Fuzzy Control Systems_(Ch5 of Dynamics of Controlled Systems)

Pilot Project No: CZ/02/B/F/PP/134001

Course on Dynamics of multidisplicinary andcontrolled Systems

Part IIISystem Control

Module Version 3.2.0

November 4, 2004

Chr. Schmid

Lehrstuhl fur Automatisierungstechnik und Prozessinformatik

Ruhr-Universitat Bochum

ii

Preface

This document contains a course both on the basics of control and on extensionsto get deeper insight into this area. The text contains many examples to illus-trate the subjects. Demonstration examples, problems, interactive questions anddemonstration examples with on-line simulations and models visualised in virtualreality are linked to the Internet. A CD version completes the set.

All the ideas, material and the facilities of this course would not have been im-plemented and completed without the hard work of the following persons, whohave directly or indirectly contributed to this course (in alphabetical order):

Abid AliDerek AthertonCarsten FritschArnd Grosse-FrintropChristoph HacksteinNorman MarkgrafAndrea MarschallTom RobertHeinz Unbehauen

Bochum, October 31, 2004Christian Schmid

iii

Contents

1 Introduction into System Control 2

1.1 Control objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Open loop vs closed loop . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 The basic structure of closed-loop systems . . . . . . . . . . . . . . 6

2 Basic Control Elements 9

2.1 The Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.2 Correspondences of the Laplace transform . . . . . . . . . . 10

2.1.3 Main theorems of the Laplace transform . . . . . . . . . . . 11

2.1.4 The inverse Laplace transform . . . . . . . . . . . . . . . . 14

2.1.5 Solving linear dierential equations using the Laplace transform 19

2.1.6 Laplace transform of the impulse (t) and step (t) . . . . 24

2.2 Transfer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.2 Interpretation of the transfer function . . . . . . . . . . . . 27

2.2.3 Realisability and properness of transfer functions . . . . . . 28

2.2.4 Transfer functions with dead time . . . . . . . . . . . . . . 28

2.2.5 Poles and zeros of the transfer function . . . . . . . . . . . 28

2.2.6 Using transfer functions for calculations . . . . . . . . . . . 30

2.3 Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.2 Nyquist plot of a frequency response . . . . . . . . . . . . . 37

2.3.3 Bode plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.4 Some important transfer function elements . . . . . . . . . 41

iv Contents

2.3.4.1 The proportional element (P element) . . . . . . . 41

2.3.4.2 The integrator (I element) . . . . . . . . . . . . . 41

2.3.4.3 The derivative element (D element) . . . . . . . . 42

2.3.4.4 The 1st-order lag element (PT1 element) . . . . . 43

2.3.4.5 The proportional plus derivative element (PD element) 48

2.3.4.6 The derivative lag element (DT1 element) . . . . . 48

2.3.4.7 The 2nd-order lag element (PT2 element and PT2S element) 50

2.3.4.8 Bandwidth of a system . . . . . . . . . . . . . . . 56

2.3.4.9 Example for the construction of a Bode plot . . . 56

2.3.5 Systems with minimum and non-minimum phase behaviour 58

2.3.6 Systems with dead time . . . . . . . . . . . . . . . . . . . . 61

2.4 Stability of linear control systems . . . . . . . . . . . . . . . . . . . 62

2.4.1 Stable and unstable systems . . . . . . . . . . . . . . . . . . 62

2.4.2 Denition of stability and stability conditions . . . . . . . . 62

2.4.3 Algebraic stability criteria . . . . . . . . . . . . . . . . . . . 64

2.4.3.1 The Hurwitz criterion . . . . . . . . . . . . . . . . 65

2.4.3.2 Routh criterion . . . . . . . . . . . . . . . . . . . . 68

2.4.3.3 Nyquist criterion . . . . . . . . . . . . . . . . . . . 70

2.4.3.4 Nyquist criterion using Nyquist plots . . . . . . . 70

2.4.3.5 Simplied forms of the Nyquist criterion . . . . . 74

2.4.3.6 The Nyquist criterion using Bode plots . . . . . . 75

2.5 The root-locus method . . . . . . . . . . . . . . . . . . . . . . . . . 81

2.5.1 Introduction and basic ideas . . . . . . . . . . . . . . . . . . 81

2.5.2 General rules for constructing root loci . . . . . . . . . . . . 86

2.5.3 Example of an application of the root-locus method . . . . 92

3 Behaviour of linear continuous-time control systems 95

3.1 Dynamical behaviour of a closed loop system . . . . . . . . . . . . 95

3.2 Static properties of the closed loop . . . . . . . . . . . . . . . . . . 100

v3.2.1 Transfer function G0(s) with delayed P behaviour . . . . . 102

3.2.2 Transfer function G0(s) with delayed I behaviour . . . . . . 103

3.2.3 Transfer function G0(s) with delayed I2 behaviour . . . . . 104

3.3 Performance indices . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.3.1 Time-response specications . . . . . . . . . . . . . . . . . . 104

