Abstract
The problem of synthesizing an \({{\mathscr {H}}_\infty }\) loop-shaping controller, but with a bound on its complexity, is shown to be a tractable optimization problem. Here complexity is defined in terms of the smoothness of the transfer function.
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Notes
- 1.
i.e. \(P^* = \left( P(-s)\right) ^T\); \(P(s) = D+C(sI-A)^{-1}B = \left[ \begin{array}{c|c} A &{} B \\ \hline C &{} D \end{array} \right] \); \({\mathscr {F}}_\ell \left( \left[ \begin{array}{cc} P_{11} &{} P_{12} \\ P_{21} &{} P_{22}\end{array}\right] ,K\right) = P_{11}+P_{12}K(I-P_{22}K)^{-1} P_{21}\).
- 2.
This ensures that \(\kappa (P_1,P_2;s_1,s_2)\) is analytic and bounded (by 1 in fact) for \(s_1,s_2\) in the closed RHP, with no requirements for either \(P_1\) or \(P_2\) themselves to be stable or minimum phase, since RHP zeros of the inverted spectral factors cancel any unstable poles of the plants.
- 3.
This is the simplest way to state the theorem. If instead continuity requirements are placed on the implied map from \(s_1\) to \(s_2\) in the definition of \(\delta _{\text {eff}}\) then an explicit path linking \(P_1\) and \(P_2\) is not required. See [7] for more details.
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Glover, K., Vinnicombe, G. (2018). Smooth Operators Enhance Robustness. In: Tempo, R., Yurkovich, S., Misra, P. (eds) Emerging Applications of Control and Systems Theory. Lecture Notes in Control and Information Sciences - Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-319-67068-3_12
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