Loop Stability Criteria

A frequency domain stability criterion developed by Nyquist (1932) is based upon Cauchy s theorem. If the function F(s) is in fact the characteristic equation of a closed-loop control system, then... 

The Nyquist stability criterion can be stated as A closed-loop control system is stable if, and only if, a contour in the G s)H s) plane describes a number of counterclockwise encirclements of the (—l,jO) point, the number of encirclements being equal to the number of poles of G s)H s) with positive real parts . 

In practice, only the frequencies lu = 0 to+oo are of interest and since in the frequency domain. v = jtu, a simplified Nyquist stability criterion, as shown in Figure 6.18 is A closed-loop system is stable if, and only if, the locus of the G(iLu)H(iuj) function does not enclose the (—l,j0) point as lu is varied from zero to infinity. Enclosing the (—1, jO) point may be interpreted as passing to the left of the point . The G(iLu)H(iLu) locus is referred to as the Nyquist Diagram. 

The program REFRIG 1 calculates the steady-state heat generation and heat loss quantities, HG and HL, as functions of the reactor temperature, TR, over the range 320 to 410 K. It is shown that, according to the van Heerden steady-state stability criterion that the simple loop control response, KP=0 is unstable. 

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. 

With frequency response analysis, we can derive a general relative stability criterion. The result is apphcable to systems with dead time. The analysis of the closed-loop system can be reduced to using only the open-loop transfer functions in the computation. 

This equation, of course, contains information regarding stability, and as it is written, implies that one may match properties on the LHS with the point (-1,0) on the complex plane. The form in (7-2a) also imphes that in the process of analyzing the closed-loop stability property, the calculation procedures (or computer programs) only require the open-loop transfer functions. For complex problems, this fact eliminates unnecessary algebra. We just state the Nyquist stability criterion here.1... 

Nyquist stability criterion Given the closed-loop equation 1 + Gol (joi) = 0, if the function G0l(J ) has P open-loop poles and if the polar plot of GOL(](o) encircles the (-1,0) point... 

Do not panic Without the explanation in our Web Support, this statement makes little sense. On the other hand, we do not really need this full definition because we know that just one unstable closed-loop pole is bad enough. Thus the implementation of the Nyquist stability criterion is much simpler than the theory. 

Once we understand the origin of Nyquist stability criterion, putting it to use is easy. Suppose we have a closed-loop system with characteristic equation 1 + GCGP = 0. With the point (-1,0) as a reference and the Gc(jco)Gp(jco) curve on a... 

Check the system (or loop) instability by using the Ledinegg criterion with an average lumped channel pressure drop. If it does not satisfy the Ledinegg stability criterion, one or more of the three remedies can be taken orifice the inlet, increase the steepness of the pump head-versus-flow curve or increase the resistance of the downcomer of a natural-circulation loop. 

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. 

An inherently stable process can be destabilised by the addition of a feedback control loop—particularly where integral action is included. This is illustrated in the following example using the characteristic equation and the Routh-Hurwitz stability criterion. 

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. 

The stability criterion is therefore equivalent to saying that the phase shift (of the open-loop transfer function) should not be equal to 180° (or —180°) at the crossover frequency. But... 

The stability criterion stated above secures stable response of a feedback system independently if the input changes are in the set point or the load. The reason is that the roots of the characteristic equation are the common poles of the two transfer functions, GSp and Gioad, which determine the stability of the closed loop with respect to changes in the set point and the load, respectively. 

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. 

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. 

Using the Nyquist stability criterion, show that feedback systems with first-and second-order open-loop responses are always stable. 

IV.58 Using the Bode stability criterion, find which of the control systems with the following open-loop transfer functions are stable and which are unstable ... 

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. 

Plot the frequency response of L(/ft>) in Bode diagram (Fig. 12.6). For open-loop stable systems ZLijw) falls with frequency such that ZL ja ) crosses -180° only once at the fi equency i8o. Bode s stability criterion says that the closed-loop system is stable if and only if the loop gain L is less than 1 at this frequency. 

If the pressure is reduced to 30 bar, below the mechanical critical point, the loop in the fugacity becomes more pronounced. Here violates the diffusional stability criterion dfildxi > 0 for stability) over only a small range of Xj but the mechanical stability criterion (k > 0) is also violated at states between the extrema in fp Finally, at the lowest pressure (10 bar), the loop in has completely closed, dividing into two parts a vapor part that is linear and obeys the ideal-gas law, and a fluid part that includes stable and metastable liquid states at small x values plus mechanically unstable fluid states at higher x values. The broken horizontal lines in Figure 10.1 are vapor-liquid tie lines, computed by solving the phi-phi equations (10.1.3) simultaneously for both components. 

The general iterative scheme is illustrated by the flowchart in Fig. 3. Note the iteration loop on the determination of the coupled fields < >/ and u. The equilibrium potential is determined first. Then Eqs. (46) and (47) are solved alternately until convergence of the global electric and mass fluxes I and U. An estimate of the velocity is given (zero velocity when the routine enters the loop) an estimate of l is determined by solving (46) the velocity field is then updated by solving Eqs. (47). The stabilization criterion for the fluxes is generally set to 10. ... 

Figure 11.25 provides a graphical interpretation of this stability criterion. Note that all of the roots of the characteristic equation must he to the left of the imaginary axis in the complex plane for a stable system to exist. The qualitative effects of these roots on the transient response of the closed-loop system are shown in Fig. 11.26. The left portion of each part of Fig. 11.26 shows representative root locations in the complex plane. The corresponding figure on the right shows the contributions these poles make to the closed-loop response for a step change in set point. Similar responses would occur for a step change in a disturbance. A system that has all negative real roots will have a stable.

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 ... 

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. 

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

Next we state one of the most important results of frequency response analysis, the Bode stability criterion. It allows the stability of a closed-loop system to be determined from the open-loop transfer function. 

---