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[Solution]Computer-Controlled Systems 3ed by Karl J. Astrom 등록

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Computer-Controlled Systems 3판 Karl J.Astrom저 1997년판 챕터2 ~ 12장까지

Computer-Controlled Systems

Third Edition

Solutions Manual

Karl J. Astrom Bjorn Wittenmark

Department of Automatic Control Lund Institute of Technology October 1997

Preface

This Solutions Manual contains solutions to most of the problems in the third edition of Astrom, K. J. and
B. Wittenmark (1997): Computer controlled Systems Theory and Applications, Prentice Hall Inc., Englewood Cliffs, N. J. Many of the problems are intentionally made such that the students have to use a simulation program to verify the analytical solutions. This is important since it gives a feeling for the relation between the pulse transfer function and the time domain. In the book and the solutions we have used Matlab/Simulink. Information about macros used to generate the illustrations can be obtained by writing to us. It is also important that a course in digital control includes laboratory exercises. The contents in the laboratory experiments are of course dependent on the available equipment. Examples of experiments are Illustration of aliasing Comparison between continuous time and discrete time controllers State feedback control. Redesign of continuous time controllers as well as controllers based on discrete time synthesis Controllers based on input-output design Control of systems subject to stochastic disturbances. Finally we would like to thank collegues and students who have helped us to test the book and the solutions.

Karl J. Astrom Bjorn Wittenmark

Department of Automatic Control Lund Institute of Technology Box 118 S-220 00 Lund, Sweden

i

Solutions to Chapter 2

Problem 2.1 The system is described by ˙ x y

ax + bu

cx

Sampling the system using (2.4) and (2.5) gives x( kh + h) y( kh) e ah x( kh) + 1 cx( kh)

e

ah

b u( kh) a

The pole of the sampled system is exp( ah). The pole is real. For small values of h the pole is close to
1. If a > 0 then the pole moves towards the origin when h increases. If a < 0 then the pole moves along the positive real axis. Problem 2.2
a. Using the Laplace transform method we nd that



Φ

e

Ah

L 1

sI

A

1

L 1

Γ

0

cos h sin h sin h cos h h h sin s As ds e B ds cos s

1 s s2 + 1 1



1 s



1

cos h

sin h 2 s+1 1 s+2

0

b. The system has the transfer function G ( s) s2 s+3 + 3s + 2 s+3 ( s + 1)( s + 2)

Using the Table 2.1 gives H ( q) 2 1 q

e 1 e e 2 q e

h h

2h 2h

c.

One state space realization of the system is 0 0 0 1 ˙ 1 0 0 x + 0 u x 0 1 0 0 y 0 0 1 x Now A2

0 0 0 0 0 0 1 0 0

A3

0

1

then e Ah

h

I + Ah + A2 h2 /2 +

h

0 0 1 h 1 0 2 h /2 h 1 h

h2 /2

Γ

0

e B ds

0

As

1

s

T s2 /2 ds 0 0 1 (q

x

T h3 /6

and H ( q) C ( qI

Φ) Γ

1



h h2 /2 x x x 3 2 2 h ( q + 1)/2 h( q 1) ( q 1) h /6

x x

1)

3





h3 ( q2 + 4q + 1) 6( q 1)3



Problem 2.3

Φ

a.

e Ah y( k)

⇒

A

1 ln Φ h 6u( k 6u ˙ x( t) ax( t) + bu( t) y( t)… 등록



..... (중략)






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