On-Line Estimation of All Electrical Parameters in Symmetrical and Faulty Induction Machines: Stator Fault Case

Identification of all the electrical parameters of Induction Machines (IMs) is a prerequisite for successful model-based control and/or fault detection. Various approaches have been proposed previously: off-line experiments under alternating stationary conditions, least squares batch and on-line estimation, to name only a few. Both continuous and discrete time have been considered, all of them utilizing models for symmetrical IMs. Existing methods for identification of the electrical parameters of the IM are strictly speaking valid only at stationary conditions, which limitation results in either poor dynamics excitation or gross approximation errors, depending on the working conditions of the motor. Unrealistic assumptions about the rotor’s angular acceleration are also often employed. The topic of identification of all electrical parameters of faulty IMs has not been addressed so far. In this paper, a method for on-line observer-based estimation of all electrical parameters of symmetrical IMs as well as IMs with faulty stator is presented aiming primarily at fault tolerant control and fault detection. The approach does not rely on any assumptions regarding the rotor’s angular speed and/or acceleration. Nomenclature Variables (normalized) ψ magnetic flux ı electrical current u stator feeding voltage ω rotor speed Parameters R′ A, RS electrical resistances LA, L ′ S lumped inductances M ′ Ar,M mutual inductances εA, εS , εr electrical parameters μA, μ goodness factors KA,KS normalization factors For the variables, ·̄ denotes a normalized (dimensionless) variable and subscripts α and β refers to the corresponding axis in a two-axis system, respectively. For both the variables and parameters, capital letter subscript A refers to stator axis A in a conventional three-axis system. Similarly, subscript S refers to stator axis B or C. In the same manner, a small letter subscript r refers to rotor axis a, b or c. The units used for the variables and parameters are SI-units throughout the paper. Bold capital letters stand for matrices, whereas small and capital letters denote respectively scalars and vectors. Identity and zero matrices of appropriate dimension (mostly 2× 2) are denoted as I and 0. The unitary matrix J = [ 0 −1 1 0 ] is frequently used in the sequel. The complex conjugate transpose of a matrix is denoted by ·H and ·∗ means complex conjugate of a scalar. The determinant of a matrix is denoted by | · |. The transition matrix of matrix A is denoted by ΦA(t, t0). The notation θij stands for element j in vector θi. The maximum and minimum singular values of a matrix X are denoted by σ{X} and σ{X}, respectively. A vector norm is denoted by ‖ · ‖ and the induced matrix p-norm by ‖ · ‖p. The real and imaginary parts of a scalar are denoted by R{·} and I{·}, respectively. Finally, differentiation with respect to time is denoted by ẋ = dx dt .