Strong Static Stability Research Articles

Necessary and sufficient conditions for strong stability of distributed parameter systems

In this paper we propose definitions for strong stabilizability and strong detectability of infinite-dimensional control systems. We show that these definitions are appropriate by showing that they can be used to give necessary and sufficient conditions for the strong stability of a system in terms of its input-output stability. As an application, we discuss the strong stability of the feedback interconnection of two strongly stabilizable and strongly detectable systems.

Gravity Waves Driven by Diurnal Fluctuations in Mesoscale Heating

Observed convective systems, such as mesoscale convective complexes (MCCs), often undergo repeated cycles of nocturnal growth and daytime decay especially during the summer. The gravity wave response to these pulsations is poorly understood. The motivation for this study is to understand this response and especially its sensitivity to environmental factors such as horizontal wind and static stability. A semianalytical approach is used that focuses on the roles of singularities in a complex horizontal wavenumber space. The model is linear, Boussinesq, hydrostatic, and rotating with uniform ambient conditions. Prescribed, 2D, pulsating, upright, convective heating drives the waves. The Fourier transform technique is used to unravel the response into a discrete and continuous horizontal spectrum. Many features of the response depend on a Froude number, F 5 pU/(DN) where U 5 background wind, D 5 depth of source region, and N 5 buoyancy frequency. The most efficient forcing of the gravity wave field occurs near criticality (F 5 1). For typical values of U and D associated with midlatitude convective systems, the critical value of N is about one-fourth of average tropospheric values. The resulting enhanced loss of energy from the convective system due to gravity waves could limit the intensity of convective systems near criticality. At subcriticality (F , 1), the pulsating upstream response is dominated by unbalanced ageostrophic propagating modes upstream of the source region and by a balanced geostrophic mode downstream of the source. The latter advances with a speed equal to the background flow. No far-upstream response occurs for supercriticality ( F . 1). The downstream response for F . 1 is dominated by the geostrophic mode in the pressure, temperature, and source-parallel wind. In addition, a vigorous ageostrophic mode advances downstream from the subcritical source region giving rise to alternating regions of rising motion and subsidence. It is hypothesized that the latter could trigger new lines of convection in the downstream subcrticial Froude number regime. Mature MCCs often develop low-level cooling in response to evaporative cooling. This cooling primarily triggers the advective inertial gravity wave mode, which propagates downstream at the background wind speed. It is shown that subcritical flows ( F , 1; weak ambient flow and/or strong static stability) are tuned to strongly respond to this mode. It is suggested that the development of new convection might be suppressed near the system under subcritical conditions. Low-level cooling has little effect on the supercritical response.

Modeling clouds and radiation for the November 1997 period of SHEBA using a column climate model

A column version of the Arctic regional climate system model (ARCSYM) has been developed for testing general circulation model parameterizations in the Arctic. The ARCSYM column model has been employed for a 23‐day period in November to simulate conditions over a multiyear ice floe that has been the site of intensive observations as part of the Surface Heat Budget of the Arctic (SHEBA) project. The large‐scale tendencies of temperature, moisture, and wind are specified with values obtained from a special column data set obtained from the European Centre for Medium‐Range Weather Forecasting. Comparisons between the ARCSYM column simulations and SHEBA data reveal that modeled temperature profiles are too cold aloft and generally too warm in the boundary layer. The occurrence of low clouds is severely underpredicted while the high cloud fraction is over predicted. The modeled longwave radiative cooling at the surface is 1.5–3 times as large as that observed. Much of this bias is related to problems with the treatment of clear‐sky radiative transfer and in the simulated cloud optical properties. At the same time, the magnitude of modeled downward sensible heat flux at the surface is much too large. This has been related, in part, to the method for scaling temperature at the lowest modeled level to its surface air value under conditions of strong static stability. The importance of properly treating longwave radiative transfer under extremely cold, clear‐sky conditions is evident in the sensitivity studies. The best simulation of cloud properties was achieved by assuming liquid cloud processes and properties at temperatures above 255 K. This temperature is significantly colder than that used in many climate models. The occurrence of supercooled clouds in the simulation dramatically reduced longwave cooling at the surface due to increases in the optical depth and fractional coverage of clouds. Results from a coupled sea ice‐atmosphere simulation reveal that improvements in the atmospheric parameterizations are enhanced when the system is coupled.

