Richard Harrison wrote:
K7ITM wrote:
"If you say that in the system (y = mx + b) that b is the response to
zero input (x=0) then you will conclude the system is nonlinear."

Why? the factor (b) is a constant, a value to be added to (mx) to total
a value for (y).


Because, from 3.3, 'Network Theorems,' in "The Electrical Engineering
Handbook,"
Ed. Richard C. Dorf, Boca Raton: CRC Press LLC, 2000 (you can find it
online),

"The superposition condition: If the input to the system, e1, causes a
response, r1, and if an input to the system, e2, causes a response, r2,
then a response, r1 + r2, will occur when the input is e1 + e2."

If you take the response to x0=0 to be b, then the response to, say,
x1=1 must be m+b, and to x2=2 must be 2*m+b. Then for superposition,
and therefore for linearity, the response to (x1+x2)=3 must be
m+b+2*m+b = 3*m+2*b, which it is not: it is 3*m+b. I do not suggest a
change in the definition of superposition or of homogeneity (which
seems to be simply a subset of superpostion anyway) or of linearity. I
only suggest that "response" be interpreted as the change in output
which occurs when you go from input a to input b. I've generally not
seen "response" to be very well defined in those texts which define
linearity, though it's quite possible I've missed it. In the PDF file
for all of Chapter 3 of the book quoted above, it is certainly not.

I guess I'd prefer, basically, to say that the following relationship
defines a linear system (MIMO even) -- but it is NOT sufficient to
describe all linear systems:

dx/dt = Ax + Bu
y = Cx + Du
where u is a vector of independent input variables, y is a vector of
dependent output variables, x represents state variables for the
system, and A, B, C and C are matrices of coefficients which are
constant in a time-invariant system, but which may be variable with
time in a system which is time variant.

Cheers,
Tom



(mx) is a straight line. Every value of (b) produces a straight lline
parallel with lhe line y=mx when b=0. Factor (b) is merely the offset
value of the sloped line in the x direction.

y=mx+b is listed in math books as a defining example of a linear
equation. (When plotted, a linear equation produces a straight line.)
y=mx+b has a special name: "The point slope formula". Perfectly
descriptive, too.

To clarify everything, graph a few values for yourself.

Best regards, Richard Harrison, KB5WZI