1,335 research outputs found
Expensive control of long-time averages using sum of squares and Its application to a laminar wake flow
The paper presents a nonlinear state-feedback con- trol design approach for long-time average cost control, where the control effort is assumed to be expensive. The approach is based on sum-of-squares and semi-definite programming techniques. It is applicable to dynamical systems whose right-hand side is a polynomial function in the state variables and the controls. The key idea, first described but not implemented in (Chernyshenko et al. Phil. Trans. R. Soc. A, 372, 2014), is that the difficult problem of optimizing a cost function involving long-time averages is replaced by an optimization of the upper bound of the same average. As such, controller design requires the simultaneous optimization of both the control law and a tunable function, similar to a Lyapunov function. The present paper introduces a method resolving the well-known inherent non-convexity of this kind of optimization. The method is based on the formal assumption that the control is expensive, from which it follows that the optimal control is small. The resulting asymptotic optimization problems are convex. The derivation of all the polynomial coefficients in the controller is given in terms of the solvability conditions of state-dependent linear and bilinear inequalities. The proposed approach is applied to the problem of designing a full-information feedback controller that mitigates vortex shedding in the wake of a circular cylinder in the laminar regime via rotary oscillations. Control results on a reduced-order model of the actuated wake and in direct numerical simulation are reported
Sum-of-Squares approach to feedback control of laminar wake flows
A novel nonlinear feedback control design methodology for incompressible
fluid flows aiming at the optimisation of long-time averages of flow quantities
is presented. It applies to reduced-order finite-dimensional models of fluid
flows, expressed as a set of first-order nonlinear ordinary differential
equations with the right-hand side being a polynomial function in the state
variables and in the controls. The key idea, first discussed in Chernyshenko et
al. 2014, Philos. T. Roy. Soc. 372(2020), is that the difficulties of treating
and optimising long-time averages of a cost are relaxed by using the
upper/lower bounds of such averages as the objective function. In this setting,
control design reduces to finding a feedback controller that optimises the
bound, subject to a polynomial inequality constraint involving the cost
function, the nonlinear system, the controller itself and a tunable polynomial
function. A numerically tractable approach to the solution of such optimisation
problems, based on Sum-of-Squares techniques and semidefinite programming, is
proposed.
To showcase the methodology, the mitigation of the fluctuation kinetic energy
in the unsteady wake behind a circular cylinder in the laminar regime at
Re=100, via controlled angular motions of the surface, is numerically
investigated. A compact reduced-order model that resolves the long-term
behaviour of the fluid flow and the effects of actuation, is derived using
Proper Orthogonal Decomposition and Galerkin projection. In a full-information
setting, feedback controllers are then designed to reduce the long-time average
of the kinetic energy associated with the limit cycle. These controllers are
then implemented in direct numerical simulations of the actuated flow. Control
performance, energy efficiency, and physical control mechanisms identified are
analysed. Key elements, implications and future work are discussed
Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization
We describe methods for proving upper and lower bounds on infinite-time
averages in deterministic dynamical systems and on stationary expectations in
stochastic systems. The dynamics and the quantities to be bounded are assumed
to be polynomial functions of the state variables. The methods are
computer-assisted, using sum-of-squares polynomials to formulate sufficient
conditions that can be checked by semidefinite programming. In the
deterministic case, we seek tight bounds that apply to particular local
attractors. An obstacle to proving such bounds is that they do not hold
globally; they are generally violated by trajectories starting outside the
local basin of attraction. We describe two closely related ways past this
obstacle: one that requires knowing a subset of the basin of attraction, and
another that considers the zero-noise limit of the corresponding stochastic
system. The bounding methods are illustrated using the van der Pol oscillator.
