An Optimal Control Formulation of Pulse-Based Control Using Koopman Operator

In many applications, and in systems/synthetic biology, in particular, it is desirable to compute control policies that force the trajectory of a bistable system from one equilibrium (the initial point) to another equilibrium (the target point), or in other words to solve the switching problem. It was recently shown that, for monotone bistable systems, this problem admits easy-to-implement open-loop solutions in terms of temporal pulses (i.e., step functions of fixed length and fixed magnitude). In this paper, we develop this idea further and formulate a problem of convergence to an equilibrium from an arbitrary initial point. We show that this problem can be solved using a static optimization problem in the case of monotone systems. Changing the initial point to an arbitrary state allows to build closed-loop, event-based or open-loop policies for the switching/convergence problems. In our derivations we exploit the Koopman operator, which offers a linear infinite-dimensional representation of an autonomous nonlinear system. One of the main advantages of using the Koopman operator is the powerful computational tools developed for this framework. Besides the presence of numerical solutions, the switching/convergence problem can also serve as a building block for solving more complicated control problems and can potentially be applied to non-monotone systems. We illustrate this argument on the problem of synchronizing cardiac cells by defibrillation. Potentially, our approach can be extended to problems with different parametrizations of control signals since the only fundamental limitation is the finite time application of the control signal.Comment: corrected typo

Reduced Order Optimal Control of the Convective FitzHugh-Nagumo Equation

In this paper, we compare three model order reduction methods: the proper orthogonal decomposition (POD), discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD) for the optimal control of the convective FitzHugh-Nagumo (FHN) equations. The convective FHN equations consists of the semi-linear activator and the linear inhibitor equations, modeling blood coagulation in moving excitable media. The semilinear activator equation leads to a non-convex optimal control problem (OCP). The most commonly used method in reduced optimal control is POD. We use DEIM and DMD to approximate efficiently the nonlinear terms in reduced order models. We compare the accuracy and computational times of three reduced-order optimal control solutions with the full order discontinuous Galerkin finite element solution of the convection dominated FHN equations with terminal controls. Numerical results show that POD is the most accurate whereas POD-DMD is the fastest

Modeling Nonlinear Control Systems via Koopman Control Family: Universal Forms and Subspace Invariance Proximity

This paper introduces the Koopman Control Family (KCF), a mathematical framework for modeling general discrete-time nonlinear control systems with the aim of providing a solid theoretical foundation for the use of Koopman-based methods in systems with inputs. We demonstrate that the concept of KCF can completely capture the behavior of nonlinear control systems on a (potentially infinite-dimensional) function space. By employing a generalized notion of subspace invariance under the KCF, we establish a universal form for finite-dimensional models, which encompasses the commonly used linear, bilinear, and linear switched models as specific instances. In cases where the subspace is not invariant under the KCF, we propose a method for approximating models in general form and characterize the model's accuracy using the concept of invariance proximity. The proposed framework naturally lends itself to the incorporation of data-driven methods in modeling and control.Comment: 16 page

Quantum Computing for Fusion Energy Science Applications

This is a review of recent research exploring and extending present-day quantum computing capabilities for fusion energy science applications. We begin with a brief tutorial on both ideal and open quantum dynamics, universal quantum computation, and quantum algorithms. Then, we explore the topic of using quantum computers to simulate both linear and nonlinear dynamics in greater detail. Because quantum computers can only efficiently perform linear operations on the quantum state, it is challenging to perform nonlinear operations that are generically required to describe the nonlinear differential equations of interest. In this work, we extend previous results on embedding nonlinear systems within linear systems by explicitly deriving the connection between the Koopman evolution operator, the Perron-Frobenius evolution operator, and the Koopman-von Neumann evolution (KvN) operator. We also explicitly derive the connection between the Koopman and Carleman approaches to embedding. Extension of the KvN framework to the complex-analytic setting relevant to Carleman embedding, and the proof that different choices of complex analytic reproducing kernel Hilbert spaces depend on the choice of Hilbert space metric are covered in the appendices. Finally, we conclude with a review of recent quantum hardware implementations of algorithms on present-day quantum hardware platforms that may one day be accelerated through Hamiltonian simulation. We discuss the simulation of toy models of wave-particle interactions through the simulation of quantum maps and of wave-wave interactions important in nonlinear plasma dynamics.Comment: 42 pages; 12 figures; invited paper at the 2021-2022 International Sherwood Fusion Theory Conferenc

TOWARDS OPTIMAL OPERATION AND CONTROL OF EMERGING ELECTRIC DISTRIBUTION NETWORKS

The growing integration of power-electronics converters enabled components causes low inertia in the evolving electric distribution networks, which also suffer from uncertainties due to renewable energy sources, electric demands, and anomalies caused by physical or cyber attacks, etc. These issues are addressed in this dissertation. First, a virtual synchronous generator (VSG) solution is provided for solar photovoltaics (PVs) to address the issues of low inertia and system uncertainties. Furthermore, for a campus AC microgrid, coordinated control of the PV-VSG and a combined heat and power (CHP) unit is proposed and validated. Second, for islanded AC microgrids composed of SGs and PVs, an improved three-layer predictive hierarchical power management framework is presented to provide economic operation and cyber-physical security while reducing uncertainties. This scheme providessuperior frequency regulation capability and maintains low system operating costs. Third, a decentralized strategy for coordinating adaptive controls of PVs and battery energy storage systems (BESSs) in islanded DC nanogrids is presented. Finally, for transient stability evaluation (TSE) of emerging electric distribution networks dominated by EV supercharging stations, a data-driven region of attraction (ROA) estimation approach is presented. The proposed data-driven method is more computationally efficient than traditional model-based methods, and it also allows for real-time ROA estimation for emerging electric distribution networks with complex dynamics