3.3.2 Integral performance indices . . . . . . . . . . . . . . . . . . 107

3.3.3 Determination of quadratic performance indices . . . . . . . 109

3.4 PID control and associated controller types . . . . . . . . . . . . . 111

3.4.1 The classical three-term PID controller . . . . . . . . . . . 111

3.4.2 Optimal tuning of PID controllers . . . . . . . . . . . . . . 113

3.4.3 Advantages and disadvantages of the dierent types of controllers116

3.4.4 Empirical tuning rules according to Ziegler and Nichols . . 117

3.5 Design of controllers using pole-zero compensators . . . . . . . . . 120

3.5.1 Characteristics in frequency and time domain . . . . . . . . 120

3.5.2 Controller design using frequency domain characteristics . . 131

3.5.3 Application of the design using frequency domain characteristics138

3.5.4 Controller design using the root-locus method . . . . . . . . 142

3.6 Compensator design methods . . . . . . . . . . . . . . . . . . . . . 148

3.6.1 Basic ideas of compensator design . . . . . . . . . . . . . . 148

3.6.2 Design by specifying the closed-loop transfer function . . . 149

3.6.3 The method of Truxal and Guillemin . . . . . . . . . . . . . 150

3.6.4 Generalised compensator design method . . . . . . . . . . . 156

3.6.4.1 The basic idea . . . . . . . . . . . . . . . . . . . . 156

3.6.4.2 Zeros of the closed loop . . . . . . . . . . . . . . . 158

3.6.4.3 The synthesis equations . . . . . . . . . . . . . . . 161

3.6.4.4 Application of the method . . . . . . . . . . . . . 163

3.6.5 Compensator design for reference and disturbances . . . . . 168

3.6.5.1 Structure of the closed loop . . . . . . . . . . . . . 168

3.6.5.2 The design procedure . . . . . . . . . . . . . . . . 169

vi Contents

3.6.5.3 Design of the pre-lter . . . . . . . . . . . . . . . 175

3.6.5.4 Application of the design method . . . . . . . . . 179

3.7 Improving the control behaviour by more complex loop structures 184

3.7.1 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

3.7.2 Disturbance feed-forward control . . . . . . . . . . . . . . . 184

3.7.2.1 Disturbance feed-forward on the controller . . . . 184

3.7.2.2 Disturbance feed-forward on the manipulated variable186

3.7.3 Control systems with an auxiliary manipulated variable . . 188

3.7.4 Cascade control systems . . . . . . . . . . . . . . . . . . . . 190

3.7.5 Control system with auxiliary manipulated variable . . . . 192

3.7.6 Control system with anti-windup measure . . . . . . . . . . 193

4 State variable techniques 197

4.1 State-space representation . . . . . . . . . . . . . . . . . . . . . . . 197

4.1.1 State-space representation of single-input-single-output systems197

4.1.2 State-space representation of multi-input-multi-output systems199

4.1.3 The relationship between transfer functions and the state-space representation199

4.1.4 State-space vs transfer function approach . . . . . . . . . . 201

4.1.5 Uniqueness of the state variables . . . . . . . . . . . . . . . 201

4.1.6 Controllability and observability . . . . . . . . . . . . . . . 203

4.2 Design of state-feedback control systems . . . . . . . . . . . . . . . 204

4.2.1 Structures and properties of state-feedback control systems 204

4.2.1.1 State-feedback control in the frequency domain . . 205

4.2.1.2 Steady state of the closed-loop system . . . . . . . 207

4.2.2 State-feedback control with integrator . . . . . . . . . . . . 207

4.2.3 Design of state-feedback controllers by pole placement . . . 209

4.2.3.1 Design of a system in controller canonical form . . 210

4.2.3.2 Design of a system not in a canonical form . . . . 211

4.2.3.3 Design using Ackermanns formula . . . . . . . . . 213

vii

4.2.4 State reconstruction using observers . . . . . . . . . . . . . 214

4.2.4.1 Structure of an observer . . . . . . . . . . . . . . . 215

4.2.4.2 Design of observers . . . . . . . . . . . . . . . . . 216

4.2.5 Combined observer-controllers . . . . . . . . . . . . . . . . 217

4.2.6 Example of a state-feedback control system . . . . . . . . . 219

5 Fuzzy control systems 225

5.1 Introduction to fuzzy techniques . . . . . . . . . . . . . . . . . . . 225

5.1.1 Crisp and fuzzy logic . . . . . . . . . . . . . . . . . . . . . . 225

5.1.2 Why use fuzzy logic for control ? . . . . . . . . . . . . . . . 227

5.1.3 Ideas of the fuzzy control methodology . . . . . . . . . . . . 229

5.2 Basics of fuzzy sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

5.2.1 Fuzzy sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

5.2.2 Membership functions . . . . . . . . . . . . . . . . . . . . . 232

5.2.3 Elementary operators for fuzzy sets . . . . . . . . . . . . . . 235

5.2.4 Fuzzy relations . . . . . . . . . . . . . . . . . . . . . . . . . 237

5.2.5 Fuzzy composition . . . . . . . . . . . . . . . . . . . . . . . 240

5.3 Fuzzy systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

5.3.1 Fuzzication . . . . . . . . . . . . . . . . . . . . . . . . . . 243

5.3.2 Fuzzy inference machine . . . . . . . . . . . . . . . . . . . . 243

5.3.3 Defuzzication . . . . . . . . . . . . . . . . . . . . . . . . . 247

5.3.3.1 Centre of gravity method (COG) . . . . . . . . . . 249

5.3.3.2 Centre of singleton method (COS) . . . . . . . . . 249

5.3.3.3 Maximum methods . . . . . . . . . . . . . . . . . 250

5.3.3.4 Margin properties of the centroid methods . . . . 250

5.3.4 The Takagi-Sugeno fuzzy system . . . . . . . . . . . . . . . 250

5.3.5 The components of a fuzzy system . . . . . . . . . . . . . . 251

5.4 Fuzzy control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

5.4.1 Basic structure of a fuzzy controller . . . . . . . . . . . . . 252

viii Contents

5.4.2 Transfer behaviour of fuzzy controllers . . . . . . . . . . . . 253

5.4.2.1 Representation using 2D characteristics . . . . . . 254

5.4.2.2 Inuence of the membership functions and rule base on the characteristic256

5.4.2.3 Representation using 3D characteristics . . . . . . 260

5.4.3 Example of a fuzzy control system . . . . . . . . . . . . . . 261

5.4.3.1 Loading crane plant model . . . . . . . . . . . . . 261

5.4.3.2 Fuzzy control system design . . . . . . . . . . . . 263

5.4.4 Contribution of fuzzy control . . . . . . . . . . . . . . . . . 265

A Mathematical and table appendix 269

A.1 The Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . 269

A.1.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 269

A.1.2 The inverse Laplace transform . . . . . . . . . . . . . . . . 270

A.1.3 Main theorems of the Laplace transform . . . . . . . . . . . 270

A.1.3.1 Derivative theorem . . . . . . . . . . . . . . . . . 270

A.1.3.2 Integral theorem . . . . . . . . . . . . . . . . . . . 271

A.1.3.3 Convolution in the time domain . . . . . . . . . . 271

A.1.3.4 Convolution in the frequency domain . . . . . . . 272

A.1.3.5 Initial value theorem . . . . . . . . . . . . . . . . . 273

A.1.3.6 Final value theorem . . . . . . . . . . . . . . . . . 274

A.2 The complex G-plane . . . . . . . . . . . . . . . . . . . . . . . . . 275

A.3 Detailed analysis of 2nd-order lag elements . . . . . . . . . . . . . 277

A.3.1 Determining resonances of 2nd-order lag elements . . . . . 277

A.3.2 Poles and step responses of 2nd-order lag elements . . . . . 278

A.4 The law of Bode and the Hilbert transformation . . . . . . . . . . 281

A.5 Stability considerations using the weighting function . . . . . . . . 281

A.6 Equivalence of the Hurwitz and Routh criteria . . . . . . . . . . . 282

A.7 Determination of JISE using determinants . . . . . . . . . . . . . . 283

A.8 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

1Index 295

List of Tables 308

List of Figures 309

21 Introduction into System Control

1.1 Control objectives

The use of automatic control systems permeates life in all advanced societiestoday. Such systems act as a catalyst for promoting progress and development.Control systems are an integral component of any industrial society and arenecessary for the production of goods. Technological developments have madeit possible to travel to the moon and outer space. The successful production ofchemical components depends on the proper functioning of a large number ofcontrol systems used in lines for their production. As this fact is seldom apparentcontrol engineering is often called a hidden technology.

Control engineering deals with the task of aecting a temporally changing processin such a way that the process behaves in a given way. Such tasks are not onlyfound in technology, but also in daily life in very large number. For examplethe ambient temperature in a room must be held between given limits, despitetemporal changes due to sun exposure and other inuences. The grip arm of arobot must move along the edge of a workpiece or be led as fast as possible fromone point to another in order to grip a workpiece. The same applies to the griparm of a crane, which is to carry bricks to a certain place on the building site.

In all of these cases, a manipulated variable must be selected in such a waythat the given goal is achieved. As this selection depends on how well the goal isreached, a control loop arises that consists of the given process and a new feature,the controller. In the rst example, the room was the process and the thermalvalve the automatic controller, which measures the current air temperature andlets more or less heat into the heater depending on the deviation from the targettemperature. In the robot example the control equipment has the task of steeringthe grip arm on a given course and/or to a given point, whereby the control isbased on information that is supplied by the sensors installed on the grip arm. Inthe third example, the automatic controller is the crane operator, who determinesthe current grip arm position by sight and steers the crane.

1.2 Open loop vs closed loop 3

1.2 Open loop vs closed loop

The terms open-loop control and closed-loop control are often not clearly dis-tinguished. Therefore, the dierence between open-loop control and closed-loopcontrol is demonstrated in the following example of a room heating system. In thecase of open-loop control of the room temperature R according to Figure 1.2.1the outdoor temperature A will be measured by a temperature sensor and fed

Figure 1.2.1. Open-loop control of a room heating system

into a control device. In the case of changes in the outdoor temperature A (=disturbance z2) the control device adjusts the heating ow Q according to thecharacteristic Q = f(A) of Figure 1.2.2 using the motor M and the valve V.The slope of this characteristic can be tuned at the control device. If the roomtemperature R is changed by opening a window (= disturbance z1) this will notinuence the position of the valve, because only the outdoor temperature willinuence the heating ow. This control principle will not compensate the eectsof all disturbances.

Figure 1.2.2. Characteristic of a heating control device for three dierent tuning sets(1, 2, 3)

In the case of closed-loop control of the room temperature as shown in Figure 1.2.3

4 1 Introduction into System Control

the room temperature R is measured and compared with the set-point value w,(e.g. w = 20C). If the room temperature deviates from the given set-point value,a controller (C) alters the heat ow Q. All changes of the room temperature R,e.g. caused by opening the window or by solar radiation, are detected by thecontroller and removed.

Figure 1.2.3. Closed-loop control of a room heating system

The block diagrams of the open-loop and the closed-loop temperature control sys-tems are shown in Figures 1.2.4 and 1.2.5, and from these the dierence betweenopen- and closed-loop control is readily apparent.

Figure 1.2.4. Block diagram of the open-loop control of the heating system

The order of events to organise a closed-loop control is characterised by thefollowing steps:

Measurement of the controlled variable y, Calculation of the control error e = w y (comparison of the controlledvariable y with the set-point value w),

Processing of the control error such that by changing the manipulated vari-able u the control error is reduced or removed.

1.2 Open loop vs closed loop 5

Figure 1.2.5. Block diagram of the closed-loop control of the heating system

Comparing open-loop control with closed-loop control the following dierencesare seen:

Closed-loop control

shows a closed-loop action (closed control loop); can counteract against disturbances (negative feedback); can become unstable, i.e. the controlled variable does not fade away, butgrows (theoretically) to an innite value.

Open-loop control

shows an open-loop action (controlled chain); can only counteract against disturbances, for which it has been designed;other disturbances cannot be removed;

cannot become unstable as long as the controlled object is stable.

Summarising these properties we can dene:

Systems in which the output quantity has no eect upon the process input quan-tity are called open-loop control systems.

Systems in which the output has an eect upon the process input quantity in sucha manner as to maintain the desired output value are called closed-loop controlsystems.


1.3 The basic structure of closed-loop systems

In this section the general structure of control systems having a closed loop will beanalysed in more detail. According to Figure 1.3.1 a closed-loop system consistsof the following four main components:

plant, measurement device, controller and actuator.

The signals in the closed loop will be denoted by symbols. It means:

y controlled variable (actual value) u manipulated variablew command variable (set point), z disturbance.e control error (deviation)

Figure 1.3.1. Basic block diagram of a control system

From this block diagram it can be realised that the task of controlling a process(plant) consists of holding the controlled value y(t), acquired by the measurementdevice, either on a constant set point w(t) = const (xed command control)or tracking a time-varying reference variable w(t) = const (variable commandcontrol), independent of external disturbances z(t). This task is performed by acontroller. The controller processes the control error e(t) = w(t) y(t), which isthe dierence between the set point w(t) and the actual value y(t) of the controlledvariable. The control signal uC(t) generated by the controller will act via theactuator as the manipulated variable u(t) on the plant, such that it counteractsin the case of xed command control against the disturbance z(t). A closed-loopcontrol system is characterised by this closed signal path, whereby the controllerfunction consists in cancelling the occurring control error e(t) or at least holdingit very small.

Closed-loop control problems can be reduced to this basic structure. In most casesit is not possible to identify all the basic functions clearly. It is therefore properto aggregate a control loop only into two blocks. Hereby we distinguish between

1.3 The basic structure of closed-loop systems 7

! " #

# " ! " #

"

Figure 1.3.2. Simplied block diagram of a closed-loop control system

the plant, which may also aggregate the measurement device, and the controllingsystem that usually contains the actuator, as shown in Figure 1.3.2.

From Figures 1.3.1 and 1.3.2 it becomes obvious that the comparison of the set-point value w and the actual value y of the controlled variable for generatingthe control error e will become possible just through the negative feedback of thecontrolled variable y. Only because of this negative sign at the summing pointof both signals is the control error generated, which is used by the controller tobuild the control signal u using special mathematical functions (e.g. proportional,integrating, dierentiating). The principle of negative feedback, shortly also calledthe feedback principle, is a characteristic for every control loop.

The principles of closed-loop control are demonstrated by the following examples.Click on the links to start the animation with your Web browser.

Demonstration Example 1.3.1Water tank level without control and with disturbances

Demonstration Example 1.3.2Water tank level without control and with set point

Demonstration Example 1.3.3Water tank level manual control

Demonstration Example 1.3.4Water tank level min/max control

Demonstration Example 1.3.5Water tank level closed-loop control

Demonstration Example 1.3.6Main components of a control system

92 Basic Control Elements

2.1 The Laplace transform

2.1.1 Denition

The Laplace transform is an important tool for solving systems of linear dieren-tial equations with constant coecients. The dierential equations to be solvedfor control tasks normally full the conditions that must be met for taking theLaplace transform. The Laplace transform is an integral transformation, whichmaps a large class of original functions f(t) in the time domain unambiguouslyreversible into image functions F (s) in the s domain. This mapping is performedvia the Laplace integral of f(t), that is

F (s) =

0

f(t) estdt , (2.1.1)

where in the argument of the Laplace transform F (s) the complex variable s = + j appears. For the application of Eq. (2.1.1) to causal systems consideredhere the following two conditions for the time function f(t) must be met:

1. f(t) = 0 for t < 0 ;

2. the integral in Eq. (2.1.1) must converge.

