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Almost Feedback Linearization via Dynamic Extension: a Paradigm for Robust Semiglobal Stabilization of Nonlinear MIMO Systems

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Trends in Nonlinear and Adaptive Control

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 488))

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Abstract

It is well known that  invertible MIMO nonlinear systems, can be input–output linearized via dynamic state feedback (augmentation of the dynamics and memoryless state feedback from the augmented state). The procedures for the design of such feedback, developed in the late 1980s for nonlinear systems, typically are recursive procedures that involve state-dependent transformations in the input space and cancelation of nonlinear terms. As such, they are fragile. In a recent work of Wu-Isidori-Lu-Khalil, a method has been proposed, consisting of interlaced design of dynamic extensions and extended observers, that provides a robust version of those feedback-linearizing procedures. The method in question can be used as a systematic tool for robust semiglobal stabilization of invertible and strongly minimum-phase MIMO nonlinear systems. The present paper provides a review of the method in question, with an application to the design of a robust output regulator.

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Notes

  1. 1.

    For convenience, it is assumed that \(\mathrm{dim}(y_i)=1\) for all \(i=1,\ldots , m\) and \(r_1<r_2<\ldots <r_m\). In general, one should consider y split into q blocks \(y_1,\ldots , y_q\), with \(\mathrm{dim}(y_i)=m_i\ge 1\) and \(\sum _{i=1}^q m_i =m\). The structure of the equations remains the same.

  2. 2.

    Note that, if any of such multipliers is nonzero, the system fails to have a vector relative degree. Nevertheless input–output linearization is achieved (see [10, pp. 280–287]).

  3. 3.

    It is easy to show that the Assumption in question is compatible with the assumption of uniform invertibility, i.e., that if in a normal form like (1.1) such Assumption holds, and the matrix B(z, x) is nonsingular, the system is uniformly invertible in the sense of Singh. However, it must be stressed that the necessity of such triangular dependence has been proven only for systems having \(m=2\) and a trivial dynamics of z.

  4. 4.

    A property that implies, but is not implied by, the property that the system is globally minimum-phase.

  5. 5.

    A smooth function, characterized as follows: \(\mathrm{sat}_\ell (s)= s\) if \(|s|\le \ell \), \(\mathrm{sat}_\ell (s)\) is odd and monotonically increasing, with \(0< \mathrm{sat}_\ell ^\prime (s)\le 1\), and \(\lim _{s\rightarrow \infty }\mathrm{sat}_\ell (s)=\ell (1+c)\) with \(0<c\ll 1\).

  6. 6.

    The number \(b_0\) here is any number for which condition (1.7) holds.

  7. 7.

    The proof provided in the reference [6] addresses the case in which the dynamics of z are trivial. If this is not the case, appropriate modifications are needed, taking into account the Assumption of strong minimum-phase.

  8. 8.

    See [21] for a more detailed presentation.

  9. 9.

    Note that the actual values of \(\varGamma _1\) and \(\varGamma _2\) are not needed in the sequel.

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Authors and Affiliations

  1. Department of Computer, Control and Management Engineering, University of Rome “Sapienza”, Rome, Italy

    Alberto Isidori

  2. School of Automation, Guangdong University of Technology, Guangzhou, China

    Yuanqing Wu

Authors
  1. Alberto Isidori
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  2. Yuanqing Wu
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Correspondence to Alberto Isidori .

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Editors and Affiliations

  1. Department of Electrical and Computer Engineering, New York University, Brooklyn, NY, USA

    Zhong-Ping Jiang

  2. Automatic Control, CNRS, Saint Martin d’Heres, France

    Christophe Prieur

  3. Department of Electrical and Computer Engineering, Imperial College London, London, UK

    Alessandro Astolfi

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Isidori, A., Wu, Y. (2022). Almost Feedback Linearization via Dynamic Extension: a Paradigm for Robust Semiglobal Stabilization of Nonlinear MIMO Systems. In: Jiang, ZP., Prieur, C., Astolfi, A. (eds) Trends in Nonlinear and Adaptive Control. Lecture Notes in Control and Information Sciences, vol 488. Springer, Cham. https://doi.org/10.1007/978-3-030-74628-5_1

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