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PID

Controllerand Root

Locus

Jin Oh Jung

Meaning of SystemDynamics

The LaplaceTransfromMethod

TransferFunctions

Block

Diagrams

Loops

PID Control

Root LocusMethod

Outline

1 Meaning of System Dynamics

2 The Laplace Transfrom Method

3 Transfer Functions

4 Block Diagrams

5 Loops

6 PID Control

7 Root Locus Method

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PID

Controllerand Root

Locus

Jin Oh Jung



TransferFunctions

Block

Diagrams

Loops

PID Control

Root LocusMethod

What is a System?

Definition

System: Combination of elements acting together

Input: Cause of the system

Output: Effect due to the input

Model: Mathematical description of the system

Inputs Outputs

Boundary

System

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PID

Controllerand Root

Locus

Jin Oh Jung



TransferFunctions

Block

Diagrams

Loops

PID Control

Root LocusMethod

We will deal with...

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PID

Controllerand Root

Locus

Jin Oh Jung



TransferFunctions

Block

Diagrams

Loops

PID Control

Root LocusMethod


ODE Models

Inital conditions at t = 0

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PID

Controllerand Root

Locus

Jin Oh Jung



TransferFunctions

Block

DiagramsLoops

PID Control

Root LocusMethod


ODE Models

Inital conditions at t = 0

Use Laplace Transform!

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PID

Controllerand Root

Locus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Why are we doing this?

The real world looks simple...

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PID

Controllerand Root

Locus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Why are we doing this?

But it’s not!


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PID

Controllerand Root

Locus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

The Laplace Transform Method


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PID

Controllerand Root

Locus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod


Describes system’s motion in frequency domain

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PID

Controllerand Root

Locus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod



Ex) Damped harmonic oscillator


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PID

Controllerand Root

Locus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod




ẍ + 4ẋ + 53x = 0

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PID

Controllerand RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod




ẍ + 4ẋ + 53x = 0

s 2X (s )− s x (0)− ẋ (0) + 4 [s X (s )− x (0)] + 53X (s ) = 0

(s 2 + 4s + 53)X (s ) = 8s + 13

X (s ) = 8s

+ 13s 2 + 4s + 53

x (t ) = 8e −2t cos 7t −3

7e −2t

sin7t


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PID


Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod


Thus, if s is the root of the characteristic equation

s 2 + 4s + 53 = 0

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PID


Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod



s 2 + 4s + 53 = 0

Re s → ζ

(Damping ratio)


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PID


Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod



s 2 + 4s + 53 = 0

Re s → ζ

(Damping ratio)

Im s → ωd

(Damped natural frequency)

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PID


Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Transfer Functions

In general,A(s )X (s ) = B (s )F (s )

T (s ) = X (s )F (s )

, means OutputInput

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PID


Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Transfer Functions


T (s ) = X (s )F (s )

, means OutputInput

T (s ) is called the Transfer Function


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PID


Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Transfer Functions


T (s ) = X (s )F (s )

, means OutputInput

T (s ) is called the Transfer Function

Property of the system model only


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PID


Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Block Diagrams

Visual representations with transfer functions

X(s)T(s)

F(s)T(s)

X(s)

Y(s)

X(s) Z(s)+

X(s)

X(s)

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PIDControllerand Root

Locus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Block Diagrams


X(s)T(s)

Multiply...

F(s)T(s)

X(s)

Y(s)

X(s) Z(s)+

X(s)

X(s)


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Locus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Block Diagrams


X(s)

T(s)

F(s)

T(s)

X(s)

Y(s)

X(s) Z(s)+

X(s)

X(s)

B D


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Locus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Block Diagrams


X(s)

T(s)

F(s)

T(s)

X(s)

Y(s)

X(s) Z(s)+

X(s)

X(s)

B D

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Locus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Block Diagrams


X(s)

T(s)

F(s)

T(s)

X(s)

Subtract... Y(s)

X(s) Z(s)+

X(s)

X(s)

B D

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Locus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Block Diagrams


X(s)

T(s)

F(s)

T(s)

X(s)

Y(s)

X(s) Z(s)+

X(s)

X(s)

B D

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Locus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Block Diagrams


X(s)

T(s)

F(s)

T(s)

X(s)

Extract...

Y(s)

X(s) Z(s)+

X(s)

X(s)

Block Diagrams

http://goforward/http://find/http://goback/

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Locus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Block Diagrams


X(s)

T(s)

F(s)

T(s)

X(s)

Y(s)

X(s) Z(s)+

X(s)

X(s)

Loops

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Locus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Loops

Open Loop Control

Closed Loop Control

Loops

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Locus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Loops

In the closed loop,

T (s ) = Y (s )

R

(s

)

= G (s )D cl (s )

1 +G

(s

)D

cl (s

)

Loops

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Loops

In the closed loop,

T (s ) = Y (s )

R

(s

)

= G (s )D cl (s )

1 +G

(s

)D

cl (s

)

Set the denominator to 0

Loops

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Loops

In the closed loop,

T (s ) = Y (s )

R

(s

)

= G (s )D cl (s )

1 +G

(s

)D

cl (s

)

Set the denominator to 0

Shows how the system responses!

PID Controller

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

PID Controller

Proportional-Integral-Derivative Controller

Highly used in Mechanical, Aerospace Engineering

PID Controller

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

PID Controller



PID Controller

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

PID Controller



Aileron

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Aileron

Example of a PID controller

Aileron

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Aileron

Ailerons are marked with red

Aileron

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Aileron

If tilted...

Aileron

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod The angle is measured

Aileron

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod PID Controller’s signal → Aileron

Aileron

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

The aileron actives!

Aileron

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod PID Controller’s signal → aileron

Aileron

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control


Aileron

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control


Aileron

http://find/

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod Finally, at the right angle!

Aileron

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod Opps! Tilted again!

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Aileron

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control


Aileron

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control


Aileron

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control


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P Control

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Control signal is proportional to the error

P Control

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Control signal is proportional to the error

t (sec )

x (m)

0 1 2 3 4−2

−1

0

1

2

I Control

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Only P control: Constant error

I control: Accumulates error and gives feedback

I Control

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Only P control: Constant error

I control: Accumulates error and gives feedback

t (sec )

x (m)

0 1 2 3 4−2

−1

0

1

2

D Control

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Integral action produces oscillations

D control: Error approaches zero

D Control

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Integral action produces oscillations

D control: Error approaches zero

t (sec )

x (m)

0 1 2 3 4−2

−1

0

1

2

PID Control

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Altogether,

MV (t ) = K p e (t ) + K i

t 0

e (τ ) d τ + K d d

dx e (t )

In the frequency domain,

D cl (t ) = K p + K i

s + K d s

To obtain the gains → Use Root Locus method!

Root Locus Method

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PIDController

and RootLocus

Jin Oh Jung



TransferFunctions

BlockDiagrams

Loops

PID Control

Root LocusMethod

Set transfer function’s denominator zero

Solve varying K p from 0 to ∞Find the position you want!
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20080833 Jin Oh Jung.pdf

Documents

Transcript of 20080833 Jin Oh Jung.pdf