emph{{Statistics of random processes I and II}}.
\newblock Berlin, Springer, 1977.

\bibitem[Mavelli et al.(2007)]{Mave:07}
G. Mavelli and P. Palumbo.
\newblock A Carleman approximation scheme for a stochastic optimal nonlinear control problem. In Proc. of
\newblock \emph{9th European Control Conf.\ (ECC07)}, 3672--3678, 2007.

\bibitem[Serrani et al. (2001)]{Serrani:01}
A. Serrani, A. Isidori, L. Marconi.
\newblock Semiglobal nonlinear output regulation with adaptive internal model.
\newblock \emph {IEEE Trans. Autom. Contr.}, 46:\penalty0 1178--1194, 2001.

\bibitem[Tang (2005)]{TangSCL:05}
G.Y. Tang.
\newblock Suboptimal control for nonlinear systems: a successive
approximation approach.
\newblock \emph {Systems and Control Letters}, 54:\penalty0 429--434, 2005.