Value iteration and optimization of multiclass queueing networks

This paper considers in parallel the scheduling problem for multiclass queueing networks, and optimization of Markov decision processes. It is shown that the value iteration algorithm may perform poorly when the algorithm is not initialized properly. The most typical case where the initial value function is taken to be zero may be a particularly bad choice. In contrast, if the value iteration algorithm is initialized with a stochastic Lyapunov function, then the following hold: (i) a stochastic Lyapunov function exists for each intermediate policy, and hence each policy is regular (a strong stability condition), (ii) intermediate costs converge to the optimal cost, and (iii) any limiting policy is average cost optimal. It is argued that a natural choice for the initial value function is the value function for the associated deterministic control problem based upon a fluid model, or the approximate solution to Poisson’s equation obtained from the LP of Kumar and Meyn. Numerical studies show that either choice may lead to fast convergence to an optimal policy.

Strongly stable gyroscopic systems

Here, gyroscopic systems are time-invariant systems for which motions can be char- acterizedby properties of a matrix pencil L(λ )= λ 2 I + λG − C ,w hereG T = −G and C> 0. A strong stability condition is known which depends only on |G| (= (G T G) 1/2 ≥ 0) and C. If a system with coefficients G0 and C satisfies this condition then all systems with the same C andwith a G satisfying |G |≥| G0| are also strongly stable. In order to develop a sense of those variations in G0 which are admissible (preserve strong stability), the class of real skew-symmetric matrices G for which this inequality holds is investigated, and also those G for which |G| = |G0|.

On strongly stable normal operators in Krein spaces

For bounded normal operators in Krein spaces we give a necessary and sufficient condition for strong stability. The same result for unitary operators was obtained by M.G.Krein [1] (see also [2]). For selfadjoint operators we refer to the papers of P.Jonas, H.Langer [3] and H.Langer [4].

On Strongly Stable Normal Operators in Krein Spaces

We give a necessary and sufficient condition for strong stability of bounded normal operators in Krein spaces. A similar result was obtained for unitary operators by M. G. Krein [1], and for selfadjoint operators by H. Langer [2], [3].

On the stability and convergence of discretizations of initial value p.d.e.s

This paper examines the stability and convergence of discretizations of initial value p.d.e.s using spatial discretization followed by time integration with an explicit one-step method. A Cauchy integral representation is used to bound the growth in the discrete solution. New results are obtained regarding sufficient conditions for both algebraic and strong stability. Sufficient conditions are also derived for convergence on a finite time interval.

Multigrid Computation of Stratified Flow over Two-Dimensional Obstacles

A robust multigrid method for the incompressible Navier–Stokes equations is presented and applied to the computation of viscous flow over obstacles in a bounded domain under conditions of neutral stability and stable density stratification. Two obstacle shapes have been used, namely a vertical barrier, for which the grid is Cartesian, and a smooth cosine-shaped obstacle, for which a boundary-conforming transformation is incorporated. Results are given for laminar flows at low Reynolds numbers and turbulent flows at a high Reynolds number, when a simple mixing length turbulence model is included. The multigrid algorithm is used to compute steady flows for each obstacle at low and high Reynolds numbers in conditions of weak static stability, defined byK=ND/πU≤ 1, whereU,N, andDare the upstream velocity, bouyancy frequency, and domain height respectively. Results are also presented for the vertical barrier at low and high Reynolds number in conditions of strong static stability,K> 1, when lee wave motions ensure that the flow is unsteady, and the multigrid algorithm is used to compute the flow at each timestep.

