## Publications

63 results found

Arslan A, Fantuzzi G, Craske J,
et al., 2021, Bounds on internally heated convection with fixed boundary heat flux, *Journal of Fluid Mechanics*, Vol: 992, Pages: R1-R1, ISSN: 0022-1120

We prove a new rigorous bound for the mean convective heat transport ⟨wT⟩, where w and T are the non-dimensional vertical velocity and temperature, in internally heated convection between an insulating lower boundary and an upper boundary with a fixed heat flux. The quantity ⟨wT⟩ is equal to half the ratio of convective to conductive vertical heat transport, and also to 12 plus the mean temperature difference between the top and bottom boundaries. An analytical application of the background method based on the construction of a quadratic auxiliary function yields ⟨wT⟩≤12(12+13√)−1.6552R−(1/3) uniformly in the Prandtl number, where R is the non-dimensional control parameter measuring the strength of the internal heating. Numerical optimisation of the auxiliary function suggests that the asymptotic value of this bound and the −1/3 exponent are optimal within our bounding framework. This new result halves the best existing (uniform in R) bound (Goluskin, Internally Heated Convection and Rayleigh–Bénard Convection, Springer, 2016, table 1.2), and its dependence on R is consistent with previous conjectures and heuristic scaling arguments. Contrary to physical intuition, however, it does not rule out a mean heat transport larger than 12 at high R, which corresponds to the top boundary being hotter than the bottom one on average.

Arslan A, Fantuzzi G, Craske J,
et al., 2021, Bounds on heat transport for convection driven by internal heating, *Journal of Fluid Mechanics*, Vol: 919, Pages: 1-34, ISSN: 0022-1120

The mean vertical heat transport ⟨wT⟩ in convection between isothermal plates driven by uniform internal heating is investigated by means of rigorous bounds. These are obtained as a function of the Rayleigh number R by constructing feasible solutions to a convex variational problem, derived using a formulation of the classical background method in terms of quadratic auxiliary functions. When the fluid's temperature relative to the boundaries is allowed to be positive or negative, numerical solution of the variational problem shows that best previous bound ⟨wT⟩≤1/2 can only be improved up to finite R. Indeed, we demonstrate analytically that ⟨wT⟩≤2−21/5R1/5 and therefore prove that ⟨wT⟩<1/2 for R<65536. However, if the minimum principle for temperature is invoked, which asserts that internal temperature is at least as large as the temperature of the isothermal boundaries, then numerically optimised bounds are strictly smaller than 1/2 until at least R=3.4×105. While the computational results suggest that the best bound on ⟨wT⟩ approaches 1/2 asymptotically from below as R→∞, we prove that typical analytical constructions cannot be used to prove this conjecture.

Artola M, Goizueta N, Wynn A,
et al., 2021, Aeroelastic control and estimation with a minimal nonlinear modal description, *AIAA Journal: devoted to aerospace research and development*, Vol: 59, Pages: 2697-2713, ISSN: 0001-1452

Modal-based, nonlinear Moving Horizon Estimation (MHE) and Model Predictive Control(MPC) strategies for very flexible aeroelastic systems are presented. They are underpinned by an aeroelastic model built from a 1D intrinsic (based on strains and velocities) description of geometrically-nonlinear beams and an unsteady Vortex Lattice aerodynamic model. Construction of a nonlinear, modal-based, reduced order model of the aeroelastic system, employing a state-space realisation of the linearised aerodynamics around an arbitrary reference point, allows us to capture the main nonlinear geometrical couplings at a very low computational cost. Embedding this model in both MHE and MPC strategies, which solve the system continuous-time adjoints efficiently to compute sensitivities, lays the foundations for real-time estimation and control of highly flexible aeroelastic systems. Finally, the performance and versatility of the framework operating in the nonlinear regime is demonstrated on two very flexible wing models, with notably different dynamics, and on two different control setups: a gust-load alleviation problem on a very high aspect ratio wing with slower dynamics, which involves substantial deflections; and flutter suppression on a flexible wing with significantly faster dynamics, where an unconventional nonlinear stabilisation mechanism is unveiled.

