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Cleveland State University

MCE441: Intr. Linear Control Systems

Lecture 8: Final Value Theorem

Block Diagram Algebra

Prof. Richter

The Final Value Theorem

⊲The Final ValueTheorem

Block DiagramAlgebra

Block DiagramAlgebra

Block DiagramManipulations

Example

Solution

Solution

Example

Solution

Solution

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� Provides information about the value of f(t) at steady state

(t = ∞)� The steady-state may not exist: the poles of sF (s) must lie

on the open left half of C (details later)� If limt→∞ f(t) exists, then

limt→∞ f(t) = lims→0 sF (s)

� Example: For F (s) = 1s(s+1) , we see that the only pole of

sF (s) lies on the left half plane. So

limt→∞

f(t) = lims→0

sF (s) = 1

� Check by inversion of F (s)

Block Diagram Algebra

The Final ValueTheorem

⊲Block DiagramAlgebra

Block DiagramAlgebra

Block DiagramManipulations

Example

Solution

Solution

Example

Solution

Solution

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� The previous result allows us to multiply TF and input to getoutput in the Laplace domain

� Block diagrams are convenient representations of controlsystem structure

� The following basic operations are allowed in block diagramalgebra:1. Addition between signals. Result: another signal.2. Multiplication between a signal and a system. Result:another signal.

� Systems (TFs) are represented by blocks.� Signals are represented by arrows.

Block Diagram Algebra

The Final ValueTheoremBlock DiagramAlgebra

⊲Block DiagramAlgebra

Block DiagramManipulations

Example

Solution

Solution

Example

Solution

Solution

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� A Summing junction or summing point is the symbol thatrepresents signal addition

� A branch point indicates that the output from a block isapplied to other blocks or summing junctions.

G(s)E(s) Y (s) = G(s)E(s)R(s)

+−

Summing point

Branch point

E(s) = R(s) − Y (s)

Y (s)

Block Diagram Manipulations

The Final ValueTheoremBlock DiagramAlgebra

Block DiagramAlgebra

⊲Block DiagramManipulations

Example

Solution

Solution

Example

Solution

Solution

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Example

The Final ValueTheoremBlock DiagramAlgebra

Block DiagramAlgebra

Block DiagramManipulations

⊲ Example

Solution

Solution

Example

Solution

Solution

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G1 G2

G4

G3

R(s) Y (s)

+−

++

+

Solution

The Final ValueTheoremBlock DiagramAlgebra

Block DiagramAlgebra

Block DiagramManipulations

Example

⊲ Solution

Solution

Example

Solution

Solution

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Solution

The Final ValueTheoremBlock DiagramAlgebra

Block DiagramAlgebra

Block DiagramManipulations

Example

Solution

⊲ Solution

Example

Solution

Solution

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Check that the required TF is(G1+1)G2

(G1+1)G2G4+1−G2G3

Example

The Final ValueTheoremBlock DiagramAlgebra

Block DiagramAlgebra

Block DiagramManipulations

Example

Solution

Solution

⊲ Example

Solution

Solution

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G1 G3

G5 G4

R(s) Y (s)

+−

+−

G2

++

Solution

The Final ValueTheoremBlock DiagramAlgebra

Block DiagramAlgebra

Block DiagramManipulations

Example

Solution

Solution

Example

⊲ Solution

Solution

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Solution

The Final ValueTheoremBlock DiagramAlgebra

Block DiagramAlgebra

Block DiagramManipulations

Example

Solution

Solution

Example

Solution

⊲ Solution

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Check that the required TF isG1H1

1+G5(G1H1G4+G2), where H1 =

G31+G3

.