Linear control system analysis and design

contents

preface

1 introduction

1 .l introdaction

l .2 introduction to control systems

l .3 definitions

l .4 historical background

l .5 digital control development

1 .6 mathematical backgmund

l .7 general nature of the engineering control problem

l .8 computer literacy

1 .9 outline of text

2 writing system equations

2.l introduction

2.2 elecuic circuits and components

2.3 basic linear matrix algebra

2.4 state concepts
2.5 transfer function and block diagram

2.6 mechanical translation systems

2.7 analogous circuits

. 2.8 mechanical rotational systems

2.9 thermal systems

2.1o hydraulic linear actuator

2.11 liquid-level system

2.l2 rotating power amplifiers

2.13 dc servomotor

2.14 ac servomotor

2.l5 lagrange's equation

2.l6 summary

3 solution of differential equations

3.1 introduction

3.2 standard inputs to control systems

3.3 steady-state response: sinasoidal input

3.4 steady-state response: polynomial input

3.5 transient response: classical method

3.6 definition of time constant

3.7 example: second-order system-mechanical

3.8 example: second-order system-electrical

3.9 second-order transients

3. 10 time-response specifications

3. 11 cad accuracy checks (cadac)

3. l2 state-variable equations

3. l3 characteristic values

3. 14 evaluating the state transition matrix

3. l5 complete solution qf the state equation

3. i6 summary

4 laplace transform

4.1 introduction

4.2 definition of the laplace transform

4.3 deriva!ion of laplace transforms of simple functions

4.4 laplace transform theorems

4.5 cad accuracy checks: cadac

4.6 application of the laplace transform to differential equations

4.7 inverse transformation

4.8 heaviside panial-fraction expansion theorems

4.9 matlab panial-fraction example

4.10 pahial-fraction shoncuts

4.ll graphical interpretation of panial-fraction coefficients

4.l2 frequency response from the pole-zero diagram

4.13 location of poles and stability

4.14 laplace transform of the impulse function

4.15 second-order system with impulse excitation

4.16 additional matrix operations and propenies

4.17 solution of state equation
4.18 evaluation of the transfer-function matrix

4.19 summary

5 system representation

5.1 introduction

5.2 block diagrams

5.3 determination of the overall transfer fanction

5.4 standard block diagram terminology

5.5 position control system

5.6 simulation diagrams

5.7 signal flow graphs

5.8 state transition signal flow graph

5.9 parallel state diagrams from transfer functions

5.1o diagonalizing the a matrix

5.ll use of state transformation for the state equation solution
5.l2 transforming a matrix with complex eigenvalues

5.l3 transforming an a matrix into companion form

5.14 summary

6 control-system characteristics

6.1 introduction
6.2 routh's stability criterion

6.3 mathematical and physical forms

6.4 feedback system types

6.5 analysis of system types

6.6 example: type 2 system

6.7 steady-state ehof coefficients

6.8 cad accuracy checks: cadac

6.9 use of steady-state enor coefficients

6.1o nonunity-feedback system

6.ll summary

7 root locus

7.1 introduction

7.2 plotting roots of a characteristic equation

7.3 qualitative analysis of the root locus

7.4 procedure outline

7.5 open-loop transfer function

7.6 poles of the control ratio c(s)/r(s)

7.7 application of the magqitude and angle conditions

7.8 geometrical propehies (construction rules)

7.9 cad accuracy checks (cadac)

7.10 examples

7.l1 example l : matlab root locus

7.12 pe40rmance characteristics

7.l3 transpoh lag

7.14 synthesis
7.15 summary of root-locus construction rules for negative feedback

7.l6 summary

8 frequency response

8.1 introduction

8.2 comelation of the sinusoidal and time responses

8.3 frequency-response curves

8.4 bode plots (logarithmic plots)

8.5 general frequency-transfer-function relationships

8.6 drawing the bode plots

8.7. example of drawing a bode plot

8.8 system type and gain as related to log magnitude curves

8.9 cad accuracy check (cadac)

8.10 experimental determination of transfer functions

8.1l direct polar plots
8.l2 summary: direct polar plots

8.l3 nyquist's stability criterion

8.14 examples of nyquist's criterion using direct polar plot

8.15 nyquist's stability criterion applied to systems haviog dead time
8.16 definitions of phase margin and gain margin and their relation to stability

8.17 stability characteristics of the log magnitude altd phase diagram

8.18 stability from the nichols plot (log magnitude-angle diagram)

8.l9 summary

9 closed-loop tracking performance based on the frequency response

9.1 introduction

9.2 direct polar plot

9.3 determination of rlm and a}m fof a simple second-order system

9.4 correlation of sinusoidal and time responses

9.5 constant m(w) and a(w) contours of c(jw)/r(jw) on the complex plane (direct plot)

9.6 constant l/m and a contonrs (unity feedback) in the inverse polar plane
9.7 gain adjustment for a desired mm of a unity-feedback system:direct polar plot

9.8 coostant m and a curves on the log magnitude-angle diagram (nichols chart)

9.9 generation of matlab ( 1992 student version) bode and nyqoist plots

9.1o aajustment of gain by use of the log magnitude-angle diagram

9.ll comelation of pole-zero diagram with freqnency and time responses

9.l2 summary

1o root-locus compensation: design

l0.l introduction to design

1o.2 transient response: dominant complex poles

1o.3 additiooal significant poles

1o.4 root-locus design considerations

1o.5 reshaping the root locus

1o.6 cad accuracy checks (cadac)

