Part I

Question 1

Prove the least squares estimates are unbiased, and what conditions this is true.

Answer 1

16

Question 2

Show that instrumental variable estimates can be unbiased.

Answer 2

17

Question 3

Show that estimates can be biased if corrupted by coloured noise

Answer 3

18

Question 4

Determine if t1 t2 and t3 are unbiased estimators

Answer 4

19

Question 5

How to determine the most efficient estimator?

Answer 5

20

Question 6

Determine the relative degree of stability of the system: x = … y= … etc

Answer 6

21

Question 7

Find an input linearising control law for a system

Answer 7

Perform as 21, then set ydotdot = v, and rearrange for u = …
Write ‘such that ydotdot = v’

Question 8

How can you design a minimum variance controller?

Answer 8

22

Question 9

How can you find the variance of a minimum variance controller?

Answer 9

Variance = J = (1+f1^2 + f2^2 …)*variance of noise (sigma squared)

Question 10

How can you design a control Lyapunov function (clf) controller?

Answer 10

Find Vdot(x)
Select u such that Vdot(x) = - Q(x)
Use the chain rule when differentiating. x1^3 goes to 3x1^2x1dot . Differentiate the whole thing by x1 and then by x2. Add the answers

Question 11

How can you design a feedback control law to ensure a system is stable given the Lyapunov function?

Answer 11

Find Vdot(x). Need to ensure that Vdot(x) < 0 for system to be stable.
Task is to select u such that this is the case. Can write constraints for other parameters

Question 12

How can you work out the zero dynamics. of a system? When will the system be a minimum phase system?

Answer 12

Set y = 0 and find out what states are.
Set dot = 0 and find out what states are.
Set v = 0.
Remaining dynamics are whatever x3 dot =. (substitute in control law to u = …)
Minimum phase system when x3dot is negative.