Artola M, Wynn A, Palacios R, 2021, Modal-based model predictive control of multibody very flexible structures, 21st IFAC World Congress on Automatic Control - Meeting Societal Challenges, Publisher: Elsevier, Pages: 7472-7478, ISSN: 2405-8963

A model predictive control strategy for flexible multibody structures undergoing large deformations is presented. The dynamics of such structures are highly nonlinear, with local effects introduced by the joint constraints and distributed effects arising from the structure’s increased flexibility, from which arbitrary large deflections and rotations can be expected. A modal-based nonlinear reduced order model of an intrinsic description (based on velocities and strains) of geometrically-exact beams is used to underpin the internal model. This low-order model, constructed using the linearised eigenfunctions of the constrained structures, is a set of nonlinear ordinary differential equations in time (i.e. no algebraic equations are present) thus facilitating analysis and demonstrating successful control. Numerical examples are presented based on a very flexible hinged two-link manipulator.

Artola M, Wynn A, Palacios R, 2021, Modal-based nonlinear model predictive control for 3D very flexible structures, *IEEE Transactions on Automatic Control*, ISSN: 0018-9286

In this paper a novel NMPC scheme is derived, which is tailored to the underlying structure of the intrinsic description of geometrically exact nonlinear beams (in which velocities and strains are primary variables). This is an important class of PDE models whose behaviour is fundamental to the performance of flexible structural systems (e.g., wind turbines, High-Altitude Long-Endurance aircraft). Furthermore, this class contains the much-studied Euler-Bernoulli and Timoshenko beam models, but has significant additional complexity (to capture 3D effects and arbitrarily large displacements) and requires explicit computation of rotations in the PDE dynamics to account for orientation-dependent forces such as gravity. A challenge presented by this formulation is that uncontrollable modes necessarily appear in any finite dimensional approximation to the PDE dynamics. We show, however, that an NMPC scheme can be constructed in which the error introduced by the uncontrollable modes can be explicitly controlled. Furthermore, in challenging numerical examples exhibiting considerable deformation and nonlinear effects, it is demonstrated that the asymptotic error can be made insignificant (from a practical perspective) usingour NMPC scheme and excellent performance is obtained evenwhen applied to a highly resolved numerical simulation of thePDEs. We also present a generalisation of Kelvin-Voigt dampingto the intrinsic description of geometrically-exact beams. Finally,special emphasis is placed on constructing a framework suitablefor real-time NMPC control, where the particular structure ofthe underlying PDEs is exploited to obtain both efficient finite-dimensional models and numerical schemes.

Goizueta N, Wynn A, Palacios R, 2021, Parametric Krylov-based order reduction of aircraft aeroelastic models, AIAA Scitech 2021 Forum, Publisher: American Institute of Aeronautics and Astronautics, Pages: 1-25

We present an interpolation-based scheme to obtain in real-time linearised aeroelastic models of very flexible aircraft at any flight condition within the flight envelope. First, in an offline phase, a library of reduced-order linear aeroelastic models is computed by finding the nonlinear equilibrium (trimmed aircraft) using a nonlinear, geometrically-exact beam formulation coupled with an Unsteady Vortex Lattice Method. Subsequent linearisation is achieved analytically and followed by reduction using Krylov subspace methods. The interpolation is performed directly on the reduced-order system matrices, which are projected onto a congruent coordinate system for compatibility. Any interpolation scheme of choice can be used, and the size of the models permits real-time computation.

Goizueta N, Wynn A, Palacios R, et al., 2021, Flutter prediction for a very flexible wing wind tunnel test, AIAA Scitech 2021 Forum, Pages: 1-17

Two different nonlinear aeroelastic tool sets, SHARPy and the Modal Rotation Method (MRM), have been employed to predict and design a wind tunnel flutter test campaign of a very flexible wing, the Pazy Wing, as part of the 3rd Aeroelastic Prediction Workshop. The first method, SHARPy, uses geometrically exact beams coupled with an Unsteady Vortex Lattice, which is linearised about a deformed configuration, reduced by means of Krylov subspaces and analysed to compute the stability boundaries of the wing. The MRM is based on structural modal data, from either beam models or finite element models, coupled with a doublet-lattice aerodynamic model from ZAERO of the straight wing configuration. The excellent agreement between numerical and experimental data for structural-only and static aeroelastic analyses paves the way for predicting the stability boundaries of the Pre-Pazy wing with sufficient confidence for the safe design of a flutter wind tunnel test campaign.

