On model of cyber-physical system with state and measurement attacks on the basis of stochastic difference equations

The article is devoted to questions of simulation of cyber-physical systems taking into account information threats. The principles of organization of structure and reliable functioning of the studied systems are analyzed. The conceptual model of cyber-physical systems taking into account information threats is presented. The constructive algorithm for finding an estimation in a cyberphysical model during attacks on a state and observation on the basis of measuring system indices is proposed in the work. The model is presented as a non-stationary descriptor system, which includes differential equations for node state variables and algebraic equations for measurements. State variables and measurements are considered as random vectors. The information criterion is used to find the optimal estimate of the scalar product that characterizes the state of the system. The algorithm is based on the construction of the task of estimation to the problem of optimal control. The value of the mean square error that does not depend on attacks is obtained. The evaluation was carried out in the form of a scalar product, which includes a measurement vector. An algorithm for constructing an optimal estimation for a scalar product is proposed, which includes the state of the network model during an attack. Mutual information between random vectors was used as a quality criterion for optimization. The evaluation was carried out in the form of a scalar product, which includes a measurement vector. Considering the effects of attacks on this estimate, it should be noted that the build estimate depends on such attacks only through the multiplier. The average square error of such an estimate is independent of the attack. The reason for the absence of impact of the attack on the mean square error is the quadratic form of the sets that are selected for state attacks. This assumption makes it possible to apply a linear-quadratic optimization device, which reduces the problem of evaluation to the problem of optimal control. Received results illustrated in the numerical example for the stage of system without noise and with attacks and for the stage of system with noise and attacks. In this case , a three-node cyberphysical system was used which could be used for planning technical service and optimized management for achievement more high totalperformance and safety of the studied systems .