Immersion-based model predictive control of constrained nonlinear systems: Polyflow approximation

In the framework of Model Predictive Control (MPC), the control input is typically computed by solving optimization problems repeatedly online. For general nonlinear systems, the online optimization problems are non-convex and computationally expensive or even intractable. In this paper, we propose to circumvent this issue by computing a high-dimensional linear embedding of discrete-time nonlinear systems. The computation relies on an algebraic condition related to the immersibility property of nonlinear systems and can be implemented offline. With the high-dimensional linear model, we then define and solve a convex online MPC problem. We also provide an interpretation of our approach under the Koopman operator framework.Comment: Accepted to the European Control Conferenc

Computation of the maximal invariant set of discrete-time linear systems subject to a class of non-convex constraints

We consider the problem of computing the maximal invariant set of discrete-time linear systems subject to a class of non-convex constraints that admit quadratic relaxations. These non-convex constraints include semialgebraic sets and other smooth constraints with Lipschitz gradient. With these quadratic relaxations, a sufficient condition for set invariance is derived and it can be formulated as a set of linear matrix inequalities. Based on the sufficient condition, a new algorithm is presented with finite-time convergence to the actual maximal invariant set under mild assumptions. This algorithm can be also extended to switched linear systems and some special nonlinear systems. The performance of this algorithm is demonstrated on several numerical examples.Comment: Accepted in Automatic

Non-local Linearization of Nonlinear Differential Equations via Polyflows

Motivated by the mathematics literature on the algebraic properties of so-called polynomial vector flows, we propose a technique for approximating nonlinear differential equations by linear differential equations. Although the idea of approximating nonlinear differential equations with linear ones is not new, we propose a new approximation scheme that captures both local as well as global properties. This is achieved via a hierarchy of approximations, where the Nth degree of the hierarchy is a linear differential equation obtained by globally approximating the Nth Lie derivatives of the trajectories. We show how the proposed approximation scheme has good approximating capabilities both with theoretical results and empirical observations. In particular, we show that our approximation has convergence range at least as large as a Taylor approximation while, at the same time, being able to account for asymptotic stability (a nonlocal behavior). We also compare the proposed approach with recent and classical work in the literature