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Page 22 of 31 Songthumjitti et al. Intell Robot 2023;3(3):306-36 I http://dx.doi.org/10.20517/ir.2023.20

Figure 27. Feedback system block diagram.

To get better robustness to the change in structure characteristics, we need to make sensitivity small around
the resonance frequency. The sensitivity of this system can be calculated using Equation (15).

Δ
Δ
= = (15)
Δ Δ

The experiment system with a feedback controller, which can be expressed in Figure 27, can be derived into
Equation (16). To find the sensitivity of the system, we must derivate Equation by the structure transfer
function, as shown in Equation (17), , and then multiply with , as shown in Equation (18).

( + 1)
= = 2 (16)
+ 1

2
(1 − )
= (17)
2
( + 1) 2
1 − 2

= = 2 (18)
1 + 1 +

We need to design the compensator, , that makes the sensitivity small and also stabilizes the feedback
system. We can use the Youla parameterization method [20] , which is a design technique for the stable plant.
Considering the feedback system shown in Figure 28, the Youla parameterization can be expressed in terms of
a transfer function as Equation (19), where is the transfer function from to . Using this stable parameter,
, all stabilizing compensators, , are parameterized as Equation (20).

= (19)
1 +