Proportional-plus-Integral PI Control

The PI controller gain has an effect not only on the error, but also on the integral action. When we compare the equation for a PI controller (Equation 4.14) with that for a P-only controller (Equation 4.11) we see that the bias term in the P-only controller has been replaced by the integral term in the PI controller. Thus, the bias term for PI 

Therefore, the integral action provides a bias that is automatically adjusted to eliminate any error. The PI controller is faster in response than the I-only controller because of the addition of the proportional action, as illustrated in Eigure 4.12. 

As can be seen from Equation 4.16, pi, the gain of the PI controller, is the sum of the two component gains. These component gains are proportional action Kf and integral action K.  

The Kc and 71 are used to adjust the PI controller gain to give the loop a desired response. Suppose 71 = oo, which would result in = 0, regardless of the value of Kc- In effect, the response would be that of a P-only controller with a period equal to and a sustained error. Although 71 = oo is not realizable, it can be set to a very large number in min/repeat to minimize the integral action. 

suppose 71 were set to a very small value. In this case, the PI controller gain would approach that of an integral-only controller, since Ki p. The control action in the loop would now be that of an I-only controller with a return to the set point, but with a long response period. 

Proportional-plus-integral (PI) Control Integral action eliminates the offset described above by moving the controller output at a 

8-23 Both load regulation and set-point response require high gains for the feedback controller 

Because the integral term lags the proportional term by 90° in phase, the PI controller then always produces a phase lag between 0° and 90°  

In equation (4.68), T is called the integral action time, and is formally defined as The time interval in which the part of the control signal due to integral action increases by an amount equal to the part of the control signal due to proportional action when the error is unchanging . (BS 1523). 

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

The denominator is now in the standard second-order system form of equation (3.42). The steady-state response may be obtained using the final value theorem given in equation (3.10). 

For a first-order plant, PI eontrol will produee a seeond-order response. There will be zero steady-state errors if the referenee and disturbanee inputs r (t) and f2(f) are either unehanging or have step ehanges. The proeess of ineluding an integrator within the eontrol loop to reduee or eliminate steady-state errors is diseussed in more detail in Chapter 6 under system type elassifieation . 

The measured head h it) is obtained from the pressure transducer 

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Choose the best value of Kp and compare the behavior with the behavior for the open loop system without controls, i.e., for Kp = 0.0. Show that it is possible to remove the offset by using a proportional plus integral (PI) controller and find the best value of Kj that can be used with the best value for Kp as obtained above. Plot the change of height with time and compare it with the results of the open-loop system in the case without control and also when the system has only proportional control. 

Proportional-plus-Integral (PI) Control Integral action eliminates me 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 ... 

Chapter 22 provides equations for typical process controllers and control valve dynamics. The controllers considered are the proportional controller, the proportional plus integral (PI) controller and the proportional plus integral plus derivative (PID) controller. Integral desaturation is an important feature of PI controllers, and mathematical mc els are produced for three different types in industrial use. The control valve is almost always the final actuator in process plan. A simple model for the transient response of the control valve is given, which makes allowance for limitations on the maximum velocity of movement. In addition, backlash and velocity deadband methods are presented to model the nonlinear effect of static friction on the valve. 

Combining with Equation (3.5) gives proportional plus integral (PI) control... 

The combination of the two control modes is called the proportional plus reset (PI) control mode. It combines the immediate output characteristics of a proportional control mode with the zero residual offset characteristics of the integral mode. 

An example of a typical commercial controller is shown in Figure 24.8. The Foxboro SPEC 200 PID Controller and Display provides proportional-plus-integral (PI) or proportional-plus-integral-plus-derivative (PID) control. The set-point dial on the... 

We have just described proportional (P) control and proportional plus derivative (PD) control. Integration can be added to a controller, which not only gives it reset action, but also can exacerbate instability. There are proportional plus integral (PI) and proportional-integral-derivative (PID) controllers. These classical types are used where the system dynamics (the Plant) are well defined. 

Controller modes—settings and functions that include proportional (P), proportional plus integral (PI), proportional plus derivative (PD), and proportbnal-integral-derivative (PID). Proportional control is primarily used to provide gain where little or no load change typically occurs in the process. Proportional plus integral is used to eliminate offset between the setpoint and process variables PI works best where... 

The controller can be proportional plus integral (PI) or integral only (1-only) and is tuned similarly to the flow controller. 

A proportional plus integral controller will give a response period that is longer than a P-only controller but much shorter than an 1-only controller. Typically, the response period of the process variable PV under PI control is approximately 50 per cent longer than for the P-only (1.5th, Figure 4.11). Since this response is much faster than I-only, and only somewhat longer than P-only control, the majority (>90 per cent) of controllers found in plants are PI controllers. The equation for a PI controller is... 

Proportional-integral control. The proportional-integral (or PI) mode is also often referred to as proportional plus reset since this control mode eliminates the offset associated with proportional control alone. The... 

A set of Internal Model Control (IMC) tuning rules were established by Rivera, Morari, and Skogestad for a first-order plus dead time (FOPDT) open-loop process response that simply involves the adjustment of the proportional gain in the controller, K, for tuning. The integral time constant, 7, is set equal to the first-order time constant, TFO, for PI controllers (Table 10.5). 

The most common type of controller used in the chemical and petroleum industries was once called proportional plus automatic reset, later shortened to proportional reset. Today it is more common to use PI, which stands far proportional-integral. It is also becoming common today to speak of controller gain rather than proportional band (PB = IQO/K ). We will also use reset time, usually in minutes, rather than its older reciprocal, repeats per minute. ... 

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