Bode stability criterion

As discussed in Section 7.10.4, Volume 3, the Bode stability criterion states that the total open loop phase shift is —n radians at the limit of stability of the closed loop system. 

This heuristic argument forms the basis of the Bode stability criterion(22,24) which states that a control system is unstable if its open-loop frequency response exhibits an AR greater than unity at the frequency for which the phase shift is —180°. This frequency is termed the cross-over frequency (coco) for reasons which become evident when using the Bode diagram (see Example 7.7). Thus if the open-loop AR is unity when i/r = —180°, then the closed-loop control system will oscillate with constant amplitude, i.e. it will be on the verge of instability. The greater the difference between the open-loop AR (< I) at coc and AR = 1, the more stable the closed-loop... 

In the following example the effect of the various fixed parameter control modes on the stability of a simple feedback control loop are examined using the Bode stability criterion and the concept of gain and phase margins. 

All systems in Example 18.1 have an important common feature The AR and of the corresponding open-loop transfer functions decrease continuously as co increases. This is also true for the large majority of chemical processing systems. For such systems the Bode stability criterion leads to rigorous conclusions. Thus it constitutes a very useful tool for the stability analysis of most control systems of interest to a chemical engineer. 

IV.61 Repeat Problem IV.25 using the Bode stability criterion. 

IV.63 Using the Bode stability criterion, find the range of Kc values that stabilize the unstable process... 

We know (see Section 17.3) that the phase lag for a first-order system is between 0 and 90°. Therefore, according to the Bode stability criterion, the system above is always stable since there is no crossover frequency. 

The Bode stability criterion indicates how we can establish a rational method for tuning the feedback controllers in order to avoid unstable behavior by the closed-loop response of a process. 

As we pointed out in Section 18.1, the Bode stability criterion is valid for systems with AR and monotonically decreasing with a). For feedback systems with open-loop Bode plots like those of Figure 18.4 the more general Nyquist criterion is employed. In this section we present a simple outline of this criterion and its usage. For more details on the theoretical background of the methodology, the reader can consult Refs. 13 and 14. 

IV.59 Consider the processes with the transfer functions given in Problem IV.23. Each of these processes is feedback controlled with a proportional controller. Assume that Gm = Gf = 1. Using the Bode stability criterion, find the range of values of the proportional gain Kc which produce stable (if it is possible) closed-loop responses. 

An exact time-delay analysis, without the Fade approximation and based on the Bode stability criterion (see Example 14.6), indicates that the actual upper limit on Kc is 4.25, which is 23% lower than the approximate value of... 

The Bode stability criterion has two important advantages in comparison with the Routh stability criterion of Chapter 11 ... 

The Bode stability criterion provides a measure of the relative stability rather than merely a yes or no answer to the question Is the closed-loop system stable ... 

Before considering the basis for the Bode stability criterion, it is useful to review the General Stability Criterion of Section 11.1 A feedback control system is stable if and only if all roots of the characteristic equation lie to the left of the imaginary axis in the complex plane. 

Some of the important properties of the Bode stability criterion are ... 

The Bode stability criterion is very useful for a wide variety of process control problems. However, for any Gql ) that does not satisfy the required conditions, the Nyquist stabihty criterion discussed in Appendix J can be applied. 

For systems with multiple (o or Wg, the Bode stability criterion has been modified by Hahn et al. (2001) to provide a sufficient condition for stability. 

Also, Gy = 0.1 and G = 10. For a proportional controller, evaluate the stability of the closed-loop control system using the Bode stability criterion and three values of Kc . 1, 4, and 20. 

Figure 14.7 shows the open-loop amplitude ratio and phase angle plots for Gql- Note that the phase angle crosses -180° at three points. Because there is more than one value of coc, the Bode stability criterion cannot be applied. 

According to the Bode stability criterion, ARc must be less than one for closed-loop stability. An equivalent stability requirement is that GM > 1. The gain margin provides a measure of relative stability, because it indicates how much any gain in the feedback loop component can increase before instability occurs. For example, if GM = 2.1, either process gain Kp or controller gain Kc could be doubled, and the closed-loop system would still be stable, although probably very oscillatory. 

Frequency response techniques are powerful tools for the design and analysis of feedback control systems. The frequency response characteristics of a process, its amplitude ratio AR and phase angle, characterize the dynamic behavior of the process and can be plotted as functions of frequency in Bode diagrams. The Bode stability criterion provides exact stability results for a... 

J.l Closed-Loop Behavior J.2 Bode Stability Criterion J.3 Nyquist Stability Criterion J.4 Gain and Phase Margins... 

In the next section, we consider one of the most important and useful frequency response results, the Bode stability criterion. 

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