Classical Control - Lecture 3 - homes.et.aau.dk

Outline

Classical ControlLecture 3

Lecture 3 Classical Control

Outline

Outline

1 Properties of PID Control

2 Tuning Methods of PID Control

3 Antiwindup Technique


Properties of PID ControlTuning Methods of PID Control

Antiwindup Technique

P-ControlPI-ControlPID-ControlPD-Control

Outline

1 Properties of PID ControlP-ControlPI-ControlPID-ControlPD-Control

2 Tuning Methods of PID ControlQuarter Decay MethodUltimate Sensitivity Method

3 Antiwindup TechniqueEffect of SaturationIntroducing Dead ZoneUsing the Non-Linearity





PID Feedback Controllers

PID means:

P : Proportional (control) u(t) = K · e(t)

I : Integral (control) u(t) =K

TI

∫ t

t0

e(τ)dτ

D : Derivative (control) u(t) = K · TD · e(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?





Proportional Feedback Control

Control Structure

Time Domain: u(t) = K · e(t)

Frequency Domain: D(s) = U(s)E(s) = K

Closed Loop Control System

r(t)+

−

e(t)P-Controller: K

PlantG (s)

w(t)y(t)

Gcl(s) =D(s)G (s)

1 + D(s)G (s)=

KG (s)

1 + KG (s)





Proportional Feedback Control

Advantages:

A simple controller

Disadvantages:

Steady state offset/error problem

Unity feedback system:

Kp = lims→0

Gcl(s) ess =1

1 + Kp

Kv = lims→0

sGcl(s) ess =1

Kv

Ka = lims→0

s2Gcl(s) ess =1

Ka

Disturbance rejection problem





Example: Speed Control of a DC Motor (FC p. 41-43)

Working mechanism of a DC motor:

T = Kt · ia

e = Ke · θm

Kt : Torque constant

ia: Armature current

Ke : Electromotive force (emf) constant






Differential equation description:

Jmθm + bθm = Kt ia

va − La

dia

dt− Raia = Ke θm






Obtaining a transfer function: Taking the Laplace transform andisolating I (s) and V (s):

I (s) =Jm

Kt

s2Θ(s) +b

Kt

sΘ(s)

V (s) = LasI (s) + RaI (s) + KesΘ(s)

= Las

(

Jm

Kt

s2Θ(s) +b

Kt

sΘ(s)

)

+ Ra

(

Jm

Kt

s2Θ(s) +b

Kt

sΘ(s)

)

+ KesΘ(s)

=LaJ

Kt

s3Θ(s) +Lab + RaJ

Kt

s2Θ(s) +

(

Rab

Kt

+ Ke

)

sΘ(s)






Obtaining a transfer function: The transfer function is thengiven by:

Θ(s)

V (s)=

1

LaJKt

s3 + Lab+RaJKt

s2 +(

RabKt

+ Ke

)

s

Using speed as a reference we have:

Ω(s)

V (s)= s

Θ(s)

V (s)

=1

LaJKt

s2 + Lab+RaJKt

s +(

RabKt

+ Ke

)

A second order system!





P-Control for the DC Motor

Download ”motorP.mdl”:

time

t

disturbance 10

Transfer Fcn1

1

0.1 s+1

Transfer Fcn

1

s+1To Workspace

y_P

Scope

K

u*K

Clock

1

const

DC Motor Model:

G (s) =A

(τ1s + 1)(τ2s + 1)τ1 = 1, τ2 = 0.1, A = 1





P-Control for the DC Motor: Result

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

In order to eliminate steady-state offset: Integral control





PI Feedback Control

Control StructureTime Domain: u(t) = K

(

e(t) + 1TI

∫ t

t0e(τ)dτ)

)


(

1 + 1TI s

)


r(t)+

−

e(t)K + K

TI sPlantG (s)

w(t)y(t)

Gcl(s) =D(s)G (s)

1 + D(s)G (s)=

K(

1 + 1TI s

)

G (s)

1 + K(

1 + 1TI s

)

G (s)

=K (TI s + 1)G (s)

TI s + K (TI s + 1)G (s)Lecture 3 Classical Control




PI Feedback Control

Advantages:

Eliminates steady state offset/error

Good steady state disturbance rejection

What about the transient response?





PI-Control for the DC Motor

Download ”motorPI.mdl”:

time

t

disturbance10

Transfer Fcn1

1

0.1 s+1

Transfer Fcn

1

s+1To Workspace

y_PI

Scope

PI Controller

In1 Out1

Constant

const

Clock

Out1

1

K/Ti

u*K

K

u*K

Integrator

1/s

In1

1





PI-Control for the DC Motor: Result

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5





PID Feedback Control

Control StructureTime Domain: u(t) = K

(

e(t) + 1TI

∫ t

t0e(τ)dτ) + TD e(t)

)


(

1 + 1TI s

+ TDs)


r(t)+

−

e(t)K

(

1 + 1TI s

+ TDs)

PlantG (s)

w(t)y(t)

Gcl(s) =D(s)G (s)

1 + D(s)G (s)

=K

(

TDTI s2 + TI s + 1

)

G (s)

TI s + K (TDTI s2 + TI s + 1) G (s)





PID Feedback Control

Advantages:

Increases the damping → Improves the stability

Good transient and steady state disturbance rejection

The most popular control technique used in industry!





