Modern Control Engineering

700 Chapter 9 / Control Systems Analysis in State Space

wherekis a vector whose components are the magnitudes of rstep functions applied at

t=0. The solution to the step input at t=0is given by

IfAis nonsingular, then this last equation can be simplified to give

(9–96)

(c)Ramp response:Let us write the ramp input u(t)as

wherevis a vector whose components are magnitudes of ramp functions applied at t=0.The

solution to the ramp input tvgiven at t=0is

IfAis nonsingular, then this last equation can be simplified to give

(9–97)

A–9–7. Obtain the response y(t)of the following system:

whereu(t)is the unit-step input occurring at t=0,or

u(t)=1(t)

Solution.For this system

The state transition matrix can be obtained as follows:

(t)=eAt=l-^1 C(s I-A)-^1 D

(t)=eAt

A= B

- 1

1

- 0.5

0

R, B= B

0.5

0

R

y =[1 0]C

x 1

x 2

S

B

x# 1

x# 1

R = B

- 1

1

- 0.5

0

RB

x 1

x 2

R+ B

0.5

0

Ru, B

x 1 (0)

x 2 (0)

R = B

0

0

R

=eAt x( 0 )+ CA-^2 AeAt-IB-A-^1 tD Bv

x(t)=eAt x( 0 )+AA-^2 BAeAt-I-AtB Bv

=eAt x( 0 )+eAta

I

2

t^2 -

2 A

3!

t^3 +

3 A^2

4!

t^4 -

4 A^3

5!

t^5 +pb Bv

=eAt x( 0 )+eAt

3

t

0

e-Attdt Bv

x(t)=eAt x( 0 )+

3

t

0

eA(t-t) Bt vdt

u(t)=t v

=eAt x(0)+A-^1 AeAt-IB Bk

x(t)=eAt x(0)+eAtC-AA-^1 BAe-At-IBD Bk

=eAt x(0)+eAtaIt-

At^2

2!

+

A^2 t^3

3!

• pb Bk

=eAt x(0)+eAtc

3

t

0

aI-At+

A^2 t^2

2!

• pbdtd Bk

x(t)=eAt x(0)+

3

t

0

eA(t-t) Bkdt

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