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PID Compensator Design

Using Root Locus Methods

ELEC 312

Objectives

Improve transient response & steady-

state error

Reshaping the root locus of an

uncompensated system to include a

specified point s = s1 (usually complex) in

the s-plane as a closed-loop pole

Assumption: s1 is a dominant closed-loop

pole such that time-domain transient

specifications (% overshoot & settling

time) and steady-state error are satisfied

2

Compensation Configurations

Cascade

Parallel or feedback

o An example of improving transient response by

decreasing the response time (peak time)

while keeping %OS the same.

o Proportional action alone will not suffice.

3

A system with desired transient response defined by

the dominant pole at A, but with unacceptable ess.

Compensating the system with an ideal integrator to

increase system type and drive ess to 0 resulting in

driving the dominant pole at A off the root locus.

4

Adding an OL zero close to the OL pole at the origin

reshapes the root locus to approximately pass through

point A to achieve desired transient response as before.

Proportional Controller

Its output directly proportional to the error

signal (kp =proportional gain)

Increase oscillation

pi

oc k

RRRR

sVsV

sG 13

24

)(

)()(

5

Integral Controller

Its output directly proportional to the integral

of the error signal (ki = integral gain)

Drive steady-state error for step input to 0

Slow to respond and the system may become

unstable for plant with complex poles close

to the j-axis

sk

CsRRR

sVsV

sG

i

i

oc

213

4 1

)(

)()(

Derivative Controller

Its output directly proportional to the slope

or derivative of error (kd = derivative gain)

Anticipate the direction of the error, and thus

fast to respond

Very difficult to compute derivatives for real

systems seldom used alone

Use in combination with lowpass filter to

smooth the signal and filter out noise before

the derivative is computed

sksG dc )(

6

Function: Improve steady-state error

Compensator: Proportional + Integral (PI)

Transfer function:

Characteristics:

1. Increases system type

2. Error becomes zero

3. Zero at –zc is small and negative

4. Active circuits are required to implement

s

kk

s

kszs

KsGp

i

pc

c )(

7

Control Action: Proportional+Integral (PI)

22

22

1

2

3

4 1

)(

)()(

CsRCsR

RR

RR

sVsV

sGi

oc

Function: Improve transient response

Compensator: Proportional + Derivative (PD)

Transfer function:

Characteristics:

1. Zero at –zc is close to and to the left of the pole

2. Active circuits are required to implement

3. Can cause noise and saturation; implement

with rate feedback or with a pole (lead)

d

pdcc k

kskzsKsG )()(

8

Control Action: Proportional + Derivative (PD)

)1()(

)()( 11

1

2

3

4 CsRR

R

R

R

sVsV

sGi

oc

Function: Improve steady-state error and

transient response

Compensator: Proportional+Integral+Derivative (PID)

Transfer function:

Characteristics:

1. Lag zero at –zlag and pole at the origin improve

steady-state error.

2. Lead zero at –zlead improve transient response

3. Lag zero at –zlag is close to and to the left of origin

4. Lead zero at –zlead is selected to put design point

on root locus.

5. Active circuits are required to implement

6. Can cause noise and saturation; implement

with rate feedback or with additional pole

s

zszsKsGc

))(()(

leadlag

9

s

kk

sk

ksk

s

zszsKsG d

i

d

pd

c

2

leadlag ))(()(

Control Action: Proportional+Integral+Derivative (PID)

22

2211

1

2

3

4 )1)(1(

)(

)()(

CsRCsRCsR

RR

RR

sVsV

sGi

oc

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PID Design Steps

Evaluate performance of the uncompensated system to determine how much improvement is required

Design the PD controller to meet transient response specifications

Simulate the resulting system. Redesign if necessary

Design the PI controller to yield the required steady-state error

Determine values of the three gains

Simulate the resulting system. Redesign if necessary

Example 9.5

Design a PID controller such that the given

system can operate with a peak time that is

two-thirds that of the uncompensated

system at 20% overshoot and zero steady-

state error for a step input.

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