PI controllers

ProportionaJ-plus-Integral (PI) Control Integral action eliminates the offset described above by moving the controller output at a rate proportional to the deviation from set point. Although available alone in an integral controller, it is most often combined with proportional action in a PI controller ... 

The PI controller is by far the most commonly used controller in the process industries. The summation of the deviation with its integral in the above equation can be interpreted in terms of frequency response of the controller (Seborg, Edgar, and Melhchamp, Process Dynamics and Control, Wiley, New York, 1989). The PI controller produces a phase lag between zero and 90 degrees ... 

An example of a pneumatic PI controller is shown in Fig. 8-64 7. This controller has two stages of pneumatic amphfication and a Bourdon tube input element that measures process pressure. The Bourdon tube element is a flattened tube that has been formed into a cui ve so that changes in pressure inside the tube cause vertical motions to occur at the ungrounded end. This motion is transferred to the left end of the beam, as shown. 

The PI controller, even when optimally tuned, is also unable to prevent surge. Furthermore, it is unable to stop surge once it occurs. In the above situation, the operator would correctly identify the problem as instability of the closed-loop PI controller. The only viable action would be to open the closed control loop by placing the controller in manual, thereby freezing the valve open. In this scenario, open-loop control will stop surge. 

It has been observed that the value of the combined closed-loop PI control and the open-loop recycle trip control justifies its use on all turbocompressors. 

Inserting the PI control law given in equation (4.68) into the first-order plant transfer function shown in equation (4.60) gives... 

Fig. 4.25 Step response of a first-order plant using PI control. When there are step ehanges in r t) and riit). ...

Fig. 4.28 Response of the PI controlled liquid-level system shown in Figure 4.26 to a step change in ha t) from 0 to 4 m.

Ivinelic for a Single Ideal SluTed-Tank Flow Reactor under Transient Closed loop Liquid-Level PI Control... 

Proportional integral (PI) control A control algorithm that combines the proportional response and integral response control algorithms. 

Proportional integral derivative (PID) control A control algorithm that enhances the PI control algorithm by adding a component that is proportional to the rate of change of the deviation of the controlled variables. 

Fig. 3. PI control results when EC feed change form 187.56 g/hr to 202.56 g/hr...

For the operating conditions, the set-points of EC and DMC compositions at the top and bottom are 0.01 and 0.2996, respectively, and the bottom temperature should not exceed 140°C to prevent the decomposition of reactants. From these plots, it can be concluded that the MPC outperforms the PI controller in terms of response speed in disturbance rejection, maintaining the variables at set points, and optimization capability. Especially, the PI controller failed to maintain the DMC composition set-point due to the slow long-term dynamics caused by the interaction between the RD column and azeotropic recovery column. 

Obtain the process reaction curve for the process with disconnected controller, as explained in Sec. 2.3.3. Analyse this curve to obtain the parameters for the Ziegler-Nichols Method. Use Table 2.2 to obtain the best controller settings for P and PI control. Try these out in a simulation. 

In practice, integral action is never used by itself. The norm is a proportional-integral (PI) controller. The time-domain equation and the transfer function are ... 

PI control can eliminate offset. We must use a PI controller in our design if the offset is unacceptably large. 

The sign of the rate of change in the error could be opposite that of the proportional or integral terms. Thus adding derivative action to PI control may counteract the overcompensation of the integrating action. PD control may improve system response while reducing oscillations and overshoot. (Formal analysis later will show that the problem is more complex than this simple statement.)... 

If simple proportional control works fine (in the sense of acceptable offset), we may try PD control. Similarly, we may try PID on top of PI control. The additional stabilizing action allows us to use a larger proportional gain and obtain a faster system response. 

Another implementation of the actual PID control is to introduce the derivative control in series with PI control ... 

If we have a second order system, we can derive an analytical relation for the controller. If we have a proportional controller with a second order process as in Example 5.2, the solution is unique. However, if we have, for example, a PI controller (2 parameters) and a first order process, there are no unique answers since we only have one design equation. We must specify one more design constraint in order to have a well-posed problem. 

There are two noteworthy items. First, the closed-loop system is now second order. The integral action adds another order. Second, the system steady state gain is unity and it will not have an offset. This is a general property of using PI control. (If this is not immediately obvious, try take R = 1/s and apply the final value theorem. We should find the eventual change in the controlled variable to be c (°°) =1.)... 

We can see quickly that the system has unity gain and there should be no offset. The point is that integral action can be introduced by the process and we do not need PI control under such circumstances. We come across processes with integral action in the control of rotating bodies and liquid levels in tanks connected to pumps (Example 3.1, p. 3-4). 

The PI controller function % The process function Unity closed loop function GcGp/... 

We could also modify the M-file by changing the PI controller to a PD or PID controller to observe the effects of changing the derivative time constant. (Help is in MATLAB Session 5.) We ll understand the features of these dynamic simulations better when we cover later chapters. For now, the simulations should give us a qualitative feel on the characteristics of a PID controller and (hopefully) also the feeling that we need a better way to select controller settings. 

PI controllers are most common. They eliminate offsets and have acceptable speeds of response in most industrial settings. We usually pick a low to intermediate gain (wide proportional band, PB 150) to reduce the effect of noisy signals (from flow turbulence also why we do not use D control). We also use a low reset time ( 0.1 min/repeat i.e. relatively large I action) to get fast set-point tracking. 

For the vapor pressure in a flash drum (and thus also vapor flow rate), we need a fast and tight response loop. We need at least a PI controller (c.f. the flow control). 

Heat transfer lags can be significant and the nature of the problem can be quite different in various processes. If there is a sensor lag, it is mostly due to heat transfer between the sensor and the fluid medium. (Thermocouples, depending on how we make them, can have very fast response times.) The overall response is sluggish and PI control will make it more so. It is unlikely we can live with any offsets. PID control is the appropriate choice. 

In terms of the situation, if we use a PI controller on a slow multi-capacity process, the resulting system response will be even more sluggish. We should use PID control to increase the speed of the closed-loop response (being able to use a higher proportional gain) while maintaining stability and robustness. This comment applies to other cases such as temperature control as well. 

All tuning relations provide different results. Generally, the Cohen and Coon relation has the largest proportional gain and the dynamic response tends to be the most underdamped. The Ciancone-Marlin relation provides the most conservative setting, and it uses a very small derivative time constant and a relatively large integral time constant. In a way, their correlation reflects a common industrial preference for PI controllers. 

---