Action integrals

The main purpose of integral action is to eliminate offset. Sometimes called reset action it continues to change the controller output for as long as an error exists. It does this by making the rate of change of output proportional to the error, i.e. 

Ti is known as the integral time and is the means by which the engineer can dictate how much integral action is taken. Equation (3.10) is already in the velocity form, integrating gives us the form that gives the action its name. 

Converting Equation (3.10) to its discrete form (where ts is the controller scan interval) gives 

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

For most situations a PI controller is adequate. Indeed many engineers will elect not to include derivative action to simplify tuning the controller by trial-and-error. A two-dimensional search for optimum parameters is considerably easier than a three-dimensional one. However in most situations the performance of even an optimally tuned PI controller can be substantially improved. 

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The integral action of a PID algorithm is designed to remove this offset. The integral action term is given by... 

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 ... 

Eodt (13-174) where V and are initial values, Kc and T are respectively feed-back-controller gain and feedback-reset time for integr action, and E is the error or deviation from the set point as given by ... 

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). 

Proportional plus integral plus derivative action Proportional action provides a controller output proportional to the error signal. Integral action supplies a controller output which changes in the direction to reduce a constant error. Derivative action provides a controller output determined by the direction and rate of change of the deviation. When all these are combined into one controller (three-term or PID), there is an automatic control facility to correct any process changes. 

Integral action is brought in with high Xi values. These are reduced by factors of 2 until the response is oscillatory, and Tj is set at 2 times this value. 

Information flow in model 7 Integral action time 182 Integral control 97, 547 Integral control constant 97, 507 Integrated extraction 335 Integration... 

The exception is when a process contains integrating action, i.e., 1/s in the transfer functions—a point that we will illustrate later. 

To eliminate offset, we can introduce integral action in the controller. In other words, we use a compensation that is related to the history of the error ... 

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 ... 

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.)... 

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 result is an ideal PD controller with the choice of xD = xp. See that you can obtain the same result with IMC too. Here, take the process function as the approximate model and it has no parts that we need to consider as having positive zeros. There is no offset the integrating action is provided by Gp. 

You may notice that nothing that we have covered so far does integral control as in a PID controller. To implement integral action, we need to add one state variable as in Fig. 9.2. Here, we integrate the error [r(t) -, (t) to generate the new variable xn+1. This quantity is multiplied by the additional feedback gain Kn+1 before being added to the rest of the feedback data. 

Example 4.7C Add integral action to the system in Example 4.7B so we can eliminate the steady state error. 

The modest 10% offset that we have in the slave loop is acceptable under most circumstances. As long as we have integral action in the outer loop, the primary controller can make necessary adjustments in its output and ensure that there is no steady state error in the controlled variable (e.g., the furnace temperature). 

A proportional plus integral controller is used to control the level in the reflux accumulator of a distillation column by regulating the top product flowrate. At time t = 0, the desired value of the flow controller which is controlling the reflux is increased by 3 x 10-4 m3/s. If the integral action time of the level controller is half the value which would give a critically damped response and the proportional band is 50 per cent, obtain an expression for the resulting change in level. 

If there had been no integral action, what would have been the offset in the level in the accumulator ... 

The feedback controller has proportional and integral action. It changes Cam based on the magnitude of the error (the difference between the setpoint and Ca3) and the integral of this error. 

B. INTEGRAL ACTION (RESET). Proportional action moves the control valve in direct proportion to the magnitude of the error. Integral action moves the control valve based on the time integral of the error, as sketched in Fig. 7.9b. 

Most controllers are calibrated in minutes (or minutes/repeat, a term that comes from the test of putting into the controller a fixed error and seeing how long it takes the integral action to ramp up the controller output to produce the same change that a proportional controller would make when its gain is 1 the integral repeats the action of the proportional controller). 

The basic purpose of integral action is to drive the process back to its setpoint when it has been disturbed. A proportional controller will not usually return the controlled variable to the setpoint when a load or setpoint disturbance occurs. This permanent error (SP — PM) is called steadystate error or offset. Integral action reduces the offset to zero. 

Integral action degrades the dynamic response of a control loop. We will demonstrate this quantitatively in Chap. 10. It makes the control loop more oscillatory and moves it toward instability. But integral action is usually needed if it is desirable to have zero offset. This is another example of an engineering trade-off that must be made between dynamic performance and steadystate performance. 

B. PROPORTIONAL CONTROLLER. The output of a proportional controller changes only if the error signal changes. Since a load change requires a new control-valve position, the controller must end up with a new error signal. This means that a proportional controller usually gives a steadystate error or offset. This is an inherent limitation of P controllers and why integral action is usually added. 

C. PROPORTIONAL-INTEGRAL (PI) CONTROLLER. Most control loops use PI controllers. The integral action eliminates steadystate error in T (see Fig. 7.11c). The smaller the integral time r, the faster the error is reduced. But the system becomes more underdamped as t( is reduced. If it is made too small, the loop becomes unstable. 

The common types of control loops are level, flow, temperature, and pressure. The type of controller and the settings used for any one type are sometimes pretty much the same from one application to another. For example, most flow control loops use PI controllers with wide proportional band and fast integral action. 

Now start bringing in integral action by reducing by factors of 2, making small disturbances at each value of i to see the eifect. 

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