Derivative action

Derivative action is intended to be anticipatory in nature indeed, in older texts, it was called this. It anticipates by taking action if it detects a rapid change in error. The error may be very small (even zero) but, if changing quickly, will surely be large in the future. Derivative action attempts to prevent this by changing the output in proportion to the rate of change of error, i.e. 

Ta is known as the derivative time and is the means by which the engineer can dictate how much derivative action is taken. Converting Equation (3.14) to its discrete form, 

In order to demonstrate the effect of derivative action we will formulate a proportional plus derivative (PD) controller. This probably has no practical application but including integral action would make the trends very difficult to interpret. Combining Equations (3.5) 

Processes with a long deadtime, more specifically processes with a large dh ratio, are relatively few. Thus most controllers, when responding to a change in SP, do not obviously benefit firom the addition of derivative action. Indeed, if the 6/z ratio is small, instability can be caused by relatively small amounts of derivative action. However we will demonstrate later that, even as the 9/x ratio approaches zero, a little derivative action can be very useful in speeding the recovery from a process disturbance. 

While there are temperatures with very short deadtimes there will be other measurements that, under certain circumstances, show long deadtimes. In Chapter 4 we include a level control configuration that is likely to benefit from derivative action. In Chapter 7 we describe a composition control strategy with a very large OIr ratio. 

The derivative (rate or preact) action in a controller is for responding to the rate of change of the error (Equation 8.3). 

The derivative time constant, 7, determines the amount of change in controller output as the rate of change of the error increases or decreases. When the controller output responds directly to the error signal as shown in Equation 8.3, a spike or noisy movement in the process variable can be translated into an immediate movement in the controller output. 

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The derivative action assists with settling out these oscillations faster. The derivative action term is as follows ... 

Process-variable feedback for the controller is achieved by one of two methods. The process variable can (I) be measured and transmitted to the controller by using a separate measurement transmitter with a 0.2-I.0-bar (3-15-psi pneumatic output, or (2) be sensed directly by the controller, which contains the measurement sensor within its enclosure. Controllers with integral sensing elements are available that sense pressure, differential pressure, temperature, and level. Some controller designs have the set point adjustment knob in the controller, making set point adjustment a local and manual operation. Other types receive a set point from a remotely located pneumatic source, such as a manual air set regulator or another controller, to achieve set point adjustment. There are versions of the pneumatic controller that support the useful one-, two-, and three-mode combinations of proportional, integral, and derivative actions. Other options include auto/manual transfer stations, antireset windup circuitry, on/off control, and process-variable and set point indicators. 

In equation (4.91), is called the derivative action time, and is formally defined as The time interval in which the part of the control signal due to proportional action increases by an amount equal to the part of the control signal due to derivative action when the error is changing at a constant rate (BS 1523). 

Derivative mode This improves on the proportional-only control by responding solely to the rate of change of the deviation but not in any way to the actual value of the deviation. Derivative action is always used with proportional control. 

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. 

Include derivative action and increase Xq until noise develops. Set Tq at 1/2 this value. 

Derivative action is never used by itself. The simplest implementation is a proportional-derivative (PD) controller. The time-domain equation and the transfer function of an "ideal" PD controller are ... 

In practice, we cannot build a pneumatic device or a passive circuit which provides ideal derivative action. Commercial (real ) PD controllers are designed on the basis of a lead-lag element ... 

To reduce derivative kick (the sudden jolt in response to set point changes), the derivative action can be based on the rate of change of the measured (controlled) variable instead of the rate of change of the error. One possible implementation of this idea is in Fig. 5.3. This way, the derivative control action ignores changes in the reference and just tries to keep the measured variable constant.2... 

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.)... 

In real life, different manufacturers implement the real PID controller slightly differently. 1 One possibility is to modify the derivative action as... 

We can build our own controllers, but two simple ones are available an ideal PID and a PID with approximate derivative action. 

Derivative cannot be used alone as a control mode. This is because a steady-state input produces a zero output in a differentiator. If the differentiator were used as a controller, the input signal it would receive is the error signal. As just described, a steady-state error signal corresponds to any number of necessary output signals for the positioning of the final control element. Therefore, derivative action is combined with proportional action in a manner such that the proportional section output serves as the derivative section input. 

