LOOP-SHAPING H∞ CONTROL FOR A DOUBLY FED INDUCTION MOTOR SALLOUM Georges 1, MBAYED Rita 2, PIETRZAK-DAVID Maria 1, DE FORNEL Bernard 1 1 INP-ENSEEIHT, Laboratoire Plasma et Conversion d’Energie – UMR N° 5213 au CNRS 2, rue Charles Camichel, BP 7122 - 31071 Toulouse Cedex 7 – FRANCE 2 Lebanese University – Engineering Faculty – Branch II Roumieh – Beirut – LEBANON 1 Phone (33) 5 61 58 82 08 Fax (33) 5 61 63 88 75 2 Phone (961) 4 872 209 Fax (961) 4 872 208 1 E-Mail :
[email protected],
[email protected], bernard.de.fornel@laplace. univ-tlse.fr 2 E-Mail :
[email protected]
Keywords « Doubly fed induction motor », « Variable speed drive », « Robust control », « Non-linear control », « Robustness ».
Abstract This paper deals with a Doubly Fed Induction Machine (DFIM) where both sides are supplied by two PWM inverters. The study consists in elaborating a robust vector control of the machine by the loopshaping H∞ approach. The H∞ loop-shaping theory is presented in brief; then it is applied to the control of the DFIM. The simulation results prove that the implementation of the H∞ controllers for currents, fluxes and speed loops leads to excellent robustness in stability and good dynamic performance even with large electrical and mechanical parameters uncertainty.
Introduction There was much progress accomplished in the few past decades, in power electronics and digital control fields. Thanks to its low cost and high reliability, DFIM became an industrial standard, for many applications. However, DIFM controls are based on a stationary model which is subject to many constraints, such as parameters uncertainty, temperature, saturation, that might divert the system from its optimal operation. For this reason, the control should be designed to offer satisfying robustness in stability and in performance. In this paper, we present the design method of the loop-shaping H∞ control. This method was applied to all the control loops (currents, flux and speed) but we will expose in details only the speed controller study in order to achieve robust stability, low sensitivity to exogenous disturbances and parameters uncertainty. Using Linear Fractional Transformation (LFT) and structured uncertainties representation [6] will allow us, via the µ-analysis, to quantify stability and performance robustness.
Overview of the H∞ control theory H∞ Analysis
The standard setup of the H∞ control problem, as introduced by Zames [1] and Doyle [2], [3] is given in figure 1. P(s) represents the generalized plant, K(s) is the controller, w is the exogenous inputs, z denotes the output signals to be minimized, y is the measurement outputs and u is the control signals.
The interaction between inputs and outputs leads to a number of characteristic transfer matrices. We note particularly: - S = ( I + GK )−1 : the sensitivity function, which represents the perturbation influence on the measured outputs. - T = GK ( I + GK )−1 : the complementary sensitivity function, which represents the influence of the noise on the measured outputs. - KS : represents the impact of the perturbation on the control signals. The H∞ problem consists in minimising the effect of perturbation on the system, i.e. minimising the ratio
z w 2
(energy less than one).
The analysis results demonstrate that, in order to obtain nominal performance and robust stability, the three matrices S, T and KS have to be minimised. This synthesis may seem contradictory but it is not since these minimisations are imposed in different frequency ranges. Weighting matrices W1 ( s ) , W2 ( s ) , W3 ( s ) are introduced and the above results are written in terms of three inequalities : W1 S
