Function lnyquist1:
plotting the Nyquist on a log-base-2 scale
Below is the function lnyquist1.m.
This function is a modified version of Matlab's nyquist command, and
has the same attributes as the original, with a few improvements.
Lnyquist1.m plots
(log2(1 + |G(jw)|), angle(G(jw)))
in polar
coordinates. By taking the log of the magnitude, the magnitude scale is
compressed and the
overall shape of the Nyquist plot is
easier to see on the screen. We use log base 2 and add one to the
magnitude so as to preserve the key attributes of the -1 point for the
Nyquist plot.
Lnyquist1 also takes poles on the imaginary axis into
account when creating the Nyquist plot, and plots around them.
Furthermore, this function outputs the number of open-loop right hand
plane poles, the number of anti-clockwise encirclements, and the number of
closed-loop right half plane poles onto your screen.
Copy the following text into a file lnyquist1.m by viewing the document source. (We have experienced some very peculiar bugs by just copying the text appearing on in the browser window.) , Put it in the same
directory as the Matlab software, or in a
directory which is contained in Matlab's search path.
<![CDATA[
function [reout,imt,w] = lnyquist1(a,b,c,d,iu,w)
%LNYQUIST1 Nyquist frequency response for continuous-time linear systems.
%
% This Verison of the LNYQUIST Command takes into account poles at the
% jw-axis and loops around them when creating the frequency vector in order
% to produce the appropriate Nyquist Diagram (The NYQUIST command does
% not do this and therefore produces an incorrect plot when we have poles in the
% jw axis). As an added feature, this function outputs the number of open loop right
% hand plane poles, the number of anti-clockwise encirclements, and the number of
% closed loop right half plane poles on your screen.
%
% NOTE: This version of LNYQUIST1 does not account for pole-zero
% cancellation. Therefore, the user must simplify the transfer function before using
% this command.
%LNYQUIST1 Nyquist frequency response for continuous-time linear systems.
% NYQUIST(A,B,C,D,IU) produces a Nyquist plot from the single input
% IU to all the outputs of the system:
% . -1
% x = Ax + Bu G(s) = C(sI-A) B + D
% y = Cx + Du RE(w) = real(G(jw)), IM(w) = imag(G(jw))
%
% The frequency range and number of points are chosen automatically.
%
% LNYQUIST1(NUM,DEN) produces the Nyquist plot for the polynomial
% transfer function G(s) = NUM(s)/DEN(s) where NUM and DEN contain
% the polynomial coefficients in descending powers of s.
%
% LNYQUIST1(A,B,C,D,IU,W) or LNYQUIST(NUM,DEN,W) uses the user-supplied
% freq. vector W which must contain the frequencies, in radians/sec,
% at which the Nyquist response is to be evaluated. When invoked
% with left hand arguments,
% [RE,IM,W] = LNYQUIST(A,B,C,D,...)
% [RE,IM,W] = LNYQUIST(NUM,DEN,...)
% returns the frequency vector W and matrices RE and IM with as many
% columns as outputs and length(W) rows. No plot is drawn on the
% screen.
% See also: LOGSPACE,MARGIN,BODE, and NICHOLS.
% J.N. Little 10-11-85
% Revised ACWG 8-15-89, CMT 7-9-90, ACWG 2-12-91, 6-21-92,
% AFP 2-23-93
% WCM 8-30-97
%
% ********************************************************************
% Modifications made to the nyquist - takes into account poles on jw axis.
% then goes around these to make up frequency vector
%
if nargin==0, eval('exresp(''nyquist'')'), return, end
% --- Determine which syntax is being used ---
if (nargin==1),
error('Wrong number of input arguments.');
elseif (nargin==2), % Transfer function form without frequency vector
num = a; den = b;
w = freqint2(num,den,30);
[ny,nn] = size(num); nu = 1;
elseif (nargin==3), % Transfer function form with frequency vector
num = a; den = b;
w = c;
[ny,nn] = size(num); nu = 1;
elseif (nargin==4), % State space system, w/o iu or frequency vector
error(abcdchk(a,b,c,d));
w = freqint2(a,b,c,d,30);
[iu,nargin,re,im]=mulresp('nyquist',a,b,c,d,w,nargout,0);
if ~iu, if nargout, reout = re; end, return, end
[ny,nu] = size(d);
elseif (nargin==5), % State space system, with iu but w/o freq. vector
error(abcdchk(a,b,c,d));
w = freqint2(a,b,c,d,30);
[ny,nu] = size(d);
else
error(abcdchk(a,b,c,d));
[ny,nu] = size(d);
end
if nu*ny==0, im=[]; w=[]; if nargout~=0, reout=[]; end, return, end
% *********************************************************************
% depart from the regular nyquist program here
% now we have a frequency vector, a numerator and denominator
% now we create code to go around all poles and zeroes here.
