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ELEC9731 ASSIGNMENT
submission – please submit as a SINGLE pdf file by
email
use subject: ELEC 9731-Assignment 2
Late assignments will be penalised at 10% of the max-
imum value per day late
Question 1 (8 marks). A linear control system is described by the equations
x˙(t) =
3.8 0 7aa 3a3 6.0
−5.0 0 4.2
x(t) +
−12
−3
u(t),
y(t) =
−a
2 0 3.1
4.5− a a+ 7.5 3.7
0.8 0 6.5
x(t),
where a varies from −∞ to ∞. Determine the range of a for observability.
Question 2 (9 marks). Consider the following systems of two linear second order
ODEs:
z¨1(t) = −z1(t) + az2(t) + u1(t),
z¨2(t) = −z2(t) + az1(t)− 2u2(t),
where a is some parameter.
(i) Re-write this system in a state space form.
(ii) For what values of a this system is controllable?
(iii) For what values of a this system is controllable from u1 alone?
(iv) For what values of a this system is controllable from u2 alone?
(v) If the controls are constrained to be equal, u1 = u2, for what values of a this system
is controllable?
(vi) Find the transfer function from u to z =
(
z1
z2
)
. Hence find the the Smith-
McMillan form.
Question 3 (8 marks). The characteristic polynomial of a control system is the
following uncertain polynomial:
s3 + a2s
2 + a1s+ 1.2
where a1 ∈ [1, K], a2 ∈ [1, K] and a1+a2 ≥ 2.3 and K is a parameter such that K ∈ [1,∞).
For what values of K is this uncertain polynomial stable?