Adaptive State Observer Synthesis Based On Instrumental Variables Method
Abstract
The present paper presents non-recurrent adaptive observation algorithm for SISO linear time-invariant discrete systems. The algorithm is based on the instrumental variables method and the adaptive state observer estimates the parameters, the initial and the current state vectors of discrete systems. The algorithm workability is proved by using simulation data in the MATLAB/Simulink environment.
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Introduction
State feedback control system design is often related to reconstruction of the state vector by measurements of the output variable and the input signal of the open loop system.
The reconstruction of the state vector is only possible by implication of state observer and the adaptive observation problem is related to observer synthesis with parameter estimator [6,7]. The matrices 𝑨 and 𝒃 or 𝒄 (depending on the canonical form chosen for state space representation) are considered unknown.
The parameters are being estimated and the unknown matrices are determined during the observation process and the state vector is reconstructed.
The present paper investigates a non-recurrent algorithm for adaptive observation of single input single output (SISO) linear time invariant (LTI) discrete systems developed on the basis of the instrumental variables (IV) method [2].
The parameters estimator built-in in the adaptive observer is based on a simplified calculation procedure which also includes inversion of the informative matrix [5].
Conclusion
The algorithm suggested for open loop system parameter estimations which serve as a basis for further reconstruction of the current state vector implements the method of the instrumental variables excluding the zero iteration which only uses the least squares method (steps 1 to 4 of the suggested calculation procedure).
The algorithm proposed estimates as well the initial state vector x0 which allows the forming of the instrumental variables matrix even for nonzero initial conditions.
The results delivered show that the number of the input output data measurements (N) is of high significance in relation to accuracy of estimations in the case of noise corrupted output. The highest accuracy is to be expected for highest counts of N (see Fig.2 and Fig.3).
The method of the IV method gives best results in case of estimation of a-priori collection of data [3,8], however in relation to the closed loop system the added noise fk is transferred to the input signal through the feedback channel. Thus invariance between the instrumental matrices and the added noise is not possible; it is only possible that the estimates are unbiased and significant in presence of a white noise however the real systems do not allow such solution of the problem.
The used of IV method for investigation of the closed loop system is only applicable if additional input signal is implemented [8]. For this reason the implementation of the algorithm suggested above is not recommended for closed systems implications.
The algorithm for adaptive observation based on the instrumental variables (IV) method introduced in the present paper is developed on the basis of non-recurrent method which ensures the convergence of the iterative procedure [1,2].
The most positive feature of this algorithm however is related to the method used for informative matrix formation. It is formed through the four sub-matrices Y11, Y21, U12 and U22 which reduces the calculation complexity of the procedure for inversion of matrix G formed by the sub-matrices G11, G12, G21, G22. Independently of the N count in numbers for the estimation of the coefficients bi and ai is only needed to be inverted the matrices G11 and G22 − G21M1G12 which are always guaranteed n × n dimensions. In all other cases this procedure is related to inversion of a matrix at least N − n × N − n dimensional.