The passive transport of NO x emissions from aircraft studied with a hierarchy of models

The passive transport of aircraft emissions of nitrogen oxides (NO x = NO + NO 2) has been studied with a hierarchy of models ranging from two-dimensional and three-dimensional chemistry transport models up to three-dimensional models of the general circulation. The sink of NO x was parameterized by an exponential decay process with a globally constant half-lifetime of 10 days. By performing a simple experiment the importance of the various transport processes has been studied. The three-dimensional models show that the monthly mean volume mixing ratio of NO x varies by a factor of three in the longitudinal direction and the temporal variability is of the order of 30%. In view of the nonlinearity of the chemical processes leading to ozone formation in the presence of NO x this implies that the assessment of the effects of subsonic aircraft emissions of NO x should be done with three-dimensional models. Vertical redistribution by convection strongly affects the maximum NO x mixing ratio at cruise altitudes, but due to the limited lifetime of NO x of the order of ten days the most important contribution to the mixing ratio at a certain level usually stems from emissions around that level. The strong static stability in the stratosphere hampers significant dispersion of the subsonic aircraft emissions above the height where the emissions take place for the lifetimes considered here. Some model deficiencies and biases have been identified and discussed. Examples are the oscillatory signature of NO x distributions obtained with a spectral advection scheme, the strong diffusion of one of the GCMs into the polar regions, and the too intense interhemispheric exchange of one of the two-dimensional CTMs. For the vertical redistribution of the emissions it may be necessary to include not only updrafts but also downdrafts in the convective parametrization of the transport model.

New sufficient conditions for strong simultaneous stabilization

We present sufficient conditions for the existence of a single stable controller to stabilize a set of n SISO plants: P 1, P 2, …, P n (strong simultaneous stabilization). As is well known, this is equivalent to the existence of a single controller, not necessarily stable, to stabilize n + 1 plants (simultaneous stabilization). The basic assumption required in this paper is that all the plants have the same unstable zeros. A practical example of where such an assumption holds is when each plant has the same pure delay, which is approximated by a non-minimum-phase rational function. The results here are applied to examples of this type.

ANALYSIS OF THE THERMO STABILITY CONDITIONS OF SOLID STATE LASER MODES

The mode thermo stability conditions for the modes of a conventional linear resonator are developed to apply to a general resonator. The concept of weak thermo stability is introduced, and a new design method,which satisfies both the weak thermo stability conditions for the resonator modes and the thermo stability conditions for the output power, is proposed. With this method, one can make the output power thermo stable and the resonator modes partially thermo stable (for example, the divergence angle can be made thermo stable). Furthermore, analytic solutions to the strong and weak thermo stability conditions for several kinds of resonators are obtained.

Bhopal Gas Leak: A Numerical Investigation of the Prevailing Meteorological Conditions

Abstract A three-dimensional mesoscale model was used to understand the meteorological conditions and the influence of the terrain on the local flow pattern during the Bhopal methyl isocyanate (MIC) gas leak. The study reveals that under the prevailing conditions of weak wind and strong stability the lakes in Bhopal influenced the local circulation significantly and caused northwesterly flow near the surface. The modified flow pattern resulted in the transport of MIC into the city area of Bhopal. However, with the increase in the ambient synoptic wind, the role of the lakes was found to diminish. Further, the other topographical features such as the hillocks in and around the city and the gently rolling terrain toward the southeastern sector of the city seem to have played a secondary role in influencing the meteorological conditions on the episodic night.

2-delay出力制御による零点消去条件と強安定化則

2-delay出力制御による零点消去条件と強安定化則

The stability of linear periodic Hamiltonian systems under non-Hamiltonian perturbations

A definition of strong stability and strong instability is proposed for a linear periodic Hamiltonian system of differential equations under a given non-Hamiltonian perturbation. Such a system is subject to the action of periodic perturbations: an arbitrary Hamiltonian perturbation and a given non-Hamiltonian one. Sufficient conditions for strong stability and strong instability are established. Using the linear periodic Lagrange equations of the second kind, the effect of gyroscopic forces and specified dissipative and non-conservative perturbing forces on strong stability and strong instability is investigated on the assumption that the critical relations of combined resonances are satisfied.

