Modern Control Engineering

aa

Section 5–8 / Steady-State Errors in Unity-Feedback Control Systems 229

is given by

The static acceleration error constant Kais defined by the equation

The steady-state error is then

Note that the acceleration error, the steady-state error due to a parabolic input, is an

error in position.

The values of Kaare obtained as follows:

For a type 0 system,

For a type 1 system,

For a type 2 system,

For a type 3 or higher system,

forN 3

Thus, the steady-state error for the unit parabolic input is

for type 0 and type 1 systems

for type 2 systems

ess=0, for type 3 or higher systems

ess=

1

K

,

ess=q,

Ka=lim

sS 0

s^2 KATa s+ 1 BATb s+ 1 Bp

sNAT 1 s+ 1 BAT 2 s+ 1 Bp

=q,

Ka=lim

sS 0

s^2 KATa s+ 1 BATb s+ 1 Bp

s^2 AT 1 s+ 1 BAT 2 s+ 1 Bp

=K

Ka=lim

sS 0

s^2 KATa s+ 1 BATb s+ 1 Bp

sAT 1 s+ 1 BAT 2 s+ 1 Bp

= 0

Ka=lim

sS 0

s^2 KATa s+ 1 BATb s+ 1 Bp

AT 1 s+ 1 BAT 2 s+ 1 Bp

= 0

ess=

1

Ka

Ka=slimS 0 s^2 G(s)

=

1

limsS 0 s^2 G(s)

ess=limsS 0

s

1 +G(s)

1

s^3