Chapter 6 Position and Speed Regulation in DC Servomechanism

§ 6.1 DC Servo System

§ 6.2 Servo Components

§ 6.3 Component Model

§ 6.4 System Control

§ 6.1 DC Servo System (1) • Fundamental Operations of DC Servo:

(1)Speed servo

Speed regulation, speed tracking

(2)Position servo (Servomechanism)

Position control, position tracking

§ 6.1 DC Servo System (2)

Stable operation

Fast transient response

Small steady-state error

System robustness

Command following

Disturbance rejection

• Control Objective:

Implies

§ 6.1 DC Servo System (3)

PC VC CC Amp Motorv(t)i(t) y(t)

ysp(t)Current Velocity

( )dt

• Generic Motor Position Control:

PC: Position controller

VC: Velocity controller

CC: Current controller

§ 6.1 DC Servo System (4)

• Mechatronic View Point:

Power Amp&

DC ServoMotor

DC PowerSupply

(External)

Speed &

PositionSensor

MotorGear Train

& Load

ServoController

Ex: DC servo control of liquid-level process

Amplifier

Valvepositioner

Pump

ea(t)

p(t)ee(t)

K (s)v

xv(t) Potentiometerbridge

Gc(s)

Gm(s)

x(t)

eb(t)er(t)

hc(t)pu(t) qu(t)

qm(t)

hv(t)

§ 6.1 DC Servo System (5)

Gm(s): Servo motor and gear train

Gc(s)Kh KvGm(s)

Kh 1.0

hv(t) eR(t) eE(t) p(t) xV(t)

x(t)

qM(t)

hC(t)

pu(t) qu(t)+ -

+

+

+-

1Rf

Rf/γ

τps+1

§ 6.1 DC Servo System (6)

§ 6.2 Servo Components (1)

DC motor http://www.shangyi.com.tw/c_produact_c.htm

Rotary Encoders

Resolvers

http://www.encodersindia.com/products.html

http://www.encodersindia.com/products.html

• Hardware Components:

§ 6.2 Servo Components (2)

Optical Encoders http://www.usdigital.com/products/hd25a/

http://www.mi-technologies.com/catalog/antennas/indencdr.pdf

Inductosyn Encoders

Potentiometerhttp://www.islproducts.com/prod/pots.htm

§ 6.2 Servo Components (3)

Series

Printedcircuitmotor

DC motor

Commutator typeBrushless

type

Shuntwound

Permanentmagnet

Compoundwound

Fieldmotor

Moving coilor ironlessarmature

Shell typearmature(Ironless

shell or rotor)

Torque motor

Servo motor

Cumulativecompound

Differentialcompound

Permanentmagnet

Switchedreluctance

Stepper

Variablereluctance

Permanentmagnet

Hybrid

• Classifications of DC Motor:

§ 6.2 Servo Components (4)Feedback Devices

Analog Digital

PositionVelocity(output proportional

to velocity)

TachometerPotentiometers

Linear Rotary Rotary Linear

Synchro Resolver Resolver DifferentialTransformer

Incremental(train of pulses)

Absolute(pattern of pulses)

RotaryOptical

Rotary Linear

Quantizer OpticalInduction

• Feedback Devices:

§ 6.2 Servo Components (5)• Resolution of Displacement Sensors:

Opticalencoders

Inductosyn

Resolvers

Synchros

Potentiometers

Angularresolution

• DC Servomotor: Ex: Armature Control

(1) Static model

tT

max

maxv

tmaxT (Stall)

tT BT K v K

t

tmax

T maxmax 0

max Bb

max T maxT 0

tmax

i a T maxmax

T K (Stall)

v

v K K (No load)

K

T K R K

i

La

ia(t)if+

eb(t)

Ra

va(t) M

-

=const.

,Ttm , Jm, Bm,mθ mω

or Permanent magnet

Ttd

§ 6.3 Component Model (1)

(2) Dynamic model

§ 6.3 Component Model (2)

M

E M

E

BmRa

Kb

Ki

+

-

+

-a

1

L sm

1

J s1

s

m(s)ai m(s)

Electronics Mechatronics Mechanics

tdT (s) Load disturbance

aV (s) -

K

i1

Td

(s)+-

mTm

Va(s) 1Ra

1sJms+Bm

1N

m

(s)

(s)(s)

§ 6.3 Component Model (3)

• Gear Train

(3) Simplified Servo Motor

(with gear train)

t im a

a

KT (t) v (t)

R t t

m m m m m dJ B T (t) T (t)

m(s) 1N

(s)

(Motor) (Load)

1

m

(t)

(t) N

mT (s)N

T (s)

(Motor) (Load)

t

tm

T (t)N

T (t)

§ 6.3 Component Model (4)

• Tachometer:

Θ(s) V(s)sKv

Ω(s) V(s)Kvor

vK θ(t) = v(t)

• Potentiometer:

Θ(s) V(s)Ks

s

v(t)K

(t)

§ 6.4 System Control (1)

• Plant Dynamics:

(1) Dynamics

m mI C

0

u(t) w(t)

Servo Motor Command Disturbance

(2) States m

m

Position

Speed

§ 6.4 System Control (2)

(1) Position feedback

Control law: 0 c mu(t) u (t) a ,a

By potentiometer

m m m cI C a u (t) w(t)

F.B. increases stiffness

• Closed-loop Control:

Position feedback is to increase response speed.

Closed-loop:

(2) Velocity feedback

Control law: 0 mcu(t) u (t) b , b

By tachometer

m m cI (c b) u (t) w(t)

F.B. increases damping

Velocity feedback is to decrease response overshoot.

Closed-loop:

§ 6.4 System Control (3)

(3) Position and Velocity feedback

Control law: 0 m mcu(t) u (t) a b , a,b

By tachometerBy potentiometer

,m m

m m m cI (c b) a u (t) w(t)

Optimal control of fast speed and minimum overshoot by optimal design of a and b, respectively.

Optimal control law is realized by state equations with states feedback in modern control theory.

Closed-loop:

Control response speedControl overshoot

Feedback States feedback

§ 6.4 System Control (4)

Ki1

m mJ s BPI

Td(s)

m(s)+ +

-

-(s) IR(s)

Ki

Kb

GCI(s)1

m mJ s B

1

a aL s R

IR(s) Ia(s)

Td(s)

m(s)

+

-+ +

-

-

• Speed Control with Current Loop:

p i

CIi

K (1 Ts)For G (s) , PI control

TspK 1

§ 6.4 System Control (5)

ZOH A/DDigital

computationD/AHold

G(s)

H(s)

Command

ContinuousSys.

Sample & Hold Number Seq. + Algorithm

+

-

• Digital Control Configuration:

(1) System structure

§ 6.4 System Control (6)

wait for clockinterrupt

read analoginput

compute control singal

set analog output

update controllervariable

tsampling time, h

(2) Procedure in real-time sampling and control

§ 6.4 System Control (7)

(3) Discrete PID

§ 6.4 System Control (8)

• Continuous form

t

P I D0

de(t)u(t) =K e(t)+K e(τ)dτ +K

dt

• Discretization

• Discrete form

e(t) e(kh),

0

t

n

i=0

e(τ)dτ h e(i)

de(t) e(k) - e(k -1)

dt h

n

P I Di=0

e(k) - e(k -1)u(k) =K e(k)+K h e(i)+K

h

e(kh) is represented as e(k), h is sampling time