John McKernan

On the Minimization of Maximum Transient Energy Growth

The problem of minimizing the maximum transient energy growth is considered. This problem has importance in some fluid flow control problems and other classes of nonlinear systems. Conditions for the existence of static controllers that ensure strict dissipativity of the transient energy are established and an explicit parametrization of all such controllers is provided. It also is shown that by means of a Q-parametrization, the problem of minimizing the maximum transient energy growth can be posed as a convex optimization problem that can be solved by means of a Ritz approximation of the free parameter. By considering the transient energy growth at an appropriate sequence of discrete time points, the minimal maximum transient energy growth problem can be posed as a semidefinite program. The theoretical developments are demonstrated on a numerical example.

Linear quadratic control of plane Poiseuille flow-the transient behaviour.

This paper describes the design of optimal linear quadratic controllers for single wavenumber-pair periodic 2-D disturbances in plane Poiseuille flow, and subsequent verification using a finite-volume full Navier–Stokes solver, at both linear and non-linear levels of initial conditions selected to produce the largest linear transient energy growth. For linear magnitude initial conditions, open and closed-loop finite-volume solver results agree well with a linear simulation. Transient energy growth is an important performance measure in fluid flow problems. The controllers reduce the transient energy growth, and the non-linear effects are generally seen to keep energy levels below the scaled linear values, although they do cause instability in one simulation. Comparatively large local quantities of transpiration fluid are required. The modes responsible for the transient energy growth are identified. Modes are shown not to become significantly more orthogonal by the application of control. The synthesis of state estimators is shown to require higher levels of discretiation than the synthesis of state-feedback controllers. A simple tuning of the estimator weights is presented with improved convergence over uniform weights from zero initial estimates. Nomenclature Greek symbols α  streamwise (x) wave number, cycles per 2π distance β  spanwise (z) wave number, cycles per 2π distance ε(t)  synchronic transient energy bound at time t  synchronic error energy bound at time t ζ  eigenvalue in synchronic transient energy bound eigensystem η(x,y,z,t)  wall-normal vorticity perturbation  η Fourier coefficient at wavenumber pair α,β θ  diachronic transient energy bound θ Error  diachronic error energy bound θ Est  estimated energy bound  ith eigenvalue  diagonal eigenvalue matrix μ  dynamic viscosity ρ  fluid density  ith singular value of A (A)  spectral norm or largest singular value of A  modal amplitude vector, χ0  initial χ, at time t = 0 Ψ  matrix of right eigenvectors ψ i  ith right eigenvector  frequency Roman symbols  system matrix  input matrix  output matrix  multiplying co-efficient for nth Chebyshev polynomial c i  amplitude of mode i E(t)  transient energy, , at time t E Est(t)  estimated transient energy, , at time t E Error (t)  error energy , at time t E 0  E of worst open-loop perturbation of , at t = 0 (2.26 × 10− 9) E pair,bound   upper bound on mode pair energy growth h  channel wall separation  identity matrix j    state feedback gain matrix  estimator gain matrix N  highest Chebyshev polynomial degree used, final collocation point index P  pressure P b  steady base flow pressure p  pressure perturbation  state variable weighting (energy) matrix  control weighting matrix R  Reynolds number r  control weight multiplier s  measurement noise weight multiplier  invertible matrix for conversion between state variables and , excludes next-to-wall velocities and wall vorticities t  time x,y,z  streamwise, wall-normal and spanwise co-ordinates  flow velocity vector  steady base flow velocity U cl  Ub at centreline  velocity perturbation vector  u,v,w Fourier coefficients at wavenumber pair α,β  control vector  measurement noise power spectral density  process noise power spectral density  state variable vector  state estimates vector  estimate error vector,   which generates θ   which generates θ Error   transformed to values at collocation points  state variables transformed to , thus  measurement vector y n  y at nth Chebyshev–Gauss–Lobatto collocation point

A linear state-space representation of plane Poiseuille flow for control design: a tutorial

A method for the incorporation of wall transpiration into a model of linearised plane Poiseuille flow is presented, with the aim of producing a state-space model suitable for the development of feedback control of transition to turbulence in channel flow. The system state is observed via wall shear-stress measurements and controlled by wall transpiration. The streamwise discretisation in the linearised model is by Fourier series and the wall-normal discretisation is by a Chebyshev polynomial basis, which is modified to conform to the control boundary conditions. This paper is intended as a tutorial on the addition of boundary control to a spectral model of a fluid continuum, to form a state-space model, as used in the emerging multidisciplinary field of flow control by means of Microelectrical Machines (MEMs). The ultimate aim of such flow control is the reduction of skin-friction drag on moving bodies.

