Reliability of nonlinear stochastic controlled systems considering the dynamics of sensors and actuators

A closed-loop controlled system usually consists of the main structure, sensors, and actuators. The dynamics of sensors and actuators may influence the motion of the main structure. This article presents an analytical study on the first-passage reliability of a nonlinear stochastic controlled system under the consideration of the dynamics of sensors and actuators. The coupled dynamic equations of the controlled systems with sensors and actuators are first given, which are further integrated into a controlled, randomly excited, dissipated Hamiltonian system. By applying the stochastic averaging method for quasi-Hamiltonian systems, a one-dimensional averaged differential equation for the Hamiltonian function is obtained. The backward Kolmogorov equation associated with the averaged equation is then derived for the first-passage reliability analysis, from which the approximate reliability function and probability density of first-passage time are obtained. The accuracy of the proposed procedure is demonstrated by an example. A comparative analysis of the reliability of the system with/without sensors and actuators is carried out, which indicates that ignoring sensors and actuators will make underestimation of the reliability of the closed-loop system with small time. However, when time increases, there appears the opposite trend. Our findings provide a reference for control strategy design.

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Construction of first-passage-time densities for a diffusion process which is not necessarily time-homogeneous

Journal of Applied Probability ◽

10.2307/3214694 ◽

1991 ◽

Vol 28 (4) ◽

pp. 903-909 ◽

Cited By ~ 8

Author(s):

R. Gutiérrez Jáimez ◽

A. Juan Gonzalez ◽

P. Román Román

Keyword(s):

Diffusion Process ◽

Probability Density ◽

First Passage Time ◽

Passage Time ◽

Probability Density Functions ◽

New Method ◽

Kolmogorov Equation ◽

Density Functions ◽

First Passage ◽

Time Transformations

In Giorno et al. (1988) a new method for constructing first-passage-time probability density functions is outlined. This rests on the possibility of constructing the transition p.d.f. of a new time-homogeneous diffusion process in terms of a preassigned transition p.d.f. without making use of the classical space-time transformations of the Kolmogorov equation (Ricciardi (1976)).In the present paper we give an extension of this result to the case of a diffusion process X(t) which is not necessarily time-homogeneous, and a few examples are presented.

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Construction of first-passage-time densities for a diffusion process which is not necessarily time-homogeneous

Journal of Applied Probability ◽

10.1017/s0021900200042820 ◽

1991 ◽

Vol 28 (04) ◽

pp. 903-909 ◽

Cited By ~ 2

Author(s):

R. Gutiérrez Jáimez ◽

A. Juan Gonzalez ◽

P. Román Román

Keyword(s):

Diffusion Process ◽

Probability Density ◽

First Passage Time ◽

Passage Time ◽

Probability Density Functions ◽

New Method ◽

Kolmogorov Equation ◽

Density Functions ◽

First Passage ◽

Time Transformations

In Giorno et al. (1988) a new method for constructing first-passage-time probability density functions is outlined. This rests on the possibility of constructing the transition p.d.f. of a new time-homogeneous diffusion process in terms of a preassigned transition p.d.f. without making use of the classical space-time transformations of the Kolmogorov equation (Ricciardi (1976)). In the present paper we give an extension of this result to the case of a diffusion process X(t) which is not necessarily time-homogeneous, and a few examples are presented.

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Sufficient Conditions for a First Passage Time Process to be that of Brownian Motion

Journal of Applied Probability ◽

10.1017/s0021900200032678 ◽

1969 ◽

Vol 6 (01) ◽

pp. 218-223

Author(s):

M.T. Wasan

Keyword(s):

Differential Equation ◽

Brownian Motion ◽

First Passage Time ◽

Sufficient Conditions ◽

Passage Time ◽

Kolmogorov Equation ◽

First Passage ◽

State Variable ◽

Strong Markov ◽

Passage Times

In this paper we assign a set of conditions to a strong Markov process and arrive at a differential equation analogous to the Kolmogorov equation. However, in this case the duration variable is the net distance travelled and the state variable is a time, a situation precisely opposite to that of Brownian motion. Solving this differential equation under certain boundary conditions produces the density function of the first passage times of Brownian motion with positive drift (see [1]), with the aid of which we define a new stochastic process.

