Expression for Position, Velocity and Acceleration Error Constants
Given Below are the Expression for Position, Velocity and Acceleration error Constants for Type 0, Type 1 and Type 2 Control Systems
Position Error Constant for Unit Step Unit
We know that for a unit step input signal, r(t) = A u(t)
Taking Laplace Transform on both sides we get, R(s)= A/s
According to final Value theorem; ess = lim s→0 [ s*R(s)/1+G(s)]
or, ess = lim s→0 [s*A/s]/ 1+G(s)
= A lim s→0 [ 1/1+G(s)]
= A / 1 + lim s→0 G(s)
= A/ 1+ Kp
Where Kp = lim s→0 G(s) , Static Position error coefficient
Position Error For Type 0, Type 1 and Type 2 System
We know that transfer function of a control system is the ratio of Laplace transform of the output to Laplace transform of the input: G(s)= N(s)/ D(s)
Type 0 order system : ( System with No poles at origin ⇒ D(s) ≠ 0 : lim s→0 s* G(s) = Kp ≠ 0 ⇒ Static position error coefficient is Finite vale
⇒ ess = A/ 1+Kp ⇒ means the steady state position error is finite value which is not desirable.
Type 1 order System or higher order system : G(s) = N(s)/ D(s)
lim s→0 G(s) = lim s→0 N(s)/ D(s)
= A/ sn ( 1+sT) , as s→0; G(s)→∞
Hence Kp= lim s→0 G(s) = ∞
ess = A/1+Kp = 1/∞ = 0 , means the position error is zero for type 1 or higher order system, thus it is suitable for designing.
Velocity Error Constant for Ramp Input
We know that for a ramp input signal, r(t) = A t u(t)
Taking Laplace Transform on both sides we get, R(s)= A/s2
According to final Value theorem; ev = lim s→0 [ s*R(s)/ s2 (1+G(s))]
or, ev = lim s→0 [s*A/s2]/ 1+G(s)
= A lim s→0 [ 1/ s (1+G(s))]
= A / 1 + lim s→0 s *G(s)
= A/ Kv
Where Kv = lim s→0 s*G(s) , Velocity error coefficient
Velocity Error For Type 0, Type 1 and Type 2 System
We know that transfer function of a control system is the ratio of Laplace transform of the output to Laplace transform of the input: G(s)= N(s)/ D(s)
Type 0 order system : ( System with No poles at origin ⇒ D(s) ≠ 0 : lim s→0 s* G(s) = Kv = 0 ⇒ velocity error coefficient is zero
⇒ ev = A/ Kv = A/0 = ∞, means the steady state velocity error is infinite which is not desirable
Type 1 order System : G(s) = N(s)/ D(s) = A/ s*(1+sT)
Kv= lim s→0 [ s * (A/ s* (1+sT))] = A
⇒ Kv = Finite value
Hence, ev = Finite error
Type 2 order System : G(s) = N(s)/ D(s) = A/ s2 *(1+sT)
Kv= lim s→0 [ s * (A/ s2 (1+sT))] = A/0= ∞
⇒ Kv = Infinite Value
Hence, ev = A/Kv = A/∞ = 0, The velocity error for a type 2 system is zero therefore it is desirable
Acceleration Error Constant for Parabolic Input
We know that for a Parabolic input signal, r(t) = A t2 u(t)
Taking Laplace Transform on both sides we get, R(s)= A/s3
According to final Value theorem; ev = lim s→0 [ s*R(s)/ s3 (1+G(s))]
or, ea = lim s→0 [s*A/s3]/ 1+G(s)
= A lim s→0 [ 1/ s2 (1+G(s))]
= A / 1 + lim s→0 s2 *G(s)
= A/ Ka
Where Ka = lim s→0 s2 *G(s) , Acceleration error coefficient
Acceleration error for Type 0, Type 1 and Type 2 System
We know that transfer function of a control system is the ratio of Laplace transform of the output to Laplace transform of the input: G(s)= N(s)/ D(s)
Type 0 order system : ( System with No poles at origin ⇒ D(s) ≠ 0 : lim s→0 s2 * G(s) = Ka = 0 ⇒ velocity error coefficient is zero
⇒ ea = A/ Ka = A/0 = ∞, means the steady state acceleration error is infinite which is not desirable
Type 1 order System : G(s) = N(s)/ D(s) = A/ s *(1+sT)
Ka= lim s→0 [ s2 * (A/ s (1+sT))] = 0
⇒ Ka = Zero acceleration coefficient
Hence, ea = A/Ka = A/0 = ∞, means the steady state acceleration error is infinite which is not desirable.
Type 2 order System : G(s) = N(s)/ D(s) = A/ s *(1+sT)
Ka= lim s→0 [ s2 * (A/ s2 (1+sT))] = A
⇒ Ka = Finite Value
Hence, ea = Finite error for type 2 order system.
Steady State Errors for Unity Feedback Systems
In this Article, We will be going through Steady State Errors for Unity Feedback Systems in control systems, First, we will start our Article with an introduction to Steady State Errors, then we will through its two types, and then we will see mathematical Expression for calculating the Steady-State Error, At last, we will conclude our Article with its Advantages, Disadvantages, Applications and Some FAQs.
Table of Content
- What is Steady State Errors?
- Types of Steady State Errors For a Unity Feedback System
- Expression for Position, Velocity and Acceleration Error Constants
- Applications
- Advantages
- Disadvantages