Chapter 8 Steady-State Errorsfivedots.coe.psu.ac.th/Software.coe/240-209/Slides/... · 2006-11-17 · 240-209: Steady-State Errors ผเรยบเรยง ธเนศ เคารพาพงศ

ผ��เร�ยบเร�ยง ธเนศ เคารพาพงศ� แก�ไข 12 มกราคม 2548 หน�า 1240-209 : Steady-State Errors

Chapter 8

Steady-State Errors


Outline

1 Definition2 Test input 2.1 Step function 2.2 Ramp function 2.3 Parabola function3 Steady-state errors4 Steady-state errors for disturbances


Definition and test input

R(s) E(s) C(s)G(s)

H(s)

+-

e ∞= limt∞

e t

e t =L−1[E s ]



Steady-state error

Step input


Ramp input

Steady-state error


Closed-loop control system error

(a) general representation;(b) representation for unity feedback systems


Steady-State Error for Unity Feedback Systems

E s = R s 1G s

e ∞= limt∞

e t =lims 0

sE s


Step input

e ∞=e step ∞=lims 0

s 1 /s 1G s

= 11lim

s0G s

e ∞ 0 When lims 0

G s =∞


Ramp input

When

e ∞=e ramp ∞=lims 0

s 1/ s21G s

= 1lims 0

sG s

e ∞ 0 lims 0

sG s =∞


Parabolic input

When

e ∞=e para ∞=lims 0

s 1/s31G s

= 1lims 0

s2G s

e ∞ 0 lims 0

s2G s =∞


Static Error Constants and System type

Position constant, Kp

K p=lims 0

G s

Velocity constant, Kv

Acceleration constant, Ka

K v=lims 0

sG s

K a=lims 0

s 2G s


Feedback control system for definingsystem type



Steady-State Error Specifications

If a control system has the specification Kv=1000, we can draw a conclusions :1. The system is stable.2. The system is of type 1.3. A ramp input is the test signal.4. The steady-state error between the input ramp and the output ramp is 1/Kv per unit of input slope.


Steady-state Error for Disturbances

Feedback control system showing disturbance


System rearranged to show disturbanceas input and error as output, with R(s) = 0


Forming an equivalent unity feedback system from a general nonunity feedback system

Steady-state Error for Non-unity Feedback Systems


G(s) = G1(s)G2(s)H(s) = H1(s)/G1(s)

When





Nonunity feedback control system withdisturbance


Sensitivity

SF : P= lim P0

Fractional change in the function , FFractional change in the parameter , P

= lim P0

F / FP / P

= lim P 0

PFFP

S F : P=PF F P


Example (Nise) Calculate the sensitivity of the closed-loop transfer function to changes in parameter “a”


Steady-state Error for System in State space

x=A xB ry=C x

E s =R s [1−C sI−A −1B ]

lims 0

sE s =lims 0

sR s [1−C s I−A −1 B ]


Example (Nise) Find the steady-state errors for inputs of 5u(t), 5tu(t) and 5t2�u(t) of the system shown below.


Example (Nise) Find the steady-state errors for inputs of 5u(t), 5tu(t) and 5t2�u(t) of the system shown below.


Example (Nise) Find the value of K so that there are 10% error in the steady state for the system shown below.


Example (Nise) Find the steady-state error for a unit step input of the system shown below.

Documents

Chapter 8 Steady-State Errorsfivedots.coe.psu.ac.th/Software.coe/240-209/Slides/... · 2006-11-17 · 240-209: Steady-State Errors ผเรยบเรยง ธเนศ เคารพาพงศ