Decomposition Principle in the Problem of Synthesis of State Observers for SISO Systems Under the Action of External Disturbances

Abstract

The problem of reference signal tracking of system's output is considered for a linear SISO system in the case of incomplete measurements and the action of external unmatched disturbances. Under the assumption of smoothness of external disturbances, the original mathematical model is presented in the equivalent quasi-canonical form of the input-output relative to the tracking error. In terms of the mixed variables of the new coordinate basis, which are linear combinations of state variables, external disturbances and their derivatives, the basic combined control law is synthesized, providing asymptotic stabilization of the tracking error invariant with respect to external disturbances. The main result of the paper is estimation method of linear combinations of unmeasured variables for feedback law under the assumption that only tracking error is measured. The problem is solved in a “narrow” setting and there is no extension of the state space, except for the observation subsystem, and the introduction of exogenous dynamic models simulating external disturbances is not required. The developed original method of synthesis of the full-dimensional observer of the state without corrective actions realizes the principle of a full decomposition. The observer setting requires the direct assignment of eigenvalues of the system coefficients matrix of observation errors. In this case the given accuracy of estimation of mixed variables and external disturbances for the basic control law is provided.