To show the correspondence between the original and mapped functions it isuseful to use the operator notation

F (s) = [f(t)] .

Another possibility of correspondence is to use the sign in the following way:F (s) f(t) .

During the treatment of control systems usually the original function f(t) isa function of time. As the complex variable s contains the frequency , theimage function F (s) will often be called a frequency function. Therefore, theLaplace transform allows one to make a transition from the time domain intothe frequency domain according to Eq. (2.1.1).

10 2 Basic Control Elements

2.1.2 Correspondences of the Laplace transform

The so called back transformation or inverse Laplace transformation, i.e. thedetermination of the original function from the mapped function, is described bythe inverse integral shown in section A.1.2. For this inverse Laplace transform anoperator notation in the form

f(t) = 1[F (s)]

can be used.

The Laplace transformation is an unambiguously reversible mapping betweenthe original function and the mapped function. f(t) and F (s) are referred to astransform pairs and have a unique correspondence. This is the reason why inmost cases one does not need to use the inverse integral. A Correspondence table,as shown in Table 2.1.1, will suce. For the inverse transformation case onesgoes from the right column to the left column. In addition some theorems on theLaplace transform given in the next section may be useful.

Table 2.1.1: Corresponding elements of the Laplace transform

Nr. time response f(t), f(t) = 0 for t < 0 Laplace transformed F (s)

1 pulse (t) 1

2 unit step (t)1s

3 t1s2

4 t22s3

5tn

n!1

sn+1

6 eat1

s + a

7 teat1

(s + a)2

8 t2eat2

(s + a)3

9 tneatn!

(s + a)n+1

2.1 The Laplace transform 11

Table 2.1.1 continued

10 1 eat as(s+ a)

111a2

(eat 1 + at) 1s2(s + a)

12 (1 at) eat s(s + a)2

13 sin0t0

s2 + 20

14 cos0ts

s2 + 20

15 eat sin0t0

(s + a)2 + 20

16 eat cos0ts + a

(s + a)2 + 20

171af

(t

a

)F (as) (a > 0)

18 eatf(t) F (s a)

19f(t a) for t > a 0

0 for t < a easF (s)

20 t f(t) dF (s)ds

21 (t)n f(t) dnF (s)dsn

22 f1(t) f2(t)1

2j

c+jcj

F1(p)F2(s p) dp

2.1.3 Main theorems of the Laplace transform

a) Superposition theorem:

For arbitrary constants a1 and a2 it follows that

{a1f1(t) + a2f2(t)} = a1F1(s) + a2F2(s) . (2.1.2)


The Laplace transformation is a linear integral transformation.

b) Similarity theorem:

For an arbitrary constant a > 0

{f(at)} = 1aF(sa

)(2.1.3)

is valid. This follows from Eq. (2.1.1) by the substitution of = at.

c) Real Shifting theorem:


{f(t a)} = easF (s) (2.1.4)

is valid. This follows directly from Eq. (2.1.1) by the substitution of = ta.d) Complex Shifting theorem:


{eatf(t)

}= F (s + a) (2.1.5)

is valid. This follows directly from Eq. (2.1.1).

e) Derivative theorem:

For a causal function of time, f(t), for which the derivative for t > 0 exists,then as shown in section A.1.3.1, one obtains

{d f(t)dt

}= s F (s) f(0+) , (2.1.6)

and in the case of multiple dierentiation

{dnf(t)d tn

}= snF (s)

ni=1

snid(i1)f(t)d t(i1)

t=0+

. (2.1.7)

f) Complex dierentiation theorem:

This theorem shows that a dierentiation of the mapped function F (s) cor-responds to a multiplication with the time t in the time domain:

{tk f(t)

}= (1)k d

kF (s)dsk

. (2.1.8)


g) Integral theorem:

The integral of a function is mapped by

{ t0

f() d}

=1sF (s) . (2.1.9)

as shown in section A.1.3.2.

h) Convolution in the time domain:

The convolution of two functions of time f1(t) and f2(t), presented by thesymbolic notation f1(t) f2(t), is dened as

f1(t) f2(t) =t

0

f1() f2(t ) d . (2.1.10)

In section A.1.3.3 it is shown that the convolution of the two original func-tions corresponds to the multiplication of the related mapped functions, thatis

{f1(t) f2(t)} = F1(s)F2(s) . (2.1.11)

i) Convolution in the frequency domain:

Whereas in h) the convolution of two functions of time was given, a similarresult for the convolution of two functions in the frequency domain existsand is given by

{f1(t) f2(t)} = 12j

c+jcj

F1(p)F2(s p) dp . (2.1.12)

Here F1(s)f1(t) and F2(s)f2(t) is valid. Furthermore, p is the complexvariable of integration. According to this theorem the Laplace transform ofthe product of two functions of time is equal to the convolution of F1(s) andF2(s) in the mapped domain. This is shown in detail in section A.1.3.4.

j) Initial and nal value theorems:

The theorem of the initial condition allows the direct calculation of the func-tion value f(0+) of a causal function of time f(t) from the Laplace transformF (s). If the Laplace transform of f(t) and f(t) exist, then

f(0+) = limt0+

f(t) = lims s F (s) (2.1.13)

is valid if the limt0

f(t) exists, see section A.1.3.5.


Using the theorem of the nal value the value of f(t) for t can bedetermined from F (s), if the Laplace transform of f(t) and f(t) exist andthe limit lim

t f(t) also exists. Then it follows from section A.1.3.6 that

f() = limt f(t) = lims0

s F (s) . (2.1.14)

One has to observe that

limt f(t) or limt0

f(t)

can be calculated only from the corresponding Laplace transform {f(t)}by application of the theorems of the initial or nal value, if the existence ofthe related limit in the time domain is a priori assured. The following twoexamples should explain this:

Example 2.1.1

f(t) = et( > 0) F (s) = 1s

The limit limt e

t does not exist so that the nal value theorem may not beapplied.

Example 2.1.2

f(t) = cos0t F (s) = ss2 + 20

The limit limt cos0t does not exist and therefore the nal value theorem may

not be applied.

It can be concluded from the last two examples that the following general state-ment is valid: If the Laplace transform F (s) has, apart from a single pole at theorigin s = 0, poles on the imaginary axis or in the right-half s plane, then theinitial or nal value theorems cannot be applied.

2.1.4 The inverse Laplace transform

The inverse Laplace transform is described by Eq. (A.1.2). As already mentionedin section 2.1.2 in many cases a direct evaluation of the complex inverse inte-gral is not necessary, as the most important elementary functions are given inTable 2.1.1. A complicated function, F (s), not given in Table 2.1.1 must be de-composed into a sum of simple functions of s that is


F (s) = F1(s) + F2(s) + . . . + Fn(s) , (2.1.15)

which have a known inverse Laplace transform:

1 {F (s)} = 1 {F1(s)}+ 1 {F2(s)}+ . . . + 1 {F3(s)}= f1(t) + f2(t) + . . . + fn(t) = f(t) .

(2.1.16)

For many problems in control the function F (s) is a ratio of polynomials in s,known as rational fraction, that is

Fs) =n0 + n1s+ . . . nmsm

d0 + d1s + . . . + sn=

N(s)D(s)

, (2.1.17)

where N(s) and D(s) are the numerator and the denominator, respectively.

If m > n, then N(s) is divided by D(s), where a polynomial in s and a ratio ofpolynomials are obtained. The numerator of the fraction N1(s) has a lower orderthan n. E.g, if m = n + 2, then

N(s)D(s)

= k2s2 + k1s+ k0 +N1(s)D(s)

, (2.1.18)

whereby degreeN1(s) < n and k0, k1 and k2 are constants.

A rational fraction F (s) given in Eq. (2.1.17) can be decomposed into moresimple functions by application of partial fraction decomposition, as shown inEq. (2.1.15). In order to perform this decomposition the denominator polynomialD(s) must be factorised into the form

F (s) =N(s)

(s s 1) (s s2) . . . (s sn) . (2.1.19)

For a denominator polynomial of n-th order one obtains n roots or zeros s =s1,s2, . . . ,sn. The zeros of D(s) are also known as the poles of F (s), since theydene where F (s) is innite. The partial fraction decomposition for dierent typesof poles is shown in the following.

Case 1: F (s) has only single poles.

Here F (s) can be expanded into the form

F (s) =n

k=1

cks sk , (2.1.20)

where the residuals ck are real or complex constants. Using the table of corre-spondences one immediately can obtain the corresponding function of time

f(t) =n

k=1

ckeskt for t > 0 . (2.1.21)


The values ck can be determined either by comparing the coecients or by usingthe theorem of residuals from the theory of functions according to

ck =N(sk)D(sk)

= (s sk)N(s)D(s)

s=sk

(2.1.22)

for k = 1,2, . . . ,n with D(sk) = dD/ds |s=sk .Case 2: F (s) has multiple poles.

For multiple poles of F (s) each with multiplicity rk(k = 1,2, . . . ,l) the corre-sponding partial fraction decomposition is

F (s) =l

k=1

rk=1

ck(s sk) with n =

lk=1

rk . (2.1.23)

The back transformation of Eq. (2.1.23) into the time domain is

f(t) =l

k=1

esktrk

=1

ckt1

( 1)! for t > 0 . (2.1.24)

The real or complex coecients ck for = 1,2, . . . ,rk determined by the theoremof residuals are

ck =1

(rk )!

{d(rk)

ds(rk)![F (s) (s sk)rk ]

}s=sk

. (2.1.25)

This general relation also contains the case of single poles of F (s). The poles maybe real or complex.

Case 3: F (s) has also conjugate complex poles.

