INGENIERÍA DE

CONTROL
UNIDAD I: FUNDAMENTOS DE LA
INGENIERÍA DE CONTROL

Prof. Ing. Jack Cutipa

Docente de Ing. Mecatrónica e Ing. Electrónica
vidis.cutipa@upn.pe
Introducción a los sistemas de control.

SESIÓN 7

LOGRO DE LA SESION

Al finalizar la sesión, el estudiante analiza el error en estado estable

utilizando la función de transferencia y señales estandar de prueba,
empleando los conocimientos adquiridos en clase, con claridad y
criterio.

https://www.youtube.com/watch?v=qhXPb-cJ94c
ERROR en ESTADO ESTACIONARIO
CONTENIDOS

1. Error en estado estable

 Introduction
• Any physical control system inherently suffers
steady-state error in response to certain types of
inputs.

• A system may have no steady-state error to a step

input, but the same system may exhibit nonzero
steady-state error to a ramp input.

• Whether a given system will exhibit steady-state

error for a given type of input depends on the type
of open-loop transfer function of the system.
 Classification of Control Systems
• Control systems may be classified according to
their ability to follow step inputs, ramp inputs,
parabolic inputs, and so on.

• The magnitudes of the steady-state errors due

to these individual inputs are indicative of the
goodness of the system.
 Classification of Control Systems
• Consider the unity-feedback control system
with the following open-loop transfer function

• It involves the term sN in the denominator,

representing N poles at the origin.

• A system is called type 0, type 1, type 2, ... , if

N=0, N=1, N=2, ... , respectively.
 Classification of Control Systems
• As the type number is increased, accuracy is
improved.

• However, increasing the type number

aggravates the stability problem.

• A compromise between steady-state accuracy

and relative stability is always necessary.
 Steady State Error of Unity Feedback Systems

• Consider the system shown in following figure.

• The closed-loop transfer function is

 Steady State Error of Unity Feedback Systems
• The transfer function between the error signal E(s) and the
input signal R(s) is
E( s ) 1
=
R( s ) 1 + G( s )
• The final-value theorem provides a convenient way to find
the steady-state performance of a stable system.

• Since E(s) is

• The steady state error is

 Static Error Constants
• The static error constants are figures of merit of
control systems. The higher the constants, the
smaller the steady-state error.
• In a given system, the output may be the position,
velocity, pressure, temperature, or the like.
• Therefore, in what follows, we shall call the output
“position,” the rate of change of the output
“velocity,” and so on.
• This means that in a temperature control system
“position” represents the output temperature,
“velocity” represents the rate of change of the
output temperature, and so on.
 Static Position Error Constant (Kp)
• The steady-state error of the system for a unit-step input is

• The static position error constant Kp is defined by

• Thus, the steady-state error in terms of the static position

error constant Kp is given by
 Static Position Error Constant (Kp)
• For a Type 0 system

• For Type 1 or higher systems

• For a unit step input the steady state error ess is

 Static Velocity Error Constant (Kv)
• The steady-state error of the system for a unit-ramp input is

• The static position error constant Kv is defined by

• Thus, the steady-state error in terms of the static velocity

error constant Kv is given by
 Static Velocity Error Constant (Kv)
• For a Type 0 system

• For Type 1 systems

• For type 2 or higher systems

 Static Velocity Error Constant (Kv)
• For a ramp input the steady state error ess is
 Static Acceleration Error Constant (Ka)
• The steady-state error of the system for parabolic input is

• The static acceleration error constant Ka is defined by

• Thus, the steady-state error in terms of the static acceleration

error constant Ka is given by
 Static Acceleration Error Constant (Ka)
• For a Type 0 system

• For Type 1 systems

• For type 2 systems

• For type 3 or higher systems

 Static Acceleration Error Constant (Ka)
• For a parabolic input the steady state error ess is
Summary
 Example#1
• For the system shown in figure below evaluate the static
error constants and find the expected steady state errors
for the standard step, ramp and parabolic inputs.

100( s + 2)( s + 5)
R(S) C(S)
2
s ( s + 8)( s + 12)
-
 Example#1 (evaluation of Static Error Constants)
100( s + 2)( s + 5)
G( s ) =
s 2 ( s + 8)( s + 12)
K p = lim G( s )
s →0 K v = lim sG( s )
s →0
 100( s + 2)( s + 5) 
K p = lim  2   100s( s + 2)( s + 5) 
s →0  s ( s + 8)( s + 12)  K v = lim  2 
s →0  s ( s + 8)( s + 12) 
Kp = 
Kv = 

K a = lim s 2 G( s )  100s 2 ( s + 2)( s + 5) 

K a = lim  2 
s →0  
s →0
 s ( s + 8 )( s + 12 ) 
 100( 0 + 2)(0 + 5) 
K a =   = 10.4
 ( 0 + 8)(0 + 12) 
Example#1 (Steady Sate Errors)
Kp =  Kv =  K a = 10.4

=0

=0

= 0.09
Bibliografia

Katsuhiko Ogata, Ingeniería de control moderna, 5ta Edición

Benjamin Kuo, Sistemas de control automatico, 7ma Edición