8. Control System Stability Analysis 23.1

8.1. Static Stability

Static stability is defined as the stability of the control system under small disturbances, i.e., disturbances under which the system remains linear.

The analysis is performed based on the control system block diagram presented in Section 7. The coefficients K1 … K5 of the block diagram depend on the load. In [L5], it is shown that the greatest variations in the coefficients occur in the region of the rated load mode. From the curves presented in [L5          ], it is evident that when the load changes by 10%, the coefficients change by no more than 10% and can be considered constant; thus, we can utilize the system of linearized equations presented in Section 7.

In accordance with [L2], the stability of the control system can be evaluated by the profile of the open-loop system Logarithmic Amplitude-Frequency Characteristic (LAFC).

If at the "cutoff frequency" (ωc, at which K = 1) the phase margin (Δφc = 180° + φ) is greater than zero, the system can be considered stable. Here, φ is the phase shift at ωc, i.e., the electrical angle by which the output signal lags behind the input signal.  According to Fig. 11, ωc = 40.

φ = –2 · arctg(40 · 0.03) = –2 · arctg(1.2) = –100°      (See [L2])
Engineering Note on Characteristic Slope: Since the cutoff frequency Wc = 40 s-1 lies immediately past the corner frequency of the second-order voltage filtering block (1/0.03 = 33.3 s-1), the open-loop LAFC baseline transitions to a slope of -40 dB/dec in this zone. However, due to the close proximity of these frequencies, the cumulative phase lag has only accumulated to -100 degrees. This yields a highly resilient phase margin of Δφ= 180 - 100 = 80 degrees, guaranteeing exceptional steady-state stability before the high-frequency attenuation takes full effect.