Identification of electric circuits described by ill-conditioned mathematical models

This paper treats the problem of identification of a linear electric circuit described by an ill-conditioned mathematical model. The identification problem is considered as the problem of model parameters determination by means of processing experimental data measured for the objective circuit. Topological singularities (low-admittance cutsets and low-impedance loops) in a circuit are found to be origins of ill-conditionality of a circuit model. For more in-depth investigation the classification of electric circuits is made in respect to singularities position. It is shown that the first set of experimental data obtained for an ill-conditioned model is useless for getting the required solution of the identification problem. In this case a solution error amounts to a huge value that exponentially increases with growth in condition number of a model matrix. It is found that linear relations between model parameters can be determined in an ill-conditioned problem. Accuracy of these relations does not depend on condition number, but is defined only by measurement precision. An approach named as repeat measurements principle (RMP) and based on linear relations is suggested to solve an ill-conditioned identification problem. A new RMP-based algorithm of linear circuit identification is developed. The algorithm shows a high efficiency and allows us to determine model parameters accurate to measurement precision as applied to any type of reciprocal and nonreciprocal linear circuits.     

Using Linear Relations Between Experimental Characteristics in Stiff Identification Problems of Linear Circuit Theory

The paper treats an inverse problem posed in the time domain for a circuit described by a stiff system of ordinary differential equations (ODEs). Identification of model parameters (circuit elements) is performed by processing transient characteristics measured in the circuit. An illustrative example discussed throughout the paper shows that reducing the identification problem to curve fitting, which is the most general way, may be hardly used for a stiff model due to a ravine shape of the objective functional. Moreover, a necessity of solving the stiff ODE system to get a functional value makes the inverse problem be practically unsolvable. It has been shown that the initial inverse problem may be simplified significantly by taking into account linear relations which are observed between the experimental characteristics measured for a stiff system. The relation coefficients are discussed in the paper with regard to the accuracy of the approximation. Finally the initial identification problem has been reduced to a nonlinear system of algebraic equations which may be easily solved considering different sensitivity of the relation coefficients to the model parameters. The final solution is presented for different levels of "measurement" error involved in the simulation.