K
(QX
k
)
T
Q Q X
k
6
iii)
For the function
f(X) = x
1
2
+ 4x
1
x
2
+ x
2
2
Use the optimum step length derived in a)ii) to calculate
X
1
given that:
X
0
= [
2
2
]
T
5
c)
i)
Given the quadratic function
f(X) = C
T
X +
1
2
X
T
Q X
Show that the optimum solution can be sought in a single iteration by using the
Newton- Raphson method (refer to Appendix) such that the optimum solution
X* = -Q
-1
C.
5
ii)
A system has two state equations such that:
w
̇
= x
1
+ 5x
2
+ 1
v
̈
= x
1
+ 2x
2
By using the result of 4c i) calculate the values of x
1
and x
2
that minimise the
system property f(X) given by:
f(X) = w
̇
v
̈
−
12x
1
x
2
Prove that the solution is a minimum by evaluating if Q is positive definite,
negative definite or indefinite.
6
Total 25