Estimating the domain of attraction for uncertain polynomial systems.

*(English)*Zbl 1067.93055The problem of estimating the robust domain of attraction is studied for a nonlinear system of the form
\[
\dot x= f(x,\phi),
\]
where \(x\in\mathbb{R}^n\), \(\phi\) is a vector of uncertain parameter, \(\phi\in\Phi\), and \(\Phi\) is a polytope of \(\mathbb{R}^z\). The map \(f\) is assumed to be a polynomial with respect to both \(x\) and \(\phi\). The approach relies on polynomial Lyapunov functions, whose coefficients may be either constant (when the family of systems admits a common Lyapunov function) or polynomially dependent on \(\phi\). The robust domain of attraction is defined as the intersection of the domains of attraction of each individual system, as \(\phi\) varies on \(\Phi\). In order to estimate it, the author computes lower bounds for the level sets of the Lyapunov functions contained in the region where their derivatives are negative. This is done by an appropriate transformation of the system and by solving a linear matrix inequality optimization problem and an eigenvalue problem.

Reviewer: Andrea Bacciotti (Torino)

##### MSC:

93D30 | Lyapunov and storage functions |

93D20 | Asymptotic stability in control theory |

93B35 | Sensitivity (robustness) |

93C40 | Adaptive control/observation systems |

15A39 | Linear inequalities of matrices |

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

93B60 | Eigenvalue problems |

##### Keywords:

domain of attraction; uncertain systems; Lyapunov functions; polynomial nonlinearity; level sets; linear matrix inequality; eigenvalue problem##### Software:

LEDA
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