Abstract
This chapter describes the ellipsoidal techniques for control problems introduced in earlier chapters. We derive formulas for reachability sets using the properties of ellipsoids and relations from convex analysis. The formulas are derived through inductive procedures. They allow calculation of both external and internal ellipsoidal approximations of forward and backward reachability sets with any desired level of accuracy. The approximations are illustrated on examples explained in detail, then followed by ellipsoid-based formulas for problems of reachability and control synthesis.
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Notes
- 1.
In the following formulas symbol p[⋅ ] stands for the pair {p 0, p(s), s ∈ [t 0, t]}.
- 2.
Here we also take into account that \((l^{{\prime}}(t)A(t) + dl(t)/dt)Y _{l}(t)\) is bounded with p 0(t) > 0 and p t (τ) > 0 almost everywhere due to the controllability assumption.
- 3.
Here and in the sequel the terms of type o i (ε) are assumed to be such that o i (ε)ε−1 → 0 with ε → 0.
- 4.
Note that for \(l_{2}^{0}/l_{1}^{0} > 0\) or \(l_{2}^{0}/l_{1}^{0} < -t\) we have \(\upsigma \not\in [-t, 0]\). For such vectors the point of support x l[t] will be at either of the vertices of set \(\mathcal{X}[t]\).
- 5.
Thus example was worked out by P. Gagarinov.
- 6.
If \(\mathcal{W}[t]\) is already calculated, the procedure is simple.
- 7.
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Kurzhanski, A.B., Varaiya, P. (2014). Ellipsoidal Techniques: Reachability and Control Synthesis. In: Dynamics and Control of Trajectory Tubes. Systems & Control: Foundations & Applications, vol 85. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-10277-1_3
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DOI: https://doi.org/10.1007/978-3-319-10277-1_3
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