Liapunov criteria for weak stochastic stability.
Sufficient conditions for recurrence and positivity of diffusion process defined by stochastic differential equation, using Liapunov function
DIFFERENTIAL EQUATION↗
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Applications of Liapunov stability theory to control systems.
Liapunov stability theory applied to control system, noting stability of function and stochastic differential equations
LIAPUNOV FUNCTION↗
A compilation of results pertaining to the behavior of phase locked loops
State-of-the art on phase locked loops PLL is reported by summarizing some specific results. Following a statement of the overall analysis and design objectives, results are presented in a format identifying working terminology, inherent assumptions, and references for each result. The use of PLL in tracking, synchronization, and demodulation is reemphasized, as well as the mathematical challenge involved in solving nonlinear stochastic differential equations.
Gleicher, N.↗
A stochastic model for eye movements during fixation on a stationary target.
A stochastic model describing small eye movements occurring during steady fixation on a stationary target is presented. Based on eye movement data for steady gaze, the model has a hierarchical structure; the principal level represents the random motion of the image point within a local area of fixation, while the higher level mimics the jump processes involved in transitions from one local area to another. Target image motion within a local area is described by a Langevin-like stochastic differential equation taking into consideration the microsaccadic jumps pictured as being due to point processes and the high frequency muscle tremor, represented as a white noise. The transform of the probability density function for local area motion is obtained, leading to explicit expressions for their means and moments. Evaluation of these moments based on the model is comparable with experimental results.
Vasudevan, R.↗
Approximations to and local properties of diffusions with discontinuous controls
The paper discusses several properties of control systems defined by stochastic differential equations, which are defined by the method of Girsanov, using a transformation of measures, and where the controls are discontinuous. Uniqueness of the multivariate distributions of the process is proved, and it is shown that the process is a limit, in a natural sense, of a certain discrete time approximation. Other questions, concerning the effects on the distributions of the paths, and of the cost of approximating the control by a smooth control and concerning local properties of the solution, are discussed.
Kushner, H. J.↗
Position accuracy of aircraft area navigation systems and the effect of system parameters
The steady-state solution to the stochastic differential equation describing the error covariance matrix for a simplified area navigation system has been obtained. The solution shows that nominal error deviations of less than 1.0 nmi are feasible for a DME system. The manner in which estimates are affected by range, air data system accuracy, measurement time interval, and gust deviations, and their impacts on the area navigation system are discussed.
Foudriat, E. C.↗
Structural Properties and Estimation of Delay Systems
Two areas in the theory of delay systems were studied: structural properties and their applications to feedback control, and optimal linear and nonlinear estimation. The concepts of controllability, stabilizability, observability, and detectability were investigated. The property of pointwise degeneracy of linear time-invariant delay systems is considered. Necessary and sufficient conditions for three dimensional linear systems to be made pointwise degenerate by delay feedback were obtained, while sufficient conditions for this to be possible are given for higher dimensional linear systems. These results were applied to obtain solvability conditions for the minimum time output zeroing control problem by delay feedback. A representation theorem is given for conditional moment functionals of general nonlinear stochastic delay systems, and stochastic differential equations are derived for conditional moment functionals satisfying certain smoothness properties.
Kwong, R. H. S.↗
Estimation and filter stability of stochastic delay systems
Linear and nonlinear filtering for stochastic delay systems are studied. A representation theorem for conditional moment functionals is obtained, which, in turn, is used to derive stochastic differential equations describing the optimal linear or nonlinear filter. A complete characterization of the optimal filter is given for linear systems with Gaussian noise. Stability of the optimal filter is studied in the case where there are no delays in the observations. Using the duality between linear filtering and control, asymptotic stability of the optimal filter is proved. Finally, the cascade of the optimal filter and the deterministic optimal quadratic control system is shown to be asymptotically stable as well.
Kwong, R. H.↗
Towards sub-optimal stochastic control of partially observable stochastic systems
A class of multidimensional stochastic control problems with noisy data and bounded controls encountered in aerospace design is examined. The emphasis is on suboptimal design, the optimality being taken in quadratic mean sense. To that effect the problem is viewed as a stochastic version of the Lurie problem known from nonlinear control theory. The main result is a separation theorem (involving a nonlinear Kalman-like filter) suitable for Lurie-type approximations. The theorem allows for discontinuous characteristics. As a byproduct the existence of strong solutions to a class of non-Lipschitzian stochastic differential equations in dimensions is proven.
Ruzicka, G. J.↗
Towards sub-optimal stochastic control of partially observable stochastic systems
The paper deals with a class of multidimensional stochastic control problems with noisy data and bounded controls encountered in aerospace design. The emphasis is on suboptimal design, the optimality being taken in quadratic mean sense. To that effect the problem is viewed as a stochastic version of the Lurie problem known from nonlinear control theory. The main result is a separation theorem (involving a nonlinear Kalman-like filter) suitable for Lurie-type approximations. The theorem allows for discontinuous characteristics. As a byproduct the existence of strong solutions to a class of non-Lipschitzian stochastic differential equations in n dimensions is proved.
Ruzicka, G. J.↗
A statistical rain attenuation prediction model with application to the advanced communication technology satellite project. Part 2: Theoretical development of a dynamic model and application to rain fade durations and tolerable control delays for fade countermeasures
A dynamic rain attenuation prediction model is developed for use in obtaining the temporal characteristics, on time scales of minutes or hours, of satellite communication link availability. Analagous to the associated static rain attenuation model, which yields yearly attenuation predictions, this dynamic model is applicable at any location in the world that is characterized by the static rain attenuation statistics peculiar to the geometry of the satellite link and the rain statistics of the location. Such statistics are calculated by employing the formalism of Part I of this report. In fact, the dynamic model presented here is an extension of the static model and reduces to the static model in the appropriate limit. By assuming that rain attenuation is dynamically described by a first-order stochastic differential equation in time and that this random attenuation process is a Markov process, an expression for the associated transition probability is obtained by solving the related forward Kolmogorov equation. This transition probability is then used to obtain such temporal rain attenuation statistics as attenuation durations and allowable attenuation margins versus control system delay.
