Digital Control Systems

Design via Frequency Response Technique

Controller Types

• Lead compensator

when is very small, can be approximated as PD control

• Lag compensator

For large , can be approximated as PI control

• Lead-Lag compensator

Can be approximated as PID control

11

1( ) , 0<

wC w

wK

1( ) , 1

1ww

wC K

Lead Compensator Transfer function:

Frequency response:

If is given, pole and zero of is obtained as

The gain will be determined from the specification

11

1( ) , 0<

wC w

wK

max max, ( )C w

max

max

1 sin

1 sin

max

1

KssE

1 1 1

20log 20log(1/ )K

20log K

Lead Characteristic

Comparison:

Key Idea: The lead compensator will enlarge the gain crossover

frequency and increase the PM. However, the GM can not be designed from

the lead compensator

maxnew oldPM PM

max (new)gc

Design Example (1)

Design a lead compensator for the digital control system below

so that the PM is 50, the GM is at least 10 DB

and the static velocity error

Sol. Obtain the discrete-time plant (by Matlab or by hand)

C(z) ZOH

-

( )R s

+ 0.2T

( )Y s

( 1)

K

s s

2vK

1 1

2{ { }}

( 1( ) )

)(1G z

K

s sz

Z L

2

(0.01873 0.01752)

1.8187 0.8187

K z

z

Design Example (2)

By using the bilinear mapping, we obtain

The controller is in the form of

1 ( /2) 1 0.1

1 ( /2) 1 0.1

( ) ( ) |w w T wz

w T w

G w G z

2

2

( 0.000333 0.09633 0.9966)

0.9969

K w w

w w

11(

1) , 0<w

wC

ww

Design Example (3) Design technique:

Step 1 Compute the gain that satisfies the required

Step 2 Set , find Bode plot of

Matlab command: bode(2*Gw)

Step 3 Determine PM from the Bode diagram

Here, we get PM=30 (approximately)

Matlab command: [Gm,Pm,Wcg,Wcp] = margin(2*Gw)

vK

0lim ( ) ( ) 2wvw

wK wC w G w K

2K ( )wG w

2

2

2( 0.000333 0.09633 0.9966)( )

0.9969w

w wG w

w w

Design Example (4)

Design Example (5)

Step 4 Estimate required phase lead

Step 5 Compute the lead factor

Step 6 Find the new “gain cross over freq.” from

By reading the Bode diagram, we get

max new oldPM PM

30 8 25 80

(1 sin 28 ) / (1 sin 28 ) 0.361

| ( ) | 10log( ) dB 4.4251/ dBw gcG j

1.7 rad/sgc

Design Example (6)

Step 7 Get corner frequencies for zero and pole

The lead compensator is

Step 8 Verify margins from the Bode plot of

Step 9 If everything is OK, obtain the controller in z-plane

1 10.9790

1.7 0.361

0.3534

gc

0.9790 1(

1

1)

0.3534 1w

wC w

w

w

w

( ) ( )w wC w G w

2 1

1

2.3798 1.9387( ) ( ) |

0.5589w z

wT z

zC z C w

z

Design Example (7)

Discussion (Lead Comp.)

Advantage

• Improve phase margin

• Improve high-frequency performance

• Improve the speed of the response

Disadvantage

• May have effects from high-frequency noise

• Generate large signals which may damage the system

Lag Compensator Transfer function:

Frequency response:

The gain will be determined from the specification

1( ) , 1

1ww

wC K

KssE

20log K

20log 20logK

1

1

Lag Characteristic

Comparison:

Key Idea: The lag compensator will

reduce the gain crossover frequency to where the phase margin is satisfied

The zero corner frequency is set to 1 decade

below the new gain cross over frequency

new oldPhase Phase

5 -10 deg.

1/

Design Procedure (Lag Comp.)

1. Determine the gain to satisfy the requirement on

2. Find the new gain cross-over frequency such that

3. Choose the corner frequency one decade below

4. Magnitude reduction from the lag comp. at is equal to , then is determined from

5. Obtain the lag comp.

6. If all the requirements are satisfied,

KssE

gc

gc desiredPM PM

1

gc

gc

20log

20log | ( 20log) |gcKG j

1(

1)

wK

wC w

2 1

1

( ) ( ) | zw

T z

wC z C w

Discussion (Lag Comp.)

Advantage

• Low-frequency characteristics is improved or maintained

• Stability margins are improved

• BW is reduced reduce effects from high-freq. noise

Disadvantage

• BW is reduced slower rise time

• Numerical problems with controller coefficients may result in bad control performance

Lag-Lead Compensator

Objective : Cascade a phase-lag compensator with a phase-

lead compensator to change the overall system characteristics

lead sectionLag section

1 1

1 1

w wC w K K

w w

Lag compensator :

increase the low-frequency gain

Lead compensator :

increase BW and stability margin

Approximation of a PID controller

Reading Materials

K. Ogata, “Discrete-time Control Systems” , Chapter 4,

pp. 225-242 . See also problems A-4-10, A-4-11, and A-4-12