Stochastic process

About: Stochastic process is a research topic. Over the lifetime, 31227 publications have been published within this topic receiving 898736 citations. The topic is also known as: random process & stochastic processes.

Random processes with independent increments

Abstract: 0. Preliminary Informationh.- 0.1 Probability Space.- 0.2 Random Functions and Processes.- 0.3 Conditional Probabilities.- 0.4 Independence.- 1. Sums of Independent Random Variables.- 1.1 Main Inequalities.- 1.2 Renewal Scheme.- 1.3 Random Walks. Recurrence.- 1.4 Distribution of Ladder Functions.- 2. General Processes with Independent Increments (Random Measures).- 2.1 Nonnegative Random Measures with Independent Values (r.m.i.v.).- 2.2 Random Measures with Alternating Signs.- 2.3 Stochastic Integrals and Countably Additive r.m.i.v.- 2.4 Random Linear Functional and Generalized Functions.- 3. Processes with Independent Increments. General Properties.- 3.1 Decomposition of a Process. Properties of Sample Functions.- 3.2 Stochastically Continuous Processes.- 3.3 Properties of Sample Functions.- 3.4 Locally Homogeneous Processes with Independent Increments.- 4. Homogeneous Processes.- 4.1 General Properties.- 4.2 Additive Functionals.- 4.3 Composed Poisson Process.- 4.4 Homogeneous Processes in R.- 5. Multiplicative Processes.- 5.1 Definition and General Properties.- 5.2 Multiplicative Processes in Abelian Groups.- 5.3 Stochastic Semigroups of Linear Operators in Rd.- Notes.- References.

On adaptive control processes

Abstract: One of the most challenging areas in the field of automatic control is the design of automatic control devices that 'learn' to improve their performamce based upon experience, i.e., that can adapt themselves to circumstances as they find them. The military and commercial implications of such devices are impressive, and interest in the two main areas of research in the field of control, the USA and the USSR, runs high. Unfortunately, though, both theory and construction of adaptive controllers are in their infancy, and some time may pass before they are commonplace. Nonetheless, development at this time of adequate theories of processes of this nature is essential. The purpose of our paper is to show how the functional equation technique of a new mathematical discipline, dynamic programming, can be used in the formulation and solution of a variety of optimization problems concerning the design of adaptive devices. Although, occasionally, a solution in closed form can be obtained, in general, numerical solution via the use of high-speed digital computers is contemplated. We discuss here the closely allied problems of formulating adaptive control processes in precise mathematical terms and of presenting feasible computational algoritbms for determining numerical solutioms. To illustrate the general concepts, consider a system which is governed by the inhomogeneous Van der Pol equation \ddot{x} + \mu(x^{2} - 1) \dot{x} + x = r(t), 0 \leq t \leq T , where r(t) is a random function whose statistical properties are only partially known to a feedback control device which seeks to keep the system near the unstable equilibrium state x = 0, \dot{x} = 0 . It proposes to do this by selecting the value of μ as a function of the state of the system at time t , and the time t itself. By observing the random process r(t) , the controller may, with the passage of time, infer more and more concerning the statistical properties of the function r(t) and thus may be expected to improve the quality of its control decisions. In this way the controller adapts itself to circumstances as it finds them. The process is thus an interesting example of adaptive control, and, conceivably, with some immediate applications. Lastly, some areas of this general domain requiring additional research are indicated.

Stochastic analysis of structural and mechanical vibrations

Abstract: 1. Introduction. 2. Analysis of Stochastic Processes. 3. Time Domain Linear Vibration Analysis. 4. Frequency Domain Analysis. 5. Gaussian and Non-Gaussian Stochastic Processes. 6. Occurrence Rates and Distributions of Extremes. 7. Linear Systems with Multiple Inputs and Outputs. 8. State-Space Analysis. 9. Introduction to Nonlinear Stochastic Vibration. 10. Stochastic Analysis of Fatigue Damage. Appendix A. Analysis of Random Variables. Appendix B. Gaussian Random Variables. Appendix C. Dirac Delta Functions. Appendix D. Fourier Analysis. References.

Adaptive Event-Triggered SMC for Stochastic Switching Systems With Semi-Markov Process and Application to Boost Converter Circuit Model

Abstract: In this article, the sliding mode control (SMC) design is studied for a class of stochastic switching systems subject to semi-Markov process via an adaptive event-triggered mechanism. Network-induced communication constraints, semi-Markov switching parameters, and uncertain parameters are considered in a unified framework for the SMC design. Due to the constraint of measuring transducers, the system states always appear with unmeasurable characteristic. Compared with the traditional event-triggered mechanism, the adaptive event-triggered mechanism can effectively reduce the number of triggering than the static event-triggered mechanism. During the data transmission of network communication systems, network-induced delays are characterized from the event trigger to the zero-order holder. The aim of this work is to design an appropriate SMC law based on an adaptive event-triggered communication scheme such that the resulting closed-loop system could realize stochastic stability and reduce communication burden. By introducing the stochastic semi-Markov Lyapunov functional, sojourn-time-dependent sufficient conditions are established for stochastic stability. Then, a suitable SMC law is designed such that the system state can be driven onto the specified sliding surface in a finite-time region. Finally, the simulation study on boost converter circuit model (BCCM) illustrates the effectiveness of the theoretical findings.