Theory and methods to obtain reduced order models by moment matching from input/output data are presented. Algorithms for the estimation of the moments of linear and nonlinear systems are proposed. The estimates are exploited to construct families of reduced order models. These models asymptotically match the moments of the unknown system to be reduced. Conditions to enforce additional properties, e.g. matching with prescribed eigenvalues, upon the reduced order model are provided and discussed. The computational complexity of the algorithms is analyzed and their use is illustrated by two examples: we compute converging reduced order models for a linear system describing the model of a building and we provide, exploiting an approximation of the moment, a nonlinear planar reduced order model for a nonlinear DC-to-DC converter

A B S T R A C TWave Energy Converters (WECs) have to be controlled to ensure maximum energy extraction from waves while considering, at the same time, physical constraints on the motion of the real device and actuator characteristics. Since the control objective for WECs deviates significantly from the traditional reference ''tracking'' problem in classical control, the specification of an optimal control law, that optimises energy absorption under different sea-states, is non-trivial. Different approaches based on optimal control methodologies have been proposed for this energy-maximising objective, with considerable diversity on the optimisation formulation. Recently, a novel mathematical tool to compute the steady-state response of a system has been proposed: the moment-based phasor transform. This mathematical framework is inspired by the theory of model reduction by moment-matching and considers both continuous and discontinuous inputs, depicting an efficient and closed-form method to compute such a steady-state behaviour. This study approaches the design of an energy-maximising optimal controller for a single WEC device by employing the moment-based phasor transform, describing a pioneering application of this novel moment-matching mathematical scheme to an optimal control problem. Under this framework, the energy-maximising optimal control formulation is shown to be a strictly concave quadratic program, allowing the application of well-known efficient real-time algorithms. (N. Faedo). minimise the risk of damage, such an optimisation strategy must take into account the physical limitations of the whole conversion chain. Such an optimisation procedure can be achieved by designing an optimal controller that accomplishes such objectives.A considerable number of optimal control formulations and methods have been studied and developed to maximise the energy extraction process from WECs, with extensive reviews available, for example in Ringwood, Bacelli, and Fusco (2014). One particular popular wave energy control strategy is Model Predictive Control (MPC). The success of MPC on the energy-maximising control is mainly due to its ability to handle physical constraints systematically and within a finite horizon optimisation process. While MPC applied to WECs also involves a mathematical model, a typical receding horizon strategy, and can deal with system constraints, the objective function contrasts significantly with the one related to the usual set-point tracking objective. Rather, a converted energy-maximising objective, consistent with the definition https://doi.

Mathematical models are at the core of modern science and technology. An accurate description of behaviors, systems and processes often requires the use of complex models which are difficult to analyze and control. To facilitate analysis of and design for complex systems, model reduction theory and tools allow determining “simpler” models which preserve some of the features of the underlying complex description. A large variety of techniques, which can be distinguished depending on the features which are preserved in the reduction process, has been proposed to achieve this goal. One such a method is the moment matching approach. This monograph focuses on the problem of model reduction by moment matching for nonlinear systems. The central idea of the method is the preservation, for a prescribed class of inputs and under some technical assumptions, of the steady-state output response of the system to be reduced. We present the moment matching approach from this vantage point, covering the problems of model reduction for nonlinear systems, nonlinear time-delay systems, data-driven model reduction for nonlinear systems and model reduction for “discontinuous” input signals. Throughout the monograph linear systems, with their simple structure and strong properties, are used as a paradigm to facilitate understanding of the theory and provide foundation of the terminology and notation. The text is enriched by several numerical examples, physically motivated examples and with connections to well-established notions and tools, such as the phasor transform

Linear dynamics are virtually always assumed when designing optimal controllers for wave energy converters (WECs), motivated by both their simplicity and computational convenience. Nevertheless, unlike traditional tracking control applications, the assumptions under which the linearisation of WEC models is performed are challenged by the energy-maximising controller itself, which intrinsically enhances device motion to maximise power extraction from incoming ocean waves. In this paper, we present a moment-based energy-maximising control strategy for WECs subject to nonlinear dynamics. We develop a framework under which the objective function (and system variables) can be mapped to a finite-dimensional tractable nonlinear program, which can be efficiently solved using state-of-the-art nonlinear programming solvers. Moreover, we show that the objective function belongs to a class of generalised convex functions when mapped to the moment-domain, guaranteeing the existence of a global energy-maximising solution, and giving explicit conditions for when a local solution is, effectively, a global maximiser. The performance of the strategy is demonstrated through a case study, where we consider (state and input-constrained) energymaximisation for a state-of-the-art CorPower-like WEC, subject to different hydrodynamic nonlinearities.

The persistence of excitation of signals generated by time-invariant, autonomous, linear, and nonlinear systems is studied using a geometric approach. A rank condition is shown to be equivalent, under certain assumptions, to the persistence of excitation of the solutions of the class of systems considered, both in the discrete-time and in the continuous-time settings. The rank condition is geometric in nature and can be checked a priori, i.e. without knowing explicitly the solutions of the system, for almost periodic systems. The significance of the ideas and tools presented is illustrated by means of simple examples. Applications to model reduction from input-output data and stability analysis of skew-symmetric systems are also discussed.