Steady State Error (page 4)

Besides system type, the input function type is needed to determine steady state error.  The three input types covered in Table 7.2 are step (u(t)), ramp (t*u(t)), and parabola (0.5*t2*u(t)).  The system type and the input function type are used in Table 7.2 to get the proper static error constant.  There are three of these: Kp (position error constant), Kv (velocity error constant), and Ka (acceleration error constant).  Once you have the proper static error constant, you can find ess.

The static error constants are found from the following formulae:

Now use Table 7.2 to find ess.

Table 7.2

    Type 0   Type 1   Type 2  

Input

ess Static
Error
Constant
ess Static
Error
Constant
ess Static
Error
Constant
ess
u(t) Kp = Constant Kp = Infinity 0 Kp = Infinity 0
t*u(t) Kv = 0 Infinity

Kv = Constant

Kv = Infinity 0
0.5*t2*u(t) Ka = 0 Infinity Ka = 0 Infinity Ka = Constant

Note that ess has one of three values: 0, a constant, infinity.  Notice how these values are distributed in the table.  Also note the aberration in the formula for ess using the position error constant.  ess is not equal to 1/Kp.

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