Modern Control Engineering

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Section 5–5 / Transient-Response Analysis with MATLAB 185

EXAMPLE 5–3 Consider the following system:

Obtain the unit-step response curves.

Although it is not necessary to obtain the transfer-matrix expression for the system to obtain

the unit-step response curves with MATLAB, we shall derive such an expression for reference.

For the system defined by

the transfer matrix G(s)is a matrix that relates Y(s)andU(s)as follows:

Taking Laplace transforms of the state-space equations, we obtain

(5–37)

(5–38)

In deriving the transfer matrix, we assume that Then, from Equation (5–37), we get

(5–39)

Substituting Equation (5–39) into Equation (5–38), we obtain

Thus the transfer matrix G(s)is given by

The transfer matrix G(s)for the given system becomes

Hence

Since this system involves two inputs and two outputs, four transfer functions may be defined,

depending on which signals are considered as input and output. Note that, when considering the

B

Y 1 (s)

Y 2 (s)

R = ≥

s- 1

s^2 +s+6.5

s+7.5

s^2 +s+6.5

s

s^2 +s+6.5

6.5

s^2 +s+6.5

¥B

U 1 (s)

U 2 (s)

R

=

1

s^2 +s+6.5

B

s- 1

s+7.5

s

6.5

R

=

1

s^2 +s+6.5

B

s

6.5

- 1

s+ 1

RB

1

1

1

0

R

= B

1

0

0

1

RB

s+ 1

• 6.5

1

s

R

• 1
  B

1

1

1

0

R

G(s)=C(s I-A)-^1 B

G(s)=C(s I-A)-^1 B+D

Y(s)=CC(s I-A)-^1 B+DD U(s)

X(s)=(s I-A)-^1 BU(s)

x(0)= 0.

Y(s)=CX(s)+DU(s)

s X(s)-x(0)=AX(s)+BU(s)

Y(s)=G(s) U(s)

y=Cx+Du

x# =Ax+Bu

B

y 1

y 2

R= B

1

0

0

1

RB

x 1

x 2

R + B

0

0

0

0

RB

u 1

u 2

R

B

x# 1

x# 2

R= B

- 1

6.5

- 1

0

RB

x 1

x 2

R+ B

1

1

1

0

RB

u 1

u 2

R