1. Sliding Mode Control of a Non-Collocated Flexible System
Aimee Beargie
November 13, 2002
Committee
Dr. Wayne Book, Advisor
Dr. Nader Sadegh
Dr. Stephen Dickerson
Sponsor
CAMotion, Inc.

2. Problem Statement
Develop an algorithm to control the tip position of a mechanism that is actuated at the base (non-collocated problem)
Recently developed algorithms generally deal with collocated problems
Sensors: Encoder, Accelerometer, Machine Vision
State Feedback Control
Kalman Filter
Robust to parameter variations

3. Variable Structure Control Research
Model using Assumed Mode Method
Qian & Ma – Tracking Control
Chang & Chen – Force Control
Comparison to other Methods
Hisseine & Lohmann – Singular Perturbation
Chen & Zhai – Pole Placement
Robustness
Iordanou & Surgenor – using inverted pendulum
Combined with Other methods
Romano, Agrawal, & Bernelli-Zazzera – Input Shaping
Li, Samali, & Ha – Fuzzy Logic

4. System Model

5. System Model
Equations of Motion
Small Angle Approximation

6. System Model

7. System Model System Parameters: m1 = 8 kg m2 = 2.55 kg L = 0.526 m
r = m
I = kg-m2
k = 32,199 N-m
b = N-m-s

8. Variable Structure Control (VSC)
Also called Sliding Mode Control
Switched feedback control method that drives a system trajectory to a specified sliding surface in the state space.
Two Part Design Process
Sliding Surface (s) ® desired dynamics
Controller ® Lyapunov analysis

9. VSC: Sliding Surface Design
Regular Form
Dynamics of state feedback structure

10. VSC: Sliding Surface Design
Transformation to Regular Form

11. VSC: Control Design Use Lyapunov stability theory
Positive Definite Lyapunov Function
Want Derivative to be Negative Definite for Stability

12. VSC: Control Design
Control Structure
Resulting Equation

13. VSC: Generalizing Gain Calculation

14. Control System Overview
Desired Trajectory
System Dynamics
Control Algorithm
RASID
Motor & Amp
Kalman Filter
Encoder Meas.
Accelerometer Meas.
Vision Meas.
1kHz
RASID: internal PID 10kHz

15. Outer Loop Simulation Used LQR for Sliding Surface Design
Error used in Control Calculation

16. Outer Loop Simulation
Max error: 0.015mm

17. Inner Loop Simulation Force converted into Position Signal
PD Equations
Discrete Position Calculation

18. Inner Loop Simulation
Max error: 0.02mm

19. Simulation using Estimated States
Developed by Mashner
Vision
Measurement Frequency of 30 Hz
Delay of 5 ms
Covariance
Accelerometer: std. deviation squared
Vision/Encoder:

20. Simulation using Estimated States
Max error: 0.2mm

21. Simulation: Penalty on xtip and vbase
Max error: 0.5mm

22. Robustness Simulation: 50% of mtip
Max error: 0.3mm

23. Robustness Simulation: 110% of mtip
Max error: 0.5mm

24. Experimental Set-up

25. Experimental Results: VSC w/ Kalman Filter
MSE = e-6 m2

26. Experimental Results: Robustness
Mean Squared Error
0%: e-6 m2
10%: e-6 m2
16%: e-6 m2

27. Experimental Results: Comparison of Control Methods
Mean Squared Error
PD: e-7 m2
LQR: e-5 m2
VSC: e-6 m2

28. Conclusions
Developed method results in acceptable tracking of tip position
Verified through simulations and experiments
Method generalized for LTI systems
Better performance than other control methods
Robust to parameter variations
Choice of Cost function critical
Verified experimentally for tip mass

29. Further Work Desired Trajectory Adaptive Learning Input Shaping
Currently designed for rigid system
Possible use trajectory that is continuous in fourth derivative
Adaptive Learning
Input Shaping

30. JLKJLKJLKJLKJLKJLKJLKJLKJLKJLKJLKJLKJLKJLKJLKJLK
QUESTIONS
JLKJLKJLKJLKJLKJLKJLKJLKJLKJLKJLKJLKJLKJLKJLKJLK ?