Stationary Disturbances Research Articles

Convergence Analysis of the Multi-Variable Filtered-X LMS Algorithm

The filtered-X LMS algorithm is one of the most widely used algorithms for the adaptive feedforward control problem for disturbance rejection. It is known that the behavior of a single-input/single-output (SISO) system which uses this algorithm, is equivalent to that of an linear time-invariant (LTI) system, when the disturbance is a single- frequency sinusoid. However, the behavior of a multiple-input/multiple-output (MIMO) system with non-sinusoidal disturbances is not well understood. In this paper, averaging analysis is used to study the asymptotic behavior of the filtered-X LMS algorithm, considering general stationary disturbances in both SISO and MIMO systems. A certain multi-variable strictly positive real (SPR) condition on the model error is shown to be sufficient for convergence but not sufficient for disturbance attenuation. The practical implications of these conditions are explored in the context of the adaptive active noise cancelling problem.

Internal solitary waves moving over the low slopes of topographies

Internal solitary waves moving over uneven bottoms are analyzed based on the reductive perturbation method, in which the amplitude, slope and horizontal lengthscale of a topography on the bottom are of the orders of ε, ε 5 2 and ε −3 2 , respectively, where the small parameter ε is also a measure of the wave amplitude. A free surface condition is adopted at the top of the fluid layer. That condition contains two parameters, δ and Δ, the first of which concerns the discontinuity of the basic density between the outer layer and the inner one; the second concerns the discontinuity of the mean density between them. An amplitude equation for the disturbance of order ε decomposes into a Korteweg-de Vries (KdV) equation and a system of algebraic equations for a stationary disturbance around a topography on the bottom. Solitary waves moving over a localized hill are studied in a simple case where both the basic flow speed and the Brunt-Vaisalla frequency are constant over the fluid layer. For this case, the expression for the amplitude of the stationary disturbance contains singular points with respect to basic flow speed. These singularities correspond to the resonant conditions modified by the free surface condition. The advancing speeds of solitary waves are changed by the influence of bottom topography, in a case where the long internal waves propagate in the direction opposite to the basic flow, but their waveforms remain almost unchanged.

Transition in flow over a swept cylinder: Correlation with experimental data

Linear stability analysis, which includes the effects of nonparallelism and body curvature, is used to calculate the N factors of stationary and traveling disturbances for the same experimental conditions measured by Poll for flow over a long swept cylinder. The correlating N factors for traveling disturbances have less scatter than the correlating N factors of stationary disturbances. The mean value of the correlating N factor for traveling disturbances is 14.9 and 7.4 for stationary disturbances. Therefore, for an initial amplitude of a stationary disturbance that is 100 times larger than that of a traveling disturbance, the amplitude of the traveling disturbance is 18 times larger than the amplitude of a stationary disturbance at the location of transition onset. The calculations indicate that transition is due to traveling disturbances; this result is in agreement with the experimental observations.

Results analysis for real-time optimization (RTO): Deciding when to change the plant operation

After the automation goals of safety, product quality and production rate have been achieved, opportunity exists in many plants to further increase profit by adjusting selected operating variables. When external conditions change frequently, incentive exists for the optimization to be performed in real time. However, real-time optimization might change conditions too frequently when responding to measurement noise and high frequency stationary disturbances. A results analysis method is proposed to decide when a significant change in plant operation has been proposed and should be implemented. The method is based on a knowledge of the common cause measurement variability and the transmission of this variability in the RTO loop to the recommended operating conditions. The basic expressions for the results analysis method are developed, and sample results from a case study are presented. Extensions are also briefly discussed.

Calculation of circulating flows in bubble columns

Bubble columns are widely used as chemical reactors. The liquid phase is well mixed due to a liquid circulation present in practically all technical applications. The liquid circulation is shown experimentally (Lippert, 1982) to contain several circulation cells. Each cell is about one bubble column diameter high and undergoes fluctuations in position and circulation velocity. The liquid circulation is suppressed if the gas phase is distributed very evenly over the cross-sectional area of the bubble column as shown experimentally by Molerus and Kurtin (1985, Chem. Engng Sci. 40, 647–652). Thus, the liquid circulation is a flow instability caused by a disturbance of gas distribution. The onset of the liquid circulation is investigated in the present paper by applying an analytical method. The flow field of several consecutive circulation cells is obtained applying a numerical method, which is specially adapted for this kind of flow instability. Our focus is on understanding the mechanism of the circulation cells. Thus, all terms of smaller orders of magnitudes are neglected in the mass and momentum balances for both phases. The main terms, which cause the flow instability, are extensively discussed. We assumed a small but stationary disturbance of gas distribution in order to simplify the calculations. The general trend in two-phase modeling, even though there are some remarkable exceptions, is to implement more and more forces and higher sophisticated turbulence models. But circulation cells are an example that some flow phenomena may not be described following strictly that way. Basically, a very high spatial resolution and a very stable numerical method are required in order to predict circulation cells.

