Modern Control Engineering

we have

(10–170)

Note that

where

Then, Equation (10–170) can be written as

(10–171)

Since, from Equation (10–94), we have

(10–172)

by combining Equations (10–171) and (10–172) into one equation, we have

The state matrix here is a 6*6matrix. The response of the system to the given initial condition

can be obtained easily with MATLAB. (See MATLAB Program 10–27.) The resulting response

curves are shown in Figure 10–50. The response curves seem to be acceptable.

c

x#

e#

d =c

A-BK

0

BKF

Abb-Ke Aab

dc

x

e

d

e# =AAbb-Ke AabB e

x# =(A-BK) x+BKFe

F= c

0

I

d

x-x= c

xa

xb

d -c

xa

xb

d= c

0

xb- xb

d =c

0

e

d = c

0

I

de=Fe

x# =Ax-BK x =(A-BK) x+BKAx-xB

838 Chapter 10 / Control Systems Design in State Space

x^1

01234

t (sec)

01234

t (sec)

01234

t (sec)

01234

t (sec)

01234

t (sec)

−0.05

0

0.1

0.05

0.15

e^1

0

0.05

0.1

01234

t (sec)

e^2

0

0.02

0.04

0.06

x^2

−0.02

0.02

0

0.04

0.06

x^3

−0.6

−0.2

−0.4

0

0.2

x^4

−0.2

0.1

0

−0.1

0.2

Response to initial condition Response to initial condition

Figure 10–50

Response curves to

initial condition.

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