Modern Control Engineering

Section 10–8 / Quadratic Optimal Regulator Systems 803

x^1 x^2

x^3

0.6

0.8

1

1.2

0.4

0.2

0

02468

02468

02468

−0.2

− 3

− 2

− 1

0

1

2

0

0.2

−0.2

−0.4

−0.8

−0.6

− 1

−1.2

t (sec)

t (sec)

t (sec)

Figure 10–38
Response curves to
initial condition.

In determining an optimal control law, we assume that the input is zero, or r=0.

Let us determine the state-feedback gain matrix K, where

such that the following performance index is minimized:

where

Q=C

q 11

0

0

0

q 22

0

0

0

q 33

S, R=1, x=C

x 1

x 2

x 3

S = C

y

y#

y

$

S

J=

3

q

0

(x¿ Qx+u¿ Ru)dt

K=Ck 1 k 2 k 3 D

+– +

• x=Ax+Bu

.

k 2

k 3

k 1 y=Cx

ru

x

x 2

x 3

y=x 1

Figure 10–39
Control system.