IJPAM: Volume 50, No. 2 (2009)

EXTERNAL POLYHEDRAL ESTIMATES FOR REACHABLE
SETS OF LINEAR DISCRETE-TIME SYSTEMSWITH
INTEGRAL BOUNDS ON CONTROLS

Elena K. Kostousova
Institute of Mathematics and Mechanics
Ural Branch of Russian Academy of Sciences
16, S. Kovalevskaja Str., Ekaterinburg GSP-384, 620219, RUSSIA
e-mail: [email protected]


Abstract.The approach for estimating the reachable sets $\XX[k]$ of the linear discrete-time systems with integral bounds on controls is presented. It is based on considering reachable sets $\ZZ[k]$ in the ``extended" phase space and allows to construct estimates for systems without and with state constraints. We construct the external estimates of $\ZZ[k]$ in the form of special polytopes. The specific cross-sections of them provide the parallelepiped-valued estimates of the reachable sets $\XX[k]$ in the ``initial" phase space. The whole families of estimates are introduced. Evolution of estimates is determined by recurrence relations. The families of touching estimates which ensure the exact representations of the reachable sets are described for systems without state constraints. The novel family of estimates for time-invariant systems is introduced. It is efficient for systems without and with small number of state constraints. The results of numerical simulations are presented.

Received: August 14, 2008

AMS Subject Classification: 93B03, 93C41, 93C55, 52B12

Key Words and Phrases: control theory, reachable sets, state estimation, polyhedral estimates, parallelepipeds, linear discrete-time systems

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 50
Issue: 2