# MM7. PID Control Design

10/19/2004 Process Control 1 MM7. PID Control Design ... cascade control ... Example: Speed Control of a DC Motor...

MM7. PID Control Design Reading Material: FC: p.179-200, DC: p.66-68

1. 2. 3. 4. 10/19/2004

Properties of PID control Tuning Methods of PID Control Antiwindup Technique A real case study – BO9000 Process Control

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1. PID Feedback Controllers PID Means: P: Proportional (control) I : Integral (control) D: Derivative (control)

u(t ) = Ke(t ) K u (t ) = TI

t t0

e(τ )dτ

u(t ) = KT De(t )

PID Control System Structure: cascade control r(t)

+

e(t) -

PID Controller

Plant G(s)

y(t)

What are the characteristics of PID control? 10/19/2004

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1.1 Proportional Feedback Control Control Structure

Time Domain : u(t ) = Ke(t )

Frequency Domain : D(s) =

U ( s) =K E ( s)

Closed loop Control System r(t)

+

e(t) -

P-Controller: K

Gcl ( s ) =

Plant G(s)

D( s )G ( s ) KG ( s ) = 1 + D( s )G ( s ) 1 + KG ( s ) unity

y(t)

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K

p

feedback

sys

= lim G o ( s ) s→ 0

K

v

= lim sG

K

a

= lim s 2 G o ( s )

s→ 0

s→ 0

o

(s)

1 1+ K

e ss = e ss = e ss =

p

1 Kv

1 Ka

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Example: Speed Control of a DC Motor Working mechanism of a DC motor T = K tia e = K

e

θ

. m

Kt torque constant ia armature current Ke electromotive force (emf) constant Differential equation description J

..

m

θ

m

+ b θ

. m

= K tia

di a + R a ia = v La dt : simplified J

m

θ

.. m

+ (b +

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K

t

K

R

a

a

e

− K

. m

e

θ

=

. m

K R

t

v

a

a

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See FC p.47-49

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Simulink has become the most widely used software package in academia and industry for modelling and simulating dynamic systems It turns your computer into a lab for modeling and analyzing systems that simply wouldn't be possible or practical otherwise

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DC Motor Model : Kp =

,

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ess =

A G(s) = (τ 1s + 1)(τ 2 s + 1)

τ 1 = 1, τ 2 = 0.1, A = 1

In order to eliminate the steady-state offset, introduce integral control !... 6 Process Control

1.2 PI Feedback Control Control Structure

1 Time Domain : u(t ) = K ( e(t ) + TI

t t0

TI – integral/reset time

e(τ )dτ )

Frequency Domain : D(s) =

1 U ( s) = K (1 + ) E ( s) TI s

Closed loop Control System r(t)

+

e(t) -

Plant G(s)

K+K/TIs

y(t)

1 )G ( s ) D ( s )G ( s ) K ( T I s + 1) G ( s ) TI s G cl ( s ) = = = T I s + K ( T I s + 1) G ( s ) 1 + D ( s ) G ( s ) 1 + K (1 + 1 ) G ( s ) TI s K (1 +

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1.3 PID Feedback Control Control Structure u ( t ) = K ( e( t ) +

1 TI

t t0

TD – Derivative/rate time e(τ )dτ + TD e(t)

D(s) =

Closed loop Control System r(t)

+

e(t) -

K(1+1/Tis+ TDs)

1 U ( s) = K (1 + + TD s ) E ( s) TI s

Plant G(s)

y(t)

D ( s )G ( s ) K ( T D T I s 2 + T I s + 1) G ( s ) G cl ( s ) = = 1 + D ( s ) G ( s ) T I s + K ( T D T I s 2 + T I s + 1) G ( s )

Advantages: Increase the damping Improve the stability Good transient and steady disturbance rejection The most popular control technique used in industry! 10/19/2004

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PID-Control for the DC Motor

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Effect of constant Disturbance

