Matlab, Simulink - Using Simulink and Stateflow in Automotive Applications
Enhancing Linear System Theory with an Pendulum...
Transcript of Enhancing Linear System Theory with an Pendulum...
Enhancing Linear System Theory Curriculum with an Inverted
Pendulum Robot
Brian HowardGeneral Electric
Dr. Linda BushnellUniversity of Washington, EE
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• LST of today similar to that of 50 years ago.• Applications, though, are so much broader:
– Driverless cars– Financial models– Electric grid– Drones
• Provide a platform and environment that lets students apply and extend theory.
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Why enhance LST?
• Lecture component includes:– Block diagrams, modelling, and system properties– Causal and LTI systems– State space and transfer function relationships– Lyapunov and BIBO stability– Controllability and observability– Closed loop control, estimators, and observers
• Class (PMP) was enhanced by adding:– Matlab/Simulink projects to practice the theory– Work on the robot to apply control theory to a specific platform– Balancing of robot was bonus for the course.
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LST Structure
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Physical system (“Plant”) Model
MinSeg model development
Improved engagement:• Parameters can be hard to measure. • This naturally leads to questions about accuracy, uncertainty,
and how the parameters impact controller performance.
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MinSeg Model
α
, α
∗ 0. 1. 0. 0.62.1 44.6 0. 2120. 0. 0. 1.6.10 10.2 0. 485
090.0020.6
1 0 0 00 1 0 00 0 1 00 0 0 1
0000
State-space
Model Behavior
• Eigenvalues (Unstable):
• Controllability (System is controllable):
• Observability (System is observable).
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rank rank0.00 90.0 47700 2.53 1090.0 47700 2.53 10 1.34 100.00 20.6 10900 5.77 1020.6 10900 5.77 10 3.06 10
4,
rank rank
1.00 0.00 0.00 0.000.00 1.00 0.00 0.00… … … …
1.90x10 2.86 10 0.00 1.36 10
4,
LQR Feedback Controller• An LQR (Linear‐Quadratic Regulator) controller applies gains to all the state variables:
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MinSeg with LQR impulse responseLQR Feedback Controller
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, 93.0864 18.0634 9.93776 52.6578
Feedback Controller DevelopmentP Controller
• Begin with proportional, “P” feedback. Block diagram and modified transfer function:
• With Matlab a gain of is found.
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PI Feedback Controller• Add integral, “I” control:
• Gain of and .
• This is a problem: Excessive control effort.
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PI controller impulse response
• Lowered required response time and found more realistic gains:
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Pend
ulum
ang
le, α
Slow ripple
About those wheels...PID Feedback Controller
• Added differential gain, , to eliminate ripple and wheels need controller:
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PID + P controller impulse response
• Found and ,adjusted , and added gain to achieve desired response:
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Pend
ulum
ang
le, α
Summary
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• Class– Enhanced engagement– Robot improved understanding of concepts (pole placement, observers)
• Lab– Expanded learning (complimentary filter design)– Introduced additional depth to control theory (Motor controls and ultrasonic distance sensor)
Conclusion
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• Successful experiment!
• Next steps:– Explore Kalman filtering–Apply lab structure to other topics