Nikola Šerman, prof. emeritus                                                                                  Linear Theory                                                                                                         Transfer Function Concept Digest


Closed-loop dynamics is optimized to achieve the closed-loop stability with a satisfactory transients damping and a transients duration to be as short as possible.  In terms of the Bode diagram stability margins this means that the stability margins should be:

  1. positive (provides stability as such),
  2. sufficiently large (provides adequate transients damping),
  3. within relevant frequency range (provides faster transients vanishing).

To comply with #3 the shaded area in fig. 11.1 should be as far as its achievable to the right side.

Figure 11.1 Stability margins and relevant frequency range (shaded area) in an open-loop frequency characteristics Bode plot.


Frequency characteristics of a serial connection in Bode diagram are made by a simple summation of the respective block characteristics.  This offers a rational ground for qualitatively analyzing of what are the effects of additional block inclusion into the open loop.

Figure 11.2 Closed-loop system under consideration – open loop W(s) = H(s) H0(s) encircled by negative unity feed-back

Transfer function H(s) in fig 11.2 is an additional block.  Fig 11.3 shows what would be the effects if that block is a static gain while fig 11.4 shows the same in a case of an integrator.

Figure 11.3 Effect of adding static gain H(s) = K to the component H0(s) when K > 1.

Adding a static gain to the existing component H0(s) results in an amplitude frequency characteristic vertical translation.  The transition is  upwards for K > 1  and downwards for K < 1, thus directly influencing the stability margins.  The red arrows are the stability margins after the inclusion of H(s).

Figure 11.4 Adding of an integrator s  to the existing component H0(s)

Adding an integrator to the existing component H0(s) rotates the amplitude frequency characteristic around its point at frequency ω0. The phase characteristic is shifted 90° downwards.