Derivative kick

Here, the controller output is zero as long as the error stays constant. That is, even if the error is not zero. Because of the proportionality to the rate of change, the controller response is very sensitive to noise. If there is a sudden change in error, especially when we are just changing the set point, the controller response can be unreasonably large—leading to what is called a derivative kick. 

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... 

This configuration is also referred to as interacting PID, series PID, or rate-before-reset. To eliminate derivative kick, the derivative lead-lag element is implemented on the measured (controlled) variable in the feedback loop. 

When a setpoint change is made using the forms given by Equations (15.1) and (15.2), a spike in the calculated value of de(t)ldt will occur, causing a spike in c(t). This behavior is called derivative kick and can be eliminated by replacing de(t)ldt with -dy, t)ldt, yielding... 

The PID controller may also use the rate of change of the measured variable (for example, h(t)) instead of the error e(t) to eliminate set point change derivative kick. Derivative kick is mitigated if a derivative mode filter is used. 

The type of control signal response to a step setpoint change encountered with a stable process under feedback control would be familiar to a process engineer. The first key feature is the immediate step change in the control signal when the setpoint value is changed. It is common practice to place the derivative term of the PID controller in the feedback loop so that it only acts on the filtered process variable to avoid derivative kick (see Figure 6.2). 

One disadvantage of the previous PID controllers is that a sudden change in set point (and hence the error, e) will cause the derivative term momentarily to become very large and thus provide a derivative kick to the final control element. This sudden spike is undesirable and can be avoided by basing the derivative action on the measurement, y, rather than on the error signal, e. To illustrate the elimination of derivative kick, consider the parallel form of PID control in Eq. 8-13. Replacing del dt by -dy ldt gives... 

This method of eliminating derivative kick is a standard feature in most commercial controllers. For a series-form PID controller, it can be implemented quite easily by placing the PD element in the feedback path, as shown in Fig. 8.10. Note that the elimination of... 

Figure 8.10 Block diagram of the series form of PID control that eliminates derivative kick.

Elimination of Derivative Kick. When a sudden set-point change is made, the PID control algorithms in Eq. 8-26 or Eq. 8-28 will produce a large immediate change in the output due to the derivative control action. For digital control algorithms. 

Find an expression for the amount of derivative kick that will be applied to the process when using the position form of the PID digital algorithm (Eq. 8-26) if a set-point change of magnitude is made between the k - 1 and k sampling instants. 

The Simulink diagram for this example is quite simple, as shown in Fig. 12.5. (See Appendix C.) However, the simulation results for Figs, 12.3 and 12.4 were generated using a modified controller that eliminated derivative kick (see Chapter 8). 

This is a mode of control that anticipates when a process variable will reach its desired control point by sensing its rate of change. This allows a control change to take place before the process variable overshoots the desired control point. You might say that derivative control gives you a little kick ahead. 

And other heuristic guides could kick in to give plausible, but never deductively derived, predictions . (So, for example, Paneth in the 1920s used simple reasoning based on the Periodic Table to predict that then undiscovered hafnium would occur in the same ores as zirconium. This is nonetheless a considerably looser notion of prediction than applies to standard cases from physics. See Scerri, 1994). 

This chapter is organized as follows. In Section II, we show how quantum chaos systems can be controlled under the optimal fields obtained by OCT. The examples are a random matrix system and a quantum kicked rotor. (The former is considered as a strong-chaos-limit case, and the latter is considered as mixed regular-chaotic cases.) In Section III, a coarse-grained Rabi state is introduced to analyze the controlled dynamics in quantum chaos systems. We numerically obtain a smooth transition between time-dependent states, which justifies the use of such a picture. In Section IV, we derive an analytic expression for the optimal field under the assumption of the CG Rabi state, and we numerically show that the field can really steer an initial state to a target state in random matrix systems. Finally, we summarize the chapter and discuss further aspects of controlling quantum chaos. 

