The use of a negative definite state-weighting matrix in linear optimal aircraft stability augmentation system problems

Most of the published work on the Linear Quadratic Regulator (LQR) theory states it is necessary to restrict the state-weighting matrix in the quadratic performance index to be at least positive semi-definite (P.S.D). In this paper, a method of obtaining specified closed-loop eigenvalues is described which uses a procedure that results in a corresponding state-weighting matrix which can be negative definite (N.D). The value of this method for the design of aircraft Stability Augmentation Systems (S.A.S) is that it permits a designer to use a set of specified closed-loop eigenvalues which correspond to parameters given in those aircraft flying qualities specifications published by aviation authorities. Because these flying qualities are based on low order mathematical models corresponding to particular modes of flight, the choice of appropriate closed-loop eigenvalues is direct. The control law obtained from this method not only provides considerable robustness, but also results in the prescribed closed-loop dynamics. The method is illustrated by presenting the results of two examples. The effectiveness of the method is shown from the results obtained from digital simulation of the S.A.S for both aircraft.

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