I am doing loop shaping using $H_{\infty }$ approch, and I have a plant with an unstable pole:

G = 10/((s+10)*(s-1));
K1 = 4*(s+1)/s;

where G is the plant, and K1 is the controller. I am trying to do a confrontation between the approch with $H_{\infty }$ and the one without, in particular I am placing a weight only for the sensitivity function, and designing a controller using the mixed sensitivity design.

What I get if I plot the frequency response of the system in the two cases is the following:

enter image description here

in which the yellow line is the frequency response of the system in which I applied the $H_{\infty }$ and the red line is the one in which I used G and K1 as they are, so without mixed sensitivity and weights.

What I don't understand is why the yellow line has this strange behaviour, while the red line decrease smoothly.

I see that the bandwidth of the yellow line is lower the the other one, so maybe this fact is due to the fact that I have a RHP pole which imposes a lower limitation on the bandwidth of the system. But, if I plot the sensitivity functions of these two systems, I have:

enter image description here

so the yellow line behaves better with respect to disturbance rejection with respect to the red line, as expected.

I don't understand how to interpret the behaviour of the yellow line of the complementary sensitivity function.

Can somebody help me?


1 Answer 1


It appears to have -40 dB/decade roll-off in magnitude starting at 1 rad/sec and transitioning to -120 dB/decade roll-off in magnitude at approximately 300 rad/sec (this suggests you have 2 dominant poles that begin to be dominant when f is at 1 rad/sec and 4 additional poles that begin to contribute at f> 300 rad/sec since each pole would contribute -20 dB/decade to frequency roll-off.

In comparison, red is rolling off at -80 dB/decade consistent with 4 poles. The poles would actually be complex given the frequency peaking in all cases.

  • $\begingroup$ Thanks for answering. Can I ask you if this will affect noise rejection? What I see from the two plots is that the yellow line has an higher sensitivity bandwidth and a loweer complementary sensitivity bandwidth. But I am not sure this is possible, because it would mean that it gets better in disturbance rejection and also better in noise rejection. But as far as I know this is not possible. So the fact that the yellow line does not roll-off smoothly maybe has some implication with noise rejection? Thanks in advance. $\endgroup$
    – J.D.
    Dec 22, 2019 at 16:00
  • 2
    $\begingroup$ I am not sure I follow- doesn't the yellow line has inferior disturbance rejection at all frequencies below 2 rad/sec, and inferior noise rejection from 10 to 50,000 rad/sec when compared to blue line? $\endgroup$ Dec 22, 2019 at 16:06

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