Numerical simulation of pollutant dispersion in a small valley under conditions with supercritical Richardson numbers

The numerical drainage wind model of Wonget al. (1987) is used together with a Lagrangian particle model in the simulation of carbon monoxide (CO) dispersion within a small urban valley in Edmonton, Alberta, Canada. The conditions studied are those of strong static stability when vertical mixing is suppressed. These are conditions with the Richardson number exceeding its critical value (hereafter referred to as supercritical conditions). Observations showed that under such conditions, vertical turbulence is suppressed but horizontal turbulence still exists. The effects of turbulence in the dispersion and transport of pollutants under such conditions are small. However, in the present simulation, a simple turbulence parameterization based on observations is used for supercritical conditions. Some field experiments were performed and the observations are compared with model results. For a location downwind of the CO source, two peaks can be observed during the course of the drainage flow regime. The model results suggest that these represent an initial flux from the drainage flow and a second flux later from drainage wind recirculation. Another main feature of the model-predicted concentration field is zones of maximum concentration at and above the valley floor. There is a drainage wind cell on each side of the valley slope and the cells are effectively decoupled from the prevailing wind above. The present modelling results show that when the prevailing wind exists before the development of the drainage wind, it can be instrumental in transporting CO from one drainage wind cell to the other. Otherwise, the CO released within one drainage wind cell is well contained.

Direct Evidence of “Sheets” in the Atmospheric Temperature Field

This paper presents experimental evidence showing the ubiquitous presence in the lower atmosphere (at least up to 25 km) of very strong (positive) temperature gradients within very thin layers. The presence of such “sheets” in the temperature field of the free atmosphere was frequently hypothesized in order to account for the aspect sensitivity of VHF radar measurements. Owing to their high vertical resolution (20 cm) and to the fast-response thermometers used, the in situ balloon measurements discussed in this paper constitute the first direct evidence of their true existence. Statistical study of the properties of the sheets results in the following typical values: thickness 3–20 m, temperature increase 0.2–0.8 K, gradient 30–100 K/km. The sheets are frequently observed in groups, associated with and taking part in regions of high static stability. Local measurements using two pairs of sensors one meter apart indicate that the sheets are not flat and horizontal. Sometimes, clear evidence of ongoing or recent mixing, despite the strong local static stability, can account for such distortions. Observations of the same sheets by two thermometers 65 m apart (vertically) indicate vertical distortions of the sheets up to 10 m and horizontal extensions larger than 100 m. The possible contribution of the observed sheets to the VHF radar vertical reflectivity is estimated and compared with simultaneous reflectivity profiles measured by the PROVENCE radar. Without any adjustment, these profiles compare favorably, both in shape and in level, thus suggesting that the sheets can account for a significant fraction of the VHF radar vertical echoes.