Adaptive feedforward control of non-minimum phase structural systems

Abstract Adaptive feedforward control algorithms have been effectively applied to stationary disturbance rejection. For structural systems, the ideal feedforward compensator is a recursive filter which is a function of the transfer functions between the disturbance and control inputs and the error sensor output. Unfortunately, most control configurations result in a non-minimum phase control path; even a collocated control actuator and error sensor will not necessarily produce a minimum phase control path in the discrete domain. Therefore, the common practice is to choose a suitable approximation of the ideal compensator. In particular, all-zero finite impulse response (FIR) filters are desirable because of their inherent stability for adaptive control approaches. However, for highly resonant systems, large order filters are required for broadband applications. In this work, a control configuration is investigated for controlling non-minimum phase lightly damped structural systems. The control approach uses low order FIR filters as feedforward compensators in a configuration that has one more control actuator than error sensors. The performance of the controller was experimentally evaluated on a simply supported plate under white noise excitation for a two-input, one-output (2I1O) system. The results show excellent error signal reduction, attesting to the effectiveness of the method.

Active damping of enclosed sound fields through direct rate feedback control

An alternative approach to the control of enclosed sound fields subjected to impulsive or broadband disturbances is presented based upon direct rate-feedback control. Results from this preliminary study suggest that if a pressure transducer is collocated with a volumetric source in an enclosed sound field and rate-feedback control is employed, then the global properties of the sound field can be modified, specifically the modal damping ratio of the acoustic modes. Guaranteed asymptotic stability of the control approach was demonstrated for positive feedback gains, making it model insensitive and thus ideal for actively tuning the transient characteristics (i.e., the reverberation time) of the enclosure. Significant levels of global reduction in the acoustic potential energy can be achieved with the proposed control approach for stationary random disturbances or impulsive disturbances.

Experimental study of the transition to turbulence at a single stationary disturbance in the boundary layer of an oblique aerofoil

Experimental study of the transition to turbulence at a single stationary disturbance in the boundary layer of an oblique aerofoil

The stability of natural convection in a vertical layer of electrically conducting fluid in the presence of a transverse magnetic field

The stability of natural convection of an electrically conducting fluid which is confined between two parallel vertical plates maintained at constant and different temperatures and is permeated by a transverse magnetic field is analyzed using linear stability theory. In deriving the equations governing the stability, a simplification is made using the fact that the magnetic Prandtl number Pm for most electrically conducting fluids is extremely small. The Chebyshev collocation method is used to obtain the eigenvalue equation, which is then solved numerically. The critical Grashof number Gc, the critical wavenumber ac, and the critical wave speed cc are obtained for a wide range of the Prandtl number P and for several selected values of the Hartmann number M. It is found that the magnetic field has a stabilizing effect on the convection flow against both stationary and travelling-wave disturbances. The detailed values of P, Gc, ac, and cc, at the point of transition from stationary to travelling-wave mode are also obtained for several selected values of M.

An unbiased equation error identifier and reduced-order approximations

The equation error (EE) identification technique is modified to remove the parameter bias problem induced by uncorrelated measurement errors. The modification replaces a constraint with a constraint; the asymptotic solution replaces a normal equation with an eigenequation. The resulting algorithm is simpler than previous schemes, while at the same time preserving the desirable properties of the conventional EE method: simplicity of an on-line algorithm, unimodality of the performance surface, and consistent identification in the sufficient-order case. In the more realistic undermodeled case, a robustness result shows that the mean optimal parameter values of both the monic and unit-norm EE schemes correspond to a stable transfer function for all degrees of undermodeling, and for all stationary output disturbances, provided the input sequence satisfies an autoregressive constraint; otherwise an unstable model may result. Model approximation properties for the undermodeled case are exposed in detail for the case of autoregressive inputs; although both the monic and unit-norm variants provide Pade approximation properties, the unit-norm version is capable of autocorrelation matching properties as as well, and yields the optimal solution to a first- and second-order interpolation problem. Finally, the mismodeling error for the undermodeled case is shown to be a well-behaved function of the Hankel singular values of the unknown system. This modification allows EE methods to be admitted to the class of unbiased identification and approximation techniques. >

