Feedback Damping

Electro chemists are aware of the annoying residual uncompensated solution resistance Ru between the Luggin probe and the working electrode, see for example [74]. Although it is possible in principle to compensate fully for the iR error thus introduced [131,132], this is rarely done, as it introduces, in practice, undesirable instrumental oscillations or, in the case of damped feedback [132], sluggish potentiostat response. 

Example 4.7B Let us revisit the two CSTR-in-series problem in Example 4.7 (p. 4-5). Use the inlet concentration as the input variable and check that the system is controllable and observable. Find the state feedback gain such that the reactor system is very slightly underdamped with a damping ratio of 0.8, which is equivalent to about a 1.5% overshoot. 

Although, strictly speaking, total compensation cannot be achieved, partial compensation may well lead to a negligible residual ohmic drop, although the presence of damped oscillations does not yet prevent detection of the Faradaic current. Such a situation is typically reached for ARu = 417 in the system shown in Figure 1.8. This figure illustrates how the positive feedback ohmic drop compensation should be carried out in practice. The procedure may be summarized as follows ... 

With real cells, the resistance Ru is measured by augmenting positive feedback until sustained oscillations are observed. Then R = Re and the value of Ru is obtained by a simple reading of Re. The amount of positive feedback is then decreased back to a new value of Re so as to obtain damped oscillations compatible with the measure of the Faradaic component of the current, as in Figure 1.8 for ARu =412. The remaining resistance is thus obtained as ARu = R — Re. In a number of cases, the residual ohmic drop is negligible. If not, it may be taken into account in a simulation of the voltammograms, as depicted below. 

Figure 29 Bifurcation diagram of the minimal model of glycolysis as a function of feedback strength and saturation 6 of the ATPase reaction. Shown are the transitions to instability via a saddle node (SN) and a Hopf (HO) bifurcation (solid lines). In the regions (i) and (iv), the largest real part with in the spectrum of eigenvalues is positive > 0. Within region (ii), the metabolic state is a stable node, within region (iii) a stable focus, corresponding to damped transient oscillations.

This step response is sketched in Fig. 6.7 for several values of the damping coefTi-cient. Note that the amount of overshoot of the final sleadystate value increases as the damping coeflicient decreases. The system also becomes more oscillatory. In Chap. 7 we will tune feedback controllers so that we get a reasonable amount of overshoot by selecting a damping coefficient in the 0.3 to 0.S range. 

Normally we design the feedback controller flj,) to give some desire closed-loop performance. For example, we might specify a desired closedloop damping coefficient. 

Find the value of feedback controller gain that gives a closedloop damping coefficient of 0.8 for the system with a proportional controller and an openloop transfer function ... 

If a proportional feedback temperature controller is used, calculate the con> troller gain that yields a closedloop damping coefficient of 0.707 and calculate the closedloop time constant of the system when (u) Jacket water only is used. 

Find the value of feedback controller gain K, that gives a closedloop system with a damping coeflicient of 0.707 for a second-order openloop unstable process with... 

Note the very unique shape of the log modulus curves in Fig. 12.19. The lower the damping coefficient, the higher the peak in the L curve. A damping coefficient of about 0.4 gives a peak of about +2 dB, We will use this property extensively in our tuning of feedback controllers. We will adjust the controller gain to give a maximum peak of +2 dB in the log modulus curve for the closedloop servo transfer function X/X. ... 

Taking into account these considerations it is possible to obtain a set of transfer functions. Nevertheless, if the range of stable gains K ax/Ko (see Fig. 4) is computed for all the reactors, it appears that as the reactor s volume increases lower jacket temperatures are required, and the range of operation of the feedback controller decreases. Similar results can be obtained using Eq.(34) even, for instance, considering a fixed value of the damping ratio. The transfer function obtained for the small reactor of volume V = 0.0126 m is ... 

Further extensions of the model are required to address the dynamical consequences of these additional regulatory loops and of the indirect nature of the negative feedback on gene expression. Such extended models have been proposed for Drosophila [112, 113] and mammals [113]. The model for the circadian clock mechanism in mammals is schematized in Fig. 3C. The presence of additional mRNA and protein species, as well as of multiple complexes formed between the various clock proteins, complicates the model, which is now governed by a system of 16 or 19 kinetic equations. Sustained or damped oscillations can occur in this model for parameter values corresponding to continuous darkness. As observed in the experiments on the mammalian clock. Email mRNA oscillates in opposite phase with respect to Per and Cry mRNAs [97]. The model displays the property of entrainment by the ED cycle... 

Oscillations of the segmentation clock with a period of 2 h have also been observed in fibroblast cell cultures following serum shock. There also, oscillations in the expression of the gene Hesl related to the Notch pathway have been attributed to negative feedback on transcription [171]. The periodic operation of the segmentation clock was recently demonstrated in cells of the PSM, where intercellular coupling is needed to prevent damping of the oscillations [172]. 

J. R. Pomerening, S. Y. Kim, and J. E. Eerrell, Jr. Systems-level dissection of the cell-cycle oscillator Bypassing positive feedback produces damped oscillations. Cell 122, 565-578 (2005). 

Fig. 11.7. Transient response of the STM feedback system. Three different values of the loop gain G give different results. The response is overdamped with a gain of 100, critically damped with 200, and underdamped with 1000. (After Kuk and Silverman, 1989.)...

In this section we consider a model of interactions between the Kerr oscillators applied by J. Fiurasek et al. [139] and Perinova and Karska [140]. Each Kerr oscillator is externally pumped and damped. If the Kerr nonlinearity is turned off, the system is linear. This enables us to perform a simple comparison of the linear and nonlinear dynamics of the system, and we have found a specific nonlinear version of linear filtering. We study numerically the possibility of synchronization of chaotic signals generated by the Kerr oscillators by employing different feedback methods. 

In feedback control, after an offset of the controlled variable from a preset value has been generated, the controller acts to eliminate or reduce the offset. Usually there is produced an oscillation in the value of the controlled variable whose amplitude, period, damping and permanent offset depend on the nature of the system and the... 

The Altex liquid chromatograph (Fig.3.29) utilizes a dual piston approach similar to that in the Waters ALC 200 instrument. The pistons are 180° out of phase. The special cam shape results in a steady flow which is interrupted by minor fluctuations. These are damped by a flow-feedback system which is incorporated into the pump. This system has been rated at a pressure of 7000 p.s.i. 

Therefore we conclude not only that feedback control is useful to stabilize an optimal unstable steady state such as depicted in Figures 4.34 to 4.37 for the original set of parameter data, but feedback control can also help ensure the robustness of an otherwise stable optimal steady state over a larger region of parameters and system perturbations. Proper feedback control is also helpful in damping temperature explosions. 

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