We bound deterministic averages on the attracting limit cycle above and below
to within 1%, which requires a lower bound that does not hold for the unstable
fixed point at the origin. We obtain similarly tight upper and lower bounds on
stochastic expectations for a range of noise amplitudes. Limitations of our
methods for certain types of deterministic systems are discussed, along with
prospects for improvement.Comment: 25 pages; Added new Section 7.2; Added references; Corrected typos;
Submitted to SIAD
Moored acoustic travel time (ATT) current meters : evolution, performance, and future designs
New laboratory measurements and numeric model studies show
the present folded-path ATT current meters are stable and
sensitive, but are not well suited for mean flow observations
in surface gravity waves. Alternate designs which reduce
unwanted wake effects are proposed. ATT flowmeter history,
principles of acoustic flow sensors, mean flow near cylinders,
and the need for linear flow sensors are reviewed.Prepared for the Office of Naval Research under
Contract Number N00014-76-C-0197; NR083-400 to
the Woods Hole Oceanographic Institution
Experiments in a turbine cascade for the validation of turbulence and transition models
This thesis presents a detailed investigation of the secondary flow and boundary layers in a large scale, linear cascade of high pressure turbine rotor blades. The puropose of the data is to provide a suitable test case to aid the design and validation of the turbulence and transition models used in computational fluid dynamics. Hot-wire measurements have been made on a number of axial planes upstream, within and downstream of the blades to give both the mean flow conditions and all six components of Reynolds stress. Suitable inlet conditions have been defined at one axial chord upstream of the blade leading edge where the velocity and turbulence have been measured in both the freestream and endwall boundary layer. The turbulence dissipation rate has also been measured in order to define fully the inlet flow, a quantity that is usually missing in other data. Measurements through the blade show that the turbulence generation associated with the secondary flows is considerable and that all three shear stress components are significant. Intermittency measurements close to the endwall and blade surfaces show that the boundary layers are mostly laminar or transitional. The new endwall boundary layer, that forms behind the separation line, was found to be initially laminar. On the suction surface transition occurs over the latter part of the blade and on the pressure surface the accelerating flow causes relaminarisation. A number of calculations using a mixing length and high and low Reynolds number k-ϵ calculations show that reasonable overall results may be obtained. The lack, or failure, of transition modelling caused profile losses to be generally overpredicted and there was little evidence that the more sophisticated models produced better results. No model accurately predicted the individual turbulence quantities largely due to the inadequacy of the Boussinesq assumption for this type of flow. Good transition modelling appears to be more important than turbulence modelling in terms of the overall results
Finding unstable periodic orbits for nonlinear dynamical systems using polynomial optimisation
Computing unstable periodic orbits (UPOs) for systems governed by ordinary differential equations (ODEs) is a fundamental problem in the study of nonlinear dynamical systems that exhibit chaotic dynamics. Success of any existing method to compute UPOs relies on the availability of very good initial guesses for both the UPO and its time period. This thesis presents a computational framework for computing UPOs that are extremal, in the sense that they optimise the infinite-time average of a certain observable. Constituting this framework are two novel techniques.
The first is a method to localise extremal UPOs for polynomial ODE systems that does not rely on numerical integration. The UPO search procedure relies on polynomial optimisation to construct nonnegative polynomials whose sublevel sets approximately localise parts of the extremal periodic orbit. Points inside the relevant sublevel sets can then be computed efficiently through direct nonlinear optimisation. Such points provide good initial conditions for UPO computations with existing algorithms.
The second technique involves the addition of a control term to the original polynomial ODE system to reduce the instability of the extremal UPO, and, in some cases, to provably stabilise it. This control methodology produces a family of controlled systems parametrised by a control amplitude, to which existing UPO-finding algorithms are often more easily applied.
The practical potential of these techniques is demonstrated by applying them to find extremal UPOs for a nine-dimensional model of sinusoidally forced shear flow, an extended version of the Lorenz system, and two different three-dimensional chaotic ODE systems. Extensions of the framework to non-polynomial and Hamiltonian ODE systems are also discussed.Open Acces
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