Computer Controlled Systems, EEE 591

I am currently taking Computer Controlled Systems this Spring 2014 that is being 

taught by professor Armando Rodriguez.

His website: http://aar.faculty.asu.edu/

http://aar.faculty.asu.edu/classes/eee481S12/

aar@asu.edu

Professor

Sch Elect Comptr & Energy Engr

Faculty

Mail Code: 5706

GWC 352 (Map)

(480)965-3712

The following are notes for this course:

% why would you want to use the 

% home command?

clear all

close all

clear 

clc

clf

%

% 3-6-2014

%

% M.J. Thompson

%

% GOAL: PLOT ERRORS ASSOCIATED WITH 3 MOST COMMONLY USED

%       DISCRETIZATION METHODS

%

% 1) Forward difference backward Euler 

%    s1=(z-1) / T

% 2) Backward difference forward Euler

%    s2= s1/z

% 3) Bilinear Tustin trapezoilda method

%    s3= 2/T [z-1] / [z+1]

%

%

%

w=[0:0.001:20]                                  % Vector of discrete-time

temp1 = (exp(j.*w)-1);

e1    = abs(j.*w-temp1);                        % FDBE

temp2 = j.*w.*temp1;

e2    = abs(j.*w-temp2);                        % BDFE

temp3 = 2.*temp1./(exp(j.*w)+1);

e3    = abs(j.*w-temp3)                         % BTT

figure(10)

subplot(4,1,1)

plot(w,e1,'b-',w,e2,'r-',w,e3,'k-')             % plot 3 errors vs w           

axis([0 20, 0 20])                              % set ranges on x and y axes

grid minor                                      % set grid on

title('Discretization Error vs Frequency')

xlabel(' Discrete frequency: \omega (rad/sec)')

ylabel('Magnitude of Error')

legend('e1','e2','e3')

subplot(4,1,2)

plot(w,e1,'b-',w,e2,'r-',w,e3,'k-')             % plot 3 errors vs w           

axis([0 4, 0 5])                                % set ranges on x and y axes

grid minor                                      % set grid on

title('Discretization Error vs Frequency')

xlabel(' Discrete frequency: \omega (rad/sec)')

ylabel('Magnitude of Error')

legend('e1','e2','e3')

subplot(4,1,3)

plot(w,e1,'b-',w,e2,'r-',w,e3,'k-')             % plot 3 errors vs w           

axis([0 2, 0 0.5])                                % set ranges on x and y axes

grid minor                                      % set grid on

title('Discretization Error vs Frequency')

xlabel(' Discrete frequency: \omega (rad/sec)')

ylabel('Magnitude of Error')

legend('e1','e2','e3')

subplot(4,1,4)

plot(w,e1,'b-',w,e2,'r-',w,e3,'k-')             % plot 3 errors vs w           

axis([0 1.2, 0 0.1])                                % set ranges on x and y axes

grid minor                                      % set grid on

title('Discretization Error vs Frequency')

xlabel(' Discrete frequency: \omega (rad/sec)')

ylabel('Magnitude of Error')

legend('e1','e2','e3')

% why would you want to use return command?

-----------------------------------------------------------------------

Matlab: Ways to integrate under the curve with Forward Euler and backward Euler methods

% Setup: solve y' = f(t,y(t)), y(0) = y0, t <= T, with

g = @(t)(t.^4 - 6*t.^3 + 12*t.^2 - 14*t + 9)./(1+t).^2;

f = @(t,y)(y.^2 - g(t));

y0 = 2;

T = 2.2;

% Exact solution

yexact = @(t)(1-t).*(2-t)./(1+t);

hfine = 1e-3;

tfine = 0:hfine:T;

yfine = yexact(tfine);

figure(1); clf;

plot(tfine,yfine,'r-','linewidth',2)

axis([0 T -1 2]);

set(gca,'fontsize',16);

%% Numerical solution: Forward Euler (FE)

for h = [.2 .1 .05]

    t = 0:h:T;

    nsteps = length(t)-1;

    yFE = [y0; zeros(nsteps, 1)];    

    for n = 1:nsteps

        yFE(n+1) = yFE(n) + h*f(t(n),yFE(n));        

    end

    figure(1); hold on;

    plot(t,yFE,'b-x','linewidth',1);

    axis([0 T -1 2]);

end

xlabel('t'); ylabel('y'); title('Forward Euler')

legend('Exact','h = .2', 'h = .1', 'h = .05');

%% Numerical solution: Runge-Kutta 2 (RK)

figure(1); clf;

plot(tfine,yfine,'r-','linewidth',2)

set(gca,'fontsize',16);

for h = [.2 .1 .05]

    t = 0:h:T;

    nsteps = length(t)-1;

    yRK = [y0; zeros(nsteps, 1)];    

    for n = 1:nsteps

        yFE = yRK(n) + h*f(t(n),yRK(n));

        yRK(n+1) = yRK(n) + h/2*...

            ( f(t(n),yRK(n)) + f(t(n+1), yFE) );        

    end

    figure(1); hold on;

    plot(t,yRK,'b-x','linewidth',1);

    axis([0 T -1 2]);

end

xlabel('t'); ylabel('y'); title('Runge-Kutta 2')

legend('Exact','h = .2', 'h = .1', 'h = .05');

%% Numerical solution: Backward Euler (BE)

figure(1); clf;

plot(tfine,yfine,'r-','linewidth',2)

set(gca,'fontsize',16);

for h = [.1 .05]

    t = 0:h:T;

    nsteps = length(t)-1;

    yBE = [y0; zeros(nsteps, 1)];    

    for n = 1:nsteps

        F = @(x)(x - yBE(n) - h*f(t(n+1),x));        

        yBE(n+1) = fzero(F,yBE(n));

    end

    figure(1); hold on;

    plot(t,yBE,'b-x','linewidth',1);

    axis([0 T -1 2]);

end

xlabel('t'); ylabel('y'); title('Backward Euler')

legend('Exact','h = .1','h = .05');

%% Numerical solution: Midpoint method (MM)

figure(1); clf;

plot(tfine,yfine,'r-','linewidth',2)

set(gca,'fontsize',16);

for h = [.2 .1 .05]

    t = 0:h:T;

    nsteps = length(t)-1;

    yMM = [y0; zeros(nsteps, 1)];    

    for n = 1:nsteps

        F = @(x)(x - yMM(n) - h/2*...

            ( f(t(n),yMM(n)) + f(t(n+1),x)) );        

        yMM(n+1) = fzero(F,yMM(n));

    end

    figure(1); hold on;

    plot(t,yMM,'b-x','linewidth',1);

    axis([0 T -1 2]);

end

xlabel('t'); ylabel('y'); title('Midpoint method')

legend('Exact','h = .2', 'h = .1','h = .05');

------------------------------------------------------------------------

The least significant bit

http://en.wikipedia.org/wiki/Least_significant_bit

Harry Nyquist

http://en.wikipedia.org/wiki/Harry_Nyquist

Paper about Harry Nyquist

http://www.dsif.fee.unicamp.br/~moschim/cursos/simulation/nyquist28.pdf

Resolution = voltage difference / 2^n

Drawing root locus by hand