As illustrated in Figure 29, the proportional only control mode responds to the decrease in demand, but because of the inherent characteristics of proportional control, a residual offset error remains. Adding the derivative action affects the response by allowing only one small overshoot and a rapid stabilization to the new control point. Thus, derivative action provides increased stability to the system, but does not eliminate offset error. 

Rate action is not usually employed with fast responding processes such as flow control or noisy processes because derivative action responds to any rate of change in the error signal, including the noise. 

Derivative action cannot be used as a control mode alone. 

Proportional plus reset plus rate controllers combine proportional control actions with integral and derivative actions. 

When an error is introduced to a PID controller, the controller s response is a combination of the proportional, integral, and derivative actions, as shown in Figure 30. 

The derivative action provides additional stability to the process. 

C DERIVATIVE ACTION. The purpose of derivative action (also called rate or preact) is to anticipate where the process is heading by looking at the time rate of change of the controlled variable (its derivative). If we were able to take the derivative of the error signal (which we cannot do perfectly, as we will explain more fully in Chap. 10), we would have ideal derivative action. 

In theory, derivative action should always improve dynamic response, and it does in many loops. In others, however, the problem of noisy signals (fluctuating process-measurement signals) maJces the use of derivative action undesirable. 

D. PROPORTIONAI INTEGRAI DERIVATIVE (PID) controller. PID controllers are used in loops where signals are not noisy and where tight dynamic response is important. The derivative action helps to compensate for tags in the... 

Derivative aetion can be used on either the error signal (SP — PM) or just the process measurement (PM). If it is on the error signal, step changes in set-point will produce large bumps in the control valve. Therefore, in most process control applications, the derivative action is applied only to the PM signal as it enters the controller. The P and I action is then applied to the difference between the setpoint and the output signal from the derivative unit (see Fig. 7.12). 

With the controller on manual, take all the integral and derivative action out of the controller, i.e., set tj at maximum minutes per repeat and minimum minutes. 

Control is improved when the PID controller is used. There is less deviation in the controlled variable because the manipulated variable changes more quickly. As discussed above, if rapid and large changes in the manipulated variable cannot be tolerated, derivative action cannot be used to improve the control performance. 

Therefore, if < tpi a proportional controller cannot make the system closed-loop stable. A controller with derivative action might be able to stabilize the system. Figure 11.9fi,c gives the root locus plots for the two cases and... 

Up to this point we have looked at using proportional controllers on openloop unstable systems. The controllability can often be improved by using derivative action in the controller. An example will illustrate the point. 

The BLT procedure discussed above was applied with PI controllers. The method can be extended to include derivative action (PID controllers) by using two detuning factors F detunes the ZN reset and gain values and Fp detunes the ZN derivative value. The optimum value of Fg is that which gives the minimum... 

A capsule summary of the merits of the three kinds of corrective action can be made. The proportional action is rapid but has a permanent offset that increases as the action speeds up. The addition of integral action reduces or entirely eliminates the offset but has a more sluggish response. The further addition of derivative action speeds up the correction. The action of a three-mode PID controller can be made rapid and without offset. These effects are illustrated in Figure 3.3 for a process subjected to a unit step upset, in this case a change in the pressure of the control air. The ordinate is the ratio of the displacements of the response and upset from the set point. 

Derivative action (often termed rate control) gives an output which is proportional to the derivative of the error. Hence, for PD control ... 

Control action due to the derivative mode occurs only when the error is changing (equation 7.4). The presence of the derivative mode contributes an additional output, KD(de/dt), to the final control element as soon as there is any change in error. When the error ceases to change, derivative action no longer occurs (Fig. 7.8). The effect of this is similar to having a proportional controller with a high gain... 

Such an element provides the high frequency roll-off that is necessary with derivative action (i.e. it avoids the tendency of the ideal derivative mode to amplify noise in the error signal). The inclusion of such an element leads to the transfer function of the relevant industrial controller as being ... 

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