Closed-loop modulus

Figure 6.29 (see also Appendix 1, fig629.m) shows the closed-loop modulus frequency response. Curve (a) is the best flatband response, curve (b) is the response with Mp set to 3dB. 

Design a cascade lead compensator that will ensure stability and provide a phase margin of at least 30°, a bandwidth greater than 5rad/s and a peak closed-loop modulus Mp of less than 6dB. 

The Nichols chart shown in Figure 6.26 is a rectangular plot of open-loop phase on the x-axis against open-loop modulus (dB) on the jr-axis. M and N contours are superimposed so that open-loop and closed-loop frequency response characteristics can be evaluated simultaneously. Like the Bode diagram, the effect of increasing the open-loop gain constant K is to move the open-loop frequency response locus in the y-direction. The Nichols chart is one of the most useful tools in frequency domain analysis. 

Maximum closed-loop log modulus A plot of the magnitude vs. frequency of the closed-loop transfer function. 

Here the 2 in the denominator is to avoid double counting because it takes two sites to form one link. However closed loops and chain entanglements are both possibilities and Equation (2.57) must be modified for these effects. We can write the network modulus as... 

Equation (13.30) says that we want the output to track the setpoint perfectly for all frequencies, and we want the output to be unaffected by the load disturbance for ail frequencies. Log modulus curves for these ideal (but unattainable) closed-loop systems are shown in Fig. 13.10b. 

Elastic modulus and yield stress were measured by compression tests on 10x5x4 mm prisms cut out from hot pressed plates. The J-integral measurements (J-Aa curve) were performed from tensile tests on CT specimens of 48 x 40 x 4 mm size, with a closed loop controlled machine (Schenk-Trebel). The starting crack was made using a razor blade (notch radius < 50 pm) up to a depth ao of 10 mm. 

The detuning parameter F is increased from one until the biggest log-modulus reaches a specified value. The biggest log modulus is a measure of how far the closed-loop system is from being unstable (Luyben, 1986). 

The sources of the discrepancies between the experimental data and the theoretical curves need some discussion. First, the model assumes a monodisperse polymer. This affects the curve shape after the upper ionic plateau. Second, the model assumes that all ions (except for a few very close together) are coupled into multiplets with ions from other chains. There are few loops. The analysis of Equation 4 suggests that perhaps most of the ions are coupled to adjacent ions in the same chain. This would change the nature of the relaxations considerably and affect the modulus at the ionic plateau. Such a model could lead to semi-permanent entanglements, kep in place by the ionic multiplets. The retardation of all relaxations of small segments at higher ionic content, even that fraction which occurs in ion-free segments is not considered in this model. 

Unstable poles conversely imply a certain minimal bandwidth (2-5 times larger than the modulus of the poles) of the controlled system. Clearly, the sets of controlled and manipulated variables must be chosen such that the unstable or marginally stable plant dynamics can be stabilized. In particular, all variables that are not self-regulating, mostly inventories, must be controlled. The control structure selection problem is often simplified by first closing these basic loops and then analyzing the remaining quality control problem [33]. 

The calculated distributions of crystallinity and orientation functions can be used in conjunction with models of the type developed by Seferis and Samuels [43] to predict quantities such as the Young s modulus. Conceptually, this closes the loop among polymer processing, polymer structure, and polymer properties. 

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