Artola M, Goizueta N, Wynn A, et al., 2021, Proof of concept for a hardware-in-the-loop nonlinear control framework for very flexible aircraft, AIAA Scitech 2021 Forum, Publisher: https://arc.aiaa.org/doi/abs/10.2514/6.2021-1392, Pages: 1-26

Nonlinear Moving Horizon Estimation (MHE) and Model Predictive Control (MPC) strategies for very flexible aircraft are presented. They are underpinned by a nonlinear reduced-order model built upon the structure’s natural modes of vibration. This internal model aims for a minimal realisation of the aircraft which retains sufficient information to enable efficient real-time estimation and control. It is based on a modal intrinsic description of geometrically-nonlinear beams and a linearised unsteady vortex lattice aerodynamic model. Numerical evidence has shown that models of this form are able to capture the main nonlinear geometrical couplings at a very low computational cost. This opens the door to MHE and MPC strategies, which are naturally more computationally demanding than other linear conventional strategies, but are more versatile and able to provide control in the usually neglected nonlinear regime. The proposed control framework is tested on models built in an in-house open-source nonlinear aeroelasticity simulation and analysis package, to emulate the controller performance on a realistic plant model. Very satisfactory results are obtained in a flutter suppression problem involving a very flexible clamped wing, where the nonlinearity of the problem is leveraged by the internal model to achieve stabilisation, and a payload drop control of a very flexible HALE aircraft.

Artola M, Wynn A, Palacios R, 2021, Generalized Kelvin–Voigt damping models for geometrically nonlinear beams, *AIAA Journal: devoted to aerospace research and development*, Vol: 59, Pages: 356-365, ISSN: 0001-1452

Strain-rate-based damping is investigated in the strong form of the intrinsic equations of three-dimensional geometrically exact beams. Kelvin–Voigt damping, often limited in the literature to linear or two-dimensional beam models, is generalized to the three-dimensional case, including rigid-body motions. The result is an elegant infinite-dimensional description of geometrically exact beams that facilitates theoretical analysis and sets the baseline for any chosen numerical implementation. In particular, the dissipation rates and equilibrium points of the system are derived for the most general case and for one in which a first-order approximation of the resulting damping terms is taken. Finally, numerical examples are given that validate the resulting model against a nonlinear damped Euler–Bernoulli beam (where detail is given on how an equivalent description using our intrinsic formulation is obtained) and support the analytical results of energy decay rates and equilibrium solutions caused by damping. Throughout the paper, the relevance of damping higher-order terms, arising from the geometrically exact description, to the accurate prediction of its effect on the dynamics of highly flexible structures is highlighted.

Muñoz-Simón A, Palacios R, Wynn A, 2020, Benchmarking different fidelities in wind turbine aerodynamics under yaw, The Science of Making Torque from Wind (TORQUE 2020), Publisher: IOP Publishing, Pages: 1-11, ISSN: 1742-6588

This paper analyses the aerodynamics of wind turbines under yaw with different modelling fidelities (BEM, BEM with skewed wake model, UVLM and LES-AL). First of all, models are compared in a zero-yaw case to demonstrate their accuracy in prediction of out-of-plane loads and the discrepancy of UVLM in the in-plane loads due to the lack of viscous drag. Secondly, the yaw aerodynamics are described through the advancing/retreating and skewed wake effects, which are appropriately captured by UVLM and LES-AL and lead to an incorrect prediction of the location of maximum and minimum loading along a revolution by BEM. Further, when a skew-wake model is included in BEM, it predicts the correct locations but exhibits overly large loading variations. These predictions are consistent for all yaw angles studied (γ = 10° − 30°). All solvers predict similar decrease of root-bending moments, rotor power and thrust coefficients up to a yaw angle of 10°. However, at larger yaw angles, BEM overpredicts this decrease of coefficients with the yaw angle due to the unsuccessful performance of yaw corrections as opposed to UVLM that inherently accounts for three-dimensional effects. This study demonstrates the need to use computational models that can account for three-dimensional effects in the computation of aerodynamic loads for yaw angles above 10°.

