Closedloop Systems

Now let us make life a little more interesting. The system considered above is an openloop system, i.e no feedback control is used. If we add a feedback controller, we have a closedloop system. The controller looks at the product concentration leaving the third tank and makes adjustments in the inlet concentration to the first reactor C>o in order to keep near its desired setpoint value C 3. The variable is a disturbance concentration and the variable is a manipulated concentration that is changed by the controller. We... 

Then in Chaps. 7 and 8 we will look at closedloop systems. Instrumentation hardware, controller types and performance, controller tuning, and various types of control systems structures will be discussed. 

One of the most important parameters that we will use in the remaining sections of this book is the damping coefiident of the closedloop system. We typically tune a controller to give a closedloop system that has a damping coefficient of about 0.3. 

Let us now consider a process with a feedback controller in service. This closedloop system can experience disturbances at two difierent spots in the feedback loop load disturbances and setpoint disturbances. 

There are a number of criteria by which the desired performance of a closedloop system can be spedlied in the time domain. For example, we could specify that the closedloop system be critically damped so that there is no overshoot or oscillation. We must then select the type of controller and set its tuning constants so that it will give, when coupled with the process, the desired closedloop response. Naturally the control specification must be physically attainable. We cannot make a Boeing 747 jumbo jet airplane behave like an F-IS fighter. We cannot... 

Notice that the first five of these assume an underdamped closedloop system, i.e., one that has some oscillatory nature. 

The isotherma] three-CSTR process of Sec 5.2 has, as we will prove in Chap. 10, an ultimate gain of 64 and an ultimate period of 3.63 minutes. The ZN settings for this system are given in Table 7.2. The response of the closedloop system to a step load disturbance in Cas is shown in Fig. 7.16 with P, PI, and PID controllers and the ZN settings. 

In this chapter we will demonstrate the signiScant computational and nota-tional advantages of LaplaTce transforms. The techniques involve finding the transfer function of the openloop process, specifying the desired performance of the closedloop system (process plus controller) and finding the feedback controller transfer function that is.required to do the job. 

Equation 10.5 gives the transfer functions describing the closedloop system, so these are closedloop transfer functions. The two inputs are the load L, and the setpoint The controlled variable is Note that the denominators of both of these closedloop transfer functions are identical. 

Then the closedloop block diagram, shown in Fig. 10.3, becomes more simple. The equation describing this simplified closedloop system is... 

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. 

In order to design feedback controllers, we must have some way to evaluate their effect on the performance of the closedloop system, both dynamically and at steadystate. 

The dynamic performance of a system can be deduced by merely observing the location of the roots of the system characteristic equation in the s plane. The time-domain specifications of time constants and damping coefficients for a closedloop system can be used directly in the Laplace domain. 

The examples above have illustrated a very important point the higher the order of the system, the worse the dynamic response of the closedloop system. The hrst-order system is never underdamped and cannot be made closedloop unstable for any value of gain. The second-order system becomes underdamped as gain is increased but never goes unstable. Third-order (and higher) systems can be made closedloop unstable. 

Find the ultimate gain and period of a closedloop system with a proportional controller and an openloop transfer function ... 

Make a root locus plot of the closedloop system with a value of integral time Tj = 10 minutes. 

Repeat Prob. 10.6 using a proportional feedback controller [parts (h) and (d)]. Will there be a steadystate error in the closedloop system for (n) a step change in setpoint or (h) a step change in feed rate Fq ... 

Derive an expression relating the controller gain to the parameters t, and K, such that the closedloop system damping coelTicient is 0.707. What happens to as T gels very small or very large What is the value of that provides the smallest value of ... 

Remember that in Example 10.8 adding a lead or a negative zero made the closedloop system more stable. In this example we have shown that adding a positive zero has just the reverse effect. 

Find the value of feedback controller gain K, that gives a closedloop system with a damping coeflicient of 0.707 for a second-order openloop unstable process with... 

The Nyquist stability ciiterion is a method for determining the stability of systenas in the frequency domain. It is almost always applied to closedloop systems. A working, but not completely general, statement of the Nyquist stabiU ity criterion is ... 

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. 

B, APPLICATION OF THEOREM TO CLOSEDLOOP STABILITY. To check the stability of a system, we are interested in the roots or zeros of the characteristic equation. If any of them lie in the right half of the s plane, the system is unstable. For a closedloop system, the characteristic equation is... 

SiAble and unstable closedloop systems in Nyquist, Bode, and Nichols plots. 

Figure 13.6 illustrates stable and unstable closedloop systems on the three types of plots. 

All the Nyquist, Bode, and Nichols plots discussed in previous sections have been for openloop system transfer functions B(j ). Frequency-response plots can be made for any type of system, openloop or closedloop. The two closedloop transfer functions that we derived in Chap. 10 show how the output is affected in a closedloop system by a setpoint input and by a load. Equation (13.28) gives the closedloop servo transfer function. Equation (13.29) gives the closedloop load transfer function. 

In most systems, the closedloop servo log modulus curves move out to higher frequencies as the gain of the feedback controller is increased. This is desirable since it means a faster closedloop system. Remember, the breakpoint frequency is the reciprocal of the closedloop time constant. 

But the height of the resonant peak also increases as the controller gain is increased. This means that the closedloop system becomes more underdamped. The effects of increasing controller gain are sketched in Fig. 13.10c. 

The lines of constant closedloop log modulus L, are part of the Nichols chart. If we are designing a closedloop system for an L specification, we merely have to adjust the controller type and settings so that the openloop B curve is tangent to the desired line on the Nichols chart. For example, the G B curve in Fig. 13,11b with X, = 20 is just tangent to the +2 dB line of the Nichols chart. The value of frequency at the point of tangency, 1.1 radians per minute, is the closedloop resonant frequency aif. The peak in the log modulus plot is clearly seen in the closedloop curves given in Fig. 13.12. 

---