Linear feedback control of transient energy growth and control performance limitations in subcritical plane Poiseuille flow

Suppression of the transient energy growth in subcritical plane Poiseuille flow via feedback control is addressed. It is assumed that the time derivative of any of the velocity components can be imposed at the walls as control input, and that full-state information is available. We show that it is impossible to design a linear state-feedback controller that leads to a closed-loop flow system without transient energy growth. In a subsequent step, full-state feedback controllers – directly targeting the transient growth mechanism – are designed, using a procedure based on a Linear Matrix Inequalities approach. The performance of such controllers is analyzed first in the linear case, where comparison to previously proposed linear-quadratic optimal controllers is made; further, transition thresholds are evaluated via Direct Numerical Simulations of the controlled three-dimensional Poiseuille flow against different initial conditions of physical interest, employing different velocity components as wall actuation. The present controllers are effective in increasing the transition thresholds in closed loop, with varying degree of performance depending on the initial condition and the actuation component employed.Suppression of the transient energy growth in subcritical plane Poiseuille flow via feedback control is addressed. It is assumed that the time derivative of any of the velocity components can be imposed at the walls as control input and that full-state information is available. We show that it is impossible to design a linear state-feedback controller that leads to a closed-loop flow system without transient energy growth. In a subsequent step, state-feedback controllers—directly targeting the transient growth mechanism—are designed using a procedure based on a linear matrix inequalities approach. The performance of such controllers is analyzed first in the linear case, where comparison to previously proposed linear-quadratic optimal controllers is made; further, transition thresholds are evaluated via direct numerical simulations of the controlled three-dimensional Poiseuille flow against different initial conditions of physical interest, employing different velocity components as wall actuation. The pre...

Optimal Low-Frequency Filter Design for Uncertain 2-1 Sigma-Delta Modulators

Variability in the analogue components of integrators in cascaded 2-1 sigma-delta modulators causes imperfect cancellation of first stage quantization noise, and reduced signal-to-noise ratio in analogue-to-digital converters. Design of robust matching filters based on low-frequency weighted convex optimization over uncertain linearized representations are mathematically very complex and computationally intensive, and offer little insight into the solution. This letter describes a design method based on formal optimization of a low-frequency uncertain linearized model of the modulator, and leads to a simple intuitive result which can shed light on the more complex models. Simulation results confirm the optimal properties of the filter.

Minimizing transient energy growth in plane Poiseuille flow

The feedback control of laminar plane Poiseuille flow is considered. In common with many flows, the dynamics of plane Poiseuille flow is very non-normal. Consequently, small perturbations grow rapidly with a large transient that may trigger non-linearities and lead to turbulence, even though such perturbations would, in a linear flow, eventually decay. This sensitivity can be measured using the maximum transient energy growth. The linearized flow equations are discretized using spectral methods and then considered at one wave-number pair in order to obtain a model of the flow dynamics in a form suitable for advanced control design. State feedback controllers that minimize an upper bound on the maximum transient energy growth are obtained by the repeated solution of a set of linear matrix inequalities. The controllers are tested using a full Navier—Stokes solver, and the transient energy response magnitudes are significantly reduced compared with the uncontrolled case.

Robust Filter Design for Uncertain 2-1 Sigma-Delta Modulators via the Central Polynomial Method

Uncertainty in the integrators of 2-1 sigma-delta modulators causes imperfect cancellation of first stage quantization noise, and reduces signal-to-noise ratio in analogue-to-digital converters. Design of robust matching filters based on convex optimization over uncertain linearized state-space representations gives complicated models and high-order designs. This letter describes a polynomial design method leading to simpler multilinear models and fixed-order filters. The modulators are cast as a polynomial polytope, and filters satisfying an Hinfin bound arise from solving linear matrix inequalities (LMIs). Results at low frequency show the proposed filter outperforming the nominal one, with a performance close to the estimated optimum.

MINIMIZATION OF MAXIMUM TRANSIENT ENERGY GROWTH BY OUTPUT FEEDBACK

Abstract The the problem of minimizing the transient energy of a linear system following a unit energy initial disturbance is considered in this paper. This paper extends previous results on the state feedback case to the output feedback case. Furthermore, it is shown that the problem can be solved by convex optimization of the free parameter following a Q -parametrization. The techniques are illustrated by numerical examples.

MINIMISATION OF TRANSIENT PERTURBATION GROWTH IN LINEARISED LORENZ EQUATIONS

Abstract This paper describes the LMI synthesis of feedback controllers which minimise closed loop transient perturbation growth with limited control effort. Controllers are synthesized for the linearised Lorenz equations, and their performance is compared to that of LQR controllers. At low control effort the controllers behave similarly but the LMI based controllers are able to produce an almost monotonically falling transient with increasing control effort, whereas LQR controllers have a distinct minimum transient. Evidence is found that controllers which produce the lowest transients do not necessarily have the most orthogonal system eigenvectors, and an explanation in terms of modal and non-modal growth components is presented. Both LMI and LQR controllers are able to stabilise the full Lorenz equations for limited initial conditions.

Kolmogorov-Chaitin complexity of linear digital controllers implemented using fixed-point arithmetic

The complexity of linear, fixed-point arithmetic, digital controllers is investigated from a Kolmogorov-Chaitin perspective. Based on the idea of Kolmogorov-Chaitin complexity, practical measures of complexity are developed for both state-space realizations, and for parallel and cascade realizations. The complexity of solutions to a restricted complexity controller benchmark problem is investigated using this measure. The results show that, from a Kolmogorov-Chaitin viewpoint, higher-order controllers with a shorter word-length may have a lower complexity but a better performance than lower-order controllers with longer word-length.