Root Locus Design Outline Block Diagram PI Control

Outline • PI control system design. • PD control system design. • PID control system design.

Root Locus Design M. Sami Fadali Professor of Electrical Engineering University of Nevada

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PI Control

Block Diagram  R(s)

E(s)

Compensator



• Add integral control to improve steady-state error, (increases type number one)  worse transient response or instability. • Add a proportional control  controller has pole at origin and zero. • Cascade compensation: integral term in feedback path is equivalent to a differentiator in the forward path.

C(s) Plant

Cascade Compensation

R(s)





2

C(s) Plant

Compensator Feedback Compensation 3

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PI Design Procedure 1. Design a proportional controller to meet the transient response specifications, i.e. place the dominant closed-loop system poles at a desired location 2. Add a PI controller with a zero at /10 3. Tune the gain of the system to move the closed-loop pole closer to . 4. Check the time response and modify the design until it is acceptable.

Example: PI Design Design a controller for the system for percentage overshoot less than 5% and zero steady-state error due to step.

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Step 1: Proportional Control Design proportional control to meet transient response specifications.

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Root Locus for PI Example

Percentage overshoot less than 5%

c.l. characteristic polynomial Equate coefficients 7

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Step 2: Add PI Compensator

RL for PI Compensated System

• For this example, an easier design canceling the pole at –3 may be superior. 9

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Increase gain for faster response

Time Response for PI Example

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PD Design

PD Control • Can be used in cascade or feedback control. • Zero pulls RL to left: improves the transient response. • Transfer Function of PD Controller

• Obtain the desired dominant pair from the design specifications. • Angle of PD controller

j

scl

Compensator

d c

 n a

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PD Design Procedure

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PD Design Procedure (cont.) 3. Calculate the new loop gain function including the compensator then calculate the gain (magnitude condition) with a calculator or

1. Calculate   angle[ L( scl ) ] using a calculator or MATLAB » scl=-zeta*wn+j*wd; » thetac= piangle(evalfr(g, scl) ) 2. Calculate the zero location using

» L = g*tf( [1, a],1) » K = 1/abs( evalfr( L, scl) ) 4. Check the time response of the PD compensated system and modify the design to meet the desired specifications if necessary.

» a=wd/tan(thetac)+zeta*wn 15

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P-Control

Example

• Percentage overshoot less than 10%

• Design a P-controller for % OS < 10% • Design a PD controller to reduce to 1 s.

• Find the intersection of constant line with the root locus using the angle condition by trial and error. • The intersection is easy to find with MATLAB. 17

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PD Design

Root Locus for P-Control

• Reduce

to 1 s

• Find the c.l. pole location

• Angle of PD controller 19

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PD Controller

Modified PD Design

• Compensator zero

Loop gain

We need to speed up the time response. 1. Move zero closer to the j-axis. 2. Increase the gain.

.

• Calculate the gain

Simulation results show that the system is too slow 21

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Step Response K = 38.8

Compensated Root Locus

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PID Control

Step Response K = 70

zeros can be complex conjugate

Procedure 1- Design a PD controller to meet the transient response specifications. 2- Add a PI controller to meet the steady-state specifications without appreciably affecting the transient response. 25

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PID Design

Example: PID Design •Design a PID controller for the system to meet the PD specifications ( with zero steady-state error due to ramp.

• PD specifications met with earlier PD design • PI design

• Combine for PID .

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Root Locus with PID

PID Design with K = 38

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PID with K=70, zero at 0.2

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PD Design: Pole-Zero Cancellation • • • • •

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Cancel pole with PD zero. Do not cancel RHP poles. Do not cancel pole at origin. Cancel closest real pole to the j-axis. Imperfect implementation: almost cancel.

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RHP Pole

Example

• Zeros do not change the response due to the ICs. • Imperfect cancellation: RHP closed-loop pole. Example

For the closed-loop system with unity feedback, design a controller to reduce the percentage overshoot to less than 5% and the settling time to less than 1.5 second.

• Form of complete step response

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Step Response

Root Locus: Uncompensated

• Uncompensated closed-loop system

With 10 included in MATLAB transfer function

Step Response

1.4 System: untitled1 Peak amplitude: 1.39 Overshoot (%): 39.3 At time (sec): 2.69

Root Locus 8

1.2 6

System: g Gain: 1 Pole: -0.348 + 1.21i Damping: 0.276 Overshoot (%): 40.5 Frequency (rad/sec): 1.26

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System: untitled1 Settling Time (sec): 11.1

1

Amplitude

Imaginary Axis

4

0

0.8

0.6

>> step(feedback(g,1))

-2

0.4

-4 -6 -8 -8

0.2 -7

-6

-5

-4

-3

-2

-1

0

1

2

Real Axis

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0

0

2

4

6

8

10 Time (sec)

12

14

16

18

20

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Design

Equating Coefficients • Loop gain

• Cancel pole at 1 with a zero • C.l. characteristic polynomial • Choose K for

(gives