Adaptive State Observer for Linear Time-Varying System with Partially Unknown State Matrix and Input Matrix Parameters

In this paper the problem of adaptive state observer synthesis for linear time-varying SISO (single-input-single-output) dynamical system with partially unknown  parameters was considered. It is assumed that the input signal and output variable of the system are measurable.  It is also assumed  that the state matrix  of the plant contains known  variables and unknown  constants when the input matrix (vector) is unknown. Observer synthesis is based on GPEBO  (generalized parameter estimation based observer) method proposed in [1]. Observer synthesis provides preliminary parametrization  of the initial system and its conversion to a linear regression model with further unknown  parameters identification.  For identification  of the unknown  constant parameters classical estimation algorithm — least squares method with forgetting factor — was used. This approach works well in cases, when the known regressor is " frequency poor" (i.e. the regressor spectrum contains r/2 harmonics,  where r is a value of the unknown  parameters) or does not meet PE (persistent excitation)  condition.  To illustrate performance of the proposed method, an example is provided in this paper. A time-varying  second-order  plant with four unknown  parameters was considered. Parametrization of the initial dynamical  model was made. A linear static regression with six unknown  parameters (including unknown  state initial conditions vector) was obtained. An adaptive observer was synthesized and the simulation results were provided to illustrate the purpose reached. The main difference with the results, that were published earlier in [2], is the new assumption that not only does the state matrix of the linear time-varying system contain unknown  parameters, but input matrix (vector) contains unknown  constant coefficients.

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