Class Of Stochastic Systems Research Articles

On Stochastic Analysis of a Class of Chemical Systems in Very Far Away from Equilibrium

The stochastic system of Schlogl's bimolecular reaction model has only a trivial stationary distribution in very far away from equilibrium and has been considered not to cause nonequilibrium transition. However in the same manner there exist a class of stochastic systems which have only a trivial stationary distribution too in very far away from equilibrium. This has its origin in strong dependency of stationary distribution on both the size of system and the distance from equilibrium. Then it is pointed out that reasonable results are obtained by the joint infinite limit under a relation between the size of system and the distance from equilibrium.

Schwingungsisolation stochastisch gestörter systeme mit methiden der optimalen steuerung

In many practical problems it is necessary to stabilize stochastic disturbed dynamic systems. In mechanics these are for instance problems of active vibration isolation. This paper is a contribution to this problem by means of an optimal control algorithm for the following class of stochastic systems: A non-linear dynamical system is subjected to additive white noise. Amplitude bounds are imposed on the component of the control vector. The cost functional is the expectation of a quadratic function of state and control variables under stationary stochastic conditions. By combination of the statistical linearization method with dynamic programming we obtain an iterative procedure for computing the sub-optimal control. This iterative method starts with the solution of the standard (deterministic) linear regulator problem (nonconstrained control). In order to find a sub-optimal control we approximate the solution of the BELLMAN's equation by a quadratic form. To be consistent with this approximation the control and the equation of motion must be linear. Therefore, statistical linearization is used to achieve consistency of the nonlinear control with the quadratic approximation. As an example we consider a problem of vehicle dynamics. For calculating the vertical vibration of a car the three-mass-model is a suitable approximation. The nonlinearities to be taken into account are the following: • - nonlinearity of the sat-function-type due to bounded control • - nonlinearity of a cubic type describing the stiffness of the wheel • - nonlinearity of a piecewise linear type. Furthermore we assume that the vibrations are only caused by the roughness of the road. The spectral density of the road can be approximated by formfilter or - in a more simple manner - by calculating the mean integral value between the minimal and the maximal frequencies. The variance E[x̋ A 2] and E[c R 2(c R−x E) 2] are used as criteria for the quality of the vibration isolation. A well-known fact occuring in vehicle dynamic is an increasing of E[c R 2(x R−x E) 2] after a minimization of E[x̋ A 2] and vice versa. In this situation it is advantageous to use bounded control. Some numerical results show the influence of the velocity and control force to the variances R yi yI . Finally, a short discussion of the controllability is given in the case of a three-mass-model.

Off-line parameter identification of stochastic systems

The main objective of the present paper is to extend some previously reported results (Kudva-Narandra 1974) to a class of stochastic system. The parameter identification scheme suggested in this paper requires two different measurements associated with an open-loop and a closed-loop systems. The stability of the closed-loop system along with the knowledge on the order of system are assumed. The stochastic version of Lyapunov's second method, along with a modified Meyer-Kalman-Yakubovich (M.K.Y.) lemma is used in deriving the stability properties of the parameter identification scheme.

Optimal control of systems governed by stochastic McShane differential equations with fixed terminal time

In this paper we consider a class of stochastic systems governed by McShane stochastic differential equations for which both parameters and controls are to be chosen optimally with respect to a certain performance criterion over a fixed time interval. It is shown that this problem can be converted into an equivalent optimization problem of distributed parameter systems of parabolic type with a Cauchy boundary condition. For this reduced problem a necessary condition for optimality is given. This result is then used to obtain the individual necessary conditions for optimal controls and optimal parameters. Finally, a bang—bang principle is shown to hold true for n class of linear optimal control problems.

Stability criteria for stochastic systems with colored multiplicative noise

In this paper a particular class of stochastic systems is considered, described by linear differential equations with multiplicative colored noise parameters; it is assumed that a certain commutativity condition is true. For these systems explicit criteria are derived for the asymptotic stability of the moments of any order and also for almost sure asymptotic stability.

Frequency stability criteria for non-linear stochastic systems

In this paper general classes of stochastic systems described by stochastic integrals of the Volterra type are considered. Existence and uniqueness of random solutions as well as stochastic stability criteria of the Popov frequency form are established. These frequency stability criteria relate not only to the linear system dynamics (or the kernel of the Volterra integral) of the system, but also to the characteristics of the additive and multiplicative random processes that govern the system.

Optimal control of stochastic Ito differential systems by fixed terminal time

In this paper, the authors consider a class of stochastic systems described by Ito differential equations for which both controls and parameters are to be chosen optimally with respect to a certain performance index over a fixed time interval. The controls to be optimized depend only on partially observed current states as in a work of Fleming. However, he considered, instead, a problem of optimal control of systems governed by stochastic Ito differential equations with Markov terminal time. The fixed time problems usually give rise to the Cauchy problems (unbounded domain) whereas the Markov time problems give rise to the first boundary value problems (bounded domain). This fact makes the former problems relatively more involved than the latter. For the latter problems, Fleming has reported a necessary condition for optimality and an existence theorem of optimal controls. In this paper, a necessary condition for optimality for both controls and parameters combined together is presented for the former problems.