As both, the numerator N(s) and the denominator D(s) of the function F (s) arerational algebraic functions, complex factors always arise as conjugate complexpairs. If F (s) has a conjugate complex pair of poles s1,2 = 1 j1, then for thefunction F1,2(s) in the partial fraction decomposition of

F (s) =N(s)D(s)

= F1,2(s) + F3(s) + . . . + Fn(s)

Eq. (2.1.20) can be applied to give

F1,2(s) =c1

s (1 + j1) +c2

s (1 j1) , (2.1.26)

where the residualsc1,2 = 1 j1


are also a conjugate complex pair. Therefore, both fractions of F1,2(s) can becombined, and one obtains

F1,2(s) =0 + 1s

0 + 1s + s2(2.1.27)

with the real coecients

0 = 21 + 21 ; 1 = 21

0 = 2(21 + 11) ; 1 = 21}

. (2.1.28)

The determination of the coecients 0 und 1 is performed again using thetheorem of residuals by

(0 + 1s)s=s1

= (s s1) (s s2) N(s)D(s)

s=s1

. (2.1.29)

As s1 is complex, both sides of this equation are complex. Comparing the realand imaginary parts of both sides one gets two equations for the calculation of0 und 1. This procedure is demonstrated now using the following example.

Example 2.1.3Find the inverse Laplace transform f(t) of

F (s) =1

(s2 + 2s + 2) (s + 2).

The partial fraction decomposition of F (s) is

F (s) = F1,2(s) + F3(s) =0 + 1s

s2 + 2s + 2+

c3s + 2

,

where the function F1,2(s) contains the conjugate pair of poles

s1,2 = 1 j .

In addition the third pole of F (s) is

s3 = 2 .

For the coecients 0 and 1 it follows from Eq. (2.1.29)

(0 + 1s)s=s1

=1

s + 2

s=s1

(0 1) + j1 = 11 + j + 2 =12 j1

2.


Comparing the real and imaginary parts on both sides one obtains

0 1 = 12 and 1 = 12

and from this nally 0 = 0.

Using Eq. (2.1.22) the residual is

c3 = (s + 2)1

(s2 + 2s + 2) (s + 2)

s=s3

=12

.

The partial fraction decomposition of F (s) is thus

F (s) = 12

[s

s2 + 2s + 2

]+

12

1s + 2

,

which can be rearranged in the form

F (s) = 12

[s + 1

(s + 1)2 + 1 1

(s + 1)2 + 1 1

s + 2

]

such that correspondences given in Table 2.1.1 can be directly applied to nd theinverse transformation. Using the correspondences 16, 15 and 6 of this table, itfollows that

f(t) = 12[et cos t et sin t e2t] for t > 0 ,

which can be rearranged as

f(t) =12et[et + sin t cos t] for t > 0 .

The graphical representation of f(t) is shown in Figure 2.1.1a. Figure 2.1.1b showsthe corresponding poles, marked by a x, for this F (s) in the complex s plane.

It can be seen from this example that the position of the poles s1,s2 and s3 aectsthe shape of the graph of f(t). In this case all poles of F (s) have negative realparts, therefore the graph of f(t) shows a damped behaviour, i.e. it decreases tozero for t . If the real part of one pole be positive, then the graph of f(t)would be innitely large for t.Since in control problems the original function f(t) always represents the timebehaviour of a system variable, the behaviour of this system variable f(t) canbe judged to a large extent by investigation of the positions of the poles of thecorresponding mapped function F (s). This will be further commented on in latersections.


$

$

$

% & ' ( )

* &

*

* & "

Figure 2.1.1. (a) Graph of the original function f(t) (function in the time domain)and (b) position of the poles of F (s) in the s plane

2.1.5 Solving linear dierential equations using the Laplace trans-form

The Laplace transform, the basics of which have been introduced in the sectionsabove, is an elegant way for fast and schematic solving of linear dierential equa-tions with constant coecients. In the following the importance of this approachis demonstrated. Instead of solving the dierential equation with the initial condi-tions directly in the original domain, the detour via a mapping into the frequencydomain is taken, where only an algebraic equation has to be solved. Thus solvingdierential equations is performed according to Figure 2.1.2 in the following threesteps:

1. Transformation of the dierential equation into the mapped space ,

2. Solving the algebraic equation in the mapped space,

3. Back transformation of the solution into the original space.

+ + " , # "

" " , # "

" + "

#

#

" + "

Figure 2.1.2. Schema for solving dierential equations using the Laplace transforma-tion


Demonstration Example 2.1.1Here the same in animated form

Whereas the rst two steps are trivial, the third step usually demands more eort.The procedure will be demonstrated by the following two examples.

Example 2.1.4Consider the dierential equation

f(t) + 3f(t) + 2f(t) = et

with the initial conditions f(0+) = f(0+) = 0.Proceeding using the steps given above one has

Step 1:

s2F (s) + 3sF (s) + 2F (s) =1

s + 1

Step 2:

F (s) =1

s + 11

s2 + 3s + 2

Step 3:

The complex function F (s) must be decomposed into partial fractions inorder to use the tables of correspondences. This gives

F (s) =1

s + 2 1

s + 1+

1(s + 1)2

.

By means of the correspondences 6 and 7 of Table 2.1.1 it follows fromthe inverse Laplace transformation that the solution of the given dierentialequation is

f(t) = e2t et + t et .

Example 2.1.5Given the dierential equation

x + a1x + a0x = 0 , (2.1.30)

where a0 and a1 are constants and the initial conditions x(0+) and x(0+) areknown. Then

Step 1:

s2X(s) s x(0+) x(0+) + a1[sX(s) x(0+)] + a0X(s) = 0


Step 2:

X(s) =s + a1

s2 + a1s + a0x(0+) +

1s2 + a1s + a0

x(0+) , (2.1.31)

X(s) = L0(s)x(0+) + L(s) x(0+)

with the abbreviation

L0(s) =N0(s)D(s)

=s+ a1

s2 + a1s + a0and L(s) =

N(s)D(s)

=1

s2 + a1s + a0.

Step 3:

Case a): two single real zeros of the denominator :

This meansD(s) = s2 + a1s + a0 = (s 1) (s 2) .

For both rational expressions L0(s) and L(s) it follows by partial fractiondecomposition that

L0(s) =A1

s 1 +A2

s 2 and L(s) =B1

s 1 +B2

s 2 .

The coecients Ai and Bi can now be determined by comparing coecientsor by applying Eq. (2.1.22):

Ai =N0(i)D(i)

; Bi =N(i)D(i)

for i = 1, 2 .

Thus for Eq. (2.1.31) follows

X(s) =[

A1s 1 +

A2s 2

]x(0+) +

[B1

s 1 +B2

s 2

]x(0+) ,

and by applying the correspondence 6 from Table 2.1.1 the solution of thedierential equation is

x(t) =[A1e1t + A2e2t

]x(0+) +

[B1e1t + B2e2t

]x(0+)

= [A1x(0+) +B1x(0+)] e1t + [A2x(0+) + B2x(0+)] e2t . (2.1.32)

Case b): One double real zero of the denominator :

Here isD(s) = (s )2 .

For the two rational expressions L0(s) and L(s) of Eq. (2.1.31) the partialfraction decomposition is


L0(s) =A1

s +A2

(s )2 and L(s) =B1

s +B2

(s )2 .

The coecients Ai and Bi are determined by comparing both sides or byevaluation of Eq. (2.1.25):

A1 ={

dds

[N0(s)D(s)

(s )2]}

s=

= 1, A2 =[N0(s)D(s)

(s )2]s=

= + a1

and

B1 ={

dds

[N(s)D(s)

(s )2]}

s=

= 0, B2 =[N(s)D(s)

(s )2]s=

= 1 .

From these results one obtains the solution

X(s) =x(0+)s +

( + a1)x(0+) + x(0+)(s )2

in the mapped space. By applying the inverse Laplace transformation therequired solution of the dierential equation is

x(t) = x(0+) et + [( + a1)x(0+) + x(0+)] t et . (2.1.33)

Case c): Two conjugate complex zeroes of the denominator :

HereD(s) = (s 1) (s 2) with 1,2 = 1 j1 .

Introducing the values of 1 and 2 and after multiplication of this expressionone obtains

D(s) = (s 1)2 + 21 .

Comparison with the denominator of the original relation, Eq. (2.1.31), givesaccording to Eq. (2.1.28)

a0 = 21 + 21 and a1 = 21 .

With these coecients Eq. (2.1.31) is in the form

X(s) =s 21

(s 1)2 + 21x(0+) +

1(s 1)2 + 21

x(0+)

=[

s 1(s 1)2 + 21

11

1(s 1)2 + 21

]x(0+)

+11

1(s 1)2 + 21

x(0+) ,


and from this one gets by applying the correspondences 15 and 16 of Ta-ble 2.1.1 to X(s) the corresponding time function

x(t) = e1t[cos1t 1

1sin1t

]x(0+) +

11

e1tx(0+) sin1t

or rearranged

x(t) = e1t{x(0+) cos 1t +

[11

x(0+) 11

x(0+)]sin1t

}. (2.1.34)

Also from Eq. (2.1.31) of this example the importance of the position of the zerosof D(s), the poles of X(s), on the solution is clear. For all three cases the solutionof the dierential equation according to Eqs. (2.1.32), (2.1.33) and (2.1.34) ismainly inuenced by the position of the poles of X(s). These poles of X(s)are as one can see from the two examples only depend on the left side ofthe corresponding dierential equation, i.e. the homogeneous part of it. As isgenerally known the solution of the homogeneous dierential equation describesthe modes of the system, that is the behaviour, which depends only on the initialconditions. Therefore, consider for the general case only the homogeneous partof an nth-order ordinary homogeneous linear dierential equation with constantcoecients that is

ni=0

aidixa(t)

dti= 0 (2.1.35)

with all n initial conditions

(dixa(t)/dti) |t=0+ for i = 0,1, . . . ,n 1 .One obtains by Laplace transformation

Xa(s)

[n

i=0

aisi

][

ni=1

ai

i=1

sid1

dt1xa(t)

t=0+

]= 0

and a form according to Eq. (2.1.31)

Xa(s) =

ni=1

aii

=1si

d1

dt1xa(t)

t=0+

ni=0

aisi=

N(s)D(s)

, (2.1.36)

where N(s) and D(s) are polynomials in s and the initial conditions are only inthe numerator polynomial N(s). The poles sk(k = 1,2, . . . ,n) of Xa(s) can bedetermined directly from the solution of the equation


ni=0

aisi = 0 . (2.1.37)

After factorisation of this equation one obtains

an(s s1) (s s2) . . . (s sn) = 0 . (2.1.38)The poles sk of Xa(s) make it possible to perform a partial fraction decompositionof Xa(s), e.g. for the case of single poles according to Eq. (2.1.20). For this caseone obtains following Eq. (2.1.21) the solution of the homogeneous dierentialequation, Eq. (2.1.35), in the form

xa(t) =n

k=1

ckeskt for t > 0 .

