The Generalized Nyquist Criterion and Robustness Margins with Applications

Multivariable Nyquist eigenloci provide a much richer family of curves as compared to the SISO case. The eigenloci may be computed symbolically in many simple cases. Inspection of the generalized Nyquist eigenloci plot allows the determination of the exact values of the multivariable gain margin (GM), phase margin (PM), and the complex margin (CM). Hence there is no need to compute estimates of GM and PM from the singular values of the sensitivity and the complementary sensitivity functions as they are usually very conservative. Furthermore, the complex margin allows the computation of the minimum additive equal diagonal complex perturbation, and the associated multiplicative complex perturbation that drives the system unstable. All the computations mirror the SISO case. The ideas are applied to semiconductor wafer manufacturing process control and aerospace examples.