Certification of fixed computation time first-order optimization-based controllers for a class of nonlinear dynamical systems

Abstract

Abstract-This paper proposes a stability verification method for systems controlled by an early terminated first-order method (e.g., an MPC problem approximately solved by a fixed number of iterations of the fast gradient method).The method is based on the observation that each step of the vast majority of first-order methods is characterized by a Karush-Kuhn-Tucker (KKT) system which (provided that all data are polynomial) is a basic semialgebraic set; M steps of a first-order method is then characterized by a basic semialgebraic set given by the intersection of M coupled KKT systems. Using sum-of-squares techniques, one can then search for a polynomial Lyapunov function that decreases between two consecutive time instances for all control inputs belonging to this coupled KKT system.The proposed method applies to nonlinear dynamical systems described by polynomial (or trigonometric) data affected by a (possibly state-dependent) disturbance; in particular the method is not restricted to linear systems and/or convex cost functions. To the best of the authors' knowledge, this is the first verification approach for early terminated optimization schemes with this level of generality.

Stability and performance verification of optimization-based controllers

“…This example shows how to model within the presented framework a nonlinear MPC controller with state estimation, a model mismatch 2 , soft constraints and no a priori stability guarantees (enforced, e.g., using a terminal penalty and/or terminal set) and possibly only locally optimal solutions delivered by the optimization algorithm. In this case the system (8) is an estimator of the state of the dynamical system (5) and in each time step the following optimization problem is solved…”

“…In this case the system ( 8) is an estimator of the state of the dynamical system (5) and in each time step the following optimization problem is solved minimize û,x,ε…”

“…The presented framework can also handle the situation where the optimization problem ( 16) is solved using an iterative optimization method, each step of which is either an optimization problem or a polynomial mapping. This scenario was elaborated on in detail in [5], where it was shown that the vast majority of first order optimization methods fall into this category. Here we present the basic idea on one of the simplest optimization algorithms, the projected gradient method.…”

“…This work is a continuation of [5] where the approach was used to analyze the stability of optimization-based controllers where the optimization problem is solved approximately by a fixed number of iterations of a first order method. The results of [5] are summarized in Section 4.4 of this paper as one of the examples that fit in the presented framework.…”

Relay-type safeguarding regulator for MPC with unknown computational time

“…In recent times, great efforts have been devoted to MPC adaptation to a faster range of plants, such as future cars or UAVs (see e.g. [2], [3], [4], [5], [6]). The main problem in this case seems to be the sertification of finite computational time for such algorithms.…”