From this one can realise that the position of the poles sk of Xa(s) in the s planecompletely characterises the modes or inherent behaviour of the system describedby Eq. (2.1.35). Thus one obtains for Re sk < 0 (left-half s plane) a decreasingand for Re sk > 0 (right-half s plane) an increasing behaviour of xa(t), while forpairs of poles with Re sk = 0 permanent oscillations occur. Therefore, Eq. (2.1.37)or equivalently Eq. (2.1.38), is called the characteristic equation and the poles skof Xa(s) are often called eigenvalues of the equation. Therefore investigation ofthe characteristic equation provides the most important information about theoscillating behaviour of a system.

2.1.6 Laplace transform of the impulse (t) and step (t)

The impulse function (t) is not a function in the sense of classical analysis,but a distribution (pseudo-function). Therefore without entering the theory ofdistributions the integral

{(t)} =0

(t) estdt (2.1.39)

is not dened. The singularity exactly matches with the lower integration limit.The impulse function can be approximately described by the limit

(t) = lim0

r(t)

with the rectangular impulse function

r =

{1/ for 0 t 0 otherwise .

(2.1.40)


Strictly speaking this representation of (t) is not a distribution, as r(t) for0 t is not arbitrarily often dierentiable. Because of the simple descriptioncompared with other functions (e.g. Gaussian functions) this approach is preferredhere. From Eq. (2.1.39) it follows that

{(t)} =0

[lim0

r(t)]estdt . (2.1.41)

As Eq. (2.1.40) can also be represented in the form

r(t) =1[(t) (t )] , (2.1.42)

where (t) is the unit step. Since the integration is independent of , the limitand integration can be permuted so that

{(t)} = lim0

{1

0

[(t) (t )] estdt}

{(t)} = lim0

{1

1s

(1 es)} .

By applying lHospitals rule one obtains

{(t)} = lim0

s es

s= 1 . (2.1.43)

As the impulse (t) has an area of unity it is also called unit impulse.

Example 2.1.6Given the dierential equation

dydt

= (t) .

Find solution y(t).

Remark: The derivative theorem according to Eq. (2.1.6) is as mentioned insection A.1.3.1 valid only for classical functions. If, however, a signal consistsof a function at t = 0, then the lower integration limit of Eq. (2.1.1) must bechosen equal to t = 0 and also in Eq. (A.1.11) the left-hand initial condition toy(0). According to the denition of Eq. (2.1.1) all left-hand initial conditionsare always zero.

The solution can be determined in the following three steps:


Step 1:The Laplace transform of the given dierential equation is:

s Y (s) y(0) = 1 withy(0) = 0 .Step 2:The solution of the algebraic equation is:

Y (s) =1s

.

Step 3:From the back transformation the solution follows as

y(t) = (t) ,

where (t) is the unit step function.

2.2 Transfer functions

2.2.1 Denition

Linear, continuous-time time-invariant systems with lumped parameters as ini-tially a dead time is not being taken into account will be described by theordinary dierential equation

ni=0

aidixa(t)

dti=

mj=0

bjdjxe(t)dtj

. (2.2.1)

If all initial conditions are set to zero and the Laplace transformation is appliedto both sides of this equation, one obtains

Xa(s)n

i=0

aisi = Xe(s)

mj=0

bjsj ,

or reorderingXa(s)Xe(s)

=b0 + b1s + . . . + bmsm

a0 + a1a+ . . . + ansn= G(s) =

N(s)D(s)

, (2.2.2)

where N(s) and D(s) describe the numerator and denominator polynomials, re-spectively. The quotient of the Laplace-transformed output and input of sucha type of system is a rational fraction. The coecients of this fraction dependonly on the structure and parameters of the system. Such a type of functionG(s), which describes completely the transfer behaviour of a system, is called thetransfer function of the system. With such a transfer function the output

Xa(s) = G(s)Xe(s) (2.2.3)

can be immediately calculated for a known input signal xe(t), and therefore Xe(s).

2.2 Transfer functions 27

2.2.2 Interpretation of the transfer function

Comparing Eq. (2.2.3) with the convolution theorem from section 2.1.3, Eqs. (2.1.10)and (2.1.11), it then follows for the representation of Eq. (2.2.3) in the time do-main that

xa(t) =

t0

g(t )xe() d , (2.2.4)

where obviously the inverse Laplace transform of G(s) is the function g(t). Thisfunction is generally known as the weighting function of the system. In otherwords, the transfer function is the Laplace-transformed weighting function ac-cording to

G(s) = {g(t)} . (2.2.5)

If the unit impulse (t) is taken as the input signal xa(t) for a system describedby the transfer function G(s), one obtains according to Eq. (2.2.3)

Xa(s) = G(s) {(t)}and after using the Laplace transform of the unit impulse (t) from Eq. (2.1.43)

Xa(s) = G(s)

or from Eq. (2.2.5)xa(t) = g(t) .

This shows that the response to an unit impulse (t) is the weighting function.Therefore the weighting function is also called the impulse response.

Another interpretation is when the system is excited by the input signal

xe(t) = xe et sin(t) .

For the steady-state case one obtains for the output signal

xas(t) = |G( + j)| xe et sin(t + ( + j)) . (2.2.6)This shows that the modulus |G( + j)| of the transfer function describes thegain, and that ( + j) = argG( + j) describes the phase shift of a sinu-soidal function with the frequency and with increasing or decreasing amplitudeaccording to et.

Interactive Questions 2.2.1Test your knowledge about impulse and step response

Interactive Questions 2.2.2Test your knowledge about convolution


2.2.3 Realisability and properness of transfer functions

It must be mentioned that a transfer function with m > n is physically notrealisable. The transfer function of an ideal dierentiator is described by G(s) = saccording to Eq. (2.1.6). Any transfer function with m > n can be decomposedinto

G(s) =N(s)D(s)

=N1(s)D(s)

+ k0 + k1s+ . . . + kmmsmn ,

where degreeN1(s) = n 1 and terms in s with positive powers also occur. Suchderivative elements would deliver for input signals of arbitrary high frequencycorresponding output signals of arbitrary high amplitude, which are physicallynot realisable. The condition of realisability of the transfer function according toEq. (2.2.2) is

degreeN(s) degreeD(s) or m n . (2.2.7)This condition is also called as the condition of properness. If a transfer functiondoes not follow this condition, it is called an improper transfer function. It hasthe property that G(s) as s . A realisable transfer function is calledas proper and it always follows G(s) k0 as s . A transfer function withk0 = 0 is called as strictly proper.

2.2.4 Transfer functions with dead time

If a time delay or dead time Tt is introduced in the input signal xe(t) , one obtainsinstead of Eq. (2.2.1) the dierential equation

ni=0

aidixa(t)

dti=

mj=0

bjdjxe(t Tt)

dtj. (2.2.8)

In this case taking the Laplace transformation gives the transcendental transferfunction

G(s) =N(s)D(s)

esTt . (2.2.9)

2.2.5 Poles and zeros of the transfer function

In some cases (e.g. stability analysis) it is expedient to represent the rationaltransfer function G(s) according to Eq. (2.2.2) in the factorised form

G(s) =N(s)D(s)

= k0(s sZ1) (s sZ2) . . . (s sZm)(s sP1) (s sP2) . . . (s sPn)

. (2.2.10)


For physical reasons only real coecients ai,bj occur. Therefore the poles sPi andthe zeros sZj of G(s), respectively, can be real or complex conjugate pairs. Theterms zeros and poles are chosen, because the transfer function is zero at sZj andinnite at sPi . Zeros and poles can be graphically represented in the complex splane as shown in Figure 2.2.1. A linear time-invariant system without dead timeis described completely by the distribution of its poles and zeros and the gainfactor k0.

$ -

Figure 2.2.1. Example of the pole and zero distribution of a rational transfer functionin the complex s plane

Moreover, the poles and zeros of a transfer function have a further signicance.Observing a system without input (xe(t) 0) according to Eq. (2.2.1) and de-termining the time response xa(t) for the given n initial conditions, one has tosolve the associated homogeneous dierential equation

ni=0

aidixa(t)

dti= 0 , (2.2.11)

which corresponds exactly to Eq. (2.1.35). For the approach xa(t) = est ofEq. (2.2.11) one obtains for the solution in s the characteristic equation

P (s) =n

i=0

aisi = 0 , (2.2.12)

which was already mentioned in Eq. (2.1.37). This relation can be directly deter-mined by setting the denominator of G(s) to zero (D(s) = 0), as long as D(s)and N(s) have no common factor. The zeros sk of the characteristic equationare the poles sPi of the transfer function. As already shown in section 2.1.5 themodes (i.e. xe(t) 0) are described by the characteristic equation, so that thepoles sPi of a transfer function contain all of this information.

The zeros of a transfer function are those values s = sZj for which |G(sZj )| = 0.This means that the output signal Xa(s) does not contain any components whichdepend on sZj . In order to explain this in more detail a stable system with atransfer function according to Eq. (2.2.10) is excited by the input signal


xe(t) = esZj t.

First for simplication the zero sZj = Zj is assumed to be real. For this case theinput signal is xe(t) = e

Zj t. Because |G(sZj )| = 0 one obtains from Eq. (2.2.6)the steady-state output signal as

xas(t) = 0.