Zheng Y, Fantuzzi G, Papachristodoulou A,
et al., 2020, Chordal decomposition in operator-splitting methods for sparse semidefinite programs, *Mathematical Programming*, Vol: 180, Pages: 489-532, ISSN: 0025-5610

We employ chordal decomposition to reformulate a large and sparse semidefinite program (SDP), either in primal or dual standard form, into an equivalent SDP with smaller positive semidefinite (PSD) constraints. In contrast to previous approaches, the decomposed SDP is suitable for the application of first-order operator-splitting methods, enabling the development of efficient and scalable algorithms. In particular, we apply the alternating direction method of multipliers (ADMM) to solve decomposed primal- and dual-standard-form SDPs. Each iteration of such ADMM algorithms requires a projection onto an affine subspace, and a set of projections onto small PSD cones that can be computed in parallel. We also formulate the homogeneous self-dual embedding (HSDE) of a primal-dual pair of decomposed SDPs, and extend a recent ADMM-based algorithm to exploit the structure of our HSDE. The resulting HSDE algorithm has the same leading-order computational cost as those for the primal or dual problems only, with the advantage of being able to identify infeasible problems and produce an infeasibility certificate. All algorithms are implemented in the open-source MATLAB solver CDCS. Numerical experiments on a range of large-scale SDPs demonstrate the computational advantages of the proposed methods compared to common state-of-the-art solvers.

Fantuzzi G, Nobili C, Wynn A, 2020, New bounds on the vertical heat transport for Bénard-Marangoni convection at infinite Prandtl number, *Journal of Fluid Mechanics*, Vol: 885, Pages: R4-1-R4-12, ISSN: 0022-1120

We prove a new rigorous upper bound on the vertical heat transport for Bénard–Marangoni convection of a two- or three-dimensional fluid layer with infinite Prandtl number. Precisely, for Marangoni number 𝑀𝑎≫1 the Nusselt number 𝑁𝑢 is bounded asymptotically by 𝑁𝑢⩽const.×𝑀𝑎2/7(ln𝑀𝑎)−1/7 . Key to our proof are a background temperature field with a hyperbolic profile near the fluid’s surface and new estimates for the coupling between temperature and vertical velocity.

Artola M, Goizueta N, Wynn A, et al., 2020, Modal-based nonlinear estimation and control for highly flexible aeroelastic systems, AIAA Scitech 2020 Forum, Publisher: American Institute of Aeronautics and Astronautics, Pages: 1-23

Modal-based, nonlinear Moving Horizon Estimation (MHE) and Model Predictive Control (MPC) strategies for highly flexible aeroelastic systems are presented. The aeroelastic model is built from a 1D intrinsic (based on strains and velocities) description of geometrically-nonlinear beams and an unsteady Vortex Lattice aerodynamic model. Construction of a nonlinear modal-based reduced order model of the aeroelastic system, employing a state-space realisation of the linearised aerodynamics around an arbitrary equilibrium point, allows us to capture the main nonlinear geometrical couplings at a very low computational cost. Embedding this model in both MHE and MPC strategies, which solve the system's continuous-time adjoints efficiently to compute sensitivities, lays the foundations for real-time estimation and control of highly flexible aeroelastic systems.

Muñoz-Simón A, Wynn A, Palacios R, 2020, Unsteady and three-dimensional aerodynamic effects on wind turbine rotor loads, AIAA Scitech 2020 Forum, Publisher: American Institute of Aeronautics and Astronautics, Pages: 1-22

The paper investigates the use of vortex methods in the computation of important flows for wind turbine aerodynamics. These flows are characterised by inflow velocity unsteadiness, spatial variations and non-zero spanwise component. First, computational vortex methods are shown to match analytical solutions on simple geometries. Second, these geometries will be studied to reveal the effects of these flows on airfoil aerodynamics: unsteady damping and load reduction when subjected to side slip or spanwise inflow velocity variations. Finally, vortex methods will be compared with traditional BEM methods in full wind turbine configurations under real operating inflows such as shear, yaw and turbulence which are characterised by unsteadiness and three-dimensional effects. Vortex methods allow quantifying the error of BEM methods for these conditions.

