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Convex computation of the region of attraction of polynomial control systems

Didier Henrion(henrion***at***laas.fr)
Milan Korda(milan.korda***at***epfl.ch)

Abstract: We address the long-standing problem of computing the region of attraction (ROA) of a target set (typically a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving a convex linear programming (LP) problem over the space of measures. In turn, this problem can be solved approximately via a classical converging hierarchy of convex finite-dimensional linear matrix inequalities (LMIs). Our approach is genuinely primal in the sense that convexity of the problem of computing the ROA is an outcome of optimizing directly over system trajectories. The dual LP on nonnegative continuous functions (approximated by polynomial sum-of-squares) allows us to generate a hierarchy of semialgebraic outer approximations of the ROA at the price of solving a sequence of LMI problems with asymptotically vanishing conservatism. This sharply contrasts with the existing literature which follows an exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix inequalities or conservative LMI conditions.

Keywords: Region of attraction, polynomial control systems, occupation measures, linear matrix inequalities (LMIs), convex optimization, viability theory, capture basin.

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Applications -- Science and Engineering (Control Applications )

Category 3: Convex and Nonsmooth Optimization (Convex Optimization )

Citation:

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Entry Submitted: 08/07/2012
Entry Accepted: 08/07/2012
Entry Last Modified: 08/07/2012

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