Derivative mode, controllers

Proportional-plus-Integral-plus-Derivative (PID) Control The derivative mode moves the controller output as a function of the rate-of-change of the controlled variable, which adds phase lead to the controller, increasing its speed of response. It is normally combined... 

The pressure controller (controller block) amplifies the transmitter signal and sends a modified signal to the final element. Depending on the system requirements, the controller block may include additional correction factors, integral and derivative (reset and rate). This is called a three-mode controller. 

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. 

Three principal functional control modes are proportional (P), integral (I) and derivative (D) control. These are performed by the ideal three-mode controller (PID), described by the equation... 

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... 

This is essentially a compromise between the advantages and disadvantages of PI and PD control. Offset is eliminated by the presence of integral action and the derivative mode reduces the maximum deviation and time of oscillation, although the latter are still greater than with PD control alone (Fig. 7.5). 

Normal commercial PID controllers are generally constructed by adding a lead compensator (Section 7.12.2) as the derivative mode to a PI controller. This type of derivative module typically has the transfer function ... 

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 ... 

The computational power and flexibility of the computer is much used now to simulate controllers having characteristics other than the standard P, PI, etc., modes. Controllers are described in the following for which the design algorithm is derived directly from a specification of the discrete time character of the response of the controlled variable to a given change in set point. 

A controller compares its measurement (y) to its set point (r), and based on the difference (e = error) sends an output signal (m - manipulated variable) to the final control element (e.g., control valve) to eliminate the error. The control mode options include on-off, floating, proportional (P), integral (I), differential (D), and many others. The proportional mode considers the present state of the process error, the integral mode looks at the past history of the error, and the derivative mode anticipates the future values of the error... 

The derivative mode anticipates the future values of the error and acts on that prediction. This third control mode became necessary as the size of processing equipment increased and, correspondingly, the mass and the thermal inertia of such equipment also became greater. The purpose of the derivative mode is to predict the process errors before they have evolved and take corrective action in advance. The derivative action is not used by itself, but only in combination with the proportional mode. The output of a PD controller is as follows ... 

The derivative (or rate) settings are in units of time and can be adjusted from a few seconds to up to 10 h or more. Because the derivative mode acts on the rate at which the error signal changes, it can also cause unnecessary upsets because, for example, it will react to the sudden set point changes made by the operator. It will also amplify noise, and will cause upsets when the measurement signal changes occur in steps, as in case of periodic measurements. Therefore, in such situations it should either be avoided or the controller be reconfigured so that the D-mode acts only on the measurement and not the error. 

A special-purpose control action used on extremely fast processes is the so-called inverse derivative mode. The output of the inverse derivative mode is inversely proportional to the error s rate of change. It is used to reduce the gain of a controller at high frequencies and is therefore useful in stabilizing a flow loop. Inverse derivative can also be added to a proportional-plus-inte-gral controller to stabilize flow and other loops requiring very low proportional gain for stability. Because inverse derivative is available in a separate unit, it can be added to the loop when stability problems are encountered. 

The design of the valve, process, and measurement should be made such as to minimize deadtime in the loop while providing a reliable, more linear response, then the controller can be tuned to provide the best performance, with an acceptable operating margin for robustness. The PID controller is the most widespread and applicable control algorithm, which can be tuned to provide near optimal responses to load disturbances. PID is an acronym for Proportional, Integral, and Derivative modes of control. 

The Derivative mode is sometimes referred to as rate because it applies control action proportional to the rate of change of its input. Most controllers use the process measurement, rather than the error, for this input in order to prevent an exaggerated response to step changes in the setpoint. Also, noise in the process measurement is attenuated by an inherent filter on the Derivative term, which has a time constant 1/8 to 1/10 of the Derivative time. Even with these considerations, process noise is a major deterrent to the use of Derivative mode. 

The same ideas extend straightforwardly to deal with property surfaces describing the dependence of a molecular property on geometry, for example, the dipole moment and polarisability derivatives that control the activity of a vibrational mode in IR and Raman spectroscopy. Extension to the case of redundant internal coordinates, the typical situation for polyatomic molecules, is also straightforward. 

With the exception of derivative action any of these control modes may be used alone in certain applications. Integral and derivative actions are most usually combined with proportional control to give proportional plus integral control (proportional control with automatic reset) proportional plus derivative control or three-mode control, which is proportional plus integral plus derivative. 

It can be seen from the ideal equation that the reset rate, /TiJA, is the number of times per minute that the integral action repeats the proportional action, and the rate time, Td, is the time that the derivative mode advances the control action over that of the proportional mode alone. 

The same result obtains with three-mode control. On the other hand the combination of proportional and derivative control gives the same steady state error as proportional alone, since the derivative contribution disappears at low frequency. 

In voltage mode control, the ramp applied to the PWM comparator is derived from an internal (fixed) clock. However in current mode control, it is derived from the inductor current (or switch current). And the latter leads to a rather odd situation where even a slight disturbance in the inductor current waveform can become worse in the next cycle (see upper half of Figure 2-10). 

In current mode control (CMC) --- ramp is derived from switch/inductor current... 

The output of the error amplifier (sometimes called COMP, sometimes EA-out, sometimes control voltage ) is applied to one of the inputs of the pulse width modulator ( PWM ) comparator. On the other input of this comparator, we have an applied sawtooth voltage ramp — either internally generated from the clock when using voltage mode control, or derived from the current ramp when using current mode control. Thereafter, by normal comparator action, we get pulses of desired width, with which to drive the switch. 

The plant transfer functions presented earlier were only for voltage mode control. In current mode control, the ramp to the pulse width (duty cycle) modulator is derived from the... 

Although proportional control can be used alone, this is almost never the case for integral or derivative control actions. Instead, proportional-integral (PI) and proportional-integral-derivative (PID) are the usual controllers employing integral and derivative modes of control. 

The most commonly used analog controller is the three-mode proportional-integral-derivative (PID) controller. Its general form is given by eq. (13.6) ... 

It is easy to develop the digital approximation of the PID controller. This was done in Example 27.5, using rectangular integration to approximate the integral control mode and first-order difference to approximate the derivative mode. The resulting discrete-time approximation is given by eq. (27.7) ... 

An ideal three-mode PID (proportional-integral-derivative) feedback controller is described by the equation... 

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