Mahfoze O, Moody A, Wynn A,
et al., 2019, Reducing the skin-friction drag of a turbulent boundary-layer flow with low-amplitude wall-normal blowing within a Bayesian optimisation framework, *Physical Review Fluids*, Vol: 4, Pages: 094601-1-094601-23, ISSN: 2469-990X

A Bayesian optimisation framework is developed to optimise low-amplitude wall-normal blowing control of a turbulent boundary-layer flow. The Bayesian optimisation framework determines the optimum blowing amplitude and blowing coverage to achieve up to a 5% net-power saving solution within 20 optimisation iterations, requiring 20 Direct Numerical Simulations (DNS). The power input required to generate the low-amplitude wall-normal blowing is measured experimentally for two different types of blowing device, and is used in the simulations to assess control performance. Wall-normal blowing with amplitudes of less than 1% of the free-stream velocity generate a skin-friction drag reduction of up to 76% over the control region, with a drag reduction which persists for up to 650δ0 downstream of actuation (where δ0 is the boundary-layer thickness at the start of the simulation domain). It is shown that it is the slow spatial recovery of the turbulent boundary-layer flow downstream of control which generates the net-power savings in this study. The downstream recovery of the skin-friction drag force is decomposed using the Fukagata-Iwamoto-Kasagi (FIK) identity, which shows that the generation of the net-power savings is due to changes in contributions to both the convection and streamwise development terms of the turbulent boundary-layer flow.

Beit-Sadi M, Krol J, Wynn A, 2019, Data driven feature identification and sparse representation of turbulent flows, Eleventh International Symposium on Turbulence and Shear Flow Phenomena (TSFP11)

dentifying coherent structures of fluid flows is of greatimportance for reduced order modelling and flow control.Finding such structures in a turbulent flow, however, canbe challenging. A number of modal decomposition algo-rithms have been proposed in recent years which decom-pose snapshots of data into spatial modes, each associatedwith a single frequency and growth-rate, most prominentlydynamic mode decomposition (DMD). However, the num-ber of modes that DMD-like algorithms construct may beunrelated to the number of significant degrees of freedomof the underlying system. This provides a difficulty if onewants to create a low-order model of a flow. In this work,we present a method of post-processing DMD modes forextracting a small number of dynamically relevant modes.This is achieved by first ranking the DMD modes, then us-ing an iterative approach based on the graph-theoretic no-tion of maximal cliques to identify clusters of modes and,finally, by replacing each cluster with a single (pair of)modes.

Artola M, Wynn A, Palacios R, 2019, A nonlinear modal-based framework for low computational cost optimal control of 3D very flexible structures, 2019 18th European Control Conference (ECC), Publisher: Institute of Electrical and Electronics Engineers

A nonlinear modal-based reduced-order model, equipped with an efficient adjoint-sensitivity analysis, is presented as a low computational cost framework for optimal control of very flexible structures, with particular focus on efficiently computing finite rotations. Multiple shooting is shown to improve convergence of a highly nonlinear problem when compared to the single shooting case, with optimisation further accelerated via parallelisation, which suggests the presented approach may be employed for real-time control of very flexible structures.

Mahfoze OA, Wynn A, Whalley RD, et al., 2019, Bayesian optimisation of intermittent wall blowing in a turbulent boundary layer for net power saving

A Bayesian optimisation framework is used to optimise low-amplitude wall-normal blowing control of a turbulent boundary-layer (TBL) flow in order to achieve skin-friction drag reduction and net-power saving. The study is carried out using Direct Numerical Simulations (DNS) and Implicit Large Eddy Simulations (ILES). Control performance is assessed by using the power consumption from two different sets of experimental data from two different types of blowing device. The simulations demonstrate that wall-normal blowing control can generate a local skin-friction drag reduction of up to 75%, which persists far downstream of the control. This slow spatial recovery of the skin-friction coefficient back to its canonical counterpart can generate net-power savings up to 5% in the present study. When combined with DNS or ILES, Bayesian optimisation, with its fast convergence (within a dozen iterations with three parameters to optimise) is an ideal tool to find the optimal set of parameters to maximise net-power saving. The evolution of the skin-friction coefficient is decomposed using the Fukagata-Iwamoto-Kasagi (FIK) identity, which shows that the generation of the net-power savings is due to changes in contributions to both the convection and streamwise development terms of the turbulent boundary-layer flow.

