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Item type:Publication, Access status: without files , Turnpike properties in optimal control: An overview of discrete-time and continuous-time resultsThe turnpike property refers to the phenomenon that in many optimal control problems, the solutions for different initial conditions and varying horizons approach a neighborhood of a specific steady state, then stay in this neighborhood for the major part of the time horizon, until they may finally depart. While early observations of the phenomenon can be traced back to works of Ramsey and von Neumann on problems in economics in 1928 and 1938, the turnpike property received continuous interest in economics since the 1960s and recent interest in systems and control. The present chapter provides an introductory overview of discrete-time and continuous-time results in finite and infinite-dimensions. We comment on dissipativity-based approaches and infinite-horizon results, which enable the exploitation of turnpike properties for the numerical solution of problems with long and infinite horizons. After drawing upon numerical examples, the chapter concludes with an outlook on time-varying, discounted, and open problems.Publicationtype: Book Part29 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , On continuous-time infinite horizon optimal control—Dissipativity, stability, and transversality(2021-12); This paper analyzes the interplay between dissipativity and stability properties in continuous-time infinite-horizon Optimal Control Problems (OCPs). We establish several relations between these properties, which culminate in a set of equivalence conditions. Moreover, we investigate convergence and stability of the infinite-horizon optimal adjoint trajectories. The workhorse for our investigations is a notion of strict dissipativity in OCPs, which has been coined in the context of economic model predictive control. With respect to the link between stability and dissipativity, the present paper can be seen as an extension of the seminal work on least squares optimal control by Jan C. Willems (1971). Furthermore, we show that strict dissipativity provides a conclusive answer to the question of adjoint transversality conditions in infinite-horizon optimal control which has been raised by Hubert Halkin (1974). Put differently, we establish conditions under which the adjoints converge to their optimal steady-state value. We draw upon several examples to illustrate our findings. Moreover, we discuss the relation of our findings to results available in the literature.Publicationtype: Journal Article9 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Control of port-Hamiltonian systems with minimal energy supply(Elsevier, 2021-11-01); ; ; ; We investigate optimal control of linear port-Hamiltonian systems with control constraints, in which one aims to perform a state transition with minimal energy supply. Decomposing the state space into dissipative and non-dissipative (i.e. conservative) subspaces, we show that the set of reachable states is bounded w.r.t. the dissipative subspace. We prove that the optimal control problem exhibits the turnpike property with respect to the non-dissipative subspace, i.e., for varying initial conditions and time horizons optimal state trajectories evolve close to the conservative subspace most of the time. We analyze the corresponding steady-state optimization problem and prove that all optimal steady states lie in the non-dissipative subspace. We conclude this paper by illustrating these results by a numerical example from mechanics.Publicationtype: Journal Article6 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , First results on turnpike bounds for stabilizing horizons in NMPCThe stability analysis of NMPC schemes has seen significant progress in recent years, which includes schemes with and without terminal ingredients such as penalties and constraints. In the context of economic MPC, turnpike properties, which are closely related to dissipativity properties of the underlying optimal control problem, have enabled novel insights on stability conditions. In the present note, we show that turnpike properties naturally enable to bound the required stabilizing horizon length in MPC. The main idea is to define the turnpike with respect to a level-set of the terminal penalty. This way we derive a bound on the stabilizing horizon which guarantees that a terminal constraint is satisfied without being explicitly stated in the underlying optimal control problem. A numerical example indicates that if the terminal set is not too small, the horizon bound is not overly conservative.Publicationtype: Conference Paper8 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Towards necessary and sufficient stability conditions for NMPCThe analysis of convergence and stability conditions for Non-linear Model Predictive Control (NMPC) schemes has seen significant progress during the last decades. Numerous results in the literature state sufficient conditions for convergence and/or stability of NMPC. Yet, necessary and sufficient conditions are—to the best of the author's knowledge—not known and, given the variety of choices for MPC design, seemingly appear to be difficult. This paper analyzes the convergence of NMPC of continuous-time systems in time-invariant settings. We introduce the receding-horizon Hamiltonian—i.e., the value of the optimal control Hamiltonian along the sequence of OCPs—as a novel tool for stability/convergence analysis. Based on the assumption of strict dissipativity, we prove necessary and sufficient conditions for asymptotic convergence of the closed loop to the optimal steady state. Numerical results obtained for the Van de Vusse reactor illustrate the proposed conditions.Publicationtype: Conference Paper19 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , A dissipativity characterization of velocity turnpikes in optimal control problems for mechanical systemsTurnpikes have recently gained significant research interest in optimal control, since they allow for pivotal insights into the structure of solutions to optimal control problems. So far, mainly steady state solutions which serve as optimal operation points, are studied. This is in contrast to time-varying turnpikes, which are in the focus of this paper. More concretely, we analyze symmetry-induced velocity turnpikes, i.e. controlled relative equilibria, called trim primitives, which are optimal operation points regarding the given cost criterion. We characterize velocity turnpikes by means of dissipativity inequalities. Moreover, we study the equivalence between optimal control problems and steady-state problems via the corresponding necessary optimality conditions. An academic example is given for illustration.Publicationtype: Conference Paper30 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Inferring the adjoint turnpike property from the primal turnpike propertyThis paper investigates an interval turnpike result for the adjoints/costates of finite-and infinite-dimensional nonlinear optimal control problems under the assumption of an interval turnpike on states and controls. We consider stabilizable dynamics governed by a generator of a semigroup with finite-dimensional unstable part satisfying a spectral decomposition condition and show the desired turnpike property under continuity assumptions on the first-order optimality conditions. We further provide a numerical example with a semilinear heat equation to illustrate the results.Publicationtype: Conference Paper12 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Minimizing the energy supply of infinite-dimensional linear port-Hamiltonian systems(IFAC, 2021); ; ; ; We consider the problem of minimizing the supplied energy of infinite-dimensional linear port-Hamiltonian systems and prove that optimal trajectories exhibit the turnpike phenomenon towards certain subspaces induced by the dissipation of the dynamics. The theoretical foundations are illustrated by means of numerical examples concerning a Timoshenko beam and the heat equation.Publicationtype: Conference Paper14 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Turnpike properties in discrete-time mixed-integer optimal controlThis note discusses properties of parametric discrete-time Mixed-Integer Optimal Control Problems (MIOCPs) as they often arise in model predictive control with discrete controls. We argue that, in want for a handle on similarity properties of parametric MIOCPs, the turnpike phenomenon known in optimal control is helpful. We provide sufficient turnpike conditions based on a dissipativity notion of MIOCPs, and we prove that the turnpike phenomenon allows specific and accurate guesses for the discrete controls. We also derive an easily checkable sufficient condition for dissipativity of linear-quadratic MIOCPs. Moreover, we show how the turnpike property can be used to derive efficient node-weighted branch-and-bound schemes tailored to parametric MIOCPs. We draw upon numerical examples to illustrate our findings.Publicationtype: Journal Article4 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Towards velocity turnpikes in optimal control of mechanical systemsThe paper proposes first steps towards the formalization and characterization of time-varying turnpikes in optimal control of mechanical systems. We propose the concepts of velocity steady states, which can be considered as partial steady states, and hyperbolic velocity turnpike properties for analysis and control. We show for a specific example that, for all finite horizons, both the (essential part of the) optimal solution and the orbit of the time-varying turnpike correspond to (optimal) trim solutions. Hereby, the present paper appears to be the first to combine the concepts of trim primitives and time-varying turnpike properties.Publicationtype: Conference Paper8