6.22.  WHAT IS THE “BEST” STABILITY ANALYSIS TECHNIQUE? GUIDELINES FOR USING THE ANALYSIS TECHNIQUES PRESENTED

We have presented the following methods in this chapter for analyzing linear control systems: Routh–Hurwitz method, Nyquist diagram, Bode diagram, Nichols chart, and the root-locus method. Which of these stability analyses methods presented is the “best” to use? This is a logical question to ask at this point in the presentation of this chapter. For a particular application, one of these methods may be better than the others. However, in general, each method presented in this chapter complements the other. In general, there is no “best” method. The reader should regard all of the methods presented in this chapter as control-system analysis tools that can be used. The practicing control-system engineer should not restrict himself or herself to one method.

Guidelines are presented here for selecting the best method for particular cases [32]. For systems of first or second order, all of the methods presented reveal about the same amount of information, and they are of similar complexity. For systems from about the third to seventh order, there are two primary approaches, and these two approaches complement each other. One approach uses the Bode-diagram method in conjunction with the Nichols chart. The Routh–Hurwitz method can be used as a quick check. The other method uses the root locus. Using the first approach, the control-system engineer can determine relative stability from the phase and gain margins of the Bode diagram. The closed-loop frequency response can then be found from the Nichols chart, and bounds on the time-domain response can be determined as was illustrated in Section 6.12. The root-locus method is the basis of the second approach. Using this method, the control-system engineer can determine relative stability, and the transient response can be found from the location of the closed-loop roots of the characteristic equation—in particular, the location of the dominant pair of complex-conjugate roots. In conclusion, the information provided by these two basic approaches complement each other.

For systems greater than the seventh order, I have found that the root-locus method is more complex than the Bode-diagram–Nichols-chart approach. This guideline, however, is general, and as in all general statements is not true all of the time. For example, we would not use the Bode-diagram–Nichols-chart approach if the open-loop control system did not satisfy the minimum-phase requirement regardless of the system order.

I want to make one additional important point before leaving this discussion. The Bode-diagram approach has the great advantage that it is a fequency-response approach that can be easily measured in the laboratory using readily available instruments. The procedure involves opening the feed-back loop and applying a sinusoidal signal to the input of the open-loop system, and measuring the magnitude of the input and output quantities and the phase relationship between them. (Be careful of avoiding saturation when testing the open-loop control system.) From this procedure, the phase and gain margins can be easily measured. Therefore, the control-system engineer can check his or her design directly using experimental techniques.

On the other hand, an experimental direct determination of the closed-loop roots to check the root-locus method is not easy. We can only test the root locus indirectly in the laboratory based on system performance measures such as the control system’s transient response, and then infer the location of the dominant pair of complex-conjugate roots. Therefore, many practicing control-system engineers prefer the Bode-diagram–Nichols-chart approach over the root-locus method in cases where either approach is applicable. However, in the final analysis, all methods of stability analysis should be considered before selecting the technique or techniques to be used in the final design.