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    Linking Approximate Dynamic Programming, Predictive Control, and Dissipativity: Novel Adaptive Control Approaches for Nonlinear Systems
    Approximate Dynamic Programming (ADP), Model Predictive Control (MPC), and dissipativity are established concepts, which are closely related through optimality considerations. Indeed, there is mature understanding of many aspects of MPC such as, e.g., nominal stability and performance. Beyond nominal considerations, robust designs are often intrinsically conservative and stochastic methods tend to induce a substantial mathematical overhead. Hence, in the face of uncertainty, control approaches for nonlinear systems which allow for online adaptation of models, feedback laws, or both are of growing interest. In the context of MPC, ADP concepts are promising guiding principles for online adaptation due to their close mutual relation. Yet, established analysis techniques for MPC often use upper-bounds on the cost-to-go (e.g. in tracking MPC with terminal ingredients) or they rely on lower-bounds on the cost-to-go (e.g. in dissipativity-based designs). Conceptually, we aim to broaden the scope, i.e., we consider terminal penalties that are mere approximations of the cost-to-go. Specifically, this project explores new approaches to adaptive control of nonlinear systems linking ADP, MPC, and dissipativity. We focus on two fundamental research questions: i) How to leverage ADP for the design of MPC controllers for nonlinear systems subject to model uncertainty? ii) How to exploit dissipativity concepts in ADP? The investigations with respect to i) target the understanding of model uncertainty in ADP and the analysis of approximations errors of ADP. For this, we investigate the interplay of closed-loop stability and approximation errors and the effect of model errors on ADP control policies. In turn, these considerations provide the basis to analyze the robustness of MPC wherein the terminal penalty is not an upper-bound on the cost-to-go but a mere approximation of it. Moreover, we are interested in analyzing the closed-loop performance of MPC with approximated cost-to-go. At the core of our investigations of ii) is the deep relation between optimal control and the dissipativity notion of open systems. Specifically, we leverage the links between storage functions needed for dissipativity and infinite-horizon value functions of optimal control problems to develop novel schemes for dissipativity-informed online ADP. The helpful relations between storage and value functions include gradient approximation properties and the fact that the storage provides a lower bound on the value function. Moreover, we combine the insights of our investigations on i) with the above towards dissipativity-informed online adaptation in MPC. Our findings are validated on case studies from biogas reactors, which is subject to growing research attention due to the increasing need for renewable energy sources.
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    Exploiting nonlinear port-Hamiltonian structures for optimization-based and data-driven control
    Port-Hamiltonian (pH) systems are ubiquitous in engineering. Application domains encompass mechanics, electrical circuits, thermodynamics, and multi-energy systems to name but a few. Key advantages of the pH perspective are balance equations for physical quantities like energy and entropy, and closedness of pH systems under power-conserving interconnections. This project concerns optimal control of nonlinear port-Hamiltonian systems. We consider dissipative, reversible, and irreversible pH systems and we investigate physically-motivated optimal control problems (OCPs), e.g., state transition with minimal energy supply and/or entropy growth. We exploit the port-Hamiltonian structure intertwined in dynamics and cost functions to thoroughly analyze and to efficiently solve such OCPs. To this end, we proceed in three main steps. First, we analyze port-Hamiltonian OCPs using dissipativity-based methods invoking the nonlinear balance equations for energy and entropy. Furthermore, we consider multi-objective OCPs in view of minimizing the supplied energy and the generated entropy while simultaneously maximizing a produced quantity of interest. Finally, we broaden the analysis towards optimal control of cyclo-passive port-Hamiltonian systems. Second, we tailor Newton homotopy methods to solve the intrinsically-motivated but singular port-Hamiltonian OCPs. To this end, we approximate the solution by a sequence of regularized problems. Further, we construct preconditioners consistent with the pH structure to efficiently conduct the Newton step. Third, we propose novel variants of SINDy (Sparse Identification of Nonlinear Dynamics) and eDMD (extended Dynamic Mode Decomposition) tailored to nonlinear pH systems: On the one hand, we aim to reduce data requirements by leveraging the pH structures. On the other hand, we ensure that key quantities, e.g., the energy balance, are properly taken into account in the data-driven surrogate models while providing rigorous bounds on the approximation error. In conclusion, we analyze physically-motivated OCPs of pH systems and their numerical solution. We develop novel algorithms to efficiently control nonlinear pH systems, e.g., w.r.t. the required energy supply or data requirements, while providing guarantees on stability, numerical efficiency, and approximation accuracy of data-driven surrogate models.
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