  The article deals with the problem of synthesis of an adaptive observer of
state variables of a linear time-varying SISO dynamic system. It is assumed
that the control signal and the output variable are measurable. It is assumed
that the state matrix of the plant contains known variables and unknown
constant parameters, and the control matrix (vector) is unknown. The synthesis
of the observer is based on the GPEBO method (generalized parameter based
observer) proposed in $[1]$. Synthesis of adaptive provides for preliminary
parametrization of the initial system and its transformation to a linear
regression model with further identification of unknown parameters. To identify
unknown constant parameters a classical estimation algorithm was used (the
least squares method with a forgetting factor). This approach has proven itself
well in cases where the known regressor is frequency poor (that is, the
spectral composition of the regressor contains less than $r/2$ harmonics, where
r is the number of unknown parameters) or does not satisfy the so-called
undamped excitation condition. To illustrate the efficiency of the proposed
method an example is presented in the article. A time-varying second-order
object with four unknown parameters was considered. Parameterization of the
initial dynamic model was performed and a linear static regression containing
six unknowns parameters was obtained (including the vector of unknown initial
conditions of system state variables). An adaptive observer was synthesized and
the results of computer modeling illustrating the achievement of a given goal
were presented. The main difference from the results published earlier in $[2]$
is the new assumption that the linear a time-varing system contains not only
unknown parameters in the state matrix, but also the matrix (vector) for
control contains unknown constant coefficients.