Manning, Robert M.↗
Stochastic time-optimal control problems
Two types of stochastic time-optimal controls in a one-dimensional setting are considered. Multidimensional problems, in the case of complete state information available and the system modeled by stochastic differential equations, are studied under the formulation of minimizing the expected transient-response time. The necessary condition of optimality is the satisfaction for the value function of a parabolic partial differential equation with boundary conditions. The sufficient condition of optimality is also provided, based on Dynkin's formula. Finally, three examples are given.
Zhang, W.↗
Performance evaluation of a Doppler radar system for wind shear detection
Nonlinear stochastic differential equations are used to model wind shear, and extended Kalman filters are used to generate state estimates from measurements received from a Doppler radar onboard an aircraft. Likelihood-ratio tests are then used to detect the presence of wind shear. The performance of the system is evaluated by deriving theoretical expressions for the false alarm and miss error probabiilties. The approach uses a Fokker-Planck equation. The overall methodology is general and should be of interest in other applications.
Khalaf, Camille S.↗
Dynamics of growing interfaces
We propose a stochastic differential equation for the growth of interfaces that is invariant under reparametrization and thus incorporates the change in local time scales resulting from nonlinear distortions. In its most general form, the equation accommodates overhanging configurations and in the nearly planar limit it reduces to previously proposed interface evolution models. The new features are relevant and lead to qualitatively new behavior at long times.
Maritan, Amos↗
Complexity of life via collective mind
e mind is introduced as a set of simple intelligent units (say, neurons, or interacting agents), which can communicate by exchange of information without explicit global control. Incomplete information is compensated by a sequence of random guesses symmetrically distributed around expectations with prescribed variances. Both the expectations and variances are the invariants characterizing the whole class of agents. These invariants are stored as parameters of the collective mind, while they contribute into dynamical formalism of the agents' evolution, and in particular, into the reflective chains of their nested abstract images of the selves and non-selves. The proposed model consists of the system of stochastic differential equations in the Langevin form representing the motor dynamics, and the corresponding Fokker-Planck equation representing the mental dynamics (Motor dynamics describes the motion in physical space, while mental dynamics simulates the evolution of initial errors in terms of the probability density). The main departure of this model from Newtonian and statistical physics is due to a feedback from the mental to the motor dynamics which makes the Fokker-Planck equation nonlinear. Interpretation of this model from mathematical and physical viewpoints, as well as possible interpretation from biological, psychological, and social viewpoints are discussed. The model is illustrated by the dynamics of a dialog.
collective intelligence↗
A Stochastic Fractional Dynamics Model of Space-time Variability of Rain
Rainfall varies in space and time in a highly irregular manner and is described naturally in terms of a stochastic process. A characteristic feature of rainfall statistics is that they depend strongly on the space-time scales over which rain data are averaged. A spectral model of precipitation has been developed based on a stochastic differential equation of fractional order for the point rain rate, that allows a concise description of the second moment statistics of rain at any prescribed space-time averaging scale. The model is thus capable of providing a unified description of the statistics of both radar and rain gauge data. The underlying dynamical equation can be expressed in terms of space-time derivatives of fractional orders that are adjusted together with other model parameters to fit the data. The form of the resulting spectrum gives the model adequate flexibility to capture the subtle interplay between the spatial and temporal scales of variability of rain but strongly constrains the predicted statistical behavior as a function of the averaging length and times scales. We test the model with radar and gauge data collected contemporaneously at the NASA TRMM ground validation sites located near Melbourne, Florida and in Kwajalein Atoll, Marshall Islands in the tropical Pacific. We estimate the parameters by tuning them to the second moment statistics of radar data. The model predictions are then found to fit the second moment statistics of the gauge data reasonably well without any further adjustment.
TRMM↗
Convergence of the discretization equations which arise in stochastic control
A problem is considered which arises in the numerical analysis of the degenerate partial differential equations of stochastic control theory. The relationship between diffusions and finite state Markov chains is considered. The role of probability theory in numerical analysis is illustrated.
Kushner, H. J.↗
A Fast Monte Carlo Method for Model-Based Prognostics Based on Stochastic Calculus
This work proposes a fast Monte Carlo method to solve differential equations utilized in model-based prognostics. The methodology is derived from the theory of stochastic calculus, and the goal of such a method is to speed up the estimation of the probability density functions describing the independent variable evolution over time. In the prognostic scenarios presented in this paper, the stochastic differential equations describe variables directly or indirectly related to the degradation of a monitored system. The method allows the estimation of the probability density functions by solving the deterministic equation and approximating the stochastic integrals using samples of the model noise. By so doing, the prognostic problem is solved without the Monte Carlo simulation based on Euler's forward method, which is typically the most time consuming task of the prediction stage. Three different prognostic scenarios are presented as proof of concept: (i) life prediction of electrolytic capacitors, (ii) remaining time to discharge of Lithium-ion batteries, and (iii) prognostic of cracked structures under fatigue loading. The paper shows how the method produces probability density functions that are statistically indistinguishable from the distributions estimated with Euler's forward Monte Carlo simulation. However, the proposed solution is orders of magnitude faster when computing the time-to-failure distribution of the monitored system. The approach may enable complex real-time prognostics and health management solutions with limited computing power.
stochastic calculus↗