Mahfoze OA, Wynn A, Whalley RD, et al., 2019, Bayesian optimisation of intermittent wall blowing in a turbulent boundary layer for net power saving

© 2019 International Symposium on Turbulence and Shear Flow Phenomena, TSFP. All rights reserved. A Bayesian optimisation framework is used to optimise low-amplitude wall-normal blowing control of a turbulent boundary-layer (TBL) flow in order to achieve skin-friction drag reduction and net-power saving. The study is carried out using Direct Numerical Simulations (DNS) and Implicit Large Eddy Simulations (ILES). Control performance is assessed by using the power consumption from two different sets of experimental data from two different types of blowing device. The simulations demonstrate that wall-normal blowing control can generate a local skin-friction drag reduction of up to 75%, which persists far downstream of the control. This slow spatial recovery of the skin-friction coefficient back to its canonical counterpart can generate net-power savings up to 5% in the present study. When combined with DNS or ILES, Bayesian optimisation, with its fast convergence (within a dozen iterations with three parameters to optimise) is an ideal tool to find the optimal set of parameters to maximise net-power saving. The evolution of the skin-friction coefficient is decomposed using the Fukagata-Iwamoto-Kasagi (FIK) identity, which shows that the generation of the net-power savings is due to changes in contributions to both the convection and streamwise development terms of the turbulent boundary-layer flow.

Brackston R, Wynn A, Stumpf MPH, 2018, Construction of quasi-potentials for stochastic dynamical systems: An optimization approach, *Physical Review E*, Vol: 98, ISSN: 1539-3755

The construction of effective and informative landscapes for stochastic dynamical systems has proven a long-standing and complex problem. In many situations, the dynamics may be described by a Langevin equation while constructing a landscape comes down to obtaining the quasipotential, a scalar function that quantifies the likelihood of reaching each point in the state space. In this work we provide a novel method for constructing such landscapes by extending a tool from control theory: the sum-of-squares method for generating Lyapunov functions. Applicable to any system described by polynomials, this method provides an analytical polynomial expression for the potential landscape, in which the coefficients of the polynomial are obtained via a convex optimization problem. The resulting landscapes are based on a decomposition of the deterministic dynamics of the original system, formed in terms of the gradient of the potential and a remaining “curl” component. By satisfying the condition that the inner product of the gradient of the potential and the remaining dynamics is everywhere negative, our derived landscapes provide both upper and lower bounds on the true quasipotential; these bounds becoming tight if the decomposition is orthogonal. The method is demonstrated to correctly compute the quasipotential for high-dimensional linear systems and also for a number of nonlinear examples.

Wang Y, Wynn A, Palacios R, 2018, Nonlinear aeroelastic control of very flexible aircraft using modelupdating, *Journal of Aircraft*, Vol: 55, Pages: 1551-1563, ISSN: 0021-8669

A nonlinear modeling strategy is introduced to drive online optimization-based controllers for trajectory tracking and stabilization of very flexible aircraft. This is achieved thanks to a compact residualized modal projection of the aeroelastic equations of motion, including large (geometrically nonlinear) wing deflections, that is used to describe the vehicle dynamics. This study shows that a model-predictive control system can then be built with an internal model based on successive linearization of the equations of motion around the instantaneous vehicle geometry. Significant improvements in stability against large disturbances and maneuverability are demonstrated on numerical simulations of a very flexible flying wing when geometrically nonlinear effects are included in the internal model of the controller.

