Position Control by means of a Flexible Transmission

Table 8.7. Specifications, and achieved modulus margin, delay margin and maximum |Sup| for various controllers

	HS(q^-1)	HR(q^-1)	Closed loop poles			Modulus margin (dB)		Delay Margin (s)		|Sup| Max (dB)
			Dominant	Auxiliary
A	1-q^-1		0 = 11.94  = 0.8		0.498 (-6.06)		0.043		18.43
B	1-q^-1		0 = 11.94  = 0.8	0 = 31.46  = 0.15 (1-0.2q^-1)^4	0.522 (-5.65)		0.062		6.24
C	1-q^-1	1+q^-1	0 = 11.94  = 0.8	0 = 31.46  = 0.15 (1-0.2q^-1)^4	0.544 (-5.29)		0.057		1.5

is not critical). For the remaining poles to be assigned, as indicated in Chapter 3, Section 3.6, it is wise to select aperiodic poles located on the real axis between
0.05 and 0.5. These poles will have a beneficial effect upon the maximum value of |S_up| at high frequencies. The remaining four poles have been set to 0.2 (since four poles have been already assigned). The frequency characteristics of S_yp and S_up for the new controller are represented in Figures 8.34 and 8.35 (curves B). The new controller satisfies the imposed specifications (see Table 8.7).
Figures 8.36 and 8.37 show the real time results obtained with controller B. Figure
8.36 corresponds to a step change on position reference. One observes that the rise time of the real system is close to the specified one (6T_s plus the delay of 2T_s).
Figure 8.37 illustrates the behavior in regulation for a position disturbance
followed by an instantaneous release (it is the return to the equilibrium
position that is interesting for performance evaluation in regulation).
A detailed examination of the control signal (see Figures 8.36 and 8.37) shows
the presence of a high frequency component without effect on the output (the
position of the third pulley), since this frequency region is far beyond the band pass
of the closed loop.
This phenomenon is even visible on the real system, where the position of the
first pulley will follow to a large extent the control signal (the local position control
of the first pulley has a high band pass). The input sensitivity function, in fact, will amplify by a few dB the high frequency measurement noise (which is interpreted
as a disturbance). To counteract this effect it is sufficient to open the loop at high frequencies, by introducing in the controller a fixed part of the form H_R =1 + q^-1
(the controller gain will be zero at 0.5f_s), and to re-compute the controller with the same specifications.