Convex Computation of the Region of Attraction of Polynomial Control Systems

By Didier Henrion, Milan Korda. IEEE Transactions on Automatic Control, 2012

330 citations recorded by the source corpus.

We address the long-standing problem of computing the region of attraction (ROA) of a target set (e.g., 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 an infinite-dimensional 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 infinite-dimensional 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. The approach is simple and readily applicable as the outer approximations are the outcome of a single semidefinite program with no additional data required besides the problem description. The approach is demonstrated on several numerical examples.

Subfields: Control and Systems Engineering, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/9d7fc71ffb6cb475f11e8faf05f7dcfc6241da7f

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Citations in the corpus

Set-Theoretic Approaches in Analysis, Estimation and Control of Nonlinear Systems (2015, background): “Related to reachability analysis, the problems of computing the region of attraction of a target set or the maximum invariant set are also essential and long-standing challenges in dynamic system and control theory (Henrion and Korda, 2014; Korda et al., 2014).”
A fast sampling method for estimating the domain of attraction (2016, background): “Non-Lyapunovmethods include, for instance, trajectory reversing [11,24], determining reachable sets of the system [2], and occupation measures [14,18].”
Hamilton-Jacobi reachability: A brief overview and recent advances (2017, neutral): “For example, [24], [40]–[42] present scalable methods for affine systems that rely on polytopic or ellipsoidal representation of sets, while the methods presented in [43]–[45] are wellsuited to systems with polynomial dynamics.”