Qi P, Zhao X, Wang Y, et al., 2018, Automatic Landing Control of a Very Flexible Flying Wing, 2018 Annual American Control Conference (ACC), Publisher: IEEE

Qi P, Zhao X, Wang Y,
et al., 2018, Aeroelastic and Trajectory Control of High Altitude Long Endurance Aircraft, *IEEE Transactions on Aerospace and Electronic Systems*, ISSN: 0018-9251

IEEE We investigate the aeroelastic and trajectory control of an High Altitude Long Endurance (HALE) aircraft model in the presence of gust and turbulence disturbances. The model is derived from geometrically-nonlinear beam theory using intrinsic degrees of freedom and linear unsteady aerodynamics, which results in a coupled structural dynamics, aerodynamics and flight dynamics description. The control design employs a two-loop PI/LADRC (linear active disturbance rejection control) and H<formula><tex>$\infin$</tex></formula> control scheme in both the longitudinal and lateral channels, based on a reduced-order linearised model. In each channel, the outer loop (position control) employs a PI/LADRC technique to track the desired flight routes and generate attitude command to the inner loop, while the inner loop (attitude control) uses H<formula><tex>$\infin$</tex></formula> control to track the attitude command generated from the outer loop and computes the control inputs to the corresponding control surfaces. A Particle Swarm Optimization algorithm is employed for parameter optimization of the weighting matrices in the H<formula><tex>$\infin$</tex></formula> control design. The simulation tests conducted on the full-order nonlinear model show that the aeroelastic and trajectory control system achieves good performance with respect to robustness, trajectory tracking and disturbance rejection

Brackston RD, Wynn A, Morrison JF, 2018, Modelling and feedback control of vortex shedding for drag reduction of a turbulent bluff body wake, *International Journal of Heat and Fluid Flow*, Vol: 71, Pages: 127-136, ISSN: 0142-727X

Three-dimensional bluff body wakes are of key importance due to their relevance to the automotive industry. Such wakes have both large pressure drag and a number of coherent flow structures associated with them. Depending on the geometry, these structures may include both a bistability resulting from a spatial symmetry breaking (SB), and a quasi-periodic vortex shedding. The authors have recently shown that the bistability may be modelled by a Langevin equation and that this model enables the design of a feedback control strategy that efficiently reduces the drag through suppression of asymmetry. In this work the stochastic modelling approach is extended to the vortex shedding, capturing qualitatively both the forced and unforced behaviour. A control strategy is then presented that makes use of the frequency response of the wake, and aims to reduce the measured fluctuations associated with the vortex shedding. The strategy proves to be effective at suppressing fluctuations within specific frequency ranges but, due to amplification of disturbances at other frequencies, is unable to give drag reduction.

Broughton-Venner JJ, Wynn A, Palacios R, 2018, Aeroservoelastic Optimisation of Aerofoils with Compliant Flaps via Reparameterization and Variable Selection, 58th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Publisher: AMER INST AERONAUTICS ASTRONAUTICS, Pages: 1146-1157, ISSN: 0001-1452

Fantuzzi G, Pershin A, Wynn A, 2018, Bounds on heat transfer for Bénard-Marangoni convection at infinite Prandtl number, *Journal of Fluid Mechanics*, Vol: 837, Pages: 562-596, ISSN: 0022-1120

The vertical heat transfer in Bénard–Marangoni convection of a fluid layer with infinite Prandtl number is studied by means of upper bounds on the Nusselt number 𝑁𝑢 as a function of the Marangoni number 𝑀𝑎 . Using the background method for the temperature field, it has recently been proved by Hagstrom & Doering (Phys. Rev. E, vol. 81, 2010, art. 047301) that 𝑁𝑢⩽0.838𝑀𝑎2/7 . In this work we extend previous background method analysis to include balance parameters and derive a variational principle for the bound on 𝑁𝑢 , expressed in terms of a scaled background field, that yields a better bound than Hagstrom & Doering’s formulation at a given 𝑀𝑎 . Using a piecewise-linear, monotonically decreasing profile we then show that 𝑁𝑢⩽0.803𝑀𝑎2/7 , lowering the previous prefactor by 4.2 %. However, we also demonstrate that optimisation of the balance parameters does not affect the asymptotic scaling of the optimal bound achievable with Hagstrom & Doering’s original formulation. We subsequently utilise convex optimisation to optimise the bound on 𝑁𝑢 over all admissible background fields, as well as over two smaller families of profiles constrained by monotonicity and convexity. The results show that 𝑁𝑢⩽𝑂(𝑀𝑎2/7(ln𝑀𝑎)−1/2) when the background field has a non-monotonic boundary layer near the surface, while a power-law bound with exponent 2/7 is optimal within the class of monotonic background fields. Further analysis of our upper-bounding principle reveals the role of non-monotonicity, and how it may be exploited in a rigorous mathematical argument.