The magnitude of the critical control parameter Kc can be estimated analytically with the help of Chirikov s criterion. This criterion is naturally derived in the context of the kicked rotor, and is introduced in the following section. The Chirikov criterion is also the basis for estimating the onset of chaos in many other chaotic atomic physics systems. Examples are presented in Chapters 6 and 7. 

In order to develop the criterion more quantitatively, consider the sequence of phase-space portraits shown in Figs. 5.4(a) - (d). This sequence suggests that, as the control parameter K increases, the diameter of the resonance islands at Z = 0 mod 27t grows in action. In order to predict the touching point of the resonances, we need the widths of the resonances as a function of K. The width of the resonances is derived on the basis of the Hamiltonian (5.2.1). Since the dynamics induced by H is equivalent to the chaotic mapping (5.1.6), the Hamiltonian H itself cannot be treated analytically and has to be simplified. One way is to consider only the average effect of the periodic 6 kicks in (5.2.1). The average perturbation... 

The electron is restricted to move in the half-space x > 0. There is a totally reflecting wall at x = 0. Since the Hamiltonian (8.1.1) of the kicked hydrogen atom and the Hamiltonian of microwave-driven surface state electrons are so similar, we can use many of the results that were derived in Chapter 6. The most important result is the transformation to action and angle variables I and 6, respectively, defined in (6.1.18). The... 

The most common representatives of the L-C=Y class of electron sinks are the carboxyl derivatives with Y equal to oxygen. In basic media there is only one pathway the addition-elimination path, path Ad y + Ep (see Section 4.5.1). The leaving group should be a more stable anion than the nucleophile, or the reaction will reverse at the tetrahedral intermediate. A follow-up reaction of a second addition to the polarized multiple bond occasionally occurs. With lone pair sources a second addition is rare because the nucleophile is usually a relatively stable species the second tetrahedral intermediate tends to kick it back out (see Section 9.2). 

For high density system, the enhancement factor becomes unity, and exchange effects dominate over the correlation effects. When the density becomes lower, the enhancement factor kicks in and includes correlation effects into the exchange energies. The enhancement factor is not unique, but can be derived differently in different approximations. The most reliable ones are parameterizations of molecular Monte-Carlo data. Some well known, and regularly used, parameterizations have been made by Hedin and Lundqvist [29], von Barth and Hedin [22], Gun-narsson and Lundqvist [30], Ceperly and Adler [31], Vosko, Wilk, and Nusair [32], and Perdew and Zunger [27]. 

Formation pressure data were obtained from repeat formation tester (RFT) or modular dynamic tester (MDT) measurements of numerous deep wells in the Central North Sea. These data were used as the primary pressure dataset as they are the most accurate pressure measurements possible down-hole. The MDT/RFT wireline tool takes a pressure reading within a permeable formation, by setting a seal at a precise depth determined by using an accompanying gamma ray tool for depth correlation. Drill stem test (DST), mudweight data and kick (influxes of pore fluids into the wellbore) information was also used where RFT or MDT data were not available or of very poor quality. A summary of the various approaches used to derive formation pressures is provided by Holm (1998). 

A further idea introduced in [184] was to use a force decoupling strategy to reduce the cost of computations. The idea is to exploit the observation that impulsive forces ( kicks ) can be supplied without reducing the order of accuracy as long as these have a vanishing component in the direction of the collision vector that is if F = —VUiqc) u = 0. This can be achieved in systems of spheres with pah-potentials (fij only by writing hybrid method that uses only the first part a to define the quadratic Verlet paths for the collision detection scheme, whereas fi is introduced as a standard kick at collision points. 

Similar behavior is exhibited using the splitting strategy in (7.5), when interposing the kick term B between two O steps, such as in the schemes [AOBOA or BOAOB. By avoiding OBO updates, we can derive schemes that are consistent even at infinite friction (the BAOAB and ABOBA schemes are examples of such methods). 

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