Less conservative conditions for robust LQR-state-derivative controller design: an LMI approach

This study proposes less conservative conditions for robust linear quadratic regulator controllers using state-derivative feedback (SDF). The algebraic Ricatti equation was formulated using the SDF, and its solution was obtained by linear matrix inequalities. SDF was chosen owing to the presence of accelerometers as sensors. Since accelerometers are the main sensors in mechanical systems, the proposed technique may be used to control/attenuate their vibrations/oscillations. Moreover, to formulate the less conservative conditions, some methods in the specialised literature were used, such as, for example, slack variables by Finler's Lemma. The paper also offers necessary and sufficient conditions for an arbitrary convex combination of square real matrices $ A_1, A_2,\ldots , A_r $ A1,A2,…,Ar to be a nonsingular matrix, and thus an invertible one: $ A_1 $ A1 must be nonsingular and all the real eigenvalues of $ A_1^{-1}A_2, A_1^{-1}A_3,\ldots , A_1^{-1}A_r $ A1−1A2,A1−1A3,…,A1−1Ar must be positive. This result is important in the formulation of the proposed less conservative conditions since it was assumed that a given convex combination is nonsingular. A feasibility analysis demonstrates that the proposed conditions reduce the conservatism. Thereby, it is possible to stabilise a higher number of systems and to reduce the guaranteed cost. Furthermore, a practical implementation illustrated the application of the proposed conditions.