Design of reduced-order observers with precise loop transfer recovery

This paper concerns the design of reduced-order observers for systems in which the number of measurements is more than the number of controls. We develop an algorithm that applies to regular systems that have no transmission zeros. The algorithm uses eigenstructure assignment whereas other approaches use Kalman filter methods. The advantages of this approach are the following: 1) precise loop transfer recovery rather than approximate loop transfer recovery, 2) finite observer gain rather than asymptotic observer gain, and 3) modest computational tools and operations counts. Case studies are presented illustrating these features. T HE problem of designing an observer that can achieve loop transfer recovery (LTR) has received continuous at- tention since Doyle1 presented an example with a Kalman-fil- ter-based observer (linear quadratic Gaussian (LQG)) design lacking robustness even though the full-state feedback (linear quadratic regulator (LQR)) controller had impressive robust- ness properties, namely gain margins of - 6 dB to + oo dB and phase margins of ±60 deg.2 To alleviate this problem, Doyle and Stein3 developed a robustness recovery procedure in which fictitious process noise is added to the input in the design model. The LQR robustness properties are preserved with the loop open at the input since the loop transfer function is re- covered asymptotically as the intensity of the fictitious pro- cess noise is increased. Stein and Athans4 call this procedure LQG/LTR. There have been further developments and appli- cations of LQG/LTR by a number of workers. Madiwale and Williams5 extended the theory to reduced-order observer- based LQG designs for nonsquare, minimum phase, and left invertible plants. Calise and Prasad6 developed an approach for designing a fixed-order compensator and obtained an ap- proximate LTR for nonsquare, minimum phase systems. It is similar to the full-order compensator design of ordinary LQG/ LTR. Fu7 developed the necessary and sufficient condition for exact LTR employing a general feedback structure for model matching control that includes the observer-based state feedback control as a special case. He formulated the prob- lem of model matching by assigning certain stable matrices such that the desired loop transfer function became equal to the associated loop transfer function of the closed-loop input- output transfer function. He claimed that exact LTR is achiev- able under the conditions for strong stabilization of systems. Tsui8'9 introduced a theoretical analysis of an entirely new approach to the problem of loop transfer recovery. His ap- proach was to minimize the observer gain to the system input by observer pole selection. Furthermore, he claimed that this new approach aimed directly at achieving the necessary and sufficient condition of LTR. The purpose of this paper is to present a computational algorithm for the solution of the constrained matrix Sylves- ter equation that arises in Tsui's approach for designing ro- bust reduced-order Luenberger observers.10 Precise LTR is achieved with finite observer gain for regular nonsquare plants with no transmission zeros, for which the number of sensors is greater than the number of controls. Other approaches have been unable to attain these results. The method developed here provides freedom to select eigenvalues and eigenvectors for the observer.

Some measurements of turbulence over an antarctic ice shelf

AbstractProfiles of wind speed, temperature and turbulent fluxes of heat and momentum in the lowest 32 m have been measured at a station on an antarctic ice shelf. During the antarctic winter the surface layer often shows strong static stability, with temperature gradients as large as 1 Km−1 in the lowest few metres. the surface inversion is destroyed during periods of high wind speed but the wind profile shows significant deviation from the expected logarithmic form under such conditions. Measurements of stress at 5 m indicate that the roughness length of the snow surface is about lO−4 m.At 5 m height, the variations of the dimensionless wind shear, ϕm, and potential temperature gradient, ϕT, agree with previously determined forms of the Monin‐Obukhov similarity functions. Above 5 m, the behaviour of ϕm and ϕT is only qualitatively similar and surface‐layer similarity theory does not provide a good description of the profiles.Turbulence length scales have been deduced from vertical velocity power spectra. Under near‐neutral conditions, the ratio of turbulence length scale to measurement height is observed to decrease with increasing height of measurement. This observation is consistent with the variation of turbulence length scale with height implied by the measurements of ϕm. It is suggested that the stability of the overlying atmosphere restricts the depth of the turbulent boundary layer and hence the length scales of turbulence within this layer. Increasing stability causes a decrease of turbulence length scales at all levels.The ratios of turbulence kinetic energy to stress and temperature variance to heat flux are examined. Measurements are somewhat scattered, but the distribution of values varies little with height or stability. the form of the distribution suggests that large‐scale motions, possibly internal gravity waves, may be playing an important role in boundary‐layer processes.

Sufficient conditions for the strong stability of the differential equation [p(D)+f(t)q(D)]y= 0

AbstractA number of sufficient conditions for stability or strong stability, as used in the context of Hamiltonian systems, are found for the differential equationwhere the continuous functionf(t)is periodic ω int, D =d/dtandp(s),q(s)are real monic polynomials having special properties which allow the differential equation to be transformed into a canonical system ofksecond order equations.