1o.7 ideal integral cascade compensation (pi controller)

1o.8 cascade lag compensation design using passive elements

1o.9 ideal derivative cascade compensation (pd conuoller)

1o.1o lead compensation design using passive elements

1o.l1 general lead-compensator design

1o.l2 lag-lead cascade compensation design

1o.l3 comparison of cascade compensators

1o.14 pid controller

1o.15 introduction to feedback compensation

10.16 feedback compensation: design procedures

10.l7 simplified rate feedback compensation: a design approach

1o.18. design of rate feedback

1o.19 design: feedback of second derivative of ootput

1o.20 results of feedback compensation design

1o.21 rate feedback: plants with dominant complex poles

1o.22 summary

11 frequency-response compensation design

ll.l introductiqn to feedback compensation design

ll.2 selection of a cascade compensator

1l.3 cascade lag compensator

1l.4 design example: cascade lag compensation

1l.5 lead compensator

11.6 design example: cascade lead compensation

ll.7 lag-lead compensator

11.8 design example: cascade lag-lead compensation

11.9 feedback compensation design using log plots

11.1o design example: feegback compensatioo (log plots)

11.ll application goidelines: basic minor-loop feedback compensators

11.12 summary

12 control-ratio modeling
l2.l inuoduction

l2.2 modeling a desired tracking conuol ratio

l2.3 guillen}in-truxal design procedure

l2.4 inuoduction to distorbance rejection

l2.5 a second-order disturbance-rejection model

l2.6 disturbance-rejection design principles for siso systems

12.7 disturbance-rejection design example

12.s disturbance-rejection models

l2.9 summary

13 design: closed-loop pole-zero assignment(state-variable feedback)
l3.1 introduction

13.2 controllability and observability

13.3 state feedback for siso systems

13.4 state-feedback design for siso systems using the control canonical (phase-variables) form

l3.5 state-variable feedback (physical variables)

i3.6 general profenies of state feedback (using phase variables)

13.7 state-variable feedback: steady-state ehof analysis

l3.8 use of steady-state enor coefficients

l3.9 state-variable feedback: aii-pole plant

13.1o plants with complex poles

13.l1 compensator containing a zero

13.l2 state-variable feedback: pole-zero plant

13.13 summary

14 parameter sensitivity and state space trajectories

l4.1 introduction

14.2 sensitivity

14.3 sensitivity analysis

14.4 parameter sensitivity examples

14.5 inaccessible st'ates

14.6 state-space trajectories

14.7 linearization (jacobian m'atrix)

14.8 summary

15 digital control systems

15.l introduction

l5.2 sampling

15.3 ideal sampling .

15.4 z-transform theorems

15.5 synthesis in the z domain (direct method)

15.6 the inverse z transform

l5.7 zero-order hold

l5.8 limitations

15.9 tustin transformation

15.l0 tustin transformation propenies

15.11 pseudo-continuous-time (pct) control system (dig method)

15.12 analysis of a basic (unc'ompensated) system

15.l3 design of digitai control systems

15.14 direct (dir) design technique

i5.15 lead controller (compensator): dir design method

15.16 lag and lag-lead controllers: dir design method

15.l7 digitization (dig) design technique

15.18 summary

16 entire eigenstructure assignment for multivariable systems
16.l introduction

16.2 effect of eigenstructure on time response

16.3 entire eigenstruc!ure assignment

16.4 examples of entire eigenstructure assignment for regulators

l6.5 matlab eigenvectors

16.6 uncontrollable systems

16.7 tracking systems

16.8 tracking-system design example

16.9 matlab example of tracker design in sec. 16.8

16.1o summary

17 design of tracking systems using output feedback

17. i introduction

17.2 outpat .feedback tracking system

17.3 block diagonalization

17.4 analysis of closed-loop system performance

17.5 design procedure for regular plants

17.6 regular system design example

l7.7 lrregular plant characteristics

17.8 irregular system performance

l7.9 design of the measarement matrix m

17. io irregutar system design example

17.1l tracker simulation '

17.12 summary

18 quantitative feedback theory (qrr) technique

18.1 introduction -

l8.2 frequency responses with parameter variations

18.3 introduction to the qrr method (single-loop system)

18.4 minimum-phase system performance specifications

18.5 multiple-input multiple-output (mimo) uncenain plants

l8.6 plant templates of p(s), sp(jwi)

18.7 u-contour

18.8 tracking bounds lm br(jw) on the nc

18.9 disturbance bounds bo(jwi): case l [d2(t) = dou-1(t), dl(t) = 0]

18.1o disturbance bounds bo(jaji): case 2 fdl(r) = dou_1 (f),d2(t) = 0]

18.1l the composite boundary bo(jml)

l8.12 shaping of lo(jw)

l8.l3 ouidelines tbr shaping lo(jal)

18.14 design ot the prefilter f(s)

18.15 basic design procedure for a miso system

18.16 design example 1

18.17 design example 2

18.18 template generation for unstable plants

18.19 summary

appendixes

a table of laplace transform pairs

b interactive comput.er-aided design programs for digital and continuous control-system analysis and synthesis

b.1 introduction

b.2 overview of icecap-pc and total-p6

b.3 overview of matlab

b.4 qrr cad packages
b.5 computer-aided design accuracy checks (cadac)

b.6 other computer-aided design packages

problems

answers to selected problems
index