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Stability Analysis and Stabilization—Polytopic Representation Approach

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Stability and Stabilization of Linear Systems with Saturating Actuators

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Abstract

Considering polytopic differential inclusions to model the saturation effects on the closed-loop system dynamics, this chapter addresses the stability analysis and stabilization of systems presenting control saturation. First, conditions for the regional asymptotic stability of the closed-loop system are derived. In particular, it is deeply discussed how to obtain estimates of the basin of attraction from the conditions. Similarly, conditions to ensure the external stability of the system with saturating inputs are stated considering that the system is subject to the action of amplitude or energy bounded exogenous signals. Secondly, the problem of designing a control law taking explicitly into account the possibility of actuator saturation is addressed. Although the results are mainly focused on the state feedback control laws, results regarding the design of dynamic output feedback control are briefly presented. The extension of the approach to cope with model uncertainties is also briefly discussed. Finally, the discrete-time counterpart of the results are presented. In this case, some particularities regarding the determination of polyhedral regions of stability are considered.

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Notes

  1. 1.

    p(t) and p(t)2 are considered independently to satisfy the linearity condition of uncertain parameters in the polytopic modeling, i.e. two uncertain parameters are considered. This leads to a polytope of matrices with four vertices.

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Authors and Affiliations

  1. Laboratoire Analyse et Architecture des Systèmes (LAAS), CNRS, av. du Colonel Roche 7, 31077, Toulouse CX 4, France

    Sophie Tarbouriech & Isabelle Queinnec

  2. Laboratoire Analyse et Architecture des Systèmes (LAAS), CNRS, av. du Colonel Roche, 31077, Toulouse CX 4, France

    Germain Garcia

  3. Departamento de Engenharia Elétrica, Universidade Federal do Rio Grande do Sul, av. Osvaldo Aranha 103, 90035-190, Porto Alegre, Rio Grande do Sul, Brazil

    João Manoel Gomes da Silva Jr.

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  1. Sophie Tarbouriech
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  2. Germain Garcia
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  3. João Manoel Gomes da Silva Jr.
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  4. Isabelle Queinnec
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Correspondence to Sophie Tarbouriech .

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Tarbouriech, S., Garcia, G., Gomes da Silva, J.M., Queinnec, I. (2011). Stability Analysis and Stabilization—Polytopic Representation Approach. In: Stability and Stabilization of Linear Systems with Saturating Actuators. Springer, London. https://doi.org/10.1007/978-0-85729-941-3_2

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  • DOI: https://doi.org/10.1007/978-0-85729-941-3_2

  • Publisher Name: Springer, London

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  • Online ISBN: 978-0-85729-941-3

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