Process open-loop

Example 11JL Let s take the same process as studied in Example 11.7. The process open loop transfer function is... 

The transfer functions for the different parts can be written as follows Process, Open-Loop Equation... 

Process Monitoring Process Open-Loop Control Process Closed-Loop Control... 

An open-loop system positions the manipulated variable either manually or on a programmed basis, without using any process measurements. This operation is acceptable for well-defined processes without disturbances. An automanual transfer switch is provided to allow manual adjustment of the manipulated variable in case the process or the control system is not performing satisfac torily. 

Open-Loop versus Closed-Loop Dynamics It is common in industry to manipulate coolant in a jacketed reacdor in order to control conditions in the reacdor itself. A simplified schematic diagram of such a reactor control system is shown in Fig. 8-2. Assume that the reacdor temperature is adjusted by a controller that increases the coolant flow in proportion to the difference between the desired reactor temperature and the temperature that is measured. The proportionality constant is K. If a small change in the temperature of the inlet stream occurs, then depending on the value or K, one might observe the reactor temperature responses shown in Fig. 8-3. The top plot shows the case for no control (K = 0), which is called the open loop, or the normal dynamic response of the process by itself. As increases, several effects can be noted. First, the reactor temperature responds faster and faster. Second, for the initial increases in K, the maximum deviation in the reactor temperature becomes smaller. Both of these effects are desirable so that disturbances from normal operation have... 

A process plant has an open-loop transfer function... 

As stated in the previous section, the major reactant feed was chosen as the manipulated variable. In the trial this feed was subjected to a pseudo-random binary sequence (PRBS) signal in an open loop operation of the process. The results of the trial, plotted in Fig. 2, show a strong -- but delayed -- cross-correlation between the manipulated feed rate and the reactor temperature. Using techniques described by Box and Jenkins (2), a transfer function relating the manipulated variable to the control variable of interest can be developed. The general form of this transfer function is... 

The identification of plant models has traditionally been done in the open-loop mode. The desire to minimize the production of the off-spec product during an open-loop identification test and to avoid the unstable open-loop dynamics of certain systems has increased the need to develop methodologies suitable for the system identification. Open-loop identification techniques are not directly applicable to closed-loop data due to correlation between process input (i.e., controller output) and unmeasured disturbances. Based on Prediction Error Method (PEM), several closed-loop identification methods have been presented Direct, Indirect, Joint Input-Output, and Two-Step Methods. 

The PBL reactor considered in the present study is a typical batch process and the open-loop test is inadequate to identify the process. We employed a closed-loop subspace identification method. This method identifies the linear state-space model using high order ARX model. To apply the linear system identification method to the PBL reactor, we first divide a single batch into several sections according to the injection time of initiators, changes of the reactant temperature and changes of the setpoint profile, etc. Each section is assumed to be linear. The initial state values for each section should be computed in advance. The linear state models obtained for each section were evaluated through numerical simulations. 

The key is to recognize that the system may exhibit underdamped behavior even though the open-loop process is overdamped. The closed-loop characteristic polynomial can have either real or complex roots, depending on our choice of Kc. (This is much easier to see when we work with... 

To make use of empirical tuning relations, one approach is to obtain the so-called process reaction curve. We disable the controller and introduce a step change to the actuator. We then measure the open-loop step response. This practice can simply be called an open-loop step test. Although we disconnect the controller in the schematic diagram (Fig. 6.1), we usually only need to turn the controller to the manual mode in reality. As shown in the block diagram, what we measure is a lumped response, representing the dynamics of the blocks Ga,... 

Figure 6.2. Illustration of fitting Eq. (6-2, solid curve) to open-loop step test data representative of self-regulating and multi-capacity processes (dotted curve). The time constant estimation shown here is based on the initial slope and a visual estimation of dead time. The Ziegler-Nichols tuning relation (Table 6.1) also uses the slope through the inflection point of the data (not shown). Alternative estimation methods are provided on our Web Support.

One reason why this approximation works is that process unit operations are generally open-loop stable, and many are multi-capacity in nature. Reminder Underdamped response of the system is due to the controller, which is taken out in the open-loop step test. 

Even though this result is based on what we say is a process function, we could apply (E6-4) as if the derivation is for the first order with dead time function GPRC obtained from an open-loop step test. 

The idea is that we may cancel the (undesirable open-loop) poles of our process and replace them with a desirable closed-loop pole. Recall in Eq. (6-20) that Gc is sort of the reciprocal of Gp. The zeros of Gc are by choice the poles of Gp. The product of GcGp cancels everything out—hence the term pole-zero cancellation. To be redundant, we can rewrite the general design equation as... 

Under normal circumstances, we would pick a x which we deem appropriate. Now if we pick x to be identical to xp, the zero of the controller function cancels the pole of the process function. We are left with only one open-loop pole at the origin. Eq. (6-29), when x = xp, is reduced to... 

Say we model our process (read fitting the open-loop step test data) as a first order function with time delay, and expecting experimental errors or uncertainties, our measured or approximate model... 

Empirical tuning with open-loop step test Measure open-loop step response, the so-called process reaction curve. Fit data to first order with dead-time function. 

Addition of a feedback control loop can stabilize or destabilize a process. We will see plenty examples of the latter. For now, we use the classic example of trying to stabilize an open-loop unstable process. 

Note that with this very specific case by choosing x = 1, the open-loop zero introduced by the PI controller cancels one of the open-loop poles of the process function at -1. If we do a root locus plot later, we d see how the root loci change to that of a purely second order system. With respect to this example, the value is not important as long as x > 1/2. 

With only open-loop poles, examples (a) to (c) can only represent systems with a proportional controller. In case (a), the system contains a first orders process, and in (b) and (c) are overdamped and critically damped second order processes. 

Another advantage of frequency response analysis is that one can identify the process transfer function with experimental data. With either a frequency response experiment or a pulse experiment with proper Fourier transform, one can construct the Bode plot using the open-loop transfer functions and use the plot as the basis for controller design.1... 

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... 

No doubt there are interactions from other loops, and from (10-37), some of the process gains must have opposite signs (or act in different directions). When Ay = 0.5, we can interpret that the effect of the interactions is identical to the open-loop gain—recall statement after (10-36). When Ay > 0.5, the interaction is less than the main effect of nij on Cj. However, when Ay < 0.5, the interactive effects predominate and we want to avoid pairing nij with c . 

We will ignore the values of any gains. We focus only on the probable open-loop pole and zero positions introduced by a process or by a controller, or in other words, the shape of the root locus plots. 

Let s consider an overdamped process with two open-loop poles at -1 and -2 (time constants at 1 and 0.5 time units). A system with a proportional controller would have a root locus plot as follows. We stay with tf (), but you can always use zpk (). 

When the new product to be manufactured is the same as what it started as, for example a new bottle made from bottle scrap, the recycling is referred to as closed-loop. When the new application is different from the starting one, the process is referred to as open-loop recycling, as is the case when the polyethylene terephthalate bottle is used to produce polyester fiber for carpeting. 

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