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MM7. PID Control Design 1. 2. 3. 4. 10/19/2004

Properties of PID control Tuning Methods of PID Control Antiwindup Technique A real case study: BO9000 Process Control

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2. Tuning the PID Controllers Principle: 1 u ( t ) = K ( e( t ) + TI

t t0

e(τ )dτ + TD e(t)

D(s) =

1 U ( s) = K (1 + + TD s ) E ( s) TI s

Ziegler-Nichols Tuning Method(1942-1943) Quarter decay ratio method: Tuning by decaying ratio of 0.25 (step response: Process reaction curve) Slope rate R=K/ττ Lag time L 10/19/2004

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Exercise 1: Design a P, PI, PID controller for the DC motor example, According to quarter decay m.

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Tuning the PID Controllers (continue...) Ziegler-Nichols Tuning Method (1942-1943) Ultimate Sensitivity method r(t)

+

e(t) -

Ultimate: Ku

Plant G(s)

y(t)

•Impulse response •Ultimate gain: Ku •Ultimate period: Pu

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MM7. PID Control Design 1. Properties of PID control 2. Tuning Methods of PID Control 3. Antiwindup Technique

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3. Integrator Antiwindup Motivation: actuator saturation phenomena integration integrator windup Antiwindup technique: Turn off the integral action as soon as the actuator saturates

•Implement with a dead zone •Implement with a nonlinearity

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Output responses

Control effort

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MM7. PID Control Design 1. 2. 3. 4. 10/19/2004

Properties of PID control Tuning Methods of PID Control Antiwindup Technique A real case study: BO9000 Process Control

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BO9000 Control: BO9000

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BO9000 Control: Problem (I) " , -# ./#01 " \$ 23 " " 5 23 ! , !

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BO9000 Control: Problem (II)

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BO9000 Control: Problem (III)

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BO9000 Control: Reuirements

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BO9000 Control: Modelling (I) S le d g e d ire c tio n to C D 6

W ire

S le d g e d ire c tio n to C D 1

S le d g e

P u lle y 1

P u lle y 2

M o to r

W ire w in d s u p a ro u n d th e m o to rs ’ a x e l

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BO9000 Control: Modelling (II) TFT2 = Friction moment for pulley 2

FFS

TFT1 = Friction moment for pulley 1

FT2S = The force on the sledge affected by pulley 2

TM = The moment of the motor (electromagnetic moment)

FT1S = The force on the sledge affected by pulley 1

TFM = The friction moment of the motor

FMT1 = The force on pulley 1 affected by the motor

FT2M = The force on the motor affected by pulley 2

FMT2 = The force on pulley 2 affected by the motor

FT1M = The force on the motor affected by pulley 1

FST1 = The force on pulley 1 affected by the sledge

= The friction force of the sledge

FST2 = The force on pulley 2 affected by the sledge

The positive orbital direction

FT1S

Sledge

FT2S

FST1

FST2

Pulley 1

TFT1

FMT1

Pulley 2

FFS FT1M

FT2M

FMT2

Motor

TFT2

TM TFM

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BO9000 Control: Modelling (III) Fgx

Ua + -

1 SLa + Ra

i

Kt

TM

1 rM

-

+

Ms + -

-

1 2 ⋅ IT rT 2

+

IM

x

rM 2

1 s

x

1 s

x

f f

Ka

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C

1 rM

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BO9000 Control: PID Controller (I)

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BO9000 Control: PID Controller (II)

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BO9000 Control: PID Controller (III)

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BO9000 Control: PID Controller (IV)

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BO9000 Control: Root Locus Design with Antiwindup

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Exercise One Exercise 1: Design a P, PI, PID controller for the following DC motor speed control, According to quarter decay method. Download ZN_tuning_motor.mdl

Exercise 2: Implement the above system with an actuator saturation in simulink model with umax=2, umin=-2. Design an integrator antiwindup strategy for your designed PI controller.

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