How Matlab calculate MARGIN (Problem with system with time-delay)

Question

0 votes

Hi,

I'm calculating dependence of critical parameters (for Ziegler-Nichols method) on the size of the time-delay. For example I have first order system sys=3/(s+1) and I'm adding different time-delays like this sys.iodelay=Ts/2. I have vector of these delays Ts=logspace(-3,1, 10000) (10000 values between 10^-3 and 10^1). I was using command [Gm,Pm,Wgm,Wpm] = margin(sys) to calculate dependence of GM and Tkrit=2*PI/Wgm on the size of the time-delay. I got something like this:

First graph is ZOOMED dependence of GM on time-delay Ts/2 and second is dependence of Tkrit=2*PI/Wgm on time-delay. You can see this steps/peaks in the graph. I found that this is because of MATLAB calculate more than one GainMargin for higher Ts (This is due to repeating -180° phase every 2PI or 360° cycle, because of infinity phase caused by time-delay). Before that peak MATLAB takes first GM in array but after the peak it is taking second GM of the array. I bypassed it using this code:

        Fs.iodelay=ts/2;
        M = allmargin(Fs);
        KK(i)=M.GainMargin(1);
        TK(i)=2*pi/M.GMFrequency(1);

Where I require to take only first GM (which I believe is on frequency when phase is -180°). This solved this one problem, but still in particular time-delay Ts=T*PI/100 (where T is time constant of system)i have one little peak (see picture):

What I found was: before the step MATLAB found only one GM (at the required frequency when phase is -180°), in the peak MATLAB found again only one GM (BUT ON HIGHER FREQUENCY THAN ITS WHEN PHASE IS CROSSING -180°!!!), after the peak it found 2 GainMargin's (one on frequency for -180° and one on higher frequency) and because of my code it took first (which is for -180°phase), so my question is: WHY MATLAB CANT SEE GAINMARGIN AT FREQUENCY WHEN PHASE CROSS -180° FOR THIS INTERVAL OF TIME-DELAYS=Ts/2? (interval starts with Ts=T*PI/100 and end approximately after 1.2ms.