In the case of complex conjugate pairs of zeros sZj , sZj+1 = sZj

both zeros haveto be taken into consideration in the input signal

xe(t) = esZj t + e

sZj t

= 2eZj t cost

= 2eZj t sin(t +

2) .

Eq. (2.2.6) leads also to the result xas(t) = 0. This shows that a zero sZj of asystem blocks the transmission of the input signal eZj t.

Example 2.2.1The mass-spring-damper mechanical system in Figure 2.2.2 with the mechanicalconstants c1=1, c2=2, d = 1.5, m1=1 and m2=4 is excited by the force xe. Thetransfer function between the force xe and the position xa can be shown to be

G(s) =s2 + 1

s4 + 0.5s3 + 1.75s2 + 0.5s + 0.5which has the zeros sZ1,2 = j. If this system is excited by the sinusoidal inputsignal

Xe(s) =1

(s sZ1)(s sZ2)=

1s2 + 1

1 {Xe(s)} = sin t ,which is derived from this pair of zeros, the output signal xa(t) decays to zeroas shown in Figure 2.2.3b even though the input signal is a sinusoidal signal andthe mass m1 shows an undamped oscillation (see Figure 2.2.3c). The system doesnot pass this oscillation to the mass m2 when the frequency matches the zerossZ1,2 .

2.2.6 Using transfer functions for calculations

For combinations of transfer functions simple rules for determining the resultingtransfer function can be derived. The combinations are of the type that transferfunction blocks are connected. In making any transfer function block connectionit is assumed that the connection does not load the block to which the connectionis being made.


"

Figure 2.2.2. Mass-spring-damper mechanical system used for the interpretation ofzeros

a) Series connection

From the diagram in Figure 2.2.4 it follows that

Y (s) = G2(s)Xe2(s)Xe2(s) = Xa1(s) = G1(s)U(s)Y (s) = G2(s)G1(s)U(s) .

The total transfer function of this series connection is

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

= G1(s)G2(s) . (2.2.13)

b) Parallel connection

For the output of both functions it follows from Figure 2.2.5 that

Xa1(s) = G1(s)U(s)Xa2(s) = G2(s)U(s) .

The output of the total system is

Y (s) = Xa(s) = Xa1(s) + Xa2(s) = [G1(s) +G2(s)]U(s),

and from this the transfer function of a parallel connection is

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

= G1(s) + G2(s) . (2.2.14)


.

. / &

/ &

. &

.

. / &

/ &

"

&

Figure 2.2.3. Response to the sinusoidal input signal xa(t) = sin t (a), (b) position ofthe mass m2 and (c) position of the mass m1

0

" 0

" 0

Figure 2.2.4. Series connection of two transfer functions

c) Feedback loop

From Figure 2.2.6 the output is

Y (s) = Xa(s) = [U(s) (+)Xa2(s)]G1(s) .

SinceXa2(s) = G2(s)Y (s)

one obtainsY (s) = [U(s)

(+)G2(s)Y (s)]G1(s) ,

and from thisY (s) =

G1(s)1 +()G1(s)G2(s)

U(s) .


0 " 0

"

"

Figure 2.2.5. Parallel connection of two transfer functions

The total transfer function of the feedback loop is

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

=G1(s)

1 +()G1(s)G2(s)

. (2.2.15)

0

"0

"

Figure 2.2.6. Feedback using two elements

As the output of G1(s) is fed back via G2(s) to the input, this is called feedback.One has to distinguish between a positive feedback for positive adding of Xa2(s)and a negative feedback for negative adding of Xa2(s).

Example 2.2.2For the special case of G1(s) being a pure amplier with a high gain K ,one obtains for negative feedback

G(s) =K

1 + K G2(s)=

11K

+ G2(s) 1

G2(s).

The entire technique of operational ampliers is based on this principle. In afeedback loop for which G1(s) is an amplier with K then an elementG2(s) can be used to realise any transfer function within certain limits.

Interactive Questions 2.2.3Test your basic knowledge about transfer functions


Interactive Questions 2.2.4Test your knowledge about transfer functions and stability

Interactive Questions 2.2.5Test your knowledge about transfer functions and electrical circuits

Interactive Questions 2.2.6Test your knowledge about transfer functions and electrical elements

Interactive Questions 2.2.7Test your knowledge about simple combinations of transfer functions

Interactive Questions 2.2.8Test your knowledge about complicated combinations of transfer functions

Interactive Questions 2.2.9Test your knowledge about transfer functions with dead time

2.3 Frequency Response

2.3.1 Denitions

In order to represent a transfer function G(s) in graphical form, there are dierentpossibilities. Representing |G(s)| and argG(s) over the complex plane s = + jtwo three-dimensional diagrams are needed. Figure 2.3.1 shows an example forthe magnitude of the transfer function

G(s) =s 1

s2 + s+ 1.25=

s 1(s + 0.5 j)(s + 0.5 + j) .

The two yellow peaks are at the pole positions where |G(0.5 j)| andthe blue negative peak is at the position of the zero s = 1 where |G(1)| = 0or |G(1)|dB , respectively. This three-dimensional representation clearlyshows the inuence of the poles and zeros on the magnitude of the transfer func-tion. Such diagrams are dicult to handle and are fortunately not necessary forthe analysis and design of control systems, because many properties can be in-vestigated by using the frequency response G(j). The frequency response is justthe cut through the three-dimensional diagrams of G(s) at the imaginary axis( = 0). For the magnitude diagram in Figure 2.3.1 the intersection of this cutand the surface is at the red curve, which represents the magnitude of the fre-quency response. This curve is symmetric with respect to and therefore thefrequency response G(j) is only considered for positive frequencies .

2.3 Frequency Response 35

Figure 2.3.1. Magnitude in dB of a transfer function over the complex plane s = +j

Another interesting possibility to represent a transfer function G(s) in graphicalform, is to map the s plane into the complex G plane as shown in section A.2.The result is a cartesian representation of a band of curves G( + j) as shownin Figure A.2.1. For = 0, i.e. the special case s = j, the transfer function G(s)migrates into the frequency response G(j) and we obtain the Nyquist plot ofthe frequency response, which will be discussed later in section 2.3.2.

The frequency response G(j) is an immediately physically interpretable andmeasurable quantity. To show this, the frequency response will be represented bythe complex entity

G(j) = R() + jI() (2.3.1)

with the real part R() and the imaginary part I(), and by the amplituderesponse A() and phase response () in polar notation as

G(j) = A() ej() . (2.3.2)

If the system is excited by a sinusoidal input xe(t) with amplitude xe and fre-quency , i.e.

xe(t) = xe sint , (2.3.3)

then in the case of a linear continuous-time system the output signal will oscillatein the steady state with the same frequency , but with another amplitude xa


"

"

0 1

" 0 " 1

"

Figure 2.3.2. (a) Sinusoidal input signal xe(t) and corresponding output signal xa(t)of a linear element in steady state (b) vector representation of both signals

and with a certain phase shift = t (Figure 2.3.2a). Both oscillations xe(t) andxa(t) can be represented by two vectors of length proportional to their amplitudesxe and xa according to Figure 2.3.2b. They are rotating with the phase shift and the same speed . For the system output one gets

xa(t) = xa sin(t + ) . (2.3.4)

If this experiment is conducted for dierent frequencies ( = 0,1,2, . . .) withxe = const, then one notices that the amplitude xa and the phase of the outputsignal depend on the frequency . Therefore for each frequency one gets

xa, = xa() and = () .

Now from the ratio of the values xe and xa() the amplitude of the frequencyresponse

A() =xa()xe

= |G(j)| =

R2() + I2() (2.3.5)

can be represented as a function of frequency. Furthermore, the phase shift ()will be represented as the phase of the frequency response. For the phase

() = argG(j) = tan1I()R()

(2.3.6)

is valid, where the value of the tan1 function in the range 0 to 360 must befound through the signs of R() and I().


From this experiment it is obvious that the amplitude response A() and phaseresponse () of the frequency response G(j) can be directly measured by ap-plying sinusoidal input signals xe(t) of dierent frequencies. The total frequencyresponse G(j) for all frequencies = 0 to describes completely thebehaviour of a linear continuous-time system, like the the transfer function G(s)or the step response h(t). In some cases only some partial information is sucientin order to reconstruct the total frequency response. As will be shown later onlya knowledge of the real part R() or of the amplitude response A() may benecessary.

Between the representations of a linear system in the time domain and frequencydomain there are some general and simple relationships. E.g. on the basis ofthe initial and nal value theorems two important relations between the transferfunction G(s), the Frequency response G(j) and the step response h(t) or H(s)are valid:

limt0+

h(t) = lims sH(s) = limsG(s) = limj

G(j) , (2.3.7a)

limth(t) = lims0

sH(s) = lims0

G(s) = limj0

G(j) . (2.3.7b)

The suppositions for the application of the initial and nal value theorems arethe existence of the related limits in the time domain according to section 2.1.3.

2.3.2 Nyquist plot of a frequency response

If in the experiment described above for each value of the value of

G(j) = A() ej()

is plotted in the complex G plane, one obtains the Nyquist plot of the fre-quency response, which is parameterised by the frequency . Figure 2.3.3 showssuch a curve drawn from measurements of eight frequencies. Using Eqs. (2.3.7a)and (2.3.7b) the initial and nal values of the step response h(t) can be esti-mated from a Nyquist plot based on such measurements (Figure 2.3.4). If G(j)is known analytically the step response can be calculated from G(s) by using

h(t) = 1{1sG(s)

}. (2.3.8)

The graphical representation of a frequency response using a Nyquist plot hasamong other things the advantage that the frequency responses, both of a seriesconnection and a parallel connection of two systems, can be simply constructedgraphically. Here the vectors of the Nyquist plots for the same values of mustbe taken. For the case of a parallel connection the vectors will be added, and forthe case of a series connection the vectors will be multiplied by multiplying thelengths of the vectors and adding their angles according to Figure 2.3.5.


