A Case Study on Designing a Sliding Mode Controller to Stabilize the Stochastic Effect of Noise on Mechanical Structures: Residential Buildings Equipped with ATMD

This study aims to stabilize the unwanted fluctuation of buildings as mechanical structures subjected to earth excitation as the noise. In this study, the ground motion is considered as a Wiener process, in which the governing stochastic differential equations have been presented in the form of Ito equation. To stabilize the vibration of the system, the ATMD system is considered and located on the upmost story of the building. A sliding mode controller has been utilized to control the ATMD system, which is a robust controller in the presence of uncertainty. For this purpose, the design of a sliding mode controller for the general dynamic system with Lipschitz nonlinearity and considering the Ito relations has been accomplished. The mentioned design has been implemented considering the presence of the Weiner process and existence of uncertainty in the structure and actuator. Then, the obtained general control law has been generalized to control the ATMD system. The results show that the designed controller is effective to reduce the effect of the unwanted impused vibrations on the building.

Mean Square Synchronization of Stochastic Nonlinear Delayed Coupled Complex Networks

We investigate the problem of adaptive mean square synchronization for nonlinear delayed coupled complex networks with stochastic perturbation. Based on the LaSalle invariance principle and the properties of the Weiner process, the controller and adaptive laws are designed to ensure achieving stochastic synchronization and topology identification of complex networks. Sufficient conditions are given to ensure the complex networks to be mean square synchronization. Furthermore, numerical simulations are also given to demonstrate the effectiveness of the proposed scheme.

On Durbin's Series for the Density of First Passage Times

Durbin (1992) derived a convergent series for the density of the first passage time of a Weiner process to a curved boundary. We show that the successive partial sums of this series can be expressed as the iterates of the standard substitution method for solving an integral equation. The calculation is thus simpler than it first appears. We also show that, under a certain condition, the series converges uniformly. This strengthens Durbin's result of pointwise convergence. Finally, we present a modified procedure, based on scaling, which sometimes works better. These approaches cover some cases that Durbin did not.

On Durbin's Series for the Density of First Passage Times

Durbin (1992) derived a convergent series for the density of the first passage time of a Weiner process to a curved boundary. We show that the successive partial sums of this series can be expressed as the iterates of the standard substitution method for solving an integral equation. The calculation is thus simpler than it first appears. We also show that, under a certain condition, the series converges uniformly. This strengthens Durbin's result of pointwise convergence. Finally, we present a modified procedure, based on scaling, which sometimes works better. These approaches cover some cases that Durbin did not.

On some stochastic parabolic differential equations in a Hilbert space

We consider some stochastic difference partial differential equations of the form du(x,t,c)=L(x,t,D)u(x,t,c)dt+M(x,t,D)u(x,t−a,c)dw(t), where L(x,t,D) is a linear uniformly elliptic partial differential operator of the second order, M(x,t,D) is a linear partial differential operator of the first order, and w(t) is a Weiner process. The existence and uniqueness of the solution of suitable mixed problems are studied for the considered equation. Some properties are also studied. A more general stochastic problem is considered in a Hilbert space and the results concerning stochastic partial differential equations are obtained as applications.

First passage to a general threshold for a process corresponding to sampling at Poisson times

The problem of computing the distribution of the time of first passage to a constant threshold for special classes of stochastic processes has been the subject of considerable study. For example, Baxter and Donsker (1957) have considered the problem for processes with stationary, independent increments, Darling and Siegert (1953) for continuous Markov processes, Mehr and McFadden (1965) for Gauss-Markov processes, and Stone (1969) for semi-Markov processes. The results, however, generally express the first passage distribution in terms of transforms which can be inverted only in a relatively few special cases, such as in the classical case of the Weiner process and for certain stable and compound Poisson processes. For linear threshold functions and processes with non-negative interchangeable increments the first passage problem has been studied by Takács (1957) (an explicit result was obtained by Pyke (1959) in the special case of a Poisson process). Again in the case of a linear threshold, an explicit form for the first passage distribution was found by Slepian (1961) for the Weiner process. For the Ornstein-Uhlenbeck process and certain U-shaped thresholds the problem has recently been studied by Daniels (1969).

First passage to a general threshold for a process corresponding to sampling at Poisson times

The problem of computing the distribution of the time of first passage to a constant threshold for special classes of stochastic processes has been the subject of considerable study. For example, Baxter and Donsker (1957) have considered the problem for processes with stationary, independent increments, Darling and Siegert (1953) for continuous Markov processes, Mehr and McFadden (1965) for Gauss-Markov processes, and Stone (1969) for semi-Markov processes. The results, however, generally express the first passage distribution in terms of transforms which can be inverted only in a relatively few special cases, such as in the classical case of the Weiner process and for certain stable and compound Poisson processes. For linear threshold functions and processes with non-negative interchangeable increments the first passage problem has been studied by Takács (1957) (an explicit result was obtained by Pyke (1959) in the special case of a Poisson process). Again in the case of a linear threshold, an explicit form for the first passage distribution was found by Slepian (1961) for the Weiner process. For the Ornstein-Uhlenbeck process and certain U-shaped thresholds the problem has recently been studied by Daniels (1969).