- #1

Ethers0n

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I"m using a polynomial equations approach to the design problem. It seems that the Z transform of the controller's transfer function has near zero values in the numerator. I'm not sure if this is the actual problem, but, it's the only thing I've been able to come up with. I just need a fresh pair of eyes to take a look at it.

There are two different files here. The first produces the transfer function of the plant. The second has the coefficients of the plant transfer function hard coded into it, and is suppossed to produce the transfer function of the controller.

It seems that the Sylvester Matrix in the second file has zeros in, giving me a very small numerator for the controller...giving near infinite gain = bad.

thanks for the help.

And I apologize in advance if the post is long...

first file

%Servo Position Controller

%Using Polynomial Equation Method

%Defining the vairables in the TF

clear all;

Rm = 2.6; %armature resistance

Km = 0.00767; %Back-emf constant

Kt = 0.00767; %motor torque constant

Jm = 3.87e-7; %motor moment of inertia

Jeq = 2.0e-3; %equivalent moment of inertia at the laod

Beq = 4.0e-3; %equivalent viscous damping coefficent

Kg = 70; %srv02 system gear ratio (motor->load) (14X5)

Ng = 0.9; %gearbox efficiency

Nm = 0.69; %motor efficiency

Ts = .001; %sampling time is 1 mili-second

a0 = Jeq*Rm; %defining the terms of plant denominator,

a1 = (Beq*Rm + (Ng*Nm*Km*Kt*(Kg)^2)); %the denominator must be monic

a2 = 0;

b0 = (Ng*Nm*Kt*Kg); %defining terms of plant Numerator

PlantNum = [b0]; %plant numerator

PlantDenom = [a0 a1 a2]; %plant denominator

PlantTF = tf(PlantNum, PlantDenom) %making the plant transfer function

PlantTFz = c2d(PlantTF, Ts) %this produces the Z-Transform of the

%plant transfer function

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

second file

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%This part of the file uses the Z-Transformed Plant Transfer Function

%solved for in SRV02part1.m and from the hard-coded variables (of the

%numerator and denominator coefficients, solves the Diophontine Equation

%used to produce a controller using the Polynomial Equation Approach

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% B(z) Y(z)

%------ = ------

% A(z) U(z)

%where A(z) = z^n + a1*z^(n-1) + ...+ a(n-1)*z + a(n)

clear all;

plantnum = [0.3334];

plantden = [0.0052 0.1894 0];

Plant = tf(plantnum, plantden);

Ts = .001; %this is the sampling time

Plantz = c2d(Plant,Ts)

a0 = 1;

a1 = - 1.964;

a2 = 0.9642;

%where B(z) = b0*z^n + b1*z^(n-1) + ...+ b(n-1)*z + b(n)

b0 = 0;

b1 = 3.167e-005;

b2 = 3.129e-005;

E = [a2, 0, b2, 0; %the Sylvester Matrix E for a plant with the equation

a1, a2, b1, b2; %like this (0*z^2) + (w*z) + X

1, a1, b0, b1; % ---------------------

0, 1, 0, b0;] % (q^2) + (g*z) + (y)

%becuase a second order system is desired (from the lab handout)

%the closed loop poles will be at z^2 + 2z + 1 ie at +-45 degrees from the

%axis...

% D = F*H, D = (z^3) + (2z^2) + z + 0 = (d0*z^3)+(d1*z^2)+(d2*z)+d3

% D = [d3

% d2

% d1

% d0]

D = [0;

1;

2;

1;];

M = (inv(E))*D %inverting the Sylvester Matrix and multiplying by D

%this gives the coefficients which will produce the desired

%controller...

% M = [alpha1

% alpha0

% beta1

% beta0]

%The desired controller is of the form

% beta0*z + beta1 Beta(z)

% --------------- = -----------

% alpha0*z + alpha1 Alpha(z)

Alpha = [M(2,1) M(1,1)]; %the alpha(z) term of controller

Beta = [M(4,1) M(3,1)]; %the beta(z) term of the controller

DioControTF = tf(Beta, Alpha, Ts); %feedback regulator or designed controller

%Tz = minreal((Plantz*DioControTF)/(1 + Plantz*DioControTF))

%figure

%step(Tz, 3)