Diagonal dominance and integrity

The diagonal dominance is the property of matrices that the diagonal elements are relatively larger than the off-diagonal elements. It is an important property for multivariable feedback systems, especially with diagonal controllers. In this paper, a new diagonal dominance defined by 2-norm is proposed. This diagonal dominance is less conservative (i.e. more relaxed) condition than the other dominance. This paper also clarifies the relation between the diagonal dominance and strictly positive realness. Integrity is the property that closed-loop systems remain stable in the presence of failures of sensors and/or actuators. This property is also important for multivariable control systems. The conventional integrity conditions are the diagonal dominance and the positive realness of the return difference transfer function of closed-loop systems. This paper clarifies that the essential condition is the positive realness. This paper also shows that the direct Nyquist array method is still a good design method from the viewpoint of integrity.