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2014 | OriginalPaper | Buchkapitel

7. Dynamics and Control Under State Constraints

verfasst von : Alexander B. Kurzhanski, Pravin Varaiya

Erschienen in: Dynamics and Control of Trajectory Tubes

Verlag: Springer International Publishing

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Abstract

The topics of this chapter are problems of reachability and system dynamics under state constraints in the form of reach tubes. Indicated are general approaches based on the Hamiltonian formalism and a related Comparison Principle. Further emphasis is on the dynamics of linear systems under hard bounds on the controls and system trajectories. A detailed solution is presented based on ellipsoidal approximations of bounded trajectory tubes.

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Fußnoten
1
The boundary \(\partial \mathcal{X}[\uptau ]\) of \(\mathcal{X}[\uptau ]\) may be defined as the set \(\partial \mathcal{X}[\uptau ] = \mathcal{X}[\uptau ]\setminus \mathrm{int}\mathcal{X}[\uptau ]\). Under the controllability assumption, \(\mathcal{X}[\uptau ]\) has a nonempty interior, \(\mathrm{int}\mathcal{X}[\uptau ]\not =\varnothing,\;\uptau> t_{0}\).
 
2
In the general case, under Assumption 7.2.1, the optimal trajectory may visit the smooth boundary of the state constraint for a countable set of closed intervals, and function L 0(⋅ ) allows not more than a countable set of discontinuities of the first order.
 
3
The same minimum value is also attained here in classes of functions broader than \(\mathcal{C}_{00}\).
 
4
The computer illustrations for this subsection were made by M.N. Kirilin.
 
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Metadaten
Titel
Dynamics and Control Under State Constraints
verfasst von
Alexander B. Kurzhanski
Pravin Varaiya
Copyright-Jahr
2014
DOI
https://doi.org/10.1007/978-3-319-10277-1_7