The influence of discretization methods on the conditionality of the information matrix and the nature of the formation of error hyperspheres in problems of parameter estimation based on a small number of observations

The paper considers the issues of application of discrete models of dynamic systems based on multi-step methods for identification and parameter estimation problems based on a small number of observations using the example of a first-order aperiodic unit. The problem of identification of parameters of dynamic systems is relevant in control problems of stationary and non-stationary dynamic systems, for example, in sensorless control systems of synchronous or asynchronous electric drives of submersible centrifugal electric pumps operating in intermittent operation mode. The problem of parameter estimation was solved in the classical formulation of the least squares method without taking into account a priori information about the nature of noise in the measuring channels. It was shown that the accuracy of parameter estimation depends not only on the norm of the interference vector, but also on its orientation relative to the column space of the information matrix, as well as on the condition number of the information matrix. The matrix formation structure and the problem conditionality depend on the selected approximation method of the state variable derivative. It has been clarified that the obtained parameter estimation results require additional post-filtering for use in closed-loop control systems for reasons of stability. According to the obtained results, implicit multi-step methods can be recommended for most engineering applications in the problems of estimating the parameters of linear and nonlinear systems. The highest accuracy of estimating the parameters of the aperiodic link in terms of the moving average for the entire considered section of the simulation was achieved using the Tustin and Milne methods for transition to discrete time, which allowed for a higher degree of conditionality of the problem compared to other methods.