Sliding surface design for discrete VSS using LQR technique with a preset real eigenvalue

This paper presents an LQR-based sliding surface design procedure that takes, in addition to the desired weighting matrix, a preset real eigenvalue consequential in discrete VSS as inputs. A weighting matrix that simultaneously stays "closest" to the desired one and yields the preset eigenvalue is sought. Sliding surface is then determined from the optimal gain matrix and the preset eigenvalue. The existence and non-uniqueness of feasible weighting matrices, the uniqueness of solution and convexity of the constrained optimization problem, as well as the use of least-squares to simplify and solve the problem are studied. The paper ends with two MATLAB examples that illustrate the design procedure.