Robust mixed control and linear parameter-varying control with full block scalings.

*(English)*Zbl 0952.93025
El Ghaoui, Laurent (ed.) et al., Advances in linear matrix inequality methods in control. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. Adv. Des. Control. 2, 187-207 (2000).

Mixed control problems and their solution via the linear matrix inequality (LMI) approach have been the subject of numerous papers in recent years. Concerning the design of output controllers, a nonlinear change of controller parameters allows to pass directly from analysis specifications formulated in terms of matrix inequalities to the corresponding synthesis inequalities. In this work the above technique is applied to robust performance specifications when the system is subject to time-varying parameter uncertainties. A general approach is presented which allows to translate robust performance objectives formulated in terms of Lyapunov functions into an analysis test with multipliers. For this purpose full block multipliers are used which reduce the conservatism typical of the so-called \({\mathcal S}\)-procedure. This technique is called a full block \({\mathcal S}\)-procedure. It is a generalization of a similar technique, used before for robust stability specifications with affine dependence on the uncertainties, to the case when the system matrices are rational functions of the parameters.

The robust performance analysis via Lyapunov quadratic forms with constant matrices guarantees well posedness of the linear fractional transformation used to describe the system, uniform exponential stability relative to the given class of uncertainties and robust performance, specified by \(L_2\) and \(H_2\) criteria. Finally, a single-objective linear parameter-varying control problem is considered under the assumption that the parameters are not unknown but rather that they can be measured online. In this case a subclass of full block scalings is employed. The justification for the full block \({\mathcal S}\)-procedure is provided in an appendix, where the number of auxiliary results on quadratic forms and inequalities is proved.

For the entire collection see [Zbl 0932.00034].

The robust performance analysis via Lyapunov quadratic forms with constant matrices guarantees well posedness of the linear fractional transformation used to describe the system, uniform exponential stability relative to the given class of uncertainties and robust performance, specified by \(L_2\) and \(H_2\) criteria. Finally, a single-objective linear parameter-varying control problem is considered under the assumption that the parameters are not unknown but rather that they can be measured online. In this case a subclass of full block scalings is employed. The justification for the full block \({\mathcal S}\)-procedure is provided in an appendix, where the number of auxiliary results on quadratic forms and inequalities is proved.

For the entire collection see [Zbl 0932.00034].

Reviewer: Michael M.Konstantinov (Sofia)

##### MSC:

93B35 | Sensitivity (robustness) |

93B40 | Computational methods in systems theory (MSC2010) |

93B50 | Synthesis problems |

93B51 | Design techniques (robust design, computer-aided design, etc.) |

93B36 | \(H^\infty\)-control |

15A39 | Linear inequalities of matrices |