**Nikola Šerman,**prof. emeritus

*Linear Theory*nserman@fsb.hr

*Transfer Function Concept Digest*

**11. BASICS OF CLOSED LOOP DYNAMICS QUALITATIVE ANALYSIS**

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:

- positive (provides stability as such),
- sufficiently large (provides adequate transients damping),
- 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) H _{0}(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 *H _{0}(s)* when

*K > 1*.

Adding a static gain to the existing component *H _{0}(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 *H _{0}(s)*

Adding an integrator to the existing component *H _{0}(s) *rotates the amplitude frequency characteristic around its point at frequency

*ω*. The phase characteristic is shifted 90° downwards.

_{0}