PERSONALIZED EXPERIMENTATION IN CLASSICAL CONTROLS WITH MATLAB REAL TIME WINDOWS TARGET AND PORTABLE AEROPENDULUM KIT

Size: px

Start display at page:

Download "PERSONALIZED EXPERIMENTATION IN CLASSICAL CONTROLS WITH MATLAB REAL TIME WINDOWS TARGET AND PORTABLE AEROPENDULUM KIT"

Opal Marion Bates
5 years ago
Views:

1 Eniko T. Enikov, University of Arizona Estelle Eke, California State University Sacramento PERSONALIZED EXPERIMENTATION IN CLASSICAL CONTROLS WITH MATLAB REAL TIME WINDOWS TARGET AND PORTABLE AEROPENDULUM KIT

2 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

3 Motivation Develop an portable lowcost 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.

4 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

5 Data Flow Diagram

6 Real Time Windows Target Environment

7 Modeling Tasks

8 Experiment 1: Parameter Extraction Using the steady-state response, find the parameters K/mg

9 Challenge 1: Dealing with Dead Zone

10 Challenge II: Feedback Linearization Result Type 1 System

11 Experiment II: Weightless Pendulum (K=0)

12 Experiment III: Parameter Identification (Kp=1)

13 Challenge III: Stability and Root Locus. What is wrong? Kp>3 (unstable)

14 Model Correction: Motor Dynamics

15 Experiment IV: Controller Design Using Bode Plots

16 Evaluation

17 Hands-On Activities Open Loop Response

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

19 Gather Data u_ss=[ ]; theta_ss=[ ]; sine_ss=sind(theta_ss); plot(sine_ss,u_ss) ylabel('pwm Input') xlabel('sin theta_{ss}')

20 Steady State Data PWM Input SIN theta ss

21 Fit a Line on Points 3-13 P=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))) shg legend('experiment', 'Linear fit')

22 Slope and Offset P = Experiment Linear fit X: 0 Y:

23 Project Installment # 2 Update Model mg S K u u0=24 Slope=92.5

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

25 Check System Type using Kp=1 Theta=30

26 plot(t,theta) degrees error (linearization Is not perfect)

27 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 K p to the point when you start getting an overshoot.

Using Kp=2 50 40 30 20 10 0 Tr=0.4 sec=>wn=4.5 PO=(48-32)/32=50% K/mL=4.5^2/Kp=10 c/ml^2=2*dzeta*wn/kp=0.

28 Using Kp= Tr=0.4 sec=>wn=4.5 PO=(48-32)/32=50% K/mL=4.5^2/Kp=10 c/ml^2=2*dzeta*wn/kp= sec Matlab for Damping > zeta=fzero(@(x)exp(-pi*x/sqrt(1-x^2))-0.5,0) zeta=0.22

System Identified g=tf(10,[1 0.97 0]) 10 +0.97 0.4 Root Locus 0.

29 System Identified g=tf(10,[ ]) Root Locus 0.3 Imaginary Axis (seconds -1 ) Real Axis (seconds -1 )

30 Test Stability with Kp=1, 2, 3 Kp=2.4 ->stability limit

31 Modify The Model Kp ( s s) 1 s 3 Root Locus g=tf(10,conv([ ],[1 4.5])) rlocus(g) Imaginary Axis (seconds -1 ) System: g Gain: 2.3 Pole: i Damping: Overshoot (%): 98 Frequency (rad/s): Real Axis (seconds -1 )

32 Adjust for Steady State Error Kp 10 2 ( s 0.97s) s Kp 10 2 ( s 0.97s b) s 1 e ss 10 1 b b

33 Lag Compensator s z s p 10 2 ( s 0.97s 2) s Imaginary Axis (seconds -1 ) s 5 s 1 Root Locus System: untitled1 Gain: 0.46 Pole: i Damping: Overshoot (%): 117 Frequency (rad/s): 1.81 Imaginary Axis (seconds -1 ) s 0.5 s 0.1 Root Locus System: untitled1 Gain: 1.34 Pole: i Damping: Overshoot (%): 88 Frequency (rad/s): Real Axis (seconds -1 ) Real Axis (seconds -1 )

34 Step Response of Lag Compensation X: Y: 30.8 c=tf([1.5],[1.1]) clagd=c2d(c,0.01)

35 Prof. Eke s Slides on Implementation

36 Optional Activities Lead Compensator Lead-Lag Compensator On/Off Controller

Readings: FC: p : lead compensation. 9/9/2011 Classical Control 1

Readings: FC: p : lead compensation. 9/9/2011 Classical Control 1 MM0 Frequency Response Design Readings: FC: p389-407: lead compensation 9/9/20 Classical Control What Have We Talked about in MM9? Control design based on Bode plot Stability margins (Gain margin and phase