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?