" $ 2 3 4

3 4

Figure 2.3.3. Example of a Nyquist plot drawn from measurements

$ 2 3 4

3 4

$

Figure 2.3.4. Relations between initial and nal value of the frequency response G(j)and the step response h(t)

Interactive Questions 2.3.1Test your knowledge about frequency response and Nyquist plots

Interactive Questions 2.3.2Test your knowledge about frequency response and Nyquist plots with other questions

Interactive Questions 2.3.3Test your knowledge about frequency response and Nyquist plots with more demanding questions

2.3.3 Bode plot

If the absolute value A() and the phase () of the frequency responseG(j) =A() ej() are separately plotted over the frequency according to Figure 2.3.6,one obtains the amplitude response and the phase response. Both together


" " $

$

$ $

$

$ $ $

$

$ $ $

$ 2 3 4 $ 2 3 4

3 4 3 4

Figure 2.3.5. Addition (a) and multiplication (b) of frequency responses in Nyquistplots

% )

5

. 6 5

) 5

( 5

.

* 3 . 4

( 5

) 5

6 5

5 *

%

' 3 . 4

)

7

.

3 . 4' 3 . 4

Figure 2.3.6. Plot of a frequency response: (a) linear, (b) logarithmic presentation (on a logarithmic scale) (Bode plot)

are the frequency response characteristics. A() and are normally drawn witha logarithm and () with a linear scale. This representation is called a Bodediagram or Bode plot. Usually A() will be specied in decibels [dB]. By denitionthis is

A()dB = 20 log10A() [dB] . (2.3.9)

The logarithmic representation of the amplitude response A()dB has conse-quently a linear scale in this diagram and is called the magnitude.


The logarithmic representation has some advantages for series connections oftransfer functions. For complicated frequency responses, e.g. with

G(s) = K(s sZ1) . . . (s sZm)(s sP1) . . . (s sPn)

(2.3.10)

for s = j, it can be represented as series connections of the frequency responsesof simple elements of the form

Gi(j) =

{K = const for i = 0(j sZi) for i = 1,2, . . . ,m

(2.3.11)

and

Gi(j) =1

j sPfor

i = m + 1,m + 2, . . . ,m + n , = 1,2, . . . ,n . (2.3.12)

From this it follows that

G(j) = KG1(j) . . . Gm+n(j) , (2.3.13)

withGi(j) = Ai() eji() for i = 1, . . . ,m + n .

From the representation

G(j) = KA1()A2() . . . Am+n() ej[0() + 1() + 2() + . . . + m+n()]

(2.3.14)and

A() = |K| |G1(j)| |G2(j)| . . . |Gm+n(j)| = |K|A1()A2() . . . Am+n() ,

respectively, one obtains the logarithmic characteristic of the magnitude

A()dB = 20 log10 [|K|A1()A2() . . . Am+n()] (2.3.15)= |K|dB + A1()dB + A2()dB + . . . + Am+n()dB

and the phase characteristic

() = 0() + 1() + 2() + . . . + m+n() (2.3.16)

with 0() = 0 for K > 0 and 0() = 180 for K < 0. Thus, the frequencyresponse of a series connection is obtained by addition of the individual frequencyresponse characteristics.

A further advantage of this logarithmic representation is for the determinationof the inverse of a frequency response, that is for 1/G(j) = G1(j). Here

20 log10[|G(j)|1] = 20 log10 |G(j)| = 20 log10A()


andarg[G1(j)] = arg[G(j)] (2.3.17)

are valid, the curves of A() and () need only to be mirrored at the axes20 log10A = 0 (0-dB line) and = 0.

Because of the double logarithmic and of the single logarithmic scale of A() and(), respectively, the curve of A() and that of () can be approximated by linesegments. This approximation by lines allows the analysis and synthesis of controlsystems using simple geometric constructions. They are important concepts forthe control engineer.

2.3.4 Some important transfer function elements

In the following the transfer function G(s), frequency response G(j), Nyquistplot and the Bode diagram for some important elements will be derived andshown.

2.3.4.1 The proportional element (P element)

The P element describes a pure proportional relationship between input andoutput:

xa(t) = K xe(t) , (2.3.18)

with the arbitrary positive or negative constant K. K is called the gain of the Pelement. The transfer function of this system is

G(s) = K . (2.3.19)

The locus of the frequency response

G(j) = K (2.3.20)

is for all frequencies a point on the real axis with distance K from the origin.This means that the phase response () is zero for K > 0 or 180 for K < 0.The characteristic of the magnitude is

A()dB = 20 log10K = KdB = const .

2.3.4.2 The integrator (I element)

The dynamical behaviour of this element is described in the time domain by


xa(t) =1TI

t0

xe() d + xa(0) (2.3.21)

with input and output signals xe(t) and xa(t), respectively, and with the time con-stant TI, which has the dimension time. This element integrates the input signal.Setting xa(0) = 0 one obtains by taking the Laplace transform of Eq. (2.3.21)the transfer function of the I element as

G(s) =1

s TI, (2.3.22)

and with s = j the frequency response

G(j) =1

j TI=

1 TI

ej2 . (2.3.23)

From this the amplitude and phase responses

A() =1

TIand () =

2

follow. For the magnitude characteristic one obtains

A()dB = +20 log101

TI= 20 log10 TI . (2.3.24)

The graphical representation of Eq. (2.3.24) gives a line with a slope of -20dB/decade,or equivalently -6dB/octave, in the Bode diagram, Figure 2.3.7a. The phase re-sponse is independent of the frequency. The Nyquist plot of the frequency re-sponse

G(j) = j 1 TI

coincides with the negative imaginary axis, as shown in Figure 2.3.7b.

2.3.4.3 The derivative element (D element)

The relationship between the input xe(t) and output xa(t) of a derivative elementis described by

xa(t) = TDddt

xe(t) . (2.3.25)

This element dierentiates the input signal xe(t) and therefore is called a deriva-tive element, or in short D element. The associated transfer function is

G(s) = s TD , (2.3.26)


2

2

*

5

6 5

7

3 . 4

3 . 4

$ 2 3 4

3 4

$

Figure 2.3.7. (a) Magnitude and phase response (b) Nyquist plot of the frequencyresponse of an integrator

and with s = j it follows that the frequency response

G(j) = j TD = TDej2 , (2.3.27)

from which the magnitude

A()dB = 20 log10 T (2.3.28)

and the phase response() =

2(2.3.29)

follow. It can easily be seen, that the I and D elements are related by an inversion.Therefore the curves of the magnitude and phase response of the D element canbe found as shown above by mirroring those of the I element at the 0-dBline and = 0 line, respectively. This is obvious from Eqs. (2.3.28) and (2.3.29).Figure 2.3.8 shows the Bode diagram and the Nyquist plot of the frequencyresponse of the D element. The slope of the line A() is +20dB/decade and thephase response is independent of the frequency.

The D element discussed here is as already mentioned in section 2.2.3 an ideali-sation and therefore not a physically realisable element. For practical applicationsthe D element will be approximated by the DT1 element (see section 2.3.4.6).

2.3.4.4 The 1st-order lag element (PT1 element)

The 1st-order lag element or in short PT1 element is an element with an outputsignal xa(t) that for a step input xe(t) has a certain initial slope and approaches


2

2

*

5

6 5

7

3 . 4

3 . 4

$ 2 3 4

3 4

$

Figure 2.3.8. (a) Magnitude and phase response (b) Nyquist plot of the frequencyresponse of a D element

asymptotically the nal value. An example of such an element is the simple RClag circuit shown in Figure 2.3.9. When at time t = 0 a voltage ue = 2V isapplied, the voltage ua at the output will approach exponentially with the timeconstant T = RC the nal value ua = 2V.

" 0

"

!

" 0

" 0

Figure 2.3.9. Simple RC lag as an example of a 1st-order lag element

The circuit involves a single energy storage element, the capacitor C. The dier-ential equation of this RC lag is

xa(t) + RCxa(t) = xe(t) . (2.3.30)

For the general notation of a PT1 element one obtains the dierential equation

xa(t) + T xa(t) = K xe(t) . (2.3.31)

If the initial condition xa is set to zero, on taking the Laplace transform thetransfer function is

G(s) =K

1 + s T, (2.3.32)


and it follows that with s = j the frequency response is

G(j) = K1

1 + j T. (2.3.33)

With the breakpoint frequency B =1T

one obtains

G(j) = K1

1 + j

B

= K1 j

B

1 +(

B

)2 . (2.3.34)

The amplitude response is

A() = |G(j)| = K 11 +

(

B

)2 (2.3.35)

and the phase response is

() = tan1I()R()

= tan1 B

. (2.3.36)

The magnitude characteristic derived from Eq. (2.3.35) is

A()dB = 20 log10K 20 log10

1 +(

B

)2. (2.3.37)

Eq. (2.3.37) can be asymptotically approximated by lines for:

a)

B> 1 by

A()dB 20 log10K 20 log10

B(nal asymptote) ,

with()

2.


In the Bode diagram A()dB can be consequently approximated by two lines. Theprogression of the initial asymptote is horizontal, whereas the nal asymptoteshows a slope of -20dB/decade. The intersection of both lines (breakpoint) canbe determined from

20 log10K = 20 log10K 20 log10

B

and provides the frequency = B .

Therefore = B is called the breakpoint frequency. As can be easily seen fromFigure 2.3.10a, the deviation between A()dB and the asymptotes has a maximum

7

7

7#

"

. 7 8 "

7

6 5

% & 5

5

#

7

$ 2 3 4

3 4

$

7

/

Figure 2.3.10. (a) Magnitude and phase response (b) Nyquist plot of the frequencyresponse of a PT1 element

at the breakpoint for B. The exact values are

A(B) = K12

and (B) = 4 .

The deviation of the magnitude characteristic from the asymptotes for = B is

A(B)dB = 20 log102 dB 3 dB .

The deviations at the other frequencies are symmetrical from the breakpoint onthe logarithmic scale, as can be seen directly from Table 2.3.1. This is the reasonwhy the magnitude and phase characteristics can be easily constructed on a Bodediagram. The phase curve is approximately zero up to one tenth of the breakpointfrequency, and 90 beyond 10 time the breakpoint frequency. In between, it isapproximately a straight line with slope 45 per decade through 45 at thebreakpoint frequency.

As already shown in section A.2 the locus of the frequency response of a PT1element is a semicircle, which starts for = 0 at K on the real axis and stops for at the origin, as shown in Figure 2.3.10b.


