Stability, openloop

We want to look at the stability of the closedloop system with a proportional controller 0, . First, however, let us check the openloop stability of this system. 

This stability requirement applies to any system, openloop or closedloop. The stability of an openloop process depends upon the location of the poles of its openloop transfer function. The stability of a closedloop process depends upon the location of the poles of its closedloop transfer function. These closedloop poles will naturally be different from the openloop poles. 

Thus the criteria for openloop and closedloop stability are different. Most systems are openloop stable but can be either closedloop stable or unstable, depending on the values of the controller parameters. We will show that any real process can be made closedloop unstable by making the gain of the feedback controller high enough. 

In the last chapter we used Laplace-domain techniques to study the dynamics and stability of simple closedloop control systems. In this chapter we want to apply these same methods to more complex systems cascade control, feedforward control, openloop unstable processes, and processes with inverse response. Finally we will discuss an alternative way to look at controller design that is called model-based control. 

Nevertheless linear techniques are very useful in looking at stability near some operating level. Mathematically, if the system is openloop unstable, it must have an openloop transfer function that has at least one pole in the RHP. 

Thus the stability of the system depends on the location of the pole 022 If this pole is positive, the system is openloop unstable. The value of is given in Eqs. (9.82). 

Keep in mind that the positive zero does not make the system openloop unstable. Stability depends on the poles of the transfer function, not on the zeros. Positive zeros in a system do, however, affect closedloop stability as the example below illustrates. 

Find the closedloop stability requirements for a third-order openloop unstable process with a proportional controller ... 

We will show in Sec. 13.1 that chsedhop stability can be determined from the frequency-response plot of the total openloop transfer function of the system (process openloop transfer function and feedback controller This... 

The Nyquist stability criterion is, on the surface, quite remarkable. We are able to deduce something about the stability of the closedloop system by making a frequency response plot of the openloop system And the encirclement of the mystical, magical (— 1, 0) point somehow tells us that the system is closedloop unstable. This all looks like blue smoke and mirrors However, as we will prove below, it all goes back to finding out if there are any roots of the closedloop characteristic equation in the RHP. 

If the process is openloop unstMe, Gm<.) will have one or more poles in the RHP, so F(,y = 1 + Gm( , B( will also have one or more poles in the RHP. We can find out how many poles there are by solving for the roots of the openloop characteristic equation or by using the Routh stability criterion on the openloop characteristic equation (the denominator of Gj, ). Once the number of poles P is known, the number of zeros can be found from Eq. (13.6). 

Example 13.4. Figure 13.Sa shows the polar plot of an interesting system that has conditional stability. The system openloop transfer function has the form... 

Keep in mind that we are talking about closedloop stability and that we are studying it by making frequency-response plots of the total openlaop system transfer function. We are also considering openlocp stable systems most of the time. We will show how to deal with openloop unstable processes in Sec. 13.4. 

An openloop unstable, second-order process has one positive pole at + 1/ti and one negative pole at — I/tj. If a proportional controller is used and if ti < show by using a root locus plot and then by using the Nyquist stability criterion that the system is always unstable. 

Thus the Nyquist stability criterion for a multivariable openloop-stable process is ... 

The stability of this openloop system will depend on the values of the poles of the openloop transfer function. If all the p, lie inside the unit circle, the system is openloop stable. 

Processes with RHP poles in their openloop transfer functions are openloop unstable. The irreversible, exothermic chemical reactor is the classical chemical engineering example. These systems can sometimes be stabilized by using... 

Of course, if b is too small (so that h Ms greater than 2), one of the paths will never enter the unit circle. Thus a proportional controller is unable to stabilize this openloop unstable system if... 

The reactor temperature used in the results discussed above is 350 K. If a lower reactor temperature is used, the reactor will be larger for the same conversion, which will give larger heat transfer area. We would expect the controllability of the reactor to improve. This is quantitatively confirmed by the location of the openloop poles, as shown in Table 3.2, and also in the root locus plots shown in Figure 3.3. Now the 85% conversion case is still openloop-unstable but has a range of controller gains that will stabilize the closedloop system. 

The first reactor in the 3-CSTR process has a conversion rate of 72.8%, and the reactant concentration in this first reactor is 2.18 kmol/m3. The reactor volume is low (14.3 m3), and the jacket heat transfer area is only 24.5 m2. The resulting jacket temperature (300 K) is almost down to the inlet cooling water temperature of 294 K. Linear analysis gives a Nyquist plot that never drops into the third quadrant, so the critical (—1,0) point cannot be encircled in a counterclockwise direction. This is required for closedloop stability because the openloop system is unstable and has a positive pole. Thus a proportional controller cannot stabilize this first reactor. 

These results show that aggressive tuning (ZN) is required to stabilize this openloop unstable system. Production rate can be effectively changed by changing either reactor inlet temperature or recycle flow. 

Figure 7.5b gives Nyquist plots for the process with the furnace (FS2). First, note that the system is now openloop-sfaWe for values of reactor gain KR = 2, 3, which was not the case for the FS 1 flowsheet. Second, observe that even for reactor gains up to about KR = 8 it is possible to use a P controller to stabilize the system. Remember that we are talking about the GYi(s) controller, with the G<2ls) controller on automatic (Kcl = 2.13 and rn = 1-94). 

Most processes are openloop stable, i.e., stable with no controllers on the system. One important and very interesting exception that we will study in some detail is the exothermic chemical reactor, which can be openloop unstable. All real processes can be made closedloop unstable (unstable when a feedback controller is in the system) if the controller gain is made large enough. Thus, stability is of vital concern in feedback control systems. 

This stability requirement applies to any system, openloop or closedloop. The stability of an openloop process depends on the location of the poles of its openloop transfer function. The. stability of a closedloop process depends on the location of the poles of its closedloop transfer function. These closedloop poles will naturally be different from the openloop poles because of the introduction of the feedback loop with the controller. Thus, the criteria for openloop and closedloop stability are different. Most systems are openloop stable but can be either closedloop stable or unstable, depending on the values of the controller parameters. We will show that any real process can be made closedloop unstable by making the gain of the feedback controller large enough. There are some processes that are openloop unstable. We will show that these systems can usually be made closedloop stable by the correct choice of the type of controller and its settings. 

On the surface, the Nyquist stability criterion is quite remarkable. We are able to deduce something about the stability of the closedloop system by making a frequency response plot of the openloop system And the encirclement of the mystical. 

There are two basic types of specifications commonly used in the frequency domain. The first type, phase margin and gain margin, specifies how near the openloop GM iu)Gc ia)) polar plot is to the critical (- 1,0) point. The second type, maximum closedloop log modulus, specifies the height of the resonant peak on the log modulus Bode plot of the closedloop servo transfer function. So keep the apples and the oranges straight. We make openloop transfer function plots and look at the (- 1, 0) point. We make closedloop servo transfer function plots and look at the peak in the log modulus curve (indicating an underdamped system). But in both cases we are concerned with closedloop stability. 

The Nyquist stability criterion can be used for openloop-unstable processes, but we have to use the complete, rigorous version with P (the number of poles of the closedloop characteristic equation in the RHP) no longer equal to zero. 

---