a seeps w 2012 slides

PERSONALIZED EXPERIMENTATION IN

CLASSICAL CONTROLS WITH MATLAB

REAL‐TIME WINDOWS TARGET AND

PORTABLE AEROPENDULUM KIT

Eniko T. Enikov, University of ArizonaEstelle Eke, California State University Sacramento

www.aeropendulum.arizona.edu

Outline Motivation The Aeropendulum Apparatus Real-Time vs. Soft Real Time Student Design Activities◦ Plant modeling, ◦ parameter identification, identification of non-linearities◦ Feedback linearization, steady-state error and system types◦ parameter identification ◦ Matlab's pem() prediction-error minimization function (time

permitting)◦ closed-loop control experiments: proportional, phase lag,◦ phase lead and on/off (bang-bang) control (time permitting)

Results from implementation at CSUS and Univ. of Arizona

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Motivation

Develop an portable low-cost apparatus that illustrates classical control systems course with a hands-on experimentation.

Eliminate the need for lab space, teaching assistant.

Provide a quick pathway from controller design to implementation for mechanical engineering students.

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Experimental Apparatus Acrylic stand Pendulum with angle

sensing potentiometer, DC-motor and propeller

Target circuit board driving the propeller with different PWM ratios in forward and reverse direction

MATLAB Simulink Real Time Windows Target GUI for controller implementation

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Data Flow Diagram

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Real Time Windows Target Environment

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Modeling Tasks

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Experiment 1: Parameter Extraction

Using the steady-state response, find the parameters K/mg

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Challenge 1: Dealing with Dead Zone

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Challenge II: Feedback Linearization

Result Type 1 System

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Experiment II: Weightless Pendulum (K=0)

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Experiment III: Parameter Identification (Kp=1)

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Challenge III: Stability and Root Locus. What is wrong?

Kp>3 (unstable)

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Model Correction: Motor Dynamics

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Experiment IV: Controller Design Using Bode Plots

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Evaluation

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Hands-On Activities Open Loop Response

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plot(t,theta,t,PWM); grid minor

0 10 20 30 40 50 60-20

0

20

40

60

80

100

120

PWM

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Gather Data

u_ss=[0 10 20 30 40 50 60 70 80 90 100 110 120]; theta_ss=[0 0 0 0 8 16 24 32 40 48 56 68 73]; sine_ss=sind(theta_ss); plot(sine_ss,u_ss) ylabel('PWM Input') xlabel('SIN theta_{ss}')

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Steady State Data

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

PW

M In

put

SIN thetass

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Fit a Line on Points 3-13P=polyfit(sine_ss(3:13),u_ss(3:13),1) ;plot(sine_ss,u_ss,sine_ss(3:13), polyval(P,sine_ss(3:13))) shglegend('Experiment', 'Linear fit')

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Slope and Offset P = 92.4879 24.1771

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

X: 0Y: 24.18

ExperimentLinear fit

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Project Installment # 2 Update Model

24

5.92

0

uK

mgS

u0=24Slope=92.5

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Test Using Closed Loop with Zero Gain (Slope needs adjustment, c. a 80)

0 5 10 15 20 25 30-40

-20

0

20

40

60

80

100

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Check System Type using Kp=1Theta=30

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plot(t,theta)

0 2 4 6 8 10 12-5

0

5

10

15

20

25

30

35

40

4-5 degrees error(linearizationIs not perfect)

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Identify Parameters From the response extract approximate values

for and , then calculate and . (Use formulas for overshoot, peak time, rise time etc. to find and and then related to the physical parameters).

The plot achieved for proportional controller may produce an over-damped relation in which case you will not be able to find out the parameters by using the above formulas.

Just try increasing the proportional gain Kp to the point when you start getting an overshoot.

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Using Kp=2

0 1 2 3 4 5 6 7 8 9-10

0

10

20

30

40

50

sec

Tr=0.4 sec=>wn=4.5PO=(48-32)/32=50%K/mL=4.5^2/Kp=10c/mL^2=2*dzeta*wn/Kp=0.97

Matlab for Damping > zeta=fzero(@(x)exp(-pi*x/sqrt(1-x^2))-0.5,0)zeta=0.22

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System Identified

g=tf(10,[1 0.97 0])

10

+0.97

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4Root Locus

Real Axis (seconds-1)

Imag

inar

y A

xis

(sec

onds

-1)

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Test Stability with Kp=1, 2, 3…

Kp=2.4 ->stability limit

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-10

0

10

20

30

40

50

60

Modify The Model

sss 1)97.0(10

2Kp

g=tf(10,conv([1 0.97 0],[1 4.5]))rlocus(g)

-7 -6 -5 -4 -3 -2 -1 0 1-3

-2

-1

0

1

2

3

System: gGain: 2.3Pole: -0.0129 + 2.06iDamping: 0.00627Overshoot (%): 98Frequency (rad/s): 2.06

Root Locus

Real Axis (seconds-1)

Imag

inar

y A

xis

(sec

onds

-1)

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Adjust for Steady State Error

sss 5.41)97.0(10

2 Kp

sbss 5.41)97.0(10

2 Kp

2

305

101

1

bb

ess

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Lag Compensator

-12 -10 -8 -6 -4 -2 0 2 4-10

-5

0

5

10System: untitled1Gain: 1.34Pole: -0.0834 + 2.05iDamping: 0.0407Overshoot (%): 88Frequency (rad/s): 2.0

Root Locus

Real Axis (seconds-1)

Imag

inar

y A

xis

(sec

onds

-1)

sss 5.41)297.0(10

2 pszs

1.05.0

ss

-15 -10 -5 0 5 10-15

-10

-5

0

5

10

15

System: untitled1Gain: 0.46Pole: 0.092 + 1.81iDamping: -0.0509Overshoot (%): 117Frequency (rad/s): 1.81

Root Locus

Real Axis (seconds-1)

Imag

inar

y A

xis

(sec

onds

-1)

15

ss

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Step Response of Lag Compensation

0 5 10 15 20-5

0

5

10

15

20

25

30

35

40

45

X: 15.39Y: 30.8

c=tf([1 .5],[1 .1])

clagd=c2d(c,0.01)

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Prof. Eke’s Slides on Implementation

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Optional Activities

Lead Compensator Lead-Lag Compensator On/Off Controller