Plotting Inverse Nyquist Diagram with Gershgorin Bands for System Analysis

Visual representations serve as powerful tools for understanding system behavior. The inverse Nyquist plot provides valuable insights into system stability and performance characteristics. However, interpreting standard inverse Nyquist plots can be challenging without supplementary analytical frameworks. The Gershgorin theorem offers a mathematical foundation to enhance this visualization by generating Gershgorin bands around the system's transfer function. These bands delineate regions of uncertainty and stability margins in the complex frequency domain. In computational implementation, this involves calculating the system's frequency response using functions like freqresp() in MATLAB or equivalent methods in other programming environments. The Gershgorin bands are computed by determining the radius of influence around each eigenvalue of the system matrix, typically achieved through matrix norm calculations and eigenvalue decomposition algorithms. The resulting Gershgorin inverse Nyquist plot superimposes these confidence bands onto the traditional inverse Nyquist trajectory, enabling engineers to assess robustness against parameter variations and model uncertainties. This enhanced visualization technique facilitates more accurate stability analysis, particularly for multivariable systems where interactions between different channels must be considered. Implementation typically requires: 1. Defining the system transfer function matrix 2. Computing frequency response data across relevant frequency ranges 3. Calculating Gershgorin disk radii using matrix norm properties 4. Plotting the inverse Nyquist curve with superimposed confidence bands 5. Analyzing band intersections with critical points (-1 point) for stability assessment This comprehensive approach provides deeper system insights by combining graphical analysis with rigorous mathematical bounds, making it particularly valuable for control system design and robustness verification.