Automatic Control II, 2016 Assignment One
Due: 5pm Friday week 7 (9 Sept 2016)
1 Magnetic levitation
Consider the classic magnetic levitation system of a coil supporting a steel ball (Figure 1). In such systems, the electromagnetic force 𝐹𝑒𝑚 is often described by the following empirical equation related to coil current 𝐼 and displacement 𝑥 (defined positive up):
𝑎𝑖
𝐹𝑒𝑚(𝑥, 𝑖) = 2 (1)
(𝑏 + 𝑥)
where 𝑎 and 𝑏 are constants dependent on parameters such as the ball permeability, the dimensions and properties of the coil, and so on. Also consider eddy current damping in the form
𝐹𝑒𝑑𝑑𝑦(𝑥,? 𝑖) = -h𝑥𝑖?2 . (2)
The negative sign indicates a force in the opposite direction to 𝐹𝑒𝑚.
Since the coil has electrical dynamics of its own, we cannot use current as an input into the overall system. Generally a control system will send out a control voltage. The dynamic equation for such a coil is
𝑑
𝑉 = 𝐿 𝑖 + 𝑅𝑖 + 𝐺𝑥? (3)
𝑑𝑡
where 𝐼 is current, 𝑉 is input voltage, and 𝑅, 𝐿 are resistance and inductance, respectively, and 𝐺𝑥? is back-EMF.
The magnetic levitation problem has the coil supporting the weight of the ball with a constant current, with a control system used to maintain the desired height below coil. The weight of the ball is chosen to match up with a specified equilibrium position 𝑥0 and nominal current 𝑖0. ?O

|
a??????? |? ?|
Figure 1: Magnetic levitation system. The coil creates an electromagnetic force 𝐹𝑒𝑚 on the steel ball, balancing the gravity force 𝐹𝑔. Since displacement is positive upwards, the ball as shown has a negative value of displacement 𝑥.
1. Explain intuitively why the system is unstable for constant current 𝑖 = 𝑖0. [marks: 1]
2. Write out the complete nonlinear dynamic equation (i.e., sum of forces) for the ball shown in
Figure 1. [marks: 2]
3. Hence, write the nonlinear dynamic equations for this system in standard form for states 𝑥, 𝑥?, and 𝑖. (Hint: one of the equations will be simply 𝑥? = 𝑥?.) [marks: 2]
4. Calculate a linear approximation 𝐹𝑒𝑚~ of equation (1) around the point 𝑥0. Express this in the form 𝐹𝑒𝑚~ (𝑥, 𝐼) = 𝑐0 + 𝑐1(𝑥 - 𝑥0) + 𝑐2(𝐼 - 𝐼0) where 𝑐0, 𝑐1, 𝑐2 are constants. Hint: linearise around two variables using: [marks: 4]
𝑓 (𝑥, 𝑦) ˜ 𝑓 (𝑥, 𝑦) = 𝑓 (𝑥~ 0, 𝑦0) + (𝑥 - 𝑥0) · 𝜕𝑓 | + (𝑦 - 𝑦0) · 𝜕𝑓 | (4)
𝜕𝑥 𝑥=𝑥0 𝜕𝑦 𝑥=𝑥0
𝑦=𝑦0 𝑦=𝑦0
4b. Linearise 𝐹𝑒𝑑𝑑𝑦(𝑥,? 𝑖) of equation (2). Express it in the form 𝐹𝑒𝑑𝑑𝑦~ (𝑥,? 𝑖) = 𝑐3 + 𝑐4(𝑥? - 𝑥?0) +
𝑐5(𝑖 - 𝑖0). [marks: 2]
5. Using parameters shown in Table 1, evaluate values for 𝑐0, 𝑐1, 𝑐2 and 𝑐3, 𝑐4, 𝑐5. [marks: 2]
6. Use Matlab to print a well-drawn and labelled graph of 𝐹𝑒𝑚 and 𝐹𝑒𝑚~ for constant 𝑖 = 𝑖0 and
𝑥 ? [-0.025m, 0m]. Include your code. [marks: 4]
7. Whatmasscanthesystemsupportatthelinearisedequilibriumposition 𝑥0 forconstantcurrent
𝑖 = 𝑖0? [marks: 2]
8. Write out the linearised equation for 𝑥¨ in symbolic form. [marks: 2]
9. Now we complete the linearisation. Replace 𝑥 - 𝑥0 by 𝑥, 𝑥? - 𝑥?0 by 𝑥?, and 𝑖 - 𝑖0 by 𝑖 in the expression from question 8. (You can imagine the new 𝑥, 𝑖 with invisible accents on them, 𝑥^,
𝑖, if you prefer.) You should notice it is now a strictly linear equation with no constant term.^
[marks: 1]
10. Using state vector 𝐱 = [𝑥, 𝑥,? 𝑖]T, write out the matrix state equation for this system to define the 𝐴 and 𝐵 state matrices. Use symbolic entries with 𝑐1, 𝑐2, 𝑚, 𝑅, 𝐿, etc.; i.e., do not substitute values in yet! [marks: 4]
11. Now substitute values into the 𝐴 and 𝐵 matrices and express them numerically. [marks: 2]
12. Assume two laser sensors are used to measure the displacement and velocity of the ball, respectively. Write out the matrix output equation for this system to define the 𝐶 and 𝐷 matrices. [marks: 2]
𝑎 = 0.002 𝑏 = -0.04 h = 0.01kg/s/A2
𝑥0 = -0.008m 𝑥?0 = 0m/s 𝑖0 = 3A
𝑅 = 2O 𝐿 = 0.003H 𝐺 = 0.001Vs/m
Table 1: Operating parameters of the system.
13. Using values as above, calculate the characteristic equation for the state matrix by hand andthen find the poles of the system. (You can use a calculating device to solve the resulting cubic.)
[marks: 4]
14. Check the calculation of the poles using Matlab by first defining a state space system in termsof its state matrices. Include your code. [marks: 4]
15. Do the poles indicate that the system is stable? [marks: 1]
16. Calculate by hand the controllability and observability matrices for the system described above.
Is the system controllable? Is the system observable? [marks: 4]
17. Now consider the system as if we could only measure current instead. Is the system still controllable and observable? If so, use Matlab to calculate the condition number of the relevant matrices and indicate whether the system is more or less controllable and/or observable.
[marks: 4]
[Total marks: 47]