Modern Control Engineering

54 Chapter 2 / Mathematical Modeling of Control Systems

A–2–8. Obtain a state-space model of the system shown in Figure 2–26.

Solution.The system involves one integrator and two delayed integrators. The output of each

integrator or delayed integrator can be a state variable. Let us define the output of the plant as

x 1 ,the output of the controller as x 2 ,and the output of the sensor as x 3 .Then we obtain

Y(s)=X 1 (s)

X 3 (s)

X 1 (s)

=

1

s+ 1

X 2 (s)

U(s)-X 3 (s)

=

1

s

X 1 (s)

X 2 (s)

=

10

s+ 5

U(s) 1 Y(s)

s

Controller Plant

Sensor

10

s+ 5

1

s+ 1

+

• 
  

Figure 2–26

Control system.

MATLAB Program 2–4

num = [2 1 1 2];

den = [1 4 5 2];

[A,B,C,D] = tf2ss(num,den)

A=

-4 -5 -2

100

010

B =

1

0

0

C =

-7 -9 -2

D =

2

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