Active damping of enclosed sound fields through direct rate feedback control

A new method of controlling enclosed sound fields subjected to impulsive or broadband disturbances is presented based upon direct rate-feedback control. Results from this preliminary study suggest that if a pressure transducer is collocated with a volumetric source in an enclosed sound field and rate-feedback control is employed, then the global properties of the sound field can be modified, specifically the modal damping ratio of the acoustic modes. Asymptotic stability of the control approach was demonstrated for positive feedback gains (neglecting specific sensor and actuator dynamics), making it model insensitive and thus ideal for actively tuning the transient characteristics and thus the reverberation time of the enclosure. Significant levels of global reduction in the acoustic potential energy can be achieved with the proposed control approach for stationary random disturbances or impulsive disturbances.

Feedforward controller design by eigenvalue assignment

Active feedforward control algorithms have proved to be successful in the attenuation of stationary disturbances such as single, multiple frequencies, and random inputs. In feedforward control, the mean square value of a measurable response quantity induced by the disturbance input is minimized by coherent ‘‘secondary’’ control inputs. The traditional design procedure is to choose the location/number of control forces and error sensors upon a physical understanding of the response of the uncontrolled and controlled systems. Thus an analytical design procedure for feedforward controlled systems is nonexistent in contrast to feedback control where pole allocation, optimal control, etc. are extensively used. Recent analytical work has demonstrated that feedforward controlled systems have new eigenproperties. The controlled eigenvalues and eigenfunctions are only functions of the control input location and the error quantity to be minimized. Here, a design methodology for feedforward controllers is presented ...

The Instability of Long's Stationary Solution and the Evolution toward Severe Downslope Windstorm Flow. Part II: The Application of Finite-Amplitude Local Wave-Activity Flow Diagnostics

Abstract An analysis of severe downslope windstorm evolution, from flow conditions initially described by Long's stationary solution, is considered through the application of finite-amplitude wave-activity diagnostics. Such quantities satisfy a local conservation law of the form At + ∇ · F = 0. In particular, an analysis of the pseudoenergy density A and flux F is undertaken since the corresponding conservation law may be employed to describe disturbances to basic states that are nonparallel. In this study, such diagnostics are primarily enlisted to identify and chronicle the life cycle of normal-mode instability in the breakdown of Long's solution. As a result of this investigation, it is concluded that the large-amplitude stationary disturbance, which defined the second stage of windstorm evolution in Part I of this study, was the result of a normal-mode instability of Long's solution. The development of this mode, its breakdown via secondary shear instability, and the subsequent evolution toward the ma...

The Instability of Long's Stationary Solution and the Evolution toward Severe Downslope Windstorm Flow. Part I: Nested Grid Numerical Simulations

Through direct numerical simulation, the instability of Long's exact finite-amplitude steady-state solution to the problem of stratified flow over topography and the subsequent evolution towards severe downslope windstorm flow is investigated. The integrations are initialized with Long's analytical solution and are calculated in a model domain that employs three levels of interactive grid nesting. In this manner, the resolution achieved is approximately a factor of 10 greater than that previously employed. As a result of this increased resolution, three distinct stages of windstorm development are explicitly identified. In the first, convection acts to neutralize the region of overturned isentropes. During the next stage, a large-amplitude stationary disturbance develops above the lee slope of the topography. In time, small-scale secondary shear instability develops in local regions of enhanced shear associated with flow perturbations caused by the large-amplitude disturbance. In the final stage of development, these modes of shear instability evolve to larger spatial scale and come to dominate the flow in the mature windstorm state. In our analysis, it is furthermore demonstrated that these stages of development can be qualitatively and, to some extent, quantitatively reproduced in a parallel flow extracted from a cross section through Long's solution if a horizontally localized forcing, designed to enhance the vertical shear in the background wind field, is imposed.

