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Suppose we re given a point P = (xp,yp,zp) in Cartesian three-space Prove that if we negate any one, any two, or all three of the coordinates, the resulting point is the same distance from the origin as P

For the point P, the distance c from the origin is c = (xp2 + yp2 + zp2)1/2 The square of any real number is always the same as the square of its negative That tells us three things: ( xp)2 = xp2 ( yp)2 = yp2 ( zp)2 = zp2 By substitution, all these quantities are identical: (xp2 + yp2 + zp2)1/2 [( xp)2 + yp2 + zp2]1/2 [xp2 + ( yp)2 + zp2]1/2 [xp2 + yp2 + ( zp)2]1/2 [( xp)2 + ( yp)2 + zp2]1/2 [( xp)2 + yp2 + ( zp)2]1/2 [xp2 + ( yp)2 + ( zp)2]1/2 [( xp)2 + ( yp)2 + ( zp)2]1/2

These quantities represent the distances of the following points from the origin, respectively: (xp,yp,zp) ( xp,yp,zp) (xp, yp,zp) (xp,yp, zp) ( xp, yp,zp) ( xp,yp, zp) (xp, yp, zp) ( xp, yp, zp) That s all the points we can get, in addition to P itself, by negating any one, any two, or all three of the coordinates of P They re all the same distance c from the origin, where c = (xp2 + yp2 + zp2)1/2

If 2 < ~0/7~~ < 4, dead time is dominant If T /T~ = 4, t here is a single dominant capacity If r /ird > 4, more than one capacity is present l urthcrmorc, the setting of proport,ional band responsible for uniform oscillation equals the gain product of the other elements in the loop at TV When these bits of information are combined with the characteristics of the known elements, a remarkably accurate picture of the process can he asscmblcd Icor example, if the process is known t,o contain one principal capacity, and TJ~,~ = 4, no other time constants need be sought If t he time constant of this capacity is known, its dynamic gain G1 at 7O can be calculated Combining this with known values of transmitter and valve gain, together with the controller proportional band, yields.

If we change the scope, we are really starting a whole new, di erent project, and we should start again In the analysis stage, all of the details of the customer s requirements are exible The customers are learning about the technology, and we are learning about the customers wants and needs But once we agree on what we are doing, we write it down in the project overview, in the requirements speci cation, and in support documents From the end of analysis forward, all changes to the requirements speci cation should be in writing, following the procedures of written change control It is essential to lock down these documents, as they are the input for the work of the design stage.

When we want to determine the distance between any two points in Cartesian three-space, we can expand the formula from Cartesian two-space that we learned in Chap 1 into an extra dimension

The general formula Imagine two different points in Cartesian three-space, after the fashion of Fig 7-5 Let s call the points and their coordinates

Any change in the description of the purpose and function of the system we are making will force a restart to the design process, where we de ne the product or service that will achieve that purpose These restarts mean that much work done up to that point was wasted, and we begin to see the losses resulting from inadequate early planning In the design stage, the technical speci cation is very open, and can be changed But, at the end of the design stage, the technical speci cation is locked down, and the project plan is in place The design documents are the input for development work Changes during development require a restart on the development work, with much of the earlier work being wasted To avoid the costs and risks of these changes, at the end of design, all future changes require written change control.