Solving the problem of stability of the stationary quality indicator of complex technical systems

Abstract

Aim. To define and solve the problem of stability of the stationary quality indicator of a complex technical system. The random process of the system’s walking by its possible states is described with a homogeneous Markov process. The stability problem consists in defining such stationary (final) probabilities of the Markov process that implement the maximum and minimal values of the quality indicator, provided that the rate of state transitions have interval estimations and the rates are conditioned by the process of failures and recoveries of the system’s elements. The stationary quality indicator has a fairly standard form and is a scalar product of the final probability vector of the Markov process and the vector that characterises the “weight” of each state, where “weight” may be understood as various contensive characteristics of states.

Methods. The paper uses mathematical methods of optimal control of a Markov process using an income vector of a special form and linear programming.

Results. A method is proposed and substantiated for solving the problem of stability of a stationary indicator of the quality of operation of a complex technical system. A numerical algorithm for solving the above problem is presented as well. The paper gives an example of the solution of a problem of stability with quality indicator that is a “penalty” function.

Conclusion. The paper discusses the problem of numerical solution of the stability problem of a large dimensionality.