Abstract
In this chapter, we study the discrete-time minimax controller design problem, as formulated by (1.6)-(1.7), when the controller is allowed to have perfect access to the system state, either without or with one step delay. We first consider the case when the controlled output is a concatenation of the system state and the current value of the control, and the initial state is zero; that is, the system dynamics and the performance index are (without any loss of generality in this class):
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Here, even though the sets U and W are not compact, it follows from the quadratic nature of the objective function that u and w can be restricted to closed and bounded (hence compact) subsets of finite-dimensional spaces, without affecting the saddle-point solution. Hence, Theorem 2.3 can be used.
For this terminology, see Section 2.2.
This result follows readily from Remark 3.1.
This bound is the largest root of the polynomial 5s3 — 1382 + 7s — 1.
This bound is the largest root of the polynomial 13s3 — 19s2 + 8s — 1.
We will shortly see that this is true also for the infinite-horizon case.
This condition holds when ry° ry*, which (intuitively) is generally the case, due to loss of information to the controller under delayed state information.
Henceforth, in this section, we take all the system matrices, in (3.1) and (3.2a), as constant matrices.
This fact has already been used in the proof of Theorem 3.2.
Here the gain coefficient of the controller remains bounded as -y y, by (the infinite-horizon version of) Lemma 3.2.
One such sufficient condition is Q 0, under which boundedness of upper value implies input-output stability under controller (3.59a).
For a related transformation in the time-invariant continuous-time case, see [74] which uses a loop-shifting method.
This follows from Theorem 2.5.
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© 1991 Springer Science+Business Media New York
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Başar, T., Bernhard, P. (1991). The Discrete-Time Minimax Design Problem With Perfect State Measurements. In: H∞-Optimal Control and Related. Systems & Control: Foundations & Applications. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-3561-8_3
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DOI: https://doi.org/10.1007/978-1-4899-3561-8_3
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4899-3563-2
Online ISBN: 978-1-4899-3561-8
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