PD control

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 ... 

In effect, we are adding a very large real pole to the derivative transfer function. Later, after learning root locus and frequency response analysis, we can make more rational explanations, including why the function is called a lead-lag element. We ll see that this is a nice strategy which is preferable to using the ideal PD controller. 

PD control is not useful for systems with large dead time or noisy signals. 

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. 

The system steady state gain is the same as that with proportional control in Example 5.1. We, of course, expect the same offset with PD control too. The system time constant depends on various parameters. Again, we defer this analysis to when we discuss root locus. 

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. 

If you cannot follow the fancy generalization, think of a simple problem such as a unity feedback loop with a PD controller and a first order process. The closed-loop characteristic equation is... 

Systems (e) and (f) would contain a third order process. Of course, we can only have a proportional control in case (e), while (f) represents one probable scenario of using an ideal PD controller. 

Example 8.11. Derive the magnitude and phase lag of the transfer function of an ideal PD controller. 

The nice feature of the phase-lead compensator, and for that matter a real PD controller, is that it limits the high frequency magnitude. In contrast, an ideal PD controller has no upper limit and would amplify high frequency input noises much more significantly. 

The root locus plot resembles that of a real PD controller. The system remains overdamped with... 

To implement an ideal PD controller, we ll have an additional open-loop zero. Two (of infinite) possibilities are... 

We ll finish with implementing the P, PI and PD controllers on a second order overdamped process. As in the exercise above, try to calculate the derivative or integral time constants, and take a minute to observe the plots and see what may lead to better controller designs. 

This control mode is called proportional plus rate (PD) control because the derivative section responds to the rate of change of the error signal. 

It is proposed to use one of two types of controller in this control loop, either (a) a PD controller whose action approximates to ... 

Having regard to the form of the equation describing the control action in each case, under what general circumstances would inverse rate control be better than normal PD control ... 

Now wc will use a PD controller with Tq set equal to 0.5 minutes (j st to make the algebra work out nicely this is not necessarily the optimum value oftj>). The closcdloop characteristic equation becomes <... 

The response of to a step change in setpoint will be a deadtime of D minutes followed by an exponential rise. The IMC controller becomes a PD controller... 

A perspective based on kinetics leads to a better understanding of the adsorption mechanism of both ionic and nonionic compounds. Boyd et al. (1947) stated that the ion exchange process is diffusion controlled and the reaction rate is limited by mass transfer phenomena that are either film diffusion (FD) or particle diffusion (PD) controlled. Sparks (1988) and Pignatello (1989) provide a comprehensive overview on this topic. 

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

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). 

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