Modern Control Engineering

834 Chapter 10 / Control Systems Design in State Space

Taking the conjugate transpose of both sides of Equation (10–146), we have

SinceKe=K*, this last equation is the same as Equation (10–159). Thus, we obtained Equation

(10–159) by considering the dual problem.

A–10–11. Consider an observed-state feedback control system with a minimum-order observer described

by the following equations:

(10–162)

(10–163)

where

Axais the state variable that can be directly measured, and corresponds to the observed state

variables.B

Show that the closed-loop poles of the system comprise the closed-loop poles due to pole

placementCthe eigenvalues of matrix (A-BK)] and the closed-loop poles due to the minimum-

order observer [the eigenvalues of matrix

Solution.The error equation for the minimum-order observer may be derived as given by

Equation (10–94), rewritten thus:

(10–164)

where

From Equations (10–162) and (10–163), we obtain

(10–165)

Combining Equations (10–164) and (10–165) and writing

we obtain

(10–166)

Equation (10–166) describes the dynamics of the observed-state feedback control system with a

minimum-order observer. The characteristic equation for this system is

or

@s I-A+BK@@s I-Abb+Ke Aab@ = 0

2 s^ I-A+BK

0

• BKb
  s I-Abb+Ke Aab

(^2) = 0
B
x#
e
#R = B

A-BK

0

BKb

Abb-Ke Aab

RB

x

e

R

K= CKaKbD

=Ax-BKex- c

0

e

df=(A-BK) x+BKc

0

e

d

x# =Ax-BK x =Ax-BKc

xa

xb

d=Ax-BKc

xa

xb-e

d

e=xb-xb

e# =AAbb-Ke AabB e

AAbb-Ke AabBD

xb

x= c

xa

xb

d, x= c

xa

xb

d

u =-Kx

y =Cx

x# =Ax+Bu

K=AT-^1 BF

an-an

an- 1 - an- 1







a 1 - a 1

V =(T*)-^1 F

an-an

an- 1 - an- 1







a 1 - a 1

V =(WN*)-^1 F

an-an

an- 1 - an- 1







a 1 - a 1

V

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