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Wave Energy Converters (WECs) are designed to be deployed in arrays, usually in a limited space, to minimize the cost of installation, mooring, and maintenance. Control methods that attempt to maximize the harvested power often lead to power flow from the WEC to the ocean, at times, to maximize the overall harvested power from the ocean over a longer period. The Power Take-Off (PTO) units that can provide power to the ocean (reactive power) are usually more expensive and complex. In this work, an optimal control formulation is presented using Pontryagin’s minimum principle that aims to maximize the harvested energy subject to constraints on the maximum PTO force and power flow direction. An analytical formulation is presented for the optimal control of an array of WECs, assuming irregular wave input. Three variations of the developed control are tested: a formulation without power constraints, a formulation that only allows for positive power, and finally, a formulation that allows for finite reactive power. The control is compared with optimally tuned damping and bang–bang control. 

This paper develops a control co-design (CCD) framework to simultaneously optimize the spacecraft’s trajectory and onboard system (rocket engine) and quantify its benefit. An open-loop optimal control problem (two-finite burn Mars missions) is used as the benchmark, and the engine design considers the combustion equilibrium and nozzle geometry. The objective function is the fuel burn. The design variables are the trajectory control parameters (such as burn times, burn directions, and time of flight), initial fuel mass, and engine design parameters (such as throat area, mixture ratio, and chamber pressure). The constraints include the final velocities and positions of spacecraft. Single-point optimizations are conducted for three departure dates in May, July, and September 2020. A multipoint optimization is also performed to balance the engine performance for these dates with 49 design variables and 20 constraints. It is found that the CCD optimizations exhibit 22–28% more fuel burn reduction than the trajectory-only optimization with fixed engine parameters and 16–20% more fuel burn reduction than the decoupled trajectory-engine optimization. The proposed CCD optimization framework can be extended to more spacecraft trajectory control parameters and onboard systems and has the potential to design more efficient spacecraft missions. 

Wave energy converters typically use various control methods to extract energy from ocean waves. The objective of the control system is to optimize the energy extraction process, taking into account the dynamics of the system and the wave conditions. The task of deriving the optimal control laws of wave energy converter arrays for regular and irregular waves using the Pontryagin minimum principle was previously investigated in the literature. The result is a combination between the singular arc and bang-bang control laws. For irregular waves, some complexity arises due to the radiation state-space representation, which requires ignoring the hydrodynamic coupling terms related to the added mass and radiation-damping coefficients; this reduces the computational complexity of the control force but adversely affects the solution's accuracy. Also, the derived control laws are specific to a particular wave condition. Recently, the optimal control of a flexible buoy wave energy converter was derived using the convolution representation for the radiation force. In this work, the optimal control laws of flexible buoy wave energy converters are modified to simulate wave energy converter arrays; then, the results are compared to those obtained by dropping the hydrodynamic radiation coupling terms. Although using a convolution representation adds computational complexity to the optimal control problem, it generates an equation that is generic to any wave condition, can be used with any wave spectrum, and provides an expression for the switching condition.