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Linear Robust Control Design

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Randomized Algorithms for Analysis and Control of Uncertain Systems

Part of the book series: Communications and Control Engineering ((CCE))

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Abstract

In this chapter, we focus on robust controller synthesis in the \(\mathcal{H}_{\infty}\) and structured singular value setting. The design solution is formulated in the form of linear matrix inequality conditions. Linear quadratic regulator, guaranteed-cost, and \(\mathcal{H}_{2}\) controller design problems are also discussed. The chapter ends with some historical notes on robustness and related discussions.

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Notes

  1. 1.

    To this end, take , where \(D_{21}^{T\bot}\) is the orthogonal complement of \(D_{21}^{T}\), i.e. \(D_{21}D_{21}^{T\bot}=0\) and \(D_{21}^{T\bot T}D_{21}^{T\bot}=I\). Since \(D_{21}B_{1}^{T}=0\) we may write \(B_{1}^{T}=D_{21}^{T\bot}Z\) for some matrix Z and, therefore, \(B_{1}D_{21}^{T\bot}D_{21}^{T\bot T}B_{1}^{T}=B_{1}B_{1}^{T}\). Everything follows in a similar way for the first inequality, choosing .

  2. 2.

    \(X\in{ \mathbb{S}^{n} } \) is a stabilizing solution of the ARE A T X+XA+XRX+Q=0 if it satisfies the equation and A+RX is stable.

  3. 3.

    The asterisks indicate elements whose values are easily inferred by symmetry.

  4. 4.

    T.J. Sargent was awarded the Nobel Memorial Prize in Economic Sciences in 2011 together with C.A. Sims “for their empirical research on cause and effect in the macroeconomy.”

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Authors and Affiliations

  1. CNR - IEIIT, Politecnico di Torino, Turin, Italy

    Roberto Tempo & Fabrizio Dabbene

  2. Dip. Automatica e Informatica, Politecnico di Torino, Turin, Italy

    Giuseppe Calafiore

Authors
  1. Roberto Tempo
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  2. Giuseppe Calafiore
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  3. Fabrizio Dabbene
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Tempo, R., Calafiore, G., Dabbene, F. (2013). Linear Robust Control Design. In: Randomized Algorithms for Analysis and Control of Uncertain Systems. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-1-4471-4610-0_4

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  • DOI: https://doi.org/10.1007/978-1-4471-4610-0_4

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  • Online ISBN: 978-1-4471-4610-0

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