Fantuzzi G, Wynn A, Goulart P,
et al., 2017, Optimization with affine homogeneous quadratic integral inequality constraints, *IEEE Transactions on Automatic Control*, Vol: 62, Pages: 6221-6236, ISSN: 0018-9286

We introduce a new technique to optimize a linear cost function subject to an affine homogeneous quadratic integral inequality, i.e. the requirement that a homogeneous quadratic integral functional affine in the optimization variables is non-negative over a space of functions defined by homogeneous boundary conditions. Such problems arise in control and stability or input-to-state/output analysis of systems governed by partial differential equations (PDEs), particularly fluid dynamical systems. We derive outer approximations for the feasible set of a homogeneous quadratic integral inequality in terms of linear matrix inequalities (LMIs), and show that a convergent sequence of lower bounds for the optimal cost can be computed with a sequence of semidefinite programs (SDPs). We also obtain inner approximations in terms of LMIs and sum-of-squares constraints, so upper bounds for the optimal cost and strictly feasible points for the integral inequality can be computed with SDPs. We present QUINOPT, an open-source add-on to YALMIP to aid the formulation and solution of our SDPs, and demonstrate our techniques on problems arising from the stability analysis of PDEs.

Zheng Y, Fantuzzi G, Papachristodoulou A, et al., 2017, Fast ADMM for homogeneous self-dual embedding of sparse SDPs, 20th IFAC World Congress, Publisher: Elsevier, Pages: 8411-8416, ISSN: 1474-6670

We propose an efficient first-order method, based on the alternatingdirection method of multipliers (ADMM), to solve the homogeneous self-dual embedding problem for a primal-dual pair of semidefinite programs (SDPs) with chordal sparsity. Using a series of block eliminations, the per-iteration cost of our method is the same as applying a splitting method to the primal or dual alone. Moreover, our approach is more efficient than other first-order methods for generic sparse conic programs since we work with smaller semidefinite cones. In contrast to previous first-order methods that exploit chordal sparsity, our algorithm returns both primal and dual solutions when available, and a certificate of infeasibility otherwise. Our techniques are implemented in the open-source MATLAB solver CDCS. Numerical experiments on three sets of benchmark problems from the library SDPLIB show speed-ups compared to some common state-of-the-art software packages.

Krol J, Wynn A, 2017, Dynamic reconstruction and data reconstruction for subsampled or irregularly sampled data, *Journal of Fluid Mechanics*, Vol: 825, Pages: 133-166, ISSN: 0022-1120

The Nyquist–Shannon criterion indicates the sample rate necessary to identify information with particular frequency content from a dynamical system. However, in experimental applications such as the interrogation of a flow field using particle image velocimetry (PIV), it may be impracticable or expensive to obtain data at the desired temporal resolution. To address this problem, we propose a new approach to identify temporal information from undersampled data, using ideas from modal decomposition algorithms such as dynamic mode decomposition (DMD) and optimal mode decomposition (OMD). The novel method takes a vector-valued signal, such as an ensemble of PIV snapshots, sampled at random time instances (but at sub-Nyquist rate) and projects onto a low-order subspace. Subsequently, dynamical characteristics, such as frequencies and growth rates, are approximated by iteratively approximating the flow evolution by a low-order model and solving a certain convex optimisation problem. The methodology is demonstrated on three dynamical systems, a synthetic sinusoid, the cylinder wake at Reynolds number and turbulent flow past the axisymmetric bullet-shaped body. In all cases the algorithm correctly identifies the characteristic frequencies and oscillatory structures present in the flow.

Qi P, Wang Y, Zhao X, et al., 2017, Trajectory Control of a Very Flexible Flying Wing, American Control Conference (ACC), Publisher: IEEE, Pages: 4480-4485, ISSN: 0743-1619

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