Simulation of the global structure of stationary planetary waves in the mesosphere and lower thermosphere

When simulating the global structure of stationary planetary waves (SPW) the problem of obtaining the numerical solution in the equatorial region appears. It results from the presence of apparent singularities in the operator of the SPW latitudinal structure when the Coriolis parameter is small. The new method based on SPW latitudinal operator inversion is proposed. This method permits the difficulties arising from the simulation of stationary large scale disturbances at low latitudes to be avoided. The global structure of SPW with zonal wave number m = 1 at the mesosphere and lower thermosphere heights has been calculated for the background zonal wind distribution representing a climatic picture of the solstice conditions. In the region of the mean zonal westerlies the SPW penetration across the equator is obtained. The SPW at low latitudes are shown to appear most significantly in the zonal component of the wind velocity. The influence of planetary wave motions on the distribution of longlived species in the ionospheric D-region and at the heights of lower thermosphere are discussed.

Crossflow vortex and transition measurements by use of multielement hot films

An experiment on a 45-deg swept wing was conducted to study three-dimensional boundary-layer characteristics using surface-mounted, microthin, multielement hot-film sensors. A large hot-film gauge was fabricated that combined three arrays of 64 crossflow sensors each and a single array of 32 transition sensors. Two such gauges were mounted along the model chord. Crossflow vortex structure and boundary-layer transition were measured from the simultaneously acquired signals of the hot films. Stationary and nonstationary crossflow disturbances were measured from hot-film sensor arrays

Structure and performance of discreteH∞ optimal control

In recent years a good deal of attention has been focused on the development of model-based controllers optimizing infinity-norm performance objectives. The motivation behind this is a perceived need for controllers which are more robust to modelling errors than those which result from conventional LQ designs. At the present time, however, numerical solution of the H∞ problems remains very much an open issue. This paper examines Grimble's approach to the design of discrete-time H∞ control laws for univariate plants subject to stationary stochastic disturbances. The structure of the controller is analysed using the internal model control concept of Garcia and Morari and is compared with that obtained for a polynomial LQ scheme. Simulation results are presented for both minimum phase and non-minimum phase systems in which the H∞ technique is seen to be inferior to LQ in terms of performance, robustness to parametric mismatch and computational complexity.

The viscous-diffusion nonlinear critical layer in a stratified shear flow

Stationary finite-amplitude wave disturbances in a stratified shear flow with Richardson number larger than ¼ are investigated for large Reynolds numbers when viscosity and thermal conductivity, as well as nonlinearity, are essential factors in the critical layer. The jumps across the critical layer in average vorticity, reflection and transmission coefficients are calculated as functions of the local Reynolds number determined by the amplitude of the incident wave. With the increase of the incident wave amplitude the asymptotic value of the Richardson number on the same side of critical layer as the incident wave tends to 1/4 the reflection coefficient tends to unity and the transmission coefficient to zero.

Primary and secondary stability analysis of a three-dimensional boundary-layer flow

A three-dimensional, incompressible boundary-layer flow is investigated theoretically with respect to primary and secondary instability. These investigations accompany a basic transition experiment, which is being performed at the DLR in Göttingen [B. Müller, in Laminar-Turbulent Transition, edited by D. Arnal and R. Michel (Springer-Verlag, Berlin, 1990), p. 489]. Primary stationary and secondary nonstationary disturbances are used to model the mean flow and the fluctuations of a measured (quasi-) saturation state. The analysis is based on a Falkner–Skan–Cooke approximation of the undisturbed flow. Good agreement of the secondary stability model with this experiment is obtained, especially concerning the spatial distribution of the rms fluctuation. A striking change of the vortex pattern due to secondary instability has not been observed.

Feedforward controller design by eigenvalue assignment.

Active feedforward control algorithms have proved to be successful in the attenuation of stationary disturbances such as single, multiple frequencies, and random inputs. In feedforward control, the mean square value of a measurable response quantity induced by the disturbance input is minimized by coherent ‘‘secondary’’ control inputs. The traditional design procedure is to choose the location/number of control forces and error sensors upon a physical understanding of the response of the uncontrolled and controlled systems. Thus an analytical design procedure for feedforward controlled systems is nonexistent in contrast to feedback control where pole allocation, optimal control, etc. are extensively used. Recent analytical work has demonstrated that feedforward controlled systems have new eigenproperties. The controlled eigenvalues and eigenfunctions are only functions of the control input location and the error quantity to be minimized. Here, a design methodology for feedforward controllers is presented that is based on designing an optimal error sensor such that the controlled system eigenvalues have predetermined values. The formulation is experimentally verified on a simply supported beam with an optimally shaped PVDF film sensor. [Work supported by ONR/DARPA.]