Modern Control Engineering

Section 10–5 / State Observers 765

MATLAB Program 10–8

% Obtaining transfer function of observer controller --- full-order observer

A = [0 1;20.6 0];

B = [0;1];

C = [1 0];

K = [29.6 3.6];

Ke = [16;84.6];

AA = A-KeC-BK;

BB = Ke;

CC = K;

DD = 0;

[num,den] = ss2tf(AA,BB,CC,DD)

num =

1.0e+003*

0 0.7782 3.6907

den =

1.0000 19.6000 151.2000

The dynamics of the observed-state feedback control system just designed can be described
by the following equations: For the plant,

For the observer,

The system, as a whole, is of fourth order. The characteristic equation for the system is

The characteristic equation can also be obtained from the block diagram for the system shown in
Figure 10–14(b). Since the closed-loop transfer function is

Y(s)

R(s)

=

778.2s+3690.7

As^2 +19.6s+151.2BAs^2 - 20.6B+778.2s+3690.7

=s^4 +19.6s^3 +130.6s^2 +374.4s+ 576 = 0

@s I-A+BK@@s I-A+Ke C@ =As^2 +3.6s+ 9 BAs^2 + 16 s+ 64 B

u =-[29.6 3.6]B

x 1

x 2

R

B

x 1

x 2

R =B

- 16

- 93.6

1

- 3.6

RB

x 1

x 2

R +B

16

84.6

Ry

y =[1 0]B

x 1

x 2

R

B

x# 1

x# 2

R= B

0

20.6

1

0

RB

x 1

x 2

R + B

0

1

Ru