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5.2.4 R ELATIONSHIP BETWEEN P OLES AND THE C LOSED -L OOP S YSTEM R ESPONSE
The closed-loop system response is directly related to the locations of the poles in the
complex plane. To illustrate the relationship between the poles and the system
response, we consider a second-order dynamic system with zero input. Assume
that the closed-loop second-order system is given by
x ( k þ
1
) ¼ Ax ( k )
(
5
:
41
)
where matrix A is given in the diagonalized form as
1
1
1
2
l 1
0
2
5
l 1 þ
6
l 2
2
l 1
2
l 2
A ¼
¼
(
:
)
5
42
l 2
l 1 þ
l 2
l 1
l 2
3
5
0
3
5
15
15
6
5
Here
l 2 are eigenvalues (poles) of the closed-loop dynamic system which can be
assigned arbitrarily. Assume that the system is initially at x 1 (
l 1 ,
0
) ¼
1 and x 2 (
0
) ¼
2.
The time response of the system for the given initial condition is given by
k
5
l 1 þ
6
l 2
2
l 1
2
l 2
1
2
x ( k ) ¼ A k x (
0
) ¼
(
5
:
43
)
15
l 1 þ
15
l 2
6
l 1
5
l 2
This can be simpli
ed as
1 þ
2 ] u ( k )
x 1 ( k ) ¼ [l
2
l
(
5
:
44
)
and
1
2
x 2 ( k ) ¼ [
3
l
þ
5
l
] u ( k )
(
5
:
45
)
For the system to be stable the poles are chosen to be inside the unit circle in complex
plane. This means that
1.
The response of the system for different values of
jl 1 j <
1, and
jl 2 j <
l 1 , and
l 2 are shown in
Table 5.1.
5.3 LQR DESIGN
5.3.1 I NTRODUCTION
The LQR is a well-known optimal control design technique that results in a feedback
gain similar to the state feedback design [4,5]. It provides an optimal design based on
a speci
figure of merit which is quadratic in both states as well as in control. In
the derivation of the LQR, we assume that all the states of the system are available to
the controller.
Let the plant to be controlled be given in state space as
ed
x ( k þ
1
) ¼ Ax ( k ) þ Bu ( k )
(
5
:
46
)
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