PID-Control for the DC Motor

Download ”motorPID.mdl”:

time

t

disturbance 10

Transfer Fcn1

1

0.1 s+1

Transfer Fcn

1

s+1To Workspace

y_PID

Scope

PID Controller

In1 Out1

Constant

const

Clock

Out1

1

K/Ti

u*K

K*Td

u*K

K

u*K

Integrator

1/s

Derivative

du/dt

In1

1





PID-Control for the DC Motor: Result

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8





PD Feedback Control

Control Structure

Time Domain: u(t) = K (e(t) + TD e(t))

Frequency Domain: D(s) = U(s)E(s) = K (1 + TDs)


r(t)+

−

e(t)K + KTDs Plant

G (s)

w(t)y(t)

Gcl(s) =D(s)G (s)

1 + D(s)G (s)=

K (1 + TDs)G (s)

1 + K (1 + TDs) G (s)





PD Feedback Control

Advantages:

Increases damping

Good transient disturbance rejection





PD-Control for the DC Motor

Download ”motorPD.mdl”:

time

t

disturbance 10

Transfer Fcn1

1

0.1 s+1

Transfer Fcn

1

s+1To Workspace

y_PD

Scope

PD Controller

In1 Out1

Clock

1

const

Out1

1

K*Td

u*K

K

u*K

Derivative

du/dt

In1

1





PD-Control for the DC Motor: Result

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2





Control for the DC Motor: Results

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

PPIPIDPD





Effect of Constant Disturbance

0 1 2 3 4 5 6 7 8 9 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

PPIPIDPD




Quarter Decay MethodUltimate Sensitivity Method

Outline








Tuning the PID Controllers

Principle

u(t) = K

(

e(t) +1

TI

∫ t

t0

e(τ)dτ) + TD e(t)

)

D(s) =U(s)

E (s)= K

(

1 +1

TI s+ TDs

)

Increasing K and 1TI

tends to reduce system errorsIncreasing TD tends to improve stability

Ziegler-Nichols Tuning Methods

Quarter decay ratioUltimate sensitivity





Quarter Decay

Tuning by decaying ratio of 0.25:Step response → Process reaction curve





Quarter Decay

Tuning by decaying ratio of 0.25:





Ultimate Sensitivity

r(t)+

−

e(t)Ultimate: Ku

PlantG (s)

w(t)y(t)




Effect of SaturationDead ZoneNon-Linearity

Outline








Integrator Antiwindup

Motivation: Actuator saturation phenomena

Integrator → Integrator windup

Antiwindup Technique: Turn off the integral action as soonas the actuator saturates

Implement using dead zoneImplement using non-linearity





Saturation Model

Download ”motorPIsaturation.mdl”:

time

t

disturbance10

Transfer Fcn1

1

0.1 s+1

Transfer Fcn

1

s+1

To Workspace1

u_PIS

To Workspace

y_PIS

Scope1

Scope

Saturation

K/Ti

u*K

K

u*K

Integrator

1/s

Constant

1

Clock





Effect of Saturation

0 1 2 3 4 5 6 7 8 9 10−5

0

5

10

15

20

25

without satwith sat

Control Effort

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

without satwith sat

Output





Integrator Antiwindup: Dead Zone





Integrator Antiwindup: Non-Linearity





Integrator Antiwindup Equivalent

When saturated the anti-windup turns the integrator into a lag:





Antiwindup Model

Download ”motorPIantiwind.mdl”:

time

t

disturbance10

Transfer Fcn1

1

0.1 s+1

Transfer Fcn

1

s+1

To Workspace1

u_PIW

To Workspace

y_PIW

Scope1

Scope

Saturation

Ka

u*K

K/Ti

u*K

K

u*K

Integrator

1/s

Dead Zone

Constant

1

Clock





Effect of Antiwindup

0 1 2 3 4 5 6 7 8 9 10

0.8

1

1.2

1.4

1.6

1.8

2

without antiwwith antiw

Control Effort

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

without antiwwith antiw

Output





Exercises

1 Design a P, PI, PID controller for speed control of thefollowing DC motor according to the quarter decay method(Download ”ZNtuning motor.mdl”).

2 Implement the above system with an actuator saturation inthe simulink model with umax=2 and umin=-2. Design anintegrator antiwindup strategy for your designed PI controller.


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Classical Control - Lecture 3 - homes.et.aau.dk