Abstract
The paper deals with the state estimation problem in control theory under set-membership uncertainty. We consider linear systems of ordinary differential equations (ODE) with parallelepiped-valued uncertainties in initial states and interval uncertainties in coefficients of the system. As a result we have the uncertainty of the bilinear type and essentially nonlinear problem. We construct internal and external estimates for trajectory tubes of such systems. Using discrete-time approximations and techniques of the “polyhedral calculus” and passing to the limit in the discrete-time estimates, we obtain nonlinear ODE systems which describe the evolution of the parallelotope-valued estimates for reachable sets (time cross-sections of the trajectory tubes). The main results are obtained for internal estimates. The properties of the obtained ODE systems are investigated; existence and uniqueness of solutions and also nondegeneracy of estimates are established. Results of numerical simulations are presented.
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Notes
- 1.
The normality condition \(\|{p}^{i}\|_{2} = 1\) may be omitted to simplify formulas (particulary, it ensures the uniqueness of the representation of a parallelepiped with nonzero values of semi-axes).
- 2.
Our estimates will satisfy the generalised semigroup property [16] which is analogues to the well-known semigroup property for \(\mathcal{X}(t)\).
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Acknowledgements
The work was supported by the Program of the Presidium of the Russian Academy of Sciences “Mathematical Theory of Control” under the support of the Ural Branch of RAS (project 09-P-1-1014) and by the Russian Foundation for Basic Research (grant 09-01-00223).
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Kostousova, E.K. (2013). On Polyhedral Estimates for Trajectory Tubes of Differential Systems with a Bilinear Uncertainty. In: Pinelas, S., Chipot, M., Dosla, Z. (eds) Differential and Difference Equations with Applications. Springer Proceedings in Mathematics & Statistics, vol 47. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7333-6_41
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