Table 2.3.1 Magnitude, phase response and deviation A() of the exact magnitudefrom the asymptotes for a PT1 element with K = 1

BA()dB () A()dB

0.03 0.0 - 2 0.000.1 0.04 - 6 0.04

0.25 0.26 - 14 0.260.5 0.97 - 27 0.97

0.76 2.00 - 37 2.001.0 3.00 - 45 3.00

1.31 4.35 - 53 2.002.0 6.99 - 63 0.974.0 12 - 76 0.26

10.0 20 - 84 0.0430.0 30 - 88 0.00

The constant T = 1/B in the transfer function and on the frequency response,respectively, is usually called the time constant of the PT1 element. It can bedetermined also by the point of intersection of the line with the initial slope andthe horizontal line of the nal asymptote, h(), of the step response h(t) asshown in Figure 2.3.11. This time constant can also be physically interpreted. It

/ '

Figure 2.3.11. Graphical representation of the step response, h(t), of a PT1 element

is the time when the step response has reached approx. 63% of the nal value,h(). K is similar to the P element called the gain of the PT1 element. It isdened as the value of the frequency response at = 0.

DYNAST study example 2.3.1Simple automobile model

DYNAST study example 2.3.2D.C. motor - open loop


2.3.4.5 The proportional plus derivative element (PD element)

The PD element shows both proportional and derivative behaviour and is de-scribed by the transfer function

G(s) = K(1 + s T ) . (2.3.38)

Apart from the gain factor K this element is the inverse of the PT1 element.Hence for K = 1 one obtains the magnitude and phase response by mirroring atthe 0-dB axis and the () = 0 line, respectively (compare Figure 2.3.12 with2.3.10). The locus of the frequency response

G(j) = K(1 + j T ) (2.3.39)

is a semi-line, which starts for = 0 on the real axis at K and progresses parallelto the imaginary axis for increasing values of .

7

7#

7 8 "

7

6 5

% & 5

5

7

. .

$ 2 3 4

3 4

$

#

/

Figure 2.3.12. (a) Magnitude and phase response (b) Nyquist plot of the frequencyresponse of a PD element

2.3.4.6 The derivative lag element (DT1 element)

This element has a step response which initially contains a step and then decreasesexponential to zero with a characteristic time constant as shown in Figure 2.3.13.Figure 2.3.14 shows an example of such a system; a simple RC high pass lter.The dierential equation of this circuit is

Cd(ue ua)

dt=

uaR

,


#

Figure 2.3.13. Graphical representation of the step response, h(t), of a DT1 element

" 0 " 0

" " 0

!

Figure 2.3.14. Simple RC high-pass circuit as an example of a DT1 element

which can be written as

ua + RCduadt

= RCdueR

.

Applying the Laplace transform one obtains the transfer function

G(s) =Ua(s)Ue(s)

=RCs

1 + RCs. (2.3.40)

The generalised notation of the DT1 element is

G(s) = KTs

1 + Ts. (2.3.41)

For constructing the Bode plot one starts with the frequency response

G(j) = Kj T

1 + j T, (2.3.42)

which on substituting B =1T

gives

G(j) = Kj

B

1

1 + j

B

=

BK

B+ j

1 +(

B

)2 . (2.3.43)


From Eq. (2.3.43) it follows that

A() = |G(j)| = B

K1

1 +(

B

)2 ,

and

A()dB = 20 log10

B+ 20 log10K 20 log10

1 +

(

B

)2. (2.3.44)

After some calculations the phase response can be shown to be

() =

2 tan1

(

e

). (2.3.45)

Comparing Eq. (2.3.44) with Eqs. (2.3.37) and (2.3.40) shows that the magnitudecharacteristic of the DT1 element can be obtained by adding the correspondingcurves of a PT1 element and a D element. The same also holds for the phaseresponse (). With this information the curves of the frequency response char-acteristics and of the Nyquist diagram can be simply constructed according toFigure 2.3.15.

2.3.4.7 The 2nd-order lag element (PT2 element and PT2S ele-ment)

A 2nd-order lag element is characterised by two independent energy storages.Depending on the damping properties and the position of the poles of G(s),respectively, one distinguishes between oscillating and aperiodic behaviour. If a2nd-order lag element has a conjugate complex pair of poles, then it shows anoscillating behaviour (PT2S behaviour). If both poles are on the negative realaxis, then the element has a lag behaviour (PT2 behaviour).

The RLC network of Figure 2.3.16 is an example of such an element. From theequation of the mesh

ieR + Ldiedt

+ ua = ue (2.3.46)

withie = C

duadt

(2.3.47)

the dierential equation becomes

LCd2uadt2

+ RCduadt

+ ua(t) = ue(t) . (2.3.48)


7

7

6 5

% & 5

5

9

9 :

; :

"

/

7

7

7#

#

$ 2 3 4 0 7 0 8

Figure 2.3.15. (a) Magnitude and phase responses (b) Nyquist plot of the frequencyresponse of a DT1 element

Interactive Questions 2.3.4Test using other example

The transfer function is

G(s) =Ua(s)Ue(s)

=1

1 + RC s+ LC s2. (2.3.49)

For the 2nd-order lag element the general notation of the transfer function

$ !

" 0

" 0

" 0 "

Figure 2.3.16. Simple RLC network as an example of a 2nd-order lag element


G(s) =K

1 + T1s+ T 22 s2(2.3.50)

is chosen. Introducing terms which characterise the time behaviour, that is thedamping ratio

=12T1T2

(2.3.51)

and the natural frequency (frequency of the undamped oscillation)

0 =1T2

, (2.3.52)

one obtains from Eq. (2.3.50)

G(s) =K

1 +20

s +120

s2=

K

D(s). (2.3.53)

For s = j the frequency response

G(j) =K

1 + j2

0

2

20

= K

[1

(

0

)2] j2

0[1

(

0

)2]2+[2

0

]2 . (2.3.54)

results. The amplitude response is

A() =K[

1(

0

)2]2+(2

0

)2 (2.3.55)

and the phase response

() = tan12

0

1(

0

)2 . (2.3.56)

Here the ambivalence of the tan1 function has to be observed. For the magnitudecharacteristic one has from Eq. (2.3.55)

A()dB = 20 log10K 20 log10[

1(

0

)2]2+(2

0

)2. (2.3.57)

The progression of A()dB can be approximated by the following asymptotes:


a) For

0> 1 by

A()dB 20 log10K 20 log10(

0

)2 20 log10K 40 log10

(

0

)(nal asymptote) ,

with() .

In the Bode diagram the nal asymptote is a line with a slope of -40dB/decade.The point of intersection of both asymptotes follows from

20 log10K = 20 log10K 40 log10(

0

)to be the normalised frequency 0 . The exact value of A()dB may deviate con-siderably from the point of intersection at = 0, because it is

A(0)dB = 20 log10K 20 log10 2according to Eq. (2.3.55). For < 0.5 this value is above, for > 0.5 below theasymptotes.

Figure 2.3.17 shows for 0 < 2.5 and K = 1 the magnitude and phase responsesin a Bode diagram. This graphical representation contains the cases of PT2S andPT2 behaviour, which will be discussed in section A.3.2. From Figure 2.3.17 itcan be seen that a maximum magnitude exists for some values of the dampingratio . This maximum occurs at the so called resonant peak frequency p. Adetailed analysis of this resonance can be found in section A.3.1.

Figure 2.3.18 shows the Nyquist plots of 2nd-order lag elements with high andlow damping. From Eq. (2.3.54) it follows that for = 0 the real part of G(j) iszero. Therefore the locus of the frequency response intersects with the imaginaryaxis at = 0 independent of the values of , as can be seen in Figure 2.3.18.

The modes of a dynamical system are determined according to Eq. (2.2.12) bythe roots of the characteristic equation or by the poles of the transfer function,respectively. From the characteristic equation of the 2nd-order lag element

P (s) D(s) = 1 + 20

s +120

s2 = 0 (2.3.58)


&

%

% / / / / % / ' / ) % ' )

/ / / / % / ' / ) % ' )

5

5

' 5

6 5

5

& 5

) 5

0 / & 0 / & 0 / & 0 / & 0 / &

0 / & 0 / & 0 / & 0 / & 0 / & 0 / 0 / & 0 / 0 / &

7

0 / 0 / &0 / 0 / &

Figure 2.3.17. Bode diagram of a 2nd-order lag element with the transfer functionG(s) = 1/

[1 + s2/0 + (s/0)2

]

one obtains the poles of the transfer function as

s1,2 = 0 0

2 1 . (2.3.59)

The oscillating behaviour of a 2nd-order lag element is dependent on the positionof the poles in the s plane. For a graphical analysis the step response h(t) isa useful tool. Table 2.3.2 shows step responses and the corresponding poles fordierent values of . These ve cases will be analysed and discussed in more detailin section A.3.2.


Table 2.3.2 Pole positions in the s plane and step responses for elements with thetransfer function G(s) = 1/

[1 + s2/0 + (s/0)2

] "

&

&

&

$

$

%

$

$

$

$

$

0

<

0

= =

&

0 0 &

&

0 /

0 / &

8


&

$ 2 3 4

3 4

$ 2 3 4

3 4

$ $

Figure 2.3.18. Nyquist plots of 2nd-order lag elements, (a) PT2 element, (b) PT2Selement

2.3.4.8 Bandwidth of a system

An important term that has not been dened so far is the bandwidth of a system.Lag elements with a proportional behaviour, e.g. PT1, PT2 and PT2S elements aswell as PTn elements (n PT1 elements in series connection), show a so-called low-pass property. This means that they pass low frequencies whereas high frequenciesin signals are attenuated by the strongly decreasing amplitude response of thefrequency response. In order to describe this behaviour the concept of bandwidthis introduced. This is the frequency b at which the magnitude of the frequencyresponse is decreased by 3dB from the value of the initial hori

Fuzzy Control Systems_(Ch5 of Dynamics of Controlled Systems)

Documents

Transcript of Fuzzy Control Systems_(Ch5 of Dynamics of Controlled Systems)