In this paper control of high index or even non-regular linear descriptor systems subject to general quadratic constraints is considered. This setup especially includes the H_∞ control problem and the design for strict passivity. In a previous paper [9] this problem was addressed by a linearizing change of variables approach under a certain assumption on the structure of the algebraic equations within the system description. This restriction is overcome in the paper at hand. By means of a modified linearizing change of variables approach the synthesis conditions (given as linear matrix inequalities (LMIs)) not only become structurally simpler, but, in contrast to [9], also include the case without algebraic constraints. The approach is constructive: controller parameterizations can be computed on the basis of the solution of the LMI synthesis conditions.