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Songthumjitti et al. Intell Robot 2023;3(3):306-36 I http://dx.doi.org/10.20517/ir.2023.20 Page 23 of 31

Figure 28. Fundamental Youla block diagram.

= (20)
1 −

In Equation (18), can be replaced from Equation (20), and the plant of the system, , is equal to ,
2
and we will get the result shown in Equation (21).

2
= (1 − (1 + )) (21)
1 +

Similarly, can be expressed with the parameter as Equation (22).

2
= (1 + )(1 − ) (22)

Equation (21) shows that if is expressed as Equation (23), the sensitivity of the system, , will be 0. Fur-
thermore, from Equation (22), will be just with the same ; this means that the effect of the robot
structure can be completely eliminated.

1
= (23)
2
(1 + )
But in reality, we know that has an unstable zero that cause by dead time and also that 1 is unstable itself.
2
Therefore, the parameter in Equation (23) is not stable. So, is divided into two parts, which consist of
stable zero and unstable zero, as shown in Equation (24), and we only use 0, as shown in Equation (25),
which is the stable zero part. The result of is shown in Equation (26).

= 0 ∗ . (24)

2
0 = (25)
2 2
+ 2 +

1
= (26)
2
0 (1 + )