outline introduction reaction wheels modelling control system real time issues questions conclusions
TRANSCRIPT
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Outline
• Introduction
• Reaction Wheels
• Modelling
• Control System
• Real Time Issues
• Questions
• Conclusions
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The Plant
• Pendulum
• Reaction Wheel
• Motor
• Encoders
• TI Digital Signal Processor
• PWM Motor Driver
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Reaction Wheel
• Wheel acceleration by torque from motor
• Torque on motor from wheel inertia
• Torque is transferred to the whole pendulum
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• Satellite adjustment
• Motorcycle mid-jumpcorrection
Applications of Reaction Wheels
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• Three states– – –
• Model derived by laws of physics and measurements
Model Derivation
rr
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Validating the Model
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Hybrid Automaton
• Two discrete states– Swinging State– Balancing State
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Swing Up Controller
• Bang-bang energy control
• Energy of pendulum• Wtotal = Wpotential + Wkinetic
• Reference value is the potential energy at the upright position
• The pendulum will reach the catch angle with the right amount of speed
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Two Approaches of Controller Design
1. Design in Continuous Time
2. Design in Discrete Time
Continuous time Plant
Discretized Plant
Continuous time
Controller
Discrete Controller
h
h
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1. Design in Continuous Time• Design of a State Feedback Controller
• Investigate PD controller:
kkv
v
ry
Controller)(yCv
ProcessBvAxx
Cxy
Trx )(sF)(sC
)(sP ,r
,,r
State observerState Feedback
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1. Analysis of the Root Locus• Root locus : closed-loop pole
trajectories as a function of
15100 v
),( kk
)(sF
kk
)(sP
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1. A Stable Closed-loop System
necessity of a feedback on r
)(sP
rkkk )(sF
rv 1.060400
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1. Sampling of the controller
• Discrete transformation of the derivatives in using backward difference
• Filtering of the velocities and
First order low pass filter
h
zs
1
)(sF
)(zF)(zC
)(sP ,r
,,r
State observerState Feedback
hold sample
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1. Performance of the PD-controller
• Higher overshoot in reality(Nonlinearities such as dry friction)
• High rising time (>1.5s)
• Open loop plant has 3 poles : 8.82, -8,72, 7.64
• 2 turns around -1 stable closed loop
dBGm 93.3
1.41m
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1. LQ-controller
• How to choose for optimal results?
Computed from the continuous plant state matrices
With , and
gives optimal solution
kkk ,,
0
2
),0[:)2(min dtNvxRvQxx TT
IRv
BvAxx
Lxv
78.9278 688.1082 0.1311 v
000
010
001
Q 100R 0N
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1. Performance of the LQ-controller
dBGm 6
60m
• No overshoot.
• Phase margin 60 degrees
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1. Performance of the LQ-controller
Demo of the continuous LQ…
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2. Design in Discrete Time
• Plant is sampled with a zero-order hold approximation.
• LQ controller derived with the discrete plant state matrices :
• with
• Gives optimum solution for any sampling period h :
)(zC
)(zF)(zP ,r ,,r
State observer
Discrete Plant
(h)k (h)k (h)k v
Ahe dsBeh
As0
dIRN
NQ
IRN
NQT
h
T
T
dTd
dd
0
)()(
)(
0)(
0
])[][2][][][(min 2
1),0[:
nNvnxnvRnxQnx Tdd
n
T
IRv
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2. Performance of the Discrete LQ-controller
Demo of continuous and discrete LQ…
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2. Deadbeat Control
• Use a state feedback
• The strategy: drive the state into the origin in at most 3 steps
• Possible if
• Cayley-Hamilton theorem states that if the desired closed loop poles are put at the origin,
0)0()3( 3 xx c
)()()()1( kxkxLkx c
03 c
0)()( 33 ccpzzp
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2. Performance of the Deadbeat Controller
Demo of the Deadbeat Controller…
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Embedded behaviour
• CPU time 2% no pb with deadlines not met
• Sampling frequency VS control performance
• Maximum sampling period h=150 ms according to rising time of motor
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Conclusion
• Energy controller for swinging up the pendulum gives good results.
• Continuous LQ works fine with high sampling frequency
• For lower sampling frequencies, discrete design of controller needed.
• Deadbeat controller does not work because of voltage limitations
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Questions?
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