Optimal control of stochastic Ito differential systems by fixed terminal time

In this paper, the authors consider a class of stochastic systems described by Ito differential equations for which both controls and parameters are to be chosen optimally with respect to a certain performance index over a fixed time interval. The controls to be optimized depend only on partially observed current states as in a work of Fleming. However, he considered, instead, a problem of optimal control of systems governed by stochastic Ito differential equations with Markov terminal time. The fixed time problems usually give rise to the Cauchy problems (unbounded domain) whereas the Markov time problems give rise to the first boundary value problems (bounded domain). This fact makes the former problems relatively more involved than the latter. For the latter problems, Fleming has reported a necessary condition for optimality and an existence theorem of optimal controls. In this paper, a necessary condition for optimality for both controls and parameters combined together is presented for the former problems.

Solutions of generalized stochastic optimization problems

Abstract In this paper we consider the problems of existence of optimal controls for a large class of stochastic systems whose responses are Markor processes with jumps. The usual control problem is deparametrized and the problem is reformulated in such a way that one chooses the drift coefficient from a suitable class in some function space. The associated advantage is that existence theorems are obtained for openloop controls, feed-back controls, as well as controls considered as measures on the Borel algebra of subsets of the control restraint set. The results also cover parameter optimization.

Testing hypotheses for a certain class of stochastic systems with distributed parameters

Testing hypotheses for a certain class of stochastic systems with distributed parameters

Stability analysis of a particular class of stochastic system

In the letter, a particular class of linear system with multiplicative coloured noise is considered. It is shown that the stability of the moments can be analysed by means of an equivalent ltô equation involving white noise. The necessary and sufficient condition for almost sure asymptotic stability is also given.

Optimal feedback control for a class of stochastic systems

Abstract In this paper it ia shown that certain problems of optimal (feedback) control of stochastic systems can be reduced to an optimal control problem of a class of nonstandard distributed parameter systems (with controls appearing in the coefficients). Fixed time problems reduce to optimal control problems of distributed parameter system with Canchy condition and Markov time problems reduce to those with first boundary condition. (Il'In, Kalashnikov and Oleinik 1962, p. 40.)

Modelling a particular class of stochastic systems†

In this paper a method is given for obtaining a mathematical model of a class of black boxes having multiple inputs and multiple outputs in terms of Ito stochastic integral equations. This method is applicable to the class of black boxes having ergodic correlation functions when there is zero applied input. The point of view adopted in this paper is phenomenological in that it is desired that calculations made using the mathematical model should be 'close' to what is actually observed at the output of the black box.

Infinite time optimal control for a class of stochastic systems

Optimization results developed by the author (Brandeberry and Wu 1970) for a class of stochastic regulator problems will be extended from the finite time interval to the infinite time interval. Conditions are given for the existence and stability of the infinite constant feedback of the system state; necessary conditions will also be obtained for the inverse problem (i.e. conditions for a linear constant control law to be optimal for some cost functional).

Stronger form for the position-momentum uncertainty relation

A stronger and more general form of the position-momentum uncertainty relation is derived on the basis of stochastic theory; this new relation applies equally well to classical stochastic and quantum-mechanical systems.

Identification and control of a class of stochastic systems with unknown parameters†

Abstract We consider the recursive estimation of the unknown parameters in the stochastic differential equation describing the system utilizing the observations of the state of the system. We show that the unknown parameters can be divided into two groups so that the true values of the members of the first group can be recovered instantaneously whereas the true values of the members of the second group cannot be recovered in finite time. We derive the recursive equations for the conditional mean and covariance of the unknown parameters of the second. Then we consider the determination of the optimal feedback control function for minimizing a suitable criterion function. We will show that the optimal feedback control is a function of the current state measurement, the conditional mean and variance of the unknown parameters. We mention a method of approximation of the optimal feedback control function and the results of the simulation of the control scheme.

Modelling and simulation of stochastic systems†

In the field of automatic control, systems with stochastic inputs are frequently encountered. The purpose of this paper is twofold: (I) to treat the problem of mathematically modelling a general class of stochastic systems, and (2) to simulate the optimal control for a special case of this class of stochastic systems, specifically a linear time-invariant system with stochastic inputs and stochastically perturbed observations. The general class of systems considered in the modelling includes systems having non-linear and time-varying state dependence and additive white noise which is added in a non-linear and time-varying manner.

On stochastic optimal control

It is shown that, for a class of stochastic systems, i.e., those in which the cost increases as the distance between the stochastic and the deterministic controls increases, the optimal stochastic control is the conditional expectation of the deterministic control, given the measurement history.

Optimal Regulation of a Class of Linear Stochastic Systems Relative to Quadratic Criteria

ABSTRACT A method of optimally regulating a class of stochastic linear systems relative to quadratic performance criteria is presented. The class of stochastic systems considered are those whose dynamics are described by an nth order stochastic linear differential equation. The system input is passed through a randomly varying gain. The control function is taken to be unconstrained in magnitude. The stochastic processes treated are increments of Brownian motion and generalized Poisson. A deterministic optimal feedback control is obtained. This control is a function of time, the state vector and the system parameters (both deterministic and stochastic). All results are derived using Bellman's continuous dynamic programming.

Approach to Equilibrium of a Relaxing System. I. Classical 3-Level System

A complete investigation of the properties of a classical stochastic three-level system is made. The existence of a circulatory probability current is deduced for the system considered. A geometrical representation for the probability vector in the case of complex eigenvectors is given, and it is further shown that the probability vector reaches the equilibrium state asymptotically, sweeping smaller and smaller areas of a spiral as $t\ensuremath{\rightarrow}\ensuremath{\infty}$, irrespective of what "state" we start with.