if (nargin==5) | (nargin ==4) | (nargin == 6)
[num,den]=ss2tf(a,b,c,d)
end
tol = 1e-6; %defined tolerance for finding imaginary poles
z = roots(num);
p = roots(den);
% ***** If all of the poles are at the origin, just move them a tad to the left***
if norm(p) == 0
length_p = length(p);
p = -tol*ones(length_p,1);
den = den(1,1)*[1 tol];
for ii = 2:length_p
den = conv(den,[1 tol])
end
end
zp = [z;p]; % combine the zeros and poles of the system
nzp = length(zp); % number of zeros and poles
ones_zp=ones(nzp,1);
%Ipo = find((abs(real(p))<1e-6) & (imag(p)>=0)) %index poles with zero real part + non-neg imag part
Ipo = find((abs(real(p)) < tol) & (imag(p)>=0)); %index poles with zero real part + non-neg imag part
if ~isempty(Ipo) %
% **** only if we have such poles do we do the following:*************************
po = p(Ipo); % poles with 0 real part and non-negative imag part
% check for distinct poles
[y,ipo] = sort(imag(po)); % sort imaginary parts
po = po(ipo);
dpo = diff(po);
idpo = find(abs(dpo)>tol);
idpo = [1;idpo+1]; % indexes of the distinct poles
po = po(idpo); % only distinct poles are used
nIpo = length(idpo); % # of such poles
originflag = find(imag(po)==0); % locate origin pole
s = []; % s is our frequency response vector
w = sqrt(-1)*w; % create a jwo vector to evaluate all frequencies with
for ii=1:nIpo % for all Ipo poles
[nrows,ncolumns]=size(w);
if nrows == 1
w = w.'; % if w is a row, make it a column
end;
if nIpo == 1
r(ii) = .1;
else % check distances to other poles and zeroes
pdiff = zp-po(ii)*ones_zp % find the differences between
% poles you are checking and other poles and zeros
ipdiff = find(abs(pdiff)>tol) % ipdiff is all nonzero differences
r(ii)=0.2*min(abs(pdiff(ipdiff))) % take half this difference
r(ii)=min(r(ii),0.1) % take the minimum of this diff.and .1
end;
t = linspace(-pi/2,pi/2,25);
if (ii == originflag)
t = linspace(0,pi/2,25);
end; % gives us a vector of points around each Ipo
s1 = po(ii)+r(ii)*(cos(t)+sqrt(-1)*sin(t)); % detour here
s1 = s1.'; % make sure it is a column
% Now here I reconstitute s - complex frequency - and
% evaluate again.
bottomvalue = po(ii)-sqrt(-1)*r(ii); % take magnitude of imag part
if (ii == originflag) % if this is an origin point
bottomvalue = 0;
end;
topvalue = po(ii)+sqrt(-1)*r(ii); % the top value where detour stops
nbegin = find(imag(w) < imag(bottomvalue)); %
nnbegin = length(nbegin); % find all the points less than encirclement
if (nnbegin == 0)& (ii ~= originflag) % around jw root
sbegin = 0
else sbegin = w(nbegin);
end;
nend = find(imag(w) > imag(topvalue)); % find all points greater than
nnend = length(nend); % encirclement around jw root
if (nnend == 0)
send = 0
else send = w(nend);
end
w = [sbegin; s1; send]; % reconstituted half of jw axis
end
else
w = sqrt(-1)*w;
end
%end % this ends the loop for imaginary axis poles
% *************************************************************
% back to the regular nyquist program here
% Compute frequency response
if (nargin==2)|(nargin==3)
gt = freqresp(num,den,w);
else
gt = freqresp(a,b,c,d,iu,w);
end
% ***********************************************************
nw = length(gt);
mag = abs(gt); % scaling factor added
ang = angle(gt);
mag = log2(mag+1); % scale by log2(mag) throughout
for n = 1:nw
h(n,1) = mag(n,1)*(cos(ang(n,1))+sqrt(-1)*sin(ang(n,1)));
end; % recalculate G(jw) with scaling factor
gt = h;
% ***********************************************************
ret=real(gt);
imt=imag(gt);
% If no left hand arguments then plot graph.
