Bode criterion

In order to facilitate the application of the Bode criterion the system frequency response may be represented graphically in the form of a Bode diagram or plot. This consists of two graphs which are normally drawn with the axes ... 

The Nyquist stability criterion may be employed in cases where the Bode criterion is not applicable (e.g. where the phase shift has a value of -180° for more than one value of frequency). It is usually stated in the form071 ... 

The open-loop Bode diagram for all the components in the control loop, excepting the controller, is plotted and the cross-over frequency determined. If the total open-loop amplitude ratio at proportional controller which would cause the system to be on the verge of instability will be ... 

It is possible, though, that the AR or of an open-loop transfer function may not be decreasing continuously with co. In Figure 18.4 we see the Bode plots of an open-loop transfer function where AR and Nyquist criterion which will be discussed in Section 18.4. Fortunately, systems with AR or like those of Figure 18.4 are very few, and consequently the Bode criterion will be applicable in most cases. 

To use the Bode criterion, we need the Bode plots for the open-loop transfer function of the controlled system. These can be constructed in two ways (a) numerically, if the transfer functions of the process, measuring device, controller, and final control element are known and (b) experimentally, if all or some of the transfer functions are unknown. In the second case the system is disturbed with a sinusoidal input at various frequencies, and the amplitude and phase lag of the open-loop response are recorded. From these data we can construct the Bode plots. 

Let M be the amplitude ratio at the crossover frequency (see Figure 18.5). According to the Bode criterion ... 

What is the basis of the Bode criterion Why is it not generally rigorous ... 

Example 18.1 Stability Characteristics of Some Typical Dynamic Systems Using the Bode Criterion... 

The last equation shows that phase lag is between 0° and -oo. Consequently, there exists a crossover frequency coCo where = -180°, and according to the Bode criterion the system may become unstable for a large Kc which leads to AR > 1 at this frequency. This example demonstrates a very important characteristic for the stability of chemical processes ... 

Do you think that the following modified statement of the Bode criterion is generally rigorous Explain. 

From Eqs. (15)-(16) it is easily seen that the feedback will induce instability if the characteristic equation 1 —g2i s)gR s) = 0 has any roots in the RHP. Provided the open loop system is stable, the Bode criterion applies, i.e., the feedback will induce instability provided there exist some frequency (0 for which the loop-gain... 

When the phase shift is -180° at co=coco, the crossover frequency, and the amplitude ratio is M, then, according to the Bode criterion, the control system is stable when M< 1 and the system is unstable when M>1. The crossover frequency and amplitude ratio Mhave been... 

The Nyquist stability criterion is similar to the Bode criterion in that it determines closed-loop stability from the open-loop frequency response characteristics. Both criteria provide convenient measures of relative stability, the gain and phase margins, which will be introduced in Section J.4. As the name implies, the Nyquist stability criterion is based on the Nyquist plot for GqiXs), a polar plot of its frequency response characteristics (see Chapter 14). The Nyquist stability criterion does not have the same restrictions as the Bode stability criterion, because it is applicable to open-loop unstable systems and to systems with multiple values of co or cOg. The Nyquist stability criterion is the most powerful stability test that is available for linear systems described by transfer function models. 

A preview We can derive the ultimate gain and ultimate period (or frequency) with stability analyses. In Chapter 7, we use the substitution s = jco in the closed-loop characteristic equation. In Chapter 8, we make use of what is called the Nyquist stability criterion and Bode plots. 

The Nyquist stability criterion can be applied to Bode plots. In fact, the calculation using the Bode plot is much easier. To obtain the gain margin, we find the value of GCGP which corresponds to a phase lag of-180°. To find the phase margin, we look up the phase lag corresponding to when GCGP is 1. 

The concept of gain and phase margins derived from the Nyquist criterion provides a general relative stability criterion. Frequency response graphical tools such as Bode, Nyquist and Nichols plots can all be used in ensuring that a control system is stable. As in root locus plots, we can only vary one parameter at a time, and the common practice is to vary the proportional gain. 

For a first order function with deadtime, the proportional gain, integral and derivative time constants of an ideal PID controller. Can handle dead-time easily and rigorously. The Nyquist criterion allows the use of open-loop functions in Nyquist or Bode plots to analyze the closed-loop problem. The stability criteria have no use for simple first and second order systems with no positive open-loop zeros. 

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. 

In Chap. 12 we presented three different kinds of graphs that were used to represent the frequency response of a system Nyquist, Bode, and Nichols plots. The Nyquist stability criterion was developed in the previous section for Nyquist or polar plots. The critical point for closedloop stability was shown to be the 1,0) point on the Nyquist plot. 

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. 

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