Stability locus

Pig. 8. A schematic representation of a stability locus phase diagram with parameters chosen to correspond to a cluster of Arss. 

In practice, only the frequencies lu = 0 to+oo are of interest and since in the frequency domain. v = jtu, a simplified Nyquist stability criterion, as shown in Figure 6.18 is A closed-loop system is stable if, and only if, the locus of the G(iLu)H(iuj) function does not enclose the (—l,j0) point as lu is varied from zero to infinity. Enclosing the (—1, jO) point may be interpreted as passing to the left of the point . The G(iLu)H(iLu) locus is referred to as the Nyquist Diagram. 

Sketch the root locus diagram for Example 7.4, shown in Figure 7.14. Determine the breakaway points, the value of K for marginal stability and the unit circle crossover. 

Case study Example 5.11 uses root locus to design a ship roll stabilization system. The script file exampSll.m considers a combined PD and PID (PIDD) controller of the form... 

Chaib, J. et al., Stability over genetic backgrounds, generations and years of quantitative trait locus (QTLs) for organoleptic quality in tomato, Theoret. Appl. Genet. 112, 934, 2006. 

For a more complex problem, the characteristic polynomial will not be as simple, and we need tools to help us. The two techniques that we will learn are the Routh-Hurwitz criterion and root locus. Root locus is, by far, the more important and useful method, especially when we can use a computer. Where circumstances allow (/.< ., the algebra is not too ferocious), we can also find the roots on the imaginary axis—the case of marginal stability. In the simple example above, this is where Kc = a/K. Of course, we have to be smart enough to pick Kc > a/K, and not Kc < a/K. 

The entire range of stability for x = 0.1 is 0 < Kc < 0.25. We will revisit this problem when we cover root locus plots we can make much better sense without doing any algebraic work ... 

To begin with, this is a second order system with no positive zeros and so stability is not an issue. Theoretically speaking, we could have derived and proved all results with the simple second order characteristic equation, but we take the easy way out with root locus plots. 

The concept of gain and phase margins derived from the Nyquist criterion provides a general relative stability criterion. Frequency response graphical tools such as Bode, Nyquist and Nichols plots can all be used in ensuring that a control system is stable. As in root locus plots, we can only vary one parameter at a time, and the common practice is to vary the proportional gain. 

Do the root locus plots in Example 10-l(d). Confirm the stability analysis in Example 10-... 

Figure 2.4 Flip-flop switch model of wake and slow wave sleep active systems. Mutually inhibitory connections exist between GABAergic/Galaninergic slow wave sleep active neurons in the ventrolateral preoptic area (VLPO) of the anterior hypothalamus and aminergic neurons in the hypothalamus (histamine (HA) neurons in the tuberomammillary nucleus (TMN)) and brainstem (serotonin (5-HT) neurons in the dorsal raphe (DR) and noradrenaline (NA) neurons in the locus coeruleus (LC)). Orexinergic neurons in the perifornical hypothalamus (PFH) stabilize the waking state via excitation of the waking side of the flip-flop switch (aminergic neurons).

Zhang, Y., Y. Xiong, and W. G. Yarbrough, ARF promotes MDM2 degradation and stabilizes p53 ARF-INK4a locus deletion impairs both the Rb and p53 tumor suppression pathways. Cell, 1998, 92(6), 725-34. 

Therefore, if < tpi a proportional controller cannot make the system closed-loop stable. A controller with derivative action might be able to stabilize the system. Figure 11.9fi,c gives the root locus plots for the two cases and... 

Figure 1 l.9d gives a sketch of a typical root locus plot for this type of system. We now have a case of coiufitioRul stability. Below the system is closedloop unstable. Above the system is again closedloop unstable. A range of stable values of controller gain exists between these limits. 

There is some critical value of gain at which the G, B plot goes right through the (—1, 0) point. This is the limit of closedloop stability. See Fig. 13.3e. The value of K, at this limit should be the ultimate gain that we have dealt with before in making root locus plots of this system. We found in Chap. 10 that = 64 and Let us see if the frequency-domain Nyquist stability... 

This conditional stability is shown on a root locus plot for this system sketched in Fig. 13.5h. 

An openloop unstable, second-order process has one positive pole at + 1/ti and one negative pole at — I/tj. If a proportional controller is used and if ti < show by using a root locus plot and then by using the Nyquist stability criterion that the system is always unstable. 

First we will look at the question of stability in the z plane. Then root locus and frequency response methods will be used to analyze sampled-data systems. Various types of processes and controllers will be studied. 

With continuous systems we made root locus plots in the s plane. Controller gain was varied from zero to infinity, and the roots of the closedloop characteristic equation were plotted. Time constants, damping coefficients, and stability could be easily determined from the positions of the roots in the s plane. The limit of stability was the imaginary axis. Lines of constant closedloop damping coefficient were radial straight lines from the origin. The closedloop time constant was the reciprocal of the distance from the origin. 

The bilinear transformation is another change of variables. We convert from the z variable into the lU variable. The transformation maps the unit circle in the z plane into the left half of the ID plane. This mapping converts the stability region back to the familiar LHP region. The Routh criterion can then be used. Root locus plots can be made in the 11 plane with the system going closedloop unstable when the loci cross over into the RHP. 

We could make a root locus plot in the U) plane. Or we could use the direct-substitution method (let U) = iv) to find the maximum stable value of. Let us use the Routh stability criterion. This criterion cannot be applied in the z plane because it gives the number of positive roots, not the number of roots outside the unit circle. The Routh array is... 

The reaction described in this example is carried out in miniemulsion.Miniemulsions are dispersions of critically stabilized oil droplets with a size between 50 and 500 nm prepared by shearing a system containing oil, water,a surfactant and a hydrophobe. In contrast to the classical emulsion polymerization (see 5ect. 2.2.4.2), here the polymerization starts and proceeds directly within the preformed micellar "nanoreactors" (= monomer droplets).This means that the droplets have to become the primary locus of the nucleation of the polymer reaction. With the concept of "nanoreactors" one can take advantage of a potential thermodynamic control for the design of nanoparticles. Polymerizations in such miniemulsions, when carefully prepared, result in latex particles which have about the same size as the initial droplets.The polymerization of miniemulsions extends the possibilities of the widely applied emulsion polymerization and provides advantages with respect to copolymerization reactions of monomers with different polarity, incorporation of hydrophobic materials, or with respect to the stability of the formed latexes. 

Suspension polymerization may be the most important particle-forming polymerization from an industrial viewpoint. The system is very simple, composed of monomer, initiator, stabilizer, and medium (water in most cases). The monomer droplets with dissolving initiator are dispersed in water and the stabilizer exists at the interface. But suspension polymerization is regarded as a kind of homogeneous polymerization because the polymerization occurs only in monomer droplets and water does not affect the polymerization. Water contributes only to dividing the polymerization locus into small droplets and absorbing the heat evolved by polymerization. On the contrary, in emulsion polymerization, which is another type of polymerization performed in water and as practically important as suspension polymerization, water affects the polymerization significantly. In this section, emulsion polymerization is first discussed, and then some modified emulsion polymerizations such as soap-free emulsion polymerization and micro and mini emulsion polymerizations are described. 

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