Control Designof

Positioning Servo Systems

Shuo Zhang

Application of positioning servo systems

Hardware architecture

Control design requirements

I As fast as possibleI Little/No overshootI Accuracy

Switching control strategy

I Large position error (before P ): saturation controlI Medium error: decelerate with the maximal acceleration +

feed-forwardI Small error (linear system): PID + feed-forward

Linear control design: oversee

I GW (s) = 400.0584s+1 , 1

is = 1400s , KDA = 25.5V/rad

I First: linear feedback designI Then: determine feedforward gains by tuning

Linear feedback design 1: PD

I Set KD = 0, KP low. Choose ωBW = 10rad/sI KP increases⇒ bandwidth increases but phase margin

decreases⇒ faster response but larger overshootI KD increases (not too much)⇒ eliminate overshootI KP = 4, KD = 0.15 (green: command, red:

uncompensated, blue: PD cascade control)

0 1 2 3 4 5 6 7 8 9 10−1.5

−1

−0.5

0

0.5

1

1.5

Linear feedback design 2: state feedback

I K = 1/0.0584, KG = 2.55K

I Desired control ratio: M(s) = A0KG(KDs+KP)[s2+(KGKD+K)s+KGKP](s+p)

,where A0 = p = 100

I Controller parameters: k1 = 1, a = KPKD

, A = A0KD,

k2 =p(KGKD+K−1)+KG(KP−KD)−AKGk1

(a−1)AKG= 0.1059,

b = KGKD + p−AKGk2 > a

Step response

red: uncompensated, blue: PD cascade, green: state feedback

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (seconds)

Am

plitu

de

Why state feedback?red: uncompensated, blue: PD cascade, green: state feedback

−120 −100 −80 −60 −40 −20 0 20−20

−15

−10

−5

0

5

10

15

20

Root Locus

Real Axis (seconds−1)

Imag

inar

y A

xis

(sec

onds

−1 )

I System output of state feedback with high forward gain isinsensitive to gain/parameter variations in the forward path

More design insights from Bode plots

red: uncompensated, blue: PD cascade, green: state feedback,black: PID cascade

I Tuning method: simple and effective (PD and PID)I Lack of integral: no overshoot but bad disturbance

rejection (PD and PID)

−150

−100

−50

0

50

100

Mag

nitu

de (

dB)

10−2

10−1

100

101

102

103

104

−180

−135

−90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Final control design strategy

I Determine KP and KD

I Tune velocity feedforward gain KVFI increase closed-loop bandwidth (faster response) but more

overshoot⇒ requires reduction of KP ⇒ bad fordisturbance rejection

I Tune acceleration feedforward gain KAFI eliminate overshoot without reducing KP

I Increase KI (until overshoot occurs)I State feedback design

Design resultsgreen: command, red: PID cascade without feedforwardblue: final strategy (step by step)KP = 4, KD = 0.15, KVF = KAF = 0

0 2 4 6 8 10 12

0

0.2

0.4

0.6

0.8

1

1.2

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

KP = 4, KD = 0.15, KVF = 2KD, KAF = 0

0 2 4 6 8 10 12−0.2

0

0.2

0.4

0.6

0.8

1

1.2

3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 50.85

0.9

0.95

1

1.05

Design results Ctd.

KP = 4, KD = 0.15, KVF = 2KD, KAF = 0.015

0 2 4 6 8 10 12−0.2

0

0.2

0.4

0.6

0.8

1

1.2

3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

KP = 4, KD = 0.15, KVF = 2KD, KAF = 0.015, KI = 0.15

0 2 4 6 8 10 12

0

0.2

0.4

0.6

0.8

1

1.2

2.5 3 3.5 4 4.5 5 5.5

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2