Position Control by means of a Flexible Transmission

Figure 8.34. Output sensitivity functions for various controllers




Figure 8.35. Input sensitivity functions for various controllers

The resulting modulus margin ∆M = 0.498 is slightly less than the desired

value. The delay margin ∆τ = 0.043s is less than the desired value and the maximum of |S_up(q^-1)| at high frequencies is over the required value 10 dB (see Table 8.7).

The number of closed loop poles that can be specified for a minimum size of the controller (n_R = n_S = 4) is in this case

n_P = n_A + n_B + n_Hs + d – 1 = 8

Since one has specified only a pair of poles corresponding to the first vibration mode but with ζ = 0.8, it results that all the other poles for the closed loop have been implicitly set to zero (i.e. aperiodic poles at 0.5 f_s ). Therefore controller (A) will damp the poles corresponding to the first vibration mode without changing the frequency but, in the mean time, it will accelerate and strongly damp the second vibration mode (z=0 corresponding to ω₀=62.8 rad/s and ζ =1). This requires an important control effort in a frequency region where the gain of the system is low and, consequently, a high value of |S_up| at high frequencies will result.

In order to avoid this phenomenon, it is necessary to specify a second pair of poles corresponding to the second vibration mode, with a damping equal or higher to the open loop value. Increasing the desired value of the damping will induce an increasing of |S_up| in this frequency region. Thus a second pair of desired closed

loop poles is selected with ω₀=31.46 rad/s and ζ = 0.15 (this value of the damping