1. ## Diophant equation

Hi !

I'm trying to solve a problem concerning a RST regulator. In this context, I need to solve a Diophant Equation.

this equation is :
A(z)*R(z) + B(z)*S(z) = C(z)

where :
A(z) = (z-1)*(z-0.513)
B(z) = 0.648*(z+0.801)
R(z) = r0*z + r1
S(z) = (z-1)*(s0*z + s1)
C(z) = z*(z²-0.7497*z+0.2432)

I need to find r0, r1, s0 and s1 with a minimal solution for S(z)

I have begun with the rewriting of the equation in a matrix :
A*X = C

where (Matlab code) :
Code:
A =
1.0000         0    0.6480         0
-1.5130    1.0000   -0.1290    0.6480
0.5130   -1.5130   -0.5190   -0.1290
0    0.5130         0   -0.5190
C =
1.0000
-0.7497
0.2432
0
X =
r0
r1
s0
s1
For information, A is a Sylvester matrix

I don't know how to do. can you help me ?
(if it is easyer for you, I have Matlab for my disposal)

Thanks !

2. I have a few questions, just for clarification:

1. What is an RST Regulator? I've heard of an LQR (Linear Quadratic Regulator) - it shows up in control theory. Is this the same thing?

2. Do you really mean a Diophantine equation?

3. A clarification about your equations: you say that A is a matrix, for example. And yet you have A(z) = (z-1)*(z-0.513). What kind of multiplication is that? And what kind of addition? Or is your matrix A different from A(z)?

4. The determinant of A is very close to zero. Is it supposed to be exactly zero? I would assume so, since otherwise your system would have only one solution, and there'd be no room for manoeuver. What do you get for the row reduction?

Cheers.