TY - JOUR

T1 - Dissipativity properties in constrained optimal control

T2 - A computational approach

AU - Berberich, Julian

AU - Köhler, Johannes

AU - Allgöwer, Frank

AU - Müller, Matthias A.

N1 - Funding information: The authors thank the German Research Foundation (DFG) for support of this work within grants AL 316/12-2 and MU 3929/1-2 . The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Gabriele Pannocchia under the direction of Editor Ian R. Petersen

PY - 2020/4

Y1 - 2020/4

N2 - In this paper, we consider discrete-time nonlinear optimal control problems with possibly non-convex cost subject to constraints on states and inputs. For these problems, we present a computational approach for the verification of strict dissipativity properties, which have recently been employed to study optimal system operation, turnpike phenomena, and closed-loop properties of economic model predictive control schemes. Based on a non-strict dissipation inequality, we provide necessary and sufficient conditions for strict dissipativity by explicitly computing the set w.r.t. which the system is strictly dissipative as well as the corresponding storage function. We focus on strict dissipativity w.r.t. periodic orbits and steady-states, being the most relevant cases of strict dissipativity, although our approach can be directly extended to strict dissipativity w.r.t. more general sets. For polynomial optimal control problems, the presented approach leads to a polynomial optimization problem, which is solved via sum-of-squares programming. The optimal periodic orbit can then be constructed, without a priori knowledge of its period length, as the set of points for which a suitable non-strict dissipation inequality holds with equality. In addition, we consider the important special case of indefinite linear quadratic optimal control problems subject to quadratic constraints, for which an optimal periodic orbit resulting from our construction is always located on the boundary of the constraint set.

AB - In this paper, we consider discrete-time nonlinear optimal control problems with possibly non-convex cost subject to constraints on states and inputs. For these problems, we present a computational approach for the verification of strict dissipativity properties, which have recently been employed to study optimal system operation, turnpike phenomena, and closed-loop properties of economic model predictive control schemes. Based on a non-strict dissipation inequality, we provide necessary and sufficient conditions for strict dissipativity by explicitly computing the set w.r.t. which the system is strictly dissipative as well as the corresponding storage function. We focus on strict dissipativity w.r.t. periodic orbits and steady-states, being the most relevant cases of strict dissipativity, although our approach can be directly extended to strict dissipativity w.r.t. more general sets. For polynomial optimal control problems, the presented approach leads to a polynomial optimization problem, which is solved via sum-of-squares programming. The optimal periodic orbit can then be constructed, without a priori knowledge of its period length, as the set of points for which a suitable non-strict dissipation inequality holds with equality. In addition, we consider the important special case of indefinite linear quadratic optimal control problems subject to quadratic constraints, for which an optimal periodic orbit resulting from our construction is always located on the boundary of the constraint set.

KW - Control of constrained systems

KW - Economic model predictive control

KW - Nonlinear model predictive control

KW - Optimal periodic control

UR - http://www.scopus.com/inward/record.url?scp=85078093879&partnerID=8YFLogxK

U2 - 10.1016/j.automatica.2020.108840

DO - 10.1016/j.automatica.2020.108840

M3 - Article

VL - 114

JO - Automatica

JF - Automatica

SN - 0005-1098

M1 - 108840

ER -