Graphical Dominance Analysis for Linear Systems: A Frequency-Domain Approach

We propose a frequency-domain approach to dominance analysis for multi-input multi-output (MIMO) linear time-invariant systems. The dominance of a MIMO system is defined to be the number of its poles in the open right half-plane. Our approach is graphical: we define a frequency-wise notion of the recently-introduced scaled graph of a MIMO system plotted in a complex plane. The scaled graph provides a bound of the eigenloci of the system, which can be viewed as a robust MIMO extension of the classical Nyquist plot. Our main result characterizes sufficient conditions for quantifying the dominance of a closed-loop system based upon separation of the scaled graphs of two open-loop systems in a frequency-wise manner. The result reconciles existing small gain, small phase and passivity theorems for feedback dominance analysis.