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Outline
• Introduction
• Reaction Wheels
• Modelling
• Control System
• Real Time Issues
• Questions
• Conclusions
The Plant
• Pendulum
• Reaction Wheel
• Motor
• Encoders
• TI Digital Signal Processor
• PWM Motor Driver
Reaction Wheel
• Wheel acceleration by torque from motor
• Torque on motor from wheel inertia
• Torque is transferred to the whole pendulum
• Satellite adjustment
• Motorcycle mid-jumpcorrection
Applications of Reaction Wheels
• Three states– – –
• Model derived by laws of physics and measurements
Model Derivation
rr
Validating the Model
Hybrid Automaton
• Two discrete states– Swinging State– Balancing State
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
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
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
1. Analysis of the Root Locus• Root locus : closed-loop pole
trajectories as a function of
15100 v
),( kk
)(sF
kk
)(sP
1. A Stable Closed-loop System
necessity of a feedback on r
)(sP
rkkk )(sF
rv 1.060400
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
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
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
1. Performance of the LQ-controller
dBGm 6
60m
• No overshoot.
• Phase margin 60 degrees
1. Performance of the LQ-controller
Demo of the continuous LQ…
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
2. Performance of the Discrete LQ-controller
Demo of continuous and discrete LQ…
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
2. Performance of the Deadbeat Controller
Demo of the Deadbeat Controller…
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
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
Questions?