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First-Passage Failure of Quasi-Integrable Hamiltonian Systems

Journal of Applied Mechanics ◽

10.1115/1.1460912 ◽

2002 ◽

Vol 69 (3) ◽

pp. 274-282 ◽

Cited By ~ 40

Author(s):

W. Q. Zhu ◽

M. L. Deng ◽

Z. L. Huang

Keyword(s):

Hamiltonian Systems ◽

First Passage Time ◽

Digital Simulation ◽

Reliability Function ◽

Passage Time ◽

Kolmogorov Equation ◽

Conditional Probability Density ◽

Integrable Hamiltonian Systems ◽

First Passage ◽

First Passage Failure

The first-passage failure of quasi-integrable Hamiltonian systems (multidegree-of-freedom integrable Hamiltonian systems subject to light dampings and weakly random excitations) is investigated. The motion equations of such a system are first reduced to a set of averaged Ito^ stochastic differential equations by using the stochastic averaging method for quasi-integrable Hamitonian systems. Then, a backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function, and the conditional probability density and moments of first-passage time are obtained by solving these equations with suitable initial and boundary conditions. Two examples are given to illustrate the proposed procedure and the results from digital simulation are obtained to verify the effectiveness of the procedure.

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Reliability of Strongly Nonlinear Single Degree of Freedom Dynamic Systems by the Path Integration Method

Journal of Applied Mechanics ◽

10.1115/1.2967896 ◽

2008 ◽

Vol 75 (6) ◽

Cited By ~ 29

Author(s):

Daniil Iourtchenko ◽

Eirik Mo ◽

Arvid Naess

Keyword(s):

Path Integration ◽

Integration Method ◽

First Passage Time ◽

Degree Of Freedom ◽

First Passage ◽

Controlled System ◽

Single Degree Of Freedom ◽

Strongly Nonlinear ◽

Single Degree ◽

Path Integration Method

This paper presents a first passage type reliability analysis of strongly nonlinear stochastic single-degree-of-freedom systems. Specifically, the systems considered are a dry friction system, a stiffness controlled system, an inertia controlled system, and a swing. These systems appear as a result of implementation of the quasioptimal bounded in magnitude control law. The path integration method is used to obtain the reliability function and the first passage time.

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Remaining Useful Lifetime Prognosis of Controlled Systems: A Case of Stochastically Deteriorating Actuator

Mathematical Problems in Engineering ◽

10.1155/2015/356916 ◽

2015 ◽

Vol 2015 ◽

pp. 1-16 ◽

Cited By ~ 13

Author(s):

Danh Ngoc Nguyen ◽

Laurence Dieulle ◽

Antoine Grall

Keyword(s):

Particle Filtering ◽

Closed Loop ◽

Health Condition ◽

Second Step ◽

Proportional Integral Derivative ◽

Level Control ◽

Piecewise Deterministic Markov Process ◽

Controlled System ◽

Controlled Systems ◽

Useful Lifetime

This paper addresses the case of automatic controlled system which deteriorates during its operation because of components’ wear or deterioration. Depending on its specific closed-loop structure, the controlled system has the ability to compensate for disturbances affecting the actuators which can remain partially hidden. The deterioration modeling and the Remaining Useful Lifetime (RUL) estimation for such closed-loop dynamic system have not been addressed extensively. In this paper, we consider a controlled system with Proportional-Integral-Derivative controller. It is assumed that the actuator is subject to shocks that occur randomly in time. An integrated model is proposed to jointly describe the state of the controlled process and the actuator deterioration. Only the output of the controlled system is available to assess its health condition. By considering a Piecewise Deterministic Markov Process, the RUL of the system can be estimated by a two-step approach. In the first step referred as the “Diagnosis” step, the system state is estimated online from the available monitoring observations by using a particle filtering method. In the second step referred as the “Prognosis” step, the RUL is estimated as a conditional reliability by Monte Carlo simulation. To illustrate the approach, a simulated tank level control system is used.

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Sufficient Conditions for a First Passage Time Process to be that of Brownian Motion

Journal of Applied Probability ◽

10.2307/3212290 ◽

1969 ◽

Vol 6 (1) ◽

pp. 218-223 ◽

Cited By ~ 4

Author(s):

M.T. Wasan

Keyword(s):

Differential Equation ◽

Brownian Motion ◽

First Passage Time ◽

Sufficient Conditions ◽

Passage Time ◽

Kolmogorov Equation ◽

First Passage ◽

State Variable ◽

Strong Markov ◽

Passage Times

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First passage time for the state of exact chemical equilibrium

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19800777 ◽

1980 ◽

Vol 45 (3) ◽

pp. 777-782 ◽

Cited By ~ 1

Author(s):

Milan Šolc

Keyword(s):

Chemical Equilibrium ◽

First Passage Time ◽

Passage Time ◽

The State ◽

Recurrence Time ◽

First Passage ◽

First Order ◽

The Mean ◽

Passage Times ◽

Number Of Particles

The establishment of chemical equilibrium in a system with a reversible first order reaction is characterized in terms of the distribution of first passage times for the state of exact chemical equilibrium. The mean first passage time of this state is a linear function of the logarithm of the total number of particles in the system. The equilibrium fluctuations of composition in the system are characterized by the distribution of the recurrence times for the state of exact chemical equilibrium. The mean recurrence time is inversely proportional to the square root of the total number of particles in the system.

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