Reliability Estimation of the Discrete-time Control Systems with Random Parameters

In many different industries, such as chemical industry, plastic industry and especially in rubber industry, there are systems with random parameters. It is well-known in case of the systems with determined parameters that the systems can be either stable or unstable. But, in case of the random parameters the systems can be stable, unstable, or stable with some probability. The stability problem of the systems with geometrical imperfection is very well known. In this paper we are considering the stability problem of the linear discrete-time system with a parametric imperfection. The main result in this paper is the estimated reliability of systems with random parameters. There are several modes of stochastic stability: stability of probability, stability of the K-th moment, almost certain stability, Lyapunov average stability, exponential stability in the K-th moment, monotonic entropy stability, asymptotic entropy stability, etc. For all definitions of stochastic stability it is necessary for probability stability to be p=1. Caughey and Gray [2] determined almost certain stability of linear dynamic systems with stochastic coefficients. Khasminski [5], Kozin [6, 7] and Pinsky [10] gave various definitions and properties of stochastic stability of ordinary differential equations. Necessary and sufficient conditions guaranteeing the average square stability were obtained by Sawaragi [9]. In researching stochastic stability of systems with random parameters, we have also considered the systems with random timevarying parameters by Gaussian distribution [4]. Stochastic stability of systems with random imperfection is considered in [11, 12]. Probability stability estimation of the linear systems with random parameters is considered in [1] for continuous systems, and in [3, 14] for discrete-time systems. In [13], the geometrical imperfection is interpreted as having spatially fluctuating structural properties with respect to perfect geometry. In [8], the failure probability of the systems has been considered. As for the stability probability of the systems with imperfect parameters, this probability is computed by integration of the probability density of the random parameters over the stability domain S in the parametric space: