Image Processing Reference
In-Depth Information
The state feedback control law is given by
u ( k ) ¼ K ( k ) x ( k )
(
5
:
47
)
where K ( k )
is the controller feedback gain at time k. The closed-loop system
dynamic is governed by the state equation given as
x ( k þ
1
) ¼ ( A BK ( k )) x ( k )
(
5
:
48
)
Let us form a quadratic performance function as follows
2 P
N 1
k ¼ 0 x T ( k ) Qx ( k ) þ u T ( k ) Ru ( k )
1
2 x T ( N ) Sx ( N ) þ
1
J ¼
(
5
:
49
)
where
S and Q are positive or positive semide
nite matrices
R is a positive de
nite matrix
The
first term in the performance index penalizes the deviation of the
final state from
the origin. The term x T ( k ) Qx ( k )
penalizes deviation of the states from the origin during
the control process and the term u T ( k ) Qu ( k )
is a measure of energy used by the control
signal. The goal is to minimize performance index J with respect to the control signal
u ( k )
subject to the constraints given by the state equation of the plant (Equation 5.46).
5.3.2 S OLUTION OF THE LQR P ROBLEM
The optimal control problem is to minimize the objective function
2 P
N 1
1
2 x T ( N ) Sx ( N ) þ
1
0 x T ( k ) Qx ( k ) þ u T ( k ) Ru ( k )
J ¼
(
5
:
50
)
k ¼
subject to the constraint given by the state equations
x ( k þ
1
) ¼ Ax ( k ) þ Bu ( k )
(
5
:
51
)
Using Lagrange multipliers, we form a new performance index H as follows
2 P
N 1
k ¼ 0 x T ( k ) Qx ( k ) þ u T ( k ) Ru ( k )
1
2 x T ( N ) Sx ( N ) þ
1
H ¼
þ l T ( k þ
1
) x ( k þ
½
1
) Ax ( k ) Bu ( k )
(
5
:
52
)
T .
where
l( k ) ¼ l 1 ( k ) l 2 ( k )
½
l N ( k )
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