1. State Space Control of a Magnetic Suspension System Margaret Glavin Supervisor: Prof. Gerard Hurley

2. Introduction Proportional and Derivative Control PWM Control State Space Control Applications of the Suspension System

3. State Space Background Developed since 1960’s Modern control theory State variable method of describing differential equations Not one unique set of state variables to describe the state space of the system

4. State Equations dx/dt = Ax + Bu y = Cx + Du A – State Matrix B – Input Matrix C – Output Matrix D – Direct transmission Matrix

5. Block Diagram B 1/sC D A u + + + + x’x y

6. Steps for State Space Design State Matrices Controllability and feedback gain Observability and observer gain Combine both Introducing reference input

7. Equations Differential equation for system Transfer function

8. Methods to Calculate Space State Matrices from Transfer Functions Canonical forms Controllable canonical form Observable canonical form Jordan canonical form Modal canonical form Diagonal canonical form MatLab

9. State Space Matrices

10. Controllability Controllability matrix Matrix rank is n or n linearly independent column vectors If determinant is non zero system is controllable

11. Feedback Gain Matrix Used to place the poles If controllable poles placed at any location Methods to calculate matrix Direct substitution method Transformation matrix Ackermann’s formula

12. Reference Input K matrix calculated with input set to zero Kc input gain Overcomes steady state error Kc=(1/(C*(-1/(A-Bk))*B)

13. Observer State variables not always available Observer designed to estimate the state variables Full state observer Reduced state observer

14. Observability Observability matrix Matrix rank is n or has n linearly independent column vectors Determinant is a non zero value

15. Observer Gain Used to place the observer poles Poles two to five times faster than controller poles Same methods of calculation used as for feedback gain matrix

16. Simulink Part of the Matlab Program Used to draw and simulate block diagrams Graphs at different points in the system can be plotted

18. PSpice MicroSim Corporation Designing and simulating circuits Schematic capture or netlist Libraries Modelling transfer function Saves time and money

20. Hardware Building circuit Testing circuit Fault finding Part of circuit already built

21. Applications MagLev train Floats above guide way Two types Reach speeds of 310 mph (500 kph) Frictionless bearings

22. Questions