Loop stability

We should remember that phase angle can start changing gradually — starting at a frequency even 10 times lower than where the pole or zero may actually reside. We have also seen that a second-order double-pole (—2 slope with two reactive components) can cause a very sudden phase shift of about 180° at the resonant frequency if the Q is very high. Therefore, in practice, it is almost impossible to estimate the phase at a certain frequency, with certainty — nor therefore the phase margin — unless a certain strategy is followed  

Therefore, one of the most popular (and simple) approaches to ensuring loop stability is as follows  

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Commercial Ar+ lasers may incorporate a light regulator accessory (also called a servo loop stabilizer) which minimizes fluctuations in the laser output power thus improving long term power stability to better than 0.5% rms. 

Conserved sequences that fold into large loops stabilized by three disulfide linkages. The name Kringle comes from the Scandinavian pastry that these structures resemble. They can mediate certain protein-protein interactions. 

This equation, of course, contains information regarding stability, and as it is written, implies that one may match properties on the LHS with the point (-1,0) on the complex plane. The form in (7-2a) also imphes that in the process of analyzing the closed-loop stability property, the calculation procedures (or computer programs) only require the open-loop transfer functions. For complex problems, this fact eliminates unnecessary algebra. We just state the Nyquist stability criterion here.1... 

O Neill PR, Velazquez LR, Dunn DG, Gwinn EG, Fygenson DK (2009) Hairpins with poly-C loops stabilize four types of fluorescent Ag DNA. J Phys Chem C 113 4229-4233... 

The first objective can be reached by using a general property of discretized system namely, if for the system (23) there exists a matrix Kd such that Ad + BdKd has all its eigenvalues inside the unit circle, then the controller u k) = Kdx(k) will also stabilize the continuous system. Thus, the closed-loop stability can be then assured by properly assigning the discrete poles inside the unit circle. 

In order to measure the control loop stability of the regulator, a simple modification was made to the circuit. LI and C3 were added to the circuit to effectively open the control loop of the regulator. This allows AC phase and gain measurements to be made. The ESR of capacitor C2... 

The output of the element represented by equation 7.155 lags the input. However, the destabilising effect of this additional lag is more than offset by an associated decrease in amplitude ratio. This decrease is more pronounced as the difference between r, and tj is increased. Lag compensators can be designed to produce different total open-loop stability specifications (e.g. in terms of allowable gain margin, phase margin, etc.) in a manner similar to that for lead compensators. 

Recent structural comparisons between enzymes from mesophiles and thermo-philes have validated numerous protein-stablizing effects, including hydrophobic interactions, packing efficiency, salt bridges, hydrogen bonds, loop stabilization... 

Beside the classical secondary structure elements occurring at the surface of proteins such as y- and P-tums, 3,0- and a-helices, and P-sheets, loops of variable size seem to play a crucial role in many protein-protein interactions (see Section I.A). Several amino acid-derived templates have been proposed to stabilize P-tums as external templates (see Section II.B) and a compilation of loop-stabilizing templates is shown in Figure 25.189-194... 

The controller has tuning parameters related to proportional, integral, derivative, lag, dead time, and sampling functions. A negative feedback loop will oscillate if the controller gain is too high but if it is too low, control will be ineffective. The controller parameters must be properly related to the process parameters to ensure closed-loop stability while still providing effective control. This relationship is accomplished, first,... 

The most relevant structural difference between the a and p monomers is located in the loop S9-S10, which is 8 residues longer in a-tub than in p-tub owing to an insertion. In the a-subunit, the long S9-S10 loop stabilizes the M loop. In the P-subunit, the equivalent position is occupied by the PTX binding site. This suggests that PTX mimics the 8-residue insertion of the a-subunit loop. In fact, the atomic model shows that the PTX molecule acts as a linker between helix H7 and the M loop, possibly stabilizing an M loop conformation that favors the lateral interprotofilament interaction and contributes to MT stabilization [24, 25], Both side chains and the baccatin core of PTX participate in these interactions [25],... 

Fig. 5. Schematic representation of the collagen IV molecule, which consists of two al(IV) chains and one a2(IV) chain. The non-triple-helical interruptions of the triple helix are indicated by black bars. The cysteine residues (C) and lysine or hydroxylysine (K) residues putatively involved in intra- or intermolecular bonds are shown. CHO designates a N-giycosidically bound oligosaccharide chain. The subscript numerals indicate the number of residues in a distinct region, summarized for all three a-chains. P designates a main pepsin cleavage site. In interruption 13, the a2(IV) chain forms a 21-residue-long loop, stabilized by an interchain disulfide bridge. NCI, Noncollagenous domain 1 TH, triple-helical domain 7 S, carboxyl-terminal domain.

Fig. 13.4 Blue CM/Pf-striatum-GPi-CM/Pf a positive FB loop and red CM/Pf-STN-GPi-CM/Pf a negative FB loop. Somatotopically organized parallel projections form closed loops (vertical, in black) between the motor cortex, the basal ganglia and the motor cortex. These loops signal movement preparation and execution. Concomitantly feedback (horizontal) loops stabilize basal ganglia activity. (From [50] with permission)...

The challenge posed by establishing and maintaining communication between distributed controllers has also stimulated research in the area of networked process control (El-Farra et al. 2005, Mhaskar et al. 2007, Sun and El-Farra 2008, 2010). The central issue of maintaining closed-loop stability in the presence of bandwidth constraints and limitations in transmitter battery longevity is typically addressed by a judicious distribution of computation and communication burdens between local/distributed control systems and a centralized supervisory controller. 

We call this high sensitivity of the recycle flowrates to small disturbances the snowball effect. We illustrate its occurrence in the simple example below, It is important to note that this is not a dynamic effect it is a steady-state phenomenon. But it does have dynamic implications for disturbance propagation and for inventory control. It has nothing to do with closed-loop stability. However, this does not imply that it is independent of the plant s control structure. On the contrary, the extent of the snowball effect is very strongly dependent upon the control structure used. 

The vinyl acetate reactor we use in Chap. 11 has been designed to be insensitive to parameter variations under normal operating conditions. The hot-spot temperature is only 162°C with an exit temperature of 159°C. It is adequate to control the exit temperature instead of the hot spot. Since multiplicity, open-loop stability, and sensitivity are of no concern for this reactor, we can focus our attention on the open-loop characteristics relevant, to the control of exit temperature with jacket cooling. 

When we insert the proper terms from the Jacobian matrix we find the following set of inequalities must be satisfied for open-loop stability. 

Start with one isolated unit operation. Get its control system installed, tuned, and tested for closed-loop stability and robustness. Then move on to the next unit and repeat. Build up the entire plant... 

Example 6—Closed-loop stability for a nonminimum-phase process. 

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