- ME 279 Automatic Control of Dynamic Systems Dr Robert G Landers Steady State Error Steady State Error Dr Robert G Landers Definition 2

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Steady State Error ME 279 Automatic Control of Dynamic Systems Dr. Robert G. Landers

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Definition 2 Steady–state error is the difference between the reference and the output as t → ∞. The methods we will use only apply to stable systems. Steady State Error Dr. Robert G. Landers e 1 ( ) = 0 e 2 ( ) e 3 ( ) = time output r(t) y 1 (t) y 2 (t) y 3 (t) e 1 ( ) = 0 e 2 ( ) time r(t) y 1 (t) y 2 (t)
Final Value Theorem 3 Steady State Error Dr. Robert G. Landers Final Value Theorem The Final Value Theorem (FVT) does not apply to unstable systems. The poles of Y(s) must be in the left s–plane (i.e., the must have negative real parts). The exception is that one pole of Y(s) can be at the origin (i.e., s = 0). ( 29 ( 29 ( 29 0 lim lim t s y y t sY s →∞ ∞ = =

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Example 1 4 Steady State Error Dr. Robert G. Landers Determine y(∞) for the three outputs below. ( 29 ( 29 ( 29 ( 29 ( 29 1 2 3 5 20 5 20 5 20 s Y s s s Y s s s s Y s s s + = + + = + + = -
Example 1 5 Steady State Error Dr. Robert G. Landers Cannot evaluate y3(∞) since Y3(s) is unstable ( 29 1 0 5 lim 0 20 s s y s s + ∞ = = + ( 29 ( 29 2 0 5 lim 0.25 20 s s y s s s + ∞ = = +

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Steady State Error in Terms of F(s) 6 Steady State Error Dr. Robert G. Landers F(s) Y(s) E(s) + - R(s) A block diagram of a closed–loop system is given to the right. The transfer function F(s) is known as the forward loop transfer function. The error signal is The steady state error is Assume F(s) can be written as ( 29 ( 29 ( 29 1 1 E s R s F s = + ( 29 ( 29 ( 29 0 1 lim 1 s e s R s F s ∞ = + ( 29 ( 29 ( 29 ( 29 ( 29 1 2 1 2 n s z s z F s s s p s p + + = + + L L
Steady State Error in Terms of F(s) 7 Steady State Error Dr. Robert G. Landers For a unit step input (i.e., r(t) = 1), R(s) = 1/s and the steady state error is e(∞) = 0 only if For a unit ramp input (i.e., r(t) = t), R(s) = 1/s2 and the steady state error is Therefore, n ≥ 1. In this case, at least one pure integrator must exist in the forward loop transfer function. ( 29 ( 29 ( 29 0 0 1 1 1 lim 1 1 lim s s e s F s s F s ∞ = × = + + ( 29 0 lim s F s = ∞ ( 29 ( 29 ( 29 2 0 0 1 1 1 lim lim 1 s s e s F s s sF s ∞ = × = +

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Steady State Error in Terms of F(s) 8 Steady State Error Dr. Robert G. Landers For a unit parabola input (i.e., r(t) = t2), R(s) = 2/s3 and the steady state error is e(∞) = 0 only if Therefore, n ≥ 3. In this case, at least three pure integrators must exist in the forward loop transfer function. e(∞) = 0 only if Therefore, n ≥ 2. In this case, at least two pure integrators must exist in the forward loop transfer function. ( 29 ( 29 ( 29 3 2 0 0 1 2 2 lim lim 1 s s e s F s s s F s ∞ = × = + ( 29 2 0 lim s s F s = ∞ ( 29 0 lim s sF s = ∞
9 Steady State Error Dr. Robert G. Landers For the system below determine the steady state errors for r(t) = 7, r(t) = 7t, and r(t) = 7t2. F(s)

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This note was uploaded on 02/01/2012 for the course MECH ENG 279 taught by Professor Landers during the Spring '11 term at Missouri S&T.

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- ME 279 Automatic Control of Dynamic Systems Dr Robert G Landers Steady State Error Steady State Error Dr Robert G Landers Definition 2

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