Response contours of constant

Figure 12.15 Contours of constant response as functions of jr, and for the response surface of Figure 12.14.

Figure 12.17 Upper left panel contours of constant response in two-dimensional factor space. Upper right panel a subset of the contours of constant response. Lower left panel canonical axes translated to stationary point of response surface. Lower right panel canonical axes rotated to coincide with principal axes of response surface.

However, it is not possible to add °C and min In a normalized factor space the factors are unitless and there is no difficulty with calculating distances. Coded rotatable designs do produce contours of constant response in the uncoded factor space, but in the uncoded factor space the contours are usually elliptical, not circular. 

Figure 11.17a. Contours of constant response in two-dimensional factor space.

Point M in the figure is the optimum that by one definition of the research problem objective should be determined. Each intersection line in the plane is a line of constant response values and is called contour lines-contour diagram. 

Figure 4.3 shows some contours of constant error density p(e 27) for systems with two responses. In all four cases the responses are linearly independent, but in the last two they are statistically correlated because the off-diagonal elements of 27 are nonzero. 

We can obtain a two-dimensional representation of the modeled surface by drawing its response contour lines, that is, lines of constant response. The contour lines of a plane are line segments. For example, if we let y = 70 in Eq. (6.3) we arrive at the expression... 

The Nichols chart in Fig. J.15 provides a graphical display of the closed-loop frequency response characteristics for set-point changes when G is) = Kjn Contours of constant AR l and ql are shown on a plot of ARol vs. OL- In a typical Nichols chart application, ARq and ql are calculated from Gol s) and plotted on the Nichols chart as a series of points. Then AR l and CL are obtained by interpolation. For example, if ARol 1 and ql = -100° at a certain frequency, then interpolation of Fig. J.15 gives ARcl = 0.76 and CL = 50° for the same frequency. The Nichols chart can be generated in the MATLAB Control Toolbox by a single command, nichols. 

Figures 5 and 6 show the response surfaces plotted for Property A and Property B, respectively. Note that two variables are plotted at once, with the values of the other variables fixed at levels chosen by the experimenter. The contours in the graph represent constant levels of the response. Fortunately, the computer allows rapid replotting for various levels of the fixed variables, as well as changing the identities of the fixed and floating variables, so that the entire design space can be investigated.

When the response surface has an extreme, then all coefficients of a canonic equation have the same signs and the center of the figure is close to the center of experiment. A saddle-type surface has a canonic equation where all coefficients have different signs. In a crest-type surface some canonic equation coefficients are insignificant and the center of the figure is far away from the center of experiment. To obtain a surface approximated by a second-order model for two factors, it is possible to get four kinds of contour curves-graphs of constant values ... 

In this section, we look at methods of obtaining a mathematical model that can be used for qualitative predictions of a response over the whole of the experimental domain. If the model depends on two factors, the response may be considered a topographical surface, drawn as contours or in 3D (Fig. 4). For more factors, we can visualize the surface by taking slices at constant values of all but two factors. These methods allow both process and formulation optimization. 

When the response of an analytical system depends on two factors which are continuous variables, the relationship between the response and the levels of the two factors can be represented by a surface in three dimensions as shown in Figure 7.4. This surface is known as the response surface, with the target optimum being the top of the mountain. A more convenient representation is a contour diagram (Figure 7.5). Here the response on each contour is constant, and the target optimum is close to the... 

Bishopp et al. [31] show a contour plot, or response curve, for the variation in bead size and pump rate against the D0.9 value for the particle size (D0.9 is the particle size, in p.m, below which 90% of the particles fall) constants are bead volume at 75% and mill speed at 4000 rpm. 

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. 

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