Robust Controllers, Singular Value Analysis

The analysis of robust controllers took a different tack from adaptive controls with the work of J. C. Doyle and his associates, starting around 1980. The key to the new approach is a generalization of system gain using the singular values of a matrix. Matrix singular values are another term for the matrix norm, defined as the square root of the sum of the squares of the absolute values of the elements. The matrix norm is the trace of A* A, where A is the given matrix and A* is the Hermitian conjugate of A (or the transpose if A is real).

According to the singular value approach, control system robustness against uncertainties in mechanical and aerodynamic properties is assured if the amplitude of the maximum expected uncertainty is less than the minimum system gain at all frequencies.

A simpler, but equally important application of singular value analysis is to system stabil­ity margins, without considering uncertainties. Stability margins are guaranteed if the mini­mum singular values of the system’s return difference matrix are all positive (Mukhopadhyay and Newsom, 1984). The system return difference matrix I + G is a matrix generalization of the closed-loop transfer function denominator for a single-input single-output system. This stability margin application of singular value analysis was made for the X-29A research airplane (Clarke et al., 1994).