if nargout==0,
status = ishold;
plot(ret,imt,'r-',ret,-imt,'g--')
% plot(real(w),imag(w))
% modifications added here
% ****************************************
% modifications added here to count encirclements
[numc,denc] = tfchk(num,den);
% create the + and - reflection of G(jw) on imag axis
gw = [(ret + j*imt); numc(1)/denc(1); (flipud(ret) - j*flipud(imt))]; %
%look at G(jw)
[Ngw,Mgw] = size(gw); % size of evaluated G(jw)
gwp1 = gw + ones(Ngw,Mgw);
% moves origin from 0 to -1, so we
% can count encirclements around -1 now
initial_angle = rem(180/pi*angle(gwp1(1)) + 360, 360);
% define initial angle
angle_gwp1 = rem(180/pi*angle(gwp1) - initial_angle + 720,360);
% angle from origin for all points
dagw = diff(angle_gwp1);
% subtract off initial angle find degrees of encirclement
tolerance = 180;
% define tolerance - where encirclement counter "flips" over
Ipd = find(dagw < -tolerance);
% number of anti-clockwise encirclements
Ind = find(dagw > tolerance);
% number of clockwise encirclemtents
Nacw = max(size(Ipd)) - max(size(Ind)); % number of encirclements
% Nyquist Criterion says Z = P - N
P = length(find(p>0))
disp('P = number of Open loop poles in rhp');
N = Nacw
disp('N = number of anti-clockwise encirclements')
Z = P - N %
disp('Z = number of closed loop poles in rhp')
%*******************************************
set(gca, 'YLimMode', 'auto')
limits = axis;
% Set axis hold on because next plot command may rescale
set(gca, 'YLimMode', 'auto')
set(gca, 'XLimMode', 'manual')
hold on
% Make arrows
for k=1:size(gt,2),
g = gt(:,k);
re = ret(:,k);
im = imt(:,k);
sx = limits(2) - limits(1);
[sy,sample]=max(abs(2*im));
arrow=[-1;0;-1] + 0.75*sqrt(-1)*[1;0;-1];
sample=sample+(sample==1);
reim=diag(g(sample,:));
d=diag(g(sample+1,:)-g(sample-1,:));
% Rotate arrow taking into account scaling factors sx and sy
d = real(d)*sy + sqrt(-1)*imag(d)*sx;
rot=d./abs(d); % Use this when arrow is not horizontal
arrow = ones(3,1)*rot'.*arrow;
scalex = (max(real(arrow)) - min(real(arrow)))*sx/50;
scaley = (max(imag(arrow)) - min(imag(arrow)))*sy/50;
arrow = real(arrow)*scalex + sqrt(-1)*imag(arrow)*scaley;
xy =ones(3,1)*reim' + arrow;
xy2=ones(3,1)*reim' - arrow;
[m,n]=size(g);
hold on
plot(real(xy),-imag(xy),'r-',real(xy2),imag(xy2),'g-')
end
xlabel('Real Axis'), ylabel('Imag Axis')
limits = axis;
% Make cross at s = -1 + j0, i.e the -1 point
if limits(2) >= -1.5 & limits(1) <= -0.5 % Only plot if -1 point is not far out.
line1 = (limits(2)-limits(1))/50;
line2 = (limits(4)-limits(3))/50;
plot([-1+line1, -1-line1], [0,0], 'w-', [-1, -1], [line2, -line2], 'w-')
end
% Axis
plot([limits(1:2);0,0]',[0,0;limits(3:4)]','w:');
if ~status, hold off, end % Return hold to previous status
return % Suppress output
end
%reout = ret;
% plot(real(p),imag(p),'x',real(z),imag(z),'o');
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8/29/96 LO
8/30/97 WCM
