MATLAB: Mathematically, what is the source of the lines of constant damping factors that are generated by ZGRID in Control System Toolbox 9.4 (R2012b)

Control System Toolboxdiscretelaplacelocusroottransformz transform

I am using the ZGRID command in Control System Toolbox 9.4 (R2012b). I see that this command plots lines of constant damping ratio zeta in the z-plane. I am curious to know the source of these curves, from a mathematical standpoint.

Best Answer

The shapes of the curves of constant damping factors are a result from control theory. Essentially, the equivalent characteristics in the z-plane are related to those in the s-plane by z = exp(s*T), with T being the sample period, and so the diagonal lines of constant zeta in the s-plane get mapped to spirals starting from (1,0) in the z-plane. A control theory textbook would be a good reference for the mathematical details.

For example, see pp.602-603 of Feedback Control of Dynamic Systems, by Franklin, Powell, and Emami-Naeini.

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MATLAB: Is it possible to allow switching between cartesian grid and plot type grids in the Control System Toolbox 8.3 (R2009a)

The ability to do switch between cartesian and plot type grids using SGRID or GRID is not available in the Control System Toolbox 8.3 (R2009a).

To work around this issue, you can superimpose horizontal and vertical lines on the plot as given below. The following example is for vertical grid lines and can be extended to horizontal lines as well.

ax = gca;
rlocus(zpk([],[-2,-3,-5],1))
hold on
yminmax = get(ax,'ylim'); % min max of vertical grid lines
xdata = get(ax,'xtick'); % Location of x grid lines eg xdata = [1,2,3,4]
lxdata = [];
lydata = [];
for ct = 1:length(xdata)
lxdata = [lxdata; xdata(ct)*ones(2,1); NaN];
lydata = [lydata; yminmax(:); NaN];
end
vertgridline = line(lxdata,lydata,'color',[.8,.8,.8],'linestyle',':');
% to delete these gridlines 
%remove grid lines
delete(vertgridline)

MATLAB: A matlab app for S-Z plane mapping

If you

plot(z)

where z is complex, and you are not specifying an x, y pair, then this is treated as

plot(real(z), imag(z))

which of course you could do explicitly anyhow.

z = exp(s.*T);

Best Answer

Related Solutions

MATLAB: Is it possible to allow switching between cartesian grid and plot type grids in the Control System Toolbox 8.3 (R2009a)

MATLAB: A matlab app for S-Z plane mapping

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