Algorithms of Stable Adaptive Observation of a Multidimensional Undefinite Object

This article presents an algorithm for simultaneously estimating the parameters and state coordinates of a multidimensional control object when some of its state variables are not directly measured. The inability to measure all state variables (coordinates) of an object is a well-known drawback of identification schemes. Such conditions require the construction of adaptive state observers. This work demonstrates that when identifying the parameters of a mathematical model for an uncertain multidimensional object, the asymptotic stability of the object and the convergence of its parameters to the model parameters are ensured, provided the input vector is sufficiently informative. The construction of adaptive tracking algorithms for both single-dimensional and multidimensional control objects is also proposed. The observer system operates in discrete time, and its adaptive parameters are optimized to minimize the maximum eigenvalue of the transfer matrix. During this modeling, it was found that convergence was slow in most scenarios. This is because the largest eigenvalues are located close to the unit circle. It was also discovered that using time-varying adaptive gains accelerates convergence compared to using fixed gains.