Random Initial State Research Articles

A Stochastic-Dynamical Approach to the Study of the Natural Variability of the Climate

A method, suggested by Leith (1975), which employed stochastic-dynamic forecasts obtained from a general circulation model in such a way as to satisfy the definition of climatic noise, was used to validate assumptions accounting for the effects of external influences in estimating the climatic noise. Two assumptions were investigated: (1) that the weather fluctuations can be represented as a Markov process, and (2) that changing external conditions do not influence the atmosphere's statistical properties on short time scales. The general circulation model's simulation of the daily weather fluctuations was generated by performing integrations with prescribed climatological boundary conditions for random initial atmospheric states, with resulting dynamical forecasts providing an ensemble of simulated data for the autoregressive modeling of weather fluctuations. To estimate the climatic noise from the observational data (consisting of hourly values of sea level pressure and surface temperature at 54 U.S. stations for the month of January for the years 1949-1975) use of the short time-scale assumption is made. The simulated and observed data were found not to be consistent with either white noise or a Markov process of weather fluctuations. Good agreement was found between the results of the hypothetical testing of the simulated and the observed surface temperatures; and only partial support was found for the short time-scale assumption, i.e., for sea level pressure.

A further note on backwards Markovian models (Corresp.)

Given a wide-sense forwards Markovian model, namely a state-space system driven by a white-noise process that is uncorrelated with the random initial state of the system, several papers have shown how to obtain a corresponding {\sl backwards} Markovian model: a system driven backwards in time from a 'terminal' state by a white-noise process that is uncorrelated with this state, and such that the joint second-order statistics of the backwards-model state and output processes equal those of the corresponding forwards-model processes. A very direct derivation is presented that shows in effect how a simple rearrangement of the forwards

Invariants of optimal minimal-order observer-based compensators

The relations Characterizing the optimal minimal-order observer-based compensators for a linear time-invariant multivariable system in a certain canonical form, with a random initial state (or equivalently, known initial state with white driving noise) have been reported by Miller [1] and independently by others [2], [3]. In this note, we establish in general that the plant transfer function uniquely determines the optimal compensator transfer function, and that this characterizes precisely the degrees of freedom in the compensator design. Any two realizations of the optimal compensator dynamics yield the same performance.

Linear systems with non-stationary random inputs†

This paper concerns the response correlation matrix of a linear time variable system with stationary or non-stationary stochastic inputs modulated by linear homogeneous operators. By adapting Lanczos' ‘selected points’ method an algorithm is introduced to compute the state covariance matrix of systems with deterministic initial state. Certain computational features of response correlation matrices of systems with random initial state and random inputs are also outlined. Limited numerical studies indicate that the proposed method is efficient to treat uni-dimensional and small-order multi-dimensional systems. Mean square response comparisons with some of the previously studied systems indicate good agreement and the saving in machine time appears to be substantial.

Design of control systems with random parameters†

In this paper a method for designing control systems with random parameters and random initial state is presented. The observed signal available for feedback is a function of the state and system parameters corrupted by additive white Gaussian noise with zero mean. The proposed control consists of an open-loop term, which is the optimal control for the parameters and initial state equal to their mean values, plus a feedback correction term. The feedback correction term is a linear function of the estimated values of the deviation in initial state and system parameters from their mean values. The control tends to the optimally sensitive control when the measurement noise tends to zero, and the feedback correction tends to zero when the measurement noise tends to infinity. A numerical experiment with a simplified model for nuclear reactor control shows that in spite of fairly large measurement noise, significant improvement in performance can be achieved using the proposed control compared to open-loop control.

Dynamic systems with random initial state

A perturbation scheme is described to treat time variable cryptodeterministic systems. According to Moyal's degree of randomness criteria the method provides a complete stochastic characterization of the system response. Certain digital computational features, when the perturbation scheme is not applicable, are also outlined. For an assumed random initial state, the results are then applied to describe the transient flapping oscillations of a helicopter blade which in forward flight has periodically varying aerodynamic damping and spring parameters.

Stress Softening in Natural Rubber Vulcanizates. IV. Unfilled Vulcanizates

Abstract Stress softening (Mullins effect) in gum natural rubber vulcanizates is similar in magnitude to that in carbon black filled vulcanizates. The amount of stress softening is slightly greater in vulcanizates cured to produce predominantly polysulfide crosslinks than in those containing monosulfide or carbon to carbon crosslinks. The total recovery of stress softening in the vulcanizates containing monosulfide or carbon to carbon crosslinks suggests that the phenomenon is attributable to a quasiirreversible rearrangement of molecular networks due to localized non-affine deformation resulting from short chains reaching the limit of their extensibility. This nonaffine deformation results in a displacement of the network junctions from their initial random state.

Stress Softening in Natural Rubber Vulcanizates III. Carbon Black Filled Vulcanizates

Abstract This paper has confirmed the conclusions of the previous paper that the stress softening (Mullins effect) of a black-loaded vulcanizate is similar in magnitude to the stress softening of a gum rubber if the two vulcanizates are stretched initially to the same stress. The initial stress used in the present work was 180 kg/cm2, which is very near to the breaking stress of these vulcanizates. The similarity of the normalized stress-strain curves for all the vulcanizates, both gum and loaded with 60 phr of different types of black, suggests that the main difference between the stress-strain characteristics of a filled and a pure gum rubber, after the initial stressing cycle, can be accounted for by the strain amplification factor X. The more reinforcing blacks possess the higher X factors, i.e., they stiffen the rubber more than, for example, a fine thermal black. It is concluded that the black is acting mainly in a stiffening capacity due to the hydrodynamic effects of the degenerate carbon black networks. For sulfur crosslinked pure gum vulcanizates, in which the crosslinks are polysulfidic, the stress softening is partly associated with the breakage of polysulfide linkages. These reform in the extended condition and produce a real permanent set, but the major stress softening is attributed to the incomplete recovery of the crosslinked network to its initial random state due to network junctions or similar associations being displaced in a nonaffine way during extension. For example, junctions at the ends of chains which become fully extended at relatively low extensions will be displaced in this way. Thus when the rubber is subsequently strained, the network is already in a preferred disposition.

Stress softening in natural rubber vulcanizates. Part IV. Unfilled vulcanizates

AbstractStress softening (Mullins effect) is observed in gum natural rubber vulcanizates. The magnitude of this softening is similar to that in carbon black‐filled vulcanizates. The amount of stress softening is slightly greater in vulcanizates cured to produce predominantly polysulfide crosslinks than in those containing monosulfide or carbon–carbon crosslinks. The total recovery of stress softening in the vulcanizates containing monosulfide or carbon–carbon crosslinks suggests that the phenomenon is attributable to a quasi‐irreversible rearrangement of molecular networks due to localized nonaffine deformation resulting from short chains reaching the limit of their extensibility. This non‐affine deformation results in a displacement of the network junctions from their initial random state.